{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_73877/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_73877","id":"gutenberg_73877","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/73877","summary":null,"tags":["Category: British Literature","Category: Philosophy & Ethics","Philosophy, English -- 17th century"],"text":"------------------------------------------------------------------------\n\n Transcriber\u2019s Note:\n\nThis version of the text cannot represent certain typographical effects.\nItalics are delimited with the \u2018_\u2019 character as _italic_.\n\nThis is a very visual work, with numerous diagrams and meticulous\ndiscussion of them. The HTML or eBook versions would be more suitable if\nthose aspects of the work are of interest.\n\nThere is a single footnote following the Translator\u2019s note.\n\nIf an article spans several pages, marginal paragraph notes were\nrepeated. These redundant notes have been removed.\n\nMinor errors, attributable to the printer, have been corrected. Please\nsee the transcriber\u2019s note at the end of this text for details regarding\nthe handling of any textual issues encountered during its preparation.\n\n The English Works of Thomas Hobbes: Volume 1\n\n[Illustration:\n\n THOMAS HOBBES.\n _W Humphrys sc._\n _London. Published March 1st 1839, by John Bohn. 17, Henrietta Street,\n Covent Garden._\n]\n\n THE\n\n ENGLISH WORKS\n\n OF\n\n THOMAS HOBBES\n\n OF MALMESBURY;\n\n NOW FIRST COLLECTED AND EDITED\n\n BY\n\n SIR WILLIAM MOLESWORTH, BART.\n\n ---\n\n VOL. I.\n\n ---\n\n\n\n\n LONDON:\n JOHN BOHN,\n HENRIETTA STREET, COVENT GARDEN.\n\n ---\n\n MDCCCXXXIX.\n\n\n\n\n LONDON:\n C. RICHARDS, PRINTER, ST. MARTIN'S LANE.\n\n\n\n\n TO\n\n GEORGE GROTE, ESQ.\n\n M.P. FOR THE CITY OF LONDON.\n\n --------------\n\nDEAR GROTE,\n\n I dedicate to you this edition of the Works of Hobbes; first,\nbecause I know you will be well pleased to see a complete collection of\nall the writings of an Author for whom you have so high an admiration.\nSecondly, because I am indebted to you for my first acquaintance with\nthe speculations of one of the greatest and most original thinkers in\nthe English language, whose works, I have often heard you regret, were\nso scarce, and so much less read and studied than they deserved to be.\nIt now, therefore, gives me great satisfaction to be able to gratify a\nwish, you have frequently expressed, that some person, who had time and\ndue reverence for that illustrious man, would undertake to edite his\nworks, and bring his views again before his countrymen, who have so long\nand so unjustly neglected him. And likewise, I am desirous, in some way,\nto express the sincere regard and respect that I feel for you, and the\ngratitude that I owe you for the valuable instruction, that I have\nobtained from your society, and from the friendship with which you have\nhonoured me, during the many years we have been companions in political\nlife.\n\n Yours, truly,\n WILLIAM MOLESWORTH.\n\n February 25th, 1839.\n79, Eaton Square, London.\n\n------------------------------------------------------------------------\n\n ELEMENTS OF PHILOSOPHY.\n\n THE FIRST SECTION,\n\n CONCERNING BODY,\n\n WRITTEN IN LATIN\n\n BY\n\n THOMAS HOBBES OF MALMESBURY\n\n AND\n\n TRANSLATED INTO ENGLISH.\n\n\n\n\n THE\n\n TRANSLATOR TO THE READER.\n\nIf, when I had finished my translation of this first section of the\nElements of Philosophy, I had presently committed the same to the press,\nit might have come to your hands sooner than now it doth. But as I\nundertook it with much diffidence of my own ability to perform it well;\nso I thought fit, before I published it, to pray Mr. Hobbes to view,\ncorrect, and order it according to his own mind and pleasure. Wherefore,\nthough you find some places enlarged, others altered, and two chapters,\nXVIII and XX, almost wholly changed, you may nevertheless remain\nassured, that as now I present it to you, it doth not at all vary from\nthe author's own sense and meaning. As for the Six Lessons to the\nSavilian Professors at Oxford, they are not of my translation, but were\nwritten, as here you have them in English, by Mr. Hobbes himself; and\nare joined to this book, because they are chiefly in defence of the\nsame.[A]\n\nFootnote A:\n\n They will be published in a separate volume, with other works of a\n similar description. W. M.\n\n\n THE AUTHOR'S EPISTLE DEDICATORY,\n\n TO THE\n\n RIGHT HONORABLE, MY MOST HONORED LORD,\n\n WILLIAM, EARL OF DEVONSHIRE.\n\n -------\n\nThis first section of the Elements of Philosophy, the monument of my\nservice and your Lordship's bounty, though, after the Third Section\npublished, long deferred, yet at last finished, I now present, my most\nexcellent Lord, and dedicate to your Lordship. A little book, but full;\nand great enough, if men count well for great; and to an attentive\nreader versed in the demonstrations of mathematicians, that is, to your\nLordship, clear and easy to understand, and almost new throughout,\nwithout any offensive novelty. I know that that part of philosophy,\nwherein are considered lines and figures, has been delivered to us\nnotably improved by the ancients; and withal a most perfect pattern of\nthe logic by which they were enabled to find out and demonstrate such\nexcellent theorems as they have done. I know also that the hypothesis of\nthe earth's diurnal motion was the invention of the ancients; but that\nboth it, and astronomy, that is, celestial physics, springing up\ntogether with it, were by succeeding philosophers strangled with the\nsnares of words. And therefore the beginning of astronomy, except\nobservations, I think is not to be derived from farther time than from\nNicolaus Copernicus; who in the age next preceding the present revived\nthe opinion of Pythagoras, Aristarchus, and Philolaus. After him, the\ndoctrine of the motion of the earth being now received, and a difficult\nquestion thereupon arising concerning the descent of heavy bodies,\nGalileus in our time, striving with that difficulty, was the first that\nopened to us the gate of natural philosophy universal, which is the\nknowledge of the nature of _motion_. So that neither can the age of\nnatural philosophy be reckoned higher than to him. Lastly, the science\nof _man's body_, the most profitable part of natural science, was first\ndiscovered with admirable sagacity by our countryman Doctor Harvey,\nprincipal Physician to King James and King Charles, in his books of the\n_Motion of the Blood_, and of the _Generation of Living Creatures_; who\nis the only man I know, that conquering envy, hath established a new\ndoctrine in his life-time. Before these, there was nothing certain in\nnatural philosophy but every man's experiments to himself, and the\nnatural histories, if they may be called certain, that are no certainer\nthan civil histories. But since these, astronomy and natural philosophy\nin general have, for so little time, been extraordinarily advanced by\nJoannes Keplerus, Petrus Gassendus, and Marinus Mersennus; and the\nscience of human bodies in special by the wit and industry of\nphysicians, the only true natural philosophers, especially of our most\nlearned men of the College of Physicians in London. Natural Philosophy\nis therefore but young; but Civil Philosophy yet much younger, as being\nno older (I say it provoked, and that my detractors may know how little\nthey have wrought upon me) than my own book _De Cive_. But what? were\nthere no philosophers natural nor civil among the ancient Greeks? There\nwere men so called; witness Lucian, by whom they are derided; witness\ndivers cities, from which they have been often by public edicts\nbanished. But it follows not that there was _philosophy_. There walked\nin old Greece a certain phantasm, for superficial gravity, though full\nwithin of fraud and filth, a little like philosophy; which unwary men,\nthinking to be it, adhered to the professors of it, some to one, some to\nanother, though they disagreed among themselves, and with great salary\nput their children to them to be taught, instead of wisdom, nothing but\nto dispute, and, neglecting the laws, to determine every question\naccording to their own fancies. The first doctors of the Church, next\nthe Apostles, born in those times, whilst they endeavoured to defend the\nChristian faith against the Gentiles by natural reason, began also to\nmake use of philosophy, and with the decrees of Holy Scripture to mingle\nthe sentences of heathen philosophers; and first some harmless ones of\nPlato, but afterwards also many foolish and false ones out of the\nphysics and metaphysics of Aristotle; and bringing in the enemies,\nbetrayed unto them the citadel of Christianity. From that time, instead\nof the worship of God, there entered a thing called _school divinity_,\nwalking on one foot firmly, which is the Holy Scripture, but halted on\nthe other rotten foot, which the Apostle Paul called _vain_, and might\nhave called _pernicious philosophy_; for it hath raised an infinite\nnumber of controversies in the Christian world concerning religion, and\nfrom those controversies, wars. It is like that _Empusa_ in the Athenian\ncomic poet, which was taken in Athens for a ghost that changed shapes,\nhaving one brazen leg, but the other was the leg of an ass, and was\nsent, as was believed, by Hecate, as a sign of some approaching evil\nfortune. Against this _Empusa_ I think there cannot be invented a better\nexorcism, than to distinguish between the rules of religion, that is,\nthe rules of honouring God, which we have from the laws, and the rules\nof philosophy, that is, the opinions of private men; and to yield what\nis due to religion to the Holy Scripture, and what is due to philosophy\nto natural reason. And this I shall do, if I but handle the Elements of\nPhilosophy truly and clearly, as I endeavour to do. Therefore having in\nthe Third Section, which I have published and dedicated to your\nLordship, long since reduced all power ecclesiastical and civil by\nstrong arguments of reason, without repugnance to God's word, to one and\nthe same sovereign authority; I intend now, by putting into a clear\nmethod the true foundations of natural philosophy, to fright and drive\naway this metaphysical _Empusa_; not by skirmish, but by letting in the\nlight upon her. For I am confident, if any confidence of a writing can\nproceed from the writer's fear, circumspection, and diffidence, that in\nthe three former parts of this book all that I have said is sufficiently\ndemonstrated from definitions; and all in the fourth part from\nsuppositions not absurd. But if there appear to your Lordship anything\nless fully demonstrated than to satisfy every reader, the cause was\nthis, that I professed to write not all to all, but some things to\ngeometricians only. But that your Lordship will be satisfied, I cannot\ndoubt.\n\nThere remains the second section, which is concerning _Man_. That part\nthereof, where I handle the _Optics_, containing six chapters, together\nwith the tables of the figures belonging to them, I have already written\nand engraven lying by me above these six years. The rest shall, as soon\nas I can, be added to it; though by the contumelies and petty injuries\nof some unskilful men, I know already, by experience, how much greater\nthanks will be due than paid me, for telling men the truth of what men\nare. But the burthen I have taken on me I mean to carry through; not\nstriving to appease, but rather to revenge myself of envy, by encreasing\nit. For it contents me that I have your Lordship's favour, which, being\nall you require, I acknowledge; and for which, with my prayers to\nAlmighty God for your Lordship's safety, I shall, to my power, be always\nthankful.\n\n Your Lordship's most humble servant,\n\n THOMAS HOBBES.\n\n London,\nApril 23, 1655.\n\n THE\n AUTHOR'S EPISTLE TO THE READER.\n\nThink not, Courteous Reader, that the philosophy, the elements whereof I\nam going to set in order, is that which makes philosophers' stones, nor\nthat which is found in the metaphysic codes; but that it is the natural\nreason of man, busily flying up and down among the creatures, and\nbringing back a true report of their order, causes and effects.\nPhilosophy, therefore, the child of the world and your own mind, is\nwithin yourself; perhaps not fashioned yet, but like the world its\nfather, as it was in the beginning, a thing confused. Do, therefore, as\nthe statuaries do, who, by hewing off that which is superfluous, do not\nmake but find the image. Or imitate the creation: if you will be a\nphilosopher in good earnest, let your reason move upon the deep of your\nown cogitations and experience; those things that lie in confusion must\nbe set asunder, distinguished, and every one stamped with its own name\nset in order; that is to say, your method must resemble that of the\ncreation. The order of the creation was, _light_, _distinction of day\nand night_, the _firmament_, the _luminaries_, _sensible creatures_,\n_man_; and, after the creation, the _commandment_. Therefore the order\nof contemplation will be, _reason_, _definition_, _space_, the _stars_,\n_sensible quality_, _man_; and after man is grown up, _subjection to\ncommand_. In the first part of this section, which is entitled Logic, I\nset up the light of reason. In the second, which hath for title the\nGrounds of Philosophy, I distinguish the most common notions by accurate\ndefinition, for the avoiding of confusion and obscurity. The third part\nconcerns the expansion of space, that is Geometry. The fourth contains\nthe Motion of the Stars, together with the doctrine of sensible\nqualities.\n\nIn the second section, if it please God, shall be handled _Man_. In the\nthird section, the doctrine of _Subjection_ is handled already. This is\nthe method I followed; and if it like you, you may use the same; for I\ndo but propound, not commend to you anything of mine. But whatsoever\nshall be the method you will like, I would very fain commend philosophy\nto you, that is to say, the study of wisdom, for want of which we have\nall suffered much damage lately. For even they, that study wealth, do it\nout of love to wisdom; for their treasures serve them but for a\nlooking-glass, wherein to behold and contemplate their own wisdom. Nor\ndo they, that love to be employed in public business, aim at anything\nbut place wherein to show their wisdom. Neither do voluptuous men\nneglect philosophy, but only because they know not how great a pleasure\nit is to the mind of man to be ravished in the vigorous and perpetual\nembraces of the most beauteous world. Lastly, though for nothing else,\nyet because the mind of man is no less impatient of empty time than\nnature is of empty place, to the end you be not forced for want of what\nto do, to be troublesome to men that have business, or take hurt by\nfalling into idle company, but have somewhat of your own wherewith to\nfill up your time, I recommend unto you to study philosophy. Farewell.\n\n T. H.\n\n TITLES OF THE CHAPTERS.\n\n\n PART FIRST,\n\n OR LOGIC.\n\n\n CHAP. PAGE.\n\n 1. Of Philosophy 1\n\n 2. Of Names 13\n\n 3. Of Proposition 29\n\n 4. Of Syllogism 44\n\n 5. Of Erring, Falsity, and Captions 55\n\n 6. Of Method 65\n\n\n PART SECOND,\n\n OR THE FIRST GROUNDS OF PHILOSOPHY.\n\n 7. Of Place and Time 91\n\n 8. Of Body and Accident 101\n\n 9. Of Cause and Effect 120\n\n 10. Of Power and Act 127\n\n 11. Of Identity and Difference 132\n\n 12. Of Quantity 138\n\n 13. Of Analogism, or the Same Proportion 144\n\n 14. Of Strait and Crooked, Angle and Figure 176\n\n\n PART THIRD,\n\n OF THE PROPORTIONS OF MOTIONS AND MAGNITUDES.\n\n 15. Of the Nature, Properties, and divers 203\n Considerations of Motion and Endeavour\n\n 16. Of Motion Accelerated and Uniform, and of Motion 218\n by Concourse\n\n 17. Of Figures Deficient 246\n\n 18. Of the Equation of Strait Lines with the Crooked 268\n Lines of Parabolas, and other Figures made in\n imitation of Parabolas\n\n 19. Of Angles of Incidence and Reflection, equal by 273\n Supposition\n\n 20. Of the Dimension of a Circle, and the Division of 287\n Angles or Arches\n\n 21. Of Circular Motion 317\n\n 22. Of other Variety of Motions 333\n\n 23. Of the Centre of Equiponderation of Bodies 350\n pressing downwards in Strait Parallel Lines\n\n 24. Of Refraction and Reflection 374\n\n\n PART FOURTH,\n\n OF PHYSICS, OR THE PHENOMENA OF NATURE.\n\n 25. Of Sense and Animal Motion 387\n\n 26. Of the World and of the Stars 410\n\n 27. Of Light, Heat, and of Colours 445\n\n 28. Of Cold, Wind, Hard, Ice, Restitution of Bodies 466\n bent, Diaphanous, Lightning and Thunder, and of\n the Heads of Rivers\n\n 29. Of Sound, Odour, Savour, and Touch 485\n\n 30. Of Gravity 508\n\n------------------------------------------------------------------------\n\n COMPUTATION OR LOGIC.\n\n\n -------\n\n\n CHAPTER I.\n\n OF PHILOSOPHY.\n\n1. The Introduction.\u20142. The Definition of Philosophy explained.\u20143.\n Ratiocination of the Mind.\u20144. Properties, what they are.\u20145. How\n Properties are known by Generation, and contrarily.\u20146. The Scope of\n Philosophy.\u20147. The Utility of it.\u20148. The Subject.\u20149. The Parts of\n it.\u201410. The Epilogue.\n\n\n[Sidenote: Introduction.]\n\nPhilosophy seems to me to be amongst men now, in the same manner as corn\nand wine are said to have been in the world in ancient time. For from\nthe beginning there were vines and ears of corn growing here and there\nin the fields; but no care was taken for the planting and sowing of\nthem. Men lived therefore upon acorns; or if any were so bold as to\nventure upon the eating of those unknown and doubtful fruits, they did\nit with danger of their health. In like manner, every man brought\nPhilosophy, that is, Natural Reason, into the world with him; for all\nmen can reason to some degree, and concerning some things: but where\nthere is need of a long series of reasons, there most men wander out of\nthe way, and fall into error for want of method, as it were for want of\nsowing and planting, that is, of improving their reason. And from hence\nit comes to pass, that they who content themselves with daily\nexperience, which may be likened to feeding upon acorns, and either\nreject, or not much regard philosophy, are commonly esteemed, and are,\nindeed, men of sounder judgment than those who, from opinions, though\nnot vulgar, yet full of uncertainty, and carelessly received, do nothing\nbut dispute and wrangle, like men that are not well in their wits. I\nconfess, indeed, that that part of philosophy by which magnitudes and\nfigures are computed, is highly improved. But because I have not\nobserved the like advancement in the other parts of it, my purpose is,\nas far forth as I am able, to lay open the few and first Elements of\nPhilosophy in general, as so many seeds from which pure and true\nPhilosophy may hereafter spring up by little and little.\n\nI am not ignorant how hard a thing it is to weed out of men's minds such\ninveterate opinions as have taken root there, and been confirmed in them\nby the authority of most eloquent writers; especially seeing true (that\nis, accurate) Philosophy professedly rejects not only the paint and\nfalse colours of language, but even the very ornaments and graces of the\nsame; and the first grounds of all science are not only not beautiful,\nbut poor, arid, and, in appearance, deformed. Nevertheless, there being\ncertainly some men, though but few, who are delighted with truth and\nstrength of reason in all things, I thought I might do well to take this\npains for the sake even of those few. I proceed therefore to the matter,\nand take my beginning from the very definition of philosophy, which is\nthis.\n\n[Sidenote: Definition of Philosophy explained.]\n\n2. PHILOSOPHY _is such knowledge of effects or appearances, as we\nacquire by true ratiocination from the knowledge we have first of their\ncauses or generation; And again, of such causes or generations as may be\nfrom knowing first their effects_.\n\nFor the better understanding of which definition, we must consider,\nfirst, that although Sense and Memory of things, which are common to man\nand all living creatures, be knowledge, yet because they are given us\nimmediately by nature, and not gotten by ratiocination, they are not\nphilosophy.\n\nSecondly, seeing Experience is nothing but memory; and Prudence, or\nprospect into the future time, nothing but expectation of such things as\nwe have already had experience of, Prudence also is not to be esteemed\nphilosophy.\n\nBy RATIOCINATION, I mean _computation_. Now to compute, is either to\ncollect the sum of many things that are added together, or to know what\nremains when one thing is taken out of another. _Ratiocination_,\ntherefore, is the same with _addition_ and _substraction_; and if any\nman add _multiplication_ and _division_, I will not be against it,\nseeing multiplication is nothing but addition of equals one to another,\nand division nothing but a substraction of equals one from another, as\noften as is possible. So that all ratiocination is comprehended in these\ntwo operations of the mind, addition and substraction.\n\n[Sidenote: Ratiocination of the Mind.]\n\n3. But how by the _ratiocination_ of our mind, we add and substract in\nour silent thoughts, without the use of words, it will be necessary for\nme to make intelligible by an example or two. If therefore a man see\nsomething afar off and obscurely, although no appellation had yet been\ngiven to anything, he will, notwithstanding, have the same idea of that\nthing for which now, by imposing a name on it, we call it _body_. Again,\nwhen, by coming nearer, he sees the same thing thus and thus, now in one\nplace and now in another, he will have a new idea thereof, namely, that\nfor which we now call such a thing _animated_. Thirdly, when standing\nnearer, he perceives the figure, hears the voice, and sees other things\nwhich are signs of a rational mind, he has a third idea, though it have\nyet no appellation, namely, that for which we now call anything\n_rational_. Lastly, when, by looking fully and distinctly upon it, he\nconceives all that he has seen as one thing, the idea he has now is\ncompounded of his former ideas, which are put together in the mind in\nthe same order in which these three single names, _body_, _animated_,\n_rational_, are in speech compounded into this one name,\n_body-animated-rational_, or _man_. In like manner, of the several\nconceptions of _four sides, equality of sides, and right angles_, is\ncompounded the conception of a _square_. For the mind may conceive a\nfigure of four sides without any conception of their equality, and of\nthat equality without conceiving a right angle; and may join together\nall these single conceptions into one conception or one idea of a\nsquare. And thus we see how the conceptions of the mind are compounded.\nAgain, whosoever sees a man standing near him, conceives the whole idea\nof that man; and if, as he goes away, he follow him with his eyes only,\nhe will lose the idea of those things which were signs of his being\nrational, whilst, nevertheless, the idea of a body-animated remains\nstill before his eyes, so that the idea of rational is substracted from\nthe whole idea of man, that is to say, of body-animated-rational, and\nthere remains that of body-animated; and a while after, at a greater\ndistance, the idea of animated will be lost, and that of body only will\nremain; so that at last, when nothing at all can be seen, the whole idea\nwill vanish out of sight. By which examples, I think, it is manifest\nenough what is the internal ratiocination of the mind without words.\n\nWe must not therefore think that computation, that is, ratiocination,\nhas place only in numbers, as if man were distinguished from other\nliving creatures (which is said to have been the opinion of\n_Pythagoras_) by nothing but the faculty of numbering; for _magnitude,\nbody, motion, time, degrees of quality, action, conception, proportion,\nspeech and names_ (in which all the kinds of philosophy consist) are\ncapable of addition and substraction. Now such things as we add or\nsubstract, that is, which we put into an account, we are said to\n_consider_, in Greek \u03bb\u03bf\u03b3\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9, in which language also \u03c3\u03c5\u03bb\u03bb\u03bf\u03b3\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9\nsignifies to _compute_, _reason_, or _reckon_.\n\n[Sidenote: Properties, what they are.]\n\n4. But _effects_ and the _appearances_ of things to sense, are faculties\nor powers of bodies, which make us distinguish them from one another;\nthat is to say, conceive one body to be equal or unequal, like or unlike\nto another body; as in the example above, when by coming near enough to\nany body, we perceive the motion and going of the same, we distinguish\nit thereby from a tree, a column, and other fixed bodies; and so that\nmotion or going is the _property_ thereof, as being proper to living\ncreatures, and a faculty by which they make us distinguish them from\nother bodies.\n\n[Sidenote: How Properties are known by Generation, and contrarily.]\n\n5. How the knowledge of any effect may be gotten from the knowledge of\nthe generation thereof, may easily be understood by the example of a\ncircle: for if there be set before us a plain figure, having, as near as\nmay be, the figure of a circle, we cannot possibly perceive by sense\nwhether it be a true circle or no; than which, nevertheless, nothing is\nmore easy to be known to him that knows first the generation of the\npropounded figure. For let it be known that the figure was made by the\ncircumduction of a body whereof one end remained unmoved, and we may\nreason thus; a body carried about, retaining always the same length,\napplies itself first to one _radius_, then to another, to a third, a\nfourth, and successively to all; and, therefore, the same length, from\nthe same point, toucheth the circumference in every part thereof, which\nis as much as to say, as all the _radii_ are equal. We know, therefore,\nthat from such generation proceeds a figure, from whose one middle point\nall the extreme points are reached unto by equal _radii_. And in like\nmanner, by knowing first what figure is set before us, we may come by\nratiocination to some generation of the same, though perhaps not that by\nwhich it was made, yet that by which it might have been made; for he\nthat knows that a circle has the property above declared, will easily\nknow whether a body carried about, as is said, will generate a circle or\nno.\n\n[Sidenote: Scope of Philosophy.]\n\n6. The _end_ or _scope_ of philosophy is, that we may make use to our\nbenefit of effects formerly seen; or that, by application of bodies to\none another, we may produce the like effects of those we conceive in our\nmind, as far forth as matter, strength, and industry, will permit, for\nthe commodity of human life. For the inward glory and triumph of mind\nthat a man may have for the mastering of some difficult and doubtful\nmatter, or for the discovery of some hidden truth, is not worth so much\npains as the study of Philosophy requires; nor need any man care much to\nteach another what he knows himself, if he think that will be the only\nbenefit of his labour. The end of knowledge is power; and the use of\ntheorems (which, among geometricians, serve for the finding out of\nproperties) is for the construction of problems; and, lastly, the scope\nof all speculation is the performing of some action, or thing to be\ndone.\n\n[Sidenote: Utility of Philosophy.]\n\n7. But what the _utility_ of philosophy is, especially of natural\nphilosophy and geometry, will be best understood by reckoning up the\nchief commodities of which mankind is capable, and by comparing the\nmanner of life of such as enjoy them, with that of others which want the\nsame. Now, the greatest commodities of mankind are the arts; namely, of\nmeasuring matter and motion; of moving ponderous bodies; of\narchitecture; of navigation; of making instruments for all uses; of\ncalculating the celestial motions, the aspects of the stars, and the\nparts of time; of geography, &c. By which sciences, how great benefits\nmen receive is more easily understood than expressed. These benefits are\nenjoyed by almost all the people of Europe, by most of those of Asia,\nand by some of Africa: but the Americans, and they that live near the\nPoles, do totally want them. But why? Have they sharper wits than these?\nHave not all men one kind of soul, and the same faculties of mind? What,\nthen, makes this difference, except philosophy? Philosophy, therefore,\nis the cause of all these benefits. But the utility of moral and civil\nphilosophy is to be estimated, not so much by the commodities we have by\nknowing these sciences, as by the calamities we receive from not knowing\nthem. Now, all such calamities as may be avoided by human industry,\narise from war, but chiefly from civil war; for from this proceed\nslaughter, solitude, and the want of all things. But the cause of war is\nnot that men are willing to have it; for the will has nothing for object\nbut good, at least that which seemeth good. Nor is it from this, that\nmen know not that the effects of war are evil; for who is there that\nthinks not poverty and loss of life to be great evils? The cause,\ntherefore, of civil war is, that men know not the causes neither of war\nnor peace, there being but few in the world that have learned those\nduties which unite and keep men in peace, that is to say, that have\nlearned the rules of civil life sufficiently. Now, the knowledge of\nthese rules is moral philosophy. But why have they not learned them,\nunless for this reason, that none hitherto have taught them in a clear\nand exact method? For what shall we say? Could the ancient masters of\nGreece, Egypt, Rome, and others, persuade the unskilful multitude to\ntheir innumerable opinions concerning the nature of their gods, which\nthey themselves knew not whether they were true or false, and which were\nindeed manifestly false and absurd; and could they not persuade the same\nmultitude to civil duty, if they themselves had understood it? Or shall\nthose few writings of geometricians which are extant, be thought\nsufficient for the taking away of all controversy in the matters they\ntreat of, and shall those innumerable and huge volumes of _ethics_ be\nthought unsufficient, if what they teach had been certain and well\ndemonstrated? What, then, can be imagined to be the cause that the\nwritings of those men have increased science, and the writings of these\nhave increased nothing but words, saving that the former were written by\nmen that knew, and the latter by such as knew not, the doctrine they\ntaught only for ostentation of their wit and eloquence? Nevertheless, I\ndeny not but the reading of some such books is very delightful; for they\nare most eloquently written, and contain many clear, wholesome and\nchoice sentences, which yet are not universally true, though by them\nuniversally pronounced. From whence it comes to pass, that the\ncircumstances of times, places, and persons being changed, they are no\nless frequently made use of to confirm wicked men in their purposes,\nthan to make them understand the precepts of civil duties. Now that\nwhich is chiefly wanting in them, is a true and certain rule of our\nactions, by which we might know whether that we undertake be just or\nunjust. For it is to no purpose to be bidden in every thing to do right,\nbefore there be a certain rule and measure of right established, which\nno man hitherto hath established. Seeing, therefore, from the not\nknowing of civil duties, that is, from the want of moral science,\nproceed civil wars, and the greatest calamities of mankind, we may very\nwell attribute to such science the production of the contrary\ncommodities. And thus much is sufficient, to say nothing of the praises\nand other contentment proceeding from philosophy, to let you see the\nutility of the same in every kind thereof.\n\n[Sidenote: Subject of Philosophy.]\n\n8. The _subject_ of Philosophy, or the matter it treats of, is every\nbody of which we can conceive any generation, and which we may, by any\nconsideration thereof, compare with other bodies, or which is capable of\ncomposition and resolution; that is to say, every body of whose\ngeneration or properties we can have any knowledge. And this may be\ndeduced from the definition of philosophy, whose profession it is to\nsearch out the properties of bodies from their generation, or their\ngeneration from their properties; and, therefore, where there is no\ngeneration or property, there is no philosophy. Therefore it excludes\n_Theology_, I mean the doctrine of God, eternal, ingenerable,\nincomprehensible, and in whom there is nothing neither to divide nor\ncompound, nor any generation to be conceived.\n\nIt excludes the doctrine of _angels_, and all such things as are thought\nto be neither bodies nor properties of bodies; there being in them no\nplace neither for composition nor division, nor any capacity of more and\nless, that is to say, no place for ratiocination.\n\nIt excludes _history_, as well _natural_ as _political_, though most\nuseful (nay necessary) to philosophy; because such knowledge is but\nexperience, or authority, and not ratiocination.\n\nIt excludes all such knowledge as is acquired by Divine inspiration, or\nrevelation, as not derived to us by reason, but by Divine grace in an\ninstant, and, as it were, by some sense supernatural.\n\nIt excludes not only all doctrines which are false, but such also as are\nnot well-grounded; for whatsoever we know by right ratiocination, can\nneither be false nor doubtful; and, therefore, _astrology_, as it is now\nheld forth, and all such divinations rather than sciences, are excluded.\n\nLastly, the doctrine of _God's worship_ is excluded from philosophy, as\nbeing not to be known by natural reason, but by the authority of the\nChurch; and as being the object of faith, and not of knowledge.\n\n[Sidenote: Parts of Philosophy.]\n\n9. The principal parts of philosophy are two. For two chief kinds of\nbodies, and very different from one another, offer themselves to such as\nsearch after their generation and properties; one whereof being the work\nof nature, is called a _natural body_, the other is called a\n_commonwealth_, and is made by the wills and agreement of men. And from\nthese spring the two parts of philosophy, called _natural_ and _civil_.\nBut seeing that, for the knowledge of the properties of a commonwealth,\nit is necessary first to know the dispositions, affections, and manners\nof men, civil philosophy is again commonly divided into two parts,\nwhereof one, which treats of men's dispositions and manners, is called\n_ethics_; and the other, which takes cognizance of their civil duties,\nis called _politics_, or simply _civil philosophy_. In the first place,\ntherefore (after I have set down such premises as appertain to the\nnature of philosophy in general), I will discourse of _bodies natural_;\nin the second, of the _dispositions and manners of men_; and in the\nthird, of the _civil duties of subjects_.\n\n[Sidenote: Epilogue.]\n\n10. To conclude; seeing there may be many who will not like this my\ndefinition of philosophy, and will say, that, from the liberty which a\nman may take of so defining as seems best to himself, he may conclude\nany thing from any thing (though I think it no hard matter to\ndemonstrate that this definition of mine agrees with the sense of all\nmen); yet, lest in this point there should be any cause of dispute\nbetwixt me and them, I here undertake no more than to deliver the\nelements of that science by which the effects of anything may be found\nout from the known generation of the same, or contrarily, the generation\nfrom the effects; to the end that they who search after other\nphilosophy, may be admonished to seek it from other principles.\n\n\n -------\n\n\n CHAPTER II.\n\n OF NAMES.\n\n1. The necessity of sensible Moniments or Marks for the help of Memory:\n a Mark defined.\u20142. The necessity of Marks for the signification of\n the conceptions of the Mind.\u20143. Names supply both those\n necessities.\u20144. The Definition of a Name.\u20145. Names are Signs not of\n Things, but of our Cogitations.\u20146. What it is we give Names to.\u20147.\n Names Positive and Negative.\u20148. Contradictory Names.\u20149. A Common\n Name.\u201410. Names of the First and Second Intention.\u201411. Universal,\n Particular, Individual, and Indefinite Names.\u201412. Names Univocal and\n Equivocal.\u201413. Absolute and Relative Names.\u201414. Simple and\n Compounded Names.\u201415. A Predicament described.\u201416. Some things to be\n noted concerning Predicaments.\n\n\n[Sidenote: Necessity of sensible Moniments or Marks for the help of\n Memory.]\n\n1. How unconstant and fading men's thoughts are, and how much the\nrecovery of them depends upon chance, there is none but knows by\ninfallible experience in himself. For no man is able to remember\nquantities without sensible and present measures, nor colours without\nsensible and present patterns, nor number without the names of numbers\ndisposed in order and learned by heart. So that whatsoever a man has put\ntogether in his mind by ratiocination without such helps, will presently\nslip from him, and not be revocable but by beginning his ratiocination\nanew. From which it follows, that, for the acquiring of philosophy, some\nsensible moniments are necessary, by which our past thoughts may be not\nonly reduced, but also registered every one in its own order.[Sidenote:\nA Mark defined] These moniments I call MARKS, namely, sensible things\ntaken at pleasure, that, by the sense of them, such thoughts may be\nrecalled to our mind as are like those thoughts for which we took them.\n\n[Sidenote: Necessity of Marks for the signification of the conceptions\n of the Mind.]\n\n2. Again, though some one man, of how excellent a wit soever, should\nspend all his time partly in reasoning, and partly in inventing marks\nfor the help of his memory, and advancing himself in learning; who sees\nnot that the benefit he reaps to himself will not be much, and to others\nnone at all? For unless he communicate his notes with others, his\nscience will perish with him. But if the same notes be made common to\nmany, and so one man's inventions be taught to others, sciences will\nthereby be increased to the general good of mankind. It is therefore\nnecessary, for the acquiring of philosophy, that there be certain signs,\nby which what one man finds out may be manifested and made known to\nothers. Now, those things we call SIGNS are the _antecedents of their\nconsequents, and the consequents of their antecedents, as often as we\nobserve them to go before or follow after in the same manner_. For\nexample, a thick cloud is a sign of rain to follow, and rain a sign that\na cloud has gone before, for this reason only, that we seldom see clouds\nwithout the consequence of rain, nor rain at any time but when a cloud\nhas gone before. And of signs, some are _natural_, whereof I have\nalready given an example, others are _arbitrary_, namely, those we make\nchoice of at our own pleasure, as a bush hung up, signifies that wine is\nto be sold there; a stone set in the ground signifies the bound of a\nfield; and words so and so connected, signify the cogitations and\nmotions of our mind. The difference, therefore, betwixt marks and signs\nis this, that we make those for our own use, but these for the use of\nothers.\n\n[Sidenote: Names supply both those necessities.]\n\n3. Words so connected as that they become signs of our thoughts, are\ncalled SPEECH, of which every part is a _name_. But seeing (as is said)\nboth marks and signs are necessary for the acquiring of philosophy,\n(marks by which we may remember our own thoughts, and signs by which we\nmay make our thoughts known to others), names do both these offices; but\nthey serve for marks before they be used as signs. For though a man were\nalone in the world, they would be useful to him in helping him to\nremember; but to teach others, (unless there were some others to be\ntaught) of no use at all. Again, names, though standing singly by\nthemselves, are marks, because they serve to recal our own thoughts to\nmind; but they cannot be signs, otherwise than by being disposed and\nordered in speech as parts of the same. For example, a man may begin\nwith a word, whereby the hearer may frame an idea of something in his\nmind, which, nevertheless, he cannot conceive to be the idea which was\nin the mind of him that spake, but that he would say something which\nbegan with that word, though perhaps not as by itself, but as part of\nanother word. So that the nature of a name consists principally in this,\nthat it is a mark taken for memory's sake; but it serves also by\naccident to signify and make known to others what we remember ourselves,\nand, therefore, I will define it thus:\n\n[Sidenote: Definition of a Name.]\n\n4. _A_ NAME _is a word taken at pleasure to serve for a mark, which may\nraise in our mind a thought like to some thought we had before, and\nwhich being pronounced to others, may be to them a sign of what thought\nthe speaker had, or had not before in his mind_. And it is for brevity's\nsake that I suppose the original of names to be arbitrary, judging it a\nthing that may be assumed as unquestionable. For considering that new\nnames are daily made, and old ones laid aside; that diverse nations use\ndifferent names, and how impossible it is either to observe similitude,\nor make any comparison betwixt a name and a thing, how can any man\nimagine that the names of things were imposed from their natures? For\nthough some names of living creatures and other things, which our first\nparents used, were taught by God himself; yet they were by him\narbitrarily imposed, and afterwards, both at the Tower of Babel, and\nsince, in process of time, growing everywhere out of use, are quite\nforgotten, and in their room have succeeded others, invented and\nreceived by men at pleasure. Moreover, whatsoever the common use of\nwords be, yet philosophers, who were to teach their knowledge to others,\nhad always the liberty, and sometimes they both had and will have a\nnecessity, of taking to themselves such names as they please for the\nsignifying of their meaning, if they would have it understood. Nor had\nmathematicians need to ask leave of any but themselves to name the\nfigures they invented, _parabolas_, _hyperboles_, _cissoeides_,\n_quadratices_, &c. or to call one magnitude A, another B.\n\n[Sidenote: Names are signs not of things, but of our cogitations.]\n\n5. But seeing names ordered in speech (as is defined) are signs of our\nconceptions, it is manifest they are not signs of the things themselves;\nfor that the sound of this word _stone_ should be the sign of a stone,\ncannot be understood in any sense but this, that he that hears it\ncollects that he that pronounces it thinks of a stone. And, therefore,\nthat disputation, whether names signify the matter or form, or something\ncompounded of both, and other like subtleties of the _metaphysics_, is\nkept up by erring men, and such as understand not the words they dispute\nabout.\n\n[Sidenote: What it is we give names to.]\n\n6. Nor, indeed, is it at all necessary that every name should be the\nname of something. For as these, a _man_, a _tree_, a _stone_, are the\nnames of the things themselves, so the images of a man, of a tree, and\nof a stone, which are represented to men sleeping, have their names\nalso, though they be not things, but only fictions and phantasms of\nthings. For we can remember these; and, therefore, it is no less\nnecessary that they have names to mark and signify them, than the things\nthemselves. Also this word _future_ is a name, but no future thing has\nyet any being, nor do we know whether that which we call future, shall\never have a being or no. Nevertheless, seeing we use in our mind to knit\ntogether things past with those that are present, the name _future_\nserves to signify such knitting together. Moreover, that which neither\nis, nor has been, nor ever shall, or ever can be, has a name, namely,\n_that which neither is nor has been_, &c.; or more briefly this,\n_impossible_. To conclude; this word _nothing_ is a name, which yet\ncannot be the name of any thing: for when, for example, we substract 2\nand 3 from 5, and so nothing remaining, we would call that substraction\nto mind, this speech _nothing remains_, and in it the word _nothing_ is\nnot unuseful. And for the same reason we say truly, _less than nothing_\nremains, when we substract more from less; for the mind feigns such\nremains as these for doctrine's sake, and desires, as often as is\nnecessary, to call the same to memory. But seeing every name has some\nrelation to that which is named, though that which we name be not always\na thing that has a being in nature, yet it is lawful for doctrine's sake\nto apply the word _thing_ to whatsoever we name; as if it were all one\nwhether that thing be truly existent, or be only feigned.\n\n[Sidenote: Names Positive and Negative.]\n\n7. The first distinction of names is, that some are _positive_, or\n_affirmative_, others _negative_, which are also called _privative_ and\n_indefinite_. _Positive_ are such as we impose for the likeness,\nequality, or identity of the things we consider; _negative_, for the\ndiversity, unlikeness, or inequality of the same. Examples of the former\nare, _a man_, _a philosopher_; for a _man_ denotes any one of a\nmultitude of men, and _a philosopher_, any one of many philosophers, by\nreason of their similitude; also, _Socrates_ is a positive name, because\nit signifies always one and the same man. Examples of _negatives_ are\nsuch positives as have the negative particle _not_ added to them, as\n_not-man_, _not-philosopher_. But positives were before negatives; for\notherwise there could have been no use at all of these. For when the\nname of _white_ was imposed upon certain things, and afterwards upon\nother things, the names of _black_, _blue_, _transparent_, _&c._ the\ninfinite dissimilitudes of these with _white_ could not be comprehended\nin any one name, save that which had in it the negation of white, that\nis to say, the name _not-white_, or some other equivalent to it, in\nwhich the word _white_ is repeated, such as _unlike to white_, _&c._ And\nby these _negative names_, we take notice ourselves, and signify to\nothers what we have not thought of.\n\n[Sidenote: Contradictory names.]\n\n8. Positive and negative names are _contradictory_ to one another, so\nthat they cannot both be the name of the same thing. Besides, of\ncontradictory names, one is the name of anything whatsoever; for\nwhatsoever is, is either man, or not-man, white or not-white, and so of\nthe rest. And this is so manifest, that it needs no farther proof or\nexplication; for they that say _the same thing cannot both be, and not\nbe_, speak obscurely; but they that say, _whatsoever is, either is, or\nis not_, speak also absurdly and ridiculously. The certainty of this\naxiom, viz. _of two contradictory names, one is the name of anything\nwhatsoever, the other not_, is the original and foundation of all\nratiocination, that is, of all philosophy; and therefore it ought to be\nso exactly propounded, that it may be of itself clear and perspicuous to\nall men; as indeed it is, saving to such, as reading the long discourses\nmade upon this subject by the writers of _metaphysics_ (which they\nbelieve to be some egregious learning) think they understand not, when\nthey do.\n\n[Sidenote: A common name.]\n\n9. Secondly, of names, some are _common_ to many things, as a _man_, _a\ntree_; others _proper_ to one thing, as _he that writ the Iliad_,\n_Homer_, _this man_, _that man_. And a common name, being the name of\nmany things severally taken, but not collectively of all together (as\nman is not the name of all mankind, but of every one, as of Peter, John,\nand the rest severally) is therefore called an _universal name_; and\ntherefore this word _universal_ is never the name of any thing existent\nin nature, nor of any idea or phantasm formed in the mind, but always\nthe name of some word or name; so that when _a living creature_, _a\nstone_, _a spirit_, or any other thing, is said to be _universal_, it is\nnot to be understood, that any man, stone, &c. ever was or can be\nuniversal, but only that these words, _living creature_, _stone_, _&c._\nare _universal names_, that is, names common to many things; and the\nconceptions answering them in our mind, are the images and phantasms of\nseveral living creatures, or other things. And therefore, for the\nunderstanding of the extent of an universal name, we need no other\nfaculty but that of our imagination, by which we remember that such\nnames bring sometimes one thing, sometimes another, into our mind. Also\nof common names, some are more, some less common. _More common_, is that\nwhich is the name of more things; _less common_, the name of fewer\nthings; as _living creature_ is more common than _man_, or _horse_, or\n_lion_, because it comprehends them all: and therefore a more common\nname, in respect of a less common, is called the _genus_, or a _general\nname_; and this in respect of that, the _species_, or a _special name_.\n\n[Sidenote: Names of the first and second intention.]\n\n10. And from hence proceeds the third distinction of names, which is,\nthat some are called names of the _first_, others of the _second\nintention_. Of the _first intention_ are the names of things, a _man_,\n_stone_, _&c._: of the _second_ are the names of names and speeches, as\n_universal_, _particular_, _genus_, _species_, _syllogism_, and the\nlike. But it is hard to say why those are called names of the _first_,\nand these of the _second intention_, unless perhaps it was first\nintended by us to give names to those things which are of daily use in\nthis life, and afterwards to such things as appertain to science, that\nis, that our second intention was to give names to names. But whatsoever\nthe cause hereof may be, yet this is manifest, that _genus_, _species_,\n_definition_, _&c._ are names of words and names only; and therefore to\nput _genus_ and _species_ for things, and _definition_ for the nature of\nany thing, as the writers of _metaphysics_ have done, is not right,\nseeing they be only significations of what we think of the nature of\nthings.\n\n[Sidenote: Universal, particular, individual, and indefinite names.]\n\n11. Fourthly, some names are of _certain_ and _determined_, others of\n_uncertain_ and _undetermined_ signification. Of _determined_ and\n_certain_ signification is, first, that name which is given to any one\nthing by itself, and is called an _individual name_; as _Homer_, _this\ntree_, _that living creature_, &c. Secondly that which has any of these\nwords, _all_, _every_, _both_, _either_, or the like added to it; and it\nis therefore called an universal name, because it signifies every one of\nthose things to which it is common; and of _certain_ signification for\nthis reason, that he which hears, conceives in his mind the same thing\nthat he which speaks would have him conceive. Of _indefinite_\nsignification is, first, that name which has the word _some_, or the\nlike added to it, and is called a _particular name_; secondly, a common\nname set by itself without any note either of universality or\nparticularity, as _man_, _stone_, and is called an _indefinite name_;\nbut both _particular_ and _indefinite_ names are of uncertain\nsignification, because the hearer knows not what thing it is the speaker\nwould have him conceive; and therefore in speech, particular and\nindefinite names are to be esteemed equivalent to one another. But these\nwords, _all_, _every_, _some_, _&c._ which denote universality and\nparticularity, are not names, but parts only of names; so that _every\nman_, and _that man which the hearer conceives in his mind_, are all\none; and _some man_, and _that man which the speaker thought of_,\nsignify the same. From whence it is evident, that the use of signs of\nthis kind, is not for a man's own sake, or for his getting of knowledge\nby his own private meditation (for every man has his own thoughts\nsufficiently determined without such helps as these) but for the sake of\nothers; that is, for the teaching and signifying of our conceptions to\nothers; nor were they invented only to make us remember, but to make us\nable to discourse with others.\n\n[Sidenote: Names univocal and equivocal.]\n\n12. Fifthly, names are usually distinguished into _univocal_ and\n_equivocal_. _Univocal_ are those which in the same train of discourse\nsignify always the same thing; but _equivocal_ those which mean\nsometimes one thing and sometimes another. Thus, the name _triangle_ is\nsaid to be _univocal_, because it is always taken in the same sense; and\n_parabola_ to be _equivocal_, for the signification it has sometimes of\nallegory or similitude, and sometimes of a certain geometrical figure.\nAlso every _metaphor_ is by profession _equivocal_. But this distinction\nbelongs not so much to names, as to those that use names, for some use\nthem properly and accurately for the finding out of truth; others draw\nthem from their proper sense, for ornament or deceit.\n\n[Sidenote: Absolute and relative names.]\n\n13. Sixthly, of names, some are _absolute_, others _relative_.\n_Relative_ are such as are imposed for some comparison, as _father_,\n_son_, _cause_, _effect_, _like_, _unlike_, _equal_, _unequal_,\n_master_, _servant_, _&c._ And those that signify no comparison at all\nare _absolute names_. But, as it was noted above, that universality is\nto be attributed to words and names only, and not to things, so the same\nis to be said of other distinctions of names; for no things are either\n_univocal_ or _equivocal_, or _relative_ or _absolute_. There is also\nanother distinction of names into _concrete_ and _abstract_; but because\nabstract names proceed from proposition, and can have no place where\nthere is no affirmation, I shall speak of them hereafter.\n\n[Sidenote: Simple and compounded names.]\n\n14. Lastly, there are _simple_ and _compounded names_. But here it is to\nbe noted, that a name is not taken in philosophy, as in grammar, for one\nsingle word, but for any number of words put together to signify one\nthing; for among philosophers _sentient animated body_ passes but for\none name, being the name of every living creature, which yet, among\ngrammarians, is accounted three names. Also a _simple name_ is not here\ndistinguished from a _compounded name_ by a preposition, as in grammar.\nBut I call a _simple name_, that which in every kind is the most common\nor most universal; and that a _compounded name_, which by the joining of\nanother name to it, is made less universal, and signifies that more\nconceptions than one were in the mind, for which that latter name was\nadded. For example, in the conception of _man_ (as is shown in the\nformer chapter.) First, he is conceived to be something that has\nextension, which is marked by the word _body_. _Body_, therefore, is a\n_simple name_, being put for that first single conception; afterwards,\nupon the sight of such and such motion, another conception arises, for\nwhich he is called an _animated body_; and this I here call a\n_compounded name_, as I do also the name _animal_, which is equivalent\nto an _animated body_. And, in the same manner, an _animated rational\nbody_, as also a _man_, which is equivalent to it, is a more compounded\nname. And by this we see how the composition of conceptions in the mind\nis answerable to the composition of names; for, as in the mind one idea\nor phantasm succeeds to another, and to this a third; so to one name is\nadded another and another successively, and of them all is made one\n_compounded_ name. Nevertheless we must not think bodies which are\nwithout the mind, are compounded in the same manner, namely, that there\nis in nature a body, or any other imaginable thing existent, which at\nthe first has no magnitude, and then, by the addition of magnitude,\ncomes to have quantity, and by more or less quantity to have density or\nrarity; and again, by the addition of figure, to be figurate, and after\nthis, by the injection of light or colour, to become lucid or coloured;\nthough such has been the philosophy of many.\n\n[Sidenote: A predicament described.]\n\n15. The writers of _logic_ have endeavoured to digest the names of all\nthe kinds of things into certain _scales_ or degrees, by the continual\nsubordination of names less common, to names more common. In the scale\nof _bodies_ they put in the first and highest place _body_ simply, and\nin the next place under it less common names, by which it may be more\nlimited and determined, namely _animated_ and _inanimated_, and so on\ntill they come to _individuals_. In like manner, in the scale of\n_quantities_, they assign the first place to _quantity_, and the next to\n_line_, _superficies_, and _solid_, which are names of less latitude;\nand these orders or scales of names they usually call _predicaments_ and\n_categories_. And of this ordination not only positive, but negative\nnames also are capable; which may be exemplified by such forms of the\n_predicaments_ as follow:\n\n THE FORM OF THE PREDICAMENT OF BODY.\n\n Not-Body, or\n Accident.\n { Not animated.\n Body { { Not living Creature.\n { Animated {\n { Living { Not Man.\n { Creature { { Not Peter.\n { Man {\n { Peter.\n\n { { Quantity, or so much.\n Both Accident and Body { Absolutely, as {\n are considered { or { Quality, or such.\n { Comparatively, which is called their Relation.\n\n THE FORM OF THE PREDICAMENT OF QUANTITY.\n\n { Not continual,\n { as Number. { Line.\n Quantity. { { Of itself, as { Superficies.\n { { { Solid.\n { Continual {\n { { Time, by Line.\n { { Motion, by Line and\n { By accident, as { Time.\n { Force, by Motion and\n { Solid.\n\nWhere, it is to be noted, that _line_, _superficies_, and _solid_, may\nbe said to be of such and such quantity, that is, to be originally and\nof their own nature capable of equality and inequality; but we cannot\nsay there is either majority or minority, or equality, or indeed any\nquantity at all, in _time_, without the help of _line_ and _motion_; nor\nin _motion_, without _line_ and _time_; nor in _force_, otherwise than\nby _motion_ and _solid_.\n\n THE FORM OF THE PREDICAMENT OF QUALITY.\n\n { Seeing.\n { Hearing.\n { Primary { Smelling.\n { { Tasting.\n { Perception { { Touching.\n { by Sense {\n { { { Imagination.\n { { Secondary { Affection { pleasant.\n Quality { { { unpleasant.\n { { By Seeing, as Light and Colour.\n { { By Hearing, as Sound.\n { Sensible { By Smelling, as Odours.\n { Quality { By Tasting, as Savours.\n { By Touching, as Hardness, Heat,\n { Cold, &c.\n\n THE FORM OF THE PREDICAMENT OF RELATION.\n\n { Magnitudes, as Equality and Inequality.\n { Qualities, as Likeness and Unlikeness.\n Relation of { { Together { In Place.\n { { { In Time.\n { Order {\n { { In Place { Former.\n { Not together { { Later\n { In Time { Former.\n { Later\n\n[Sidenote: Some things to be noted concerning predicaments.]\n\n16. Concerning which predicaments it is to be noted, in the first place,\nthat as the division is made in the first predicament into contradictory\nnames, so it might have been done in the rest. For, as there, _body_ is\ndivided into _animated_ and _not-animated_, so, in the second\npredicament, _continual quantity_ may be divided into _line_ and\n_not-line_, and again, _not-line_ into _superficies_ and\n_not-superficies_, and so in the rest; but it was not necessary.\n\nSecondly, it is to be observed, that of _positive_ names the former\ncomprehends the latter; but of _negatives_ the former is comprehended by\nthe latter. For example, _living-creature_ is the name of every man, and\ntherefore it comprehends the name _man_; but, on the contrary, _not-man_\nis the name of everything which is not-living-creature, and therefore\nthe name _not-living-creature_, which is put first, is comprehended by\nthe latter name, _not-man_.\n\nThirdly, we must take heed that we do not think, that as names, so the\ndiversities of things themselves may be searched out and determined by\nsuch distinctions as these; or that arguments may be taken from hence\n(as some have done ridiculously) to prove that the kinds of things are\nnot infinite.\n\nFourthly, I would not have any man think I deliver the forms above for a\ntrue and exact ordination of names; for this cannot be performed as long\nas philosophy remains imperfect; nor that by placing (for example)\n_light_ in the predicament of _qualities_, while another places the same\nin the predicament of bodies, I pretend that either of us ought for this\nto be drawn from his opinion; for this is to be done only by arguments\nand ratiocination, and not by disposing of words into _classes_.\n\nLastly, I confess I have not yet seen any great use of the predicaments\nin philosophy. I believe _Aristotle_ when he saw he could not digest the\nthings themselves into such orders, might nevertheless desire out of his\nown authority to reduce words to such forms, as I have done; but I do it\nonly for this end, that it may be understood what this ordination of\nwords is, and not to have it received for true, till it be demonstrated\nby good reason to be so.\n\n\n -------\n\n\n CHAPTER III.\n\n OF PROPOSITION.\n\n1. Divers kinds of speech.\u20142. Proposition defined.\u20143. Subject,\n predicate, and copula, what they are; and abstract and concrete\n what. 4. The use and abuse of names abstract.\u20145. Proposition,\n universal and particular.\u20146. Affirmative and negative.\u20147. True and\n false.\u20148. True and false belongs to speech, and not to things.\u20149.\n Proposition, primary, not primary, definition, axiom, petition.\u201410.\n Proposition, necessary and contingent.\u201411. Categorical and\n hypothetical.\u201412. The same proposition diversely pronounced.\u201413.\n Propositions that may be reduced to the same categorical\n proposition, are equipollent.\u201414. Universal propositions converted\n by contradictory names, are equipollent.\u201415. Negative propositions\n are the same, whether negation be before or after the copula.\u201416.\n Particular propositions simply converted, are equipollent.\u201417. What\n are subaltern, contrary, subcontrary, and contradictory\n propositions.\u201418. Consequence, what it is.\u201419. Falsity cannot follow\n from truth.\u201420. How one proposition is the cause of another.\n\n\n[Sidenote: Divers kinds of speech.]\n\n1. From the connexion or contexture of names arise divers kinds of\nspeech, whereof some signify the desires and affections of men; such\nare, first, _interrogations_, which denote the desire of knowing: as,\n_Who is a good man?_ In which speech there is one name expressed, and\nanother desired and expected from him of whom we ask the same. Then\n_prayers_, which signify the desire of having something; _promises_,\n_threats_, _wishes_, _commands_, _complaints_, and other significations\nof other affections. Speech may also be absurd and insignificant; as\nwhen there is a succession of words, to which there can be no succession\nof thoughts in mind to answer them; and this happens often to such, as,\nunderstanding nothing in some subtle matter, do, nevertheless, to make\nothers believe they understand, speak of the same incoherently; for the\nconnection of incoherent words, though it want the end of speech (which\nis signification) yet it is speech; and is used by writers of\n_metaphysics_ almost as frequently as speech significative. In\nphilosophy, there is but one kind of speech useful, which some call in\nLatin _dictum_, others _enuntiatum et pronunciatum_; but most men call\nit _proposition_, and is the speech of those that affirm or deny, and\nexpresseth truth or falsity.\n\n[Sidenote: Proposition defined.]\n\n2. _A_ PROPOSITION _is a speech consisting of two names copulated, by\nwhich he that speaketh signifies he conceives the latter name to be the\nname of the same thing whereof the former is the name; or_ (which is all\none) _that the former name is comprehended by the latter_. For example,\nthis speech, _man is a living creature_, in which two names are\ncopulated by the verb _is_, is a _proposition_, for this reason, that he\nthat speaks it conceives both _living creature_ and _man_ to be names of\nthe same thing, or that the former name, _man_, is comprehended by the\nlatter name, _living creature_. Now the former name is commonly called\nthe _subject_, or _antecedent_, or the _contained name_, and the latter\n_predicate_, _consequent_, or _containing name_. The sign of connection\namongst most nations is either some word, as the word _is_ in the\nproposition _man is a living creature_, or some case or termination of a\nword, as in this proposition, _man walketh_ (which is equivalent to\nthis, _man is walking_); the termination by which it is said he\n_walketh_, rather than he _is walking_, signifieth that those two are\nunderstood to be copulated, or to be names of the same thing.\n\nBut there are, or certainly may be, some nations that have no word which\nanswers to our verb _is_, who nevertheless form propositions by the\nposition only of one name after another, as if instead of _man is a\nliving creature_, it should be said _man a living creature_; for the\nvery order of the names may sufficiently show their connection; and they\nare as apt and useful in philosophy, as if they were copulated by the\nverb _is_.\n\n[Sidenote: Subject, predicate, and copula, what they are, and abstract\n and concrete what.]\n\n3. Wherefore, in every proposition three things are to be considered,\n_viz._ the two names, which are the _subject_, and the _predicate_, and\ntheir _copulation_; both which names raise in our mind the thought of\none and the same thing; but the copulation makes us think of the cause\nfor which those names were imposed on that thing. As, for example, when\nwe say _a body is moveable_, though we conceive the same thing to be\ndesigned by both those names, yet our mind rests not there, but searches\nfarther what it is _to be a body_, or _to be moveable_, that is, wherein\nconsists the difference betwixt these and other things, for which these\nare so called, others are not so called. They, therefore, that seek what\nit is _to be_ any thing, as _to be moveable_, _to be hot_, _&c._ seek in\nthings the causes of their names.\n\nAnd from hence arises that distinction of names (touched in the last\nchapter) into _concrete_ and _abstract_. For _concrete_ is the name of\nany thing which we suppose to have a being, and is therefore called the\n_subject_, in Latin _suppositum_, and in Greek \u1f51\u03c0\u03bf\u03ba\u03ad\u03b9\u03bc\u03b5\u03bd\u03bf\u03bd; as _body_,\n_moveable_, _moved_, _figurate_, _a cubit high_, _hot_, _cold_, _like_,\n_equal_, _Appius_, _Lentulus_, and the like; and, _abstract_ is that\nwhich in any subject denotes the cause of the concrete name, as _to be a\nbody_, _to be moveable_, _to be moved_, _to be figurate_, _to be of such\nquantity_, _to be hot_, _to be cold_, _to be like_, _to be equal_, _to\nbe Appius_, _to be Lentulus_, _&c._ Or names equivalent to these, which\nare most commonly called _abstract_ names, as _corporiety_, _mobility_,\n_motion_, _figure_, _quantity_, _heat_, _cold_, _likeness_, _equality_,\nand (as Cicero has it) _Appiety_ and _Lentulity_. Of the same kind also\nare infinitives; for _to live_ and _to move_ are the same with _life_\nand _motion_, or _to be living_ and _to be moved_. But _abstract names_\ndenote only the causes of _concrete names_, and not the things\nthemselves. For example, when we see any thing, or conceive in our mind\nany visible thing, that thing appears to us, or is conceived by us, not\nin one point, but as having parts distant from one another, that is, as\nbeing extended and filling some space. Seeing therefore we call the\nthing so conceived _body_, the cause of that name is, that that thing is\n_extended_, or the _extension_ or _corporiety_ of it. So when we see a\nthing appear sometimes here, sometimes there, and call it _moved_ or\n_removed_, the cause of that name is that it _is moved_ or the _motion_\nof the same.\n\nAnd these causes of names are the same with the causes of our\nconceptions, namely, some power of action, or affection of the thing\nconceived, which some call the manner by which any thing works upon our\nsenses, but by most men they are called _accidents_; I say accidents,\nnot in that sense in which accident is opposed to necessary; but so, as\nbeing neither the things themselves, nor parts thereof, do nevertheless\naccompany the things in such manner, that (saving extension) they may\nall perish, and be destroyed, but can never be abstracted.\n\n[Sidenote: The use and abuse of names abstract.]\n\n4. There is also this difference betwixt _concrete_ and _abstract_\nnames, that those were invented before propositions, but these after;\nfor these could have no being till there were propositions, from whose\n_copula_ they proceed. Now in all matters that concern this life, but\nchiefly in philosophy, there is both great use and great abuse of\n_abstract names_; and the use consists in this, that without them we\ncannot, for the most part, either reason, or compute the properties of\nbodies; for when we would multiply, divide, add, or substract heat,\nlight, or motion, if we should double or add them together by concrete\nnames, saying (for example) hot is double to hot, light double to light,\nor moved double to moved, we should not double the properties, but the\nbodies themselves that are hot, light, moved, &c. which we would not do.\nBut the abuse proceeds from this, that some men seeing they can\nconsider, that is (as I said before) bring into account the increasings\nand decreasings of quantity, heat and other accidents, without\nconsidering their bodies or subjects (which they call _abstracting_, or\nmaking to exist apart by themselves) they speak of accidents, as if they\nmight be separated from all bodies. And from hence proceed the gross\nerrors of writers of metaphysics; for, because they can consider thought\nwithout the consideration of body, they infer there is no need of a\nthinking-body; and because quantity may be considered without\nconsidering body, they think also that quantity may be without body, and\nbody without quantity; and that a body has quantity by the addition of\nquantity to it. From the same fountain spring those insignificant words,\n_abstract substance_, _separated essence_, and the like; as also that\nconfusion of words derived from the Latin verb _est_, as _essence_,\n_essentiality_, _entity_, _entitative_; besides _reality_,\n_aliquiddity_, _quiddity_, _&c._ which could never have been heard of\namong such nations as do not copulate their names by the verb _is_, but\nby adjective verbs, as runneth, readeth, &c. or by the mere placing of\none name after another; and yet seeing such nations compute and reason,\nit is evident that philosophy has no need of those words _essence_,\n_entity_, and other the like barbarous terms.\n\n[Sidenote: Proposition, universal and particular.]\n\n5. There are many distinctions of propositions, whereof the first is,\nthat some are _universal_, others _particular_, others _indefinite_, and\nothers _singular_; and this is commonly called the distinction of\n_quantity_. An _universal_ proposition is that whose subject is affected\nwith the sign of an universal name, as _every man is a living creature_.\n_Particular_, that whose subject is affected with the sign of a\nparticular name, as _some man is learned_. An _indefinite_ proposition\nhas for its subject a common name, and put without any sign, as _man is\na living creature_, _man is learned_. And a _singular_ proposition is\nthat whose subject is a singular name, as _Socrates is a philosopher_,\n_this man is black_.\n\n[Sidenote: Affirmative and negative.]\n\n6. The second distinction is into _affirmative_ and _negative_, and is\ncalled the distinction of _quality_. An _affirmative_ proposition is\nthat whose predicate is a positive name, as _man is a living creature_.\n_Negative_, that whose predicate is a negative name, as _man is not a\nstone_.\n\n[Sidenote: True & false.]\n\n7. The third distinction is, that one is _true_, another _false_. A\n_true_ proposition is that, whose predicate contains, or comprehends its\nsubject, or whose predicate is the name of every thing, of which the\nsubject is the name; as _man is a living creature_ is therefore a true\nproposition, because whatsoever is called _man_, the same is also called\n_living creature_; and _some man is sick_, is true, because _sick_ is\nthe name of _some man_. That which is not true, or that whose predicate\ndoes not contain its subject, is called a _false_ proposition, as _man\nis a stone_.\n\nNow these words _true_, _truth_, and _true proposition_, are equivalent\nto one another; for truth consists in speech, and not in the things\nspoken of; and though _true_ be sometimes opposed to _apparent_ or\n_feigned_, yet it is always to be referred to the truth of proposition;\nfor the image of a man in a glass, or a ghost, is therefore denied to be\na very man, because this proposition, _a ghost is a man_, is not true;\nfor it cannot be denied but that a ghost is a very ghost. And therefore\ntruth or verity is not any affection of the thing, but of the\nproposition concerning it. As for that which the writers of metaphysics\nsay, that _a thing_, _one thing_, and _a very thing_, are equivalent to\none another, it is but trifling and childish; for who does not know,\nthat _a man_, _one man_, and _a very man_, signify the same.\n\n[Sidenote: True & false belongs to speech, and not to things.]\n\n8. And from hence it is evident, that truth and falsity have no place\nbut amongst such living creatures as use speech. For though some brute\ncreatures, looking upon the image of a man in a glass, may be affected\nwith it, as if it were the man himself, and for this reason fear it or\nfawn upon it in vain; yet they do not apprehend it as true or false, but\nonly as like; and in this they are not deceived. Wherefore, as men owe\nall their true ratiocination to the right understanding of speech; so\nalso they owe their errors to the misunderstanding of the same; and as\nall the ornaments of philosophy proceed only from man, so from man also\nis derived the ugly absurdity of false opinions. For speech has\nsomething in it like to a spider's web, (as it was said of old of\n_Solon's_ laws) for by contexture of words tender and delicate wits are\nensnared and stopped; but strong wits break easily through them.\n\nFrom hence also this may be deduced, that the first truths were\narbitrarily made by those that first of all imposed names upon things,\nor received them from the imposition of others. For it is true (for\nexample) that _man is a living creature_, but it is for this reason,\nthat it pleased men to impose both those names on the same thing.\n\n[Sidenote: Proposition, primary, not primary, definition, axiom,\n petition.]\n\n9. Fourthly, propositions are distinguished into _primary_ and _not\nprimary_. _Primary_ is that wherein the subject is explicated by a\npredicate of many names, as _man is a body_, _animated_, _rational_; for\nthat which is comprehended in the name _man_, is more largely expressed\nin the names _body_, _animated_, and _rational_, joined together; and it\nis called _primary_, because it is first in ratiocination; for nothing\ncan be proved, without understanding first the name of the thing in\nquestion. Now _primary_ propositions are nothing but definitions, or\nparts of definitions, and these only are the principles of\ndemonstration, being truths constituted arbitrarily by the inventors of\nspeech, and therefore not to be demonstrated. To these propositions,\nsome have added others, which they call _primary_ and _principles_,\nnamely, _axioms_, and _common notions_; which, (though they be so\nevident that they need no proof) yet, because they may be proved, are\nnot truly principles; and the less to be received for such, in regard\npropositions not intelligible, and sometimes manifestly false, are\nthrust on us under the name of _principles_ by the clamour of men, who\nobtrude for evident to others, all that they themselves think true. Also\ncertain petitions are commonly received into the number of principles;\nas, for example, _that a straight line may be drawn between two points_,\nand other petitions of the writers of geometry; and these are indeed the\nprinciples of art or construction, but not of science and demonstration.\n\n[Sidenote: Proposition necessary & contingent.]\n\n10. Fifthly, propositions are distinguished into _necessary_, that is,\nnecessarily true; and true, but not necessarily, which they call\n_contingent_. A _necessary_ proposition is when nothing can at any time\nbe conceived or feigned, whereof the subject is the name, but the\npredicate also is the name of the same thing; as _man is a living\ncreature_ is a necessary proposition, because at what time soever we\nsuppose the name _man_ agrees with any thing, at that time the name\n_living-creature_ also agrees with the same. But a _contingent_\nproposition is that, which at one time may be true, at another time\nfalse; as _every crow is black_; which may perhaps be true now, but\nfalse hereafter. Again, in every _necessary_ proposition, the predicate\nis either equivalent to the subject, as in this, _man is a rational\nliving creature_; or part of an equivalent name, as in this, _man is a\nliving creature_, for the name _rational-living-creature_, or _man_, is\ncompounded of these two, _rational_ and _living-creature_. But in a\n_contingent_ proposition this cannot be; for though this were true,\n_every man is a liar_, yet because the word _liar_ is no part of a\ncompounded name equivalent to the name _man_, that proposition is not to\nbe called _necessary_, but _contingent_, though it should happen to be\ntrue always. And therefore those propositions only are _necessary_,\nwhich are of sempiternal truth, that is, true at all times. From hence\nalso it is manifest, that truth adheres not to things, but to speech\nonly, for some truths are eternal; for it will be eternally true, _if\nman, then living-creature_; but that any _man_, or _living-creature_,\nshould exist eternally, is not necessary.\n\n[Sidenote: Categorical & hypothetical.]\n\n11. A sixth distinction of propositions is into _categorical_ and\n_hypothetical_. A _categorical_ proposition is that which is simply or\nabsolutely pronounced, as _every man is a living-creature_, _no man is a\ntree_; and _hypothetical_ is that which is pronounced conditionally, as,\n_if any thing be a man, the same is also a living-creature_, _if\nanything be a man, the same is also not-a-stone_.\n\nA _categorical_ proposition, and an _hypothetical_ answering it, do both\nsignify the same, if the propositions be necessary; but not if they be\ncontingent. For example, if this, _every man is a living-creature_, be\ntrue, this also will be true, _if any thing be a man, the same is also a\nliving-creature_; but in contingent propositions, though this be true,\n_every crow is black_, yet this, _if any thing be a crow, the same is\nblack_, is false. But an _hypothetical_ proposition is then rightly said\nto be true, when the consequence is true, as _every man is a\nliving-creature_, is rightly said to be a true proposition, because of\nwhatsoever it is truly said _that is a man_, it cannot but be truly said\nalso, _the same is a living creature_. And therefore whensoever an\n_hypothetical_ proposition is true, the _categorical_ answering it, is\nnot only true, but also necessary; which I thought worth the noting, as\nan argument, that philosophers may in most things reason more solidly by\n_hypothetical_ than _categorical_ propositions.\n\n[Sidenote: The same proposition diversely pronounced.]\n\n12. But seeing every proposition may be, and uses to be, pronounced and\nwritten in many forms, and we are obliged to speak in the same manner as\nmost men speak, yet they that learn philosophy from masters, had need to\ntake heed they be not deceived by the variety of expressions. And\ntherefore, whensoever they meet with any obscure proposition, they ought\nto reduce it to its most simple and categorical form; in which the\ncopulative word _is_ must be expressed by itself, and not mingled in any\nmanner either with the subject or predicate, both which must be\nseparated and clearly distinguished one from another. For example, if\nthis proposition, _man can not sin_, be compared with this, _man cannot\nsin_, their difference will easily appear if they be reduced to these,\n_man is able not to sin_, and, _man is not able to sin_, where the\npredicates are manifestly different. But they ought to do this silently\nby themselves, or betwixt them and their masters only; for it will be\nthought both ridiculous and absurd, for a man to use such language\npublicly. Being therefore to speak of _equipollent_ propositions, I put\nin the first place all those for _equipollent_, that may be reduced\npurely to one and the same categorical proposition.\n\n[Sidenote: Propositions that may be reduced to the same categorical\n proposition, are equipollent.]\n\n13. Secondly, that which is categorical and necessary, is _equipollent_\nto its hypothetical proposition; as this categorical, _a right-lined\ntriangle has its three angles equal to two right angles_, to this\nhypothetical, _if any figure be a right-lined triangle, the three angles\nof it are equal to two right angles_.\n\n[Sidenote: Universal propositions converted by contradictory names, are\n equipollent.]\n\n14. Also, any two universal propositions, of which the terms of the one\n(that is, the subject and predicate) are to the terms of the other, and\ntheir order inverted, as these, _every man is a living creature_, and\n_every thing that is not a living-creature is not a man_, are\n_equipollent_. For seeing _every man is a living creature_ is a true\nproposition, the name _living creature_ contains the name _man_; but\nthey are both positive names, and therefore (by the last article of the\nprecedent chapter) the negative name _not man_, contains the negative\nname _not living-creature_; wherefore _every thing that is not a\nliving-creature, is not a man_, is a true proposition. Likewise these,\n_no man is a tree_, _no tree is a man_, are _equipollent_. For if it be\ntrue that _tree_ is not the name of any _man_, then no one thing can be\nsignified by the two names _man_ and _tree_, wherefore _no tree is a\nman_ is a true proposition. Also to this, _whatsoever is not a\nliving-creature is not a man_, where both the terms are negative, this\nother proposition is equipollent, _only a living creature is a man_.\n\n[Sidenote: Negative propositions are the same, whether the negation be\n before or after the copula.]\n\n15. Fourthly, negative propositions, whether the particle of negation be\nset after the copula as some nations do, or before it, as it is in Latin\nand Greek, if the terms be the same, are equipollent: as, for example,\n_man is not a tree_, and, _man is not-a-tree_, are equipollent, though\nAristotle deny it. Also these, _every man is not a tree_, and _no man is\na tree_, are equipollent, and that so manifestly, as it needs not be\ndemonstrated.\n\n[Sidenote: Particular propositions simply converted, are equipollent.]\n\n16. Lastly, all particular propositions that have their terms inverted,\nas these, _some man is blind, some blind thing is a man_, are\nequipollent; for either of the two names, is the name of some one and\nthe same man; and therefore in which soever of the two orders they be\nconnected, they signify the same truth.\n\n[Sidenote: What are subaltern, contrary, subcontrary, and contradictory\n propositions.]\n\n17. Of propositions that have the same terms, and are placed in the same\norder, but varied either by quantity or quality, some are called\n_subaltern_, others _contrary_, others _subcontrary_, and others\n_contradictory_.\n\n_Subaltern_, are universal and particular propositions of the same\nquality; as, _every man is a living creature_, _some man is a living\ncreature_; or, _no man is wise_, _some man is not wise_. Of these, if\nthe universal be true, the particular will be true also.\n\n_Contrary_, are universal propositions of different quality; as, _every\nman is happy_, _no man is happy_. And of these, if one be true, the\nother is false: also, they may both be false, as in the example given.\n\n_Subcontrary_, are particular propositions of different quality; as,\n_some man is learned_, _some man is not learned_; which cannot be both\nfalse, but they may be both true.\n\n_Contradictory_ are those that differ both in quantity and quality; as,\n_every man is a living creature_, _some man is not a living-creature_;\nwhich can neither be both true, nor both false.\n\n[Sidenote: Consequence, what it is.]\n\n18. A proposition is said to _follow_ from two other propositions, when\nthese being granted to be true, it cannot be denied but the other is\ntrue also. For example, let these two propositions, _every man is a\nliving creature_, and, _every living creature is a body_, be supposed\ntrue, that is, that _body_ is the name of _every living creature_, and\n_living creature_ the name of _every man_. Seeing therefore, if these be\nunderstood to be true, it cannot be understood that _body_ is not the\nname of _every man_, that is, that _every man is a body_ is false, this\nproposition will be said to _follow_ from those two, or to be\nnecessarily _inferred_ from them.\n\n[Sidenote: Falsity cannot follow from truth.]\n\n19. That a true proposition may follow from false propositions, may\nhappen sometimes; but false from true, never. For if these, _every man\nis a stone_, and _every stone is a living creature_, (which are both\nfalse) be granted to be true, it is granted also that _living creature_\nis the name of _every stone_, and _stone_ of _every man_, that is, that\n_living creature_ is the name of _every man_; that is to say, this\nproposition _every man is a living creature_, is true, as it is indeed\ntrue. Wherefore a true proposition may sometimes follow from false. But\nif any two propositions be true, a false one can never follow from them.\nFor if true follow from false, for this reason only, that the false are\ngranted to be true, then truth from two truths granted will follow in\nthe same manner.\n\n[Sidenote: How one proposition is the cause of another.]\n\n20. Now, seeing none but a true proposition will follow from true, and\nthat the understanding of two propositions to be true, is the cause of\nunderstanding that also to be true which is deduced from them; the two\nantecedent propositions are commonly called the causes of the inferred\nproposition, or conclusion. And from hence it is that logicians say, the\n_premises_ are causes of the _conclusion_; which may pass, though it be\nnot properly spoken; for though understanding be the cause of\nunderstanding, yet speech is not the cause of speech. But when they say,\nthe cause of the properties of any thing, is the thing itself, they\nspeak absurdly. For example, if a figure be propounded which is\ntriangular; seeing every triangle has all its angles together equal to\ntwo right angles, from whence it follows that all the angles of that\nfigure are equal to two right angles, they say, for this reason, that\nthat figure is the cause of that equality. But seeing the figure does\nnot itself make its angles, and therefore cannot be said to be the\n_efficient-cause_, they call it the _formal-cause_; whereas indeed it is\nno cause at all; nor does the property of any figure follow the figure,\nbut has its being at the same time with it; only the knowledge of the\nfigure goes before the knowledge of the properties; and one knowledge is\ntruly the cause of another knowledge, namely the _efficient cause_.\n\nAnd thus much concerning _proposition_; which in the progress of\nphilosophy is the first step, like the moving towards of one foot. By\nthe due addition of another step I shall proceed to _syllogism_, and\nmake a complete pace. Of which in the next chapter.\n\n\n -------\n\n\n CHAPTER IV.\n\n OF SYLLOGISM.\n\n1. The definition of syllogism.\u20142. In a syllogism there are but three\n terms.\u20143. Major, minor, and middle term; also major and minor\n proposition, what they are.\u20144. The middle term in every syllogism\n ought to be determined in both the propositions to one and the same\n thing.\u20145. From two particular propositions nothing can be\n concluded.\u20146. A syllogism is the collection of two propositions into\n one sum.\u20147. The figure of a syllogism, what it is.\u20148. What is in the\n mind answering to a syllogism.\u20149. The first indirect figure, how it\n is made.\u201410. The second indirect figure, how made.\u201411. How the third\n indirect figure is made.\u201412. There are many moods in every figure,\n but most of them useless in philosophy.\u201413. An hypothetical\n syllogism when equipollent to a categorical.\n\n\n[Sidenote: Definition of syllogism.]\n\n1. A SPEECH, consisting of three propositions, from two of which the\nthird follows, is called a SYLLOGISM; and that which follows is called\nthe _conclusion_; the other two _premises_. For example, this speech,\n_every man is a living creature_, _every living creature is a body_,\ntherefore, _every man is a body_, is a _syllogism_, because the third\nproposition follows from the two first; that is, if those be granted to\nbe true, this must also be granted to be true.\n\n[Sidenote: In a syllogism there are but three terms.]\n\n2. From two propositions which have not one term common, no conclusion\ncan follow; and therefore no _syllogism_ can be made of them. For let\nany two premises, _a man is a living creature_, _a tree is a plant_, be\nboth of them true, yet because it cannot be collected from them that\n_plant_ is the name of a _man_, or _man_ the name of a _plant_, it is\nnot necessary that this conclusion, _a man is a plant_, should be true.\nCorollary: therefore, in the _premises_ of a _syllogism_ there can be\nbut three _terms_.\n\nBesides, there can be no term in the _conclusion_, which was not in the\n_premises_. For let any two premises be, _a man is a living creature_,\n_a living creature is a body_, yet if any other term be put in the\nconclusion, as _man is two-footed_; though it be true, it cannot follow\nfrom the premises, because from them it cannot be collected, that the\nname _two-footed_ belongs to a _man_; and therefore, again, in every\n_syllogism_ there can be but three _terms_.\n\n[Sidenote: Major, minor and middle term; also major and minor\n proposition, what they are.]\n\n3. Of these terms, that which is the _predicate_ in the conclusion, is\ncommonly called the _major_; that which is the _subject_ in the\nconclusion, the _minor_, and the other is the _middle term_; as in this\nsyllogism, _a man is a living creature_, _a living creature is a body_,\ntherefore, _a man is a body_, _body_ is the _major_, _man_ the _minor_,\nand _living creature_ the _middle term_. Also of the premises, that in\nwhich the _major term_ is found, is called the _major proposition_, and\nthat which has the _minor term_, the _minor proposition_.\n\n[Sidenote: The middle term in every syllogism to be determined in both\n propositions to one and the same thing.]\n\n4. If the middle term be not in both the premises determined to one and\nthe same singular thing, no conclusion will follow, nor syllogism be\nmade. For let the minor term be _man_, the middle term _living\ncreature_, and the major term _lion_; and let the premises be, _man is a\nliving creature_, _some living creature is a lion_, yet it will not\nfollow that _every_ or _any man is a lion_. By which it is manifest,\nthat in every syllogism, that proposition which has the middle term for\nits _subject_, ought to be either _universal_ or _singular_, but not\n_particular_ nor _indefinite_. For example, this syllogism, _every man\nis a living creature_, _some living creature is four-footed_, therefore\n_some man is four-footed_, is therefore faulty, because the middle term,\n_living creature_, is in the first of the premises determined only to\n_man_, for there the name of _living creature_ is given to man only, but\nin the latter premise it may be understood of some other living creature\nbesides man. But if the latter premise had been _universal_, as here,\n_every man is a living creature_, _every living creature is a body_,\ntherefore _every man is a body_, the syllogism had been true; for it\nwould have followed that _body_ had been the name of _every living\ncreature_, that is of _man_; that is to say, the conclusion _every man\nis a body_ had been true. Likewise, when the middle term is a _singular_\nname, a syllogism may be made, I say a true syllogism, though useless in\nphilosophy, as this, _some man is Socrates_, _Socrates is a\nphilosopher_, therefore, _some man is a philosopher_; for the premises\nbeing granted, the conclusion cannot be denied.\n\n[Sidenote: From two particular propositions nothing can be concluded.]\n\n5. And therefore of two premises, in both which the middle term is\nparticular, a syllogism cannot be made; for whether the middle term be\nthe _subject_ in both the premises, or the _predicate_ in both, or the\n_subject_ in one, and the _predicate_ in the other, it will not be\nnecessarily determined to the same thing. For let the premises be,\n\n _Some man is blind_, In both which the middle\n _Some man is learned_, } term is the _subject_,\n\nit will not follow that _blind_ is the name of any _learned man_, or\n_learned_ the name of any _blind man_, seeing the name _learned_ does\nnot contain the name _blind_, nor this that; and therefore it is not\nnecessary that both should be names of the same man. So from these\npremises,\n\n _Every man is a living-creature_, } In both which the middle\n _Every horse is a living-creature_, } term is the _predicate_,\n\nnothing will follow. For seeing _living creature_ is in both of them\n_indefinite_, which is equivalent to _particular_, and that _man_ may be\none kind of _living creature_, and _horse_ another kind, it is not\nnecessary that _man_ should be the name of _horse_, or _horse_ of _man_.\nOr if the premises be,\n\n _Every man is a In one of which the\n living-creature_, } middle-term\n _Some living creature is\n four-footed_,\n\n } is the _subject_, and in the\n\n } other the _predicate_,\n\nthe conclusion will not follow, because the name _living creature_ being\nnot determined, it may in one of them be understood of _man_, in the\nother of _not-man_.\n\n[Sidenote: A syllogism is the collection of two propositions into one\n sum.]\n\n6. Now it is manifest from what has been said, that a syllogism is\nnothing but a collection of the sum of two propositions, joined together\nby a common term, which is called the _middle term_. And as proposition\nis the addition of two names, so syllogism is the adding together of\nthree.\n\n[Sidenote: The figure of a syllogism what it is.]\n\n7. Syllogisms are usually distinguished according to their diversity of\n_figures_, that is, by the diverse position of the middle term. And\nagain in figure there is a distinction of certain _moods_, which consist\nof the differences of propositions in _quantity_ and _quality_. The\nfirst figure is that, in which the terms are placed one after another\naccording to their latitude of signification; in which order the _minor\nterm_ is first, the _middle term_ next, and the _major_ last; as, if the\nminor term be _man_, the middle term, _living creature_, and the major\nterm, _body_, then, _man is a living-creature, is a body_, will be a\nsyllogism in the first figure: in which, _man is a living creature_ is\nthe minor proposition; the major, _living creature is a body_, and the\nconclusion, or sum of both, _man is a body_. Now this figure is called\n_direct_, because the terms stand in direct order; and it is varied by\n_quantity_ and _quality_ into four _moods_: of which the first is that\nwherein all the terms are _positive_, and the minor term _universal_, as\n_every man is a living creature_, _every living creature is a body_: in\nwhich all the propositions are affirmative, and universal. But if the\nmajor term be a negative name, and the minor an universal name, the\n_figure_ will be in the second _mood_, as, _every man is a living\ncreature_, _every living creature is not a tree_, in which the major\nproposition and conclusion are both universal and negative. To these\ntwo, are commonly added two more, by making the minor term particular.\nAlso it may happen that both the major and middle terms are negative\nterms, and then there arises another _mood_, in which all the\npropositions are negative, and yet the syllogism will be good; as, if\nthe minor term be _man_, the middle term _not a stone_, and the major\nterm _not a flint_, this syllogism, _no man is a stone_, _whatsoever is\nnot a stone is not a flint_, therefore, _no man is a flint_, is true,\nthough it consist of three negatives. But in philosophy, the profession\nwhereof is to establish universal rules concerning the properties of\nthings, seeing the difference betwixt negatives and affirmatives is only\nthis, that in the former the subject is affirmed by a negative name, and\nby a positive in the latter, it is superfluous to consider any other\n_mood_ in direct _figure_, besides that, in which all the propositions\nare both universal and affirmative.\n\n[Sidenote: What is in the mind answering to a syllogism.]\n\n8. The thoughts in the mind answering to a direct syllogism, proceed in\nthis manner; first, there is conceived a phantasm of the thing named,\nwith that accident or quality thereof, for which it is in the minor\nproposition called by that name which is the subject; next, the mind has\na phantasm of the same thing with that accident, or quality, for which\nit hath the name, that in the same proposition is the predicate;\nthirdly, the thought returns of the same thing as having that accident\nin it, for which it is called by the name, that is the predicate of the\nmajor proposition; and lastly, remembering that all those are the\naccidents of one and the same thing, it concludes that those three names\nare also names of one and the same thing; that is to say, the conclusion\nis true. For example, when this syllogism is made, _man is a living\ncreature_, _a living creature is a body_, therefore, _man is a body_,\nthe mind conceives first an image of a man speaking or discoursing, and\nremembers that that, which so appears, is called _man_; then it has the\nimage of the same man moving, and remembers that that, which appears so,\nis called _living creature_; thirdly, it conceives an image of the same\nman, as filling some place or space, and remembers that what appears so\nis called _body_; and lastly, when it remembers that that thing, which\nwas extended, and moved and spake, was one and the same thing, it\nconcludes that the three names, _man_, _living creature_, and _body_,\nare names of the same thing, and that therefore _man is a living\ncreature_ is a true proposition. From whence it is manifest, that living\ncreatures that have not the use of speech, have no conception or thought\nin the mind, answering to a syllogism made of universal propositions;\nseeing it is necessary to think not only of the thing, but also by turns\nto remember the divers names, which for divers considerations thereof\nare applied to the same.\n\n[Sidenote: The first indirect figure how made.]\n\n9. The rest of the figures arise either from the inflexion, or inversion\nof the first or direct figure; which is done by changing the major, or\nminor, or both the propositions, into converted propositions equipollent\nto them.\n\nFrom whence follow three other figures; of which, two are _inflected_,\nand the third _inverted_. The first of these three is made by the\nconversion of the major proposition. For let the minor, middle, and\nmajor terms stand in direct order, thus, _man is a living creature_, _is\nnot a stone_, which is the first or direct figure; the inflection will\nbe by converting the major proposition in this manner, _man is a living\ncreature_, _a stone is not a living creature_; and this is the second\nfigure, or the first of the indirect figures; in which the conclusion\nwill be, _man is not a stone_. For (having shown in the last chapter,\nart. 14, that universal propositions, converted by contradiction of the\nterms, are equipollent) both those syllogisms conclude alike; so that if\nthe major be read (like Hebrew) backwards, thus, _a living creature is\nnot a stone_, it will be direct again, as it was before. In like manner\nthis direct syllogism, _man is not a tree_, _is not a pear-tree_, will\nbe made indirect by converting the major proposition (by contradiction\nof the terms) into another equipollent to it, thus, _man is not a tree_,\n_a pear-tree is a tree_; for the same conclusion will follow, _man is\nnot a pear-tree_.\n\nBut for the conversion of the direct figure into the first indirect\nfigure, the major term in the direct figure ought to be negative. For\nthough this direct, _man is a living creature_, _is a body_, be made\nindirect, by converting the major proposition, thus,\n\n _Man is a living creature_,\n _Not a body is not a living creature_,\n Therefore, _Every man is a body_;\n\nYet this conversion appears so obscure, that this mood is of no use at\nall. By the conversion of the major proposition, it is manifest, that in\nthis figure, the middle term is always the predicate in both the\npremises.\n\n[Sidenote: Second indirect figure how made.]\n\n10. The second indirect figure is made by converting the minor\nproposition, so as that the middle term is the subject in both. But this\nnever concludes universally, and therefore is of no use in philosophy.\nNevertheless I will set down an example of it; by which this direct\n\n _Every man is a living creature_,\n _Every living creature is a body_,\n\nby conversion of the minor proposition, will stand thus,\n\n _Some living creature is a man_,\n _Every living creature is a body_,\n Therefore, _Some man is a body_.\n\nFor _every man is a living creature_ cannot be converted into this,\n_every living creature is a man_: and therefore if this syllogism be\nrestored to its direct form, the minor proposition will be _some man is\na living creature_, and consequently the conclusion will be _some man is\na body_, seeing the minor term _man_, which is the subject in the\nconclusion, is a particular name.\n\n[Sidenote: How the third indirect figure is made.]\n\n11. The third indirect or inverted figure, is made by the conversion of\nboth the premises. For example, this direct syllogism,\n\n _Every man is a living creature_,\n _Every living creature is not a stone_,\n Therefore, _Every man is not a stone_,\n\nbeing inverted, will stand thus,\n\n _Every stone is not a living creature_,\n _Whatsoever is not a living creature, is not a man_,\n Therefore, _Every stone is not a man_;\n\nwhich conclusion is the converse of the direct conclusion, and\nequipollent to the same.\n\nThe figures, therefore, of syllogisms, if they be numbered by the\ndiverse situation of the middle term only, are but three; in the first\nwhereof, the middle term has the middle place; in the second, the last;\nand in the third, the first place. But if they be numbered according to\nthe situation of the terms simply, they are four; for the first may be\ndistinguished again into two, namely, into direct and inverted. From\nwhence it is evident, that the controversy among logicians concerning\nthe fourth figure, is a mere \u03bb\u03bf\u03b3\u03cc\u03bc\u03b1\u03c7\u03b9\u03b1, or contention about the name\nthereof; for, as for the thing itself, it is plain that the situation of\nthe terms (not considering the quantity or quality by which the moods\nare distinguished) makes four differences of syllogisms, which may be\ncalled figures, or have any other name at pleasure.\n\n[Sidenote: There are many moods in every figure, but most of them\n useless in philosophy.]\n\n12. In every one of these figures there are many moods, which are made\nby varying the premises according to all the differences they are\ncapable of, by quantity and quality; as namely, in the direct figure\nthere are six moods; in the first indirect figure, four; in the second,\nfourteen; and in the third, eighteen. But because from the direct figure\nI rejected as superfluous all moods besides that which consists of\nuniversal propositions, and whose minor proposition is affirmative, I\ndo, together with it, reject the moods of the rest of the figures which\nare made by conversion of the premises in the direct figure.\n\n[Sidenote: An hypothetical syllogism when equipollent to a categorical.]\n\n13. As it was showed before, that in necessary propositions a\ncategorical and hypothetical proposition are equipollent; so likewise it\nis manifest that a categorical and hypothetical syllogism are\nequivalent. For every categorical syllogism, as this,\n\n _Every man is a living creature_,\n _Every living creature is a body_,\n Therefore, _Every man is a body_,\n\nis of equal force with this hypothetical syllogism:\n\n _If any thing be a man, the same is also a living creature_,\n _If any thing be a living creature, the same is a body_,\n Therefore, _If any thing be a man, the same is a body_.\n\nIn like manner, this categorical syllogism in an indirect figure,\n\n _No stone is a living creature_,\n _Every man is a living creature_,\n Therefore, _No man is a stone_,\n Or, _No stone is a man_,\n\nis equivalent to this hypothetical syllogism:\n\n _If any thing be a man, the same is a living creature_,\n _If any thing be a stone, the same is not a living creature_,\n Therefore, _If any thing be a stone, the same is not a man_,\n Or, _If any thing be a man, the same is not a stone_.\n\nAnd thus much seems sufficient for the nature of syllogisms; (for the\ndoctrine of moods and figures is clearly delivered by others that have\nwritten largely and profitably of the same). Nor are precepts so\nnecessary as practice for the attaining of true ratiocination; and they\nthat study the demonstrations of mathematicians, will sooner learn true\nlogic, than they that spend time in reading the rules of syllogizing\nwhich logicians have made; no otherwise than little children learn to\ngo, not by precepts, but by exercising their feet. This, therefore, may\nserve for the first pace in the way to Philosophy.\n\nIn the next place I shall speak of the faults and errors into which men\nthat reason unwarily are apt to fall; and of their kinds and causes.\n\n\n -------\n\n\n CHAPTER V.\n\n OF ERRING, FALSITY, AND CAPTIONS.\n\n1. Erring and falsity how they differ. Error of the mind by itself\n without the use of words, how it happens.\u20142. A sevenfold incoherency\n of names, every one of which makes always a false proposition.\u20143.\n Examples of the first manner of incoherency.\u20144. Of the second.\u20145. Of\n the third.\u20146. Of the fourth.\u20147. Of the fifth.\u20148. Of the sixth.\u20149. Of\n the seventh.\u201410. Falsity of propositions detected by resolving the\n terms with definitions continued till they come to simple names, or\n names that are the most general of their kind.\u201411. Of the fault of a\n syllogism consisting in the implication of the terms with the\n copula.\u201412. Of the fault which consists in equivocation.\u201413.\n Sophistical captions are oftener faulty in the matter than in the\n form of syllogisms.\n\n\n[Sidenote: Erring & falsity how they differ. Error of the mind by\n itself, without the use of words, how it happens.]\n\n1. Men are subject to _err_ not only in affirming and denying, but also\nin perception, and in silent cogitation. In affirming and denying, when\nthey call any thing by a name, which is not the name thereof; as if from\nseeing the sun first by reflection in water, and afterwards again\ndirectly in the firmament, we should to both those appearances give the\nname of sun, and say there are two suns; which none but men can do, for\nno other living creatures have the use of names. This kind of error only\ndeserves the name of _falsity_, as arising, not from sense, nor from the\nthings themselves, but from pronouncing rashly; for names have their\nconstitution, not from the species of things, but from the will and\nconsent of men. And hence it comes to pass, that men pronounce\n_falsely_, by their own negligence, in departing from such appellations\nof things as are agreed upon, and are not deceived neither by the\nthings, nor by the sense; for they do not perceive that the thing they\nsee is called sun, but they give it that name from their own will and\nagreement. Tacit errors, or the errors of sense and cogitation, are\nmade, by passing from one imagination to the imagination of another\ndifferent thing; or by feigning that to be past, or future, which never\nwas, nor ever shall be; as when, by seeing the image of the sun in\nwater, we imagine the sun itself to be there; or by seeing swords, that\nthere has been or shall be fighting, because it uses to be so for the\nmost part; or when from promises we feign the mind of the promiser to be\nsuch and such; or lastly, when from any sign we vainly imagine something\nto be signified, which is not. And errors of this sort are common to all\nthings that have sense; and yet the deception proceeds neither from our\nsenses, nor from the things we perceive; but from ourselves while we\nfeign such things as are but mere images to be something more than\nimages. But neither things, nor imaginations of things, can be said to\nbe false, seeing they are truly what they are; nor do they, as signs,\npromise any thing which they do not perform; for they indeed do not\npromise at all, but we from them; nor do the clouds, but we, from seeing\nthe clouds, say it shall rain. The best way, therefore, to free\nourselves from such errors as arise from natural signs, is first of all,\nbefore we begin to reason concerning such conjectural things, to suppose\nourselves ignorant, and then to make use of our ratiocination; for these\nerrors proceed from the want of ratiocination; whereas, errors which\nconsist in affirmation and negation, (that is, the falsity of\npropositions) proceed only from reasoning amiss. Of these, therefore, as\nrepugnant to philosophy, I will speak principally.\n\n[Sidenote: A sevenfold incoherency of names, all of which make always a\n false proposition.]\n\n2. Errors which happen in reasoning, that is, in syllogizing, consist\neither in the falsity of the premises, or of the inference. In the first\nof these cases, a syllogism is said to be faulty in the _matter_ of it;\nand in the second case, in the _form_. I will first consider the matter,\nnamely, how many ways a proposition may be false; and next the form, and\nhow it comes to pass, that when the premises are true, the inference is,\nnotwithstanding, false.\n\nSeeing, therefore, that proposition only is true, (chap, III, art. 7) in\nwhich are copulated two names of one and the same thing; and that always\nfalse, in which names of different things are copulated, look how many\nways names of different things may be copulated, and so many ways a\nfalse proposition may be made.\n\nNow, all things to which we give names, may be reduced to these four\nkinds, namely, _bodies_, _accidents_, _phantasms_, and _names_\nthemselves; and therefore, in every true proposition, it is necessary\nthat the names copulated, be both of them names of _bodies_, or both\nnames of _accidents_, or both names of _phantasms_, or both names of\n_names_. For names otherwise copulated are incoherent, and constitute a\nfalse proposition. It may happen, also, that the name of a _body_, of an\n_accident_, or of a _phantasm_, may be copulated with the name of a\n_speech_. So that copulated names may be incoherent seven manner of\nways.\n\n 1. If the name of a Body } { the name of an Accident.\n 2. If the name of a Body } { the name of a Phantasm.\n 3. If the name of a Body } be { the name of a Name.\n 4. If the name of an Accident } copulated { the name of a Phantasm.\n 5. If the name of an Accident } with { the name of a Name.\n 6. If the name of a Phantasm } { the name of a Name.\n 7. If the name of a Body, } { the name of a Speech.\n Accident, or Phantasm } {\n\nOf all which I will give some examples.\n\n[Sidenote: Examples of the first manner of incoherency.]\n\n3. After the first of these ways propositions are false, when abstract\nnames are copulated with concrete names; as (in Latin and Greek) _esse\nest ens_, _essentia est ens_, \u03c4\u1f78 \u03c4\u03af \u1f26\u03bd \u03b5\u03b9\u03bd\u03b1\u1f76 (_i._); _quidditas est\nens_, and many the like, which are found in Aristotle's _Metaphysics_.\nAlso, the _understanding worketh_, the _understanding understandeth_,\nthe _sight seeth_; a _body_ is _magnitude_, a _body_ is _quantity_, a\n_body_ is _extension_; to _be a man_ is a _man_, _whiteness_ is a _white\nthing_, &c.; which is as if one should say, the _runner_ is the\n_running_, or the _walk walketh_. Moreover, _essence_ is _separated_,\n_substance_ is _abstracted_: and others like these, or derived from\nthese, (with which common philosophy abounds.) For seeing no _subject_\nof an _accident_ (that is, no _body_) is an _accident_: no name of an\n_accident_ ought to be given to a _body_, nor of a _body_ to an\n_accident_.\n\n[Sidenote: The second.]\n\n4. False, in the second manner, are such propositions as these; a _ghost\nis a body_, or a _spirit_, that is, a thin body; _sensible species fly\nup and down in the air_, or _are moved hither and thither_, which is\nproper to bodies; also, a _shadow is moved_, or _is a body_; _light is\nmoved_, or _is a body_; _colour is the object of sight_, _sound of\nhearing_; _space_ or _place is extended_; and innumerable others of this\nkind. For seeing ghosts, sensible species, a shadow, light, colour,\nsound, space, &c. appear to us no less sleeping than waking, they cannot\nbe things without us, but only phantasms of the mind that imagines them;\nand therefore the names of these, copulated with the names of bodies,\ncannot constitute a true proposition.\n\n[Sidenote: The third.]\n\n5. False propositions of the third kind, are such as these; _genus est\nens_, _universale est ens_, _ens de ente pr\u00e6dicatur_. For _genus_, and\n_universale_, and _predicare_, are names of names, and not of things.\nAlso, _number is infinite_, is a false proposition; for no number can be\ninfinite, but only the word _number_ is then called an indefinite name\nwhen there is no determined number answering to it in the mind.\n\n[Sidenote: The fourth.]\n\n6. To the fourth kind belong such false propositions as these, _an\nobject is of such magnitude or figure as appears to the beholders_;\n_colour, light, sound, are in the object_; and the like. For the same\nobject appears sometimes greater, sometimes lesser, sometimes square,\nsometimes round, according to the diversity of the distance and medium;\nbut the true magnitude and figure of the thing seen is always one and\nthe same; so that the magnitude and figure which appears, is not the\ntrue magnitude and figure of the object, nor anything but phantasm; and\ntherefore, in such propositions as these, the names of accidents are\ncopulated with the names of phantasms.\n\n[Sidenote: The fifth.]\n\n7. Propositions are false in the fifth manner, when it is said that _the\ndefinition is the essence of a thing_; _whiteness_, or some other\naccident, _is the genus, or universal_. For definition is not the\nessence of any thing, but a speech signifying what we conceive of the\nessence thereof; and so also not whiteness itself, but the word\nwhiteness, is a genus, or an universal name.\n\n[Sidenote: The sixth.]\n\n8. In the sixth manner they err, that say the _idea of anything is\nuniversal_; as if there could be in the mind an image of a man, which\nwere not the image of some one man, but a man simply, which is\nimpossible; for every idea is one, and of one thing; but they are\ndeceived in this, that they put the _name_ of the thing for the _idea_\nthereof.\n\n[Sidenote: The seventh.]\n\n9. They err in the seventh manner, that make this distinction between\nthings that have being, that some of them _exist by themselves_, others\nby _accident_; namely, because _Socrates is a man_ is a necessary\nproposition, and _Socrates is a musician_ a contingent proposition,\ntherefore they say some things exist necessarily or by themselves,\nothers contingently or by accident; whereby, seeing _necessary_,\n_contingent_, _by itself_, _by accident_, are not names of things, but\nof propositions, they that say _any thing that has being, exists by\naccident_, copulate the name of a proposition with the name of a thing.\nIn the same manner also, they err, which place some ideas in the\nunderstanding, others in the fancy; as if from the understanding of this\nproposition, _man is a living creature_, we had one idea or image of a\nman derived from sense to the memory, and another to the understanding;\nwherein that which deceives them is this, that they think one idea\nshould be answerable to a name, another to a proposition, which is\nfalse; for proposition signifies only the order of those things one\nafter another, which we observe in the same idea of man; so that this\nproposition, _man is a living creature_ raises but one idea in us,\nthough in that idea we consider that first, for which he is called man,\nand next that, for which he is called living creature. The falsities of\npropositions in all these several manners, is to be discovered by the\ndefinitions of the copulated names.\n\n[Sidenote: Falsity of propositions detected by resolving the terms with\n definitions.]\n\n10. But when names of bodies are copulated with names of bodies, names\nof accidents with names of accidents, names of names with names of\nnames, and names of phantasms, with names of phantasms, if we,\nnevertheless, remain still doubtful whether such propositions are true,\nwe ought then in the first place to find out the definition of both\nthose names, and again the definitions of such names as are in the\nformer definition, and so proceed by a continual resolution till we come\nto a simple name, that is, to the most general or most universal name of\nthat kind; and if after all this, the truth or falsity thereof be not\nevident, we must search it out by philosophy, and ratiocination,\nbeginning from definitions. For every proposition, universally true, is\neither a definition, or part of a definition, or the evidence of it\ndepends upon definitions.\n\n[Sidenote: Of the fault of a syllogism consisting in implication of the\n terms with the copula.]\n\n11. That fault of a syllogism which lies hid in the form thereof, will\nalways be found either in the implication of the copula with one of the\nterms, or in the equivocation of some word; and in either of these ways\nthere will be four terms, which (as I have shewn) cannot stand in a true\nsyllogism. Now the implication of the copula with either term, is easily\ndetected by reducing the propositions to plain and clear predication; as\n(for example) if any man should argue thus,\n\n _The hand toucheth the pen_,\n _The pen toucheth the paper_,\n Therefore, _The hand toucheth the paper_;\n\nthe fallacy will easily appear by reducing it, thus:\n\n _The hand, is, touching the pen_,\n _The pen, is, touching the paper_,\n Therefore, _The hand, is, touching the paper_;\n\nwhere there are manifestly these four terms, _the hand_, _touching the\npen_, _the pen_, and _touching the the paper_. But the danger of being\ndeceived by sophisms of this kind, does not seem to be so great, as that\nI need insist longer upon them.\n\n[Sidenote: Of the fault which consists in equivocation.]\n\n12. And though there may be fallacy in equivocal terms, yet in those\nthat be manifestly such, there is none at all; nor in metaphors, for\nthey profess the transferring of names from one thing to another.\nNevertheless, sometimes equivocals (and those not very obscure) may\ndeceive; as in this argumentation:--_It belongs to metaphysics to treat\nof principles; but the first principle of all, is, that the same thing\ncannot both exist and not exist at the same time; and therefore it\nbelongs to metaphysics to treat whether the same thing may both exist\nand not exist at the same time_; where the fallacy lies in the\nequivocation of the word principle; for whereas Aristotle in the\nbeginning of his _Metaphysics_, says, that _the treating of principles\nbelongs to primary science_, he understands by principles, causes of\nthings, and certain existences which he calls primary; but where he says\n_a primary proposition is a principle_, by principle, there, he means\nthe beginning and cause of knowledge, that is, the understanding of\nwords, which, if any man want, he is incapable of learning.\n\n[Sidenote: Sophistical captions are oftener faulty in the matter than in\n the form of syllogisms.]\n\n13. But the _captions_ of sophists and sceptics, by which they were\nwont, of old, to deride and oppose truth, were faulty for the most part,\nnot in the form, but in the matter of syllogism; and they deceived not\nothers oftener than they were themselves deceived. For the force of that\nfamous argument of Zeno against motion, consisted in this proposition,\n_whatsoever may be divided into parts, infinite in number, the same is\ninfinite_; which he, without doubt, thought to be true, yet nevertheless\nis false. For to be divided into infinite parts, is nothing else but to\nbe divided into as many parts as any man will. But it is not necessary\nthat a line should have parts infinite in number, or be infinite,\nbecause I can divide and subdivide it as often as I please; for how many\nparts soever I make, yet their number is finite; but because he that\nsays parts, simply, without adding how many, does not limit any number,\nbut leaves it to the determination of the hearer, therefore we say\ncommonly, a line may be divided infinitely; which cannot be true in any\nother sense.\n\n[Sidenote: Conclusion.]\n\nAnd thus much may suffice concerning syllogism, which is, as it were,\nthe first pace towards philosophy; in which I have said as much as is\nnecessary to teach any man from whence all true argumentation has its\nforce. And to enlarge this treatise with all that may be heaped\ntogether, would be as superfluous, as if one should (as I said before)\ngive a young child precepts for the teaching of him to go; for the art\nof reasoning is not so well learned by precepts as by practice, and by\nthe reading of those books in which the conclusions are all made by\nsevere demonstration. And so I pass on to the way of philosophy, that\nis, to the method of study.\n\n\n -------\n\n\n CHAPTER VI.\n\n OF METHOD.\n\n1. Method and science defined.\u20142. It is more easily known concerning\n singular, than universal things, that they are; and contrarily, it\n is more easily known concerning universal, than singular things, why\n they are, or what are their causes.\u20143. What it is philosophers seek\n to know.\u20144. The first part, by which principles are found out, is\n purely analytical.\u20145. The highest causes, and most universal in\n every kind, are known by themselves.\u20146. Method from principles found\n out, tending to science simply, what it is.\u20147. That method of civil\n and natural science, which proceeds from sense to principles, is\n analytical; and again, that, which begins at principles, is\n synthetical.\u20148. The method of searching out, whether any thing\n propounded be matter or accident.\u20149. The method of seeking whether\n any accident be in this, or in that subject.\u201410. The method of\n searching after the cause of any effect propounded.\u201411. Words serve\n to invention, as marks; to demonstration, as signs.\u201412. The method\n of demonstration is synthetical.\u201413. Definitions only are primary\n and universal propositions.\u201414. The nature and definition of a\n definition.\u201415. The properties of a definition.\u201416. The nature of a\n demonstration.\u201417. The properties of a demonstration, and order of\n things to be demonstrated.\u201418. The faults of a demonstration.\u201419.\n Why the analytical method of geometricians cannot be treated of in\n this place.\n\n\n[Sidenote: Method and science defined.]\n\n1. For the understanding of _method_, it will be necessary for me to\nrepeat the definition of philosophy, delivered above (Chap. I, art. 2.)\nin this manner, _Philosophy is the knowledge we acquire, by true\nratiocination, of appearances, or apparent effects, from the knowledge\nwe have of some possible production or generation of the same; and of\nsuch production, as has been or may be, from the knowledge we have of\nthe effects_. METHOD, therefore, in the study of philosophy, _is the\nshortest way of finding out effects by their known causes, or of causes\nby their known effects_. But we are then said to know any effect, when\nwe know _that there be causes of the same_, and _in what subject those\ncauses are_, and _in what subject they produce that effect_, and _in\nwhat manner they work the same_. And this is the science of causes, or,\nas they call it, of the \u03b4\u03b9\u03cc\u03c4\u03b9. All other science, which is called the\n\u1f44\u03c4\u03b9, is either perception by sense, or the imagination, or memory\nremaining after such perception.\n\nThe first beginnings, therefore, of knowledge, are the phantasms of\nsense and imagination; and that there be such phantasms we know well\nenough by nature; but to know why they be, or from what causes they\nproceed, is the work of ratiocination; which consists (as is said above,\nin the 1st Chapter, Art. 2) in _composition_, and _division_ or\n_resolution_. There is therefore no method, by which we find out the\ncauses of things, but is either _compositive_ or _resolutive_, or\n_partly compositive_, and _partly resolutive_. And the resolutive is\ncommonly called _analytical_ method, as the compositive is called\n_synthetical_.\n\n[Sidenote: It is easier known concerning singular than universal things,\n that they are; and contrarily it is easier known concerning\n universal than singular things, why they are, or what are\n their causes.]\n\n2. It is common to all sorts of method, to proceed from known things to\nunknown; and this is manifest from the cited definition of philosophy.\nBut in knowledge by sense, the whole object is more known, than any part\nthereof; as when we see a man, the conception or whole idea of that man\nis first or more known, than the particular ideas of his being\n_figurate_, _animate_, and _rational_; that is, we first see the whole\nman, and take notice of his being, before we observe in him those other\nparticulars. And therefore in any knowledge of the \u1f45\u03c4\u03b9, or that any\nthing _is_, the beginning of our search is from the whole idea; and\ncontrarily, in our knowledge of the \u03b4\u03b9\u03cc\u03c4\u03b9, or of the causes of any\nthing, that is, in the sciences, we have more knowledge of the causes of\nthe parts than of the whole. For the cause of the whole is compounded of\nthe causes of the parts; but it is necessary that we know the things\nthat are to be compounded, before we can know the whole compound. Now,\nby parts, I do not here mean parts of the thing itself, but parts of its\nnature; as, by the parts of man, I do not understand his head, his\nshoulders, his arms, &c. but his figure, quantity, motion, sense,\nreason, and the like; which accidents being compounded or put together,\nconstitute the whole nature of man, but not the man himself. And this is\nthe meaning of that common saying, namely, that some things are more\nknown to us, others more known to nature; for I do not think that they,\nwhich so distinguish, mean that something is known to nature, which is\nknown to no man; and therefore, by those things, that are more known to\nus, we are to understand things we take notice of by our senses, and, by\nmore known to nature, those we acquire the knowledge of by reason; for\nin this sense it is, that the _whole_, that is, those things that have\nuniversal names, (which, for brevity's sake, I call _universal_) are\nmore known to us than the _parts_, that is, such things as have names\nless universal, (which I therefore call _singular_); and the causes of\nthe parts are more known to nature than the cause of the whole; that is,\nuniversals than singulars.\n\n[Sidenote: What it is philosophers seek to know.]\n\n3. In the study of philosophy, men search after science either simply or\nindefinitely; that is, to know as much as they can, without propounding\nto themselves any limited question; or they enquire into the cause of\nsome determined appearance, or endeavour to find out the certainty of\nsomething in question, as what is the cause of _light_, of _heat_, of\n_gravity_, of a _figure_ propounded, and the like; or in what _subject_\nany propounded _accident_ is inherent; or what may conduce most to the\n_generation_ of some propounded _effect_ from many _accidents_; or in\nwhat manner particular causes ought to be compounded for the production\nof some certain effect. Now, according to this variety of things in\nquestion, sometimes the _analytical method_ is to be used, and sometimes\nthe _synthetical_.\n\n[Sidenote: The first part, by which principles are discovered, is purely\n analytical.]\n\n4. But to those that search after science indefinitely, which consists\nin the knowledge of the causes of all things, as far forth as it may be\nattained, (and the causes of singular things are compounded of the\ncauses of universal or simple things) it is necessary that they know the\ncauses of universal things, or of such accidents as are common to all\nbodies, that is, to all matter, before they can know the causes of\nsingular things, that is, of those accidents by which one thing is\ndistinguished from another. And, again, they must know what those\nuniversal things are, before they can know their causes. Moreover,\nseeing universal things are contained in the nature of singular things,\nthe knowledge of them is to be acquired by reason, that is, by\nresolution. For example, if there be propounded a conception or _idea_\nof some singular thing, as of a _square_, this square is to be resolved\ninto a _plain, terminated with a certain number of equal and straight\nlines and right angles_. For by this resolution we have these things\nuniversal or agreeable to all matter, namely, _line_, _plain_, (which\ncontains _superficies_) _terminated_, _angle_, _straightness_,\n_rectitude_, and _equality_; and if we can find out the causes of these,\nwe may compound them altogether into the cause of a square. Again, if\nany man propound to himself the conception of _gold_, he may, by\nresolving, come to the ideas of _solid_, _visible_, _heavy_, (that is,\ntending to the centre of the earth, or downwards) and many other more\nuniversal than gold itself; and these he may resolve again, till he come\nto such things as are most universal. And in this manner, by resolving\ncontinually, we may come to know what those things are, whose causes\nbeing first known severally, and afterwards compounded, bring us to the\nknowledge of singular things. I conclude, therefore, that the method of\nattaining to the universal knowledge of things, is purely _analytical_.\n\n[Sidenote: The highest causes, and most universal in every kind, are\n known by themselves.]\n\n5. But the causes of universal things (of those, at least, that have any\ncause) are manifest of themselves, or (as they say commonly) known to\nnature; so that they need no method at all; for they have all but one\nuniversal cause, which is motion. For the variety of all figures arises\nout of the variety of those motions by which they are made; and motion\ncannot be understood to have any other cause besides motion; nor has the\nvariety of those things we perceive by sense, as of _colours_, _sounds_,\n_savours_, &c. any other cause than motion, residing partly in the\nobjects that work upon our senses, and partly in ourselves, in such\nmanner, as that it is manifestly some kind of motion, though we cannot,\nwithout ratiocination, come to know what kind. For though many cannot\nunderstand till it be in some sort demonstrated to them, that all\nmutation consists in motion; yet this happens not from any obscurity in\nthe thing itself, (for it is not intelligible that anything can depart\neither from rest, or from the motion it has, except by motion), but\neither by having their natural discourse corrupted with former opinions\nreceived from their masters, or else for this, that they do not at all\nbend their mind to the enquiring out of truth.\n\n[Sidenote: Method from principles found out, tending to science simply,\n what it is.]\n\n6. By the knowledge therefore of universals, and of their causes (which\nare the first principles by which we know the \u03b4\u03b9\u03cc\u03c4\u03b9 of things) we have\nin the first place their definitions, (which are nothing but the\nexplication of our simple conceptions.) For example, he that has a true\nconception of _place_, cannot be ignorant of this definition, _place is\nthat space which is possessed or filled adequately by some body_; and\nso, he that conceives _motion_ aright, cannot but know that _motion is\nthe privation of one place, and the acquisition of another_. In the next\nplace, we have their generations or descriptions; as (for example) that\n_a line is made by the motion of a point, superficies by the motion of a\nline_, and _one motion by another motion_, &c. It remains, that we\nenquire what motion begets such and such effects; as, what motion makes\na straight line, and what a circular; what motion thrusts, what draws,\nand by what way; what makes a thing which is seen or heard, to be seen\nor heard sometimes in one manner, sometimes in another. Now the method\nof this kind of enquiry, is _compositive_. For first we are to observe\nwhat effect a body moved produceth, when we consider nothing in it\nbesides its motion; and we see presently that this makes a line, or\nlength; next, what the motion of a long body produces, which we find to\nbe superficies; and so forwards, till we see what the effects of simple\nmotion are; and then, in like manner, we are to observe what proceeds\nfrom the addition, multiplication, subtraction, and division, of these\nmotions, and what effects, what figures, and what properties, they\nproduce; from which kind of contemplation sprung that part of philosophy\nwhich is called _geometry_.\n\nFrom this consideration of what is produced by simple motion, we are to\npass to the consideration of what effects one body moved worketh upon\nanother; and because there may be motion in all the several parts of a\nbody, yet so as that the whole body remain still in the same place, we\nmust enquire first, what motion causeth such and such motion in the\nwhole, that is, when one body invades another body which is either at\nrest or in motion, what way, and with what swiftness, the invaded body\nshall move; and, again, what motion this second body will generate in a\nthird, and so forwards. From which contemplation shall be drawn that\npart of philosophy which treats of motion.\n\nIn the third place we must proceed to the enquiry of such effects as are\nmade by the motion of the parts of any body, as, how it comes to pass,\nthat things when they are the same, yet seem not to be the same, but\nchanged. And here the things we search after are sensible qualities,\nsuch as _light_, _colour_, _transparency_, _opacity_, _sound_, _odour_,\n_savour_, _heat_, _cold_, and the like; which because they cannot be\nknown till we know the causes of sense itself, therefore the\nconsideration of the causes of _seeing_, _hearing_, _smelling_,\n_tasting_, and _touching_, belongs to this third place; and all those\nqualities and changes, above mentioned, are to be referred to the fourth\nplace; which two considerations comprehend that part of philosophy which\nis called _physics_. And in these four parts is contained whatsoever in\nnatural philosophy may be explicated by demonstration, properly so\ncalled. For if a cause were to be rendered of natural appearances in\nspecial, as, what are the motions and influences of the heavenly bodies,\nand of their parts, the reason hereof must either be drawn from the\nparts of the sciences above mentioned, or no reason at all will be\ngiven, but all left to uncertain conjecture.\n\nAfter _physics_ we must come to _moral philosophy_; in which we are to\nconsider the motions of the mind, namely, _appetite_, _aversion_,\n_love_, _benevolence_, _hope_, _fear_, _anger_, _emulation_, _envy_,\n_&c._; what causes they have, and of what they be causes. And the reason\nwhy these are to be considered after _physics_ is, that they have their\ncauses in sense and imagination, which are the subject of _physical_\ncontemplation. Also the reason, why all these things are to be searched\nafter in the order above-said, is, that physics cannot be understood,\nexcept we know first what motions are in the smallest parts of bodies;\nnor such motion of parts, till we know what it is that makes another\nbody move; nor this, till we know what simple motion will effect. And\nbecause all appearance of things to sense is determined, and made to be\nof such and such quality and quantity by compounded motions, every one\nof which has a certain degree of velocity, and a certain and determined\nway; therefore, in the first place, we are to search out the ways of\nmotion simply (in which geometry consists); next the ways of such\ngenerated motions as are manifest; and, lastly, the ways of internal and\ninvisible motions (which is the enquiry of natural philosophers). And,\ntherefore, they that study natural philosophy, study in vain, except\nthey begin at geometry; and such writers or disputers thereof, as are\nignorant of geometry, do but make their readers and hearers lose their\ntime.\n\n[Sidenote: That method of civil and natural science, proceeding from\n sense to principles, is analytical; and again, that which\n begins at principles is synthetical.]\n\n7. _Civil_ and _moral philosophy_ do not so adhere to one another, but\nthat they may be severed. For the causes of the motions of the mind are\nknown, not only by ratiocination, but also by the experience of every\nman that takes the pains to observe those motions within himself. And,\ntherefore, not only they that have attained the knowledge of the\npassions and perturbations of the mind, by the _synthetical method_, and\nfrom the very first principles of philosophy, may by proceeding in the\nsame way, come to the causes and necessity of constituting\ncommonwealths, and to get the knowledge of what is natural right, and\nwhat are civil duties; and, in every kind of government, what are the\nrights of the commonwealth, and all other knowledge appertaining to\ncivil philosophy; for this reason, that the principles of the politics\nconsist in the knowledge of the motions of the mind, and the knowledge\nof these motions from the knowledge of sense and imagination; but even\nthey also that have not learned the first part of philosophy, namely,\n_geometry_ and _physics_, may, notwithstanding, attain the principles of\ncivil philosophy, by the _analytical method_. For if a question be\npropounded, as, _whether such an action be just or unjust_; if that\n_unjust_ be resolved into _fact against law_, and that notion _law_ into\nthe _command_ of him or them that have _coercive power_; and that\n_power_ be derived from the _wills_ of men that constitute such power,\nto the end they may live in peace, they may at last come to this, that\nthe appetites of men and the passions of their minds are such, that,\nunless they be restrained by some power, they will always be making war\nupon one another; which may be known to be so by any man's experience,\nthat will but examine his own mind. And, therefore, from hence he may\nproceed, by compounding, to the determination of the justice or\ninjustice of any propounded action. So that it is manifest, by what has\nbeen said, that the method of philosophy, to such as seek science\nsimply, without propounding to themselves the solution of any particular\nquestion, is partly analytical, and partly synthetical; namely, that\nwhich proceeds from sense to the invention of principles, analytical;\nand the rest synthetical.\n\n[Sidenote: The method of searching out, whether anything propounded be\n matter or accident.]\n\n8. To those that seek the cause of some certain and propounded\nappearance or effect, it happens, sometimes, that they know not whether\nthe thing, whose cause is sought after, be matter or body, or some\naccident of a body. For though in geometry, when the cause is sought of\nmagnitude, or proportion, or figure, it be certainly known that these\nthings, namely magnitude, proportion, and figure, are accidents; yet in\nnatural philosophy, where all questions are concerning the causes of the\nphantasms of sensible things, it is not so easy to discern between the\nthings themselves, from which those phantasms proceed, and the\nappearances of those things to the sense; which have deceived many,\nespecially when the phantasms have been made by light. For example, a\nman that looks upon the sun, has a certain shining idea of the magnitude\nof about a foot over, and this he calls the sun, though he know the sun\nto be truly a great deal bigger; and, in like manner, the phantasm of\nthe same thing appears sometimes round, by being seen afar off, and\nsometimes square, by being nearer. Whereupon it may well be doubted,\nwhether that phantasm be matter, or some body natural, or only some\naccident of a body; in the examination of which doubt we may use this\nmethod. The properties of matter and accidents already found out by us,\nby the synthetical method, from their definitions, are to be compared\nwith the idea we have before us; and if it agree with the properties of\nmatter or body, then it is a body; otherwise it is an accident. Seeing,\ntherefore, matter cannot by any endeavour of ours be either made or\ndestroyed, or increased, or diminished, or moved out of its place,\nwhereas that idea appears, vanishes, is increased and diminished, and\nmoved hither and thither at pleasure; we may certainly conclude that it\nis not a body, but an accident only. And this method is _synthetical_.\n\n[Sidenote: The method of seeking whether any accident be in this or in\n that subject.]\n\n9. But if there be a doubt made concerning the subject of any known\naccident (for this may be doubted sometimes, as in the precedent\nexample, doubt may be made in what subject that splendour and apparent\nmagnitude of the sun is), then our enquiry must proceed in this manner.\nFirst, matter in general must be divided into parts, as, into object,\nmedium, and the sentient itself, or such other parts as seem most\nconformable to the thing propounded. Next, these parts are severally to\nbe examined how they agree with the definition of the subject; and such\nof them as are not capable of that accident are to be rejected. For\nexample, if by any true ratiocination the sun be found to be greater\nthan its apparent magnitude, then that magnitude is not in the sun; if\nthe sun be in one determined straight line, and one determined distance,\nand the magnitude and splendour be seen in more lines and distances than\none, as it is in reflection or refraction, then neither that splendour\nnor apparent magnitude are in the sun itself, and, therefore, the body\nof the sun cannot be the subject of that splendour and magnitude. And\nfor the same reasons the air and other parts will be rejected, till at\nlast nothing remain which can be the subject of that splendour and\nmagnitude but the sentient itself. And this method, in regard the\nsubject is divided into parts, is analytical; and in regard the\nproperties, both of the subject and accident, are compared with the\naccident concerning whose subject the enquiry is made, it is\nsynthetical.\n\n[Sidenote: Method of searching for the cause of any effect, propounded.]\n\n10. But when we seek after the cause of any propounded effect, we must\nin the first place get into our mind an exact notion or idea of that\nwhich we call cause, namely, that _a cause is the sum or aggregate of\nall such accidents, both in the agents and the patient, as concur to the\nproducing of the effect propounded; all which existing together, it\ncannot be understood but that the effect existeth with them; or that it\ncan possibly exist if any one of them be absent_. This being known, in\nthe next place we must examine singly every accident that accompanies or\nprecedes the effect, as far forth as it seems to conduce in any manner\nto the production of the same, and see whether the propounded effect may\nbe conceived to exist, without the existence of any of those accidents;\nand by this means separate such accidents, as do not concur, from such\nas concur to produce the said effect; which being done, we are to put\ntogether the concurring accidents, and consider whether we can possibly\nconceive, that when these are all present, the effect propounded will\nnot follow; and if it be evident that the effect will follow, then that\naggregate of accidents is the entire cause, otherwise not; but we must\nstill search out and put together other accidents. For example, if the\ncause of light be propounded to be sought out; first, we examine things\nwithout us, and find that whensoever light appears, there is some\nprincipal object, as it were the fountain of light, without which we\ncannot have any perception of light; and, therefore, the concurrence of\nthat object is necessary to the generation of light. Next we consider\nthe medium, and find, that unless it be disposed in a certain manner,\nnamely, that it be transparent, though the object remain the same, yet\nthe effect will not follow; and, therefore, the concurrence of\ntransparency is also necessary to the generation of light. Thirdly, we\nobserve our own body, and find that by the indisposition of the eyes,\nthe brain, the nerves, and the heart, that is, by obstructions,\nstupidity, and debility, we are deprived of light, so that a fitting\ndisposition of the organs to receive impressions from without is\nlikewise a necessary part of the cause of light. Again, of all the\naccidents inherent in the object, there is none that can conduce to the\neffecting of light, but only action (or a certain motion), which cannot\nbe conceived to be wanting, whensoever the effect is present; for, that\nanything may shine, it is not requisite that it be of such or such\nmagnitude or figure, or that the whole body of it be moved out of the\nplace it is in (unless it may perhaps be said, that in the sun, or other\nbody, that which causes light is the light it hath in itself; which yet\nis but a trifling exception, seeing nothing is meant thereby but the\ncause of light; as if any man should say that the cause of light is that\nin the sun which produceth it); it remains, therefore, that the action,\nby which light is generated, is motion only in the parts of the object.\nWhich being understood, we may easily conceive what it is the medium\ncontributes, namely, the continuation of that motion to the eye; and,\nlastly, what the eye and the rest of the organs of the sentient\ncontribute, namely, the continuation of the same motion to the last\norgan of sense, the heart. And in this manner the cause of light may be\nmade up of motion continued from the original of the same motion, to the\noriginal of vital motion, light being nothing but the alteration of\nvital motion, made by the impression upon it of motion continued from\nthe object. But I give this only for an example, for I shall speak more\nat large of light, and the generation of it, in its proper place. In the\nmean time it is manifest, that in the searching out of causes, there is\nneed partly of the analytical, and partly of the synthetical method; of\nthe analytical, to conceive how circumstances conduce severally to the\nproduction of effects; and of the synthetical, for the adding together\nand compounding of what they can effect singly by themselves. And thus\nmuch may serve for the method of invention. It remains that I speak of\nthe method of teaching, that is, of demonstration, and of the means by\nwhich we demonstrate.\n\n[Sidenote: Words serve to invention as marks; to demonstration as\n signs.]\n\n11. In the method of invention, the use of words consists in this, that\nthey may serve for marks, by which, whatsoever we have found out may be\nrecalled to memory; for without this all our inventions perish, nor will\nit be possible for us to go on from principles beyond a syllogism or\ntwo, by reason of the weakness of memory. For example, if any man, by\nconsidering a triangle set before him, should find that all its angles\ntogether taken are equal to two right angles, and that by thinking of\nthe same tacitly, without any use of words either understood or\nexpressed; and it should happen afterwards that another triangle, unlike\nthe former, or the same in different situation, should be offered to his\nconsideration, he would not know readily whether the same property were\nin this last or no, but would be forced, as often as a different\ntriangle were brought before him (and the difference of triangles is\ninfinite) to begin his contemplation anew; which he would have no need\nto do if he had the use of names, for every universal name denotes the\nconceptions we have of infinite singular things. Nevertheless, as I said\nabove, they serve as _marks_ for the help of our memory, whereby we\nregister to ourselves our own inventions; but not as _signs_ by which we\ndeclare the same to others; so that a man may be a philosopher alone by\nhimself, without any master; Adam had this capacity. But to teach, that\nis, to demonstrate, supposes two at the least, and syllogistical speech.\n\n[Sidenote: The method of demonstration is synthetical.]\n\n12. And seeing teaching is nothing but leading the mind of him we teach,\nto the knowledge of our inventions, in that track by which we attained\nthe same with our own mind; therefore, the same method that served for\nour invention, will serve also for demonstration to others, saving that\nwe omit the first part of method which proceeded from the sense of\nthings to universal principles, which, because they are principles,\ncannot be demonstrated; and seeing they are known by nature, (as was\nsaid above in the 5th article) they need no demonstration, though they\nneed explication. The whole method, therefore, of demonstration, is\n_synthetical_, consisting of that order of speech which begins from\nprimary or most universal propositions, which are manifest of\nthemselves, and proceeds by a perpetual composition of propositions into\nsyllogisms, till at last the learner understand the truth of the\nconclusion sought after.\n\n[Sidenote: Definitions only are primary, & universal propositions.]\n\n13. Now, such principles are nothing but definitions, whereof there are\ntwo sorts; one of names, that signify such things as have some\nconceivable cause, and another of such names as signify things of which\nwe can conceive no cause at all. Names of the former kind are, _body_,\nor _matter_, _quantity_, or _extension_, _motion_, and whatsoever is\ncommon to all matter. Of the second kind, are _such a body_, _such and\nso great motion_, _so great magnitude_, _such figure_, and whatsoever we\ncan distinguish one body from another by. And names of the former kind\nare well enough defined, when, by speech as short as may be, we raise in\nthe mind of the hearer perfect and clear ideas or conceptions of the\nthings named, as when we define motion to be _the leaving of one place,\nand the acquiring of another continually_; for though no thing moved,\nnor any cause of motion be in that definition, yet, at the hearing of\nthat speech, there will come into the mind of the hearer an _idea_ of\nmotion clear enough. But definitions of things, which may be understood\nto have some cause, must consist of such names as express the cause or\nmanner of their generation, as when we define a circle to be a figure\nmade by the circumduction of a straight line in a plane, &c. Besides\ndefinitions, there is no other proposition that ought to be called\nprimary, or (according to severe truth) be received into the number of\nprinciples. For those _axioms of Euclid_, seeing they may be\ndemonstrated, are no principles of demonstration, though they have by\nthe consent of all men gotten the authority of principles, because they\nneed not be demonstrated. Also, those _petitions_, or _postulata_, (as\nthey call them) though they be principles, yet they are not principles\nof demonstration, but of construction only; that is, not of science, but\nof power; or (which is all one) not of _theorems_, which are\nspeculations, but of _problems_, which belong to practice, or the doing\nof something. But as for those common received opinions, _Nature abhors\nvacuity_, _Nature doth nothing in vain_, and the like, which are neither\nevident in themselves, nor at all to be demonstrated, and which are\noftener false than true, they are much less to be acknowledged for\nprinciples.\n\nTo return, therefore, to definitions; the reason why I say that the\ncause and generation of such things, as have any cause or generation,\nought to enter into their definitions, is this. The end of science is\nthe demonstration of the causes and generations of things; which if they\nbe not in the definitions, they cannot be found in the conclusion of the\nfirst syllogism, that is made from those definitions; and if they be not\nin the first conclusion, they will not be found in any further\nconclusion deduced from that; and, therefore, by proceeding in this\nmanner, we shall never come to science; which is against the scope and\nintention of demonstration.\n\n[Sidenote: The nature & definition of a definition.]\n\n14. Now, seeing definitions (as I have said) are principles, or primary\npropositions, they are therefore speeches; and seeing they are used for\nthe raising of an _idea_ of some thing in the mind of the learner,\nwhensoever that thing has a name, the definition of it can be nothing\nbut the explication of that name by speech; and if that name be given it\nfor some compounded conception, the definition is nothing but a\nresolution of that name into its most universal parts. As when we define\nman, saying _man is a body animated, sentient, rational_, those names,\n_body animated_, _&c._ are parts of that whole name _man_; so that\ndefinitions of this kind always consist of _genus_ and _difference_; the\nformer names being all, till the last, _general_; and the last of all,\n_difference_. But if any name be the most universal in its kind, then\nthe definition of it cannot consist of _genus_ and _difference_, but is\nto be made by such circumlocution, as best explicateth the force of that\nname. Again, it is possible, and happens often, that the _genus_ and\n_difference_ are put together, and yet make no definition; as these\nwords, _a straight line_, contain both the _genus_ and _difference_; but\nare not a definition, unless we should think a straight line may be thus\ndefined, _a straight line is a straight line_: and yet if there were\nadded another name, consisting of different words, but signifying the\nsame thing which these signify, then these might be the definition of\nthat name. From what has been said, it may be understood how a\ndefinition ought to be defined, namely, _that it is a proposition, whose\npredicate resolves the subject, when it may; and when it may not, it\nexemplifies the same_.\n\n[Sidenote: Properties of a definition.]\n\n15. The properties of a definition are:\n\nFirst, that it takes away equivocation, as also all that multitude of\ndistinctions, which are used by such as think they may learn philosophy\nby disputation. For the nature of a definition is to define, that is, to\ndetermine the signification of the defined name, and to pare from it all\nother signification besides what is contained in the definition itself;\nand therefore one definition does as much, as all the distinctions (how\nmany soever) that can be used about the name defined.\n\nSecondly, that it gives an universal notion of the thing defined,\nrepresenting a certain universal picture thereof, not to the eye, but to\nthe mind. For as when one paints a man, he paints the image of some man;\nso he, that defines the name man, makes a representation of some man to\nthe mind.\n\nThirdly, that it is not necessary to dispute whether definitions are to\nbe admitted or no. For when a master is instructing his scholar, if the\nscholar understand all the parts of the thing defined, which are\nresolved in the definition, and yet will not admit of the definition,\nthere needs no further controversy betwixt them, it being all one as if\nhe refused to be taught. But if he understand nothing, then certainly\nthe definition is faulty; for the nature of a definition consists in\nthis, that it exhibit a clear idea of the thing defined; and principles\nare either known by themselves, or else they are not principles.\n\nFourthly, that, in philosophy, definitions are before defined names. For\nin teaching philosophy, the first beginning is from definitions; and all\nprogression in the same, till we come to the knowledge of the thing\ncompounded, is compositive. Seeing, therefore, definition is the\nexplication of a compounded name by resolution, and the progression is\nfrom the parts to the compound, definitions must be understood before\ncompounded names; nay, when the names of the parts of any speech be\nexplicated, it is not necessary that the definition should be a name\ncompounded of them. For example, when these names, _equilateral_,\n_quadrilateral_, _right-angled_, are sufficiently understood, it is not\nnecessary in geometry that there should be at all such a name as\n_square_; for defined names are received in philosophy for brevity's\nsake only.\n\nFifthly, that compounded names, which are defined one way in some one\npart of philosophy, may in another part of the same be otherwise\ndefined; as a _parabola_ and an _hyperbole_ have one definition in\ngeometry, and another in rhetoric; for definitions are instituted and\nserve for the understanding of the doctrine which is treated of. And,\ntherefore, as in one part of philosophy, a definition may have in it\nsome one fit name for the more brief explanation of some proposition in\ngeometry; so it may have the same liberty in other parts of philosophy;\nfor the use of names is particular (even where many agree to the\nsettling of them) and arbitrary.\n\nSixthly, that no name can be defined by any one word; because no one\nword is sufficient for the resolving of one or more words.\n\nSeventhly, that a defined name ought not to be repeated in the\ndefinition. For a defined name is the whole compound, and a definition\nis the resolution of that compound into parts; but no total can be part\nof itself.\n\n[Sidenote: Nature of a demonstration.]\n\n16. Any two definitions, that may be compounded into a syllogism,\nproduce a conclusion; which, because it is derived from principles, that\nis, from definitions, is said to be demonstrated; and the derivation or\ncomposition itself is called a demonstration. In like manner, if a\nsyllogism be made of two propositions, whereof one is a definition, the\nother a demonstrated conclusion, or neither of them is a definition, but\nboth formerly demonstrated, that syllogism is also called a\ndemonstration, and so successively. The definition therefore of a\ndemonstration is this, _a demonstration is a syllogism, or series of\nsyllogisms, derived and continued, from the definitions of names, to the\nlast conclusion_. And from hence it may be understood, that all true\nratiocination, which taketh its beginning from true principles,\nproduceth science, and is true demonstration. For as for the original of\nthe name, although that, which the Greeks called \u1f00\u03c0\u03bf\u03b4\u03ad\u03b9\u03be\u03b9\u03c2, and the\nLatins _demonstratio_, was understood by them for that sort only of\nratiocination, in which, by the describing of certain lines and figures,\nthey placed the thing they were to prove, as it were before men's eyes,\nwhich is properly \u1f00\u03c0\u03bf\u03b4\u03b5\u03b9\u03ba\u03bd\u03cd\u03b5\u03b9\u03bd, or to _shew_ by the figure; yet they\nseem to have done it for this reason, that unless it were in geometry,\n(in which only there is place for such figures) there was no\nratiocination certain, and ending in science, their doctrines concerning\nall other things being nothing but controversy and clamour; which,\nnevertheless, happened, not because the truth to which they pretended\ncould not be made evident without figures, but because they wanted true\nprinciples, from which they might derive their ratiocination; and,\ntherefore, there is no reason but that if true definitions were premised\nin all sorts of doctrines, the demonstrations also would be true.\n\n[Sidenote: Properties of a demonstration, and order of things to be\n demonstrated.]\n\n17. It. is proper to methodical demonstration,\n\nFirst, that there be a true succession of one reason to another,\naccording to the rules of syllogizing delivered above.\n\nSecondly, that the premises of all syllogisms be demonstrated from the\nfirst definitions.\n\nThirdly, that after definitions, he that teaches or demonstrates any\nthing, proceed in the same method by which he found it out; namely, that\nin the first place those things be demonstrated, which immediately\nsucceed to universal definitions (in which is contained that part of\nphilosophy which is called _philosophia prima_). Next, those things\nwhich may be demonstrated by simple motion (in which geometry consists).\nAfter geometry, such things as may be taught or shewed by manifest\naction, that is, by thrusting from, or pulling towards. And after these,\nthe motion or mutation of the invisible parts of things, and the\ndoctrine of sense and imaginations, and of the internal passions,\nespecially those of men, in which are comprehended the grounds of civil\nduties, or civil philosophy; which takes up the last place. And that\nthis method ought to be kept in all sorts of philosophy, is evident from\nhence, that such things as I have said are to be taught last, cannot be\ndemonstrated, till such as are propounded to be first treated of, be\nfully understood. Of which method no other example can be given, but\nthat treatise of the elements of philosophy, which I shall begin in the\nnext chapter, and continue to the end of the work.\n\n[Sidenote: Faults of a demonstration.]\n\n18. Besides those _paralogisms_, whose fault lies either in the falsity\nof the premises, or the want of true composition, of which I have spoken\nin the precedent chapter, there are two more, which are frequent in\ndemonstration; one whereof is commonly called _petitio principii_; the\nother is the supposing of a _false cause_; and these do not only deceive\nunskilful learners, but sometimes masters themselves, by making them\ntake that for well demonstrated, which is not demonstrated at all.\n_Petitio principii_ is, when the conclusion to be proved is disguised in\nother words, and put for the definition or principle from whence it is\nto be demonstrated; and thus, by putting for the cause of the thing\nsought, either the thing itself or some effect of it, they make a circle\nin their demonstration. As for example, he that would demonstrate that\nthe earth stands still in the centre of the world, and should suppose\nthe earth's gravity to be the cause thereof, and define gravity to be a\nquality by which every heavy body tends towards the centre of the world,\nwould lose his labour; for the question is, what is the cause of that\nquality in the earth? and, therefore, he that supposes gravity to be the\ncause, puts the thing itself for its own cause.\n\nOf a _false cause_ I find this example in a certain treatise where the\nthing to be demonstrated is the motion of the earth. He begins,\ntherefore, with this, that seeing the earth and the sun are not always\nin the same situation, it must needs be that one of them be locally\nmoved, which is true; next, he affirms that the vapours, which the sun\nraises from the earth and sea, are, by reason of this motion,\nnecessarily moved, which also is true; from whence he infers the winds\nare made, and this may pass for granted; and by these winds he says, the\nwaters of the sea are moved, and by their motion the bottom of the sea,\nas if it were beaten forwards, moves round; and let this also be\ngranted; wherefore, he concludes, the earth is moved; which is,\nnevertheless, a paralogism. For, if that wind were the cause why the\nearth was, from the beginning, moved round, and the motion either of the\nsun or the earth were the cause of that wind, then the motion of the sun\nor the earth was before the wind itself; and if the earth were moved,\nbefore the wind was made, then the wind could not be the cause of the\nearth's revolution; but, if the sun were moved, and the earth stand\nstill, then it is manifest the earth might remain unmoved,\nnotwithstanding that wind; and therefore that motion was not made by the\ncause which he allegeth. But paralogisms of this kind are very frequent\namong the writers of _physics_, though none can be more elaborate than\nthis in the example given.\n\n[Sidenote: Why the analytical method of geometricians cannot be treated\n of in this place.]\n\n19. It may to some men seem pertinent to treat in this place of that art\nof the geometricians, which they call _logistica_, that is, the art, by\nwhich, from supposing the thing in question to be true, they proceed by\nratiocination, till either they come to something known, by which they\nmay demonstrate the truth of the thing sought for; or to something which\nis impossible, from whence they collect that to be false, which they\nsupposed true. But this art cannot be explicated here, for this reason,\nthat the method of it can neither be practised, nor understood, unless\nby such as are well versed in geometry; and among geometricians\nthemselves, they, that have most theorems in readiness, are the most\nready in the use of this _logistica_; so that, indeed, it is not a\ndistinct thing from geometry itself; for there are, in the method of it,\nthree parts; the first whereof consists in the finding out of equality\nbetwixt known and unknown things, which they call equation; and this\nequation cannot be found out, but by such as know perfectly the nature,\nproperties, and transpositions of proportion, as also the addition,\nsubtraction, multiplication, and division of lines and superficies, and\nthe extraction of roots; which are the parts of no mean geometrician.\nThe second is, when an equation is found, to be able to judge whether\nthe truth or falsity of the question may be deduced from it, or no;\nwhich yet requires greater knowledge. And the third is, when such an\nequation is found, as is fit for the solution of the question, to know\nhow to resolve the same in such manner, that the truth or falsity may\nthereby manifestly appear; which, in hard questions, cannot be done\nwithout the knowledge of the nature of crooked-lined figures; but he\nthat understands readily the nature and properties of these, is a\ncomplete geometrician. It happens besides, that for the finding out of\nequations, there is no certain method, but he is best able to do it,\nthat has the best natural wit.\n\n\n\n\n PART II.\n\n THE\n FIRST GROUNDS OF PHILOSOPHY.\n\n\n -------\n\n\n CHAPTER VII.\n\n OF PLACE AND TIME.\n\n1. Things that have no existence, may nevertheless be understood and\n computed.\u20142. What is Space.\u20143. Time.\u20144. Part.\u20145. Division.\u20146.\n One.\u20147. Number.\u20148. Composition.\u20149. The whole.\u201410. Spaces and times\n contiguous, and continual.\u201411. Beginning, end, way, finite,\n infinite.\u201412. What is infinite in power. Nothing infinite can be\n truly said to be either whole, or one; nor infinite spaces or times,\n many.\u201413. Division proceeds not to the least.\n\n\n[Sidenote: Things that have no existence, may nevertheless be understood\n and computed.]\n\n1. In the teaching of natural philosophy, I cannot begin better (as I\nhave already shewn) than from _privation_; that is, from feigning the\nworld to be annihilated. But, if such annihilation of all things be\nsupposed, it may perhaps be asked, what would remain for any man (whom\nonly I except from this universal annihilation of things) to consider as\nthe subject of philosophy, or at all to reason upon; or what to give\nnames unto for ratiocination's sake.\n\nI say, therefore, there would remain to that man ideas of the world, and\nof all such bodies as he had, before their annihilation, seen with his\neyes, or perceived by any other sense; that is to say, the memory and\nimagination of magnitudes, motions, sounds, colours, &c. as also of\ntheir order and parts. All which things, though they be nothing but\nideas and phantasms, happening internally to him that imagineth; yet\nthey will appear as if they were external, and not at all depending upon\nany power of the mind. And these are the things to which he would give\nnames, and subtract them from, and compound them with one another. For\nseeing, that after the destruction of all other things, I suppose man\nstill remaining, and namely that he thinks, imagines, and remembers,\nthere can be nothing for him to think of but what is past; nay, if we do\nbut observe diligently what it is we do when we consider and reason, we\nshall find, that though all things be still remaining in the world, yet\nwe compute nothing but our own phantasms. For when we calculate the\nmagnitude and motions of heaven or earth, we do not ascend into heaven\nthat we may divide it into parts, or measure the motions thereof, but we\ndo it sitting still in our closets or in the dark. Now things may be\nconsidered, that is, be brought into account, either as internal\naccidents of our mind, in which manner we consider them when the\nquestion is about some faculty of the mind; or as species of external\nthings, not as really existing, but appearing only to exist, or to have\na being without us. And in this manner we are now to consider them.\n\n[Sidenote: What is Space.]\n\n2. If therefore we remember, or have a phantasm of any thing that was in\nthe world before the supposed annihilation of the same; and consider,\nnot that the thing was such or such, but only that it had a being\nwithout the mind, we have presently a conception of that we call\n_space_: an imaginary space indeed, because a mere phantasm, yet that\nvery thing which all men call so. For no man calls it space for being\nalready filled, but because it may be filled; nor does any man think\nbodies carry their places away with them, but that the same space\ncontains sometimes one, sometimes another body; which could not be if\nspace should always accompany the body which is once in it. And this is\nof itself so manifest, that I should not think it needed any explaining\nat all, but that I find space to be falsely defined by certain\nphilosophers, who infer from thence, one, that the world is infinite\n(for taking _space_ to be the extension of bodies, and thinking\nextension may encrease continually, he infers that bodies may be\ninfinitely extended); and, another, from the same definition, concludes\nrashly, that it is impossible even to God himself to create more worlds\nthan one; for, if another world were to be created, he says, that seeing\nthere is nothing without this world, and therefore (according to his\ndefinition) no space, that new world must be placed in nothing; but in\nnothing nothing can be placed; which he affirms only, without showing\nany reason for the same; whereas the contrary is the truth: for more\ncannot be put into a place already filled, so much is empty space fitter\nthan that, which is full, for the receiving of new bodies. Having\ntherefore spoken thus much for these men's sakes, and for theirs that\nassent to them, I return to my purpose, and define _space_ thus: SPACE\n_is the phantasm of a thing existing without the mind simply_; that is\nto say, that phantasm, in which we consider no other accident, but only\nthat it appears without us.\n\n[Sidenote: Time.]\n\n3. As a body leaves a phantasm of its magnitude in the mind, so also a\nmoved body leaves a phantasm of its motion, namely, an idea of that body\npassing out of one space into another by continual succession. And this\nidea, or phantasm, is that, which (without receding much from the common\nopinion, or from _Aristotle's_ definition) I call _Time_. For seeing all\nmen confess a year to be time, and yet do not think a year to be the\naccident or affection of any body, they must needs confess it to be, not\nin the things without us, but only in the thought of the mind. So when\nthey speak of the times of their predecessors, they do not think after\ntheir predecessors are gone, that their times can be any where else than\nin the memory of those that remember them. And as for those that say,\ndays, years, and months are the motions of the sun and moon, seeing it\nis all one to say, motion _past_ and motion _destroyed_, and that\n_future_ motion is the same with motion which _is not yet begun_, they\nsay that, which they do not mean, that there neither is, nor has been,\nnor shall be any time: for of whatsoever it may be said, _it has been_\nor _it shall be_, of the same also it might have been said heretofore,\nor may be said hereafter, _it is_. What then can days, months, and\nyears, be, but the names of such computations made in our mind? _Time_\ntherefore is a phantasm, but a phantasm of motion, for if we would know\nby what moments time passes away, we make use of some motion or other,\nas of the sun, of a clock, of the sand in an hour-glass, or we mark some\nline upon which we imagine something to be moved, there being no other\nmeans by which we can take notice of any time at all. And yet, when I\nsay _time_ is a phantasm of motion, I do not say this is sufficient to\ndefine it by; for this word _time_ comprehends the notion of _former_\nand _latter_, or of _succession_ in the motion of a body, in as much as\nit is first _here_ then _there_. Wherefore a complete definition of\n_time_ is such as this, TIME _is the phantasm of before and after in\nmotion_; which agrees with this definition of _Aristotle_, _time is the\nnumber of motion according to former and latter_; for that numbering is\nan act of the mind; and therefore it is all one to say, _time is the\nnumber of motion according to former and latter_; and _time is a\nphantasm of motion numbered_. But that other definition, _time is the\nmeasure of motion_, is not so exact, for we measure time by motion and\nnot motion by time.\n\n[Sidenote: Part.]\n\n4. One space is called _part_ of another space, and one time _part_ of\nanother time, when this contains that and something besides. From whence\nit may be collected, that nothing can rightly be called a PART, but that\nwhich is compared with something that contains it.\n\n[Sidenote: Division.]\n\n5. And therefore to _make parts_, or to _part_ or DIVIDE space or\n_time_, is nothing else but to consider one and another within the same;\nso that if any man _divide_ space or time, the diverse conceptions he\nhas are more, by one, than the parts he makes; for his first conception\nis of that which is to be divided, then of some part of it, and again of\nsome other part of it, and so forwards as long as he goes on in\ndividing.\n\nBut it is to be noted, that here, by _division_, I do not mean the\nsevering or pulling asunder of one space or time from another (for does\nany man think that one hemisphere may be separated from the other\nhemisphere, or the first hour from the second?) but diversity of\nconsideration; so that division is not made by the operation of the\nhands but of the mind.\n\n[Sidenote: One.]\n\n6. When space or time is considered among other spaces or times, it is\nsaid to be ONE, namely _one of them_; for except one space might be\nadded to another, and subtracted from another space, and so of time, it\nwould be sufficient to say space or time simply, and superfluous to say\none space or one time, if it could not be conceived that there were\nanother. The common definition of _one_, namely, that _one is that which\nis undivided_, is obnoxious to an absurd consequence; for it may thence\nbe inferred, that whatsoever is divided is many things, that is, that\nevery divided thing, is divided things, which is insignificant.\n\n[Sidenote: Number.]\n\n7. NUMBER is _one_ and _one_, or _one one_ and _one_, and so forwards;\nnamely, _one_ and _one_ make the number _two_, and _one one_ and _one_\nthe number _three_; so are all other numbers made; which is all one as\nif we should say, _number is unities_.\n\n[Sidenote: Composition.]\n\n8. To COMPOUND space of spaces, or time of times, is first to consider\nthem one after another, and then altogether as one; as if one should\nreckon first the head, the feet, the arms, and the body, severally, and\nthen for the account of them all together put _man_. And that which is\nso put for all the severals of which it consists, is called the WHOLE;\nand those severals, when by the division of the whole they come again to\nbe considered singly, are parts thereof; and therefore the _whole_ and\n_all the parts taken together_ are the same thing. And as I noted above,\nthat in _division_ it is not necessary to pull the parts asunder; so in\n_composition_, it is to be understood, that for the making up of a whole\nthere is no need of putting the parts together, so as to make them touch\none another, but only of collecting them into one sum in the mind. For\nthus all men, being considered together, make up the whole of mankind,\nthough never so much dispersed by time and place; and twelve hours,\nthough the hours of several days, may be compounded into one number of\ntwelve.\n\n[Sidenote: The whole.]\n\n9. This being well understood, it is manifest, that nothing can rightly\nbe called a whole, that is not conceived to be compounded of parts, and\nthat it may be divided into parts; so that if we deny that a thing has\nparts, we deny the same to be a whole. For example, if we say the soul\ncan have no parts, we affirm that no soul can be a whole soul. Also it\nis manifest, that nothing has parts till it be divided; and when a thing\nis divided, the parts are only so many as the division makes them.\nAgain, that a part of a part is a part of the whole; and thus any part\nof the number _four_, as _two_, is a part of the number _eight_; for\n_four_ is made of _two_ and _two_; but _eight_ is compounded of _two_,\n_two_, and _four_, and therefore _two_, which is a part of the part\n_four_, is also a part of the whole _eight_.\n\n[Sidenote: Spaces and times contiguous and continual.]\n\n10. Two spaces are said to be CONTIGUOUS, when there is no other space\nbetwixt them. But two times, betwixt which there is no other time, are\ncalled immediate, as A B, B C.[Sidenote: A B C] And any two\nspaces, as well as times, are said to be CONTINUAL, when they have one\ncommon part, as A C, B D, where the part B C is common; [Sidenote:\nA B C D] and more spaces and times are continual, when every two\nwhich are next one another are continual.\n\n A B C D\n -------------\n\n A B C\n ------------\n\n[Sidenote: Beginning, end, way, finite, infinite.]\n\n11. That part which is between two other parts, is called a MEAN; and\nthat which is not between two other parts, an EXTREME. And of extremes,\nthat which is first reckoned is the BEGINNING, and that which last, the\nEND; and all the means together taken are the WAY. Also, _extreme parts_\nand _limits_ are the same thing. And from hence it is manifest, that\n_beginning_ and _end_ depend upon the order in which we number them; and\nthat to _terminate_ or _limit_ space and time, is the same thing with\n_imagining their beginning and end_; as also that every thing is FINITE\nor INFINITE, according as we imagine or not imagine it _limited_ or\n_terminated_ every way; and that the _limits_ of any number are\n_unities_, and of these, that which is the first in our numbering is the\n_beginning_, and that which we number last, is the _end_. When we say\nnumber is _infinite_, we mean only that no number is expressed; for when\nwe speak of the numbers _two_, _three_, _a thousand_, &c. they are\nalways _finite_. But when no more is said but this, _number is\ninfinite_, it is to be understood as if it were said, this name _number_\nis an _indefinite_ name.\n\n[Sidenote: What is infinite in power. Nothing infinite can be truly said\n to be either whole or one; nor infinite spaces or times,\n many.]\n\n12. Space or time is said to _finite in power_, or _terminable_, when\nthere may be assigned a number of finite spaces or times, as of paces or\nhours, than which there can be no greater number of the same measure in\nthat space or time; and _infinite in power_ is that space or time, in\nwhich a greater number of the said paces or hours may be assigned, than\nany number that can be given. But we must note, that, although in that\nspace or time which is infinite in power, there may be numbered more\npaces or hours than any number that can be assigned, yet their number\nwill always be finite; for every number is finite. And therefore his\nratiocination was not good, that undertaking to prove the world to be\nfinite, reasoned thus; _If the world be infinite, then there may be\ntaken in it some part which is distant from us an infinite number of\npaces: but no such part can be taken; wherefore the world is not\ninfinite_; because that consequence of the major proposition is false;\nfor in an infinite space, whatsoever we take or design in our mind, the\ndistance of the same from us is a finite space; for in the very\ndesigning of the place thereof, we put an end to that space, of which we\nourselves are the beginning; and whatsoever any man with his mind cuts\noff both ways from infinite, he determines the same, that is, he makes\nit finite.\n\nOf infinite space or time, it cannot be said that it is a _whole_ or\n_one_: not a _whole_, because not compounded of parts; for seeing parts,\nhow many soever they be, are severally finite, they will also, when they\nare all put together, make a whole finite: nor _one_, because nothing\ncan be said to be one, except there be another to compare it with; but\nit cannot be conceived that there are two spaces, or two times,\ninfinite. Lastly, when we make question whether the world be finite or\ninfinite, we have nothing in our mind answering to the name _world_; for\nwhatsoever we imagine, is therefore finite, though our computation reach\nthe fixed stars, or the ninth or tenth, nay, the thousandth sphere. The\nmeaning of the question is this only, whether God has actually made so\ngreat an addition of body to body, as we are able to make of space to\nspace.\n\n[Sidenote: Division proceeds not to the least.]\n\n13. And, therefore, that which is commonly said, that space and time may\nbe divided infinitely, is not to be so understood, as if there might be\nany infinite or eternal division; but rather to be taken in this sense,\n_whatsoever is divided, is divided into such parts as may again be\ndivided_; or thus, _the least divisible thing is not to be given_; or,\nas geometricians have it, _no quantity is so small, but a less may be\ntaken_; which may easily be demonstrated in this manner. Let any space\nor time, that which was thought to be the least divisible, be divided\ninto two equal parts, A and B. I say either of them, as A, may be\ndivided again. For suppose the part A to be contiguous to the part B of\none side, and of the other side to some other space equal to B. This\nwhole space, therefore, being greater than the space given, is\ndivisible. Wherefore, if it be divided into two equal parts, the part in\nthe middle, which is A, will be also divided into two equal parts; and\ntherefore A was divisible.\n\n\n -------\n\n\n CHAPTER VIII.\n\n OF BODY AND ACCIDENT.\n\n1. Body defined.\u20142. Accident defined.\u20143. How an accident may be\n understood to be in its subject.\u20144. Magnitude, what it is.\u20145. Place,\n what it is, and that it is immovable.\u20146. What is full and empty.\u20147.\n Here, there, somewhere, what they signify.\u20148. Many bodies cannot be\n in one place, nor one body in many places.\u20149. Contiguous and\n continual, what they are.\u201410. The definition of motion. No motion\n intelligible but with time.\u201411. What it is to be at rest, to have\n been moved, and to be moved. No motion to be conceived, without the\n conception of past and future.\u201412. A point, a line, superficies and\n solid, what they are.\u201413. Equal, greater, and less in bodies and\n magnitudes, what they are.\u201414. One and the same body has always one\n and the same magnitude.\u201415. Velocity, what it is.\u201416. Equal,\n greater, and less in times, what they are.\u201417. Equal, greater, and\n less, in velocity, what.\u201418. Equal, greater, and less, in motion,\n what.\u201419. That which is at rest, will always be at rest, except it\n be moved by some external thing; and that which is moved, will\n always be moved, unless it be hindered by some external thing.\u201420.\n Accidents are generated and destroyed, but bodies not so.\u201421. An\n accident cannot depart from its subject.\u201422. Nor be moved.\u201423.\n Essence, form, and matter, what they are.\u201424. First matter,\n what.\u201425. That the whole is greater than any part thereof, why\n demonstrated.\n\n\n[Sidenote: Body defined.]\n\n1. Having understood what imaginary space is, in which we supposed\nnothing remaining without us, but all those things to be destroyed,\nthat, by existing heretofore, left images of themselves in our minds;\nlet us now suppose some one of those things to be placed again in the\nworld, or created anew. It is necessary, therefore, that this\nnew-created or replaced thing do not only fill some part of the space\nabove mentioned, or be coincident and coextended with it, but also that\nit have no dependance upon our thought. And this is that which, for the\nextension of it, we commonly call _body_; and because it depends not\nupon our thought, we say is _a thing subsisting of itself_; as also\n_existing_, because without us; and, lastly, it is called the _subject_,\nbecause it is so placed in and _subjected_ to imaginary space, that it\nmay be understood by reason, as well as perceived by sense. The\ndefinition, therefore, of _body_ may be this, a _body is that, which\nhaving no dependance upon our thought, is coincident or coextended with\nsome part of space_.\n\n[Sidenote: Accident defined.]\n\n2. But what an _accident_ is cannot so easily be explained by any\ndefinition, as by examples. Let us imagine, therefore, that a body fills\nany space, or is coextended with it; that coextension is not the\ncoextended body: and, in like manner, let us imagine that the same body\nis removed out of its place; that removing is not the removed body: or\nlet us think the same not removed; that not removing or rest is not the\nresting body. What, then, are these things? They are _accidents_ of that\nbody. But the thing in question is, _what is an accident?_ which is an\nenquiry after that which we know already, and not that which we should\nenquire after. For who does not always and in the same manner understand\nhim that says any thing is extended, or moved, or not moved? But most\nmen will have it be said that _an accident is something_, namely, some\npart of a natural thing, when, indeed, it is no part of the same. To\nsatisfy these men, as well as may be, they answer best that define an\n_accident_ to be _the manner by which any body is conceived_; which is\nall one as if they should say, _an accident is that faculty of any body,\nby which it works in us a conception of itself_. Which definition,\nthough it be not an answer to the question propounded, yet it is an\nanswer to that question which should have been propounded, namely,\n_whence does it happen that one part of any body appears here, another\nthere?_ For this is well answered thus: _it happens from the extension\nof that body_. Or, _how comes it to pass that the whole body, by\nsuccession, is seen now here, now there?_ and the answer will be, _by\nreason of its motion_. Or, lastly, _whence is it that any body\npossesseth the same space for sometime?_ and the answer will be,\n_because it is not moved_. For if concerning the name of a body, that\nis, concerning a concrete name, it be asked, _what is it?_ the answer\nmust be made by definition; for the question is concerning the\nsignification of the name. But if it be asked concerning an abstract\nname, _what is it?_ the cause is demanded why a thing appears so or so.\nAs if it be asked, _what is hard?_ The answer will be, hard is that,\nwhereof no part gives place, but when the whole gives place. But if it\nbe demanded, _what is hardness?_ a cause must be shewn why a part does\nnot give place, except the whole give place. Wherefore, I define an\n_accident_ to be _the manner of our conception of body_.\n\n[Sidenote: How an accident may be understood to be in its subject.]\n\n3. When an _accident_ is said _to be in a body_, it is not so to be\nunderstood, as if any thing were contained in that body; as if, for\nexample, redness were in blood, in the same manner, as blood is in a\nbloody cloth, that is, as a part in the whole; for so, an accident would\nbe a body also. But, as magnitude, or rest, or motion, is in that which\nis great, or which resteth, or which is moved, (which, how it is to be\nunderstood, every man understands) so also, it is to be understood, that\nevery other accident _is in_ its subject. And this, also, is explicated\nby _Aristotle_ no otherwise than negatively, namely, that _an accident\nis in its subject, not as any part thereof, but so as that it may be\naway, the subject still remaining_; which is right, saving that there\nare certain accidents which can never perish except the body perish\nalso; for no body can be conceived to be without extension, or without\nfigure. All other accidents, which are not common to all bodies, but\npeculiar to some only, as _to be at rest_, _to be moved_, _colour_,\n_hardness_, and the like, do perish continually, and are succeeded by\nothers; yet so, as that the body never perisheth. And as for the opinion\nthat some may have, that all other accidents are not in their bodies in\nthe same manner that extension, motion, rest, or figure, are in the\nsame; for example, that colour, heat, odour, virtue, vice, and the like,\nare otherwise in them, and, as they say, _inherent_; I desire they would\nsuspend their judgment for the present, and expect a little, till it be\nfound out by ratiocination, whether these very accidents are not also\ncertain motions either of the mind of the perceiver, or of the bodies\nthemselves which are perceived; for in the search of this, a great part\nof natural philosophy consists.\n\n[Sidenote: Magnitude, what it is.]\n\n4. The _extension_ of a body, is the same thing with the _magnitude_ of\nit, or that which some call _real space_. But this _magnitude_ does not\ndepend upon our cogitation, as imaginary space doth; for this is an\neffect of our imagination, but _magnitude_ is the cause of it; this is\nan accident of the mind, that of a body existing out of the mind.\n\n[Sidenote: Place, what it is, and that it is immovable.]\n\n5. That space, by which word I here understand imaginary space, which is\ncoincident with the magnitude of any body, is called the _place_ of that\nbody; and the body itself is that which we call the _thing placed_. Now\n_place_, and the _magnitude_ of the thing placed, differ. First in this,\nthat a body keeps always the same _magnitude_, both when it is at rest,\nand when it is moved; but when it is moved, it does not keep the same\n_place_. Secondly in this, that _place_ is a phantasm of any body of\nsuch and such quantity and figure; but _magnitude_ is the peculiar\naccident of every body; for one body may at several times have several\nplaces, but has always one and the same magnitude. Thirdly in this, that\n_place_ is nothing out of the mind, nor _magnitude_ any thing within it.\nAnd lastly, _place_ is feigned extension, but _magnitude_ true\nextension; and a placed body is not extension, but a thing extended.\nBesides, _place is immovable_; for, seeing that which is moved, is\nunderstood to be carried from place to place, if place were moved, it\nwould also be carried from place to place, so that one place must have\nanother place, and that place another place, and so on infinitely, which\nis ridiculous. And as for those, that, by making _place_ to be of the\nsame nature with _real space_, would from thence maintain it to be\nimmovable, they also make place, though they do not perceive they make\nit so, to be a mere phantasm. For whilst one affirms that place is\ntherefore said to be immovable, because space in general is considered\nthere; if he had remembered that nothing is general or universal besides\nnames or signs, he would easily have seen that that space, which he says\nis considered in general, is nothing but a phantasm, in the mind or the\nmemory, of a body of such magnitude and such figure. And whilst another\nsays: real space is made immovable by the understanding; as when, under\nthe superficies of running water, we imagine other and other water to\ncome by continual succession, that superficies fixed there by the\nunderstanding, is the _immovable place_ of the river: what else does he\nmake it to be but a phantasm, though he do it obscurely and in perplexed\nwords? Lastly, the nature of _place_ does not consist in the\n_superficies of the ambient_, but in _solid space_; for the whole placed\nbody is coextended with its whole place, and every part of it with every\nanswering part of the same place; but seeing every placed body is a\nsolid thing, it cannot be understood to be coextended with superficies.\nBesides, how can any whole body be moved, unless all its parts be moved\ntogether with it? Or how can the internal parts of it be moved, but by\nleaving their place? But the internal parts of a body cannot leave the\nsuperficies of an external part contiguous to it; and, therefore, it\nfollows, that if place be the superficies of the ambient, then the parts\nof a body moved, that is, bodies moved, are not moved.\n\n[Sidenote: What is full and empty.]\n\n6. Space, or place, that is possessed by a body, is called _full_, and\nthat which is not so possessed, is called _empty_.\n\n[Sidenote: Here, there, somewhere, what they signify.]\n\n7. _Here_, _there_, _in the country_, _in the city_, and other the like\nnames, by which answer is made to the question _where is it?_ are not\nproperly names of place, nor do they of themselves bring into the mind\nthe place that is sought; for _here_ and _there_ signify nothing, unless\nthe thing be shewn at the same time with the finger or something else;\nbut when the eye of him that seeks, is, by pointing or some other sign,\ndirected to the thing sought, the place of it is not hereby defined by\nhim that answers, but found out by him that asks the question. Now such\nshewings as are made by words only, as when we say, _in the country_, or\n_in the city_, are some of greater latitude than others, as when we say,\n_in the country_, _in the city_, _in such a street_, _in a house_, _in\nthe chamber_, _in bed_, &c. For these do, by little and little, direct\nthe seeker nearer to the proper place; and yet they do not determine the\nsame, but only restrain it to a lesser space, and signify no more, than\nthat the place of the thing is within a certain space designed by those\nwords, as a part is in the whole. And all such names, by which answer is\nmade to the question _where?_ have, for their highest _genus_, the name\n_somewhere_. From whence it may be understood, that whatsoever is\nsomewhere, is in some place properly so called, which place is part of\nthat greater space that is signified by some of these names, _in the\ncountry_, _in the city_, or the like.\n\n[Sidenote: Many bodies cannot be in one place, nor one body in many\n places.]\n\n8. A body, and the magnitude, and the place thereof, are divided by one\nand the same act of the mind; for, to divide an extended body, and the\nextension thereof, and the idea of that extension, which is place, is\nthe same with dividing any one of them; because they are coincident, and\nit cannot be done but by the mind, that is by the division of space.\nFrom whence it is manifest, that neither two bodies can be together in\nthe same place, nor one body be in two places at the same time. Not two\nbodies in the same place; because when a body that fills its whole place\nis divided into two, the place itself is divided into two also, so that\nthere will be two places. Not one body in two places; for the place that\na body fills being divided into two, the placed body will be also\ndivided into two; for, as I said, a place and the body that fills that\nplace, are divided both together; and so there will be two bodies.\n\n[Sidenote: Contiguous and continual, what they are.]\n\n9. Two bodies are said to be _contiguous_ to one another, and\n_continual_, in the same manner as spaces are; namely, _those are\ncontiguous, between which there is no space_. Now, by space I\nunderstand, here as formerly, an idea or phantasm of a body. Wherefore,\nthough between two bodies there be put no other body, and consequently\nno magnitude, or, as they call it, real space, yet if another body may\nbe put between them, that is, if there intercede any imagined space\nwhich may receive another body, then those bodies are not contiguous.\nAnd this is so easy to be understood, that I should wonder at some men,\nwho being otherwise skilful enough in philosophy, are of a different\nopinion, but that I find that most of those that affect metaphysical\nsubtleties wander from truth, as if they were led out of their way by an\n_ignis fatuus_. For can any man that has his natural senses, think that\ntwo bodies must therefore necessarily touch one another, because no\nother body is between them? Or that there can be no _vacuum_, because\n_vacuum_ is nothing, or as they call it, _non ens_? Which is as\nchildish, as if one should reason thus; no man can fast, because to fast\nis to eat nothing; but nothing cannot be eaten. _Continual, are any two\nbodies that have a common part; and more than two are continual, when\nevery two, that are next to one another, are continual._\n\n[Sidenote: The definition of motion. No motion intelligible but with\n time.]\n\n10. MOTION _is a continual relinquishing of one place, and acquiring of\nanother_; and that place which is relinquished is commonly called the\n_terminus a quo_, as that which is acquired is called the _terminus ad\nquem_; I say a continual relinquishing, because no body, how little\nsoever, can totally and at once go out of its former place into another,\nso, but that some part of it will be in a part of a place which is\ncommon to both, namely, to the relinquished and the acquired places.\n[Sidenote: A G B I E +-----+------+ : : : : +-----+------+ C H D K F]\nFor example, let any body be in the place A C B D; the same body cannot\ncome into the place B D E F, but it must first be in G H I K, whose part\nG H B D is common to both the places A C B D, and G H I K, and whose\npart B D I K, is common to both the places G H I K, and B D E F. Now it\ncannot be conceived that any thing can be moved without time; for time\nis, by the definition of it, a phantasm, that is, a conception of\nmotion; and, therefore, to conceive that any thing may be moved without\ntime, were to conceive motion without motion, which is impossible.\n\n[Sidenote: What it is to be at rest, to have been moved, and to be\n moved. No motion to be conceived without the conception of\n past and future.]\n\n11. _That is said to be at rest, which, during any time, is in one\nplace; and that to be moved, or to have been moved, which, whether it be\nnow at rest or moved, was formerly in another place than that which it\nis now in._ From which definitions it may be inferred, first, that\n_whatsoever is moved, has been moved_; for if it be still in the same\nplace in which it was formerly, it is at rest, that is, it is not moved,\nby the definition of _rest_; but if it be in another place, it has been\nmoved, by the definition of _moved_. Secondly, that _what is moved, will\nyet be moved_; for that which is moved, leaveth the place where it is,\nand therefore will be in another place, and consequently will be moved\nstill. Thirdly, that _whatsoever is moved, is not in one place during\nany time, how little soever that time be_; for by the definition of\nrest, that which is in one place during any time, is at rest.\n\nThere is a certain sophism against motion, which seems to spring from\nthe not understanding of this last proposition. For they say, that, _if\nany body be moved, it is moved either in the place where it is, or in\nthe place where it is not; both which are false; and therefore nothing\nis moved_. But the falsity lies in the major proposition; for that which\nis moved, is neither moved in the place where it is, nor in the place\nwhere is not; but from the place where it is, to the place where it is\nnot. Indeed it cannot be denied but that whatsoever is moved, is moved\nsomewhere, that is, within some space; but then the place of that body\nis not that whole space, but a part of it, as is said above in the\nseventh article. From what is above demonstrated, namely, that\nwhatsoever is moved, has also been moved, and will be moved, this also\nmay be collected, that there can be no conception of motion, without\nconceiving past and future time.\n\n[Sidenote: A point, a line, superficies, and solid, what they are.]\n\n12. Though there be no body which has not some magnitude, yet if, when\nany body is moved, the magnitude of it be not at all considered, the way\nit makes is called a _line_, or one single dimension; and the space,\nthrough which it passeth, is called _length_; and the body itself, a\n_point_; in which sense the earth is called a _point_, and the way of\nits yearly revolution, the _ecliptic line_. But if a body, which is\nmoved, be considered as _long_, and be supposed to be so moved, as that\nall the several parts of it be understood to make several lines, then\nthe way of every part of that body is called _breadth_, and the space\nwhich is made is called _superficies_, consisting of two dimensions, one\nwhereof to every several part of the other is applied whole. Again, if a\nbody be considered as having _superficies_, and be understood to be so\nmoved, that all the several parts of it describe several lines, then the\nway of every part of that body is called _thickness_ or _depth_, and the\nspace which is made is called _solid_, consisting of three dimensions,\nany two whereof are applied whole to every several part of the third.\n\nBut if a body be considered as _solid_, then it is not possible that all\nthe several parts of it should describe several lines; for what way\nsoever it be moved, the way of the following part will fall into the way\nof the part before it, so that the same solid will still be made which\nthe foremost superficies would have made by itself. And therefore there\ncan be no other dimension in any body, as it is a body, than the three\nwhich I have now described; though, as it shall be shewed hereafter,\n_velocity_, which is motion according to _length_, may, by being applied\nto all the parts of a _solid_, make a magnitude of motion, consisting of\nfour dimensions; as the goodness of gold, computed in all the parts of\nit, makes the price and value thereof.\n\n[Sidenote: Equal, great, greater, and less, in bodies and magnitudes,\n what they are.]\n\n13. _Bodies_, how many soever they be, that can fill every one the place\nof every one, are said to be _equal_ every one to every other. Now, one\nbody may fill the same place which another body filleth, though it be\nnot of the same figure with that other body, if so be that it may be\nunderstood to be reducible to the same figure, either by flexion or\ntransposition of the parts. And _one body is greater than another body,\nwhen a part of that is equal to all this; and less, when all that is\nequal to a part of this_. Also, _magnitudes_ are _equal_, or _greater_,\nor _lesser_, than one another, for the same consideration, namely, when\nthe bodies, of which they are the magnitudes, are either _equal_, or\n_greater_, or _less_, &c.\n\n[Sidenote: One and the same body has always one and the same magnitude.]\n\n14. One and the same body is always of one and the same magnitude. For\nseeing a body and the magnitude and place thereof cannot be comprehended\nin the mind otherwise than as they are coincident, if any body be\nunderstood to be at rest, that is, to remain in the same place during\nsome time, and the magnitude thereof be in one part of that time\ngreater, and in another part less, that body's place, which is one and\nthe same, will be coincident sometimes with greater, sometimes with less\nmagnitude, that is, the same place will be greater and less than itself,\nwhich is impossible. But there would be no need at all of demonstrating\na thing that is in itself so manifest, if there were not some, whose\nopinion concerning bodies and their magnitudes is, that a body may exist\nseparated from its magnitude, and have greater or less magnitude\nbestowed upon it, making use of this principle for the explication of\nthe nature of _rarum_ and _densum_.\n\n[Sidenote: Velocity, what it is.]\n\n15. Motion, in as much as a certain length may in a certain time be\ntransmitted by it, is called VELOCITY or _swiftness_: &c. For though\n_swift_ be very often understood with relation to _slower_ or _less\nswift_, as great is in respect of less, yet nevertheless, as magnitude\nis by philosophers taken absolutely for extension, so also _velocity_ or\n_swiftness_ may be put absolutely for motion according to length.\n\n[Sidenote: Equal, greater, and less, in times, what they are.]\n\n16. Many motions are said to be made in equal times, when every one of\nthem begins and ends together with some other motion, or if it had begun\ntogether, would also have ended together with the same. For time, which\nis a phantasm of motion, cannot be reckoned but by some exposed motion;\nas in dials by the motion of the sun or of the hand; and if two or more\nmotions begin and end with this motion, they are said to be made in\nequal times; from whence also it is easy to understand what it is to be\nmoved in greater or longer time, and in less time or not so long;\nnamely, that that is longer moved, which beginning with another, ends\nlater; or ending together, began sooner.\n\n[Sidenote: Equal, greater, and less, in velocity, what.]\n\n17. Motions are said to be equally swift, when equal lengths are\ntransmitted in equal times; and greater swiftness is that, wherein\ngreater length is passed in equal time, or equal length in less time.\nAlso that swiftness by which equal lengths are passed in equal parts of\ntime, is called _uniform_ swiftness or motion; and of motions _not\nuniform_, such as become swifter or slower by equal increasings or\ndecreasings in equal parts of time, are said to be _accelerated_ or\n_retarded uniformly_.\n\n[Sidenote: Equal, greater, and less, in motion, what.]\n\n18. But motion is said to be greater, less, and equal, not only in\nregard of the length which is transmitted in a certain time, that is, in\nregard of swiftness only, but of swiftness applied to every smallest\nparticle of magnitude; for when any body is moved, every part of it is\nalso moved; and supposing the parts to be halves, the motions of those\nhalves have their swiftness equal to one another, and severally equal to\nthat of the whole; but the motion of the whole is equal to those two\nmotions, either of which is of equal swiftness with it; and therefore it\nis one thing for two motions to be equal to one another, and another\nthing for them to be equally swift. And this is manifest in two horses\nthat draw abreast, where the motion of both the horses together is of\nequal swiftness with the motion of either of them singly; but the motion\nof both is greater than the motion of one of them, namely, double.\nWherefore _motions are said to be simply equal to one another, when the\nswiftness of one, computed in every part of its magnitude, is equal to\nthe swiftness of the other computed also in every part of its magnitude:\nand greater than one another, when the swiftness of one computed as\nabove, is greater than the swiftness of the other so computed; and less,\nwhen less_. Besides, the magnitude of motion computed in this manner is\nthat which is commonly called FORCE.\n\n[Sidenote: That which is at rest will always be at rest, except it be\n moved by some external thing.]\n\n19. _Whatsoever is at rest, will always be at rest, unless there be some\nother body besides it, which, by endeavouring to get into its place by\nmotion, suffers it no longer to remain at rest._ For suppose that some\nfinite body exist and be at rest, and that all space besides be empty;\nif now this body begin to be moved, it will certainly be moved some way;\nseeing therefore there was nothing in that body which did not dispose it\nto rest, the reason why it is moved this way is in something out of it;\nand in like manner, if it had been moved any other way, the reason of\nmotion that way had also been in something out of it; but seeing it was\nsupposed that nothing is out of it, the reason of its motion one way\nwould be the same with the reason of its motion every other way,\nwherefore it would be moved alike all ways at once; which is impossible.\n\n[Sidenote: That which is moved will always be moved, unless it be\n hindered by some external thing.]\n\nIn like manner, _whatsoever is moved, will always be moved, except there\nbe some other body besides it, which causeth it to rest_. For if we\nsuppose nothing to be without it, there will be no reason why it should\nrest now, rather than at another time; wherefore its motion would cease\nin every particle of time alike; which is not intelligible.\n\n[Sidenote: Accidents are generated and destroyed, but bodies not so.]\n\n20. When we say a living creature, a tree, or any other specified body\nis _generated_ or _destroyed_, it is not to be so understood as if there\nwere made a body of that which is not-body, or not a body of a body, but\nof a living creature not a living creature, of a tree not a tree, &c.\nthat is, that those accidents for which we call one thing a living\ncreature, another thing a tree, and another by some other name, are\ngenerated and destroyed; and that therefore the same names are not to be\ngiven to them now, which were given them before. But that magnitude for\nwhich we give to any thing the name of body is neither generated nor\ndestroyed. For though we may feign in our mind that a point may swell to\na huge bulk, and that this may again contract itself to a point; that\nis, though we may imagine something to arise where before was nothing,\nand nothing to be there where before was something, yet we cannot\ncomprehend in our mind how this may possibly be done in nature. And\ntherefore philosophers, who tie themselves to natural reason, suppose\nthat a body can neither be generated nor destroyed, but only that it may\nappear otherwise than it did to us, that is, under different _species_,\nand consequently be called by other and other names; so that that which\nis now called man, may at another time have the name of not-man; but\nthat which is once called body, can never be called not-body. But it is\nmanifest, that all other accidents besides magnitude or extension may be\ngenerated and destroyed; as when a white thing is made black, the\nwhiteness that was in it perisheth, and the blackness that was not in it\nis now generated; and therefore bodies, and the accidents under which\nthey appear diversely, have this difference, that bodies are things, and\nnot generated; accidents are generated, and not things.\n\n[Sidenote: An accident cannot depart from its subject]\n\n21. And therefore, when any thing appears otherwise than it did by\nreason of other and other accidents, it is not to be thought that an\naccident goes out of one subject into another, (for they are not, as I\nsaid above, in their subjects as a part in the whole, or as a contained\nthing in that which contains it, or as a master of a family in his\nhouse,) but that one accident perisheth, and another is generated. For\nexample, when the hand, being moved, moves the pen, motion does not go\nout of the hand into the pen; for so the writing might be continued\nthough the hand stood still; but a new motion is generated in the pen,\nand is the pen's motion.\n\n[Sidenote: Nor be moved.]\n\n22. And therefore also it is improper to say, an accident is moved; as\nwhen, instead of saying, _figure is an accident of a body carried away_,\nwe say, _a body carries away its figure_.\n\n[Sidenote: Essence, form, and matter, what they are.]\n\n23. Now that accident for which we give a certain name to any body, or\nthe accident which denominates its subject, is commonly called the\nESSENCE thereof; as rationality is the essence of a man; whiteness, of\nany white thing, and extension the essence of a body. And the same\nessence, in as much as it is generated, is called the FORM. Again, a\nbody, in respect of any accident, is called the SUBJECT, and in respect\nof the form it is called the MATTER.\n\nAlso, the production or perishing of any accident makes its subject be\nsaid _to be changed_; only the production or perishing of form makes it\nbe said it is _generated_ or _destroyed_; but in all generation and\nmutation, the name of _matter_ still remains. For a table made of wood\nis not only wooden, but wood; and a statue of brass is brass as well as\nbrazen; though Aristotle, in his _Metaphysics_, says, that whatsoever is\nmade of any thing ought not to be called \u1f10\u03ba\u03b5\u03b9\u03bd\u1f78, but \u1f10\u03ba\u03ad\u03b9\u03bd\u03b9\u03bd\u03bf\u03bd; as that\nwhich is made of wood, not \u03be\u03cd\u03bb\u03bf\u03bd, but \u03be\u03cd\u03bb\u03b9\u03bd\u03bf\u03bd, that is, not wood, but\nwooden.\n\n[Sidenote: First matter, what.]\n\n24. And as for that matter which is common to all things, and which\nphilosophers, following Aristotle, usually call _materia prima_, that\nis, _first matter_, it is not any body distinct from all other bodies,\nnor is it one of them. What then is it? A mere name; yet a name which is\nnot of vain use; for it signifies a conception of body without the\nconsideration of any form or other accident except only magnitude or\nextension, and aptness to receive form and other accident. So that\nwhensoever we have use of the name _body in general_, if we use that of\n_materia prima_, we do well. For as when a man not knowing which was\nfirst, water or ice, would find out which of the two were the matter of\nboth, he would be fain to suppose some third matter which were neither\nof these two; so he that would find out what is the matter of all\nthings, ought to suppose such as is not the matter of anything that\nexists. Wherefore _materia prima_ is nothing; and therefore they do not\nattribute to it either form or any other accident besides quantity;\nwhereas all singular things have their forms and accidents certain.\n\n_Materia prima_, therefore, is body in general, that is, body considered\nuniversally, not as having neither form nor any accident, but in which\nno form nor any other accident but quantity are at all considered, that\nis, they are not drawn into argumentation.\n\n[Sidenote: That the whole is greater than any part thereof, why\n demonstrated.]\n\n25. From what has been said, those axioms may be demonstrated, which are\nassumed by Euclid in the beginning of his first element, about the\nequality and inequality of magnitudes; of which, omitting the rest, I\nwill here demonstrate only this one, _the whole is greater than any part\nthereof_; to the end that the reader may know that those axioms are not\nindemonstrable, and therefore not principles of demonstration; and from\nhence learn to be wary how he admits any thing for a principle, which is\nnot at least as evident as these are. _Greater_ is defined to be that,\nwhose part is equal to the whole of another. Now if we suppose any whole\nto be A, and a part of it to be B; seeing the whole B is equal to\nitself, and the same B is a part of A; therefore a part of A will be\nequal to the whole B. Wherefore, by the definition above, A is greater\nthan B; which was to be proved.\n\n\n -------\n\n\n CHAPTER IX.\n\n OF CAUSE AND EFFECT.\n\n1. Action and passion, what they are.\u20142. Action and passion mediate and\n immediate.\u20143. Cause simply taken. Cause without which no effect\n follows, or cause necessary by supposition.\u20144. Cause efficient and\n material.\u20145. An entire cause is always sufficient to produce its\n effect. At the same instant that the cause is entire, the effect is\n produced. Every effect has a necessary cause.\u20146. The generation of\n effects is continual. What is the beginning in causation.\u20147. No\n cause of motion but in a body contiguous and moved.\u20148. The same\n agents and patients, if alike disposed, produce like effects though\n at different times.\u20149. All mutation is motion.\u201410. Contingent\n accidents, what they are.\n\n\n[Sidenote: Action and passion, what they are.]\n\n1. A body is said to work upon or _act_, that is to say, _do_ something\nto another body, when it either generates or destroys some accident in\nit: and the body in which an accident is generated or destroyed is said\nto _suffer_, that is, to have something _done_ to it by another body; as\nwhen one body by putting forwards another body generates motion in it,\nit is called the AGENT; and the body in which motion is so generated, is\ncalled the PATIENT; so fire that warms the hand is the _agent_, and the\nhand, which is warmed, is the _patient_. That accident, which is\ngenerated in the patient, is called the EFFECT.\n\n[Sidenote: Action and passion, mediate, and immediate.]\n\n2. When an agent and patient are contiguous to one another, their action\nand passion are then said to be _immediate_, otherwise, _mediate_; and\nwhen another body, lying betwixt the agent and patient, is contiguous to\nthem both, it is then itself both an agent and a patient; an agent in\nrespect of the body next after it, upon which it works, and a patient in\nrespect of the body next before it, from which it suffers. Also, if many\nbodies be so ordered that every two which are next to one another be\ncontiguous, then all those that are betwixt the first and the last are\nboth agents and patients, and the first is an agent only, and the last a\npatient only.\n\n[Sidenote: Cause simply taken.]\n\n3. An agent is understood to _produce_ its determined or certain effect\nin the patient, according to some certain accident or accidents, with\nwhich both it and the patient are affected; that is to say, the agent\nhath its effect precisely such, not because it is a body, but because\nsuch a body, or so moved. For otherwise all agents, seeing they are all\nbodies alike, would produce like effects in all patients. And therefore\nthe fire, for example, does not warm, because it is a body, but because\nit is hot; nor does one body put forward another body because it is a\nbody, but because it is moved into the place of that other body.\n[Sidenote: Cause without which no effect follows, or cause necessary by\nsupposition.]The cause, therefore, of all effects consists in certain\naccidents both in the agents and in the patients; which when they are\nall present, the effect is produced; but if any one of them be wanting,\nit is not produced; and that accident either of the agent or patient,\nwithout which the effect cannot be produced, is called _causa sine qua\nnon_, or _cause necessary by supposition_, as also the _cause requisite\nfor the production of the effect_. But a CAUSE simply, or _an entire\ncause, is the aggregate of all the accidents both of the agents how many\nsoever they be, and of the patient, put together; which when they are\nall supposed to be present, it cannot be understood but that the effect\nis produced at the same instant; and if any one of them be wanting, it\ncannot be understood but that the effect is not produced_.\n\n[Sidenote: Cause efficient, and material.]\n\n4. The aggregate of accidents in the agent or agents, requisite for the\nproduction of the effect, the effect being produced, is called the\n_efficient cause_ thereof; and the aggregate of accidents in the\npatient, the effect being produced, is usually called the _material\ncause_; I say the effect being produced; for where there is no effect,\nthere can be no cause; for nothing can be called a cause, where there is\nnothing that can be called an effect. But the efficient and material\ncauses are both but partial causes, or parts of that cause, which in the\nnext precedent article I called an entire cause. And from hence it is\nmanifest, that the effect we expect, though the agents be not defective\non their part, may nevertheless be frustrated by a defect in the\npatient; and when the patient is sufficient, by a defect in the agents.\n\n[Sidenote: An entire cause is always sufficient to produce its effect.]\n\n5. An entire cause is always sufficient for the production of its\neffect, if the effect be at all possible. For let any effect whatsoever\nbe propounded to be produced; if the same be produced, it is manifest\nthat the cause which produced it was a sufficient cause; but if it be\nnot produced, and yet be possible, it is evident that something was\nwanting either in some agent, or in the patient, without which it could\nnot be produced; that is, that some accident was wanting which was\nrequisite for its production; and therefore, that cause was not\n_entire_, which is contrary to what was supposed.\n\n[Sidenote: At the same instant that the cause is entire, the effect is\n produced.]\n\nIt follows also from hence, that in whatsoever instant the cause is\nentire, in the same instant the effect is produced. For if it be not\nproduced, something is still wanting, which is requisite for the\nproduction of it; and therefore the cause was not entire, as was\nsupposed.\n\n[Sidenote: Every effect has a necessary cause.]\n\nAnd seeing a necessary cause is defined to be that, which being\nsupposed, the effect cannot but follow; this also may be collected, that\nwhatsoever effect is produced at any time, the same is produced by a\nnecessary cause. For whatsoever is produced, in as much as it is\nproduced, had an entire cause, that is, had all those things, which\nbeing supposed, it cannot be understood but that the effect follows;\nthat is, it had a necessary cause. And in the same manner it may be\nshewn, that whatsoever effects are hereafter to be produced, shall have\na necessary cause; so that all the effects that have been, or shall be\nproduced, have their necessity in things antecedent.\n\n[Sidenote: The generation of effects is continual. What is the beginning\n in causation.]\n\n6. And from this, that whensoever the cause is entire, the effect is\nproduced in the same instant, it is manifest that causation and the\nproduction of effects consist in a certain continual progress; so that\nas there is a continual mutation in the agent or agents, by the working\nof other agents upon them, so also the patient, upon which they work, is\ncontinually altered and changed. For example: as the heat of the fire\nincreases more and more, so also the effects thereof, namely, the heat\nof such bodies as are next to it, and again, of such other bodies as are\nnext to them, increase more and more accordingly; which is already no\nlittle argument that all mutation consists in motion only; the truth\nwhereof shall be further demonstrated in the ninth article. But in this\nprogress of causation, that is, of action and passion, if any man\ncomprehend in his imagination a part thereof, and divide the same into\nparts, the first part or beginning of it cannot be considered otherwise\nthan as action or cause; for, if it should be considered as effect or\npassion, then it would be necessary to consider something before it, for\nits cause or action; which cannot be, for nothing can be before the\nbeginning. And in like manner, the last part is considered only as\neffect; for it cannot be called cause, if nothing follow it; but after\nthe last, nothing follows. And from hence it is, that in all action the\nbeginning and cause are taken for the same thing. But every one of the\nintermediate parts are both action and passion, and cause and effect,\naccording as they are compared with the antecedent or subsequent part.\n\n[Sidenote: No cause of motion but in a body contiguous and moved.]\n\n7. There can be no cause of motion, except in a body contiguous and\nmoved. For let there be any two bodies which are not contiguous, and\nbetwixt which the intermediate space is empty, or, if filled, filled\nwith another body which is at rest; and let one of the propounded bodies\nbe supposed to be at rest; I say it shall always be at rest. For if it\nshall be moved, the cause of that motion, by the 8th chapter, article\n19, will be some external body; and, therefore, if between it and that\nexternal body there be nothing but empty space, then whatsoever the\ndisposition be of that external body or of the patient itself, yet if it\nbe supposed to be now at rest, we may conceive it will continue so till\nit be touched by some other body. But seeing cause, by the definition,\nis the aggregate of all such accidents, which being supposed to be\npresent, it cannot be conceived but that the effect will follow, those\naccidents, which are either in external bodies, or in the patient\nitself, cannot be the cause of future motion. And in like manner, seeing\nwe may conceive that whatsoever is at rest will still be at rest, though\nit be touched by some other body, except that other body be moved;\ntherefore in a contiguous body, which is at rest, there can be no cause\nof motion. Wherefore there is no cause of motion in any body, except it\nbe contiguous and moved.\n\nThe same reason may serve to prove that whatsoever is moved, will always\nbe moved on in the same way and with the same velocity, except it be\nhindered by some other contiguous and moved body; and consequently that\nno bodies, either when they are at rest, or when there is an\ninterposition of vacuum, can generate or extinguish or lessen motion in\nother bodies. There is one that has written that things moved are more\nresisted by things at rest, than by things contrarily moved; for this\nreason, that he conceived motion not to be so contrary to motion as\nrest. That which deceived him was, that the words _rest_ and _motion_\nare but contradictory names; whereas motion, indeed, is not resisted by\nrest, but by contrary motion.\n\n[Sidenote: The same agents and patients, if alike disposed, produce like\n effects, though at different times.]\n\n8. But if a body work upon another body at one time, and afterwards the\nsame body work upon the same body at another time, so that both the\nagent and patient, and all their parts, be in all things as they were;\nand there be no difference, except only in time, that is, that one\naction be former, the other later in time; it is manifest of itself,\nthat the effects will be equal and like, as not differing in anything\nbesides time. And as effects themselves proceed from their causes, so\nthe diversity of them depends upon the diversity of their causes also.\n\n[Sidenote: All mutation is motion.]\n\n9. This being true, it is necessary that mutation can be nothing else\nbut motion of the parts of that body which is changed. For first, we do\nnot say anything is changed, but that which appears to our senses\notherwise than it appeared formerly. Secondly, both those appearances\nare effects produced in the sentient; and, therefore, if they be\ndifferent, it is necessary, by the preceding article, that either some\npart of the agent, which was formerly at rest, is now moved, and so the\nmutation consists in this motion; or some part, which was formerly\nmoved, is now otherwise moved, and so also the mutation consists in this\nnew motion; or which, being formerly moved, is now at rest, which, as I\nhave shewn above, cannot come to pass without motion; and so again,\nmutation is motion; or lastly, it happens in some of these manners to\nthe patient, or some of its parts; so that mutation, howsoever it be\nmade, will consist in the motion of the parts, either of the body which\nis perceived, or of the sentient body, or of both. Mutation therefore is\nmotion, namely, of the parts either of the agent or of the patient;\nwhich was to be demonstrated. And to this it is consequent, that rest\ncannot be the cause of anything, nor can any action proceed from it;\nseeing neither motion nor mutation can be caused by it.\n\n[Sidenote: Contingent accidents.]\n\n10. Accidents, in respect of other accidents which precede them, or are\nbefore them in time, and upon which they do not depend as upon their\ncauses, are called _contingent_ accidents; I say, in respect of those\naccidents by which they are not generated; for, in respect of their\ncauses, all things come to pass with equal necessity; for otherwise they\nwould have no causes at all; which, of things generated, is not\nintelligible.\n\n\n -------\n\n\n CHAPTER X.\n\n OF POWER AND ACT.\n\n1. Power and cause are the same thing.\u20142. An act is produced at the same\n instant in which the power is plenary.\u20143. Active and passive power\n are parts only of plenary power.\u20144. An act, when said to be\n possible.\u20145. An act necessary and contingent, what.\u20146. Active power\n consists in motion.\u20147. Cause, formal and final, what they are.\n\n\n[Sidenote: Power and cause are the same thing.]\n\n1. Correspondent to _cause_ and _effect_, are POWER and ACT; nay, those\nand these are the same things; though, for divers considerations, they\nhave divers names. For whensoever any agent has all those accidents\nwhich are necessarily requisite for the production of some effect in the\npatient, then we say that agent has _power_ to produce that effect, if\nit be applied to a patient. But, as I have shewn in the precedent\nchapter, those accidents constitute the efficient cause; and therefore\nthe same accidents, which constitute the efficient cause, constitute\nalso the _power_ of the agent. Wherefore the _power of the agent_ and\nthe _efficient cause_ are the same thing. But they are considered with\nthis difference, that _cause_ is so called in respect of the effect\nalready produced, and power in respect of the same effect to be produced\nhereafter; so that _cause_ respects the past, _power_ the future time.\nAlso, the _power of the agent_ is that which is commonly called _active\npower_.\n\nIn like manner, whensoever any patient has all those accidents which it\nis requisite it should have, for the production of some effect in it, we\nsay it is in the _power_ of that patient to produce that effect, if it\nbe applied to a fitting agent. But those accidents, as is defined in the\nprecedent chapter, constitute the material cause; and therefore the\n_power of the patient_, commonly called _passive power_, and _material\ncause_, are the same thing; but with this different consideration, that\nin cause the past time, and in power the future, is respected. Wherefore\nthe power of the agent and patient together, which may be called entire\nor _plenary power_, is the same thing with _entire cause_; for they both\nconsist in the sum or aggregate of all the accidents, as well in the\nagent as in the patient, which are requisite for the production of the\neffect. Lastly, as the accident produced is, in respect of the cause,\ncalled an effect, so in respect of the power, it is called an _act_.\n\n[Sidenote: An act is produced at the same instant in which the power is\n plenary.]\n\n2. As therefore the effect is produced in the same instant in which the\ncause is entire, so also every act that may be produced, is produced in\nthe same instant in which the power is plenary. And as there can be no\neffect but from a sufficient and necessary cause, so also no act can be\nproduced but by sufficient power, or that power by which it could not\nbut be produced.\n\n[Sidenote: Active and passive power are parts only of plenary power.]\n\n3. And as it is manifest, as I have shewn, that the efficient and\nmaterial causes are severally and by themselves parts only of an entire\ncause, and cannot produce any effect but by being joined together, so\nalso power, active and passive, are parts only of plenary and entire\npower; nor, except they be joined, can any act proceed from them; and\ntherefore these powers, as I said in the first article, are but\nconditional, namely, _the agent has power, if it be applied to a\npatient; and the patient has power, if it be applied to an agent_;\notherwise neither of them have power, nor can the accidents, which are\nin them severally, be properly called powers; nor any action be said to\nbe possible for the power of the agent alone or of the patient alone.\n\n[Sidenote: An act, when said to be possible.]\n\n4. For that is an impossible act, for the production of which there is\nno power plenary. For seeing plenary power is that in which all things\nconcur, which are requisite for the production of an act, if the power\nshall never be plenary, there will always be wanting some of those\nthings, without which the act cannot be produced; wherefore that act\nshall never be produced; that is, that act is IMPOSSIBLE: and every act,\nwhich is not impossible, is POSSIBLE. Every act, therefore, which is\npossible, shall at some time be produced; for if it shall never be\nproduced, then those things shall never concur which are requisite for\nthe production of it; wherefore that act is _impossible_, by the\ndefinition; which is contrary to what was supposed.\n\n[Sidenote: An act necessary and contingent, what.]\n\n5. A _necessary act_ is that, the production whereof it is impossible to\nhinder; and therefore every act, that shall be produced, shall\nnecessarily be produced; for, that it shall not be produced, is\nimpossible; because, as is already demonstrated, every possible act\nshall at some time be produced; nay, this proposition, _what shall be,\nshall be_, is as necessary a proposition as this, _a man is a man_.\n\nBut here, perhaps, some man may ask whether those future things, which\nare commonly called _contingents_, are necessary. I say, therefore, that\ngenerally all contingents have their necessary causes, as is shewn in\nthe preceding chapter; but are called contingents in respect of other\nevents, upon which they do not depend; as the rain, which shall be\ntomorrow, shall be necessary, that is, from necessary causes; but we\nthink and say it happens by chance, because we do not yet perceive the\ncauses thereof, though they exist now; for men commonly call that\n_casual_ or _contingent_, whereof they do not perceive the necessary\ncause; and in the same manner they used to speak of things past, when\nnot knowing whether a thing be done or no, they say it is possible it\nnever was done.\n\nWherefore, all propositions concerning future things, contingent or not\ncontingent, as this, _it will rain tomorrow_, or this, _tomorrow the sun\nwill rise_, are either necessarily true, or necessarily false; but we\ncall them contingent, because we do not yet know whether they be true or\nfalse; whereas their verity depends not upon our knowledge, but upon the\nforegoing of their causes. But there are some, who though they confess\nthis whole proposition, _tomorrow it will either rain, or not rain_, to\nbe true, yet they will not acknowledge the parts of it, as, _tomorrow it\nwill rain_, or, _tomorrow it will not rain_, to be either of them true\nby itself; because they say neither this nor that is true\n_determinately_. But what is this _determinately true_, but true upon\nour knowledge, or evidently true? And therefore they say no more but\nthat it is not yet known whether it be true or no; but they say it more\nobscurely, and darken the evidence of the truth with the same words,\nwith which they endeavour to hide their own ignorance.\n\n[Sidenote: Active power consists in motion.]\n\n6. In the 9th article of the preceding chapter, I have shewn that the\nefficient cause of all motion and mutation consists in the motion of the\nagent, or agents; and in the first article of this chapter, that the\npower of the agent is the same thing with the efficient cause. From\nwhence it may be understood, that all active power consists in motion\nalso; and that power is not a certain accident, which differs from all\nacts, but is, indeed, an act, namely, motion, which is therefore called\npower, because another act shall be produced by it afterwards. For\nexample, if of three bodies the first put forward the second, and this\nthe third, the motion of the second, in respect of the first which\nproduceth it, is the act of the second body; but, in respect of the\nthird, it is the active power of the same second body.\n\n[Sidenote: Cause, formal and final, what they are.]\n\n7. The writers of metaphysics reckon up two other causes besides the\n_efficient_ and _material_, namely, the ESSENCE, which some call the\n_formal cause_, and the END, or _final cause_; both which are\nnevertheless efficient causes. For when it is said the essence of a\nthing is the cause thereof, _as to be rational is the cause of man_, it\nis not intelligible; for it is all one, as if it were said, _to be a man\nis the cause of man_; which is not well said. And yet the knowledge of\nthe _essence_ of anything, is the cause of the knowledge of the thing\nitself; for, if I first know that a thing is _rational_, I know from\nthence, that the same is _man_; but this is no other than an efficient\ncause. A _final cause_ has no place but in such things as have sense and\nwill; and this also I shall prove hereafter to be an efficient cause.\n\n\n -------\n\n\n CHAPTER XI.\n\n OF IDENTITY AND DIFFERENCE.\n\n1. What it is for one thing to differ from another.\u20142. To differ in\n number, magnitude, species, and genus, what.\u20143. What is relation,\n proportion, and relatives.\u20144. Proportionals, what.\u20145. The proportion\n of magnitudes to one another, wherein it consists.\u20146. Relation is no\n new accident, but one of those that were in the relative before the\n relation or comparison was made. Also the causes of accidents in the\n correlatives, are the cause of relation.\u20147. Of the beginning of\n individuation.\n\n\n[Sidenote: What it is for one thing to differ from another.]\n\n1. Hitherto I have spoken of body simply, and accidents common to all\nbodies, as _magnitude_, _motion_, _rest_, _action_, _passion_, _power_,\n_possible_, _&c._; and I should now descend to those accidents by which\none body is distinguished from another, but that it is first to be\ndeclared what it is to be _distinct_ and _not distinct_, namely, what\nare the SAME and DIFFERENT; for this also is common to all bodies, that\nthey may be distinguished and differenced from one another. Now, two\nbodies are said to _differ_ from one another, when something may be said\nof one of them, which cannot be said of the other at the same time.\n\n[Sidenote: To differ in number, magnitude, species, and genus, what.]\n\n2. And, first of all, it is manifest that no two bodies are the _same_;\nfor seeing they are two, they are in two places at the same time; as\nthat, which is the _same_, is at the same time in one and the same\nplace. All bodies therefore differ from one another in _number_, namely,\nas one and another; so that the _same_ and _different in number_, are\nnames opposed to one another by contradiction.\n\nIn _magnitude_ bodies differ when one is greater than another, as _a\ncubit long_, and _two cubits long_, of _two pound weight_, and of _three\npound weight_. And to these, _equals_ are opposed.\n\nBodies, which differ more than in magnitude, are called _unlike_; and\nthose, which differ only in magnitude, _like_. Also, of unlike bodies,\nsome are said to differ in the _species_, others in the _genus_; in the\n_species_, when their difference is perceived by one and the same sense,\nas _white_ and _black_; and in the _genus_, when their difference is not\nperceived but by divers senses, as _white_ and _hot_.\n\n[Sidenote: What is relation, proportion, and relatives.]\n\n3. And the _likeness_, or _unlikeness_, _equality_, or _inequality_ of\none body to another, is called their RELATION; and the bodies themselves\n_relatives_ or _correlatives_; _Aristotle_ calls them \u03c4\u1f70 \u03c0\u03c1\u1f78\u03c2 \u03c4\u03af; the\nfirst whereof is usually named the _antecedent_, and the second the\n_consequent_; and the _relation_ of the antecedent to the consequent,\naccording to magnitude, namely, the equality, the excess or defect\nthereof, is called the PROPORTION of the antecedent to the consequent;\nso that _proportion_ is nothing but the equality or inequality of the\nmagnitude of the antecedent compared to the magnitude of the consequent\nby their difference only, or compared also with their difference. For\nexample, the _proportion_ of three to two consists only in this, that\nthree _exceeds_ two by unity; and the proportion of two to five in this,\nthat two, compared with five, is _deficient_ of it by three, either\nsimply, or compared with the numbers different; and therefore in the\nproportion of unequals, the proportion of the less to the greater, is\ncalled DEFECT; and that of the greater to the less, EXCESS.\n\n[Sidenote: Proportionals, what.]\n\n4. Besides, of unequals, some are more, some less, and some equally\nunequal; so that there is _proportion of proportions_, as well as of\n_magnitudes_; namely, where two unequals have relation to two other\nunequals; as, when the inequality which is between 2 and 3, is compared\nwith the inequality which is between 4 and 5. In which comparison there\nare always four magnitudes; or, which is all one, if there be but three,\nthe middlemost is twice numbered; and if the proportion of the first to\nthe second, be equal to the proportion of the third to the fourth, then\nthe four are said to be _proportionals_; otherwise they are not\nproportionals.\n\n[Sidenote: The proportion of magnitudes to one another, wherein it\n consists.]\n\n5. The proportion of the antecedent to the consequent consists in their\ndifference, not only simply taken, but also as compared with one of the\nrelatives; that is, either in that part of the greater, by which it\nexceeds the less, or in the remainder, after the less is taken out of\nthe greater; as the proportion of two to five consists in the three by\nwhich five exceeds two, not in three simply only, but also as compared\nwith five or two. For though there be the same difference between two\nand five, which is between nine and twelve, namely three, yet there is\nnot the same inequality; and therefore the proportion of two to five is\nnot in all relation the same with that of nine to twelve, but only in\nthat which is called arithmetical.\n\n[Sidenote: Relation is no new accident, but one of those that were in\n the relative, before the relation or comparison was made.\n Also the causes of accidents in correlatives are the cause of\n relation.]\n\n6. But we must not so think of relation, as if it were an accident\ndiffering from all the other accidents of the relative; but one of them,\nnamely, that by which the comparison is made. For example, the likeness\nof one _white_ to another _white_, or its unlikeness to _black_, is the\nsame accident with its _whiteness_; and _equality_ and _inequality_, the\nsame accident with the _magnitude_ of the thing compared, though under\nanother name: for that which is called _white_ or _great_, when it is\nnot compared with something else, the same when it is compared, is\ncalled _like_ or _unlike_, _equal_ or _unequal_. And from this it\nfollows that the causes of the accidents, which are in relatives, are\nthe causes also of _likeness_, _unlikeness_, _equality_ and\n_inequality_; namely, that he, that makes two unequal bodies, makes also\ntheir inequality; and he, that makes a rule and an action, makes also,\nif the action be congruous to the rule, their congruity; if incongruous,\ntheir incongruity. And thus much concerning _comparison_ of one body\nwith another.\n\n[Sidenote: Of the beginning of individuation.]\n\n7. But the same body may at different times be compared with itself. And\nfrom hence springs a great controversy among philosophers about the\n_beginning of individuation_, namely, in what sense it may be conceived\nthat a body is at one time the same, at another time not the same it was\nformerly. For example, whether a man grown old be the same man he was\nwhilst he was young, or another man; or whether a city be in different\nages the same, or another city. Some place _individuity_ in the unity of\n_matter_; others, in the unity of _form_; and one says it consists in\nthe unity of the _aggregate of all the accidents together_. For\n_matter_, it is pleaded that a lump of wax, whether it be spherical or\ncubical, is the same wax, because the same matter. For _form_, that when\na man is grown from an infant to be an old man, though his matter be\nchanged, yet he is still the same numerical man; for that _identity_\nwhich cannot be attributed to the matter, ought probably to be ascribed\nto the form. For the _aggregate of accidents_, no instance can be made;\nbut because, when any new accident is generated, a new name is commonly\nimposed on the thing, therefore he, that assigned this cause of\n_individuity_, thought the thing itself also was become another thing.\nAccording to the first opinion, he that sins, and he that is punished,\nshould not be the same man, by reason of the perpetual flux and change\nof man's body; nor should the city, which makes laws in one age and\nabrogates them in another, be the same city; which were to confound all\ncivil rights. According to the second opinion, two bodies existing both\nat once, would be one and the same numerical body. For if, for example,\nthat ship of Theseus, concerning the difference whereof made by\ncontinual reparation in taking out the old planks and putting in new,\nthe sophisters of Athens were wont to dispute, were, after all the\nplanks were changed, the same numerical ship it was at the beginning;\nand if some man had kept the old planks as they were taken out, and by\nputting them afterwards together in the same order, had again made a\nship of them, this, without doubt, had also been the same numerical ship\nwith that which was at the beginning; and so there would have been two\nships numerically the same, which is absurd. But, according to the third\nopinion, nothing would be the same it was; so that a man standing would\nnot be the same he was sitting; nor the water, which is in the vessel,\nthe same with that which is poured out of it. Wherefore the beginning of\n_individuation_ is not always to be taken either from matter alone, or\nfrom form alone.\n\nBut we must consider by what name anything is called, when we inquire\nconcerning the _identity_ of it. For it is one thing to ask concerning\nSocrates, whether he be the same man, and another to ask whether he be\nthe same body; for his body, when he is old, cannot be the same it was\nwhen he was an infant, by reason of the difference of magnitude; for one\nbody has always one and the same magnitude; yet, nevertheless, he may be\nthe same man. And therefore, whensoever the name, by which it is asked\nwhether a thing be the same it was, is given it for the matter only,\nthen, if the matter be the same, the thing also is _individually_ the\nsame; as the water, which was in the sea, is the same which is\nafterwards in the cloud; and any body is the same, whether the parts of\nit be put together, or dispersed; or whether it be congealed, or\ndissolved. Also, if the name be given for such form as is the beginning\nof motion, then, as long as that motion remains, it will be the same\n_individual_ thing; as that man will be always the same, whose actions\nand thoughts proceed all from the same beginning of motion, namely, that\nwhich was in his generation; and that will be the same river which flows\nfrom one and the same fountain, whether the same water, or other water,\nor something else than water, flow from thence; and that the same city,\nwhose acts proceed continually from the same institution, whether the\nmen be the same or no. Lastly, if the name be given for some accident,\nthen the _identity_ of the thing will depend upon the matter; for, by\nthe taking away and supplying of matter, the accidents that were, are\ndestroyed, and other new ones are generated, which cannot be the same\nnumerically; so that a ship, which signifies matter so figured, will be\nthe same as long as the matter remains the same; but if no part of the\nmatter be the same, then it is numerically another ship; and if part of\nthe matter remain and part be changed, then the ship will be partly the\nsame, and partly not the same.\n\n\n -------\n\n\n CHAPTER XII.\n\n OF QUANTITY.\n\n1. The definition of quantity.\u20142. The exposition of quantity, what it\n is.\u20143. How line, superficies, and solid, are exposed.\u20144. How time is\n exposed.\u20145. How number is exposed.\u20146. How velocity is exposed.\u20147.\n How weight is exposed.\u20148. How the proportion of magnitudes is\n exposed.\u20149. How the proportion of times and velocities is exposed.\n\n\n[Sidenote: Definition of quantity.]\n\n1. What and how manifold _dimension_ is, has been said in the 8th\nchapter, namely, that there are three dimensions, line or length,\nsuperficies, and solid; every one of which, if it be determined, that\nis, if the limits of it be made known, is commonly called _quantity_;\nfor by _quantity_ all men understand that which is signified by that\nword, by which answer is made to the question, _How much is it?_\nWhensoever, therefore, it is asked, for example, _How long is the\njourney?_ it is not answered indefinitely, _length_; nor, when it is\nasked, _How big is the field?_ is it answered indefinitely,\n_superficies_; nor, if a man ask, _How great is the bulk?_ indefinitely,\n_solid_: but it is answered determinately, the journey is a hundred\nmiles; the field is a hundred acres; the bulk is a hundred cubical feet;\nor at least in some such manner, that the magnitude of the thing\nenquired after may by certain limits be comprehended in the mind.\nQUANTITY, therefore, cannot otherwise be defined, than to be _a\ndimension determined_, or _a dimension, whose limits are set out, either\nby their place, or by some comparison_.\n\n[Sidenote: The exposition of quantity, what it is.]\n\n2. And _quantity_ is determined two ways; one, by the sense, when some\nsensible object is set before it; as when a line, a superficies or\nsolid, of a foot or cubit, marked out in some matter, is objected to the\neyes; which way of determining, is called _exposition_, and the quantity\nso known is called _exposed quantity_; the other by memory, that is, by\ncomparison with some exposed quantity. In the first manner, when it is\nasked of what quantity a thing is, it is answered, of such quantity as\nyou see exposed. In the second manner, answer cannot be made but by\ncomparison with some exposed quantity; for if it be asked, how long is\nthe way? the answer is, so many thousand paces; that is, by comparing\nthe way with a pace, or some other measure, determined and known by\nexposition; or the quantity of it is to some other quantity known by\nexposition, as the diameter of a square is to the side of the same, or\nby some other the like means. But it is to be understood, that the\nquantity exposed must be some standing or permanent thing, such as is\nmarked out in consistent or durable matter; or at least something which\nis revocable to sense; for otherwise no comparison can be made by it.\nSeeing, therefore, by what has been said in the next preceding chapter,\ncomparison of one magnitude with another is the same thing with\nproportion; it is manifest, that quantity determined in the second\nmanner is nothing else but the proportion of a dimension not exposed to\nanother which is exposed; that is, the comparison of the equality or\ninequality thereof with an exposed quantity.\n\n[Sidenote: How line, superficies, and solids, are exposed.]\n\n3. _Lines_, _superficies_, and _solids_, are exposed, first, by\n_motion_, in such manner as in the 8th chapter I have said they are\ngenerated; but so as that the marks of such motion be permanent; as when\nthey are designed upon some matter, as a line upon paper; or graven in\nsome durable matter. Secondly, by _apposition_; as when one line or\nlength is applied to another line or length, one breadth to another\nbreadth, and one thickness to another thickness; which is as much as to\ndescribe a line by points, a superficies by lines, and a solid by\nsuperficies; saving that by points in this place are to be understood\nvery short lines; and, by superficies, very thin solids. Thirdly, lines\nand superficies may be exposed by _section_, namely, a line may be made\nby cutting an exposed superficies; and a superficies, by the cutting of\nan exposed solid.\n\n[Sidenote: How time is exposed.]\n\n4. _Time_ is exposed, not only by the exposition of a line, but also of\nsome moveable thing, which is moved uniformly upon that line, or at\nleast is supposed so to be moved. For, seeing time is an idea of motion,\nin which we consider former and latter, that is succession, it is not\nsufficient for the exposition of time that a line be described; but we\nmust also have in our mind an imagination of some moveable thing passing\nover that line; and the motion of it must be uniform, that time may be\ndivided and compounded as often as there shall be need. And, therefore,\nwhen philosophers, in their demonstrations, draw a line, and say, _Let\nthat line be time_, it is to be understood as if they said, _Let the\nconception of uniform motion upon that line, be time_. For though the\ncircles in dials be lines, yet they are not of themselves sufficient to\nnote time by, except also there be, or be supposed to be, a motion of\nthe shadow or the hand.\n\n[Sidenote: How number is exposed.]\n\n5. _Number_ is exposed, either by the exposition of points, or of the\nnames of number, _one_, _two_, _three_, _&c._; and those points must not\nbe contiguous, so as that they cannot be distinguished by notes, but\nthey must be so placed that they may be _discerned_ one from another;\nfor, from this it is, that number is called _discreet quantity_, whereas\nall quantity, which is designed by motion, is called _continual\nquantity_. But that number may be exposed by the names of number, it is\nnecessary that they be recited by heart and in order, as one, two,\nthree, &c.; for by saying one, one, one, and so forward, we know not\nwhat number we are at beyond two or three; which also appear to us in\nthis manner, not as number, but as figure.\n\n[Sidenote: How velocity is exposed.]\n\n6. For the exposition of _velocity_, which, by the definition thereof,\nis a motion which, in a certain time, passeth over a certain space, it\nis requisite, not only that time be exposed, but that there be also\nexposed that space which is transmitted by the body, whose velocity we\nwould determine; and that a body be understood to be moved in that space\nalso; so that there must be exposed two lines, upon one of which uniform\nmotion must be understood to be made, that the time may be determined;\nand, upon the other, the velocity is to be computed. As if we would\nexpose the velocity of the we would expose the velocity of the body A,\nwe draw two lines A B and C D, and place a body in C also; which done,\nwe say the velocity of the body A is so great, that it passeth over the\nline A B in the same time in which the body C passeth over the line C D\nwith uniform motion.\n\n A B\n ------\n C D\n ------\n\n[Sidenote: How weight is exposed.]\n\n7. _Weight_ is exposed by any heavy body, of what matter soever, so it\nbe always alike heavy.\n\n[Sidenote: How the proportion of magnitudes is exposed.]\n\n8. The _proportion_ of two magnitudes is then exposed, when the\nmagnitudes themselves are exposed, namely, the proportion of equality,\nwhen the magnitudes are equal; and of inequality, when they are unequal.\nFor seeing, by the 5th article of the preceding chapter, the proportion\nof two unequal magnitudes consists in their difference, compared with\neither of them; and when two unequal magnitudes are exposed, their\ndifference is also exposed; it follows, that when magnitudes, which have\nproportion to one another, are exposed, their proportion also is exposed\nwith them; and, in like manner, the proportion of equals, which consists\nin this, that there is no difference of magnitude betwixt them, is\nexposed at the same time when the equal magnitudes themselves are\nexposed. For example, if the exposed lines A B and C D be equal, the\nproportion of equality is exposed in them; and if the exposed lines, E F\nand E G be unequal, the proportion which E F has to E G, and that which\nE G has to E F are also exposed in them; for not only the lines\nthemselves, but also their difference, G F, is exposed. The proportion\nof unequals is quantity; for the difference, G F, in which it consists,\nis quantity. But the proportion of equality is not quantity; because,\nbetween equals, there is no difference; nor is one equality greater than\nanother, as one inequality is greater than another inequality.\n\n A B\n -------\n C D\n -------\n E F F\n -------\n\n[Sidenote: How the proportion of times and velocities is exposed.]\n\nThe proportion of two _times_, or of two _uniform velocities_, is then\nexposed, when two lines are exposed by which two bodies are understood\nto be moved uniformly; and therefore the same two lines serve to exhibit\nboth their own proportion, and that of the times and velocities,\naccording as they are considered to be exposed for the magnitudes\nthemselves, or for the times or velocities. For let the two lines A and\nB be exposed; their proportion therefore (by the last foregoing article)\nis exposed; and if they be considered as drawn with equal and uniform\nvelocity, then, seeing their times are greater, or equal, or less,\naccording as the same spaces are transmitted in greater, or equal, or\nless time, the lines A and B will exhibit the equality or inequality,\nthat is, the proportion of the times. To conclude, if the same lines, A\nand B, be considered as drawn in the same time, then, seeing their\nvelocities are greater, or equal, or less, according as they pass over\nin the same time longer, or equal, or shorter lines, the same lines, A\nand B, will exhibit the equality, or inequality, that is, the proportion\nof their velocities.\n\n A\n ---\n B\n\n\n -------\n\n\n CHAPTER XIII.\n\n OF ANALOGISM, OR THE SAME PROPORTION.\n\n1, 2, 3, 4. The nature and definition of proportion, arithmetical and\n geometrical.\u20145. The definition, and some properties of the same\n arithmetical proportion.\u20146, 7. The definition and transmutations of\n analogism, or the same geometrical proportion.\u20148, 9.. The\n definitions of hyperlogism and hypologism, that is, of greater and\n less proportion, and their transmutations.\u201410, 11, 12. Comparison of\n analogical quantities, according to magnitude.\u201413, 14, 15.\n Composition of proportions.\u201416, 17, 18, 19, 20, 21, 22, 23, 24, 25.\n The definition and properties of continual proportion.\u201426, 27, 28,\n 29. Comparison of arithmetical and geometrical proportions.\n\n [Note, that in this chapter the sign + signifies that the quantities\n betwixt which it is put, are added together; and this sign - the\n remainder after the latter quantity is taken out of the former. So\n that A + B is equal to both A and B together; and where you see A - B,\n there A is the whole, B the part taken out of it, and A - B the\n remainder. Also, two letters, set together without any sign, signify,\n unless they belong to a figure, that one of the quantities is\n multiplied by the other; as A B signifies the product of A multiplied\n by B.]\n\n\n[Sidenote: The nature and definition of proportion, arithmetical &\n geometrical.]\n\n1. Great and little are not intelligible, but by comparison. Now that,\nto which they are compared, is something exposed; that is, some\nmagnitude either perceived by sense, or so defined by words, that it may\nbe comprehended by the mind. Also that, to which any magnitude is\ncompared, is either greater or less, or equal to it. And therefore\nproportion (which, as I have shewn, is the estimation or comprehension\nof magnitudes by comparison,) is threefold, namely, proportion of\n_equality_, that is, of equal to equal; or of _excess_, which is of the\ngreater to the less; or of _defect_, which is the proportion of the less\nto the greater.\n\nAgain, every one of these proportions is two-fold; for if it be asked\nconcerning any magnitude given, how great it is, the answer may be made\nby comparing it two ways; first, by saying it is greater or less than\nanother magnitude, by so much; as seven is less than ten, by three\nunities; and this is called _arithmetical proportion_. Secondly, by\nsaying it is greater or less than another magnitude, by such a part or\nparts thereof; as seven is less than ten, by three tenth parts of the\nsame ten. And though this proportion be not always explicable by number,\nyet it is a determinate proportion, and of a different kind from the\nformer, and called _geometrical proportion_, and most commonly\n_proportion simply_.\n\n2. Proportion, whether it be arithmetical or geometrical, cannot be\nexposed but in two magnitudes, (of which the former is commonly called\nthe _antecedent_, and the latter the _consequent_ of the proportion) as\nI have shewn in the 8th article of the preceding chapter. And,\ntherefore, if two proportions be to be compared, there must be four\nmagnitudes exposed, namely, two antecedents and two consequents; for\nthough it happen sometimes that the consequent of the former proportion\nbe the same with the antecedent of the latter, yet in that double\ncomparison it must of necessity be twice numbered; so that there will be\nalways four terms.\n\n3. Of two proportions, whether they be arithmetical or geometrical, when\nthe magnitudes compared in both (which Euclid, in the fifth definition\nof his sixth book, calls the _quantities of proportions_,) are equal,\nthen one of the proportions cannot be either greater or less than the\nother; for one equality is neither greater nor less than another\nequality. But of two proportions of inequality, whether they be\nproportions of excess or of defect, one of them may be either greater or\nless than the other, or they may both be equal; for though there be\npropounded two magnitudes that are unequal to one another, yet there may\nbe other two more, unequal, and other two equally unequal, and other two\nless unequal than the two which were propounded. And from hence it may\nbe understood, that the proportions of excess and defect are quantity,\nbeing capable of more and less; but the proportion of equality is not\nquantity, because not capable neither of _more_, nor of _less_. And\ntherefore proportions of inequality may be added together, or subtracted\nfrom one another, or be multiplied or divided by one another, or by\nnumber; but proportions of equality not so.\n\n4. Two equal proportions are commonly called _the same proportion_; and,\nit is said, that the proportion of the first antecedent to the first\nconsequent is the _same_ with that of the second antecedent to the\nsecond consequent. And when four magnitudes are thus to one another in\ngeometrical proportion, they are called _proportionals_; and by some,\nmore briefly, _analogism_. And _greater proportion_ is the proportion of\na greater antecedent to the same consequent, or of the same antecedent\nto a less consequent; and when the proportion of the first antecedent to\nthe first consequent is greater than that of the second antecedent to\nthe second consequent, the four magnitudes, which are so to one another,\nmay be called _hyperlogism_.\n\n_Less proportion_ is the proportion of a less antecedent to the same\nconsequent, or of the same antecedent to a greater consequent; and when\nthe proportion of the first antecedent to the first consequent is less\nthan that of the second to the second, the four magnitudes may be called\n_hypologism_.\n\n[Sidenote: The definition and some properties of the same arithmetical\n proportion.]\n\n5. One arithmetical proportion is the _same_ with another arithmetical\nproportion, when one of the antecedents exceeds its consequent, or is\nexceeded by it, as much as the other antecedent exceeds its consequent,\nor is exceeded by it. And therefore, in four magnitudes, arithmetically\nproportional, the sum of the extremes is equal to the sum of the means.\nFor if A. B :: C. D be arithmetically proportional, and the difference\non both sides be the same excess, or the same defect, E, then B + C (if\nA be greater than B) will be equal to A - E + C; and A + D will be equal\nto A + C - E; but A - E + C and A + C - E are equal. Or if A be less\nthan B, then B + C will be equal to A + E + C; and A + D will be equal\nto A + C + E; but A + E + C and A + C + E are equal.\n\nAlso, if there be never so many magnitudes, arithmetically proportional,\nthe sum of them all will be equal to the product of half the number of\nthe terms multiplied by the sum of the extremes.\n\nFor if A. B :: C. D :: E. F be arithmetically proportional, the couples\nA + F, B + E, C + D will be equal to one another; and their sum will be\nequal to A + F, multiplied by the number of their combinations, that is,\nby half the number of the terms.\n\nIf, of four unequal magnitudes, any two, together taken, be equal to the\nother two together taken, then the greatest and the least of them will\nbe in the same combination. Let the unequal magnitudes be A, B, C, D;\nand let A + B be equal to C + D; and let A be the greatest of them all;\nI say B will be the least. For, if it may be, let any of the rest, as D,\nbe the least. Seeing therefore A is greater than C, and B than D, A+B\nwill be greater than C + D; which is contrary to what was supposed.\n\nIf there be any four magnitudes, the sum of the greatest and least, the\nsum of the means, the difference of the two greatest, and the difference\nof the two least, will be arithmetically proportional. For, let there be\nfour magnitudes, whereof A is the greatest, D the least, and B and C the\nmeans; I say A + D. B + C :: A - B. C - D are arithmetically\nproportional. For the difference between the first antecedent and its\nconsequent is this, A + D - B - C; and the difference between the second\nantecedent and its consequent this, A - B - C + D; but these two\ndifferences are equal; and therefore, by this 5th article, A + D. B + C\n:: A - B. C - D are arithmetically proportional.\n\nIf, of four magnitudes, two be equal to the other two, they will be in\nreciprocal arithmetical proportion. For let A + B be equal to C + D, I\nsay A. C :: D. B are arithmetically proportional. For if they be not,\nlet A. C :: D. E (supposing E to be greater or less than B) be\narithmetically proportional, and then A + E will be equal to C + D;\nwherefore A + B and C + D are not equal; which is contrary to what was\nsupposed.\n\n[Sidenote: The definition and transmutations of analogism, or the same\n geometrical proportion.]\n\n6. One geometrical proportion is the same with another geometrical\nproportion; when the same cause, producing equal effects in equal times,\ndetermines both the proportions.\n\nIf a point uniformly moved describe two lines, either with the same, or\ndifferent velocity, all the parts of them which are contemporary, that\nis, which are described in the same time, will be two to two, in\ngeometrical proportion, whether the antecedents be taken in the same\nline, or not. For, from the point A (in the 10th figure at the end of\nthe 14th chapter) let the two lines, A D, A G, be described with uniform\nmotion; and let there be taken in them two parts A B, A E, and again,\ntwo other parts, A C, A F; in such manner, that A B, A E, be\ncontemporary, and likewise A C, A F contemporary. I say first (taking\nthe antecedents A B, A C in the line A D, and the conquents A E, A F in\nthe line A G) that A B. A C :: A E. A F are proportionals. For seeing\n(by the 8th chap, and the 15th art.) velocity is motion considered as\ndetermined by a certain length or line, in a certain time transmitted by\nit, the quantity of the line A B will be determined by the velocity and\ntime by which the same A B is described; and for the same reason, the\nquantity of the line A C will be determined by the velocity and time, by\nwhich the same A C is described; and therefore the proportion of A B to\nA C, whether it be proportion of equality, or of excess or defect, is\ndetermined by the velocities and times by which A B, A C are described;\nbut seeing the motion of the point A upon A B and A C is uniform, they\nare both described with equal velocity; and therefore whether one of\nthem have to the other the proportion of majority or of minority, the\nsole cause of that proportion is the difference of their times; and by\nthe same reason it is evident, that the proportion of A E to A F is\ndetermined by the difference of their times only. Seeing therefore A B,\nA E, as also A C, A F are contemporary, the difference of the times in\nwhich A B and A C are described, is the same with that in which A E and\nA F are described. Wherefore the proportion of A B to A C, and the\nproportion of A E to A F are both determined by the same cause. But the\ncause, which so determines the proportion of both, works equally in\nequal times, for it is uniform motion; and therefore (by the last\nprecedent definition) the proportion of A B to A C is the same with that\nof A E to A F; and consequently A B. A C :: A E. A F are proportionals;\nwhich is the first.\n\nSecondly, (taking the antecedents in different lines) I say, A B. A E ::\nA C. A F are proportionals; for seeing A B, A E are described in the\nsame time, the difference of the velocities in which they are described\nis the sole cause of the proportion they have to one another. And the\nsame may be said of the proportion of A C to A F. But seeing both the\nlines A D and A G are passed over by uniform motion, the difference of\nthe velocities in which A B, A E are described, will be the same with\nthe difference of the velocities, in which A C, A F are described.\nWherefore the cause which determines the proportion of A B to A E, is\nthe same with that which determines the proportion of A C to A F; and\ntherefore A B. A E :: A C. A F, are proportionals; which remained to be\nproved.\n\nCoroll. I. If four magnitudes be in geometrical proportion, they will\nalso be proportionals by _permutation_, that is, by transposing the\nmiddle terms. For I have shown, that not only A B. A C :: A E. A F, but\nalso that, by _permutation_, A B. A E :: A C. A F are proportionals.\n\nCoroll. II. If there be four proportionals, they will also be\nproportionals by _inversion_ or _conversion_, that is, by turning the\nantecedents into consequents. For if in the last _analogism_, I had for\nA B, A C, put by inversion A C, A B, and in like manner converted A E, A\nF into A F, A E, yet the same demonstration had served. For as well A C,\nA B, as A B, A C are of equal velocity; and A C, A F, as well as A F, A\nC are contemporary.\n\nCoroll. III. If proportionals be added to proportionals, or taken from\nthem, the aggregates, or remainders, will be proportionals. For\ncontemporaries, whether they be added to contemporaries, or taken from\nthem, make the aggregates or remainders contemporary, though the\naddition or subtraction be of all the terms, or of the antecedents\nalone, or of the consequents alone.\n\nCoroll. IV. If both the antecedents of four proportionals, or both the\nconsequents, or all the terms, be multiplied or divided by the same\nnumber or quantity, the products or quotients will be proportionals. For\nthe multiplication and division of proportionals, is the same with the\naddition and subtraction of them.\n\nCoroll. V. If there be four proportionals, they will also be\nproportionals by _composition_, that is, by compounding an antecedent of\nthe antecedent and consequent put together, and by taking for consequent\neither the consequent singly, or the antecedent singly. For this\ncomposition is nothing but addition of proportionals, namely, of\nconsequents to their own antecedents, which by supposition are\nproportionals.\n\nCoroll. VI. In like manner, if the antecedent singly, or consequent\nsingly, be put for antecedent, and the consequent be made of both put\ntogether, these also will be proportionals. For it is the _inversion of\nproportion by composition_.\n\nCoroll. VII. If there be four proportionals, they will also be\nproportionals by division, that is, by taking the remainder after the\nconsequent is subtracted from the antecedent, or the difference between\nthe antecedent and consequent for antecedent, and either the whole or\nthe subtracted for consequent; as if A. B :: C. D be proportionals, they\nwill by division be A - B. B :: C - D. D, and A - B. A :: C - D. C; and\nwhen the consequent is greater than the antecedent, B - A. A :: D - C.\nC, and B - A. B :: D - C. D. For in all these divisions, proportionals\nare, by the very supposition of the analogism A. B :: C. D, taken from A\nand B, and from C and D.\n\nCoroll. VIII. If there be four proportionals, they will also be\nproportionals by the _conversion of proportion_, that is, by inverting\nthe divided proportion, or by taking the whole for antecedent, and the\ndifference or remainder for consequent.\n\nAs, if A. B :: C. D be proportionals, then A. A - B :: C. C - D, as also\nB. A - B :: D. C - D will be proportionals. For seeing these inverted be\nproportionals, they are also themselves proportionals.\n\nCoroll. IX. If there be two analogisms which have their quantities\nequal, the second to the second, and the fourth to the fourth, then\neither the sum or difference of the first quantities will be to the\nsecond, as the sum or difference of the third quantities is to the\nfourth. Let A. B :: C. D and E. B :: F. D be analogisms; I say A + E. B\n:: C + F. D are proportionals. For the said analogisms will by\npermutation be A. C :: B. D, and E. F :: B. D; and therefore A. C :: E.\nF will be proportionals, for they have both the proportion of B to D\ncommon. Wherefore, if in the permutation of the first analogism, there\nbe added E and F to A and C, which E and F are proportional to A and C,\nthen (by the third coroll.) A + E. B :: C + F. D will be proportionals;\nwhich was to be proved.\n\nAlso in the same manner it may be shown, that A - E. B :: C - F. D are\nproportionals.\n\n7. If there be two analogisms, where four antecedents make an analogism,\ntheir consequents also shall make an analogism; as also the sums of\ntheir antecedents will be proportional to the sums of their consequents.\nFor if A. B :: C. D and E. F :: G. H be two analogisms, and A. E :: C. G\nbe proportionals, then by permutation A. C :: E. G, and E. G :: F. H,\nand A. C :: B. D will be proportionals; wherefore B. D :: E. G, that is,\nB. D :: F. H, and by permutation B. F :: D. H are proportionals; which\nis the first. Secondly, I say A + E. B + F :: C + G. D + H will be\nproportionals. For seeing A. E :: C. G are proportionals, A + E. E :: C\n+ G. G will also by composition be proportionals, and by permutation A +\nE. C + G :: E. G will be proportionals; wherefore, also A + E. C + G ::\nB + F. D + H will be proportionals. Again, seeing, as is shown above, B.\nF :: D. H are proportionals, B + F. F :: D + H. H will also by\ncomposition be proportionals; and by permutation B + F. D + H :: F. H\nwill also be proportionals; wherefore A + E. C + G :: B + F. D + H are\nproportionals; which remained to be proved.\n\nCoroll. By the same reason, if there be never so many analogisms, and\nthe antecedents be proportional to the antecedents, it may be\ndemonstrated also that the consequents will be proportional to the\nconsequents, as also the sum of the antecedents to the sum of the\nconsequents.\n\n[Sidenote: The definition of hyperlogism and hypologism, that is, of\n greater and less proportion, and their transmutations.]\n\n8. In an hyperlogism, that is, where the proportion of the first\nantecedent to its consequent is greater than the proportion of the\nsecond antecedent to its consequent, the permutation of the\nproportionals, and the addition of proportionals to proportionals, and\nsubstraction of them from one another, as also their composition and\ndivision, and their multiplication and division by the same number,\nproduce always an hyperlogism. For suppose A. B :: C . D and A. C :: E.\nF be analogisms, A + E. B :: C + F . D will also be an analogism; but A\n+ E. B :: C. D will be an hyperlogism; wherefore by permutation, A + E.\nC :: B. D is an hyperlogism, because A. B :: C. D is an analogism.\nSecondly, if to the hyperlogism A + E. B :: C. D the proportionals G and\nH be added, A + E + G. B :: C + H. D will be an hyperlogism, by reason A\n+ E + G. B :: C + F + H. D is an analogism. Also, if G and H be taken\naway, A + E - G. B :: C - H. D will be an hyperlogism; for A + E - G. B\n:: C + F - H. D is an analogism. Thirdly, by composition A + E + B. B ::\nC + D. D will be an hyperlogism, because A + E + B. B :: C + F + D. D is\nah analogism, and so it will be in all the varieties of composition.\nFourthly, by division, A + E - B. B :: C - D. D will by an hyperlogism,\nby reason A E - B. B :: C + F - D. D is an analogism. Also A + E - B. A\n+ E :: C - D. C is an hyperlogism; for A + E - B. A + E :: C + F - D. C\nis an analogism. Fifthly, by multiplication 4 A + 4 E. B :: 4 C. D is an\nhyperlogism, because 4 A. B :: 4 C. D is an analogism; and by division \u00bc\nA + \u00bc E. B :: \u00bc C.D is an hyperlogism, because \u00bc A. B :: \u00bc C. D is an\nanalogism.\n\n9. But if A + E. B :: C. D be an hyperlogism, then by inversion B. A + E\n:: D. C will be an hypologism, because B. A :: D. C being an analogism,\nthe first consequent will be too great. Also, by conversion of\nproportion, A + E. A + E - B :: C. C - D is an hypologism, because the\ninversion of it, namely A + E - B. A + E :: C - D. C is an hyperlogism,\nas I have shown but now. So also B. A + E - B :: D. C - D is an\nhypologism, because, as I have newly shown, the inversion of it, namely\nA + E - B. B :: C - D. D is an hyperlogism. Note that this hypologism A\n+ E. A + E - B :: C. C - D is commonly thus expressed; if the proportion\nof the whole, (A + E) to that which is taken out of it (B), be greater\nthan the proportion of the whole (C) to that which is taken out of it\n(D), then the proportion of the whole (A + E) to the remainder (A + E -\nB) will be less than the proportion of the whole (C) to the remainder (C\n- D).\n\n[Sidenote: Comparison of analogical quantities, according to magnitude.]\n\n10. If there be four proportionals, the difference of the two first, to\nthe difference of the two last, will be as the first antecedent is to\nthe second antecedent, or as the first consequent to the second\nconsequent. For if A. B :: C. D be proportionals, then by division A -\nB. B :: C - D. D will be proportionals; and by permutation A - B. C - D\n:: B. D; that is, the differences are proportional to the consequents,\nand therefore they are so also to the antecedents.\n\n11. Of four proportionals, if the first be greater than the second, the\nthird also shall be greater than the fourth. For seeing the first is\ngreater than the second, the proportion of the first to the second is\nthe proportion of excess; but the proportion of the third to the fourth\nis the same with that of the first to the second; and therefore also the\nproportion of the third to the fourth is the proportion of excess;\nwherefore the third is greater than the fourth. In the same manner it\nmay be proved, that whensoever the first is less than the second, the\nthird also is less than the fourth; and when those are equal, that these\nalso are equal.\n\n12. If there be four proportionals whatsoever, A. B :: C.D, and the\nfirst and third be multiplied by any one number, as by 2; and again the\nsecond and fourth be multiplied by any one number, as by 3; and the\nproduct of the first 2 A, be greater than the product of the second 3 B;\nthe product also of the third 2 C, will be greater than the product of\nthe fourth 3 D. But if the product of the first be less than the product\nof the second, then the product of the third will be less than that of\nthe fourth. And lastly, if the products of the first and second be\nequal, the products of the third and fourth shall also be equal. Now\nthis theorem is all one with Euclid's definition of _the same\nproportion_; and it may be demonstrated thus. Seeing A. B :: C. D are\nproportionals, by permutation also (art. 6, coroll. I.) A. C :: B . D\nwill be proportionals; wherefore (by coroll. IV. art. 6) 2 A. 2 C :: 3\nB. 3 D will be proportionals; and again, by permutation, 2 A. 3 B :: 2\nC. 3 D will be proportionals; and therefore, by the last article, if 2 A\nbe greater than 3 B, then 2 C will be greater than 3 D; if less, less;\nand if equal, equal; which was to be demonstrated.\n\n[Sidenote: Composition of proportions.]\n\nIf any three magnitudes be propounded, or three things whatsoever that\nhave any proportion one to another, as three numbers, three times, three\ndegrees, &c.; the proportions of the first to the second, and of the\nsecond to the third, together taken, are equal to the proportion of the\nfirst to the third. Let there be three lines, for any proportion may be\nreduced to the proportion of lines, A B, A C, A D; and in the first\nplace, let the proportion as well of the first A B to the second A C, as\nof the second A C to the third AD, be the proportion of defect, or of\nless to greater; I say the proportions together taken of A B to A C, and\nof A C to A D, are equal to the proportion of A B to A D. Suppose the\npoint A to be moved over the whole line A D with uniform motion; then\nthe proportions as well of A B to A C, as of A C to A D, are determined\nby the difference of the times in which they are described; that is, A B\nhas to A C such proportion as is determined by the different times of\ntheir description; and A C to AD such proportion as is determined by\ntheir times. But the proportion of A B to A D is such as is determined\nby the difference of the times in which A B and A D are described; and\nthe difference of the times in which A B and A C are described, together\nwith the difference of the times in which A C and A D are described, is\nthe same with the difference of the times in which A B and A D are\ndescribed. And therefore, the same cause which determines the two\nproportions of A B to A C and of A C to A D, determines also the\nproportion of A B to A D. Wherefore, by the definition of the same\nproportion, delivered above in the 6th article, the proportion of A B to\nA C together with the proportion of A C to A D, is the same with the\nproportion of A B to A D.\n\n A B C D\n -------------\n\nIn the second place, let A D be the first, A C the second, and A B the\nthird, and let their proportion be the proportion of excess, or the\ngreater to less; then, as before, the proportions of A D to A C, and of\nA C to A B, and of A D to A B, will be determined by the difference of\ntheir times; which in the description of A D and A C, and of A C and A B\ntogether taken, is the same with the difference of the times in the\ndescription of A D and A B. Wherefore the proportion of A D to A B is\nequal to the two proportions of A D to A C and of A C to A B.\n\nIn the last place. If one of the proportions, namely of A D to A B, be\nthe proportion of excess, and another of them, as of A B to A C be the\nproportion of defect, thus also the proportion of A D to A C will be\nequal to the two proportions together taken of A D to A B, and of A B to\nA C. For the difference of the times in which AD and AB are described,\nis excess of time; for there goes more time to the description of A D\nthan of A B; and the difference of the times in which A B and A C are\ndescribed, is defect of time, for less time goes to the description of A\nB than of A C; but this excess and defect being added together, make D B\n- B C, which is equal to D C, by which the first A D exceeds the third A\nC; and therefore the proportions of the first A D to the second A B, and\nof the second A B to the third A C, are determined by the same cause\nwhich determines the proportion of the first A D to the third A C.\nWherefore, if any three magnitudes, &c.\n\nCoroll. I. If there be never so many magnitudes having proportion to one\nanother, the proportion of the first to the last is compounded of the\nproportions of the first to the second, of the second to the third, and\nso on till you come to the last; or, the proportion of the first to the\nlast is the same with the sum of all the intermediate proportions. For\nany number of magnitudes having proportion to one another, as A, B, C,\nD, E being propounded, the proportion of A to E, as is newly shown, is\ncompounded of the proportions of A to D and of D to E; and again, the\nproportion of A to D, of the proportions of A to C, and of C to D; and\nlastly, the proportion of A to C, of the proportions of A to B, and of B\nto C.\n\nCoroll. II. From hence it may be understood how any two proportions may\nbe compounded. For if the proportions of A to B, and of C to D, be\npropounded to be added together, let B have to something else, as to E,\nthe same proportion which C has to D, and let them be set in this order,\nA, B, E; for so the proportion of A to E will evidently be the sum of\nthe two proportions of A to B, and of B to E, that is, of C to D. Or let\nit be as D to C, so A to something else, as to E, and let them be\nordered thus, E, A, B; for the proportion of E to B will be compounded\nof the proportions of E to A, that is, of C to D, and of A to B. Also,\nit may be understood how one proportion may be taken out of another. For\nif the proportion of C to D be to be subtracted out of the proportion of\nA to B, let it be as C to D, so A to something else, as E, and setting\nthem in this order, A, E, B, and taking away the proportion of A to E,\nthat is, of C to D, there will remain the proportion of E to B.\n\nCoroll. III. If there be two orders of magnitudes which have proportion\nto one another, and the several proportions of the first order be the\nsame and equal in number with the proportions of the second order; then,\nwhether the proportions in both orders be successively answerable to one\nanother, which is called _ordinate proportion_, or not successively\nanswerable, which is called _perturbed proportion_, the first and the\nlast in both will be proportionals. For the proportion of the first to\nthe last is equal to all the intermediate proportions; which being in\nboth orders the same, and equal in number, the aggregates of those\nproportions will also be equal to one another; but to their aggregates,\nthe proportions of the first to the last are equal; and therefore the\nproportion of the first to the last in one order, is the same with the\nproportion of the first to the last in the other order. Wherefore the\nfirst and the last in both are proportionals.\n\n[Sidenote: Composition of proportions.]\n\n14. If any two quantities be made of the mutual multiplication of many\nquantities, which have proportion to one another, and the efficient\nquantities on both sides be equal in number, the proportion of the\nproducts will be compounded of the several proportions, which the\nefficient quantities have to one another.\n\n A B. A.\n C B. C.\n C D. E.\n\nFirst, let the two products be A B and C D, whereof one is made of the\nmultiplication of A into B, and the other of the multiplication of C\ninto D. I say the proportion of A B to C D is compounded of the\nproportions of the efficient A to the efficient C, and of the efficient\nB to the efficient D. For let A B, C B and C D be set in order; and as B\nis to D, so let C be to another quantity as E; and let A, C, E be set\nalso in order. Then (by coroll. IV. of the 6th art.) it will be as A B\nthe first quantity to C B the second quantity in the first order, so A\nto C in the second order; and again, as C B to C D in the first order,\nso B to D, that is, by construction, so C to E in the second order; and\ntherefore (by the last corollary) A B. C D :: A. E will be\nproportionals. But the proportion of A to E is compounded of the\nproportions of A to C, and of B to D; wherefore also the proportion of A\nB to C D is compounded of the same.\n\nSecondly, let the two products be A B F, and C D G, each of them made of\nthree efficients, the first of A, B and F, and the second of C, D and G;\nI say, the proportion of A B F to C D G is compounded of the proportions\nof A to C, of B to D, and of F to G. For let them be set in order as\nbefore; and as B is to D, so let C be to another quantity E; and again,\nas F is to G, so let E be to another, H; and let the first order stand\nthus, ABF, CBF, CDF and CDG; and the second order thus, A, C, E, H. Then\nthe proportion of A B F to C B F in the first order, will be as A to C\nin the second; and the proportion of CBF to CDF in the first order, as B\nto D, that is, as C to E (by construction) in the second order; and the\nproportion of CDF to CDG in the first, as F to G, that is, as E to H (by\nconstruction) in the second order; and therefore A B F. C D G:: A. H\nwill be proportionals. But the proportion of A to H is compounded of the\nproportions of A to C, B to D, and F to G. Wherefore the proportion of\nthe product A B F to C D G is also compounded of the same. And this\noperation serves, how many soever the efficients be that make the\nquantities given.\n\n ABF. A.\n CBF. C.\n CDF. E.\n CDG. H.\n\nFrom hence ariseth another way of compounding many proportions into one,\nnamely, that which is supposed in the 5th definition of the 6th book of\nEuclid; which is, by multiplying all the antecedents of the proportions\ninto one another, and in like manner all the consequents into one\nanother. And from hence also it is evident, in the first place, that the\ncause why parallelograms, which are made by the duction of two straight\nlines into one another, and all solids which are equal to figures so\nmade, have their proportions compounded of the proportions of the\nefficients; and in the second place, why the multiplication of two or\nmore fractions into one another is the same thing with the composition\nof the proportions of their several numerators to their several\ndenominators. For example, if these fractions 1\u20442, 2\u20443, 3\u20444 be to be\nmultiplied into one another, the numerators 1, 2, 3, are first to be\nmultiplied into one another, which make 6; and next the denominators 2,\n3, 4, which make 24; and these two products make the fraction 6\u204424. In\nlike manner, if the proportions of 1 to 2, of 2 to 3, and of 3 to 4, be\nto be compounded, by working as I have shown above, the same proportion\nof 6 to 24 will be produced.\n\n15. If any proportion be compounded with itself inverted, the compound\nwill be the proportion of equality. For let any proportion be given, as\nof A to B, and let the inverse of it be that of C to D; and as C to D,\nso let B be to another quantity; for thus they will be compounded (by\nthe second coroll. of the 12th art.) Now seeing the proportion of C to D\nis the inverse of the proportion of A to B, it will be as C to D, so B\nto A; and therefore if they be placed in order, A, B, A, the proportion\ncompounded of the proportions of A to B, and of C to D, will be the\nproportion of A to A, that is, the proportion of equality. And from\nhence the cause is evident why two equal products have their efficients\nreciprocally proportional. For, for the making of two products equal,\nthe proportions of their efficients must be such, as being compounded\nmay make the proportion of equality, which cannot be except one be the\ninverse of the other; for if betwixt A and A any other quantity, as C,\nbe interposed, their order will be A, C, A, and the later proportion of\nC to A will be the inverse of the former proportion of A to C.\n\n[Sidenote: The definition and properties of continual proportion.]\n\n16. A proportion is said to be multiplied by a number, when it is so\noften taken as there be unities in that number; and if the proportion be\nof the greater to the less, then shall also the quantity of the\nproportion be increased by the multiplication; but when the proportion\nis of the less to the greater, then as the number increaseth, the\nquantity of the proportion diminisheth; as in these three numbers, 4, 2,\n1, the proportion of 4 to 1 is not only the duplicate of 4 to 2, but\nalso twice as great; but inverting the order of those numbers thus, 1,\n2, 4, the proportion of 1 to 2 is greater than that of 1 to 4; and\ntherefore though the proportion of 1 to 4 be the duplicate of 1 to 2,\nyet it is not twice so great as that of 1 to 2, but contrarily the half\nof it. In like manner, a proportion is said to be divided, when between\ntwo quantities are interposed one or more means in continual proportion,\nand then the proportion of the first to the second is said to be\nsubduplicate of that of the first to the third, and subtriplicate of\nthat of the first to the fourth, &c.\n\nThis mixture of proportions, where some are proportions of excess,\nothers of defect, as in a merchant's account of debtor and creditor, is\nnot so easily reckoned as some think; but maketh the composition of\nproportions sometimes to be addition, sometimes substraction; which\nsoundeth absurdly to such as have always by composition understood\naddition, and by diminution substraction. Therefore to make this account\na little clearer, we are to consider (that which is commonly assumed,\nand truly) that if there be never so many quantities, the proportion of\nthe first to the last is compounded of the proportions of the first to\nthe second, and of the second to the third, and so on to the last,\nwithout regarding their equality, excess, or defect; so that if two\nproportions, one of inequality, the other of equality, be added\ntogether, the proportion is not thereby made greater nor less; as for\nexample, if the proportions of A to B and of B to B be compounded, the\nproportion of the first to the second is as much as the sum of both,\nbecause proportion of equality, being not quantity, neither augmenteth\nquantity nor lesseneth it. But if there be three quantities, A, B, C,\nunequal, and the first be the greatest, the last least, then the\nproportion of B to C is an addition to that of A to B, and makes it\ngreater; and on the contrary, if A be the least, and C the greatest\nquantity, then doth the addition of the proportion of B to C make the\ncompounded proportion of A to C less than the proportion of A to B, that\nis, the whole less than the part. The composition therefore of\nproportions is not in this case the augmentation of them, but the\ndiminution; for the same quantity (Euclid V. 8) compared with two other\nquantities, hath a greater proportion to the lesser of them than to the\ngreater. Likewise, when the proportions compounded are one of excess,\nthe other of defect, if the first be of excess, as in these numbers, 8,\n6, 9, the proportion compounded, namely, of 8 to 9, is less than the\nproportion of one of the parts of it, namely, of 8 to 6; but if the\nproportion of the first to the second be of defect, and that of the\nsecond to the third be of excess, as in these numbers, 6, 8, 4, then\nshall the proportion of the first to the third be greater than that of\nthe first to the second, as 6 hath a greater proportion to 4 than to 8;\nthe reason whereof is manifestly this, that the less any quantity is\ndeficient of another, or the more one exceedeth another, the proportion\nof it to that other is the greater. Suppose now three quantities in\ncontinual proportion, A B 4, A C 6, A D 9. Because therefore A D is\ngreater than A C, but not greater than A D, the proportion of A D to AC\nwill be (by Euclid, V. 8) greater than that of A D to A D; and likewise,\nbecause the proportions of A D to A C, and of A C to A B are the same,\nthe proportions of A D to A C and of A C to A B, being both proportions\nof excess, make the whole proportion of A D to A B, or of 9 to 4, not\nonly the duplicate of A D to A C, that is, of 9 to 6, but also the\ndouble, or twice so great. On the other side, because the proportion of\nA D to A D, or 9 to 9, being proportion of equality, is no quantity, and\nyet greater than that of A C to A D, or 6 to 9, it will be as 0 - 9 to 0\n- 6, so A C to A D, and again, as 0 - 9 to 0 - 6, so 0 - 6 to 0 - 4; but\n0 - 4, 0 - 6, 0 - 9 are in continual proportion; and because 0 - 4 is\ngreater than 0 - 6, the proportion of 0 - 4 to 0 - 6 will be double to\nthe proportion of 0 - 4 to 0 - 9, double I say, and yet not duplicate,\nbut subduplicate.\n\nIf any be unsatisfied with this ratiocination, let him first consider\nthat (by Euclid V. 8) the proportion of A B to A C is greater than that\nof A B to A D, wheresoever D be placed in the line A C prolonged; and\nthe further off the point\n\nD is from C, so much the greater is the proportion of A B to A C than\nthat of A B to A D. There is therefore some point (which suppose be E)\nin such distance from C, as that the proportion of A B to A C will be\ntwice as great as that of A B to A E. That considered, let him determine\nthe length of the line A E, and demonstrate, if he can, that A E is\ngreater or less than A D.\n\n \u1e04 C\u0323 \u1e0c\n A\u2014\u2014\u2014\u2014\u2014E\n\nBy the same method, if there be more quantities than three, as A, B, C,\nD, in continual proportion, and A be the least, it may be made appear\nthat the proportion of A to B is triple magnitude, though subtriple in\nmultitude, to the proportion of A to D.\n\n17. If there be never so many quantities, the number whereof is odd, and\ntheir order such, that from the middlemost quantity both ways they\nproceed in continual proportion, the proportion of the two which are\nnext on either side to the middlemost is subduplicate to the proportion\nof the two which are next to these on both sides, and subtriplicate of\nthe proportion of the two which are yet one place more remote, &c. For\nlet the magnitudes be C, B, A, D, E, and let A, B, C, as also A, D, E be\nin continual proportion; I say the proportion of D to B is subduplicate\nof the proportion of E to C. For the proportion of D to B is compounded\nof the proportions of D to A, and of A to B once taken; but the\nproportion of E to C is compounded of the same twice taken; and\ntherefore the proportion of D to B is subduplicate of the proportion of\nE to C. And in the same manner, if there were three terms on either\nside, it might be demonstrated that the proportion of D to B would be\nsubtriplicate of that of the extremes, &c.\n\n18. If there be never so many continual proportionals, as the first,\nsecond, third, &c. their differences will be proportional to them. For\nthe second, third, &c. are severally consequents of the preceding, and\nantecedents of the following proportion. But (by art. 10) the difference\nof the first antecedent and consequent, to difference of the second\nantecedent and consequent, is as the first antecedent to the second\nantecedent, that is, as the first term to the second, or as the second\nto the third, &c. in continual proportionals.\n\n19. If there be three continual proportionals, the sum of the extremes,\ntogether with the mean twice taken, the sum of the mean and either of\nthe extremes, and the same extreme, are continual proportionals. For let\nA. B. C be continual proportionals. Seeing, therefore, A. B :: B. C are\nproportionals, by composition also A + B. B :: B + C. C will be\nproportionals; and by permutation A + B. B + C :: B. C will also be\nproportionals; and again, by composition A + 2B + C. B + C :: B + C. C;\nwhich was to be proved.\n\n20. In four continual proportionals, the greatest and the least put\ntogether is a greater quantity than the other two put together. Let A. B\n:: C. D be continual proportionals; whereof let the greatest be A, and\nthe least be D; I say A + D is greater than B + C. For by art. 10, A -\nB. C - D :: A. C are proportionals; and therefore A - B is, by art. 11,\ngreater than C - D. Add B on both sides, and A will be greater than C +\nB - D. And again, add D on both sides, and A + D will be greater than B\n+ C; which was to be proved.\n\n[Sidenote: The definition and properties of continual proportion.]\n\n21. If there be four proportionals, the extremes multiplied into one\nanother, and the means multiplied into one another, will make equal\nproducts. Let A. B :: C. D be proportionals; I say A D is equal to B C.\nFor the proportion of A D to B C is compounded, by art. 13, of the\nproportions of A to B, and D to C, that is, its inverse B to A; and\ntherefore, by art. 14, this compounded proportion is the proportion of\nequality; and therefore also, the proportion of A D to B C is the\nproportion of equality. Wherefore they are equal.\n\n22. If there be four quantities, and the proportion of the first to the\nsecond be duplicate of the proportion of the third to the fourth, the\nproduct of the extremes to the product of the means, will be as the\nthird to the fourth. Let the four quantities be A, B, C and D; and let\nthe proportion of A to B be duplicate of the proportion of C to D, I say\nA D, that is, the product of A into D is to B C, that is, to the product\nof the means, as C to D. For seeing the proportion of A to B is\nduplicate of the proportion of C to D, if it be as C to D, so D to\nanother, E, then A. B :: C. E will be proportionals; for the proportion\nof A to B is by supposition duplicate of the proportion of C to D; and C\nto E duplicate also of that of C to D by the definition, art. 15.\nWherefore, by the last article, A E or A into E is equal to B C or B\ninto C; but, by coroll. IV. art. 6, A D is to A E as D to E, that is, as\nC to D; and therefore A D is to B C, which as I have shown is equal to A\nE, as C to D; which was to be proved.\n\nMoreover, if the proportion of the first A to the second B be triplicate\nof the proportion of the third C to the fourth D, the product of the\nextremes to the product of the means will be duplicate of the proportion\nof the third to the fourth. For if it be as C to D so D to E, and again,\nas D to E so E to another, F, then the proportion of C to F will be\ntriplicate of the proportion of C to D; and consequently, A. B :: C. F\nwill be proportionals, and A F equal to B C. But as A D to A F, so is D\nto F; and therefore, also, as A D to B C, so D to F, that is, so C to E;\nbut the proportion of C to E is duplicate of the proportion of C to D;\nwherefore, also, the proportion of A D to B C is duplicate of that of C\nto D, as was propounded.\n\n23. If there be four proportionals, and a mean be interposed betwixt the\nfirst and second, and another betwixt the third and fourth, the first of\nthese means will be to the second, as the first of the proportionals is\nto the third, or as the second of them is to the fourth. For let A. B ::\nC. D be proportionals, and let E be a mean betwixt A and B, and F a mean\nbetwixt C and D; I say A. C :: E. F are proportionals. For the\nproportion of A to E is subduplicate of the proportion of A to B, or of\nC to D. Also, the proportion of C to F is subduplicate of that of C to\nD; and therefore A. E :: C. F are proportionals; and by permutation A. C\n:: E. F are also proportionals; which was to be proved.\n\n24. Any thing is said to be divided into extreme and mean proportion,\nwhen the whole and the parts are in continual proportion. As for\nexample, when A + B. A. B are continual proportionals; or when the\nstraight line A C is so divided in B, that A C. A B. B C are in\ncontinual proportion. And if the same line A C be again divided in D, so\nas that A C. C D. A D be continual proportionals; then also A C. A B. A\nD will be continual proportionals; and in like manner, though in\ncontrary order, C A. C D. C B will be continual proportionals; which\ncannot happen in any line otherwise divided.\n\n A B C\n ----|----|----\n D\n\n25. If there be three continual proportionals, and again, three other\ncontinual proportions, which have the same middle term, their extremes\nwill be in reciprocal proportion. For let A. B. C and D. B. E be\ncontinual proportionals, I say A. D :: E. C shall be proportionals. For\nthe proportion of A to D is compounded of the proportions of A to B, and\nof B to D; and the proportion of E to C is compounded of those of E to\nB, that is, of B to D, and of B to C, that is, of A to B. Wherefore, by\nequality, A. D :: E. C are proportionals.\n\n[Sidenote: Comparison of arithmetical and geometrical proportion.]\n\nIf any two unequal quantities be made extremes, and there be interposed\nbetwixt them any number of means in geometrical proportion, and the same\nnumber of means in arithmetical proportion, the several means in\ngeometrical proportion will be less than the several means in\narithmetical proportion. For betwixt A the lesser, and E the greater\nextreme, let there be interposed three means, B, C, D, in geometrical\nproportion, and as many more, F, G, H, in arithmetical proportion; I say\nB will be less than F, C than G, and D than H. For first, the difference\nbetween A and F is the same with that between F and G, and with that\nbetween G and H, by the definition of arithmetical proportion; and\ntherefore, the difference of the proportionals which stand next to one\nanother, to the difference of the extremes, is, when there is but one\nmean, half their difference; when two, a third part of it; when three, a\nquarter, &c.; so that in this example it is a quarter. But the\ndifference between D and E, by art. 17, is more than a quarter of the\ndifference between the extremes, because the proportion is geometrical,\nand therefore the difference between A and D is less than three quarters\nof the same difference of the extremes. In like manner, if the\ndifference between A and D be understood to be divided into three equal\nparts, it may be proved, that the difference between A and C is less\nthan two quarters of the difference of the extremes A and E. And lastly,\nif the difference between A and C be divided into two equal parts, that\nthe difference between A and B is less than a quarter of the difference\nof the extremes A and E.\n\n A A\n - -\n B F\n --- ---\n C G\n ----- -----\n D H\n ------- -------\n E E\n --------- ---------\n\nFrom the consideration hereof, it is manifest, that B, that is A\ntogether with something else which is less than a fourth part of the\ndifference of the extremes A and E, is less than F, that is, than the\nsame A with something else which is equal to the said fourth part. Also,\nthat C, that is A with something else which is less than two fourth\nparts of the said difference, is less than G, that is, than A together\nwith the said two-fourths. And lastly, that D, which exceeds A by less\nthan three-fourths of the said difference, is less than H, which exceeds\nthe same A by three entire fourths of the said difference. And in the\nsame manner it would be if there were four means, saving that instead of\nfourths of the difference of the extremes we are to take fifth parts;\nand so on.\n\n27. LEMMA. If a quantity being given, first one quantity be both added\nto it and subtracted from it, and then another greater or less, the\nproportion of the remainder to the aggregate, is greater where the less\nquantity is added and substracted, than where the greater quantity is\nadded and substracted. Let B be added to and substracted from the\nquantity A; so that A - B be the remainder, and A + B the aggregate; and\nagain, let C, a greater quantity than B, be added to and substracted\nfrom the same A, so that A - C be the remainder and A + C the aggregate;\nI say A - B. A + B :: A - C. A + C will be an hyperlogism. For A - B. A\n:: A - C. A is an hyperlogism of a greater antecedent to the same\nconsequent; and therefore A - B. A + B :: A - C. A + C is a much greater\nhyperlogism, being made of a greater antecedent to a less consequent.\n\n28. If unequal parts be taken from two equal quantities, and betwixt the\nwhole and the part of each there be interposed two means, one in\ngeometrical, the other in arithmetical proportion; the difference\nbetwixt the two means will be greatest, where the difference betwixt the\nwhole and its part is greatest. For let A B and A B be two equal\nquantities, from which let two unequal parts be taken, namely, A E the\nless, and A F the greater; and betwixt A B and A E let A G be a mean in\ngeometrical proportion, and A H a mean in arithmetical proportion. Also\nbetwixt A B and A F let A I be a mean in geometrical proportion, and A K\na mean in arithmetical proportion; I say H G is greater than K I.\n\n A E G H B\n ---+---+---+----\n ----+---+---+---\n A F I K B\n\n For in the first place we have this A B. A G :: B G. G E, by\n analogism article 18.\n\n\n Then by composition we have this A B + A G. A B :: B G + G E\n that is, B E. B G.\n\n\n And by taking the halves of the \u00bdA B + \u00bdA G. A B :: \u00bdB G + \u00bdG E,\n antecedents this third that is, B H. B G.\n\n\n And by conversion a fourth A B. \u00bdA B + \u00bdA G :: B G. B H.\n\n\n And by division this fifth \u00bdA B - \u00bdA G. \u00bdA B + \u00bdA G\n :: H G. B H.\n\n\n And by doubling the first\n antecedent and the first\n consequent A B - A G. A B + A G :: H G. B H.\n\n\n Also by the same method may be\n found out this analogism A B - A I. A B + AI :: K I. B K.\n\nNow seeing the proportion of A B to A E is greater than that of A B to A\nF, the proportion of A B to A G, which is half the greater proportion,\nis greater than the proportion of A B to A I the half of the less\nproportion; and therefore A I is greater than A G. Wherefore the\nproportion of A B - A G to A B + A G, by the precedent lemma, will be\ngreater than the proportion of A B - A I to A B + A I; and therefore\nalso the proportion of H G to B H will be greater than that of K I to B\nK, and much greater than the proportion of K I to B H, which is greater\nthan B K; for B H is the half of B E, as B K is the half of B F, which,\nby supposition, is less than B E. Wherefore H G is greater than K I;\nwhich was to be proved.\n\nCoroll. It is manifest from hence, that if any quantity be supposed to\nbe divided into equal parts infinite in number, the difference between\nthe arithmetical and geometrical means will be infinitely little, that\nis, none at all. And upon this foundation, chiefly, the art of making\nthose numbers, which are called Logarithms, seems to have been built.\n\n29. If any number of quantities be propounded, whether they be unequal,\nor equal to one another; and there be another quantity, which multiplied\nby the number of the propounded quantities, is equal to them all; that\nother quantity is a mean in arithmetical proportion to all those\npropounded quantities.\n\n\n -------\n\n\n CHAP. XIV.\n\n OF STRAIT AND CROOKED, ANGLE AND\n FIGURE.\n\n1. The definition and properties of a strait line.\u20142. The definition and\n properties of a plane superficies.\u20143. Several sorts of crooked\n lines.\u20144. The definition and properties of a circular line.\u20145. The\n properties of a strait line taken in a plane.\u20146.. The definition of\n tangent lines.\u20147. The definition of an angle, and the kinds\n thereof.\u20148. In concentric circles, arches of the same angle are to\n one another, as the whole circumferences are.\u20149. The quantity of an\n angle, in what it consists. \u201410. The distinction of angles, simply\n so called.\u201411. Of strait lines from the centre of a circle to a\n tangent of the same. \u201412. The general definition of parallels, and\n the properties of strait parallels.\u201413. The circumferences of\n circles are to one another, as their diameters are.\u201414. In\n triangles, strait lines parallel to the bases are to one another, as\n the parts of the sides which they cut off from the vertex.\u201415. By\n what fraction of a strait line the circumference of a circle is\n made.\u201416. That an angle of contingence is quantity, but of a\n different kind from that of an angle simply so called; and that it\n can neither add nor take away any thing from the same.\u201417. That the\n inclination of planes is angle simply so called.\u201418. A solid angle\n what it is.\u201419. What is the nature of asymptotes.\u201420. Situation, by\n what it is determined.\u201421. What is like situation; what is figure;\n and what are like figures.\n\n\n[Sidenote: The definition end properties of a strait line.]\n\n1. Between two points given, the shortest line is that, whose extreme\npoints cannot be drawn farther asunder without altering the quantity,\nthat is, without altering the proportion of that line to any other line\ngiven. For the magnitude of a line is computed by the greatest distance\nwhich may be between its extreme points; so that any one line, whether\nit be extended or bowed, has always one and the same length, because it\ncan have but one greatest distance between its extreme points.\n\nAnd seeing the action, by which a strait line is made crooked, or\ncontrarily a crooked line is made strait, is nothing but the bringing of\nits extreme points nearer to one another, or the setting of them further\nasunder, a _crooked line_ may rightly be defined to be _that, whose\nextreme points may be understood to be drawn further asunder_; and a\n_strait line_ to be _that, whose extreme points cannot be drawn further\nasunder; and comparatively, a more crooked, to be that line whose\nextreme points are nearer to one another than those of the other,\nsupposing both the lines to be of equal length_. Now, howsoever a line\nbe bowed, it makes always a _sinus_ or cavity, sometimes on one side,\nsometimes on another; so that the same crooked line may either have its\nwhole cavity on one side only, or it may have it part on one side and\npart on the other side. Which being well understood, it will be easy to\nunderstand the following comparisons of strait and crooked lines.\n\nFirst, if a strait and a crooked line have their extreme points common,\nthe crooked line is longer than the strait line. For if the extreme\npoints of the crooked line be drawn out to their greatest distance, it\nwill be made a strait line, of which that, which was a strait line from\nthe beginning, will be but a part; and therefore the strait line was\nshorter than the crooked line, which had the same extreme points. And\nfor the same reason, if two crooked lines have their extreme points\ncommon, and both of them have all their cavity on one and the same side,\nthe outermost of the two will be the longest line.\n\nSecondly, a strait line and a perpetually crooked line cannot be\ncoincident, no, not in the least part. For if they should, then not only\nsome strait line would have its extreme points common with some crooked\nline, but also they would, by reason of their coincidence, be equal to\none another; which, as I have newly shown, cannot be.\n\nThirdly, between two points given, there can be understood but one\nstrait line; because there cannot be more than one least interval or\nlength between the same points. For if there may be two, they will\neither be coincident, and so both of them will be one strait line; or if\nthey be not coincident, then the application of one to the other by\nextension will make the extended line have its extreme points at greater\ndistance than the other; and consequently, it was crooked from the\nbeginning.\n\nFourthly, from this last it follows, that two strait lines cannot\ninclude a superficies. For if they have both their extreme points\ncommon, they are coincident; and if they have but one or neither of them\ncommon, then at one or both ends the extreme points will be disjoined,\nand include no superficies, but leave all open and undetermined.\n\nFifthly, every part of a strait line is a strait line. For seeing every\npart of a strait line is the least that can be drawn between its own\nextreme points, if all the parts should not constitute a strait line,\nthey would altogether be longer than the whole line.\n\n[Sidenote: The definition and properties of a plane superficies.]\n\n2. A _plane_ or _a plane superficies, is that which is described by a\nstrait line so moved, that all the several points thereof describe\nseveral strait lines_. A strait line, therefore, is necessarily all of\nit in the same plane which it describes. Also the strait lines, which\nare made by the points that describe a plane, are all of them in the\nsame plane. Moreover, if any line whatsoever be moved in a plane, the\nlines, which are described by it, are all of them in the same plane.\n\nAll other superficies, which are not plane, are crooked, that is, are\neither concave or convex. And the same comparisons, which were made of\nstrait and crooked lines, may also be made of plane and crooked\nsuperficies.\n\nFor, first, if a plane and crooked superficies be terminated with the\nsame lines, the crooked superficies is greater than the plane\nsuperficies. For if the lines, of which the crooked superficies\nconsists, be extended, they will be found to be longer than those of\nwhich the plane superficies consists, which cannot be extended, because\nthey are strait.\n\nSecondly, two superficies, whereof the one is plane, and the other\ncontinually crooked, cannot be coincident, no, not in the least part.\nFor if they were coincident, they would be equal; nay, the same\nsuperficies would be both plane and crooked, which is impossible.\n\nThirdly, within the same terminating lines there can be no more than one\nplane superficies; because there can be but one least superficies within\nthe same.\n\nFourthly, no number of plane superficies can include a solid, unless\nmore than two of them end in a common vertex. For if two planes have\nboth the same terminating lines, they are coincident, that is, they are\nbut one superficies; and if their terminating lines be not the same,\nthey leave one or more sides open.\n\nFifthly, every part of a plane superficies is a plane superficies. For\nseeing the whole plane superficies is the least of all those, that have\nthe same terminating lines; and also every part of the same superficies\nis the least of all those, that are terminated with the same lines; if\nevery part should not constitute a plane superficies, all the parts put\ntogether would not be equal to the whole.\n\n[Sidenote: Several sorts of crooked lines.]\n\n3. Of straitness, whether it be in lines or in superficies, there is but\none kind; but of crookedness there are many kinds; for of crooked\nmagnitudes, some are congruous, that is, are coincident when they are\napplied to one other; others are incongruous. Again, some are\n\u1f41\u03bc\u03bf\u03b9\u03bf\u03bc\u03b5\u03c1\u03b5\u1fd6\u03c2 or uniform, that is, have their parts, howsoever taken,\ncongruous to one another; others are \u1f00\u03bd\u03bf\u03bc\u03bf\u03b9\u03bf\u03bc\u03b5\u03c1\u03b5\u1fd6\u03c2 or of several forms.\nMoreover, of such as are crooked, some are continually crooked, others\nhave parts which are not crooked.\n\n[Sidenote: Definition and properties of a circular line.]\n\n4. If a strait line be moved in a plane, in such manner, that while one\nend of it stands still, the whole line be carried round about till it\ncome again into the same place from whence it was first moved, it will\ndescribe a plane superficies, which will be terminated every way by that\ncrooked line, which is made by that end of the strait line which was\ncarried round. Now this superficies is called a CIRCLE; and of this\ncircle, the unmoved point is the _centre_; the crooked line which\nterminates it, the _perimeter_; and every part of that crooked line, a\n_circumference_ or _arch_; the strait line, which generated the\n_circle_, is the _semidiameter_ or _radius_; and any strait line, which\npasseth through the centre and is terminated on both sides in the\ncircumference, is called the _diameter_. Moreover, every point of the\nradius, which describes the circle, describes in the same time its own\nperimeter, terminating its own circle, which is said to be _concentric_\nto all the other circles, because this and all those have one common\ncentre.\n\nWherefore in every circle, all strait lines from the centre to the\ncircumference are equal. For they are all coincident with the radius\nwhich generates the circle.\n\nAlso the diameter divides both the perimeter and the circle itself into\ntwo equal parts. For if those two parts be applied to one another, and\nthe semiperimeters be coincident, then, seeing they have one common\ndiameter, they will be equal; and the semicircles will be equal also;\nfor these also will be coincident. But if the semiperimeters be not\ncoincident, then some one strait line, which passes through the centre,\nwhich centre is in the diameter, will be cut by them in two points.\nWherefore, seeing all the strait lines from the centre to the\ncircumference are equal, a part of the same strait line will be equal to\nthe whole; which is impossible.\n\nFor the same reason the perimeter of a circle will be uniform, that is,\nany one part of it will be coincident with any other equal part of the\nsame.\n\n[Sidenote: The properties of a strait line taken in a plane.]\n\n5. From hence may be collected this property of a strait line, namely,\nthat it is all contained in that plane which contains both its extreme\npoints. For seeing both its extreme points are in the plane, that strait\nline, which describes the plane, will pass through them both; and if one\nof them be made a centre, and at the distance between both a\ncircumference be described, whose radius is the strait line which\ndescribes the plane, that circumference will pass through the other\npoint. Wherefore between the two propounded points, there is one strait\nline, by the definition of a circle, contained wholly in the propounded\nplane; and therefore if another strait line might be drawn between the\nsame points, and yet not be contained in the same plane, it would\nfollow, that between two points two strait lines may be drawn; which has\nbeen demonstrated to be impossible.\n\nIt may also be collected, that if two planes cut one another, their\ncommon section will be a strait line. For the two extreme points of the\nintersection are in both the intersecting planes; and between those\npoints a strait line may be drawn; but a strait line between any two\npoints is in the same plane, in which the points are; and seeing these\nare in both the planes, the strait line which connects them will also be\nin both the same planes, and therefore it is the common section of both.\nAnd every other line, that can be drawn between those points, will be\neither coincident with that line, that is, it will be the same line; or\nit will not be coincident, and then it will be in neither, or but in one\nof those planes.\n\nAs a strait line may be understood to be moved round about whilst one\nend thereof remains fixed, as the centre; so in like manner it is easy\nto understand, that a plane may be circumduced about a strait line,\nwhilst the strait line remains still in one and the same place, as the\n_axis_ of that motion. Now from hence it is manifest, that any three\npoints are in some one plane. For as any two points, if they be\nconnected by a strait line, are understood to be in the same plane in\nwhich the strait line is; so, if that plane be circumduced about the\nsame strait line, it will in its revolution take in any third point,\nhowsoever it be situate; and then the three points will be all in that\nplane; and consequently the three strait lines which connect those\npoints, will also be in the same plane.\n\n[Sidenote: Definition of tangent lines.]\n\n6. Two lines are said to _touch_ one another, which being both drawn to\none and the same point, will not cut one another, though they be\nproduced, produced, I say, in the same manner in which they were\ngenerated. And therefore if two strait lines touch one another in any\none point, they will be contiguous through their whole length. Also two\nlines continually crooked will do the same, if they be congruous and be\napplied to one another according to their congruity; otherwise, if they\nbe incongruously applied, they will, as all other crooked lines, touch\none another, where they touch, but in one point only. Which is manifest\nfrom this, that there can be no congruity between a strait line and a\nline that is continually crooked; for otherwise the same line might be\nboth strait and crooked. Besides, when a strait line touches a crooked\nline, if the strait line be never so little moved about upon the point\nof contact, it will cut the crooked line; for seeing it touches it but\nin one point, if it incline any way, it will do more than touch it; that\nis, it will either be congruous to it, or it will cut it; but it cannot\nbe congruous to it; and therefore it will cut it.\n\n[Sidenote: The definition of an angle, and the kinds thereof.]\n\n7. An angle, according to the most general acceptation of the word, may\nbe thus defined; _when two lines, or many superficies, concur in one\nsole point, and diverge every where else, the quantity of that\ndivergence is an_ ANGLE. And an angle is of two sorts; for, first, it\nmay be made by the concurrence of lines, and then it is a _superficial\nangle_; or by the concurrence of superficies, and then it is called a\n_solid angle_.\n\nAgain, from the two ways by which two lines may diverge from one\nanother, superficial angles are divided into two kinds. For two strait\nlines, which are applied to one another, and are contiguous in their\nwhole length, may be separated or pulled open in such manner, that their\nconcurrence in one point will still remain; and this separation or\nopening may be either by circular motion, the centre whereof is their\npoint of concurrence, and the lines will still retain their straitness,\nthe quantity of which separation or divergence is an _angle_ simply so\ncalled; or they may be separated by continual flexion or curvation in\nevery imaginable point; and the quantity of this separation is that,\nwhich is called an _angle of contingence_.\n\nBesides, of superficial angles simply so called, those, which are in a\nplane superficies, are plane; and those, which are not plane, are\ndenominated from the superficies in which they are.\n\nLastly, those are _strait-lined angles_, which are made by strait lines;\nas those which are made by crooked lines are _crooked-lined_; and those\nwhich are made both of strait and crooked lines, are _mixed angles_.\n\n[Sidenote: In concentric circles, arches of the same angle are to one\n another, as the whole circumferences are.]\n\n8. Two arches intercepted between two radii of concentric circles, have\nthe same proportion to one another, which their whole perimeters have to\none another. For let the point A (in the first figure) be the centre of\nthe two circles B C D and E F G, in which the radii A E B and A F C\nintercept the arches B C and E F; I say the proportion of the arch B C\nto the arch E F is the same with that of the perimeter B C D to the\nperimeter E F G. For if the radius A F C be understood to be moved about\nthe centre A with circular and uniform motion, that is, with equal\nswiftness everywhere, the point C will in a certain time describe the\nperimeter B C D, and in a part of that time the arch B C; and because\nthe velocities are equal by which both the arch and the whole perimeter\nare described, the proportion of the magnitude of the perimeter B C D to\nthe magnitude of the arch BC is determined by nothing but the difference\nof the times in which the perimeter and the arch are described. But both\nthe perimeters are described in one and the same time, and both the\narches in one and the same time; and therefore the proportions of the\nperimeter B C D to the arch B C, and of the perimeter E F G to the arch\nE F, are both determined by the same cause. Wherefore B C D. B C :: E F\nG. E F are proportionals (by the 6th art. of the last chapter), and by\npermutation B C D. E F G :: B C. E F will also be proportionals; which\nwas to be demonstrated.\n\n[Sidenote: The quantity of an angle, in what it consists.]\n\n9. Nothing is contributed towards the quantity of an angle, neither by\nthe length, nor by the equality, nor by the inequality of the lines\nwhich comprehend it. For the lines A B and A C comprehend the same angle\nwhich is comprehended by the lines A E and A F, or A B and A F. Nor is\nan angle either increased or diminished by the absolute quantity of the\narch, which subtends the same; for both the greater arch B C and the\nlesser arch E F are subtended to the same angle. But the quantity of an\nangle is estimated by the quantity of the subtending arch compared with\nthe quantity of the whole perimeter. And therefore the quantity of an\nangle simply so called may be thus defined: _the quantity of an angle is\nan arch or circumference of a circle, determined by its proportion to\nthe whole perimeter_. So that when an arch is intercepted between two\nstrait lines drawn from the centre, look how great a portion that arch\nis of the whole perimeter, so great is the angle. From whence it may be\nunderstood, that when the lines which contain an angle are strait lines,\nthe quantity of that angle may be taken at any distance from the centre.\nBut if one or both of the containing lines be crooked, then the quantity\nof the angle is to be taken in the least distance from the centre, or\nfrom their concurrence; for the least distance is to be considered as a\nstrait line, seeing no crooked line can be imagined so little, but that\nthere may be a less strait line. And although the least strait line\ncannot be given, because the least given line may still be divided, yet\nwe may come to a part so small, as is not at all considerable; which we\ncall a point. And this point may be understood to be in a strait line\nwhich touches a crooked line; for an angle is generated by separating,\nby circular motion, one strait line from another which touches it, as\nhas been said above in the 7th article. Wherefore an angle, which two\ncrooked lines make, is the same with that which is made by two strait\nlines which touch them.\n\n[Sidenote: The distinction of angles, simply so called.]\n\n10. From hence it follows, that _vertical angles_, such as are A B C, D\nB F in the second figure, are equal to one another. For if, from the two\nsemiperimeters D A C, F D A, which are equal to one another, the common\narch D A be taken away, the remaining arches A C, D F will be equal to\none another.\n\nAnother distinction of angles is into _right_ and _oblique_. _A right\nangle is that, whose quantity is the fourth part of the perimeter._ And\nthe lines, which make a right angle, are said to be _perpendicular_ to\none another. Also, of oblique angles, that which is greater than a\nright, is called an _obtuse angle_; and that which is less, an _acute\nangle_. From whence it follows, that all the angles that can possibly be\nmade at one and the same point, together taken, are equal to four right\nangles; because the quantities of them all put together make the whole\nperimeter. Also, that all the angles, which are made on one side of a\nstrait line, from any one point taken in the same, are equal to two\nright angles; for if that point be made the centre, that strait line\nwill be the diameter of a circle, by whose circumference the quantity of\nan angle is determined; and that diameter will divide the perimeter into\ntwo equal parts.\n\n[Sidenote: Of strait lines from the centre of a circle to a tangent of\n the same.]\n\n11. If a tangent be made the diameter of a circle, whose centre is the\npoint of contact, a strait line drawn from the centre of the former\ncircle to the centre of the latter circle, will make two angles with the\ntangent, that is, with the diameter of the latter circle, equal to two\nright angles, by the last article. And because, by the 6th article, the\ntangent has on both sides equal inclination to the circle, each of them\nwill be a right angle; as also the semidiameter will be perpendicular to\nthe same tangent. Moreover, the semidiameter, inasmuch as it is the\nsemidiameter, is the least strait line which can be drawn from the\ncentre to the tangent; and every other strait line, that reaches the\ntangent, will pass out of the circle, and will therefore be greater than\nthe semidiameter. In like manner, of all the strait lines, which may be\ndrawn from the centre to the tangent, that is the greatest which makes\nthe greatest angle with the perpendicular; which will be manifest, if\nabout the same centre another circle be described, whose semidiameter is\na strait line taken nearer to the perpendicular, and there be drawn a\nperpendicular, that is, a tangent, to the same.\n\nFrom whence it is also manifest, that if two strait lines, which make\nequal angles on either side of the perpendicular, be produced to the\ntangent, they will be equal.\n\n[Sidenote: The general definition of parallels; the properties of strait\n parallels.]\n\n12. There is in Euclid a definition of strait-lined parallels; but I do\nnot find that parallels in general are anywhere defined; and therefore\nfor an universal definition of them, I say that _any two lines\nwhatsoever, strait or crooked, as also any two superficies, are_\nPARALLEL; _when two equal strait lines, wheresoever they fall upon them,\nmake always equal angles with each of them_.\n\nFrom which definition it follows; first, that any two strait lines, not\ninclined opposite ways, falling upon two other strait lines, which are\nparallel, and intercepting equal parts in both of them, are themselves\nalso equal and parallel. As if A B and C D (in the third figure),\ninclined both the same way, fall upon the parallels A C and B D, and A C\nand B D be equal, A B and C D will also be equal and parallel. For the\nperpendiculars B E and D F being drawn, the right angles E B D and F D H\nwill be equal. Wherefore, seeing E F and B D are parallel, the angles E\nB A and F D C will be equal. Now if D C be not equal to B A, let any\nother strait line equal to B A be drawn from the point D; which, seeing\nit cannot fall upon the point C, let it fall upon G. Wherefore A G will\nbe either greater or less than B D; and therefore the angles E B A and F\nD C are not equal, as was supposed. Wherefore A B and C D are equal;\nwhich is the first.\n\nAgain, because they make equal angles with the perpendiculars B E and D\nF; therefore the angle C D H will be equal to the angle A B D, and, by\nthe definition of parallels, A B and C D will be parallel; which is the\nsecond.\n\n_That plane, which is included both ways within parallel lines, is\ncalled a_ PARALLELOGRAM.\n\nCoroll. I. From this last it follows, that the angles A B D and C D H\nare equal, that is, that a strait line, as B H, falling upon two\nparallels, as A B and C D, makes the internal angle A B D equal to the\nexternal and opposite angle C D H.\n\nCoroll. II. And from hence again it follows, that a strait line falling\nupon two parallels, makes the alternate angles equal, that is, the angle\nA G F, in the fourth figure, equal to the angle G F D. For seeing G F D\nis equal to the external opposite angle E G B, it will be also equal to\nits vertical angle A G F, which is alternate to G F D.\n\nCoroll. III. That the internal angles on the same side of the line F G\nare equal to two right angles. For the angles at F, namely, G F C and G\nF D, are equal to two right angles. But G F D is equal to its alternate\nangle A G F. Wherefore both the angles G F C and A G F, which are\ninternal on the same side of the line F G, are equal to two right\nangles.\n\nCoroll. IV. That the three angles of a strait-lined plain triangle are\nequal to two right angles; and any side being produced, the external\nangle will be equal to the two opposite internal angles. For if there be\ndrawn by the vertex of the plain triangle A B C (fig. 5) a parallel to\nany of the sides, as to A B, the angles A and B will be equal to their\nalternate angles E and F, and the angle C is common. But, by the 10th\narticle, the three angles E, C and F, are equal to two right angles; and\ntherefore the three angles of the triangle are equal to the same; which\nis the first. Again, the two angles B and D are equal to two right\nangles, by the 10th article. Wherefore taking away B, there will remain\nthe angles A and C, equal to the angle D; which is the second.\n\nCoroll. V. If the angles A and B be equal, the sides A C and C B will\nalso be equal, because A B and E F are parallel; and, on the contrary,\nif the sides A C and C B be equal, the angles A and B will also be\nequal. For if they be not equal, let the angles B and G be equal.\nWherefore, seeing G B and E F are parallels, and the angles G and B\nequal, the sides G C and C B will also be equal; and because C B and A C\nare equal by supposition, C G and C A will also be equal; which cannot\nbe, by the 11th article.\n\nCoroll. VI. From hence it is manifest, that if two radii of a circle be\nconnected by a strait line, the angles they make with that connecting\nline will be equal to one another; and if there be added that segment of\nthe circle, which is subtended by the same line which connects the\nradii, then the angles, which those radii make with the circumference,\nwill also be equal to one another. For a strait line, which subtends any\narch, makes equal angles with the same; because, if the arch and the\nsubtense be divided in the middle, the two halves of the segment will be\ncongruous to one another, by reason of the uniformity both of the\ncircumference of the circle, and of the strait line.\n\n[Sidenote: The circumferences of circles are to one another as their\n diameters are.]\n\n13. Perimeters of circles are to one another, as their semidiameters\nare. For let there be any two circles, as, in the first figure, B C D\nthe greater, and E F G the lesser, having their common centre at A; and\nlet their semidiameters be A C and A E. I say, A C has the same\nproportion to A E, which the perimeter B C D has to the perimeter E F G.\nFor the magnitude of the semidiameters A C and A E is determined by the\ndistance of the points C and E from the centre A; and the same distances\nare acquired by the uniform motion of a point from A to C, in such\nmanner, that in equal times the distances acquired be equal. But the\nperimeters B C D and E F G are also determined by the same distances of\nthe points C and E from the centre A; and therefore the perimeters B C D\nand E F G, as well as the semidiameters A C and A E, have their\nmagnitudes determined by the same cause, which cause makes, in equal\ntimes, equal spaces. Wherefore, by the 13th chapter and 6th article, the\nperimeters of circles and their semidiameters are proportionals; which\nwas to be proved.\n\n[Sidenote: In triangles strait lines parallel to the bases are to one\n another, as the parts of the sides which they cut off from\n the vertex.]\n\n14. If two strait lines, which constitute an angle, be cut by\nstrait-lined parallels, the intercepted parallels will be to one\nanother, as the parts which they cut off from the vertex. Let the strait\nlines A B and A C, in the 6th figure, make an angle at A, and be cut by\nthe two strait-lined parallels B C and D E, so that the parts cut off\nfrom the vertex in either of those lines, as in A B, may be A B and A D.\nI say, the parallels B C and D E are to one another, as the parts A B\nand A D. For let A B be divided into any number of equal parts, as into\nA F, F D, D B; and by the points F and D, let F G and D E be drawn\nparallel to the base B C, and cut A C in G and E; and again, by the\npoints G and E, let other strait lines be drawn parallel to A B, and cut\nB C in H and I. If now the point A be understood to be moved uniformly\nover A B, and in the same time B be moved to C, and all the points F, D,\nand B be moved uniformly and with equal swiftness over F G, D E, and B\nC; then shall B pass over B H, equal to F G, in the same time that A\npasses over A F; and A F and F G will be to one another, as their\nvelocities are; and when A is in F, D will be in K; when A is in D, D\nwill be in E; and in what manner the point A passes by the points F, D,\nand B, in the same manner the point B will pass by the points H, I, and\nC; and the strait lines F G, D K, K E, B H, H I, and I C, are equal, by\nreason of their parallelism; and therefore, as the velocity in A B is to\nthe velocity in B C, so is A D to D E; but as the velocity in A B is to\nthe velocity in B C, so is A B to B C; that is to say, all the parallels\nwill be severally to all the parts cut off from the vertex, as A F is to\nF G. Wherefore, A F. G F :: A D. D E :: A B. B C are proportionals.\n\nThe subtenses of equal angles in different circles, as the strait lines\nB C and F E (in fig. 1), are to one another as the arches which they\nsubtend. For (by art. 8) the arches of equal angles are to one another\nas their perimeters are; and (by art. 13) the perimeters as their\nsemidiameters; but the subtenses B C and F E are parallel to one another\nby reason of the equality of the angles which they make with the\nsemidiameters; and therefore the same subtenses, by the last precedent\narticle, will be proportional to the semidiameters, that is, to the\nperimeters, that is, to the arches which they subtend.\n\n[Sidenote: By what fraction of a strait line the circumference of a\n circle is made.]\n\n15. If in a circle any number of equal subtenses be placed immediately\nafter one another, and strait lines be drawn from the extreme point of\nthe first subtense to the extreme points of all the rest, the first\nsubtense being produced will make with the second subtense an external\nangle double to that, which is made by the same first subtense, and a\ntangent to the circle touching it in the extreme points thereof; and if\na strait line which subtends two of those arches be produced, it will\nmake an external angle with the third subtense, triple to the angle\nwhich is made by the tangent with the first subtense; and so\ncontinually. For with the radius A B (in fig. 7) let a circle be\ndescribed, and in it let any number of equal subtenses, B C, C D, and D\nE, be placed; also let B D and B E be drawn; and by producing B C, B D\nand B E to any distance in G, H and I, let them make angles with the\nsubtenses which succeed one another, namely, the external angles G C D,\nand H D E. Lastly, let the tangent K B be drawn, making with the first\nsubtense the angle K B C. I say the angle G C D is double to the angle K\nB C, and the angle H D E triple to the same angle K B C. For if A C be\ndrawn cutting B D in M, and from the point C there be drawn L C\nperpendicular to the same A C, then C L and M D will be parallel, by\nreason of the right angles at C and M; and therefore the alterne angles\nL C D and B D C will be equal: as also the angles B D C and C B D will\nbe equal, because of the equality of the strait lines B C and C D.\nWherefore the angle G C D is double to either of the angles C B D or C D\nB; and therefore also the angle G C D is double to the angle L C D, that\nis, to the angle K B C. Again, C D is parallel to B E, by reason of the\nequality of the angles C B E and D E B, and of the strait lines C B and\nD E; and therefore the angles G C D and G B E are equal; and\nconsequently G B E, as also D E B is double to the angle K B C. But the\nexternal angle H D E is equal to the two internal D E B and D B E; and\ntherefore the angle H D E is triple to the angle K B C, &c.; which was\nto be proved.\n\nCoroll. I. From hence it is manifest, that the angles K B C and C B D,\nas also, that all the angles that are comprehended by two strait lines\nmeeting in the circumference of a circle and insisting upon equal\narches, are equal to one another.\n\nCoroll. II. If the tangent B K be moved in the circumference with\nuniform motion about the centre B, it will in equal times cut off equal\narches; and will pass over the whole perimeter in the same time in which\nitself describes a semiperimeter about the centre B.\n\nCoroll. III. From hence also we may understand, what it is that\ndetermines the bending or curvation of a strait line into the\ncircumference of a circle; namely, that it is fraction continually\nincreasing in the same manner, as numbers, from one upwards, increase by\nthe continual addition of unity. For the indefinite strait line K B\nbeing broken in B according to any angle, as that of K B C, and again in\nC according to a double angle, and in D according to an angle which is\ntriple, and in E according to an angle which is quadruple to the first\nangle, and so continually, there will be described a figure which will\nindeed be rectilineal, if the broken parts be considered as having\nmagnitude; but if they be understood to be the least that can be, that\nis, as so many points, then the figure described will not be\nrectilineal, but a circle, whose circumference will be the broken line.\n\nCoroll. IV. From what has been said in this present article, it may also\nbe demonstrated, that an angle in the centre is double to an angle in\nthe circumference of the same circle, if the intercepted arches be\nequal. For seeing that strait line, by whose motion an angle is\ndetermined, passes over equal arches in equal times, as well from the\ncentre as from the circumference; and while that, which is from the\ncircumference, is passing over half its own perimeter, it passes in the\nsame time over the whole perimeter of that which is from the centre, the\narches, which it cuts off in the perimeter whose centre is A, will be\ndouble to those, which it makes in its own semiperimeter, whose centre\nis B. But in equal circles, as arches are to one another, so also are\nangles.\n\nIt may also be demonstrated, that the external angle made by a subtense\nproduced and the next equal subtense is equal to an angle from the\ncentre insisting upon the same arch; as in the last diagram, the angle G\nC D is equal to the angle C A D; for the external angle G C D is double\nto the angle C B D; and the angle C A D insisting upon the same arch C D\nis also double to the same angle C B D or K B C.\n\n[Sidenote: That an angle of contingence is quantity, but of a different\n kind from that of an angle simply so called; and that it can\n neither add nor take away anything from the same.]\n\n16. An angle of contingence, if it be compared with an angle simply so\ncalled, how little soever, has such proportion to it as a point has to a\nline; that is, no proportion at all, nor any quantity. For first, an\nangle of contingence is made by continual flexion; so that in the\ngeneration of it there is no circular motion at all, in which consists\nthe nature of an angle simply so called; and therefore it cannot be\ncompared with it according to quantity. Secondly, seeing the external\nangle made by a subtense produced and the next subtense is equal to an\nangle from the centre insisting upon the same arch, as in the last\nfigure the angle G C D is equal to the angle C A D, the angle of\ncontingence will be equal to that angle from the centre, which is made\nby A B and the same A B; for no part of a tangent can subtend any arch;\nbut as the point of contact is to be taken for the subtense, so the\nangle of contingence is to be accounted for the external angle, and\nequal to that angle whose arch is the same point B.\n\nNow, seeing an angle in general is defined to be the opening or\ndivergence of two lines, which concur in one sole point; and seeing one\nopening is greater than another, it cannot be denied, but that by the\nvery generation of it, an angle of contingence is quantity; for\nwheresoever there is greater and less, there is also quantity; but this\nquantity consists in greater and less flexion; for how much the greater\na circle is, so much the nearer comes the circumference of it to the\nnature of a strait line; for the circumference of a circle being made by\nthe curvation of a strait line, the less that strait line is, the\ngreater is the curvation; and therefore, when one strait line is a\ntangent to many circles, the angle of contingence, which it makes with a\nless circle, is greater than that which it makes with a greater circle.\n\nNothing therefore is added to or taken from an angle simply so called,\nby the addition to it or taking from it of never so many angles of\ncontingence. And as an angle of one sort can never be equal to an angle\nof the other sort, so they cannot be either greater or less than one\nanother.\n\nFrom whence it follows, that an angle of a segment, that is, the angle,\nwhich any strait line makes with any arch, is equal to the angle which\nis made by the same strait line, and another which touches the circle in\nthe point of their concurrence; as in the last figure, the angle which\nis made between G B and B K is equal to that which is made between G B\nand the arch B C.\n\n[Sidenote: That the inclination of planes is angle simply so called.]\n\n17. An angle, which is made by two planes, is commonly called the\ninclination of those planes; and because planes have equal inclination\nin all their parts, instead of their inclination an angle is taken,\nwhich is made by two strait lines, one of which is in one, the other in\nthe other of those planes, but both perpendicular to the common section.\n\n[Sidenote: A solid angle what it is.]\n\n18. A solid angle may be conceived two ways. First, for the aggregate of\nall the angles, which are made by the motion of a strait line, while one\nextreme point thereof remaining fixed, it is carried about any plain\nfigure, in which the fixed point of the strait line is not contained.\nAnd in this sense, it seems to be understood by Euclid. Now it is\nmanifest, that the quantity of a solid angle so conceived is no other,\nthan the aggregate of all the angles in a superficies so described, that\nis, in the superficies of a pyramidal solid. Secondly, when a pyramis or\ncone has its vertex in the centre of a sphere, a solid angle may be\nunderstood to be the proportion of a spherical superficies subtending\nthat vertex to the whole superficies of the sphere. In which sense,\nsolid angles are to one another as the spherical bases of solids, which\nhave their vertex in the centre of the same sphere.\n\n[Sidenote: What is the nature of asymptotes.]\n\n19. All the ways, by which two lines respect one another, or all the\nvariety of their position, may be comprehended under four kinds; for any\ntwo lines whatsoever are either parallels, or being produced, if need\nbe, or moved one of them to the other parallelly to itself, they make an\nangle; or else, by the like production and motion, they touch one\nanother; or lastly, they are _asymptotes_. The nature of parallels,\nangles, and tangents, has been already declared. It remains that I speak\nbriefly of the nature of asymptotes.\n\nAsymptosy depends upon this, that quantity is infinitely divisible. And\nfrom hence it follows, that any line being given, and a body supposed to\nbe moved from one extreme thereof towards the other, it is possible, by\ntaking degrees of velocity always less and less, in such proportion as\nthe parts of the line are made less by continual division, that the same\nbody may be always moved forwards in that line, and yet never reach the\nend of it. For it is manifest, that if any strait line, as A F, (in the\n8th figure) be cut anywhere in B, and again B F be cut in C, and C F in\nD, and D F in E, and so eternally, and there be drawn from the point F,\nthe strait line F F at any angle A F F; and lastly, if the strait lines\nA F, B F, C F, D F, E F, &c., having the same proportion to one another\nwith the segments of the line A F, be set in order and parallel to the\nsame A F, the crooked line A B C D E, and the strait line F F, will be\n_asymptotes_, that is, they will always come nearer and nearer together,\nbut never touch one another. Now, because any line may be cut eternally\naccording to the proportions which the segments have to one another,\ntherefore the divers kinds of asymptotes are infinite in number, and not\nnecessary to be further spoken of in this place. In the nature of\nasymptotes in general there is no more, than that they come still nearer\nand nearer, but never touch. But in special in the asymptosy of\nhyperbolic lines, it is understood they should approach to a distance\nless than any given quantity.\n\n[Sidenote: Situation, by what it is determined.]\n\n20. SITUATION _is the relation of one place to another_; and where there\nare many places, their situation is determined by four things; by _their\ndistances from one another_; by _several distances from a place\nassigned_; by _the order of strait lines drawn from a place assigned to\nthe places of them all_; and by _the angles which are made by the lines\nso drawn_. For if their distances, order, and angles, be given, that is,\nbe certainly known, their several places will also be so certainly\nknown, as that they can be no other.\n\n[Sidenote: What is like situation; what is figure; and what are like\n figures.]\n\n21. Points, how many soever they be, have _like situation_ with an equal\nnumber of other points, when all the strait lines, that are drawn from\nsome one point to all these, have severally the same proportion to\nthose, that are drawn in the same order and at equal angles from some\none point to all those. For let there be any number of points as A, B,\nand C, (in the 9th figure) to which from some one point D let the strait\nlines D A, D B, and D C be drawn; and let there be an equal number of\nother points, as E, F, and G, and from some point H let the strait lines\nH E, H F, and H G be drawn, so that the angles A D B and B D C be\nseverally and in the same order equal to the angles E H F and F H G, and\nthe strait lines D A, D B, and D C proportional to the strait lines H E,\nH F, and H G; I say, the three points A, B, and C, have like situation\nwith the three points E, F, and G, or are placed alike. For if H E be\nunderstood to be laid upon D A, so that the point H be in D, the point F\nwill be in the strait line D B, by reason of the equality of the angles\nA D B and E H F; and the point G will be in the strait line D C, by\nreason of the equality of the angles B D C and F H G; and the strait\nlines A B and E F, as also B C and F G, will be parallel, because A D. E\nH :: B D. F H :: C D. G H are proportionals by construction; and\ntherefore the distances between the points A and B, and the points B and\nC, will be proportional to the distances between the points E and F, and\nthe points F and G. Wherefore, in the situation of the points A, B, and\nC, and the situation of the points E, F and G, the angles in the same\norder are equal; so that their situations differ in nothing but the\ninequality of their distances from one another, and of their distances\nfrom the points D and H. Now, in both the orders of points, those\ninequalities are equal; for A B. B C :: E F. F G, which are their\ndistances from one another, as also D A. D B. D C :: H E. H F. H G,\nwhich are their distances from the assumed points D and H, are\nproportionals. Their difference, therefore, consists solely in the\nmagnitude of their distances. But, by the definition of _like_, (chapter\nI. article 2) those things, which differ only in magnitude, are like.\nWherefore the points A, B, and C, have to one another like situation\nwith the points E, F, and G, or are placed alike; which was to be\nproved.\n\nFIGURE _is quantity, determined by the situation or placing of all its\nextreme points_. Now I call those points extreme, which are contiguous\nto the place which is without the figure. In lines therefore and\nsuperficies, all points may be called extreme; but in solids only those\nwhich are in the superficies that includes them.\n\n_Like figures_ are those, whose extreme points in one of them are all\nplaced like all the extreme points in the other; for such figures differ\nin nothing but magnitude.\n\nAnd like figures are _alike placed_, when in both of them the homologal\nstrait lines, that is, the strait lines which connect the points which\nanswer one another, are parallel, and have their proportional sides\ninclined the same way.\n\nAnd seeing every strait line is like every other strait line, and every\nplane like every other plane, when nothing but planeness is considered;\nif the lines, which include planes, or the superficies, which include\nsolids, have their proportions known, it will not be hard to know\nwhether any figure be like or unlike to another propounded figure.\n\nAnd thus much concerning the _first grounds of philosophy_. The next\nplace belongs to _geometry_; in which the quantities of figures are\nsought out from the proportions of lines and angles. Wherefore it is\nnecessary for him, that would study geometry, to know first what is the\nnature of quantity, proportion, angle and figure. Having therefore\nexplained these in the three last chapters, I thought fit to add them to\nthis part; and so pass to the next.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C. XIV.\n _Fig. 1-10_\n]\n\n------------------------------------------------------------------------\n\n\n\n\n PART III\n\n PROPORTIONS OF MOTIONS\n AND MAGNITUDES.\n\n\n -------\n\n\n CHAPTER XV.\n\n OF THE NATURE, PROPERTIES, AND DIVERS\n CONSIDERATIONS OF MOTION AND\n ENDEAVOUR.\n\n1.. Repetition of some principles of the doctrine of motion formerly set\n down.\u20142. Other principles added to them.\u20143. Certain theorems\n concerning the nature of motion.\u20144. Divers considerations of\n motion.\u20145. The way by which the first endeavour of bodies moved\n tendeth.\u20146. In motion which is made by concourse, one of the movents\n ceasing, the endeavour is made by the way by which the rest tend.\u20147.\n All endeavour is propagated in infinitum.\u20148. How much greater the\n velocity or magnitude is of a movent, so much the greater is the\n efficacy thereof upon any other body in its way.\n\n\n[Sidenote: Repetition of some principles of the doctrine of motion\n formerly set down.]\n\n1. The next things in order to be treated of are MOTION and MAGNITUDE,\nwhich are the most common accidents of all bodies. This place therefore\nmost properly belongs to the elements of geometry. But because this part\nof philosophy, having been improved by the best wits of all ages, has\nafforded greater plenty of matter than can well be thrust together\nwithin the narrow limits of this discourse, I thought fit to admonish\nthe reader, that before he proceed further, he take into his hands the\nworks of Euclid, Archimedes, Apollonius, and other as well ancient as\nmodern writers. For to what end is it, to do over again that which is\nalready done? The little therefore that I shall say concerning geometry\nin some of the following chapters, shall be such only as is new, and\nconducing to natural philosophy.\n\nI have already delivered some of the principles of this doctrine in the\neighth and ninth chapters; which I shall briefly put together here, that\nthe reader in going on may have their light nearer at hand.\n\nFirst, therefore, in chap. VIII. art. 10, _motion_ is defined to be _the\ncontinual privation of one place, and acquisition of another_.\n\nSecondly, it is there shown, that _whatsoever is moved is moved in\ntime_.\n\nThirdly, in the same chapter, art. 11, I have defined _rest to be when a\nbody remains for some time in one place_.\n\nFourthly, it is there shown, that _whatsoever is moved is not in any\ndetermined place_; as also that the same _has been moved, is still\nmoved, and will yet be moved_; so that in every part of that space, in\nwhich motion is made, we may consider three times, namely, the _past_,\nthe _present_, and _future time_.\n\nFifthly, in art. 15 of the same chapter, I have defined _velocity_ or\n_swiftness to be motion considered as power, namely, that power by which\na body moved may in a certain time transmit a certain length_; which\nalso may more briefly be enunciated thus, _velocity is the quantity of\nmotion determined by time and line_.\n\nSixthly, in the same chapter, art. 16, I have shown that _motion is the\nmeasure of time_.\n\nSeventhly, in the same chapter, art. 17, I have defined motions to be\nequally swift, when in equal times equal lengths are transmitted by\nthem.\n\nEighthly, in art. 18 of the same chapter, _motions_ are defined to be\n_equal, when the swiftness of one moved body, computed in every part of\nits magnitude, is equal to the swiftness of another, computed also in\nevery part of its magnitude_. From whence it is to be noted, that\n_motions equal to one another, and motions equally swift_, do not\nsignify the same thing; for when two horses draw abreast, the motion of\nboth is greater than the motion of either of them singly; but the\nswiftness of both together is but equal to that of either.\n\nNinthly, in art. 19 of the same chapter, I have shown, that _whatsoever\nis at rest will always be at rest, unless there be some other body\nbesides it, which by getting into its place suffers it no longer to\nremain at rest_. And that _whatsoever is moved, will always be moved,\nunless there be some other body besides it, which hinders its motion_.\n\nTenthly, in chap. IX. art. 7, I have demonstrated, that _when any body\nis moved which was formerly at rest, the immediate efficient cause of\nthat motion is in some other moved and contiguous body_.\n\nEleventhly, I have shown in the same place, that _whatsoever is moved,\nwill always be moved in the same way, and with the same swiftness, if it\nbe not hindered by some other moved and contiguous body_.\n\n[Sidenote: Other principles\n added to them.]\n\n2. To which principles I shall here add those that follow. First, I\ndefine ENDEAVOUR _to be motion made in less space and time than can be\ngiven_; that is, _less than can be determined or assigned by exposition\nor number_; that is, _motion made through the length of a point, and in\nan instant or point of time_. For the explaining of which definition it\nmust be remembered, that by a point is not to be understood that which\nhas no quantity, or which cannot by any means be divided; for there is\nno such thing in nature; but that, whose quantity is not at all\nconsidered, that is, whereof neither quantity nor any part is computed\nin demonstration; so that a point is not to be taken for an indivisible,\nbut for an undivided thing; as also an instant is to be taken for an\nundivided, and not for an indivisible time.\n\nIn like manner, endeavour is to be conceived as motion; but so as that\nneither the quantity of the time in which, nor of the line in which it\nis made, may in demonstration be at all brought into comparison with the\nquantity of that time, or of that line of which it is a part. And yet,\nas a point may be compared with a point, so one endeavour may be\ncompared with another endeavour, and one may be found to be greater or\nless than another. For if the vertical points of two angles be compared,\nthey will be equal or unequal in the same proportion which the angles\nthemselves have to one another. Or if a strait line cut many\ncircumferences of concentric circles, the inequality of the points of\nintersection will be in the same proportion which the perimeters have to\none another. And in the same manner, if two motions begin and end both\ntogether, their endeavours will be equal or unequal, according to the\nproportion of their velocities; as we see a bullet of lead descend with\ngreater endeavour than a ball of wool.\n\nSecondly, I define IMPETUS, _or quickness of motion, to be the swiftness\nor velocity of the body moved, but considered in the several points of\nthat time in which it is moved. In which sense_ impetus _is nothing else\nbut the quantity or velocity of endeavour. But considered with the whole\ntime, it is the whole velocity of the body moved taken together\nthroughout all the time, and equal to the product of a line representing\nthe time, multiplied into a line representing the arithmetically mean_\nimpetus _or quickness_. Which arithmetical mean, what it is, is defined\nin the 29th article of chapter XIII.\n\nAnd because in equal times the ways that are passed are as the\nvelocities, and the _impetus_ is the velocity they go withal, reckoned\nin all the several points of the times, it followeth that during any\ntime whatsoever, howsoever the _impetus_ be increased or decreased, the\nlength of the way passed over shall be increased or decreased in the\nsame proportion; and the same line shall represent both the way of the\nbody moved, and the several _impetus_ or degrees of swiftness wherewith\nthe way is passed over.\n\nAnd if the body moved be not a point, but a strait line moved so as that\nevery point thereof make a several strait line, the plane described by\nits motion, whether uniform, accelerated, or retarded, shall be greater\nor less, the time being the same, in the same proportion with that of\nthe _impetus_ reckoned in one motion to the _impetus_ reckoned in the\nother. For the reason is the same in parallelograms and their sides.\n\nFor the same cause also, if the body moved be a plane, the solid\ndescribed shall be still greater or less in the proportions of the\nseveral _impetus_ or quicknesses reckoned through one line, to the\nseveral _impetus_ reckoned through another.\n\nThis understood, let A B C D, (in figure 1, chap. XVII.) be a\nparallelogram; in which suppose the side A B to be moved parallelly to\nthe opposite side C D, decreasing all the way till it vanish in the\npoint C, and so describing the figure A B E F C; the point B, as A B\ndecreaseth, will therefore describe the line B E F C; and suppose the\ntime of this motion designed by the line C D; and in the same time C D,\nsuppose the side A C to be moved parallel and uniformly to B D. From the\npoint O taken at adventure in the line C D, draw O R parallel to B D,\ncutting the line B E F C in E, and the side A B in R. And again, from\nthe point Q taken also at adventure in the line C D, draw Q S parallel\nto B D, cutting the line B E F C in F, and the side A B in S; and draw E\nG and F H parallel to C D, cutting A C in G and H. Lastly, suppose the\nsame construction done in all the points possible of the line B E F C. I\nsay, that as the proportions of the swiftness wherewith Q F, O E, D B,\nand all the rest supposed to be drawn parallel to D B and terminated in\nthe line B E F C, are to the proportions of their several times designed\nby the several parallels H F, G E, A B, and all the rest supposed to be\ndrawn parallel to the line of time C D and terminated in the line B E F\nC, the aggregate to the aggregate, so is the area or plane D B E F C to\nthe area or plane A C F E B. For as A B decreasing continually by the\nline B E F C vanisheth in the time C D into the point C, so in the same\ntime the line D C continually decreasing vanisheth by the same line C F\nE B into the point B; and the point D describeth in that decreasing\nmotion the line D B equal to the line A C described by the point A in\nthe decreasing motion of A B; and their swiftnesses are therefore equal.\nAgain, because in the time G E the point O describeth the line OE, and\nin the same time the point S describeth the line S E, the line O E shall\nbe to the line SE, as the swiftness wherewith OE is described to the\nswiftness wherewith SE is described. In like manner, because in the same\ntime H F the point Q describeth the line Q F, and the point R the line R\nF, it shall be as the swiftness by which Q F is described to the\nswiftness by which R F is described, so the line itself Q F to the line\nitself R F; and so in all the lines that can possibly be drawn parallel\nto B D in the points where they cut the line B E F C. But all the\nparallels to B D, as S E, R F, A C, and the rest that can possibly be\ndrawn from the line A B to the line B E F C, make the area of the plane\nA B E F C; and all the parallels to the same B D, as Q F, O E, D B and\nthe rest drawn to the points where they cut the same line B E F C, make\nthe area of the plane B E F C D. As therefore the aggregate of the\nswiftnesses wherewith the plane B E F C D is described, is to the\naggregate of the swiftnesses wherewith the plane A C F E B is described,\nso is the plane itself B E F C D to the plane itself A C F E B. But the\naggregate of the times represented by the parallels A B, G E, H F and\nthe rest, maketh also the area A C F E B. And therefore, as the\naggregate of all the lines Q F, O E, D B and all the rest of the lines\nparallel to B D and terminated in the line B E F C, is to the aggregate\nof all the lines H F, G E, A B and all the rest of the lines parallel to\nC D and terminated in the same line B E F C; that is, as the aggregate\nof the lines of swiftness to the aggregate of the lines of time, or as\nthe whole swiftness in the parallels to D B to the whole time in the\nparallels to C D, so is the plane B E F C D to the plane A C F E B. And\nthe proportions of Q F to F H, and of O E to E G, and of D B to B A, and\nso of all the rest taken together, are the proportions of the plane D B\nE F C to the plane A B E F C. But the lines Q F, O E, D B and the rest\nare the lines that design the swiftness; and the lines H F, G E, A B and\nthe rest are the lines that design the times of the motions; and\ntherefore the proportion of the plane D B E F C to the plane A B E F C\nis the proportion of all the velocities taken together to all the times\ntaken together. Wherefore, as the proportions of the swiftnesses, &c.;\nwhich was to be demonstrated.\n\nThe same holds also in the diminution of the circles, whereof the lines\nof time are the semidiameters, as may easily be conceived by imagining\nthe whole plane A B C D turned round upon the axis B D; for the line B E\nF C will be everywhere in the superficies so made, and the lines H F, G\nE, A B, which are here parallelograms, will be there cylinders, the\ndiameters of whose bases are the lines H F, G E, A B, &c. and the\naltitude a point, that is to say, a quantity less than any quantity that\ncan possibly be named; and the lines Q F, O E, D B, &c. small solids\nwhose lengths and breadths are less than any quantity that can be named.\n\nBut this is to be noted, that unless the proportion of the sum of the\nswiftnesses to the proportion of the sum of the times be determined, the\nproportion of the figure D B E F C to the figure A B E F C cannot be\ndetermined.\n\nThirdly, I define RESISTANCE _to be the endeavour of one moved body\neither wholly or in part contrary to the endeavour of another moved\nbody, which toucheth the same_. I say, wholly contrary, when the\nendeavour of two bodies proceeds in the same strait line from the\nopposite extremes, and contrary in part, when two bodies have their\nendeavour in two lines, which, proceeding from the extreme points of a\nstrait line, meet without the same.\n\nFourthly, that I may define what it is to PRESS, I say, that _of two\nmoved bodies one presses the other, when with its endeavour it makes\neither all or part of the other body to go out of its place_.\n\nFifthly, _a body, which is pressed and not wholly removed, is said to_\nRESTORE _itself, when, the pressing body being taken away, the parts\nwhich were moved do, by reason of the internal constitution of the\npressed body, return every one into its own place_. And this we may\nobserve in springs, in blown bladders, and in many other bodies, whose\nparts yield more or less to the endeavour which the pressing body makes\nat the first arrival; but afterwards, when the pressing body is removed,\nthey do, by some force within them, _restore_ themselves, and give their\nwhole body the same figure it had before.\n\nSixthly, I define FORCE _to be the_ impetus _or quickness of motion\nmultiplied either into itself, or into the magnitude of the movent, by\nmeans whereof the said movent works more or less upon the body that\nresists it_.\n\n[Sidenote: Certain theorems concerning the nature of motion.]\n\n3. Having premised thus much, I shall now demonstrate, first, that if a\npoint moved come to touch another point which is at rest, how little\nsoever the impetus or quickness of its motion be, it shall move that\nother point. For if by that impetus it do not at all move it out of its\nplace, neither shall it move it with double the same impetus. For\nnothing doubled is still nothing; and for the same reason it shall never\nmove it with that impetus, how many times soever it be multiplied,\nbecause nothing, however it be multiplied, will for ever be nothing.\nWherefore, when a point is at rest, if it do not yield to the least\nimpetus, it will yield to none; and consequently it will be impossible\nthat that, which is at rest, should ever be moved.\n\nSecondly, that when a point moved, how little soever the impetus thereof\nbe, falls upon a point of any body at rest, how hard soever that body\nbe, it will at the first touch make it yield a little. For if it do not\nyield to the impetus which is in that point, neither will it yield to\nthe impetus of never so many points, which have all their impetus\nseverally equal to the impetus of that point. For seeing all those\npoints together work equally, if any one of them have no effect, the\naggregate of them all together shall have no effect as many times told\nas there are points in the whole body, that is, still no effect at all;\nand by consequent there would be some bodies so hard that it would be\nimpossible to break them; that is, a finite hardness, or a finite force,\nwould not yield to that which is infinite; which is absurd.\n\nCoroll. It is therefore manifest, that rest does nothing at all, nor is\nof any efficacy; and that nothing but motion gives motion to such things\nas be at rest, and takes it from things moved.\n\nThirdly, that cessation in the movent does not cause cessation in that\nwhich was moved by it. For (by number 11 of art. 1 of this chapter)\nwhatsoever is moved perseveres in the same way and with the same\nswiftness, as long as it is not hindered by something that is moved\nagainst it. Now it is manifest, that cessation is not contrary motion;\nand therefore it follows that the standing still of the movent does not\nmake it necessary that the thing moved should also stand still.\n\nCoroll. They are therefore deceived, that reckon the taking away of the\nimpediment or resistance for one of the causes of motion.\n\n[Sidenote: Divers considerations of motions.]\n\n4. Motion is brought into account for divers respects; first, as in a\nbody _undivided_, that is, considered as a point; or, as in a _divided_\nbody. In an undivided body, when we suppose the way, by which the motion\nis made, to be a line; and in a divided body, when we compute the motion\nof the several parts of that body, as of parts.\n\nSecondly, from the diversity of the regulation of motion, it is in body,\nconsidered as undivided. sometimes _uniform_ and sometimes _multiform_.\n_Uniform_ is that by which equal lines are always transmitted in equal\ntimes; and _multiform_, when in one time more, in another time less\nspace is transmitted. Again, of multiform motions, there are some in\nwhich the degrees of acceleration and retardation proceed in the same\nproportions, which the spaces transmitted have, whether duplicate, or\ntriplicate, or by whatsoever number multiplied; and others in which it\nis otherwise.\n\nThirdly, from the number of the movents; that is, one motion is made by\none movent only, and another by the concourse of many movents.\n\nFourthly, from the position of that line in which a body is moved, in\nrespect of some other line; and from hence one motion is called\n_perpendicular_, another _oblique_, another _parallel_.\n\nFifthly, from the position of the movent in respect of the moved body;\nfrom whence one motion is _pulsion_ or driving, another _traction_ or\ndrawing. _Pulsion_, when the movent makes the moved body go before it;\nand _traction_, when it makes it follow. Again, there are two sorts of\n_pulsion_; one, when the motions of the movent and moved body begin both\ntogether, which may be called _trusion_ or _thrusting_ and _vection_;\nthe other, when the movent is first moved, and afterwards the moved\nbody, which motion is called _percussion_ or _stroke_.\n\nSixthly, motion is considered sometimes from the effect only which the\nmovent works in the moved body, which is usually called _moment_. Now\n_moment is the excess of motion which the movent has above the motion or\nendeavour of the resisting body_.\n\nSeventhly, it may be considered from the diversity of the _medium_; as\none motion may be made in _vacuity_ or _empty place_; another in a\n_fluid_; another in a _consistent medium_, that is, a _medium_ whose\nparts are by some power so _consistent_ and _cohering_, that no part of\nthe same will yield to the movent, unless the whole yield also.\n\nEighthly, when a moved body is considered as having parts, there arises\nanother distinction of motion into _simple_ and _compound_. _Simple_,\nwhen all the several parts describe several equal lines; _compounded_,\nwhen the lines described are unequal.\n\n[Sidenote: The way by which the first endeavour of bodies moved\n tendeth.]\n\n5. All endeavour tends towards that part, that is to say, in that way\nwhich is determined by the motion of the movent, if the movent be but\none; or, if there be many movents, in that way which their concourse\ndetermines. For example, if a moved body have direct motion, its first\nendeavour will be in a strait line; if it have circular motion, its\nfirst endeavour will be in the circumference of a circle.\n\n[Sidenote: In motion, which it made by concourse, one of the movents\n ceasing, the endeavour is made by the way by which the rest\n tend.]\n\n6. And whatsoever the line be, in which a body has its motion from the\nconcourse of two movents, as soon as in any point thereof the force of\none of the movents ceases, there immediately the former endeavour of\nthat body will be changed into an endeavour in the line of the other\nmovent.\n\nWherefore, when any body is carried on by the concourse of two winds,\none of those winds ceasing, the endeavour and motion of that body will\nbe in that line, in which it would have been carried by that wind alone\nwhich blows still. And in the describing of a circle, where that which\nis moved has its motion determined by a movent in a tangent, and by the\nradius which keeps it in a certain distance from the centre, if the\nretention of the radius cease, that endeavour, which was in the\ncircumference of the circle, will now be in the tangent, that is, in a\nstrait line. For, seeing endeavour is computed in a less part of the\ncircumference than can be given, that is, in a point, the way by which a\nbody is moved in the circumference is compounded of innumerable strait\nlines, of which every one is less than can be given; which are therefore\ncalled points. Wherefore when any body, which is moved in the\ncircumference of a circle, is freed from the retention of the radius, it\nwill proceed in one of those strait lines, that is, in a tangent.\n\n[Sidenote: All endeavour is propagated in infinitum.]\n\n7. All endeavour, whether strong or weak, is propagated to infinite\ndistance; for it is motion. If therefore the first endeavour of a body\nbe made in space which is empty, it will always proceed with the same\nvelocity; for it cannot be supposed that it can receive any resistance\nat all from empty space; and therefore, (by art. 7, chap, IX) it will\nalways proceed in the same way and with the same swiftness. And if its\nendeavour be in space which is filled, yet, seeing endeavour is motion,\nthat which stands next in its way shall be removed, and endeavour\nfurther, and again remove that which stands next, and so infinitely.\nWherefore the propagation of endeavour, from one part of full space to\nanother, proceeds infinitely. Besides, it reaches in any instant to any\ndistance, how great soever. For in the same instant in which the first\npart of the full _medium_ removes that which is next it, the second also\nremoves that part which is next to it; and therefore all endeavour,\nwhether it be in empty or in full space, proceeds not only to any\ndistance, how great soever, but also in any time, how little soever,\nthat is, in an instant. Nor makes it any matter, that endeavour, by\nproceeding, grows weaker and weaker, till at last it can no longer be\nperceived by sense; for motion may be insensible; and I do not here\nexamine things by sense and experience, but by reason.\n\n[Sidenote: How much greater the velocity or magnitude is of a movent, so\n much the greater is the efficacy thereof upon any other body\n in its way.]\n\n8. When two movents are of equal magnitude, the swifter of them works\nwith greater force than the slower, upon a body that resists their\nmotion. Also, if two movents have equal velocity, the greater of them\nworks with more force than the less. For where the magnitude is equal,\nthe movent of greater velocity makes the greater impression upon that\nbody upon which it falls; and where the velocity is equal, the movent of\ngreater magnitude falling upon the same point, or an equal part of\nanother body, loses less of its velocity, because the resisting body\nworks only upon that part of the movent which it touches, and therefore\nabates the impetus of that part only; whereas in the mean time the\nparts, which are not touched, proceed, and retain their whole force,\ntill they also come to be touched; and their force has some effect.\nWherefore, for example, in batteries a longer than a shorter piece of\ntimber of the same thickness and velocity, and a thicker than a\nslenderer piece of the same length and velocity, work a greater effect\nupon the wall.\n\n\n -------\n\n\n CHAPTER XVI.\n\n OF MOTION ACCELERATED AND UNIFORM, AND\n OF MOTION BY CONCOURSE.\n\n1. The velocity of any body, in what time soever it be computed, is that\n which is made of the multiplication of the impetus, or quickness of\n its motion into the time.\u20142-5. In all motion, the lengths which are\n passed through are to one another, as the products made by the\n impetus multiplied into the time.\u20146. If two bodies be moved with\n uniform motion through two lengths, the proportion of those lengths\n to one another will be compounded of the proportions of time to\n time, and impetus to impetus, directly taken.\u20147. If two bodies pass\n through two lengths with uniform motion, the proportion of their\n times to one another will be compounded of the proportions of length\n to length, and impetus to impetus reciprocally taken; also the\n proportion of their impetus to one another will be compounded of the\n proportions of length to length, and time to time reciprocally\n taken.\u20148. If a body be carried on with uniform motion by two movents\n together, which meet in an angle, the line by which it passes will\n be a strait line, subtending the complement of that angle to two\n right angles.\u20149, &c. If a body be carried by two movents together,\n one of them being moved with uniform, the other with accelerated\n motion, and the proportion of their lengths to their times being\n explicable in numbers, how to find out what line that body\n describes.\n\n\n[Sidenote: The velocity of any body, in what time soever it be computed,\n is that which is made of the multiplication of the impetus or\n quickness of its motion into the time.]\n\n1. The velocity of any body, in whatsoever time it be moved, has its\nquantity determined by the sum of all the several quicknesses or\nimpetus, which it hath in the several points of the time of the body's\nmotion. For seeing velocity, (by the definition of it, chap, VIII, art.\n15) is that power by which a body can in a certain time pass through a\ncertain length; and quickness of motion or impetus, (by chap. XV, art.\n2, num. 2) is velocity taken in one point of time only, all the impetus,\ntogether taken in all the points of time, will be the same thing with\nthe mean impetus multiplied into the whole time, or which is all one,\nwill be the velocity of the whole motion.\n\nCoroll. If the impetus be the same in every point, any strait line\nrepresenting it may be taken for the measure of time: and the\nquicknesses or impetus applied ordinately to any strait line making an\nangle with it, and representing the way of the body's motion, will\ndesign a parallelogram which shall represent the velocity of the whole\nmotion. But if the impetus or quickness of motion begin from rest and\nincrease uniformly, that is, in the same proportion continually with the\ntimes which are passed, the whole velocity of the motion shall be\nrepresented by a triangle, one side whereof is the whole time, and the\nother the greatest impetus acquired in that time; or else by a\nparallelogram, one of whose sides is the whole time of motion, and the\nother, half the greatest impetus; or lastly, by a parallelogram having\nfor one side a mean proportional between the whole time and the half of\nthat time, and for the other side the half of the greatest impetus. For\nboth these parallelograms are equal to one another, and severally equal\nto the triangle which is made of the whole line of time, and of the\ngreatest acquired impetus; as is demonstrated in the elements of\ngeometry.\n\n[Sidenote: In all motion, the lengths which are passed through are to\n one another, as the products made by the impetus multiplied\n into time.]\n\n2. In all uniform motions the lengths which are transmitted are to one\nanother, as the product of the mean impetus multiplied into its time, to\nthe product of the mean impetus multiplied also into its time.\n\nFor let A B (in fig. 1) be the time, and A C the impetus by which any\nbody passes with uniform motion through the length D E; and in any part\nof the time A B, as in the time A F, let another body be moved with\nuniform motion, first, with the same impetus A C. This body, therefore,\nin the time A F with the impetus A C will pass through the length A F.\nSeeing, therefore, when bodies are moved in the same time, and with the\nsame velocity and impetus in every part of their motion, the proportion\nof one length transmitted to another length transmitted, is the same\nwith that of time to time, it followeth, that the length transmitted in\nthe time A B with the impetus A C will be to the length transmitted in\nthe time A F with the same impetus A C, as A B itself is to A F, that\nis, as the parallelogram A I is to the parallelogram A H, that is, as\nthe product of the time A B into the mean impetus A C is to the product\nof the time A F into the same impetus A C. Again, let it be supposed\nthat a body be moved in the time A F, not with the same but with some\nother uniform impetus, as A L. Seeing therefore, one of the bodies has\nin all the parts of its motion the impetus A C, and the other in like\nmanner the impetus A L, the length transmitted by the body moved with\nthe impetus A C will be to the length transmitted by the body moved with\nthe impetus A L, as A C itself is to A L, that is, as the parallelogram\nA H is to the parallelogram F L. Wherefore, by ordinate proportion it\nwill be, as the parallelogram A I to the parallelogram F L, that is, as\nthe product of the mean impetus into the time is to the product of the\nmean impetus into the time, so the length transmitted in the time A B\nwith the impetus A C, to the length transmitted in the time A F with the\nimpetus A L; which was to be demonstrated.\n\nCoroll. Seeing, therefore, in uniform motion, as has been shown, the\nlengths transmitted are to one another as the parallelograms which are\nmade by the multiplication of the mean impetus into the times, that is,\nby reason of the equality of the impetus all the way, as the times\nthemselves, it will also be, by permutation, as time to length, so time\nto length; and in general, to this place are applicable all the\nproperties and transmutations of analogisms, which I have set down and\ndemonstrated in chapter XIII.\n\n3. In motion begun from rest and uniformly accelerated, that is, where\nthe impetus increaseth continually according to the proportion of the\ntimes, it will also be, as one product made by the mean impetus\nmultiplied into the time, to another product made likewise by the mean\nimpetus multiplied into the time, so the length transmitted in the one\ntime to the length transmitted in the other time.\n\nFor let A B (in fig. 1) represent a time; in the beginning of which time\nA, let the impetus be as the point A; but as the time goes on, so let\nthe impetus increase uniformly, till in the last point of that time A B,\nnamely in B, the impetus acquired be B I. Again, let A F represent\nanother time, in whose beginning A, let the impetus be as the point\nitself A; but as the time proceeds, so let the impetus increase\nuniformly, till in the last point F of the time A F the impetus acquired\nbe F K; and let D E be the length passed through in the time A B with\nimpetus uniformly increased. I say, the length D E is to the length\ntransmitted in the time A F, as the time A B multiplied into the mean of\nthe impetus increasing through the time A B, is to the time A F\nmultiplied into the mean of the impetus increasing through the time A F.\n\nFor seeing the triangle A B I is the whole velocity of the body moved in\nthe time A B, till the impetus acquired be B I; and the triangle A F K\nthe whole velocity of the body moved in the time A F with impetus\nincreasing till there be acquired the impetus F K; the length D E to the\nlength acquired in the time A F with impetus increasing from rest in A\ntill there be acquired the impetus F K, will be as the triangle A B I to\nthe triangle A F K, that is, if the triangles A B I and A F K be like,\nin duplicate proportion of the time A B to the time A F; but if unlike,\nin the proportion compounded of the proportions of A B to A F and of B I\nto F K. Wherefore, as A B I is to A F K, so let D E be to D P; for so,\nthe length transmitted in the time A B with impetus increasing to B I,\nwill be to the length transmitted in the time A F with impetus\nincreasing to F K, as the triangle A B I is to the triangle A F K; but\nthe triangle A B I is made by the multiplication of the time A B into\nthe mean of the impetus increasing to B I; and the triangle A F K is\nmade by the multiplication of the time A F into the mean of the\n_impetus_ increasing to F K; and therefore the length D E which is\ntransmitted in the time A B with impetus increasing to B I, to the\nlength D P which is transmitted in the time A F with impetus increasing\nto F K, is as the product which is made of the time A B multiplied into\nits mean impetus, to the product of the time A F multiplied also into\nits mean impetus; which was to be proved.\n\nCoroll. I. In motion uniformly accelerated, the proportion of the\nlengths transmitted to that of their times, is compounded of the\nproportions of their times to their times, and impetus to impetus.\n\nCoroll. II. In motion uniformly accelerated, the lengths transmitted in\nequal times, taken in continual succession from the beginning of motion,\nare as the differences of square numbers beginning from unity, namely,\nas 3, 5, 7, &c. For if in the first time the length transmitted be as 1,\nin the first and second times the length transmitted will be as 4, which\nis the square of 2, and in the three first times it will be as 9, which\nis the square of 3, and in the four first times as 16, and so on. Now\nthe differences of these squares are 3, 5, 7, &c.\n\nCoroll. III. In motion uniformly accelerated from rest, the length\ntransmitted is to another length transmitted uniformly in the same time,\nbut with such impetus as was acquired by the accelerated motion in the\nlast point of that time, as a triangle to a parallelogram, which have\ntheir altitude and base common. For seeing the length D E (in fig. 1) is\npassed through with velocity as the triangle A B I, it is necessary that\nfor the passing through of a length which is double to D E, the velocity\nbe as the parallelogram A I; for the parallelogram A I is double to the\ntriangle A B I.\n\n4. In motion, which beginning from rest is so aclerated, that the\nimpetus thereof increases continually in proportion duplicate to the\nproportion of the times in which it is made, a length transmitted in one\ntime will be to a length transmitted in another time, as the product\nmade by the mean impetus multiplied into the time of one of those\nmotions, to the product of the mean impetus multiplied into the time of\nthe other motion.\n\nFor let A B (in fig. 2) represent a time, in whose first instant A let\nthe impetus be as the point A; but as the time proceeds, so let the\nimpetus increase continually in duplicate proportion to that of the\ntimes, till in the last point of time B the impetus acquired be B I;\nthen taking the point F anywhere in the time A B, let the impetus F K\nacquired in the time A F be ordinately applied to that point F. Seeing\ntherefore the proportion of F K to B I is supposed to be duplicate to\nthat of A F to A B, the proportion of A F to A B will be subduplicate to\nthat of F K to B I; and that of A B to A F will be (by chap. XIII. art.\n16) duplicate to that of B I to F K; and consequently the point K will\nbe in a parabolical line, whose diameter is A B and base B I; and for\nthe same reason, to what point soever of the time A B the impetus\nacquired in that time be ordinately applied, the strait line designing\nthat impetus will be in the same parabolical line A K I. Wherefore the\nmean impetus multiplied into the whole time A B will be the parabola A K\nI B, equal to the parallelogram A M, which parallelogram has for one\nside the line of time A B and for the other the line of the impetus A L,\nwhich is two-thirds of the impetus B I; for every parabola is equal to\ntwo-thirds of that parallelogram with which it has its altitude and base\ncommon. Wherefore the whole velocity in A B will be the parallelogram A\nM, as being made by the multiplication of the impetus A L into the time\nA B. And in like manner, if F N be taken, which is two-thirds of the\nimpetus F K, and the parallelogram F O be completed, F O will be the\nwhole velocity in the time A F, as being made by the uniform impetus A O\nor F N multiplied into the time A F. Let now the length transmitted in\nthe time A B and with the velocity A M be the strait line D E; and\nlastly, let the length transmitted in the time A F with the velocity A N\nbe D P; I say that as A M is to A N, or as the parabola A K I B to the\nparabola A K F, so is D E to D P. For as A M is to F L, that is, as A B\nis to A F, so let D E be to D G. Now the proportion of A M to A N is\ncompounded of the proportions of A M to F L, and of F L to A N. But as A\nM to F L, so by construction is D E to D G; and as F L is to A N (seeing\nthe time in both is the same, namely, A F), so is the length D G to the\nlength D P; for lengths transmitted in the same time are to one another\nas their velocities are. Wherefore by ordinate proportion, as A M is to\nA N, that is, as the mean impetus A L multiplied into its time A B, is\nto the mean impetus A O multiplied into A F, so is D E to D P; which was\nto be proved.\n\nCoroll. I. Lengths transmitted with motion so accelerated, that the\nimpetus increase continually in duplicate proportion to that of their\ntimes, if the base represent the impetus, are in triplicate proportion\nof their impetus acquired in the last point of their times. For as the\nlength D E is to the length D P, so is the parallelogram A M to the\nparallelogram A N, and so the parabola A K I B to the parabola A K F.\nBut the proportion of the parabola A K I B to the parabola A K F is\ntriplicate to the proportion which the base B I has to the base F K.\nWherefore also the proportion of D E to D P is triplicate to that of B I\nto F K.\n\nCoroll. II. Lengths transmitted in equal times succeeding one another\nfrom the beginning, by motion so accelerated, that the proportion of the\nimpetus be duplicate to the proportion of the times, are to one another\nas the differences of cubic numbers beginning at unity, that is as 7,\n19, 37, &c. For if in the first time the length transmitted be as 1, the\nlength at the end of the second time will be as 8, at the end of the\nthird time as 27, and at the end of the fourth time as 64, &c.; which\nare cubic numbers, whose differences are 7, 19, 37, &c.\n\nCoroll. III. In motion so accelerated, as that the length transmitted be\nalways to the length transmitted in duplicate proportion to their times,\nthe length uniformly transmitted in the whole time, and with impetus all\nthe way equal to that which is last acquired, is as a parabola to a\nparallelogram of the same altitude and base, that is, as 2 to 3. For the\nparabola A K I B is the impetus increasing in the time A B; and the\nparallelogram A I is the greatest uniform impetus multiplied into the\nsame time A B. Wherefore the lengths transmitted will be as a parabola\nto a parallelogram, &c., that is, as 2 to 3.\n\n5. If I should proceed to the explication of such motions as are made by\nimpetus increasing in proportion triplicate, quadruplicate,\nquintuplicate, &c., to that of their times, it would be a labour\ninfinite and unnecessary. For by the same method by which I have\ncomputed such lengths, as are transmitted with impetus increasing in\nsingle and duplicate proportion, any man may compute such as are\ntransmitted with impetus increasing in triplicate, quadruplicate, or\nwhat other proportion he pleases.\n\nIn making which computation he shall find, that where the impetus\nincrease in proportion triplicate to that of the times, there the whole\nvelocity will be designed by the first parabolaster (of which see the\nnext chapter); and the lengths transmitted will be in proportion\nquadruplicate to that of the times. And in like manner, where the\nimpetus increase in quadruplicate proportion to that of the times, that\nthere the whole velocity will be designed by the second parabolaster,\nand the lengths transmitted will be in quintuplicate proportion to that\nof the times; and so on continually.\n\n[Sidenote: If two bodies be moved with uniform motion through two\n lengths, the proportion of those lengths to one another, will\n be compounded of the proportions of time to time, and impetus\n to impetus, directly taken.]\n\n6. If two bodies with uniform motion transmit two lengths, each with its\nown impetus and time, the proportion of the lengths transmitted will be\ncompounded of the proportions of time to time, and impetus to impetus,\ndirectly taken.\n\nLet two bodies be moved uniformly (as in fig. 3), one in the time A B\nwith the impetus A C, the other in the time A D with the impetus A E. I\nsay the lengths transmitted have their proportion to one another\ncompounded of the proportions of A B to A D, and of A C to A E. For let\nany length whatsoever, as Z, be transmitted by one of the bodies in the\ntime A B with the impetus A C; and any other length, as X, be\ntransmitted by the other body in the time A D with the impetus A E; and\nlet the parallelograms A F and A G be completed. Seeing now Z is to X\n(by art. 2) as the impetus A C multiplied into the time A B is to the\nimpetus A E multiplied into the time A D, that is, as A F to A G; the\nproportion of Z to X will be compounded of the same proportions, of\nwhich the proportion of A F to A G is compounded; but the proportion of\nA F to A G is compounded of the proportions of the side A B to the side\nA D, and of the side A C to the side A E (as is evident by the Elements\nof Euclid), that is, of the proportions of the time A B to the time A D,\nand of the impetus A C to the impetus A E. Wherefore also the proportion\nof Z to X is compounded of the same proportions of the time A B to the\ntime A D, and of the impetus A C to the impetus A E; which was to be\ndemonstrated.\n\nCoroll. I. When two bodies are moved with uniform motion, if the times\nand impetus be in reciprocal proportion, the lengths transmitted shall\nbe equal. For if it were as A B to A D (in the same fig. 3) so\nreciprocally A E to A C, the proportion of A F to A G would be\ncompounded of the proportions of A B to A D, and of A C to A E, that is,\nof the proportions of A B to A D, and of A D to A B. Wherefore, A F\nwould be to A G as A B to A B, that is, equal; and so the two products\nmade by the multiplication of impetus into time would be equal; and by\nconsequent, Z would be equal to X.\n\nCoroll. II. If two bodies be moved in the same time, but with different\nimpetus, the lengths transmitted will be as impetus to impetus. For if\nthe time of both of them be A D, and their different impetus be A E and\nA C, the proportion of A G to D C will be compounded of the proportions\nof A E to A C and of A D to A D, that is, of the proportions of A E to A\nC and of A C to A C; and so the proportion of A G to D C, that is, the\nproportion of length to length, will be as A E to A C, that is, as that\nof impetus to impetus. In like manner, if two bodies be moved uniformly,\nand both of them with the same impetus, but in different times, the\nproportion of the lengths transmitted by them will be as that of their\ntimes. For if they have both the same impetus A C, and their different\ntimes be A B and A D, the proportion of A F to D C will be compounded of\nthe proportions of A B to A D and of A C to A C; that is, of the\nproportions of A B to A D and of A D to A D; and therefore the\nproportion of A F to D C, that is, of length to length, will be the same\nwith that of A B to A D, which is the proportion of time to time.\n\n[Sidenote: If two bodies pass through two lengths with uniform motion,\n the proportion of their times to one another, will be\n compounded of the proportions of length to length, and\n impetus to impetus reciprocally taken; also the proportion of\n their impetus to one another, will be compounded of the\n proportions of length to length, and time to time\n reciprocally taken.]\n\n7. If two bodies pass through two lengths with uniform motion, the\nproportion of the times in which they are moved will be compounded of\nthe proportions of length to length and impetus to impetus reciprocally\ntaken.\n\nFor let any two lengths be given, as (in the same fig. 3) Z and X, and\nlet one of them be transmitted with the impetus A C, the other with the\nimpetus A E. I say the proportion of the times in which they are\ntransmitted, will be compounded of the proportions of Z to X, and of A\nE, which is the impetus with which X is transmitted, to A C, the impetus\nwith which Z is transmitted. For seeing A F is the product of the\nimpetus A C multiplied into the time A B, the time of motion through Z\nwill be a line, which is made by the application of the parallelogram A\nF to the strait line A C, which line is A B; and therefore A B is the\ntime of motion through Z. In like manner, seeing A G is the product of\nthe impetus A E multiplied into the time A D, the time of motion through\nX will be a line which is made by the application of A G to the strait\nline A D; but A D is the time of motion through X. Now the proportion of\nA B to A D is compounded of the proportions of the parallelogram A F to\nthe parallelogram A G, and of the impetus A E to the impetus A C; which\nmay be demonstrated thus. Put the parallelograms in order A F, A G, D C,\nand it will be manifest that the proportion of A F to D C is compounded\nof the proportions of A F to A G and of A G to D C; but A F is to D C as\nA B to A D; wherefore also the proportion of A B to A D is compounded of\nthe proportions of A F to A G and of A G to D C. And because the length\nZ is to the length X as A F is to A G, and the impetus A E to the\nimpetus A C as A G to D C, therefore the proportion of A B to A D will\nbe compounded of the proportions of the length Z to the length X, and of\nthe impetus A E to the impetus A C; which was to be demonstrated.\n\nIn the same manner it may be proved, that in two uniform motions the\nproportion of the impetus is compounded of the proportions of length to\nlength and of time to time reciprocally taken.\n\nFor if we suppose A C (in the same fig. 3) to be the time, and A B the\nimpetus with which the length Z is passed through; and A E to be the\ntime, and A D the impetus with which the length X is passed through, the\ndemonstration will proceed as in the last article.\n\n[Sidenote: If a body be carried on with uniform motion by two movents\n together, which meet in an angle, the line by which it passes\n will be a strait line, subtending the complement of that\n angle to 2 right angles.]\n\n8. If a body be carried by two movents together, which move with strait\nand uniform motion, and concur in any given angle, the line by which\nthat body passes will be a strait line.\n\nLet the movent A B (in fig. 4) have strait and uniform motion, and be\nmoved till it come into the place C D; and let another movent A C,\nhaving likewise strait and uniform motion, and making with the movent A\nB any given angle C A B, be understood to be moved in the same time to D\nB; and let the body be placed in the point of their concourse, A. I say\nthe line which that body describes with its motion is a strait line. For\nlet the parallelogram A B D C be completed, and its diagonal A D be\ndrawn; and in the strait line A B let any point E be taken; and from it\nlet E F be drawn parallel to the strait lines A C and B D, cutting A D\nin G; and through the point G let H I be drawn parallel to the strait\nlines A B and C D; and lastly, let the measure of the time be A C.\nSeeing therefore both the motions are made in the same time, when A B is\nin C D, the body also will be in C D; and in like manner, when A C is in\nB D, the body will be in B D. But A B is in C D at the same time when A\nC is in B D; and therefore the body will be in C D and B D at the same\ntime; wherefore it will be in the common point D. Again, seeing the\nmotion from A C to B D is uniform, that is, the spaces transmitted by it\nare in proportion to one another as the times in which they are\ntransmitted, when A C is in E F, the proportion of A B to A E will be\nthe same with that of E F to E G, that is, of the time A C to the time A\nH. Wherefore A B will be in H I in the same time in which A C is in E F,\nso that the body will at the same time be in E F and H I, and therefore\nin their common point G. And in the same manner it will be, wheresoever\nthe point E be taken between A and B. Wherefore the body will always be\nin the diagonal A D; which was to be demonstrated.\n\nCoroll. From hence it is manifest, that the body will be carried through\nthe same strait line A D, though the motion be not uniform, provided it\nhave like acceleration; for the proportion of A B to A E will always be\nthe same with that of A C to A H.\n\n[Sidenote: If a body be carried by two movents together, one of them\n being moved with uniform, the other with accelerated motion,\n and the proportion of their lengths to their times being\n explicable in numbers, how to find out what line that body\n describes.]\n\n9. If a body be carried by two movents together, which meet in any given\nangle, and are moved, the one uniformly, the other with motion uniformly\naccelerated from rest, that is, that the proportion of their impetus be\nas that of their times, that is, that the proportion of their lengths be\nduplicate to that of the lines of their times, till the line of greatest\nimpetus acquired by acceleration be equal to that of the line of time of\nthe uniform motion; the line in which the body is carried will be the\ncrooked line of a semiparabola, whose base is the impetus last acquired,\nand vertex the point of rest.\n\nLet the straight line A B (in fig. 5) be understood to be moved with\nuniform motion to C D; and let another movent in the strait line A C be\nsupposed to be moved in the same time to B D, but with motion uniformly\naccelerated, that is, with such motion, that the proportion of the\nspaces which are transmitted be always duplicate to that of the times,\ntill the impetus acquired be B D equal to the strait line A C; and let\nthe semiparabola A G D B be described. I say that by the concourse of\nthose two movents, the body will be carried through the semiparabolical\ncrooked line A G D. For let the parallelogram A B D C be completed; and\nfrom the point E, taken anywhere in the strait line A B, let E F be\ndrawn parallel to A C and cutting the crooked line in G; and lastly,\nthrough the point G let H I be drawn parallel to the strait lines A B\nand C D. Seeing therefore the proportion of A B to A E is by supposition\nduplicate to the proportion of E F to E G, that is, of the time A C to\nthe time A H, at the same time when A C is in E F, A B will be in H I;\nand therefore the moved body will be in the common point G. And so it\nwill always be, in what part soever of A B the point E be taken.\nWherefore the moved body will always be found in the parabolical line A\nG D; which was to be demonstrated.\n\n10. If a body be carried by two movents together, which meet in any\ngiven angle, and are moved the one uniformly, the other with impetus\nincreasing from rest, till it be equal to that of the uniform motion,\nand with such acceleration, that the proportion of the lengths\ntransmitted be every where triplicate to that of the times in which they\nare transmitted; the line, in which that body is moved, will be the\ncrooked line of the first semiparabolaster of two means, whose base is\nthe impetus last acquired.\n\nLet the strait line A B (in the 6th figure) be moved uniformly to C D;\nand let another movent A C be moved at the same time to B D with motion\nso accelerated, that the proportion of the lengths transmitted be\neverywhere triplicate to the proportion of their times; and let the\nimpetus acquired in the end of that motion be B D, equal to the strait\nline A C; and lastly, let A G D be the crooked line of the first\nsemiparabolaster of two means. I say, that by the concourse of the two\nmovents together, the body will be always in that crooked line A G D.\nFor let the parallelogram A B D C be completed; and from the point E,\ntaken anywhere in the strait line A B, let E F be drawn parallel to A C,\nand cutting the crooked line in G; and through the point G let H I be\ndrawn parallel to the strait lines A B and C D. Seeing therefore the\nproportion of A B to A E is, by supposition, triplicate to the\nproportion of E F to E G, that is, of the time A C to the time A H, at\nthe same time when A C is in E F, A B will be in H I; and therefore the\nmoved body will be in the common point G. And so it will always be, in\nwhat part soever of A B the point E be taken; and by consequent, the\nbody will always be in the crooked line A G D; which was to be\ndemonstrated.\n\n11. By the same method it may be shown, what line it is that is made by\nthe motion of a body carried by the concourse of any two movents, which\nare moved one of them uniformly, the other with acceleration, but in\nsuch proportions of spaces and times as are explicable by numbers, as\n_duplicate_, _triplicate_, &c., or such as may be designed by any broken\nnumber whatsoever. For which this is the rule. Let the two numbers of\nthe length and time be added together; and let their sum be the\ndenominator of a fraction, whose numerator must be the number of the\nlength. Seek this fraction in the table of the third article of the\nXVIIth chapter; and the line sought will be that, which denominates the\nthree-sided figure noted on the left hand; and the kind of it will be\nthat, which is numbered above over the fraction. For example, let there\nbe a concourse of two movents, whereof one is moved uniformly, the other\nwith motion so accelerated, that the spaces are to the times as 5 to 3.\nLet a fraction be made whose denominator is the sum of 5 and 3, and the\nnumerator 5, namely the fraction 5\u20448. Seek in the table, and you will\nfind 5\u20448 to be the third in that row, which belongs to the three-sided\nfigure of four means. Wherefore the line of motion made by the concourse\nof two such movents, as are last of all described, will be the crooked\nline of the third parabolaster of four means.\n\n12. If motion be made by the concourse of two movents, whereof one is\nmoved uniformly, the other beginning from rest in the angle of concourse\nwith any acceleration whatsoever; the movent, which is moved uniformly,\nshall put forward the moved body in the several parallel spaces, less\nthan if both the movents had uniform motion; and still less and less, as\nthe motion of the other movent is more and more accelerated.\n\nLet the body be placed in A, (in the 7th figure) and be moved by two\nmovents, by one with uniform motion from the strait line A B to the\nstrait line C D parallel to it; and by the other with any acceleration,\nfrom the strait line A C to the strait line B D parallel to it; and in\nthe parallelogram A B D C let a space be taken between any two parallels\nE F and G H. I say, that whilst the movent A C passes through the\nlatitude which is between E F and G H, the body is less moved forwards\nfrom A B towards C D, than it would have been, if the motion from A C to\nB D had been uniform.\n\nFor suppose that whilst the body is made to descend to the parallel E F\nby the power of the movent from A C towards B D, the same body in the\nsame time is moved forwards to any point F in the line E F, by the power\nof the movent from A B towards C D; and let the strait line A F be drawn\nand produced indeterminately, cutting G H in H. Seeing therefore, it is\nas A E to A G, so E F to G H; if A C should descend towards B D with\nuniform motion, the body in the time G H, (for I make A C and its\nparallels the measure of time,) would be found in the point H. But\nbecause A C is supposed to be moved towards B D with motion continually\naccelerated, that is, in greater proportion of space to space, than of\ntime to time, in the time G H the body will be in some parallel beyond\nit, as between G H and B D. Suppose now that in the end of the time G H\nit be in the parallel I K, and in I K let I L be taken equal to G H.\nWhen therefore the body is in the parallel I K, it will be in the point\nL. Wherefore when it was in the parallel G H, it was in some point\nbetween G and H, as in the point M; but if both the motions had been\nuniform, it had been in the point H; and therefore whilst the movent A C\npasses over the latitude which is between E F and G H, the body is less\nmoved forwards from A B towards C D, than it would have been, if both\nthe motions had been uniform; which was to be demonstrated.\n\n13. Any length being given, which is passed through in a given time with\nuniform motion, to find out what length shall be passed through in the\nsame time with motion uniformly accelerated, that is, with such motion\nthat the proportion of the lengths passed through be continually\nduplicate to that of their times, and that the line of the impetus last\nacquired be equal to the line of the whole time of the motion.\n\nLet A B (in the 8th figure) be a length, transmitted with uniform motion\nin the time A C; and let it be required to find another length, which\nshall be transmitted in the same time with motion uniformly accelerated,\nso that the line of the impetus last acquired be equal to the strait\nline A C.\n\nLet the parallelogram A B D C be completed; and let B D be divided in\nthe middle at E; and between B E and B D let B F be a mean proportional;\nand let A F be drawn and produced till it meet with C D produced in G;\nand lastly, let the parallelogram A C G H be completed. I say, A H is\nthe length required.\n\nFor as duplicate proportion is to single proportion, so let A H be to A\nI, that is, let A I be the half of A H; and let I K be drawn parallel to\nthe strait line A C, and cutting the diagonal A D in K, and the strait\nline A G in L. Seeing therefore A I is the half of A H, I L will also be\nthe half of B D, that is, equal to B E; and I K equal to B F; for B D,\nthat is, G H, B F, and B E, that is, I L, being continual proportionals,\nA H, A B and A I will also be continual proportionals. But as A B is to\nA I, that is, as A H is to A B, so is B D to I K, and so also is G H,\nthat is, B D to B F; and therefore B F and I K are equal. Now the\nproportion of A H to A I is duplicate to the proportion of A B to A I,\nthat is, to that of B D to I K, or of G H to I K. Wherefore the point K\nwill be in a parabola, whose diameter is A H, and base G H, which G H is\nequal to A C. The body therefore proceeding from rest in A, with motion\nuniformly accelerated in the time A C, when it has passed through the\nlength A H, will acquire the impetus G H equal to the time A C, that is,\nsuch impetus, as that with it the body will pass through the length A C\nin the time A C. Wherefore any length being given, &c., which was\npropounded to be done.\n\n14. Any length being given, which in a given time is transmitted with\nuniform motion, to find out what length shall be transmitted in the same\ntime with motion so accelerated, that the lengths transmitted be\ncontinually in triplicate proportion to that of their times, and the\nline of the impetus last of all acquired be equal to the line of time\ngiven.\n\nLet the given length A B (in the 9th figure) be transmitted with uniform\nmotion in the time A C; and let it be required to find what length shall\nbe transmitted in the same time with motion so accelerated, that the\nlengths transmitted be continually in triplicate proportion to that of\ntheir times, and the impetus last acquired be equal to the time given.\n\nLet the parallelogram A B D C be completed; and let B D be so divided in\nE, that B E be a third part of the whole B D; and let B F be a mean\nproportional between B D and B E; and let A F be drawn and produced till\nit meet the strait line C D in G; and lastly, let the parallelogram A C\nG H be completed. I say, A H is the length required.\n\nFor as triplicate proportion is to single proportion, so let A H be to\nanother line, A I, that is, make A I a third part of the whole A H; and\nlet I K be drawn parallel to the strait line A C, cutting the diagonal A\nD in K, and the strait line A G in L; then, as A B is to A I, so let A I\nbe to another, A N; and from the point N let N Q be drawn parallel to A\nC, cutting A G, A D, and F K produced in P, M, and O; and last of all,\nlet F O and L M be drawn, which will be equal and parallel to the strait\nlines B N and I N. By this construction, the lengths transmitted A H, A\nB, A I, and A N, will be continual proportionals; and, in like manner,\nthe times G H, B F, I L and N P, that is, N Q, N O, N M and N P, will be\ncontinual proportionals, and in the same proportion with A H, A B, A I\nand A N. Wherefore the proportion of A H, A B, A I and A N. Wherefore\nthe proportion of A H to A N is the same with that of B D, that is, of N\nQ to N P; and the proportion of N Q to N P triplicate to that of N Q to\nN O, that is, triplicate to that of B D to I K; wherefore also the\nlength A H is to the length A N in triplicate proportion to that of the\ntime B D, to the time I K; and therefore the crooked line of the first\nthree-sided figure of two means whose diameter is A H, and base G H\nequal to A C, shall pass through the point O; and consequently, A H\nshall be transmitted in the time A C, and shall have its last acquired\nimpetus G H equal to A C, and the proportions of the lengths acquired in\nany of the times triplicate to the proportions of the times themselves.\nWherefore A H is the length required to be found out.\n\nBy the same method, if a length be given which is transmitted with\nuniform motion in any given time, another length may be found out which\nshall be transmitted in the same time with motion so accelerated, that\nthe lengths transmitted shall be to the times in which they are\ntransmitted, in proportion quadruplicate, quintuplicate, and so on\ninfinitely. For if B D be divided in E, so that B D be to B E as 4 to 1;\nand there be taken between B D and B E a mean proportional F B; and as A\nH is to A B, so A B be made to a third, and again so that third to a\nfourth, and that fourth to a fifth, A N, so that the proportion of A H\nto A N be quadruplicate to that of A H to A B, and the parallelogram N B\nF O be completed, the crooked line of the first three-sided figure of\nthree means will pass through the point O; and consequently, the body\nmoved will acquire the impetus G H equal to A C in the time A C. And so\nof the rest.\n\n15. Also, if the proportion of the lengths transmitted be to that of\ntheir times, as any number to any number, the same method serves for the\nfinding out of the length transmitted with such impetus, and in such\ntime.\n\nFor let A C (in the 10th figure) be the time in which a body is\ntransmitted with uniform motion from A to B; and the parallelogram A B D\nC being completed, let it be required to find out a length in which that\nbody may be moved in the same time A C from A, with motion so\naccelerated, that the proportion of the lengths transmitted to that of\nthe times be continually as 3 to 2.\n\nLet B D be so divided in E, that B D be to B E as 3 to 2; and between B\nD and B E let B F be a mean proportional; and let A F be drawn and\nproduced till it meet with C D produced in G; and making A M a mean\nproportional between A H and A B, let it be as A M to A B, so A B to A\nI; and so the proportion of A H to A I will be to that of A H to A B as\n3 to 2; for of the proportions, of which that of A H to A M is one, that\nof A H to A B is two, and that of A H to A I is three; and consequently,\nas 3 to 2 to that of G H to B F, and (F K being drawn parallel to B I\nand cutting A D in K) so likewise to that of G H or B D to I K.\nWherefore the proportion of the length A H to A I is to the proportion\nof the time B D to I K as 3 to 2; and therefore if in the time A C the\nbody be moved with accelerated motion, as was propounded, till it\nacquire the impetus H G equal to A C, the length transmitted in the same\ntime will be A H.\n\n16. But if the proportion of the lengths to that of the times had been\nas 4 to 3, there should then have been taken two mean proportionals\nbetween A H and A B, and their proportion should have been continued one\nterm further, so that A H to A B might have three of the same\nproportions, of which A H to A I has four; and all things else should\nhave been done as is already shown. Now the way how to interpose any\nnumber of means between two lines given, is not yet found out.\nNevertheless this may stand for a general rule; _if there be a time\ngiven, and a length be transmitted in that time with uniform motion; as\nfor example, if the time be_ A C, _and the length_ A B, _the strait\nline_ A G, _which determines the length_ C G _or_ A H, _transmitted in\nthe same time_ A C _with any accelerated motion, shall so cut_ B D _in_\nF, _that_ B F _shall be a mean proportional between_ B D _and_ B E, B E\n_being so taken in_ B D, _that the proportion of length to length be\neverywhere to the proportion of time to time, as the whole_ B D _is to\nits part_ B E.\n\n17. If in a given time two lengths be transmitted, one with uniform\nmotion, the other with motion accelerated in any proportion of the\nlengths to the times; and again, in part of the same time, parts of the\nsame lengths be transmitted with the same motions, the whole length will\nexceed the other length in the same proportion in which one part exceeds\nthe other part.\n\nFor example, let A B (in the 8th figure) be a length transmitted in the\ntime A C, with uniform motion; and let A H be another length transmitted\nin the same time with motion uniformly accelerated, so that the impetus\nlast acquired be G H equal to A C; and in A H let any part A I be taken,\nand transmitted in part of the time A C with uniform motion; and let\nanother part A B be taken and transmitted in the same part of the time A\nC with motion uniformly accelerated; I say, that as A H is to A B, so\nwill A B be to A I.\n\nLet B D be drawn parallel and equal to H G, and divided in the midst at\nE, and between B D and B E let a mean proportional be taken as B F; and\nthe strait line A G, by the demonstration of art. 13, shall pass through\nF. And dividing A H in the midst at I, A B shall be a mean proportional\nbetween A H and A I. Again, because A I and A B are described by the\nsame motions, if I K be drawn parallel and equal to B F or A M, and\ndivided in the midst at N, and between I K and I N be taken the mean\nproportional I L, the strait line A F will, by the demonstration of the\nsame art. 13, pass through L. And dividing A B in the midst at O, the\nline A I will be a mean proportional between A B and A O. Where A B is\ndivided in I and O, in like manner as A H is divided in B and I; and as\nA H to A B, so is A B to A I. Which was to be proved.\n\nCoroll. Also as A H to A B, so is H B to B I; and so also B I to I O.\n\nAnd as this, where one of the motions is uniformly accelerated, is\nproved out of the demonstration of art. 13; so, when the accelerations\nare in double proportion to the times, the same may be proved by the\ndemonstration of art. 14; and by the same method in all other\naccelerations, whose proportions to the times are explicable in numbers.\n\n18. If two sides, which contain an angle in any parallelogram, be moved\nin the same time to the sides opposite to them, one of them with uniform\nmotion, the other with motion uniformly accelerated; that side, which is\nmoved uniformly, will affect as much with its concourse through the\nwhole length transmitted, as it would do if the other motion were also\nuniform, and the length transmitted by it in the same time were a mean\nproportional between the whole and the half.\n\nLet the side A B of the parallelogram A B D C, (in the 11th figure) be\nunderstood to be moved with uniform motion till it be coincident with C\nD; and let the time of that motion be A C or B D. Also in the same time\nlet the side A C be understood to be moved with motion uniformly\naccelerated, till it be coincident with B D; then dividing A B in the\nmiddle in E, let A F be made a mean proportional between A B and A E;\nand drawing F G parallel to A C, let the side A C be understood to be\nmoved in the same time A C with uniform motion till it be coincident\nwith F G. I say, the whole A B confers as much to the velocity of the\nbody placed in A, when the motion of A C is uniformly accelerated till\nit comes to B D, as the part A F confers to the same, when the side A C\nis moved uniformly and in the same time to F G.\n\nFor seeing A F is a mean proportional between the whole A B and its half\nA E, B D will (by the 13th article) be the last impetus acquired by A C,\nwith motion uniformly accelerated till it come to the same B D; and\nconsequently, the strait line F B will be the excess, by which the\nlength, transmitted by A C with motion uniformly accelerated, will\nexceed the length transmitted by the same A C in the same time with\nuniform motion, and with impetus every where equal to B D. Wherefore, if\nthe whole A B be moved uniformly to C D in the same time in which A C is\nmoved uniformly to F G, the part F B, seeing it concurs not at all with\nthe motion of the side A C which is supposed to be moved only to F G,\nwill confer nothing to its motion. Again, supposing the side A C to be\nmoved to B D with motion uniformly accelerated, the side A B with its\nuniform motion to C D will less put forwards the body when it is\naccelerated in all the parallels, than when it is not at all\naccelerated; and by how much the greater the acceleration is, by so much\nthe less it will put it forwards, as is shown in the 12th article. When\ntherefore A C is in F G with accelerated motion, the body will not be in\nthe side C D at the point G, but at the point D; so that G D will be the\nexcess, by which the length transmitted with accelerated motion to B D\nexceeds the length transmitted with uniform motion to F G; so that the\nbody by its acceleration avoids the action of the part A F, and comes to\nthe side C D in the time A C, and makes the length C D, which is equal\nto the length A B. Wherefore uniform motion from A B to C D in the time\nA C, works no more in the whole length A B upon the body uniformly\naccelerated from A C to B D, than if A C were moved in the same time\nwith uniform motion to F G; the difference consisting only in this, that\nwhen A B works upon the body uniformly moved from A C to F G, that, by\nwhich the accelerated motion exceeds the uniform motion, is altogether\nin F B or G D; but when the same A B works upon the body accelerated,\nthat, by which the accelerated motion exceeds the uniform motion, is\ndispersed through the whole length A B or C D, yet, so that if it were\ncollected and put together, it would be equal to the same F B or G D.\nWherefore, if two sides which contain an angle, &c.; which was to be\ndemonstrated.\n\n19. If two transmitted lengths have to their times any other proportion\nexplicable by number, and the side A B be so divided in E, that A B be\nto A E in the same proportion which the lengths transmitted have to the\ntimes in which they are transmitted, and between A B and A E there be\ntaken a mean proportional A F; it may be shown by the same method, that\nthe side, which is moved with uniform motion, works as much with its\nconcourse through the whole length A B, as it would do if the other\nmotion were also uniform, and the length transmitted in the same time A\nC were that mean proportional A F.\n\nAnd thus much concerning motion by concourse.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C. XVI.\n _Fig. 1-11_\n]\n\n------------------------------------------------------------------------\n\n\n -------\n\n\n CHAP. XVII.\n\n OF FIGURES DEFICIENT.\n\n1. Definitions of a deficient figure; of a complete figure; of the\n complement of a deficient figure; and of proportions which are\n proportional and commensurable to one another.\u20142. The proportion of\n a deficient figure to its complement.\u20143. The proportions of\n deficient figures to the parallelograms in which they are described,\n set forth in a table.\u20144. The description and production of the same\n figures.\u20145. The drawing of tangents to them.\u20146. In what proportion\n the same figures exceed a strait-lined triangle of the same altitude\n and base.\u20147. A table of solid deficient figures described in a\n cylinder.\u20148. In what proportion the same figures exceed a cone of\n the same altitude and base.\u20149. How a plain deficient figure may be\n described in a parallelogram, so that it be to a triangle of the\n same base and altitude, as another deficient figure, plain or solid,\n twice taken, is to the same deficient figure, together with the\n complete figure in which it is described.\u201410. The transferring of\n certain properties of deficient figures described in a parallelogram\n to the proportions of the spaces transmitted with several degrees of\n velocity.\u201411. Of deficient figures described in a circle.\u201412. The\n proposition demonstrated in art. 2 confirmed from the elements of\n philosophy.\u201413. An unusual way of reasoning concerning the equality\n between the superficies of a portion of a sphere and a circle.\u201414.\n How from the description of deficient figures in a parallelogram,\n any number of mean proportionals may be found out between two given\n strait lines.\n\n\n[Sidenote: Definition of a deficient figure.]\n\n1. I call those _deficient figures_ which may be understood to be\ngenerated by the uniform motion of some quantity, which decreases\ncontinually, till at last it have no magnitude at all.\n\n[Sidenote: Definitions of a complete figure; of the complement of a\n deficient figure; and of proportions which are proportional &\n commensurable to one another.]\n\nAnd I call that a _complete figure_, answering to a deficient figure,\nwhich is generated with the same motion and in the same time, by a\nquantity which retains always its whole magnitude.\n\nThe _complement_ of a deficient figure is that which being added to the\ndeficient figure makes it complete.\n\nFour proportions are said to be _proportional_, when the first of them\nis to the second as the third is to the fourth. For example, if the\nfirst proportion be duplicate to the second, and again, the third be\nduplicate to the fourth, those proportions are said to be\n_proportional_.\n\nAnd _commensurable_ proportions are those, which are to one another as\nnumber to number. As when to a proportion given, one proportion is\nduplicate, another triplicate, the duplicate proportion will be to the\ntriplicate proportion as 2 to 3; but to the given proportion it will be\nas 2 to 1; and therefore I call those three proportions _commensurable_.\n\n[Sidenote: The proportion of a deficient figure to its complement.]\n\n2. A deficient figure, which is made by a quantity continually\ndecreasing to nothing by proportions everywhere proportional and\ncommensurable, is to its complement, as the proportion of the whole\naltitude to an altitude diminished in any time is to the proportion of\nthe whole quantity, which describes the figure, to the same quantity\ndiminished in the same time.\n\nLet the quantity A B (in fig. 1), by its motion through the altitude A\nC, describe the complete figure A D; and again, let the same quantity,\nby decreasing continually to nothing in C, describe the deficient figure\nA B E F C, whose complement will be the figure B D C F E. Now let A B be\nsupposed to be moved till it lie in G K, so that the altitude diminished\nbe G C, and A B diminished be G E; and let the proportion of the whole\naltitude A C to the diminished altitude G C, be, for example, triplicate\nto the proportion of the whole quantity A B or G K to the diminished\nquantity G E. And in like manner, let H I be taken equal to G E, and let\nit be diminished to H F; and let the proportion of G C to H C be\ntriplicate to that of H I to H F; and let the same be done in as many\nparts of the strait line A C as is possible; and a line be drawn through\nthe points B, E, F and C. I say the deficient figure A B E F C is to its\ncomplement B D C F E as 3 to 1, or as the proportion of A C to G C is to\nthe proportion of A B, that is, of G K to G E.\n\nFor (by art. 2, chapter XV.) the proportion of the complement B E F C D\nto the deficient figure A B E F C is all the proportions of D B to B A,\nO E to E G, Q F to F H, and of all the lines parallel to D B terminated\nin the line B E F C, to all the parallels to A B terminated in the same\npoints of the line B E F C. And seeing the proportions of D B to O E,\nand of D B to Q F &c. are everywhere triplicate of the proportions of A\nB to G E, and of A B to H F &c. the proportions of H F to A B, and of G\nE to A B &c. (by art. 16, chap. XIII.), are triplicate of the\nproportions of Q F to D B, and of O E to D B &c. and therefore the\ndeficient figure A B E F C, which is the aggregate of all the lines H F,\nG E, A B, &c. is triple to the complement B E F C D made of all the\nlines Q F, O E, D B, &c.; which was to be proved.\n\nIt follows from hence, that the same complement B E F C D is 1\u20444 of the\nwhole parallelogram. And by the same method may be calculated in all\nother deficient figures, generated as above declared, the proportion of\nthe parallelogram to either of its parts; as that when the parallels\nincrease from a point in the same proportion, the parallelogram will be\ndivided into two equal triangles; when one increase is double to the\nother, it will be divided into a semiparabola and its complement, or\ninto 2 and 1.\n\nThe same construction standing, the same conclusion may otherwise be\ndemonstrated thus.\n\nLet the strait line C B be drawn cutting G K in L, and through L let M N\nbe drawn parallel to the strait line A C; wherefore the parallelograms G\nM and L D will be equal. Then let L K be divided into three equal parts,\nso that it may be to one of those parts in the same proportion which the\nproportion of A C to G C, or of G K to G L, hath to the proportion of G\nK to G E. Therefore L K will be to one of those three parts as the\narithmetical proportion between G K and G L is to the arithmetical\nproportion between G K and the same G K wanting the third part of L K;\nand K E will be somewhat greater than a third of L K. Seeing now the\naltitude A G or M L is, by reason of the continual decrease, to be\nsupposed less than any quantity that can be given; L K, which is\nintercepted between the diagonal B C and the side B D, will be also less\nthan any quantity that can be given; and consequently, if G be put so\nnear to A in _g_, as that the difference between C _g_ and C A be less\nthan any quantity that can be assigned, the difference also between C\n_l_ (removing L to _l_) and C B, will be less than any quantity that can\nbe assigned; and the line _g l_ being drawn and produced to the line B D\nin _k_, cutting the crooked line in _e_, the proportion of G _k_ to G\n_l_ will still be triplicate to the proportion of G _k_ to G _e_, and\nthe difference between _k_ and _e_, the third part of _k l_, will be\nless than any quantity that can be given; and therefore the\nparallelogram _e_ D will differ from a third part of the parallelogram A\n_e_ by a less difference than any quantity that can be assigned. Again,\nlet H I be drawn parallel and equal to G E, cutting C B in P, the\ncrooked line in F, and O E in I, and the proportion of C _g_ to C H will\nbe triplicate to the proportion of H F to H P, and I F will be greater\nthan the third part of P I. But again, setting H in _h_ so near to _g_,\nas that the difference between C _h_ and C _g_ may be but as a point,\nthe point P will also in _p_ be so near to _l_, as that the difference\nbetween C _p_ and C _l_ will be but as a point; and drawing _h p_ till\nit meet with B D in _i_, cutting the crooked line in _f_ and having\ndrawn _e o_ parallel to B D, cutting D C in _o_, the parallelogram _f o_\nwill differ less from the third part of the parallelogram _g f_, than by\nany quantity that can be given. And so it will be in all other spaces\ngenerated in the same manner. Wherefore the differences of the\narithmetical and geometrical means, which are but as so many points B,\n_e_, _f_, &c. (seeing the whole figure is made up of so many indivisible\nspaces) will constitute a certain line, such as is the line B E F C,\nwhich will divide the complete figure A D into two parts, whereof one,\nnamely, A B E F C, which I call a deficient figure, is triple to the\nother, namely, B D C F E, which I call the complement thereof. And\nwhereas the proportion of the altitudes to one another is in this case\neverywhere triplicate to that of the decreasing quantities to one\nanother; in the same manner, if the proportion of the altitudes had been\neverywhere quadruplicate to that of the decreasing quantities, it might\nhave been demonstrated that the deficient figure had been quadruple to\nits complement; and so in any other proportion. Wherefore, a deficient\nfigure, which is made, &c. which was to be demonstrated.\n\nThe same rule holdeth also in the diminution of the bases of cylinders,\nas is demonstrated in the second article of chapter XV.\n\n[Sidenote: The proportion of deficient figures to the parallelograms in\n which they are described, set forth in a table.]\n\n3. By this proposition, the magnitudes of all deficient figures, when\nthe proportions by which their bases decrease continually are\nproportional to those by which their altitudes decrease, may be compared\nwith the magnitudes of their complements; and consequently, with the\nmagnitudes of their complete figures. And they will be found to be, as I\nhave set them down in the following tables; in which I compare a\nparallelogram with three-sided figures; and first, with a strait-lined\ntriangle, made by the base of the parallelogram continually decreasing\nin such manner, that the altitudes be always in proportion to one\nanother as the bases are, and so the triangle will be equal to its\ncomplement; or the proportions of the altitudes and bases will be as 1\nto 1, and then the triangle will be half the parallelogram. Secondly,\nwith that three-sided figure which is made by the continual decreasing\nof the bases in subduplicate proportion to that of the altitudes; and so\nthe deficient figure will be double to its complement, and to the\nparallelogram as 2 to 3. Then, with that where the proportion of the\naltitudes is triplicate to that of the bases; and then the deficient\nfigure will be triple to its complement, and to the parallelogram as 3\nto 4. Also the proportion of the altitudes to that of the bases may be\nas 3 to 2; and then the deficient figure will be to its complement as 3\nto 2, and to the parallelogram as 3 to 5; and so forwards, according as\nmore mean proportionals are taken, or as the proportions are more\nmultiplied, as may be seen in the following table. For example, if the\nbases decrease so, that the proportion of the altitudes to that of the\nbases be always as 5 to 2, and it be demanded what proportion the figure\nmade has to the parallelogram, which is supposed to be unity; then,\nseeing that where the proportion is taken five times, there must be four\nmeans; look in the table amongst the three-sided figures of four means,\nand seeing the proportion was as 5 to 2, look in the uppermost row for\nthe number 2, and descending in the second column till you meet with\nthat three-sided figure, you will find 5\u20447; which shows that the\ndeficient figure is to the parallelogram as 5\u20447 to 1, or as 5 to 7.\n\n | 1 | 2 3 4 5 6 7\nParallelogram | 1 | : : : : : :\n +----+\nStrait-sided triangle | 1/2| : : : : : :\n |----+\nThree-sided figure of 1 mean | 2/3| : : : : : :\n |----+----+\nThree-sided figure of 2 means | 3/4| 3/5| : : : : :\n |----+----+----+\nThree-sided figure of 3 means | 4/5| 4/6| 4/7| : : : :\n |----+----+----+----+\nThree-sided figure of 4 means | 5/6| 5/7| 5/8|5/9 | : : :\n |----+----+----+----+----+\nThree-sided figure of 5 means | 6/7| 6/8| 6/9|6/10|6/11| : :\n |----+----+----+----+----+----+\nThree-sided figure of 6 means | 7/8| 7/9|7/11|7/11|7/12|7/13| `` :\n |----+----+----+----+----+----+----+\nThree-sided figure of 7 means | 8/9|8/10|8/11|8/12|8/13|8/14|8/15|\n +----+----+----+----+----+----+----+\n\n[Sidenote: Description & production of the same figures.]\n\n4. Now for the better understanding of the nature of these three-sided\nfigures, I will show how they may be described by points; and first,\nthose which are in the first column of the table. Any parallelogram\nbeing described, as A B C D (in figure 2) let the diagonal B D be drawn;\nand the strait-lined triangle B C D will be half the parallelogram; then\nlet any number of lines, as E F, be drawn parallel to the side B C, and\ncutting the diagonal B D in G; and let it be everywhere, as E F to E G,\nso E G to another, E H; and through all the points H let the line B H H\nD be drawn; and the figure B H H D C will be that which I call a\nthree-sided figure of one mean, because in three proportionals, as E F,\nE G and E H, there is but one mean, namely, E G; and this three-sided\nfigure will be 2\u20443 of the parallelogram, and is called a _parabola_.\nAgain, let it be as E G to E H, so E H to another, E I, and let the line\nB I I D be drawn, making the three-sided figure B I I D C; and this will\nbe 3\u20444 of the parallelogram, and is by many called a _cubic parabola_.\nIn like manner, if the proportions be further continued in E F, there\nwill be made the rest of the three-sided figures of the first column;\nwhich I thus demonstrate. Let there be drawn strait lines, as H K and G\nL, parallel to the base D C. Seeing therefore the proportion of E F to E\nH is duplicate to that of E F to E G, or of B C to B L, that is, of C D\nto L G, or of K M (producing K H to A D in M) to K H, the proportion of\nB C to B K will be duplicate to that of K M to K H; but as B C is to B\nK, so is D C or K M to K N, and therefore the proportion of K M to K N\nis duplicate to that of K M to K H; and so it will be wheresoever the\nparallel K M be placed. Wherefore the figure B H H D C is double to its\ncomplement B H H D A, and consequently 2\u20443 of the whole parallelogram.\nIn the same manner, if through I be drawn O P I Q parallel and equal to\nC D, it may be demonstrated that the proportion of O Q to O P, that is,\nof B C to B O, is triplicate that of O Q to O I, and therefore that the\nfigure B I I D C is triple to its complement B I I D A, and consequently\n\u00be of the whole parallelogram, &c.\n\nSecondly, such three-sided figures as are in any of the transverse rows,\nmay be thus described. Let A B C D (in fig. 3) be a parallelogram, whose\ndiagonal is B D. I would describe in it such figures, as in the\npreceding table I call three-sided figures of three means. Parallel to D\nC, I draw E F as often as is necessary, cutting B D in G; and between E\nF and E G, I take three proportionals E H, E I and E K. If now there be\ndrawn lines through all the points H, I and K, that through all the\npoints H will make the figure B H D C, which is the first of those\nthree-sided figures; and that through all the points I, will make the\nfigure B I D C, which is the second; and that which is drawn through all\nthe points K, will make the figure B K D C the third of those\nthree-sided figures. The first of these, seeing the proportion of E F to\nE G is quadruplicate of that E F to E H, will be to its complement as 4\nto 1, and to the parallelogram as 4 to 5. The second, seeing the\nproportion of E F to E G is to that of E F to E I as 4 to 2, will be\ndouble to its complement, and 4\u20446 or 2\u20443 of the parallelogram. The\nthird, seeing the proportion of E F to E G is that of E F to E K as 4 to\n3, will be to its complement as 4 to 3, and to the parallelogram as 4 to\n7.\n\nAny of these figures being described may be produced at pleasure, thus;\nlet A B C D (in fig. 4) be a parallelogram, and in it let the figure B K\nD C be described, namely, the third three-sided figure of three means.\nLet B D be produced indefinitely to E, and let E F be made parallel to\nthe base D C, cutting A D produced in G, and B C produced in F; and in G\nE let the point H be so taken, that the proportion of F E to F G may be\nquadruplicate to that of F E to F H, which may be done by making F H the\ngreatest of three proportionals between F E and F G; the crooked line B\nK D produced, will pass through the point H. For if the strait line B H\nbe drawn, cutting C D in I, and H L be drawn parallel to G D, and\nmeeting C D produced in L; it will be as F E to F G, so C L to C I, that\nis, in quadruplicate proportion to that of F E to F H, or of C D to C I.\nWherefore if the line B K D be produced according to its generation, it\nwill fall upon the point H.\n\n[Sidenote: The drawing of tangents to them.]\n\n5. A strait line may be drawn so as to touch the crooked line of the\nsaid figure in any point, in this manner. Let it be required to draw a\ntangent to the line B K D H (in fig. 4) in the point D. Let the points B\nand D be connected, and drawing D A equal and parallel to B C, let B and\nA be connected; and because this figure is by construction the third of\nthree means, let there be taken in A B three points, so, that by them\nthe same A B be divided into four equal parts; of which take three,\nnamely, A M, so that A B may be to A M, as the figure B K D C is to its\ncomplement. I say, the strait line M D will touch the figure in the\npoint given D. For let there be drawn anywhere between A B and D C a\nparallel, as R Q, cutting the strait line B D, the crooked line B K D,\nthe strait line M D, and the strait line A D, in the points P, K, O and\nQ. R K will therefore, by construction, be the least of three means in\ngeometrical proportion between R Q and R P. Wherefore (by coroll. of\nart. 28, chapter XIII.) R K will be less than R O; and therefore M D\nwill fall without the figure. Now if M D be produced to N, F N will be\nthe greatest of three means in arithmetical proportion between F E and F\nG; and F H will be the greatest of three means in geometrical proportion\nbetween the same F E and F G. Wherefore (by the same coroll. of art. 28,\nchapter XIII.) F H will be less than F N; and therefore D N will fall\nwithout the figure, and the strait line M N will touch the same figure\nonly in the point D.\n\n[Sidenote: In what proportion the same figures exceed a strait-lined\n triangle of the same altitude and base.]\n\n6. The proportion of a deficient figure to its complement being known,\nit may also be known what proportion a strait-lined triangle has to the\nexcess of the deficient figure above the same triangle; and these\nproportions I have set down in the following table; where if you seek,\nfor example, how much the fourth three-sided figure of five means\nexceeds a triangle of the same altitude and base, you will find in the\nconcourse of the fourth column with the three-sided figures of five\nmeans 2\u204410; by which is signified, that that three-sided figure exceeds\nthe triangle by two-tenths or by one-fifth part of the same triangle.\n\n 1 2 3 4 5 6 7\n The triangle | 1 : : : : : :\n +---+\n {A three-sided fig. of 1 mean |1/3| : : : : : :\n { +---+----+\n {A three-sided fig. of 2 means|2/4| 1/5| : : : : :\n { +---+----+----+\nThe {A three-sided fig. of 3 means|3/5| 2/6| 1/7| : : : :\n { +---+----+----+----+\nexcess{A three-sided fig. of 4 means|4/6| 3/7| 2/8| 1/9| : : :\n { +---+----+----+----+----+\nof {A three-sided fig. of 5 means|5/7| 4/8| 3/9|2/10|1/11| : :\n { +---+----+----+----+----+----+\n {A three-sided fig. of 6 means|6/8| 5/9|4/10|3/11|2/12|1/13| :\n { +---+----+----+----+----+----+----+\n {A three-sided fig. of 7 means|7/9|6/10|5/11|4/12|3/13|2/14|1/15|\n +---+----+----+----+----+----+----+\n\n[Sidenote: A table of solid deficient figures described in a cylinder.]\n\n7. In the next table are set down the proportion of a cone and the\nsolids of the said three-sided figures, namely, the proportions between\nthem and a cylinder. As for example, in the concourse of the second\ncolumn with the three-sided figures of four means, you have 5\u20449; which\ngives you to understand, that the solid of the second three-sided figure\nof four means is to the cylinder as 5\u20449 to 1, or as 5 to 9.\n\n | 1 2 3 4 5 6 7\n { A cylinder | 1 | : : : : : :\n { +----+ : : : : : :\n { A cone |1/3 | : : : : : :\n { +----+----+ : : : : :\n { A three-sided fig. of 1 mean |2/4 | | : : : : :\n { +----|----+ : : : : :\nThe { A three-sided fig. of 2 means |3/5 |3/7 | : : : : :\nsolids{ +----+----+----+----+ : : :\nof { A three-sided fig. of 3 means |4/6 |4/8 |4/10| | : : :\n { +----+----+----+----+----+ : :\n { A three-sided fig. of 4 means |5/7 |5/9 |5/11|5/13| | : :\n { +----+----+----+----+----+----+ :\n { A three-sided fig. of 5 means |6/8 |6/10|6/12|6/14|6/16| | :\n { +----+----+----+----+----+----+----|\n { A three-sided fig. of 6 means |7/9 |7/11|7/13|7/15|7/17|7/19| |\n { +----+----+----+----+----+----+----+\n { A three-sided fig. of 7 means |8/10|8/12|8/14|8/16|8/18|8/20|8/22|\n +----+----+----+----+----+----+----+\n\n[Sidenote: In what proportion the same figures exceed a cone of the same\n altitude and base.]\n\n8. Lastly, the excess of the solids of the said three-sided figures\nabove a cone of the same altitude and base, are set down in the table\nwhich follows:\n\n 1 2 3 4 5 6 7\n The Cone | 1 : : : : : : :\n +-----+\n {Of the solid of a three-}| 6/12| : : : : : :\n { sided figure of 1 mean}| |\nThe ex- { +-----+-----+\ncesses {Ditto ditto, 2 means |12/15| 6/21| : : : : :\nof the { +-----+-----+-----+\nsolids {Ditto ditto, 3 means |18/18|12/24| 6/30| : : : :\nof these{ +-----+-----+-----+-----+\nthree- {Ditto ditto, 4 means |24/21|18/27|12/33| 6/39| : : :\nsided { +-----+-----+-----+-----+-----+\nfigures {Ditto ditto, 5 means |30/24|24/30|18/36|12/42| 6/48| : :\nabove a { +-----+-----+-----+-----+-----+-----+\ncone. {Ditto ditto, 6 means |36/27|30/33|24/39|18/45|12/51| 6/57| :\n { +-----+-----+-----+-----+-----+-----+-----+\n {Ditto ditto, 7 means |42/30|36/36|30/42|24/48|18/54|12/60| 6/66|\n +-----+-----+-----+-----+-----+-----+-----+\n\n[Sidenote: How a plain deficient figure may be described in a\n parallelogram, so that it be to a triangle of the same base\n and altitude, as another deficient figure, plain or solid,\n twice taken, is to the same deficient figure, together with\n the complete figure, in which it is described.]\n\n9. If any of these deficient figures, of which I have now spoken, as A B\nC D (in the 5th figure) be inscribed within the complete figure B E,\nhaving A D C E for its complement; and there be made upon C B produced\nthe triangle A B I; and the parallelogram A B I K be completed; and\nthere be drawn parallel to the strait line C I, any number of lines, as\nM F, cutting every one of them the crooked line of the deficient figure\nin D, and the strait lines A C, A B and A I in H, G, and L; and as G F\nis to G D, so G L be made to another, G N; and through all the points N\nthere be drawn the line A N I: there will be a deficient figure A N I B,\nwhose complement will be A N I K. I say, the figure A N I B is to the\ntriangle A B I, as the deficient figure A B C D twice taken is to the\nsame deficient figure together with the complete figure B E.\n\nFor as the proportion of A B to A G, that is, of G M to G L, is to the\nproportion of G M to G N, so is the magnitude of the figure A N I B to\nthat of its complement A N I K, by the second article of this chapter.\n\nBut, by the same article, as the proportion of A B to A G, that is, of G\nM to G L, is to the proportion of G F to G D, that is, by construction,\nof G L to G N, so is the figure A B C D to its complement A D C E.\n\nAnd by composition, as the proportion of G M to G L, together with that\nof G L to G N, is to the proportion of G M to G L, so is the complete\nfigure B E to the deficient figure A B C D.\n\nAnd by conversion, as the proportion of G M to G L is to both the\nproportions of G M to G L and of G L to G N, that is, to the proportion\nof G M to G N, which is the proportion compounded of both, so is the\ndeficient figure A B C D to the complete figure B E.\n\nBut it was, as the proportion of G M to G L to that of G M to G N, so\nthe figure A N I B to its complement A N I K. And therefore, A B C D. B\nE :: A N I B. A N I K are proportionals. And by composition, A B C D + B\nE. A B C D :: B K. A N I B are proportionals.\n\nAnd by doubling the consequents, A B C D + B E. 2 A B C D :: B K. 2 A N\nI B are proportionals.\n\nAnd by taking the halves of the third and the fourth, A B C D + B E. 2 A\nB C D :: A B I. A N I B are also proportionals; which was to be proved.\n\n[Sidenote: The transferring of certain properties of deficient figures\n described in a parallelogram to the proportions of spaces\n transmitted with several degrees of velocity.]\n\n10. From what has been said of deficient figures described in a\nparallelogram, may be found out what proportions spaces, transmitted\nwith accelerated motion in determined times, have to the times\nthemselves, according as the moved body is accelerated in the several\ntimes with one or more degrees of velocity.\n\nFor let the parallelogram A B C D, in the 6th figure, and in it the\nthree-sided figure D E B C be described; and let F G be drawn anywhere\nparallel to the base, cutting the diagonal B D in H, and the crooked\nline B E D in E; and let the proportion of B C to B F be, for example,\ntriplicate to that of F G to F E; whereupon the figure D E B C will be\ntriple to its complement B E D A; and in like manner, I E being drawn\nparallel to B C, the three-sided figure E K B F will be triple to its\ncomplement B K E I. Wherefore the parts of the deficient figure cut off\nfrom the vertex by strait lines parallel to the base, namely, D E B C\nand E K B F, will be to one another as the parallelograms A C and I F;\nthat is, in proportion compounded of the proportions of the altitudes\nand bases. Seeing therefore the proportion of the altitude B C to the\naltitude B F is triplicate to the proportion of the base D C to the base\nF E, the figure D E B C to the figure E K B F will be quadruplicate to\nthe proportion of the same D C to F E. And by the same method, may be\nfound out what proportion any of the said three-sided figures has to any\npart of the same, cut off from the vertex by a strait line parallel to\nthe base.\n\nNow as the said figures are understood to be described by the continual\ndecreasing of the base, as of C D, for example, till it end in a point,\nas in B; so also they may be understood to be described by the continual\nincreasing of a point, as of B, till it acquire any magnitude, as that\nof C D.\n\nSuppose now the figure B E D C to be described by the increasing of the\npoint B to the magnitude C D. Seeing therefore the proportion of B C to\nB F is triplicate to that of C D to F E, the proportion of F E to C D\nwill, by conversion, as I shall presently demonstrate, be triplicate to\nthat B F to B C. Wherefore if the strait line B C be taken for the\nmeasure of the time in which the point B is moved, the figure E K B F\nwill represent the sum of all the increasing velocities in the time B F;\nand the figure D E B C will in like manner represent the sum of all the\nincreasing velocities in the time B C. Seeing therefore the proportion\nof the figure E K B F to the figure D E B C is compounded of the\nproportions of altitude to altitude, and base to base; and seeing the\nproportion of F E to C D is triplicate to that of B F to B C; the\nproportion of the figure E K B F to the figure D E B C will be\nquadruplicate to that of B F to B C; that is, the proportion of the sum\nof the velocities in the time B F, to the sum of the velocities in the\ntime B C, will be quadruplicate to the proportion of B F to B C.\nWherefore if a body be moved from B with velocity so increasing, that\nthe velocity acquired in the time B F be to the velocity acquired in the\ntime B C in triplicate proportion to that of the times themselves B F to\nB C, and the body be carried to F in the time B F; the same body in the\ntime B C will be carried through a line equal to the fifth proportional\nfrom B F in the continual proportion of B F to B C. And by the same\nmanner of working, we may determine what spaces are transmitted by\nvelocities increasing according to any other proportions.\n\nIt remains that I demonstrate the proportion of F E to C D to be\ntriplicate to that of B F to B C. Seeing therefore the proportion of C\nD, that is, of F G to F E is subtriplicate to that of B C to B F; the\nproportion of F G to F E will also be subtriplicate to that of F G to F\nH. Wherefore the proportion of F G to F H is triplicate to that of F G,\nthat is, of C D to F E. But in four continual proportionals, of which\nthe least is the first, the proportion of the first to the fourth, (by\nthe 16th article of chapter XIII.), is subtriplicate to the proportion\nof the third to the same fourth. Wherefore the proportion of F H to G F\nis subtriplicate to that of F E to C D; and therefore the proportion of\nF E to C D is triplicate to that of F H to F G, that is, of B F to B C;\nwhich was to be proved.\n\nIt may from hence be collected, that when the velocity of a body is\nincreased in the same proportion with that of the times, the degrees of\nvelocity above one another proceed as numbers do in immediate succession\nfrom unity, namely, as 1, 2, 3, 4, &c. And when the velocity is\nincreased in proportion duplicate to that of the times, the degrees\nproceed as numbers from unity, skipping one, as 1, 3, 5, 7, &c. Lastly,\nwhen the proportions of the velocities are triplicate to those of the\ntimes, the progression of the degrees is as that of numbers from unity,\nskipping two in every place, as 1, 4, 7, 10, &c., and so of other\nproportions. For geometrical proportionals, when they are taken in every\npoint, are the same with arithmetical proportionals.\n\n[Sidenote: Of deficient figures described in a circle.]\n\n11. Moreover, it is to be noted that as in quantities, which are made by\nany magnitudes decreasing, the proportions of the figures to one another\nare as the proportions of the altitudes to those of the bases; so also\nit is in those, which are made with motion decreasing, which motion is\nnothing else but that power by which the described figures are greater\nor less. And therefore in the description of _Archimedes' spiral_, which\nis done by the continual diminution of the semidiameter of a circle in\nthe same proportion in which the circumference is diminished, the space,\nwhich is contained within the semidiameter and the spiral line, is a\nthird part of the whole circle. For the semidiameters of circles,\ninasmuch as circles are understood to be made up of the aggregate of\nthem, are so many sectors; and therefore in the description of a spiral,\nthe sector which describes it is diminished in duplicate proportions to\nthe diminutions of the circumference of the circle in which it is\ninscribed; so that the complement of the spiral, that is, that space in\nthe circle which is without the spiral line, is double to the space\nwithin the spiral line. In the same manner, if there be taken a mean\nproportional everywhere between the semidiameter of the circle, which\ncontains the spiral, and that part of the semidiameter which is within\nthe same, there will be made another figure, which will be half the\ncircle. And to conclude, this rule serves for all such spaces as may be\ndescribed by a line or superficies decreasing either in magnitude of\npower; so that if the proportions, in which they decrease, be\ncommensurable to the proportions of the times in which they decrease,\nthe magnitudes of the figures they describe will be known.\n\n[Sidenote: The proposition demonstrated in art. 2 confirmed from the\n elements of philosophy.]\n\n12. The truth of that proposition, which I demonstrated in art. 2, which\nis the foundation of all that has been said concerning deficient\nfigures, may be derived from the elements of philosophy, as having its\noriginal in this; _that all equality and inequality between two effects,\nthat is, all proportion, proceeds from, and is determined by, the equal\nand unequal causes of those effects, or from the proportion which the\ncauses, concurring to one effect, have to the causes which concur to the\nproducing of the other effect_; and that therefore the proportions of\nquantities are the same with the proportions of their causes. Seeing,\ntherefore, two deficient figures, of which one is the complement of the\nother, are made, one by motion decreasing in a certain time and\nproportion, the other by the loss of motion in the same time; the\ncauses, which make and determine the quantities of both the figures, so\nthat they can be no other than they are, differ only in this, that the\nproportions by which the quantity which generates the figure proceeds in\ndescribing of the same, that is, the proportions of the remainders of\nall the times and altitudes, may be other proportions than those by\nwhich the same generating quantity decreases in making the complement of\nthat figure, that is, the proportions of the quantity which generates\nthe figure continually diminished. Wherefore, as the proportion of the\ntimes in which motion is lost, is to that of the decreasing quantities\nby which the deficient figure is generated, so will the defect or\ncomplement be to the figure itself which is generated.\n\n[Sidenote: An unusual way of reasoning concerning the equality between\n the superficies of a portion of a sphere and a circle.]\n\n13. There are also other quantities which are determinable from the\nknowledge of their causes, namely, from the comparison of the motions by\nwhich they are made; and that more easily than from the common elements\nof geometry. For example, that the superficies of any portion of a\nsphere is equal to that circle, whose radius is a strait line drawn from\nthe pole of the portion to the circumference of its base, I may\ndemonstrate in this manner. Let B A C (in fig. 7) be a portion of a\nsphere, whose axis is A E, and whose base is B C; and let A B be the\nstrait line drawn from the pole A to the base in B; and let A D, equal\nto A B, touch the great circle B A C in the pole A. It is to be proved\nthat the circle made by the radius A D is equal to the superficies of\nthe portion B A C. Let the plain A E B D be understood to make a\nrevolution about the axis A E; and it is manifest that by the strait\nline A D a circle will be described; and by the arch A B the superficies\nof a portion of a sphere; and lastly, by the subtense A B the\nsuperficies of a right cone. Now seeing both the strait line A B and the\narch A B make one and the same revolution, and both of them have the\nsame extreme points A and B, the cause why the spherical superficies,\nwhich is made by the arch, is greater than the conical superficies,\nwhich is made by the subtense, is, that A B the arch is greater than A B\nthe subtense; and the cause why it is greater consists in this, that\nalthough they be both drawn from A to B, yet the subtense is drawn\nstrait, but the arch angularly, namely, according to that angle which\nthe arch makes with the subtense, which angle is equal to the angle D A\nB (for an angle of contingence adds nothing to an angle of a segment, as\nhas been shown in chapter XIV, article 16.) Wherefore the magnitude of\nthe angle D A B is the cause why the superficies of the portion,\ndescribed by the arch A B, is greater than the superficies of the right\ncone described by the subtense A B.\n\nAgain, the cause why the circle described by the tangent A D is greater\nthan the superficies of the right cone described by the subtense A B\n(notwithstanding that the tangent and the subtense are equal, and both\nmoved round in the same time) is this, that A D stands at right angles\nto the axis, but A B obliquely; which obliquity consists in the same\nangle D A B. Seeing therefore the quantity of the angle D A B is that\nwhich makes the excess both of the superficies of the portion, and of\nthe circle made by the radius A D, above the superficies of the right\ncone described by the subtense A B; it follows, that both the\nsuperficies of the portion and that of the circle do equally exceed the\nsuperficies of the cone. Wherefore the circle made by A D or A B, and\nthe spherical superficies made by the arch A B, are equal to one\nanother; which was to be proved.\n\n[Sidenote: How from the description of deficient figures in a\n parallelogram, any number of mean proportionals may be found\n out between two given strait lines.]\n\n14. If these deficient figures, which I have described in a\nparallelogram, were capable of exact description, then any number of\nmean proportionals might be found out between two strait lines given.\nFor example, in the parallelogram A B C D, (in figure 8) let the\nthree-sided figure of two means be described (which many call a _cubical\nparabola_); and let R and S be two given strait lines; between which, if\nit be required to find two mean proportionals, it may be done thus. Let\nit be as R to S, so B C to B F; and let F E be drawn parallel to B A,\nand cut the crooked line in E; then through E let G H be drawn parallel\nand equal to the strait line A D, and cut the diagonal B D in I; for\nthus we have G I the greatest of two means between G H and G E, as\nappears by the description of the figure in article 4. Wherefore, if it\nbe as G H to G I, so R to another line, T, that T will be the greatest\nof two means between R and S. And therefore if it be again as R to T, so\nT to another line, X, that will be done which was required.\n\nIn the same manner, four mean proportionals may be found out, by the\ndescription of a three-sided figure of four means; and so any other\nnumber of means, &c.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C. XVII.\n _Fig. 1-8_\n]\n\n------------------------------------------------------------------------\n\n\n -------\n\n\n CHAPTER XVIII.\n\n OF THE EQUATION OF STRAIT LINES WITH THE\n CROOKED LINES OF PARABOLAS AND OTHER\n FIGURES MADE IN IMITATION OF PARABOLAS.\n\n1. To find the strait line equal to the crooked line of a\n semiparabola.\u20142. To find a strait line equal to the crooked line of\n the first semiparabolaster, or to the crooked line of any other of\n the deficient figures of the table of the 3d article of the\n precedent chapter.\n\n\n[Sidenote: To find a strait line equal to the crooked line of a\n semiparabola.]\n\n1. A parabola being given, to find a strait line equal to the crooked\nline of the semiparabola.\n\nLet the parabolical line given be A B C (in figure 1), and the diameter\nfound be A D, and the base drawn D C; and the parallelogram A D C E\nbeing completed, draw the strait line A C. Then dividing A D into two\nequal parts in F, draw F H equal and parallel to D C, cutting A C in K,\nand the parabolical line in O; and between F H and F O take a mean\nproportional F P, and draw A O, A P and P C. I say that the two lines A\nP and P C, taken together as one line, are equal to the parabolical line\nA B O C.\n\nFor the line A B O C being a parabolical line, is generated by the\nconcourse of two motions, one uniform from A to E, the other in the same\ntime uniformly accelerated from rest in A to D. And because the motion\nfrom A to E is uniform, A E may represent the times of both those\nmotions from the beginning to the end. Let therefore A E be the time;\nand consequently the lines ordinately applied in the semiparabola will\ndesign the parts of time wherein the body, that describeth the line A B\nO C, is in every point of the same; so that as at the end of the time A\nE or D C it is in C, so at the end of the time F O it will be in O. And\nbecause the velocity in A D is increased uniformly, that is, in the same\nproportion with the time, the same lines ordinately applied in the\nsemiparabola will design also the continual augmentation of the impetus,\ntill it be at the greatest, designed by the base D C. Therefore\nsupposing uniform motion in the line A F, in the time F K the body in A\nby the concourse of the two uniform motions in A F and F K will be moved\nuniformly in the line A K; and K O will be the increase of the impetus\nor swiftness gained in the time F K; and the line A O will be uniformly\ndescribed by the concourse of the two uniform motions in A F and F O in\nthe time F O. From O draw O L parallel to E C, cutting A C in L; and\ndraw L N parallel to D C, cutting E C in N, and the parabolical line in\nM; and produce it on the other side to A D in I; and I N, I M and I L\nwill be, by the construction of a parabola, in continual proportion, and\nequal to the three lines F H, F P and F O; and a strait line parallel to\nE C passing through M will fall on P; and therefore O P will be the\nincrease of impetus gained in the time F O or I L. Lastly, produce P M\nto C D in Q; and Q C or M N or P H will be the increase of impetus\nproportional to the time F P or I M or D Q. Suppose now uniform motion\nfrom H to C in the time P H. Seeing therefore in the time F P with\nuniform motion and the impetus increased in proportion to the times, is\ndescribed the straight line A P; and in the rest of the time and\nimpetus, namely, P H, is described the line C P uniformly; it followeth\nthat the whole line A P C is described with the whole impetus, and in\nthe same time wherewith is described the parabolical line A B C; and\ntherefore the line A P C, made of the two strait lines A P and P C, is\nequal to the parabolical line A B C; which was to be proved.\n\n[Sidenote: To find a strait line equal to the crooked line of the first\n semiparabolaster or to the crooked line of any other of the\n deficient figures of the table of art. 3 of the preceding\n chapter.]\n\n2. To find a strait line equal to the crooked line of the first\nsemiparabolaster.\n\n[Sidenote: [Discussion of Figure 18.2]]\n\nLet A B C be the crooked line of the first semiparabolaster; A D the\ndiameter; D C the base; and let the parallelogram completed be A D C E,\nwhose diagonal is A C. Divide the diameter into two equal parts in F,\nand draw F H equal and parallel to D C, cutting A C in K, the crooked\nline in O, and E C in H. Then draw O L parallel to E C, cutting A C in\nL; and draw L N parallel to the base D C, cutting the crooked line in M,\nand the strait line E C in N; and produce it on the other side to A D in\nI. Lastly, through the point M draw P M Q parallel and equal to H C,\ncutting F H in P; and join C P, A P and A O. I say, the two strait lines\nA P and P C are equal to the crooked line A B O C.\n\nFor the line A B O C, being the crooked line of the first\nsemiparabolaster, is generated by the concourse of two motions, one\nuniform from A to E, the other in the same time accelerated from rest in\nA to D, so as that the impetus increaseth in proportion perpetually\ntriplicate to that of the increase of the time, or which is all one, the\nlengths transmitted are in proportion triplicate to that of the times of\ntheir transmission; for as the impetus or quicknesses increase, so the\nlengths transmitted increase also. And because the motion from A to E is\nuniform, the line A E may serve to represent the time, and consequently\nthe lines, ordinately drawn in the semiparabolaster, will design the\nparts of time wherein the body, beginning from rest in A, describeth by\nits motion the crooked line A B O C. And because D C, which represents\nthe greatest acquired impetus, is equal to A E, the same ordinate lines\nwill represent the several augmentations of the impetus increasing from\nrest in A. Therefore, supposing uniform motion from A to F, in the time\nF K there will be described, by the concourse of the two uniform motions\nA F and F K, the line A K uniformly, and K O will be the increase of\nimpetus in the time F K; and by the concourse of the two uniform motions\nin A F and F O will be described the line A O uniformly. Through the\npoint L draw the strait line L M N parallel to D C, cutting the strait\nline A D in I, the crooked line A B C in M, and the strait line E C in\nN; and through the point M the strait line P M Q parallel and equal to H\nC, cutting D C in Q and F H in P. By the concourse therefore of the two\nuniform motions in A F and F P in the time F P will be uniformly\ndescribed the strait line A P; and L M or O P will be the increase of\nimpetus to be added for the time F O. And because the proportion of I N\nto I L is triplicate to the proportion of I N to I M, the proportion of\nF H to F O will also be triplicate to the proportion of F H to F P; and\nthe proportional impetus gained in the time F P is P H. So that F H\nbeing equal to D C, which designed the whole impetus acquired by the\nacceleration, there is no more increase of impetus to be computed. Now\nin the time P H suppose an uniform motion from H to C; and by the two\nuniform motions in C H and H P will be described uniformly the strait\nline P C. Seeing therefore the two strait lines A P and P C are\ndescribed in the time A E with the same increase of impetus, wherewith\nthe crooked line A B O C is described in the same time A E, that is,\nseeing the line A P C and the line A B O C are transmitted by the same\nbody in the same time and with equal velocities, the lines themselves\nare equal; which was to be demonstrated.\n\nBy the same method (if any of the semiparabolasters in the table of art.\n3 of the precedent chapter be exhibited) may be found a strait line\nequal to the crooked line thereof, namely, by dividing the diameter into\ntwo equal parts, and proceeding as before. Yet no man hitherto hath\ncompared any crooked with any strait line, though many geometricians of\nevery age have endeavoured it. But the cause, why they have not done it,\nmay be this, that there being in Euclid no definition of equality, nor\nany mark by which to judge of it besides congruity (which is the 8th\naxiom of the first Book of his Elements) a thing of no use at all in the\ncomparing of strait and crooked; and others after Euclid (except\nArchimedes and Apollonius, and in our time Bonaventura) thinking the\nindustry of the ancients had reached to all that was to be done in\ngeometry, thought also, that all that could be propounded was either to\nbe deduced from what they had written, or else that it was not at all to\nbe done: it was therefore disputed by some of those ancients themselves,\nwhether there might be any equality at all between crooked and strait\nlines; which question Archimedes, who assumed that some strait line was\nequal to the circumference of a circle, seems to have despised, as he\nhad reason. And there is a late writer that granteth that between a\nstrait and a crooked line there is equality; but now, says he, since the\nfall of Adam, without the special assistance of Divine Grace it is not\nto be found.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C. XVIII.\n _Fig. 1-2_\n]\n\n------------------------------------------------------------------------\n\n\n -------\n\n\n CHAPTER XIX.\n\n OF ANGLES OF INCIDENCE AND REFLECTION,\n EQUAL BY SUPPOSITION.\n\n1. If two strait lines falling upon another strait line be parallel, the\n lines reflected from them shall also be parallel.\u20142. If two strait\n lines drawn from one point fall upon another strait line, the lines\n reflected from them, if they be drawn out the other way, will meet\n in an angle equal to the angle made by the lines of incidence.\u20143. If\n two strait parallel lines, drawn not oppositely, but from the same\n parts, fall upon the circumference of a circle, the lines reflected\n from them, if produced they meet within the circle, will make an\n angle double to that which is made by two strait lines drawn from\n the centre to the points of incidence.\u20144. If two strait lines drawn\n from the same point without a circle fall upon the circumference,\n and the lines reflected from them being produced meet within the\n circle, they will make an angle equal to twice that angle, which is\n made by two strait lines drawn from the centre to the points of\n incidence, together with the angle which the incident lines\n themselves make.\u20145. If two strait lines drawn from one point fall\n upon the concave circumference of a circle, and the angle they make\n be less than twice the angle at the centre, the lines reflected from\n them and meeting within the circle will make an angle, which being\n added to the angle of the incident lines will be equal to twice the\n angle at the centre.\u20146. If through any one point two unequal chords\n be drawn cutting one another, and the centre of the circle be not\n placed between them, and the lines reflected from them concur\n wheresoever, there cannot through the point, through which the two\n former lines were drawn, be drawn any other strait line whose\n reflected line shall pass through the common point of the two former\n lines reflected.\u20147. In equal chords the same is not true.\u20148. Two\n points being given in the circumference of a circle, to draw two\n strait lines to them, so that their reflected lines may contain any\n angle given.\u20149. If a strait line falling upon the circumference of a\n circle be produced till it reach the semidiameter, and that part of\n it, which is intercepted between the circumference and the\n semidiameter, be equal to that part of the semidiameter which is\n between the point of concourse and the centre, the reflected line\n will be parallel to the semidiameter.\u201410. If from a point within a\n circle, two strait lines be drawn to the circumference, and their\n reflected lines meet in the circumference of the same circle, the\n angle made by the reflected lines will be a third part of the angle\n made by the incident lines.\n\n\n[Sidenote: Angles of incidence and reflection.]\n\nWhether a body falling upon the superficies of another body and being\nreflected from it, do make equal angles at that superficies, it belongs\nnot to this place to dispute, being a knowledge which depends upon the\nnatural causes of reflection; of which hitherto nothing has been said,\nbut shall be spoken of hereafter.\n\nIn this place, therefore, let it be supposed that the angle of incidence\nis equal to the angle of reflection; that our present search may be\napplied, not to the finding out of the causes, but some consequences of\nthe same.\n\nI call an _angle of incidence_, that which is made between a strait line\nand another line, strait or crooked, upon which it falls, and which I\ncall the _line reflecting_; and an _angle of reflection_ equal to it,\nthat which is made at the same point between the strait line which is\nreflected and the line reflecting.\n\n[Sidenote: If two strait lines falling upon another strait line be\n parallel, the lines reflected from them shall also be\n parallel.]\n\n1. If two strait lines, which fall upon another strait line, be\nparallel, their reflected lines shall be also parallel.\n\nLet the two strait lines A B and C D (in fig. 1), which fall upon the\nstrait line E F, at the points B and D, be parallel; and let the lines\nreflected from them be B G and D H. I say, B G and D H are also\nparallel.\n\nFor the angles A B E and C D E are equal by reason of the parallelism of\nA B and C D; and the angles G B F and H D F are equal to them by\nsupposition; for the lines B G and D H are reflected from the lines A B\nand C D. Wherefore B G and D H are parallel.\n\n[Sidenote: If two strait lines drawn from one point fall upon another\n strait line, the lines reflected from them, if they be drawn\n out the other way, will meet in an angle equal to the angle\n made by the lines of incidence.]\n\n2. If two strait lines drawn from the same point fall upon another\nstrait line, the lines reflected from them, if they be drawn out the\nother way, will meet in an angle equal to the angle of the incident\nlines.\n\nFrom the point A (in fig. 2) let the two strait lines A B and A D be\ndrawn; and let them fall upon the strait line E K at the points B and D;\nand let the lines B I and D G be reflected from them. I say, I B and G D\ndo converge, and that if they be produced on the other side of the line\nE K, they shall meet, as in F; and that the angle B F D shall be equal\nto the angle B A D.\n\nFor the angle of reflection I B K is equal to the angle of incidence A B\nE; and to the angle I B K its vertical angle E B F is equal; and\ntherefore the angle A B E is equal to the angle E B F. Again, the angle\nA D E is equal to the angle of reflection G D K, that is, to its\nvertical angle E D F; and therefore the two angles A B D and A D B of\nthe triangle A B D are one by one equal to the two angles F B D and F D\nB of the triangle F B D; wherefore also the third angle B A D is equal\nto the third angle B F D; which was to be proved.\n\nCoroll. I. If the strait line A F be drawn, it will be perpendicular to\nthe strait line E K. For both the angles at E will be equal, by reason\nof the equality of the two angles A B E and F B E, and of the two sides\nA B and F B.\n\nCoroll. II. If upon any point between B and D there fall a strait line,\nas A C, whose reflected line is C H, this also produced beyond C, will\nfall upon F; which is evident by the demonstration above.\n\n[Sidenote: If two strait parallel lines, drawn not oppositely, but from\n the same parts, fall upon the circumference of a circle, the\n lines reflected from them, if produced they meet within the\n circle, will make an angle double to that which is made by\n two strait lines drawn from the centre to the points of\n incidence.]\n\n3. If from two points taken without a circle, two strait parallel lines,\ndrawn not oppositely, but from the same parts, fall upon the\ncircumference; the lines reflected from them, if produced they meet\nwithin the circle, will make an angle double to that which is made by\ntwo strait lines drawn from the centre to the points of incidence.\n\nLet the two strait parallels A B and D C (in fig. 3) fall upon the\ncircumference B C at the points B and C; and let the centre of the\ncircle be E; and let A B reflected be B F, and D C reflected be C G; and\nlet the lines F B and G C produced meet within the circle in H; and let\nE B and E C be connected. I say the angle F H G is double to the angle B\nE C.\n\nFor seeing A B and D C are parallels, and E B cuts A B in B, the same E\nB produced will cut D C somewhere; let it cut it in D; and let D C be\nproduced howsoever to I, and let the intersection of D C and B F be at\nK. The angle therefore I C H, being external to the triangle C K H, will\nbe equal to the two opposite angles C K H and C H K. Again, I C E being\nexternal to the triangle C D E, is equal to the two angles at D and E.\nWherefore the angle I C H, being double to the angle I C E, is equal to\nthe angles at D and E twice taken; and therefore the two angles C K H\nand C H K are equal to the two angles at D and E twice taken. But the\nangle C K H is equal to the angles D and A B D, that is, D twice taken;\nfor A B and D C being parallels, the altern angles D and A B D are\nequal. Wherefore C H K, that is the angle F H G is also equal to the\nangle at E twice taken; which was to be proved.\n\nCoroll. If from two points taken within a circle two strait parallels\nfall upon the circumference, the lines reflected from them shall meet in\nan angle, double to that which is made by two strait lines drawn from\nthe centre to the points of incidence. For the parallels A B and I C\nfalling upon the points B and C, are reflected in the lines B H and C H,\nand make the angle at H double to the angle at E, as was but now\ndemonstrated.\n\n[Sidenote: If two strait lines drawn from the same point without a circle fall upon the circumference, and the lines reflected from them being produced meet within\n the circle, they will make an angle equal to twice that angle, which is made by two strait lines drawn from the centre to the points of incidence, together with the angle which the incident lines themselves make.]\n\n4. If two strait lines drawn from the same point without a circle fall\nupon the circumference, and the lines reflected from them being produced\nmeet within the circle, they will make an angle equal to twice that\nangle, which is made by two strait lines drawn from the centre to the\npoints of incidence, together with the angle which the incident lines\nthemselves make.\n\nLet the two strait lines A B and A C (in fig. 4) be drawn from the point\nA to the circumference of the circle, whose centre is D; and let the\nlines reflected from them be B E and C G, and, being produced, make\nwithin the circle the angle H; also let the two strait lines D B and D C\nbe drawn from the centre D to the points of incidence B and C. I say,\nthe angle H is equal to twice the angle at D together with the angle at\nA.\n\nFor let A C be produced howsoever to I. Therefore the angle I C H, which\nis external to the triangle C K H, will be equal to the two angles C K H\nand C H K. Again, the angle I C D, which is external to the triangle C L\nD, will be equal to the two angles C L D and C D L. But the angle I C H\nis double to the angle I C D, and is therefore equal to the angles C L D\nand C D L twice taken. Wherefore the angles C K H and C H K are equal to\nthe angles C L D and C D L twice taken. But the angle C L D, being\nexternal to the triangle A L B, is equal to the two angles L A B and L B\nA; and consequently C L D twice taken is equal to L A B and L B A twice\ntaken. Wherefore C K H and C H K are equal to the angle C D L together\nwith L A B and L B A twice taken. Also the angle C K H is equal to the\nangle L A B once and A B K, that is, L B A twice taken. Wherefore the\nangle C H K is equal to the remaining angle C D L, that is, to the angle\nat D, twice taken, and the angle L A B, that is, the angle at A, once\ntaken; which was to be proved.\n\nCoroll. If two strait converging lines, as I C and M B, fall upon the\nconcave circumference of a circle, their reflected lines, as C H and B\nH, will meet in the angle H, equal to twice the angle D, together with\nthe angle at A made by the incident lines produced. Or, if the incident\nlines be H B and I C, whose reflected lines C H and B M meet in the\npoint N, the angle C N B will be equal to twice the angle D, together\nwith the angle C K H made by the lines of incidence. For the angle C N B\nis equal to the angle H, that is, to twice the angle D, together with\nthe two angles A, and N B H, that is, K B A. But the angles K B A and A\nare equal to the angle C K H. Wherefore the angle C N B is equal to\ntwice the angle D, together with the angle C K H made by the lines of\nincidence I C and H B produced to K.\n\n[Sidenote: If two strait lines drawn from one point fall upon the\n concave circumference of a circle, and the angle they make be\n less than twice the angle at the centre, the lines reflected\n from them and meeting within the circle will make an angle,\n which being added to the angle of the incident lines will be\n equal to twice the angle at the centre.]\n\n5. If two strait lines drawn from one point fall upon the concave\ncircumference of a circle, and the angle they make be less than twice\nthe angle at the centre, the lines reflected from them and meeting\nwithin the circle will make an angle, which being added to the angle of\nthe incident lines, will be equal to twice the angle at the centre.\n\nLet the two lines A B and A C (in fig. 5), drawn from the point A, fall\nupon the concave circumference of the circle whose centre is D; and let\ntheir reflected lines B E and C E meet in the point E; also let the\nangle A be less than twice the angle D. I say, the angles A and E\ntogether taken are equal to twice the angle D.\n\nFor let the strait lines A B and E C cut the strait lines D C and D B in\nthe points G and H; and the angle B H C will be equal to the two angles\nE B H and E; also the same angle B H C will be equal to the two angles D\nand D C H; and in like manner the angle B G C will be equal to the two\nangles A C D and A, and the same angle B G C will be also equal to the\ntwo angles D B G and D. Wherefore the four angles E B H, E, A C D and A,\nare equal to the four angles D, D C H, D B G and D. If, therefore,\nequals be taken away on both sides, namely, on one side A C D and E B H,\nand on the other side D C H and D B G, (for the angle E B H is equal to\nthe angle D B G, and the angle A C D equal to the angle D C H), the\nremainders on both sides will be equal, namely, on one side the angles A\nand E, and on the other the angle D twice taken. Wherefore the angles A\nand E are equal to twice the angle D.\n\nCoroll. If the angle A be greater than twice the angle D, their\nreflected lines will diverge. For, by the corollary of the third\nproposition, if the angle A be equal to twice the angle D, the reflected\nlines B E and C E will be parallel; and if it be less, they will concur,\nas has now been demonstrated. And therefore, if it be greater, the\nreflected lines B E and C E will diverge, and consequently, if they be\nproduced the other way, they will concur and make an angle equal to the\nexcess of the angle A above twice the angle D; as is evident by art. 4.\n\n[Sidenote: If through any one point two unequal chords be drawn cutting\n one another, and the centre of the circle be not placed\n between them, and the lines reflected from them concur\n wheresoever, there cannot through the point, through which\n the two former lines were drawn, be drawn any other strait\n line whose reflected line shall pass through the common point\n of the two former lines reflected.]\n\n6. If through any one point two unequal chords be drawn cutting one\nanother, either within the circle, or, if they be produced, without it,\nand the centre of the circle be not placed between them, and the lines\nreflected from them concur wheresoever; there cannot, through the point\nthrough which the former lines were drawn, be drawn another strait line,\nwhose reflected line shall pass through the point where the two former\nreflected lines concur.\n\nLet any two unequal chords, as B K and C H (in fig. 6), be drawn through\nthe point A in the circle B C; and let their reflected lines B D and C E\nmeet in F; and let the centre not be between A B and A C; and from the\npoint A let any other strait line, as A G, be drawn to the circumference\nbetween B and C. I say, G N, which passes through the point F, where the\nreflected lines B D and C E meet, will not be the reflected line of A G.\n\nFor let the arch B L be taken equal to the arch B G, and the strait line\nB M equal to the strait line B A; and L M being drawn, let it be\nproduced to the circumference in O. Seeing therefore B A and B M are\nequal, and the arch B L equal to the arch B G, and the angle M B L equal\nto the angle A B G, A G and M L will also be equal, and, producing G A\nto the circumference in I, the whole lines L O and G I will in like\nmanner be equal. But L O is greater than G F N, as shall presently be\ndemonstrated; and therefore also G I is greater than G N. Wherefore the\nangles N G C and I G B are not equal. Wherefore the line G F N is not\nreflected from the line of incidence A G, and consequently no other\nstrait line, besides A B and A C, which is drawn through the point A,\nand falls upon the circumference B C, can be reflected to the point F;\nwhich was to be demonstrated.\n\nIt remains that I prove L O to be greater than G N; which I shall do in\nthis manner. L O and G N cut one another in P; and P L is greater than P\nG. Seeing now L P. P G :: P N. P O are proportionals, therefore the two\nextremes L P and P O together taken, that is L O, are greater than P G\nand P N together taken, that is, G N; which remained to be proved.\n\n[Sidenote: In equal chords the same is not true.]\n\n7. But if two equal chords be drawn through one point within a circle,\nand the lines reflected from them meet in another point, then another\nstrait line may be drawn between them through the former point, whose\nreflected line shall pass through the latter point.\n\nLet the two equal chords B C and E D (in the 7th figure) cut one another\nin the point A within the circle B C D; and let their reflected lines C\nH and D I meet in the point F. Then dividing the arch C D equally in G,\nlet the two chords G K and G L be drawn through the points A and F. I\nsay, G L will be the line reflected from the chord K G. For the four\nchords B C, C H, E D and D I are by supposition all equal to one\nanother; and therefore the arch B C H is equal to the arch E D I; as\nalso the angle B C H to the angle E D I; and the angle A M C to its\nverticle angle F M D; and the strait line D M to the strait line G M;\nand, in like manner, the strait line A C to the strait line F D; and the\nchords C G and G D being drawn, will also be equal; and also the angles\nF D G and A C G, in the equal segments G D I and G C B. Wherefore the\nstrait lines F G and A G are equal; and, therefore, the angle F G D is\nequal to the angle A G C, that is, the angle of incidence equal to the\nangle of reflection. Wherefore the line G L is reflected from the\nincident line C G; which was to be proved.\n\nCoroll. By the very sight of the figure it is manifest, that if G be not\nthe middle point between C and D, the reflected line G L will not pass\nthrough the point F.\n\n[Sidenote: Two points being given in the circumference of a circle, to\n draw two strait lines to them so as that their reflected\n lines may contain any angle given.]\n\n8. Two points in the circumference of a circle being given to draw two\nstrait lines to them, so as that their reflected lines may be parallel,\nor contain any angle given.\n\nIn the circumference of the circle, whose centre is A, (in the 8th\nfigure) let the two points B and C be given; and let it be required to\ndraw to them from two points taken without the circle two incident\nlines, so that their reflected lines may, first, be parallel.\n\nLet A B and A C be drawn; as also any incident line D C, with its\nreflected line C F; and let the angle E C D be made double to the angle\nA; and let H B be drawn parallel to E C, and produced till it meet with\nD C produced in I. Lastly, producing A B indefinitely to K, let G B be\ndrawn so that the angle G B K may be equal to the angle H B K, and then\nG B will be the reflected line of the incident line H B. I say, D C and\nH B are two incident lines, whose reflected lines C F and B G are\nparallel.\n\nFor seeing the angle E C D is double to the angle B A C, the angle H I C\nis also, by reason of the parallels E C and H I, double to the same B A\nC; therefore also F C and G B, namely, the lines reflected from the\nincident lines D C and H B, are parallel. Wherefore the first thing\nrequired is done.\n\nSecondly, let it be required to draw to the points B and C two strait\nlines of incidence, so that the lines reflected from them may contain\nthe given angle Z.\n\nTo the angle E C D made at the point C, let there be added on one side\nthe angle D C L equal to half Z, and on the other side the angle E C M\nequal to the angle D C L; and let the strait line B N be drawn parallel\nto the strait line C M; and let the angle K B O be made equal to the\nangle N B K; which being done, B O will be the line of reflection from\nthe line of incidence N B. Lastly, from the incident line L C, let the\nreflected line C O be drawn, cutting B O at O, and making the angle C O\nB. I say, the angle C O B is equal to the angle Z.\n\nLet N B be produced till it meet with the strait line L C produced in P.\nSeeing, therefore, the angle L C M is, by construction, equal to twice\nthe angle B A C, together with the angle Z; the angle N P L, which is\nequal to L C M by reason of the parallels N P and M C, will also be\nequal to twice the same angle B A C, together with the angle Z. And\nseeing the two strait lines O C and O B fall from the point O upon the\npoints C and B; and their reflected lines L C and N B meet in the point\nP; the angle N P L will be equal to twice the angle B A C together with\nthe angle C O B. But I have already proved the angle N P L to be equal\nto twice the angle B A C together with the angle Z. Therefore the angle\nC O B is equal to the angle Z; wherefore, two points in the\ncircumference of a circle being given, I have drawn, &c.; which was to\nbe done.\n\nBut if it be required to draw the incident lines from a point within the\ncircle, so that the lines reflected from them may contain an angle equal\nto the angle Z, the same method is to be used, saving that in this case\nthe angle Z is not to be added to twice the angle B A C, but to be taken\nfrom it.\n\n[Sidenote: If a strait line falling upon the circumference of a circle\n be produced till it reach the semidiameter, and that part of\n it, which is intercepted between the circumference and the\n semidiameter, be equal to that part of the semidiameter which\n is between the point of concourse and the centre, the\n reflected line will be parallel to the semidiameter.]\n\n9. If a strait line, falling upon the circumference of a circle, be\nproduced till it reach the semidiameter, and that part of it which is\nintercepted between the circumference and the semidiameter be equal to\nthat part of the semidiameter which is between the point of concourse\nand the centre, the reflected line will be parallel to the semidiameter.\n\nLet any line A B (in the 9th figure) be the semidiameter of the circle\nwhose centre is A; and upon the circumference B D let the strait line C\nD fall, and be produced till it cut A B in E, so that E D and E A may be\nequal; and from the incident line C D let the line D F be reflected. I\nsay, A B and D F will be parallel.\n\nLet A G be drawn through the point D. Seeing, therefore, E D and E A are\nequal, the angles E D A and E A D will also be equal. But the angles F D\nG and E D A are equal; for each of them is half the angle E D H or F D\nC. Wherefore the angles F D G and E A D are equal; and consequently D F\nand A B are parallel; which was to be proved.\n\nCoroll. If E A be greater then E D, then D F and A B being produced will\nconcur; but if E A be less than E D, then B A and D H being produced\nwill concur.\n\n[Sidenote: If from a point within a circle two strait lines be drawn to\n the circumference, and their reflected lines meet in the\n circumference of the same circle, the angle made by the\n reflected lines will be a third part of the angle made by the\n incident lines.]\n\n10. If from a point within a circle two strait lines be drawn to the\ncircumference, and their reflected lines meet in the circumference of\nthe same circle, the angle made by the lines of reflection will be a\nthird part of the angle made by the lines of incidence.\n\nFrom the point B (in the 10th figure) taken within the circle whose\ncentre is A, let the two strait lines B C and B D be drawn to the\ncircumference; and let their reflected lines C E and D E meet in the\ncircumference of the same circle at the point E. I say, the angle C E D\nwill be a third part of the angle C B D.\n\nLet A C and A D be drawn. Seeing, therefore, the angles C E D and C B D\ntogether taken are equal to twice the angle C A D (as has been\ndemonstrated in the 5th article); and the angle C A D twice taken is\nquadruple to the angle C E D; the angles C E D and C B D together taken\nwill also be equal to the angle C E D four times taken; and therefore if\nthe angle C E D be taken away on both sides, there will remain the angle\nC B D on one side, equal to the angle C E D thrice taken on the other\nside; which was to be demonstrated.\n\nCoroll. Therefore a point being given within a circle, there may be\ndrawn two lines from it to the circumference, so as their reflected\nlines may meet in the circumference. For it is but trisecting the angle\nC B D, which how it may be done shall be shown in the following chapter.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C. XIX.\n _Fig. 1-10_\n]\n\n------------------------------------------------------------------------\n\n\n -------\n\n\n CHAPTER XX.\n\n OF THE DIMENSION OF A CIRCLE, AND THE\n DIVISION OF ANGLES OR ARCHES.\n\n1. The dimension of a circle never determined in numbers by Archimedes\n and others.\u20142. The first attempt for the finding out of the\n dimension of a circle by lines.\u20143. The second attempt for the\n finding out of the dimension of a circle from the consideration of\n the nature of crookedness.\u20144. The third attempt; and some things\n propounded to be further searched into.\u20145. The equation of the\n spiral of Archimedes with a strait line.\u20146. Of the analysis of\n geometricians by the powers of lines.\n\n\n[Sidenote: The dimension of a circle never determined in numbers by\n Archimedes and others.]\n\n1. In the comparing of an arch of a circle with a strait line, many and\ngreat geometricians, even from the most ancient times, have exercised\ntheir wits; and more had done the same, if they had not seen their\npains, though undertaken for the common good, if not brought to\nperfection, vilified by those that envy the praises of other men.\nAmongst those ancient writers whose works are come to our hands,\nArchimedes was the first that brought the length of the perimeter of a\ncircle within the limits of numbers very little differing from the\ntruth; demonstrating the same to be less than three diameters and a\nseventh part, but greater than three diameters and ten seventy-one parts\nof the diameter. So that supposing the radius to consist of 10,000,000\nequal parts, the arch of a quadrant will be between 15,714,285 and\n15,704,225 of the same parts. In our times, Ludovicus Van Cullen and\nWillebrordus Snellius, with joint endeavour, have come yet nearer to the\ntruth; and pronounced from true principles, that the arch of a quadrant,\nputting, as before, 10,000,000 for radius, differs not one whole unity\nfrom the number 15,707,963; which, if they had exhibited their\narithmetical operations, and no man had discovered any error in that\nlong work of theirs, had been demonstrated by them. This is the furthest\nprogress that has been made by the way of numbers; and they that have\nproceeded thus far deserve the praise of industry. Nevertheless, if we\nconsider the benefit, which is the scope at which all speculation should\naim, the improvement they have made has been little or none. For any\nordinary man may much sooner and more accurately find a strait line\nequal to the perimeter of a circle, and consequently square the circle,\nby winding a small thread about a given cylinder, than any geometrician\nshall do the same by dividing the radius into 10,000,000 equal parts.\nBut though the length of the circumference were exactly set out, either\nby numbers, or mechanically, or only by chance, yet this would\ncontribute no help at all towards the section of angles, unless happily\nthese two problems, _to divide a given angle according to any proportion\nassigned_, and _to find a strait line equal to the arch of a circle_,\nwere reciprocal, and followed one another. Seeing therefore the benefit\nproceeding from the knowledge of the length of the arch of a quadrant\nconsists in this, that we may thereby divide an angle according to any\nproportion, either accurately, or at least accurately enough for common\nuse; and seeing this cannot be done by arithmetic, I thought fit to\nattempt the same by geometry, and in this chapter to make trial whether\nit might not be performed by the drawing of strait and circular lines.\n\n[Sidenote: The first attempt for the finding out of the dimension of a\n circle by lines.]\n\n2. Let the square A B C D (in the first figure) be described; and with\nthe radii A B, B C, and D C, the three arches B D, C A, and A C; of\nwhich let the two B D and C A cut one another in E, and the two B D and\nA C in F. The diagonals therefore B D and A C being drawn will cut one\nanother in the centre of the square G, and the two arches B D and C A in\ntwo equal parts in H and Y; and the arch B H D will be trisected in F\nand E. Through the centre G let the two strait lines K G L and M G N be\ndrawn parallel and equal to the sides of the square A B and A D, cutting\nthe four sides of the same square in the points K, L, M, and N; which\nbeing done, K L will pass through F, and M N through E. Then let O P be\ndrawn parallel and equal to the side B C, cutting the arch B F D in F,\nand the sides A B and D C in O and P. Therefore O F will be the sine of\nthe arch B F, which is an arch of 30 degrees; and the same O F will be\nequal to half the radius. Lastly, dividing the arch B F in the middle in\nQ, let R Q, the sine of the arch B Q, be drawn and produced to S, so\nthat Q S be equal to R Q, and consequently R S be equal to the chord of\nthe arch B F; and let F S be drawn and produced to T in the side B C. I\nsay, the strait line B T is equal to the arch B F; and consequently that\nB V, the triple of B T, is equal to the arch of the quadrant B F E D.\n\nLet T F be produced till it meet the side B A produced in X; and\ndividing O F in the middle in Z, let Q Z be drawn and produced till it\nmeet with the side B A produced. Seeing therefore the strait lines R S\nand O F are parallel, and divided in the midst in Q and Z, Q Z produced\nwill fall upon X, and X Z Q produced to the side B C will cut B T in the\nmidst in \u03b1.\n\nUpon the strait line F Z, the fourth part of the radius A B, let the\nequilateral triangle _a_ Z F be constituted; and upon the centre _a_,\nwith the radius _a_ Z, let the arch Z F be drawn; which arch Z F will\ntherefore be equal to the arch Q F, the half of the arch B F. Again, let\nthe strait line Z O be cut in the midst in _b_, and the strait line _b_\nO in the midst in _c_; and let the bisection be continued in this manner\ntill the last part O _c_ be the least that can possibly be taken; and\nupon it, and all the rest of the parts equal to it into which the strait\nline O F may be cut, let so many equilateral triangles be understood to\nbe constituted; of which let the last be _d_ O _c_. If, therefore, upon\nthe centre _d_, with the radius _d_ O, be drawn the arch O _c_, and upon\nthe rest of the equal parts of the strait line O F be drawn in like\nmanner so many equal arches, all those arches together taken will be\nequal to the whole arch B F, and the half of them, namely, those that\nare comprehended between O and Z, or between Z and F, will be equal to\nthe arch B Q or Q F, and in sum, what part soever the strait line O _c_\nbe of the strait line O F, the same part will the arch O _c_ be of the\narch B F, though both the arch and the chord be infinitely bisected. Now\nseeing the arch O _c_ is more crooked than that part of the arch B F\nwhich is equal to it; and seeing also that the more the strait line X\n_c_ is produced, the more it diverges from the strait line X O, if the\npoints O and _c_ be understood to be moved forwards with strait motion\nin X O and X _c_, the arch O _c_ will thereby be extended by little and\nlittle, till at the last it come somewhere to have the same crookedness\nwith that part of the arch B F which is equal to it. In like manner, if\nthe strait line X _b_ be drawn, and the point _b_ be understood to be\nmoved forwards at the same time, the arch _c b_ will also by little and\nlittle be extended, till its crookedness come to be equal to the\ncrookedness of that part of the arch B F which is equal to it. And the\nsame will happen in all those small equal arches which are described\nupon so many equal parts of the strait line O F. It is also manifest,\nthat by strait motion in X O and X Z all those small arches will lie in\nthe arch B F, in the points B, Q and F. And though the same small equal\narches should not be coincident with the equal parts of the arch B F in\nall the other points thereof, yet certainly they will constitute two\ncrooked lines, not only equal to the two arches B Q and Q F, and equally\ncrooked, but also having their cavity towards the same parts; which how\nit should be, unless all those small arches should be coincident with\nthe arch B F in all its points, is not imaginable. They are therefore\ncoincident, and all the strait lines drawn from X, and passing through\nthe points of division of the strait line O F, will also divide the arch\nB F into the same proportions into which O F is divided.\n\nNow seeing X _b_ cuts off from the point B the fourth part of the arch B\nF, let that fourth part be B _e_; and let the sine thereof, _f e_, be\nproduced to F T in _g_, for so _f e_ will be the fourth part of the\nstrait line _f g_, because as O _b_ is to O F, so is _f e_ to _f g_. But\nB T is greater than _f g_; and therefore the same B T is greater than\nfour sines of the fourth part of the arch B F. And in like manner, if\nthe arch B F be subdivided into any number of equal parts whatsoever, it\nmay be proved that the strait line B T is greater than the sine of one\nof those small arches, so many times taken as there be parts made of the\nwhole arch B F. Wherefore the strait line B T is not less than the arch\nB F. But neither can it be greater, because if any strait line\nwhatsoever, less than B T, be drawn below B T, parallel to it, and\nterminated in the strait lines X B and X T, it would cut the arch B F;\nand so the sine of some one of the parts of the arch B F, taken so often\nas that small arch is found in the whole arch B F, would be greater than\nso many of the same arches; which is absurd. Wherefore the strait line B\nT is equal to the arch B F; and the strait line B V equal to the arch of\nthe quadrant B F D; and B V four times taken, equal to the perimeter of\nthe circle described with the radius A B. Also the arch B F and the\nstrait line B T are everywhere divided into the same proportions; and\nconsequently any given angle, whether greater or less than B A F, may be\ndivided into any proportion given.\n\nBut the strait line B V, though its magnitude fall within the terms\nassigned by Archimedes, is found, if computed by the canon of signs, to\nbe somewhat greater than that which is exhibited by the Rudolphine\nnumbers. Nevertheless, if in the place of B T, another strait line,\nthough never so little less, be substituted, the division of angles is\nimmediately lost, as may by any man be demonstrated by this very scheme.\n\nHowsoever, if any man think this my strait line B V to be too great,\nyet, seeing the arch and all the parallels are everywhere so exactly\ndivided, and B V comes so near to the truth, I desire he would search\nout the reason, why, granting B V to be precisely true, the arches cut\noff should not be equal.\n\nBut some man may yet ask the reason why the strait lines, drawn from X\nthrough the equal parts of the arch B F, should cut off in the tangent B\nV so many strait lines equal to them, seeing the connected straight line\nX V passes not through the point D, but cuts the strait line A D\nproduced in _l_; and consequently require some determination of this\nproblem. Concerning which, I will say what I think to be the reason,\nnamely, that whilst the magnitude of the arch doth not exceed the\nmagnitude of the radius, that is, the magnitude of the tangent B C, both\nthe arch and the tangent are cut alike by the strait lines drawn from X;\notherwise not. For A V being connected, cutting the arch B H D in I, if\nX C being drawn should cut the same arch in the same point I, it would\nbe as true that the arch B I is equal to the radius B C, as it is true\nthat the arch B F is equal to the strait line B T; and drawing X K it\nwould cut the arch B I in the midst in _i_; also drawing A _i_ and\nproducing it to the tangent B C in _k_, the strait line B _k_ will be\nthe tangent of the arch B _i_, (which arch is equal to half the radius)\nand the same strait line B _k_ will be equal to the strait line _k_ I. I\nsay all this is true, if the preceding demonstration be true; and\nconsequently the proportional section of the arch and its tangent\nproceeds hitherto. But it is manifest by the golden rule, that taking B\n_h_ double to B T, the line X _h_ shall not cut off the arch B E, which\nis double to the arch B F, but a much greater. For the magnitude of the\nstraight lines X M, X B, and M E, being known (in numbers), the\nmagnitude of the strait line cut off in the tangent by the strait line X\nE produced to the tangent, may also be known; and it will be found to be\nless than B _h_; Wherefore the strait line X _h_ being drawn, will cut\noff a part of the arch of the quadrant greater than the arch B E. But I\nshall speak more fully in the next article concerning the magnitude of\nthe arch B I.\n\nAnd let this be the first attempt for the finding out of the dimension\nof a circle by the section of the arch B F.\n\n[Sidenote: The second attempt for the finding out of the dimension of a\n circle from the consideration of the nature of crookedness.]\n\n3. I shall now attempt the same by arguments drawn from the nature of\nthe crookedness of the circle itself; but I shall first set down some\npremises necessary for this speculation; and\n\nFirst, if a strait line be bowed into an arch of a circle equal to it,\nas when a stretched thread, which toucheth a right cylinder, is so bowed\nin every point, that it be everywhere coincident with the perimeter of\nthe base of the cylinder, the flexion of that line will be equal, in all\nits points; and consequently the crookedness of the arch of a circle is\neverywhere uniform; which needs no other demonstration than this, that\nthe perimeter of a circle is an uniform line.\n\nSecondly, and consequently: if two unequal arches of the same circle be\nmade by the bowing of two strait lines equal to them, the flexion of the\nlonger line, whilst it is bowed into the greater arch, is greater than\nthe flexion of the shorter line, whilst it is bowed into the lesser\narch, according to the proportion of the arches themselves; and\nconsequently, the crookedness of the greater arch is to the crookedness\nof the lesser arch, as the greater arch is to the lesser arch.\n\nThirdly: if two unequal circles and a strait line touch one another in\nthe same point, the crookedness of any arch taken in the lesser circle,\nwill be greater than the crookedness of an arch equal to it taken in the\ngreater circle, in reciprocal proportion to that of the radii with which\nthe circles are described; or, which is all one, any strait line being\ndrawn from the point of contact till it cut both the circumferences, as\nthe part of that strait line cut off by the circumference of the greater\ncircle to that part which is cut off by the circumference of the lesser\ncircle.\n\nFor let A B and A C (in the second figure) be two circles, touching one\nanother, and the strait line A D in the point A; and let their centres\nbe E and F; and let it be supposed, that as A E is to A F, so is the\narch A B to the arch A H. I say the crookedness of the arch A C is to\nthe crookedness of the arch A H, as A E is to A F. For let the strait\nline A D be supposed to be equal to the arch A B, and the strait line A\nG to the arch A C; and let A D, for example, be double to A G.\nTherefore, by reason of the likeness of the arches A B and A C, the\nstrait line A B will be double to the strait line A C, and the radius A\nE double to the radius A F, and the arch A B double to the arch A H. And\nbecause the strait line A D is so bowed to be coincident with the arch A\nB equal to it, as the strait line A G is bowed to be coincident with the\narch A C equal also to it, the flexion of the strait line A G into the\ncrooked line A C will be equal to the flexion of the strait line A D\ninto the crooked line A B. But the flexion of the strait line A D into\nthe crooked line A B is double to the flexion of the strait line A G\ninto the crooked line A H; and therefore the flexion of the strait line\nA G into the crooked line A C is double to the flexion of the same\nstrait line A G into the crooked line A H. Wherefore, as the arch A B is\nto the arch A C or A H; or as the radius A E is to the radius A F; or as\nthe chord A B is to the chord A C; so reciprocally is the flexion or\nuniform crookedness of the arch A C, to the flexion or uniform\ncrookedness of the arch A H, namely, here double. And this may by the\nsame method be demonstrated in circles whose perimeters are to one\nanother triple, quadruple, or in whatsoever given proportion. The\ncrookedness therefore of two equal arches taken in several circles are\nin proportion reciprocal to that of their radii, or like arches, or like\nchords; which was to be demonstrated.\n\nLet the square A B C D be again described (in the third figure), and in\nit the quadrants A B D, B C A and D A C; and dividing each side of the\nsquare A B C D in the midst in E, F, G and H, let E G and F H be\nconnected, which will cut one another in the centre of the square at I,\nand divide the arch of the quadrant A B D into three equal parts in K\nand L. Also the diagonals A C and B D being drawn will cut one another\nin I, and divide the arches B K D and C L A into two equal parts in M\nand N. Then with the radius B F let the arch F E be drawn, cutting the\ndiagonal B D in O; and dividing the arch B M in the midst in P, let the\nstrait line E _a_ equal to the chord B P be set off from the point E in\nthe arch E F, and let the arch _a b_ be taken equal to the arch O _a_,\nand let B _a_ and B _b_ be drawn and produced to the arch A N in _c_ and\n_d_; and lastly, let the strait line A _d_ be drawn. I say the strait\nline A _d_ is equal to the arch A N or B M.\n\nI have proved in the preceding article, that the arch E O is twice as\ncrooked as the arch B P, that is to say, that the arch E O is so much\nmore crooked than the arch B P, as the arch B P is more crooked than the\nstrait line E _a_. The crookedness therefore of the chord E _a_, of the\narch B P, and of the arch E O, are as 0, 1, 2. Also the difference\nbetween the arches E O and E O, the difference between the arches E O\nand E _a_, and the difference between the arches E O and E _b_, are as\n0, 1, 2. So also the difference between the arches A N and A N, the\ndifference between the arches A N and A _c_, and the difference between\nthe arches A N and A _d_, are as 0, 1, 2; and the strait line A _c_ is\ndouble to the chord B P or E _a_, and the strait line A _d_ double to\nthe chord E _b_.\n\nAgain, let the strait line B F be divided in the midst in Q, and the\narch B P in the midst in R; and describing the quadrant B Q S (whose\narch Q S is a fourth part of the arch of the quadrant B M D, as the arch\nB R is a fourth part of the arch B M, which is the arch of the\nsemiquadrant A B M) let the chord S _e_ equal to the chord B R be set\noff from the point S in the arch S Q; and let B _e_ be drawn and\nproduced to the arch A N in _f_; which being done, the strait line A _f_\nwill be quadruple to the chord B R or S _e_. And seeing the crookedness\nof the arch S _e_, or of the arch A _c_, is double to the crookedness of\nthe arch B R, the excess of the crookedness of the arch A _f_ above the\ncrookedness of the arch A _c_ will be subduple to the excess of the\ncrookedness of the arch A _c_ above the crookedness of the arch A N; and\ntherefore the arch N _c_ will be double to the arch _c f_. Wherefore the\narch _c d_ is divided in the midst in _f_, and the arch N _f_ is \u00be of\nthe arch N _d_. And in like manner if the arch B R be bisected in V, and\nthe strait line B Q in X, and the quadrant B X Y be described, and the\nstrait line Y _g_ equal to the chord B V be set off from the point Y in\nthe arch Y X, it may be demonstrated that the strait line B _g_ being\ndrawn and produced to the arch A N, will cut the arch _f d_ into two\nequal parts, and that a strait line drawn from A to the point of that\nsection, will be equal to eight chords of the arch B V, and so on\nperpetually; and consequently, that the strait line A _d_ is equal to so\nmany equal chords of equal parts of the arch B M, as may be made by\ninfinite bisections. Wherefore the strait line A _d_ is equal to the\narch B M or A N, that is, to half the arch of the quadrant A B D or B C\nA.\n\nCoroll. An arch being given not greater than the arch of a quadrant (for\nbeing made greater, it comes again towards the radius B A produced, from\nwhich it receded before) if a strait line double to the chord of half\nthe given arch be adapted from the beginning of the arch, and by how\nmuch the arch that is subtended by it is greater than the given arch, by\nso much a greater arch be subtended by another strait line, this strait\nline shall be equal to the first given arch.\n\nSupposing the strait line B V (in fig. 1) be equal to the arch of the\nquadrant B H D, and A V be connected cutting the arch B H D in I, it may\nbe asked what proportion the arch B I has to the arch I D. Let therefore\nthe arch A Y be divided in the midst in _o_, and in the strait line A D\nlet A _p_ be taken equal, and A _q_ double to the drawn chord A _o_.\nThen upon the centre A, with the radius A _q_, let an arch of a circle\nbe drawn cutting the arch A Y in _r_, and let the arch Y _r_ be doubled\nat _t_; which being done, the drawn strait line A _t_ (by what has been\nlast demonstrated) will be equal to the arch A Y. Again, upon the centre\nA with the radius A _t_ let the arch _t u_ be drawn cutting A D in _u_;\nand the strait line A _u_ will be equal to the arch A Y. From the point\n_u_ let the strait line _u s_ be drawn equal and parallel to the strait\nline A B, cutting M N in _x_, and bisected by M N in the same point _x_.\nTherefore the strait line A _x_ being drawn and produced till it meet\nwith B C produced in V, it will cut off B V double to B _s_, that is,\nequal to the arch B H D. Now let the point, where the strait line A V\ncuts the arch B H D, be I; and let the arch D I be divided in the midst\nin _y_; and in the strait line D C, let D _z_ be taken equal, and D \u03b4\ndouble to the drawn chord D _y_; and upon the centre D with the radius D\n\u03b4 let an arch of a circle be drawn cutting the arch B H D in the point\n_n_; and let the arch _n m_ be taken equal to the arch I _n_; which\nbeing done, the strait line D _m_ will (by the last foregoing corollary)\nbe equal to the arch D I. If now the strait lines D _m_ and C V be\nequal, the arch B I will be equal to the radius A B or B C; and\nconsequently X C being drawn, will pass through the point I. Moreover,\nif the semicircle B H D \u03d0 being completed, the strait lines \u03d0 I and B I\nbe drawn, making a right angle (in the semicircle) at I, and the arch B\nI be divided in the midst at _i_, it will follow that A _i_ being\nconnected will be parallel to the strait line \u03d0 I, and being produced to\nB C in _k_, will cut off the strait line B _k_ equal to the strait line\n_k_ I, and equal also to the strait line A \u03b3 cut off in A D by the\nstrait line \u03d0 I. All which is manifest, supposing the arch B I and the\nradius B C to be equal.\n\nBut that the arch B I and the radius B C are precisely equal, cannot\n(how true soever it be) be demonstrated, unless that be first proved\nwhich is contained in art. 1, namely, that the strait lines drawn from X\nthrough the equal parts of O F (produced to a certain length) cut off so\nmany parts also in the tangent B C severally equal to the several arches\ncut off; which they do most exactly as far as B C in the tangent, and B\nI in the arch B E; insomuch that no inequality between the arch B I and\nthe radius B C can be discovered either by the hand or by ratiocination.\nIt is therefore to be further enquired, whether the strait line A V cut\nthe arch of the quadrant in I in the same proportion as the point C\ndivides the strait line B V, which is equal to the arch of the quadrant.\nBut however this be, it has been demonstrated that the strait line B V\nis equal to the arch B H D.\n\n[Sidenote: The third attempt; and some things propounded to be further\n searched into.]\n\n4. I shall now attempt the same dimension of a circle another way,\nassuming the two following lemmas.\n\nLemma I. If to the arch of a quadrant, and the radius, there be taken in\ncontinual proportion a third line Z; then the arch of the semiquadrant,\nhalf the chord of the quadrant, and Z, will also be in continual\nproportion.\n\nFor seeing the radius is a mean proportional between the chord of a\nquadrant and its semichord, and the same radius a mean proportional\nbetween the arch of the quadrant and Z, the square of the radius will be\nequal as well to the rectangle made of the chord and semichord of the\nquadrant, as to the rectangle made of the arch of the quadrant and Z;\nand these two rectangles will be equal to one another. Wherefore, as the\narch of a quadrant is to its chord, so reciprocally is half the chord of\nthe quadrant to Z. But as the arch of the quadrant is to its chord, so\nis half the arch of the quadrant to half the chord of the quadrant.\nWherefore, as half the arch of the quadrant is to half the chord of the\nquadrant (or to the sine of 45 degrees), so is half the chord of the\nquadrant to Z; which was to be proved.\n\nLemma II. The radius, the arch of the semiquadrant, the sine of 45\ndegrees, and the semiradius, are proportional.\n\nFor seeing the sine of 45 degrees is a mean proportional between the\nradius and the semiradius; and the same sine of 45 degrees is also a\nmean proportional (by the precedent lemma) between the arch of 45\ndegrees and Z; the square of the sine of 45 degrees will be equal as\nwell to the rectangle made of the radius and semiradius, as to the\nrectangle made of the arch of 45 degrees and Z. Wherefore, as the radius\nis to the arch of 45 degrees, so reciprocally is Z to the semiradius;\nwhich was to be demonstrated.\n\nLet now A B C D (in fig. 4) be a square; and with the radii A B, B C and\nD A, let the three quadrants A B D, B C A and D A C, be described; and\nlet the strait lines E F and G H, drawn parallel to the sides B C and A\nB, divide the square A B C D into four equal squares. They will\ntherefore cut the arch of the quadrant A B D into three equal parts in I\nand K, and the arch of the quadrant B C A into three equal parts in K\nand L. Also let the diagonals A C and B D be drawn, cutting the arches B\nI D and A L C in M and N. Then upon the centre H with the radius H F\nequal to half the chord of the arch B M D, or to the sine of 45 degrees,\nlet the arch F O be drawn cutting the arch C K in O; and let A O be\ndrawn and produced till it meet with B C produced in P; also let it cut\nthe arch B M D in Q, and the strait line D C in R. If now the strait\nline H Q be equal to the strait line D R, and being produced to D C in\nS, cut off D S equal to half the strait line B P; I say then the strait\nline B P will be equal to the arch B M D.\n\nFor seeing P B A and A D R are like triangles, it will be as P B to the\nradius B A or A D, so A D to D R; and therefore as well P B, A D and D\nR, as P B, A D (or A Q) and Q H are in continual proportion; and\nproducing H O to D C in T, D T will be equal to the sine of 45 degrees,\nas shall by and by be demonstrated. Now D S, D T and D R are in\ncontinual proportion by the first lemma; and by the second lemma D C. D\nS:: D R. D F are proportionals. And thus it will be, whether B P be\nequal or not equal to the arch of the quadrant B M D. But if they be\nequal, it will then be, as that part of the arch B M D which is equal to\nthe radius, is to the remainder of the same arch B M D; so A Q to H Q,\nor so B C to C P. And then will B P and the arch B M D be equal. But it\nis not demonstrated that the strait lines H Q and D R are equal; though\nif from the point B there be drawn (by the construction of fig. 1) a\nstrait line equal to the arch B M D, then D R to H Q, and also the half\nof the strait line B P to D S, will always be so equal, that no\ninequality can be discovered between them. I will therefore leave this\nto be further searched into. For though it be almost out of doubt, that\nthe strait line B P and the arch B M D are equal, yet that may not be\nreceived without demonstration; and means of demonstration the circular\nline admitteth none that is not grounded upon the nature of flexion, or\nof angles. But by that way I have already exhibited a strait line equal\nto the arch of a quadrant in the first and second aggression.\n\nIt remains that I prove D T to be equal to the sine of 45 degrees.\n\nIn B A produced let A V be taken equal to the sine of 45 degrees; and\ndrawing and producing V H, it will cut the arch of the quadrant C N A in\nthe midst in N, and the same arch again in O, and the strait line D C in\nT, so that D T will be equal to the sine of 45 degrees, or to the strait\nline A V; also the strait line V H will be equal to the strait line H I,\nor the sine of 60 degrees.\n\nFor the square of A V is equal to two squares of the semiradius; and\nconsequently the square of V H is equal to three squares of the\nsemiradius. But H I is a mean proportional between the semiradius and\nthree semiradii; and, therefore, the square of H I is equal to three\nsquares of the semiradius. Wherefore H I is equal to H V. But because A\nD is cut in the midst in H, therefore V H and H T are equal; and,\ntherefore, also D T is equal to the sine of 45 degrees. In the radius B\nA let B X be taken equal to the sine of 45 degrees; for so V X will be\nequal to the radius; and it will be as V A to A H the semiradius, so V X\nthe radius to X N the sine of 45 degrees. Wherefore V H produced passes\nthrough N. Lastly, upon the centre V with the radius V A let the arch of\na circle be drawn cutting V H in Y; which being done, V Y will be equal\nto H O (for H O is, by construction, equal to the sine of 45 degrees)\nand Y H will be equal to O T; and, therefore, V T passes through O. All\nwhich was to be demonstrated.\n\nI will here add certain problems, of which if any analyst can make the\nconstruction, he will thereby be able to judge clearly of what I have\nnow said concerning the dimension of a circle. Now these problems are\nnothing else (at least to sense) but certain symptoms accompanying the\nconstruction of the first and third figure of this chapter.\n\nDescribing, therefore, again, the square A B C D (in fig. 5) and the\nthree quadrants A B D, B C A and D A C, let the diagonals A C and B D be\ndrawn, cutting the arches B H D and C I A in the middle in H and I; and\nthe strait lines E F and G L, dividing the square A B C D into four\nequal squares, and trisecting the arches B H D and C I A, namely, B H D\nin K and M, and C I A in M and O. Then dividing the arch B K in the\nmidst in P, let Q P the sine of the arch B P, be drawn and produced to\nR, so that Q R be double to Q P; and, connecting K R, let it be produced\none way to B C in S, and the other way to B A produced in T. Also let B\nV be made triple to B S, and consequently, (by the second article of\nthis chapter) equal to the arch B D. This construction is the same with\nthat of the first figure, which I thought fit to renew discharged of all\nlines but such as are necessary for my present purpose.\n\nIn the first place, therefore, if A V be drawn, cutting the arch B H D\nin X, and the side D C in Z, I desire some analyst would, if he can,\ngive a reason why the strait lines T E and T C should cut the arch B D,\nthe one in Y, the other in X, so as to make the arch B Y equal to the\narch Y X; or if they be not equal, that he would determine their\ndifference.\n\nSecondly, if in the side D A, the strait line D _a_ be taken equal to D\nZ, and V _a_ be drawn; why V _a_ and V B should be equal; or if they be\nnot equal, what is the difference.\n\nThirdly, drawing Z _b_ parallel and equal to the side C B, cutting the\narch B H D in _c_, and drawing the strait line A _c_, and producing it\nto B V in _d_; why A _d_ should be equal and parallel to the strait line\n_a_ V, and consequently equal also to the arch B D.\n\nFourthly, drawing _e_ K the sine of the arch B K, and taking (in _e_ A\nproduced) _e f_ equal to the diagonal A C, and connecting _f_ C; why _f_\nC should pass through _a_ (which point being given, the length of the\narch B H D is also given) and _c_; and why _f e_ and _f c_ should be\nequal; or if not, why unequal.\n\nFifthly, drawing _f_ Z, I desire he would show, why it is equal to B V,\nor to the arch B D; or if they be not equal, what is their difference.\n\nSixthly, granting _f_ Z to be equal to the arch B D, I desire he would\ndetermine whether it fall all without the arch B C A, or cut the same,\nor touch it, and in what point.\n\nSeventhly, the semicircle B D _g_ being completed, why _g_ I being drawn\nand produced, should pass through X, by which point X the length of the\narch B D is determined. And the same _g_ I being yet further produced to\nD C in _h_, why A _d_, which is equal to the arch B D, should pass\nthrough that point _h_.\n\nEighthly, upon the centre of the square A B C D, which let be _k_, the\narch of the quadrant E _i_ L being drawn, cutting _e_ K produced in _i_,\nwhy the drawn strait line _i_ X should be parallel to the side C D.\n\nNinthly, in the sides B A and B C taking _g l_ and B _m_ severally equal\nto half B V, or to the arch B H, and drawing _m n_ parallel and equal to\nthe side B A, cutting the arch B D in _o_, why the strait line which\nconnects V _l_ should pass through the point _o_.\n\nTenthly, I would know of him why the strait line which connects _a_ H\nshould be equal to B _m_; or if not, how much it differs from it.\n\nThe analyst that can solve these problems without knowing first the\nlength of the arch B D, or using any other known method than that which\nproceeds by perpetual bisection of an angle, or is drawn from the\nconsideration of the nature of flexion, shall do more than ordinary\ngeometry is able to perform. But if the dimension of a circle cannot be\nfound by any other method, then I have either found it, or it is not at\nall to be found.\n\nFrom the known length of the arch of a quadrant, and from the\nproportional division of the arch and of the tangent B C, may be deduced\nthe section of an angle into any given proportion; as also the squaring\nof the circle, the squaring of a given sector, and many the like\npropositions, which it is not necessary here to demonstrate. I will,\ntherefore, only exhibit a strait line equal to the spiral of Archimedes,\nand so dismiss this speculation.\n\n[Sidenote: The equation of the spiral of Archimedes with a strait line.]\n\n5. The length of the perimeter of a circle being found, that strait line\nis also found, which touches a spiral at the end of its first\nconversion. For upon the centre A (in fig. 6) let the circle B C D E be\ndescribed; and in it let Archimedes' spiral A F G H B be drawn,\nbeginning at A and ending at B. Through the centre A let the strait line\nC E be drawn, cutting the diameter B D at right angles; and let it be\nproduced to I, so that A I be equal to the perimeter B C D E B.\nTherefore I B being drawn will touch the spiral A F G H B in B; which is\ndemonstrated by Archimedes in his book _De Spiralibus_.\n\nAnd for a strait line equal to the given spiral A F G H B, it may be\nfound thus.\n\nLet the strait line A I, which is equal to the perimeter B C D E, be\nbisected in K; and taking K L equal to the radius A B, let the rectangle\nI L be completed. Let M L be understood to be the axis, and K L the base\nof a parabola, and let M K be the crooked line thereof. Now if the point\nM be conceived to be so moved by the concourse of two movents, the one\nfrom I M to K L with velocity encreasing continually in the same\nproportion with the times, the other from M L to I K uniformly, that\nboth those motions begin together in M and end in K; Galil\u00e6us has\ndemonstrated that by such motion of the point M, the crooked line of a\nparabola will be described. Again, if the point A be conceived to be\nmoved uniformly in the strait line A B, and in the same time to be\ncarried round upon the centre A by the circular motion of all the points\nbetween A and B; Archimedes has demonstrated that by such motion will be\ndescribed a spiral line. And seeing the circles of all these motions are\nconcentric in A; and the interior circle is always less than the\nexterior in the proportion of the times in which A B is passed over with\nuniform motion; the velocity also of the circular motion of the point A\nwill continually increase proportionally to the times. And thus far the\ngenerations of the parabolical line M K, and of the spiral line A F G H\nB, are like. But the uniform motion in A B concurring with circular\nmotion in the perimeters of all the concentric circles, describes that\ncircle, whose centre is A, and perimeter B C D E; and, therefore, that\ncircle is (by the coroll. of art. 1, chap, XVI) the aggregate of all the\nvelocities together taken of the point A whilst it describes the spiral\nA F G H B. Also the rectangle I K L M is the aggregate of all the\nvelocities together taken of the point M, whilst it describes the\ncrooked line M K. And, therefore the whole velocity by which the\nparabolical line M K is described, is to the whole velocity with which\nthe spiral line A F G H B is described in the same time, as the\nrectangle I K L M is to the circle B C D E, that is to the triangle A I\nB. But because A I is bisected in K, and the strait lines I M and A B\nare equal, therefore the rectangle I K L M and the triangle A I B are\nalso equal. Wherefore the spiral line A F G H B, and the parabolical\nline M K, being described with equal velocity and in equal times, are\nequal to one another. Now, in the first article of chap. XVIII, a strait\nline is found out equal to any parabolical line. Wherefore also a strait\nline is found out equal to a given spiral line of the first revolution\ndescribed by Archimedes; which was to be done.\n\n[Sidenote: Of the analysis of geometricians by the powers of lines.]\n\n6. In the sixth chapter, which is of _Method_, that which I should there\nhave spoken of the analytics of geometricians I thought fit to defer,\nbecause I could not there have been understood, as not having then so\nmuch as named _lines_, _superficies_, _solids_, _equal_ and _unequal_,\n_&c._ Wherefore I will in this place set down my thoughts concerning it.\n\n_Analysis_ is continual reasoning from the definitions of the terms of a\nproposition we suppose true, and again from the definitions of the terms\nof those definitions, and so on, till we come to some things known, the\ncomposition whereof is the demonstration of the truth or falsity of the\nfirst supposition; and this composition or demonstration is that we call\n_Synthesis_. _Analytica_, therefore, is that art, by which our reason\nproceeds from something supposed, to principles, that is, to prime\npropositions, or to such as are known by these, till we have so many\nknown propositions as are sufficient for the demonstration of the truth\nor falsity of the thing supposed. _Synthetica_ is the art itself of\ndemonstration. Synthesis, therefore, and analysis, differ in nothing,\nbut in proceeding forwards or backwards; and _Logistica_ comprehends\nboth. So that in the analysis or synthesis of any question, that is to\nsay, of any problem, the terms of all the propositions ought to be\nconvertible; or if they be enunciated hypothetically, the truth of the\nconsequent ought not only to follow out of the truth of its antecedent,\nbut contrarily also the truth of the antecedent must necessarily be\ninferred from the truth of the consequent. For otherwise, when by\nresolution we are arrived at principles, we cannot by composition return\ndirectly back to the thing sought for. For those terms which are the\nfirst in analysis, will be the last in synthesis; as for example, when\nin _resolving_, we say, these two rectangles are equal, and therefore\ntheir sides are reciprocally proportional, we must necessarily in\n_compounding_ say, the sides of these rectangles are reciprocally\nproportional, and therefore the rectangles themselves are equal; which\nwe could not say, unless _rectangles have their sides reciprocally\nproportional_, and _rectangles are equal_, were terms convertible.\n\nNow in every analysis, that which is sought is the proportion of two\nquantities; by which proportion, a figure being described, the quantity\nsought for may be exposed to sense. And this _exposition_ is the end and\nsolution of the question, or the construction of the problem.\n\nAnd seeing analysis is reasoning from something supposed, till we come\nto principles, that is, to definitions, or to theorems formerly known;\nand seeing the same reasoning tends in the last place to some equation,\nwe can therefore make no end of resolving, till we come at last to the\ncauses themselves of equality and inequality, or to theorems formerly\ndemonstrated from those causes; and so have a sufficient number of those\ntheorems for the demonstration of the thing sought for.\n\nAnd seeing also, that the end of the analytics is either the\nconstruction of such a problem as is possible, or the detection of the\nimpossibility thereof; whensoever the problem may be solved, the analyst\nmust not stay, till he come to those things which contain the efficient\ncause of that whereof he is to make construction. But he must of\nnecessity stay, when he comes to prime propositions; and these are\ndefinitions. These definitions therefore must contain the efficient\ncause of his construction; I say of his construction, not of the\nconclusion which he demonstrates; for the cause of the conclusion is\ncontained in the premised propositions; that is to say, the truth of the\nproposition he proves is drawn from the propositions which prove the\nsame. But the cause of his construction is in the things themselves, and\nconsists in motion, or in the concourse of motions. Wherefore those\npropositions, in which analysis ends, are definitions, but such as\nsignify in what manner the construction or generation of the thing\nproceeds. For otherwise, when he goes back by synthesis to the proof of\nhis problem, he will come to no demonstration at all; there being no\ntrue demonstration but such as is scientifical; and no demonstration is\nscientifical, but that which proceeds from the knowledge of the causes\nfrom which the construction of the problem is drawn. To collect\ntherefore what has been said into few words; ANALYSIS _is ratiocination\nfrom the supposed construction or generation of a thing to the efficient\ncause or coefficient causes of that which is constructed or generated.\nAnd_ SYNTHESIS _is ratiocination from the first causes of the\nconstruction, continued through all the middle causes till we come to\nthe thing itself which is constructed or generated_.\n\nBut because there are many means by which the same thing may be\ngenerated, or the same problem be constructed, therefore neither do all\ngeometricians, nor doth the same geometrician always, use one and the\nsame method. For, if to a certain quantity given, it be required to\nconstruct another quantity equal, there may be some that will inquire\nwhether this may not be done by means of some motion. For there are\nquantities, whose equality and inequality may be argued from motion and\ntime, as well as from congruence; and there is motion, by which two\nquantities, whether lines or superficies, though one of them be crooked,\nthe other strait, may be made congruous or coincident. And this method\nArchimedes made use of in his book _De Spiralibus_. Also the equality or\ninequality of two quantities may be found out and demonstrated from the\nconsideration of weight, as the same Archimedes did in his quadrature of\nthe parabola. Besides, equality and inequality are found out often by\nthe division of the two quantities into parts which are considered as\nindivisable; as Cavallerius Bonaventura has done in our time, and\nArchimedes often. Lastly, the same is performed by the consideration of\nthe powers of lines, or the roots of those powers, and by the\nmultiplication, division, addition, and subtraction, as also by the\nextraction of the roots of those powers, or by finding where strait\nlines of the same proportion terminate. For example, when any number of\nstrait lines, how many soever, are drawn from a strait line and pass all\nthrough the same point, look what proportion they have, and if their\nparts continued from the point retain everywhere the same proportion,\nthey shall all terminate in a strait line. And the same happens if the\npoint be taken between two circles. So that the places of all their\npoints of termination make either strait lines, or circumferences of\ncircles, and are called _plane places_. So also when strait parallel\nlines are applied to one strait line, if the parts of the strait line to\nwhich they are applied be to one another in proportion duplicate to that\nof the contiguous applied lines, they will all terminate in a conical\nsection; which section, being the place of their termination, is called\na _solid place_, because it serves for the finding out of the quantity\nof any equation which consists of three dimensions. There are therefore\nthree ways of finding out the cause of equality or inequality between\ntwo given quantities; namely, first, by the computation of _motions_;\nfor by equal motion, and equal time, equal spaces are described; and\nponderation is motion. Secondly, by _indivisibles_: because all the\nparts together taken are equal to the whole. And thirdly, by the\n_powers_: for when they are equal, their roots also are equal; and\ncontrarily, the powers are equal, when their roots are equal. But if the\nquestion be much complicated, there cannot by any of these ways be\nconstituted a certain rule, from the supposition of which of the unknown\nquantities the analysis may best begin; nor out of the variety of\nequations, that at first appear, which we were best to choose; but the\nsuccess will depend upon dexterity, upon formerly acquired science, and\nmany times upon fortune.\n\nFor no man can ever be a good analyst without being first a good\ngeometrician; nor do the rules of analysis make a geometrician, as\nsynthesis doth; which begins at the very elements, and proceeds by a\nlogical use of the same. For the true teaching of geometry is by\nsynthesis, according to Euclid's method; and he that hath Euclid for his\nmaster, may be a geometrician without Vieta, though Vieta was a most\nadmirable geometrician; but he that has Vieta for his master, not so,\nwithout Euclid.\n\nAnd as for that part of analysis which works by the powers, though it be\nesteemed by some geometricians, not the chiefest, to be the best way of\nsolving all problems, yet it is a thing of no great extent; it being all\ncontained in the doctrine of rectangles, and rectangled solids. So that\nalthough they come to an equation which determines the quantity sought,\nyet they cannot sometimes by art exhibit that quantity in a plane, but\nin some conic section; that is, as geometricians say, not geometrically,\nbut mechanically. Now such problems as these, they call _solid_; and\nwhen they cannot exhibit the quantity sought for with the help of a\nconic section, they call it a _lineary_ problem. And therefore in the\nquantities of angles, and of the arches of circles, there is no use at\nall of the analytics which proceed by the powers; so that the ancients\npronounced it impossible to exhibit in a plane the division of angles,\nexcept bisection, and the bisection of the bisected parts, otherwise\nthan mechanically. For Pappus, (before the 31st proposition of his\nfourth book) distinguishing and defining the several kinds of problems,\nsays that \"some are _plane_, others _solid_, and others _lineary_.\nThose, therefore, which may be solved by strait lines and the\ncircumferences of circles, (that is, which may be described with the\nrule and compass, without any other instrument), are fitly called\n_plane_; for the lines, by which such problems are found out, have their\ngeneration in a plane. But those which are solved by the using of some\none or more conic sections in their construction, are called _solid_,\nbecause their construction cannot be made without using the superficies\nof solid figures, namely, of cones. There remains the third kind, which\nis called _lineary_, because other lines besides those already mentioned\nare made use of in their construction, &c.\" And a little after he says,\n\"of this kind are the _spiral_ lines, the _quadratrices_, the\n_conchoeides_, and the _cissoeides_, And geometricians think it no small\nfault, when for the finding out of a plane problem any man makes use of\nconics, or new lines.\" Now he ranks the trisection of an angle among\nsolid problems, and the quinquesection among lineary. But what! are the\nancient geometricians to be blamed, who made use of the quadratrix for\nthe finding out of a strait line equal to the arch of a circle? And\nPappus himself, was he faulty, when he found out the trisection of an\nangle by the help of an hyperbole? Or am I in the wrong, who think I\nhave found out the construction of both these problems by the rule and\ncompass only? Neither they, nor I. For the ancients made use of this\nanalysis which proceeds by the powers; and with them it was a fault to\ndo that by a more remote power, which might be done by a nearer; as\nbeing an argument that they did not sufficiently understand the nature\nof the thing.\n\nThe virtue of this kind of analysis consists in the changing and turning\nand tossing of rectangles and analogisms; and the skill of analysts is\nmere logic, by which they are able methodically to find out whatsoever\nlies hid either in the subject or predicate of the conclusion sought\nfor. But this doth not properly belong to algebra, or the analytics\nspecious, symbolical, or cossick; which are, as I may say, the\n_brachygraphy_ of the analytics, and an art neither of teaching nor\nlearning geometry, but of registering with brevity and celerity the\ninventions of geometricians. For though it be easy to discourse by\nsymbols of very remote propositions; yet whether such discourse deserve\nto be thought very profitable, when it is made without any ideas of the\nthings themselves, I know not.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n C. XX.\n _of\n the translation_.\n _Fig. 1-2_\n]\n\n[Illustration:\n\n C. XX.\n _of\n the translation_.\n _Fig. 3_\n]\n\n[Illustration:\n\n C. XX.\n _of\n the translation_.\n _Fig. 4_\n]\n\n[Illustration:\n\n C. XX.\n _of\n the translation_.\n _Fig. 5-6_\n]\n\n------------------------------------------------------------------------\n\n\n -------\n\n\n CHAPTER XXI.\n\n OF CIRCULAR MOTION.\n\n1. In simple motion, every strait line taken in the body moved is so\n carried, that it is always parallel to the places in which it\n formerly was.\u20142. If circular motion be made about a resting centre,\n and in that circle there be an epicycle, whose revolution is made\n the contrary way, in such manner that in equal times it make equal\n angles, every strait line taken in that epicycle will be so carried,\n that it will always be parallel to the places in which it formerly\n was.\u20143. The properties of simple motion.\u20144. If a fluid be moved with\n simple circular motion, all the points taken in it will describe\n their circles in times proportional to the distances from the\n centre.\u20145. Simple motion dissipates heterogeneous and congregates\n homogeneous bodies.\u20146. If a circle made by a movent moved with\n simple motion be commensurable to another circle made by a point\n which is carried about by the same movent, all the points of both\n the circles will at some time return to the same situation.\u20147.. If a\n sphere have simple motion, its motion will more dissipate\n heterogeneous bodies by how much it is more remote from the\n poles.\u20148. If the simple circular motion of a fluid body be hindered\n by a body which is not fluid, the fluid body will spread itself upon\n the superficies of that body.\u20149. Circular motion about a fixed\n centre casteth off by the tangent such things as lie upon the\n circumference and stick not to it.\u201410. Such things, as are moved\n with simple circular motion, beget simple circular motion.\u201411. If\n that which is so moved have one side hard and the other side fluid,\n its motion will not be perfectly circular.\n\n\n[Sidenote: In simple motion, every strait line taken in the body moved\n is so carried, that it is always parallel to the places in\n which it formerly was.]\n\n1. I have already defined _simple motion_ to be that, in which the\nseveral points taken in a moved body do in several equal times describe\nseveral equal arches. And therefore in simple circular motion it is\nnecessary that every strait line taken in the moved body be always\ncarried parallel to itself; which I thus demonstrate.\n\nFirst, let A B (in the first figure) be any strait line taken in any\nsolid body; and let A D be any arch drawn upon any centre C and radius\nCA. Let the point B be understood to describe towards the same parts the\narch B E, like and equal to the arch A D. Now in the same time in which\nthe point A transmits the arch A D, the point B, which by reason of its\nsimple motion is supposed to be carried with a velocity equal to that of\nA, will transmit the arch B E; and at the end of the same time the whole\nA B will be in D E; and therefore A B and D E are equal. And seeing the\narches A D and B E are like and equal, their subtending strait lines AD\nand BE will also be equal; and therefore the four-sided figure A B D E\nwill be a parallelogram. Wherefore A B is carried parallel to itself.\nAnd the same may be proved by the same method, if any other strait line\nbe taken in the same moved body in which the strait line A B was taken.\nSo that all strait lines, taken in a body moved with simple circular\nmotion, will be carried parallel to themselves.\n\nCoroll. I. It is manifest that the same will also happen in any body\nwhich hath simple motion, though not circular. For all the points of any\nstrait line whatsoever will describe lines, though not circular, yet\nequal; so that though the crooked lines A D and B E were not arches of\ncircles, but of parabolas, ellipses, or of any other figures, yet both\nthey, and their subtenses, and the strait lines which join them, would\nbe equal and parallel.\n\nCoroll. II. It is also manifest, that the radii of the equal circles A D\nand B E, or the axis of a sphere, will be so carried, as to be always\nparallel to the places in which they formerly were. For the strait line\nB F drawn to the centre of the arch B E being equal to the radius A C,\nwill also be equal to the strait line F E or C D; and the angle B F E\nwill be equal to the angle A C D. Now the intersection of the strait\nlines C A and B E being at G, the angle C G E (seeing B E and A D are\nparallel) will be equal to the angle D A C. But the angle E B F is equal\nto the same angle D A C; and therefore the angles C G E and E B F are\nalso equal. Wherefore A C and B F are parallel; which was to be\ndemonstrated.\n\n[Sidenote: If circular motion be made about a resting centre, and in\n that circle there be an epicycle whose revolution is made the\n contrary way, in such manner that in equal times it make\n equal angles, every strait line taken in that epicycle will\n be so carried, that it will always be parallel to the places\n in which it formerly was.]\n\n2. Let there be a circle given (in the second figure) whose centre is A,\nand radius A B; and upon the centre B and any radius B C let the\nepicycle C D E be described. Let the centre B be understood to be\ncarried about the centre A, and the whole epicycle with it till it be\ncoincident with the circle F G H, whose centre is I; and let B A I be\nany angle given. But in the time that the centre B is moved to I, let\nthe epicycle C D E have a contrary revolution upon its own centre,\nnamely from E by D to C, according to the same proportions; that is, in\nsuch manner, that in both the circles, equal angles be made in equal\ntimes. I say E C, the axis of the epicycle, will be always carried\nparallel to itself. Let the angle F I G be made equal to the angle B A\nI; I F and A B will therefore be parallel; and how much the axis A G has\ndeparted from its former place A C (the measure of which progression is\nthe angle C A G, or C B D, which I suppose equal to it) so much in the\nsame time has the axis I G, the same with B C, departed from its own\nformer situation. Wherefore, in what time B C comes to I G by the motion\nfrom B to I upon the centre A, in the same time G will come to F by the\ncontrary motion of the epicycle; that is, it will be turned backwards to\nF, and I G will lie in I F. But the angles F I G and G A C are equal;\nand therefore A C, that is, B C, and I F, (that is the axis, though in\ndifferent places) will be parallel. Wherefore, the axis of the epicycle\nE D C will be carried always parallel to itself; which was to be proved.\n\nCoroll. From hence it is manifest, that those two annual motions which\nCopernicus ascribes to the earth, are reducible to this one circular\nsimple motion, by which all the points of the moved body are carried\nalways with equal velocity, that is, in equal times they make equal\nrevolutions uniformly.\n\nThis, as it is the most simple, so it is the most frequent of all\ncircular motions; being the same which is used by all men when they turn\nanything round with their arms, as they do in grinding or sifting. For\nall the points of the thing moved describe lines which are like and\nequal to one another. So that if a man had a ruler, in which many pens'\npoints of equal length were fastened, he might with this one motion\nwrite many lines at once.\n\n[Sidenote: Properties of simple motion.]\n\n3. Having shown what simple motion is, I will here also set down some\nproperties of the same.\n\nFirst, when a body is moved with simple motion in a fluid medium which\nhath no vacuity, it changes the situation of all the parts of the fluid\nambient which resist its motion; I say there are no parts so small of\nthe fluid ambient, how far soever it be continued, but do change their\nsituation in such manner, as that they leave their places continually to\nother small parts that come into the same.\n\nFor (in the same second figure) let any body, as K L M N, be understood\nto be moved with simple circular motion; and let the circle, which every\npoint thereof describes, have any determined quantity, suppose that of\nthe same K L M N. Wherefore the centre A and every other point, and\nconsequently the moved body itself, will be carried sometimes towards\nthe side where is K, and sometimes towards the other side where is M.\nWhen therefore it is carried to K, the parts of the fluid medium on that\nside will go back; and, supposing all space to be full, others on the\nother side will succeed. And so it will be when the body is carried to\nthe side M, and to N, and every way. Now when the nearest parts of the\nfluid medium go back, it is necessary that the parts next to those\nnearest parts go back also; and supposing still all space to be full,\nother parts will come into their places with succession perpetual and\ninfinite. Wherefore all, even the least parts of the fluid medium,\nchange their places, &c. Which was to be proved.\n\nIt is evident from hence, that simple motion, whether circular or not\ncircular, of bodies which make perpetual returns to their former places,\nhath greater or less force to dissipate the parts of resisting bodies,\nas it is more or less swift, and as the lines described have greater or\nless magnitude. Now the greatest velocity that can be, may be understood\nto be in the least circuit, and the least in the greatest; and may be so\nsupposed, when there is need.\n\n[Sidenote: If a fluid be moved with simple circular motion, all the\n points taken in it will describe their circles in times\n proportional to the distances from the centre.]\n\n4. Secondly, supposing the same simple motion in the air, water, or\nother fluid medium; the parts of the medium, which adhere to the moved\nbody, will be carried about with the same motion and velocity, so that\nin what time soever any point of the movent finishes its circle, in the\nsame time every part of the medium, which adheres to the movent, shall\nalso describe such a part of its circle, as is equal to the whole circle\nof the movent; I say, it shall describe a part, and not the whole\ncircle, because all its parts receive their motion from an interior\nconcentric movent, and of concentric circles the exterior are always\ngreater than the interior; nor can the motion imprinted by any movent be\nof greater velocity than that of the movent itself. From whence it\nfollows, that the more remote parts of the fluid ambient shall finish\ntheir circles in times, which have to one another the same proportion\nwith their distances from the movent. For every point of the fluid\nambient, as long as it toucheth the body which carries it about, is\ncarried about with it, and would make the same circle, but that it is\nleft behind so much as the exterior circle exceeds the interior. So that\nif we suppose some thing, which is not fluid, to float in that part of\nthe fluid ambient which is nearest to the movent, it will together with\nthe movent be carried about. Now that part of the fluid ambient, which\nis not the nearest but almost the nearest, receiving its degree of\nvelocity from the nearest, which degree cannot be greater than it was in\nthe giver, doth therefore in the same time make a circular line, not a\nwhole circle, yet equal to the whole circle of the nearest. Therefore in\nthe same time that the movent describes its circle, that which doth not\ntouch it shall not describe its circle; yet it shall describe such a\npart of it, as is equal to the whole circle of the movent. And after the\nsame manner, the more remote parts of the ambient will describe in the\nsame time such parts of their circles, as shall be severally equal to\nthe whole circle of the movent; and, by consequent, they shall finish\ntheir whole circles in times proportional to their distances from the\nmovent; which was to be proved.\n\n[Sidenote: Simple motion dissipates heterogeneous and congregates\n homogeneous bodies.]\n\n5. Thirdly, the same simple motion of a body placed in a fluid medium,\ncongregates or gathers into one place such things as naturally float in\nthat medium, if they be homogeneous; and if they be heterogeneous, it\nseparates and dissipates them. But if such things as be heterogeneous do\nnot float, but settle, then the same motion stirs and mingles them\ndisorderly together. For seeing bodies, which are unlike to one another,\nthat is, heterogeneous bodies, are not unlike in that they are bodies;\nfor bodies, as bodies, have no difference; but only from some special\ncause, that is, from some internal motion, or motions of their smallest\nparts (for I have shown in chap. IX, art. 9, that all mutation is such\nmotion), it remains that heterogeneous bodies have their unlikeness or\ndifference from one another from their internal or specifical motions.\nNow bodies which have such difference receive unlike and different\nmotions from the same external common movent; and therefore they will\nnot be moved together, that is to say, they will be dissipated. And\nbeing dissipated they will necessarily at some time or other meet with\nbodies like themselves, and be moved alike and together with them; and\nafterwards meeting with more bodies like themselves, they will unite and\nbecome greater bodies. Wherefore homogeneous bodies are congregated, and\nheterogeneous dissipated by simple motion in a medium where they\nnaturally float. Again, such as being in a fluid medium do not float,\nbut sink, if the motion of the fluid medium be strong enough, will be\nstirred up and carried away by that motion, and consequently they will\nbe hindered from returning to that place to which they sink naturally,\nand in which only they would unite, and out of which they are\npromiscuously carried; that is, they are disorderly mingled.\n\nNow this motion, by which homogeneous bodies are congregated and\nheterogeneous are scattered, is that which is commonly called\n_fermentation_, from the Latin _fervere_; as the Greeks have their \u0396\u03cd\u03bc\u03b7,\nwhich signifies the same, from \u0396\u03ad\u03c9 _ferveo_. For seething makes all the\nparts of the water change their places; and the parts of any thing, that\nis thrown into it, will go several ways according to their several\nnatures. And yet all _fervour_ or seething is not caused by fire; for\nnew wine and many other things have also their fermentation and fervour,\nto which fire contributes little, and sometimes nothing. But when in\nfermentation we find heat, it is made by the fermentation.\n\n[Sidenote: If a circle made by a movent moved with simple motion, be\n commensurable to another circle made by a point which is\n carried about by the same movent, all the points of both the\n circles will at some time return to the same situation.]\n\n6. Fourthly, in what time soever the movent, whose centre is A (in fig.\n2) moved in K L N, shall, by any number of revolutions, that is, when\nthe perimeters B I and K L N be commensurable, have described a line\nequal to the circle which passes through the points B and I; in the same\ntime all the points of the floating body, whose centre is B, shall\nreturn to have the same situation in respect of the movent, from which\nthey departed. For seeing it is as the distance B A, that is, as the\nradius of the circle which passes through B I is to the perimeter itself\nB I, so the radius of the circle K L N is to the perimeter K L N; and\nseeing the velocities of the points B and K are equal, the time also of\nthe revolution in I B to the time of one revolution in K L N, will be as\nthe perimeter B I to the perimeter K L N; and therefore so many\nrevolutions in K L N, as together taken are equal to the perimeter B I,\nwill be finished in the same time in which the whole perimeter B I is\nfinished; and therefore also the points L, N, F and H, or any of the\nrest, will in the same time return to the same situation from which they\ndeparted; and this may be demonstrated, whatsoever be the points\nconsidered. Wherefore all the points shall in that time return to the\nsame situation; which was to be proved.\n\nFrom hence it follows, that if the perimeters B I and L K N be not\ncommensurable, then all the points will never return to have the same\nsituation or configuration in respect of one another.\n\n[Sidenote: If a sphere have simple motion, its motion will more\n dissipate heterogeneous bodies by how much it is more remote\n from the poles.]\n\n7. In simple motion, if the body moved be of a spherical figure, it hath\nless force towards its poles than towards its middle to dissipate\nheterogeneous, or to congregate homogeneous bodies.\n\nLet there be a sphere (as in the third figure) whose centre is A and\ndiameter B C; and let it be conceived to be moved with simple circular\nmotion; of which motion let the axis be the strait line D E, cutting the\ndiameter B C at right angles in A. Let now the circle, which is\ndescribed by any point B of the sphere, have B F for its diameter; and\ntaking F G equal to B C, and dividing it in the middle in H, the centre\nof the sphere A will, when half a revolution is finished, lie in H. And\nseeing H F and A B are equal, a circle described upon the centre H with\nthe radius H F or H G, will be equal to the circle whose centre is A and\nradius A B. And if the same motion be continued, the point B will at the\nend of another half revolution return to the place from whence it began\nto be moved; and therefore at the end of half a revolution, the point B\nwill be carried to F, and the whole hemisphere D B E into that\nhemisphere in which are the points L, K and F. Wherefore that part of\nthe fluid medium, which is contiguous to the point F, will in the same\ntime go back the length of the strait line B F; and in the return of the\npoint F to B, that is, of G to C, the fluid medium will go back as much\nin a strait line from the point C. And this is the effect of simple\nmotion in the middle of the sphere, where the distance from the poles is\ngreatest. Let now the point I be taken in the same sphere nearer to the\npole E, and through it let the strait line I K be drawn parallel to the\nstrait line B F, cutting the arch F L in K, and the axis H L in M; then\nconnecting H K, upon H F let the perpendicular K N be drawn. In the same\ntime therefore that B comes to F the point I will come to K, B F and I K\nbeing equal and described with the same velocity. Now the motion in I K\nto the fluid medium upon which it works, namely, to that part of the\nmedium which is contiguous to the point K, is oblique, whereas if it\nproceeded in the strait line H K it would be perpendicular; and\ntherefore the motion which proceeds in I K has less power than that\nwhich proceeds in H K with the same velocity. But the motions in H K and\nH F do equally thrust back the medium; and therefore the part of the\nsphere at K moves the medium less than the part at F, namely, so much\nless as K N is less than H F. Wherefore also the same motion hath less\npower to disperse heterogeneous, and to congregate homogeneous bodies,\nwhen it is nearer, than when it is more remote from the poles; which was\nto be proved.\n\nCoroll. It is also necessary, that in planes which are perpendicular to\nthe axis, and more remote than the pole itself from the middle of the\nsphere, this simple motion have no effect. For the axis D E with simple\nmotion describes the superficies of a cylinder; and towards the bases of\nthe cylinder there is in this motion no endeavour at all.\n\n[Sidenote: If a simple circular motion of a fluid body be hindered by a\n body which is not fluid, the fluid body will spread itself\n upon the superficies of that body.]\n\n8. If in a fluid medium moved about, as hath been said, with simple\nmotion, there be conceived to float some other spherical body which is\nnot fluid, the parts of the medium, which are stopped by that body, will\nendeavour to spread themselves every way upon the superficies of it. And\nthis is manifest enough by experience, namely, by the spreading of water\npoured out upon a pavement. But the reason of it may be this. Seeing the\nsphere A (in fig. 3) is moved towards B, the medium also in which it is\nmoved will have the same motion. But because in this motion it falls\nupon a body not liquid, as G, so that it cannot go on; and seeing the\nsmall parts of the medium cannot go forwards, nor can they go directly\nbackwards against the force of the movent; it remains, therefore, that\nthey diffuse themselves upon the superficies of that body, as towards O\nand P; which was to be proved.\n\n[Sidenote: Circular motion about a fixed centre casteth off by the\n tangent such things as lie upon the circumference & stick\n not to it.]\n\n9. Compounded circular motion, in which all the parts of the moved body\ndo at once describe circumferences, some greater, others less, according\nto the proportion of their several distances from the common centre,\ncarries about with it such bodies, as being not fluid, adhere to the\nbody so moved; and such as do not adhere, it casteth forwards in a\nstrait line which is a tangent to the point from which they are cast\noff.\n\nFor let there be a circle whose radius is A B (in fig. 4); and let a\nbody be placed in the circumference in B, which if it be fixed there,\nwill necessarily be carried about with it, as is manifest of itself. But\nwhilst the motion proceeds, let us suppose that body to be unfixed in B.\nI say, the body will continue its motion in the tangent B C. For let\nboth the radius A B and the sphere B be conceived to consist of hard\nmatter; and let us suppose the radius A B to be stricken in the point B\nby some other body which falls upon it in the tangent D B. Now,\ntherefore, there will be a motion made by the concourse of two things,\nthe one, endeavour towards C in the strait line D B produced, in which\nthe body B would proceed, if it were not retained by the radius A B; the\nother, the retention itself. But the retention alone causeth no\nendeavour towards the centre; and, therefore, the retention being taken\naway, which is done by the unfixing of B, there will remain but one\nendeavour in B, namely, that in the tangent B C. Wherefore the motion of\nthe body B unfixed will proceed in the tangent B C; which was to be\nproved.\n\nBy this demonstration it is manifest, that circular motion about an\nunmoved axis shakes off and puts further from the centre of its motion\nsuch things as touch, but do not stick fast to its superficies; and the\nmore, by how much the distance is greater from the poles of the circular\nmotion; and so much the more also, by how much the things, that are\nshaken off, are less driven towards the centre by the fluid ambient, for\nother causes.\n\n[Sidenote: Such things as are moved with simple circular motion, beget\n simple circular motion.]\n\n10. If in a fluid medium a spherical body be moved with simple circular\nmotion, and in the same medium there float another sphere whose matter\nis not fluid, this sphere also shall be moved with simple circular\nmotion.\n\nLet B C D (in fig. 5) be a circle, whose centre is A, and in whose\ncircumference there is a sphere so moved, that it describes with simple\nmotion the perimeter B C D. Let also E F G be another sphere of\nconsistent matter, whose semidiameter is E H, and centre H; and with the\nradius A H let the circle H I be described. I say, the sphere E F G\nwill, by the motion of the body in B C D, be moved in the circumference\nH I with simple motion.\n\nFor seeing the motion in B C D (by art. 4 of this chapter) makes all the\npoints of the fluid medium describe in the same time circular lines\nequal to one another, the points E, H and G of the strait line E H G\nwill in the same time describe with equal radii equal circles. Let E B\nbe drawn equal and parallel to the strait line A H; and let A B be\nconnected, which will therefore be equal and parallel to E H; and\ntherefore also, if upon the centre B and radius B E the arch E K be\ndrawn equal to the arch H I, and the strait lines A I, B K and I K be\ndrawn, B K and A I will be equal; and they will also be parallel,\nbecause the two arches E K and H I, that is, the two angles K B E and I\nA H are equal; and, consequently, the strait lines A B and K I, which\nconnect them, will also be equal and parallel. Wherefore K I and E H are\nparallel. Seeing, therefore, E and H are carried in the same time to K\nand I, the whole strait line I K will be parallel to E H, from whence it\ndeparted. And, therefore, seeing the sphere E F G is supposed to be of\nconsistent matter, so as all its points keep always the same situation,\nit is necessary that every other strait line, taken in the same sphere,\nbe carried always parallel to the places in which it formerly was.\nWherefore the sphere E F G is moved with simple circular motion; which\nwas to be demonstrated.\n\n[Sidenote: If that which is so moved have one side hard and the other\n side fluid, its motion will not be perfectly circular.]\n\n11. If in a fluid medium, whose parts are stirred by a body moved with\nsimple motion, there float another body, which hath its superficies\neither wholly hard, or wholly fluid, the parts of this body shall\napproach the centre equally on all sides; that is to say, the motion of\nthe body shall be circular, and concentric with the motion of the\nmovent. But if it have one side hard, and the other side fluid, then\nboth those motions shall not have the same centre, nor shall the\nfloating body be moved in the circumference of a perfect circle.\n\nLet a body be moved in the circumference of the circle K L M N (in fig\n2.) whose centre is A. And let there be another body at I, whose\nsuperficies is either all hard or all fluid. Also let the medium, in\nwhich both these bodies are placed, be fluid. I say, the body at I will\nbe moved in the circle I B about the centre A. For this has been\ndemonstrated in the last article.\n\nWherefore let the superficies of the body at I be fluid on one side, and\nhard on the other. And first, let the fluid side be towards the centre.\nSeeing, therefore, the motion of the medium is such, as that its parts\ndo continually change their places, (as hath been shown in art 5); if\nthis change of place be considered in those parts of the medium which\nare contiguous to the fluid superficies, it must needs be that the small\nparts of that superficies enter into the places of the small parts of\nthe medium which are contiguous to them; and the like change of place\nwill be made with the next contiguous parts towards A. And if the fluid\nparts of the body at I have any degree at all of tenacity (for there are\ndegrees of tenacity, as in the air and water) the whole fluid side will\nbe lifted up a little, but so much the less, as its parts have less\ntenacity; whereas the hard part of the superficies, which is contiguous\nto the fluid part, has no cause at all of elevation, that is to say, no\nendeavour towards A.\n\nSecondly, let the hard superficies of the body at I be towards A. By\nreason, therefore, of the said change of place of the parts which are\ncontiguous to it, the hard superficies must, of necessity, seeing by\nsupposition there is no empty space, either come nearer to A, or else\nits smallest parts must supply the contiguous places of the medium,\nwhich otherwise would be empty. But this cannot be, by reason of the\nsupposed hardness; and, therefore, the other must needs be, namely, that\nthe body come nearer to A. Wherefore the body at I has greater endeavour\ntowards the centre A, when its hard side is next it, than when it is\naverted from it. But the body in I, while it is moving in the\ncircumference of the circle I B, has sometimes one side, sometimes\nanother, turned towards the centre; and, therefore, it is sometimes\nnearer, sometimes further off from the centre A. Wherefore the body at I\nis not carried in the circumference of a perfect circle; which was to be\ndemonstrated.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C. XXI.\n _Fig. 1-5_\n]\n\n------------------------------------------------------------------------\n\n\n -------\n\n\n CHAPTER XXII.\n\n OF OTHER VARIETY OF MOTION.\n\n1. Endeavour and pressure how they differ.\u20142. Two kinds of mediums in\n which bodies are moved.\u20143. Propagation of motion, what it is.\u20144.\n What motion bodies have, when they press one another.\u20145. Fluid\n bodies, when they are pressed together, penetrate one another.\u20146.\n When one body presseth another and doth not penetrate it, the action\n of the pressing body is perpendicular to the superficies of the body\n pressed.\u20147. When a hard body, pressing another body, penetrates the\n same, it doth not penetrate it perpendicularly, unless it fall\n perpendicularly upon it.\u20148. Motion sometimes opposite to that of the\n movent.\u20149. In a full medium, motion is propagated to any\n distance.\u201410. Dilatation and contraction what they are.\u201411.\n Dilatation and contraction suppose mutation of the smallest parts in\n respect of their situation.\u201412. All traction is pulsion.\u201413. Such\n things as being pressed or bent restore themselves, have motion in\n their internal parts.\u201414. Though that which carrieth another be\n stopped, the body carried will proceed.\u201415, 16. The effects of\n percussion not to be compared with those of weight.\u201417, 18. Motion\n cannot begin first in the internal parts of a body.\u201419. Action and\n reaction proceed in the same line.\u201420. Habit, what it is.\n\n\n[Sidenote: Endeavour and pressure how they differ.]\n\n1. I have already (chapter XV. art. 2) defined endeavour to be motion\nthrough some length, though not considered as length, but as a point.\nWhether, therefore, there be resistance or no resistance, the endeavour\nwill still be the same. For simply to endeavour is to go. But when two\nbodies, having opposite endeavours, press one another, then the\nendeavour of either of them is that which we call pressure, and is\nmutual when their pressures are opposite.\n\n[Sidenote: Two kinds of mediums in which bodies are moved.]\n\n2. Bodies moved, and also the mediums in which they are moved, are of\ntwo kinds. For either they have their parts coherent in such manner, as\nno part of the moved body will easily yield to the movent, except the\nwhole body yield also, and such are the things we call _hard_: or else\ntheir parts, while the whole remains unmoved, will easily yield to the\nmovent, and these we call _fluid_ or _soft_ bodies. For the words\n_fluid_, _soft_, _tough_, and _hard_, in the same manner as _great_ and\n_little_, are used only comparatively; and are not different kinds, but\ndifferent degrees of quality.\n\n[Sidenote: Propagation of motion, what it is.]\n\n3. To _do_, and to _suffer_, is to move and to be moved; and nothing is\nmoved but by that which toucheth it and is also moved, as has been\nformerly shown. And how great soever the distance be, we say the first\nmovent moveth the last moved body, but mediately; namely so, as that the\nfirst moveth the second, the second the third, and so on, till the last\nof all be touched. When therefore one body, having opposite endeavour to\nanother body, moveth the same, and that moveth a third, and so on, I\ncall that action _propagation of motion_.\n\n[Sidenote: What motion bodies have when they press one another.]\n\n4. When two fluid bodies, which are in a free and open space, press one\nanother, their parts will endeavour, or be moved, towards the sides; not\nonly those parts which are there where the mutual contact is, but all\nthe other parts. For in the first contact, the parts, which are pressed\nby both the endeavouring bodies, have no place either forwards or\nbackwards in which they can be moved; and therefore they are pressed out\ntowards the sides. And this expressure, when the forces are equal, is in\na line perpendicular to the bodies pressing. But whensoever the foremost\nparts of both the bodies are pressed, the hindermost also must be\npressed at the same time; for the motion of the hindermost parts cannot\nin an instant be stopped by the resistance of the foremost parts, but\nproceeds for some time; and therefore, seeing they must have some place\nin which they may be moved, and that there is no place at all for them\nforwards, it is necessary that they be moved into the places which are\ntowards the sides every way. And this effect follows of necessity, not\nonly in fluid, but in consistent and hard bodies, though it be not\nalways manifest to sense. For though from the compression of two stones\nwe cannot with our eyes discern any swelling outwards towards the sides,\nas we perceive in two bodies of wax; yet we know well enough by reason,\nthat some tumour must needs be there, though it be but little.\n\n[Sidenote: Fluid bodies, when they are pressed together, penetrate one\n another.]\n\n5. But when the space is enclosed, and both the bodies be fluid, they\nwill, if they be pressed together, penetrate one another, though\ndifferently, according to their different endeavours. For suppose a\nhollow cylinder of hard matter, well stopped at both ends, but filled\nfirst, below with some heavy fluid body, as quicksilver, and above with\nwater or air. If now the bottom of the cylinder be turned upwards, the\nheaviest fluid body, which is now at the top, having the greatest\nendeavour downwards, and being by the hard sides of the vessel hindered\nfrom extending itself sideways, must of necessity either be received by\nthe lighter body, that it may sink through it, or else it must open a\npassage through itself, by which the lighter body may ascend. For of the\ntwo bodies, that, whose parts are most easily separated, will be the\nfirst divided; which being done, it is not necessary that the parts of\nthe other suffer any separation at all. And therefore when two liquors,\nwhich are enclosed in the same vessel, change their places, there is no\nneed that their smallest parts should be mingled with one another; for a\nway being opened through one of them, the parts of the other need not be\nseparated.\n\nNow if a fluid body, which is not enclosed, press a hard body, its\nendeavour will indeed be towards the internal parts of that hard body;\nbut being excluded by the resistance of it, the parts of the fluid body\nwill be moved every way according to the superficies of the hard body,\nand that equally, if the pressure be perpendicular; for when all the\nparts of the cause are equal, the effects will be equal also. But if the\npressure be not perpendicular, then the angles of the incidence being\nunequal, the expansion also will be unequal, namely, greater on that\nside where the angle is greater, because that motion is most direct\nwhich proceeds by the directest line.\n\n[Sidenote: When one body presseth another and doth not penetrate it, the\n action of the pressing body is perpendicular to the\n superficies of the body pressed.]\n\n6. If a body, pressing another body, do not penetrate it, it will\nnevertheless give to the part it presseth an endeavour to yield, and\nrecede in a strait line perpendicular to its superficies in that point\nin which it is pressed.\n\nLet A B C D (in fig. 1) be a hard body, and let another body, falling\nupon it in the strait line E A, with any inclination or without\ninclination, press it in the point A. I say the body so pressing, and\nnot penetrating it, will give to the part A an endeavour to yield or\nrecede in a strait line perpendicular to the line A D.\n\nFor let A B be perpendicular to A D, and let B A be produced to F. If\ntherefore A F be coincident with A E, it is of itself manifest that the\nmotion in E A will make A to endeavour in the line A B. Let now E A be\noblique to A D, and from the point E let the strait line E C be drawn,\ncutting A D at right angles in D, and let the rectangles A B C D and A D\nE F be completed. I have shown (in the 8th article of chapter XVI) that\nthe body will be carried from E to A by the concourse of two uniform\nmotions, the one in E F and its parallels, the other in E D and its\nparallels. But the motion in E F and its parallels, whereof D A is one,\ncontributes nothing to the body in A to make it endeavour or press\ntowards B; and therefore the whole endeavour, which the body hath in the\ninclined line E A to pass or press the strait line A D, it hath it all\nfrom the perpendicular motion or endeavour in F A. Wherefore the body E,\nafter it is in A, will have only that perpendicular endeavour which\nproceeds from the motion in F A, that is, in A B; which was to be\nproved.\n\n[Sidenote: When a hard body, pressing another body, penetrates the same,\n it doth not penetrate it perpendicularly, unless it fall\n perpendicularly upon it.]\n\n7. If a hard body falling upon or pressing another body penetrate the\nsame, its endeavour after its first penetration will be neither in the\ninclined line produced, nor in the perpendicular, but sometimes betwixt\nboth, sometimes without them.\n\nLet E A G (in the same fig. 1) be the inclined line produced; and first,\nlet the passage through the medium, in which E A is, be easier than the\npassage through the medium in which A G is. As soon therefore as the\nbody is within the medium in which is A G, it will find greater\nresistance to its motion in D A and its parallels, than it did whilst it\nwas above A D; and therefore below A D it will proceed with slower\nmotion in the parallels of D A, than above it. Wherefore the motion\nwhich is compounded of the two motions in E F and E D will be slower\nbelow A D than above it; and therefore also, the body will not proceed\nfrom A in E A produced, but below it. Seeing, therefore, the endeavour\nin A B is generated by the endeavour in F A; if to the endeavour in F A\nthere be added the endeavour in D A, which is not all taken away by the\nimmersion of the point A into the lower medium, the body will not\nproceed from A in the perpendicular A B, but beyond it; namely, in some\nstrait line between A B and A G, as in the line A H.\n\nSecondly, let the passage through the medium E A be less easy than that\nthrough A G. The motion, therefore, which is made by the concourse of\nthe motions in E F and F B, is slower above A D than below it; and\nconsequently, the endeavour will not proceed from A in E A produced, but\nbeyond it, as in A I. Wherefore, if a hard body falling, &c.; which was\nto be proved.\n\nThis divergency of the strait line A H from the strait line A G is that\nwhich, the writers of optics commonly called _refraction_, which, when\nthe passage is easier in the first than in the second medium, is made by\ndiverging from the line of inclination towards the perpendicular; and\ncontrarily, when the passage is not so easy in the first medium, by\ndeparting further from the perpendicular.\n\n[Sidenote: Motion sometimes opposite to that of the movent.]\n\n8. By the 6th theorem it is manifest, that the force of the movent may\nbe so placed, as that the body moved by it may proceed in a way almost\ndirectly contrary to that of the movent, as we see in the motion of\nships.\n\nFor let A B (in fig. 2) represent a ship, whose length from the prow to\nthe poop is A B, and let the wind lie upon it in the strait parallel\nlines C B, D E and F G; and let D E and F G be cut in E and G by a\nstrait line drawn from B perpendicular to A B; also let B E and E G be\nequal, and the angle A B C any angle how small soever. Then between B C\nand B A let the strait line B I be drawn; and let the sail be conceived\nto be spread in the same line B I, and the wind to fall upon it in the\npoints L, M and B; from which points, perpendicular to B I, let B K, M Q\nand L P be drawn. Lastly, let E N and G O be drawn perpendicular to B G,\nand cutting B K in H and K; and let H N and K O be made equal to one\nanother, and severally equal to B A. I say, the ship B A, by the wind\nfalling upon it in C B, D E, F G, and other lines parallel to them, will\nbe carried forwards almost opposite to the wind, that is to say, in a\nway almost contrary to the way of the movent.\n\nFor the wind that blows in the line C B will (as hath been shown in art.\n6) give to the point B an endeavour to proceed in a strait line\nperpendicular to the strait line B I, that is, in the strait line B K;\nand to the points M and L an endeavour to proceed in the strait lines M\nQ and L P, which are parallel to B K. Let now the measure of the time be\nB G, which is divided in the middle in E; and let the point B be carried\nto H in the time B E. In the same time, therefore, by the wind blowing\nin D M and F L, and as many other lines as may be drawn parallel to\nthem, the whole ship will be applied to the strait line H N. Also at the\nend of the second time E G, it will be applied to the strait line K O.\nWherefore the ship will always go forward; and the angle it makes with\nthe wind will be equal to the angle A B C, how small soever that angle\nbe; and the way it makes will in every time be equal to the strait line\nE H. I say, thus it would be, if the ship might be moved with as great\ncelerity sideways from B A towards K O, as it may be moved forwards in\nthe line B A. But this is impossible, by reason of the resistance made\nby the great quantity of water which presseth the side, much exceeding\nthe resistance made by the much smaller quantity which presseth the prow\nof the ship; so that the way the ship makes sideways is scarce sensible;\nand, therefore, the point B will proceed almost in the very line B A,\nmaking with the wind the angle A B C, how acute soever; that is to say,\nit will proceed almost in the strait line B C, that is, in a way almost\ncontrary to the way of the movent; which was to be demonstrated.\n\nBut the sail in B I must be so stretched as that there be left in it no\nbosom at all; for otherwise the strait lines L P, M Q and B K will not\nbe perpendicular to the plane of the sail, but falling below P, Q and K,\nwill drive the ship backwards. But by making use of a small board for a\nsail, a little waggon with wheels for the ship, and of a smooth pavement\nfor the sea, I have by experience found this to be so true, that I could\nscarce oppose the board to the wind in any obliquity, though never so\nsmall, but the waggon was carried forwards by it.\n\nBy the same 6th theorem it may be found, how much a stroke, which falls\nobliquely, is weaker than a stroke falling perpendicularly, they being\nlike and equal in all other respects.\n\nLet a stroke fall upon the wall A B obliquely, as for example, in the\nstrait line C A (in fig. 3.) Let C E be drawn parallel to A B, and D A\nperpendicular to the same A B and equal to C A; and let both the\nvelocity and time of the motion in C A be equal to the velocity and time\nof the motion in D A. I say, the stroke in C A will be weaker than that\nin D A, in the proportion of E A to D A. For producing D A howsoever to\nF, the endeavour of both the strokes will (by art. 6) proceed from A in\nthe perpendicular A F. But the stroke in C A is made by the concourse of\ntwo motions in C E and E A, of which that in C E contributes nothing to\nthe stroke in A, because C E and B A are parallels; and, therefore, the\nstroke in C A is made by the motion which is in E A only. But the\nvelocity or force of the perpendicular stroke in E A, to the velocity or\nforce of the stroke in D A, is as E A to D A. Wherefore the oblique\nstroke in C A is weaker than the perpendicular stroke in D A, in the\nproportion of E A to D A or C A; which was to be proved.\n\n[Sidenote: In a full medium, motion is propagated to any distance.]\n\n9. In a full medium, all endeavour proceeds as far as the medium itself\nreacheth; that is to say, if the medium be infinite, the endeavour will\nproceed infinitely.\n\nFor whatsoever endeavoureth is moved, and therefore whatsoever standeth\nin its way it maketh it yield, at least a little, namely, so far as the\nmovent itself is moved forwards. But that which yieldeth is also moved,\nand consequently maketh that to yield which is in its way, and so on\nsuccessively as long as the medium is full; that is to say, infinitely,\nif the full medium be infinite; which was to be proved.\n\nNow although endeavour thus perpetually propagated do not always appear\nto the senses as motion, yet it appears as action, or as the efficient\ncause of some mutation. For if there be placed before our eyes some very\nlittle object, as for example, a small grain of sand, which at a certain\ndistance is visible; it is manifest that it may be removed to such a\ndistance as not to be any longer seen, though by its action it still\nwork upon the organs of sight, as is manifest from that which was last\nproved, that all endeavour proceeds infinitely. Let it be conceived\ntherefore to be removed from our eyes to any distance how great soever,\nand a sufficient number of other grains of sand of the same bigness\nadded to it; it is evident that the aggregate of all those sands will be\nvisible; and though none of them can be seen when it is single and\nsevered from the rest, yet the whole heap or hill which they make will\nmanifestly appear to the sight; which would be impossible, if some\naction did not proceed from each several part of the whole heap.\n\n[Sidenote: Dilatation & contraction what they are.]\n\n10. Between the degrees of hard and soft are those things which we call\n_tough_, tough being that which may be bent without being altered from\nwhat it was; and the bending of a line is either the adduction or\ndiduction of the extreme parts, that is, a motion from straitness to\ncrookedness, or contrarily, whilst the line remains still the same it\nwas; for by drawing out the extreme points of a line to their greatest\ndistance, the line is made strait, which otherwise is crooked. So also\nthe bending of a superficies is the diduction or adduction of its\nextreme lines, that is, their dilatation and contraction.\n\n[Sidenote: Dilatation & contraction suppose mutation of the smallest\n parts in respect of their situation.]\n\n11. _Dilatation_ and _contraction_, as also all _flexion_, supposes\nnecessarily that the internal parts of the body bowed do either come\nnearer to the external parts, or go further from them. For though\n_flexion_ be considered only in the length of a body, yet when that body\nis bowed, the line which is made on one side will be convex, and the\nline on the other side will be concave; of which the concave, being the\ninterior line, will, unless something be taken from it and added to the\nconvex line, be the more crooked, that is, the greater of the two. But\nthey are equal; and, therefore, in flexion there is an accession made\nfrom the interior to the exterior parts; and, on the contrary, in\ntension, from the exterior to the interior parts. And as for those\nthings which do not easily suffer such transposition of their parts,\nthey are called brittle; and the great force they require to make them\nyield, makes them also with sudden motion to leap asunder, and break in\npieces.\n\n[Sidenote: All traction is pulsion.]\n\n12. Also motion is distinguished into _pulsion_ and _traction_. And\npulsion, as I have already defined it, is when that which is moved goes\nbefore that which moveth it. But contrarily, in traction the movent goes\nbefore that which is moved. Nevertheless, considering it with greater\nattention, it seemeth to be the same with pulsion. For of two parts of a\nhard body, when that which is foremost drives before it the medium in\nwhich the motion is made, at the same time that which is thrust forwards\nthrusteth the next, and this again the next, and so on successively. In\nwhich action, if we suppose that there is no place void, it must needs\nbe, that by continual pulsion, namely, when that action has gone round,\nthe movent will be behind that part, which at the first seemed not to be\nthrust forwards, but to be drawn; so that now the body, which was drawn,\ngoes before the body which gives it motion; and its motion is no longer\ntraction, but pulsion.\n\n[Sidenote: Such things as being pressed or bent restore themselves, have\n motion in their internal parts.]\n\n13. Such things as are removed from their places by forcible compression\nor extension, and, as soon as the force is taken away, do presently\nreturn and restore themselves to their former situation, have the\nbeginning of their restitution within themselves, namely, a certain\nmotion in their internal parts, which was there, when, before the taking\naway of the force, they were compressed, or extended. For that\nrestitution is motion, and that which is at rest cannot be moved, but by\na moved and a contiguous movent. Nor doth the cause of their restitution\nproceed from the taking away of the force by which they were compressed\nor extended; for the removing of impediments hath not the efficacy of a\ncause, as has been shown at the end of the 3rd article of chap. XV. The\ncause therefore of their restitution is some motion either of the parts\nof the ambient, or of the parts of the body compressed or extended. But\nthe parts of the ambient have no endeavour which contributes to their\ncompression or extension, nor to the setting of them at liberty, or\nrestitution. It remains therefore that from the time of their\ncompression or extension there be left some endeavour or motion, by\nwhich, the impediment being removed, every part resumes its former\nplace; that is to say, the whole restores itself.\n\n[Sidenote: Though that which carrieth another be stopped, the body\n carried will proceed.]\n\n14. In the carriage of bodies, if that body, which carries another, hit\nupon any obstacle, or be by any means suddenly stopped, and that which\nis carried be not stopped, it will go on, till its motion be by some\nexternal impediment taken away.\n\nFor I have demonstrated (chap. VIII, art. 19) that motion, unless it be\nhindered by some external resistance, will be continued eternally with\nthe same celerity; and in the 7th article of chap. IX, that the action\nof an external agent is of no effect without contact. When therefore\nthat, which carrieth another thing, is stopped, that stop doth not\npresently take away the motion of that which is carried. It will\ntherefore proceed, till its motion be by little and little extinguished\nby some external resistance: which was to be proved; though experience\nalone had been sufficient to prove this.\n\nIn like manner, if that body which carrieth another be put from rest\ninto sudden motion, that which is carried will not be moved forwards\ntogether with it, but will be left behind. For the contiguous part of\nthe body carried hath almost the same motion with the body which carries\nit; and the remote parts will receive different velocities according to\ntheir different distances from the body that carries them; namely, the\nmore remote the parts are, the less will be their degrees of velocity.\nIt is necessary, therefore, that the body, which is carried, be left\naccordingly more or less behind. And this also is manifest by\nexperience, when at the starting forward of the horse the rider falleth\nbackwards.\n\n[Sidenote: The effects of percussion not to be compared with those of\n weight.]\n\n15. In _percussion_, therefore, when one hard body is in some small part\nof it stricken by another with great force, it is not necessary that the\nwhole body should yield to the stroke with the same celerity with which\nthe stricken part yields. For the rest of the parts receive their motion\nfrom the motion of the part stricken and yielding, which motion is less\npropagated every way towards the sides, than it is directly forwards.\nAnd hence it is, that sometimes very hard bodies, which being erected\ncan hardly be made to stand, are more easily broken than thrown down by\na violent stroke; when, nevertheless, if all their parts together were\nby any weak motion thrust forwards, they would easily be cast down.\n\n16. Though the difference between _trusion_ and _percussion_ consist\nonly in this, that in trusion the motion both of the movent and moved\nbody begin both together in their very contact; and in percussion the\nstriking body is first moved, and afterwards the body stricken; yet\ntheir effects are so different, that it seems scarce possible to compare\ntheir forces with one another. I say, any effect of percussion being\npropounded, as for example, the stroke of a beetle of any weight\nassigned, by which a pile of any given length is to be driven into earth\nof any tenacity given, it seems to me very hard, if not impossible, to\ndefine with what weight, or with what stroke, and in what time, the same\npile may be driven to a depth assigned into the same earth. The cause of\nwhich difficulty is this, that the velocity of the percutient is to be\ncompared with the magnitude of the ponderant. Now velocity, seeing it is\ncomputed by the length of space transmitted, is to be accounted but as\none dimension; but weight is as a solid thing, being measured by the\ndimension of the whole body. And there is no comparison to be made of a\nsolid body with a length, that is, with a line.\n\n[Sidenote: Motion cannot begin first in the internal parts of a body.]\n\n17. If the internal parts of a body be at rest, or retain the same\nsituation with one another for any time how little soever, there cannot\nin those parts be generated any new motion or endeavour, whereof the\nefficient cause is not without the body of which they are parts. For if\nany small part, which is comprehended within the superficies of the\nwhole body, be supposed to be now at rest, and by and by to be moved,\nthat part must of necessity receive its motion from some moved and\ncontiguous body. But by supposition, there is no such moved and\ncontiguous part within the body. Wherefore, if there be any endeavour or\nmotion or change of situation in the internal parts of that body, it\nmust needs arise from some efficient cause that is without the body\nwhich contains them; which was to be proved.\n\n18. In hard bodies, therefore, which are compressed or extended, if,\nthat which compresseth or extendeth them being taken away, they restore\nthemselves to their former place or situation, it must needs be that\nthat endeavour or motion of their internal parts, by which they were\nable to recover their former places or situations, was not extinguished\nwhen the force by which they were compressed or extended was taken away.\nTherefore, when the lath of a cross-bow bent doth, as soon as it is at\nliberty, restore itself, though to him, that judges by sense, both it\nand all its parts seem to be at rest; yet he, that judging by reason\ndoth not account the taking away of impediment for an efficient cause,\nnor conceives that without an efficient cause any thing can pass from\nrest to motion, will conclude that the parts were already in motion\nbefore they began to restore themselves.\n\n[Sidenote: Action and reaction proceed in the same line.]\n\n19. _Action_ and _reaction_ proceed in the same line, but from opposite\nterms. For seeing reaction is nothing but endeavour in the patient to\nrestore itself to that situation from which it was forced by the agent;\nthe endeavour or motion both of the agent and patient or reagent will be\npropagated between the same terms; yet so, as that in action the term,\n_from which_, is in reaction the term _to which_. And seeing all action\nproceeds in this manner, not only between the opposite terms of the\nwhole line in which it is propagated, but also in all the parts of that\nline, the terms _from which_ and _to which_, both of the action and\nreaction, will be in the same line. Wherefore action and reaction\nproceed in the same line, &c.\n\n[Sidenote: Habit, what it is.]\n\n20. To what has been said of motion, I will add what I have to say\nconcerning habit. _Habit_, therefore, is a generation of motion, not of\nmotion simply, but an easy conducting of the moved body in a certain and\ndesigned way. And seeing it is attained by the weakening of such\nendeavours as divert its motion, therefore such endeavours are to be\nweakened by little and little. But this cannot be done but by the long\ncontinuance of action, or by actions often repeated; and therefore\ncustom begets that facility, which is commonly and rightly called\n_habit_; and it may be defined thus: HABIT _is motion made more easy and\nready by custom; that is to say, by perpetual endeavour, or by iterated\nendeavours in a way differing from that in which the motion proceeded\nfrom the beginning, and opposing such endeavours as resist_. And to make\nthis more perspicuous by example, we may observe, that when one that has\nno skill in music first puts his hand to an instrument, he cannot after\nthe first stroke carry his hand to the place where he would make the\nsecond stroke, without taking it back by a new endeavour, and, as it\nwere beginning again, pass from the first to the second. Nor will he be\nable to go on to the third place without another new endeavour; but he\nwill be forced to draw back his hand again, and so successively, by\nrenewing his endeavour at every stroke; till at the last, by doing this\noften, and by compounding many interrupted motions or endeavours into\none equal endeavour, he be able to make his hand go readily on from\nstroke to stroke in that order and way which was at the first designed.\nNor are habits to be observed in living creatures only, but also in\nbodies inanimate. For we find that when the lath of a cross-bow is\nstrongly bent, and would if the impediment were removed return again\nwith great force; if it remain a long time bent, it will get such a\nhabit, that when it is loosed and left to its own freedom, it will not\nonly not restore itself, but will require as much force for the bringing\nof it back to its first posture, as it did for the bending of it at the\nfirst.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C. XXII.\n _Fig. 1-3_\n]\n\n------------------------------------------------------------------------\n\n\n -------\n\n\n CHAP. XXIII.\n\n OF THE CENTRE OF EQUIPONDERATION; OF\n BODIES PRESSING DOWNWARDS IN STRAIT\n PARALLEL LINES.\n\n1. Definitions and suppositions.\u20142. Two planes of equiponderation are\n not parallel.\u20143. The centre of equiponderation is in every plane of\n equiponderation.\u20144. The moments of equal ponderants are to one\n another as their distances from the centre of the scale.\u20145, 6. The\n moments of unequal ponderants have their proportion to one another\n compounded of the proportions of their weights and distances from\n the centre of the scale.\u20147. If two ponderants have their weights and\n distances from the centre of the scale in reciprocal proportion,\n they are equally poised; and contrarily.\u20148. If the parts of any\n ponderant press the beams of the scale every where equally, all the\n parts cut off, reckoned from the centre of the scale, will have\n their moments in the same proportion with that of the parts of a\n triangle cut off from the vertex by strait lines parallel to the\n base.\u20149. The diameter of equiponderation of figures, which are\n deficient according to commensurable proportions of their altitudes\n and bases, divides the axis, so that the part taken next the vertex\n is to the other part of the complete figure to the deficient\n figure.\u201410. The diameter of equiponderation of the complement of the\n half of any of the said deficient figures, divides that line which\n is drawn through the vertex parallel to the base, so that the part\n next the vertex is to the other part as the complete figure to the\n complement.\u201411. The centre of equiponderation of the half of any of\n the deficient figures in the first row of the table of art. 3, chap.\n XVII, may be found out by the numbers of the second row.\u201412. The\n centre of equiponderation of the half of any of the figures of the\n second row of the same table, may be found out by the numbers of the\n fourth row.\u201413. The centre of equiponderation of the half of any of\n the figures in the same table being known, the centre of the excess\n of the same figure above a triangle of the same altitude and base is\n also known.\u201414. The centre of equiponderation of a solid sector is\n in the axis so divided, that the part next the vertex be to the\n whole axis, wanting half the axis of the portion of the sphere, as 3\n to 4.\n\n DEFINITIONS.\n\n[Sidenote: Definitions.]\n\nI. A _scale_ is a strait line, whose middle point is immovable, all the\nrest of its points being at liberty; and that part of the scale, which\nreaches from the centre to either of the weights, is called the _beam_.\n\nII. _Equiponderation_ is when the endeavour of one body, which presses\none of the beams, resists the endeavour of another body pressing the\nother beam, so that neither of them is moved; and the bodies, when\nneither of them is moved, are said to be _equally poised_.\n\nIII. _Weight_ is the aggregate of all the endeavours, by which all the\npoints of that body, which presses the beam, tend downwards in lines\nparallel to one another; and the body which presses is called the\n_ponderant_.\n\nIV. _Moment_ is the power which the ponderant has to move the beam, by\nreason of a determined situation.\n\nV. _The plane of equiponderation_ is that by which the ponderant is so\ndivided, that the moments on both sides remain equal.\n\nVI. _The diameter of equiponderation_ is the common section of the two\nplanes of equiponderation, and is in the strait line by which the weight\nis hanged.\n\nVII. _The centre of equiponderation_ is the common point of the two\ndiameters of equiponderation.\n\n SUPPOSITIONS.\n\n[Sidenote: Suppositions.]\n\nI. When two bodies are equally poised, if weight be added to one of them\nand not to the other, their equiponderation ceases.\n\nII. When two ponderants of equal magnitude, and of the same species or\nmatter, press the beam on both sides at equal distances from the centre\nof the scale, their moments are equal. Also when two bodies endeavour at\nequal distances from the centre of the scale, if they be of equal\nmagnitude and of the same species, their moments are equal.\n\n[Sidenote: Two planes\n of equiponderation\n are\n not parallel.]\n\n2. No two planes of equiponderation are parallel.\n\nLet A B C D (in fig. 1) be any ponderant whatsoever; and in it let E F\nbe a plane of equiponderation; parallel to which, let any other plane be\ndrawn, as G H. I say, G H is not a plane of equiponderation. For seeing\nthe parts A E F D and E B C F of the ponderant A B C D are equally\npoised; and the weight E G H F is added to the part A E F D, and nothing\nis added to the part E B C F, but the weight E G H F is taken from it;\ntherefore, by the first supposition, the parts A G H D and G B C H will\nnot be equally poised; and consequently G H is not a plane of\nequiponderation. Wherefore, no two planes of equiponderation are\nparallel; which was to be proved.\n\n[Sidenote: The centre of equiponderation is in every plane of\n equiponderation.]\n\n3. The centre of equiponderation is in every plane of equiponderation.\n\nFor if another plane of equiponderation be taken, it will not, by the\nlast article, be parallel to the former plane; and therefore both those\nplanes will cut one another. Now that section (by the 6th definition) is\nthe diameter of equiponderation. Again, if another diameter of\nequiponderation be taken, it will cut that former diameter; and in that\nsection (by the 7th definition) is the centre of equiponderation.\nWherefore the centre of equiponderation is in that diameter which lies\nin the said plane of equiponderation.\n\n[Sidenote: The moments\n of equal ponderants\n are to one\n another as their\n distances from\n the centre of\n the scale.]\n\n4. The moment of any ponderant applied to one point of the beam, to the\nmoment of the same or an equal ponderant applied to any other point of\nthe beam, is as the distance of the former point from the centre of the\nscale, to the distance of the latter point from the same centre. Or\nthus, those moments are to one another, as the arches of circles which\nare made upon the centre of the scale through those points, in the same\ntime. Or lastly thus, they are as the parallel bases of two triangles,\nwhich have a common angle at the centre of the scale.\n\nLet A (in fig. 2) be the centre of the scale; and let the equal\nponderants D and E press the beam A B in the points B and C; also let\nthe strait lines B D and C E be diameters of equiponderation; and the\npoints D and E in the ponderants D and E be their centres of\nequiponderation. Let A G F be drawn howsoever, cutting D B produced in\nF, and E C in G; and lastly, upon the common centre A, let the two\narches B H and C I be described, cutting A G F in H and I. I say, the\nmoment of the ponderant D to the moment of the ponderant E is as A B to\nA C, or as B H to C I, or as B F to C G. For the effect of the ponderant\nD, in the point B, is circular motion in the arch B H; and the effect of\nthe ponderant E, in the point C, circular motion in the arch C I; and by\nreason of the equality of the ponderants D and E, these motions are to\none another as the quicknesses or velocities with which the points B and\nC describe the arches B H and C I, that is, as the arches themselves B H\nand C I, or as the strait parallels B F and C G, or as the parts of the\nbeam A B and A C; for A B. A C:: B F. C G:: B H. C I. are proportionals;\nand therefore the effects, that is, by the 4th definition, the moments\nof the equal ponderants applied to several points of the beam, are to\none another as A B and A C; or as the distances of those points from the\ncentre of the scale; or as the parallel bases of the triangles which\nhave a common angle at A; or as the concentric arches B H and C I; which\nwas to be demonstrated.\n\n[Sidenote: The moments of unequal ponderants have their proportion to\n one another compounded of the proportions of their weights\n and distances from the centre of the scale.]\n\n5. Unequal ponderants, when they are applied to several points of the\nbeam, and hang at liberty, that is, so as the line by which they hang be\nthe diameter of equiponderation, whatsoever be the figure of the\nponderant, have their moments to one another in proportion compounded of\nthe proportions of their distances from the centre of the scale, and of\ntheir weights.\n\nLet A (in fig. 3) be the centre of the scale, and A B the beam; to which\nlet the two ponderants C and D be applied at the points B and E. I say,\nthe proportion of the moment of the ponderant C to the moment of the\nponderant D, is compounded of the proportions of A B to A E, and of the\nweight C to the weight D; or, if C and D be of the same species, of the\nmagnitude C to the magnitude D.\n\nLet either of them, as C, be supposed to be bigger than the other, D.\nIf, therefore, by the addition of F, F and D together be as one body\nequal to C, the moment of C to the moment of F + D will be (by the last\narticle) as B G is to E H. Now as F + D is to D, so let E H be to\nanother E I; and the moment of F + D, that is of C, to the moment of D,\nwill be as B G to E I. But the proportion of B G to E I is compounded of\nthe proportions of B G to E H, that is, of A B to A E, and of E H to E\nI, that is, of the weight C to the weight D. Wherefore unequal\nponderants, when they are applied, &c. Which was to be proved.\n\n6. The same figure remaining, if I K be drawn parallel to the beam A B,\nand cutting A G in K; and K L be drawn parallel to B G, cutting A B in\nL, the distances A B and A L from the centre will be proportional to the\nmoments of C and D. For the moment of C is B G, and the moment of D is E\nI, to which K L is equal. But as the distance A B from the centre is to\nthe distance A L from the centre, so is B G, the moment of the ponderant\nC, to L K, or E I the moment of the ponderant D.\n\n[Sidenote: If two ponderants have their weights and distances from the\n centre of the scale in reciprocal proportion, they are\n equally poised; and contrarily.]\n\n7. If two ponderants have their weights and distances from the centre in\nreciprocal proportion, and the centre of the scale be between the points\nto which the ponderants are applied, they will be equally poised. And\ncontrarily, if they be equally poised, their weights and distances from\nthe centre of the scale will be in reciprocal proportion.\n\nLet the centre of the scale (in the same third figure) be A, the beam A\nB; and let any ponderant C, having B G for its moment, be applied to the\npoint B; also let any other ponderant D, whose moment is E I, be applied\nto the point E. Through the point I let I K be drawn parallel to the\nbeam A B, cutting A G in K; also let K L be drawn parallel to B G, K L\nwill then be the moment of the ponderant D; and by the last article, it\nwill be as B G, the moment of the ponderant C in the point B, to L K the\nmoment of the ponderant D in the point E, so A B to A L. On the other\nside of the centre of the scale, let A N be taken equal to A L; and to\nthe point N let there be applied the ponderant O, having to the\nponderant C the proportion of A B to A N. I say, the ponderants in B and\nN will be equally poised. For the proportion of the moment of the\nponderant O, in the point N, to the moment of the ponderant C in the\npoint B, is by the 5th article, compounded of the proportions of the\nweight O to the weight C, and of the distance from the centre of the\nscale A N or A L to the distance from the centre of the scale A B. But\nseeing we have supposed, that the distance A B to the distance A N is in\nreciprocal proportion of the weight O to the weight C, the proportion of\nthe moment of the ponderant O, in the point N, to the moment of the\nponderant C, in the point B, will be compounded of the proportions of A\nB to A N, and of A N to A B. Wherefore, setting in order A B, A N, A B,\nthe moment of O to the moment of C will be as the first to the last,\nthat is, as A B to A B. Their moments therefore are equal; and\nconsequently the plane which passes through A will (by the fifth\ndefinition) be a plane of equiponderation. Wherefore they will be\nequally poised; as was to be proved.\n\nNow the converse of this is manifest. For if there be equiponderation\nand the proportion of the weights and distances be not reciprocal, then\nboth the weights will always have the same moments, although one of them\nhave more weight added to it or its distance changed.\n\nCoroll. When ponderants are of the same species, and their moments be\nequal; their magnitudes and distances from the centre of the scale will\nbe reciprocally proportional. For in homogeneous bodies, it is as weight\nto weight, so magnitude to magnitude.\n\n[Sidenote: If the parts of any ponderant press the beams of the scale\n every where equally, all the parts cut off, reckoned from\n centre of the scale, will have their moments in the same\n proportion with that of the parts of a triangle cut off from\n the vertex by strait lines parallel to the base.]\n\n8. If to the whole length of the beam there be applied a parallelogram,\nor a parallelopipedum, or a prisma, or a cylinder, or the superficies of\na cylinder, or of a sphere, or of any portion of a sphere or prisma; the\nparts of any of them cut off with planes parallel to the base will have\ntheir moments in the same proportion with the parts of a triangle, which\nhas its vertex in the centre of the scale, and for one of its sides the\nbeam itself, which parts are cut off by planes parallel to the base.\n\nFirst, let the rectangled parallelogram A B C D (in figure 4) be applied\nto the whole length of the beam A B; and producing C B howsoever to E,\nlet the triangle A B E be described. Let now any part of the\nparallelogram, as A F, be cut off by the plane F G, parallel to the base\nC B; and let F G be produced to A E in the point H. I say, the moment of\nthe whole A B C D to the moment of its part A F, is as the triangle A B\nE to the triangle A G H, that is, in proportion duplicate to that of the\ndistances from the centre of the scale.\n\nFor, the parallelogram A B C D being divided into equal parts, infinite\nin number, by strait lines drawn parallel to the base; and supposing the\nmoment of the strait line C B to be B E, the moment of the strait line F\nG will (by the 7th article) be G H; and the moments of all the strait\nlines of that parallelogram will be so many strait lines in the triangle\nA B E drawn parallel to the base B E; all which parallels together taken\nare the moment of the whole parallelogram A B C D; and the same\nparallels do also constitute the superficies of the triangle A B E.\nWherefore the moment of the parallelogram A B C D is the triangle A B E;\nand for the same reason, the moment of the parallelogram A F is the\ntriangle A G H; and therefore the moment of the whole parallelogram to\nthe moment of a parallelogram which is part of the same, is as the\ntriangle A B E to the triangle A G H, or in proportion duplicate to that\nof the beams to which they are applied. And what is here demonstrated in\nthe case of a parallelogram may be understood to serve for that of a\ncylinder, and of a prisma, and their superficies; as also for the\nsuperficies of a sphere, of an hemisphere, or any portion of a sphere.\nFor the parts of the superficies of a sphere have the same proportion\nwith that of the parts of the axis cut off by the same parallels, by\nwhich the parts of the superficies are cut off, as Archimedes has\ndemonstrated; and therefore when the parts of any of these figures are\nequal and at equal distances from the centre of the scale, their moments\nalso are equal, in the same manner as they are in parallelograms.\n\nSecondly, let the parallelogram A K I B not be rectangled; the strait\nline I B will nevertheless press the point B perpendicularly in the\nstrait line B E; and the strait line L G will press the point G\nperpendicularly in the strait line G H; and all the rest of the strait\nlines which are parallel to I B will do the like. Whatsoever therefore\nthe moment be which is assigned to the strait line I B, as here, for\nexample, it is supposed to be B E, if A E be drawn, the moment of the\nwhole parallelogram A I will be the triangle A B E; and the moment of\nthe part A L will be the triangle A G H. Wherefore the moment of any\nponderant, which has its sides equally applied to the beam, whether they\nbe applied perpendicularly or obliquely, will be always to the moment of\na part of the same in such proportion as the whole triangle has to a\npart of the same cut off by a plane which is parallel to the base.\n\n[Sidenote: The diameter of equiponderation of figures which are\n deficient according to commensurable proportions of their\n altitudes and bases, divides the axis, so that the part taken\n next the vertex is to the other part as the complete figure\n to the deficient figure.]\n\n9. The centre of equiponderation of any figure, which is deficient\naccording to commensurable proportions of the altitude and base\ndiminished, and whose complete figure is either a parallelogram or a\ncylinder, or a parallelopipedum, divides the axis, so, that the part\nnext the vertex, to the other part, is as the complete figure to the\ndeficient figure.\n\nFor let C I A P E (in fig. 5) be a deficient figure, whose axis is A B,\nand whose complete figure is C D F E; and let the axis A B be so divided\nin Z, that A Z be to Z B as C D F E is to C I A P E. I say, the centre\nof equiponderation of the figure C I A P E will be in the point Z.\n\nFirst, that the centre of equiponderation of the figure C I A P E is\nsomewhere in the axis A B is manifest of itself; and therefore A B is a\ndiameter of equiponderation. Let A E be drawn, and let B E be put for\nthe moment of the strait line C E; the triangle A B E will therefore (by\nthe third article) be the moment of the complete figure C D F E. Let the\naxis A B be equally divided in L, and let G L H be drawn parallel and\nequal to the strait line C E, cutting the crooked line C I A P E in I\nand P, and the strait lines A C and A E in K and M. Moreover, let Z O be\ndrawn parallel to the same C E; and let it be, as L G to L I, so L M to\nanother, L N; and let the same be done in all the rest of the strait\nlines possible, parallel to the base; and through all the points N, let\nthe line A N E be drawn; the three-sided figure A N E B will therefore\nbe the moment of the figure C I A P E. Now the triangle A B E is (by the\n9th article of chapter XVII) to the three-sided figure A N E B, as A B C\nD + A I C B is to A I C B twice taken, that is, as C D F E + C I A P E\nis to C I A P E twice taken. But as C I A P E is to C D F E, that is, as\nthe weight of the deficient figure is to the weight of the complete\nfigure, so is C I A P E twice taken to C D F E twice taken. Wherefore,\nsetting in order C D F E + C I A P E. 2 C I A P E. 2 C D F E; the\nproportion of C D F E + C I A P E to C D F E twice taken will be\ncompounded of the proportion of C D F E + C I A P E to C I A P E twice\ntaken, that is, of the proportion of the triangle A B E to the\nthree-sided figure A N E B, that is, of the moment of the complete\nfigure to the moment of the deficient figure, and of the proportion of C\nI A P E twice taken to C D F E twice taken, that is, to the proportion\nreciprocally taken of the weight of the deficient figure to the weight\nof the complete figure.\n\nAgain, seeing by supposition A Z. Z B:: C D F E. C I A P E are\nproportionals; A B. A Z:: C D F E + C I A P E. C D F E will also, by\ncompounding, be proportionals. And seeing A L is the half of A B, A L. A\nZ:: C D F E + C I A P E. 2 C D F E will also be proportionals. But the\nproportion of C D F E + C I A P E to 2 C D F E is compounded, as was but\nnow shown, of the proportions of moment to moment, &c., and therefore\nthe proportion of A L to A Z is compounded of the proportion of the\nmoment of the complete figure C D F E to the moment of the deficient\nfigure C I A P E, and of the proportion of the weight of the deficient\nfigure C I A P E to the weight of the complete figure C D F E; but the\nproportion of A L to A Z is compounded of the proportions of A L to B Z\nand of B Z to A Z. Now the proportion of B Z to A Z is the proportion of\nthe weights reciprocally taken, that is to say, of the weight C I A P E\nto the weight C D F E. Therefore the remaining proportion of A L to B Z,\nthat is, of L B to B Z, is the proportion of the moment of the weight C\nD F E to the moment of the weight C I A P E. But the proportion of A L\nto B Z is compounded of the proportions of A L to A Z and of A Z to Z B;\nof which proportions that of A Z to Z B is the proportion of the weight\nC D F E to the weight C I A P E. Wherefore (by art. 5 of this chapter)\nthe remaining proportion of A L to A Z is the proportion of the\ndistances of the points Z and L from the centre of the scale, which is\nA. And, therefore, (by art. 6) the weight C I A P E shall hang from O in\nthe strait line O Z. So that O Z is one diameter of equiponderation of\nthe weight C I A P E. But the strait line A B is the other diameter of\nequiponderation of the same weight C I A P E. Wherefore (by the 7th\ndefinition) the point Z is the centre of the same equiponderation; which\npoint, by construction, divides the axis so, that the part A Z, which is\nthe part next the vertex, is to the other part Z B, as the complete\nfigure C D F E is to the deficient figure C I A P E; which is that which\nwas to be demonstrated.\n\nCoroll. I. The centre of equiponderation of any of those plane\nthree-sided figures, which are compared with their complete figures in\nthe table of art. 3, chap. XVII, is to be found in the same table, by\ntaking the denominator of the fraction for the part of the axis cut off\nnext the vertex, and the numerator for the other part next the base. For\nexample, if it be required to find the centre of equiponderation of the\nsecond three-sided figure of four means, there is in the concourse of\nthe second column with the row of three-sided figures of four means this\nfraction 5\u20447, which signifies that that figure is to its parallelogram\nor complete figure as 5\u20447 to unity, that is, as 5\u20447 to 7\u20447, or as 5 to\n7; and, therefore the centre of equiponderation of that figure divides\nthe axis, so that the part next the vertex is to the other part as 7 to\n5.\n\nCoroll. II. The centre of equiponderation of any of the solids of those\nfigures, which are contained in the table of art. 7 of the same chap.\nXVII, is exhibited in the same table. For example, if the centre of\nequiponderation of a cone be sought for, the cone will be found to be\n1\u20443 of its cylinder; and, therefore, the centre of its equiponderation\nwill so divide the axis, that the part next the vertex to the other part\nwill be as 3 to 1. Also the solid of a three-sided figure of one mean,\nthat is, a parabolical solid, seeing it is 2\u20444, that is \u00bd of its\ncylinder, will have its centre of equiponderation in that point, which\ndivides the axis, so that the part towards the vertex be double to the\npart towards the base.\n\n[Sidenote: The diameter of equiponderation of the complement of the half\n of any of the said deficient figures, divides that line which\n is drawn through the vertex parallel to the base, so that the\n part next the vertex is to the other part as the complete\n figure to the complement.]\n\n10. The diameter of equiponderation of the complement of the half of any\nof those figures which are contained in the table of art. 3, chap. XVII,\ndivides that line which is drawn through the vertex parallel and equal\nto the base, so that the part next the vertex will be to the other part,\nas the complete figure to the complement.\n\nFor let A I C B (in the same fig. 5) be the half of a parabola, or of\nany other of those three-sided figures which are in the table of art. 3,\nchap. XVII, whose axis is A B, and base B C, having A D drawn from the\nvertex, equal and parallel to the base B C, and whose complete figure is\nthe parallelogram A B C D. Let I Q be drawn at any distance from the\nside C D, but parallel to it; and let A D be the altitude of the\ncomplement A I C D, and Q I a line ordinately applied in it. Wherefore\nthe altitude A L in the deficient figure A I C B is equal to Q I the\nline ordinately applied in its complement; and contrarily, L I the line\nordinately applied in the figure A I C B is equal to the altitude A Q in\nits complement; and so in all the rest of the ordinate lines and\naltitudes the mutation is such, that that line, which is ordinately\napplied in the figure, is the altitude of its complement. And,\ntherefore, the proportion of the altitudes decreasing to that of the\nordinate lines decreasing, being multiplicate according to any number in\nthe deficient figure, is submultiplicate according to the same number in\nits complement. For example, if A I C B be a parabola, seeing the\nproportion of A B to A L is duplicate to that of B C to L I, the\nproportion of AD to A Q in the complement A I C D, which is the same\nwith that of B C to L I, will be subduplicate to that of C D to Q I,\nwhich is the same with that of A B to A L; and consequently, in a\nparabola, the complement will be to the parallelogram as 1 to 3; in a\nthree-sided figure of two means, as 1 to 4; in a three-sided figure of\nthree means, as 1 to 5, &c. But all the ordinate lines together in A I C\nD are its moment; and all the ordinate lines in A I C B are its moment.\nWherefore the moments of the complements of the halves of deficient\nfigures in the table of art. 3 of chap. XVII, being compared, are as the\ndeficient figures themselves; and, therefore, the diameter of\nequiponderation will divide the strait line A D in such proportion, that\nthe part next the vertex be to the other part, as the complete figure A\nB C D is to the complement A I C D.\n\nCoroll. The diameter of equiponderation of these halves may be found by\nthe table of art. 3 of chap. XVII, in this manner. Let there be\npropounded any deficient figure, namely, the second three-sided figure\nof two means. This figure is to the complete figure as 3\u20445 to 1, that is\n3 to 5. Wherefore the complement to the same complete figure is as 2 to\n5; and, therefore, the diameter of equiponderation of this complement\nwill cut the strait line drawn from the vertex parallel to the base, so\nthat the part next the vertex will be to the other part as 5 to 2. And,\nin like manner, any other of the said three-sided figures being\npropounded, if the numerator of its fraction found out in the table be\ntaken from the denominator, the strait line drawn from the vertex is to\nbe divided, so that the part next the vertex be to the other part, as\nthe denominator is to the remainder which that subtraction leaves.\n\n[Sidenote: The centre of equiponderation of the half of any of the\n deficient figures in the first row of the table of art. 3,\n chapter xvii, may be found out by the numbers of the second\n row.]\n\n11. The centre of equiponderation of the half of any of those\ncrooked-lined figures, which are in the first row of the table of art. 3\nof chap. XVII, is in that strait line which, being parallel to the axis,\ndivides the base according to the numbers of the fraction next below it\nin the second row, so that the numerator be answerable to that part\nwhich is towards the axis.\n\nFor example, let the first figure of three means be taken, whose half is\nA B C D (in fig. 6), and let the rectangle A B E D be completed. The\ncomplement therefore will be B C D E. And seeing A B E D is to the\nfigure A B C D (by the table) as 5 to 4, the same A B E D will be to the\ncomplement B C D E as 5 to 1. Wherefore, if F G be drawn parallel to the\nbase D A, cutting the axis so that A G be to G B as 4 to 5, the centre\nof equiponderation of the figure A B C D will, by the precedent article,\nbe somewhere in the same F G. Again, seeing, by the same article, the\ncomplete figure A B E D, is to the complement B C D E as 5 to 1,\ntherefore if B E and A D be divided in I and H as 5 to 1 the centre of\nequiponderation of the complement B C D E will be somewhere in the\nstrait line which connects H and I. Let now the strait line L K be drawn\nthrough M the centre of the complete figure, parallel to the base; and\nthe strait line N O through the same centre M, perpendicular to it; and\nlet the strait lines L K and F G cut the strait line H I in P and Q. Let\nP R be taken quadruple to P Q; and let R M be drawn and produced to F G\nin S. R M therefore will be to M S as 4 to 1, that is, as the figure A B\nC D to its complement B C D E. Wherefore, seeing M is the centre of the\ncomplete figure A B E D, and the distances of R and S from the centre M\nbe in proportion reciprocal to that of the weight of the complement B C\nD E to the weight of the figure A B C D, R and S will either be the\ncentres of equiponderation of their own figures, or those centres will\nbe in some other points of the diameters of equiponderation H I and F G.\nBut this last is impossible. For no other strait line can be drawn\nthrough the point M terminating in the strait lines H I and F G, and\nretaining the proportion of M R to M S, that is, of the figure A B C D\nto its complement B C D E. The centre, therefore, of equiponderation of\nthe figure A B C D is in the point S. Now, seeing P M hath the same\nproportion to Q S which R P hath to R Q, Q S will be 5 of those parts of\nwhich P M is four, that is, of which I N is four. But I N or P M is 2 of\nthose parts of which E B or F G is 6; and, therefore, if it be as 4 to\n5, so 2 to a fourth, that fourth will be 2\u00bd. Wherefore Q S is 2\u00bd of\nthose parts of which F G is 6. But F Q is 1; and, therefore, F S is 3\u00bd.\nWherefore the remaining part G S is 2\u00bd. So that F G is so divided in S,\nthat the part towards the axis is in proportion to the other part, as 2\u00bd\nto 3\u00bd, that is as 5 to 7; which answereth to the fraction 5\u20447 in the\nsecond row, next under the fraction 4\u20445 in the first row. Wherefore\ndrawing S T parallel to the axis, the base will be divided in like\nmanner.\n\nBy this method it is manifest, that the base of a semiparabola will be\ndivided into 3 and 5; and the base of the first three-sided figure of\ntwo means, into 4 and 6; and of the first three-sided figure of four\nmeans, into 6 and 8. The fractions, therefore, of the second row denote\nthe proportions, into which the bases of the figures of the first row\nare divided by the diameters of equiponderation. But the first row\nbegins one place higher than the second row.\n\n[Sidenote: The centre of equiponderation of the half of any of the\n figures of the second row of the same table may be found out\n by the numbers of the fourth row.]\n\n12. The centre of equiponderation of the half of any of the figures in\nthe second row of the same table of art. 3, chap. XVII, is in a strait\nline parallel to the axis, and dividing the base according to the\nnumbers of the fraction in the fourth row, two places lower, so as that\nthe numerator be answerable to that part which is next the axis.\n\nLet the half of the second three-sided figure of two means be taken; and\nlet it be A B C D (in fig. 7); whose complement is B C D E, and the\nrectangle completed A B E D. Let this rectangle be divided by the two\nstrait lines L K and N O, cutting one another in the centre M at right\nangles; and because A B E D is to A B C D as 5 to 3, let A B be divided\nin G, so that A G to B G be as 3 to 5; and let F G be drawn parallel to\nthe base. Also because A B E D is (by art. 9) to B C D E as 5 to 2, let\nB E be divided in the point I, so that B I be to I E as 5 to 2; and let\nI H be drawn parallel to the axis, cutting L K and F G in P and Q. Let\nnow P R be so taken, that it be to P Q as 3 to 2, and let R M be drawn\nand produced to F G in S. Seeing, therefore, R P is to P Q, that is, R M\nto M S, as A B C D is to its complement B C D E, and the centres of\nequiponderation of A B C D and B C D E are in the strait lines F G and H\nI, and the centre of equiponderation of them both together in the point\nM; R will be the centre of the complement B C D E, and S the centre of\nthe figure A B C D. And seeing P M, that is I N, is to Q S, as R P is to\nR Q; and I N or P M is 3 of those parts, of which B E, that is F G, is\n14; therefore Q S is 5 of the same parts; and E I, that is F G, 4; and F\nS, 9; and G S, 5. Wherefore the strait line S T being drawn parallel to\nthe axis, will divide the base A D into 5 and 9. But the fraction 5\u20449 is\nfound in the fourth row of the table, two places below the fraction \u2157 in\nthe second row.\n\nBy the same method, if in the same second row there be taken the second\nthree-sided figure of three means, the centre of equiponderation of the\nhalf of it will be found to be in a strait line parallel to the axis,\ndividing the base according to the numbers of the fraction 6\u204410, two\nplaces below in the fourth row. And the same way serves for all the rest\nof the figures in the second row. In like manner, the centre of\nequiponderation of the third three-sided figure of three means will be\nfound to be in a strait line parallel to the axis, dividing the base, so\nthat the part next the axis be to the other part as 7 to 13, &c.\n\nCoroll. The centres of equiponderation of the halves of the said figures\nare known, seeing they are in the intersection of the strait lines S T\nand F G, which are both known.\n\n[Sidenote: The centre of equiponderation of the half of any of the\n figures in the same table being known, the centre of the\n excess of the same figure above a triangle of the same\n altitude and base is also known.]\n\n13. The centre of equiponderation of the half of any of the figures,\nwhich (in the table of art. 3, chap. XVII) are compared with their\nparallelograms, being known; the centre of equiponderation of the excess\nof the same figure above its triangle is also known.\n\nFor example, let the semiparabola A B C D (in fig. 8) be taken, whose\naxis is A B; whose complete figure is A B E D; and whose excess above\nits triangle is B C D B. Its centre of equiponderation may be found out\nin this manner. Let F G be drawn parallel to the base, so that A F be a\nthird part of the axis; and let H I be drawn parallel to the axis, so\nthat A H be a third part of the base. This being done, the centre of\nequiponderation of the triangle A B D will be I. Again, let K L be drawn\nparallel to the base, so that A K be to A B as 2 to 5; and M N parallel\nto the axis, so that A M be to A D as 3 to 8; and let M N terminate in\nthe strait line K L. The centre, therefore, of equiponderation of the\nparabola A B C D is N; and therefore we have the centres of\nequiponderation of the semiparabola A B C D, and of its part the\ntriangle A B D. That we may now find the centre of equiponderation of\nthe remaining part B C D B, let I N be drawn and produced to O, so that\nN O be triple to I N; and O will be the centre sought for. For seeing\nthe weight of A B D to the weight of B C D B is in proportion reciprocal\nto that of the strait line N O to the strait line I N; and N is the\ncentre of the whole, and I the centre of the triangle A B D; O will be\nthe centre of the remaining part, namely, of the figure B D C B; which\nwas to be found.\n\nCoroll. The centre of equiponderation of the figure B D C B is in the\nconcourse of two strait lines, whereof one is parallel to the base, and\ndivides the axis, so that the part next the base be \u2157 or 9\u204415 of the\nwhole axis; the other is parallel to the axis, and so divides the base,\nthat the part towards the axis be \u00bd, or 12\u204424 of the whole base. For\ndrawing O P parallel to the base, it will be as I N to N O, so F K to K\nP, that is, so 1 to 3, or 5 to 15. But A F is 5\u204415, or \u2153 of the whole A\nB; and A K is 6\u204415, or \u2156; and F K 1\u204415; and KP 3\u204415; and therefore A P\nis 9\u204415 of the axis A B. Also A H is \u2153, or 8\u204424; and A M \u215c, or 9\u204424 of\nthe whole base; and therefore O Q being drawn parallel to the axis, M Q,\nwhich is triple to H M, will be 3\u204424. Wherefore A Q is 12\u204424, or \u00bd of\nthe base A D.\n\nThe excesses of the rest of the three-sided figures in the first row of\nthe table of art. 3, chap. XVII, have their centres of equiponderation\nin two strait lines, which divide the axis and base according to those\nfractions, which add 4 to the numerators of the fractions of a parabola\n9\u204415 and 12\u204424; and 6 to the denominators, in this manner:\u2014\n\n In a parabola, the axis 9\u204415, the base 12\u204424.\n In the first three-sided figure, the axis 13\u204421, the base 16\u204430.\n In the second three-sided figure, the axis 17\u204427, the base 20\u204436, &c.\n\nAnd by the same method, any man, if it be worth the pains, may find out\nthe centres of equiponderation of the excesses above their triangles of\nthe rest of the figures in the second and third row, &c.\n\n[Sidenote: The centre of equiponderation of a solid sector is in the\n axis so divided, that the part next the vertex be to the\n whole axis, wanting half the axis of the portion of the\n sphere, as 3 to 4.]\n\n14. The centre of equiponderation of the sector of a sphere, that is, of\na figure compounded of a right cone, whose vertex is the centre of the\nsphere, and the portion of the sphere whose base is the same with that\nof the cone, divides the strait line which is made of the axis of the\ncone and half the axis of the portion together taken, so that the part\nnext the vertex be triple to the other part, or to the whole strait line\nas 3 to 4.\n\nFor let A B C (in fig. 9) be the sector of a sphere, whose vertex is the\ncentre of the sphere A; whose axis is A D; and the circle upon B C is\nthe common base of the portion of the sphere and of the cone whose\nvertex is A; the axis of which portion is E D, and the half thereof F D;\nand the axis of the cone, A E. Lastly, let A G be 3\u20444 of the strait line\nA F. I say, G is the centre of equiponderation of the sector A B C.\n\nLet the strait line F H be drawn of any length, making right angles with\nA F at F; and drawing the strait line A H, let the triangle A F H be\nmade. Then upon the same centre A let any arch I K be drawn, cutting A D\nin L; and its chord, cutting A D in M; and dividing M L equally in N,\nlet N O be drawn parallel to the strait line F H, and meeting with the\nstrait line A H in O.\n\nSeeing now B D C is the spherical superficies of the portion cut off\nwith a plane passing through B C, and cutting the axis at right angles;\nand seeing F H divides E D, the axis of the portion, into two equal\nparts in F; the centre of equiponderation of the superficies B D C will\nbe in F (by art. 8); and for the same reason the centre of\nequiponderation of the superficies I L K, K being in the strait line A\nC, will be in N. And in like manner, if there were drawn, between the\ncentre of the sphere A and the outermost spherical superficies of the\nsector, arches infinite in number, the centres of equiponderation of the\nspherical superficies, in which those arches are, would be found to be\nin that part of the axis, which is intercepted between the superficies\nitself and a plane passing along by the chord of the arch, and cutting\nthe axis in the middle at right angles.\n\nLet it now be supposed that the moment of the outermost spherical\nsuperficies B D C is F H. Seeing therefore the superficies B D C is to\nthe superficies I L K in proportion duplicate to that of the arch B D C\nto the arch I L K, that is, of B E to I M, that is, of F H to N O; let\nit be as F H to N O, so N O to another N P; and again, as N O to N P, so\nN P to another N Q; and let this be done in all the strait lines\nparallel to the base F H that can possibly be drawn between the base and\nthe vertex of the triangle A F H. If then through all the points Q there\nbe drawn the crooked line A Q H, the figure A F H Q A will be the\ncomplement of the first three-sided figure of two means; and the same\nwill also be the moment of all the spherical superficies, of which the\nsolid sector A B C D is compounded; and by consequent, the moment of the\nsector itself. Let now F H be understood to be the semidiameter of the\nbase of a right cone, whose side is A H, and axis A F Wherefore, seeing\nthe bases of the cones, which pass through F and N and the rest of the\npoints of the axis, are in proportion duplicate to that of the strait\nlines F H and N O, &c., the moment of all the bases together, that is,\nof the whole cone, will be the figure itself A F H Q A; and therefore\nthe centre of equiponderation of the cone A F H is the same with that of\nthe solid sector. Wherefore, seeing A G is \u00be of the axis A F, the centre\nof equiponderation of the cone A F H is in G; and therefore the centre\nof the solid sector is in G also, and divides the part A F of the axis\nso that A G is triple to G F; that is, A G is to A F as 3 to 4; which\nwas to be demonstrated.\n\nNote, that when the sector is a hemisphere, the axis of the cone\nvanisheth into that point which is the centre of the sphere; and\ntherefore it addeth nothing to half the axis of the portion. Wherefore,\nif in the axis of the hemisphere there be taken from the centre \u00be of\nhalf the axis, that is, \u215c of the semidiameter of the sphere, there will\nbe the centre of equiponderation of the hemisphere.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C.XXIII.\n _Fig. 1-9_\n]\n\n------------------------------------------------------------------------\n\n\n\n\n CHAPTER XXIV.\n\n OF REFRACTION AND REFLECTION.\n\n1. Definitions.\u20142. In perpendicular motion there is no refraction.\u20143.\n Things thrown out of a thinner into a thicker medium are so\n refracted that the angle refracted is greater than the angle of\n inclination.\u20144. Endeavour, which from one point tendeth every way,\n will be so refracted, as that the sine of the angle refracted will\n be to the sine of the angle of inclination, as the density of the\n first medium is to the density of the second medium, reciprocally\n taken.\u20145. The sine of the refracted angle in one inclination is to\n the sine of the refracted angle in another inclination, as the sine\n of the angle of that inclination is to the sine of the angle of this\n inclination.\u20146. If two lines of incidence, having equal inclination,\n be the one in a thinner, the other in a thicker medium, the sine of\n the angle of inclination will be a mean proportional between the two\n sines of the refracted angles.\u20147. If the angle of inclination be\n semirect, and the line of inclination be in the thicker medium, and\n the proportion of their densities be the same with that of the\n diagonal to the side of a square, and the separating superficies be\n plane, the refracted line will be in the separating superficies.\u20148.\n If a body be carried in a strait line upon another body, and do not\n penetrate the same, but be reflected from it, the angle of\n reflection will be equal to the angle of incidence.\u20149. The same\n happens in the generation of motion in the line of incidence.\n\n\n DEFINITIONS.\n\n[Sidenote: Definitions.]\n\nI. Refraction is the _breaking_ of that strait line, in which a body is\nmoved or its action would proceed in one and the same medium, into two\nstrait lines, by reason of the different natures of the two mediums.\n\nII. The former of these is called the _line of incidence_; the latter\nthe _refracted line_. III. The _point of refraction_ is the common point\nof the line of incidence, and of the refracted line.\n\nIV. The _refracting superficies_, which also is the _separating\nsuperficies_ of the two mediums, is that in which is the point of\nrefraction.\n\nV. The _angle refracted_ is that, which the refracted line makes in the\npoint of refraction with that line, which from the same point is drawn\nperpendicular to the separating superficies in a different medium.\n\nVI. The _angle of refraction_ is that which the refracted line makes\nwith the line of incidence produced.\n\nVII. The _angle of inclination_ is that which the line of incidence\nmakes with that line, which from the point of refraction is drawn\nperpendicular to the separating superficies.\n\nVIII. The _angle of incidence_ is the complement to a right angle of the\nangle of inclination.\n\nAnd so, (in fig. 1) the refraction is made in A B F. The refracted line\nis B F. The line of incidence is A B. The point of incidence and of\nrefraction is B. The refracting or separating superficies is D B E. The\nline of incidence produced directly is A B C. The perpendicular to the\nseparating superficies is B H. The angle of refraction is C B F. The\nangle refracted is H B F. The angle of inclination is A B G or H B C.\nThe angle of incidence is A B D.\n\nIX. Moreover the _thinner medium_ is understood to be that in which\nthere is less resistance to motion, or to the generation of motion; and\nthe thicker that wherein there is greater resistance.\n\nX. And that medium in which there is equal resistance everywhere, is a\n_homogeneous medium_. All other mediums are _heterogeneous_.\n\n[Sidenote: In perpendicular motion there is no refraction.]\n\n If a body pass, or there be generation of motion from one medium to\nanother of different density, in a line perpendicular to the separating\nsuperficies, there will be no refraction.\n\nFor seeing on every side of the perpendicular all things in the mediums\nare supposed to be like and equal, if the motion itself be supposed to\nbe perpendicular, the inclinations also will be equal, or rather none at\nall; and therefore there can be no cause from which refraction may be\ninferred to be on one side of the perpendicular, which will not conclude\nthe same refraction to be on the other side. Which being so, refraction\non one side will destroy refraction on the other side; and consequently\neither the refracted line will be everywhere, which is absurd, or there\nwill be no refracted line at all; which was to be demonstrated.\n\nCoroll. It is manifest from hence, that the cause of refraction\nconsisteth only in the obliquity of the line of incidence, whether the\nincident body penetrate both the mediums, or without penetrating,\npropagate motion by pressure only.\n\n[Sidenote: Things thrown out of a thinner into a thicker medium are so\n refracted that the angle refracted is greater than the angle\n of inclination.]\n\n3. If a body, without any change of situation of its internal parts, as\na stone, be moved obliquely out of the thinner medium, and proceed\npenetrating the thicker medium, and the thicker medium be such, as that\nits internal parts being moved restore themselves to their former\nsituation; the angle refracted will be greater than the angle of\ninclination.\n\nFor let D B E (in the same first figure) be the separating superficies\nof two mediums; and let a body, as a stone thrown, be understood to be\nmoved as is supposed in the strait line A B C; and let A B be in the\nthinner medium, as in the air; and B C in the thicker, as in the water.\nI say the stone, which being thrown, is moved in the line A B, will not\nproceed in the line B C, but in some other line, namely, that, with\nwhich the perpendicular B H makes the refracted angle H B F greater than\nthe angle of inclination H B C.\n\nFor seeing the stone coming from A, and falling upon B, makes that which\nis at B proceed towards H, and that the like is done in all the strait\nlines which are parallel to B H; and seeing the parts moved restore\nthemselves by contrary motion in the same line; there will be contrary\nmotion generated in H B, and in all the strait lines which are parallel\nto it. Wherefore, the motion of the stone will be made by the concourse\nof the motions in A G, that is, in D B, and in G B, that is, in B H, and\nlastly, in H B, that is, by the concourse of three motions. But by the\nconcourse of the motions in A G and B H, the stone will be carried to C;\nand therefore by adding the motion in H B, it will be carried higher in\nsome other line, as in B F, and make the angle H B F greater than the\nangle H B C.\n\nAnd from hence may be derived the cause, why bodies which are thrown in\na very oblique line, if either they be any thing flat, or be thrown with\ngreat force, will, when they fall upon the water, be cast up again from\nthe water into the air.\n\nFor let A B (in fig. 2) be the superficies of the water; into which,\nfrom the point C, let a stone be thrown in the strait line C A, making\nwith the line B A produced a very little angle C A D; and producing B A\nindefinitely to D, let C D be drawn perpendicular to it, and A E\nparallel to C D. The stone therefore will be moved in C A by the\nconcourse of two motions in C D and D A, whose velocities are as the\nlines themselves C D and D A. And from the motion in C D and all its\nparallels downwards, as soon as the stone falls upon A, there will be\nreaction upwards, because the water restores itself to its former\nsituation. If now the stone be thrown with sufficient obliquity, that\nis, if the strait line C D be short enough, that is, if the endeavour of\nthe stone downwards be less than the reaction of the water upwards, that\nis, less than the endeavour it hath from its own gravity (for that may\nbe), the stone will by reason of the excess of the endeavour which the\nwater hath to restore itself, above that which the stone hath downwards,\nbe raised again above the superficies A B, and be carried higher, being\nreflected in a line which goes higher, as the line A G.\n\n[Sidenote: Endeavour, which from one point tendeth every way, will be so\n refracted, as that the sine of the angle refracted will be to\n the sine of the angle of inclination, as the density of the\n first medium is to the density of the second medium,\n reciprocally taken.]\n\n4. If from a point, whatsoever the medium be, endeavour be propagated\nevery way into all the parts of that medium; and to the same endeavour\nthere be obliquely opposed another medium of a different nature, that\nis, either thinner or thicker; that endeavour will be so refracted, that\nthe sine of the angle refracted, to the sine of the angle of\ninclination, will be as the density of the first medium to the density\nof the second medium, reciprocally taken.\n\nFirst, let a body be in the thinner medium in A (fig. 3), and let it be\nunderstood to have endeavour every way, and consequently, that its\nendeavour proceed in the lines A B and A _b_; to which let B _b_ the\nsuperficies of the thicker medium be obliquely opposed in B and _b_, so\nthat A B and A _b_ be equal; and let the strait line B _b_ be produced\nboth ways. From the points B and _b_, let the perpendiculars B C and _b\nc_ be drawn; and upon the centres B and _b_, and at the equal distances\nB A and _b_ A, let the circles A C and A _c_ be described, cutting B C\nand _b c_ in C and _c_, and the same C B and _c b_ produced in D and\n_d_, as also A B and A _b_ produced in E and _e_. Then from the point A\nto the strait lines B C and _b c_ let the perpendiculars A F and A _f_\nbe drawn. A F therefore will be the sine of the angle of inclination of\nthe strait line A B, and A _f_ the sine of the angle of inclination of\nthe strait line A _h_, which two inclinations are by construction made\nequal. I say, as the density of the medium in which are B C and _b c_ is\nto the density of the medium in which are B D and _b d_, so is the sine\nof the angle refracted, to the sine of the angle of inclination.\n\nLet the strait line F G be drawn parallel to the strait line A B,\nmeeting with the strait line _b_ B produced in G.\n\nSeeing therefore A F and B G are also parallels, they will be equal; and\nconsequently, the endeavour in A F is propagated in the same time, in\nwhich the endeavour in B G would be propagated if the medium were of the\nsame density. But because B G is in a thicker medium, that is, in a\nmedium which resists the endeavour more than the medium in which A F is,\nthe endeavour will be propagated less in B G than in A F, according to\nthe proportion which the density of the medium, in which A F is, hath to\nthe density of the medium in which B G is. Let therefore the density of\nthe medium, in which B G is, be to the density of the medium, in which A\nF is, as B G is to B H; and let the measure of the time be the radius of\nthe circle. Let H I be drawn parallel to B D, meeting with the\ncircumference in I; and from the point I let I K be drawn perpendicular\nto B D; which being done, B H and I K will be equal; and I K will be to\nA F, as the density of the medium in which is A F is to the density of\nthe medium in which is I K. Seeing therefore in the time A B, which is\nthe radius of the circle, the endeavour is propagated in A F in the\nthinner medium, it will be propagated in the same time, that is, in the\ntime B I in the thicker medium from K to I. Therefore, B I is the\nrefracted line of the line of incidence A B; and I K is the sine of the\nangle refracted; and A F the sine of the angle of inclination.\nWherefore, seeing I K is to A F, as the density of the medium in which\nis A F to the density of the medium in which is I K; it will be as the\ndensity of the medium in which is A F or B C to the density of the\nmedium in which is I K or B D, so the sine of the angle refracted to the\nsine of the angle of inclination. And by the same reason it may be\nshown, that as the density of the thinner medium is to the density of\nthe thicker medium, so will K I the sine of the angle refracted be to A\nF the sine of the angle of inclination.\n\nSecondly, let the body, which endeavoureth every way, be in the thicker\nmedium at I. If, therefore, both the mediums were of the same density,\nthe endeavour of the body in I B would tend directly to L; and the sine\nof the angle of inclination L M would be equal to I K or B H. But\nbecause the density of the medium, in which is I K, to the density of\nthe medium, in which is L M, is as B H to B G, that is, to A F, the\nendeavour will be propagated further in the medium in which L M is, than\nin the medium in which I K is, in the proportion of density to density,\nthat is, of M L to A F. Wherefore, B A being drawn, the angle refracted\nwill be C B A, and its sine A F. But L M is the sine of the angle of\ninclination; and therefore again, as the density of one medium is to the\ndensity of the different medium, so reciprocally is the sine of the\nangle refracted to the sine of the angle of inclination; which was to be\ndemonstrated.\n\nIn this demonstration, I have made the separating superficies B _b_\nplane by construction. But though it were concave or convex, the theorem\nwould nevertheless be true. For the refraction being made in the point B\nof the plane separating superficies, if a crooked line, as P Q, be\ndrawn, touching the separating line in the point B; neither the\nrefracted line B I, nor the perpendicular B D, will be altered; and the\nrefracted angle K B I, as also its sine K I, will be still the same they\nwere.\n\n[Sidenote: The sine of the refracted angle in one inclination is to the\n sine of the refracted angle in another inclination, as the\n sine of the angle of that inclination is to the sine of the\n angle of this inclination.]\n\n5. The sine of the angle refracted in one inclination is to the sine of\nthe angle refracted in another inclination, as the sine of the angle of\nthat inclination to the sine of the angle of this inclination.\n\nFor seeing the sine of the refracted angle is to the sine of the angle\nof inclination, whatsoever that inclination be, as the density of one\nmedium to the density of the other medium; the proportion of the sine of\nthe refracted angle, to the sine of the angle of inclination, will be\ncompounded of the proportions of density to density, and of the sine of\nthe angle of one inclination to the sine of the angle of the other\ninclination. But the proportions of the densities in the same\nhomogeneous body are supposed to be the same. Wherefore refracted angles\nin different inclinations are as the sines of the angles of those\ninclinations; which was to be demonstrated.\n\n[Sidenote: If two lines of incidence, having equal inclination, be one\n in a thinner the other in a thicker medium, the sine of the\n angle of inclination will be a mean proportional between the\n two sines of the refracted angles.]\n\n6. If two lines of incidence, having equal inclination, be the one in a\nthinner, the other in a thicker medium, the sine of the angle of their\ninclination will be a mean proportional between the two sines of their\nangles refracted.\n\nFor let the strait line A B (in fig. 3) have its inclination in the\nthinner medium, and be refracted in the thicker medium in B I; and let E\nB have as much inclination in the thicker medium, and be refracted in\nthe thinner medium in B S; and let R S, the sine of the angle refracted,\nbe drawn. I say, the strait lines R S, A F, and I K are in continual\nproportion. For it is, as the density of the thicker medium to the\ndensity of the thinner medium, so R S to A F. But it is also as the\ndensity of the same thicker medium to that of the same thinner medium,\nso A F to I K. Wherefore R S. A F :: A F. I K are proportionals; that\nis, R S, A F, and I K are in continual proportion, and A F is the mean\nproportional; which was to be proved.\n\n[Sidenote: If the angle of inclination be semirect, and the line of\n inclination be in the thicker medium, and the proportion of\n their densities be the same with that of the diagonal to the\n side of a square, and the separating superficies be plain,\n the refracted line will be in the separating superficies.]\n\n7. If the angle of inclination be semirect, and the line of inclination\nbe in the thicker medium, and the proportion of the densities be as that\nof a diagonal to the side of its square, and the separating superficies\nbe plain, the refracted line will be in that separating superficies.\n\nFor in the circle A C (fig. 4) let the angle of inclination A B C be an\nangle of 45 degrees. Let C B be produced to the circumference in D; and\nlet C E, the sine of the angle E B C, be drawn, to which let B F be\ntaken equal in the separating line B G. B C E F will therefore be a\nparallelogram, and F E and B C, that is F E and B G equal. Let A G be\ndrawn, namely the diagonal of the square whose side is B G, and it will\nbe, as A G to E F so B G to B F; and so, by supposition, the density of\nthe medium, in which C is, to the density of the medium in which D is;\nand so also the sine of the angle refracted to the sine of the angle of\ninclination. Drawing therefore F D, and from D the line D H\nperpendicular to A B produced, D H will be the sine of the angle of\ninclination. And seeing the sine of the angle refracted is to the sine\nof the angle of inclination, as the density of the medium, in which is\nC, is to the density of the medium in which is D, that is, by\nsupposition, as A G is to F E, that is as B G is to D H; and seeing D H\nis the sine of the angle of inclination, B G will therefore be the sine\nof the angle refracted. Wherefore B G will be the refracted line, and\nlye in the plain separating superficies; which was to be demonstrated.\n\nCoroll. It is therefore manifest, that when the inclination is greater\nthan 45 degrees, as also when it is less, provided the density be\ngreater, it may happen that the refraction will not enter the thinner\nmedium at all.\n\n[Sidenote: If a body be carried in a strait line upon another body, and\n do not penetrate it, but be reflected from it, the angle of\n reflection will be equal to the angle of incidence.]\n\n8. If a body fall in a strait line upon another body, and do not\npenetrate it, but be reflected from it, the angle of reflection will be\nequal to the angle of incidence.\n\nLet there be a body at A (in fig. 5), which falling with strait motion\nin the line A C upon another body at C, passeth no further, but is\nreflected; and let the angle of incidence be any angle, as A C D. Let\nthe strait line C E be drawn, making with D C produced the angle E C F\nequal to the angle A C D; and let A D be drawn perpendicular to the\nstrait line D F. Also in the same strait line D F let C G be taken equal\nto C D; and let the perpendicular G E be raised, cutting C E in E. This\nbeing done, the triangles A C D and E C G will be equal and like. Let C\nH be drawn equal and parallel to the strait line A D; and let H C be\nproduced indefinitely to I. Lastly let E A be drawn, which will pass\nthrough H, and be parallel and equal to G D. I say the motion from A to\nC, in the strait line of incidence A C, will be reflected in the strait\nline C E.\n\nFor the motion from A to C is made by two coefficient or concurrent\nmotions, the one in A H parallel to D G, the other in A D perpendicular\nto the same D G; of which two motions that in A H works nothing upon the\nbody A after it has been moved as far as C, because, by supposition, it\ndoth not pass the strait line D G; whereas the endeavour in A D, that is\nin H C, worketh further towards I. But seeing it doth only press and not\npenetrate, there will be reaction in H, which causeth motion from C\ntowards H; and in the meantime the motion in H E remains the same it was\nin A H; and therefore the body will now be moved by the concourse of two\nmotions in C H and H E, which are equal to the two motions it had\nformerly in A H and H C. Wherefore it will be carried on in C E. The\nangle therefore of reflection will be E C G, equal, by construction, to\nthe angle A C D; which was to be demonstrated.\n\nNow when the body is considered but as a point, it is all one whether\nthe superficies or line in which the reflection is made be strait or\ncrooked; for the point of incidence and reflection C is as well in the\ncrooked line which toucheth D G in C, as in D G itself.\n\n[Sidenote: The same happens in the generation of motion in the line of\n incidence.]\n\n9. But if we suppose that not a body be moved, but some endeavour only\nbe propagated from A to C, the demonstration will nevertheless be the\nsame. For all endeavour is motion; and when it hath reached the solid\nbody in C, it presseth it, and endeavoureth further in C I. Wherefore\nthe reaction will proceed in C H; and the endeavour in C H concurring\nwith the endeavour in H E, will generate the endeavour in C E, in the\nsame manner as in the repercussion of bodies moved.\n\nIf therefore endeavour be propagated from any point to the concave\nsuperficies of a spherical body, the reflected line with the\ncircumference of a great circle in the same sphere will make an angle\nequal to the angle of incidence.\n\nFor if endeavour be propagated from A (in fig. 6) to the circumference\nin B, and the centre of the sphere be C, and the line C B be drawn, as\nalso the tangent D B E; and lastly if the angle F B D be made equal to\nthe angle A B E, the reflection will be made in the line B F, as hath\nbeen newly shown. Wherefore the angles, which the strait lines A B and F\nB make with the circumference, will also be equal. But it is here to be\nnoted, that if C B be produced howsoever to G, the endeavour in the line\nG B C will proceed only from the perpendicular reaction in G B; and that\ntherefore there will be no other endeavour in the point B towards the\nparts which are within the sphere, besides that which tends towards the\ncentre.\n\nAnd here I put an end to the third part of this discourse; in which I\nhave considered motion and magnitude by themselves in the abstract. The\nfourth and last part, concerning the _phenomena of nature_, that is to\nsay, concerning the motions and magnitudes of the bodies which are parts\nof the world, real and existent, is that which follows.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C. XXIV.\n _Fig. 1-6_\n]\n\n------------------------------------------------------------------------\n\n\n PART IV.\n PHYSICS,\n OR THE PHENOMENA OF NATURE.\n\n\n -------\n\n\n CHAPTER XXV.\n OF SENSE AND ANIMAL MOTION.\n\n1. The connexion of what hath been said with that which followeth.\u20142.\n The investigation of the nature of sense, and the definition of\n sense.\u20143. The subject and object of sense.\u20144. The organs of\n sense.\u20145. All bodies are not indued with sense.\u20146. But one phantasm\n at one and the same time.\u20147. Imagination the remains of past sense,\n which also is memory. Of sleep.\u20148. How phantasms succeed one\n another.\u20149. Dreams, whence they proceed.\u201410. Of the senses, their\n kinds, their organs, and phantasms proper and common.\u201411. The\n magnitude of images, how and by what it is determined.\u201412. Pleasure,\n pain, appetite and aversion, what they are.\u201413. Deliberation and\n will, what.\n\n\n[Sidenote: The connexion of what hath been said with that which\n followeth.]\n\n1. I have, in the first chapter, defined philosophy to be _knowledge of\neffects acquired by true ratiocination, from knowledge first had of\ntheir causes and generation; and of such causes or generations as may\nbe, from former knowledge of their effects or appearances_. There are,\ntherefore, two methods of philosophy; one, from the generation of things\nto their possible effects; and the other, from their effects or\nappearances to some possible generation of the same. In the former of\nthese the truth of the first principles of our ratiocination, namely\ndefinitions, is made and constituted by ourselves, whilst we consent and\nagree about the appellations of things. And this part I have finished in\nthe foregoing chapters; in which, if I am not deceived, I have affirmed\nnothing, saving the definitions themselves, which hath not good\ncoherence with the definitions I have given; that is to say, which is\nnot sufficiently demonstrated to all those, that agree with me in the\nuse of words and appellations; for whose sake only I have written the\nsame. I now enter upon the other part; which is the finding out by the\nappearances or effects of nature, which we know by sense, some ways and\nmeans by which they may be, I do not say they are, generated. The\nprinciples, therefore, upon which the following discourse depends, are\nnot such as we ourselves make and pronounce in general terms, as\ndefinitions; but such, as being placed in the things themselves by the\nAuthor of Nature, are by us observed in them; and we make use of them in\nsingle and particular, not universal propositions. Nor do they impose\nupon us any necessity of constituting theorems; their use being only,\nthough not without such general propositions as have been already\ndemonstrated, to show us the possibility of some production or\ngeneration. Seeing, therefore, the science, which is here taught, hath\nits principles in the appearances of nature, and endeth in the attaining\nof some knowledge of natural causes, I have given to this part the title\nof PHYSICS, or the _Phenomena of Nature_. Now such things as appear, or\nare shown to us by nature, we call phenomena or appearances.\n\nOf all the phenomena or appearances which are near us, the most\nadmirable is apparition itself, \u03c4\u1f78 \u03c6\u03b1\u03af\u03bd\u03b5\u03c3\u03b8\u03b1\u03b9; namely, that some natural\nbodies have in themselves the patterns almost of all things, and others\nof none at all. So that if the appearances be the principles by which we\nknow all other things, we must needs acknowledge sense to be the\nprinciple by which we know those principles, and that all the knowledge\nwe have is derived from it. And as for the causes of sense, we cannot\nbegin our search of them from any other phenomenon than that of sense\nitself. But you will say, by what sense shall we take notice of sense? I\nanswer, by sense itself, namely, by the memory which for some time\nremains in us of things sensible, though they themselves pass away. For\nhe that perceives that he hath perceived, remembers.\n\nIn the first place, therefore, the causes of our perception, that is,\nthe causes of those ideas and phantasms which are perpetually generated\nwithin us whilst we make use of our senses, are to be enquired into; and\nin what manner their generation proceeds. To help which inquisition, we\nmay observe first of all, that our phantasms or ideas are not always the\nsame; but that new ones appear to us, and old ones vanish, according as\nwe apply our organs of sense, now to one object, now to another.\nWherefore they are generated, and perish. And from hence it is manifest,\nthat they are some change or mutation in the sentient.\n\n[Sidenote: The investigation of the nature of sense, and the definition\n of sense.]\n\n2. Now that all mutation or alteration is motion or endeavour (and\nendeavour also is motion) in the internal parts of the thing that is\naltered, hath been proved (in art. 9, chap. VIII) from this, that whilst\neven the least parts of any body remain in the same situation in respect\nof one another, it cannot be said that any alteration, unless perhaps\nthat the whole body together hath been moved, hath happened to it; but\nthat it both appeareth and is the same it appeared and was before.\nSense, therefore, in the sentient, can be nothing else but motion in\nsome of the internal parts of the sentient; and the parts so moved are\nparts of the organs of sense. For the parts of our body, by which we\nperceive any thing, are those we commonly call the organs of sense. And\nso we find what is the subject of our sense, namely, that in which are\nthe phantasms; and partly also we have discovered the nature of sense,\nnamely, that it is some internal motion in the sentient.\n\nI have shown besides (in chap. IX, art. 7) that no motion is generated\nbut by a body contiguous and moved: from whence it is manifest, that the\nimmediate cause of sense or perception consists in this, that the first\norgan of sense is touched and pressed. For when the uttermost part of\nthe organ is pressed, it no sooner yields, but the part next within it\nis pressed also; and, in this manner, the pressure or motion is\npropagated through all the parts of the organ to the innermost. And thus\nalso the pressure of the uttermost part proceeds from the pressure of\nsome more remote body, and so continually, till we come to that from\nwhich, as from its fountain, we derive the phantasm or idea that is made\nin us by our sense. And this, whatsoever it be, is that we commonly call\n_the object_, Sense, therefore, is some internal motion in the sentient,\ngenerated by some internal motion of the parts of the object, and\npropagated through all the media to the innermost part of the organ. By\nwhich words I have almost defined what sense is.\n\nMoreover, I have shown (art. 2, chap. XV) that all resistance is\nendeavour opposite to another endeavour, that is to say, reaction.\nSeeing, therefore, there is in the whole organ, by reason of its own\ninternal natural motion, some resistance or reaction against the motion\nwhich is propagated from the object to the innermost part of the organ,\nthere is also in the same organ an endeavour opposite to the endeavour\nwhich proceeds from the object; so that when that endeavour inwards is\nthe last action in the act of sense, then from the reaction, how little\nsoever the duration of it be, a phantasm or idea hath its being; which,\nby reason that the endeavour is now outwards, doth always appear as\nsomething situate without the organ. So that now I shall give you the\nwhole definition of sense, as it is drawn from the explication of the\ncauses thereof and the order of its generation, thus: SENSE _is a\nphantasm, made by the reaction and endeavour outwards in the organ of\nsense, caused by an endeavour inwards from the object, remaining for\nsome time more or less_.\n\n[Sidenote: The subject and object of sense.]\n\n3. The _subject_ of sense is the _sentient_ itself, namely, some living\ncreature; and we speak more correctly, when we say a living creature\nseeth, than when we say the eye seeth. The object is the thing received;\nand it is more accurately said, that we see the sun, than that we see\nthe light. For light and colour, and heat and sound, and other qualities\nwhich are commonly called sensible, are not objects, but phantasms in\nthe sentients. For a phantasm is the act of sense, and differs no\notherwise from sense than _fieri_, that is, being a doing, differs from\n_factum esse_, that is, being done; which difference, in things that are\ndone in an instant, is none at all; and a phantasm is made in an\ninstant. For in all motion which proceeds by perpetual propagation, the\nfirst part being moved moves the second, the second the third, and so on\nto the last, and that to any distance, how great soever. And in what\npoint of time the first or foremost part proceeded to the place of the\nsecond, which is thrust on, in the same point of time the last save one\nproceeded into the place of the last yielding part; which by reaction,\nin the same instant, if the reaction be strong enough, makes a phantasm;\nand a phantasm being made, perception is made together with it.\n\n[Sidenote: The organs of sense.]\n\n4. The _organs_ of sense, which are in the sentient, are such parts\nthereof, that if they be hurt, the very generation of phantasms is\nthereby destroyed, though all the rest of the parts remain entire. Now\nthese parts in the most of living creatures are found to be certain\nspirits and membranes, which, proceeding from the _pia mater_, involve\nthe brain and all the nerves; also the brain itself, and the arteries\nwhich are in the brain; and such other parts, as being stirred, the\nheart also, which is the fountain of all sense, is stirred together with\nthem. For whensoever the action of the object reacheth the body of the\nsentient, that action is by some nerve propagated to the brain; and if\nthe nerve leading thither be so hurt or obstructed, that the motion can\nbe propagated no further, no sense follows. Also if the motion be\nintercepted between the brain and the heart by the defect of the organ\nby which the action is propagated, there will be no perception of the\nobject.\n\n[Sidenote: All bodies are not endued with sense.]\n\n5. But though all sense, as I have said, be made by reaction,\nnevertheless it is not necessary that every thing that reacteth should\nhave sense. I know there have been philosophers, and those learned men,\nwho have maintained that all bodies are endued with sense. Nor do I see\nhow they can be refuted, if the nature of sense be placed in reaction\nonly. And, though by the reaction of bodies inanimate a phantasm might\nbe made, it would nevertheless cease, as soon as ever the object were\nremoved. For unless those bodies had organs, as living creatures have,\nfit for the retaining of such motion as is made in them, their sense\nwould be such, as that they should never remember the same. And\ntherefore this hath nothing to do with that sense which is the subject\nof my discourse. For by sense, we commonly understand the judgment we\nmake of objects by their phantasms; namely, by comparing and\ndistinguishing those phantasms; which we could never do, if that motion\nin the organ, by which the phantasm is made, did not remain there for\nsome time, and make the same phantasm return. Wherefore sense, as I here\nunderstand it, and which is commonly so called, hath necessarily some\nmemory adhering to it, by which former and later phantasms may be\ncompared together, and distinguished from one another.\n\nSense, therefore, properly so called, must necessarily have in it a\nperpetual variety of phantasms, that they may be discerned one from\nanother. For if we should suppose a man to be made with clear eyes, and\nall the rest of his organs of sight well disposed, but endued with no\nother sense; and that he should look only upon one thing, which is\nalways of the same colour and figure, without the least appearance of\nvariety, he would seem to me, whatsoever others may say, to see, no more\nthan I seem to myself to feel the bones of my own limbs by my organs of\nfeeling; and yet those bones are always and on all sides touched by a\nmost sensible membrane. I might perhaps say he were astonished, and\nlooked upon it; but I should not say he saw it; it being almost all one\nfor a man to be always sensible of one and the same thing, and not to be\nsensible at all of any thing.\n\n[Sidenote: But one phantasm at one and the same time.]\n\n6. And yet such is the nature of sense, that it does not permit a man to\ndiscern many things at once. For seeing the nature of sense consists in\nmotion; as long as the organs are employed about one object, they cannot\nbe so moved by another at the same time, as to make by both their\nmotions one sincere phantasm of each of them at once. And therefore two\nseveral phantasms will not be made by two objects working together, but\nonly one phantasm compounded from the action of both.\n\nBesides, as when we divide a body, we divide its place; and when we\nreckon many bodies, we must necessarily reckon as many places; and\ncontrarily, as I have shown in the seventh chapter; so what number\nsoever we say there be of times, we must understand the same number of\nmotions also; and as oft as we count many motions, so oft we reckon many\ntimes. For though the object we look upon be of divers colours, yet with\nthose divers colours it is but one varied object, and not variety of\nobjects.\n\nMoreover, whilst those organs which are common to all the senses, such\nas are those parts of every organ which proceed in men from the root of\nthe nerves to the heart, are vehemently stirred by a strong action from\nsome one object, they are, by reason of the contumacy which the motion,\nthey have already, gives them against the reception of all other motion,\nmade the less fit to receive any other impression from whatsoever other\nobjects, to what sense soever those objects belong. And hence it is,\nthat an earnest studying of one object, takes away the sense of all\nother objects for the present. For _study_ is nothing else but a\npossession of the mind, that is to say, a vehement motion made by some\none object in the organs of sense, which are stupid to all other motions\nas long as this lasteth; according to what was said by Terence,\n\u201c_Populus studio stupidus in funambulo animum occuparat._\u201d For what is\n_stupor_ but that which the Greeks call \u1f00\u03bd\u03b1\u03b9\u03c3\u03b8\u03b7\u03c3\u03af\u03b1, that is, a cessation\nfrom the sense of other things? Wherefore at one and the same time, we\ncannot by sense perceive more than one single object; as in reading, we\nsee the letters successively one by one, and not all together, though\nthe whole page be presented to our eye; and though every several letter\nbe distinctly written there, yet when we look upon the whole page at\nonce, we read nothing.\n\nFrom hence it is manifest, that every endeavour of the organ outwards,\nis not to be called sense, but that only, which at several times is by\nvehemence made stronger and more predominant than the rest; which\ndeprives us of the sense of other phantasms, no otherwise than the sun\ndeprives the rest of the stars of light, not by hindering their action,\nbut by obscuring and hiding them with his excess of brightness.\n\n[Sidenote: Imagination, the remains of past sense; which also is memory.\n Of sleep.]\n\n7. But the motion of the organ, by which a phantasm is made, is not\ncommonly called sense, except the object be present. And the phantasm\nremaining after the object is removed or past by, is called _fancy_, and\nin Latin _imaginatio_; which word, because all phantasms are not images,\ndoth not fully answer the signification of the word _fancy_ in its\ngeneral acceptation. Nevertheless I may use it safely enough, by\nunderstanding it for the Greek \u03a6\u03b1\u03bd\u03c4\u03b1\u03c3\u03af\u03b1.\n\nIMAGINATION therefore is nothing else but _sense decaying_, or\n_weakened_, by the absence of the object. But what may be the cause of\nthis decay or weakening? Is the motion the weaker, because the object is\ntaken away? If it were, then phantasms would always and necessarily be\nless clear in the imagination, than they are in sense; which is not\ntrue. For in dreams, which are the imaginations of those that sleep,\nthey are no less clear than in sense itself. But the reason why in men\nwaking the phantasms of things past are more obscure than those of\nthings present, is this, that their organs being at the same time moved\nby other present objects, those phantasms are the less predominant.\nWhereas in sleep, the passages being shut up, external action doth not\nat all disturb or hinder internal motion.\n\nIf this be true, the next thing to be considered, will be, whether any\ncause may be found out, from the supposition whereof it will follow,\nthat the passage is shut up from the external objects of sense to the\ninternal organ. I suppose, therefore, that by the continual action of\nobjects, to which a reaction of the organ, and more especially of the\nspirits, is necessarily consequent, the organ is wearied, that is, its\nparts are no longer moved by the spirits without some pain; and\nconsequently the nerves being abandoned and grown slack, they retire to\ntheir fountain, which is the cavity either of the brain or of the heart;\nby which means the action which proceeded by the nerves is necessarily\nintercepted. For action upon a patient, that retires from it, makes but\nlittle impression at the first; and at last, when the nerves are by\nlittle and little slackened, none at all. And therefore there is no more\nreaction, that is, no more sense, till the organ being refreshed by\nrest, and by a supply of new spirits recovering strength and motion, the\nsentient awaketh. And thus it seems to be always, unless some other\npreternatural cause intervene; as heat in the internal parts from\nlassitude, or from some disease stirring the spirits and other parts of\nthe organ in some extraordinary manner.\n\n[Sidenote: How phantasms succeed one another.]\n\n8. Now it is not without cause, nor so casual a thing as many perhaps\nthink it, that phantasms in this their great variety proceed from one\nanother; and that the same phantasms sometimes bring into the mind other\nphantasms like themselves, and at other times extremely unlike. For in\nthe motion of any continued body, one part follows another by cohesion;\nand therefore, whilst we turn our eyes and other organs successively to\nmany objects, the motion which was made by every one of them remaining,\nthe phantasms are renewed as often as any one of those motions comes to\nbe predominant above the rest; and they become predominant in the same\norder in which at any time formerly they were generated by sense. So\nthat when by length of time very many phantasms have been generated\nwithin us by sense, then almost any thought may arise from any other\nthought; insomuch that it may seem to be a thing indifferent and casual,\nwhich thought shall follow which. But for the most part this is not so\nuncertain a thing to waking as to sleeping men. For the thought or\nphantasm of the desired end brings in all the phantasms, that are means\nconducing to that end, and that in order backwards from the last to the\nfirst, and again forwards from the beginning to the end. But this\nsupposes both appetite, and judgment to discern what means conduce to\nthe end, which is gotten by experience; and experience is store of\nphantasms, arising from the sense of very many things. For \u03c6\u03b1\u03bd\u03c4\u03b1\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9\nand _meminisse_, _fancy_ and _memory_, differ only in this, that memory\nsupposeth the time past, which fancy doth not. In memory, the phantasms\nwe consider are as if they were worn out with time; but in our fancy we\nconsider them as they are; which distinction is not of the things\nthemselves, but of the considerations of the sentient. For there is in\nmemory something like that which happens in looking upon things at a\ngreat distance; in which as the small parts of the object are not\ndiscerned, by reason of their remoteness; so in memory, many accidents\nand places and parts of things, which were formerly perceived by sense,\nare by length of time decayed and lost.\n\nThe perpetual arising of phantasms, both in sense and imagination, is\nthat which we commonly call discourse of the mind, and is common to men\nwith other living creatures. For he that thinketh, compareth the\nphantasms that pass, that is, taketh notice of their likeness or\nunlikeness to one another. And as he that observes readily the\nlikenesses of things of different natures, or that are very remote from\none another, is said to have a good fancy; so he is said to have a good\njudgment, that finds out the unlikenesses or differences of things that\nare like one another. Now this observation of differences is not\nperception made by a common organ of sense, distinct from sense or\nperception properly so called, but is memory of the differences of\nparticular phantasms remaining for some time; as the distinction between\nhot and lucid, is nothing else but the memory both of a heating, and of\nan enlightening object.\n\n[Sidenote: Dreams, whence they proceed.]\n\n9. The phantasms of men that sleep, are _dreams_. Concerning which we\nare taught by experience these five things. First, that for the most\npart there is neither order nor coherence in them. Secondly, that we\ndream of nothing but what is compounded and made up of the phantasms of\nsense past. Thirdly, that sometimes they proceed, as in those that are\ndrowsy, from the interruption of their phantasms by little and little,\nbroken and altered through sleepiness; and sometimes also they begin in\nthe midst of sleep. Fourthly, that they are clearer than the\nimaginations of waking men, except such as are made by sense itself, to\nwhich they are equal in clearness. Fifthly, that when we dream, we\nadmire neither the places nor the looks of the things that appear to us.\nNow from what hath been said, it is not hard to show what may be the\ncauses of these phenomena. For as for the first, seeing all order and\ncoherence proceeds from frequent looking back to the end, that is, from\nconsultation; it must needs be, that seeing in sleep we lose all thought\nof the end, our phantasms succeed one another, not in that order which\ntends to any end, but as it happeneth, and in such manner, as objects\npresent themselves to our eyes when we look indifferently upon all\nthings before us, and see them, not because we would see them, but\nbecause we do not shut our eyes; for then they appear to us without any\norder at all. The second proceeds from this, that in the silence of\nsense there is no new motion from the objects, and therefore no new\nphantasm, unless we call that new, which is compounded of old ones, as a\nchimera, a golden mountain, and the like. As for the third, why a dream\nis sometimes as it were the continuation of sense, made up of broken\nphantasms, as in men distempered with sickness, the reason is manifestly\nthis, that in some of the organs sense remains, and in others it\nfaileth. But how some phantasms may be revived, when all the exterior\norgans are benumbed with sleep, is not so easily shown. Nevertheless\nthat, which hath already been said, contains the reason of this also.\nFor whatsoever strikes the _pia mater_, reviveth some of those phantasms\nthat are still in motion in the brain; and when any internal motion of\nthe heart reacheth that membrane, then the predominant motion in the\nbrain makes the phantasm. Now the motions of the heart are appetites and\naversions, of which I shall presently speak further. And as appetites\nand aversions are generated by phantasms, so reciprocally phantasms are\ngenerated by appetites and aversions. For example, heat in the heart\nproceeds from anger and fighting; and again, from heat in the heart,\nwhatsoever be the cause of it, is generated anger and the image of an\nenemy, in sleep. And as love and beauty stir up heat in certain organs;\nso heat in the same organs, from whatsoever it proceeds, often causeth\ndesire and the image of an unresisting beauty. Lastly, cold doth in the\nsame manner generate fear in those that sleep, and causeth them to dream\nof ghosts, and to have phantasms of horror and danger; as fear also\ncauseth cold in those that wake. So reciprocal are the motions of the\nheart and brain. The fourth, namely, that the things we seem to see and\nfeel in sleep, are as clear as in sense itself, proceeds from two\ncauses; one, that having then no sense of things without us, that\ninternal motion which makes the phantasm, in the absence of all other\nimpressions, is predominant; and the other, that the parts of our\nphantasms which are decayed and worn out by time, are made up with other\nfictitious parts. To conclude, when we dream, we do not wonder at\nstrange places and the appearances of things unknown to us, because\nadmiration requires that the things appearing be new and unusual, which\ncan happen to none but those that remember former appearances; whereas\nin sleep, all things appear as present.\n\nBut it is here to be observed, that certain dreams, especially such as\nsome men have when they are between sleeping and waking, and such as\nhappen to those that have no knowledge of the nature of dreams and are\nwithal superstitious, were not heretofore nor are now accounted dreams.\nFor the apparitions men thought they saw, and the voices they thought\nthey heard in sleep, were not believed to be phantasms, but things\nsubsisting of themselves, and objects without those that dreamed. For to\nsome men, as well sleeping as waking, but especially to guilty men, and\nin the night, and in hallowed places, fear alone, helped a little with\nthe stories of such apparitions, hath raised in their minds terrible\nphantasms, which have been and are still deceitfully received for things\nreally true, under the names of _ghosts_ and _incorporeal substances_.\n\n[Sidenote: Of the senses, their kinds, their organs and phantasms,\n proper and common.]\n\n10. In most living creatures there are observed five kinds of senses,\nwhich are distinguished by their organs, and by their different kinds of\nphantasms; namely, _sight_, _hearing_, _smell_, _taste_, and _touch_;\nand these have their organs partly peculiar to each of them severally,\nand partly common to them all. The organ of sight is partly animate, and\npartly inanimate. The inanimate parts are the three humours; namely, the\nwatery humour, which by the interposition of the membrane called uvea,\nthe perforation whereof is called the apple of the eye, is contained on\none side by the first concave superficies of the eye, and on the other\nside by the ciliary processes, and the coat of the crystalline humour;\nthe crystalline, which, hanging in the midst between the ciliary\nprocesses, and being almost of spherical figure, and of a thick\nconsistence, is enclosed on all sides with its own transparent coat; and\nthe vitreous or glassy humour, which filleth all the rest of the cavity\nof the eye, and is somewhat thicker then the watery humour, but thinner\nthan the crystalline. The animate part of the organ is, first, the\nmembrane _choroeides_, which is a part of the _pia mater_, saving that\nit is covered with a coat derived from the marrow of the optic nerve,\nwhich is called the _retina_; and this _choroeides_, seeing it is part\nof the _pia mater_, is continued to the beginning of the _medulla\nspinalis_ within the scull, in which all the nerves which are within the\nhead have their roots. Wherefore all the animal spirits that the nerves\nreceive, enter into them there; for it is not imaginable that they can\nenter into them anywhere else. Seeing therefore sense is nothing else\nbut the action of objects propagated to the furthest part of the organ;\nand seeing also that animal spirits are nothing but vital spirits\npurified by the heart, and carried from it by the arteries; it follows\nnecessarily, that the action is derived from the heart by some of the\narteries to the roots of the nerves which are in the head, whether those\narteries be the _plexus retiformis_, or whether they be other arteries\nwhich are inserted into the substance of the brain. And, therefore,\nthose arteries are the complement or the remaining part of the whole\norgan of sight. And this last part is a common organ to all the senses;\nwhereas, that which reacheth from the eye to the roots of the nerves is\nproper only to sight. The proper organ of hearing is the tympanum of the\near and its own nerve; from which to the heart the organ is common. So\nthe proper organs of smell and taste are nervous membranes, in the\npalate and tongue for the taste, and in the nostrils for the smell; and\nfrom the roots of those nerves to the heart all is common. Lastly, the\nproper organ of touch are nerves and membranes dispersed through the\nwhole body; which membranes are derived from the root of the nerves. And\nall things else belonging alike to all the senses seem to be\nadministered by the arteries, and not by the nerves.\n\nThe proper phantasm of sight is light; and under this name of light,\ncolour also, which is nothing but perturbed light, is comprehended.\nWherefore the phantasm of a lucid body is light; and of a coloured body,\ncolour. But the object of sight, properly so called, is neither light\nnor colour, but the body itself which is lucid, or enlightened, or\ncoloured. For light and colour, being phantasms of the sentient, cannot\nbe accidents of the object. Which is manifest enough from this, that\nvisible things appear oftentimes in places in which we know assuredly\nthey are not, and that in different places they are of different\ncolours, and may at one and the same time appear in divers places.\nMotion, rest, magnitude, and figure, are common both to the sight and\ntouch; and the whole appearance together of figure, and light or colour,\nis by the Greeks commonly called \u03b5\u1f34\u03b4\u03bf\u03c2, and \u03b5\u1f34\u03b4\u03c9\u03bb\u03bf\u03bd, and \u1f31\u03b4\u03ad\u03b1; and by\nthe Latins, _species_ and _imago_; all which names signify no more but\nappearance.\n\nThe phantasm, which is made by hearing, is sound; by smell, odour; by\ntaste, savour; and by touch, hardness and softness, heat and cold,\nwetness, oiliness, and many more, which are easier to be distinguished\nby sense than words. Smoothness, roughness, rarity, and density, refer\nto figure, and are therefore common both to touch and sight. And as for\nthe objects of hearing, smell, taste, and touch, they are not sound,\nodour, savour, hardness, &c., but the bodies themselves from which\nsound, odour, savour, hardness, &c. proceed; of the causes of which, and\nof the manner how they are produced, I shall speak hereafter.\n\nBut these phantasms, though they be effects in the sentient, as subject,\nproduced by objects working upon the organs; yet there are also other\neffects besides these, produced by the same objects in the same organs;\nnamely certain motions proceeding from sense, which are called _animal\nmotions_. For seeing in all sense of external things there is mutual\naction and reaction, that is, two endeavours opposing one another, it is\nmanifest that the motion of both of them together will be continued\nevery way, especially to the confines of both the bodies. And when this\nhappens in the internal organ, the endeavour outwards will proceed in a\nsolid angle, which will be greater, and consequently the idea greater,\nthan it would have been if the impression had been weaker.\n\n[Sidenote: The magnitude of images, how and by what it is determined.]\n\n11. From hence the natural cause is manifest, first, why those things\nseem to be greater, which, _c\u00e6teris paribus_, are seen in a greater\nangle: secondly, why in a serene cold night, when the moon doth not\nshine, more of the fixed stars appear than at another time. For their\naction is less hindered by the serenity of the air, and not obscured by\nthe greater light of the moon, which is then absent; and the cold,\nmaking the air more pressing, helpeth or strengtheneth the action of the\nstars upon our eyes; in so much as stars may then be seen which are seen\nat no other time. And this may suffice to be said in general concerning\nsense made by the reaction of the organ. For, as for the place of the\nimage, the deceptions of sight, and other things of which we have\nexperience in ourselves by sense, seeing they depend for the most part\nupon the fabric itself of the eye of man, I shall speak of them then\nwhen I come to speak of man.\n\n[Sidenote: Pleasure, pain, appetite, and aversion, what they are.]\n\n12. But there is another kind of sense, of which I will say something in\nthis place, namely, the sense of pleasure and pain, proceeding not from\nthe reaction of the heart outwards, but from continual action from the\noutermost part of the organ towards the heart. For the original of life\nbeing in the heart, that motion in the sentient, which is propagated to\nthe heart, must necessarily make some alteration or diversion of vital\nmotion, namely, by quickening or slackening, helping or hindering the\nsame. Now when it helpeth, it is pleasure; and when it hindereth, it is\npain, trouble, grief, &c. And as phantasms seem to be without, by reason\nof the endeavour outwards, so pleasure and pain, by reason of the\nendeavour of the organ inwards, seem to be within; namely, there where\nthe first cause of the pleasure or pain is; as when the pain proceeds\nfrom a wound, we think the pain and the wound are both in the same\nplace.\n\n[Sidenote: Pleasure, pain, appetite, and aversion, what they are.]\n\nNow vital motion is the motion of the blood, perpetually circulating (as\nhath been shown from many infallible signs and marks by Doctor Harvey,\nthe first observer of it) in the veins and arteries. Which motion, when\nit is hindered by some other motion made by the action of sensible\nobjects, may be restored again either by bending or setting strait the\nparts of the body; which is done when the spirits are carried now into\nthese, now into other nerves, till the pain, as far as is possible, be\nquite taken away. But if vital motion be helped by motion made by sense,\nthen the parts of the organ will be disposed to guide the spirits in\nsuch manner as conduceth most to the preservation and augmentation of\nthat motion, by the help of the nerves. And in animal motion this is the\nvery first endeavour, and found even in the embryo; which while it is in\nthe womb, moveth its limbs with voluntary motion, for the avoiding of\nwhatsoever troubleth it, or for the pursuing of what pleaseth it. And\nthis first endeavour, when it tends towards such things as are known by\nexperience to be pleasant, is called _appetite_, that is, an\napproaching; and when it shuns what is troublesome, _aversion_, or\nflying from it. And little infants, at the beginning and as soon as they\nare born, have appetite to very few things, as also they avoid very few,\nby reason of their want of experience and memory; and therefore they\nhave not so great a variety of animal motion as we see in those that are\nmore grown. For it is not possible, without such knowledge as is derived\nfrom sense, that is, without experience and memory, to know what will\nprove pleasant or hurtful; only there is some place for conjecture from\nthe looks or aspects of things. And hence it is, that though they do not\nknow what may do them good or harm, yet sometimes they approach and\nsometimes retire from the same thing, as their doubt prompts them. But\nafterwards, by accustoming themselves by little and little, they come to\nknow readily what is to be pursued and what to be avoided; and also to\nhave a ready use of their nerves and other organs, in the pursuing and\navoiding of good and bad. Wherefore appetite and aversion are the first\nendeavours of animal motion.\n\nConsequent to this first endeavour, is the impulsion into the nerves and\nretraction again of animal spirits, of which it is necessary there be\nsome receptacle or place near the original of the nerves; and this\nmotion or endeavour is followed by a swelling and relaxation of the\nmuscles; and lastly, these are followed by contraction and extension of\nthe limbs, which is animal motion.\n\n[Sidenote: Deliberation and will, what.]\n\n13. The considerations of appetites and aversions are divers. For seeing\nliving creatures have sometimes appetite and sometimes aversion to the\nsame thing, as they think it will either be for their good or their\nhurt; while that vicissitude of appetites and aversions remains in them,\nthey have that series of thoughts which is called _deliberation_; which\nlasteth as long as they have it in their power to obtain that which\npleaseth, or to avoid that which displeaseth them. Appetite, therefore,\nand aversion are simply so called as long as they follow not\ndeliberation. But if deliberation have gone before, then the last act of\nit, if it be appetite, is called _will_; if aversion, _unwillingness_.\nSo that the same thing is called both will and appetite; but the\nconsideration of them, namely, before and after deliberation, is divers.\nNor is that which is done within a man whilst he willeth any thing,\ndifferent from that which is done in other living creatures, whilst,\ndeliberation having preceded, they have appetite.\n\nNeither is the freedom of willing or not willing, greater in man, than\nin other living creatures. For where there is appetite, the entire cause\nof appetite hath preceded; and, consequently, the act of appetite could\nnot choose but follow, that is, hath of necessity followed (as is shown\nin chapter IX, article 5). And therefore such a liberty as is free from\nnecessity, is not to be found in the will either of men or beasts. But\nif by liberty we understand the faculty or power, not of willing, but of\ndoing what they will, then certainly that liberty is to be allowed to\nboth, and both may equally have it, whensoever it is to be had.\n\nAgain, when appetite and aversion do with celerity succeed one another,\nthe whole series made by them hath its name sometimes from one,\nsometimes from the other. For the same deliberation, whilst it inclines\nsometimes to one, sometimes to the other, is from appetite called\n_hope_, and from aversion, _fear_. For where there is no hope, it is not\nto be called fear, but _hate_; and where no fear, not hope, but\n_desire_. To conclude, all the passions, called passions of the mind,\nconsist of appetite and aversion, except pure pleasure and pain, which\nare a certain fruition of good or evil; as anger is aversion from some\nimminent evil, but such as is joined with appetite of avoiding that evil\nby force. But because the passions and perturbations of the mind are\ninnumerable, and many of them not to be discerned in any creatures\nbesides men; I will speak of them more at large in that section which is\nconcerning _man_. As for those objects, if there be any such, which do\nnot at all stir the mind, we are said to contemn them.\n\nAnd thus much of sense in general. In the next place I shall speak of\nsensible objects.\n\n\n -------\n\n\n CHAPTER XXVI.\n\n OF THE WORLD AND OF THE STARS.\n\n1. The magnitude and duration of the world, inscrutable.\u20142. No place in\n the world empty.\u20143. The arguments of Lucretius for vacuum,\n invalid.\u20144. Other arguments for the establishing of vacuum,\n invalid.\u20145. Six suppositions for the salving of the phenomena of\n nature.\u20146. Possible causes of the motions annual and diurnal; and of\n the apparent direction, station, and retrogradation of the\n planets.\u20147. The supposition of simple motion, why likely.\u20148. The\n cause of the eccentricity of the annual motion of the earth.\u20149. The\n cause why the moon hath always one and the same face turned towards\n the earth.\u201410. The cause of the tides of the ocean.\u201411. The cause of\n the precession of the equinoxes.\n\n\n[Sidenote: The magnitude and duration of the world, inscrutable.]\n\n1. Consequent to the contemplation of sense is the contemplation of\nbodies, which are the efficient causes or objects of sense. Now every\nobject is either a part of the whole world, or an aggregate of parts.\nThe greatest of all bodies, or sensible objects, is the world itself;\nwhich we behold when we look round about us from this point of the same\nwhich we call the earth. Concerning the world, as it is one aggregate of\nmany parts, the things that fall under inquiry are but few; and those we\ncan determine, none. Of the whole world we may inquire what is its\nmagnitude, what its duration, and how many there be, but nothing else.\nFor as for place and time, that is to say, magnitude and duration, they\nare only our own fancy of a body simply so called, that is to say, of a\nbody indefinitely taken, as I have shown before in chapter VII. All\nother phantasms are of bodies or objects, as they are distinguished from\none another; as colour, the phantasm of coloured bodies; sound, of\nbodies that move the sense of hearing, &c. The questions concerning the\nmagnitude of the world are whether it be finite or infinite, full or not\nfull; concerning its duration, whether it had a beginning, or be\neternal; and concerning the number, whether there be one or many; though\nas concerning the number, if it were of infinite magnitude, there could\nbe no controversy at all. Also if it had a beginning, then by what cause\nand of what matter it was made; and again, from whence that cause and\nthat matter had their being, will be new questions; till at last we come\nto one or many eternal cause or causes. And the determination of all\nthese things belongeth to him that professeth the universal doctrine of\nphilosophy, in case as much could be known as can be sought. But the\nknowledge of what is infinite can never be attained by a finite\ninquirer. Whatsoever we know that are men, we learn it from our\nphantasms; and of infinite, whether magnitude or time, there is no\nphantasm at all; so that it is impossible either for a man or any other\ncreature to have any conception of infinite. And though a man may from\nsome effect proceed to the immediate cause thereof, and from that to a\nmore remote cause, and so ascend continually by right ratiocination from\ncause to cause; yet he will not be able to proceed eternally, but\nwearied will at last give over, without knowing whether it were possible\nfor him to proceed to an end or not. But whether we suppose the world to\nbe finite or infinite, no absurdity will follow. For the same things\nwhich now appear, might appear, whether the Creator had pleased it\nshould be finite or infinite. Besides, though from this, that nothing\ncan move itself, it may rightly be inferred that there was some first\neternal movent; yet it can never be inferred, though some used to make\nsuch inference, that that movent was eternally immoveable, but rather\neternally moved. For as it is true, that nothing is moved by itself; so\nit is true also that nothing is moved but by that which is already\nmoved. The questions therefore about the magnitude and beginning of the\nworld, are not to be determined by philosophers, but by those that are\nlawfully authorized to order the worship of God. For as Almighty God,\nwhen he had brought his people into Jud\u00e6a, allowed the priests the first\nfruits reserved to himself; so when he had delivered up the world to the\ndisputations of men, it was his pleasure that all opinions concerning\nthe nature of infinite and eternal, known only to himself, should, as\nthe first fruits of wisdom, be judged by those whose ministry he meant\nto use in the ordering of religion. I cannot therefore commend those\nthat boast they have demonstrated, by reasons drawn from natural things,\nthat the world had a beginning. They are contemned by idiots, because\nthey understand them not; and by the learned, because they understand\nthem; by both deservedly. For who can commend him that demonstrates\nthus? \"If the world be eternal, then an infinite number of days, or\nother measures of time, preceded the birth of Abraham. But the birth of\nAbraham preceded the birth of Isaac; and therefore one infinite is\ngreater than another infinite, or one eternal than another eternal;\nwhich,\" he says, \"is absurd.\" This demonstration is like his, who from\nthis, that the number of even numbers is infinite, would conclude that\nthere are as many even numbers as there are numbers simply, that is to\nsay, the even numbers are as many as all the even and odd together.\nThey, which in this manner take away eternity from the world, do they\nnot by the same means take away eternity from the Creator of the world?\nFrom this absurdity therefore they run into another, being forced to\ncall eternity _nunc stans_, a standing still of the present time, or an\nabiding now; and, which is much more absurd, to give to the infinite\nnumber of numbers the name of unity. But why should eternity be called\nan abiding now, rather than an abiding then? Wherefore there must either\nbe many eternities, or _now_ and _then_ must signify the same. With such\ndemonstrators as these, that speak in another language, it is impossible\nto enter into disputation. And the men, that reason thus absurdly, are\nnot idiots, but, which makes the absurdity unpardonable, geometricians,\nand such as take upon them to be judges, impertinent, but severe judges\nof other men's demonstrations. The reason is this, that as soon as they\nare entangled in the words _infinite_ and _eternal_, of which we have in\nour mind no idea, but that of our own insufficiency to comprehend them,\nthey are forced either to speak something absurd, or, which they love\nworse, to hold their peace. For geometry hath in it somewhat like wine,\nwhich, when new, is windy; but afterwards though less pleasant, yet more\nwholesome. Whatsoever therefore is true, young geometricians think\ndemonstrable; but elder not. Wherefore I purposely pass over the\nquestions of infinite and eternal; contenting myself with that doctrine\nconcerning the beginning and magnitude of the world, which I have been\npersuaded to by the holy Scriptures and fame of the miracles which\nconfirm them; and by the custom of my country, and reverence due to the\nlaws. And so I pass on to such things as it is not unlawful to dispute\nof.\n\n[Sidenote: No place in the world empty.]\n\n2. Concerning the world it is further questioned, whether the parts\nthereof be contiguous to one another, in such manner as not to admit of\nthe least empty space between; and the disputation both for and against\nit is carried on with probability enough. For the taking away of vacuum,\nI will instance in only one experiment, a common one, but I think\nunanswerable.\n\nLet A B (in fig. 1) represent a vessel, such as gardeners use to water\ntheir gardens withal; whose bottom B is full of little holes; and whose\nmouth A may be stopped with one's finger, when there shall be need. If\nnow this vessel be filled with water, the hole at the top A being\nstopped, the water will not flow out at any of the holes in the bottom\nB. But if the finger be removed to let in the air above, it will run out\nat them all; and as; soon as the finger is applied to it again, the\nwater will suddenly and totally be stayed again from running out. The\ncause whereof seems to be no other but this, that the water cannot by\nits natural endeavour to descend drive down the air below it, because\nthere is no place for it to go into, unless either by thrusting away the\nnext contiguous air, it proceed by continual endeavour to the hole A,\nwhere it may enter and succeed into the place of the water that floweth\nout, or else, by resisting the endeavour of the water downwards,\npenetrate the same and pass up through it. By the first of these ways,\nwhile the hole at A remains stopped, there is no possible passage; nor\nby the second, unless the holes be so great that the water, flowing out\nat them, can by its own weight force the air at the same time to ascend\ninto the vessel by the same holes: as we see it does in a vessel whose\nmouth is wide enough, when we turn suddenly the bottom upwards to pour\nout the water; for then the air being forced by the weight of the water,\nenters, as is evident by the sobbing and resistance of the water, at the\nsides or circumference of the orifice. And this I take for a sign that\nall space is full; for without this, the natural motion of the water,\nwhich is a heavy body, downwards, would not be hindered.\n\n[Sidenote: The arguments of Lucretius for vacuum invalid.]\n\n3. On the contrary, for the establishing of vacuum, many and specious\narguments and experiments have been brought. Nevertheless there seems to\nbe something wanting in all of them to conclude it firmly. These\narguments for vacuum are partly made by the followers of the doctrine of\nEpicurus; who taught that the world consists of very small spaces not\nfilled by any body, and of very small bodies that have within them no\nempty space, which by reason of their hardness he calls _atoms_; and\nthat these small bodies and spaces are every where intermingled. Their\narguments are thus delivered by Lucretius.\n\nAnd first he says, that unless it were so, there could be no motion. For\nthe office and property of bodies is to withstand and hinder motion. If,\ntherefore, the universe were filled with body, motion would everywhere\nbe hindered, so as to have no beginning anywhere; and consequently there\nwould be no motion at all. It is true that in whatsoever is full and at\nrest in all its parts, it is not possible motion should have beginning.\nBut nothing is drawn from hence for the proving of vacuum. For though it\nshould be granted that there is vacuum, yet if the bodies which are\nintermingled with it, should all at once and together be at rest, they\nwould never be moved again. For it has been demonstrated above, in chap.\nIX, art. 7, that nothing can be moved but by that which is contiguous\nand already moved. But supposing that all things are at rest together,\nthere can be nothing contiguous and moved, and therefore no beginning of\nmotion. Now the denying of the beginning of motion, doth not take away\npresent motion, unless beginning be taken away from body also. For\nmotion may be either co-eternal, or concreated with body. Nor doth it\nseem more necessary that bodies were first at rest, and afterwards\nmoved, than that they were first moved, and rested, if ever they rested\nat all, afterwards. Neither doth there appear any cause, why the matter\nof the world should, for the admission of motion, be intermingled with\nempty spaces rather than full; I say full, but withal fluid. Nor,\nlastly, is there any reason why those hard atoms may not also, by the\nmotion of intermingled fluid matter, be congregated and brought together\ninto compounded bodies of such bigness as we see. Wherefore nothing can\nby this argument be concluded, but that motion was either coeternal, or\nof the same duration with that which is moved; neither of which\nconclusions consisteth with the doctrine of Epicurus, who allows neither\nto the world nor to motion any beginning at all. The necessity,\ntherefore, of vacuum is not hitherto demonstrated. And the cause, as far\nas I understand from them that have discoursed with me of vacuum, is\nthis, that whilst they contemplate the nature of fluid, they conceive it\nto consist, as it were, of small grains of hard matter, in such manner\nas meal is fluid, made so by grinding of the corn; when nevertheless it\nis possible to conceive fluid to be of its own nature as homogeneous as\neither an atom, or as vacuum itself.\n\nThe second of their arguments is taken from weight, and is contained in\nthese verses of Lucretius:\n\n Corporis officium est quoniam premere omnia deorsum;\n Contra autem natura manet sine pondere inanis;\n Ergo, quod magnum est \u00e6que, leviusque videtur,\n Nimirum plus esse sibi declarat inanis.--I. 363-66.\n\nThat is to say, _seeing the office and property of body is to press all\nthings downwards; and on the contrary, seeing the nature of vacuum is to\nhave no weight at all; therefore when of two bodies of equal magnitude,\none is lighter than the other, it is manifest that the lighter body hath\nin it more vacuum than the other_.\n\nTo say nothing of the assumption concerning the endeavour of bodies\ndownwards, which is not rightly assumed, because the world hath nothing\nto do with downwards, which is a mere fiction of ours; nor of this, that\nif all things tended to the same lowest part of the world, either there\nwould be no coalescence at all of bodies, or they would all be gathered\ntogether into the same place: this only is sufficient to take away the\nforce of the argument, that air, intermingled with those his atoms, had\nserved as well for his purpose as his intermingled vacuum.\n\nThe third argument is drawn from this, that lightning, sound, heat and\ncold, do penetrate all bodies, except atoms, how solid soever they be.\nBut this reason, except it be first demonstrated that the same things\ncannot happen without vacuum by perpetual generation of motion, is\naltogether invalid. But that all the same things may so happen, shall in\ndue place be demonstrated.\n\nLastly, the fourth argument is set down by the same Lucretius in these\nverses:\n\n Duo de concursu corpora lata\n Si cita dissiliant, nempe aer omne necesse est,\n Inter corpora quod fuerat, possidat inane.\n Is porro quamvis circum celerantibus auris\n Confluat, haud poterit tamen uno tempore totum\n Compleri spatium; nam primum quemque necesse est\n Occupet ille locum, deinde omnia possideantur.--I. 385-91.\n\nThat is, _if two flat bodies be suddenly pulled asunder, of necessity\nthe air must come between them to fill up the space they left empty. But\nwith what celerity soever the air flow in, yet it cannot in one instant\nof time fill the whole space, but first one part of it, then\nsuccessively all_. Which nevertheless is more repugnant to the opinion\nof Epicurus, than of those that deny vacuum. For though it be true, that\nif two bodies were of infinite hardness, and were joined together by\ntheir superficies which were most exactly plane, it would be impossible\nto pull them asunder, in regard it could not be done but by motion in an\ninstant; yet, if as the greatest of all magnitudes cannot be given, nor\nthe swiftest of all motions, so neither the hardest of all bodies; it\nmight be, that by the application of very great force, there might be\nplace made for a successive flowing in of the air, namely, by separating\nthe parts of the joined bodies by succession, beginning at the outermost\nand ending at the innermost part. He ought, therefore, first to have\nproved, that there are some bodies extremely hard, not relatively as\ncompared with softer bodies, but absolutely, that is to say, infinitely\nhard; which is not true. But if we suppose, as Epicurus doth, that atoms\nare indivisible, and yet have small superficies of their own; then if\ntwo bodies should be joined together by many, or but one only small\nsuperficies of either of them, then I say this argument of Lucretius\nwould be a firm demonstration, that no two bodies made up of atoms, as\nhe supposes, could ever possibly be pulled asunder by any force\nwhatsoever. But this is repugnant to daily experience.\n\n[Sidenote: Other arguments for the establishing of vacuum, invalid.]\n\n4. And thus much of the arguments of Lucretius. Let us now consider the\narguments which are drawn from the experiments of later writers.\n\nI. The first experiment is this: that if a hollow vessel be thrust into\nwater with the bottom upwards, the water will ascend into it; which they\nsay it could not do, unless the air within were thrust together into a\nnarrower place; and that this were also impossible, except there were\nlittle empty places in the air. Also, that when the air is compressed to\na certain degree, it can receive no further compression, its small\nparticles not suffering themselves to be pent into less room. This\nreason, if the air could not pass through the water as it ascends within\nthe vessel, might seem valid. But it is sufficiently known, that air\nwill penetrate water by the application of a force equal to the gravity\nof the water. If therefore the force, by which the vessel is thrust\ndown, be greater or equal to the endeavour by which the water naturally\ntendeth downwards, the air will go out that way where the resistance is\nmade, namely, towards the edges of the vessel. For, by how much the\ndeeper is the water which is to be penetrated, so much greater must be\nthe depressing force. But after the vessel is quite under water, the\nforce by which it is depressed, that is to say, the force by which the\nwater riseth up, is no longer increased. There is therefore such an\nequilibration between them, as that the natural endeavour of the water\ndownwards is equal to the endeavour by which the same water is to be\npenetrated to the increased depth.\n\nII. The second experiment is, that if a concave cylinder of sufficient\nlength, made of glass, that the experiment may be the better seen,\nhaving one end open and the other close shut, be filled with\nquicksilver, and the open end being stopped with one's finger, be\ntogether with the finger dipped into a dish or other vessel, in which\nalso there is quicksilver, and the cylinder be set upright, we shall,\nthe finger being taken away to make way for the descent of the\nquicksilver, see it descend into the vessel under it, till there be only\nso much remaining within the cylinder as may fill about twenty-six\ninches of the same; and thus it will always happen whatsoever be the\ncylinder, provided that the length be not less than twenty-six inches.\nFrom whence they conclude that the cavity of the cylinder above the\nquicksilver remains empty of all body. But in this experiment I find no\nnecessity at all of vacuum. For when the quicksilver which is in the\ncylinder descends, the vessel under it must needs be filled to a greater\nheight, and consequently so much of the contiguous air must be thrust\naway as may make place for the quicksilver which is descended. Now if it\nbe asked whither that air goes, what can be answered but this, that it\nthrusteth away the next air, and that the next, and so successively,\ntill there be a return to the place where the propulsion first began.\nAnd there, the last air thus thrust on will press the quicksilver in the\nvessel with the same force with which the first air was thrust away; and\nif the force with which the quicksilver descends be great enough, which\nis greater or less as it descends from a place of greater or less\nheight, it will make the air penetrate the quicksilver in the vessel,\nand go up into the cylinder to fill the place which they thought was\nleft empty. But because the quicksilver hath not in every degree of\nheight force enough to cause such penetration, therefore in descending\nit must of necessity stay somewhere, namely, there, where its endeavour\ndownwards, and the resistance of the same to the penetration of the air,\ncome to an equilibrium. And by this experiment it is manifest, that this\nequilibrium will be at the height of twenty-six inches, or thereabouts.\n\nIII. The third experiment is, that when a vessel hath as much air in it\nas it can naturally contain, there may nevertheless be forced into it as\nmuch water as will fill three quarters of the same vessel. And the\nexperiment is made in this manner. Into the glass bottle, represented\n(in figure 2) by the sphere F G, whose centre is A, let the pipe B A C\nbe so fitted, that it may precisely fill the mouth of the bottle; and\nlet the end B be so near the bottom, that there may be only space enough\nleft for the free passage of the water which is thrust in above. Let the\nupper end of this pipe have a cover at D, with a spout at E, by which\nthe water, when it ascends in the pipe, may run out. Also let H C be a\ncock, for the opening or shutting of the passage of the water between B\nand D, as there shall be occasion. Let the cover D E be taken off, and\nthe cock H C being opened, let a syringe full of water be forced in; and\nbefore the syringe be taken away, let the cock be turned to hinder the\ngoing out of the air. And in this manner let the injection of water be\nrepeated as often as it shall be requisite, till the water rise within\nthe bottle; for example, to G F. Lastly, the cover being fastened on\nagain, and the cock H C opened, the water will run swiftly out at E, and\nsink by little and little from G F to the bottom of the pipe B.\n\nFrom this phenomenon, they argue for the necessity of vacuum in this\nmanner. The bottle, from the beginning, was full of air; which air could\nneither go out by penetrating so great a length of water as was injected\nby the pipe, nor by any other way. Of necessity, therefore, all the\nwater as high as F G, as also all the air that was in the bottle before\nthe water was forced in, must now be in the same place, which at first\nwas filled by the air alone; which were impossible, if all the space\nwithin the bottle were formerly filled with air precisely, that is,\nwithout any vacuum. Besides, though some man perhaps may think the air,\nbeing a thin body, may pass through the body of the water contained in\nthe pipe, yet from that other phenomenon, namely, that all the water\nwhich is in the space B F G is cast out again by the spout at E, for\nwhich it seems impossible that any other reason can be given besides the\nforce by which the air frees itself from compression, it follows, that\neither there was in the bottle some space empty, or that many bodies may\nbe together in the same place. But this last is absurd; and therefore\nthe former is true, namely, that there was vacuum.\n\nThis argument is infirm in two places. For first, that is assumed which\nis not to be granted; and in the second place, an experiment is brought,\nwhich I think is repugnant to vacuum. That which is assumed is, that the\nair can have no passage out through the pipe. Nevertheless, we see daily\nthat air easily ascends from the bottom to the superficies of a river,\nas is manifest by the bubbles that rise; nor doth it need any other\ncause to give it this motion, than the natural endeavour downwards of\nthe water. Why, therefore, may not the endeavour upwards of the same\nwater, acquired by the injection, which endeavour upwards is greater\nthan the natural endeavour of the water downwards, cause the air in the\nbottle to penetrate in like manner the water that presseth it downwards;\nespecially, seeing the water, as it riseth in the bottle, doth so press\nthe air that is above it, as that it generateth in every part thereof an\nendeavour towards the external superficies of the pipe, and consequently\nmaketh all the parts of the enclosed air to tend directly towards the\npassage at B? I say, this is no less manifest, than that the air which\nriseth up from the bottom of a river should penetrate the water, how\ndeep soever it be. Wherefore I do not yet see any cause why the force,\nby which the water is injected, should not at the same time eject the\nair.\n\nAnd as for their arguing the necessity of vacuum from the rejection of\nthe water; in the first place, supposing there is vacuum, I demand by\nwhat principle of motion that ejection is made. Certainly, seeing this\nmotion is from within outwards, it must needs be caused by some agent\nwithin the bottle; that is to say, by the air itself. Now the motion of\nthat air, being caused by the rising of the water, begins at the bottom,\nand tends upwards; whereas the motion by which it ejecteth the water\nought to begin above, and tend downwards. From whence therefore hath the\nenclosed air this endeavour towards the bottom? To this question I know\nnot what answer can be given, unless it be said, that the air descends\nof its own accord to expel the water. Which, because it is absurd, and\nthat the air, after the water is forced in, hath as much room as its\nmagnitude requires, there will remain no cause at all why the water\nshould be forced out. Wherefore the assertion of vacuum is repugnant to\nthe very experiment which is here brought to establish it.\n\nMany other phenomena are usually brought for vacuum, as those of\n_weather-glasses_, _\u00e6olipyles_, _wind-guns_, &c. which would all be very\nhard to be salved, unless water be penetrable by air, without the\nintermixture of empty space. But now, seeing air may with no great\nendeavour pass through not only water, but any other fluid body though\nnever so stubborn, as quicksilver, these phenomena prove nothing.\nNevertheless, it might in reason be expected, that he that would take\naway vacuum, should without vacuum show us such causes of these\nphenomena, as should be at least of equal, if not greater probability.\nThis therefore shall be done in the following discourse, when I come to\nspeak of these phenomena in their proper places. But first, the most\ngeneral hypotheses of natural philosophy are to be premised.\n\nAnd seeing that suppositions are put for the true causes of apparent\neffects, every supposition, except such as be absurd, must of necessity\nconsist of some supposed possible motion; for rest can never be the\nefficient cause of anything; and motion supposeth bodies moveable; of\nwhich there are three kinds, _fluid_, _consistent_, and _mixed of both_.\n_Fluid_ are those, whose parts may by very weak endeavour be separated\nfrom one another; and _consistent_ those for the separation of whose\nparts greater force is to be applied. There are therefore degrees of\nconsistency; which degrees, by comparison with more or less consistent,\nhave the names of _hardness_ or _softness_. Wherefore a fluid body is\nalways divisible into bodies equally fluid, as quantity into quantities;\nand soft bodies, of whatsoever degree of softness, into soft bodies of\nthe same degree. And though many men seem to conceive no other\ndifference of _fluidity_, but such as ariseth from the different\nmagnitudes of the parts, in which sense dust, though of diamonds, may be\ncalled fluid; yet I understand by _fluidity_, that which is made such by\nnature equally in every part of the fluid body; not as dust is fluid,\nfor so a house which is falling in pieces may be called fluid; but in\nsuch manner as water seems fluid, and to divide itself into parts\nperpetually fluid. And this being well understood, I come to my\nsuppositions.\n\n[Sidenote: Six suppositions for the salving of the phenomena of nature.]\n\n5. First, therefore, I suppose that the immense space, which we call the\nworld, is the aggregate of all bodies which are either consistent and\nvisible, as the earth and the stars; or invisible, as the small atoms\nwhich are disseminated through the whole space between the earth and the\nstars; and lastly, that most fluid ether, which so fills all the rest of\nthe universe, as that it leaves in it no empty place at all.\n\nSecondly, I suppose with Copernicus, that the greater bodies of the\nworld, which are both consistent and permanent, have such order amongst\nthemselves, as that the sun hath the first place, Mercury the second,\nVenus the third, the Earth with the moon going about it the fourth, Mars\nthe fifth, Jupiter with his attendants the sixth, Saturn the seventh;\nand after these, the fixed stars have their several distances from the\nsun.\n\nThirdly, I suppose that in the sun and the rest of the planets there is\nand always has been a simple circular motion.\n\nFourthly, I suppose that in the body of the air there are certain other\nbodies intermingled, which are not fluid; but withal that they are so\nsmall, that they are not perceptible by sense; and that these also have\ntheir proper simple motion, and are some of them more, some less hard or\nconsistent.\n\nFifthly, I suppose with Kepler that as the distance between the sun and\nthe earth is to the distance between the moon and the earth, so the\ndistance between the moon and the earth is to the semidiameter of the\nearth.\n\nAs for the magnitude of the circles, and the times in which they are\ndescribed by the bodies which are in them, I will suppose them to be\nsuch as shall seem most agreeable to the phenomena in question.\n\n[Sidenote: Possible causes of the motions annual and diurnal; and of the\n apparent direction, station, and retrogradation of the\n planets.]\n\n6. The causes of the different seasons of the year, and of the several\nvariations of days and nights in all the parts of the superficies of the\nearth, have been demonstrated, first by Copernicus, and since by Kepler,\nGalileus, and others, from the supposition of the earth's diurnal\nrevolution about its own axis, together with its annual motion about the\nsun in the ecliptic according to the order of the signs; and thirdly, by\nthe annual revolution of the same earth about its own centre, contrary\nto the order of the signs. I suppose with Copernicus, that the diurnal\nrevolution is from the motion of the earth, by which the equinoctial\ncircle is described about it. And as for the other two annual motions,\nthey are the efficient cause of the earth's being carried about in the\necliptic in such manner, as that its axis is always kept parallel to\nitself. Which parallelism was for this reason introduced, lest by the\nearth's annual revolution its poles should seem to be necessarily\ncarried about the sun, contrary to experience. I have, in art. 10, chap.\nXXI, demonstrated, from the supposition of simple circular motion in the\nsun, that the earth is so carried about the sun, as that its axis is\nthereby kept always parallel to itself. Wherefore, from these two\nsupposed motions in the sun, the one simple circular motion, the other\ncircular motion about its own centre, it may be demonstrated that the\nyear hath both the same variations of days and nights, as have been\ndemonstrated by Copernicus.\n\nFor if the circle _a b c d_ (in fig. 3) be the ecliptic, whose centre is\n_e_, and diameter _a e c_; and the earth be placed in _a_, and the sun\nbe moved in the little circle _f g h i_, namely, according to the order\n_f_, _g_, _h_, and _i_, it hath been demonstrated, that a body placed in\n_a_ will be moved in the same order through the points of the ecliptic\n_a_, _b_, _c_, and _d_, and will always keep its axis parallel to\nitself.\n\nBut if, as I have supposed, the earth also be moved with simple circular\nmotion in a plane that passeth through _a_, cutting the plane of the\necliptic so as that the common section of both the planes be in _a c_,\nthus also the axis of the earth will be kept always parallel to itself.\nFor let the centre of the earth be moved about in the circumference of\nthe epicycle, whose diameter is _l a k_, which is a part of the strait\nline _l a c_; therefore _l a k_, the diameter of the epicycle, passing\nthrough the centre of the earth, will be in the plane of the ecliptic.\nWherefore seeing that by reason of the earth's simple motion both in the\necliptic and in its epicycle, the strait line _l a k_ is kept always\nparallel to itself, every other strait line also taken in the body of\nthe earth, and consequently its axis, will in like manner be kept always\nparallel to itself; so that in what part soever of the ecliptic the\ncentre of the epicycle be found, and in what part soever of the epicycle\nthe centre of the earth be found at the same time, the axis of the earth\nwill be parallel to the place where the same axis would have been, if\nthe centre of the earth had never gone out of the ecliptic.\n\nNow as I have demonstrated the simple annual motion of the earth from\nthe supposition of simple motion in the sun; so from the supposition of\nsimple motion in the earth may be demonstrated the monthly simple motion\nof the moon. For if the names be but changed, the demonstration will be\nthe same, and therefore need not be repeated.\n\n[Sidenote: The supposition of simple motion, why likely.]\n\n7. That which makes this supposition of the sun's simple motion in the\nepicycle _f g h i_ probable, is first, that the periods of all the\nplanets are not only described about the sun, but so described, as that\nthey are all contained within the zodiac, that is to say, within the\nlatitude of about sixteen degrees; for the cause of this seems to depend\nupon some power in the sun, especially in that part of the sun which\nrespects the zodiac. Secondly, that in the whole compass of the heavens\nthere appears no other body from which the cause of this phenomenon can\nin probability be derived. Besides, I could not imagine that so many and\nsuch various motions of the planets should have no dependance at all\nupon one another. But, by supposing motive power in the sun, we suppose\nmotion also; for power to move without motion is no power at all. I have\ntherefore supposed that there is in the sun for the governing of the\nprimary planets, and in the earth for the governing of the moon, such\nmotion, as being received by the primary planets and by the moon, makes\nthem necessarily appear to us in such manner as we see them. Whereas,\nthat circular motion, which is commonly attributed to them, about a\nfixed axis, which is called conversion, being a motion of their parts\nonly, and not of their whole bodies, is insufficient to salve their\nappearances. For seeing whatsoever is so moved, hath no endeavour at all\ntowards those parts which are without the circle, they have no power to\npropagate any endeavour to such bodies as are placed without it. And as\nfor them that suppose this may be done by magnetical virtue, or by\nincorporeal and immaterial species, they suppose no natural cause; nay,\nno cause at all. For there is no such thing as an incorporeal movent,\nand magnetical virtue is a thing altogether unknown; and whensoever it\nshall be known, it will be found to be a motion of body. It remains,\ntherefore, that if the primary planets be carried about by the sun, and\nthe moon by the earth, they have the simple circular motions of the sun\nand the earth for the causes of their circulations. Otherwise, if they\nbe not carried about by the sun and the earth, but that every planet\nhath been moved, as it is now moved, ever since it was made, there will\nbe of their motions no cause natural. For either these motions were\nconcreated with their bodies, and their cause is supernatural; or they\nare coeternal with them, and so they have no cause at all. For\nwhatsoever is eternal was never generated.\n\nI may add besides, to confirm the probability of this simple motion,\nthat as almost all learned men are now of the same opinion with\nCopernicus concerning the parallelism of the axis of the earth, it\nseemed to me to be more agreeable to truth, or at least more handsome,\nthat it should be caused by simple circular motion alone, than by two\nmotions, one in the ecliptic, and the other about the earth's own axis\nthe contrary way, neither of them simple, nor either of them such as\nmight be produced by any motion of the sun. I thought best therefore to\nretain this hypothesis of simple motion, and from it to derive the\ncauses of as many of the phenomena as I could, and to let such alone as\nI could not deduce from thence.\n\nIt will perhaps be objected, that although by this supposition the\nreason may be given of the parallelism of the axis of the earth, and of\nmany other appearances, nevertheless, seeing it is done by placing the\nbody of the sun in the centre of that orb which the earth describes with\nits annual motion, the supposition itself is false; because this annual\norb is eccentric to the sun. In the first place, therefore, let us\nexamine what that eccentricity is, and whence it proceeds.\n\n[Sidenote: The cause of the eccentricity of the annual motion of the\n earth.]\n\n8. Let the annual circle of the earth _a b c d_ (in fig. 3) be divided\ninto four equal parts by the strait lines _a c_ and _b d_, cutting one\nanother in the centre _e_; and let _a_ be the beginning of Libra, _b_ of\nCapricorn, _c_ of Aries and _d_ of Cancer; and let the whole orb _a b c\nd_ be understood, according to Copernicus, to have every way so great\ndistance from the zodiac of the fixed stars, that it be in comparison\nwith it but as a point. Let the earth be now supposed to be in the\nbeginning of Libra at _a_. The sun, therefore, will appear in the\nbeginning of Aries at _c_. Wherefore, if the earth be moved from _a_ to\n_b_, the apparent motion of the sun will be from _c_ to the beginning of\nCancer in _d_; and the earth being moved forwards from _b_ to _c_, the\nsun also will appear to be moved forwards to the beginning of Libra in\n_a_; wherefore _c d a_ will be the summer arch, and the winter arch will\nbe _a b c_. Now, in the time, of the sun's apparent motion in the summer\narch, there are numbered 186\u00be days; and, consequently, the earth makes\nin the same time the same number of diurnal conversions in the arch _a b\nc_; and, therefore, the earth in its motion through the arch _c d a_\nwill make only 178\u00bd diurnal conversions. Wherefore the arch _a b c_\nought to be greater than the arch _c d a_ by 8\u00bc days, that is to say, by\nalmost so many degrees. Let the arch _a r_, as also _c s_, be each of\nthem an arch of two degrees and 1\u204416. Wherefore the arch _r b s_ will be\ngreater than the semicircle _a b c_ by 4\u215b degrees, and greater than the\narch _s d r_ by 8\u00bc degrees. The equinoxes, therefore, will be in the\npoints _r_ and _s_; and therefore also, when the earth is in _r_, the\nsun will appear in _s_. Wherefore the true place of the sun will be in\n_t_, that is to say, without the centre of the earth's annual motion by\nthe quantity of the sine of the arch _a r_, or the sine of two degrees\nand 16 minutes. Now this sine, putting 100,000 for the radius, will be\nnear 3580 parts thereof. And so much is the eccentricity of the earth's\nannual motion, provided that that motion be in a perfect circle; and _s_\nand _r_ are the equinoctial parts. And the strait lines _s r_ and _c a_,\nproduced both ways till they reach the zodiac of the fixed stars, will\nfall still upon the same fixed stars; because the whole orb _a b c d_ is\nsupposed to have no magnitude at all in respect of the great distance of\nthe fixed stars.\n\nSupposing now the sun to be in _c_, it remains that I show the cause why\nthe earth is nearer to the sun, when in its annual motion it is found to\nbe in _d_, than when it is in _b_. And I take the cause to be this. When\nthe earth is in the beginning of Capricorn at _b_, the sun appears in\nthe beginning of Cancer at _d_; and then is the midst of summer. But in\nthe midst of summer, the northern parts of the earth are towards the\nsun, which is almost all dry land, containing all Europe and much the\ngreatest part of Asia and America. But when the earth is in the\nbeginning of Cancer at _d_, it is the midst of winter, and that part of\nthe earth is towards the sun, which contains those great seas called the\nSouth Sea and the Indian Sea, which are of far greater extent than all\nthe dry land in that hemisphere. Wherefore by the last article of\nchapter XXI, when the earth is in _d_, it will come nearer to its first\nmovent, that is, to the sun which is in _t_; that is to say, the earth\nis nearer to the sun in the midst of winter when it is in _d_, than in\nthe midst of summer when it is in _b_; and, therefore, during the winter\nthe sun is in its _Perig\u00e6um_, and in its _Apog\u00e6um_ during the summer.\nAnd thus I have shown a possible cause of the eccentricity of the earth;\nwhich was to be done.\n\nI am, therefore, of Kepler's opinion in this, that he attributes the\neccentricity of the earth to the difference of the parts thereof, and\nsupposes one part to be affected, and another disaffected to the sun.\nAnd I dissent from him in this, that he thinks it to be by magnetic\nvirtue, and that this magnetic virtue or attraction and thrusting back\nof the earth is wrought by immateriate species; which cannot be, because\nnothing can give motion but a body moved and contiguous. For if those\nbodies be not moved which are contiguous to a body unmoved, how this\nbody should begin to be moved is not imaginable; as has been\ndemonstrated in art. 7, chap. IX, and often inculcated in other places,\nto the end that philosophers might at last abstain from the use of such\nunconceivable connexions of words. I dissent also from him in this, that\nhe says the similitude of bodies is the cause of their mutual\nattraction. For if it were so, I see no reason why one egg should not be\nattracted by another. If, therefore, one part of the earth be more\naffected by the sun than another part, it proceeds from this, that one\npart hath more water, the other more dry land. And from hence it is, as\nI showed above, that the earth comes nearer to the sun when it shines\nupon that part where there is more water, than when it shines upon that\nwhere there is more dry land.\n\n[Sidenote: The cause why the moon hath always one and the same face\n turned towards the earth.]\n\n9. This eccentricity of the earth is the cause why the way of its annual\nmotion is not a perfect circle, but either an elliptical, or almost an\nelliptical line; as also why the axis of the earth is not kept exactly\nparallel to itself in all places, but only in the equinoctial points.\n\nNow seeing I have said that the moon is carried about by the earth, in\nthe same manner that the earth is by the sun; and that the earth goeth\nabout the sun in such manner as that it shows sometimes one hemisphere,\nsometimes the other to the sun; it remains to be enquired, why the moon\nhas always one and the same face turned towards the earth.\n\nSuppose, therefore, the sun to be moved with simple motion in the little\ncircle _f g h i_, (in fig. 4) whose centre is _t_; and let \u2648 \u264b \u264e \u2651 be\nthe annual circle of the earth; and _a_ the beginning of Libra. About\nthe point _a_ let the little circle _l k_ be described; and in it let\nthe centre of the earth be understood to be moved with simple motion;\nand both the sun and the earth to be moved according to the order of the\nsigns. Upon the centre _a_ let the way of the moon _m n o p_ be\ndescribed; and let _q r_ be the diameter of a circle cutting the globe\nof the moon into two hemispheres, whereof one is seen by us when the\nmoon is at the full, and the other is turned from us.\n\nThe diameter therefore of the moon _q o r_ will be perpendicular to the\nstrait line _t a_. Wherefore the moon is carried, by reason of the\nmotion of the earth, from _o_ towards _p_. But by reason of the motion\nof the sun, if it were in _p_ it would at the same time be carried from\n_p_ towards _o_; and by these two contrary movents the strait line _q r_\nwill be turned about; and, in a quadrant of the circle _m n o p_, it\nwill be turned so much as makes the fourth part of its whole conversion.\nWherefore when the moon is in _p_, _q r_ will be parallel to the strait\nline _m o_. Secondly, when the moon is in _m_, the strait line _q r_\nwill, by reason of the motion of the earth, be in _m o_. But by the\nworking of the sun's motion upon it in the quadrant _p m_, the same _q\nr_ will be turned so much as makes another quarter of its whole\nconversion. When, therefore, the moon is in _m_, _q r_ will be\nperpendicular to the strait line _o m_. By the same reason, when the\nmoon is in _n_, _q r_ will be parallel to the strait line _m o_; and,\nthe moon returning to _o_, the same _q r_ will return to its first\nplace; and the body of the moon will in one entire period make also one\nentire conversion upon her own axis. In the making of which, it is\nmanifest, that one and the same face of the moon is always turned\ntowards the earth. And if any diameter were taken in that little circle,\nin which the moon were supposed to be carried about with simple motion,\nthe same effect would follow; for if there were no action from the sun,\nevery diameter of the moon would be carried about always parallel to\nitself. Wherefore I have given a possible cause why one and the same\nface of the moon is always turned towards the earth.\n\nBut it is to be noted, that when the moon is without the ecliptic, we do\nnot always see the same face precisely. For we see only that part which\nis illuminated. But when the moon is without the ecliptic, that part\nwhich is towards us is not exactly the same with that which is\nilluminated.\n\n[Sidenote: The cause of the tides of the ocean.]\n\n10. To these three simple motions, one of the sun, another of the moon,\nand the third of the earth, in their own little circles _f g h i_, _l\nk_, and _q r_, together with the diurnal conversion of the earth, by\nwhich conversion all things that adhere to its superficies are\nnecessarily carried about with it, may be referred the three phenomena\nconcerning the tides of the ocean. Whereof the first is the alternate\nelevation and depression of the water at the shores, twice in the space\nof twenty-four hours and near upon fifty-two minutes; for so it has\nconstantly continued in all ages. The second, that at the new and full\nmoons, the elevations of the water are greater than at other times\nbetween. And the third, that when the sun is in the equinoctial, they\nare yet greater than at any other time. For the salving of which\nphenomena, we have already the four above-mentioned motions; to which I\nassume also this, that the part of the earth which is called America,\nbeing higher than the water, and extended almost the space of a whole\nsemicircle from north to south, gives a stop to the motion of the water.\n\nThis being granted, in the same 4th figure, where _l b k c_ is supposed\nto be in the plane of the moon's monthly motion, let the little circle\n_l d k e_ be described about the same centre _a_ in the plane of the\nequinoctial. This circle therefore will decline from the circle _l b k\nc_ in an angle of almost 28\u00bd degrees; for the greatest declination of\nthe ecliptic is 23\u00bd, to which adding 5 for the greatest declination of\nthe moon from the ecliptic, the sum will be 28\u00bd degrees. Seeing now the\nwaters, which are under the circle of the moon's course, are by reason\nof the earth's simple motion in the plane of the same circle moved\ntogether with the earth, that is to say, together with their own\nbottoms, neither outgoing nor outgone; if we add the diurnal motion, by\nwhich the other waters which are under the equinoctial are moved in the\nsame order, and consider withal that the circles of the moon and of the\nequinoctial intersect one another; it will be manifest, that both those\nwaters, which are under the circle of the moon, and under the\nequinoctial, will run together under the equinoctial; and consequently,\nthat their motion will not only be swifter than the ground that carries\nthem; but also that the waters themselves will have greater elevation\nwhensoever the earth is in the equinoctial. Wherefore, whatsoever the\ncause of the tides may be, this may be the cause of their augmentation\nat that time.\n\nAgain, seeing I have supposed the moon to be carried about by the simple\nmotion of the earth in the little circle _l b k c_; and demonstrated, at\nthe 4th article of chapter XXI, that whatsoever is moved by a movent\nthat hath simple motion, will be moved always with the same velocity; it\nfollows, that the centre of the earth will be carried in the\ncircumference _l b k c_ with the same velocity with which the moon is\ncarried in the circumference _m n o p_. Wherefore the time, in which the\nmoon is carried about in _m n o p_, is to the time, in which the earth\nis carried about in _l b k c_, as one circumference to the other, that\nis, as _a o_ to _a k_. But _a o_ is observed to be to the semidiameter\nof the earth as 59 to 1; and therefore the earth, if _a k_ be put for\nits semidiameter, will make fifty-nine revolutions in _l b k c_ in the\ntime that the moon makes one monthly circuit in _m n o p_. But the moon\nmakes her monthly circuit in little more than twenty-nine days.\nWherefore the earth shall make its circuit in the circumference _l b k\nc_ in twelve hours and a little more, namely, about twenty-six minutes\nmore; that is to say, it shall make two circuits in twenty-four hours\nand almost fifty-two minutes; which is observed to be the time between\nthe high-water of one day and the high-water of the day following. Now\nthe course of the waters being hindered by the southern part of America,\ntheir motion will be interrupted there; and consequently, they will be\nelevated in those places, and sink down again by their own weight, twice\nin the space of twenty-four hours and fifty-two minutes. And thus I have\ngiven a possible cause of the diurnal reciprocation of the ocean.\n\nNow from this swelling of the ocean in those parts of the earth, proceed\nthe flowings and ebbings in the Atlantic, Spanish, British, and German\nseas; which though they have their set times, yet upon several shores\nthey happen at several hours of the day. And they receive some\naugmentation from the north, by reason that the shores of China and\nTartary, hindering the general course of the waters, make them swell\nthere, and discharge themselves in part through the strait of Anian into\nthe Northern Ocean, and so into the German Sea.\n\nAs for the spring tides which happen at the time of the new and full\nmoons, they are caused by that simple motion, which at the beginning I\nsupposed to be always in the moon. For as, when I showed the cause of\nthe eccentricity of the earth, I derived the elevation of the waters\nfrom the simple motion of the sun; so the same may here be derived from\nthe simple motion of the moon. For though from the generation of clouds,\nthere appear in the sun a more manifest power of elevating the waters\nthan in the moon; yet the power of increasing moisture in vegetables and\nliving creatures appears more manifestly in the moon than in the sun;\nwhich may perhaps proceed from this, that the sun raiseth up greater,\nand the moon lesser drops of water. Nevertheless, it is more likely, and\nmore agreeable to common observation, that rain is raised not only by\nthe sun, but also by the moon; for almost all men expect change of\nweather at the time of the conjunctions of the sun and moon with one\nanother and with the earth, more than in the time of their quarters.\n\nIn the last place, the cause why the spring tides are greater at the\ntime of the equinoxes hath been already sufficiently declared in this\narticle, where I have demonstrated, that the two motions of the earth,\nnamely, its simple motion in the little circle _l b k c_, and its\ndiurnal motion in _l d k e_, cause necessarily a greater elevation of\nwaters when the sun is about the equinoxes, than when he is in other\nplaces. I have therefore given possible causes of the phenomenon of the\nflowing and ebbing of the ocean.\n\n[Sidenote: Cause of the precession of the equinoxes.]\n\n11. As for the explication of the yearly _precession of the equinoctial\npoints_, we must remember that, as I have already shown, the annual\nmotion of the earth is not in the circumference of a circle, but of an\nellipsis, or a line not considerably different from that of an ellipsis.\nIn the first place, therefore, this elliptical line is to be described.\n\nLet the ecliptic \u264e \u2651 \u2648 \u264b (in fig. 5) be divided into four equal parts by\nthe two strait lines _a b_ and \u2651 \u264b, cutting one another at right angles\nin the centre _c_. And taking the arch _b d_ of two degrees and sixteen\nminutes, let the strait line _d e_ be drawn parallel to _a b_, and\ncutting \u2651 \u264b in _f_; which being done, the eccentricity of the earth will\nbe _c f_. Seeing therefore the annual motion of the earth is in the\ncircumference of an ellipsis, of which \u2651 \u264b is the greater axis, _a b_\ncannot be the lesser axis; for _a b_ and \u2651 \u264b are equal. Wherefore the\nearth passing through _a_ and _b_, will either pass above \u2651, as through\n_g_, or passing through \u2651, will fall between _c_ and _a_; it is no\nmatter which. Let it pass therefore through _g_; and let _g l_ be taken\nequal to the strait line \u2651 \u264b; and dividing _g l_ equally in _i_, _g i_\nwill be equal to \u2651 _f_, and _i l_ equal to _f_ \u264b; and consequently the\npoint _i_ will cut the eccentricity _c f_ into two equal parts; and\ntaking _i h_ equal to _i f_, _h i_ will be the whole eccentricity. If\nnow a strait line, namely, the line \u264e _i_ \u2648, be drawn through _i_\nparallel to the strait lines _a b_ and _e d_, the way of the sun in\nsummer, namely, the arch \u264e _g_ \u2648, will be greater than his way in\nwinter, by 8\u00bc degrees. Wherefore the true equinoxes will be in the\nstrait line \u264e _i_ \u2648; and therefore the ellipsis of the earth's annual\nmotion will not pass through _a_, _g_, _b_, and _l_; but through \u264e, _g_,\n\u2648 and _l_. Wherefore the annual motion of the earth is in the ellipsis \u264e\n_g_ \u2648 _l_; and cannot be, the eccentricity being salved, in any other\nline. And this perhaps is the reason, why Kepler, against the opinion of\nall the astronomers of former time, thought fit to bisect the\neccentricity of the earth, or, according to the ancients, of the sun,\nnot by diminishing the quantity of the same eccentricity, (because the\ntrue measure of that quantity is the difference by which the summer arch\nexceeds the winter arch), but by taking for the centre of the ecliptic\nof the great orb the point _c_ nearer to _f_, and so placing the whole\ngreat orb as much nearer to the ecliptic of the fixed stars towards \u264b,\nas is the distance between _c_ and _i_. For seeing the whole great orb\nis but as a point in respect of the immense distance of the fixed stars,\nthe two strait lines \u264e \u2648 and _a b_, being produced both ways to the\nbeginnings of Aries and Libra, will fall upon the same points of the\nsphere of the fixed stars. Let therefore the diameter of the earth _m n_\nbe in the plane of the earth's annual motion. If now the earth be moved\nby the sun's simple motion in the circumference of the ecliptic about\nthe centre _i_, this diameter will be kept always parallel to itself and\nto the strait line _g l_. But seeing the earth is moved in the\ncircumference of an ellipsis without the ecliptic, the point _n_, whilst\nit passeth through \u264e \u2651 \u2648, will go in a lesser circumference than the\npoint _m_; and consequently, as soon as ever it begins to be moved, it\nwill lose its parallelism with the strait line \u2651 \u264b; so that _m n_\nproduced will at last cut the strait line _g l_ produced. And\ncontrarily, as soon as _m n_ is past \u2648, the earth making its way in the\ninternal elliptical line \u2648 _l_ \u264e, the same _m n_ produced towards _m_,\nwill cut _l g_ produced. And when the earth hath almost finished its\nwhole circumference, the same _m n_ shall again make a right angle with\na line drawn from the centre _i_, a little short of the point from which\nthe earth began its motion. And there the next year shall be one of the\nequinoctial points, namely, near the end of \u264d; the other shall be\nopposite to it near the end of \u2653. And thus the points in which the days\nand nights are made equal do every year fall back; but with so slow a\nmotion, that, in a whole year, it makes but 51 first minutes. And this\nrelapse being contrary to the order of the signs, is commonly called the\n_precession of the equinoxes_. Of which I have from my former\nsuppositions deduced a possible cause; which was to be done.\n\nAccording to what I have said concerning the cause of the eccentricity\nof the earth; and according to Kepler, who for the cause thereof\nsupposeth one part of the earth to be affected to the sun, the other\npart to be disaffected; the apog\u00e6um and perig\u00e6um of the sun should be\nmoved every year in the same order, and with the same velocity, with\nwhich the equinoctial points are moved; and their distance from them\nshould always be the quadrant of a circle; which seems to be otherwise.\nFor astronomers say, that the equinoxes are now, the one about 28\ndegrees gone back from the first star of Aries, the other as much from\nthe beginning of Libra; so that the apog\u00e6um of the sun or the aphelium\nof the earth ought to be about the 28th degree of Cancer. But it is\nreckoned to be in the 7th degree. Seeing, therefore, we have not\nsufficient evidence of the \u1f41\u03c4\u03af (that so it is,) it is in vain to seek\nfor the \u03b4\u03b9\u03cc\u03c4\u03b9 (why it is so.) Wherefore, as long as the motion of the\napog\u00e6um is not observable by reason of the slowness thereof, and as long\nas it remains doubtful whether their distance from the equinoctial\npoints be more or less than a quadrant precisely; so long it may be\nlawful for me to think they proceed both of them with equal velocity.\n\nAlso, I do not at all meddle with the causes of the eccentricities of\nSaturn, Jupiter, Mars, and Mercury. Nevertheless, seeing the\neccentricity of the earth may, as I have shewn, be caused by the unlike\nconstitution of the several parts of the earth which are alternately\nturned towards the sun, it is credible also, that like effects may be\nproduced in these other planets from their having their superficies of\nunlike parts.\n\nAnd this is all I shall say concerning Sidereal Philosophy. And, though\nthe causes I have here supposed be not the true causes of these\nphenomena, yet I have demonstrated that they are sufficient to produce\nthem, according to what I at first propounded.\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C. XXVI.\n _Fig. 1-5_\n]\n\n\n -------\n\n\n CHAPTER XXVII.\n OF LIGHT, HEAT, AND OF COLOURS.\n\n1. Of the immense magnitude of some bodies, and the unspeakable\n littleness of others.\u20142. Of the cause of the light of the sun.\u20143.\n How light heateth.\u20144. The generation of fire from the sun.\u20145. The\n generation of fire from collision.\u20146. The cause of light in\n glow-worms, rotten wood, and the Bolognan stone.\u20147. The cause of\n light in the concussion of sea water.\u20148. The cause of flame, sparks,\n and colliquation.\u20149. The cause why wet hay sometimes burns of its\n own accord; also the cause of lightning.\u201410. The cause of the force\n of gunpowder; and what is to be ascribed to the coals, what to the\n brimstone, and what to the nitre.\u201411. How heat is caused by\n attrition.\u201412. The distinction of light into first, second, &c.\u201413.\n The causes of the colours we see in looking through a prisma of\n glass, namely, of red, yellow, blue, and violet colour.\u201414. Why the\n moon and the stars appear redder in the horizon than in the midst of\n the heaven.\u201415. The cause of whiteness.\u201416. The cause of blackness.\n\n\n[Sidenote: Of the immense magnitude of some bodies, and the unspeakable\n littleness of others.]\n\n1. Besides the stars, of which I have spoken in the last chapter,\nwhatsoever other bodies there be in the world, they may be all\ncomprehended under the name of intersidereal bodies. And these I have\nalready supposed to be either the most fluid \u00e6ther, or such bodies whose\nparts have some degree of cohesion. Now, these differ from one another\nin their several _consistencies_, _magnitudes_, _motions_, and\n_figures_. In consistency, I suppose some bodies to be harder, others\nsofter through all the several degrees of _tenacity_. In magnitude, some\nto be greater, others less, and many unspeakably little. For we must\nremember that, by the understanding, quantity is divisible into\ndivisibles perpetually. And, therefore, if a man could do as much with\nhis hands as he can with his understanding, he would be able to take\nfrom any given magnitude a part which should be less than any other\nmagnitude given. But the Omnipotent Creator of the world can actually\nfrom a part of any thing take another part, as far as we by our\nunderstanding can conceive the same to be divisible. Wherefore there is\nno impossible smallness of bodies. And what hinders but that we may\nthink this likely? For we know there are some living creatures so small\nthat we can scarce see their whole bodies. Yet even these have their\nyoung ones; their little veins and other vessels, and their eyes so\nsmall as that no microscope can make them visible. So that we cannot\nsuppose any magnitude so little, but that our very supposition is\nactually exceeded by nature. Besides, there are now such microscopes\ncommonly made, that the things we see with them appear a hundred\nthousand times bigger than they would do if we looked upon them with our\nbare eyes. Nor is there any doubt but that by augmenting the power of\nthese microscopes (for it may be augmented as long as neither matter nor\nthe hands of workmen are wanting) every one of those hundred thousandth\nparts might yet appear a hundred thousand times greater than they did\nbefore. Neither is the smallness of some bodies to be more admired than\nthe vast greatness of others. For it belongs to the same Infinite Power,\nas well to augment infinitely as infinitely to diminish. To make the\ngreat orb, namely, that whose radius reacheth from the earth to the sun,\nbut as a point in respect of the distance between the sun and the fixed\nstars; and, on the contrary, to make a body so little, as to be in the\nsame proportion less than any other visible body, proceeds equally from\none and the same Author of Nature. But this of the immense distance of\nthe fixed stars, which for a long time was accounted an incredible\nthing, is now believed by almost all the learned. Why then should not\nthat other, of the smallness of some bodies, become credible at some\ntime or other? For the Majesty of God appears no less in small things\nthan in great; and as it exceedeth human sense in the immense greatness\nof the universe, so also it doth in the smallness of the parts thereof.\nNor are the first elements of compositions, nor the first beginnings of\nactions, nor the first moments of times more credible, than that which\nis now believed of the vast distance of the fixed stars.\n\nSome things are acknowledged by mortal men to be very great, though\nfinite, as seeing them to be such. They acknowledge also that some\nthings, which they do not see, may be of infinite magnitude. But they\nare not presently nor without great study persuaded, that there is any\nmean between infinite and the greatest of those things which either they\nsee or imagine. Nevertheless, when after meditation and contemplation\nmany things which we wondered at before are now grown more familiar to\nus, we then believe them, and transfer our admiration from the creatures\nto the Creator. But how little soever some bodies may be, yet I will not\nsuppose their quantity to be less than is requisite for the salving of\nthe phenomena. And in like manner I shall suppose their motion, namely,\ntheir velocity and slowness, and the variety of their figures, to be\nonly such as the explication of their natural causes requires. And\nlastly, I suppose, that the parts of the pure \u00e6ther, as if it were the\nfirst matter, have no motion at all but what they receive from bodies\nwhich float in them, and are not themselves fluid.\n\n[Sidenote: Of the cause of the light of the sun.]\n\n2. Having laid these grounds, let us come to speak of causes; and in the\nfirst place let us inquire what may be the cause of the light of the\nsun. Seeing, therefore, the body of the sun doth by its simple circular\nmotion thrust away the ambient ethereal substance sometimes one way\nsometimes another, so that those parts, which are next the sun, being\nmoved by it, do propagate that motion to the next remote parts, and\nthese to the next, and so on continually; it must needs be that,\nnotwithstanding any distance, the foremost part of the eye will at last\nbe pressed; and by the pressure of that part, the motion will be\npropagated to the innermost part of the organ of sight, namely, to the\nheart; and from the reaction of the heart, there will proceed an\nendeavour back by the same way, ending in the endeavour outwards of the\ncoat of the eye, called the _retina_. But this endeavour outwards, as\nhas been defined in chapter XXV, is the thing which is called light, or\nthe phantasm of a lucid body. For it is by reason of this phantasm that\nan object is called lucid. Wherefore we have a possible cause of the\nlight of the sun; which I undertook to find.\n\n[Sidenote: How light heateth.]\n\n3. The generation of the light of the sun is accompanied with the\ngeneration of heat. Now every man knows what heat is in himself, by\nfeeling it when he grows hot; but what it is in other things, he knows\nonly by ratiocination. For it is one thing to grow hot, and another\nthing to heat or make hot. And therefore though we perceive that the\nfire or the sun heateth, yet we do not perceive that it is itself hot.\nThat other living creatures, whilst they make other things hot, are hot\nthemselves, we infer by reasoning from the like sense in ourselves. But\nthis is not a necessary inference. For though it may truly be said of\nliving creatures, that _they heat, therefore they are themselves hot_;\nyet it cannot from hence be truly inferred that _fire heateth, therefore\nit is itself hot_; no more than this, _fire causeth pain, therefore it\nis itself in pain_. Wherefore, that is only and properly called hot,\nwhich when we feel we are necessarily hot.\n\nNow when we grow hot, we find that our spirits and blood, and whatsoever\nis fluid within us, is called out from the internal to the external\nparts of our bodies, more or less, according to the degree of the heat;\nand that our skin swelleth. He, therefore, that can give a possible\ncause of this evocation and swelling, and such as agrees with the rest\nof the phenomena of heat, may be thought to have given the cause of the\nheat of the sun.\n\nIt hath been shown, in the 5th article of chapter XXI, that the fluid\nmedium, which we call the air, is so moved by the simple circular motion\nof the sun, as that all its parts, even the least, do perpetually change\nplaces with one another; which change of places is that which there I\ncalled fermentation. From this fermentation of the air, I have, in the\n8th article of the last chapter, demonstrated that the water may be\ndrawn up into the clouds.\n\nAnd I shall now show that the fluid parts may, in like manner, by the\nsame fermentation, be drawn out from the internal to the external parts\nof our bodies. For seeing that wheresoever the fluid medium is\ncontiguous to the body of any living creature, there the parts of that\nmedium are, by perpetual change of place, separated from one another;\nthe contiguous parts of the living creature must, of necessity,\nendeavour to enter into the spaces of the separated parts. For otherwise\nthose parts, supposing there is no vacuum, would have no place to go\ninto. And therefore that, which is most fluid and separable in the parts\nof the living creature which are contiguous to the medium, will go first\nout; and into the place thereof will succeed such other parts as can\nmost easily transpire through the pores of the skin. And from hence it\nis necessary that the rest of the parts, which are not separated, must\naltogether be moved outwards, for the keeping of all places full. But\nthis motion outwards of all parts together must, of necessity, press\nthose parts of the ambient air which are ready to leave their places;\nand therefore all the parts of the body, endeavouring at once that way,\nmake the body swell. Wherefore a possible cause is given of heat from\nthe sun; which was to be done.\n\n[Sidenote: The generation of fire from the sun.]\n\n4. We have now seen how light and heat are generated; heat by the simple\nmotion of the medium, making the parts perpetually change places with\none another; and light by this, that by the same simple motion action is\npropagated in a strait line. But when a body hath its parts so moved,\nthat it sensibly both heats and shines at the same time, then it is that\nwe say fire is generated.\n\nNow by fire I do not understand a body distinct from matter combustible\nor glowing, as wood or iron, but the matter itself, not simply and\nalways, but then only when it shineth and heateth. He, therefore, that\nrenders a cause possible and agreeable to the rest of the phenomena,\nnamely, whence, and from what action, both the _shining_ and _heating_\nproceed, may be thought to have given a possible cause of the generation\nof _fire_.\n\nLet, therefore, A B C (in the first figure) be a sphere, or the portion\nof a sphere, whose centre is D; and let it be transparent and\nhomogeneous, as crystal, glass, or water, and objected to the sun.\nWherefore, the foremost part A B C will, by the simple motion of the\nsun, by which it thrusts forwards the medium, be wrought upon by the\nsunbeams in the strait lines E A, F B, and G C; which strait lines may,\nin respect of the great distance of the sun, be taken for parallels. And\nseeing the medium within the sphere is thicker than the medium without\nit, those beams will be refracted towards the perpendiculars. Let the\nstrait lines E A and G C be produced till they cut the sphere in H and\nI; and drawing the perpendiculars A D and C D, the refracted beams E A\nand G C will of necessity fall, the one between A H and A D, the other\nbetween C I and C D. Let those refracted beams be A K and C L. And\nagain, let the lines D K M and D L N be drawn perpendicular to the\nsphere; and let A K and C L be produced till they meet with the strait\nline B D produced in O. Seeing, therefore, the medium within the sphere\nis thicker than that without it, the refracted line A K will recede\nfurther from the perpendicular K M than K O will recede from the same.\nWherefore K O will fall between the refracted line and the\nperpendicular. Let, therefore, the refracted line be K P, cutting F O in\nP; and for the same reason the strait line L P will be the refracted\nline of the strait line C L. Wherefore, seeing the beams are nothing\nelse but the ways in which the motion is propagated, the motion about P\nwill be so much more vehement than the motion about A B C, by how much\nthe base of the portion A B C is greater than the base of a like portion\nin the sphere, whose centre is P, and whose magnitude is equal to that\nof the little circle about P, which comprehendeth all the beams that are\npropagated from A B C; and this sphere being much less than the sphere A\nB C, the parts of the medium, that is, of the air about P, will change\nplaces with one another with much greater celerity than those about A B\nC. If, therefore, any matter combustible, that is to say, such as may be\neasily dissipated, be placed in P, the parts of that matter, if the\nproportion be great enough between A C and a like portion of the little\ncircle about P, will be freed from their mutual cohesion, and being\nseparated will acquire simple motion. But vehement simple motion\ngenerates in the beholder a phantasm of lucid and hot, as I have before\ndemonstrated of the simple motion of the sun; and therefore the\ncombustible matter which is placed in P will be made lucid and hot, that\nis to say, will be fire. Wherefore I have rendered a possible cause of\nfire; which was to be done.\n\n[Sidenote: The generation of fire from collision.]\n\n5. From the manner by which the sun generateth fire, it is easy to\nexplain the manner by which fire may be generated by the collision of\ntwo flints. For by that collision some of those particles of which the\nstone is compacted, are violently separated and thrown off; and being\nwithal swiftly turned round, the eye is moved by them, as it is in the\ngeneration of light by the sun. Wherefore they shine; and falling upon\nmatter which is already half dissipated, such as is tinder, they\nthoroughly dissipate the parts thereof, and make them turn round. From\nwhence, as I have newly shown, light and heat, that is to say fire, is\ngenerated.\n\n[Sidenote: The cause of light in glow-worms, rotten wood, and the\n Bolognan stone.]\n\n6. The shining of glow-worms, some kinds of rotten wood, and of a kind\nof stone made at Bologna, may have one common cause, namely, the\nexposing of them to the hot sun. We find by experience that the Bologna\nstone shines not, unless it be so exposed; and after it has been exposed\nit shines but for a little time, namely, as long as it retains a certain\ndegree of heat. And the cause may be that the parts, of which it is\nmade, may together with heat have simple motion imprinted in them by the\nsun. Which if it be so, it is necessary that it shine in the dark, as\nlong as there is sufficient heat in it; but this ceasing, it will shine\nno longer. Also we find by experience that in the glow-worm there is a\ncertain thick humour, like the crystalline humour of the eye; which if\nit be taken out and held long enough in one's fingers, and then be\ncarried into the dark, it will shine by reason of the warmth it received\nfrom the fingers; but as soon as it is cold it will cease shining. From\nwhence, therefore, can these creatures have their light, but from lying\nall day in the sunshine in the hottest time of summer? In the same\nmanner, rotten wood, except it grow rotten in the sunshine, or be\nafterwards long enough exposed to the sun, will not shine. That this\ndoth not happen in every worm, nor in all kinds of rotten wood, nor in\nall calcined stones, the cause may be that the parts, of which the\nbodies are made, are different both in motion and figure from the parts\nof bodies of other kinds.\n\n[Sidenote: The cause of light in the concussion of sea water.]\n\n7. Also the sea water shineth when it is either dashed with the strokes\nof oars, or when a ship in its course breaks strongly through it; but\nmore or less, according as the wind blows from different points. The\ncause whereof may be this, that the particles of salt, though they never\nshine in the salt-pits, where they are but slowly drawn up by the sun,\nbeing here beaten up into the air in greater quantities and with more\nforce, are withal made to turn round, and consequently to shine, though\nweakly. I have, therefore, given a possible cause of this phenomenon.\n\n[Sidenote: The cause of flame, sparks, & colliquation.]\n\n8. If such matter as is compounded of hard little bodies be set on fire,\nit must needs be, that, as they fly out in greater or less quantities,\nthe flame which is made by them will be greater or less. And if the\nethereal or fluid part of that matter fly out together with them, their\nmotion will be the swifter, as it is in wood and other things which\nflame with a manifest mixture of wind. When, therefore, these hard\nparticles by their flying out move the eye strongly, they shine bright;\nand a great quantity of them flying out together, they make a great\nshining body. For flame being nothing but an aggregate of shining\nparticles, the greater the aggregate is, the greater and more manifest\nwill be the flame. I have, therefore, shown a possible cause of flame.\nAnd from hence the cause appears evidently, why glass is so easily and\nquickly melted by the small flame of a candle blown, which will not be\nmelted without blowing but by a very strong fire.\n\nNow, if from the same matter there be a part broken off, namely, such a\npart as consisteth of many of the small particles, of this is made a\nspark. For from the breaking off it hath a violent turning round, and\nfrom hence it shines. But though from this matter there fly neither\nflame nor sparks, yet some of the smallest parts of it may be carried\nout as far as to the superficies, and remain there as ashes; the parts\nwhereof are so extremely small, that it cannot any longer be doubted how\nfar nature may proceed in dividing.\n\nLastly, though by the application of fire to this matter there fly\nlittle or nothing from it, yet there will be in the parts an endeavour\nto simple motion; by which the whole body will either be melted, or,\nwhich is a degree of melting, softened. For all motion has some effect\nupon all matter whatsoever, as has been shown at art. 3, chap. XV. Now\nif it be softened to such a degree, as that the stubbornness of the\nparts be exceeded by their gravity, then we say it is melted; otherwise,\nsoftened and made pliant and ductile.\n\nAgain, the matter having in it some particles hard, others ethereal or\nwatery; if, by the application of fire, these latter be called out, the\nformer will thereby come to a more full contact with one another; and,\nconsequently, will not be so easily separated; that is to say, the whole\nbody will be made harder. And this may be the cause why the same fire\nmakes some things soft, others hard.\n\n[Sidenote: The cause why wet hay sometimes burns of its own accord; also\n the cause of lightning.]\n\n9. It is known by experience that if hay be laid wet together in a heap,\nit will after a time begin to smoke, and then burn as it were of itself.\nThe cause whereof seems to be this, that in the air, which is enclosed\nwithin the hay, there are those little bodies, which, as I have\nsupposed, are moved freely with simple motion. But this motion being by\ndegrees hindered more and more by the descending moisture, which at the\nlast fills and stops all the passages, the thinner parts of the air\nascend by penetrating the water; and those hard little bodies, being so\nthrust together that they touch and press one another, acquire stronger\nmotion; till at last by the increased strength of this motion the watery\nparts are first driven outwards, from whence appears vapour; and by the\ncontinued increase of this motion, the smallest particles of the dried\nhay are forced out, and recovering their natural simple motion, they\ngrow hot and shine, that is to say, they are set on fire.\n\nThe same also may be the cause of lightning, which happens in the\nhottest time of the year, when the water is raised up in greatest\nquantity and carried highest. For after the first clouds are raised,\nothers after others follow them; and being congealed above, they happen,\nwhilst some of them ascend and others descend, to fall one upon another\nin such manner, as that in some places all their parts are joined\ntogether, in others they leave hollow spaces between them; and into\nthese spaces, the ethereal parts being forced out by the compressure of\nthe clouds, many of the harder little bodies are so pent together, as\nthey have not the liberty of such motion as is natural to the air.\nWherefore their endeavour grows more vehement, till at last they force\ntheir way through the clouds, sometimes in one place, sometimes in\nanother; and, breaking through with great noise, they move the air\nviolently, and striking our eyes, generate light, that is to say, they\nshine. And this shining is that we call lightning.\n\n[Sidenote: The cause of the force of gunpowder; and what is to be\n ascribed to the coals, what to the brimstone, and what to the\n nitre.]\n\n10. The most common phenomenon proceeding from fire, and yet the most\nadmirable of all others, is the force of gunpowder fired; which being\ncompounded of nitre, brimstone and coals, beaten small, hath from the\ncoals its first taking fire; from the brimstone its nourishment and\nflame, that is to say, light and motion, and from the nitre the\nvehemence of both. Now if a piece of nitre, before it is beaten, be laid\nupon a burning coal, first it melts, and, like water, quencheth that\npart of the coal it toucheth. Then vapour or air, flying out where the\ncoal and nitre join, bloweth the coal with great swiftness and vehemence\non all sides. And from hence it comes to pass, that by two contrary\nmotions, the one, of the particles which go out of the burning coal, the\nother, of those of the ethereal and watery substance of the nitre, is\ngenerated that vehement motion and inflammation. And, lastly, when there\nis no more action from the nitre, that is to say, when the volatile\nparts of the nitre are flown out, there is found about the sides a\ncertain white substance, which being thrown again into the fire, will\ngrow red-hot again, but will not be dissipated, at least unless the fire\nbe augmented. If now a possible cause of this be found out, the same\nwill also be a possible cause why a grain of gunpowder set on fire doth\nexpand itself with such vehement motion, and shine. And it may be caused\nin this manner.\n\nLet the particles, of which nitre consisteth, be supposed to be some of\nthem hard, others watery, and the rest ethereal. Also let the hard\nparticles be supposed to be spherically hollow, like small bubbles, so\nthat many of them growing together may constitute a body, whose little\ncaverns are filled with a substance which is either watery, or ethereal,\nor both. As soon, therefore, as the hard particles are dissipated, the\nwatery and ethereal particles will necessarily fly out; and as they fly,\nof necessity blow strongly the burning coals and brimstone which are\nmingled together; whereupon there will follow a great expansion of\nlight, with vehement flame, and a violent dissipation of the particles\nof the nitre, the brimstone and the coals. Wherefore I have given a\npossible cause of the force of fired gunpowder.\n\nIt is manifest from hence, that for the rendering of the cause why a\nbullet of lead or iron, shot from a piece of ordnance, flies with so\ngreat velocity, there is no necessity to introduce such rarefaction, as,\nby the common definition of it, makes the same matter to have sometimes\nmore, sometimes less quantity; which is inconceivable. For every thing\nis said to be greater or less, as it hath more or less quantity. The\nviolence with which a bullet is thrust out of a gun, proceeds from the\nswiftness of the small particles of the fired powder; at least it may\nproceed from that cause without the supposition of any empty space.\n\n[Sidenote: How heat is caused by attrition.]\n\n11. Besides, by the attrition or rubbing of one body against another, as\nof wood against wood, we find that not only a certain degree of heat,\nbut fire itself is sometimes generated. For such motion is the\nreciprocation of pressure, sometimes one way, sometimes the other; and\nby this reciprocation whatsoever is fluid in both the pieces of wood is\nforced hither and thither; and consequently, to an endeavour of getting\nout; and at last by breaking out makes fire.\n\n[Sidenote: The distinction of light into first, second, &c.]\n\n12. Now light is distinguished into, first, second, third, and so on\ninfinitely. And we call that first light, which is in the first lucid\nbody; as the sun, fire, &c.: second, that which is in such bodies, as\nbeing not transparent are illuminated by the sun; as the moon, a wall,\n&c.: and third, that which is in bodies not transparent, but illuminated\nby second light, &c.\n\n[Sidenote: The causes of the colours we see in looking through a prisma\n of glass, namely, of red, yellow, blue, & violet colour.]\n\n13. _Colour_ is light, but troubled light, namely, such as is generated\nby perturbed motion; as shall be made manifest by the red, yellow, blue\nand purple, which are generated by the interposition of a diaphanous\nprisma, whose opposite bases are triangular, between the light and that\nwhich is enlightened.\n\n[Sidenote: [Discussion of Figure 27.2]]\n\nFor let there be a prisma of glass, or of any other transparent matter\nwhich is of greater density than air; and let the triangle A B C be the\nbase of this prisma. Also let the strait line D E be the diameter of the\nsun's body, having oblique position to the strait line A B; and let the\nsunbeams pass in the lines D A and E B C. And lastly, let the strait\nlines D A and E C be produced indefinitely to F and G. Seeing therefore\nthe strait line D A, by reason of the density of the glass, is refracted\ntowards the perpendicular; let the line refracted at the point A be the\nstrait line A H. And again, seeing the medium below A C is thinner than\nthat above it, the other refraction, which will be made there, will\ndiverge from the perpendicular. Let therefore this second refracted line\nbe A I. Also let the same be done at the point C, by making the first\nrefracted line to be C K, and the second C L. Seeing therefore the cause\nof the refraction in the point A of the strait line of A B is the excess\nof the resistance of the medium in A B above the resistance of the air,\nthere must of necessity be reaction from the point A towards the point\nB; and consequently the medium at A within the triangle A B C will have\nits motion troubled, that is to say, the strait motion in A F and A H\nwill be mixed with the transverse motion between the same A F and A H,\nrepresented by the short transverse lines in the triangle A F H. Again,\nseeing at the point A of the strait line A C there is a second\nrefraction from A H in A I, the motion of the medium will again be\nperturbed by reason of the transverse reaction from A towards C,\nrepresented likewise by the short transverse lines in the triangle A H\nI. And in the same manner there is a double perturbation represented by\nthe transverse lines in the triangles C G K and C K L. But as for the\nlight between A I and C G, it will not be perturbed; because, if there\nwere in all the points of the strait lines A B and A C the same action\nwhich is in the points A and C, then the plane of the triangle C G K\nwould be everywhere coincident with the plane of the triangle A F H; by\nwhich means all would appear alike between A and C. Besides, it is to be\nobserved, that all the reaction at A tends towards the illuminated parts\nwhich are between A and C, and consequently perturbeth the first light.\nAnd on the contrary, that all the reaction at C tends towards the parts\nwithout the triangle or without the prisma A B C, where there is none\nbut second light; and that the triangle A F H shows that perturbation of\nlight which is made in the glass itself; as the triangle A H I shows\nthat perturbation of light which is made below the glass. In like\nmanner, that C G K shows the perturbation of light within the glass; and\nC K L that which is below the glass. From whence there are four divers\nmotions, or four different illuminations or colours, whose differences\nappear most manifestly to the sense in a prisma, whose base is an\nequilateral triangle, when the sunbeams that pass through it are\nreceived upon a white paper. For the triangle A F H appears red to the\nsense; the triangle A H I yellow; the triangle C G K green, and\napproaching to blue; and lastly, the triangle C K L appears purple. It\nis therefore evident that when weak but first light passeth through a\nmore resisting diaphanous body, as glass, the beams, which fall upon it\ntransversely, make redness; and when the same first light is stronger,\nas it is in the thinner medium below the strait line A C, the transverse\nbeams make yellowness. Also when second light is strong, as it is in the\ntriangle C G K, which is nearest to the first light, the transverse\nbeams make greenness; and when the same second light is weaker, as in\nthe triangle C K L, they make a purple colour.\n\n[Sidenote: Why the moon and the stars appear redder in the horizon than\n in the midst of the heaven.]\n\n14. From hence may be deduced a cause, why the moon and stars appear\nbigger and redder near the horizon than in the mid-heaven. For between\nthe eye and the apparent horizon there is more impure air, such as is\nmingled with watery and earthy little bodies, than is between the same\neye and the more elevated part of heaven. But vision is made by beams\nwhich constitute a cone, whose base, if we look upon the moon, is the\nmoon's face, and whose vertex is in the eye; and therefore, many beams\nfrom the moon must needs fall upon little bodies that are without the\nvisual cone, and be by them reflected to the eye. But these reflected\nbeams tend all in lines which are transverse to the visual cone, and\nmake at the eye an angle which is greater than the angle of the cone.\nWherefore, the moon appears greater in the horizon, than when she is\nmore elevated. And because those reflected beams go transversely, there\nwill be generated, by the last article, redness. A possible cause\ntherefore is shown, why the moon as also the stars appear greater and\nredder in the horizon, than in the midst of heaven. The same also may be\nthe cause, why the sun appears in the horizon greater and of a colour\nmore degenerating to yellow, than when he is higher elevated. For the\nreflection from the little bodies between, and the transverse motion of\nthe medium, are still the same. But the light of the sun is much\nstronger than that of the moon; and therefore, by the last article, his\nsplendour must needs by this perturbation degenerate into yellowness.\n\nBut for the generation of these four colours, it is not necessary that\nthe figure of the glass be a prisma; for if it were spherical it would\ndo the same. For in a sphere the sunbeams are twice refracted and twice\nreflected. And this being observed by Des Cartes, and withal that a\nrainbow never appears but when it rains; as also, that the drops of rain\nhave their figures almost spherical; he hath shown from thence the cause\nof the colours in the rainbow; which therefore need not be repeated.\n\n[Sidenote: The cause of whiteness.]\n\n15. _Whiteness_ is light, but light perturbed by the reflections of many\nbeams of light coming to the eye together within a little space. For if\nglass or any other diaphanous body be reduced to very small parts by\ncontusion or concussion, every one of those parts, if the beams of a\nlucid body be from any one point of the same reflected to the eye, will\nrepresent to the beholder an idea or image of the whole lucid body, that\nis to say, a phantasm of white. For the strongest light is the most\nwhite; and therefore many such parts will make many such images.\nWherefore, if those parts lie thick and close together, those many\nimages will appear confusedly, and will by reason of the confused light\nrepresent a white colour. So that from hence may be deduced a possible\ncause, why glass beaten, that is, reduced to powder, looks white. Also\nwhy water and snow are white; they being nothing but a heap of very\nsmall diaphanous bodies, namely, of little bubbles, from whose several\nconvex superficies there are by reflection made several confused\nphantasms of the whole lucid body, that is to say, whiteness. For the\nsame reason, salt and nitre are white, as consisting of small bubbles\nwhich contain within them water and air; as is manifest in nitre, from\nthis, that being thrown into the fire it violently blows the same; which\nsalt also doth, but with less violence. But if a white body be exposed,\nnot to the light of the day, but to that of the fire or of a candle, it\nwill not at the first sight be easily judged whether it be white or\nyellow; the cause whereof may be this, that the light of those things,\nwhich burn and flame, is almost of a middle colour between whiteness and\nyellowness.\n\n[Sidenote: The cause of blackness.]\n\n16. As whiteness is light, so _blackness_ is the privation of light, or\ndarkness. And, from hence it is, first, that all holes, from which no\nlight can be reflected to the eye, appear black. Secondly, that when a\nbody hath little eminent particles erected straight up from the\nsuperficies, so that the beams of light which fall upon them are\nreflected not to the eye but to the body itself, that superficies\nappears black; in the same manner as the sea appears black when ruffled\nby the wind. Thirdly, that any combustible matter is by the fire made to\nlook black before it shines. For the endeavour of the fire being to\ndissipate the smallest parts of such bodies as are thrown into it, it\nmust first raise and erect those parts before it can work their\ndissipation. If, therefore, the fire be put out before the parts are\ntotally dissipated, the coal will appear black; for the parts being only\nerected, the beams of light falling upon them will not be reflected to\nthe eye, but to the coal itself. Fourthly, that burning glasses do more\neasily burn black things than white. For in a white superficies the\neminent parts are convex, like little bubbles; and therefore the beams\nof light, which fall upon them, are reflected every way from the\nreflecting body. But in a black superficies, where the eminent particles\nare more erected, the beams of light falling upon them are all\nnecessarily reflected towards the body itself; and, therefore, bodies\nthat are black are more easily set on fire by the sun beams, than those\nthat are white. Fifthly, that all colours that are made of the mixture\nof white and black proceed from the different position of the particles\nthat rise above the superficies, and their different forms of asperity.\nFor, according to these differences, more or fewer beams of light are\nreflected from several bodies to the eye. But in regard those\ndifferences are innumerable, and the bodies themselves so small that we\ncannot perceive them; the explication and precise determination of the\ncauses of all colours is a thing of so great difficulty, that I dare not\nundertake it.\n\n------------------------------------------------------------------------\n\n[Illustration:\n\n __Vol. I. Lat. & Eng.__\n C. XXVII.\n _Fig. 1-2_]\n]\n\n \u2502Fig 1 \u2502Fig 2 \u2502\n\n------------------------------------------------------------------------\n\n\n -------\n\n\n CHAPTER XXVIII.\n\n OF COLD, WIND, HARD, ICE, RESTITUTION OF\n BODIES BENT, DIAPHANOUS, LIGHTNING AND\n THUNDER; AND OF THE HEADS OF RIVERS.\n\n1. Why breath from the same mouth sometimes heats and sometimes\n cools.\u20142. Wind, and the inconstancy of winds, whence.\u20143 Why there is\n a constant, though not a great wind, from east to west, near the\n equator.\u20144. What is the effect of air pent in between the clouds.\u20145.\n No change from soft to hard, but by motion.\u20146. What is the cause of\n cold near the poles.\u20147.. The cause of ice; and why the cold is more\n remiss in rainy than in clear weather. Why water doth not freeze in\n deep wells as it doth near the superficies of the earth. Why ice is\n not so heavy as water; and why wine is not so easily frozen as\n water.\u20148. Another cause of hardness from the fuller contact of\n atoms; also, how hard things are broken.\u20149. A third cause of\n hardness from heat.\u201410. A fourth cause of hardness from the motion\n of atoms enclosed in a narrow space.\u201411. How hard things are\n softened.\u201412. Whence proceed the spontaneous restitution of things\n bent.\u201413. Diaphanous and opacous, what they are, and whence.\u201414. The\n cause of lightning and thunder.\u201415. Whence it proceeds that clouds\n can fall again after they are once elevated and frozen.\u201416. How it\n could be that the moon was eclipsed, when she was not diametrically\n opposite to the sun.\u201417. By what means many suns may appear at\n once.\u201418. Of the heads of rivers.\n\n\n[Sidenote: Why breath from the same mouth sometimes heats and sometimes\n cools.]\n\n1. As, when the motion of the ambient ethereal substance makes the\nspirits and fluid parts of our bodies tend outwards, we acknowledge\nheat; so, by the endeavour inwards of the same spirits and humours, we\nfeel cold. So that to cool is to make the exterior parts of the body\nendeavour inwards, by a motion contrary to that of calefaction, by which\nthe internal parts are called outwards. He, therefore, that would know\nthe cause of cold, must find by what motion or motions the exterior\nparts of any body endeavour to retire inwards. To begin with those\nphenomena which are the most familiar. There is almost no man but knows,\nthat breath blown strongly, and which comes from the mouth with\nviolence, that is to say, the passage being strait, will cool the hand;\nand that the same breath blown gently, that is to say, through a greater\naperture, will warm the same. The cause of which phenomenon may be this,\nthe breath going out hath two motions; the one, of the whole and direct,\nby which the foremost parts of the hand are driven inwards; the other,\nsimple motion of the small particles of the same breath, which, (as I\nhave shown in the 3rd article of the last chapter, causeth heat.\nAccording, therefore, as either of these motions is predominant, so\nthere is the sense sometimes of cold, sometimes of heat. Wherefore, when\nthe breath is softly breathed out at a large passage, that simple motion\nwhich causeth heat prevaileth, and consequently heat is felt; and when,\nby compressing the lips, the breath is more strongly blown out, then is\nthe direct motion prevalent, which makes us feel cold. For, the direct\nmotion of the breath or air is wind; and all wind cools or diminisheth\nformer heat.\n\n[Sidenote: Wind, and the inconstancy of winds, whence.]\n\n2. And seeing not only great wind, but almost any ventilation and\nstirring of the air, doth refrigerate; the reason of many experiments\nconcerning cold cannot well be given without finding first what are the\ncauses of wind. Now, wind is nothing else but the direct motion of the\nair thrust forwards; which, nevertheless, when many winds concur, may be\ncircular or otherwise indirect, as it is in whirlwinds. Wherefore, in\nthe first place we are to enquire into the causes of winds. Wind is air\nmoved in a considerable quantity, and that either in the manner of\nwaves, which is both forwards and also up and down, or else forwards\nonly.\n\nSupposing, therefore, the air both clear and calm for any time how\nlittle soever, yet, the greater bodies of the world being so disposed\nand ordered as has been said, it will be necessary that a wind presently\narise somewhere. For, seeing that motion of the parts of the air, which\nis made by the simple motion of the sun in his own epicycle, causeth an\nexhalation of the particles of water from the seas and all other moist\nbodies, and those particles make clouds; it must needs follow, that,\nwhilst the particles of water pass upwards, the particles of air, for\nthe keeping of all spaces full, be jostled out on every side, and urge\nthe next particles, and these the next; till having made their circuit,\nthere comes continually so much air to the hinder parts of the earth as\nthere went water from before it. Wherefore, the ascending vapours move\nthe air towards the sides every way; and all direct motion of the air\nbeing wind, they make a wind. And if this wind meet often with other\nvapours which arise in other places, it is manifest that the force\nthereof will be augmented, and the way or course of it changed. Besides,\naccording as the earth, by its diurnal motion, turns sometimes the\ndrier, sometimes the moister part towards the sun, so sometimes a\ngreater, sometimes a less, quantity of vapours will be raised; that is\nto say, sometimes there will be a less, sometimes a greater wind.\nWherefore, I have rendered a possible cause of such winds as are\ngenerated by vapours; and also of their inconstancy.\n\nFrom hence it follows that these winds cannot be made in any place,\nwhich is higher than that to which vapours may ascend. Nor is that\nincredible which is reported of the highest mountains, as the Peak of\nTeneriffe and the Andes of Peru, namely, that they are not at all\ntroubled with these inconstant winds. And if it were certain that\nneither rain nor snow were ever seen in the highest tops of those\nmountains, it could not be doubted but that they are higher than any\nplace to which vapours use to ascend.\n\n[Sidenote: Why there is a constant, though not a great wind, from east\n to west, near the equator.]\n\n3. Nevertheless, there may be wind there, though not that which is made\nby the ascent of vapours, yet a less and more constant wind, like the\ncontinued blast of a pair of bellows, blowing from the east. And this\nmay have a double cause; the one, the diurnal motion of the earth; the\nother, its simple motion in its own epicycle. For these mountains being,\nby reason of their height, more eminent than all the rest of the parts\nof the earth, do by both these motions drive the air from the west\neastwards. To which, though the diurnal motion contribute but little,\nyet seeing I have supposed that the simple motion of the earth, in its\nown epicycle, makes two revolutions in the same time in which the\ndiurnal motion makes but one, and that the semidiameter of the epicycle\nis double to the semidiameter of the diurnal conversion, the motion of\nevery point of the earth in its own epicycle will have its velocity\nquadruple to that of the diurnal motion; so that by both these motions\ntogether, the tops of those hills will sensibly be moved against the\nair; and consequently a wind will be felt. For whether the air strike\nthe sentient, or the sentient the air, the perception of motion will be\nthe same. But this wind, seeing it is not caused by the ascent of\nvapours, must necessarily be very constant.\n\n[Sidenote: What is the effect of air pent in between the clouds.]\n\n4. When one cloud is already ascended into the air, if another cloud\nascend towards it, that part of the air, which is intercepted between\nthem both, must of necessity be pressed out every way. Also when both of\nthem, whilst the one ascends and the other either stays or descends,\ncome to be joined in such manner as that the ethereal substance be shut\nwithin them on every side, it will by this compression also go out by\npenetrating the water. But in the meantime, the hard particles, which\nare mingled with the air and are agitated, as I have supposed, with\nsimple motion, will not pass through the water of the clouds, but be\nmore straitly compressed within their cavities. And this I have\ndemonstrated at the 4th and 5th articles of chapter XXII. Besides,\nseeing the globe of the earth floateth in the air which is agitated by\nthe sun's motion, the parts of the air resisted by the earth will spread\nthemselves every way upon the earth's superficies; as I have shown at\nthe 8th article of chapter XXI.\n\n[Sidenote: No change from soft to hard, but by motion.]\n\n5. We perceive a body to be hard, from this, that, when touching it, we\nwould thrust forwards that part of the same which we touch, we cannot do\nit otherwise than by thrusting forwards the whole body. We may indeed\neasily and sensibly thrust forwards any particle of the air or water\nwhich we touch, whilst yet the rest of its parts remain to sense\nunmoved. But we cannot do so to any part of a stone. Wherefore I define\na _hard_ body to be that whereof no part can be sensibly moved, unless\nthe whole be moved. Whatsoever therefore is soft or fluid, the same can\nnever be made hard but by such motion as makes many of the parts\ntogether stop the motion of some one part, by resisting the same.\n\n[Sidenote: What it the cause of cold near the poles.]\n\n6. Those things premised, I shall show a possible cause why there is\ngreater cold near the poles of the earth, than further from them. The\nmotion of the sun between the tropics, driving the air towards that part\nof the earth's superficies which is perpendicularly under it, makes it\nspread itself every way; and the velocity of this expansion of the air\ngrows greater and greater, as the superficies of the earth comes to be\nmore and more straitened, that is to say, as the circles which are\nparallel to the equator come to be less and less. Wherefore this\nexpansive motion of the air drives before it the parts of the air, which\nare in its way, continually towards the poles more and more strongly, as\nits force comes to be more and more united, that is to say, as the\ncircles which are parallel to the equator are less and less; that is, so\nmuch the more, by how much they are nearer to the poles of the earth. In\nthose places, therefore, which are nearer to the poles, there is greater\ncold than in those which are more remote from them. Now this expansion\nof the air upon the superficies of the earth, from east to west, doth,\nby reason of the sun's perpetual accession to the places which are\nsuccessively under it, make it cold at the time of the sun's rising and\nsetting; but as the sun comes to be continually more and more\nperpendicular to those cooled places, so by the heat, which is generated\nby the supervening simple motion of the sun, that cold is again\nremitted; and can never be great, because the action by which it was\ngenerated is not permanent. Wherefore I have rendered a possible cause\nof cold in those places that are near the poles, or where the obliquity\nof the sun is great.\n\n[Sidenote: The cause of ice; and why the cold is more remiss in rainy\n than in clear weather. Why water doth not freeze in deep\n wells, as it doth near the superficies of the earth. Why ice\n is not so heavy as water; and why wine is not so easily\n frozen as water.]\n\n7. How water may be congealed by cold, may be explained in this manner.\nLet A (in figure 1) represent the sun, and B the earth. A will therefore\nbe much greater than B. Let E F be in the plane of the equinoctial; to\nwhich let G H, I K, and L C be parallel. Lastly, let C and D be the\npoles of the earth. The air, therefore, by its action in those\nparallels, will rake the superficies of the earth; and that with motion\nso much the stronger, by how much the parallel circles towards the poles\ngrow less and less. From whence must arise a wind, which will force\ntogether the uppermost parts of the water, and withal raise them a\nlittle, weakening their endeavour towards the centre of the earth. And\nfrom their endeavour towards the centre of the earth, joined with the\nendeavour of the said wind, the uppermost parts of the water will be\npressed together and coagulated, that is to say, the top of the water\nwill be skinned over and hardened. And so again, the water next the top\nwill be hardened in the same manner, till at length the ice be thick.\nAnd this ice, being now compacted of little hard bodies, must also\ncontain many particles of air received into it.\n\nAs rivers and seas, so also in the same manner may the clouds be frozen.\nFor when, by the ascending and descending of several clouds at the same\ntime, the air intercepted between them is by compression forced out, it\nrakes, and by little and little hardens them. And though those small\ndrops, which usually make clouds, be not yet united into greater bodies,\nyet the same wind will be made; and by it, as water is congealed into\nice, so will vapours in the same manner be congealed into snow. From the\nsame cause it is that ice may be made by art, and that not far from the\nfire. For it is done by the mingling of snow and salt together, and by\nburying in it a small vessel full of water. Now while the snow and salt,\nwhich have in them a great deal of air, are melting, the air, which is\npressed out every way in wind, rakes the sides of the vessel; and as the\nwind by its motion rakes the vessel, so the vessel by the same motion\nand action congeals the water within it.\n\nWe find by experience, that cold is always more remiss in places where\nit rains, or where the weather is cloudy, things being alike in all\nother respects, than where the air is clear. And this agreeth very well\nwith what I have said before. For in clear weather, the course of the\nwind which, as I said even now, rakes the superficies of the earth, as\nit is free from all interruption, so also it is very strong. But when\nsmall drops of water are either rising or falling, that wind is\nrepelled, broken, and dissipated by them; and the less the wind is, the\nless is the cold.\n\nWe find also by experience, that in deep wells the water freezeth not so\nmuch as it doth upon the superficies of the earth. For the wind, by\nwhich ice is made, entering into the earth by reason of the laxity of\nits parts, more or less, loseth some of its force, though not much. So\nthat if the well be not deep, it will freeze; whereas if it be so deep,\nas that the wind which causeth cold cannot reach it, it will not freeze.\n\nWe find moreover by experience, that ice is lighter than water. The\ncause whereof is manifest from that which I have already shown, namely,\nthat air is received in and mingled with the particles of the water\nwhilst it is congealing.\n\nLastly, wine is not so easily congealed as water, because in wine there\nare particles, which, being not fluid, are moved very swiftly, and by\ntheir motion congelation is retarded. But if the cold prevail against\nthis motion, then the outermost parts of the wine will be first frozen,\nand afterwards the inner parts; whereof this is a sign, that the wine\nwhich remains unfrozen in the midst will be very strong.\n\n[Sidenote: Another cause of hardness from the fuller contact of atoms.\n Also how hard things are broken.]\n\n8. We have seen one way of making things hard, namely, by congelation.\nAnother way is thus. Having already supposed that innumerable atoms,\nsome harder than others and that have several simple motions of their\nown, are intermingled with the ethereal substance; it follows\nnecessarily from hence, that by reason of the fermentation of the whole\nair, of which I have spoken in chapter XXI, some of those atoms meeting\nwith others will cleave together, by applying themselves to one another\nin such manner as is agreeable to their motions and mutual contacts;\nand, seeing there is no vacuum, cannot be pulled asunder, but by so much\nforce as is sufficient to overcome their hardness.\n\nNow there are innumerable degrees of hardness. As for example, there is\na degree of it in water, as is manifest from this, that upon a plane it\nmay be drawn any way at pleasure by one's finger. There is a greater\ndegree of it in clammy liquors, which, when they are poured out, do in\nfalling downwards dispose themselves into one continued thread; which\nthread, before it be broken, will by little and little diminish its\nthickness, till at last it be so small, as that it seems to break only\nin a point; and in their separation the external parts break first from\none another, and then the more internal parts successively one after\nanother. In wax there is yet a greater degree of hardness. For when we\nwould pull one part of it from another, we first make the whole mass\nslenderer, before we can pull it asunder. And how much the harder\nanything is which we would break, so much the more force we must apply\nto it. Wherefore, if we go on to harder things, as ropes, wood, metals,\nstones, &c., reason prompteth us to believe that the same, though not\nalways sensibly, will necessarily happen; and that even the hardest\nthings are broken asunder in the same manner, namely, by solution of\ntheir continuity begun in the outermost superficies, and proceeding\nsuccessively to the innermost parts. In like manner, when the parts of\nbodies are to be separated, not by pulling them asunder, but by breaking\nthem, the first separation will necessarily be in the convex superficies\nof the bowed part of the body, and afterwards in the concave\nsuperficies. For in all bowing there is in the convex superficies an\nendeavour in the parts to go one from another, and in the concave\nsuperficies to penetrate one another.\n\nThis being well understood, a reason may be given how two bodies, which\nare contiguous in one common superficies, may by force be separated\nwithout the introduction of vacuum; though Lucretius thought otherwise,\nbelieving that such separation was a strong establishment of vacuum. For\na marble pillar being made to hang by one of its bases, if it be long\nenough, it will by its own weight be broken asunder; and yet it will not\nnecessarily follow that there should be vacuum, seeing the solution of\nits continuity may begin in the circumference, and proceed successively\nto the midst thereof.\n\n[Sidenote: A third cause of hardness, from heat.]\n\n9. Another cause of hardness in some things may be in this manner. If a\nsoft body consist of many hard particles, which by the intermixture of\nmany other fluid particles cohere but loosely together, those fluid\nparts, as hath been shown in the last article of chapter XXI, will be\nexhaled; by which means each hard particle will apply itself to the next\nto it according to a greater superficies, and consequently they will\ncohere more closely to one another, that is to say, the whole mass will\nbe made harder.\n\n[Sidenote: A fourth cause of hardness, from the motion of atoms enclosed\n in a narrow space.]\n\n10. Again, in some things hardness may be made to a certain degree in\nthis manner. When any fluid substance hath in it certain very small\nbodies intermingled, which, being moved with simple motion of their own,\ncontribute like motion to the parts of the fluid substance, and this be\ndone in a small enclosed space, as in the hollow of a little sphere, or\na very slender pipe, if the motion be vehement and there be a great\nnumber of these small enclosed bodies, two things will happen; the one,\nthat the fluid substance will have an endeavour of dilating itself at\nonce every way; the other, that if those small bodies can nowhere get\nout, then from their reflection it will follow, that the motion of the\nparts of the enclosed fluid substance, which was vehement before, will\nnow be much more vehement. Wherefore, if any one particle of that fluid\nsubstance should be touched and pressed by some external movent, it\ncould not yield but by the application of very sensible force. Wherefore\nthe fluid substance, which is enclosed and so moved, hath some degree of\nhardness. Now, greater and less degree of hardness depends upon the\nquantity and velocity of those small bodies, and upon the narrowness of\nthe place both together.\n\n[Sidenote: How hard things are softened.]\n\n11. Such things as are made hard by sudden heat, namely such as are\nhardened by fire, are commonly reduced to their former soft form by\nmaceration. For fire hardens by evaporation, and therefore if the\nevaporated moisture be restored again, the former nature and form is\nrestored together with it. And such things as are frozen with cold, if\nthe wind by which they were frozen change into the opposite quarter,\nthey will be unfrozen again, unless they have gotten a habit of new\nmotion or endeavour by long continuance in that hardness. Nor is it\nenough to cause thawing, that there be a cessation of the freezing wind;\nfor the taking away of the cause doth not destroy a produced effect; but\nthe thawing also must have its proper cause, namely, a contrary wind, or\nat least a wind opposite in some degree. And this we find to be true by\nexperience. For, if ice be laid in a place so well enclosed that the\nmotion of the air cannot get to it, that ice will remain unchanged,\nthough the place be not sensibly cold.\n\n[Sidenote: Whence proceeds the spontaneous restitution of things bent.]\n\n12. Of hard bodies, some may manifestly be bowed; others not, but are\nbroken in the very first moment of their bending. And of such bodies as\nmay manifestly be bended, some being bent, do, as soon as ever they are\nset at liberty, restore themselves to their former posture; others\nremain still bent. Now if the cause of this restitution be asked, I say,\nit may be in this manner, namely, that the particles of the bended body,\nwhilst it is held bent, do nevertheless retain their motion; and by this\nmotion they restore it as soon as the force is removed by which it was\nbent. For when any thing is bent, as a plate of steel, and, as soon as\nthe force is removed, restores itself again, it is evident that the\ncause of its restitution cannot be referred to the ambient air; nor can\nit be referred to the removal of the force by which it was bent; for in\nthings that are at rest the taking away of impediments is not a\nsufficient cause of their future motion; there being no other cause of\nmotion, but motion. The cause therefore of such restitution is in the\nparts of the steel itself. Wherefore, whilst it remains bent, there is\nin the parts, of which it consisteth, some motion though invisible; that\nis to say, some endeavour at least that way by which the restitution is\nto be made; and therefore this endeavour of all the parts together is\nthe first beginning of restitution; so that the impediment being\nremoved, that is to say, the force by which it was held bent, it will be\nrestored again. Now the motion of the parts, by which this done, is that\nwhich I called simple motion, or motion returning into itself. When\ntherefore in the bending of a plate the ends are drawn together, there\nis on one side a mutual compression of the parts; which compression is\none endeavour opposite to another endeavour: and on the other side a\ndivulsion of the parts. The endeavour therefore of the parts on one side\ntends to the restitution of the plate from the middle towards the ends;\nand on the other side, from the ends towards the middle. Wherefore the\nimpediment being taken away, this endeavour, which is the beginning of\nrestitution, will restore the plate to its former posture. And thus I\nhave given a possible cause why some bodies, when they are bent, restore\nthemselves again; which was to be done.\n\nAs for stones, seeing they are made by the accretion of many very hard\nparticles within the earth; which particles have no great coherence,\nthat is to say, touch one another in small latitude, and consequently\nadmit many particles of air; it must needs be that, in bending of them,\ntheir internal parts will not easily be compressed, by reason of their\nhardness. And because their coherence is not firm, as soon as the\nexternal hard particles are disjoined, the ethereal parts will\nnecessarily break out, and so the body will suddenly be broken.\n\n[Sidenote: Diaphanous and opacous, what they are, and whence.]\n\n13. Those bodies are called _diaphanous_, upon which, whilst the beams\nof a lucid body do work, the action of every one of those beams is\npropagated in them in such manner, as that they still retain the same\norder amongst themselves, or the inversion of that order; and therefore\nbodies, which are perfectly diaphanous, are also perfectly homogeneous.\nOn the contrary, an _opacous_ body is that, which, by reason of its\nheterogeneous nature, doth by innumerable reflections and refractions in\nparticles of different figures and unequal hardness, weaken the beams\nthat fall upon it before they reach the eye. And of diaphanous bodies,\nsome are made such by nature from the beginning; as the substance of the\nair, and of the water, and perhaps also some parts of stones, unless\nthese also be water that has been long congealed. Others are made so by\nthe power of heat, which congregates homogeneous bodies. But such, as\nare made diaphanous in this manner, consist of parts which were formerly\ndiaphanous.\n\n[Sidenote: The cause of lightning and thunder.]\n\n14. In what manner clouds are made by the motion of the sun, elevating\nthe particles of water from the sea and other moist places, hath been\nexplained in chapter XXVI. Also how clouds come to be frozen, hath been\nshown above at the 7th article. Now from this, that air may be enclosed\nas it were in caverns, and pent together more and more by the meeting of\nascending and descending clouds, may be deduced a possible cause of\n_thunder_ and _lightning_. For seeing the air consists of two parts, the\none ethereal, which has no proper motion of its own, as being a thing\ndivisible into the least parts; the other hard, namely, consisting of\nmany hard atoms, which have every one of them a very swift simple motion\nof its own: whilst the clouds by their meeting do more and more straiten\nsuch cavities as they intercept, the ethereal parts will penetrate and\npass through their watery substance; but the hard parts will in the\nmeantime be the more thrust together, and press one another; and\nconsequently, by reason of their vehement motions, they will have an\nendeavour to rebound from each other. Whensoever, therefore, the\ncompression is great enough, and the concave parts of the clouds are,\nfor the cause I have already given, congealed into ice, the cloud will\nnecessarily be broken; and this breaking of the cloud produceth the\nfirst clap of thunder. Afterwards the air, which was pent in, having now\nbroken through, makes a concussion of the air without, and from hence\nproceeds the roaring and murmur which follows; and both the first clap\nand the murmur that follows it make that noise which is called\n_thunder_. Also, from the same air breaking through the clouds and with\nconcussion falling upon the eye, proceeds that action upon our eye,\nwhich causeth in us a perception of that light, which we call\n_lightning_. Wherefore I have given a possible cause of thunder and\nlightning.\n\n[Sidenote: Whence it proceeds that clouds can fall again, after they are\n once elevated and frozen.]\n\n15. But if the vapours, which are raised into clouds, do run together\nagain into water or be congealed into ice, from whence is it, seeing\nboth ice and water are heavy, that they are sustained in the air? Or\nrather, what may the cause be, that being once elevated, they fall down\nagain? For there is no doubt but the same force which could carry up\nthat water, could also sustain it there. Why therefore being once\ncarried up, doth it fall again? I say it proceeds from the same simple\nmotion of the sun, both that vapours are forced to ascend, and that\nwater gathered into clouds is forced to descend. For in chapter XXI,\narticle 11, I have shown how vapours are elevated; and in the same\nchapter, article 5, I have also shown how by the same motion homogeneous\nbodies are congregated, and heterogeneous dissipated; that is to say,\nhow such things, as have a like nature to that of the earth, are driven\ntowards the earth; that is to say, what is the cause of the descent of\nheavy bodies. Now if the action of the sun be hindered in the raising of\nvapours, and be not at all hindered in the casting of them down, the\nwater will descend. But a cloud cannot hinder the action of the sun in\nmaking things of an earthly nature descend to the earth, though it may\nhinder it in making vapours ascend. For the lower part of a thick cloud\nis so covered by its upper part, as that it cannot receive that action\nof the sun by which vapours are carried up; because vapours are raised\nby the perpetual fermentation of the air, or by the separating of its\nsmallest parts from one another, which is much weaker when a thick cloud\nis interposed, than when the sky is clear. And therefore, whensoever a\ncloud is made thick enough, the water, which would not descend before,\nwill then descend, unless it be kept up by the agitation of the wind.\nWherefore I have rendered a possible cause, both why the clouds may be\nsustained in the air, and also why they may fall down again to the\nearth; which was propounded to be done.\n\n[Sidenote: How it could be that the moon was eclipsed, when she was not\n diametrically opposite to the sun.]\n\n16. Granting that the clouds may be frozen, it is no wonder if the moon\nhave been seen eclipsed at such time as she hath been almost two degrees\nabove the horizon, the sun at the same time appearing in the horizon;\nfor such an eclipse was observed by M\u00e6stlin, at Tubingen, in the year\n1590. For it might happen that a frozen cloud was then interposed\nbetween the sun and the eye of the observer. And if it were so, the sun,\nwhich was then almost two degrees below the horizon, might appear to be\nin it, by reason of the passing of his beams through the ice. And it is\nto be noted that those, that attribute such refractions to the\natmosphere, cannot attribute to it so great a refraction as this.\nWherefore not the atmosphere, but either water in a continued body, or\nelse ice, must be the cause of that refraction.\n\n[Sidenote: By what means many suns may appear at once.]\n\n17. Again, granting that there may be ice in the clouds, it will be no\nlonger a wonder that many suns have sometimes appeared at once. For\nlooking-glasses may be so placed, as by reflections to show the same\nobject in many places. And may not so many frozen clouds serve for so\nmany looking-glasses? And may they not be fitly disposed for that\npurpose? Besides, the number of appearances may be increased by\nrefractions also; and therefore it would be a greater wonder to me, if\nsuch phenomena as these should never happen.\n\nAnd were it not for that one phenomenon of the new star which was seen\nin Cassiopea, I should think comets were made in the same manner,\nnamely, by vapours drawn not only from the earth but from the rest of\nthe planets also, and congealed into one continued body. For I could\nvery well from hence give a reason both of their hair, and of their\nmotions. But seeing that star remained sixteen whole months in the same\nplace amongst the fixed stars, I cannot believe the matter of it was\nice. Wherefore I leave to others the disquisition of the cause of\ncomets; concerning which nothing that hath hitherto been published,\nbesides the bare histories of them, is worth considering.\n\n[Sidenote: Of the heads of rivers.]\n\n18. The heads of rivers may be deduced from rain-water, or from melted\nsnows, very easily; but from other causes, very hardly, or not at all.\nFor both rain-water and melted snows run down the descents of mountains;\nand if they descend only by the outward superficies, the showers or\nsnows themselves may be accounted the springs or fountains; but if they\nenter the earth and descend within it, then, wheresoever they break out,\nthere are their springs. And as these springs make small streams, so,\nmany small streams running together make rivers. Now, there was never\nany spring found, but where the water which flowed to it, was either\nfurther, or at least as far from the centre of the earth, as the spring\nitself. And whereas it has been objected by a great philosopher, that in\nthe top of Mount Cenis, which parts Savoy from Piedmont, there springs a\nriver which runs down by Susa; it is not true. For there are above that\nriver, for two miles length, very high hills on both sides, which are\nalmost perpetually covered with snow; from which innumerable little\nstreams running down do manifestly supply that river with water\nsufficient for its magnitude.\n\n\n -------\n\n\n CHAP. XXIX.\n\n OF SOUND, ODOUR, SAVOUR, AND TOUCH.\n\n1.. The definition of sound, and the distinctions of sounds.\u20142. The\n cause of the degrees of sounds.\u20143. The difference between sounds\n acute and grave.\u20144. The difference between clear and hoarse sounds,\n whence.\u20145. The sound of thunder and of a gun, whence it proceeds.\u20146.\n Whence it is that pipes, by blowing into them, have a clear\n sound.\u20147. Of reflected sound.\u20148. From whence it is that sound is\n uniform and lasting.\u20149. How sound may be helped and hindered by the\n wind.\u201410. Not only air, but other bodies how hard soever they be,\n convey sound.\u201411. The causes of grave and acute sounds, and of\n concent.\u201412. Phenomena for smelling.\u201413. The first organ and the\n generation of smelling.\u201414. How it is helped by heat and by\n wind.\u201415. Why such bodies are least smelt, which have least\n intermixture of air in them.\u201416. Why odorous things become more\n odorous by being bruised.\u201417. The first organ of tasting; and why\n some savours cause nauseousness.\u201418. The first organ of feeling; and\n how we come to the knowledge of such objects as are common to the\n touch and other senses.\n\n\n[Sidenote: The definition of sound, and the distinction of sounds.]\n\n1. SOUND _is sense generated by the action of the medium, when its\nmotion reacheth the ear and the rest of the organs of sense_. Now, the\nmotion of the medium is not the sound itself, but the cause of it. For\nthe phantasm which is made in us, that is to say, the reaction of the\norgan, is properly that which we call _sound_.\n\nThe principal distinctions of sounds are these; first, that one sound is\nstronger, another weaker. Secondly, that one is more grave, another more\nacute. Thirdly, that one is clear, another hoarse. Fourthly, that one is\nprimary, another derivative. Fifthly, that one is uniform, another not.\nSixthly, that one is more durable, another less durable. Of all which\ndistinctions the members may be subdistinguished into parts\ndistinguishable almost infinitely. For the variety of sounds seems to be\nnot much less than that of colours.\n\nAs vision, so hearing is generated by the motion of the medium, but not\nin the same manner. For sight is from pressure, that is, from an\nendeavour; in which there is no perceptible progression of any of the\nparts of the medium; but one part urging or thrusting on another\npropagateth that action successively to any distance whatsoever; whereas\nthe motion of the medium, by which sound is made, is a stroke. For when\nwe hear, the drum of the ear, which is the first organ of hearing, is\nstricken; and the drum being stricken, the _pia mater_ is also shaken,\nand with it the arteries which are inserted into it; by which the action\nbeing propagated to the heart itself, by the reaction of the heart a\nphantasm is made which we call sound; and because the reaction tendeth\noutwards, we think it is without.\n\n[Sidenote: The cause of the degrees of sounds.]\n\n2. And seeing the effects produced by motion are greater or less, not\nonly when the velocity is greater or less, but also when the body hath\ngreater or less magnitude though the velocity be the same; a sound may\nbe greater or less both these ways. And because neither the greatest nor\nthe least magnitude or velocity can be given, it may happen that either\nthe motion may be of so small velocity, or the body itself of so small\nmagnitude, as to produce no sound at all; or either of them may be so\ngreat, as to take away the faculty of sense by hurting the organ.\n\nFrom hence may be deduced possible causes of the strength and weakness\nof sounds in the following phenomena.\n\nThe first whereof is this, that if a man speak through a trunk which\nhath one end applied to the mouth of the speaker, and the other to the\near of the hearer, the sound will come stronger than it would do through\nthe open air. And the cause, not only the possible, but the certain and\nmanifest cause is this, that the air which is moved by the first breath\nand carried forwards in the trunk, is not diffused as it would be in the\nopen air, and is consequently brought to the ear almost with the same\nvelocity with which it was first breathed out. Whereas, in the open air,\nthe first motion diffuseth itself every way into circles, such as are\nmade by the throwing of a stone into a standing water, where the\nvelocity grows less and less as the undulation proceeds further and\nfurther from the beginning of its motion.\n\nThe second is this, that if the trunk be short, and the end which is\napplied to the mouth be wider than that which is applied to the ear,\nthus also the sound will be stronger than if it were made in the open\nair. And the cause is the same, namely, that by how much the wider end\nof the trunk is less distant from the beginning of the sound, by so much\nthe less is the diffusion.\n\nThe third, that it is easier for one, that is within a chamber, to hear\nwhat is spoken without, than for him, that stands without, to hear what\nis spoken within. For the windows and other inlets of the moved air are\nas the wide end of the trunk. And for this reason some creatures seem to\nhear the better, because nature has bestowed upon them wide and\ncapacious ears.\n\nThe fourth is this, that though he, which standeth upon the sea-shore,\ncannot hear the collision of the two nearest waves, yet nevertheless he\nhears the roaring of the whole sea. And the cause seems to be this, that\nthough the several collisions move the organ, yet they are not severally\ngreat enough to cause sense; whereas nothing hinders but that all of\nthem together may make sound.\n\n[Sidenote: The difference between sounds acute and grave.]\n\n3. That bodies when they are stricken do yield some a more grave, others\na more acute sound, the cause may consist in the difference of the times\nin which the parts stricken and forced out of their places return to the\nsame places again. For in some bodies, the restitution of the moved\nparts is quick, in others slow. And this also may be the cause, why the\nparts of the organ, which are moved by the medium, return to their rest\nagain, sometimes sooner, sometimes later. Now, by how much the\nvibrations or the reciprocal motions of the parts are more frequent, by\nso much doth the whole sound made at the same time by one stroke consist\nof more, and consequently of smaller parts. For what is acute in sound,\nthe same is subtle in matter; and both of them, namely acute sound and\nsubtle matter, consist of very small parts, that of time, and this of\nthe matter itself.\n\nThe third distinction of sounds cannot be conceived clearly enough by\nthe names I have used of _clear_ and _hoarse_, nor by any other that I\nknow; and therefore it is needful to explain them by examples. When I\nsay hoarse, I understand whispering and hissing, and whatsoever is like\nto these, by what appellation soever it be expressed. And sounds of this\nkind seem to be made by the force of some strong wind, raking rather\nthan striking such hard bodies as it falls upon. On the contrary, when I\nuse the word clear, I do not understand such a sound as may be easily\nand distinctly heard; for so whispers would be clear; but such as is\nmade by somewhat that is broken, and such as is clamour, tinkling, the\nsound of a trumpet, &c. and to express it significantly in one word,\nnoise. And seeing no sound is made but by the concourse of two bodies at\nthe least, by which concourse it is necessary that there be as well\nreaction as action, that is to say, one motion opposite to another; it\nfollows that according as the proportion between those two opposite\nmotions is diversified, so the sounds which are made will be different\nfrom one another. And whensoever the proportion between them is so\ngreat, as that the motion of one of the bodies be insensible if compared\nwith the motion of the other, then the sound will not be of the same\nkind; as when the wind falls very obliquely upon a hard body, or when a\nhard body is carried swiftly through the air; for then there is made\nthat sound which I call a hoarse sound, in Greek \u03c3\u03c5\u03c1\u03b9\u03b3\u03bc\u03bf\u03c2. Therefore the\nbreath blown with violence from the mouth makes a hissing, because in\ngoing out it rakes the superficies of the lips, whose reaction against\nthe force of the breath is not sensible. And this is the cause why the\nwinds have that hoarse sound. Also if two bodies, how hard soever, be\nrubbed together with no great pressure, they make a hoarse sound. And\nthis hoarse sound, when it is made, as I have said, by the air raking\nthe superficies of a hard body, seemeth to be nothing but the dividing\nof the air into innumerable and very small files. For the asperity of\nthe superficies doth, by the eminences of its innumerable parts, divide\nor cut in pieces the air that slides upon it.\n\n[Sidenote: The difference between clear & hoarse sounds, whence.]\n\n4. _Noise_, or that which I call clear sound, is made two ways; one, by\ntwo hoarse sounds made by opposite motions; the other, by collision, or\nby the sudden pulling asunder of two bodies, whereby their small\nparticles are put into commotion, or being already in commotion suddenly\nrestore themselves again; which motion, making impression upon the\nmedium, is propagated to the organ of hearing. And seeing there is in\nthis collision or divulsion an endeavour in the particles of one body,\nopposite to the endeavour of the particles of the other body, there will\nalso be made in the organ of hearing a like opposition of endeavours,\nthat is to say, of motions; and consequently the sound arising from\nthence will be made by two opposite motions, that is to say, by two\nopposite hoarse sounds in one and the same part of the organ. For, as I\nhave already said, a hoarse sound supposeth the sensible motion of but\none of the bodies. And this opposition of motions in the organ is the\ncause why two bodies make a noise, when they are either suddenly\nstricken against one another, or suddenly broken asunder.\n\n[Sidenote: The sound of thunder and of a gun, whence it proceeds.]\n\n5. This being granted, and seeing withal that thunder is made by the\nvehement eruption of the air out of the cavities of congealed clouds,\nthe cause of the great noise or clap may be the sudden breaking asunder\nof the ice. For in this action it is necessary that there be not only a\ngreat concussion of the small particles of the broken parts, but also\nthat this concussion, by being communicated to the air, be carried to\nthe organ of hearing, and make impression upon it. And then, from the\nfirst reaction of the organ proceeds that first and greatest sound,\nwhich is made by the collision of the parts whilst they restore\nthemselves. And seeing there is in all concussion a reciprocation of\nmotion forwards and backwards in the parts stricken; for opposite\nmotions cannot extinguish one another in an instant, as I have shown in\nthe 11th article of chapter VIII; it follows necessarily that the sound\nwill both continue, and grow weaker and weaker, till at last the action\nof the reciprocating air grow so weak, as to be imperceptible. Wherefore\na possible cause is given both of the first fierce noise of the thunder,\nand also of the murmur that follows it.\n\nThe cause of the great sound from a discharged piece of ordnance is like\nthat of a clap of thunder. For the gunpowder being fired doth, in its\nendeavour to go out, attempt every way the sides of the metal in such\nmanner, as that it enlargeth the circumference all along, and withal\nshorteneth the axis; so that whilst the piece of ordnance is in\ndischarging, it is made both wider and shorter than it was before; and\ntherefore also presently after it is discharged its wideness will be\ndiminished, and its length increased again by the restitution of all the\nparticles of the matter, of which it consisteth, to their former\nposition. And this is done with such motions of the parts, as are not\nonly very vehement, but also opposite to one another; which motions,\nbeing communicated to the air, make impression upon the organ, and by\nthe reaction of the organ create a sound, which lasteth for some time;\nas I have already shown in this article.\n\nI note by the way, as not belonging to this place, that the possible\ncause why a gun recoils when it is shot off, may be this; that being\nfirst swollen by the force of the fire, and afterwards restoring itself,\nfrom this restitution there proceeds an endeavour from all the sides\ntowards the cavity; and consequently this endeavour is in those parts\nwhich are next the breech; which being not hollow, but solid, the effect\nof the restitution is by it hindered and diverted into the length; and\nby this means both the breech and the whole gun is thrust backwards; and\nthe more forcibly by how much the force is greater, by which the part\nnext the breech is restored to its former posture, that is to say, by\nhow much the thinner is that part. The cause, therefore, why guns\nrecoil, some more some less, is the difference of their thickness\ntowards the breech; and the greater that thickness is, the less they\nrecoil; and contrarily.\n\n[Sidenote: Whence it is that pipes, by blowing into them, have a clear\n sound.]\n\n6. Also the cause why the sound of a pipe, which is made by blowing into\nit, is nevertheless clear, is the same with that of the sound which is\nmade by collision. For if the breath, when it is blown into a pipe, do\nonly rake its concave superficies, or fall upon it with a very sharp\nangle of incidence, the sound will not be clear, but hoarse. But if the\nangle be great enough, the percussion, which is made against one of the\nhollow sides, will be reverberated to the opposite side; and so\nsuccessive repercussions will be made from side to side, till at last\nthe whole concave superficies of the pipe be put into motion; which\nmotion will be reciprocated, as it is in collision; and this\nreciprocation being propagated to the organ, from the reaction of the\norgan will arise a clear sound, such as is made by collision, or by\nbreaking asunder of hard bodies.\n\nIn the same manner it is with the sound of a man's voice. For when the\nbreath passeth out without interruption, and doth but lightly touch the\ncavities through which it is sent, the sound it maketh is a hoarse\nsound. But if in going out it strike strongly upon the _larynx_, then a\nclear sound is made, as in a pipe. And the same breath, as it comes in\ndivers manners to the palate, the tongue, the lips, the teeth, and other\norgans of speech, so the sounds into which it is articulated become\ndifferent from one another.\n\n[Sidenote: Of reflected sound.]\n\n7. I call that _primary sound_, which is generated by motion from the\nsounding body to the organ in a strait line without reflection; and I\ncall that _reflected sound_, which is generated by one or more\nreflections, being the same with that we call _echo_, and is iterated as\noften as there are reflections made from the object to the ear. And\nthese reflections are made by hills, walls, and other resisting bodies,\nso placed as that they make more or fewer reflections of the motion,\naccording as they are themselves more or fewer in number; and they make\nthem more or less frequently, according as they are more or less distant\nfrom one another. Now the cause of both these things is to be sought for\nin the situation of the reflecting bodies, as is usually done in sight.\nFor the laws of reflection are the same in both, namely, that the angles\nof incidence and reflection be equal to one another. If, therefore, in a\nhollow elliptic body, whose inside is well polished, or in two right\nparabolical solids, which are joined together by one common base, there\nbe placed a sounding body in one of the burning points, and the ear in\nthe other, there will be heard a sound by many degrees greater than in\nthe open air; and both this, and the burning of such combustible things,\nas being put in the same places are set on fire by the sun-beams, are\neffects of one and the same cause. But, as when the visible object is\nplaced in one of the burning points, it is not distinctly seen in the\nother, because every part of the object being seen in every line, which\nis reflected from the concave superficies to the eye, makes a confusion\nin the sight; so neither is sound heard articulately and distinctly when\nit comes to the ear in all those reflected lines. And this may be the\nreason why in churches which have arched roofs, though they be neither\nelliptical nor parabolical, yet because their figure is not much\ndifferent from these, the voice from the pulpit will not be heard so\narticulately as it would be, if there were no vaulting at all.\n\n[Sidenote: From whence it is that sound is uniform and lasting.]\n\n8. Concerning the _uniformity_ and _duration_ of sounds, both which have\none common cause, we may observe, that such bodies as being stricken\nyield an unequal or harsh sound, are very heterogeneous, that is to say,\nthey consist of parts which are very unlike both in figure and hardness,\nsuch as are wood, stones, and others not a few. When these are stricken,\nthere follows a concussion of their internal particles, and a\nrestitution of them again. But they are neither moved alike, nor have\nthey the same action upon one another; some of them recoiling from the\nstroke, whilst others which have already finished their recoilings are\nnow returning; by which means they hinder and stop one another. And from\nhence it is that their motions are not only unequal and harsh, but also\nthat their reciprocations come to be quickly extinguished. Whensoever,\ntherefore, this motion is propagated to the ear, the sound it makes is\nunequal and of small duration. On the contrary, if a body that is\nstricken be not only sufficiently hard, but have also the particles of\nwhich it consisteth like to one another both in hardness and figure,\nsuch as are the particles of glass and metals, which being first melted\ndo afterwards settle and harden; the sound it yieldeth will, because the\nmotions of its parts and their reciprocations are like and uniform, be\nuniform and pleasant, and be more or less lasting, according as the body\nstricken hath greater or less magnitude. The possible cause, therefore,\nof sounds uniform and harsh, and of their longer or shorter duration,\nmay be one and the same likeness and unlikeness of the internal parts of\nthe sounding body, in respect both of their figure and hardness.\n\nBesides, if two plane bodies of the same matter and of equal thickness,\ndo both yield an uniform sound, the sound of that body, which hath the\ngreatest extent of length, will be the longest heard. For the motion,\nwhich in both of them hath its beginning from the point of percussion,\nis to be propagated in the greater body through a greater space, and\nconsequently that propagation will require more time; and therefore also\nthe parts which are moved, will require more time for their return.\nWherefore all the reciprocations cannot be finished but in longer time;\nand being carried to the ear, will make the sound last the longer. And\nfrom hence it is manifest, that of hard bodies which yield an uniform\nsound, the sound lasteth longer which comes from those that are round\nand hollow, than from those that are plane, if they be like in all other\nrespects. For in circular lines the action, which begins at any point,\nhath not from the figure any end of its propagation, because the line in\nwhich it is propagated returns again to its beginning; so that the\nfigure hinders not but that the motion may have infinite progression.\nWhereas in a plane, every line hath its magnitude finite, beyond which\nthe action cannot proceed. If, therefore, the matter be the same, the\nmotion of the parts of that body whose figure is round and hollow, will\nlast longer than of that which is plane.\n\nAlso, if a string which is stretched be fastened at both ends to a\nhollow body, and be stricken, the sound will last longer than if it were\nnot so fastened; because the trembling or reciprocation which it\nreceives from the stroke, is by reason of the connection communicated to\nthe hollow body; and this trembling, if the hollow body be great, will\nlast the longer by reason of that greatness. Wherefore also, for the\nreason above mentioned, the sound will last the longer.\n\n[Sidenote: How sound may be helped and hindered by the wind.]\n\n9. In hearing it happens, otherwise than in seeing, that the action of\nthe medium is made stronger by the wind when it blows the same way, and\nweaker when it blows the contrary way. The cause whereof cannot proceed\nfrom anything but the different generation of sound and light. For in\nthe generation of light, none of the parts of the medium between the\nobject and the eye are moved from their own places to other places\nsensibly distant; but the action is propagated in spaces imperceptible;\nso that no contrary wind can diminish, nor favourable wind encrease the\nlight, unless it be so strong as to remove the object further off or\nbring it nearer to the eye. For the wind, that is to say the air moved,\ndoth not by its interposition between the object and the eye work\notherwise than it would do, if it were still and calm. For, where the\npressure is perpetual, one part of the air is no sooner carried away,\nbut another, by succeeding it, receives the same impression, which the\npart carried away had received before. But in the generation of sound,\nthe first collision or breaking asunder beateth away and driveth out of\nits place the nearest part of the air, and that to a considerable\ndistance, and with considerable velocity; and as the circles grow by\ntheir remoteness wider and wider, so the air being more and more\ndissipated, hath also its motion more and more weakened. Whensoever\ntherefore the air is so stricken as to cause sound, if the wind fall\nupon it, it will move it all nearer to the ear, if it blow that way, and\nfurther from it if it blow the contrary way; so that according as it\nblows from or towards the object, so the sound which is heard will seem\nto come from a nearer or remoter place; and the reaction, by reason of\nthe unequal distances, be strengthened or debilitated.\n\nFrom hence may be understood the reason why the voice of such as are\nsaid to speak in their bellies, though it be uttered near hand, is\nnevertheless heard, by those that suspect nothing, as if it were a great\nway off. For having no former thought of any determined place from which\nthe voice should proceed, and judging according to the greatness, if it\nbe weak they think it a great way off, if strong near. These\nventriloqui, therefore, by forming their voice, not as others by the\nemission of their breath, but by drawing it inwards, do make the same\nappear small and weak; which weakness of the voice deceives those, that\nneither suspect the artifice nor observe the endeavour which they use in\nspeaking; and so, instead of thinking it weak, they think it far off.\n\n[Sidenote: Not only air, but other bodies, how hard soever they be,\n convey sound.]\n\n10. As for the medium, which conveys sound, it is not air only. For\nwater, or any other body how hard soever, may be that medium. For the\nmotion may be propagated perpetually in any hard continuous body; but by\nreason of the difficulty, with which the parts of hard bodies are moved,\nthe motion in going out of hard matter makes but a weak impression upon\nthe air. Nevertheless, if one end of a very long and hard beam be\nstricken, and the ear be applied at the same time to the other end, so\nthat, when the action goeth out of the beam, the air, which it striketh,\nmay immediately be received by the ear, and be carried to the tympanum,\nthe sound will be considerably strong.\n\nIn like manner, if in the night, when all other noise which may hinder\nsound ceaseth, a man lay his ear to the ground, he will hear the sound\nof the steps of passengers, though at a great distance; because the\nmotion, which by their treading they communicate to the earth, is\npropagated to the ear by the uppermost parts of the earth which\nreceiveth it from their feet.\n\n[Sidenote: The causes of grave and acute sounds, and of concent.]\n\n11. I have shown above, that the difference between grave and acute\nsounds consisteth in this, that by how much the shorter the time is, in\nwhich the reciprocations of the parts of a body stricken are made, by so\nmuch the more acute will be the sound. Now by how much a body of the\nsame bigness is either more heavy or less stretched, by so much the\nlonger will the reciprocations last; and therefore heavier and less\nstretched bodies, if they be like in all other respects, will yield a\ngraver sound than such as be lighter and more stretched.\n\nAs for the concent of sounds, it is to be considered that the\nreciprocation or vibration of the air, by which sound is made, after it\nhath reached the drum of the ear, imprinteth a like vibration upon the\nair that is inclosed within it; by which means the sides of the drum\nwithin are stricken alternately. Now the concent of two sounds consists\nin this, that the tympanum receives its sounding stroke from both the\nsounding bodies in equal and equally frequent spaces of time; so that\nwhen two strings make their vibrations in the same times, the concent\nthey produce is the most exquisite of all other. For the sides of the\ntympanum, that is to say of the organ of hearing, will be stricken by\nboth those vibrations together at once, on one side or other. For\nexample, if the two equal strings A B and C D be stricken together, and\nthe latitudes of their vibrations E F and G H be also equal, and the\npoints E, G, F and H be in the concave superficies of the tympanum, so\nthat it receive strokes from both the strings together in E and G, and\nagain together in F and H, the sound, which is made by the vibrations of\neach string, will be so like, that it may be taken for the same sound,\nand is called _unison_; which is the greatest concord. Again, the string\nA B retaining still its former vibration E F, let the string C D be\nstretched till its vibration have double the swiftness it had before,\nand let E F be divided equally in I. In what time therefore the string C\nD makes one part of its vibration from G to H, in the same time the\nstring A B will make one part of its vibration from E to I; and in what\ntime the string C D hath made the other part of its vibration back from\nH to G, in the same time another part of the vibration of the string A B\nwill be made from I to F. But the points F and G are both in the sides\nof the organ, and therefore they will strike the organ both together,\nnot at every stroke, but at every other stroke. And this is the nearest\nconcord to unison, and makes that sound which is called an _eighth_.\nAgain, the vibration of the string A B remaining still the same it was,\nlet C D be stretched till its vibration be swifter than that of the\nstring A B in the proportion of 3 to 2, and let E F be divided into\nthree equal parts in K and L. In what time therefore the string C D\nmakes one third part of its vibration, which third part is from G to H,\nthe string A B will make one third part of its vibration, that is to\nsay, two-thirds of E F, namely, E L. And in what time the string C D\nmakes another third part of its vibration, namely H G, the string A B\nwill make another third part of its vibration, namely from L to F, and\nback again from F to L. Lastly, whilst the string C D makes the last\nthird part of its vibration, that is from G to H, the string A B will\nmake the last third part of its vibration from L to E. But the points E\nand H are both in the sides of the organ. Wherefore, at every third\ntime, the organ will be stricken by the vibration of both the strings\ntogether, and make that concord which is called a _fifth_.\n\n A-------B\n C-------D\n G E\n | |\n | | K\n | |\n | | I\n | |\n | | L\n | |\n H F\n\n[Sidenote: Phenomena of smelling.]\n\n12. For the finding out the cause of _smells_, I shall make use of the\nevidence of these following phenomena. First, that smelling is hindered\nby cold, and helped by heat. Secondly, that when the wind bloweth from\nthe object, the smell is the stronger; and, contrarily, when it bloweth\nfrom the sentient towards the object, the weaker; both which phenomena\nare, by experience, manifestly found to be true in dogs, which follow\nthe track of beasts by the scent. Thirdly, that such bodies, as are less\npervious to the fluid medium, yield less smell than such as are more\npervious; as may be seen in stones and metals, which, compared with\nplants and living creatures, and their parts, fruits and excretions,\nhave very little or no smell at all. Fourthly, that such bodies, as are\nof their own nature odorous, become yet more odorous when they are\nbruised. Fifthly, that when the breath is stopped, at least in men,\nnothing can be smelt. Sixthly, that the sense of smelling is also taken\naway by the stopping of the nostrils, though the mouth be left open.\n\n[Sidenote: The first organ and the generation of smelling.]\n\n13. By the fifth and sixth phenomenon it is manifest, that the first and\nimmediate organ of smelling is the innermost cuticle of the nostrils,\nand that part of it, which is below the passage common to the nostrils\nand the palate. For when we draw breath by the nostrils we draw it into\nthe lungs. That breath, therefore, which conveys smells is in the way\nwhich passeth to the lungs, that is to say, in that part of the nostrils\nwhich is below the passage through which the breath goeth. For, nothing\nis smelt, neither beyond the passage of the breath within, nor at all\nwithout the nostrils.\n\nAnd seeing that from different smells there must necessarily proceed\nsome mutation in the organ, and all mutation is motion; it is therefore\nalso necessary that, in smelling, the parts of the organ, that is to say\nof that internal cuticle and the nerves that are inserted into it, must\nbe diversely moved by different smells. And seeing also, that it hath\nbeen demonstrated, that nothing can be moved but by a body that is\nalready moved and contiguous; and that there is no other body contiguous\nto the internal membrane of the nostrils but breath, that is to say\nattracted air, and such little solid invisible bodies, if there be any\nsuch, as are intermingled with the air; it follows necessarily, that the\ncause of smelling is either the motion of that pure air or ethereal\nsubstance, or the motion of those small bodies. But this motion is an\neffect proceeding from the object of smell, and, therefore, either the\nwhole object itself or its several parts must necessarily be moved. Now,\nwe know that odorous bodies make odour, though their whole bulk be not\nmoved. Wherefore the cause of odour is the motion of the invisible parts\nof the odorous body. And these invisible parts do either go out of the\nobject, or else, retaining their former situation with the rest of the\nparts, are moved together with them, that is to say, they have simple\nand invisible motion. They that say, there goes something out of the\nodorous body, call it an effluvium; which effluvium is either of the\nethereal substance, or of the small bodies that are intermingled with\nit. But, that all variety of odours should proceed from the effluvia of\nthose small bodies that are intermingled with the ethereal substance, is\naltogether incredible, for these considerations; first, that certain\nunguents, though very little in quantity, do nevertheless send forth\nvery strong odours, not only to a great distance of place, but also for\na great continuance of time, and are to be smelt in every point both of\nthat place and time; so that the parts issued out are sufficient to fill\nten thousand times more space, than the whole odorous body is able to\nfill; which is impossible. Secondly, that whether that issuing out be\nwith strait or with crooked motion, if the same quantity should flow\nfrom any other odorous body with the same motion, it would follow that\nall odorous bodies would yield the same smell. Thirdly, that seeing\nthose effluvia have great velocity of motion (as is manifest from this,\nthat noisome odours proceeding from caverns are presently smelt at a\ngreat distance) it would follow, that, by reason there is nothing to\nhinder the passage of those effluvia to the organ, such motion alone\nwere sufficient to cause smelling; which is not so; for we cannot smell\nat all, unless we draw in our breath through our nostrils. Smelling,\ntherefore, is not caused by the effluvium of atoms; nor, for the same\nreason, is it caused by the effluvium of ethereal substance; for so also\nwe should smell without the drawing in of our breath. Besides, the\nethereal substance being the same in all odorous bodies, they would\nalways affect the organ in the same manner; and, consequently, the\nodours of all things would be alike.\n\nIt remains, therefore, that the cause of smelling must consist in the\nsimple motion of the parts of odorous bodies without any efflux or\ndiminution of their whole substance. And by this motion there is\npropagated to the organ, by the intermediate air, the like motion, but\nnot strong enough to excite sense of itself without the attraction of\nair by respiration. And this is a possible cause of smelling.\n\n[Sidenote: How smelling is helped by heat and by wind.]\n\n14. The cause why smelling is hindered by cold and helped by heat may be\nthis; that heat, as hath been shown in chapter XXI, generateth simple\nmotion; and therefore also, wheresoever it is already, there it will\nincrease it; and the cause of smelling being increased, the smell itself\nwill also be increased. As for the cause why the wind blowing from the\nobject makes the smell the stronger, it is all one with that for which\nthe attraction of air in respiration doth the same. For, he that draws\nin the air next to him, draws with it by succession that air in which is\nthe object. Now, this motion of the air is wind, and, when another wind\nbloweth from the object, will be increased by it.\n\n[Sidenote: Why such bodies are least smelt, which have the least\n intermixture of air in them.]\n\n15. That bodies which contain the least quantity of air, as stones and\nmetals, yield less smell than plants and living creatures; the cause may\nbe, that the motion, which causeth smelling, is a motion of the fluid\nparts only; which parts, if they have any motion from the hard parts in\nwhich they are contained, they communicate the same to the open air, by\nwhich it is propagated to the organ. Where, therefore, there are no\nfluid parts as in metals, or where the fluid parts receive no motion\nfrom the hard parts, as in stones, which are made hard by accretion,\nthere can be no smell. And therefore also the water, whose parts have\nlittle or no motion, yieldeth no smell. But, if the same water, by seeds\nand the heat of the sun, be together with particles of earth raised into\na plant, and be afterwards pressed out again, it will be odorous, as\nwine from the vine. And as water passing through plants is by the motion\nof the parts of those plants made an odorous liquor; so also of air,\npassing through the same plants whilst they are growing, are made\nodorous airs. And thus also it is with the juices and spirits, which are\nbred in living creatures.\n\n[Sidenote: Why odorous things become more odorous, when bruised.]\n\n16. That odorous bodies may be made more odorous by contrition proceeds\nfrom this, that being broken into many parts, which are all odorous, the\nair, which by respiration is drawn from the object towards the organ,\ndoth in its passage touch upon all those parts, and receive their\nmotion. Now, the air toucheth the superficies only; and a body having\nless superficies whilst it is whole, than all its parts together have\nafter it is reduced to powder, it follows that the same odorous body\nyieldeth less smell whilst it is whole, than it will do after it is\nbroken into smaller parts. And thus much of smells.\n\n[Sidenote: The first organ of tasting; and why some savours cause\n nauseousness.]\n\n17. The taste follows; whose generation hath this difference from that\nof the sight, hearing, and smelling, that by these we have sense of\nremote objects; whereas, we taste nothing but what is contiguous, and\ndoth immediately touch either the tongue or palate, or both. From whence\nit is evident, that the cuticles of the tongue and palate, and the\nnerves inserted into them are the first organ of taste; and (because\nfrom the concussion of the parts of these, there followeth necessarily a\nconcussion of the _pia mater_) that the action communicated to these is\npropagated to the brain, and from thence to the farthest organ, namely,\nthe heart, in whose reaction consisteth the nature of sense.\n\nNow, that savours, as well as odours, do not only move the brain but the\nstomach also, as is manifest by the loathings that are caused by them\nboth; they, that consider the organ of both these senses, will not\nwonder at all; seeing the tongue, the palate and the nostrils, have one\nand the same continued cuticle, derived from the _dura mater_.\n\nAnd that effluvia have nothing to do in the sense of tasting, is\nmanifest from this, that there is no taste where the organ and the\nobject are not contiguous.\n\nBy what variety of motions the different kinds of tastes, which are\ninnumerable, may be distinguished, I know not. I might with others\nderive them from the divers figures of those atoms, of which whatsoever\nmay be tasted consisteth; or from the diverse motions which I might, by\nway of supposition, attribute to those atoms; conjecturing, not without\nsome likelihood of truth, that such things as taste sweet have their\nparticles moved with slow circular motion, and their figures spherical;\nwhich makes them smooth and pleasing to the organ; that bitter things\nhave circular motion, but vehement, and their figures full of angles, by\nwhich they trouble the organ; and that sour things have strait and\nreciprocal motion, and their figures long and small, so that they cut\nand wound the organ. And in like manner I might assign for the causes of\nother tastes such several motions and figures of atoms, as might in\nprobability seem to be the true causes. But this would be to revolt from\nphilosophy to divination.\n\n[Sidenote: The first organ of feeling; and how we came to the knowledge\n of such objects as are common to the touch and to other\n senses.]\n\n18. By the _touch_, we feel what bodies are cold or hot, though they be\ndistant from us. Others, as hard, soft, rough, and smooth, we cannot\nfeel unless they be contiguous. The organ of touch is every one of those\nmembranes, which being continued from the _pia mater_ are so diffused\nthroughout the whole body, as that no part of it can be pressed, but the\n_pia mater_ is pressed together with it. Whatsoever therefore presseth\nit, is felt as hard or soft, that is to say, as more or less hard. And\nas for the sense of rough, it is nothing else but innumerable\nperceptions of hard and hard succeeding one another by short intervals\nboth of time and place. For we take notice of rough and smooth, as also\nof magnitude and figure, not only by the touch, but also by memory. For\nthough some things are touched in one point, yet rough and smooth, like\nquantity and figure, are not perceived but by the flux of a point, that\nis to say, we have no sense of them without time; and we can have no\nsense of time without memory.\n\n\n -------\n\n\n CHAPTER XXX.\n\n OF GRAVITY.\n\n1. A thick body doth not contain more matter, unless also more place,\n than a thin.\u20142. That the descent of heavy bodies proceeds not from\n their own appetite, but from some power of the earth.\u20143. The\n difference of gravities proceedeth from the difference of the\n impetus with which the elements, whereof heavy bodies are made, do\n fall upon the earth.\u20144. The cause of the descent of heavy bodies.\u20145.\n In what proportion the descent of heavy bodies is accelerated.\u20146.\n Why those that dive do not, when they are under water, feel the\n weight of the water above them.\u20147. The weight of a body that\n floateth, is equal to the weight of so much water as would fill the\n space, which the immersed part of the body takes up within the\n water.\u20148. If a body be lighter than water, then how big soever that\n body be, it may float upon any quantity of water, how little\n soever.\u20149. How water may be lifted up and forced out of a vessel by\n air.\u201410. Why a bladder is heavier when blown full of air, than when\n it is empty.\u201411. The cause of the ejection upwards of heavy bodies\n from a wind-gun.\u201412. The cause of the ascent of water in a\n weather-glass.\u201413. The cause of motion upwards in living\n creatures.\u201414. That there is in nature a kind of body heavier than\n air, which nevertheless is not by sense distinguishable from it.\u201415.\n Of the cause of magnetical virtue.\n\n\n[Sidenote: A thick body doth not contain more matter, unless also more\n place, than a thin.]\n\n1. In chapter XXI I have defined thick and thin, as that place required,\nso, as that by thick was signified a more resisting body, and by thin, a\nbody less resisting; following the custom of those that have before me\ndiscoursed of refraction. Now if we consider the true and vulgar\nsignification of those words, we shall find them to be names collective,\nthat is to say, names of multitude; as _thick_ to be that, which takes\nup more parts of a space given, and _thin_ that, which contains fewer\nparts of the same magnitude in the same space, or in a space equal to\nit. Thick therefore is the same with frequent, as a thick troop; and\nthin the same with unfrequent, as a thin rank, thin of houses; not that\nthere is more matter in one place than in another equal place, but a\ngreater quantity of some named body. For there is not less matter or\nbody, indefinitely taken, in a desert, than there is in a city; but\nfewer houses, or fewer men. Nor is there in a thick rank a greater\nquantity of body, but a greater number of soldiers, than in a thin.\nWherefore the multitude and paucity of the parts contained within the\nsame space do constitute density and rarity, whether those parts be\nseparated by vacuum or by air. But the consideration of this is not of\nany great moment in philosophy; and therefore I let it alone, and pass\non to the search of the causes of _gravity_.\n\n[Sidenote: That the descent of heavy bodies proceeds not from their own\n appetite, but from some power of the earth.]\n\n2. Now we call those bodies _heavy_, which, unless they be hindered by\nsome force, are carried towards the centre of the earth, and that by\ntheir own accord, for aught we can by sense perceive to the contrary.\nSome philosophers therefore have been of opinion, that the descent of\nheavy bodies proceeded from some internal appetite, by which, when they\nwere cast upwards, they descended again, as moved by themselves, to such\nplace as was agreeable to their nature. Others thought they were\nattracted by the earth. To the former I cannot assent, because I think I\nhave already clearly enough demonstrated that there can be no beginning\nof motion, but from an external and moved body; and consequently, that\nwhatsoever hath motion or endeavour towards any place, will always move\nor endeavour towards that same place, unless it be hindered by the\nreaction of some external body. Heavy bodies, therefore, being once cast\nupwards, cannot be cast down again but by external motion. Besides,\nseeing inanimate bodies have no appetite at all, it is ridiculous to\nthink that by their own innate appetite they should, to preserve\nthemselves, not understanding what preserves them, forsake the place\nthey are in, and transfer themselves to another place; whereas man, who\nhath both appetite and understanding, cannot, for the preservation of\nhis own life, raise himself by leaping above three or four feet from the\nground. Lastly, to attribute to created bodies the power to move\nthemselves, what is it else than to say that there be creatures which\nhave no dependance upon the Creator? To the latter, who attribute the\ndescent of heavy bodies to the attraction of the earth, I assent. But by\nwhat motion this is done, hath not as yet been explained by any man. I\nshall therefore in this place say somewhat of the manner and of the way\nby which the earth by its action attracteth heavy bodies.\n\n[Sidenote: The difference of gravities proceedeth from the difference of\n the impetus with which the elements, whereof heavy bodies are\n made, do fall upon the earth.]\n\n3. That by the supposition of simple motion in the sun, homogeneous\nbodies are congregated and heterogeneous dissipated, has already been\ndemonstrated in the 5th article of chapter XXI. I have also supposed,\nthat there are intermingled with the pure air certain little bodies, or,\nas others call them, atoms; which by reason of their extreme smallness\nare invisible, and differing from one another in consistence, figure,\nmotion, and magnitude; from whence it comes to pass that some of them\nare congregated to the earth, others to other planets, and others are\ncarried up and down in the spaces between. And seeing those, which are\ncarried to the earth, differ from one another in figure, motion, and\nmagnitude, they will fall upon the earth, some with greater, others with\nless impetus. And seeing also that we compute the several degrees of\ngravity no otherwise than by this their falling upon the earth with\ngreater or less impetus; it follows, that we conclude those to be the\nmore heavy that have the greater impetus, and those to be less heavy\nthat have the less impetus. Our inquiry therefore must be, by what means\nit may come to pass, that of bodies, which descend from above to the\nearth, some are carried with greater, others with less impetus; that is\nto say, some are more heavy than others. We must also inquire, by what\nmeans such bodies, as settle upon the earth, may by the earth itself be\nforced to ascend.\n\n[Sidenote: The cause of the descent of heavy bodies.]\n\n4. Let the circle made upon the centre C (in fig. 2) be a great circle\nin the superficies of the earth, passing through the points A and B.\nAlso let any heavy body, as the stone A D, be placed anywhere in the\nplane of the equator; and let it be conceived to be cast up from A D\nperpendicularly, or to be carried in any other line to E, and supposed\nto rest there. Therefore, how much space soever the stone took up in A\nD, so much space it takes up now in E. And because all place is supposed\nto be full, the space A D will be filled by the air which flows into it\nfirst from the nearest places of the earth, and afterwards successively\nfrom more remote places. Upon the centre C let a circle be understood to\nbe drawn through E; and let the plane space, which is between the\nsuperficies of the earth and that circle, be divided into plane orbs\nequal and concentric; of which let that be the first, which is contained\nby the two perimeters that pass through A and D. Whilst therefore the\nair, which is in the first orb, filleth the place A D, the orb itself is\nmade so much less, and consequently its latitude is less than the strait\nline A D. Wherefore there will necessarily descend so much air from the\norb next above. In like manner, for the same cause, there will also be a\ndescent of air from the orb next above that; and so by succession from\nthe orb in which the stone is at rest in E. Either therefore the stone\nitself, or so much air, will descend. And seeing air is by the diurnal\nrevolution of the earth more easily thrust away than the stone, the air,\nwhich is in the orb that contains the stone, will be forced further\nupwards than the stone. But this, without the admission of vacuum,\ncannot be, unless so much air descend to E from the place next above;\nwhich being done, the stone will be thrust downwards. By this means\ntherefore the stone now receives the beginning of its descent, that is\nto say, of its gravity. Furthermore, whatsoever is once moved, will be\nmoved continually (as hath been shown in the 19th article of chapter\nVIII) in the same way, and with the same celerity, except it be retarded\nor accelerated by some external movent. Now the air, which is the only\nbody that is interposed between the earth A and the stone above it E,\nwill have the same action in every point of the strait line E A, which\nit hath in E. But it depressed the stone in E; and therefore also it\nwill depress it equally in every point of the strait line E A. Wherefore\nthe stone will descend from E to A with accelerated motion. The possible\ncause therefore of the descent of heavy bodies under the equator, is the\ndiurnal motion of the earth. And the same demonstration will serve, if\nthe stone be placed in the plane of any other circle parallel to the\nequator. But because this motion hath, by reason of its greater\nslowness, less force to thrust off the air in the parallel circles than\nin the equator, and no force at all at the poles, it may well be thought\n(for it is a certain consequent) that heavy bodies descend with less and\nless velocity, as they are more and more remote from the equator; and\nthat at the poles themselves, they will either not descend at all, or\nnot descend by the axis; which whether it be true or false, experience\nmust determine. But it is hard to make the experiment, both because the\ntimes of their descents cannot be easily measured with sufficient\nexactness, and also because the places near the poles are inaccessible.\nNevertheless, this we know, that by how much the nearer we come to the\npoles, by so much the greater are the flakes of the snow that falls; and\nby how much the more swiftly such bodies descend as are fluid and\ndissipable, by so much the smaller are the particles into which they are\ndissipated.\n\n[Sidenote: In what proportion the descent of heavy bodies is\n accelerated.]\n\n5. Supposing, therefore, this to be the cause of the descent of heavy\nbodies, it will follow that their motion will be accelerated in such\nmanner, as that the spaces, which are transmitted by them in the several\ntimes, will have to one another the same proportion which the odd\nnumbers have in succession from unity. For if the strait line E A be\ndivided into any number of equal parts, the heavy body descending will,\nby reason of the perpetual action of the diurnal motion, receive from\nthe air in every one of those times, in every several point of the\nstrait line E A, a several new and equal impulsion; and therefore also\nin every one of those times, it will acquire a several and equal degree\nof celerity. And from hence it follows, by that which Galileus hath in\nhis _Dialogues of Motion_ demonstrated, that heavy bodies descend in the\nseveral times with such differences of transmitted spaces, as are equal\nto the differences of the square numbers that succeed one another from\nunity; which square numbers being 1, 4, 9, 16, &c. their differences are\n3, 5, 7, &c.; that is to say, the odd numbers which succeed one another\nfrom unity. Against this cause of gravity which I have given, it will\nperhaps be objected, that if a heavy body be placed in the bottom of\nsome hollow cylinder of iron or adamant, and the bottom be turned\nupwards, the body will descend, though the air above cannot depress it,\nmuch less accelerate its motion. But it is to be considered that there\ncan be no cylinder or cavern, but such as is supported by the earth, and\nbeing so supported is, together with the earth, carried about by its\ndiurnal motion. For by this means the bottom of the cylinder will be as\nthe superficies of the earth; and by thrusting off the next and lowest\nair, will make the uppermost air depress the heavy body, which is at the\ntop of the cylinder, in such manner as is above explicated.\n\n[Sidenote: Why those that dive, do not, when they are under water, feel\n the weight of the water above them.]\n\n6. The gravity of water being so great as by experience we find it is,\nthe reason is demanded by many, why those that dive, how deep soever\nthey go under water, do not at all feel the weight of the water which\nlies upon them. And the cause seems to be this, that all bodies by how\nmuch the heavier they are, by so much the greater is the endeavour by\nwhich they tend downwards. But the body of a man is heavier than so much\nwater as is equal to it in magnitude, and therefore the endeavour\ndownwards of a man's body is greater than that of water. And seeing all\nendeavour is motion, the body also of a man will be carried towards the\nbottom with greater velocity than so much water. Wherefore there is\ngreater reaction from the bottom; and the endeavour upwards is equal to\nthe endeavour downwards, whether the water be pressed by water, or by\nanother body which is heavier than water. And therefore by these two\nopposite equal endeavours, the endeavour both ways in the water is taken\naway; and consequently, those that dive are not at all pressed by it.\n\nCoroll. From hence also it is manifest, that water in water hath no\nweight at all, because all the parts of water, both the parts above, and\nthe parts that are directly under, tend towards the bottom with equal\nendeavour and in the same strait lines.\n\n[Sidenote: The weight of a body that floateth, is equal to the weight of\n so much water as would fill the space which the immersed part\n of the body takes up within the water.]\n\n7. If a body float upon the water, the weight of that body is equal to\nthe weight of so much water as would fill the place which the immersed\npart of the body takes up within the water.\n\nLet E\u00a0F (in fig. 3) be a body floating in the water A B C D; and let the\npart E be above, and the other part F under the water. I say, the weight\nof the whole body E F is equal to the weight of so much water as the\nspace F will receive. For seeing the weight of the body E F forceth the\nwater out of the space F, and placeth it upon the superficies A B, where\nit presseth downwards; it follows, that from the resistance of the\nbottom there will also be an endeavour upwards. And seeing again, that\nby this endeavour of the water upwards, the body E F is lifted up, it\nfollows, that if the endeavour of the body downwards be not equal to the\nendeavour of the water upwards, either the whole body E F will, by\nreason of that inequality of their endeavours or moments, be raised out\nof the water, or else it will descend to the bottom. But it is supposed\nto stand so, as neither to ascend nor descend. Wherefore there is an\nequilibrium between the two endeavours; that is to say, the weight of\nthe body E F is equal to the weight of so much water as the space F will\nreceive; which was to be proved.\n\n[Sidenote: If a body be lighter than water, then how big soever that\n body be, it may float upon any quantity of water, how little\n soever.]\n\n8. From hence it follows, that any body, of how great magnitude soever,\nprovided it consist of matter less heavy than water, may nevertheless\nfloat upon any quantity of water, how little soever.\n\nLet A B C D (in fig. 4) be a vessel; and in it let E F G H be a body\nconsisting of matter which is less heavy than water; and let the space A\nG C F be filled with water. I say, the body E F G H will not sink to the\nbottom D C. For seeing the matter of the body E F G H is less heavy than\nwater, if the whole space without A B C D were fall of water, yet some\npart of the body E F G H, as E F I K, would be above the water; and the\nweight of so much water as would fill the space I G H K would be equal\nto the weight of the whole body E F G H; and consequently G H would not\ntouch the bottom D C. As for the sides of the vessel, it is no matter\nwhether they be hard or fluid; for they serve only to terminate the\nwater; which may be done as well by water as by any other matter how\nhard soever; and the water without the vessel is terminated somewhere,\nso as that it can spread no farther. The part therefore E F I K will be\nextant above the water A G C F which is contained in the vessel.\nWherefore the body E F G H will also float upon the water A G C F, how\nlittle soever that water be; which was to be demonstrated.\n\n[Sidenote: How water may be lifted up and forced out of a vessel by\n air.]\n\n9. In the 4th article of chapter XXVI, there is brought for the proving\nof vacuum the experiment of water enclosed in a vessel; which water, the\norifice above being opened, is ejected upwards by the impulsion of the\nair. It is therefore demanded, seeing water is heavier than air, how\nthat can be done. Let the second figure of the same, chapter XXVI be\nconsidered, where the water is with great force injected by a syringe\ninto the space F G B. In that injection, the air (but pure air) goeth\nwith the same force out of the vessel through the injected water. But as\nfor those small bodies, which formerly I supposed to be intermingled\nwith air and to be moved with simple motion, they cannot, together with\nthe pure air, penetrate the water; but remaining behind are necessarily\nthrust together into a narrower place, namely into the space which is\nabove the water F G. The motions therefore of those small bodies will be\nless and less free, by how much the quantity of the injected water is\ngreater and greater; so that by their motions falling upon one another,\nthe same small bodies will mutually compress each other, and have a\nperpetual endeavour of regaining their liberty, and of depressing the\nwater that hinders them. Wherefore, as soon as the orifice above is\nopened, the water which is next it will have an endeavour to ascend, and\nwill therefore necessarily go out. But it cannot go out, unless at the\nsame time there enter in as much air; and therefore both the water will\ngo out, and the air enter in, till those small bodies which were left\nwithin the vessel have recovered their former liberty of motion; that is\nto say, till the vessel be again filled with air, and no water be left\nof sufficient height to stop the passage at B. Wherefore I have shown a\npossible cause of this phenomenon, namely, the same with that of\nthunder. For as in the generation of thunder, the small bodies enclosed\nwithin the clouds, by being too closely pent together, do by their\nmotion break the clouds, and restore themselves to their natural\nliberty; so here also the small bodies enclosed within the space which\nis above the strait line F G, do by their own motion expel the water as\nsoon as the passage is opened above. And if the passage be kept stopped,\nand these small bodies be more vehemently compressed by the perpetual\nforcing in of more water, they will at last break the vessel itself with\ngreat noise.\n\n[Sidenote: Why a bladder is heavier when blown full of air, than when it\n is empty.]\n\n10. If air be blown into a hollow cylinder, or into a bladder, it will\nincrease the weight of either of them a little, as many have found by\nexperience, who with great accurateness have tried the same. And it is\nno wonder, seeing, as I have supposed, there are intermingled with the\ncommon air a great number of small hard bodies, which are heavier than\nthe pure air. For, the ethereal substance, being on all sides equally\nagitated by the motion of the sun, hath an equal endeavour towards all\nthe parts of the universe; and, therefore, it hath no gravity at all.\n\n[Sidenote: The cause of the ejection upwards of heavy bodies from a\n wind-gun.]\n\n11. We find also by experience, that, by the force of air enclosed in a\nhollow cannon, a bullet of lead may with considerable violence be shot\nout of a gun of late invention, called the wind-gun. In the end of this\ncannon there are two holes, with their valves on the inside, to shut\nthem close; one of them serving for the admission of air, and the other\nfor the letting of it out. Also, to that end which serves for the\nreceiving in of air, there is joined another cannon of the same metal\nand bigness, in which there is fitted a rammer which is perforated, and\nhath also a valve opening towards the former cannon. By the help of this\nvalve the rammer is easily drawn back, and letteth in air from without;\nand being often drawn back and returned again with violent strokes, it\nforceth some part of that air into the former cannon, so long, till at\nlast the resistance of the enclosed air is greater than the force of the\nstroke. And by this means men think there is now a greater quantity of\nair in the cannon than there was formerly, though it were full before.\nAlso, the air thus forced in, how much soever it be, is hindered from\ngetting out again by the aforesaid valves, which the very endeavour of\nthe air to get out doth necessarily shut. Lastly, that valve being\nopened which was made for the letting out of the air, it presently\nbreaketh out with violence, and driveth the bullet before it with great\nforce and velocity.\n\nAs for the cause of this, I could easily attribute it, as most men do,\nto condensation, and think that the air, which had at the first but its\nordinary degree of rarity, was afterwards, by the forcing in of more\nair, condensed, and last of all, rarified again by being let out and\nrestored to its natural liberty. But I cannot imagine how the same place\ncan be always full, and, nevertheless, contain some times a greater,\nsometimes a less quantity of matter; that is to say, that it can be\nfuller than full. Nor can I conceive how fulness can of itself be an\nefficient cause of motion. For both these are impossible. Wherefore we\nmust seek out some other possible cause of this phenomenon. Whilst,\ntherefore, the valve which serves for the letting in of air, is opened\nby the first stroke of the rammer, the air within doth with equal force\nresist the entering of the air from without; so that the endeavours\nbetween the internal and external air are opposite, that is, there are\ntwo opposite motions whilst the one goeth in and the other cometh out;\nbut no augmentation at all of air within the cannon. For there is driven\nout by the stroke as much pure air, which passeth between the rammer and\nthe sides of the cannon, as there is forced in of air impure by the same\nstroke. And thus, by many forcible strokes, the quantity of small hard\nbodies will be increased within the cannon, and their motions also will\ngrow stronger and stronger, as long as the matter of the cannon is able\nto endure their force; by which, if it be not broken, it will at least\nbe urged every way by their endeavour to free themselves; and as soon as\nthe valve, which serves to let them out, is opened, they will fly out\nwith violent motion, and carry with them the bullet which is in their\nway. Wherefore, I have given a possible cause of this phenomenon.\n\n[Sidenote: The cause of the ascent of water in a weather-glass.]\n\n12. Water, contrary to the custom of heavy bodies, ascendeth in the\nweather-glass; but it doth it when the air is cold: for when it is warm\nit descendeth again. And this organ is called a thermometer or\nthermoscope, because the degrees of heat and cold are measured and\nmarked by it. It is made in this manner. Let A B C D (in fig. 5) be a\nvessel full of water, and E F G a hollow cylinder of glass, closed at E\nand open at G. Let it be heated, and set upright within the water to F;\nand let the open end reach to G. This being done, as the air by little\nand little grows colder, the water will ascend slowly within the\ncylinder from F towards E; till at last the external and internal air\ncoming to be both of the same temper, it will neither ascend higher nor\ndescend lower, till the temper of the air be changed. Suppose it,\ntherefore, to be settled anywhere, as at H. If now the heat of the air\nbe augmented, the water will descend below H; and if the heat be\ndiminished, it will ascend above it. Which, though it be certainly known\nto be true by experience, the cause, nevertheless, hath not as yet been\ndiscovered.\n\nIn the sixth and seventh articles of chapter XXVIII, where I consider\nthe cause of cold, I have shown, that fluid bodies are made colder by\nthe pressure of the air, that is to say, by a constant wind that\npresseth them. For the same cause it is, that the superficies of the\nwater is pressed at F; and having no place, to which it may retire from\nthis pressure, besides the cavity of the cylinder between H and E, it is\ntherefore necessarily forced thither by the cold, and consequently it\nascendeth more or less, according as the cold is more or less increased.\nAnd again, as the heat is more intense or the cold more remiss, the same\nwater will be depressed more or less by its own gravity, that is to say,\nby the cause of gravity above explicated.\n\n[Sidenote: Cause of motion upwards in living creatures]\n\n13. Also living creatures, though they be heavy, can by leaping,\nswimming and flying, raise themselves to a certain degree of height. But\nthey cannot do this except they be supported by some resisting body, as\nthe earth, the water and the air. For these motions have their beginning\nfrom the contraction, by the help of the muscles, of the body animate.\nFor to this contraction there succeedeth a distension of their whole\nbodies; by which distension, the earth, the water, or the air, which\nsupporteth them, is pressed; and from hence, by the reaction of those\npressed bodies, living creatures acquire an endeavour upwards, but such\nas by reason of the gravity of their bodies is presently lost again. By\nthis endeavour, therefore, it is, that living creatures raise themselves\nup a little way by leaping, but to no great purpose: but by swimming and\nflying they raise themselves to a greater height; because, before the\neffect of their endeavour is quite extinguished by the gravity of their\nbodies, they can renew the same endeavour again.\n\nThat by the power of the soul, without any antecedent contraction of the\nmuscles or the help of something to support him, any man can be able to\nraise his body upwards, is a childish conceit. For if it were true, a\nman might raise himself to what height he pleased.\n\n[Sidenote: That there is in nature a kind of body heavier than air,\n which nevertheless is not by sense distinguishable from it.]\n\n14. The diaphanous medium, which surrounds the eye on all sides, is\ninvisible; nor is air to be seen in air, nor water in water, nor\nanything but that which is more opacous. But in the confines of two\ndiaphanous bodies, one of them may be distinguished from the other. It\nis not therefore a thing so very ridiculous for ordinary people to think\nall that space empty, in which we say is air; it being the work of\nreason to make us conceive that the air is anything. For by which of our\nsenses is it, that we take notice of the air, seeing we neither see, nor\nhear, nor taste, nor smell, nor feel it to be anything? When we feel\nheat, we do not impute it to the air, but to the fire: nor do we say the\nair is cold, but we ourselves are cold; and when we feel the wind, we\nrather think something is coming, than that any thing is already come.\nAlso, we do not at all feel the weight of water in water, much less of\nair in air. That we come to know that to be a body, which we call air,\nit is by reasoning; but it is from one reason only, namely, because it\nis impossible for remote bodies to work upon our organs of sense but by\nthe help of bodies intermediate, without which we could have no sense of\nthem, till they come to be contiguous. Wherefore, from the senses alone,\nwithout reasoning from effects, we cannot have sufficient evidence of\nthe nature of bodies.\n\nFor there is underground, in some mines of coals, a certain matter of a\nmiddle nature between water and air, which nevertheless cannot by sense\nbe distinguished from air; for it is as diaphanous as the purest air;\nand, as far as sense can judge, equally penetrable. But if we look upon\nthe effect, it is like that of water. For when that matter breaks out of\nthe earth into one of those pits, it fills the same either totally or to\nsome degree; and if a man or fire be then let down in it, it\nextinguishes them in almost as little time as water would do. But for\nthe better understanding of this phenomenon, I shall describe the 6th\nfigure. In which let A B represent the pit of the mine; and let part\nthereof, namely C B, be supposed to be filled with that matter. If now a\nlighted candle be let down into it below C, it will as suddenly be\nextinguished as if it were thrust into water. Also, if a grate filled\nwith coals thoroughly kindled and burning never so brightly, be let\ndown, as soon as ever it is below C, the fire will begin to grow pale,\nand shortly after, losing its light, be extinguished, no otherwise than\nif it were quenched in water. But if the grate be drawn up again\npresently, whilst the coals are still very hot, the fire will, by little\nand little, be kindled again, and shine as before. There is, indeed,\nbetween this matter and water this considerable difference, that it\nneither wetteth, nor sticketh to such things as are put down into it, as\nwater doth; which, by the moisture it leaveth, hindereth the kindling\nagain of the matter once extinguished. In like manner, if a man be let\ndown below C, he will presently fall into a great difficulty of\nbreathing, and immediately after into a swoon, and die unless he be\nsuddenly drawn up again. They, therefore, that go down into these pits,\nhave this custom, that as soon as ever they feel themselves sick, they\nshake the rope by which they were let down, to signify they are not\nwell, and to the end that they may speedily be pulled up again. For if a\nman be drawn out too late, void of sense and motion, they dig up a turf,\nand put his face and mouth into the fresh earth; by which means, unless\nhe be quite dead, he comes to himself again, by little and little, and\nrecovers life by breathing out, as it were, of that suffocating matter,\nwhich he had sucked in whilst he was in the pit; almost in the same\nmanner as they that are drowned come to themselves again by vomiting up\nthe water. But this doth not happen in all mines, but in some only; and\nin those not always, but often. In such pits as are subject to it, they\nuse this remedy. They dig another pit, as D E, close by it, of equal\ndepth, and joining them both together with one common channel, E B, they\nmake a fire in the bottom E, which carries out at D the air contained in\nthe pit D E; and this draws with it the air contained in the channel E\nB; which, in like manner, is followed by the noxious matter contained in\nC B; and, by this means, the pit is for that time made healthful. Out of\nthis history, which I write only to such as have had experience of the\ntruth of it, without any design to support my philosophy with stories of\ndoubtful credit, may be collected the following possible cause of this\nphenomenon; namely, that there is a certain matter fluid and most\ntransparent, and not much lighter than water, which, breaking out of the\nearth, fills the pit to C; and that in this matter, as in water, both\nfire and living creatures are extinguished.\n\n[Sidenote: Of the cause of magnetical virtue.]\n\n15. About the nature of heavy bodies, the greatest difficulty ariseth\nfrom the contemplation of those things which make other heavy bodies\nascend to them; such as jet, amber, and the loadstone. But that which\ntroubles men most is the loadstone, which is also called _Lapis\nHerculeus_; a stone, though otherwise despicable, yet of so great power\nthat it taketh up iron from the earth, and holds it suspended in the\nair, as Hercules did Ant\u00e6us. Nevertheless, we wonder at it somewhat the\nless, because we see jet draw up straws, which are heavy bodies, though\nnot so heavy as iron. But as for jet, it must first be excited by\nrubbing, that is to say, by motion to and fro; whereas the loadstone\nhath sufficient excitation from its own nature, that is to say, from\nsome internal principle of motion peculiar to itself. Now, whatsoever is\nmoved, is moved by some contiguous and moved body, as hath been formerly\ndemonstrated. And from hence it follows evidently, that the first\nendeavour, which iron hath towards the loadstone, is caused by the\nmotion of that air which is contiguous to the iron: also, that this\nmotion is generated by the motion of the next air, and so on\nsuccessively, till by this succession we find that the motion of all the\nintermediate air taketh its beginning from some motion which is in the\nloadstone itself; which motion, because the loadstone seems to be at\nrest, is invisible. It is therefore certain, that the attractive power\nof the loadstone is nothing else but some motion of the smallest\nparticles thereof. Supposing, therefore, that those small bodies, of\nwhich the loadstone is in the bowels of the earth composed, have by\nnature such motion or endeavour as was above attributed to jet, namely,\na reciprocal motion in a line too short to be seen, both those stones\nwill have one and the same cause of attraction. Now in what manner and\nin what order of working this cause produceth the effect of attraction,\nis the thing to be enquired. And first we know, that when the string of\na lute or viol is stricken, the vibration, that is, the reciprocal\nmotion of that string in the same strait line, causeth like vibration in\nanother string which hath like tension. We know also, that the dregs or\nsmall sands, which sink to the bottom of a vessel, will be raised up\nfrom the bottom by any strong and reciprocal agitation of the water,\nstirred with the hand or with a staff. Why, therefore, should not\nreciprocal motion of the parts of the loadstone contribute as much\ntowards the moving of iron? For, if in the loadstone there be supposed\nsuch reciprocal motion, or motion of the parts forwards and backwards,\nit will follow that the like motion will be propagated by the air to the\niron, and consequently that there will be in all the parts of the iron\nthe same reciprocations or motions forwards and backwards. And from\nhence also it will follow, that the intermediate air between the stone\nand the iron will, by little and little, be thrust away; and the air\nbeing thrust away, the bodies of the loadstone and the iron will\nnecessarily come together. The possible cause therefore why the\nloadstone and jet draw to them, the one iron, the other straws, may be\nthis, that those attracting bodies have reciprocal motion either in a\nstrait line, or in an elliptical line, when there is nothing in the\nnature of the attracted bodies which is repugnant to such a motion.\n\nBut why the loadstone, if with the help of cork it float at liberty upon\nthe top of the water, should from any position whatsoever so place\nitself in the plane of the meridian, as that the same points, which at\none time of its being at rest respect the poles of the earth, should at\nall other times respect the same poles, the cause may be this; that the\nreciprocal motion, which I supposed to be in the parts of the stone, is\nmade in a line parallel to the axis of the earth, and has been in those\nparts ever since the stone was generated. Seeing therefore, the stone,\nwhilst it remains in the mine, and is carried about together with the\nearth by its diurnal motion, doth by length of time get a habit of being\nmoved in a line which is perpendicular to the line of its reciprocal\nmotion, it will afterwards, though its axis be removed from the parallel\nsituation it had with the axis of the earth, retain its endeavour of\nreturning to that situation again; and all endeavour being the beginning\nof motion, and nothing intervening that may hinder the same, the\nloadstone will therefore return to its former situation. For, any piece\nof iron that has for a long time rested in the plane of the meridian,\nwhensoever it is forced from that situation and afterwards left to its\nown liberty again, will of itself return to lie in the meridian again;\nwhich return is caused by the endeavour it acquired from the diurnal\nmotion of the earth in the parallel circles which are perpendicular to\nthe meridians.\n\nIf iron be rubbed by the loadstone drawn from one pole to the other, two\nthings will happen; one, that the iron will acquire the same direction\nwith the loadstone, that is to say, that it will lie in the meridian,\nand have its axis and poles in the same position with those of the\nstone; the other, that the like poles of the stone and of the iron will\navoid one another, and the unlike poles approach one another. And the\ncause of the former may be this, that iron being touched by motion which\nis not reciprocal, but drawn the same way from pole to pole, there will\nbe imprinted in the iron also an endeavour from the same pole to the\nsame pole. For seeing the loadstone differs from iron no otherwise than\nas ore from metal, there will be no repugnance at all in the iron to\nreceive the same motion which is in the stone. From whence it follows,\nthat seeing they are both affected alike by the diurnal motion of the\nearth, they will both equally return to their situation in the meridian,\nwhensoever they are put from the same. Also, of the latter this may be\nthe cause, that as the loadstone in touching the iron doth by its action\nimprint in the iron an endeavour towards one of the poles, suppose\ntowards the North Pole; so reciprocally, the iron by its action upon the\nloadstone doth imprint in it an endeavour towards the other pole, namely\ntowards the South Pole. It happens therefore in these reciprocations or\nmotions forwards and backwards of the particles of the stone and of the\niron betwixt the north and the south, that whilst in one of them the\nmotion is from north to south, and the return from south to north, in\nthe other the motion will be from south to north, and the return from\nnorth to south; which motions being opposite to one another, and\ncommunicated to the air, the north pole of the iron, whilst the\nattraction is working, will be depressed towards the south pole of the\nloadstone; or contrarily, the north pole of the loadstone will be\ndepressed towards the south pole of the iron; and the axis both of the\nloadstone and of the iron will be situate in the same strait line. The\ntruth whereof is taught us by experience.\n\nAs for the propagation of this magnetical virtue, not only through the\nair, but through any other bodies how hard soever, it is not to be\nwondered at, seeing no motion can be so weak, but that it may be\npropagated infinitely through a space filled with body of any hardness\nwhatsoever. For in a full medium, there can be no motion which doth not\nmake the next part yield, and that the next, and so successively without\nend; so that there is no effect whatsoever, but to the production\nthereof something is necessarily contributed by the several motions of\nall the several things that are in the world.\n\n[Sidenote: Conclusion.]\n\nAnd thus much concerning the nature of body in general; with which I\nconclude this my first section of the Elements of Philosophy. In the\nfirst, second, and third parts, where the principles of ratiocination\nconsist in our own understanding, that is to say, in the legitimate use\nof such words as we ourselves constitute, all the theorems, if I be not\ndeceived, are rightly demonstrated. The fourth part depends upon\nhypotheses; which unless we know them to be true, it is impossible for\nus to demonstrate that those causes, which I have there explicated, are\nthe true causes of the things whose productions I have derived from\nthem.\n\nNevertheless, seeing I have assumed no hypothesis, which is not both\npossible and easy to be comprehended; and seeing also that I have\nreasoned aright from those assumptions, I have withal sufficiently\ndemonstrated that they may be the true causes; which is the end of\nphysical contemplation. If any other man from other hypotheses shall\ndemonstrate the same or greater things, there will be greater praise and\nthanks due to him than I demand for myself, provided his hypotheses be\nsuch as are conceivable. For as for those that say anything may be moved\nor produced by _itself_, by _species_, by _its own power_, by\n_substantial forms_, by _incorporeal substances_, by _instinct_, by\n_antiperistasis_, by _antipathy_, _sympathy_, _occult quality_, and\nother empty words of schoolmen, their saying so is to no purpose.\n\nAnd now I proceed to the phenomena of man's body; where I shall speak of\nthe _optics_, and of the _dispositions_, _affections_, and _manners_ of\nmen, if it shall please God to give me life, and show their causes.\n\n[Illustration:\n\n _Vol. 1. Lat. & Eng._\n C. XXVIII & XXX.\n _Fig. 1-6_\n]\n\n END OF VOL. I.\n\n\n\n\n\n\n\n\n LONDON:\n C. RICHARDS, ST. MARTIN'S LANE, CHARING CROSS.\n\n------------------------------------------------------------------------\n\n Transcriber\u2019s Note\n\nFigures 18.2 and 27.2 are not mentioned by name in the text. Figure\n14.10 is mentioned only in Chapter XIII.\n\nHowever, Figure 2 of Chapter XVIII would appear to be the subject of a\ndiscussion on p. 270. A sidenote has been added, and includes a\nreference to the figure.\n\nLikewise Figure 2 of Chapter XXVII, depicting a \u2018prisma\u2019, is discussed\nin detail on p. 459 _et seq._. Again, a sidenote has been interpolated\nat the opening of that description, and includes a reference to the\nfigure.\n\nThe diacritics used in the Greek phrases is followed as printed, though\nthey sometimes seem non-standard (i.e., \u03b5\u1f30\u03bd\u03b1\u03b9 rather than \u03b5\u03b9\u03bd\u03b1\u1f76).\n\nIn each chapter's summary, the article numbers are prefixed with a dash\nif they appeared midline. Dashes have been added to the text in the case\nwhere the article number happened to appear in the first column.\n\nOn rare occasions, there are errors in the numbering of the articles in\nthe chapter summary. For example, in Chapter XVII, article 7 was\nmisnumbered as a second \u20186\u2019.\n\nIn article 18 of Chapter XIII, a reference to article 10 of that chapter\nused the Roman \u2018X\u2019 which is the only instance of that usage. It is now\nreferred to as article 10.\n\nOther errors deemed most likely to be the printer\u2019s have been corrected,\nand are noted here. The references are to the page and line in the\noriginal.\n\n 52.17 _every man [i]s a living creature_ Restored.\n\n 73.14 in the first place, we [we] are to search Removed.\n\n 255.11 seeing the [pro]proportion Removed.\n\n 267.22 And therefore if it[ it] be again as R to T Removed.\n\n 307.24 the spiral of Arch[i/e]mides Replaced.\n\n 329.29 with simple motion the[ the] perimeter Removed.\n\n 339.11 be cut in E and[ and] G by a strait line Removed.\n\n 372.12 to the base F H that[ that] can possibly be Removed.\n drawn\n\n 464.23 the sea appears b[l]ack when ruffled Inserted.\n\n 439.2 will make fifty-nine revo[lu]tions Inserted.\n\n Chapter XIV\n\n[Illustration: Fig. 1.]\n\n[Illustration: Fig. 2.]\n\n[Illustration: Fig. 3.]\n\n[Illustration: Fig. 4.]\n\n[Illustration: Fig. 5.]\n\n[Illustration: Fig. 6.]\n\n[Illustration: Fig. 7.]\n\n[Illustration: Fig. 8.]\n\n[Illustration: Fig. 9.]\n\n[Illustration: Fig. 10.]\n\n CHAPTER XVI\n\n[Illustration: Fig. 1.]\n\n[Illustration: Fig. 2.]\n\n[Illustration: Fig. 3.]\n\n[Illustration: Fig. 4.]\n\n[Illustration: Fig. 5.]\n\n[Illustration: Fig. 6.]\n\n[Illustration: Fig. 7.]\n\n[Illustration: Fig. 8.]\n\n[Illustration: Fig. 9.]\n\n[Illustration: Fig. 10.]\n\n[Illustration: Fig. 11.]\n\n Chapter XVII\n\n[Illustration: Fig. 1.]\n\n[Illustration: Fig. 2.]\n\n[Illustration: Fig. 3.]\n\n[Illustration: Fig. 4.]\n\n[Illustration: Fig. 5.]\n\n[Illustration: Fig. 6.]\n\n[Illustration: Fig. 7.]\n\n[Illustration: Fig. 8.]\n\n Chapter XVIII\n\n[Illustration: Fig. 1.]\n\n[Illustration: Fig. 2.]\n\n Chapter XIX\n\n[Illustration: Fig. 1.]\n\n[Illustration: Fig. 2.]\n\n[Illustration: Fig. 3.]\n\n[Illustration: Fig. 4.]\n\n[Illustration: Fig. 5.]\n\n[Illustration: Fig. 6.]\n\n[Illustration: Fig. 7.]\n\n[Illustration: Fig. 8.]\n\n[Illustration: Fig. 9.]\n\n[Illustration: Fig. 10.]\n\n Chapter XX\n\n[Illustration: Fig. 1.]\n\n[Illustration: Fig. 2.]\n\n[Illustration: Fig. 3.]\n\n[Illustration: Fig. 4.]\n\n[Illustration: Fig. 5.]\n\n[Illustration: Fig. 6.]\n\n Chapter XXI\n\n[Illustration: Fig. 1.]\n\n[Illustration: Fig. 2.]\n\n[Illustration: Fig. 3.]\n\n[Illustration: Fig. 4.]\n\n[Illustration: Fig. 5.]\n\n Chapter XXII\n\n[Illustration: Fig. 1.]\n\n[Illustration: Fig. 2.]\n\n[Illustration: Fig. 3.]\n\n Chapter XXIII\n\n[Illustration: Fig. 1.]\n\n[Illustration: Fig. 2]\n\n[Illustration: Fig. 3.]\n\n[Illustration: Fig. 4]\n\n[Illustration: Fig. 5.]\n\n[Illustration: Fig. 6]\n\n[Illustration: Fig. 7.]\n\n[Illustration: Fig. 8]\n\n[Illustration: Fig 9.]\n\n CHAPTER XXIV.\n\n[Illustration: Fig 1.]\n\n[Illustration: Fig 2.]\n\n[Illustration: Fig 3.]\n\n[Illustration: Fig 4.]\n\n[Illustration: Fig 5.]\n\n[Illustration: Fig 6.]\n\n CHAPTER XXVI.\n\n[Illustration: Fig 1.]\n\n[Illustration: Fig 2.]\n\n[Illustration: Fig 3.]\n\n[Illustration: Fig 4.]\n\n[Illustration: Fig 5.]\n\n Chapter XXVII.\n\n[Illustration: Fig 1.]\n\n[Illustration: Fig 2.]\n\n Chapter XXX.\n\n[Illustration: Fig 1.]\n\n[Illustration: Fig 2.]\n\n[Illustration: Fig 3.]\n\n[Illustration: Fig 4.]\n\n[Illustration: Fig 5.]\n\n[Illustration: Fig 6.]","title":"The English works of Thomas Hobbes of Malmesbury, Volume 01 (of 11)","tradition":"philosophy-ethics","translated_text":null,"word_count":171569}