{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_78674/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_78674","id":"gutenberg_78674","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/78674","summary":null,"tags":["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_. Superscripted\ncharacters are prefixed with \u2018^\u2019. Multiple characters are enclosed with\n\u2018{ }\u2019.\n\nFootnotes have been moved to follow the paragraphs in which they are\nreferenced.\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------------------------------------------------------------------------\n\n\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. VII.\n\n -------\n\n\n\n\n LONDON:\n LONGMAN, BROWN, GREEN, AND LONGMANS,\n PATERNOSTER-ROW.\n\n --\n\n MDCCCXLV.\n\n\n\n\n LONDON:\n RICHARDS, PRINTER, 100, ST. MARTIN\u2019S LANE.\n\n\n\n\n\n\n\n\n CONTENTS.\n\n ---\n\n PAGE\n SEVEN PHILOSOPHICAL PROBLEMS 1\n\n DECAMERON PHYSIOLOGICUM 69\n\n PROPORTION OF A STRAIGHT LINE TO HALF THE ARC OF A QUADRANT 178\n\n SIX LESSONS TO THE SAVILIAN PROFESSORS OF THE MATHEMATICS 181\n\n \u03a3\u03a4\u0399\u0393\u039c\u0391\u0399, OR MARKS OF THE ABSURD GEOMETRY ETC. OF DR. WALLIS 357\n\n EXTRACT OF A LETTER FROM HENRY STUBBE 401\n\n THREE PAPERS PRESENTED TO THE ROYAL SOCIETY AGAINST DR. WALLIS 429\n\n CONSIDERATIONS ON THE ANSWER OF DR. WALLIS 443\n\n LETTERS AND OTHER PIECES 449\n\n\n\n\n SEVEN\n PHILOSOPHICAL PROBLEMS\n AND\n TWO PROPOSITIONS OF GEOMETRY.\n\n\n BY\n THOMAS HOBBES\n OF MALMESBURY.\n\n WITH\n AN APOLOGY FOR HIMSELF AND HIS WRITINGS.\n DEDICATED TO THE KING IN THE YEAR 1662.\n\n TO THE KING.\n\nThat which I do here most humbly present to your sacred Majesty, is the\nbest part of my meditations upon the natural causes of events, both of\nsuch as are commonly known, and of such as have been of late\nartificially exhibited by the curious.\n\nThey are ranged under seven heads. 1. Problems of gravity. 2. Problems\nof tides. 3. Problems of vacuum. 4. Problems of heat. 5. Problems of\nhard and soft. 6. Problems of wind and weather. 7. Problems of motion\nperpendicular and oblique, &c. To which I have added two propositions of\nGeometry: one is, the duplication of the cube, hitherto sought in vain;\nthe other, a detection of the absurd use of arithmetic, as it is now\napplied to geometry.\n\nThe doctrine of natural causes hath not infallible and evident\nprinciples. For there is no effect which the power of God cannot produce\nby many several ways.\n\nBut seeing all effects are produced by motion, he that supposing some\none or more motions, can derive from them the necessity of that effect\nwhose cause is required, has done all that is to be expected from\nnatural reason. And though he prove not that the thing was thus\nproduced, yet he proves that thus it may be produced when the materials\nand the power of moving are in our hands: which is as useful as if the\ncauses themselves were known. And notwithstanding the absence of\nrigorous demonstration, this contemplation of nature (if not rendered\nobscure by empty terms) is the most noble employment of the mind that\ncan be, to such as are at leisure from their necessary business.\n\nThis that I have done I know is an unworthy present to be offered to a\nking: though considered, as God considers offerings, together with the\nmind and fortune of the offerer, I hope will not be to your Majesty\nunacceptable.\n\nBut that which I chiefly consider in it is, that my writing should be\ntried by your Majesty\u2019s excellent reason, untainted with the language\nthat has been invented or made use of by men when they were puzzled; and\nwho is acquainted with all the experiments of the time; and whose\napprobation, if I have the good fortune to obtain it, will protect my\nreasoning from the contempt of my adversaries.\n\nI will not break the custom of joining to my offering a prayer; and it\nis, that your Majesty will be pleased to pardon this following short\napology for my _Leviathan_ . Not that I rely upon apologies, but upon\nyour Majesty\u2019s most gracious general pardon.\n\nThat which is in it of theology, contrary to the general current of\ndivines, is not put there as my opinion, but propounded with submission\nto those that have the power ecclesiastical.\n\nI did never after, either in writing or discourse, maintain it.\n\nThere is nothing in it against episcopacy; I cannot therefore imagine\nwhat reason any episcopal man can have to speak of me, as I hear some of\nthem do, as of an atheist, or man of no religion, unless it be for\nmaking the authority of the Church wholly upon the regal power; which I\nhope your Majesty will think is neither atheism nor heresy.\n\nBut what had I to do to meddle with matters of that nature, seeing\nreligion is not philosophy, but law?\n\nIt was written in a time when the pretence of Christ\u2019s kingdom was made\nuse of for the most horrid actions that can be imagined; and it was in\njust indignation of that, that I desired to see the bottom of that\ndoctrine of the kingdom of Christ, which divers ministers then preached\nfor a pretence to their rebellion: which may reasonably extenuate,\nthough not excuse the writing of it.\n\nThere is therefore no ground for so great a calumny in my writing. There\nis no sign of it in my life; and for my religion, when I was at the\npoint of death at St. Germain\u2019s, the Bishop of Durham can bear witness\nof it, if he be asked. Therefore I most humbly beseech your sacred\nMajesty not to believe so ill of me upon reports, that proceed often,\nand may do so now, from the displeasure which commonly ariseth from\ndifference in opinion; nor to think the worse of me, if snatching up all\nthe weapons to fight against your enemies, I lighted upon one that had a\ndouble edge.\n\n Your Majesty\u2019s poor and\n\n most loyal subject,\n\n THOMAS HOBBES.\n\n ==========\n\n[Illustration: _Seven Philosophical Problems. English Works, Vol. 7._]\n\n PHILOSOPHICAL PROBLEMS.\n\n ---\n\n\n CHAPTER I.\n PROBLEMS OF GRAVITY.\n\n_A._ What may be the cause, think you, that stones and other bodies\nthrown upward, or carried up and left to their liberty, fall down again,\nfor aught a man can see, of their own accord? I do not think with the\nold philosophers, that they have any love to the earth; or are sullen,\nthat they will neither go nor stay. And yet I cannot imagine, what body\nthere is above that should drive them back.\n\n_B._ For my part, I believe the cause of their descending is not in any\nnatural appetite of the bodies that descend; but rather that the globe\nof the earth hath some special motion, by which it more easily casteth\noff the air than it doth other bodies. And then this descent of those we\ncall heavy bodies must of necessity follow, unless there be some empty\nspaces in the world to receive them. For when the air is thrown off from\nthe earth, somewhat must come into the place of it, in case the world be\nfull: and it must be those things which are hardliest cast off, that is,\nthose things which we say are heavy.\n\n_A._ But suppose there be no place empty, (for I will defer the question\ntill anon), how can the earth cast off either the air or anything else?\n\n_B._ I shall show you how, and that by a familiar example. If you lay\nboth your hands upon a basin with water in it, how little soever, and\nmove it circularly, and continue that motion for a while; and you shall\nsee the water rise upon the sides, and fly over. By which you may be\nassured that there is a kind of circulating motion, which would cast off\nsuch bodies as are contiguous to the body so moved.\n\n_A._ I know very well there is; and it is the same motion which country\npeople use to purge their corn; for the chaff and straws, by casting the\ngrain to the sides of the sieve, will come towards the middle. But I\nwould see the figure.\n\n_B._ Here it is. There is a circle pricked out, whose centre is A, and\nthree less circles, whose centres are B, C, D. Let every one of them\nrepresent the earth, as it goeth from B to C, and from C to D, always\ntouching the uttermost circle and throwing off the air, as is marked at\nE and F. And if the world were not full, there would follow by this\nscattering of the air, a great deal of space left empty. But supposing\nthe world full, there must be a perpetual shifting of the air, one part\ninto the place of another.\n\n_A._ But what makes a stone come down, suppose from G?\n\n_B._ If the air be thrown up beyond G, it will follow that at the last,\nif the motion be continued, all the air will be above G, that is, above\nthe stone; which cannot be, till the stone be at the earth.\n\n_A._ But why comes it down still with increasing swiftness?\n\n_B._ Because as it descends and is already in motion, it receiveth a new\nimpression from the same cause, which is the air, whereof as part\nmounteth, part also must descend, supposing as we have done the\nplentitude of the world. For, as you may observe by the figure, the\nmotion of the earth, according to the diameter of the uttermost circle,\nis progressive; and so the whole motion is compounded of two motions,\none circular and the other progressive; and consequently the air ascends\nand circulates at once. And because the stone descending receiveth a new\npressure in every point of its way, the motion thereof must needs be\naccelerated.\n\n_A._ It is true; for it will be accelerated equally in equal times; and\nthe way it makes will increase in a double proportion to the times, as\nhath heretofore been demonstrated by Galileo. I see the solution now of\nan experiment, which before did not a little puzzle me. You know that if\ntwo plummets hang by two strings of equal length, and you remove them\nfrom the perpendicular equally, I mean in equal angles, and then let\nthem go, they will make their turns and returns together and in equal\ntimes; and though the arches they describe grow continually less and\nless, yet the times they spend in the greater arches will still be equal\nto the time they spend in the lesser.\n\n_B._ It is true. Do you find any experiment to the contrary?\n\n_A._ Yes; for if you remove one of the plummets from the perpendicular,\nso as, for example, to make an angle with the perpendicular of eighty\ndegrees, and the other so as to make an angle of sixty degrees; they\nwill not make their turns and returns in equal times.\n\n_B._ And what say you is the cause of this?\n\n_A._ Because the arches are the spaces which these two motions describe,\nthey must be in double proportion to their own times: which cannot be,\nunless they be let go from equal altitudes, that is, from equal angles.\n\n_B._ It is right; and the experiment does not cross, but confirm the\nequality of the times in all the arches they describe, even from ninety\ndegrees to the least part of one degree.\n\n_A._ But is it not too bold, if not extravagant an assertion, to say the\nearth is moved as a man shakes a basin or a sieve? Does not the earth\nmove from west to east every day once, upon its own centre; and in the\necliptic circle once a year? And now you give it another odd motion. How\ncan all these consist in one and the same body?\n\n_B._ Well enough. If you be a shipboard under sail, do not you go with\nthe ship? Cannot you also walk upon the deck? Cannot every drop of blood\nmove at the same time in your veins? How many motions now do you assign\nto one and the same drop of blood? Nor is it so extravagant a thing to\nattribute to the earth this kind of motion; but that I believe, if we\ncertainly knew what motion it is that causeth the descent of bodies, we\nshould find it either the same, or more extravagant. But seeing it can\nbe nothing above that worketh this effect, it must be the earth itself\nthat does it; and if the earth, then you can imagine no other motion to\ndo it withal but this. And you will wonder more, when by the same motion\nI shall give you a probable account of the causes of very many other\nworks of nature.\n\n_A._ But what part of the heaven do you suppose the poles of your\npricked circle point to?\n\n_B._ I suppose them to be the same with the poles of the ecliptic. For,\nseeing the axis of the earth in this motion and in the annual motion,\nkeeps parallel to itself, the axis must in both motions be parallel as\nto sense. For the circle which the earth describes, is not of visible\nmagnitude at the distance it is from the sun.\n\n_A._ Though I understand well enough how the earth may make a stone\ndescend very swiftly under the ecliptic, or not far from it, where it\nthrows off the air perpendicularly; yet about the poles of the circle\nmethinks, it should cast off the air very weakly. I hope you will not\nsay, that bodies descend faster in places remote from the poles, than\nnearer to them.\n\n_B._ No; but I ascribe it to the like motion in the sun and moon. For\nsuch motions meeting, must needs cast the stream of the air towards the\npoles; and then there will be the same necessity for the descent there,\nthat there is in other places, though perhaps a little more slowly. For\nyou may have observed, that when it snows in the south parts, the flakes\nof snow are not so great as in the north: which is a probable sign they\nfall in the south from a greater height, and consequently disperse\nthemselves more, as water does that falls down from a high and steep\nrock.\n\n_A._ It is not improbable.\n\n_B._ In natural causes all you are to expect, is but probability; which\nis better yet, than making gravity the cause, when the cause of gravity\nis that which you desire to know; and better than saying the earth draws\nit, when the question is, how it draws.\n\n_A._ Why does the earth cast off air more easily than it does water, or\nany other heavy bodies?\n\n_B._ It is indeed the earth that casteth off that air which is next unto\nit; but it is that air which casteth off the next air; and so\ncontinually, air moveth air; which it can more easily do than any other\nthing, because like bodies are more susceptible of one another\u2019s\nmotions: as you may see in two lute-strings equally strained, what\nmotion one string being stricken communicates to the air, the same will\nthe other receive from the air; but strained to a differing note, will\nbe less or not at all moved. For there is no body but air, that hath not\nsome internal, though invisible, motion of its parts: and it is that\ninternal motion which distinguisheth all natural bodies one from\nanother.\n\n_A._ What is the cause why certain squibs, though their substance be\neither wood or other heavy matter, made hollow and filled with\ngunpowder, which is also heavy; do nevertheless, when the gunpowder is\nkindled, fly upwards?\n\n_B._ The same that keeps a man that swims from sinking, though he be\nheavier than so much water. He keeps himself up, and goes forward, by\nbeating back the water with his feet; and so does a squib, by beating\ndown the air with the stream of fired gunpowder, that proceeding from\nits tail makes it recoil.\n\n_A._ Why does any brass or iron vessel, if it be hollow, float upon the\nwater, being so very heavy?\n\n_B._ Because the vessel and the air in it, taken as one body, is more\neasily cast off than a body of water equal to it.\n\n_A._ How comes it to pass, that a fish, (especially such a broad fish as\na turbot or a plaice, which are broad and thin), in the bottom of the\nsea, perhaps a mile deep, is not pressed to death with the weight of\nwater that lies upon the back of it?\n\n_B._ Because all heavy bodies descend towards one point, which is the\ncentre of the earth: and consequently the whole sea, descending at once,\ndoes arch itself so, as that the upper parts cannot press the parts next\nbelow them.\n\n_A._ It is evident; nor can there possibly be any weight, as some\nsuppose there is, of a cylinder of air or water or any other liquid\nthing, while it remains in its own element, or is sustained and inclosed\nin a vessel by which one part cannot press the other.\n\n\n ==========\n\n\n CHAPTER II.\n PROBLEMS OF TIDES.\n\n_A._ What makes the flux and reflux of the sea, twice in a natural day?\n\n_B._ We must come again to our basin of water; wherein you have seen,\nwhilst it was moved, how the water mounteth up by the sides, and withal\ngoes circling round about. Now if you should fasten to the inside of the\nbasin some bar from the bottom to the top, you would see the water,\ninstead of going on, go back again from that bar ebbing, and the water\non the other side of the bar to do the same, but in counter-time; and\nconsequently to be highest where the contrary streams meet together; and\nthen return again, marking out four quarters of the vessel; two by their\nmeeting, which are the high waters; and two by their retiring, which are\nthe low waters.\n\n_A._ What bar is that you find in the ocean that stops the current of\nthe water, like that you make in the basin?\n\n_B._ You know that the main ocean lies east and west, between India and\nthe coast of America; and again on the other side, between America and\nIndia. If therefore the earth have such a motion as I have supposed, it\nmust needs carry the current of the sea east and west: in which course,\nthe bar that stoppeth it, is the south part of America, which leaves no\npassage for the water but the narrow strait of Magellan. The tide rises\ntherefore upon the coast of America; and the rising of the same in this\npart of the world, proceedeth from the swelling chiefly of the water\nthere, and partly also from the North Sea; which lieth also east and\nwest, and has a passage out of the South Sea by the strait of Anian,\nbetween America and Asia.\n\n_A._ Does not the Mediterranean Sea lie also east and west? Why are\nthere not the like tides there?\n\n_B._ So there are, proportionable to their lengths and quantity of\nwater.\n\n_A._ At Genoa, at Ancona, there are none at all, or not sensible.\n\n_B._ At Venice there are, and in the bottom of the straits, and a\ncurrent all along both the Mediterranean Sea and the Gulf of Venice: and\nit is the current that makes the tides insensible at the sides; but the\ncheck makes them visible at the bottom.\n\n_A._ How comes it about that the moon hath such a stroke in the\nbusiness, as so sensibly to increase the tides at full and change?\n\n_B._ The motion I have hitherto supposed but in the earth, I suppose\nalso in the moon, and in all those great bodies that hang in the air\nconstantly, I mean the stars, both fixed and errant. And for the sun and\nmoon, I suppose the poles of their motion to be the poles of the\nequinoxial. Which supposed, it will follow (because the sun, the earth,\nand the moon, at every full and change are almost in one straight line)\nthat this motion of the earth will then be made swifter than in the\nquarters. For this motion of the sun and moon being communicated to the\nearth, that hath already the like motion, maketh the same greater; and\nmuch greater when they are all three in one straight line, which is only\nat the full and change, whose tides are therefore called spring tides.\n\n_A._ But what then is the cause that the spring tides themselves are\ntwice a-year, namely, when the sun is in the equinoxial, greater than at\nany other times?\n\n_B._ At other times of the year, the earth being out of the equinoxial,\nthe motion thereof, by which the tides are made, will be less augmented,\nby so much as a motion in the obliquity of twenty-three degrees, or\nthereabout, which is the distance between the equinoxial and ecliptic\ncircles, is weaker than the motion which is without obliquity.\n\n_A._ All this is reasonable enough, if it be possible that such motions\nas you suppose in these bodies, be really there. But that is a thing I\nhave some reason to doubt of. For the throwing off of air, consequent to\nthese motions, is the cause, you say, that other things come to the\nearth; and therefore the like motions in the sun and moon and stars,\ncasting off the air, should also cause all other things to come to every\none of them. From whence it will follow, that the sun, moon, and earth,\nand all other bodies but air, should presently come together into one\nheap.\n\n_B._ That does not follow. For if two bodies cast off the air, the\nmotion of that air will be repressed both ways, and diverted into a\ncourse towards the poles on both sides; and then the two bodies cannot\npossibly come together.\n\n_A._ It is true. And besides, this driving of the air on both sides,\nnorth and south, makes the like motion of air there also. And this may\nanswer the question, how a stone could fall to the earth under the poles\nof the ecliptic, by the only casting off of air?\n\n_B._ It follows from hence, that there is a certain and determinate\ndistance of one of these bodies, the stars, from another, without any\nvery sensible variation.\n\n_A._ All this is probable enough, if it be true that there is no vacuum,\nno place empty in all the world. And supposing this motion of the sun\nand moon to be in the plain of the equinoxial, methinks that this should\nbe the cause of the diurnal motion of the earth; and because this motion\nof the earth is, you say, in the plain of the equinoxial, the same\nshould cause also a motion in the moon on her own centre, answerable to\nthe diurnal motion of the earth.\n\n_B._ Why not? What else can you think makes the diurnal motion of the\nearth but the sun? And for the moon, if it did not turn upon its own\ncentre, we should see sometimes one, sometimes another face of the moon,\nwhich we do not.\n\n\n ==========\n\n\n CHAPTER III.\n PROBLEMS OF VACUUM.\n\n_A._ What convincing argument is there to prove, that in all the world\nthere is no empty place?\n\n_B._ Many; but I will name but one; and that is, the difficulty of\nseparating two bodies hard and flat laid one upon another. I say the\ndifficulty, not the impossibility. It is possible, without introducing\nvacuum, to pull asunder any two bodies, how hard and flat soever they\nbe, if the force used be greater than the resistance of the hardness.\nAnd in case there be any greater difficulty to part them, besides what\nproceeds from their hardness, than there is to pull them further asunder\nwhen they are parted, that difficulty is argument enough to prove there\nis no vacuum.\n\n_A._ These assertions need demonstration. And first, how does the\ndifficulty of separation argue the plenitude of all the rest of the\nworld?\n\n_B._ If two flat polished marbles lie one upon another, you see they are\nhardly separated in all points at one and the same instant; and yet the\nweight of either of them is enough to make them slide off one from the\nother. Is not the cause of this, that the air succeeds the marble that\nso slides, and fills up the place it leaves?\n\n_A._ Yes, certainly. What then?\n\n_B._ But when you pull the whole superficies asunder, not without great\ndifficulty, what is the cause of that difficulty?\n\n_A._ I think, as most men do, that the air cannot fill up the space\nbetween in an instant; for the parting is in an instant.\n\n_B._ Suppose there be vacuum in that air into which the marble you pull\noff is to succeed, shall there be no vacuum in the air that was round\nabout the two marbles when they touched? Why cannot that vacuum come\ninto the place between? Air cannot succeed in an instant, because a\nbody, and consequently cannot be moved through the least space in an\ninstant. But emptiness is not a body, nor is moved, but is made by the\nact itself of separation. There is therefore, if you admit vacuum, no\nnecessity at all for the air to fill the space left in an instant. And\ntherefore, with what ease the marble coming off presseth out the vacuum\nof the air behind it, with the same ease will the marbles be pulled\nasunder. Seeing then, if there were vacuum, there would be no difficulty\nof separation, it follows, because there is difficulty of separation,\nthat there is no vacuum.\n\n_A._ Well, now, supposing the world full, how do you prove it possible\nto pull those marbles asunder?\n\n_B._ Take a piece of soft wax; do not you think the one half touches the\nother half as close as the smoothest marbles? Yet you can pull them\nasunder. But how? Still as you pull, the wax grows continually more and\nmore slender; there being a perpetual parting or discession of the\noutermost part of the wax one from another, which the air presently\nfills; and so there is a continual lessening of the wax, till it be no\nbigger than a hair, and at last separation. If you can do the same to a\npillar of marble, till the outside give way, the effect will be the\nsame, but much quicker, after it once begins to break in the\nsuperficies; because the force that can master the first resistance of\nthe hardness, will quickly dispatch the rest.\n\n_A._ It seems so by the brittleness of some hard bodies. But I shall\nafterward put some questions to you, touching the nature of hardness.\nBut now to return to our subject. What reason can you render (without\nsupposing vacuum) of the effects produced in the engine they use at\nGresham college?\n\n_B._ That engine produceth the same effects that a strong wind would\nproduce in a narrow room.\n\n_A._ How comes the wind in? You know the engine is a hollow round pipe\nof brass, into which is thrust a cylinder of wood covered with leather,\nand fitted to the cylinder so exactly as no air can possibly pass\nbetween the leather and the brass?\n\n_B._ I know it; and that they may thrust it up, there is a hole left in\nthe cylinder to let the air out before it, which they can stop when they\nplease. There is also in the bottom of the cylinder a passage into a\nhollow globe of glass, which passage they can also open and shut at\npleasure. And at the top of that globe there is a wide mouth to put in\nwhat they please to try conclusions on, and that also to be opened and\nshut as shall be needful. It is of the nature of a pop-gun which\nchildren use, but great, costly, and more ingenious. They thrust forward\nand pull back the wooden cylinder (because it requires much strength)\nwith an iron screw. What is there in all this to prove the possibility\nof vacuum.\n\n_A._ When this wooden cylinder covered with leather, fit and close, is\nthrust home to the bottom, and the holes in the hollow cylinder of brass\nclose stopped, how can it be drawn back, as with the screw they draw it,\nbut that the space it leaves must needs be empty: for it is impossible\nthat any air can pass into the place to fill it?\n\n_B._ Truly I think it close enough to keep out straw and feathers, but\nnot to keep out air, nor yet matter. For suppose they were not so\nexactly close but that there were round about a difference for a small\nhair to lie between; then will the pulling back of the cylinder of wood\nforce so much air in, as in retiring it forces back, and that without\nany sensible difficulty. And the air will so much more swiftly enter as\nthe passage is left more narrow. Or if they touch, and the contact be in\nsome points and not in all, the air will enter as before, in case the\nforce be augmented accordingly. Lastly, though they touch exactly, if\neither the leather yield, or the brass, which it may do, to the force of\na strong screw, the air will again enter. Do you think it possible to\nmake two superficies so exquisitely touch in all points as you suppose,\nor leather so hard as not to yield to the force of a screw? The body of\nleather will give passage both to air and water, as you will confess\nwhen you ride in rainy and windy weather. You may therefore be assured\nthat in drawing out their wooden leather cylinder, they force in as much\nair as will fill the place it leaves, and that with as much swiftness as\nis answerable to the strength that drives it in. The effect therefore of\ntheir pumping is nothing else but a vehement wind, a very vehement wind,\ncoming in on all sides of the cylinder at once into the hollow of the\nbrass pipe, and into the hollow of the glass globe joined to it.\n\n_A._ I see the reason already of one of their wonders, which is, that\nthe cylinder they pump with, if it be left to itself, after it is pulled\nback, will swiftly go up again. You will say the air comes out again\nwith the same violence by reflection, and I believe it.\n\n_B._ This is argument enough that the place was not empty. For what can\nfetch or drive up the sucker, as they call it, if the place within were\nempty? For that there is any weight in the air to do it, I have already\ndemonstrated to be impossible. Besides, you know, when they have sucked\nout, as they think, all the air from the glass globe, they can\nnevertheless both see through it what is done, and hear a sound from\nwithin when there is any made; which, if there were no other, but there\nare many other, is argument enough that the place is still full of air.\n\n_A._ What say you to the swelling of a bladder even to bursting, if it\nbe a little blown when it is put into the receiver, for so they call the\nglobe of glass?\n\n_B._ The streams of air that from every side meeting together, and\nturning in an infinite number of small points, do pierce the bladder in\ninnumerable places with great violence at once, like so many invisible\nsmall wimbles, especially if the bladder be a little blown before it be\nput in, that it may make a little resistance. And when the air has once\npierced it, it is easy to conceive, that it must afterward by the same\nviolent motion be extended till it break. If before it break you let in\nfresh air upon it, the violence of the motion will thereby be tempered,\nand the bladder be less extended; for that also they have observed. Can\nyou imagine how a bladder should be extended and broken by being too\nfull of emptiness?\n\n_A._ How come living creatures to be killed in this receiver, in so\nlittle a time as three or four minutes of an hour?\n\n_B._ If they suck into their lungs so violent a wind thus made, you must\nneeds think it will presently stop the passage of their blood; and that\nis death; though they may recover if taken out before they be too cold.\nAnd so likewise will it put out fire; but the coals taken out whilst\nthey are hot will revive again. It is an ordinary thing in many\ncoal-pits, whereof I have seen the experience, that a wind proceeding\nfrom the sides of the pit every way, will extinguish any fire let down\ninto it, and kill the workmen, unless they be quickly taken out.\n\n_A._ If you put a vessel of water into the receiver, and then suck out\nthe air, the water will boil; what say you to that?\n\n_B._ It is like enough it will dance in so great a bustling of the air;\nbut I never heard it would be hot. Nor can I imagine how vacuum should\nmake anything dance. I hope you are by this time satisfied, that no\nexperiment made with the engine at Gresham College, is sufficient to\nprove that there is, or that there may be vacuum.\n\n_A._ The world you know is finite, and consequently, all that infinite\nspace without it is empty. Why may not some of that vacuum be brought\nin, and mingled with the air here?\n\n_B._ I know nothing in matters without the world.\n\n_A._ What say you to Torricellio\u2019s experiment in quicksilver, which is\nthis: there is a basin at A filled with quicksilver, suppose to B, and C\nD a hollow glass pipe filled with the same, which if you stop with your\nfinger at B, and so set it upright, and then if you take away your\nfinger, the quicksilver will fall from C downwards but not to the\nbottom, for it will stop by the way, suppose at D. Is it not therefore\nnecessary that that space between C and D be left empty? Or will you say\nthe quicksilver does not exactly touch the sides of the glass pipe?\n\n_B._ I will say neither. If a man thrust down into a vessel of\nquicksilver a blown bladder, will not that bladder come up to the top?\n\n_A._ Yes, certainly, or a bladder of iron, or anything else but gold.\n\n_B._ You see then that air can pierce quicksilver.\n\n_A._ Yes, with so much force as the weight of quicksilver comes to.\n\n_B._ When the quicksilver is fallen to D, there is so much the more in\nthe basin, and that takes up the place which so much air took up before.\nWhither can this air go if all the world without that glass pipe B C\nwere full? There must needs be the same or as much air come into that\nspace, which only is empty, between C and D: by what force? By the\nweight of the quicksilver between D and B. Which quicksilver weigheth\nnow upward, or else it could never have raised that part higher, which\nwas at first in the basin. So you see the weight of quicksilver can\npress the air through quicksilver up into the pipe, till it come to an\nequality of force as in D, where the weight of the quicksilver is equal\nto the force which is required in air to go through it.\n\n_A._ If a man suck a phial that has nothing in it but air, and presently\ndip the mouth of it into water, the water will ascend into the phial. Is\nnot that an argument that part of the air had been sucked out, and part\nof the room within the phial left empty?\n\n_B._ No. If there were empty space in the world, why should not there be\nalso some empty space in the phial before it was sucked? And then why\ndoes not the water rise to fill that? When a man sucks the phial he\ndraws nothing out, neither into his belly, nor into his lungs, nor into\nhis mouth; only he sets the air within the glass into a circular motion,\ngiving it at once an endeavour to go forth by the sucking, and an\nendeavour to go back by not receiving it into his mouth; and so with a\ngreat deal of labour glues his lips to the neck of the phial. Then\ntaking it off, and dipping the neck of the phial into the water before\nthe circulation ceases, the air, with the endeavour it hath now gotten,\npierces the water and goes out: and so much air as goes out, so much\nmatter comes up into the room of it.\n\n\n ==========\n\n\n CHAPTER IV.\n PROBLEMS OF HEAT AND LIGHT.\n\n_A._ What is the cause of heat?\n\n_B._ How know you, that any thing is hot but yourself?\n\n_A._ Because I perceive by sense it heats me.\n\n_B._ It is no good argument, the thing heats me; therefore it is hot.\nBut what alteration do you find in your body at any time by being hot?\n\n_A._ I find my skin more extended in summer than in winter; and am\nsometimes fainter and weaker then ordinary, as if my spirits were\nexhaled; and I sweat.\n\n_B._ Then that is it you would know the cause of. I have told you before\nthat by the motion I suppose both in the sun, and in the earth, the air\nis dissipated, and consequently that there would be an infinite number\nof small empty places, but that the world being full, there comes from\nthe next parts other air into the spaces they would else make empty.\nWhen therefore this motion of the sun is exercised upon the superficies\nof the earth, if there do not come out of the earth itself some corporal\nsubstance to supply that tearing of the air, we must return again to the\nadmission of vacuum. If there do, then you see how by this motion fluid\nbodies are made to exhale out of the earth. The like happens to a man\u2019s\nbody or hand, which when he perceives, he says he is hot. And so of the\nearth when it sendeth forth water and earth together in plants, we say\nit does it by heat from the sun.\n\n_A._ It is very probable, and no less probable, that the same action of\nthe sun is that which from the sea and moist places of the earth, but\nespecially from the sea, fetcheth up the water into the clouds. But\nthere be many ways of heating besides the action of the sun or of fire.\nTwo pieces of wood will take fire if in turning they be pressed\ntogether.\n\n_B._ Here again you have a manifest laceration of the air by the\nreciprocal and contrary motions of the two pieces of wood, which\nnecessarily causeth a coming forth of whatsoever is aereal or fluid\nwithin them, and (the motion pursued) a dissipation also of the other\nmore solid parts into ashes.\n\n_A._ How comes it to pass that a man is warmed even to sweating, almost\nwith every extraordinary labour of his body?\n\n_B._ It is easy to understand, how by that labour all that is liquid in\nhis body is tossed up and down, and thereby part of it also cast forth.\n\n_A._ There be some things that make a man hot without sweat or other\nevaporation, as caustics, nettles, and other things.\n\n_B._ No doubt. But they touch the part they so heat, and cannot work\nthat effect at any distance.\n\n_A._ How does heat cause light, and that partially, in some bodies more,\nin some less, though the heat be equal?\n\n_B._ Heat does not cause light at all. But in many bodies, the same\ncause, that is to say, the same motion, causeth both together; so that\nthey are not to one another as cause and effect, but are concomitant\neffects sometimes of one and the same motion.\n\n_A._ How?\n\n_B._ You know the rubbing or hard pressing of the eye, or a stroke upon\nit, makes an apparition of light without and before it, which way soever\nyou look. This can proceed from nothing else but from the restitution of\nthe organ pressed or stricken, unto its former ordinary situation of\nparts. Does not the sun by his thrusting back the air upon your eyes\npress them? Or do not those bodies whereon the sun shines, though by\nreflection, do the same, though not so strongly? And do not the organs\nof sight, the eye, the heart, and brains, resist that pressure by an\nendeavour of restitution outwards? Why then should there not be without\nand before the eye, an apparition of light in this case as well as in\nthe other?\n\n_A._ I grant there must. But what is that which appears after the\npressing of the eye? For there is nothing without that was not there\nbefore; or if there were, methinks another should see it better, or as\nwell as he; or if in the dark, methinks it should enlighten the place.\n\n_B._ It is a fancy, such as is the appearance of your face in a\nlooking-glass; such as is a dream; such as is a ghost; such as is a spot\nbefore the eye that hath stared upon the sun or fire. For all these are\nof the regiment of fancy, without any body concealed under them, or\nbehind them, by which they are produced.\n\n_A._ And when you look towards the sun or moon, why is not that also\nwhich appears before your eyes at that time a fancy?\n\n_B._ So it is. Though the sun itself be a real body, yet that bright\ncircle of about a foot diameter cannot be the sun, unless there be two\nsuns, a greater and a lesser. And because you may see that which you\ncall the sun, both above you in the sky, and before you in the water,\nand two suns, by distorting your eye, in two places in the sky, one of\nthem must needs be fancy. And if one, both. All sense is fancy, though\nthe cause be always in a real body.\n\n_A._ I see by this that those things which the learned call the\naccidents of bodies, are indeed nothing else but diversity of fancy, and\nare inherent in the sentient, and not in the objects, except motion and\nquantity. And I perceive by your doctrine you have been tampering with\n_Leviathan_ . But how comes wood with a certain degree of heat to shine,\nand iron also with a greater degree; but no heat at all to be able to\nmake water shine?\n\n_B._ That which shineth hath the same motion in its parts that I have\nall this while supposed in the sun and earth. In which motion there must\nneeds be a competent degree of swiftness to move the sense, that is, to\nmake it visible. All bodies that are not fluid will shine with heat, if\nthe heat be very great. Iron will shine and gold will shine; but water\nwill not, because the parts are carried away before they attain to that\ndegree of swiftness which is requisite.\n\n_A._ There are many fluid bodies whose parts evaporate, and yet they\nmake a flame, as oil, and wine, and other strong drinks.\n\n_B._ As for oil I never saw any inflamed by itself, how much soever\nheated, therefore I do not think they are the parts of the oil, but of\nthe combustible body oiled that shine; but the parts of wine and strong\ndrinks have partly a strong motion of themselves, and may be made to\nshine, but not with boiling, but by adding to them as they rise the\nflame of some other body.\n\n_A._ How can it be known that the particles of wine have such a motion\nas you suppose?\n\n_B._ Have you ever been so much distempered with drinking wine, as to\nthink the windows and table move?\n\n_A._ I confess, though you be not my confessor, I have; but very seldom;\nand I remember the window seemed to go and come in a kind of circling\nmotion, such as you have described. But what of that?\n\n_B._ Nothing, but that it was the wine that caused it; which having a\ngood degree of that motion before, did, when it was heated in the veins,\ngive that concussion, which you thought was in the window, to the veins\nthemselves, and, by the continuation of the parts of man\u2019s body, to the\nbrain; and that was it which made the window seem to move.\n\n_A._ What is flame? For I have often thought the flame that comes out of\na small heap of straw to be more, before it hath done flaming, than a\nhundred times the straw itself.\n\n_B._ It was but your fancy. If you take a stick in your hand by one end,\nthe other end burning, and move it swiftly, the burning end, if the\nmotion be circular, shall seem a circle; if straight, a straight line of\nfire, longer or shorter, according to the swiftness of the motion, or\nthe space it moves in. You know the cause of that.\n\n_A._ I think it is, because the impression of that visible object, which\nwas made at the first instant of the motion, did last till it was ended.\nFor then it will follow that it must be visible all the way, the\nimpressions in all points of the time being equal.\n\n_B._ The cause can be no other. The smallest spark of fire flying up\nseems a line drawn upward; and again by that swift circular motion which\nwe have supposed for the cause of light, seems also broader than it is.\nAnd consequently the flame of every thing must needs seem much greater\nthan it is.\n\n_A._ What are those sparks that fly out of the fire?\n\n_B._ They are small pieces of the wood or coals, or other fuel loosened\nand carried away with the air that cometh up with them. And being\nextinguished before their parts be quite dissipated into others, are so\nmuch soot, and black, and may be fired again.\n\n_A._ A spark of fire may be stricken out of a cold stone. It is not\ntherefore heat that makes this shining.\n\n_B._ No it is the motion that makes both the heat and shining; and the\nstroke makes the motion. For every of those sparks, is a little parcel\nof the stone, which swiftly moved, imprinteth the same motion into the\nmatter prepared, or fit to receive it.\n\n_A._ How comes the light of the sun to burn almost any combustible\nmatter by refraction through a convex glass, and by reflection from a\nconcave?\n\n_B._ The air moved by the sun presseth the convex glass in such manner\nas the action continued through it, proceedeth not in the same straight\nline by which it proceeded from the sun, but tendeth more toward the\ncentre of the body it enters. Also when the action is continued through\nthe convex body, it bendeth again the same way. By which means the whole\naction of the sun-beams are enclosed within a very small compass: in\nwhich place therefore there must be a very vehement motion; and\nconsequently, if there be in that place combustible matter, such as is\nnot very hard to kindle, the parts of it will be dissipated, and receive\nthat motion which worketh on the eye as other fire does.\n\nThe same reason is to be given for burning by reflection. For there also\nthe beams are collected into almost a point.\n\n_A._ Why may not the sun-beams be such a body as we call fire, and pass\nthrough the pores of the glass so disposed as to carry them to a point,\nor very near?\n\n_B._ Can there be a glass that is all pores? if there cannot, then\ncannot this effect be produced by the passing of fire through the pores.\nYou have seen men light their tobacco at the sun with a burning glass,\nor with a ball of crystal, held which way they will indifferently. Which\nmust be impossible, unless the ball were all pores. Again, neither you\nnor I can conceive any other fire than we have seen, nor than such as\nwater will put out. But not only a solid globe of glass or crystal will\nserve for a burning-glass, but also a hollow one filled with water. How\nthen does the fire from the sun pass through the glass of water without\nbeing put out before it come to the matter they would have it burn?\n\n_A._ I know not. There comes nothing from the sun. If there did, there\nis come so much from it already, that at this day we had had no sun.\n\n\n ==========\n\n\n CHAPTER V.\n PROBLEMS OF HARD AND SOFT.\n\n_A._ What call you hard, and what soft?\n\n_B._ That body whereof no one part is easily put out of its place,\nwithout removing the whole, is that which I and all men call hard; and\nthe contrary soft. So that they are but degrees one of another.\n\n_A._ What is the cause that makes one body harder than another, or,\nseeing you say they are but degrees of one another, what makes one body\nsofter than another, and the same body sometimes harder, sometimes\nsofter?\n\n_B._ The same motion which we have supposed from the beginning for the\ncause of so many other effects. Which motion not being upon the centre\nof the part moved, but the part itself going in another circle to and\nagain, it is not necessary that the motion be perfectly circular. For it\nis not circulation, but the reciprocation, I mean the to and again, that\ndoes cast off, and lacerate the air, and consequently produce the\nfore-mentioned effects.\n\nFor the cause therefore of hardness, I suppose the reciprocation of\nmotion in those things which are hard, to be very swift, and in very\nsmall circles.\n\n_A._ This is somewhat hard to believe. I would you could supply it with\nsome visible experience.\n\n_B._ When you see, for example, a cross-bow bent, do you think the parts\nof it stir?\n\n_A._ No. I am sure they do not.\n\n_B._ How are you sure? You have no argument for it, but that you do not\nsee the motion. When I see you sitting still, must I believe there is no\nmotion in your parts within, when there are so many arguments to\nconvince me there is.\n\n_A._ What argument have you to convince me that there is motion in a\ncross-bow when it stands bent?\n\n_B._ If you cut the string, or any way set the bow at liberty, it will\nhave then a very visible motion. What can be the cause of that?\n\n_A._ Why the setting of the bow at liberty.\n\n_B._ If the bow had been crooked before it was bent, and the string tied\nto both ends, and then cut asunder, the bow would not have stirred.\nWhere lies the difference?\n\n_A._ The bow bent has a spring; unbent it has none, how crooked soever.\n\n_B._ What mean you by spring?\n\n_A._ An endeavour of restitution to its former posture.\n\n_B._ I understand spring as well as I do endeavour.\n\n_A._ I mean a principle or beginning of motion in a contrary way to that\nof the force which bent it.\n\n_B._ But the beginning of motion is also motion, how insensible soever\nit be. And you know that nothing can give a beginning of motion to\nitself. What is it therefore that gives the bow (which you say you are\nsure was at rest when it stood bent) its first endeavour to return to\nits former posture?\n\n_A._ It was he that bent it.\n\n_B._ That cannot be. For he gave it an endeavour to come forward, and\nthe bow endeavours to go backward.\n\n_A._ Well, grant that endeavour be motion, and motion in the bow unbent,\nhow do you derive from thence, that being set at liberty it must return\nto its former posture?\n\n_B._ Thus there being within the bow a swift (though invisible) motion\nof all the parts, and consequently of the whole; the bending causeth\nthat motion, which was along the bow (that was beaten out when it was\nhot into that length) to operate across the length in every part of it,\nand the more by how much it is more bent; and consequently endeavours to\nunbend it all the while it stands bent. And therefore when the force\nwhich kept it bent is removed, it must of necessity return to the\nposture it had before.\n\n_A._ But has that endeavour no effect at all before the impediment be\nremoved? For if endeavour be motion, and every motion have some effect\nmore or less, methinks this endeavour should in time produce something.\n\n_B._ So it does. For in time (in a long time) the course of this\ninternal motion will lie along the bow, not according to the former, but\nto the new acquired posture. And then it well be as uneasy to return it\nto its former posture, as it was before to bend it.\n\n_A._ That is true. For bows long bent lose their appetite to\nrestitution, long custom becoming nature. But from this internal\nreciprocation of the parts, how do you infer the hardness of the whole\nbody?\n\n_B._ If you apply force to any single part of such a body, you must\nneeds disorder the motion of the next parts to it before it yield, and\nthere disordered, the motion of the next again must also be disordered;\nand consequently no one part can yield without force sufficient to\ndisorder all: but then the whole body must also yield. Now when a body\nis of such a nature as no single part can be removed without removing\nthe whole, men say that body is hard.\n\n_A._ Why does the fire melt divers hard bodies, and yet not all?\n\n_B._ The hardest bodies are those wherein the motion of the parts are\nthe most swift, and yet in the least circles. Wherefore if the fire, the\nmotion of whose parts are swift, and in greater circles, be made so\nswift, as to be strong enough to master the motion of the parts of the\nhard body, it will make those parts to move in a greater compass, and\nthereby weaken their resistance, that is to say, soften them, which is a\ndegree of liquefaction. And when the motion is so weakened, as that the\nparts lose their coherence by the force of their own weight, then we\ncount the body melted.\n\n_A._ Why are the hardest things the most brittle, insomuch that what\nforce soever is enough to bend them, is enough also to break them?\n\n_B._ In bending a hard body, as (for example) a rod of iron, you do not\nenlarge the space of the internal motion of the parts of iron, as the\nfire does; but you master and interrupt the motion, and that chiefly in\none place. In which place the motion that makes the iron hard being once\novercome, the prosecution of that bending must needs suddenly master the\nmotions of the parts next unto it, being almost mastered before.\n\n_A._ I have seen a small piece of glass, the figure whereof is this, A A\nB C. Which piece of glass if you bend toward the top, as in C, the whole\nbody will shatter asunder into a million of pieces, and be like to so\nmuch dust. I would fain see you give a probable reason of that.\n\n_B._ I have seen the experiment. The making of the glass is thus: they\ndip an iron rod into the molten glass that stands in a vessel within the\nfurnace. Upon which iron rod taken out, there will hang a drop of molten\nbut tough metal of the figure you have described, which they let fall\ninto the water. So that the main drop comes first to the water, and\nafter it the tail, which though straight whilst it hung on the end of\nthe rod, yet by falling into the water becomes crooked. Now you know the\nmaking of it, you may consider what must be the consequence of it.\nBecause the main drop A comes first to the water, it is therefore first\nquenched, and consequently the motion of the parts of that drop, which\nby the fire were made to be moved in a larger compass, is by the water\nmade to shrink into lesser circles towards the other end B, but with the\nsame or not much less swiftness.\n\n_A._ Why so?\n\n_B._ If you take any long piece of iron, glass, or other uniform and\ncontinued body; and having heated one end thereof, you hold the other\nend in your hand, and so quench it suddenly, though before you held it\neasily enough, yet now it will burn your fingers.\n\n_A._ It will so.\n\n_B._ You see then how the motion of the parts from A toward C is made\nmore violent and in less compass by quenching the other parts first.\nBesides, the whole motion that was in all the parts of the main drop A,\nis now united in the small end B C. And this I take to be the cause why\nthat small part B C is so exceeding stiff. Seeing also this motion in\nevery small part of the glass, is not only circular, but proceeds also\nall along the glass from A to B, the whole motion compounded will be\nsuch as the motion of spinning any soft matter into thread, and will\ndispose the whole body of the glass in threads, which in other hard\nbodies are called the grain. Therefore if you bend this body (for\nexample) in C (which to do will require more force than a man would\nthink that has not tried) those threads of glass must needs be all bent\nat the same time, and stand so, till by the breaking of the glass at C,\nthey be all at once set at liberty; and then all at once being suddenly\nunbent, like so many brittle and overbent bows, their strings breaking,\nbe shivered in pieces.\n\n_A._ It is like enough to be so. And if nature have betrayed herself in\nany thing, I think it is in this, and in that other experience of the\ncrossbow; which strongly and evidently demonstrates the internal\nreciprocation of the motion, which you suppose to be in the internal\nparts of every hard body. And I have observed somewhat in\nlooking-glasses which much confirms that there is some such motion in\nthe internal parts of glass, as you have supposed for the cause of\nhardness. For let the glass be A B, and let the object at C be a candle,\nand the eye at D. Now by divers reflections and refractions in the two\nsuperficies of the glass, if the lines of vision be very oblique, you\nshall see many images of the candle, as E, F, G, in such order and\nposition as is here described. But if you remove your eye to C, and the\ncandle to D, they will appear in a situation manifestly different from\nthis. Which you will yet more plainly perceive if the looking-glass be\ncoloured, as I have observed in red and blue glasses; and could never\nconceive any probable cause of it, till now you tell me of this secret\nmotion of the parts across the grain of the glass, acquired by cooling\nit this or that way.\n\n_B._ There be very many kinds of hard bodies, metals, stones, and other\nkinds, in the bowels of the earth, that have been there ever since the\nbeginning of the world; and I believe also many different sorts of\njuices that may be made hard. But for one general cause of hardness it\ncan be no other than such an internal motion of parts as I have already\ndescribed, whatsoever may be the cause of the several concomitant\nqualities of their hardness in particular.\n\n_A._ We see water hardened every frosty day. It is likely therefore you\nmay give a probable cause of ice. What is the cause of freezing of the\nocean towards the poles of the earth?\n\n_B._ You know the sun being always between the tropics, and (as we have\nsupposed) always casting off the air; and the earth likewise casting it\noff from itself, there must needs on both sides be a great stream of air\ntowards the poles, shaving the superficies of the earth and sea, in the\nnorthern and southern climates. This shaving of the earth and sea by the\nstream of air must needs contract and make to shrink those little\ncircles of the internal parts of earth and water, and consequently\nharden them, first at the superficies, into a thin skin, which is the\nfirst ice; and afterwards the same motion continuing, and the first ice\nco-operating, the ice becomes thicker. And this I conceive to be the\ncause of the freezing of the ocean.\n\n_A._ If that be the cause, I need not ask how a bottle of water is made\nto freeze in warm weather with snow, or ice mingled with salt. For when\nthe bottle is in the midst of it, the wind that goeth out both of the\nsalt and of the ice as they dissolve, must needs shave the superficies\nof the bottle, and the bottle work accordingly on the water without it,\nand so give it first a thin skin, and at last thicken it into a solid\npiece of ice. But how comes it to pass that water does not use to freeze\nin a deep pit?\n\n_B._ A deep pit is a very thick bottle, and such as the air cannot come\nat but only at the top, or where the earth is very loose and spungy.\n\n_A._ Why will not wine freeze as well as water?\n\n_B._ So it will when the frost is great enough. But the internal motion\nof the parts of wine and other heating liquors is in greater circles and\nstronger than the motion of the parts of water; and therefore less\neasily to be frozen, especially quite through, because those parts that\nhave the strongest motion retire to the centre of the vessel.\n\n\n ==========\n\n\n CHAPTER VI.\n PROBLEMS OF RAIN, WIND, AND OTHER WEATHER.\n\n_A._ What is the original cause of rain? And how is it generated?\n\n_B._ The motion of the air (such as I have described to you already)\ntending to the disunion of the parts of the air, must needs cause a\ncontinual endeavour (there being no possibility of vacuum) of whatsoever\nfluid parts there are upon the face of the earth and sea, to supply the\nplace which would else be empty. This makes the water, and also very\nsmall and loose parts of the earth and sea to rise, and mingle\nthemselves with the air, and to become mist and clouds. Of which the\ngreatest quantity arise there, where there is most water, namely, from\nthe large parts of the ocean; which are the South Sea, the Indian Sea,\nand the sea that divided Europe and Africa from America; over which the\nsun for the greatest part of the year is perpendicular, and consequently\nraiseth a greater quantity of water; which afterwards gathered into\nclouds, falls down in rain.\n\n_A._ If the sun can thus draw up the water, though but in small drops,\nwhy can it not as easily hold it up?\n\n_B._ It is likely it would also hold them up, if they did not grow\ngreater by meeting together, nor were carried away by the air towards\nthe poles.\n\n_A._ What makes them gather together?\n\n_B._ It is not improbable that they are carried against hills, and there\nstopt till more overtake them. And when they are carried towards the\nNorth or South where the force of the sun is more oblique and thereby\nweaker, they descend gently by their own weight. And because they tend\nall to the centre of the earth, they must needs be united in their way\nfor want of room, and so grow bigger. And then it rains.\n\n_A._ What is the reason it rains so seldom, but snows so often upon very\nhigh mountains?\n\n_B._ Because, perhaps, when the water is drawn up higher than the\nhighest mountains, where the course of the air between the equator and\nthe poles is free from stoppage, the stream of the air freezeth it into\nsnow. And it is in those places only where the hills shelter it from\nthat stream, that it falls in rain.\n\n_A._ Why is there so little rain in Egypt, and yet so much in other\nparts nearer the equinoxial, as to make the Nile overflow the country?\n\n_B._ The cause of the falling of rain I told you was the stopping, and\nconsequently the collection of clouds about great mountains, especially\nwhen the sun is near the equinoxial, and thereby draws up the water more\npotently, and from greater seas. If you consider therefore that the\nmountains in which are the springs of Nile, lie near the equinoxial and\nare exceedingly great, and near the Indian Sea, you will not think it\nstrange there should be great store of snow. This as it melts makes the\nrain of Nile to rise, which in April and May going on toward Egypt\narrive there about the time of the solstice, and overflow the country.\n\n_A._ Why should not the Nile then overflow that country twice a year,\nfor it comes twice a-year to the equinoxial.\n\n_B._ From the autumnal equinox, the sun goeth on toward the southern\ntropic, and therefore cannot dissolve the snow on that side of the hills\nthat looks towards Egypt.\n\n_A._ But then there ought to be such another inundation southward.\n\n_B._ No doubt but there is a greater descent of water there in their\nsummer than at other times, as there must be wheresoever there is much\nsnow melted. But what should that inundate, unless it should overflow\nthe sea that comes close to the foot of those mountains? And for the\ncause why it seldom rains in Egypt, it may be this, that there are no\nvery high hills near it to collect the clouds. The mountains whence Nile\nriseth being near two thousand miles off. The nearest on one side are\nthe mountains of Nubia, and on the other side Sina and the mountains of\nArabia.\n\n_A._ Whence think you proceed the winds?\n\n_B._ From the motion, I think, especially of the clouds, partly also\nfrom whatsoever is moved in the air.\n\n_A._ It is manifest that the clouds are moved by the winds; so that\nthere were winds before any clouds could be moved. Therefore I think you\nmake the effect before the cause.\n\n_B._ If nothing could move a cloud but wind, your objection were good.\nBut you allow a cloud to descend by its own weight. But when it so\ndescends, it must needs move the air before it, even to the earth, and\nthe earth again repel it, and so make lateral winds every way, which\nwill carry forward other clouds if there be any in their way, but not\nthe cloud that made them. The vapour of the water rising into clouds,\nmust needs also, as they rise, raise a wind.\n\n_A._ I grant it. But how can the slow motion of a cloud make so swift a\nwind as it does?\n\n_B._ It is not one or two little clouds, but many and great ones that do\nit. Besides, when the air is driven into places already covered, it\ncannot but be much the swifter for the narrowness of the passage.\n\n_A._ Why does the south wind more often than any other bring rain with\nit?\n\n_B._ Where the sun hath most power, and where the seas are greatest,\nthat is in the south, there is most water in the air; which a south wind\ncan only bring to us. But I have seen great showers of rain sometimes\nalso when the wind hath been north, but it was in summer, and came\nfirst, I think, from the south or west, and was brought back from the\nnorth.\n\n_A._ I have seen at sea very great waves when there was no wind at all.\nWhat was it then that troubled the water?\n\n_B._ But had you not wind enough presently after?\n\n_A._ We had a storm within a little more than a quarter of an hour\nafter.\n\n_B._ That storm was then coming and had moved the water before it. But\nthe wind you could not perceive, for it came downwards with the\ndescending of the clouds, and pressing the water bounded above your sail\ntill it came very near. And that was it that made you think there was no\nwind at all.\n\n_A._ How comes it to pass that a ship should go against the wind which\nmoves it, even almost point blank, as if it were not driven but drawn?\n\n_B._ You are to know first, that what body soever is carried against\nanother body, whether perpendicularly or obliquely, it drives it in a\nperpendicular to the superficies it lighteth on. As for example, a\nbullet shot against a flat wall, maketh the stone, or other matter it\nhits, to retire in a perpendicular to that flat; or, if the wall be\nround, towards the centre, that is to say, perpendicularly. For if the\nway of the motion be oblique to the wall, the motion is compounded of\ntwo motions, one parallel to the wall, and the other perpendicular. By\nthe former whereof the bullet is carried along the wall side, by the\nother it approacheth to it. Now the former of these motions can have no\neffect upon it; all the battery is from the motion perpendicular, in\nwhich it approacheth, and therefore the part it hits must also retire\nperpendicularly. If it were not so, a bullet with the same swiftness\nwould execute as much obliquely shot, as perpendicularly, which you know\nit does not.\n\n_A._ How do you apply this to a ship?\n\n_B._ Let A B be the ship, the head of it A. If the wind blow just from A\ntowards B, it is true the ship cannot go forward howsoever the sail be\nset. Let C D be perpendicular to the ship, and let the sail E C be never\nso little oblique to it, and F C perpendicular to E C, and then you see\nthe ship will gain the space D F to the headward.\n\n_A._ It will so; but when it is very near to the wind it will go forward\nvery slowly, and make more way with her side to the leeward.\n\n_B._ It will indeed go slower in the proportion of the line A E to the\nline C E. But the ship will not go so fast as you think sideward: one\ncause is the force of that wind which lights on the side of the ship\nitself; the other is the bellying of the sail; for the former, it is not\nmuch, because the ship does not easily put from her the water with her\nside; and bellying of the sail gives some little hold for the wind to\ndrive the ship astern.\n\n_A._ For the motion sideward I agree with you; but I had thought the\nbellying of the sail had made the ship go faster.\n\n_B._ But it does not; only in a fore wind it hinders least.\n\n_A._ By this reason a broad thin board should make the best sail.\n\n_B._ You may easily foresee the great incommodities of such a sail. But\nI have seen tried in little what such a wind can do in such a case. For\nI have seen a board set upon four truckles, with a staff set up in the\nmidst of it for a mast, and another very thin and broad board fastened\nto that staff in the stead of a sail, and so set as to receive the wind\nvery obliquely, I mean so as to be within a point of the compass\ndirectly opposite to it, and so placed upon a reasonable smooth pavement\nwhere the wind blew somewhat strongly. The event was first, that it\nstood doubting whether it should stir at all or no, but that was not\nlong, and then it ran a-head extreme swiftly, till it was overthrown by\na rub.\n\n_A._ Before you leave the ship, tell me how it comes about that so small\na thing as a rudder can so easily turn the greatest ship.\n\n_B._ It is not the rudder only, there must also be a stream to do it;\nyou shall never turn a ship with a rudder in a standing pool, nor in a\nnatural current. You must make a stream from head to stern, either with\noars or with sails; when you have made such a stream, the turning of the\nrudder obliquely holds the water from passing freely, and the ship or\nboat cannot go on directly, but as the rudder inclines to the stern, so\nwill the ship turn; but this is too well known to insist upon. You have\nobserved that the rudders of the greatest ships are not very broad, but\ngo deep into the water, whereas western barges, though but small\nvessels, have their rudders much broader, which argues that the holding\nof water from passing is the true office of a rudder; and therefore to a\nship that draws much water the rudder is made deep accordingly; and in\nbarges that draw little water, the rudders being less deep, must so much\nthe more be extended in breadth.\n\n_A._ What makes snow?\n\n_B._ The same cause which, speaking of hardness, I supposed for the\ncause of ice. For the stream of air proceeding from that both the earth\nand the sun cast off the air, consequently maketh a stream of air from\nthe equinoxial towards the poles, passing amongst the clouds, shaving\nthose small drops of water whereof the clouds consist, and congeals them\nas they do the water of the sea, or of a river. And these small frozen\ndrops are that which we call snow.\n\n_A._ But then how are great drops frozen into hailstones, and that\nespecially (as we see they are) in summer?\n\n_B._ It is especially in summer, and hot weather, that the drops of\nwater which make the clouds, are great enough; but it is then also that\nclouds are sooner and more plentifully carried up. And therefore the\ncurrent of the air strengthened between the earth and the clouds,\nbecomes more swift; and thereby freezeth the drops of water, not in the\ncloud itself, but as they are falling. Nor does it freeze them\nthoroughly, the time of their falling not permitting it, but gives them\nonly a thin coat of ice; as is manifest by their sudden dissolving.\n\n_A._ Why are not sometimes also whole clouds when pregnant and ready to\ndrop, frozen into one piece of ice?\n\n_B._ I believe they are so whensoever it thunders.\n\n_A._ But upon what ground do you believe it?\n\n_B._ From the manner or kind of noise they make, namely a crack; which I\nsee not how it can possibly be made by water or any other soft bodies\nwhatsoever.\n\n_A._ Yes, the powder they call _aurum fulminans_, when thoroughly warm,\ngives just such another crack as thunder.\n\n_B._ But why may not every small grain of that _aurum fulminans_ by\nitself be heard, though a heap of them together be soft, as is any heap\nof sand. Salts of all sorts are of the nature of ice. But gold is\ndissolved into _aurum fulminans_ by nitre and other salts. And the least\ngrain of it gives a little crack in the fire by itself. And therefore\nwhen they are so warmed by degrees, the crack cannot choose but be very\ngreat.\n\n_A._ But before it be _aurum fulminans_ they use to wash away the salt\n(which they call dulcifying it), and then they dry it gently by degrees.\n\n_B._ That is, they exhale the pure water that is left in the powder, and\nleave the salt behind to harden with drying. Other powder made of salts\nwithout any gold in them will give a crack as great as _aurum\nfulminans_. A very great chemist of our times hath written, that salt of\ntartar, saltpetre, and a little brimstone ground together into a powder,\nand dried, a few grains of that powder will be made by the fire to give\nas great a clap as a musket.\n\n_A._ Methinks it were worth your trial to see what effect a quart or a\npint of _aurum fulminans_ would produce, being put into a great gun made\nstrong enough on purpose, and the breech of the gun set in hot cinders,\nso as to heat by degrees, till the powder fly.\n\n_B._ I pray you try it yourself; I cannot spare so much money.\n\n_A._ What is it that breaketh the clouds when they are frozen?\n\n_B._ In very hot weather the sun raiseth from the sea and all moist\nplaces abundance of water, and to a great height. And whilst this water\nhangs over us in clouds, or is again descending, it raiseth other\nclouds, and it happens very often that they press the air between them,\nand squeeze it through the clouds themselves very violently; which as it\npasses shaves and hardens them in the manner declared.\n\n_A._ That has already been granted; my question is what breaks them?\n\n_B._ I must here take in one supposition more.\n\n_A._ Then your basin, it seems, holds not all you have need of.\n\n_B._ It may for all this, for the supposition I add is no more but this;\nthat what internal motion I ascribe to the earth, and the other concrete\nparts of the world, is to be supposed also in every of their parts how\nsmall soever; for what reason is there to think, in case the whole earth\nhave in truth the motion I have ascribed to it, that one part of it\ntaken away, the remaining part should lose that motion. If you break a\nloadstone, both parts will retain their virtue, though weakened\naccording to the diminution of their quantity; I suppose therefore in\nevery small part of the earth the same kind of motion, which I have\nsupposed in the whole: and so I recede not yet from my basin.\n\n_A._ Let it be supposed, and withal, that abundance of earth, (which I\nsee you aim at), be drawn up together with the water. What then?\n\n_B._ Then if many pregnant clouds, some ascending and some descending\nmeet together, and make concavities between, and by the pressing out of\nthe air, as I have said before, become ice; those atoms, as I may call\nthem, of earth will, by the straining of the air through the water of\nthe clouds, be left behind, and remain in the cavities of the clouds,\nand be more in number than for the proportion of the air therein.\nTherefore for want of liberty they must needs justle one another, and\nbecome, as they are more and more straightened of room, more and more\nswift, and consequently at last break the ice suddenly and violently,\nnow in one place, and by and by in another; and make thereby so many\nclaps of thunder, and so many flashes of lightning. For the air\nrecoiling upon our eyes, is that which maketh those flashes to our\nfancy.\n\n_A._ But I have seen lightning in a very clear evening, when there has\nbeen neither thunder nor clouds.\n\n_B._ Yes, in a clear evening; because the clouds and the rain were below\nthe horizon, perhaps forty or fifty miles off; so that you could not see\nthe clouds nor hear the thunder.\n\n_A._ If the clouds be indeed frozen into ice, I shall not wonder if they\nbe sometimes also so situated, as, like looking-glasses, to make us see\nsometimes three or more suns by refraction and reflection.\n\n\n ==========\n\n\n CHAPTER VII.\n PROBLEMS OF MOTION PERPENDICULAR, AND OBLIQUE;\n OF PRESSION AND PERCUSSION; REFLECTION AND\n REFRACTION; ATTRACTION AND REPULSION.\n\n_A._ If a bullet from a certain point given, be shot against a wall\nperpendicularly, and again from the same point obliquely, what will be\nthe proportion of the forces wherewith they urge the wall? For example,\nlet the wall be A B, a point given E, a gun C E, that carries the bullet\nperpendicularly to F, and another gun D E, that carries the like bullet\nwith the same swiftness obliquely to G; in what proportion will their\nforces be upon the wall?\n\n_B._ The force of the stroke perpendicular from E to F will be greater\nthen the oblique force from E to G, in the proportion of the line E G to\nthe line E F.\n\n_A._ How can the difference be so much? Can the bullet lose so much of\nits force in the way from E to G?\n\n_B._ No; we will suppose it loseth nothing of its swiftness. But the\ncause is, that their swiftness being equal, the one is longer in coming\nto the wall than the other, in proportion of time, as E G to E F. For\nthough their swiftness be the same, considered in themselves, yet the\nswiftness of their approach to the wall is greater in E F than in E G,\nin proportion of the lines themselves.\n\n_A._ When a bullet enters not, but rebounds from the wall, does it make\nthe same angle going off, which it did falling on, as the sun-beams do?\n\n_B._ If you measure the angles close by the wall their difference will\nnot be sensible; otherwise it will be great enough, for the motion of\nthe bullet grows continually weaker. But it is not so with the sun-beams\nwhich press continually and equally.\n\n_A._ What is the cause of reflection? When a body can go no further on,\nit has lost its motion. Whence then comes the motion by which it\nreboundeth?\n\n_B._ This motion of rebounding or reflecting proceedeth from the\nresistance. There is a difference to be considered between the\nreflection of light, and of a bullet, answerable to their different\nmotions, pressing and striking. For the action which makes reflection of\nlight, is the pressure of the air upon the reflecting body, caused by\nthe sun, or other shining body, and is but a contrary endeavour; as if\ntwo men should press with their breasts upon the two ends of a staff,\nthough they did not remove one another, yet they would find in\nthemselves a great disposition to press backward upon whatsoever is\nbehind them, though not a total going out of their places. Such is the\nway of reflecting light. Now, when the falling on of the sun-beams is\noblique, the action of them is nevertheless perpendicular to the\nsuperficies it falls on. And therefore the reflecting body, by\nresisting, turneth back that motion perpendicularly, as from F to E; but\ntaketh nothing from the force that goes on parallel in the line of E H,\nbecause the motion never presses. And thus of the two motions from F to\nE, and from E to H, is a compounded motion in the line F H, which maketh\nan angle in B G, equal to the angle F G E.\n\nBut in percussion (which is the motion of the bullet against a wall,)\nthe bullet no sooner goeth off than it loseth of its swiftness, and\ninclineth to the earth by its weight. So that the angles made in falling\non and going off, cannot be equal, unless they be measured close to the\npoint where the stroke is made.\n\n_A._ If a man set a board upright upon its edge, though it may very\neasily be cast down with a little pressure of one\u2019s finger, yet a bullet\nfrom a musket shall not throw it down, but go through it. What is the\ncause of that?\n\n_B._ In pressing with your finger you spend time to throw it down. For\nthe motion you give to the part you touch is communicated to every other\npart before it fall. For the whole cannot fall till every part be moved.\nBut the stroke of a bullet is so swift, as it breaks through, before the\nmotion of the part it hits can be communicated to all the other parts\nthat must fall with it.\n\n_A._ The stroke of a hammer will drive a nail a great way into a piece\nof wood on a sudden. What weight laid upon the head of a nail, and in\nhow much time will do the same? It is a question I have heard propounded\namongst naturalists.\n\n_B._ The different manner of the operation of weight from the operation\nof a stroke, makes it incalculable. The suddenness of the stroke upon\none point of the wood takes away the time of resistance from the rest.\nTherefore the nail enters so far as it does. But the weight not only\ngives them time, but also augments the resistance; but how much, and in\nhow much time, is, I think, impossible to determine.\n\n_A._ What is the difference between reflection and recoiling?\n\n_B._ Any reflection may, and not unproperly, be called recoiling; but\nnot contrariwise every recoiling reflection. Reflection is always made\nby the reaction of a body pressed or stricken; but recoiling not always.\nThe recoiling of a gun is not caused by its own pressing upon the\ngunpowder, but by the force of the powder itself, inflamed and moved\nevery way alike.\n\n_A._ I had thought it had been by the sudden re-entering of the air\nafter the flame and bullet were gone out. For it is impossible that so\nmuch room as is left empty by the discharging of the gun, should be so\nsuddenly filled with the air that entereth at the touchhole.\n\n_B._ The flame is nothing but the powder itself, which scattered into\nits smallest parts, seems of greater bulk by much, than in truth it is,\nbecause they shine. And as the parts scatter more and more, so still\nmore air gets between them, entering not only at the touchhole, but also\nat the mouth of the gun, which two ways being opposite, it will be much\ntoo weak to make the gun recoil.\n\n_A._ I have heard that a great gun charged too much or too little, will\nshoot, not above, nor below, but beside the mark; and charged with one\ncertain charge between both, will hit it.\n\n_B._ How that should be I cannot imagine. For when all things in the\ncause are equal, the effects cannot be unequal. As soon as fire is\ngiven, and before the bullet be out, the gun begins to recoil. If then\nthere be any unevenness or rub in the ground more on one side than on\nthe other, it shall shoot beside the mark, whether too much, or too\nlittle, or justly charged; because if the line wherein the gun recoileth\ndecline, the way of the bullet will also decline to the contrary side of\nthe mark. Therefore I can imagine no cause of this event, but either in\nthe ground it recoils on, or in the unequal weight of the parts of the\nbreech.\n\n_A._ How comes refraction?\n\n_B._ When the action is in a line perpendicular to the superficies of\nthe body wrought upon, there will be no refraction at all. The action\nwill proceed still in the same straight line, whether it be pression as\nin light, or percussion as in the shooting of a bullet. But when the\npression is oblique, then will the refraction be that way which the\nnature of the bodies through which the action proceeds shall determine.\n\n_A._ How is light refracted?\n\n_B._ If it pass through a body of less, into a body of greater\nresistance, and to the point of the superficies it falleth on, you draw\na line perpendicular to the same superficies, the action will proceed\nnot in the same line by which it fell on, but in another line bending\ntoward that perpendicular.\n\n_A._ What is the reason of that?\n\n_B._ I told you before, that the falling on worketh only in the\nperpendicular; but as soon as the action proceedeth farther inward than\na mere touch, it worketh partly in the perpendicular, and partly\nforward, and would proceed in the same line in which it fell on, but for\nthe greater resistance which now weakeneth the motion forward, and makes\nit to incline towards the perpendicular.\n\n_A._ In transparent bodies it may be so; but there be bodies through\nwhich the light cannot pass at all.\n\n_B._ But the action by which light is made, passeth through all bodies.\nFor this action is pression; and whatsoever is pressed, presseth that\nwhich is next behind, and so continually. But the cause why there is no\nlight seen through it, is the unevenness of the parts within, whereby\nthe action is by an infinite number of reflections so diverted and\nweakened, that before it hath proceeded through, it hath not strength\nleft to work upon the eye strongly enough to produce sight.\n\n_A._ If the body being transparent, the action proceed quite through,\ninto a body again of less resistance, as out of glass into the air,\nwhich way shall it then proceed in the air?\n\n_B._ From the point where it goeth forth, draw a perpendicular to the\nsuperficies of the glass, the action now freed from the resistance it\nsuffered, will go from that perpendicular, as much as it did before come\ntowards it.\n\n_A._ When a bullet from out of the air entereth into a wall of earth,\nwill that also be refracted towards the perpendicular.\n\n_B._ If the earth be all of one kind, it will. For the parallel motion,\nwill there also at the first entrance be resisted, which it was not\nbefore it entered.\n\n_A._ How then comes a bullet, when shot very obliquely into any broad\nwater, and having entered, yet to rise again into the air?\n\n_B._ When a bullet is shot very obliquely, though the motion be never so\nswift, yet the approach downwards to the water is very slow, and when it\ncometh to it, it casteth up much water before it, which with its weight\npresseth downwards again, and maketh the water to rise under the bullet\nwith force enough to master the weak motion of the bullet downwards, and\nto make it rise in such manner as bodies use to rise by reflection.\n\n_A._ By what motion (seeing you ascribe all effects to motion) can a\nloadstone draw iron to it?\n\n_B._ By the same motion hitherto supposed. But though all the smallest\nparts of the earth have this motion, yet it is not supposed that their\nmotions are in equal circles; nor that they keep just time with one\nanother; nor that they have all the same poles. If they had, all bodies\nwould draw one another alike. For such an agreement of motion, of way,\nof swiftness, and of poles, cannot be maintained, without the\nconjunction of the bodies themselves in the centre of their common\nmotion, but by violence. If therefore the iron have but so much of the\nnature of the loadstone as readily to receive from it the like motion,\nas one string of a lute doth from another string strained to the same\nnote, (as it is like enough it hath, the loadstone being but one kind of\niron ore), it must needs after that motion received from it, unless the\ngreatness of the weight hinder, come nearer to it, because at distance\ntheir motions will differ in time, and oppose each other, whereby they\nwill be forced to a common centre. If the iron be lifted up from the\nearth, the motion of the loadstone must be stronger, or the body of it\nnearer, to overcome the weight; and then the iron will leap up to the\nloadstone as swiftly, as from the same distance it would fall down to\nthe earth; but if both the stone and the iron be set floating upon the\nwater, the attraction will begin to be manifest at a greater distance,\nbecause the hindrance of the weight is in part removed.\n\n_A._ But why does the loadstone, if it float on a calm water, never fail\nto place itself at last in the meridian just north and south.\n\n_B._ Not so, just in the meridian, but almost in all places with some\nvariations. But the cause I think is, that the axis of this magnetical\nmotion is parallel to the axis of the ecliptic, which is the axis of the\nlike motion in the earth, and consequently that it cannot freely\nexercise its natural motion in any other situation.\n\n_A._ Whence may this consent of motion in the loadstone and the earth\nproceed? Do you think, as some have written, that the earth is a great\nloadstone?\n\n_B._ Dr. Gilbert, that was the first that wrote anything of this subject\nrationally, inclines to that opinion. Descartes thought the earth,\nexcepting this upper crust of a few miles depth, to be of the same\nnature with all other stars, and bright. For my part, I am content to be\nignorant; but I believe the loadstone hath been given its virtue by a\nlong habitude in the mine, the vein of it lying in the plane of some of\nthe meridians, or rather of some of the great circles that pass through\nthe poles of the ecliptic, which are the same with the poles of the like\nmotion supposed in the earth.\n\n_A._ If that be true, I need not ask why the filings of iron laid on a\nloadstone equally distant from its poles will lie parallel to the axis,\nbut on each side will incline to the pole that is next. Nor why by\ndrawing a loadstone all along a needle of iron, the needle will receive\nthe same poles. Nor why when the loadstone and iron, or two loadstones,\nare put together floating upon water, will fall one of them astern of\nthe other, that their like parts may look the same way, and their unlike\ntouch, in which action they are commonly said to repel one another. For\nall this may be derived from the union of their motions. One thing more\nI desire to know, and that is; what are those things they call spirits?\nI mean ghosts, fairies, hobgoblins, and the like apparitions.\n\n_B._ They are no part of the subject of natural philosophy.\n\n_A._ That which in all ages, and all places is commonly seen (as those\nhave been, unless a great part of mankind be liars) cannot, I think, be\nsupernatural.\n\n_B._ All this that I have hitherto said, though upon better ground than\ncan be had for a discourse of ghosts, you ought to take but for a dream.\n\n_A._ I do so. But there be some dreams more like sense then others. And\nthat which is like sense pleases me as well in natural philosophy, as if\nit were the very truth.\n\n_B._ I was dreaming also once of these things; but was wakened by their\nnoise. And they never came into any dream of mine since, unless\napparitions in dreams and ghost be all one.\n\n\n ==========\n\n\n CHAPTER VIII.\n THE DELPHIC PROBLEM, OR DUPLICATION OF THE\n CUBE.\n\n_A._ Have you seen a printed paper sent from Paris, containing the\nduplication of the cube, written in French?\n\n_B._ Yes. It was I that writ it, and sent it thither to be printed, on\npurpose to see what objections would be made to it by our professors of\nalgebra here.\n\n_A._ Then you have also seen the confutations of it by algebra.\n\n_B._ I have seen some of them; and have one by me. For there was but one\nthat was rightly calculated, and that is it which I have kept.\n\n_A._ Your demonstration then is confuted though but by one.\n\n_B._ That does not follow. For though an arithmetical calculation be\ntrue in numbers, yet the same may be, or rather must be false, if the\nunits be not constantly the same.\n\n_A._ Is their calculation so inconstant, or rather so foolish as you\nmake it?\n\n_B._ Yes. For the same number is sometimes so many lines, sometimes so\nmany planes, and sometimes so many solids; as you shall plainly see, if\nyou will take the pains to examine first a demonstration I have to prove\nthe said duplication, and after that, the algebraic calculation which is\npretended to confute it. And not only that this one is false, but also\nany other arithmetical account used in geometry, unless the numbers be\nalways so many lines, or always so many superficies, or always so many\nsolids.\n\n_A._ Let me see the geometrical demonstration.\n\n_B._ There it is. Read it.\n\n TO FIND A CUBE DOUBLE TO A CUBE GIVEN:\n\nLet the side of the cube given be V D. Produce V D to A, till A D be\ndouble to D V. Then make the square of A D, namely A B C D. Divide A B\nand C D in the middle at E and F. Draw E F. Draw also A C cutting E F in\nI. Then in the sides B C and A D take B R and A S, each of them equal to\nA I or I C.\n\nLastly, divide S D in the middle at T, and upon the centre T, with the\ndistance T V, describe a semi-circle cutting A D in Y, and D C in X.\n\nI say the cube of D X is double to the cube of D V. For the three lines\nD Y, D X, D V are in continual proportion. And continuing the\nsemi-circle V X Y till it cut the line R S, drawn and produced in Z, the\nline S Z will be equal to D X. And drawing X Z it will pass through T.\nAnd the four lines T V, T X, T Y and T Z will be equal. And therefore\njoining Y X and Y Z, the figure V X Y Z will be a rectangle.\n\n[Illustration:\n\n _Delphic Problem.\n Vol. VII. Eng. p. 60._\n]\n\nProduce C D to P so as D P be equal to A D. Now if Y Z produced fall on\nP, there will be three rectangle equiangled triangles, D P Y, D Y X, and\nD X V; and consequently four continual proportionals, D P, D Y, D X, and\nD V, whereof D X is the least of the means. And therefore the cube of D\nX will be double to the cube of D V.\n\n_A._ That is true; and the cube of D Y will be double to the cube of D\nX; and the cube of D P double to the cube of D Y. But that Y Z produced,\nfalls upon P, is the thing they deny, and which you ought to\ndemonstrate.\n\n_B._ If Y Z produced fall not on P, then draw P Y, and from V let fall a\nperpendicular upon P Y, suppose at _u._. Divide P V in the midst at\n_a._, and join _a u._; which done _a u._ will be equal to _a._ V or _a._\nP. For because V _u._ P is a right angle, the point _u._ will be in the\nsemi-circle whereof P V is the diameter.\n\nTherefore drawing V R, the angle _u._ V R will be a right angle.\n\n_A._ Why so?\n\n_B._ Because T V and T Y are equal; and T D, T S equal; S Y will also be\nequal to D V. And because D P and R S are equal and parallel, R Y will\nbe equal and parallel to P V. And therefore V R and P Y that join them\nwill be equal and parallel. And the angles P _u._ V, R V _u._ will be\nalternate, and consequently equal. But P _u._ V is a right angle;\ntherefore also R V _u._ will be a right angle.\n\n_A._ Hitherto all is evident. Proceed.\n\n_B._ From the point Y raise a perpendicular cutting V R wheresoever in\n_t._, and then (because P Y and V R are parallel) the angle Y _t_ V will\nbe a right angle. And the figure _u_ Y _t_ V a rectangle, and _u t_\nequal to Y V. But Y V is equal to Z X; and therefore Z X is equal to _u\nt_. And _u t_ must pass through the point T (for the diameters of any\nrectangle divide each other in the middle), therefore Z and _u_ are the\nsame point, and X and _t_ the same point. Therefore Y Z produced falls\nupon P. And D X is the lesser of the two means between A D and D V. And\nthe cube of D X double to the cube of D V, which was to be demonstated\n\n_A._ I cannot imagine what fault there can be in this demonstration, and\nyet there is one thing which seems a little strange to me. And it is\nthis. You take B R, which is half the diagonal, and which is the sine of\nforty-five degrees, and which is also the mean proportional between the\ntwo extremes; and yet you bring none of these proprieties into your\ndemonstration. So that though you argue from the construction, yet you\ndo not argue from the cause. And this perhaps your adversaries will\nobject, at least, against the art of your demonstration, or enquire by\nwhat luck you pitched upon half the diagonal for your foundation.\n\n_B._ I see you let nothing pass. But for answer you must know, that if a\nman argue from the negative of the truth, though he know not that it is\nthe truth which is denied, yet he will fall at last, after many\nconsequences, into one absurdity or another. For though false do often\nproduce truth, yet it produces also absurdity, as it hath done here. But\ntruth produceth nothing but truth. Therefore in demonstrations that tend\nto absurdity, it is no good logic to require all along the operation of\nthe cause.\n\n_A._ Have you drawn from hence no corollaries?\n\n_B._ No. I leave that for others that will; unless you take this for a\ncorollary, that, what arithmetical calculation soever contradicts it, is\nfalse.\n\n_A._ Let me see now the algebraical demonstration against it.\n\n_B._ Here it is:\n\n Let A B or A D be equal to 2\n Then D F or D V is equal to 1\n And B R or A S is equal to the square root of 2\n And D Y equal to 3\n want the square root of 2\n The cube of A B is equal to 8\n The cube of D Y is equal to 45\n want the square root of 1682 that is almost equal to 4\n For 45 want the square root of 1681 is equal to 4\n\nTherefore D Y is a little less then the greater of the two means between\nA D and D V.\n\n_A._ There is I see some little difference between this arithmetical and\nyour geometrical demonstration. And though it be insensible, yet if his\ncalculation be true, yours must needs be false, which I am sure cannot\nbe.\n\n_B._ His calculation is so true, that there is never a proposition in it\nfalse, till he come to the conclusion, that the cube of D Y is equal to\n45, want the square root of 1682. But that, and the rest, is false.\n\n_A._ I shall easily see that A D is certainly 2, whereof D V is 1, and A\nV is certainly 3, whereof D V is 1.\n\n_B._ Right.\n\n_A._ And B R is without doubt the square root of 2.\n\n_B._ Why, what is 2?\n\n_A._ 2 is the line A D as being double to D V which is 1.\n\n_B._ And so, the line B R is the square root of the line A D.\n\n_A._ Out upon it, it is absurd. Why do you grant it to be true in\narithmetic?\n\n_B._ In arithmetic the numbers consist of so many units, and are never\nconsidered there as nothings. And therefore every one line has some\nlatitude, and if you allow to B I, the semi-diagonal, the same latitude\nyou do to A B, or to B R, you will quickly see the square of half the\ndiagonal to be equal to twice the square of half A B.\n\n_A._ Well, but then your demonstration is not confuted; for the point Y\nwill have latitude enough to take in that little difference which is\nbetween the root of 1681 and the root of 1682. This putting off an unit\nsometimes for one line, sometimes for one square, must needs mar the\nreckoning. Again he says, the cube of A B is equal to 8; but seeing A B\nis 2, the cube of A B must be just equal to four of its own sides; so\nthat the unit which was before sometimes a line, sometimes a square, is\nnow a cube.\n\n_B._ It can be no otherwise when you so apply arithmetic to geometry, as\nto number the lines of a plane, or the planes of a cube.\n\n_A._ In the next place, I find that the cube of D Y is equal to 45, want\nthe square root of 1682. What is that 45? Lines, or squares, or cubes?\n\n_B._ Cubes; cubes of D V.\n\n_A._ Then if you add to 45 cubes of D V the square root of 1682, the sum\nwill be 45 cubes of D V; and if you add to the cube of D Y the same root\nof 1682, the sum will be the cube of D Y, plus the square root of 1682,\nand these two sums must be equal.\n\n_B._ They must so.\n\n_A._ But the square root of 1682, being a line, adds nothing to a cube;\ntherefore the cube alone of D Y, which he says is equal almost to 4\ncubes of D V, is equal to 45 cubes of the same D V.\n\n_B._ All these impossibilities do necessarily follow the confounding of\narithmetic and geometry.\n\n_A._ I pray you let me see the operation by which the cube of D Y (that\nis, the cube of 3, want the root of 2) is found equal to 45, want the\nsquare root of 1682.\n\n_B._ Here it is.\n\n A DETECTION OF THE ABSURD USE OF ARITHMETIC AS IT IS NOW APPLIED TO\n GEOMETRY.\n\n 3\u2014\u2014\u2014\u2014\u221a2\n 3\u2014\u2014\u2014\u2014\u221a2\n \u2014\u2014\u2014\u2014\u2014\u2014\u2014\n \u2014\u221a18 + 2\n 9\u2014\u221a18\n \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n 9\u2014\u221a72 + 2\n 3\u2014\u2014\u221a2\n \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n \u2014\u2014\u221a162 + 12\u2014\u2014\u221a8\n 27\u2014\u2014\u221a648 + 6\n \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\n 27\u2014\u221a658\u2014\u221a162 + 18\u2014\u221a8\n\n ======================\n\n_A._ Why for two roots of 18 do you put the root of 72.\n\n_B._ Because 2 roots of 18 are equal to one root of four times 18, which\nis 72.\n\n_A._ Next we have, that the root of 2 multiplied into 2 makes the root\nof 8. How is that true?\n\n_B._ Does it not make 2 roots of two? And is not B R the root of 2, and\n2 B R equal to the diagonal? And is not the square of the diagonal equal\nto 8 squares of D V?\n\n_A._ It is true. But here the root of 8 is put for the cube of the root\nof 2. Can a line be equal to a cube?\n\n_B._ No. But here we are in arithmetic again, and 8 is a cubic number.\n\n_A._ How does the root of 2 multiplied into the root of 72 make 12?\n\n_B._ Because it makes the root of 2 times 72, that is to say the root of\n144 which is 12.\n\n_A._ How does 9 roots of 2 make the root of 162?\n\n_B._ Because it makes the root of 2 squares of 9, that is the root of\n162.\n\n_A._ How does 3 roots of 72 make the root of 648?\n\n_B._ Because it makes the root of 9 times 72, that is of 648.\n\n_A._ For the total sum I see 27 and 18, which make 45. Therefore the\nroot of 648 together with the root of 162 and of 8, which are to be\ndeducted from 45, ought to be equal to the root of 1682.\n\n _B._ So they are. For 648 multiplied by 162 makes 104976, of\n which the double root is 648\n and 648 and 162 added together make 810\n Therefore the root of 648, added to the root of 162, makes\n the root of 1458\n Again 1458 into 8 is 11664. The double root whereof is 216\n The sum of 1458 and 8 added together 1466\n The sum of 1466 and 216 is 1682, and the root, the root of 1682\n\n_A._ I see the calculation in numbers is right, though false in lines.\nThe reason whereof can be no other than some difference between\nmultiplying numbers into lines or planes, and multiplying lines into the\nsame lines or planes.\n\n_B._ The difference is manifest. For when you multiply a number into\nlines, the product is lines; as the number 2 multiplied into 3 lines is\nno more than 3 lines 2 times told. But if you multiply lines into lines\nyou make planes, and if you multiply lines into planes you make solid\nbodies. In geometry there are but three dimensions, lengths,\nsuperficies, and body. In arithmetic there is but one, and that is\nnumber or length which you will. And though there be some numbers called\nplane, other solids, others plano-solid, others square, others cubic,\nothers square-square, others quadrato-cubic, others cubi-cubic, &c., yet\nare all these but one dimension, namely number, or a file of things\nnumbered.\n\n_A._ But seeing this way of calculation by numbers is so apparently\nfalse, what is the reason this calculation came so near the truth?\n\n_B._ It is because in arithmetic units are not nothings, and therefore\nhave breadth. And therefore many lines set together make a superficies\nthough their breadth be insensible. And the greater the number is into\nwhich you divide your line, the less sensible will be your error.\n\n_A._ Archimedes, to find a straight line equal to the circumference of a\ncircle, used this way of extracting roots. And it is the way also by\nwhich the table of sines, secants, and tangents have been calculated.\nAre they all out?\n\n_B._ As for Archimedes, there is no man that does more admire him than I\ndo: but there is no man that cannot err. His reasoning is good. But he,\nas all other geometricians before and after him, have had two principles\nthat cross one another when they are applied to one and the same\nscience. One is, that a point is no part of a line, which is true in\ngeometry, where a part of a line when it is called a point, is not\nreckoned; another is, that a unit is part of a number; which is also\ntrue; but when they reckon by arithmetic in geometry, there a unit is\nsometimes part of a line, sometimes a part of a square, and sometimes\npart of a cube. As for the table of sines, secants, and tangents, I am\nnot the first that find fault with them. Yet I deny not but they are\ntrue enough for the reckoning of acres in a map of land.\n\n_A._ What a deal of labour has been lost by them that being professors\nof geometry have read nothing else to their auditors but such stuff as\nthis you have here seen. And some of them have written great books of it\nin strange characters, such as in troublesome times, a man would suspect\nto be a cypher.\n\n_B._ I think you have seen enough to satisfy you, that what I have\nwritten heretofore concerning the quadrature of the circle, and of other\nfigures made in imitation of the parabola, has not been yet confuted.\n\n_A._ I see you have wrested out of the hands of our antagonists this\nweapon of algebra, so as they can never make use of it again. Which I\nconsider as a thing of much more consequence to the science of geometry,\nthan either of the duplication of the cube, or the finding of two mean\nproportionals, or the quadrature of a circle, or all these problems put\ntogether.\n\n FINIS.\n\n\n\n\n DECAMERON PHYSIOLOGICUM;\n\n OR,\n\n TEN DIALOGUES OF NATURAL PHILOSOPHY.\n\n BY\n\n THOMAS HOBBES\n\n OF MALMESBURY.\n\n TO WHICH IS ADDED\n\n THE PROPORTION OF A STRAIGHT LINE TO\n HALF THE ARC OF A QUADRANT,\n\n BY THE SAME AUTHOR.\n\n[Illustration]\n\n DECAMERON PHYSIOLOGICUM.\n\n\n CHAPTER I.\n OF THE ORIGINAL OF NATURAL PHILOSOPHY.\n\n_A._ I have heard exceeding highly commended a kind of thing which I do\nnot well understand, though it be much talked of, by such as have not\notherwise much to do, by the name of philosophy; and the same again by\nothers as much despised and derided: so that I cannot tell whether it be\ngood or ill, nor what to make of it, though I see many other men that\nthrive by it.\n\n_B._ I doubt not, but what so many do so highly praise must be very\nadmirable, and what is derided and scorned by many, foolish and\nridiculous. The honour and scorn falleth finally not upon philosophy,\nbut upon the professors. Philosophy is _the knowledge of natural\ncauses_. And there is no knowledge but of truth. And to know the true\ncauses of things, was never in contempt, but in admiration. Scorn can\nnever fasten upon truth. But the difference is all in the writers and\nteachers. Whereof some have neither studied, nor care for it, otherwise\nthan as a trade to maintain themselves or gain preferment; and some for\nfashion, and to make themselves fit for ingenious company: and their\nstudy hath not been meditation, but acquiescence in the authority of\nthose authors whom they have heard commended. And some, but few, there\nbe, that have studied it for curiosity, and the delight which commonly\nmen have in the acquisition of science, and in the mastery of difficult\nand subtil doctrines. Of this last sort I count Aristotle, and a few\nothers of the ancients, and some few moderns: and to these it is that\nproperly belong the praises which are given to philosophy.\n\n_A._ If I have a mind to study, for example natural philosophy, must I\nthen needs read Aristotle, or some of those that now are in request?\n\n_B._ There is no necessity of it. But if in your own meditation you\nlight upon a difficulty, I think it is no loss of time, to enquire what\nother men say of it, but to rely only upon reason. For though there be\nsome few effects of nature, especially concerning the heavens, whereof\nthe philosophers of old time have assigned very rational causes, such as\nany man may acquiesce in, as of eclipses of the sun and moon by long\nobservation, and by the calculation of their visible motions; yet what\nis that to the numberless and quotidian phenomena of nature? Who is\nthere amongst them or their successors, that has satisfied you with the\ncauses of gravity, heat, cold, light, sense, colour, noise, rain, snow,\nfrost, winds, tides of the sea, and a thousand other things which a few\nmen\u2019s lives are too short to go through, and which you and other curious\nspirits admire (as quotidian as they are), and fain would know the\ncauses of them, but shall not find them in the books of naturalists; and\nwhen you ask what are the causes of any of them, of a philosopher now,\nhe will put you off with mere words; which words, examined to the\nbottom, signify not a jot more than I cannot tell, or because it is:\nsuch as are intrinsical quality, occult quality, sympathy, antipathy,\nantiperistasis, and the like. Which pass well enough with those that\ncare not much for such wisdom, though wise enough in their own ways; but\nwill not pass with you that ask not simply what is the cause, but in\nwhat manner it comes about that such effects are produced.\n\n_A._ That is cozening. What need had they of that? When began they thus\nto play the charlatans?\n\n_B._ Need had they none. But know you not that men from their very\nbirth, and naturally, scramble for every thing they covet, and would\nhave all the world, if they could, to fear and obey them? If by fortune\nor industry one light upon a secret in nature, and thereby obtain the\ncredit of an extraordinary knowing man, should he not make use of it to\nhis own benefit? There is scarce one of a thousand but would live upon\nthe charges of the people as far as he dares. What poor geometrician is\nthere, but takes pride to be thought a conjurer? What mountebank would\nnot make a living out of a false opinion that he were a great physician?\nAnd when many of them are once engaged in the maintenance of an error,\nthey will join together for the saving of their authority to decry the\ntruth.\n\n_A._ I pray, tell me, if you can, how and where the study of philosophy\nfirst began.\n\n_B._ If we may give credit to old histories, the first that studied any\nof the natural sciences were the astronomers of Ethiopia. My author is\nDiodorus Siculus, accounted a very faithful writer, who begins his\nhistory as high as is possible, and tells us that in Ethiopia were the\nfirst astronomers; and that for their predictions of eclipses, and other\nconjunctions and aspects of the planets, they obtained of their king not\nonly towns and fields to a third part of the whole land, but were also\nin such veneration with the people, that they were thought to have\ndiscourse with their gods, which were the stars; and made their kings\nthereby to stand in awe of them, that they durst not either eat or drink\nbut what and when they prescribed; no nor live, if they said the gods\ncommanded them to die. And thus they continued in subjection to their\nfalse prophets, till by one of their kings, called Ergamenes, (about the\ntime of the Ptolemies), they were put to the sword. But long before the\ntime of Ergamenes, the race of these astrologers (for they had no\ndisciples but their own children) was so numerous, that abundance of\nthem (whether sent for or no I cannot tell) transplanted themselves into\nEgypt, and there also had their cities and lands allowed them, and were\nin request not only for astronomy and astrology, but also for geometry.\nAnd Egypt was then as it were an university to all the world, and\nthither went the curious Greeks, as Pythagoras, Plato, Thales, and\nothers, to fetch philosophy into Greece. But long before that time,\nabundance of them went into Assyria, and had their towns and lands\nassigned them also there; and were by the Hebrews called Chaldees.\n\n_A._ Why so?\n\n_B._ I cannot tell; but I find in Martinius\u2019s Lexicon they are called\nChasdim, and Chesdim, and (as he saith) from one Chesed the son of\nNachor; but I find no such man as Chesed amongst the issue of Noah in\nthe scripture. Nor do I find that there was any certain country called\nChald\u00e6a; though a town where any of them inhabited were called a town of\nthe Chaldees. Martinius saith farther, that the same word Chasdim did\nsignify also Demons.\n\n_A._ By this reckoning I should conjecture they were called Chusdim, as\nbeing a race of Ethiopians. For the land of Chus is Ethiopia; and so the\nname degenerated first into Chuldim, and then into Chaldim; so that they\nwere such another kind of people as we call gipsies; saving that they\nwere admired and feared for their knavery, and the gipsies counted\nrogues.\n\n_B._ Nay pray, except Claudius Ptolom\u00e6us, author of that great work of\nastronomy, the Almegest.\n\n_A._ I grant he was excellent both in astronomy and geometry, and to be\ncommended for his _Almegest_ ; but then for his _Judiciar Astrologie_\nannexed to it, he is again a gipsy. But the Greeks that travelled, you\nsay, into Egypt, what philosophy did they carry home?\n\n_B._ The mathematics and astronomy. But for that sublunary physics,\nwhich is commonly called natural philosophy, I have not read of any\nnation that studied it earlier than the Greeks, from whom it proceeded\nto the Romans. Yet both Greeks and Romans were more addicted to moral\nthan to natural philosophy; in which kind we have their writings, but\nloosely and incoherently, written upon no other principles than their\nown passions and presumptions, without any respect to the laws of\ncommonwealth, which are the ground and measure of all true morality. So\nthat their books tend rather to teach men to censure than to obey the\nlaws; which has been a great hindrance to the peace of the western world\never since. But they that seriously applied themselves to natural\nphilosophy were but few, as Plato and Aristotle, whose works we have;\nand Epicurus whose doctrine we have in Lucretius. The writings of\nPhilolaus and many other curious students being by fire or negligence\nnow lost: though the doctrines of Philolaus concerning the motion of the\nearth have been revived by Copernicus, and explained and confirmed by\nGalileo now of late.\n\n_A._ But methinks the natural philosophy of Plato, and Aristotle, and\nthe rest, should have been cultivated and made to flourish by their\ndisciples.\n\n_B._ Whom do you mean, the successors of Plato, Epicurus, Aristotle, and\nthe other first philosophers? It may be some of them may have been\nlearned and worthy men. But not long after, and down to the time of our\nSaviour and his Apostles, they were for the most part a sort of needy,\nignorant, impudent, cheating fellows, who by the profession of the\ndoctrine of those first philosophers got their living. For at that time,\nthe name of philosophy was so much in fashion and honour amongst great\npersons, that every rich man had a philosopher of one sect or another to\nbe a schoolmaster to his children. And these were they that feigning\nChristianity, with their disputing and readiness of talking got\nthemselves into Christian commons, and brought so many heresies into the\nprimitive Church, every one retaining still a tang of what they had been\nused to teach.\n\n_A._ But those heresies were all condemned in the first Council of Nice.\n\n_B._ Yes. But the Arian heresy for a long time flourished no less than\nthe Roman, and was upheld by divers Emperors, and never fully\nextinguished as long as there were Vandals in Christendom. Besides,\nthere arose daily other sects, opposing their philosophy to the doctrine\nof the Councils concerning the divinity of our Saviour; as how many\npersons he was, how many natures he had. And thus it continued till the\ntime of Charlemagne, when he and Pope Leo the third divided the power of\nthe empire into temporal and spiritual.\n\n_A._ A very unequal division.\n\n_B._ Why? Which of them think you had the greater share?\n\n_A._ No doubt, the Emperor: for he only had the sword.\n\n_B._ When the swords are in the hands of men, whether had you rather\ncommand the men or the swords?\n\n_A._ I understand you. For he that hath the hands of the men, has also\nthe use both of their swords and strength.\n\n_B._ The empire thus divided into spiritual and temporal, the freedom of\nphilosophy was to the power spiritual very dangerous. And for that cause\nit behoved the Pope to get schools set up not only for divinity, but\nalso for other sciences, especially for natural philosophy. Which when\nby the power of the Emperor he had effected, out of the mixture of\nAristotle\u2019s metaphysics with the Scripture, there arose a new science\ncalled School-divinity; which has been the principal learning of these\nwestern parts from the time of Charlemagne till of very late.\n\n_A._ But I find not in any of the writings of the Schoolmen in what\nmanner, from the causes they assign, the effect is naturally and\nnecessarily produced.\n\n_B._ You must not wonder at that. For you enquire not so much, when you\nsee a change of anything, what may be said to be the cause of it, as how\nthe same is generated; which generation is the entire progress of nature\nfrom the efficient cause to the effect produced. Which is always a hard\nquestion, and for the most part impossible for a man to answer to. For\nthe alterations of the things we perceive by our five senses are made by\nthe motion of bodies, for the most part, either for distance, smallness,\nor transparence, invisible.\n\n_A._ But what need had they then to assign any cause at all, seeing that\nthey could not show the effect was to follow from it?\n\n_B._ The Schools, as I said, were erected by the Pope and Emperor, but\ndirected by the Pope only, to answer and confute the heresies of the\nphilosophers. Would you have them then betray their profession and\nauthority, that is to say, their livelihood, by confessing their\nignorance? Or rather uphold the same, by putting for causes, strange and\nunintelligible words; which might serve well enough not only to satisfy\nthe people whom they relied on, but also to trouble the philosophers\nthemselves to find a fault in.\n\n_A._ Seeing you say that alteration is wrought by the motion of bodies,\npray tell me first what I am to understand by the word body.\n\n_B._ It is a hard question, though most men think they can easily answer\nit, as that it is whatsoever they can see, feel, or take notice of by\ntheir senses. But if you will know indeed what is body, we must enquire\nfirst what there is that is not body. You have seen, I suppose, the\neffects of glasses, how they multiply and magnify the object of our\nsight; as when a glass of a certain figure will make a counter or a\nshilling seem twenty, though you be well assured there is but one. And\nif you set a mark upon it, you will find the mark upon them all. The\ncounter is certainly one of those things we call bodies: are not the\nothers so too?\n\n_A._ No, without doubt. For looking through a glass cannot make them\nreally more than they are.\n\n_B._ What then be they but fancies, so many fancies of one and the same\nthing in several places?\n\n_A._ It is manifest they are so many idols, mere nothings.\n\n_B._ When you have looked upon a star or candle with both your eyes, but\none of them a little turned awry with your finger, has not there\nappeared two stars, or two candles? And though you call it a deception\nof the sight, you cannot deny but there were two images of the object.\n\n_A._ It is true, and observed by all men. And the same I say of our\nfaces seen in looking-glasses, and of all dreams, and of all apparitions\nof dead men\u2019s ghosts; and wonder, since it is so manifest, I never\nthought upon it before, for it is a very happy encounter, and such as\nbeing by everybody well understood, would utterly destroy both idolatry\nand superstition, and defeat abundance of knaves that cheat and trouble\nthe world with their devices.\n\n_B._ But you must not hence conclude that whosoever tells his dream, or\nsometimes takes his direction from it, is therefore an idolater, or\nsuperstitious, or a cheater. For God doth often admonish men by dreams\nof what they ought to do; yet men must be wary in this case that they\ntrust not dreams with the conduct of their lives farther than by the\nlaws of their country is allowed: for you know what God says, Deut.\nxiii: _If a prophet or a dreamer of dreams give thee a sign or a wonder,\nand the sign come to pass, yet if he bid thee serve other Gods let him\nbe put to death_. Here by serving other Gods (since they have chosen God\nfor their King) we are to understand revolting from their King, or\ndisobeying of his laws. Otherwise I see no idolatry nor superstition in\nfollowing a dream, as many of the Patriarchs in the Old Testament, and\nof the Saints in the New Testament did.\n\n_A._ Yes: their own dreams. But when another man shall dream, or say\nthat he has dreamed, and require me to follow that, he must pardon me if\nI ask him by what authority, especially if he look I should pay him for\nit.\n\n_B._ But if commanded by the laws you live under, you ought to follow\nit. But when there proceed from one sound divers echoes, what are those\nechoes? And when with fingers crossed you touch a small bullet, and\nthink it two; and when the same herb or flower smells well to one and\nill to another, and the same at several times, well and ill to yourself,\nand the like of tastes, what are those echoes, feelings, odours, and\ntastes?\n\n_A._ It is manifest they are all but fancies. But certainly when the sun\nseems to my eye no bigger than a dish, there is behind it somewhere\nsomewhat else, I suppose a real sun, which creates those fancies, by\nworking, one way or other, upon my eyes, and other organs of my senses,\nto cause that diversity of fancy.\n\n_B._ You say right; and that is it I mean by the word body, which\nbriefly I define to be any thing that hath a being in itself, without\nthe help of sense.\n\n_A._ Aristotle, I think, meaneth by body, _substance_, or _subjectum_,\nwherein colour, sound, and other fancies are, as he says, inherent. For\nthe word essence has no affinity with substance. And Seneca says, he\nunderstands it not. And no wonder: for essence is no part of the\nlanguage of mankind, but a word devised by philosophers out of the\ncopulation of two names, as if a man having two hounds could make a\nthird, if it were need, of their couples.\n\n_B._ It is just so. For having said in themselves, (for example): _a\ntree is a plant_, and conceiving well enough what is the signification\nof those names, knew not what to make of the word _is_, that couples\nthose names; nor daring to call it a body, they called it by a new name\n(derived from the word _est_), _essentia_, and _substantia_, deceived by\nthe idiom of their own language. For in many other tongues, and namely\nin the Hebrew, there is no such copulative. They thought the names of\nthings sufficiently connected, when they are placed in their natural\nconsequence; and were therefore never troubled with essences, nor other\nfallacy from the copulative _est_.\n\n\n ==========\n\n\n CHAPTER II.\n OF THE PRINCIPLES AND METHOD OF NATURAL\n PHILOSOPHY.\n\n_A._ This history of the old philosophers has not put me out of love,\nbut out of hope of philosophy from any of their writings. I would\ntherefore try if I could attain any knowledge therein by my own\nmeditation: but I know neither where to begin, nor which way to proceed.\n\n_B._ Your desire, you say, is to know the causes of the effects or\nphenomena of nature; and you confess they are fancies, and,\nconsequently, that they are in yourself; so that the causes you seek for\nonly are without you, and now you would know how those external bodies\nwork upon you to produce those phenomena. The beginning therefore of\nyour enquiry ought to be at; _What it is you call a cause?_ I mean an\nefficient cause: for the philosophers make four kinds of causes, whereof\nthe efficient is one. Another they call the formal cause, or simply the\nform or essence of the thing caused; as when they say, four equal angles\nand four equal sides are the cause of a square figure; or that heaviness\nis the cause that makes heavy bodies to descend; but that is not the\ncause you seek for, nor any thing but this: _It descends because it\ndescends_. The third is the material cause, as when they say, the walls\nand roof, &c. of a house are the cause of a house. The fourth is the\nfinal cause, and hath place only in moral philosophy.\n\n_A._ We will think of final causes upon some other occasion; of formal\nand material not at all: I seek only the efficient, and how it acteth\nfrom the beginning to the production of the effect.\n\n_B._ I say then, that in the first place you are to enquire diligently\ninto the nature of motion. For the variations of fancies, or (which is\nthe same thing) of the phenomena of nature, have all of them one\nuniversal efficient cause, namely the variety of motion. For if all\nthings in the world were absolutely at rest, there could be no variety\nof fancy; but living creatures would be without sense of all objects,\nwhich is little less than to be dead.\n\n_A._ What if a child new taken from the womb should with open eyes be\nexposed to the azure sky, do not you think it would have some sense of\nthe light, but that all would seem unto him darkness?\n\n_B._ Truly, if he had no memory of any thing formerly seen, or by any\nother sense perceived, (which is my supposition), I think he would be in\nthe dark. For darkness is darkness, whether it be black or blue, to him\nthat cannot distinguish.\n\n_A._ Howsoever that be, it is evident enough that whatsoever worketh is\nmoved: for action is motion.\n\n_B._ Having well considered the nature of motion, you must thence take\nyour principles for the foundation and beginning of your enquiry.\n\n_A._ As how?\n\n_B._ Explain as fully and as briefly as you can what you constantly mean\nby motion; which will save yourself as well as others from being seduced\nby equivocation.\n\n_A._ Then I say, motion is nothing but change of place for all the\neffect of a body upon the organs of our senses is nothing but fancy.\nTherefore we can fancy nothing from seeing it moved, but change of\nplace.\n\n_B._ It is right. But you must then tell me also what you understand by\nplace: for all men are not yet agreed on that.\n\n_A._ Well then; seeing we fancy a body, we cannot but fancy it\nsomewhere. And therefore I think place is the fancy of here or there.\n\n_B._ That is not enough. Here and there are not understood by any but\nyourself, except you point towards it. But pointing is no part of a\ndefinition. Besides, though it help him to find the place, it will never\nbring him to it.\n\n_A._ But seeing sense is fancy, when we fancy a body, we fancy also the\nfigure of it, and the space it fills up. And then I may define place to\nbe the precise space within which the body is contained. For space is\nalso part of the image we have of the object seen.\n\n_B._ And how define you time?\n\n_A._ As place is to a body, so, I think, is time to the motion of it;\nand consequently I take time to be our fancy or image of the motion. But\nis there any necessity of so much niceness?\n\n_B._ Yes. The want of it is the greatest, if not the only, cause of all\nthe discord amongst philosophers, as may easily be perceived by their\nabusing and confounding the names of things that differ in their nature;\nas you shall see when there is occasion to recite some of the tenets of\ndivers philosophers.\n\n_A._ I will avoid equivocation as much as I can. And for the nature of\nmotion, I suppose I understand it by the definition. What is next to be\ndone?\n\n_B._ You are to draw from these definitions, and from whatsoever truth\nelse you know by the light of nature, such general consequences as may\nserve for axioms, or principles of your ratiocination.\n\n_A._ That is hard to do.\n\n_B._ I will draw them myself, as many as for our present discourse of\nnatural causes we shall have need of; so that your part will be no more\nthan to take heed I do not deceive you.\n\n_A._ I will look to that.\n\n_B._ My first axiom then shall be this: Two bodies, at the same time,\ncannot be in one place.\n\n_A._ That is true: for we number bodies as we fancy them distinct, and\ndistinguish them by their places. You may therefore add: nor one body at\nthe same time in two places. And philosophers mean the same, when they\nsay: there is no penetration of bodies.\n\n_B._ But they understand not their own words: for penetration signifies\nit not. My second axiom is, that nothing can begin, change, or put an\nend to, its own motion. For supposing it begin just now, or being now in\nmotion, change its way or stop; I require the cause why now rather than\nbefore or after, having all that is necessary to such motion, change, or\nrest, alike at all times?\n\n_A._ I do not doubt but the argument is good in bodies inanimate; but\nperhaps in voluntary agents it does not hold.\n\n_B._ How it holds in voluntary agents we will then consider when our\nmethod hath brought us to the powers and passions of the mind. A third\naxiom shall be this: whatsoever body being at rest is afterwards moved,\nhath for its immediate movement some other body which is in motion and\ntoucheth it. For, since nothing can move itself, the movent must be\nexternal. And because motion is change of place, the movent must put it\nfrom its place, which it cannot do till it touch it.\n\n_A._ That is manifest, and that it must more than touch it; it must also\nfollow it. And if more parts of the body are moved than are by the\nmovent touched, the movent is not immediate. And by this reason, a\ncontinued body, though never so great, if the first superficies be\npressed never so little back, the motion will proceed through it.\n\n_B._ Do you think that to be impossible? I will prove it from your own\nwords: for you say that the movent does then touch the body which it\nmoveth. Therefore it puts it back; but that which is put back, puts back\nthe next behind, and that again the next; and so onward to any distance,\nthe body being continued. The same is also manifest by experience,\nseeing one that walks with a staff can distinguish, though blind,\nbetween stone and glass; which were impossible, if the parts of his\nstaff between the ground and his hand made no resistance. So also he\nthat in the silence of the night lays his ear to the ground, shall hear\nthe treading of men\u2019s feet farther than if he stood upright.\n\n_A._ This is certainly true of a staff or other hard body, because it\nkeeps the motion in a straight line from diffusion. But in such a fluid\nbody as the air, which being put back must fill an orb, and the farther\nit is put back, the greater orb, the motion will decrease, and in time,\nby the resistance of air to air, come to an end.\n\n_B._ That any body in the world is absolutely at rest, I think not true:\nbut I grant, that in a space filled everywhere with body, though never\nso fluid, if you give motion to any part thereof, that motion will by\nresistance of the parts moved, grow less and less, and at last cease;\nbut if you suppose the space utterly void, and nothing in it, then\nwhatsoever is once moved shall go on eternally: or else that which you\nhave granted is not true, viz., that nothing can put an end to its own\nmotion.\n\n_A._ But what mean you by resistance?\n\n_B._ Resistance is the motion of a body in a way wholly or partly\ncontrary to the way of its movent, and thereby repelling or retarding\nit. As when a man runs swiftly, he shall feel the motion of the air in\nhis face. But when two hard bodies meet, much more may you see how they\nabate each other\u2019s motion, and rebound from one another. For in a space\nalready full, the motion cannot, in an instant, be communicated through\nthe whole depth of the body that is to be moved.\n\n_A._ What other definitions have I need of?\n\n_B._ In all motion, as in all quantity, you must take the beginning of\nyour reckoning from the least supposed motion. And this I call the first\nendeavour of the movent; which endeavour, how weak soever, is also\nmotion. For if it have no effect at all, neither will it do anything\nthough doubled, trebled, or by what number soever multiplied: for\nnothing, though multiplied, is still nothing. Other axioms and\ndefinitions we will take in, as we need them, by the way.\n\n_A._ Is this all the preparation I am to make?\n\n_B._ No, you are to consider also the several kinds and properties of\nmotion, viz., when a body being moved by one or more movents at once, in\nwhat way it is carried, straight, circular, or otherwise crooked; and\nwhat degree of swiftness; as also the action of the movent, whether\ntrusion, vection, percussion, reflection, or refraction; and farther you\nmust furnish yourself with as many experiments (which they call\nphenomenon) as you can. And supposing some motion for the cause of your\nphenomenon, try, if by evident consequence, without contradiction to any\nother manifest truth or experiment, you can derive the cause you seek\nfor from your supposition. If you can, it is all that is expected, as to\nthat one question, from philosophy. For there is no effect in nature\nwhich the Author of nature cannot bring to pass by more ways than one.\n\n_A._ What I want of experiments you may supply out of your own store, or\nsuch natural history as you know to be true; though I can be well\ncontent with the knowledge of the causes of those things which everybody\nsees commonly produced. Let us therefore now enquire the cause of some\neffect particular.\n\n_B._ We will begin with that which is the most universal, the universe;\nand enquire in the first place, if any place be absolutely empty, that\nis to say in the language of philosophers, whether there be any vacuum\nin nature?\n\n\n ==========\n\n\n CHAPTER III.\n OF VACUUM.\n\n_A._ It is hard to suppose, and harder to believe, that the infinite and\nomnipotent Creator of all things should make a work so vast as is the\nworld we see, and not leave a few little spaces with nothing at all in\nthem; which put altogether in respect of the whole creation, would be\ninsensible.\n\n_B._ Why say you that? Do you think any argument can be drawn from it to\nprove there is vacuum?\n\n_A._ Why not? For in so great an agitation of natural bodies, may not\nsome small parts of them be cast out, and leave the places empty from\nwhence they were thrown?\n\n_B._ Because He that created them is not a fancy, but the most real\nsubstance that is; who being infinite, there can be no place empty where\nHe is, nor full where He is not.\n\n_A._ It is hard to answer this argument, because I do not remember that\nthere is any argument for the maintenance of vacuum in the writings of\ndivines: therefore I will quit that argument, and come to another. If\nyou take a glass vial with a narrow neck, and having sucked it, dip it\npresently at the neck into a basin of water, you shall manifestly see\nthe water rise into the vial. Is not this a certain sign that you had\nsucked out some of the air, and consequently that some part of the vial\nwas left empty?\n\n_B._ No; for when I am about to suck, and have air in my mouth,\ncontracting my checks I drive the same against the air in the glass, and\nthereby against every part of the sides of the hard glass. And this\ngives to the air within an endeavour outward, by which, if it be\npresently dipped into the water, it will penetrate and enter into it.\nFor air if it be pressed will enter into any fluid, much more into\nwater. Therefore there shall rise into the vial so much water as there\nwas air forced into the basin.\n\n_A._ This I confess is possible, and not improbable.\n\n_B._ If sucking would make vacuum, what would become of those women that\nare nurses? Should they not be in a very few days exhausted, were it not\nthat either the air which is in the child\u2019s mouth penetrateth the milk\nas it descends, and passeth through it, or the breast is contracted?\n\n_A._ From what experiment can you evidently infer that there is no\nvacuum?\n\n_B._ From many, and such as to almost all men are known and familiar. If\ntwo hard bodies, flat and smooth, be joined together in a common\nsuperficies parallel to the horizontal plane, you cannot without great\nforce pull them asunder, if you apply your force perpendicularly to the\ncommon superficies: but if you place that common superficies erect to\nthe horizon, they will fall asunder with their own weight. From whence I\nargue thus: since their contiguity, in what posture soever, is the same,\nand that they cannot be pulled asunder by a perpendicular force without\nletting in the ambient air in an instant, which is impossible; or almost\nin an instant, which is difficult: and on the other side, when the\ncommon superficies is erect, the weight of the same hard bodies is able\nto break the contiguity, and let in the air successively; it is manifest\nthat the difficulty of separation proceeds from this, that neither air\nnor any other body can be moved to any, how small soever, distance in an\ninstant; but may easily be moved (the hardness at the sides once\nmastered) successively. So that the cause of this difficulty of\nseparation is this, that they cannot be parted except the air or other\nmatter can enter and fill the space made by their diremption. And if\nthey were infinitely hard, not at all. And hence also you may understand\nthe cause why any hard body, when it is suddenly broken, is heard to\ncrack; which is the swift motion of the air to fill the space between.\nAnother experiment, and commonly known, is of a barrel of liquor, whose\ntap-hole is very little, and the bung so stopped as to admit no air; for\nthen the liquor will not run: but if the tap-hole be large it will,\nbecause the air pressed by a heavier body will pierce through it into\nthe barrel. The like reason holds of a gardener\u2019s watering-pot, when the\nholes in the bottom are not too great. A third experiment is this: turn\na thin brass kettle the bottom upwards, and lay it flat upon the water.\nIt will sink till the water rise within to a certain height, but no\nhigher: yet let the bottom be perforated, and the kettle will be full\nand sink, and the air rise again through the water without. But if a\nbell were so laid on, it would be filled and sink, though it were not\nperforated, because the weight is greater than the weight of the same\nbulk of water.\n\n_A._ By these experiments, without any more, I am convinced, that there\nis not actually in nature any vacuum; but I am not sure but that there\nmay be made some little place empty, and this from two experiments, one\nwhereof is Toricellius\u2019 experiment, which is this: take a cylinder of\nglass, hollow throughout, but close at the end, in form of a sack.\n\n_B._ How long?\n\n_A._ As long as you will, so it be more than twenty-nine inches.\n\n_B._ And how broad?\n\n_A._ As broad as you will, so it be broad enough to pour into it\nquicksilver. And fill it with quicksilver, and stop up the entrance with\nyour finger, so as to unstop it again at your pleasure. Then set down a\nbasin, or, if you will, a sea of quicksilver, and inverting the cylinder\nfull as it is, dip the end into the quicksilver, and remove your finger,\nthat the cylinder may empt itself. Do you conceive me? For there is so\nmany passing by, that I cannot paint it.\n\n_B._ Yes, I conceive you well enough. What follows?\n\n_A._ The quicksilver will descend in the cylinder, not till it be level\nwith that in the basin, according to the nature of heavy fluids, but\nstay and stand above it, at the height of twenty-nine inches or very\nnear it, the bottom being now uppermost, that no air can get in.\n\n_B._ What do you infer from this?\n\n_A._ That all the cavity above twenty-nine inches is filled with vacuum.\n\n_B._ It is very strange that I, from this same experiment, should infer,\nand I think evidently, that it is filled with air. I pray, tell me, when\nyou had inverted the cylinder, full as it was, and stopped with your\nfinger, dipped into the basin, if you had then removed your finger,\nwhether you think the quicksilver would not all have fallen out?\n\n_A._ No sure. The air would have been pressed upward through the\nquicksilver itself: for a man with his hand can easily thrust a bladder\nof air to the bottom of a basin of quicksilver.\n\n_B._ It is therefore manifest that quicksilver can press the air through\nthe same quicksilver.\n\n_A._ It is manifest; and also itself rise into the air.\n\n_B._ What cause then can there be, why it should stand still at twenty\nnine inches above the level of the basin, rather than any place else?\n\n_A._ It is not hard to assign the cause of that. For so much quicksilver\nas was above the twenty-nine inches, will rise the first level of that\nin the basin, as much as if you had poured it on; and thereby bring it\nto an equilibrium. So that I see plainly now, that there is no necessity\nof vacuum from this experiment. For I considered only that naturally\nquicksilver cannot ascend in air, nor air descend in quicksilver, though\nby force it may.\n\n_B._ Nor do I think that Torricellius or any other vacuist thought of it\nmore than you. But what is the second experiment?\n\n_A._ There is a sphere of glass, which they call a recipient, of the\ncapacity of three or four gallons. And there is inserted into it the end\nof a hollow cylinder of brass above a foot long; so that the whole is\none vessel, and the bore of the cylinder three inches diameter. Into\nwhich is thrust by force a solid cylinder of wood, covered with leather\nso just, as it may in every point exactly touch the concave superficies\nof the brass. There is also, to let out the air which the wooden\ncylinder as it enters (called the sucker) drives before it, a flap to\nkeep out the external air while they are pulling the sucker. Besides, at\nthe top of the recipient there is a hole to put into it anything for\nexperiment. The sucker being now forced up into the cylinder, what do\nyou think must follow?\n\n_B._ I think it will require as much strength to pull it back, as it did\nto force it in.\n\n_A._ That is not it I ask, but what would happen to the recipient?\n\n_B._ I think so much air as would fill the place the sucker leaves,\nwould descend into it out of the recipient; and also that just so much\nfrom the external air would enter into the recipient, between the brass\nand the wood, at first very swiftly, but, as the place increased, more\nleisurely.\n\n_A._ Why may not so much air rather descend into the place forsaken, and\nleave as much vacuum as that comes to in the recipient? For otherwise no\nair will be pumped out, nor can that wooden pestle be called a sucker.\n\n_B._ That is it I say. There is no air either pumped or sucked out.\n\n_A._ How can the air pass between the leather and the brass, or between\nthe leather and the wood, being so exactly contiguous, or through the\nleather itself?\n\n_B._ I conceive no such exact contiguity, nor such fastness of the\nleather: for I never yet had any that in a storm would keep out either\nair or water.\n\n_A._ But how then could there be made in the recipient such strange\nalteration both on animate and inanimate bodies?\n\n_B._ I will tell you how. The air descends out of the recipient, because\nthe air which the sucker removeth from behind itself, as it is pulling\nout, has no place to retire into without, and therefore is driven into\nthe engine between the wood of the sucker and the brass of the cylinder,\nand causes as much air to come into the place forsaken by the retiring\nsucker; which causeth, by oft repetition of the force, a violent\ncirculation of the air within the recipient, which is able quickly to\nkill anything that lives by respiration, and make all the alterations\nthat have appeared in the engine.\n\n\n ==========\n\n\n CHAPTER IV.\n OF THE SYSTEM OF THE WORLD.\n\n_B._ You are come in good time; let us therefore sit down. There is ink,\npaper, ruler, and compass. Draw a little circle to represent the body of\nthe sun.\n\n_A._ It is done. The centre is A, the circumference is L M.\n\n_B._ Upon the same centre A, draw a larger circle to stand for the\necliptic: for you know the sun is always in the plane of the ecliptic.\n\n_A._ There it is. The diameters of it at right angles are B Z.\n\n_B._ Draw the diameter of the equator.\n\n_A._ How?\n\n_B._ Through the centre A (for the earth is also always in the plane of\nthe equator or of some of its parallels) so as to be distant from B\ntwenty-three degrees and a half.\n\n_A._ Let it be H I: and let C G be equal to B H; and so C will be one of\nthe poles of the ecliptic, suppose the north-pole; and then H will be\neast, and I west. And C A produced to the circumference in E, makes E\nthe south-pole.\n\n_B._ Take C K equal to C G, and the chord G K will be the diameter of\nthe arctic circle, and parallel to H I, the diameter of the equator.\nLastly, upon the point B, draw a little circle wherein I suppose to be\nthe globe of the earth.\n\n_A._ It is drawn, and marked with _l m_. And B D and K G joined will be\nparallel; and as H and I are east and west, and so are B and D, and G\nand K.\n\n_B._ True; but producing Z B to the circumference _l m_ in _b_, the line\nB _b_ will be in the diameter of the ecliptic of the earth, and B _m_ in\nthe diameter of the equator of the earth. In like manner, if you produce\nK G cutting the circle, whose centre is G, in _d_ and _e_, and make an\nangle _n_ G _d_ equal to _b_ B _m_, the line _n_ G will be in the\necliptic of the earth, because G _d_ is in the equator of the earth. So\nthat in the annual motion of the earth through the ecliptic, every\nstraight line drawn in the earth, is perpetually kept parallel to the\nplace from whence it is removed.\n\n_A._ It is true; and it is the doctrine of Copernicus. But I cannot yet\nconceive by what one motion this circle can be described otherwise than\nwe are taught by Euclid. And then I am sure that all the diameters shall\ncross one another in the centre, which in this figure is A.\n\n_B._ I do not say that the diameters of a sphere or circle can be\nparallel; but that if a circle of a lesser sphere be moved upon the\ncircumference of a great circle of a greater sphere, that the straight\nlines that are in the lesser sphere may be kept parallel perpetually to\nthe places they proceed from.\n\n_A._ How? And by what motion?\n\n_B._ Take into your hand any straight line (as in this figure), the line\nL A M, which we suppose to be the diameter of the sun\u2019s body; and moving\nit parallelly with the ends in the circumference, so as that the end M\nmay withal describe a small circle, as M _a_. It is manifest that all\nthe other points of the same line L M will, by the same motion, at the\nsame time, describe equal circles to it. Likewise if you take in your\nhand any two diameters fastened together, the same parallel motion of\nthe line L M, shall cause all the points of the other diameter to make\nequal circles to the same M _a_.\n\n_A._ It is evident; as also that every point of the sun\u2019s body shall do\nthe like. And not only so, but also if one end describe any other\nfigure, all the other points of the body shall describe like and equal\nfigures to it.\n\n_B._ You see by this, that this parallel motion is compounded of two\nmotions, one circular upon the superficies of a sphere, the other a\nstraight motion from the centre to every point of the same superficies,\nand beyond it.\n\n_A._ I see it.\n\n_B._ It follows hence, that the sun by this motion must every way repel\nthe air; and since there is no empty place for retiring, the air must\nturn about in a circular stream; but slower or swifter according as it\nis more or less remote from the sun; and that according to the nature of\nfluids, the particles of the air must continually change place with one\nanother; and also that the stream of the air shall be the contrary way\nto that of the motion, for else the air cannot be repelled.\n\n_A._ All this is certain.\n\n_B._ Well; then if you suppose the globe of the earth to be in this\nstream which is made by the motion of the sun\u2019s body from east to west,\nthe stream of air wherein is the earth\u2019s annual motion will be from west\nto east.\n\n_A._ It is certain.\n\n_B._ Well. Then if you suppose the globe of the earth, whose circle is\nmoved annually, to be _l m_, the stream of the air without the ecliptic\nfalling upon the superficies of the earth _l m_ without the ecliptic,\nbeing slower, and the stream that falleth within swifter, the earth\nshall be turned upon its own centre proportionally to the greatness of\nthe circles; and consequently their diameters shall be parallel; as also\nare other straight lines correspondent.\n\n_A._ I deny not but the streams are as you say; and confess that the\nproportion of the swiftness without, is to the swiftness within, as the\nsun\u2019s ecliptic to the ecliptic of the earth; that is to say, as the\nangle H A B to the angle _m_ B _b_. And I like your argument the better,\nbecause it is drawn from Copernicus his foundation. I mean the\ncompounded motion of straight and circular.\n\n_B._ I think I shall not offer you many demonstrations of physical\nconclusions that are not derived from the motions supposed or proved by\nCopernicus. For those conclusions in natural philosophy I most suspect\nof falshood, which require most variety of suppositions for their\ndemonstrations.\n\n_A._ The next thing I would know, is how great or little you suppose\nthat circle _a_ M?\n\n_B._ I suppose it less than you can make it: for there appears in the\nsun no such motion sensible. It is the first endeavour of the sun\u2019s\nmotion. But for all that, as small as the circle is, the motion may be\nas swift, and of as great strength as it is possible to be named. It is\nbut a kind of trembling that necessarily happeneth in those bodies,\nwhich with great resistance press upon one another.\n\n_A._ I understand now from what cause proceedeth the annual motion. Is\nthe sun the cause also of the diurnal motion?\n\n_B._ Not the immediate cause. For the diurnal motion of the earth is\nupon its own centre, and therefore the sun\u2019s motion cannot describe it.\nBut it proceedeth as a necessary consequence from the annual motion. For\nwhich I have both experience and demonstration. The experiment is this:\ninto a large hemisphere of wood, spherically concave, put in a globe of\nlead, and with your hands hold it fast by the brim, moving your hand\ncircularly, but in a very small compass; you shall see the globe\ncirculate about the concave vessel, just in the same manner as the earth\ndoth every year in the air; and you shall see withal, that as it goes,\nit turns perpetually upon its own centre, and very swiftly.\n\n_A._ I have seen it: and it is used in some great kitchens to grind\nmustard.\n\n_B._ Is it so? Therefore take a hemisphere of gold, if you have it, the\ngreater the better, and a bullet of gold, and, without mustard, you\nshall see the same effect.\n\n_A._ I doubt it not. But the cause of it is evident. For any spherical\nbody being in motion upon the sides of a concave and hard sphere, is all\nthe way turned upon its own centre by the resistance of the hard wood or\nmetal. But the earth is a bullet without weight, and meeteth only with\nair, without any harder body in the way to resist it.\n\n_B._ Do you think the air makes no resistance, especially to so swift a\nmotion as is the annual motion of the earth? If it do make any\nresistance, you cannot doubt but that it shall turn the earth\ncircularly, and in a contrary way to its annual motion; that is to say,\nfrom east to west, because the annual motion is from west to east.\n\n_A._ I confess it. But what deduce you from these motions of the sun?\n\n_B._ I deduce, first, that the air must of necessity be moved both\ncircularly about the body of the sun according to the ecliptic, and also\nevery way directly from it. For the motion of the sun\u2019s body is\ncompounded of this circular motion upon the sphere L M, and of the\nstraight motion of its semi-diameters from the centre A to the\nsuperficies of the sun\u2019s body, which is L M. And therefore the air must\nneeds be repelled every way, and also continually change place to fill\nup the places forsaken by other parts of the air, which else would be\nempty, there being no vacuum to retire unto. So that there would be a\nperpetual stream of air, and in a contrary way to the motion of the\nsun\u2019s body, such as is the motion of water by the sides of a ship under\nsail.\n\n_A._ But this motion of the earth from west to east is only circular,\nsuch as is described by a compass about a centre; and cannot therefore\nrepel the air as the sun does. And the disciples of Copernicus will have\nit to be the cause of the moon\u2019s monthly motion about the earth.\n\n_B._ And I think Copernicus himself would have said the same, if his\npurpose had been to have shown the natural causes of the motions of the\nstars. But that was no part of his design; which was only from his own\nobservations, and those of former astronomers, to compute the times of\ntheir motions; partly to foretel the conjunctions, oppositions, and\nother aspects of the planets; and partly to regulate the times of the\nChurch\u2019s festivals. But his followers, Kepler and Galileo, make the\nearth\u2019s motion to be the efficient cause of the monthly motion of the\nmoon about the earth; which without the like motion to that of the sun\nin L M, is impossible. Let us therefore for the present take it in as a\nnecessary hypothesis; which from some experiment that I shall produce in\nour following discourses, may prove to be a certain truth.\n\n_A._ But seeing A is the centre both of the sun\u2019s body and of the annual\nmotion of the earth, how can it be (as all astronomers say it is) that\nthe orb of the annual motion of the earth should be eccentric to the\nsun\u2019s body? For you know that from the vernal equinox to the autumnal,\nthere be one hundred and eighty-seven days; but from the autumnal\nequinox to the vernal, there be but one hundred and seventy-eight days.\nWhat natural cause can you assign for this eccentricity?\n\n_B._ Kepler ascribes it to a magnetic virtue, viz. that one part of the\nearth\u2019s superficies has a greater kindness for the sun than the other\npart.\n\n_A._ I am not satisfied with that. It is magical rather than natural,\nand unworthy of Kepler. Tell me your own opinion of it.\n\n_B._ I think that the magnetical virtue he speaks of, consisteth in\nthis: that the southern hemisphere of the earth is for the greatest part\nsea, and that the greatest part of the northern hemisphere is dry land.\nBut how it is possible that from thence should proceed the eccentricity\n(the sun being nearest to the earth, when he is in the winter solstice),\nI shall show you when we come to speak of the motions of air and water.\n\n_A._ That is time enough: for I intend it for our next meeting. In the\nmean time I pray you tell me what you think to be the cause why the\nequinoctial, and consequently the solstitial, points are not always in\none and the same point of the ecliptic of the fixed stars. I know they\nare not, because the sun does not rise and set in points diametrically\nopposite: for if it did, there would be no difference of the seasons of\nthe year.\n\n_B._ The cause of that can be no other, than that the earth, which is _l\nm_, hath the like motion to that which I suppose the sun to have in L M,\ncompounded of straight and circular from west to east in a day, as the\nannual motion hath in a year; so that, not reckoning the eccentricity,\nit will be moved through the ecliptics in one revolution, as Copernicus\nproveth, about one degree. Suppose then the whole earth moved from H to\nI, (which is half the year) circularly, but falling from I to _i_ in the\nsame time about thirty minutes, and as much in the other hemisphere from\nH to _k_; then draw the line _i k_, which will be equal and parallel to\nH I, and be the diameter of the equator for the next year. But it shall\nnot cut the diameter of the ecliptic B Z in A, which was the equinoctial\nof the former year, but in _o_ thirty-six seconds from the first degree\nof Aries. Suppose the same done in the hemisphere under the plane of the\npaper, and so you have the double of thirty-six seconds, that is\nseventy-two seconds, or very near, for the progress of the vernal\nequinox in a year. The cause why I suppose the arch I _i_ to be half a\ndegree in the ecliptic of the earth, is, that Copernicus and other\nastronomers, and experience, agree in this, that the equinoctial points\nproceed according to the order of the signs, Aries, Taurus, Gemini, &c.\nfrom west to east every hundredth year one degree or very near.\n\n_A._ In what time do they make the whole revolution through the ecliptic\nof the sky?\n\n_B._ That you may reckon. For we know by experience that it hath\nproceeded about one degree, that is sixty minutes, constantly a long\ntime in a hundred years. But as one hundred years to one degree, so is\nthirty-six thousand years to three hundred and sixty degrees. Also as\none hundred years to one degree, so is one year to the hundredth part of\none degree, or sixty minutes; which is (60)/(100), or thirty-six seconds\nfor the progress of one year; which must be somewhat more than a degree\naccording to Copernicus, who, (lib. iii. cap. 2) saith, that for four\nhundred years before Ptolomy it was one degree almost constantly. Which\nis well enough as to the natural cause of the precession of the\nequinoctial points, which is the often-said compounded motion, though\nnot an exact astronomical calculation.\n\n_A._ And it is a great sign that his supposition is true. But what is\nthe cause that the obliquity of the ecliptic, that is, the distance\nbetween the equinoctial and the solstice, is not always the same?\n\n_B._ The necessity of the obliquity of the ecliptic is but a consequence\nto the precession of the equinoctial points. And therefore, if from C,\nthe north pole, you make a little circle, C _u_, equal to fifteen\nminutes of a degree upon the earth, and another, _u s_, equal to the\nsame, which will appear like this figure 8, that is, (as Copernicus\ncalls it), a circle twined, the pole C will be moved half the time of\nthe equinoctial points, in the arc C _u_, and as much in the alternate\narc _u s_ descending to _s_. But in the arc _s u_, and its alternate\nrising to C, the cause of the twining is the earth\u2019s annual motion the\nsame way in the ecliptic, and makes the four quarters of it; and makes\nalso their revolution twice as slow as that of the equinoctial points.\nAnd, therefore, the motion of it is the same compounded motion which\nCopernicus takes for his supposition, and is the cause of the precession\nof the equinoctial points, and consequently of the variation of the\nobliquity, adding to it or taking from it somewhere more, somewhere\nless; so as that one with another the addition is not much more, nor the\nsubtraction much less than thirty minutes. But as for the natural\nefficient cause of this compounded motion, either in the sun, or the\nearth, or any other natural body, it can be none but the immediate hand\nof the Creator.\n\n_A._ By this it seems that the poles of the earth are always the same,\nbut make this 8 in the sphere of the fixed stars near that which is\ncalled Cynosura.\n\n_B._ No: it is described on the earth, but the annual motion describes a\ncircle in the sphere of the fixed stars. Though I think it improper to\nsay a sphere of the fixed stars, when it is so unlikely that all the\nfixed stars should be in the superficies of one and the same globe.\n\n_A._ I do not believe they are.\n\n_B._ Nor I, since they may seem less one than another, as well by their\ndifferent distances, as by their different magnitudes. Nor is it likely\nthat the sun (which is a fixed star) is the efficient cause of the\nmotion of those remoter planets, Mars, Jupiter, and Saturn; seeing the\nwhole sphere, whose diameter is the distance between the sun and the\nearth, is but a point in respect of the distance between the sun and any\nother fixed star. Which I say only to excite those that value the\nknowledge of the cause of comets, to look for it in the dominion of some\nother sun than that which moveth the earth. For why may not there be\nsome other fixed star, nearer to some planet than is the sun, and cause\nsuch a light in it as we call a comet?\n\n_A._ As how?\n\n_B._ You have seen how in high and thin clouds above the earth, the\nsun-beams piercing them have appeared like a beard; and why might not\nsuch a beard have appeared to you like a comet, if you had looked upon\nit from as high as some of the fixed stars?\n\n_A._ But because it is a thing impossible for me to know, I will proceed\nin my own way of inquiry. And seeing you ascribe this compounded motion\nto the sun and earth, I would grant you that the earth (whose annual\nmotion is from west to east) shall give the moon her monthly motion from\neast to west. But then I ask you whether the moon have also that\ncompounded motion of the earth, and with it a motion upon its own\ncentre, as hath the earth? For seeing the moon has no other planet to\ncarry about her, she needs it not.\n\n_B._ I see reason enough, and some necessity, that the moon should have\nboth those motions. For you cannot think that the Creator of the stars,\nwhen he gave them their circular motion, did first take a centre, and\nthen describe a circle with a chain or compass, as men do? No; he moved\nall the parts of a star together and equally in the creation: and that\nis the reason I give you. The necessity of it comes from this\nphenomenon, that the moon doth turn one and the same face towards the\nearth; which cannot be by being moved about the earth parallelly, unless\nalso it turn about its own centre. Besides, we know by experience, that\nthe motion of the moon doth add not a little to the motion of the sea:\nwhich were impossible if it did not add to the stream of the air, and by\nconsequence to that of the water.\n\n_A._ If you could get a piece of the true and intimate substance of the\nearth, of the bigness of a musket-bullet, do you believe that the bullet\nwould have the like compounded motion to that which you attribute to the\nsun, earth, and moon?\n\n_B._ Yes, truly; but with less strength, according to its magnitude;\nsaving that by its gravity falling to the earth, the activity of it\nwould be unperceived.\n\n_A._ I will trouble you no more with the nature of celestial\nappearances; but I pray you tell me by what art a man may find what part\nof a circle the diameter of the sun\u2019s body doth subtend in the ecliptic\ncircle?\n\n_B._ Kepler says it subtends thirty minutes, which is half a degree. His\nway to find it is by letting in the sun-beams into a close room through\na small hole, and receiving the image of it upon a plane\nperpendicularly. For by this means he hath a triangle, whose sides and\nangles he can know by measure; and the vertical angle he seeks for, and\nthe substance of the arc of the sun\u2019s body.\n\n_A._ But I think it impossible to distinguish where the part illuminate\ntoucheth the part not illuminate.\n\n_B._ Another way is this: upon the equinoctial day, with a watch that\nshows the minutes standing by you, observe when the lower brim of the\nsun\u2019s setting first comes to the horizon, and set the index to some\nminute of the watch; and observe again the upper brim when it comes to\nthe horizon: then count the minutes, and you have what you look for.\nOther way I know none.\n\n\n ==========\n\n\n CHAPTER V.\n OF THE MOTIONS OF WATER AND AIR.\n\n_A._ I have considered, as you bad me, this compounded motion with great\nadmiration. First, it is that which makes the difference between\n_continuum_ and _contiguum_, which till now I never could distinguish.\nFor bodies that are but contiguous, with any little force are parted;\nbut by this compounded motion (because every point of the body makes an\nequal line in equal time, and every line crosses all the rest) one part\ncannot be separated from another, without disturbing the motion of all\nthe other parts at once. And is not that the cause, think you, that some\nbodies when they are pressed or bent, as soon as the force is removed,\nreturn again of themselves to their former figure?\n\n_B._ Yes, sure; saving that it is not of themselves that they return,\n(for we were agreed that nothing can move itself), but it is the motion\nof the parts which are not pressed, that delivers those that are. And\nthis restitution the learned now call the spring of a body. The Greeks\ncalled it _antitypia_.\n\n_A._ When I considered this motion in the sun and the earth and planets,\nI fancied them as so many bodies of the army of the Almighty in an\nimmense field of air, marching swiftly, and commanded (under God) by his\nglorious officer the sun, or rather forced so to keep their order in\nevery part of every of those bodies, as never to go out from the\ndistance in which he had set them.\n\n_B._ But the parts of the air and other fluids keep not their places so.\n\n_A._ No: you told me that this motion is not natural in the air, but\nreceived from the sun.\n\n_B._ True: but since we seek the natural causes of sublunary effects,\nwhere shall we begin?\n\n_A._ I would fain know what makes the sea to ebb and flow at certain\nperiods, and what causeth such variety in the tides.\n\n_B._ Remember that the earth turneth every day upon its own axis from\nwest to east; and all the while it so turneth, every point thereof by\nits compounded motion makes other circlings, but not on the same centre,\nwhich is (you know) a rising in one part of the day, and a falling in\nthe other part. What think you must happen to the sea, which resteth on\nit, and is a fluid body?\n\n_A._ I think it must make the sea rise and fall. And the same happeneth\nalso to the air, from the motion of the sun.\n\n_B._ Remember, also, in what manner the sea is situated in respect of\nthe dry land.\n\n_A._ Is not there a great sea that reacheth from the straits of Magellan\neastward to the Indies, and thence to the same straits again? And is not\nthere a great sea called the Atlantic sea that runneth northward to us?\nAnd does not the great south sea run also up into the northern seas? But\nI think the Indian and the South sea of themselves to be greater than\nall the rest of the surface of the globe.\n\n_B._ How lieth the water in those two seas?\n\n_A._ East and west, and rises and falls a little, as it is forced to do\nby this compounded motion, which is a kind of succussion of the earth,\nand fills both the Atlantic and Northern seas.\n\n_B._ All this would not make a visible difference between high and low\nwater, because this motion being so regular, the unevenness would not be\ngreat enough to be seen. For though in a basin the water would be thrown\ninto the air, yet the earth cannot throw the sea into the air.\n\n_A._ Yes; the basin, if gently moved, will make the water so move, that\nyou shall hardly see it rise.\n\n_B._ It may be so. But you should never see it rise as it doth, if it\nwere not checked. For at the straits of Magellan, the great South sea is\nchecked by the shore of the continent of Peru and Chili, and forced to\nrise to a great height, and made to run up into the northern seas on\nthat side by the coast of China; and at the return is checked again and\nforced through the Atlantic into the British and German seas. And this\nis done every day. For we have supposed that the earth\u2019s motion in the\necliptic caused by the sun is annual; and that its motion in the\nequinoctial is diurnal. It followeth therefore from this compounded\nmotion of the earth, the sea must ebb and flow twice in the space of\ntwenty-four hours, or thereabout.\n\n_A._ Has the moon nothing to do in this business?\n\n_B._ Yes. For she hath also the like motion. And is, though less swift,\nyet much nearer to the earth. And therefore when the sun and moon are in\nconjunction or opposition, the earth, as from two agents at once, must\nneeds have a greater succussion. And if it chance at the same time the\nmoon also be in the ecliptic, it will be yet greater, because the moon\nthen worketh on the earth less obliquely.\n\n_A._ But when the full or new moon happen to be then when the earth is\nin the equinoctial points, the tides are greater than ordinary. Why is\nthat?\n\n_B._ Because then the force by which they move the sea, is at that time,\nto the force by which they move the same at other times, as the\nequinoctial circle to one of its parallels, which is a lesser circle.\n\n_A._ It is evident. And it is pleasant to see the concord of so many and\nvarious motions, when they proceed from one and the same hypothesis. But\nwhat say you to the stupendous tides which happen on the coasts of\nLincolnshire on the east, and in the river of Severn on the west?\n\n_B._ The cause of that, is their proper situation. For the current of\nthe ocean through the Atlantic sea, and the current of the south sea\nthrough the northern seas, meeting together, rise the water in the Irish\nand British seas a great deal higher than ordinary. Therefore the mouth\nof the Severn being directly opposite to the current from the Atlantic\nsea, and those sands on the coast of Lincolnshire directly opposite to\nthe current of the German sea, those tides must needs fall furiously\ninto them, by this succussion of the water.\n\n_A._ Does, when the tide runs up into a river, the water all rise\ntogether, and fall together when it goes out?\n\n_B._ No: one part riseth and another falleth at the same time; because\nthe motion of the earth rising and falling, is that which makes the\ntide.\n\n_A._ Have you any experiment that shows it?\n\n_B._ Yes. You know that in the Thames, it is high water at Greenwich\nbefore it is high water at London-bridge. The water therefore falls at\nGreenwich whilst it riseth all the way to London. But except the top of\nthe water went up, and the lower part downward, it were impossible.\n\n_A._ It is certain. It is strange that this one motion should salve so\nmany appearances, and so easily. But I will produce one experiment of\nwater, not in the sea, but in a glass. If you can show me that the cause\nof it is this compounded motion, I shall go near to think it the cause\nof all other effects of nature hitherto disputed of. The experiment is\ncommon, and described by the Lord Chancellor Bacon, in the third page of\nhis natural history. Take, saith he, a glass of water, and draw your\nfinger round about the lip of the glass, pressing it somewhat hard;\nafter you have done so a few times, it will make the water frisk up into\na fine dew. After I had read this, I tried the same with all diligence\nmyself, and found true not only the frisking of the water to above an\ninch high, but also the whole superficies to circulate, and withal to\nmake a pleasant sound. The cause of the frisking he attributes to a\ntumult of the inward parts of the substance of the glass striving to\nfree itself from the pressure.\n\n_B._ I have tried and found both the sound and motion; and do not doubt\nbut the pressure of the parts of the glass was part of the cause. But\nthe motion of my finger about the glass was always parallel; and when it\nchanced to be otherwise, both sound and motion ceased.\n\n_A._ I found the same. And being satisfied, I proceed to other\nquestions. How is the water, being a heavy body, made to ascend in small\nparticles into the air, and be there for a time sustained in form of a\ncloud, and then fall down again in rain?\n\n_B._ I have shown already, that this compounded motion of the sun, in\none part of its circumlation, drives the air one way, and in the other\npart, the contrary way; and that it cannot draw it back again, no more\nthan he that sets a stone a flying can pull it back. The air therefore,\nwhich is contiguous to the water, being thus distracted, must either\nleave a vacuum, or else some part of the water must rise and fill the\nspaces continually forsaken by the air. But, that there is no vacuum,\nyou have granted. Therefore the water riseth into the air, and maketh\nthe clouds; and seeing they are very small and invisible parts of the\nwater, they are, though naturally heavy, easily carried up and down with\nthe wind, till, meeting with some mountain or other clouds, they be\npressed together into greater drops, and fall by their weight. So also\nit is forced up in moist ground, and with it many small atoms of the\nearth, which are either twisted with the rising water into plants, or\nare carried up and down in the air incertainly. But the greatest\nquantity of water is forced up from the great South and Indian Seas,\nthat lie under the tropic of capricorn. And this climate is that which\nmakes the sun\u2019s perig\u00e6um to be always on the winter-solstice. And that\nis the part of the terrestrial globe which Kepler says is kind to the\nsun; whereas the other part of the globe, which is almost all dry land,\nhas an antipathy to the sun. And so you see where this magnetical virtue\nof the earth lies. For the globe of the earth having no natural appetite\nto any place, may be drawn by this motion of the sun a little nearer to\nit, together with the water which it raiseth.\n\n_A._ Can you guess what may be the cause of wind?\n\n_B._ I think it manifest that the unconstant winds proceed from the\nuncertain motion of the clouds ascending and descending, or meeting with\none another. For the winds after they are generated in any place by the\ndescent of a cloud, they drive other clouds this way and that way before\nthem, the air seeking to free itself from being pent up in a strait. For\nwhen a cloud descendeth, it makes no wind sensible directly under\nitself. But the air between it and the earth is pressed and forced to\nmove violently outward. For it is a certain experiment of mariners, that\nif the sea go high when they are becalmed, they say they shall have more\nwind than they would; and take in their sails all but what is necessary\nfor steering. They know, it seems, that the sea is moved by the descent\nof clouds at some distance off: which presseth the water, and makes it\ncome to them in great waves. For a horizontal wind does but curl the\nwater.\n\n_A._ From whence come the rivers?\n\n_B._ From the rain, or from the falling of snow on the higher ground.\nBut when it descendeth under ground, the place where it again ariseth is\ncalled the spring.\n\n_A._ How then can there be a spring upon the top of a hill?\n\n_B._ There is no spring upon the very top of a hill, unless some natural\npipe bring it thither from a higher hill.\n\n_A._ Julius Scaliger says, there is a river, and in it a lake, upon the\ntop of Mount Cenis in Savoy; and will therefore have the springs to be\ningendered in the caverns of the earth by condensation of the air.\n\n_B._ I wonder he should say that. I have passed over that hill twice\nsince the time I read that in Scaliger, and found that river as I\npassed, and went by the side of it in plain ground almost two miles;\nwhere I saw the water from two great hills, one on one side, the other\non the other, in a thousand small rillets of melting snow fall down into\nit. Which has made me never to use any experiment the which I have not\nmyself seen. As for the conversion of air into water by condensation,\nand of water into air by rarefaction, though it be the doctrine of the\nPeripatetics, it is a thing incogitable, and the words are\ninsignificant. For by densum is signified only frequency and closeness\nof parts; and by rarum the contrary. As when we say a town is thick with\nhouses, or a wood with trees, we mean not that one house or tree is\nthicker than another, but that the spaces between are not so great. But,\nsince there is no vacuum, the spaces between the parts of air are no\nlarger than between the parts of water, or of any thing else.\n\n_A._ What think you of those things which mariners that have sailed\nthrough the Atlantic Sea, called _spouts_, which pour down water enough\nat once to drown a great ship?\n\n_B._ It is a thing I have not seen: and therefore can say nothing to it;\nthough I doubt not but when two very large and heavy clouds shall be\ndriven together by two great and contrary winds, the thing is possible.\n\n_A._ I think your reason good. And now I will propound to you another\nexperiment. I have seen an exceeding small tube of glass with both ends\nopen, set upright in a vessel of water, and that the superficies of the\nwater within the tube was higher a good deal than of that in the vessel;\nbut I see no reason for it.\n\n_B._ Was not part of the glass under water? Must not then the water in\nthe vessel rise? Must not the air that lay upon it rise with it? Whither\nshould this rising air go, since there is no place empty to receive it?\nIt is therefore no wonder if the water, pressed by the substance of the\nglass which is dipt into it, do rather rise into a very small pipe, than\ncome about a longer way into the open air.\n\n_A._ It is very probable. I observed also that the top of the inclosed\nwater was a concave superficies; which I never saw in other fluids.\n\n_B._ The water hath some degree of tenacity, though not so great but\nthat it will yield a little to the motion of the air; as is manifest in\nthe bubbles of water, where the concavity is always towards the air. And\nthis I think the cause why the air and water meeting in the tube make\nthe superficies towards the air concave, which it cannot do to a fluid\nof greater tenacity.\n\n_A._ If you put into a basin of water a long rag of cloth, first\ndrenched in water, and let the longer part of it hang out, it is known\nby experience, that the water will drop out as long as there is any part\nof the other end under water.\n\n_B._ The cause of it is, that water, as I told you, hath a degree of\ntenacity. And therefore being continued in the rag till it be lower\nwithout than within, the weight will make it continue dropping, though\nnot only because it is heavy (for if the rag lay higher without than\nwithin, and were made heavier by the breadth, it would not descend), but\nit is because all heavy bodies naturally descend with proportion of\nswiftness duplicate to that of the time; whereof I shall say more when\nwe talk of gravity.\n\n_A._ You see how despicable experiments I trouble you with. But I hope\nyou will pardon me.\n\n_B._ As for mean and common experiments, I think them a great deal\nbetter witnesses of nature, than those that are forced by fire, and\nknown but to very few.\n\n\n ==========\n\n\n CHAPTER VI.\n OF THE CAUSES AND EFFECTS OF HEAT AND COLD.\n\n_A._ It is a fine day, and pleasant walking through the fields, but that\nthe sun is a little too hot.\n\n_B._ How know you that the sun is hot?\n\n_A._ I feel it.\n\n_B._ That is to say, you know that yourself, but not that the sun is\nhot. But when you find yourself hot, what body do you feel?\n\n_A._ None.\n\n_B._ How then can you infer your heat from the sense of feeling? Your\nwalking may have made you hot: is motion therefore hot? No. You are to\nconsider the concomitants of your heat; as, that you are more faint, or\nmore ruddy, or that you sweat, or feel some endeavour of moisture or\nspirits tending outward; and when you have found the causes of those\naccidents, you have found the causes of heat, which in a living\ncreature, and especially in a man, is many times the motion of the parts\nwithin him, such as happen in sickness, anger, and other passions of the\nmind; which are not in the sun nor in fire.\n\n_A._ That which I desire now to know, is what motions and of what bodies\nwithout me are the efficient causes of my heat.\n\n_B._ I showed you yesterday, in discoursing of rain, how by this\ncompounded motion of the sun\u2019s body, the air was every way at once\nthrust off west and east; so that where it was contiguous, the small\nparts of the water were forced to rise, for the avoiding of vacuum.\nThink then that your hand were in the place of water so exposed to the\nsun. Must not the sun work upon it as it did upon the water? Though it\nbreak not the skin, yet it will give to the inner fluids and looser\nparts of your hand, an endeavour to get forth, which will extend the\nskin, and in some climates fetch up the blood, and in time make the skin\nblack. The fire also will do the same to them that often sit with their\nnaked skins too near it. Nay, one may sit so near, without touching it,\nas it shall blister or break the skin, and fetch up both spirits and\nblood mixt into a putrid oily matter, sooner than in a furnace oil can\nbe extracted out of a plant.\n\n_A._ But if the water be above the fire in a kettle, what then will it\ndo? Shall the particles of water go toward the fire, as it did toward\nthe sun?\n\n_B._ No. For it cannot. But the motion of the parts of the kettle which\nare caused by the fire, shall dissipate the water into vapour till it be\nall cast out.\n\n_A._ What is that you call fire? Is it a hard or fluid body?\n\n_B._ It is not any other body but that of the shining coal; which coal,\nthough extinguished with water, is still the same body. So also in a\nvery hot furnace, the hollow spaces between the shining coals, though\nthey burn that you put into them, are no other body than air moved.\n\n_A._ Is it not flame?\n\n_B._ No. For flame is nothing but a multitude of sparks, and sparks are\nbut the atoms of the fuel dissipated by the incredible swift motion of\nthe movent, which makes every spark to seem a hundred times greater than\nit is, as appears by this; that, when a man swings in the air a small\nstick fired at one end, though the motion cannot be very swift, yet the\nfire will appear to the eye to be a long, straight, or crooked line.\nTherefore a great many sparks together flying upward, must needs appear\nunto the sight as one continued flame. Nor are the sparks stricken out\nof a flint any thing else but small particles of the stone, which by\ntheir swift motion are made to shine. But that fire is not a substance\nof itself, is evident enough by this, that the sun-beams passing through\na globe of water will burn as other fire does. Which beams, if they were\nindeed fire, would be quenched in the passage.\n\n_A._ This is so evident, that I wonder so wise men as Aristotle and his\nfollowers, for so long a time could hold it for an element, and one of\nthe primary parts of the universe. But the natural heat of a man or\nother living creature, whence proceedeth it? Is there anything within\ntheir bodies that hath this compounded motion?\n\n_B._ At the breaking up of a deer I have seen it plainly in his bowels\nas long as they were warm. And it is called the peristaltic motion, and\nin the heart of a beast newly taken out of his body; and this motion is\ncalled systole and diastole. But they are both of them this compounded\nmotion, whereof the former causeth the food to wind up and down through\nthe guts, and the latter makes the circulation of the blood.\n\n_A._ What kind of motion is the cause of cold? Methinks it should be\ncontrary to that which causeth heat.\n\n_B._ So it is in some respect. For seeing the motion that begets heat,\ntendeth to the separation of the parts of the body whereon it acteth, it\nstands with reason, that the motion which maketh cold, should be such as\nsets them closer together. But contrary motions are, to speak properly,\nwhen upon two ends of a line two bodies move towards each other, the\neffect whereof is to make them meet. But each of them, as to this\nquestion, is the same.\n\n_A._ Do you think (as many philosophers have held and now hold), that\ncold is nothing but a privation of heat?\n\n_B._ No. Have you never heard the fable of the satyr that dwelling with\na husbandman, and seeing him blow his fingers to warm them, and his\npottage to cool it, was so scandalized, that he ran from him, saying he\nwould no longer dwell with one that could blow both hot and cold with\none breath? Yet the cause is evident enough. For the air which had\ngotten a calefactive power from his vital parts, was from his mouth and\nthroat gently diffused on his fingers, and retained still that power.\nBut to cool his pottage he straightened the passage at his lips, which\nextinguished the calefactive motion.\n\n_A._ Do you think wind the general cause of cold? If that were true, in\nthe greatest winds we should have the greatest frosts.\n\n_B._ I mean not any of those uncertain winds which, I said, were made by\nthe clouds, but such as a body moved in the air makes to and against\nitself; (for it is all one motion of the air whether it be carried\nagainst the body, or the body against it); such a wind as is constant,\nif no other be stirring, from east to west; and made by the earth\nturning daily upon its own centre; which is so swift, as, except it be\nkept off by some hill, to kill a man, as by experience hath been found\nby those who have passed over great mountains, and specially over the\nAndes which are opposed to the east. And such is the wind which the\nearth maketh in the air by her annual motion, which is so swift, as\nthat, by the calculation of astronomers, to go sixty miles in a minute\nof an hour. And therefore this must be the motion which makes it so cold\nabout the poles of the ecliptic.\n\n_A._ Does not the earth make the wind as great in one part of the\necliptic as in another?\n\n_B._ Yes. But when the sun is in Cancer, it tempers the cold, and still\nless and less, but least of all in the winter-solstice, where his beams\nare most oblique to the superficies of the earth.\n\n_A._ I thought the greatest cold had been about the poles of the\nequator.\n\n_B._ And so did I once. But the reason commonly given for it is so\nimprobable, that I do not think so now. For the cause they render of it\nis only, that the motion of the earth is swiftest in the equinoctial,\nand slowest about the poles; and consequently, since motion is the cause\nof heat, and cold is but, as it was thought, a want of the same, they\ninferred that the greatest cold must be about the poles of the\nequinoctial. Wherein they miscounted. For not every motion causeth heat,\nbut this agitation only, which we call compounded motion; though some\nhave alleged experience for that opinion; as that a bullet out of a gun\nwill with its own swiftness melt. Which I never shall believe.\n\n_A._ It is a common thing with many philosophers to maintain their\nfancies with any rash report, and sometimes with a lie. But how is it\npossible that so soft a substance as water should be turned into so hard\na substance as ice?\n\n_B._ When the air shaves the globe of the earth with such swiftness, as\nthat of sixty miles in a minute of an hour, it cannot, where it meets\nwith still water, but beat it up into small and undistinguishable\nbubbles, and involve itself in them as in so many bladders or skins of\nwater. And ice is nothing else but the smallest imaginable parts of air\nand water mixed; which is made hard by this compounded motion, that\nkeeps the parts so close together, as not to be separated in one place\nwithout disordering the motion of them all. For when a body will not\neasily yield to the impression of an external movent in one place\nwithout yielding in all, we call it hard; and when it does, we say it is\nsoft.\n\n_A._ Why is not ice as well made in a moved as in a still water? Are\nthere not great seas of ice in the northern parts of the earth?\n\n_B._ Yes, and perhaps also in the southern parts. But I cannot imagine\nhow ice can be made in such agitation as is always in the open sea, made\nby the tides and by the winds. But how it may be made at the shore, it\nis not hard to imagine. For in a river or current, though swift, the\nwater that adhereth to the banks is quiet, and easily by the motion of\nthe air driven into small insensible bubbles; and so may the water that\nadhereth to those bubbles, and so forwards till it come into a stream\nthat breaks it, and then it is no wonder though the fragments be driven\ninto the open sea, and freeze together into greater lumps. But when in\nthe open sea, or at the shore, the tide or a great wave shall arise,\nthis young and tender ice will presently be washed away. And therefore I\nthink it evident, that as in the Thames the ice is first made at the\nbanks where the tide is weak or none, and, broken by the stream, comes\ndown to London, and part goes to the sea floating till it dissolve, and\npart, being too great to pass the bridge, stoppeth there and sustains\nthat which follows, till the river be quite frozen over; so also the ice\nin the northern seas begins first at the banks of the continent and\nislands which are situated in that climate, and then broken off, are\ncarried up and down, and one against another, till they become great\nbodies.\n\n_A._ But what if there be islands, and narrow inlets of the sea, or\nrivers also about the pole of the equinoctial?\n\n_B._ If there be, it is very likely the sea may also there be covered\nall over with ice. But for the truth of this, we must stay for some\nfarther discovery.\n\n_A._ When the ice is once made and hard, what dissolves it?\n\n_B._ The principal cause of it, is the weight of the water itself; but\nnot without some abatement in the stream of the air that hardeneth it;\nas when the sunbeams are less oblique to the earth, or some contrary\nwind resisteth the stream of the air. For when the impediment is\nremoved, then the nature of the water only worketh, and, being a heavy\nbody, downward.\n\n_A._ I forgot to ask you, why two pieces of wood rubbed swiftly one\nagainst another, will at length set on fire.\n\n_B._ Not only at length, but quickly, if the wood be dry. And the cause\nis evident, viz. the compounded motion which dissipates the external\nsmall parts of the wood. And then the inner parts must of necessity, to\npreserve the plentitude of the universe, come after; first the most\nfluid, and then those also of greater consistence, which are first\nerected, and the motion continued, made to fly swiftly out; whereby the\nair driven to the eye of the beholder, maketh that fancy which is called\nlight.\n\n_A._ Yes; I remember you told me before, that upon any strong pressure\nof the eye, the resistance from within would appear a light. But to\nreturn to the enquiry of heat and cold, there be two things that beyond\nall other put me into admiration. One is the swiftness of kindling in\ngunpowder. The other is the freezing of water in a vessel, though not\nfar from the fire, set about with other water with ice and snow in it.\nWhen paper or flax is flaming, the flame creeps gently on; and if a\nhouse full of paper were to be burnt with putting a candle to it, it\nwill be long in burning; whereas a spark of fire would set on flame a\nmountain of gunpowder in almost an instant.\n\n_B._ Know you not gunpowder is made of the powder of charcoal,\nbrimstone, and saltpetre? Whereof the first will kindle with a spark,\nthe second flame as soon as touched with fire; and the third blows it,\nas being composed of many orbs of salt filled with air, and as it\ndissolveth in the flame, furiously blowing increaseth it. And as for\nmaking ice by the fireside; it is manifest that whilst the snow is\ndissolving in the external vessel, the air must in the like manner break\nforth, and shave the superficies of the inner vessel, and work through\nthe water till it be frozen.\n\n_A._ I could easily assent to this, if I could conceive how the air that\nshaves, as you say, the outside of the vessel, could work through it. I\nconceive well enough a pail of water with ice or snow dissolving in it,\nand how it causeth wind. But how that wind should communicate itself\nthrough the vessel of wood or metal, so as to make it shave the\nsuperficies of the water which is within it, I do not so well\nunderstand.\n\n_B._ I do not say the inner superficies of the vessel shaves the water\nwithin it. But it is manifest that the wind made in the pail of water by\nthe melting snow or ice presseth the sides of the vessel that standeth\nin it; and that the pressure worketh clean through, how hard soever the\nvessel be; and that again worketh on the water within, by restitution of\nits parts, and so hardeneth the water by degrees.\n\n_A._ I understand you now. The ice in the pail by its dissolution\ntransfers its hardness to the water within.\n\n_B._ You are merry. But supposing, as I do, that the ice in the pail is\nmore than the water in the vessel, you will find no absurdity in the\nargument. Besides, the experiment, you know, is common.\n\n_A._ I confess it is probable. The Greeks have the word \u03c6\u03c1\u03af\u03ba\u03b7 (whence\nthe Latins have their word _frigus_) to signify the curling of the water\nby the wind; and use the same also for horror, which is the passion of\none that cometh suddenly into a cold air, or is put into a sudden\naffright, whereby he shrinks, and his hair stands upright. Which\nmanifestly shows that the motion which causeth cold, is that which\npressing the superficies of a body, sets the parts of it closer\ntogether. But to proceed in my queries. Monsieur Des Cartes, whom you\nknow, hath written somewhere, that the noise we hear in thunder,\nproceeds from breaking of the ice in the clouds; what think you of it?\nCan a cloud be turned into ice?\n\n_B._ Why not? A cloud is but water in the air?\n\n_A._ But how? For he has not told us that.\n\n_B._ You know that it is only in summer, and in hot weather, that it\nthunders; or if in winter, it is taken for a prodigy. You know also,\nthat of clouds, some are higher, some lower, and many in number, as you\ncannot but have oftentimes observed, with spaces between them.\nTherefore, as in all currents of water, the water is there swiftest\nwhere it is straitened with islands, so must the current of air made by\nthe annual motion be swiftest there, where it is checked with many\nclouds, through which it must, as it were, be strained, and leave behind\nit many small particles of earth always in it, and in hot weather more\nthan ordinary.\n\n_A._ This I understand, and that it may cause ice. But when the ice is\nmade, how is it broken? And why falls it not down in shivers?\n\n_B._ The particles are enclosed in small caverns of the ice; and their\nnatural motion being the same which we have ascribed to the globe of the\nearth, requires a sufficient space to move in. But when it is imprisoned\nin a less room than that, then a great part of the ice breaks: and this\nis the thunder-clap. The murmur following is from the settling of the\nair. The lightning is the fancy made by the recoiling of the air against\nthe eye. The fall is in rain, not in shivers; because the prisons which\nthey break are extreme narrow, and the shivers being small, are\ndissolved by the heat. But in less heat they would fall in drops of\nhail, that is to say, half frozen by the shaving of the air as they\nfall, and be in a very little time, much less than snow or ice,\ndissolved.\n\n_A._ Will not that lightning burn?\n\n_B._ No. But it hath often killed men with cold. But this extraordinary\nswiftness of lightning consisteth not in the expansion of the air, but\nin a straight and direct stream from where it breaks forth; which is in\nmany places successively, according to the motion of the cloud.\n\n_A._ Experience tells us that. I have now done with my problems\nconcerning the great bodies of the world, the stars, and element of air\nin which they are moved, and am therein satisfied, and the rather,\nbecause you have answered me by the supposition of one only motion, and\ncommonly known, and the same with that of Copernicus, whose opinion is\nreceived by all the learned; and because you have not used any of these\nempty terms, sympathy, antipathy, antiperistasis, etc., for a natural\ncause, as the old philosophers have done to save their credit. For\nthough they were many of them wise men, as Plato, Aristotle, Seneca, and\nothers, and have written excellently of morals and politics, yet there\nis very little natural philosophy to be gathered out of their writings.\n\n_B._ Their ethics and politics are pleasant reading, but I find not any\nargument in their discourses of justice or virtue drawn from the supreme\nauthority, on whose laws all justice, virtue, and good politics depend.\n\n_A._ Concerning this cover, or, as some have called it, the scurf or\nscab of the terrestrial star, I will begin with you tomorrow. For it is\na large subject, containing animals, vegetables, metals, stones, and\nmany other kinds of bodies, the knowledge whereof is desired by most\nmen, and of the greatest and most general profit.\n\n_B._ And this is it, in which I shall give you the least satisfaction;\nso great is the variety of motion, and so concealed from human senses.\n\n\n ==========\n\n\n CHAPTER VII.\n OF HARD AND SOFT, AND OF THE ATOMS THAT FLY IN\n THE AIR.\n\n_A._ Concerning this cover of the earth, made up of an infinite number\nof parts of different natures, I had much ado to find any tolerable\nmethod of enquiry. But I resolved at last to begin with the questions\nconcerning hard and soft, and what kind of motion it is that makes them\nso. I know that in any pulsion of air, the parts of it go innumerable\nand inexplicable ways; but I ask only if every point of it be moved?\n\n_B._ No. If you mean a mathematical point, you know it is impossible.\nFor nothing is movable but body. But I suppose it divisible, as all\nother bodies, into parts divisible. For no substance can be divided into\nnothings.\n\n_A._ Why may not that substance within our bodies, which are called\nanimal spirits, be another kind of body, and more subtile than the\ncommon air?\n\n_B._ I know not why, no more than you or any man else knows why it is\nnot very air, though purer perhaps than the common air, as being\nstrained through the blood into the brain and nerves. But howsoever that\nbe, there is no doubt, but the least parts of the common air,\nrespectively to the whole, will easilier pierce, with equal motion, the\nbody that resisteth them, than the least parts of water. For it is by\nmotion only that any mutation is made in any thing; and all things\nstanding as they did, will appear as they did. And that which changeth\nsoft into hard, must be such as makes the parts not easily to be moved\nwithout being moved all together; which cannot be done but by some\nmotion compounded. And we call hard, that whereof no part can be put out\nof order without disordering all the rest; which is not easily done.\n\n_A._ How water and air beaten into extreme small bubbles is hardened\ninto ice, you have told me already, and I understand it. But how a soft\nhomogeneous body, as air or water, should be so hardened, I cannot\nimagine.\n\n_B._ There is no hard body that hath not also some degree of gravity;\nand consequently, being loose, there must be some efficient cause, that\nis, some motion, when it is severed from the earth, to bring the same to\nit again. And seeing this compounded motion gives to the air and water\nan endeavour from the earth, the motion which must hinder it, must be in\na way contrary to the compounded motion of the earth. For whatsoever,\nhaving been asunder, comes together again, must come contrary ways, as\nthose that follow one another go the same way, though both move upon the\nsame line.\n\n_A._ What experiment have you seen to this purpose?\n\n_B._ I have seen a drop of glass like that of the second figure, newly\ntaken out of the furnace, and hanging at the end of an iron rod, and yet\nfluid, and let fall into the water and hardened. The club-end of it A A\ncoming first to the water, the tail B C following it. It is proved\nbefore, that the motion that makes it is a compounded motion, and gives\nan endeavour outward to every part of it; and that the motion which\nmaketh cold, is such as shaving the body in every point of contact, and\nturning it, gives them all an endeavour inward. Such is this motion made\nby the sinking of the hot and fluid glass into the water. It is\ntherefore manifest that the motion which hardeneth a soft body, must in\nevery point of contact be in the contrary way to that which makes a hard\nbody soft. And farther, that slender tail B C shall be made much more\nhard than common glass. For towards the upper end, in C, you cannot\neasily break it, as small as it is. And when you have broken it, the\nwhole body will fall into dust, as it must do, seeing the bending is so\ndifficult. For all the parts are bent with such force, that upon the\nbreaking at D, by their sudden restitution to their liberty, they will\nbreak together. And the cause why the tail B C, being so slender,\nbecomes so hard, is, that all the endeavour in the great part A B, is\npropagated to the small part B C, in the same manner as the force of the\nsun-beams is derived almost to a point by a burning-glass. But the cause\nwhy, when it is broken in D, it breaks also in so many other places, is,\nthat the endeavour in all the other parts, which is called the spring,\nunbends it; from whence a motion is caused the contrary way, and that\nmotion continued bends it more the other way and breaks it, as a bow\nover-bent is broken into shivers by a sudden breaking of the string.\n\n_A._ I conceive now how a body which having been hard and softened\nagain, may be rehardened; but how a fluid and mere homogeneous body, as\nair or water, may be so, I see not yet. For the hardening of water is\nmaking a hard body of two fluids, whereof one, which is the water, hath\nsome tenacity; and so a man may make a bladder hard with blowing into\nit.\n\n_B._ As for mere air, which hath no natural motion of itself, but is\nmoved only by other bodies of a greater consistence, I think it\nimpossible to be hardened. For the parts of it so easily change places,\nthat they can never be fixed by any motion. No more I think can water,\nwhich though somewhat less fluid, is with an insensible force very\neasily broken.\n\n_A._ It is the opinion of many learned men, that ice, in long time, will\nbe turned into crystal; and they allege experience for it. For they say\nthat crystal is found hanging on high rocks in the Alps, like icicles on\nthe eaves of a house; and why may not that have formerly been ice, and\nin many years have lost the power of being reduced?\n\n_B._ If that were so, it would still be ice, though also crystal: which\ncannot be, because crystal is heavier than water, and therefore much\nheavier than ice.\n\n_A._ Is there then no transubstantiation of bodies but by mixture?\n\n_B._ Mixture is no transubstantiation.\n\n_A._ Have you never seen a stone that seemed to have been formerly wood,\nand some like shells, and some like serpents, and others like other\nthings?\n\n_B._ Yes. I have seen such things, and particularly I saw at Rome, in a\nstone-cutter\u2019s workhouse, a billet of wood, as I thought it, partly\ncovered with bark, and partly with the grain bare, as long as a man\u2019s\narm, and as thick as the calf of a man\u2019s leg; which handling I found\nextreme heavy, and saw a small part of it which was polished, and had a\nvery fine gloss, and thought it a substance between stone and metal, but\nnearest to stone. I have seen also a kind of slate painted naturally\nwith forest-work. And I have seen in the hands of a chemist of my\nacquaintance at Paris, a broken glass, part of a retort, in which had\nbeen the rosin of turpentine, wherein though there were left no rosin,\nyet there appeared in the piece of glass many trees; and plants in the\nground about them, such as grow in woods; and better designed than they\ncould be done by any painter; and continued so for a long time. These be\ngreat wonders of nature, but I will not undertake to show their causes.\nBut yet this is most certain, that nothing can make a hard body of a\nsoft, but by some motion of its parts. For the parts of the hardest body\nin the world can be no closer together than to touch; and so close are\nthe parts of air and water, and consequently they should be equally\nhard, if their smallest parts had not different natural motions.\nTherefore if you ask me the causes of these effects, I answer, they are\ndifferent motions. But if you expect from me how and by what motions, I\nshall fail you. For there is no kind of substance in the world now, that\nwas not at the first creation, when the Creator gave to all things what\nnatural and special motion he thought good. And as he made some bodies\nwondrous great, so he made others wondrous little. For all his works are\nwondrous. Man can but guess, nor guess farther, than he hath knowledge\nof the variety of motion. I am therefore of opinion, that whatsoever\nperfectly homogeneous is hard, consisteth of the smallest parts, or, as\nsome call them, atoms, that were made hard in the beginning, and\nconsequently by an eternal cause; and that the hardness of the whole\nbody is caused only by the contact of the parts by pressure.\n\n_A._ What motion is it that maketh a hard body to melt?\n\n_B._ The same compounded motion that heats, namely, that of fire, if it\nbe strong enough. For all motion compounded is an endeavour to\ndissipate, as I have said before, the parts of the body to be moved by\nit. If therefore hardness consist only in the pressing contact of the\nleast parts, this motion will make the same parts slide off from one\nanother, and the whole to take such a figure as the weight of the parts\nshall dispose them to, as in lead, iron, gold, and other things melted\nwith heat. But if the small parts have such figures as they cannot\nexactly touch, but must leave spaces between them filled with air or\nother fluids, then this motion of the fire, will dissipate those parts\nsome one way, some another, the hard part still hard; as in the burning\nof wood or stone into ashes or lime. For this motion is that which\nmaketh fermentation, scattering dissimilar parts, and congregating\nsimilar.\n\n_A._ Why do some hard bodies resist breaking more one way than another?\n\n_B._ The bodies that do so, are for the most part wood, and receive that\nquality from their generation. For the heat of the sun in the\nspring-time draweth up the moisture at the root, and together with it\nthe small parts of the earth, and twisteth it into a small twig, by its\nmotion upwards, to some length, but to very little other dimensions, and\nso leaves it to dry till the spring following; and then does the same to\nthat, and to every small part round about it; so that upward the\nstrength is doubled, and the next year trebled, &c. And these are called\nthe grain of the wood, and but touch one another, like sticks with\nlittle or no binding, and therefore can hardly be broken across the\ngrain, but easily all-along it. Also some other hard bodies have this\nquality of being more fragile one way than another, as we see in\nquarrels of a glass window, that are aptest many times to break in some\ncrooked line. The cause of this may be, that when the glass, hot from\nthe furnace, is poured out upon a plain, any small stones in or under it\nwill break the stream of it into divers lines, and not only weaken it,\nbut also cause it falsely to represent the object you look on through\nit.\n\n_A._ What is the cause why a bow of wood or steel, or other very hard\nbody, being bent, but not broken, will recover its former degree of\nstraightness?\n\n_B._ I have told you already, how the smallest parts of a hard body have\nevery one, by the generation of hardness, a circular, or other\ncompounded motion; such motion is that of the smallest parts of the bow.\nWhich circles in the bending you press into narrower figures, as a\ncircle into an ellipsis, and an ellipsis into a narrower but longer\nellipsis with violence; which turns their natural motion against the\noutward parts of the bow so bent, and is an endeavour to stretch the bow\ninto its former posture. Therefore if the impediment be removed, the bow\nmust needs recover its former figure.\n\n_A._ It is manifest; and the cause can be no other but that, except the\nbow have sense.\n\n_B._ And though the bow had sense, and appetite to boot, the cause will\nbe still the same.\n\n_A._ Do you think air and water to be pure and homogeneous bodies?\n\n_B._ Yes, and many bodies both hard and heavy to be so too, and many\nliquors also besides water.\n\n_A._ Why then do men say they find one air healthy, another infectious?\n\n_B._ Not because the nature of the air varies, but because there are in\nthe air, drawn, or rather, beaten up by the sun, many little bodies,\nwhereof some have such motion as is healthful, others such as is hurtful\nto the life of man. For the sun, as you see in the generation of plants,\ncan fetch up earth as well as water: and from the driest ground any kind\nof body that lieth loose, so it be small enough, rather than admit any\nemptiness. By some of these small bodies it is that we live; which being\ntaken in with our breath, pass into our blood, and cause it, by their\ncompounded motion, to circulate through the veins and arteries; which\nthe blood of itself, being a heavy body, without it cannot do. What kind\nof substance these atoms are, I cannot tell. Some suppose them to be\nnitre. As for those infectious creatures in the air, whereof so many die\nin the plague, I have heard that Monsieur Des Cartes, a very ingenious\nman, was of opinion, that they were little flies. But what grounds he\nhad for it, I know not, though there be many experiments that invite me\nto believe it. For first, we know that the air is never universally\ninfected over a whole country, but only in or near to some populous\ntown. And therefore the cause must also be partly ascribed to the\nmultitude thronged together, and constrained to carry their excrements\ninto the fields round about and near to their habitation, which in time\nfermenting breed worms, which commonly in a month or little more,\nnaturally become flies; and though engendered at one town, may fly to\nanother. Secondly, in the beginning of a plague, those that dwell in the\nsuburbs, that is to say, nearest to this corruption, are the poorest of\nthe people, that are nourished for the most part with the roots and\nherbs which grow in that corrupted dirt; so that the same filth makes\nboth the blood of poor people, and the substance of the fly. And it is\nsaid by Aristotle, that everything is nourished by the matter whereof it\nis generated. Thirdly, when a town is infected, the gentlemen, and those\nthat live on wholsomest food, scarce one of five hundred die of the\nplague. It seems therefore, whatsoever creatures they be that invade us\nfrom the air, they can discern their proper nourishment, and do not\nenter into the mouth and nostrils with the breath of every man alike, as\nthey would do if they were inanimate. Fourthly, a man may carry the\ninfection with him a great way into the country in his clothes, and\ninfect a village. Shall another man there draw the infection from the\nclothes only by his breath? Or from the hangings of a chamber wherein a\nman hath died? It is impossible. Therefore whatsoever killing thing is\nin the clothes or hangings, it must rise and go into his mouth or\nnostrils before it can do him hurt. It must therefore be a fly, whereof\ngreat numbers get into the blood, and there feeding and breeding worms,\nobstruct the circulation of blood, and kill the man.\n\n_A._ I would we knew the palate of those little animals; we might\nperhaps find some medicine to fright them from mingling with our breath.\nBut what is that which kills men that lie asleep too near a\ncharcoal-fire? Is it another kind of fly? Or is charcoal venomous?\n\n_B._ It is neither fly nor venom, but the effect of a flameless glowing\nfire, which dissipates those atoms that maintain the circulation of the\nblood; so that for want of it, by degrees they faint, and being asleep\ncannot remove, but in short time, there sleeping die; as is evident by\nthis, that being brought into the open air, without other help, they\nrecover.\n\n_A._ It is very likely. The next thing I would be informed of, is the\nnature of gravity. But for that, if you please, we will take another\nday.\n\n\n ==========\n\n\n CHAPTER VIII.\n OF GRAVITY AND GRAVITATION.\n\n_B._ What books are those?\n\n_A._ Two books written by two learned men concerning gravity. I brought\nthem with me, because they furnish me with some material questions about\nthat doctrine; though of the nature of gravity, I find no more in either\nof them than this, that gravity is an intrinsical quality, by which a\nbody so qualified descendeth perpendicularly towards the superficies of\nthe earth.\n\n_B._ Did neither of them consider that descending is local motion; that\nthey might have called it an intrinsical motion rather than an\nintrinsical quality?\n\n_A._ Yes. But not how motion should be intrinsical to the special\nindividual body moved. For how should they, when you are the first that\never sought the differences of qualities in local motion, except your\nauthority in philosophy were greater with them than it is? For it is\nhard for a man to conceive, except he see it, how there should be motion\nwithin a body, otherwise than as it is in living creatures.\n\n_B._ But it may be they never sought, or despaired of finding what\nnatural motion could make any inanimate thing tend one way rather than\nanother.\n\n_A._ So it seems. But the first of them inquires no farther than, why so\nmuch water, being a heavy body, as lies perpendicularly on a fish\u2019s back\nin the bottom of the sea, should not kill it. The other, whereof the\nauthor is Dr. Wallis, treateth universally of gravity.\n\n_B._ Well; but what are the questions which from these books you intend\nto ask me?\n\n_A._ The author of the first book tells me, that water and other fluids\nare bodies continued, and act, as to gravity, as a piece of ice would do\nof the same figure and quantity. Is that true?\n\n_B._ That the universe, supposing there is no place empty, is one entire\nbody, and also, as he saith it is, a continual body, is very true. And\nyet the parts thereof may be contiguous, without any other cohesion but\ntouch. And it is also true, that a vessel of water will descend in a\nmedium less heavy, but fluid, as ice would do.\n\n_A._ But he means that water in a tub would have the same effect upon a\nfish in the bottom of the tub, as so much ice would have.\n\n_B._ That also would be true, if the water were frozen to the sides of\nit. Otherwise the ice, if there be enough, will crush the fish to death.\nBut how applies he this, to prove that the water cannot hurt a fish in\nthe sea by its weight?\n\n_A._ It plainly appears that water does not gravitate on any part of\nitself beneath it.\n\n_B._ It appears by experience, but not by this argument, though instead\nof water the tub were filled with quicksilver.\n\n_A._ I thought so. But how it comes to pass that the fish remains\nuncrushed, I cannot tell.\n\n_B._ The endeavour of the quicksilver downward is stopped by the\nresistance of the hard bottom. But all resistance is a contrary\nendeavour; that is, an endeavour upwards, which gives the like endeavour\nto the quicksilver, which is also heavy, and thereby the endeavour of\nthe quicksilver is diverted to the sides round about, where stopped\nagain by the resistance of the sides, it receives an endeavour upwards,\nwhich carries the fish to the top, lying all the way upon a soft bed of\nquicksilver. This is the true manner how the fish is saved harmless. But\nyour author, I believe, either wanted age, or had too much business, to\nstudy the doctrine of motion; and never considered that resistance is\nnot an impediment only, or privation, but a contrary motion; and that\nwhen a man claps two pieces of wax together, their contrary endeavour\nwill turn both the pieces into one cake of wax.\n\n_A._ I know not the author; but it seems he has deeplier considered this\nquestion than other men; for in the introduction to his book he saith,\n\u201cthat men have pre-engaged themselves to maintain certain principles of\ntheir own invention, and are therefore unwilling to receive anything\nthat may render their labour fruitless;\u201d and, \u201cthat they have not\nstrictly enough considered the several interventions that abate, impede,\nadvance, or direct the gravitation of bodies.\u201d\n\n_B._ This is true enough; and he himself is one of those men, in that he\nconsidered not, that resistance is one of those interventions which\nabate, impede, and direct gravitation. But what are his suppositions for\nthe questions he handles?\n\n_A._ His first is, that as in a pyramid of brick, wherein the bricks are\nso joined that the uppermost lies everywhere over the joint or cement of\nthe two next below it, you may break down a part and leave a cavity, and\nyet the bricks above will stand firm and sustain one another by their\ncross posture: so also it is in wheat, hailshot, sand, or water; and so\nthey arch themselves, and thereby the fish is every way secured by an\narch of water over it.\n\n_B._ That the cause why fishes are not crushed nor hurt in the bottom of\nthe sea by the weight of the water, is the water\u2019s arching itself, is\nvery manifest. For if the uppermost orb of the water should descend by\nits gravity, it would tend toward the centre of the earth, and place\nitself all the way in a less and lesser orb, which is impossible. For\nthe places of the same body are always equal. But that wheat, sand,\nhailshot, or loose stones should make a firm arch, is not credible.\n\n_A._ The author therefore, it seems, quits it, and taketh a second\nhypothesis for the true cause, though the former, he saith, be not\nuseless, but contributes its part to it.\n\n_B._ I see, though he depart from his hypothesis, he looks back upon it\nwith some kindness. What is his second hypothesis?\n\n_A._ It is, that air and water have an endeavour to motion upward,\ndownward, directly, obliquely, and every way. For air, he saith, will\ncome down his chimney, and in at his door, and up his stairs.\n\n_B._ Yes, and mine too; and so would water, if I dwelt under water,\nrather than admit of vacuum. But what of that?\n\n_A._ Why then it would follow, that those several tendencies or\nendeavours would so abate, impede, and correct one another, as none of\nthem should gravitate. Which being granted, the fish can take no harm;\nwherein I find one difficulty, which is this: the water having an\nendeavour to motion every way at once, methinks it should go no way, but\nlie at rest; which, he saith, was the opinion of Stevinus, and rejecteth\nit, saying, it would crush the fish into pieces.\n\n_B._ I think the water in this case would neither rest nor crush. For\nthe endeavour being, as he saith, intrinsical, and every way, must needs\ndrive the water perpetually outward; that is to say, as to this\nquestion, upwards; and seeing the same endeavour in one individual body\ncannot be more ways at once than one, it will carry it on perpetually\nwithout limit, beyond the fixed stars; and so we shall never more have\nrain.\n\n_A._ As ridiculous as it is, it necessarily follows.\n\n_B._ What are Dr. Wallis\u2019s suppositions?\n\n_A._ He goes upon experiments. And, first, he allegeth this, that water\nleft to itself without disturbance, does naturally settle itself into a\nhorizontal plane.\n\n_B._ He does not then, as your author and all other men, take gravity\nfor that quality whereby a body tendeth to the centre of the earth.\n\n_A._ Yes, he defines gravity to be a natural propension towards the\ncentre of the earth.\n\n_B._ Then he contradicteth himself. For if all heavy bodies tend\nnaturally to one centre, they shall never settle in a plane, but in a\nspherical superficies. But against this, that such an horizontal plane\nis found in water by experience, I say it is impossible. For the\nexperiment cannot be made in a basin, but in half a mile at sea\nexperience visibly shows the contrary. According to this, he should\nthink also that a pair of scales should hang parallel.\n\n_A._ He thinks that too.\n\n_B._ Let us then leave this experiment. What says he farther concerning\ngravity?\n\n_A._ He takes for granted, not as an experiment but an axiom, that\nnature worketh not by election, but _ad ultimum virium_, with all the\npower it can.\n\n_B._ I think he means, (for it is a very obscure passage), that every\ninanimate body by nature worketh all it can without election; which may\nbe true. But it is certain that men, and beasts, work often by election,\nand often without election; as when he goes by election, and falls\nwithout it. In this sense I grant him, that nature does all it can. But\nwhat infers he from it?\n\n_A._ That naturally every body has every way, if the ways oppose not one\nanother, an endeavour to motion; and consequently, that if a vessel have\ntwo holes, one at the side, another at the bottom, the water will run\nout at both.\n\n_B._ Does he think the body of water that runs out at the side, and that\nwhich runs out at the bottom, is but one and the same body of water?\n\n_A._ No, sure; he cannot think but that they are two several parts of\nthe whole water in the vessel.\n\n_B._ What wonder is it then, if two parts of water run two ways at once,\nor a thousand parts a thousand ways? Does it follow thence that one body\ncan go more than one way at once? Why is he still meddling with things\nof such difficulty? He will find at last that he has not a genius either\nfor natural philosophy or for geometry. What other suppositions has he?\n\n_A._ My first author had affirmed, that a lighter body does not\ngravitate on a heavier; against this Dr. Wallis thus argueth: Let there\nbe a siphon, A B C D, filled with quicksilver to the level A D; if then\nyou pour oil upon A as high as to E, he asketh if the oil in A E, as\nbeing heavy, shall not press down the quicksilver a little at A, and\nmake it rise a little at D, suppose to F; and answers himself, that\ncertainly it will; so that it is neither an experiment nor an\nhypothesis, but only his opinion.\n\n_B._ Whatsoever it be, it is not true; though the doctor may be\npardoned, because the contrary was never proved.\n\n_A._ Can you prove the contrary?\n\n_B._ Yes; for the endeavour of the quicksilver both from A and D\ndownward, is stronger than that of the oil downward. If, therefore, the\nendeavour of the quicksilver were not resisted by the bottom B C, it\nwould fall so, by reason of the acceleration of heavy bodies in their\ndescending, as to leave the oil, so that it should not only not press,\nbut also not touch the quicksilver. It is true, in a pair of scales\nequally charged with quicksilver, that the addition of a little oil to\neither scale will make it preponderate. And that was it deceived him.\n\n_A._ It is evident. The last experiment he cites is the weighing of air\nin a pair of scales, where it is found manifestly that it has some\nlittle weight. For if you weigh a bladder, and put the weight into one\nscale, and then blow the bladder full of air, and put it into the other\nscale, the full bladder will outweigh the empty. Must not then the air\ngravitate?\n\n_B._ It does not follow. I have seen the experiment just as you describe\nit, but it can never be thence demonstrated that air has any weight. For\nas much air as is pressed downward by the weight of the blown bladder,\nso much will rise from below, and lay itself spherically at the altitude\nof the centre of gravity of the bladder so blown. So that all the air\nwithin the bladder above that centre is carried thither imprisoned, and\nby violence: and the force that carries it up is equal to that which\npresseth it down. There must, therefore, be allowed some little\ncounterpoise in the other scale to balance it. Therefore, the experiment\nproves nothing to his purpose. And whereas they say there be small heavy\nbodies in the air, which make it gravitate, do they think the force\nwhich brought them thither cannot hold them there?\n\n_A._ I leave this question of the fish as clearly resolved, because the\nwater tending every way to one point, which is the centre of the earth,\nmust of necessity arch itself. And now tell me your own opinion\nconcerning the cause of gravity, and why all bodies descend or ascend\nnot all alike. For there can be no more matter in one place than another\nif the places be equal.\n\n_B._ I have already showed you in general, that the difference of motion\nin the parts of several bodies makes the difference of their natures.\nAnd all the difference of motions consisteth either in swiftness, or in\nthe way, or in the duration. But to tell you in special why gold is\nheaviest, and then quicksilver, and then, perhaps, lead, is more than I\nhope to know, or mean to enquire; for I doubt not but that the species\nof heavy, hard, opaque, and diaphanous, were all made so at their\ncreation, and at the same time separated from different species. So that\nI cannot guess at any particular motions that should constitute their\nnatures, farther than I am guided by the experiments made by fire or\nmixture.\n\n_A._ You hope not then to make gold by art?\n\n_B._ No, unless I could make one and the same thing heavier than it was.\nGod hath from the beginning made all the kinds of hard, and heavy, and\ndiaphanous bodies that are, and of such figure and magnitude as he\nthought fit; but how small soever, they may by accretion become greater\nin the mine, or perhaps by generation, though we know not how. But that\ngold, by the art of man, should be made of not gold, I cannot\nunderstand; nor can they that pretend to show how. For the heaviest of\nall bodies, by what mixture soever of other bodies, will be made\nlighter, and not to be received for gold.\n\n_A._ Why, when the cause of gravity consisteth in motion, should you\ndespair of finding it?\n\n_B._ It is certain that when any two bodies meet, as the earth and any\nheavy body will, the motion that brings them to or towards one another,\nmust be upon two contrary ways; and so also it is when two bodies press\neach other in order to make them hard; so that one contrariety of motion\nmight cause both hard and heavy, but it doth not, for the hardest bodies\nare not always the heaviest; therefore I find no access that way to\ncompare the causes of different endeavours of heavy bodies to descend.\n\n_A._ But show me at least how any heavy body that is once above in the\nair, can descend to the earth, when there is no visible movent to thrust\nor pull it down.\n\n_B._ It is already granted, that the earth hath this compounded motion\nsupposed by Copernicus, and that thereby it casteth the contiguous air\nfrom itself every way round about. Which air so cast off, must\ncontinually, by its nature, range itself in a spherical orb. Suppose a\nstone, for instance, were taken up from the ground, and held up in the\nair by a man\u2019s hand, what shall come into the place it filled when it\nlay upon the earth?\n\n_A._ So much air as is equal to the stone in magnitude, must descend and\nplace itself in an orb upon the earth. But then I see that to avoid\nvacuum, another orb of air of the same magnitude must descend, and place\nitself in that, and so perpetually to the man\u2019s hand; and then so much\nair as would fill the place must descend in the same manner, and bring\nthe stone down with it. For the stone having no endeavour upward, the\nleast motion of the air, the hand being removed, will thrust it\ndownward.\n\n_B._ It is just so. And farther, the motion of the stone downward shall\ncontinually be accelerated according to the odd numbers from unity; as\nyou know hath been demonstrated by Galileo. But we are nothing the\nnearer, by this, to the knowledge of why one body should have a greater\nendeavour downward than another. You see the cause of gravity is this\ncompounded motion with exclusion of vacuum.\n\n_A._ It may be it is the figure that makes the difference. For though\nfigure be not motion, yet it may facilitate motion, as you see commonly\nthe breadth of a heavy body retardeth the sinking of it. And the cause\nof it is, that it makes the air have farther to go laterally, before it\ncan rise from under it. For suppose a body of quicksilver falling in the\nair from a certain height, must it not, going as it does toward the\ncentre of the earth, as it draws nearer and nearer to the earth, become\nmore and more slender, in the form of a solid sector? And if it have far\nto go, divide itself into drops? This figure of a solid sector is like a\nneedle with the point downward, and therefore I should think that\nfacilitating the motion of it does the same that would be done by\nincreasing the endeavour.\n\n_B._ Do not you see that this way of facilitating is the same in water,\nand in all other fluid heavy bodies? Besides, your argument ought to be\napplicable to the weighing of bodies in a pair of scales, which it is\nnot, for there they have no such figure; it should also hold in the\ncomparison of gravity in hard and fluid bodies.\n\n_A._ I had not sufficiently considered it. But supposing now, as you do,\nthat both heavy and hard bodies, in their smallest parts, were made so\nin the creation; yet, because quicksilver is harder than water, a drop\nof water shall in descending be pressed into a more slender sector than\na drop of quicksilver, and consequently the earth shall more easily cast\noff any quantity of water than the same quantity of quicksilver.\n\n_B._ This one would think were true; as also that of simple fluid\nbodies, those whose smallest parts, naturally, without the force of\nfire, do strongliest cohere, are generally the heaviest. But why then\nshould quicksilver be heavier than stone or steel? Fluidity and hardness\nare but degrees between greater fluidity and greater hardness. Therefore\nto the knowledge of what it is that causeth the difference, in different\nbodies, of their endeavour downward, there are required, if it can be\nknown at all, a great many more experiments than have been yet made. It\nis not difficult to find why water is heavier than ice, or other body\nmixed of air and water. But to believe that all bodies are heavier or\nlighter according to the quantity of air within them, is very hard.\n\n_A._ I see by this, that the Creator of the world, as by his power he\nordered it, so by his wisdom he provided it should be never disordered.\nTherefore leaving this question, I desire to know whether if a heavy\nbody were as high as a fixed star, it would return to the earth.\n\n_B._ It is hard to try. But if there be this compounded motion in the\ngreat bodies so high, such as is in the earth, it is very likely that\nsome heavy bodies will be carried to them. But we shall never know it\ntill we be at the like height.\n\n_A._ What think you is the reason why a drop of water, though heavy,\nwill stand upon a horizontal plane of dry or unctuous wood, and not\nspread itself upon it? For let A B, in the sixth figure, be the dry\nplane, D the drop of water, and D C perpendicular to A B. The drop D,\nthough higher, will not descend and spread itself upon it.\n\n_B._ The reason I think is manifest. For those bodies which are made by\nbeating of water and air together, show plainly that the parts of water\nhave a great degree of cohesion. For the skin of the bubble is water,\nand yet it can keep the air, though moved, from getting out. Therefore\nthe whole drop of water at D, hath a good deal of cohesion of parts. And\nseeing A B is an horizontal plane, the way from the contact in D either\nto A or B is upwards, and consequently there is no endeavour in D either\nof those ways, but what proceeds from so much weight of water as is able\nto break that cohesion, which so small a drop is too weak to do. But the\ncohesion being once broken, as with your finger, the water will follow.\n\n_A._ Seeing the descent of a heavy body increaseth according to the odd\nnumbers 1, 3, 5, 7, &c. and the aggregates of those numbers, viz. of 1\nand 3; and 1 and 3 and 5; and of 1 and 3 and 5 and 7, &c. are square\nnumbers, namely 4, 9, 16; the whole swiftness of the descent will be, I\nthink, to the aggregate of so many swiftnesses equal to the first\nendeavour, as square numbers are to their sides, 1, 2, 3, 4. Is it so?\n\n_B._ Yes, you know it hath been demonstrated by Galileo.\n\n_A._ Then if, for instance, you put into a pair of scales equal\nquantities of quicksilver and water, seeing they are both accelerated in\nthe same proportion, why should not the weight of quicksilver to the\nweight of water be in duplicate proportions to their first endeavours?\n\n_B._ Because they are in a pair of scales. For there the motion of\nneither of them is accelerated. And therefore it will be, as the first\nendeavour of the quicksilver to the first endeavour of the water, so the\nwhole weight to the whole weight. By which you may see, that the cause\nwhich takes away the gravitation of liquid bodies from fish or other\nlighter bodies within them, can never be derived from the weight.\n\n_A._ I have one question more to ask concerning gravity. If gravity be,\nas some define it, an intrinsical quality, whereby a body descendeth\ntowards the centre of the earth, how is it possible that a piece of iron\nthat hath this intrinsical quality should rise from the earth, to go to\na loadstone? Hath it also an intrinsical quality to go from the earth?\nIt cannot be. The cause therefore must be extrinsical. And because when\nthey are come together in the air, if you leave them to their own\nnature, they will fall down together, they must also have some like\nextrinsical cause. And so this magnetic virtue will be such another\nvirtue as makes all heavy bodies to descend, in this our world, to the\nearth. If therefore you can from this your hypothesis of compounded\nmotion, by which you have so probably salved the problem of gravity,\nsalve also this of the loadstone, I shall acknowledge both your\nhypothesis to be true, and your conclusion to be well deduced.\n\n_B._ I think it not impossible. But I will proceed no farther in it now,\nthan, for the facilitating of the demonstrations, to tell you the\nseveral proprieties of the magnet, whereof I am to show the causes. As\nfirst, that iron, and no other body, at some little distance, though\nheavy, will rise to it. Secondly, that if it be laid upon a still water\nin a floating vessel, and left to itself, it will turn itself till it\nlie in a meridian, that is to say, with one and the same line still\nnorth and south. Thirdly, if you take a long slender piece of iron, and\napply the loadstone to it, and, according to the position of the poles\nof the loadstone, draw it over to the end of the iron, the iron will\nhave the same poles with the magnet, so it be drawn with some pressure;\nbut the poles will lie in a contrary position; and also this long iron\nwill draw other iron to it as the magnet doth.\n\nFourthly, this long iron, if it be so small as that poised upon a pin,\nthe weight of it have no visible effect, the navigators use it for the\nneedle of their compass, because it points north and south; saving that\nin most places by particular accidents it is diverted; which diversion\nis called the variation of the horizontal needle. Fifthly, the same\nneedle placed in a plane perpendicular to the horizon, hath another\nmotion called the inclination. Which that you may the better conceive,\ndraw a fourth figure; wherein let there be a circle to represent the\nterrella, that is to say, a spherical magnet.\n\n_A._ Let this be it, whose centre is A, the north pole B, the south pole\nC.\n\n_B._ Join B C, and cross it at right angles with the diameter D E.\n\n_A._ It is done.\n\n_B._ Upon the point D set the needle parallel to B C, with the cross of\nthe south pole, and the barb for the north; and describe a square about\nthe circle B D C E, and divide the arch D B into four equal parts in\n_a_, _b_, _c._\n\n_A._ It is done.\n\n_B._ Then place the middle of the needle on the points _a_, _b_, _c_, so\nthat they may freely turn; and set the barb which is at D towards the\nnorth, and that which is at C towards the south. You see plainly by\nthis, that the angles of inclination through the arch D C taken\naltogether, are double to a right angle. For when the south point of the\nneedle, looking north, as at D, comes to look south, as at C, it must\nmake half a circle.\n\n_A._ That is true. And if you draw the sine of the arch D _a_, which is\n_d a_, and the sine of the arch B _a_, which is _a e_, and the sine of\nthe arch D _b_, which is _b f_, and the sine of the arch B _c_, which is\n_c g_, the needle will lie upon _b f_ with the north-point downwards, so\nthat the needle will be parallel to A D. Then from _a_ draw the line _a\nh_, making the angle _e a h_ equal to the angle D A _a_. And then the\nneedle at _a_ shall lie in the line _a h_ with the south point toward\n_h_. Finally, draw the line _c h_, which, with _c g_, will also make a\nquarter of a right angle; and therefore if the needle be placed on the\npoint _c_, it will lie in _c h_ with the south point toward _h_. And\nthus you see by what degrees the needle inclines or dips under the\nhorizon more and more from D till it come to the north pole at B; where\nit will lie parallel to the needle in D; but with their barbs looking\ncontrary ways. And this is certain by experience, and by none\ncontradicted.\n\nYou see then why the degrees of the inclinatory needle, in coming from D\nto B, are double to the degrees of a quadrant. It is found also by\nexperience, that iron both of the mine and of the furnace put into a\nvessel so as to float, will lay itself (if some accident in the earth\nhinder it not) exactly north and south. And now I am, from this\ncompounded motion supposed by Copernicus, to derive the causes why a\nloadstone draws iron; why it makes iron to do the same; why naturally it\nplaceth itself in a parallel to the axis of the earth; why by passing it\nover the needle it changes its poles; and what is the cause that it\ninclines. But it is your part to remember what I told you of motion at\nour second meeting; and what I told you of this compounded motion\nsupposed by Copernicus, at our fourth meeting.\n\n\n ==========\n\n\n CHAPTER IX.\n OF THE LOADSTONE AND ITS POLES, AND WHETHER\n THEY SHOW THE LONGITUDE OF PLACES ON THE EARTH.\n\n_A._ I come now to hear what natural causes you can assign of the\nvirtues of the magnet; and first, why it draws iron to it, and only\niron.\n\n_B._ You know I have no other cause to assign but some local motion, and\nthat I never approved of any argument drawn from sympathy, influence,\nsubstantial forms, or incorporeal effluvia. For I am not, nor am\naccounted by my antagonists for a witch. But to answer this question, I\nshould describe the globe of the earth greater than it is at B in the\nfirst figure, but that the terrella in the fourth figure will serve our\nturn. For it is but calling B and C the poles of the earth, and D E the\ndiameter of the equinoctial circle, and making D the east, and E the\nwest. And then you must remember that the annual motion of the earth is\nfrom west to east, and compounded of a straight and circular motion, so\nas that every point of it shall describe a small circle from west to\neast, as is done by the whole globe. And let the circles about _a b c_\nbe three of those small circles.\n\n_A._ Before you go any farther, I pray you show me how I must\ndistinguish east and west in every part of this figure. For wheresoever\nI am on earth, suppose at London, and see the sun rise suppose in\nCancer, is not a straight line from my eye to the sun terminated in the\neast?\n\n_B._ It is not due east, but partly east, partly south. For the earth,\nbeing but a point compared to the sun, all the parallels to D E the\nequator, such as are _e a_, _f b_, _c g_, if they be produced, will fall\nupon the body of the sun. And therefore A _b_ is north-east; A _a_ east\nnorth-east; and A _c_ north north-east.\n\n_A._ Proceed now to the cause of attraction.\n\n_B._ Suppose now that the internal parts of the loadstone had the same\nmotion with that of the internal parts of the sun which make the annual\nmotion of the earth from west to east, but in a contrary way, for\notherwise the loadstone and the iron can never be made to meet. Then set\nthe loadstone at a little distance from the earth, marked with _z_; and\nthe iron marked with _x_ upon the superficies of the earth. Now that\nwhich makes _x_ rise to _z_, can be nothing else but air; for nothing\ntouches it but air. And that which makes the air to rise, can be nothing\nbut those small circles made by the parts of the earth, such are at _a b\nc_, for nothing else touches the air. Seeing then the motion of each\npoint of the loadstone is from east to west in circles, and the motion\nof each point of the iron from west to east; it follows, that the air\nbetween the loadstone and the iron shall be cast off both east and west;\nand consequently the place left empty, if the iron did not rise up and\nfill it. Thus you see the cause that maketh the loadstone and the iron\nto meet.\n\n_A._ Hitherto I assent. But why they should meet when some heterogeneous\nbody lies in the air between them, I cannot imagine. And yet I have seen\na knife, though within the sheath, attract one end of the needle of a\nmariner\u2019s compass; and have heard it will do the same though a\nstone-wall were between.\n\n_B._ Such iron were indeed a very vigorous loadstone. But the cause of\nit is the same that causeth fire or hot water, which have the same\ncompounded motion, to work through a vessel of brass. For though the\nmotion be altered by restraint within the heterogeneous body, yet being\ncontinued quite through, it restores itself.\n\n_A._ What is the cause why the iron rubbed over by a loadstone will\nreceive the virtue which the loadstone hath of drawing iron to it?\n\n_B._ Since the motion that brings two bodies to meet must have contrary\nways, and that the motions of the internal parts of the magnet and of\nthe iron are contrary; the rubbing of them together does not give the\niron the first endeavour to rise, but multiplies it. For the iron\nuntouched will rise to a loadstone; but if touched, it becomes a\nloadstone to other iron. For when they touch a piece of iron, they pass\nthe loadstone over it only one way, viz. from pole to pole; not back\nagain, for that would undo what before had been done; also they press it\nin passing to the very end of the iron, and somewhat hard. So that by\nthis pressing motion all the small circles about the points _a b c_, are\nturned the contrary way; and the halves of those small circles made on\nthe arch D B will be taken away and the poles changed, so as that the\nnorth poles shall point south, and the south poles north, as in the\nfigure.\n\n_A._ But how comes it to pass, that when a loadstone hath drawn a piece\nof iron, you may add to it another, as if they begat one another? Is\nthere the like motion in the generation of animals?\n\n_B._ I have told you that iron of itself will rise to the loadstone;\nmuch more then will it adhere to it when it is armed with iron, and both\nit and the iron have a plain superficies. For then not only the points\nof contact will be many, which make the coherence stronger, but also the\niron wherewith it is armed is now another loadstone, differing a little,\nwhich you perhaps think, as male and female. But whether this compounded\nmotion and confrication causeth the generation of animals, how should I\nknow, that never had so much leisure as to make any observation which\nmight conduce to that?\n\n_A._ My next question is, seeing you say the loadstone, or a needle\ntouched with it, naturally respecteth the poles of the earth, but that\nthe variation of it proceedeth from some accidents in the superficies of\nthe earth; what are those accidents?\n\n_B._ Suppose there be a hill upon the earth, for example, at _r_; then\nthe stream of the air which was between _z_ and _x_ westward, coming to\nthe hill, shall go up the hill\u2019s side, and so down to the other side,\naccording to the crooked line which I have marked about the hill by\npoints; and this infallibly will turn the north point of the needle,\nbeing on the east side, more towards the east, and that on the other\nside more towards the west, than if there had been no hill. And where\nupon the earth are there not eminences and depressions, except in some\nwide sea, and a great way from land.\n\n_A._ But if that be true, the variation in the same place should be\nalways the same, for the hills are not removed.\n\n_B._ The variation of the needle at the same place is still the same;\nbut the variation of the variation is partly from the motion of the pole\nitself, which by the astronomers is called _motus trepidationis_; and\npartly from that, that the variation cannot be truly observed, for the\nhorizontal needle and the inclinatory needle incline alike, but cannot\nincline in due quantity. For whether set upon a pin or an axis, their\ninclination is hindered, in the horizontal needle, by the pin itself: if\nupon an axis, if the axis be just, it cannot move; if slack, the weight\nwill hinder it; but chiefly because the north pole of the earth draws\naway from it the north pole of the needle, for two like poles cannot\ncome together. And this is the cause why the variation in one place is\neast, and another west.\n\n_A._ This is indeed the most probable reason why the variation varies\nthat ever I heard given; and I should presently acknowledge that this\nparallel motion of the axis of the earth in the ecliptic, supposed by\nCopernicus, is the true annual motion of the earth, but that there is\nlately come forth a book called _Longitude Found_ , which makes the\nmagnetical poles distant from the poles of the earth eight degrees and a\nhalf.\n\n_B._ I have the book. It is far from being demonstrated, as you shall\nfind, if you have the patience to see it examined. For wheresoever his\ndemonstration is true, the conclusion, if rightly inferred, will be\nthis, that the poles of the loadstone and the poles of the earth are the\nsame. And where, on the contrary, his demonstrations are fallacies, it\nis because sometimes he fancieth the lines he hath drawn, not where they\nare; sometimes because he mistakes his station; and sometimes because he\ngoes on some false principle of natural philosophy; and sometimes also\nbecause he knoweth not sufficiently the doctrine of spherical triangles.\n\n_A._ I think that is the book there which lies at your elbow. Pray you\nread.\n\n_B._ I find first (p. 4), that the grounds of his argument are the two\nobservations made by Mr. Burroughs, one at Vaygates, in 1576, where the\nvariation from the pole of the earth he found to be 11 deg. 15 min.\neast; the other at Limehouse, near London, in 1580, where the variation\nfrom the pole of the earth was 8 deg. 38 min. west, by which, he saith,\nhe might _find out the magnetical pole_.\n\n_A._ Where is Vaygates?\n\n_B._ In 70 degrees of north latitude; the difference of longitude\nbetween London and it being 58 degrees.\n\n_A._ The longitude of places being yet to seek, how came he to know this\ndifference of 58 degrees, except the poles of the magnet and the earth\nbe the same?\n\n_B._ I believe he trusted to the globe for that. For the distance\nbetween the places is not above 2000 miles the nearest way. But we will\npass by that, and come to his demonstration, and to his diagram, wherein\nL is London, P the north-pole of the earth, V Vaygates. So that L P is\n38 deg. 28 min.; P V 20 deg.; the angle L P V 58 deg. for the difference\nbetween the longitudes of Vaygates and London. This is the construction.\nBut before I come to the demonstration, I have an inference to draw from\nthese observations, which is this. Because in the same year the\nvariation at London was 11 deg. 15 min. east, and at Vaygates 8 deg. 38\nmin. west; if you subtract 11 deg. 15 min. from the arc L P; and 8 deg.\n38 min. from the arc L V, the variation on both sides will be taken\naway; so that P V being the meridian of Vaygates, and L P the meridian\nof London, they shall both of them meet in P the pole of the earth. And\nif the pole of the magnet be nearer to the zenith of London than is the\npole of the earth, it shall be just as much nearer to the zenith of\nVaygates in the meridian of Vaygates, which is P V; as is manifest by\nthe diurnal motion of the earth.\n\n_A._ All this I conceive without difficulty. Proceed to the\ndemonstration.\n\n_B._ Mark well now. His words are these (page 5): From P L V subtract 11\ndeg. 15 min., and there remains the angle V L M. Consider now which is\nthe angle P L V, and which is the remaining angle V L M, and tell what\nyou understand by it.\n\n_A._ He has marked the angle P L V with two numbers, 11 deg. 15 min. and\n21 deg. 50 min., which together make 33 deg. 5 min. And the angle 11\ndeg. 15 min. being subtracted from P L V, there will remain 21 deg. 50\nmin. for the angle V L M. I know not what to say to it. For I thought\nthe arc P V, which is 20 deg., had been the arc of the spherical angle P\nL V; and that the arc L V had been 58 deg., because he says the angle L\nP V is so; and that the arc L M had been 46 deg., because the angle L P\nM is so; and lastly, that the angle P L M had been 8 deg. 30 min.,\nbecause the arc P M is so.\n\n_B._ And what you thought had been true, if a spherical angle were a\nvery angle. For all men that have written of spherical triangles take\nfor the ground of their calculation, as Regiomontanus, Copernicus, and\nClavius, that the arch of a spherical angle is the side opposite to the\nangle. You should have considered also that he makes the angle V P M 12\ndeg., but sets down no arc to answer it. But that you may find I am in\nthe right, look into the definitions which Clavius hath put down before\nhis treatise of spherical triangles, and amongst them is this; \u201cthe arc\nof a spherical triangle is a part of a great circle intercepted between\nthe two sides drawn from the pole of the said great circle.\u201d\n\n_A._ The book is nothing worth; for it is impossible to subtract an arc\nof a circle out of a spherical angle. And I see besides that he takes\nthe superficies that lieth between the sides L P and L M for an arch,\nwhich is the quantity of an angle; and is a line, and cannot be taken\nout of a superficies. I wonder how any man that pretends to mathematics\ncould be so much mistaken.\n\n_B._ It is no great wonder. For Clavius himself striving to maintain\nthat a right angle is greater than the angle made by the diameter and\nthe circumference, fell into the same error. A corner, in vulgar speech,\nand an angle, in the language of geometry, are not the same thing. But\nit is easy even for a learned man sometimes to take them for the same,\nas this author now has done; and proceeding he saith, subtract 8 deg. 38\nmin. from the angle P V L, and there remains the angle L V M.\n\n_A._ That again is false, because impossible. What was it that deceived\nhim now?\n\n_B._ The same misunderstanding of the nature of a spherical angle. Which\nappears farther in this, that when he knew the arc V P was part of a\ngreat circle, he thought V M, which he maketh 8 deg. 30 min., were also\nparts of a great circle; which is manifestly false. For two great\ncircles, because they pass through the centre, do cut each other into\nhalves. But V P is not half a circle. He sure thought himself at\nVaygates, and that P M V was equal to P V, although in the same\nhemisphere.\n\n_A._ But how proves he that the arc P M is 8 degrees 30 minutes?\n\n_B._ Thus. We have in two triangles, P L M and P V M, two sides and one\nangle included, to find P M the distance of the magnetical pole from the\npole of the earth 8 deg. 30 min.\n\n_A._ Is that all? It is very short for a demonstration of two so\ndifficult problems, as the quantity of 8 deg. 30 min.; and of the place\nof the magnetical pole. But he has proved nothing till he has showed how\nhe found it. And though P M be 8 deg. 30 min., it follows not that M is\nthe magnetical pole.\n\n_B._ Nor is it true. For if P M be 8 deg. 30 min., and V M 8 deg. 38\nmin., the whole arc P M V will be 17 deg. 8 min., which should be 20\ndeg. Besides, whereas the variations were east and west, the subtracting\nof them should be also east and west, but they are north and south.\n\n_A._ I am satisfied that the magnetical poles and the poles of the earth\nare the same. But thus much I confess, if they were not the same, the\nlongitude were found. For the difference of the latitudes of the earth\u2019s\nequator and of the magnetical equator, is the difference of the\nlongitude. But proceed.\n\n_B._ \u201cThe earth being a solid body, and the magnetic sphere that\nencompasseth the earth being a substance that hath not solidity to keep\npace with the earth, loseth in its motion: and that may be the cause of\nthe motion of the magnetic poles from east to west.\u201d\n\n_A._ This is very fine and unexpected. The magnetic sphere, which I took\nfor a globe made of a magnet, has not solidity to keep pace with the\nearth, though it be one of the hardest stones that are. It encompasseth\nthe earth; yet I thought nothing had encompassed the earth but air in\nwhich I breath and move. By this also the whole earth must be a\nloadstone. For two bodies cannot be in one place. So that he is yet no\nfarther than Dr. Gilbert whom he slights. And if the sphere be a magnet,\nthen the earth and loadstone have the same poles. See the force of\ntruth! which though it could not draw to it his reason, hath drawn his\nwords to it.\n\n_B._ But perhaps he meant that the magnetic virtue encompasseth the\nearth, and not the magnetic body.\n\n_A._ But that helpeth him not. For if the body of the magnet be not\nthere, the virtue then is the virtue of the earth; and so again the\npoles of the earth are magnetic poles.\n\n_B._ You see how unsafe it is to boast of doctrines as of God\u2019s gifts,\ntill we are sure that they are true. For God giveth and denieth as he\npleaseth, not as ourselves wish; as now to him he hath given confidence\nenough, but hath denied him, at least hitherto, the finding of the\nlongitudes. In the next place (p. 8) he seems much pleased that his\ndoctrine agrees with an opinion of Keplerus, that from the creation to\nthe year of our Lord, it is to the year 1657 now 5650 years; and with\nthat which he saith some divines have held in times past, that as this\nworld was created in six days, so it should continue six thousand years.\nBy which account the world will be at an end three hundred and fifty\nyears hence; though the Scripture tells us it shall come as a thief in\nthe night. O what advantage three hundred and forty years hence will\nthey have that know this, over them that know it not, by taking up money\nat interest, or selling lands at twenty years\u2019 purchase!\n\n_A._ But he says he will not meddle with that.\n\n_B._ Yes, when he had meddled with it too much already.\n\n_A._ But you have not told me wherein consisteth this agreement between\nhim and Keplerus.\n\n_B._ I forgot it. It is in the motion of the magnetic poles. For\nprecedently (p. 7), he had said \u201cthat their period or revolution was six\nhundred years; their yearly motion thirty-six minutes; and (p. 8) that\ntheir motion is by sixes. Six tenths of a degree in one year; six\ndegrees in ten years; sixty degrees in a hundred years; and six times\nsixty degrees in six hundred years.\u201d\n\n_A._ But what natural cause doth he assign of this revolution of six\nhundred years?\n\n_B._ None at all. For the magnet lying upon the earth, can have no\nmotion at all but what the earth and the air give it. And because it is\nalways at 8 deg. 30 min. distance from the pole of the earth, the earth\ncan give it no other motion than what it gives to its own poles by the\nprecession of the equinoctial points. Nor can the air give it any motion\nbut by its stream; which must needs vary when the stream varieth. But\nwhat a vast difference does he make between the period of the motion of\nthe equinoctial points, which is about or near thirty-six thousand years\naccording to Copernicus (lib. iii. cap. 6), which makes the annual\nprecession to be 36 seconds, and the period of the magnetical poles\u2019\nmotion, which is but six hundred years.\n\n_A_. Go on.\n\n_B_. He comes now (p. 15) to the inclinatory needle upon a spherical\nloadstone. Where he shows, by diagram, that the needle and the\ninstrument together moved towards the magnetical pole, make the sum of\nthe inclinations equal to two quadrants, setting the north-point of the\nneedle southward: which I confess is true. But, in the same page, he\nascribeth the same motion to the earth in these words: \u201cas the\nhorizontal needle hath a double motion about the round loadstone or\nterrella, so also the inclinatory needle hath a double motion about the\nearth.\u201d What is this, but a confession that the poles of the magnet and\nof the earth are the same?\n\n_A._ It is plain enough.\n\n_B._ Besides, seeing he placeth the magnetical pole at M in the meridian\nof Vaygates, the needle being touched shall incline to the pole of the\nearth which is P, as well there as at London, and make the north-pole of\nthe earth point south.\n\n_A._ It is certain, because he puts both the magnetical pole and the\npole of the earth in the same meridian of the earth. Nor see I any cause\nwhy, the needle being the same, it should not be as subject to\nvariation, and to variation of variation, and to all accidents of the\nearth there, as in any other part.\n\n_B._ He putteth (p. 16) a question, \u201cat what distance from the earth are\nthe magnetic poles? and answers to it, they are very near the earth,\nbecause the nearer the earth, the greater the strength.\u201d What think you\nof this?\n\n_A._ I think they are in the superficies of the magnet, as the pole of\nthe earth is in the superficies of the earth. And consequently, that\nthen the earth must be a part of the magnet, and their poles the same.\nFor the body of the magnet and the body of the earth, if they be two,\ncannot be in one place.\n\n_B._ His next words are, \u201csome things are to be considered concerning\nthose variations of the horizontal needle which are not according to the\nsituation of the place from the magnetic poles, but are contrary; as all\nthe West Indies according to the poles should be easterly, and they are\nwesterly. Which is by some accidental cause in the earth; and their\nmotion, as I formerly said, is a forced motion, and not natural.\u201d\n\n_A._ He has clearly overthrown his main doctrine. For to say the motion\nof the needle is forced and unnatural, is a most pitiful shift, and\nmanifestly false, no motion being more constant or less accidental,\nnotwithstanding the variation, to which the inclinatory needle is no\nless subject than the horizontal needle.\n\n_B._ That which deceived him, was, that he thought them two sorts of\nneedles, forgetting what he had said of Norman\u2019s invention of the\ninclinatory needle by the inclining of the horizontal needle (p. 11).\nFor I will show you that what he says is easterly and should be\nwesterly, should be easterly as it is. Consider the fourth figure, in\nwhich B is the north-pole, and B _c_ 11 deg. 15 min. easterly, which was\nthe variation at London in 1576 easterly. Suppose A _c_ to be the\nneedle, shall it not incline, as well here as at D _a_, and the\nvariation B _c_ be easterly? Again, D _a_ is 11 deg. 15 min., and the\nneedle in D parallel to A B, and at _a_ inclining also 11 deg. 15 min.\nwesterly. And is not the variation there D _a_ westerly, with the north\npoint of the needle in the line _a h_?\n\n_A._ But the West-Indies are not in this hemisphere B C D E. The\nvariation therefore will proceed in an arc of the opposite hemisphere,\nwhich is westerly.\n\n_B._ I believe he might think so, forgetting that he and his compass\nwere on the superficies of the earth, and fancying them in the centre at\nA.\n\n_A._ It is like enough. If we had a straight line exactly equal to the\narc of a quadrant, I think it would very much facilitate the doctrine of\nspherical triangles.\n\n_B._ When you have done with your questions of natural philosophy, I\nwill give you a clear demonstration of the equality of a straight line\nto the arc of a quadrant, which, if it satisfy you, you may carry with\nyou, and try thereby if you can find the angle of a spherical triangle\ngiven.\n\n_A._ It is time now to give over. And at our next meeting I desire your\nopinion concerning the causes of diaphaniety, and refraction. This\nCopernicus has done much more than he thought of. For he has not only\nrestored to us astronomy, but also made the way open to physiology.\n\n\n ==========\n\n\n CHAPTER X.\n OF TRANSPARENCE, REFRACTION, AND OF THE POWER\n OF THE EARTH TO PRODUCE LIVING CREATURES.\n\n_A._ Thinking upon what you said yesterday, it looked like a generation\nof living creatures. I saw the love between the loadstone and the iron\nin their mutual attraction, their engendering in their close and\ncontrary motion, and their issue in the iron, which being touched, hath\nthe same attractive virtue. Now seeing they have the same internal\nmotion of parts with that of the earth, why should not their substance\nbe the same, or very near a-kin?\n\n_B._ The most of them, if not all, that have written on this subject,\nwhen they call the loadstone a terrella, seem to think as you do. But I,\nexcept I could find proof for it, will not affirm it. For the earth\nattracteth all kind of bodies but air, and the loadstone none but iron.\nThe earth is a star, and it were too bold to pronounce any sentence of\nits substance, especially of the planets, that are so lapt up in their\nseveral coats, as that they cannot work on our eyes, or any organ of our\nother senses.\n\n_A._ I come therefore now to the business of the day. Seeing all\ngeneration, augmentation, and alteration is local motion, how can a body\nnot transparent be made transparent?\n\n_B._ I think it can never be done by the art of man. For as I said of\nhard and heavy bodies in the creation, so I think of diaphanous, that\nthe very same individual body which was not transparent then, shall\nnever be made transparent by human art.\n\n_A._ Do not you see that every day men make glass, and other diaphanous\nbodies not much inferior in beauty to the fairest gems?\n\n_B._ It is one thing to make one transparent of many by mixture, and\nanother to make transparent of not transparent. Any very hard stone, if\nit be beaten into small sands, such as is used for hour-glasses, every\none of those sands, if you look upon it with a microscope, you will find\nto be transparent; and the harder and whiter a stone is, so much the\nmore transparent, as I have seen in the stone of which are made\nmillstones, which stone is here called greet. And I doubt not but the\nsands of white marble must be more transparent. But there are no sands\nso transparent that they have not a scurf upon them, as hard, perhaps,\nas the stone itself; which they whose profession it is to make glass,\nhave the art to scour and wash away. And therefore I think it no great\nwonder to bring those sands into one lump, though I know not how they do\nit.\n\n_A._ I know they do it with lye made with a salt extracted from the\nashes of an herb, of which salt they make a strong lye, and mingle it\nwith the sand, and then bake it.\n\n_B._ Like enough. But still it is a compound of two transparent bodies,\nwhereof one is the natural stone, the other is the mortar. This\ntherefore doth not prove, that one and the same body of not transparent\ncan be made transparent.\n\n_A._ Since they can make one transparent body of many, why do they not\nof a great many small sparks of natural diamond compound one great one?\nIt would bear the charges of all the materials, and beside, enrich them.\n\n_B._ It is probable it would. But it may be they know no salt that\nhowsoever prepared, which, with how great a fire soever, can make them\nmelt. And, it may be, the true crystal of the mountain, which is found\nin great pieces in the Alps, is but a compound of many small ones, and\nmade by the earth\u2019s annual motion; for it is a very swift motion.\nSuppose now that within a very small cavern of those rocks whose\nsmallest atoms are crystal, and the cavity filled with air; and consider\nwhat a tumult would be made by the swift reciprocation of that air;\nwhether it would not in time separate those atoms from the rock, and\njumbling them together make them rub off their scurf from one another,\nand by little and little to touch one another in polished planes, and\nconsequently stick together, till in length of time they become one lump\nof clean crystal.\n\n_A._ I believe that the least parts of created substances lay mingled\ntogether at first, till it pleased God to separate all dissimilar\nnatures, and congregate the similar, to which this annual motion is\nproper. But they say that crystal is found in the open air hanging like\nicicles upon the rocks, which, if true, defeats this supposition of a\nnarrow cavern, and therefore I must have some farther experience of it\nbefore I make it my opinion. But howsoever, it still holds true that\ndiaphanous bodies of all sorts, in their least parts, were made by God\nin the beginning of the world. But it may be true, notwithstanding those\nicicles. For the force of the air that could break off those diaphanous\natoms in a cavern, can do the same in the open air. And I know that a\nless force of air can break some bodies into small pieces, not much less\nhard than crystal, by corrupting them.\n\n_B._ That which you now have said is somewhat. But I deny not the\npossibility, but only doubt of the operation. You may therefore pass to\nsome other question.\n\n_A._ Well, I will ask you then a question about refraction. I know\nalready that for the cause of refraction, when the light falleth through\na thinner medium upon a thicker, you assign the resistance of the\nthicker body; but you do not mean there, by _rarum_ and _densum_, two\nbodies whereof in equal spaces one has more substance in it than the\nother.\n\n_B._ No; for equal spaces contain equal bodies. But I mean by _densum_\nany body which more resisteth the motion of the air, and by _rarum_ that\nwhich resisteth less.\n\n_A._ But you have not declared in what that resistance consisteth.\n\n_B._ I suppose it proceedeth from the hardness.\n\n_A._ But from thence it will follow, that all transparent bodies that\nequally refract are equally hard, which I think is not true, because the\nrefraction of glass is not greater, at least in comparison of their\nhardnesses, than that of water.\n\n_B._ I confess it. Therefore I think we must take in gravity to a share\nin the production of this refraction. For I never considered refraction\nbut in glass, because my business then was only to find the causes of\nthe phenomena of telescopes and microscopes. Let therefore A B (in fig.\n7) be a hard, and consequently, a heavy body; and from above, as from\nthe sun, let C A be the line of incidence, and produced to D; and draw A\nE perpendicular to A B. It is manifest that the hardness in A B shall\nturn the stream of the light inwards toward A E, suppose in the line A\n_e_. It is also evident that the endeavour in B, which is, being heavy,\ndownward, shall turn the stream again inward, towards A E, as in A _b_.\nThus it is in refraction from the sun downwards. In like manner, if the\nlight come from below, as from a candle in the point D, the line of\nincidence will be D A, and produced will pass to C. And the resistance\nof the hardness in A will turn the stream A C inward, suppose into A\n_l_, and make C _l_ equal to D _e_. For passing into a thinner medium,\nit will depart from the perpendicular in an angle equal to the angle D A\n_e_, by which it came nearer to it in A _e_. So also the resistance of\nthe gravity in the point A shall turn the stream of the light into the\nline A _i_, and make the angle _l_ A _i_ equal to the angle _e_ A _b_.\nAnd thus you see in what manner, though not in what proportion, hardness\nand gravity conjoin their resistance in the causing of refraction.\n\n_A._ But you proved yesterday, that a heavy body does not gravitate upon\na body equally heavy. Now this A B has upper parts and lower parts; and\nif the upper parts do not gravitate upon the lower parts, how can there\nbe any endeavour at all downward to contribute to the refraction?\n\n_B._ I told you yesterday, that when a heavy body was set upon another\nbody heavier or harder than itself, the endeavour of it downward was\ndiverted another way, but not that it was extinguished. But in this\ncase, where it lieth upon air, the first endeavour of the lowest part\nworketh downward. For neither motion nor body can be utterly\nextinguished by a less than an omnipotent power. All bodies, as long as\nthey are bodies, are in motion one way or other, though the farther it\nbe communicated, so much the less.\n\n_A._ But since you hold that motion is propagated through all bodies,\nhow hard or heavy soever they be, I see no cause but that all bodies\nshould be transparent.\n\n_B._ There are divers causes that take away transparency. First, if the\nbody be not perfectly homogeneous, that is to say, if the smallest parts\nof it be not all precisely of the same nature, or do not so touch one\nanother as to leave no vacuum within it; or though they touch, if they\nbe not as hard in the contact as in any other line. For then the\nrefractions will be so changed both in their direction, and in their\nstrength, as that no light shall come through it to the eye; as in wood\nand ordinary stone and metal. Secondly, the gravity and hardness may be\nso great, as to make the angle refracted so great, as the second\nrefraction shall not direct the beam of light to the eye; as if the\nangle of refraction were D A E, the refracted line would be\nperpendicular to A B, and never come to the line A D, in which is the\neye.\n\n_A._ To know how much of the refraction is due to the hardness, and how\nmuch to the gravity, I believe it is impossible, though the quantity of\nthe whole be easily measured in a diaphanous body given. And both you\nand Mr. Warner have demonstrated, that as the sine of the angle\nrefracted in one inclination is to the sine of the angle refracted in\nanother inclination, so is the sine of one inclination to the sine of\nthe angle of the other inclination. Which demonstrations are both\npublished by Mersennus in the end of the first volume of his _Cogitata\nPhysico-Mathematica_. But since there be many bodies, through which\nthough there pass light enough, yet no object appear through them to the\neye, what is the reason of that?\n\n_B._ You mean paper. For paper windows will enlighten a room, and yet\nnot show the image of an object without the room. But it is because\nthere are in paper abundance of pores, through which the air passing\nmoveth the air within; by the reflections whereof anything within may be\nseen. And in the same paper there are again as many parts not\ntransparent, through which the air cannot pass, but must be reflected\nfirst to all parts of the object, and from them again to the paper; and\nat the paper either reflected again or transmitted, according as it\nfalls upon pores or not pores; so that the light from the object can\nnever come together at the eye.\n\n_A._ There belongs yet to this subject the causes of the diversity of\ncolours. But I am so well satisfied with that which you have written of\nit in the twenty-fourth chapter of your book _de Corpore_, that I need\nnot trouble you farther in it. And now I have but one question more to\nask you, which I thought upon last night. I have read in an ancient\nhistorian, that living creatures after a great deluge were produced by\nthe earth, which being then very soft, there were bred in it, it may be\nby the rapid motion of the sun, many blisters, which in time breaking,\nbrought forth, like so many eggs, all manner of living creatures great\nand small, which since it is grown hard it cannot do. What think you of\nit?\n\n_B._ It is true that the earth produced the first living creatures of\nall sorts but man. For God said (Gen. i. 24), _Let the earth produce\nevery living creature, cattle, and creeping thing, &c._ But then again\n(ver. 25) it is said that _God made the beast of the earth, &c._ So that\nit is evident that God gave unto the earth that virtue. Which virtue\nmust needs consist in motion, because all generation is motion. But man,\nthough the same day, was made afterward.\n\n_A._ Why hath not the earth the same virtue now? Is not the sun the same\nas it was? Or is there no earth now soft enough?\n\n_B._ Yes. And it may be the earth may yet produce some very small living\ncreatures: and perhaps male and female. For the smallest creatures which\nwe take notice of, do engender, though they do not all by conjunction;\ntherefore if the earth produce living creatures at this day, God did not\nabsolutely rest from all his works on the seventh day, but (as it is\nchap. ii. 2) _he rested from all the work he had made_. And therefore it\nis no harm to think that God worketh still, and when and where and what\nhe pleaseth. Beside, it is very hard to believe, that to produce male\nand female, and all that belongs thereto, as also the several and\ncurious organs of sense and memory, could be the work of anything that\nhad not understanding. From whence, I think we may conclude, that\nwhatsoever was made after the creation, was a new creature made by God\nno otherwise than the first creatures were, excepting only man.\n\n_A._ They are then in an error that think there are no more different\nkinds of animals in the world now, than there were in the ark of Noah.\n\n_B._ Yes, doubtless. For they have no text of Scripture from which it\ncan be proved.\n\n_A._ The questions of nature which I could yet propound are innumerable.\nAnd since I cannot go through them, I must give over somewhere, and why\nnot here? For I have troubled you enough, though I hope you will forgive\nme.\n\n_B._ So God forgive us both as we do one another. But forget not to take\nwith you the demonstration of a straight line equal to an arc of a\ncircle.\n\n\n\n\n THE PROPORTION OF A STRAIGHT LINE TO HALF THE ARC OF A QUADRANT.\n\n\n[Illustration]\n\nDescribe the square A B C D, and divide it by the diagonals A C and B D,\nas also by the straight lines E G, F H, meeting in the centre I at right\nangles, into four equal parts. Then with the radius A B describe the\nquadrant B D cutting E G in K, and the diagonal A C in L; and so B L\nwill be half the arc B D, equal to which we are to find a straight line.\nDivide I C into halves at M, and draw B M cutting E G in _a_. I say B M\nis equal to the arc B L. For the demonstration whereof we are to assume\ncertain known truths and dictates of common-sense.\n\n1. That the arc B K is the third part of the arc B D, and consequently\ntwo-thirds of the arc B L, and B K to K L as two to one.\n\n2. That if a straight line be equal to the arc B L, and one end in B,\nthe other will be somewhere in I C, and higher than the point L.\n\n3. That wheresoever it be, two-thirds of it must be equal to the arc B\nK, and one-fifth to the arc K L.\n\n4. That the arc of a quadrant described in the third part of the radius,\nor of E G, is equal to the third part of the arc B D, viz. to the arc B\nK. I may therefore call a third part of E G, the radius of B K; and a\nsixth part of E G, the radius of the arc K L, &c.\n\n5. And lastly, that any straight line drawn from B to I C, if it be\nequal to the arc B L, it must cut the half radius I G, whose quadrantal\narc is B L, into the proportion of two to one. For as the whole arc to\nthe whole E G, so are the parts of it to the parts of E G.\n\nThese premises granted, which I think cannot be denied, I say again,\nthat the straight line B M is equal to the arc B L.\n\n DEMONSTRATION.\n\n[Illustration]\n\nBecause B I is to I M, by construction, as two to one, and the line I G\ndivides the angle B I C in the midst, B _a_ will be to _a_ M as two to\none, that is to say, as the arc B K to the arc K L. From the point M to\nthe side B C erect a perpendicular M N. And because C M is half C I, the\nline M N will be half G C; and B N will be three-quarters of B C; and\nthe square of B M equal to ten squares of a quarter of B C; and because\nB M is to B _a_ as three to two, M N will be to _a_ G as three to two.\nBut M N is a quarter of E G, therefore _a_ G is two-thirds of a quarter\nof E G; that is, one-third of I G; that is, one-sixth of the whole E G.\nAnd I _a_ one-third of E G. Therefore I _a_ is the radius of the arc B\nK; and _a_ G the radius of the arc K L; and E G the radius of the whole\narc B L D. Lastly, if a straight line be drawn from B to any other point\nof the line I C, though any line may be divided into the proportion of\ntwo to one, it shall not pass through the point _a_, and therefore not\ndivide the radius of B L, which is I G, into the proportion of two to\none. Therefore no straight line can be drawn from B to I C, except B M,\nso as to be equal to the arc B L. Therefore the straight line B M and\nthe arc B L are equal.\n\nHence it follows, that seeing the square of B M is equal to ten squares\nof a quarter of B C, that a straight line equal to the quadrantal arc B\nL D is equal to ten squares of half the radius, as I have divers ways\ndemonstrated heretofore.\n\n\n\n\n SIX LESSONS\n TO THE\n PROFESSORS OF THE MATHEMATICS,\n\n ONE OF GEOMETRY, THE OTHER OF ASTRONOMY,\n IN THE CHAIRS SET UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE, IN THE\n UNIVERSITY OF OXFORD.\n\n TO THE RIGHT HONOURABLE\n\n HENRY LORD PIERREPONT,\n\n VISCOUNT NEWARK, EARL OF KINGSTON, AND\n MARQUIS OF DORCHESTER.\n\n\nMY MOST NOBLE LORD,\n\nNot knowing on my own part any cause of the favour your Lordship has\nbeen pleased to express towards me, unless it be the principles, method,\nand manners you have observed and approved in my writings; and seeing\nthese have all been very much reprehended by men, to whom the name of\npublic professors hath procured reputation in the university of Oxford,\nI thought it would be a forfeiture of your Lordship\u2019s good opinion, not\nto justify myself in public also against them, which, whether I have\nsufficiently performed or not in the six following Lessons addressed to\nthe same professors, I humbly pray your Lordship to consider. The volume\nitself is too small to be offered to you as a present, but to be brought\nbefore you as a controversy it is perhaps the better for being short. Of\narts, some are demonstrable, others indemonstrable; and demonstrable are\nthose the construction of the subject whereof is in the power of the\nartist himself, who, in his demonstration, does no more but deduce the\nconsequences of his own operation. The reason whereof is this, that the\nscience of every subject is derived from a precognition of the causes,\ngeneration, and construction of the same; and consequently where the\ncauses are known, there is place for demonstration, but not where the\ncauses are to seek for. Geometry therefore is demonstrable, for the\nlines and figures from which we reason are drawn and described by\nourselves; and civil philosophy is demonstrable, because we make the\ncommonwealth ourselves. But because of natural bodies we know not the\nconstruction, but seek it from the effects, there lies no demonstration\nof what the causes be we seek for, but only of what they may be.\n\nAnd where there is place for demonstration, if the first principles,\nthat is to say, the definitions contain not the generation of the\nsubject, there can be nothing demonstrated as it ought to be. And this\nin the three first definitions of Euclid sufficiently appeareth. For\nseeing he maketh not, nor could make any use of them in his\ndemonstrations, they ought not to be numbered among the principles of\ngeometry. And Sextus Empiricus maketh use of them (misunderstood, yet so\nunderstood as the said professors understand them) to the overthrow of\nthat so much renowned evidence of geometry. In that part therefore of my\nbook where I treat of geometry, I thought it necessary in my definitions\nto express those motions by which lines, superficies, solids, and\nfigures, were drawn and described, little expecting that any professor\nof geometry should find fault therewith, but on the contrary supposing I\nmight thereby not only avoid the cavils of the sceptics, but also\ndemonstrate divers propositions which on other principles are\nindemonstrable. And truly, if you shall find those my principles of\nmotion made good, you shall find also that I have added something to\nthat which was formerly extant in geometry.\n\nFor first, from the seventh chapter of my book _De Corpore_, to the\nthirteenth, I have rectified and explained the principles of the\nscience; _id est_, I have done that business for which Dr. Wallis\nreceives the wages. In the seventh, I have exhibited and demonstrated\nthe proportion of the parabola and parabolasters to the parallelograms\nof the same height and base; which, though some of the propositions were\nextant without that demonstration, were never before demonstrated, nor\nare by any other than this method demonstrable.\n\nIn the eighteenth, as it is now in English, I have demonstrated, for\nanything I yet perceive, equation between the crooked line of a parabola\nor any parabolaster and a straight line.\n\nIn the twenty-third I have exhibited the centre of gravity of any sector\nof a sphere.\n\nLastly, the twenty-fourth, which is of the nature of refraction and\nreflection, is almost all new.\n\nBut your Lordship will ask me what I have done in the twentieth, about\nthe quadrature of the circle. Truly, my Lord, not much more than before.\nI have let stand there that which I did before condemn, not that I think\nit exact, but partly because the division of angles may be more exactly\nperformed by it than by any organical way whatsoever; and I have\nattempted the same by another method, which seemeth to me very natural,\nbut of calculation difficult and slippery. I call them only aggressions,\nretaining nevertheless the formal manner of assertion used in\ndemonstration. For I dare not use such a doubtful word as _videtur_,\nbecause the professors are presently ready to oppose me with a _videtur\nquod non_. Nor am I willing to leave those aggressions out, but rather\nto try if it may be made pass for lawful, (in spite of them that seek\nhonour, not from their own performances, but from other men\u2019s failings),\namongst many difficult undertakings carried through at once to leave one\nand the greatest for a time behind; and partly because the method is\nsuch as may hereafter give farther light to the finding out of the exact\ntruth.\n\nBut the principles of the professors that reprehend these of mine, are\nsome of them so void of sense, that a man at the first hearing, whether\ngeometrician or not geometrician, must abhor them. As for example:\n\n1. That two equal proportions are not double to one of the same\nproportions.\n\n2. That a proportion is double, triple, &c. of a number, but not of a\nproportion.\n\n3. That the same body, without adding to it, or taking from it, is\nsometimes greater, and sometimes less.\n\n4. That a quantity may grow less and less eternally, so as at last to be\nequal to another quantity; or, which is all one, that there is a last in\neternity.\n\n5. That the nature of an angle consisteth in that which lies between the\nlines that comprehend the angle in the very point of their concourse,\nthat is to say, an angle is the superficies which lies between the two\npoints which touch, or, as they understand a point, the superficies that\nlies between the two nothings which touch.\n\n6. That the quotient is the proportion of the division to the dividend.\n\nUpon these and some such other principles is grounded all that Dr.\nWallis has said, not only in his _Elenchus_ of my geometry, but also in\nhis treatises of the _Angle of Contact_, and in his _Arithmetica\nInfinitorum_; which two last I have here in two or three leaves wholly\nand clearly confuted. And I verily believe that since the beginning of\nthe world, there has not been, nor ever shall be, so much absurdity\nwritten in geometry, as is to be found in those books of his; with which\nthere is so much presumption joined, that an \u1f00\u03c0\u03bf\u03ba\u03b1\u03c4\u03ac\u03db\u03b1\u03c3\u03b9\u03c2 of the like\nconjunction cannot be expected in less than a Platonic year. The cause\nwhereof I imagine to be this, that he mistook the study of _symbols_ for\nthe study of _geometry_, and thought symbolical writing to be a new kind\nof method, and other men\u2019s demonstrations set down in symbols new\ndemonstrations. The way of analysis by squares, cubes, &c., is very\nancient, and useful for the finding out whatsoever is contained in the\nnature and generation of rectangled planes, which also may be found\nwithout it, and was at the highest in Vieta; but I never saw anything\nadded thereby to the science of geometry, as being a way wherein men go\nround from the equality of rectangled planes to the equality of\nproportion, and thence again to the equality of rectangled planes,\nwherein the symbols serve only to make men go faster about, as greater\nwind to a windmill.\n\nIt is in sciences as in plants; growth and branching is but the\ngeneration of the root continued; nor is the invention of theorems\nanything else but the knowledge of the construction of the subject\nprosecuted. The unsoundness of the branches are no prejudice to the\nroots, nor the faults of theorems to the principles. And active\nprinciples will correct false theorems if the reasoning be good; but no\nlogic in the world is good enough to draw evidence out of false or\nunactive principles. But I detain your Lordship too long. For all this\nwill be much more manifest in the following discourses, wherein I have\nnot only explained and rectified many of the most important principles\nof geometry, but also by the examples of those errors which have been\ncommitted by my reprehenders, made manifest the evil consequence of the\nprinciples they now proceed on. So that it is not only my own defence\nthat I here bring before you, but also a positive doctrine concerning\nthe true grounds, or rather atoms of geometry, which I dare only say are\nvery singular, but whether they be very good or not, I submit to your\nLordship\u2019s judgment. And seeing you have been pleased to bestow so much\ntime, with great success, in the reading of what has been written by\nother men in all kinds of learning, I humbly pray your Lordship to\nbestow also a little time upon the reading of these few and short\nlessons; and if your Lordship find them agreeable to your reason and\njudgment, let me, notwithstanding the clamour of my adversaries, be\ncontinued in your good opinion, and still retain the honour of being,\n\n My most noble Lord,\n Your Lordship\u2019s most\n humble and obliged servant,\n THOMAS HOBBES.\n\nLONDON, _June 10, 1656_.\n\n\n\n\n LESSONS\n\n OF\n\n THE PRINCIPLES OF GEOMETRY, &c.\n\n TO THE EGREGIOUS PROFESSORS OF THE MATHEMATICS, ONE OF\n GEOMETRY, THE OTHER OF ASTRONOMY, IN THE CHAIRS SET\n UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE,\n IN THE UNIVERSITY OF OXFORD.\n\n\n LESSON I.\n\n\nI suppose, most egregious professors, you know already that by geometry,\nthough the word import no more but the measuring of land, is understood\nno less the measuring of all other quantity than that of bodies. And\nthough the definition of geometry serve not for proof, nor enter into\nany geometrical demonstration, yet for understanding of the principles\nof the science, and for a rule to judge by, who is a geometrician, and\nwho is not, I hold it necessary to begin therewith.\n\nGeometry is the science of determining the quantity of anything, not\nmeasured, by comparing it with some other quantity or quantities\nmeasured. Which science therefore whosoever shall go about to teach,\nmust first be able to tell his disciple what measuring or dimension is;\nby what each several kind of quantity is measured; what quantity is, and\nwhat are the several kinds thereof. Therefore as they, who handle any\none part of geometry, determine by definition the signification of every\nword which they make the subject or predicate of any theorem they\nundertake to demonstrate; so must he which intendeth to write a whole\nbody of geometry, define and determine the meaning of whatsoever word\nbelongeth to the whole science. The design of Euclid was to demonstrate\nthe properties of the five regular bodies mentioned by Plato; in which\ndemonstrations there was no need to allege for argument the definition\nof quantity, which it may be was the cause he hath not anywhere defined\nit, but done what he undertook without it. And though having perpetually\noccasion to speak of measure, he hath not defined measure; yet instead\nthereof he hath, in the beginning of his first elements, assumed an\naxiom which serveth his turn sufficiently as to the measure of lines,\nwhich is the eighth axiom; that those things which lie upon one another\nall the way (called by him \u1f10\u03c6\u03b1\u03c1\u03bc\u03cc\u03b6\u03bf\u03bd\u03c4\u03b1) are equal. Which axiom is\nnothing else but a description of the art of measuring length and\nsuperficies. For this \u1f10\u03c6\u03ac\u03c1\u03bc\u03bf\u03c3\u03b9\u03c2 can have no place in solid bodies,\nunless two bodies could at the same time be in one place. But amongst\nthe principles of geometry universal, the definitions are necessary,\nboth of quantity and dimensions.\n\nQuantity is that which is signified by what we answer to him that\nasketh, _how much_ any thing is? and thereby determine the magnitude\nthereof. For magnitude being a word indefinite, if a man ask of a thing,\n_quantum est?_ that is, _how much_ it is, we do not satisfy him by\nsaying it is magnitude or quantity, but by saying it is _tantum_, _so\nmuch_. And they that first called it in Greek, \u03c0\u03b7\u03bb\u03b9\u03ba\u03cc\u03c4\u03b7\u03c2, and in Latin\n_quantity_, might more properly have called it in Latin _tantity_, and\nin Greek \u03c4\u03b7\u03bb\u03b9\u03ba\u03cc\u03c4\u03b7\u03c2; and we, if we allowed ourselves the eloquence of the\nGreeks and Latins, should call it the _so-muchness_.\n\nThere is therefore to everything concerning which a man may ask without\nabsurdity, _how much it is_, a certain quantity belonging, determining\nthe magnitude to be _so much_. Also wheresoever there is _more_ and\n_less_, there is one kind of quantity or other. And first there is the\nquantity of bodies, and that of three kinds: length, which is by one way\nof measuring; superficies, made of the complication of two lengths, or\nthe measure taken two ways; and solid, which is the complication of\nthree lengths, or of the measure taken three ways, for breadth or\nthickness are but other lengths. And the science of geometry, so far\nforth as it contemplateth bodies only, is no more but by measuring the\nlength of one or more lines, and by the position of others known in one\nand the same figure, to determine by ratiocination, how much is the\nsuperficies; and by measuring length, breadth, and thickness, to\ndetermine the quantity of the whole body. Of this kind of magnitudes and\nquantities the subject is body.\n\nAnd because for the computing of the magnitudes of bodies, it is not\nnecessary that the bodies themselves should be present, the ideas and\nmemory of them supplying their presence, we reckon upon those imaginary\nbodies, which are the quantities themselves, and say the length is so\ngreat, the breadth so great, &c. which in truth is no better than to say\nthe length is so long, or the breadth so broad, &c. But in the mind of\nan intelligent man it breedeth no mistake.\n\nBesides the quantity of bodies, there is a quantity of time. For seeing\nmen, without absurdity, do ask how much it is; by answering _tantum_,\n_so much_, they make manifest there is a quantity that belongeth unto\ntime, namely, a length. And because length cannot be an accident of\ntime, which is itself an accident, it is the accident of a body; and\nconsequently the length of the time, is the length of the body; by which\nlength or line, we determine how much the time is, supposing some body\nto be moved over it.\n\nAlso we not improperly ask with _how much_ swiftness a body is moved;\nand consequently there is a quantity of motion or swiftness, and the\nmeasure of that quantity is also a line. But then again, we must suppose\nanother motion, which determineth the time of the former. Also of force,\nthere is a question of _how much_, which is to be answered by _so much_;\nthat is, by quantity. If the force consist in swiftness, the\ndetermination is the same with that of swiftness, namely, by a line; if\nin swiftness and quantity of body jointly, then by a line and a solid;\nor if in quantity of body only, as weight, by a solid only.\n\nSo also is number quantity; but in no other sense than as a line is\nquantity divided into equal parts.\n\nOf an angle, which is of two lines whatsoever they be, meeting in one\npoint, the digression or openness in other points, it may be asked how\ngreat is that digression? This question is answered also by quantity. An\nangle therefore hath quantity, though it be not the subject of quantity;\nfor the body only can be the subject, in which body those straddling\nlines are marked.\n\nAnd because two lines may be made to divaricate by two causes; one, when\nhaving one end common and immoveable, they depart one from another at\nthe other ends circularly, and this is called simply an angle; and the\nquantity thereof is the quantity of the arch, which the two lines\nintercept.\n\nThe other cause is the bending of a straight line into a circular or\nother crooked line, till it touch the place of the same line, whilst it\nwas straight, in one only point. And this is called an angle of\ncontingence. And because the more it is bent, the more it digresseth\nfrom the tangent, it may be asked _how much_ more? And therefore the\nanswer must be made by quantity; and consequently an angle of\ncontingence hath its quantity as well as that which is called simply an\nangle. And in case the digression of two such crooked lines from the\ntangent be uniform, as in circles, the quantity of their digression may\nbe determined. For, if a straight line be drawn from the point of\ncontact, the digression of the lesser circle will be to the digression\nof the greater circle, as the part of the line drawn from the point of\ncontact, and intercepted by the circumference of the greater circle is\nto the part of the same line intercepted by the circumference of the\nlesser circle, or, which is all one, as the greater radius is to the\nlesser radius. You may guess by this what will become of that part of\nyour last book, where you handle the question of the quantity of an\nangle of contingence.\n\nAlso there lieth a question of _how much comparatively_ one magnitude is\nto another magnitude, as how much water is in a tun in respect of the\nocean, how much in respect of a pint; _little_ in the first respect,\n_much_ in the latter. Therefore the answer must be made by some\nrespective quantity. This respective quantity is called _ratio_ and\nproportion, and is determined by the quantity of their differences; and\nif their differences be compared also with the quantities themselves\nthat differ, it is called simply proportion, or proportion geometrical.\nBut if the differences be not so compared, then it is called proportion\narithmetical. And where the difference is none, there is no quantity of\nthe proportion, which in this case is but a bare comparison.\n\nAlso concerning heat, light, and divers other qualities, which have\ndegrees, there lieth a question of _how much_, to be answered by a _so\nmuch_, and consequently they have their quantities, though the same with\nthe quantity of swiftness: because the intensions and remissions of such\nqualities are but the intensions and remissions of the swiftness of that\nmotion by which the agent produceth such a quality. And as quantity may\nbe considered in all the operations of nature, so also doth geometry run\nquite through the whole body of natural philosophy.\n\nTo the principles of geometry the definition appertaineth also of a\n_measure_, which is this, _one quantity is the measure of another\nquantity, when it, or the multiple of it, is coincident in all points\nwith the other quantity_. In which definition, instead of that \u1f10\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u1f74\nof Euclid, I put coincidence. For the superposition of quantities, by\nwhich they render the word \u1f10\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u1f74, cannot be understood of bodies, but\nonly of lines and superficies. Nevertheless many bodies may be\ncoincident successively with one and the same place, and that place will\nbe their measure, as we see practised continually in the measuring of\nliquid bodies, which art of measuring may properly be called \u1f10\u03c6\u03ac\u03c1\u03bc\u03bf\u03c3\u03b9\u03c2,\nbut not superposition.\n\nAlso the definitions of _greater_, _less_, and _equal_, are necessary\nprinciples of geometry. For it cannot be imagined than any geometrician\nshould demonstrate to us the equality and inequality of magnitudes,\nexcept he tell us first what those words do signify. And it is a wonder\nto me, that Euclid hath not anywhere defined what are equals, or at\nleast, what are equal bodies, but serveth his turn throughout with that\nforementioned \u1f10\u03c6\u03ac\u03c1\u03bc\u03bf\u03c3\u03b9\u03c2, which hath no place in solids, nor in time, nor\nin swiftness, nor in circular, or other crooked lines; and therefore no\nwonder to me, why geometry hath not proceeded to the calculation neither\nof crooked lines, nor sufficiently of motion, nor of many other things,\nthat have proportion to one another.\n\nEqual bodies, superficies, and lines, are those of which every one is\ncapable of being coincident with the place of every one of the rest: and\nequal times, wherein with one and the same motion equal lines are\ndescribed. And equally swift are those motions by which we run over\nequal spaces in any time determined by any other motion. And universally\nall quantities are equal, that are measured by the same number of the\nsame measures.\n\nIt is necessary also to the science of geometry, to define what\nquantities are of one and the same kind, which they call _homogeneous_,\nthe want of which definitions hath produced those wranglings (which your\nbook _De Angulo Contactus_ will not make to cease) about the angle of\ncontingence.\n\n_Homogeneous_ quantities are those which may be compared by (\u1f10\u03c6\u03ac\u03c1\u03bc\u03bf\u03c3\u03b9\u03c2)\napplication of their measures to one another; so that solids and\nsuperficies are heterogeneous quantities, because there is no\ncoincidence or application of those two dimensions.\n\nNo more is there of line and superficies, nor of line and solid, which\nare therefore heterogeneous. But lines and lines, superficies and\nsuperficies, solids and solids, are homogeneous.\n\nHomogeneous also are line, and the quantity of time; because the\nquantity of time is measured by the application of a line to a line; for\nthough time be no line, yet the quantity of time is a line, and the\nlength of two times is compared by the length of two lines.\n\nWeight and solid have their quantity homogeneous, because they measure\none another by application, to the beam of a balance. Line and angle\nsimply so called, have their quantity homogeneous, because their measure\nis an arch or arches of a circle applicable in every point to one\nanother.\n\nThe quantity of an angle simply so called, and the quantity of an angle\nof contingence are heterogeneous. For the measures by which two angles\nsimply so called are compared, are in two coincident arches of the same\ncircle; but the measure by which an angle of contingence is measured, is\na straight line intercepted between the point of contact and the\ncircumference of the circle; and therefore one of them is not applicable\nto the other; and consequently of these two sorts of angles the\nquantities are heterogeneous. The quantities of two angles of\ncontingence are homogeneous; for they may be measured by the \u1f10\u03c6\u03ac\u03c1\u03bc\u03bf\u03c3\u03b9\u03c2\nof two lines, whereof one extreme is common, namely, the point of\ncontact, the other extremes are in the arches of the two circles.\n\nBesides this knowledge of what is quantity and measure, and their\nseveral sorts, it behoveth a geometrician to know why, and of what, they\nare called principles. For not every proposition that is evident is\ntherefore a principle. A principle is the beginning of something. And\nbecause definitions are the beginnings or first propositions of\ndemonstration, they are therefore called principles, principles, I say,\nof demonstration. But there be also necessary to the teaching of\ngeometry other principles, which are not the beginnings of\ndemonstration, but of construction, commonly called petitions; as that\nit may be granted _that a man can draw a straight line, and produce it;\nand with any radius, on any centre describe a circle_, and the like. For\nthat a man may be able to describe a square, he must first be able to\ndraw a straight line; and before he can describe an equilateral\ntriangle, he must be able first to describe a circle. And these\npetitions are therefore properly called principles, not of\ndemonstration, but of operation. As for the commonly received third sort\nof principles, called _common notions_, they are principles, only by\npermission of him that is the disciple; who being ingenuous, and coming\nnot to cavil but to learn, is content to receive them, though\ndemonstrable, without their demonstrations. And though definitions be\nthe only principles of demonstration, yet it is not true that every\ndefinition is a principle. For a man may so precisely determine the\nsignification of a word as not to be mistaken, yet may his definition be\nsuch as shall never serve for proof of any theorem, nor ever enter into\nany demonstration, such as are some of the definitions of Euclid, and\nconsequently can be no beginnings of demonstration, that is to say, no\nprinciples.\n\nAll that hitherto hath been said, is so plain and easy to be understood,\nthat you cannot, most egregious professors, without discovering your\nignorance to all men of reason, though no geometricians, deny it. And\nthe same (saving that the words are all to be found in dictionaries)\nnew; also to him that means to learn, not only the practice, but also\nthe science of geometry necessary, and, though it grieve you, mine. And\nnow I come to the definitions of Euclid.\n\nThe first is of a point: \u03a3\u03b7\u03bc\u03b5\u1fd6\u03bf\u03bd, &c. \u201c_Signum est, cujus est pars\nnulla_,\u201d that is to say, _a mark is that of which there is no part_.\nWhich definition, not only to a candid, but also to a rigid construer,\nis sound and useful. But to one that neither will interpret candidly,\nnor can interpret accurately, is neither useful nor true. Theologers say\nthe soul hath no part, and that an angel hath no part, yet do not think\nthat soul or angel is a point. A mark or as some put instead of it\n\u03db\u03af\u03b3\u03bc\u03b7, which is a mark with a hot iron, is visible; if visible, then it\nhath quantity, and consequently may be divided into parts innumerable.\nThat which is indivisible is no quantity; and if a point be not\nquantity, seeing it is neither substance nor quality, it is nothing. And\nif Euclid had meant it so in his definition, as you pretend he did, he\nmight have defined it more briefly, but ridiculously, thus, _a point is\nnothing_. Sir Henry Savile was better pleased with the candid\ninterpretation of Proclus, that would have it understood respectively to\nthe matter of geometry. But what meaneth this _respectively to the\nmatter of geometry_? It meaneth this, that no argument in any\ngeometrical demonstration should be taken from the division, quantity,\nor any part of a point; which is as much as to say, a point is that\nwhose quantity is not drawn into the demonstration of any geometrical\nconclusion; or, which is all one, whose quantity is not considered.\n\nAn accurate interpreter might make good the definition thus, _a point is\nthat which is undivided_; and this is properly the same with _cujus non\nest pars_: for there is a great difference between _undivided_ and\n_indivisible_, that is, between _cujus non est pars_, and _cujus non\npotest esse pars_. Division is an act of the understanding; the\nunderstanding is therefore that which maketh parts, and there is no part\nwhere there is no consideration but of one. And consequently Euclid\u2019s\ndefinition of a point is accurately true, and the same with mine, which\nis, that _a point is that body whose quantity is not considered_. And\n_considered_ is that, as I have defined it chap. I. at the end of the\nthird article, which is not put to account in demonstration.\n\nEuclid therefore seemeth not to be of your opinion, that say a point is\nnothing. But why then doth he never use this definition in the\ndemonstration of any proposition? Whether he useth it expressly or no, I\nremember not; but the sixteenth proposition of the third book without\nthe force of this definition is undemonstrated.\n\nThe second definition is of a line: \u03b3\u03c1\u03b1\u03bc\u03bc\u1f74 \u03b4\u1f72 \u03bc\u1fc6\u03ba\u03bf\u03c2 \u1f02\u03c0\u03bb\u03b1\u03c4\u03b5\u03c2. \u201c_Linea est\nlongitudo latitudinis expers_; _a line is length which hath no\nbreadth_;\u201d and if candidly interpreted, sound enough, though rigorously\nnot so. For to what purpose is it to say _length not broad_, when there\nis no such thing as a _broad length_. One path may be broader than\nanother path, but not one mile than another mile; and it is not the path\nbut the mile which is the way\u2019s length. If therefore a man have any\ningenuity he will understand it thus, _that a line is a body whose\nlength is considered without its breadth_, else we must say absurdly a\n_broad length_; or untruly, that there be bodies which have length and\nyet no breadth; and this is the very sense which Apollonius, saith\nProclus, makes of this definition; \u201cwhen we measure,\u201d says he, \u201cthe\nlength of a way, we take not in the breadth or depth, but consider only\none dimension.\u201d See this of Proclus cited by Sir Henry Savile, where you\nshall find the very word _consider_.\n\nThe fourth definition is of a straight line, thus \u1f18\u03c5\u03b8\u03b5\u1fd6\u03b1 \u03b3\u03c1\u03b1\u03bc\u03bc\u03ae \u1f10\u03db\u03b9\u03bd,\n&c. \u201c_Recta linea est qu\u00e6 ex \u00e6quo sua ipsius puncta inter jacet._\u201d _A\nstraight line is that which lieth equally (or perhaps evenly) between\nits own points._ This definition is inexcusable. Between what points of\nits own can a straight line lie but between its extremes? And how lies\nit evenly between them, unless it swerve no more from some other line\nwhich hath the same extremes, one way than another? And then why are not\nbetween the same points both the lines straight? How bitterly, and with\nwhat insipid jests would you have reviled Euclid for this, if living now\nhe had written a _Leviathan_ . And yet there is somewhat in this\ndefinition to help a man, not only to conceive the nature of a straight\nline (for who doth not conceive it?) but also to express it. For he\nmeant perhaps to call a straight line that which is all the way from one\nextreme to another, equally distant from any two or more such lines as\nbeing like and equal have the same extremes. So the axis of the earth is\nall the way equally distant from the circumference of any two or more\nmeridians. But then before he had defined a straight line, he should\nhave defined what lines are _like_, and what are _equal_. But it had\nbeen best of all, first to have defined crooked lines, by the\npossibility of a deduction or setting further asunder of their extremes;\nand then straight lines, by the impossibility of the same.\n\nThe seventh definition, which is that of a plain superficies, hath the\nsame faults.\n\nThe eighth is of a plane angle, \u1f19\u03c0\u1f76\u03c0\u03b5\u03b4\u03bf\u03c2 \u03b3\u03c9\u03bd\u03af\u03b1 \u1f11\u03db\u1f76\u03bd \u1f21 \u1f10\u03bd \u1f10\u03c0\u03b9\u03c0\u03ad\u03b4\u03c9, &c.\n\u201c_Angulus planus est duarum linearum in plano se mutuo tangentium, et\nnon in directum jacentium, alterius ad alteram inclinatio._\u201d _A plane\nangle is the inclination one towards another of two lines that touch one\nanother in the same plane, and lie not in the same straight line._\nBesides the faults here observed by Sir Henry Savile, as the clause of\nnot lying in the same straight line, and the obscurity or equivocation\nof the word _inclination_, there is yet another, which is, that by this\ndefinition two right angles together taken, are no angle; which is a\nfault which you somewhere (asking leave to use the word _angle_,\n\u03ba\u03b1\u03c4\u03b1\u03c7\u03c1\u03b9\u03db\u03b9\u03ba\u03ce\u03c2 acknowledge, but avoid not. For in geometry, where you\nconfess there is required all possible accurateness, every \u03ba\u03b1\u03c4\u03b1\u03c7\u03c1\u1fc6\u03c3\u03b9\u03c2 is\na fault. Besides you see by this definition, that Euclid is not of your,\nbut of Clavius\u2019s opinion. For it is manifest that the two lines which\ncontain an angle of contact incline one towards another, and come\ntogether in a point, and lie not in the same straight line, and\nconsequently make an angle.\n\nThe thirteenth definition is exact, but makes against your doctrine,\nthat a point is nothing. Examine it. \u1f4d\u03c1\u03bf\u03c2 \u1f10\u03db\u1f74\u03bd \u1f45 \u03c4\u03b9\u03bd\u03cc\u03c2 \u1f10\u03db\u1fd6 \u03c0\u03ad\u03c1\u03b1\u03c2.\n\u201c_Terminus est quod alicujus extremum est._\u201d _A term or bound is that\nwhich is the extreme of anything._ We had before, _the extremes of a\nline are points_. But what is in a line the extreme, but the first or\nlast _part_, though you may make that part as small as you will? A point\nis therefore a part, and nothing is no extreme.\n\nThe fourteenth, \u03a3\u03c7\u1fc6\u03bc\u03b1 \u1f10\u03db\u1f76 \u03c4\u1f78 \u1f51\u03c0\u1f78 \u03c4\u03b9\u03bd\u03bf\u03c2 \u1f24 \u03c4\u03b9\u03bd\u1ff6\u03bd \u1f45\u03c1\u03c9\u03bd \u03c0\u03b5\u03c1\u03b9\u03b5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd.\n\u201c_Figura est (subaudi quantitas) qu\u00e6 ab aliquo, vel aliquibus terminis\nundique continetur sive clauditur._\u201d _A figure is quantity contained\nwithin some bound or bounds._ Or shortly thus, _a figure is quantity\nevery way determined_, is in my opinion as exact a definition of a\nfigure as can possibly be given, though it must not be so in yours. For\nthis _determination_ is the same thing with _circumscription_; and\nwhatsoever is anywhere _(ubicunque) definitiv\u00e8_ is there also\n_circumscriptiv\u00e8_; and by this means the distinction is lost, by which\ntheologers, when they deny God to be in any place, save themselves from\nbeing accused of saying he is nowhere; for that which is nowhere is\nnothing. This definition of Euclid cannot therefore possibly be embraced\nby you that carry double, namely, mathematics and theology. For if you\nreject it, you will be cast out of all mathematic schools; and if you\nmaintain it, from the society of all school-divines, and lose the thanks\nof the favour you have shown (you the astronomer) to Bishop Bramhall.\n\nThe fifteenth is of a circle. \u039a\u03bf\u1f50\u03ba\u03bb\u03bf\u03c2 \u1f10\u03c3\u03c4\u1f76 \u03c3\u03c7\u1fc6\u03bc\u03b1 \u1f10\u03c0\u03af\u03c0\u03b5\u03b4\u03bf\u03bd, &c. _A circle\nis a plain figure comprehended by one line which is called the\ncircumference, to which circumference all the straight lines drawn from\none of the points within the figure are equal to one another._ This is\ntrue. But if a man had never seen the generation of a circle by the\nmotion of a compass or other equivalent means, it would have been hard\nto persuade him that there was any such figure possible. It had been\ntherefore not amiss first to have let him see that such a figure might\nbe described. Therefore so much of geometry is no part of philosophy,\nwhich seeketh the proper passions of all things in the generation of the\nthings themselves.\n\nAfter the fifteenth till the last or thirty-fifth definition, all are\nmost accurate, but the last which is this, _parallel straight lines are\nthose which being in the same plane, though infinitely produced both\nways, shall never meet_. Which is less accurate. For how shall a man\nknow that there be straight lines which shall never meet, though both\nways infinitely produced? Or how is the definition of parallels, that\nis, of lines perpetually equidistant, good, wherein the nature of\nequidistance is not signified? Or if it were signified, why should it\nnot comprehend as well the parallelism of circular and other crooked\nlines, as of straight, and as well of superficies, as of lines? By\nparallels is meant equidistant both lines and superficies, and the word\nis therefore not well defined without defining first equality of\ndistance. And because the distance between two lines or superficies, is\nthe shortest line that can join them, there either ought to be in the\ndefinition the _shortest distance_, which is that of the perpendicular\nand without inclination, or the distance in equal inclination, that is,\nin equal angles. Therefore if parallels be defined to be those lines or\nsuperficies, where the lines drawn from one to another in equal angles\nbe equal, the definition, as to like lines, or like superficies, will be\nuniversal and convertible. And if we add to this definition, that the\nequal angles be drawn not opposite ways, it will be absolute, and\nconvertible in all lines and superficies; and the definition will be\nthis: _parallels are those lines and superficies between which every\nline drawn, in any angle, is equal to any other line drawn in the same\nangle the same way_. For by this definition the distance between them\nwill perpetually be equal, and consequently they will never come nearer\ntogether, how much, or which way soever they be produced. And the\nconverse of it will be also true, _if two lines, or two superficies be\nparallel, and a straight line be drawn from one to the other, any other\nstraight line, drawn from one to the other in the same angle, and the\nsame way, will be equal to it_. This is manifestly true, and, most\negregious professors, new, at least to you.\n\nAnd thus much for the definitions placed before the first of Euclid\u2019s\nElements.\n\nBefore the third of his Elements is this definition: \u201c_In circulo\n\u00e6qualiter distare a centro rect\u00e6 line\u00e6 dicuntur, cum perpendiculares qu\u00e6\na centro in ipsas ducuntur sunt \u00e6quales_.\u201d _In a circle two straight\nlines are said to be equally distant from the centre, upon which the\nperpendiculars drawn from the centre are equal._ This is true; but it is\nrather an axiom than a definition, as being demonstrable that the\nperpendicular is the measure of the distance between a point and a\nstraight or a crooked line.\n\nBefore the fifth Element the first definition is of a part: _Pars est\nmagnitudo magnitudinis, minor majoris, cum minor metitur majorem_. _A\npart is one magnitude of another, the less of the greater, when the less\nmeasureth the greater._ From which definition it followeth, that more\nthan a half is not a part of the whole. But because Euclid meaneth here\nan aliquot part, as a half, a third, or a fourth, &c., it may pass for\nthe definition of a measure under the name of part, as thus: _a measure\nis a part of the whole, when multiplied it may be equal to the whole_,\nthough properly a measure is external to the thing measured, and not the\naliquot part itself, but equal to an aliquot part.\n\nBut the third definition is intolerable; it is the definition of \u03bb\u03cc\u03b3\u03bf\u03c2,\nin Latin _ratio_, in English, _proportion_, in this manner, \u03bb\u03cc\u03b3\u03bf\u03c2 \u1f10\u03c2\u1f76\n\u03b4\u03cd\u03bf \u03bc\u03b5\u03b3\u03b5\u03b8\u1ff6\u03bd \u1f41\u03bc\u03bf\u03b3\u03b5\u03bd\u1ff6\u03bd \u1fc6 \u03ba\u03b1\u03c4\u1f70 \u03c0\u03b7\u03bb\u03b9\u03ba\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c0\u03c1\u03bf\u03c2 \u1f04\u03bb\u03bb\u03b7\u03bb\u03b1 \u03c0\u03bf\u03b9\u1f70 \u03c3\u03c7\u03ad\u03c3\u03b9\u03c2. \u201c_Ratio\nest duarum magnitudinum ejusdem generis mutua qu\u00e6dam secundum\nquantitatem habitudo._\u201d _Proportion is a certain mutual habitude in\nquantity, of two magnitudes of the same kind, one to another._ First, we\nhave here _ignotum per ignotius_; for every man understandeth better\nwhat is meant by _proportion_ than by habitude. But it was the phrase of\nthe Greeks when they named like proportions, to say, the first to the\nsecond, \u03bf\u1f55\u03c4\u03c9\u03c2 \u1f14\u03c7\u03b5\u03b9, _id est, ita se habet_, and in English, _is as_, the\nthird to the fourth. As for example, in the proportions of two to four,\nand three to six, to say two to four, \u03bf\u1f55\u03c4\u03c9\u03c2 \u1f14\u03c7\u03b5\u03b9, _id est, ita se habet,\nid est_, _is as_, three to six. From which phrase Euclid made this his\ndefinition of proportion by \u03c0\u03bf\u03b9\u1f70 \u03c3\u03c7\u03ad\u03c3\u03b9\u03c2, which the Latins translate\n_qu\u00e6dam habitudo_. _Qu\u00e6dam_ in a definition is a most certain note of\nnot understanding the word _defined_; and in Greek, \u03c0\u03bf\u03b9\u1f70 \u03c3\u03c7\u03ad\u03c3\u03b9\u03c2 is much\nworse; for to render rightly the Greek definition, we are to say in\nEnglish, that proportion is a what-shall-I-call-it-_isness_, or _soness_\nof two magnitudes, &c.; than which nothing can be more unworthy of\nEuclid. It is as bad as anything was ever said in geometry by Orontius,\nor by Dr. Wallis. That proportion is quantity compared, that is to say,\nlittle or great in respect of some other quantity, as I have above\ndefined it, is I think intelligible.\n\nThe fourth is, \u1f08\u03bd\u03b1\u03bb\u03bf\u03b3\u03af\u03b1 \u03b4\u03ad \u1f72\u03c3\u03c4\u03b9\u03bd \u1fc6 \u03c4\u03c9\u03bd \u03bb\u03cc\u03b3\u03c9\u03bd \u1f41\u03bc\u03bf\u03b9\u03cc\u03c4\u03b7\u03c2. \u201c_Proportio vero\nest rationum similitudo._\u201d Here we have no one word by which to render\n\u1f08\u03bd\u03b1\u03bb\u03bf\u03b3\u03af\u03b1; for our word _proportion_ is already bestowed upon the\nrendering of \u03bb\u03cc\u03b3\u03bf\u03c2. Nevertheless the Greek may be translated into\nEnglish thus, _iterated proportions_. But iterated proportion is the\nsame with _eadem ratio_. To what purpose then serveth the sixth\ndefinition, which is of _eadem ratio_? For \u1f08\u03bd\u03b1\u03bb\u03bf\u03b3\u03af\u03b1 and _eadem ratio_\nand _similitudo rationum_, are the same thing, as appeareth by Euclid\nhimself, where he defines those quantities, that are in the same\nproportion by \u1f00\u03bd\u03ac\u03bb\u03bf\u03b3\u03bf\u03bd. Therefore the sixth definition is but a _lemma_,\nand assumed without demonstration.\n\nThe fourteenth, \u201c_Compositio rationis est sumptio antecedentis cum\nconsequente, ceu unius, ad ipsum consequentem_,\u201d _To compound\nproportion, is to take both antecedent and consequent together as one\nmagnitude, and compare it to the consequent_, is good; though he might\nhave compared it as well with the antecedent; for both ways it had been\na composition of proportion. We are to note here, that the composition\ndefined in this place by Euclid is not adding together of proportions,\nbut of two quantities that have proportion. And therefore it is not the\nsame composition which he defineth in the fourth place before the sixth\nelement, for there he defineth the addition of one proportion to another\nproportion in this manner: \u03bb\u03cc\u03b3\u03bf\u03c2 \u1f10\u03ba \u03bb\u03cc\u03b3\u03c9\u03bd \u03c3\u03c5\u03b3\u03ba\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9 \u03bb\u03ad\u03b3\u03b5\u03c4\u03b1\u03b9, &c. _A\nproportion is said to be compounded of proportions, when their\nquantities multiplied into one another make a proportion_; as when we\nwould compound or add together the proportions of three to two, and of\nfour to five, we must multiply three and four, which maketh twelve, and\ntwo and five, which maketh ten. And then the proportion of twelve to ten\nis the sum of the proportions of three to two, and of four to five,\nwhich is true, but not a definition; for it may and ought to be\ndemonstrated. For to define what is addition of two proportions (which\nare always in four quantities, though sometimes one of them be twice\nnamed) we are to say, that they are then added together when we make the\nsecond to another in the same proportion, which the third hath to the\nfourth.\n\nAnd thus much of the definitions; of which some, very few, you see are\nfaulty; the rest either accurate, or good enough if well interpreted.\nFor the rest of the elements all are accurate, notwithstanding that you\nallow not for good any definition in geometry that hath in it the word\n_motion_, of which there be divers before the eleventh Element. But I\nmust here put you in mind, that geometry being a science, and all\nscience proceeding from a precognition of causes, the definition of a\nsphere, and also of a circle, by the generation of it, that is to say,\nby motion, is better than by the equality of distance from a point\nwithin.\n\nThe second sort of principles are those of construction, usually called\n_postulata_, or petitions. As for those _notiones communes_, called\n_axioms_, they are from the definitions of their terms demonstrable,\nthough they be so evident as they need not demonstration. These\npetitions are by Euclid called \u1f08\u03b9\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1, such as are granted by favour,\nthat is, simply petitions, whereas by axiom is understood that which is\nclaimed as due. So that between \u1f08\u03be\u03af\u03c9\u03bc\u03b1 and \u1f08\u03af\u03c4\u03b7\u03bc\u03b1 there is this other\ndifference, that this latter is simply a petition, the former a petition\nof right.\n\nOf petitions simply, the first is, _that from any point to any point may\nbe drawn a straight line_. The second, _that a finite straight line may\nbe produced_. The third, _that upon any centre at any distance may be\ndescribed a circle_. All which are both evident and necessary to be\ngranted.\n\nAnd by all these a man may easily perceive that Euclid in the\ndefinitions of a point, a line, and a superficies, did not intend that a\npoint should be nothing, or a line be without latitude, or a superficies\nwithout thickness; for if he did, his petitions are not only\nunreasonable to be granted, but also impossible to be performed. For\nlines are not drawn but by motion, and motion is of body only. And\ntherefore his meaning was, that the quantity of a point, the breadth of\na line, and the thickness of a superficies were not to be _considered_,\nthat is to say, not to be reckoned in the demonstration of any theorems\nconcerning the quantity of bodies, either in length, superficies, or\nsolid.\n\n ==========\n\n OF THE FAULTS THAT OCCUR IN\n DEMONSTRATION.\n\n TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN\n THE UNIVERSITY OF OXFORD.\n\n\n LESSON II.\n\nThere be but two causes from which can spring an error in the\ndemonstration of any conclusion in any science whatsoever; and those are\nignorance or want of understanding, and negligence. For as in the adding\ntogether of many and great numbers, he cannot fail that knoweth the\nrules of addition, and is also all the way so careful, as not to mistake\none number or one place for another; so in any other science, he that is\nperfect in the rules of logic, and is so watchful over his pen, as not\nto put one word for another, can never fail of making a true, though not\nperhaps the shortest and easiest demonstration.\n\nThe rules of demonstration are but of two kinds: one, that the\nprinciples be true and evident definitions; the other, that the\ninferences be necessary. And of true and evident definitions, the best\nare those which declare the cause or generation of that subject, whereof\nthe proper passions are to be demonstrated. For science is that\nknowledge which is derived from the comprehension of the cause. But when\nthe cause appeareth not, then may, or rather must we define some known\nproperty of the subject, and from thence derive some possible way, or\nways, of the generation. And the more ways of generation are explicated,\nthe more easy will be the derivation of the properties; whereof some are\nmore immediate to one, some to another generation. He therefore that\nproceedeth from untrue, or not understood definitions, is ignorant of\nthat he goes about; which is an ill-favoured fault, be the matter he\nundertaketh easy or difficult, because he was not forced to undergo a\ngreater charge than he could carry through. But he that from right\ndefinitions maketh a false conclusion, erreth through human frailty, as\nbeing less awake, more troubled with other thoughts, or more in haste\nwhen he was in writing. Such faults, unless they be very frequent, are\nnot attended with shame, as being common to all men, or are at least\nless ugly than the former, except then, when he that committeth them\nreprehendeth the same in other men. For that is in every man\nintolerable, which he cannot tolerate in another. But to the end that\nthe faults of both kinds may by every man be well understood, it will\nnot be amiss to examine them by some such demonstrations as are publicly\nextant. And for this purpose I will take such as are in mine and in your\nbooks, and begin with your _Elenchus_ of the geometry contained in my\nbook _De Corpore_; to which I will also join your book lately set forth\nconcerning the _Angle of Contact, Conic Sections_ , and your\n_Arithmetica Infinitorum_; and then examine the rest of my philosophy,\nand yours that oppugn it. For I will take leave to consider you both\neverywhere as one author, because you publicly declare your approbation\nof one another\u2019s doctrine.\n\nMy first definition is of a line, of length, and of a point. \u201cThe way,\u201d\nsay I, \u201cof a body moved, in which magnitude (though it always have some\nmagnitude) is not considered, is called a line; and the space gone over\nby that motion, length, or one and a simple dimension.\u201d To this\ndefinition you say, first, \u201cwhat mathematician did ever thus define a\nline or length?\u201d Whether you call in others for help or testimony, it is\nnot done like a geometrician; for they use not to prove their\nconclusions by witnesses, but rely upon the strength of their own\nreason; and when your witnesses appear, they will not take your part.\nSecondly, you grant that what I say is true, but not a definition. But\nto tell you truly what it is which we call a line, is to define a line.\nWhy then is not this a definition? \u201cBecause,\u201d say you in the first\nplace, \u201cit is not a reciprocal proposition.\u201d But by your favour it is\nreciprocal. For not only the way of a body whose quantity is not\nconsidered is a line, but also every line is, or may be conceived to be,\nthe way of a body so moved. And if you object that there is a difference\nbetween _is_ and _may be conceived to be_, Euclid, whom you call to your\naid, will be against you in the fourteenth definition before his\neleventh Element; where he defines a sphere just as convertibly as I\ndefine a line; except you think the globes of the sun and stars cannot\nbe globes, unless they were made by the circumduction of a semicircle;\nand again in the eighteenth definition, which is of a cone, unless you\nadmit no figure for a cone, which is not generated by the revolution of\na triangle; and again, in the twentieth definition, which is of a\ncylinder, except it be generated by the circumvolution of a\nparallelogram. Euclid saw that what proper passion soever should be\nderived from these his definitions, would be true of any other cylinder,\nsphere, or cone, though it were otherwise generated; and the description\nof the generation of any one being by the imagination applicable to all,\nwhich is equivalent to convertible, he did not believe that any rational\nman could be misled by learning logic to be offended with it. Therefore\nthis exception proceedeth from want of understanding, that is, from\nignorance of the nature, and use of a definition.\n\nAgain, you object and ask: \u201cWhat need is there of motion, or of body\nmoved, to make a man understand what is a line? Are not lines in a body\nat rest, as well as in a body moved? And is not the distance of two\nresting points length, as well as the measure of the passage? Is not\nlength one and a simple dimension, and one and a simple dimension line?\nWhy then is not line and length all one?\u201d See how impertinent these\nquestions are. Euclid defines a sphere to be a solid figure described by\nthe revolution of a semicircle about the unmoved diameter. Why do you\nnot ask, what need there is to the understanding of what a sphere is, to\nbring in the motion of a semicircle? Is not a sphere to be understood\nwithout such motion? Is not the figure so made a sphere without this\nmotion? And where he defines the axis of a sphere to be that unmoved\ndiameter, may not you ask, whether there be no axis of a sphere, when\nthe whole sphere, diameter and all, is in motion? But it is not to my\npurpose to defend my definition by the example of that of Euclid.\nTherefore first, I say, to me, howsoever it may be to others, it was fit\nto define a line by motion. For the generation of a line is the motion\nthat describes it. And having defined philosophy in the beginning, to be\nthe knowledge of the properties from the generation, it was fit to\ndefine it by its generation. And to your question, _is not distance\nlength?_ I answer, that though sometimes distance be equivalent to\nlength, yet certainly the distance between the two ends of a thread\nwound up into a clue is not the length of the thread; for the length of\nthe thread is equal to all the windings whereof the clue is made. But if\nyou will needs have distance and length to be all one, tell me of what\nthe distance between any two points is the length. Is it not the length\nof the way? And how is that called way, which is not defined by some\nmotion? And have not several ways between the same places, as by land\nand by water, several lengths? But they have but one distance, because\nthe distance is the shortest way. Therefore between the length of the\npath, and the distance of places, there is a real difference in this\ncase, and in all cases a difference of the consideration. Your\nobjection, that line is longitude, proceeds from want of understanding\nEnglish. Do men ever ask what is the line of a thread, or the line of a\ntable, or of any other body? Do they not always ask what is the length\nof it? And why, but because they use their own judgments, not yet\ncorrupted by the subtlety of mistaken professors. Euclid defines a line\nto _be length without breadth_. If those terms be all one, why said he\nnot that a _line is a line without breadth_? But what definition of a\nline give you? None. Be contented then with such as you receive, and\nwith this of mine, which you shall presently see is not amiss.\n\nYour next objections are to my definition of a point. Which definition\nadhereth to the former in these words, \u201cand the body itself is called a\npoint.\u201d Here again you call for help: \u201c_Quis unquum mortalium, etc._\nWhat mortal man, what sober man, did ever so define a point?\u201d It is\nwell, and I take it to be an honour to be the first that do so. But what\nobjection do you bring against it. This: \u201cThat a point added to a point,\nif it have magnitude, makes it greater.\u201d I say it doth so, but then\npresently it loseth the name of a point, which name was given to signify\nthat it was not the meaning of him that used it in demonstration to add,\nsubtract, multiply, divide, or any way compute it. Then you come in\nwith, \u201cperhaps you will say though it have magnitude, that magnitude is\nnot considered.\u201d You need not say _perhaps_. You know I affirm it; and\ntherefore your argument might have been left out, but that it gave you\nan occasion of a digression into scurvy language.\n\nAnd whereas you ask why I defined not a point thus: \u201c_Punctum est corpus\nquod non consideratur esse corpus, et magnum quod non consideratur esse\nmagnum_.\u201d I will tell you why. First, because it is not Latin. Secondly,\nbecause when I had defined it by _corpus_, there was no need to define\nit again by _magnum_. I understand very well this language, \u201c_punctum\nest corpus, quod non consideratur ut corpus_.\u201d A point is a body not\nconsidered as body. But _punctum est corpus, quod non consideratur esse\ncorpus, vel esse magnum_, is not Latin; nor the version of it, _a point\nis a body which is not considered to be a body_, English. My definition\nwas, that a point is that body whose magnitude is not considered, not\nreckoned, not put to account in demonstration. And I exemplified the\nsame by the body of the earth describing the ecliptic line; because the\nmagnitude is not there reckoned nor chargeth the ecliptic line with any\nbreadth. But I perceive you understand not what the word _consideration_\nsignifieth, but take it for comparison or relation; and say I ought to\ndefine a point simply, and not by relation to a great body; as if to\nreckon and to compare were the same thing. \u201c_Omnia mihi_,\u201d saith Cicero,\n\u201c_provisa et considerata sunt_.\u201d I have provided and reckoned\neverything. There is a great difference between reckoning and relation.\n\nAgain, you ask, why _corpus motum_, a body moved? I will tell you;\nbecause the motion was necessary for the generation of a line. And\nthough after the generation of the line the point should rest, yet it is\nnot necessary from this definition that it should be no more a point;\nnor when Euclid defines a sphere by the circumduction of a semicircle\nupon an axis that resteth, doth it follow from thence when the sphere,\naxis, centre and all, as that of the earth, is moved from place to\nplace, that it is no more an axis.\n\nLastly, you object \u201cthat motion is accidentary to a point, and\nconsequently not essential, nor to be put into the definition.\u201d And is\nnot the circumduction of a semicircle accidentary to a sphere? Or do you\nthink the sphere of the sun was generated by the revolution of a\nsemicircle? And yet it was thought no fault in Euclid to put the motion\ninto the definition of a sphere.\n\nThe conceit you have concerning definitions, that they must explicate\nthe essence of the thing defined, and must consist of a _genus_ and a\n_difference_, is not so universally true as you are made believe, or\nelse there be very many insufficient definitions that pass for good with\nyou in Euclid. You are much deceived if you think these woful notions of\nyours, and the language that doth everywhere accompany them, show\nhandsomely together. Or that such grounds as these be able to sustain so\nmany, and so haughty reproaches as you advance upon them, so as they\nfall not, as you shall see immediately, upon your own head. I say a\npoint hath quantity, but not to be reckoned in demonstrating the\nproperties of lines, solids, or superficies; you say it hath no quantity\nat all, but is plainly nothing.\n\nThe first of the petitions of Euclid is, \u201cthat a line may be drawn from\npoint to point at any distance.\u201d The second, \u201cthat a straight line may\nbe produced.\u201d The third, \u201cthat on any centre a circle may be described\nat any distance.\u201d And the eighth axiom (which Sir H. Savile observes to\nbe the foundation of all geometry) is this, \u201c_Qu\u00e6 sibi mutuo congruunt,\netc._ Those things that are applied to one another in all points are\nequal.\u201d All or any of these principles being taken away, there is not in\nEuclid one proposition demonstrated or demonstrable. If a point have no\nquantity, a line can have no latitude; and because a line is not drawn\nbut by motion, by motion of a body, and body imprinteth latitude all the\nway, it is impossible to draw or produce a straight line, or to describe\na circular line without latitude. Also if a line have no latitude, one\nstraight line cannot be applied to another. To them therefore that deny\na point to have quantity, that is, a line to have latitude, the\nforenamed principles are not possible, and consequently no proposition\nin geometry is demonstrated or demonstrable. You therefore that deny a\npoint to have quantity, and a line to have breadth, have nothing at all\nof the science of geometry. The practice you may have, but so hath any\nman that hath learned the bare propositions by heart; but they are not\nfit to be professors either of geometry or of any other science that\ndependeth on it. Some man perhaps may say that this controversy is not\nmuch worth, and that we both mean the same thing. But that man, though\nin other things prudent enough, knoweth little of science and\ndemonstration. For definitions are not only used to give us the notions\nof those things whose appellations are defined, for many times they that\nhave no science have the ideas of things more perfect than such as are\nraised by definitions. As who is there that understandeth not better\nwhat a straight line is, or what proportion is, and what many other\nthings are, without definition, than some that set down the definitions\nof them. But their use is, when they are truly and clearly made, to draw\narguments from them for the conclusions to be proved. And therefore you\nthat in your following censures of my geometry, take your argument so\noften from this, that a point is nothing, and so often revile me for the\ncontrary, are not to be allowed such an excuse. He that is here\nmistaken, is not to be called negligent in his expression, but ignorant\nof the science.\n\nIn the next place, you take exceptions to my definition of _equal\nbodies_, which is this: \u201c_Corpora \u00e6qualia sunt qu\u00e6 eundem locum\npossidere possunt_. Equal bodies are those which may have the same\nplace.\u201d To which you object impertinently, that I may as well define a\nman to be, _he that may be prince of Transylvania_, wittily, as you\ncount wit. Formerly in every definition, you exacted an explication of\nthe essence. You are therefore of opinion that the possibility of being\nprince of Transylvania is no less essential to _a man_, than the\npossibility of the being of two bodies successively in the same place,\nis essential to _bodies equal_.\n\nYou take no notice of the twenty-third article of this same chapter,\nwhere I define what it is we call essence, namely, that accident for\nwhich we give the thing its name. As the essence of a man is his\ncapacity of reasoning; the essence of a white body, whiteness, &c.,\nbecause we give the name of _man_ to such bodies as are capable of\nreasoning, for that their capacity; and the name of _white_ to such\nbodies as have that colour, for that colour. Let us now examine why it\nis that men say bodies are one to another equal; and thereby we shall be\nable to determine whether the _possibility of having the same place_ be\nessential or not to _bodies equal_, and consequently whether this\ndefinition be so like to the defining of a man by the _possibility of\nbeing prince of Transylvania_ as you say it is. There is no man, besides\nsuch egregious geometricians as yourselves, that inquireth the equality\nof two bodies, but by measure. And for liquid bodies, or the aggregates\nof innumerable small bodies, men (men, I say) measure them by putting\nthem one after another into the same vessel, that is to say, into the\nsame place, as Aristotle defines place, or into the space determined by\nthe vessel, as I define place. And the bodies that so fill the vessel,\nthey acknowledge and receive for equal. But though, when hard bodies\ncannot be so measured, without the incommodity or trouble of altering\ntheir figure, they then enquire, if the bodies are both of the same\nkind, their equality by weight, knowing, without your teaching, that\nequal bodies of the same nature weigh proportionably to their\nmagnitudes; yet they do it not for fear of missing of the equality, but\nto avoid inconvenience or trouble. But you (you, I say), that have no\ndefinition of equals, neither received from others, nor framed by\nyourselves, out of your shallow meditation and deep conceit of your own\nwits, contend against the common light of nature. So much is unheedy\nlearning a hinderance to the knowledge of the truth, and changeth into\nelves those that were beginning to be men.\n\nAgain, when men inquire the equality of two bodies in length, they\nmeasure them by a common measure; in which measure they consider neither\nbreadth nor thickness, but how the length of it agreeth, first with the\nlength of one of the bodies, then with the length of the other. And both\nthe bodies whose lengths are measured, are successively in the same\nplace under their common measure. _Place_ therefore in lines also, is\nthe proper index and discoverer of equality and inequality. And as in\nlength, so it is in breadth and thickness, which are but lengths\notherwise taken in the same solid body. But now when we come from this\nequality and inequality of lengths known by measure, to determine the\nproportions of superficies and of solids, by ratiocination, then it is\nthat we enter into geometry; for the making of definitions, in\nwhatsoever science they are to be used, is that which we call\n_philosophia prima_. It is not the work of a geometrician, as a\ngeometrician, to define what is equality, or proportion, or any other\nword he useth, though it be the work of the same man, as a man. His\ngeometrical part is, to draw from them as many true and useful theorems\nas he can.\n\nYou object secondly, that a pyramis may be equal to a cube whilst it is\na pyramis. True. And so also whilst it is a pyramis it hath a\npossibility by flexion and transposition of parts to become a cube, and\nto be put into the place where another cube equal to it was before. This\nis to argue like a child that hath not yet the perfect understanding of\nany language.\n\nIn the third and fourth objection, you teach me to define equal bodies\n(if I will needs define them by place) by the _equality of place_, and\nto say, _that bodies are equal that have equal places_. Teach others, if\nyou can, to measure their grain, not by the same, but equal bushels.\n\nIn the fifth objection, you except against the the word _can_, in that I\nsay that bodies are equal which _can_ fill the same place. For the\ngreater body _can_, you say, fill the place of the less, though not\nreciprocally the less of the greater. It is true, that though the place\nof the less can never be the place of the greater, yet it may be filled\nby a part of the greater. But it is not then the greater body that\nfilleth the place of the less, but a part of it, that is to say, a less\nbody. Howsoever, to take away from simple men this straw they stumble\nat, I have now put the definition of equal bodies into these words:\n_equal bodies are those whereof every one can fill the place of every\nother_. And if my definition displease you, propound your own, either of\n_equal bodies_, or of _equals_ simply. But you have none. Take therefore\nthis of mine.\n\nThe sixth is a very admirable exception. \u201cWhat,\u201d say you, \u201cif the same\nbody can sometimes take up a greater, sometimes a lesser place, as by\nrarefaction and condensation?\u201d I understand very well that bodies may be\nsometimes thin and sometimes thick, as they chance to stand closer\ntogether or further from one another. So in the mathematic schools, when\nyou read your learned lectures, you have a thick or thronging audience\nof disciples, which in a great church would be but a very thin company.\nI understand how thick and thin may be attributed to bodies in the\nplural, as to a company; but I understand not how any one of them is\nthicker in the school than in the church; or how any one of them taketh\nup a greater room in the school, when he can get in, than in the street.\nFor I conceive the dimensions of the body, and of the place, whether the\nplace be filled with gold or with air, to be coincident and the same;\nand consequently both the quantity of the air, and the quantity of the\ngold, to be severally equal to the quantity of the place; and therefore\nalso, by the first axiom of Euclid, equal to one another; insomuch as if\nthe same air should be by condensation contained in a part of the place\nit had, the dimensions of it would be the same with the dimensions of\npart of the place, that is, should be less than they were, and by\nconsequence the quantity less. And then either the same body must be\nless also, or we must make a difference between greater bodies and\nbodies of greater quantity; which no man doth that hath not lost his\nwits by trusting them with absurd teachers. When you receive salary, if\nthe steward give you for every shilling a piece of sixpence, and then\nsay, every shilling is condensed into the room of sixpence, I believe\nyou would like this doctrine of yours much the worse. You see how by\nyour ignorance you confound the affairs of mankind, as far forth as they\ngive credit to your opinions, though it be but little. For nature abhors\neven empty words, such as are (in the meaning you assign them),\n_rarefying_ and _condensing_. And you would be as well understood if you\nshould say (coining words by your own power), that the same body might\ntake up sometimes a greater, sometimes a lesser place, by wallifaction\nand wardensation, as by rarefaction and condensation. You see how\nadmirable this your objection is.\n\nIn the seventh objection you bewray another kind of ignorance, which is\nthe ignorance of what are the proper works of the several parts of\nphilosophy. \u201cThough it were out of doubt,\u201d say you, \u201cthat the same body\ncannot have several magnitudes, yet seeing it is matter of natural\nphilosophy, nor hath anything to do with the present business, to what\npurpose is it to mention it in a mathematical definition?\u201d It seems by\nthis, that all this while you think it is a piece of the geometry of\nEuclid, no less to make the definitions he useth, than to infer from\nthem the theorems he demonstrateth. Which is not true. For he that\ntelleth you in what sense you are to take the appellations of those\nthings which he nameth in his discourse, teacheth you but his language,\nthat afterwards he may teach you his art. But teaching of language is\nnot mathematic, nor logic, nor physic, nor any other science; and\ntherefore to call a definition, as you do, mathematical, or physical, is\na mark of ignorance, in a professor inexcusable. All doctrine begins at\nthe understanding of words, and proceeds by reasoning till it conclude\nin science. He that will learn geometry must understand the terms before\nhe begin, which that he may do, the master demonstrateth nothing, but\nuseth his natural prudence only, as all men do when they endeavour to\nmake their meaning clearly known. For words understood are but the seed,\nand no part of the harvest of philosophy. And this seed was it, which\nAristotle went about to sow in his twelve books of _metaphysics_, and in\nhis eight books concerning the hearing of _natural philosophy_. And in\nthese books he defineth time, place, substance or essence, quantity,\nrelation, &c., that from thence might be taken the definitions of the\nmost general words for principles in the several parts of science. So\nthat all definitions proceed from common understanding; of which, if any\nman rightly write, he may properly call his writing _philosophia prima_,\nthat is, the seeds, or the grounds of philosophy. And this is the method\nI have used, defining place, magnitude, and the other the most general\nappellations in that part which I entitle _philosophia prima_. But you\nnow, not understanding this, talk of mathematical definitions. You will\nsay perhaps that others do the same as well as you. It may be so. But\nthe appeaching of others does not make your ignorance the less.\n\nIn the eighth place you object not, but ask me _why I define equal\nbodies apart_? I will tell you. Because all other things which are said\nto be equal, are said to be so from the equality of bodies; as two lines\nare said to be equal, when they be coincident with the length of one and\nthe same body; and equal times, which are measured by equal lengths of\nbody, by the same motion. And the reason is, because there is no subject\nof quantity, or of equality, or of any other accident but body; all\nwhich I thought certainly was evident enough to any uncorrupted\njudgment; and therefore that I needed first to define equality in the\nsubject thereof, which is body, and then to declare in what sense it was\nattributed to time, motion, and other things that are not body.\n\nThe ninth objection is an egregious cavil. Having set down the\ndefinition of _equal bodies_, I considered that some men might not allow\nthe attribute of equality to any things but those which are the subjects\nof quantity, because there is no equality, but in respect of quantity.\nAnd to speak rigidly, _magnum et magnitudo_ are not the same thing; for\nthat which is great, is properly a body, whereof greatness is an\naccident. In what sense therefore, might you object, can an accident\nhave quantity? For their sakes therefore that have not judgment enough\nto perceive in what sense men say the length is so long, or the\nsuperficies so broad, &c. I added these words: \u201c_Eadem ratione (qua\nscilicet corpora dicuntur \u00e6qualia) magnitudo magnitudini \u00e6qualis\ndicitur_,\u201d that is, _in the same manner, as bodies are said to be equal,\ntheir magnitudes also are said to be equal_. Which is no more than to\nsay, _when bodies are equal, their magnitudes also are called equal.\nWhen bodies are equal in length, their lengths are also called equal.\nAnd when bodies are equal in superficies, their superficies are also\ncalled equal._ All which is common speech, as well amongst\nmathematicians, as amongst common people; and, though improper, cannot\nbe altered, nor needeth to be altered to intelligent men. Nevertheless I\ndid think fit to put in that clause, that men might know what it is we\ncall equality, as well in magnitudes as in _magnis_, that is, in bodies.\nWhich you so interpret, as if it bore this sense, _that when bodies are\nequal their superficies also must be equal_, contrary to your own\nknowledge, only to take hold of a new occasion of reviling. How\nunhandsome and unmanly this is, I leave to be judged by any reader that\nhath had the fortune to see the world, and converse with honest men.\n\nAgainst the fourteenth article, where I prove that the same body hath\nalways the same magnitude, you object nothing but this, _that though it\nbe granted, that the same body hath the same magnitude, while it\nresteth, yet I bring nothing to prove that when it changeth place, it\nmay not also change its magnitude by being enlarged or contracted_.\nThere is no doubt but to a body, whether at rest or in motion, more body\nmay be added, or part of it taken away. But then it is not the same\nbody, unless the whole and the part be all one. If the schools had not\nset your wit awry, you could never have been so stupid as not to see the\nweakness of such objections. That which you add in the end of your\nobjections to this eighth chapter, _that I allow not Euclid this axiom\ngratis, that the whole is greater than a part_, you know to be untrue.\n\nAt my eleventh chapter, you enter into dispute with me about the nature\nof proportion. Upon the truth of your doctrine therein, and partly upon\nthe truth of your opinions concerning the definitions of a point, and of\na line, dependeth the question whether you have any geometry or none;\nand the truth of all the demonstrations you have in your other books,\nnamely of the _Angle of Contact_ , and _Arithmetica Infinitorum_. Here I\nsay you enter, how you will get out, your reputation saved, we shall see\nhereafter.\n\nWhen a man asketh what proportion one quantity hath to another, he\nasketh how great or how little the one is comparatively to, or in\nrespect of the other. When a geometrician prefixeth before his\ndemonstrations a definition, he doth it not as a part of his geometry,\nbut of natural evidence, not to be demonstrated by argument, but to be\nunderstood in understanding the language wherein it is set down; though\nthe matter may nevertheless, if besides geometry he have wit, be of some\nhelp to his disciple to make him understand it the sooner. But when\nthere is no significant definition prefixed, as in this case, where\nEuclid\u2019s definition of proportion, that it is a _whatshicalt habitude of\ntwo quantities, &c._, is insignificant, and you allege no other, every\none that will learn geometry, must gather the definition from observing\nhow the word to be defined is most constantly used in common speech. But\nin common speech if a man shall ask how much, for example, is six in\nrespect of four, and one man answer that it is greater by two, and\nanother that it is greater by half of four, or by a third of six, he\nthat asked the question will be satisfied by one of them, though perhaps\nby one of them now, and by the other another time, as being the only man\nthat knoweth why he himself did ask the question. But if a man should\nanswer, as you would do, that the proportion of six to two is of those\nnumbers a certain quotient, he would receive but little satisfaction.\nBetween the said answers to this question, how much is six in respect of\nfour? there is this difference. He that answereth that it is more by\ntwo, compareth not two with four, nor with six, for two is the name of a\nquantity absolute. But he that answereth it is more by half of four, or\nby a third of six, compareth the difference with one of the differing\nquantities. For halfs and thirds, &c. are names of quantity compared.\n\nFrom hence there ariseth two species or kinds of (_ratio_) proportion,\ninto which the general word _proportion_ may be divided. The one\nwhereof, namely, that wherein the difference is not compared with either\nof the differing quantities, is called _ratio arithmetica_, arithmetical\nproportion; the other _ratio geometrica_, geometrical proportion; and,\nbecause this latter is only taken notice of by the name of proportion,\nsimply _proportion_. Having considered this, I defined proportion,\nchapter II. article 3, in this manner: \u201c_Ratio est relatio antecedentis\nad consequens secundum magnitudinem_:\u201d _Proportion is the relation of\nthe antecedent to the consequent in magnitude_; having immediately\nbefore defined relatives, antecedent, and consequent, in the same\narticle, and by way of explication added, that such relation was nothing\nelse but that one of the quantities was equal to the other, or exceeded\nit by some quantity, or was by some quantity exceeded by it. And for\nexemplification of the same, I added further, that the proportion of\nthree to two was, that three exceeded two by a unity; but said not that\nthe unity, or the difference whatsoever it were, was their proportion,\n_for unity, and to exceed another by unity_, is not the same thing. This\nis clear enough to others; let us therefore see why it is not so to you.\nYou say I make proportion to consist in that which remaineth after the\nlesser quantity is subtracted out of the greater; and that you make it\nto consist in the quotient, when one number is divided by the other.\nWherein you are mistaken; first, in that you say, I make the proportion\nto consist in the remainder. For I make it to consist in the act of\nexceeding, or of being exceeded, or of being equal; whereas the\nremainder is always an absolute quantity, and never a proportion. To be\nmore or less than another number by two, is not the number two; likewise\nto be equal to two, where the difference is _nothing_, is not that\n_nothing_? Again, you mistake in saying the proportion consisteth in the\nquotient. For divide twenty by five, the quotient is four. Is it not\nabsurd to say that the proportion of five to twenty, or of twenty to\nfive, is four? You may say the proportion of five to twenty, is the\nproportion of one to four. And so say I. And you may therefore also say,\nthat the proportion of one to four is a measure of the proportion of\nfive to twenty, as being equal. And so say I. But that is only in\ngeometrical proportion, and not in proportion universally. For though\nthe _species_ obtain the denomination of the genus, yet it is not the\n_genus_. But as the quotient giveth us a measure of the proportion of\nthe dividend to the divisor in geometrical proportion, so also the\nremainder after subtraction is the measure of proportion arithmetical.\n\nYou object in the next place, \u201cthat if the proportion of one quantity to\nanother be nothing but the excess or defect, then, wheresoever the\nexcess or defect is the same, there the proportion is the same.\u201d This\nyou say follows in your logic, and from thence, that the proportion of\nthree to two, and five to four is the same. But is not three to two, and\nfive to four, where the excess is the same number, the same proportion\narithmetical? And is not arithmetical proportion, proportion? You take\nhere (_ratio_) proportion, which is the _genus_, for that _species_ of\nit which is called geometrical, because usually this species has the\nname of proportion simply. Also that the proportion of three to two, is\nthe same with that of nine to six; is it not because the excesses are\none and three, the same portions of three and nine, that is to say the\nsame excesses comparatively? I wonder you ask me not what is the _genus_\nof arithmetical and geometrical proportions, and what the _difference_?\nThe _genus_ is (_ratio_) proportion, or comparison in magnitude, and the\n_difference_ is that one comparison is made by the absolute quantity,\nthe other by the comparative quantity, of the excess or defect, if there\nbe any. Can anything be clearer than this? You after come in with\n_ignosce habitudini_ to no purpose. I am not so inhuman as not to pardon\ndulness or madness: they are not voluntary faults. But when men\nadventure voluntarily to talk of that they understand not censoriously\nand scornfully, I may tell them of it.\n\nThis difference between the excesses or defects, as they are simply or\ncomparatively reckoned, being thus explained, all the rest of that you\nsay in your objections to this eleventh chapter (saving that art. 5 for\n_ratio binarii ad quinarium est superari ternario_, as it is in other\nplaces, I have put too hastily _ratio binarii ad quinarium est\nternarius_), will be understood by every reader to be frivolous, and to\nproceed from the ignorance of what proportion is.\n\nAt the twelfth chapter you only note that I say, _that the proportion of\ninequality is quantity, but the proportion of equality not quantity_,\nand refer that which you have to say against it to the chapter\nfollowing; to which place I shall also come in the following lesson.\n\n\n ==========\n\n\n OF THE FAULTS THAT OCCUR IN\n DEMONSTRATION.\n\n TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN\n THE UNIVERSITY OF OXFORD.\n\n\n LESSON III.\n\nYou begin your reprehension of my thirteenth chapter with a question;\nwhereas _I_ divide proportion into arithmetical and geometrical. You ask\nme what _proportion it is I so divide_. Euclid divides an angle into\nright, obtuse, and acute. I may ask you as pertinently, what angle it is\nhe so divides? Or, when you divide _animal_ into _homo_ and _brutum_,\nwhat animal that is, which you so divide? You see by this, how absurd\nyour question is. But you say the definition of proportion which I make\nat Chap. II. art. 3., namely, that proportion is the comparison of two\nmagnitudes, one to another, agreeth not, neither with arithmetical, nor\nwith geometrical proportion. I believe you thought so then, but having\nread what I have said in the end of the last lesson, if you think so\nstill, your fault will be too great to be pardoned easily. But why did\nyou think so before? Is it not because there was no definition in Euclid\nof proportion universal, and because he maketh no mention of proportion\narithmetical, and because you had not in your minds a sufficient notion\nthereof yourselves to supply that defect? And is not this the cause\nalso, why you put in this parenthesis (if arithmetical proportion ought\nto be called proportion)? Which is a confession that you know not\nwhether there be such a thing as arithmetical proportion or not,\nnotwithstanding that on all occasions you speak of arithmetical\nproportionals. Yes, this was it that made you think that proportion\nuniversally, and proportion geometrical, is the same, and yet to say you\ncannot tell whether they be the same or not. It is no wonder, therefore,\nif in such confusion of the understanding, you apprehend not that the\nproportions of two to five, and nine to twelve, are the same; so you are\nblinded by seeing that they are not the same proportions geometrical.\nNor doth it help you that I say the difference is the proportion; for by\ndifference you might, if you would, have understood the act of\ndiffering.\n\nAt the second article you note for a fault in method, that _after I had\nused the words antecedent and consequent of a proportion in some of the\nprecedent chapters, I define them afterwards_. I do not believe you say\nthis against your knowledge, but that the eagerness of your malice made\nyou oversee; therefore go back again to the third article of chapter II.\nwhere, having defined correlatives, I add these words, _of which the\nfirst is called the_ antecedent, _the second the_ consequent. This is\nbut an oversight, though such as in me you would not have excused.\n\nAt the thirteenth article you find fault with, that I say _that the\nproportion of inequality, whether it be of excess or of defect, is\nquantity, but the proportion of equality is not quantity_. Whether that\nwhich you say, or that which I say, be the truth, is a question worthy\nof a very strict examination. The first time I heard it argued, was in\nMersennus\u2019 chamber at Paris, at such time as the first volume of his\n_Cogitata Physico-Mathematica_ was almost printed; in which, because he\nhad not said all he would say of proportion, he was forced to put the\nrest into a general preface, which, as was his custom, he did read to\nhis friends before he sent it to the press. In that general preface,\nunder the title _De Rationibus atque Proportionibus_, at the numbers\ntwelve, thirteen, fourteen, he maintaineth against Clavius, _that the\ncomposition of proportion is_ (as of all other things) _a composition of\nthe parts to make a total_, and _that the proportion of equality\nanswereth in quantity to_ non-ens, _or nothing; the proportion of\nexcess, to_ ens, _or quantity; and the proportion of defect, to less\nthan nothing; because equality_ (he says) _is a term of middle\nsignification between excess and defect_. And there also he refuteth the\narguments which Clavius, at the end of the ninth Element of Euclid,\nbringeth to the contrary. And though this were approved by divers good\ngeometricians then present, and never gainsaid by any since, yet do not\nI say it upon the credit of them, but upon sufficient grounds. For it\nhath been demonstrated by Eutocius, that _if there be three magnitudes,\nthe proportion of the first to the third is compounded of the\nproportions of the first to the second, and of the second to the third_;\nwhich also I demonstrate in this article. And if there were never so\nmany magnitudes ranked, it might be likewise demonstrated, that the\nproportion of the first to the last is compounded of the proportions of\nthe first to the second, and of the second to the third, and of the\nthird to the fourth, and so on to the last. If, therefore, we put in\norder any three numbers, whereof the two last be equal, as four, seven,\nseven, the proportion of four the first to seven the last, will be\ncompounded of the proportions of four the first to seven the second, and\nof seven the second to seven the third. Wherefore the proportion of\nseven to seven (which is of equality) addeth nothing to the proportion\nof four the first, to seven the second; and consequently the proportion\nof seven to seven hath no quantity; but that the proportion of\ninequality hath quantity, I prove it from this, that one inequality may\nbe greater than another.\n\nBut for the clearing of this doctrine (which Mersennus calls intricate)\nof the composition of proportions, I observed, first, that any two\nquantities, being exposed to sense, their proportion was also exposed;\nwhich is not intricate. Again, I observed that if besides the two\nexposed quantities, there were exposed a third, so as the first were the\nleast, and the third the greatest, or the first the greatest, and the\nthird the least, that not only the proportions of the first to the\nsecond, but also (because the differences and the quantities proceed the\nsame way) the proportion of the first to the last is exposed by\ncomposition, or addition of the differences; nor is there any intricacy\nin this. But when the first is less than the second, and the second\ngreater than the third, or the first greater than the second, and the\nsecond less than the third, so that to make the first and second equal,\nif we use addition, we must, to make the second and third equal, use\nsubtraction; then comes in the intricacy, which cannot be extricated,\nbut by such as know the truth of this doctrine which I now delivered out\nof Mersennus, namely, that the proportions of excess, equality, and\ndefect, are as _quantity_, _not-quantity_, _nothing want quantity_; or\nas symbolists mark them 0+1 . 0 . 0-1. And upon this ground I thought\ndepended the universal truth of this proposition, that in any rank of\nmagnitudes of the same kind, the proportion of the first to the last,\nwas compounded of all the proportions (in order) of the intermediate\nquantities; the want of the proof thereof, Sir Henry Savile calls\n(_n\u00e6vus_) a mole in the body of geometry. This proposition is\ndemonstrated at the thirteenth article of this chapter.\n\nBut before we come thither, I must examine the arguments you bring to\nconfute this proposition, that the _proportion of inequality is\nquantity, of equality, not quantity_.\n\nAnd first, you object that equality and inequality are in the same\npredicament: a pretty argument to flesh a young scholar in the logic\nschool, that but now begins to learn the predicaments. But what do you\nmean by _\u00e6quale_ and _inequale_? Do you mean _corpus \u00e6quale_, and\n_corpus inequale_? They are both in the predicament of substance,\nneither of them in that of quantity. Or do you mean _\u00e6qualitas_ and\n_in\u00e6qualitas_? They are both in the predicament of relation, neither of\nthem in that of quantity; and yet both _corpus_ and _in\u00e6qualitas_,\nthough neither of them be quantity, may be _quanta_, that is, both of\nthem have quantity. And when men say body is quantity, or inequality is\nquantity, they are no otherwise understood, than if they had said\n_corpus est tantum_, and _in\u00e6qualitas tanta_, not _tantitas_; that is,\nbodies and inequalities are _so much_, not _somuchness_; and all\nintelligent men are contented with that expression, and yourselves use\nit. And the quantity of inequality is in the predicament of quantity,\nbecause the measure of it is in that line by which one quantity exceeds\nthe other. But when neither exceedeth the other, then there is no line\nof excess, or defect by which the equality can be measured, or said to\nbe _so much_, or be called quantity. Philosophy teacheth us how to range\nour words; but Aristotle\u2019s ranging them in his predicaments doth not\nteach philosophy; and therefore no argument taken from thence, can\nbecome a doctor and a professor of geometry.\n\nTo prove that the proportion of inequality was quantity, but the\nproportion of equality not quantity, my argument was this: that _because\none inequality may be greater or less than another, but one equality\ncannot be greater nor less than another: therefore inequality hath\nquantity, or is tanta, and equality not_. Here you come in again with\nyour predicaments, and object, that to be susceptible of _magis_ and\n_minus_, belongs not to quantity, but to quality; but without any proof,\nas if you took it for an axiom. But whether true or false, you\nunderstand not in what sense it is true or false. It is true that one\ninequality is inequality, _as well_ as another; as one heat is heat _as\nwell_ as another, but not _as great_. _Tam_, but not _tantus_. But so it\nis also in the predicament of quantity; one line is as well a line as\nanother, but not so great. All degrees, intentions, and remissions of\nquality, are greater or less quantity of force, and measured by lines,\nsuperficies, or solid quantity, which are properly in the predicament of\nquantity. You see how wise a thing it is to argue from the predicaments\nof Aristotle, which you understand not; and yet you pretend to be less\naddicted to the authority of Aristotle now than heretofore.\n\nIn the next place you say, I may as well conclude from the not\nsusception of _greater_ and _less_, that a right angle is not quantity,\nbut an oblique one is. Very learnedly. As if to be _greater_ or _less_,\ncould be attributed to a quantity once determined. Number (that is,\nnumber indefinitively taken) is susceptible of _greater_ and _less_,\nbecause one number may be greater than another; and this is a good\nargument to prove that number is quantity. And do you think the argument\nthe worse for this, that one six cannot be greater than another six?\nAfter all these childish arguments which you have hitherto urged, can\nyou persuade any man, or yourselves, that you are logicians?\n\nTo the fifth and sixth article you object, first, _that if I had before\nsufficiently defined_ (ratio) _proportion, I needed not again define\nwhat is_ (eadem ratio) _the same proportion_; and ask me _whether when I\nhave defined_ man, _I use to define anew what is the_ same man? You\nthink when you have the definition of _homo_, you have also the\ndefinition of _idem homo_, when it is harder to conceive what _idem_\nsignifies, than what _homo_. Besides, _idem_ hath not the same\nsignification always, and with whatsoever it be joined; it doth not\nsignify the same with _homo_, that it doth with _ratio_. For with _homo_\nit signifies the same _individual man_, but with _ratio_ it signifies a\nlike, or an equal proportion: and both (_ratio_) _proportion_ and\n(_idem_) _the same_, being defined, there will still be need of another\ndefinition for (_eadem ratio_) _the same proportion_; and this is enough\nto defend both myself and Euclid, against this objection: for Euclid\nalso, after he had defined (_ratio_) _proportion_, and that\nsufficiently, as he believed, yet he defines _the same proportion_ again\napart. I know you did not mean in this place to object anything against\nEuclid, but you saw not what you were doing. There is within you some\nspecial cause of intenebration, which you should do well to look to.\n\nIn the next place you say, when I had defined arithmetical proportions\nto be the same when the difference is the same; it was to be expected I\nshould define geometrical proportions to be then the same, when the\nantecedents are of their consequents _totuple_ or _tantuple_, that is,\nequimultiple (for _tantuplum_ signifies nothing). In plain words, you\nexpected, that as I defined one by subtraction, I should define the\nother by the quotient in division. But why should you expect a\ndefinition of the same proportion by the quotient? Neither reason nor\nthe authority of Euclid could move you to expect it. Or why should you\nsay _it was to be expected_? But it seems you have the vanity to place\nthe measure of truth in your own learning. In lines incommensurable\nthere may be the same proportion, when, nevertheless, there is no\nquotient; for setting their symbols one above another doth not make a\nquotient: for quotient there is none, but in _aliquot parts_. It is\ntherefore impossible to define proportion universally, by comparing\nquotients. This incommensurability of magnitudes was it that confounded\nEuclid in the framing of his definition of proportion at the fifth\nElement. For when he came to numbers, he defined the _same proportion_\nirreprehensibly thus: _numbers are then proportional, when the first of\nthe second and the third of the fourth are equimultiple, or the same\npart, or the same parts_; and yet there is in this definition no mention\nat all of a quotient. For though it be true, that if in dividing two\nnumbers you make the same quotient, the dividends and the divisors are\nproportional, yet that is not the definition of the same proportion, but\na theorem demonstrable from it. But this definition Euclid could not\naccommodate to proportion in general, because of incommensurability.\n\nTo supply this want, I thought it necessary to seek out some way,\nwhereby the proportion of two lines, commensurable or incommensurable,\nmight be continued perpetually the same. And this I found might be done\nby the proportion of two lines described by some uniform motion, as by\nan efficient cause both of the said lines, and also of their\nproportions; which motions continuing, the proportions must needs be all\nthe way the same. And therefore I defined those magnitudes to have the\nsame geometrical proportion, _when some cause producing in equal times\nequal effects, did determine both the proportions_. This, you say, needs\nan \u0152dipus to make it understood. You are, I see, no \u0152dipus; but I do not\nsee any difficulty, neither in the definition nor in the demonstration.\nThat which you call perplexity in the explication, is your prejudice,\narising from the symbols in your fancy. For men that pretend no less to\nnatural philosophy than to geometry, to find fault with bringing motion\nand time into a definition, when there is no effect in nature which is\nnot produced in time by motion, is a shame. But you swim upon other\nmen\u2019s bladders in the superficies of geometry, without being able to\nendure diving, which is no fault of mine; and therefore I shall, without\nyour leave, be bold to say, I am the first that hath made the grounds of\ngeometry firm and coherent. Whether I have added anything to the edifice\nor not, I leave to be judged by the readers. You see, you that profess\nwith the pricking of bladders the letting out of their vapour, how much\nyou are deceived. You make them swell more than ever.\n\nFor the corollaries that follow this sixth article, you say they contain\nnothing new. Which is not true. For the ninth is new, and the\ndemonstrations of all the rest are new, being grounded upon a new\ndefinition of proportion; and the corollaries themselves, for want of a\ngood definition of proportion, were never before exactly demonstrated.\nFor the truth of the sixth definition of the fifth Element of Euclid\ncannot be known but by this definition of mine; because it requires a\ntrial in all numbers possible, that is to say, an infinite time of\ntrial, whether the quimultiples of the first and third, and of the\nsecond and fourth, in all multiplications, do together exceed, together\ncome short, and are together equal; which trial is impossible.\n\nIn objecting against the thirteenth and sixteenth article, I observe\nthat you bewray together, both the greatest ignorance and the greatest\nmalice; and it is well, for they are suitable to one another, and fit\nfor one and the same man. In the thirteenth article my proposition is\nthis: _If there be three magnitudes that have proportion one to another,\nthe proportions of the first to the second, and of the second to the\nthird, taken together_ (as one proportion), _are equal to the proportion\nof the first to the third_. This demonstrated, there is taken away one\nof those moles which Sir Henry Savile complaineth of in the body of\ngeometry. Let us see now what you say, both against the enunciation and\nagainst the demonstration.\n\nAgainst the enunciation you object, _that other men would say_ (not the\nproportions of the first to the second, and of the second to the third,\ntaken together, &c. but) _the proportion which is compounded of the\nproportion of the first to the second, and of the second to the third_,\n&c. Is not the compounding of any two things whatsoever the finding of\nthe sum of them both, or the taking of them together as one total? This\nis that absurdity of which Mersennus, in the general preface to his\n_Cogitata Physico-Mathematica_, hath convinced Clavius, who, at the end\nof Euclid\u2019s ninth Element, denieth the composition of proportion to be a\ncomposition of parts to make a total; which, therefore, he denied,\nbecause he did not observe, that the addition of a proportion of defect\nto a proportion of excess, was a subtraction of magnitude; and because\nhe understood not that to say, composition is not the making a whole of\nparts, was contradiction; which all but too learned men would as soon as\nthey heard abhor. Therefore, in saying that other men would not speak in\nthat manner, you say in effect they would speak absurdly. You do well to\nmark what other geometricians say; but you would do better if you could\nby your own meditation upon the things themselves, examine the truth of\nwhat they say. But you have no mind, you say, to contend about the\nphrase. Let us see, therefore, what it is you contend about.\n\n_The proportion_, you say, _which is compounded of double and triple\nproportion, is not_, as I would have it, _quintuple, but sextuple_, as\nin these numbers, six, three, one; where the proportion of six to three\nis double, the proportion of three to one triple, and the proportion of\nsix to one sextuple, not quintuple. Tell me, egregious professors, how\nis six to three double proportion? Is six to three the double of a\nnumber, or the double of some proportion? All men know the number six is\ndouble to the number three, and the number three triple to an unity. But\nis the question here of compounding numbers, or of compounding\nproportions? Euclid, at the last proposition of his ninth Element, says\nindeed, that these numbers, one, two, four, eight, are \u1f10\u03bd \u03b4\u03b9\u03c0\u03bb\u03b1\u03c3\u03af\u03bf\u03bd\u03b9\n\u1f00\u03bd\u03b1\u03bb\u03bf\u03b3\u03af\u03b1, in double proportion; yet there is no man that understands it\notherwise, than if he had said in proportion of the single quantity to\nthe double quantity; and after the same rate, if he had said three,\nnine, twenty-seven, &c. had been in triple proportion, all men would\nhave understood it, of the proportion of any quantity to its triple.\nYour instance, therefore, of six, three, one, is here impertinent, there\nbeing in them no doubling, no tripling, no sextupling of proportions,\nbut of numbers. You may observe also, that Euclid never distinguished\nbetween double and duplicate, as you do. One word \u03b4\u03b9\u03c0\u03bb\u03ac\u03c3\u03b9\u03bf\u03bd serves him\nevery where for either. Though, I confess, some curious grammarians take\n\u03b4\u03b9\u03c0\u03bb\u03ac\u03c3\u03b9\u03bf\u03bd for duplicate in number, and \u03b4\u03b9\u03c0\u03bb\u03bf\u1fe6\u03bd for double in quantity;\nwhich will not serve your turn. Your geometry is not your own, but you\ncase yourselves with Euclid\u2019s; in which, as I have showed you, there be\nsome few great holes; and you by misunderstanding him, as in this place,\nhave made them greater. Though the beasts that think your railing\nroaring, have for a time admired you; yet now that through these holes\nof your case I have showed them your ears, they will be less affrighted.\nBut to exemplify the composition of proportions, take these numbers,\nthirty-two, eight, one, and then you shall see that the proportion of\nthirty-two to one is the sum of the proportions of thirty-two to eight,\nand of eight to one. For the proportion of thirty-two to eight is double\nthe proportion of thirty-two to sixteen; and the proportion of eight to\none, is triple the proportion of thirty-two to sixteen; and the\nproportion of thirty-two to one is quintuple of thirty-two to sixteen;\nbut double and triple added together maketh quintuple. What can be here\ndenied?\n\nMy demonstration consisteth of three cases: the first is when both the\nproportions are of defect, which is then when the first quantity is the\nleast; as in these three quantities, A B, A C, A D. The first case I\ndemonstrated thus: (A B C D)/(a) Let it be supposed that the point A\nwere moved uniformly through the whole line A D. The proportions,\ntherefore, of A B to A C, and of A C to A D, are determined by the\ndifference of the times in which they are described. And the proportion\nalso of A B to A D, is that which is determined by the difference of the\ntimes in which they are described; but the difference of the times in\nwhich A B and A C are described, together with the difference of the\ntimes wherein A C and A D are described, is the same with the difference\nof the times wherein are described A B and A D. The same cause,\ntherefore, which determines both the proportions of A B to A C, and of A\nC to A D, determines also the proportion of A B to A D. Wherefore, by\nthe definition of _the same proportion_, article six, the proportion of\nA B to A C, together with the proportion of A C to A D, is the same with\nthe proportion of A B to A D.\n\nConsider now your argumentation against it. \u201c_Let there be taken_,\u201d say\nyou, \u201c_between A and B the point_ a; and then in your own words, I argue\nthus: _The difference of the times wherein are described A B and A C,\ntogether with the difference of the times wherein are described A C and\nA D, is the same with the difference of the times in which are\ndescribed_ a _B and_ a _C (namely, B D, or B C + C D_); wherefore, the\nsame cause which determines the two proportions of A B to A C, and of A\nC to A D, determines also the proportion of a _B to_ a _D_.\u201d Let me ask\nyou here whether you suppose the motion from _a_ to B, or from _a_ to D,\nto have the same swiftness with the motion from A to B, or from A to D?\nIf you do not, then you deny the supposition. If you do, then B C, which\nis the difference of the times A B and A C, cannot be the difference of\nthe times in which are described _a_ B and _a_ C, except A B and _a_ B\nare equal. Let any man judge now whether there be any paralogism in\nOrontius that can equal this. And whether all that follows in the rest\nof this, and the next two whole pages, be not all a kind of raving upon\nthe ignorance of what is the meaning of proportion, which you also make\nmore ill-favoured by writing it; not in language, but in _gambols_; I\nmean in the symbols, which have made you call those demonstrations\nshort, which put into words so many as a true demonstration requires,\nwould be longer than any of those of Clavius upon the twelfth Element of\nEuclid.\n\nTo the sixteenth article you bring no argument, but fall into a loud\n_oncethmus_ (the special figure wherewith you grace your oratory),\noffended with my unexpected crossing of the doctrine you teach, that\nproportion consisteth in a quotient. For that being denied you, your\n_a/b - c/d + e/f - g/h + i/k_ comes to nothing, that is, to just as much\nas it is worth. But are not you very simple men, to say that all\nmathematicians speak so, when it is not speaking? When did you see any\nman but yourselves publish his demonstrations by signs not generally\nreceived, except it were not with intention to demonstrate, but to teach\nthe use of signs? Had Pappus no analytics? or wanted he the wit to\nshorten his reckoning by signs? Or has he not proceeded analytically in\na hundred problems (especially in his seventh book), and never used\nsymbols? Symbols are poor unhandsome, though necessary, scaffolds of\ndemonstration; and ought no more to appear in public, than the most\ndeformed necessary business which you do in your chambers. \u201c_But why_,\u201d\nsay you, \u201c_is this limitation to the proportion of the greater to the\nless?_\u201d I will tell you; because iterating of the proportion of the less\nto the greater, is a making of the proportion less, and the defect\ngreater. And it is absurd to say that the taking of the same quantity\ntwice should make it less. And thence it is, that in quantities which\nbegin with the less, as one, two, four, the proportion of one to two is\ngreater than that of one to four, as is demonstrated by Euclid, Elem. 5,\nprop. 8; and by consequent the proportion of one to four, is a\nproportion of greater littleness than that of one to two. And who is\nthere, that when he knoweth that the respective greatness of four to\none, is double to that of the respective greatness of four to two, or of\ntwo to one, will not presently acknowledge that the respective greatness\nof one to two, or two to four, is double to the respective greatness of\none to four? But this was too deep for such men as take their opinions,\nnot from weighing, but from reading.\n\nLastly you object against the corollary of art. 28; which you make\nabsurd enough by rehearsing it thus: _Si quantitas aliqua divisa\nsupponatur in partes aliquot \u00e6quales numero infinitas_, &c. Do you think\nthat of _partes aliquot_, or of _partes aliquot\u00e6_, it can be said\nwithout absurdity, that they are _numero infinit\u00e6_? And then you say I\nseem to mean, that if of the quantity A B, there be supposed a part C B,\ninfinitely little; and that between A C and A B be taken two means, one\narithmetical, A E, the other geometrical, A D, the difference between A\nD and A E, will be infinitely little. My meaning is, and is sufficiently\nexpressed, that the said means taken everywhere (not in one place only)\nwill be the same throughout: and you that say there needed not so much\npains to prove it, and think you do it shorter, prove it not at all. For\nwhy may not I pretend against your demonstration, that B E, the\narithmetical difference, is greater than B D, the geometrical\ndifference. You bring nothing to prove it; and if you suppose it, you\nsuppose the thing you are to prove. Hitherto you have proceeded in such\nmanner with your _Elenchus_, as that so many objections as you have\nmade, so many false propositions you have advanced. Which is a peculiar\nexcellence of yours, that for so great a stipend as you receive, you\nwill give place to no man living for the number and grossness of errors\nyou teach your scholars.\n\nAt the fourteenth chapter your first exception is to the second article;\nwhere I define a plane in this manner: _A plane superficies is that\nwhich is described by a straight line so moved, as that every point\nthereof describe a several straight line_. In which you require, first,\nthat instead of _describe_, I should have said _can describe_. Why do\nyou not require of Euclid, in the definition of a cone, instead of\n_continetur_, _is contained_, he say _contineri potest_, _can be\ncontained_ ? If I tell you how one plane is generated, cannot you apply\nthe same generation to any other plane? But you object, that the plane\nof a circle may be generated by the motion of the _radius_, whose every\npoint describeth, not a straight, but a crooked line, wherein you are\ndeceived; for you cannot draw a circle (though you can draw the\nperimeter of a circle) but in a plane already generated. For the motion\nof a straight line, whose one point resting, describeth with the other\npoints several perimeters of circles, may as well describe a conic\nsuperficies, as a plane. The question, therefore, is, how you will, in\nyour definition, take in the plane which must be generated before you\nbegin to describe your circle, and before you know what point to make\nyour centre. This objection, therefore, is to no purpose; and besides,\nthat it reflecteth upon the perfect definitions of Euclid before the\neleventh Element, it cannot make good his definition (which is nothing\nworth) of a plane superficies, before his first Element.\n\nIn the next place, you reprehend briefly this _corollary, that two\nplanes cannot enclose a solid_. I should, indeed, have added, _with the\nbase on whose extremes they insist_: but this is not a fault to be\nashamed of; for any man, by his own understanding, might have mended my\nexpression without departing from my meaning. But from your doctrine,\n_that a superficies has no thickness_, it is impossible to include a\nsolid, with any number of planes whatsoever, unless you say that solid\nis included which nothing at all includes.\n\nAt the third article, where I say _of crooked lines, some are everywhere\ncrooked, and some have parts not crooked_. You ask me what crooked line\nhas parts not crooked; and I answer, it is that line which with a\nstraight line makes a rectilineal triangle. But this, you say, cannot\nstand with what I said before, namely, that a straight and crooked line\ncannot be coincident; which is true, nor is there any contradiction; for\nthat part of a crooked line which is straight, may with a straight line\nbe coincident.\n\nTo the fourth article, where I define _the centre of a circle to be that\npoint of the radius, which in the description of the circle is unmoved_;\nyou object as a contradiction, that I had before defined a point to be\nthe body which is moved in the description of a line: foolishly, as I\nhave already shown at your objection to Chap. VIII. art. 12.\n\nBut at the sixth article, where I say, that _crooked and incongruous\nlines touch one another but in one point_, you make a cavil from this,\nthat _a circle may touch a parabola in two points_. Tell me truly, did\nyou read and understand these words that followed? \u201c_A crooked line\ncannot be congruent with a straight line; because if it could, one and\nthe same line should be both straight and crooked._\u201d If you did, you\ncould not but understand the sense of my words to be this: _when two\ncrooked lines which are incongruous, or a crooked and a straight line\ntouch one another, the contact is not in a line, but only in one point_;\nand then your instance of a circle and a parabola was a wilful cavil,\nnot befitting a doctor. If you either read them not, or understood them\nnot, it is your own fault. In the rest that followeth upon this article,\nwith your diagram, there is nothing against me, nor anything of use,\nnovelty, subtlety, or learning.\n\nAt the seventh article, where I define both an _angle_, simply so\ncalled, and an _angle of contingence_, by their several generations;\nnamely, that the former is generated _when two straight lines are\ncoincident, and one of them is moved, and distracted from the other by\ncircular motion upon one common point resting, &c._; you ask me \u201c_to\nwhich of these kinds of angle I refer the angle made by a straight line\nwhen it cuts a crooked line_?\u201d I answer easily and truly, To that kind\nof angle which is called simply an angle. This you understand not. \u201cFor\nhow\u201d, will you say, \u201ccan that angle which is generated by the divergence\nof two straight lines, be other than rectilineal? or how can that angle\nwhich is not comprehended by two straight lines, be other than\ncurvilineal?\u201d I see what it is that troubles you; namely, the same which\nmade you say before, that if the body which describes a line be a point,\nthen there is nothing which is not moved that can be called a point. So\nyou say here, \u201cIf an angle be generated by the motion of a straight\nline, then no angle so generated can be curvilineal;\u201d which is as well\nargued, as if a man should say, the house was built by the carriage and\nmotion of stone and timber, therefore, when the carriage and that motion\nis ended, it is no more a house. Rectilineal and curvilineal hath\nnothing to do with the nature of an angle simply so called, though it be\nessential to an angle of contact. The measure of an angle, simply so\ncalled, is a circumference of a circle; and the measure is always the\nsame kind of quantity with the thing measured. The rectitude or curvity\nof the lines, which drawn from the centre, intercept the arch, is\naccidentary to the angle, which is the same, whether it be drawn by the\nmotion circular of a straight line or of a crooked. The diameter and the\ncircumference of a circle make a right angle, and the same which is made\nby the diameter and the tangent. And because the point of contact is\nnot, as you think, nothing, but a line unreckoned, and common both to\nthe tangent and the circumference; the same angle computed in the\ntangent is rectilineal, but computed in the circumference, not\nrectilineal, but mixed: or, if two circles cut one another, curvilineal.\nFor every chord maketh the same angle with the circumference which it\nmaketh with the line that toucheth the circumference at the end of the\nchord. And, therefore, when I divide an angle, simply so called, into\nrectilineal and curvilineal, I respect no more the generation of it,\nthan when I divide it into right and oblique. I then respect the\ngeneration, when I divide an angle into an angle simply so called, and\nan angle of contact. This that I have now said, if the reader remember\nwhen he reads your objections to this, and to the ninth article, he will\nneed no more to make him see that you are utterly ignorant of the nature\nof an angle; and that if ignorance be madness, not I, but you, are mad:\nand when an angle is comprehended between a straight and a crooked line\n(if I may compute the same angle as comprehended between the same\nstraight line and the point of contact), that it is consonant to my\ndefinition of a point by a _magnitude not considered_. But when you, in\nyour treatise, _De Angulo Contactus_ (chap. III. p. 6, l. 8) have these\nwords: \u201c_Though the whole concurrent lines incline to one another, yet\nthey form no angle anywhere but in the very point of concourse_:\u201d you,\nthat deny a point to be anything, tell me how two nothings can form an\nangle; or if the angle be not formed, neither before the concurrent\nlines meet, nor in the point of concourse, how can you apprehend that\nany angle can possibly be framed? But I wonder not at this absurdity;\nbecause this whole treatise of yours is but one absurdity, continued\nfrom the beginning to the end, as shall then appear when I come to\nanswer your objections to that which I have briefly and fully said of\nthat subject in my 14th chapter.\n\nAt the twelfth article, I confess your exception to my universal\ndefinition of parallels to be just, though insolently set down. For it\nis no fault of ignorance (though it also infect the demonstration next\nit), but of too much security. The definition is this: _Parallels are\nthose lines or superficies, upon which two straight lines falling, and\nwheresoever they fall, making equal angles with them both, are equal_;\nwhich is not, as it stands, universally true. But inserting these words\n_the same way_, and making it stand thus: _parallel lines or\nsuperficies, are those upon which two straight lines falling the same\nway, and wheresoever they fall, making equal angles, are equal_, it is\nboth true and universal; and the following consectary, with very little\nchange, as you may see in the translation, perspicuously demonstrated.\nThe same fault occurreth once or twice more; and you triumph\nunreasonably, as if you had given therein a very great proof of your\ngeometry.\n\nThe same was observed also upon this place by one of the prime\ngeometricians of Paris, and noted in a letter to his friend in these\nwords (Chap. XIV. art. 12): \u201c_The definition of parallels wanteth\nsomewhat to be supplied_.\u201d And of the consectary he says, \u201c_It\nconcludeth not, because it is grounded on the definition of parallels_.\u201d\nTruly and severely enough, though without any such words as savour of\narrogance, or of malice, or of the clown.\n\nAt the thirteenth article you recite the demonstration by which I prove\nthe perimeters of two circles to be proportional to their semidiameters;\nand with _esto_, _fortasse_, _recte_, _omnino_, noddying to the several\nparts thereof, you come at length to my last inference: _Therefore, by_\nChap. XIII. art. 6, _the perimeters and semidiameters of circles are\nproportional_; which you deny; and therefore deny, because you say it\nfolloweth by the same ratiocination, that _circles also and spheres are\nproportional to their semidiameters_. \u201c_For the same distance_, you say,\n_of the perimeter from the centre which determines the magnitude of the\nsemidiameter, determines also the magnitude both of the circle and of\nthe sphere_.\u201d You acknowledge that perimeters and semidiameters have the\ncause of their determination such as in equal times make equal spaces.\nSuppose now a sphere generated by the semidiameters, whilst the\nsemicircle is turned about. There is but one _radius_ of an infinite\nnumber of _radii_, which describes a great circle; all the rest describe\nlesser circles parallel to it, in one and the same time of revolution.\nWould you have men believe, that describing greater and lesser circles,\nis according to the supposition (_temporibus \u00e6qualibus \u00e6qualia facere_)\nto make equal spaces in equal times? Or, when by the turning about of\nthe semidiameter is described the plane of a circle, does it, think you,\nin equal times make the planes of the interior circles equal to the\nplanes of the exterior? Or is the _radius_ that describes the inner\ncircles equal to the _radius_ that describes the exterior? It does not,\ntherefore, follow from anything I have said in this demonstration, that\neither spheres or planes of circles, are proportional to their _radii_;\nand consequently, all that you have said, triumphing in your own\nincapacity, is said imprudently by yourselves to your own disgrace. They\nthat have applauded you, have reason by this time to doubt of all the\nrest that follows, and if they can, to dissemble the opinion they had\nbefore of your geometry. But they shall see before I have done, that not\nonly your whole _Elenchus_ , but also your other books of the _Angle of\nContact_ , &c. are mere ignorance and gibberish.\n\nTo the fourteenth article you object, that (in the sixth figure) I\nassume gratis, that F G, D E, B C, are proportional to A F, A D, A B;\nand you refer it to be judged by the reader: and to the reader I refer\nit also. The not exact drawing of the figure (which is now amended) is\nit that deceived you. For A F, F D, D B, are equal by construction.\nAlso, A G, G E, E C, are equal by construction. And F G, D K, B H, K E,\nH I, I C, are equal by parallelism. And because A F, F G, are as the\nvelocities wherewith they are described; also 2 A F (that is A D) and 2\nF G (that is D E) are as the same velocities. And finally, 3 A F (that\nis A B) and 3 F G (that is B C) are as the same velocities. It is not\ntherefore assumed gratis, that F G, D E, B C are proportional to A F, A\nD, A B, but grounded upon the sixth article of the thirteenth chapter;\nand consequently your objection is nothing worth. You might better have\nexcepted to the placing of D E, first at adventure, and then making A D\ntwo-thirds of A B; for that was a fault, though not great enough to\ntrouble a candid reader; yet great enough to be a ground, to a malicious\nreader, of a cavil.\n\nThat which you object to the third _corollary_ of art. 15, was certainly\na dream. There is no assuming of an angle C D E, for an angle H D E, or\nB D E, neither in the demonstration, nor in any of the corollaries. It\nmay be you dreamt of somewhat in the twentieth article of chapter XVI.\nBut because that article, though once printed, was afterwards left out,\nas not serving to the use I had designed it for, I cannot guess what it\nis: for I have no copy of that article, neither printed nor written; but\nam very sure, though it were not useful, it was true.\n\nArticle the sixteenth. Here we come to the controversy concerning the\n_angle of contact, which_, you say, _you have handled, in a special\ntreatise published; and that you have clearly demonstrated, in your\npublic lectures, that Peletarius was in the right. But that I agree not\nsufficiently, neither with Peletarius nor with Clavius._ I confess I\nagree not in all points with Peletarius, nor in all points with Clavius.\nIt does not thence follow that I agree not with the truth. I am not, as\nyou, of any faction, neither in geometry nor in politics. If I think\nthat you, or Peletarius, or Clavius, or Euclid, have erred, or been too\nobscure, I see no cause for which I ought to dissemble it. And in this\nsame question I am of opinion that Peletarius did not well in denying\nthe _angle of contingence_ to be _an angle_. And that Clavius did not\nwell to say, _the angle of a semicircle_ was less than _a right-lined\nright angle_. And that Euclid did not well to leave it so obscure what\nhe meant by _inclination_ in the definition of a _plane angle_, seeing\nelsewhere he attributeth inclination only to acute angles; and scarce\nany man ever acknowledged inclination in a straight line, to any other\nline to which it was perpendicular. But you, in this question of what is\ninclination, though you pretend not to depart from Euclid, are,\nnevertheless, more obscure than he; and also are contrary to him. For\nEuclid by inclination meaneth the inclination of one line _to_ another;\nand you understand it of the inclination of one line _from_ another;\nwhich is not inclination, but declination. For you make two straight\nlines, when they lie one on another, to lie \u1f01\u03ba\u03bb\u03b9\u03bd\u1ff6\u03c2, that is, without\nany inclination (because it serves your turn); not observing that it\nfolloweth thence, that inclination is a digression of one line _from_\nanother. This is in your first argument in the behalf of Peletarius (p.\n10, l. 22), and destroys his opinion. For, according to Euclid, the\ngreatest angle is the greatest inclination; and an angle equal to two\nright angles by this \u1f00\u03ba\u03bb\u03b9\u03c3\u03af\u03b1, should not be the greatest inclination, as\nit is, but the least that can be. But if by the inclination of two\nlines, we understand that proceeding of them to a common point, which is\ncaused by their generation, which, I believe, was Euclid\u2019s meaning; then\nwill the _angle of contact_ be no less an _angle_ than a _rectilineal_\nangle, but only (as Clavius truly says it is) heterogeneous to it; and\nthe doctrine of Clavius more conformable to Euclid than that of\nPeletarius. Besides, if it be granted you, that there is no inclination\nof the circumference to the tangent, yet it does not follow that their\nconcourse doth not form some kind of angle; for Euclid defineth there\nbut one of the kinds of a plane angle. And then you may as much in vain\nseek for the proportion of such angle to the angle of contact, as seek\nfor the _focus_ or _parameter of the parabola of Dives and Lazarus_.\nYour first argument therefore is nothing worth, except you make good\nthat which in your second argument you affirm, namely, that all plane\nangles, not excepting the angle of contact, are (_homogeneous_) of the\nsame kind. You prove it well enough of other curvilineal angles; but\nwhen you should prove the same of an angle of contact, you have nothing\nto say but (p. 17, l. 15), \u201c_Unde autem illa quam somniet heterogenia\noriatur, neque potest ille ullatenus ostendere, neque ego vel\nsomniare_:\u201d \u201c_Whence should arise that diversity of kind which he dreams\nof, neither can he at all show, nor I dream_;\u201d as if you knew what he\ncould do if he were to answer you; or all were false which you cannot\ndream of. So that besides your customary vanity, here is nothing\nhitherto proved, neither for the opinion of Peletarius, nor against that\nof Clavius. I have, I think, sufficiently explicated, in the first\nlesson, that the angle of contact is quantity, namely, that it is the\nquantity of that crookedness or flexion, by which a straight line is\nbent into an arch of a circle equal to it; and that because the\ncrookedness of one arch may be greater than the crookedness of another\narch of another circle equal to it; therefore the question _quanta est\ncurvitas_, how much is the crookedness, is pertinent, and to be answered\nby _quantity_. And I have also shown you in the same lesson, that the\nquantity of one angle of contact is compared with that of another angle\nof contact by a line drawn from the point of contact, and intercepted by\ntheir circumferences; and that it cannot be compared by any measure with\na rectilineal angle.\n\n[Illustration]\n\nBut let us see how you answer to that which Clavius has objected\nalready. \u201c_They are heterogeneous_,\u201d says he, \u201c_because the angle of\ncontact, how oft soever multiplied, can never exceed a rectilineal\nangle_.\u201d To answer which, you allege _it is no angle at all; and that\ntherefore, it is no angle at all, because the lines have no inclination\none to another_. How can lines that have no inclination one to another,\never come together? But you answer, _at least they have no inclination\nin the point of contact_. And why have two straight lines inclination\nbefore they come to touch, more than a straight line and an arch of a\ncircle? And in the point of contact itself, how can it be that there is\nless inclination of the two points of a straight line and an arch of a\ncircle, than of the points of two straight lines? But the straight\nlines, you say, will cut; which is nothing to the question; and yet this\nalso is not so evident, but that it may receive an objection. Suppose\ntwo circles, A G B and C F B, to touch in B, and have a common tangent\nthrough B. Is not the line C F B G A a crooked line? and is it not cut\nby the common tangent D B E? What is the quantity of the two angles F B\nE and G B D, seeing you say neither D B G nor E B F is an angle? It is\nnot, therefore, the cutting of a crooked line, and the touching of it,\nthat distinguisheth an angle simply, from an angle of contact. That\nwhich makes them differ, and in kind, is, that the one is the quantity\nof a _revolution_, and the other, the quantity of _flexion_.\n\nIn the seventh chapter of the same treatise, you think you prove the\nangle of contact, if it be an angle, and a rectilineal angle to be\n(_homogeneous_) of the same kind; when you prove nothing but that you\nunderstand not what you say. Those quantities which can be added\ntogether, or subtracted one from another, are of the same kind; but an\nangle of contact may be subtracted from a right angle, and the remainder\nwill be the angle of a semicircle, &c. So you say, but prove it not,\nunless you think a man must grant you that the superficies contained\nbetween the tangent and the arch, which is it you subtract, is the angle\nof contact; and that the plane of the semicircle is the angle of the\nsemicircle, which is absurd; though, as absurd as it is, you say it\ndirectly in your _Elenchus_ , p. 35, l. 14, in these words: \u201c_When\nEuclid defines a plane angle to be the inclination of two lines, he\nmeaneth not their aggregate, but that which lies between them_.\u201d It is\ntrue, he meaneth not the aggregate of the two lines; but that he means\nthat which lies between them, which is nothing else but an indeterminate\nsuperficies, is false, or Euclid was as foolish a geometrician as either\nof you two.\n\nAgain, you would prove the angle of contact, if it be an angle, to be of\nthe same kind with a rectilineal angle, out of Euclid (III. 16); where\nhe says, _it is less than any acute angle_. And it follows well, that if\nit be an angle, and less than any rectilineal angle, it is also of the\nsame kind with it. But, to my understanding, Euclid meant no more, but\nthat it was neither greater nor equal; which is as truly said of\nheterogeneous, as of homogeneous quantities. If he meant otherwise, he\nconfirms the opinion of Clavius against you, or makes the quantity of an\nangle to be a superficies, and indefinite. But I wonder how you dare\nventure to determine whether two quantities be homogeneous or not,\nwithout some definition of homogeneous (which is a hard word), that men\nmay understand what it meaneth.\n\nIn your eighth chapter you have nothing but Sir H. Savile\u2019s authority,\nwho had not then resolved what to hold; but esteeming the angle of\ncontact, first, as others falsely did, by the superficies that lies\nbetween the tangent and the arch, makes the angle of contact and a\nrectilineal angle homogeneous; and afterwards, because no multiplication\nof the angle of contact can make it equal to the least rectilineal\nangle, with great ingenuity returneth to his former uncertainty.\n\nIn your ninth and tenth chapters you prove with much ado, that the\nangles of like segments are equal; as if that might not have been taken\ngratis by Peletarius, without demonstration. And yet your argument,\ncontained in the ninth chapter, is not a demonstration, but a\nconjectural discourse upon the word _similitude_. And in the eleventh\nchapter, wherein you answer to an objection, which might be made to your\nargument in the precedent page, taken from the parallelism of two\nconcentric circles, though objection be of no moment, yet you have in\nthe same treatise of yours that which is much more foolish, which is\nthis, (p. 38, l. 12): \u201c_Non enim magnitudo anguli_,\u201d _&c._ _\u201c_The_\nmagnitude of an angle is not to be estimated by that straddling of the\nlegs, which it hath without the point of concourse, but by that\nstraddling which it hath in the point of the concourse itself._\u201d I pray\nyou tell me what straddling there is of two coincident points,\nespecially such points as you say are nothing? When did you ever see two\nnothings straddle?\n\nThe arguments in your twelfth and thirteenth chapters are grounded all\non this untruth, that an angle is that which is contained between the\nlines that make it; that is to say, is a plane superficies, which is\nmanifestly false; because the measure of an angle is an arch of a\ncircle, that is to say, a line; which is no measure of a superficies.\nBesides this gross ignorance, your way of demonstration, by putting N\nfor a great number of sides of an equilateral polygon, is not to be\nadmitted; for, though you understand something by it, you demonstrate\nnothing to anybody but those who understand your symbolic tongue, which\nis a very narrow language. If you had demonstrated it in Irish or Welsh,\nthough I had not read it, yet I should not have blamed you, because you\nhad written to a considerable number of mankind, which now you do not.\n\nIn your last chapters you defend Vitellio without need; for there is no\ndoubt but that whatsoever crooked line be touched by a straight line,\nthe angle of contingence will neither add anything to, nor take anything\nfrom, a rectilineal right angle; but that it is because the angle of\ncontact is no angle, or no quantity, is not true. For it is therefore an\nangle, because an angle of contact; and therefore quantity, because one\nangle of contact may be greater than another; and therefore\nheterogeneal, because the measure of an angle of contact cannot\n(_congruere_) be applied to the measure of a rectilineal angle, as they\nthink it may, who affirm with you that the nature of an angle consisteth\nin that which is contained between the lines that comprehend it, viz.,\nin a plane superficies. And thus you see in how few lines, and without\nbrachygraphy, your treatise of the angle of contingence is discovered\nfor the greatest part to be false, and for the rest, nothing but a\ndetection of some errors of Clavius grounded on the same false\nprinciples with your own. To return now from your treatise of the angle\nof contact back again to your _Elenchus_ .\n\nThe fault you find at art. 18, is, that I understand not that Euclid\nmakes a _plane angle_ to be that which is contained between the two\nlines that form it. It is true, that I do not understand that Euclid was\nso absurd, as to think the nature of an angle to consist in superficies;\nbut I understand that you have not had the wit to understand Euclid.\n\nThe nineteenth article of mine in this fourteenth chapter, is this:\n\u201c_All respect or variety of position of two lines, seemeth to be\ncomprehended in four kinds_. For they are either _parallel_, or (_being\nif need be produced_) _make an angle_; or, (if drawn out far enough)\n_touch_; or, lastly, they are _asymptotes_\u201d; in which you are first\noffended with the word _It seems_. But I allow you, that never err, to\nbe more peremptory than I am. For to me it seemed (I say again seemed)\nthat such a phrase, in case I should leave out something in the\nenumeration of the several kinds of position, would save me from being\ncensured for untruth; and yet your instance of two straight lines in\ndivers planes, does not make my enumeration insufficient. For those\nlines, though not parallels, nor cutting both the planes, yet being\nmoved parallelly from one plane to another, will fall into one or other\nof the kinds of position by me enumerated; and consequently, are as much\nthat position, as two straight lines in the same plane, not parallel,\nmake the same angle, though not produced till they meet, which they\nwould make if they were so produced: for you have nowhere proved, nor\ncan prove, that two such lines do not make an angle. It is not the\nactual concurrence of the lines, but the arch of a circle, drawn upon\nthat point for centre, in which they would meet if they were produced,\nand intercepted between them, that constitutes the angle.\n\nAlso your objection concerning asymptotes _in general_ is absurd. You\nwould have me add, that _their distance shall at last be less than any\ndistance that can be assigned_; and so make the definition of the\n_genus_ the same with that of the _species_. But because you are not\nprofessors of logic, it is not necessary for me to follow your counsel.\nIn like manner, if we understand one line to be moved towards another\nalways parallelly to itself, which is, though not actually, yet\npotentially the same position, all the rest of your instances will come\nto nothing.\n\nAt the two-and-twentieth article you object to me the use of the word\n_figure_, before I had defined it: wherein also you do absurdly; for I\nhave nowhere before made such use of the word _figure_, as to argue\nanything from it; and therefore your objection is just as wise as if you\nhad found fault with putting the word figure in the titles of the\nchapters placed before the book. If you had known the nature of\ndemonstration, you had not objected this.\n\nYou add further, that by my definition of _figure_, a solid sphere, and\na sphere made hollow within, is the same figure; but you say not why,\nnor can you derive any such thing from my definition. That which\ndeceived your shallowness, is, that you take those points that are in\nthe concave superficies of a hollowed sphere, not to be contiguous to\nanything without it, because that whole concave superficies is within\nthe whole sphere. Lastly, for the fault you find with the definition of\n_like figures in like positions_, I confess there wants the same word\nwhich was wanting in the definition of parallels; namely, _ad easdem\npartes_ (_the same way_) which should have been added in the end of the\ndefinition of like figures, &c., and may easily be supplied by any\nstudent of geometry, that is not otherwise a fool.\n\nAt the fifteenth chapter, art. 1, number 6, you object as a\ncontradiction, that _I make motion to be the measure of time; and yet,\nin other places, do usually measure motion and the affections thereof by\ntime_. If your thoughts were your own, and not taken rashly out of\nbooks, you could not but, (with all men else that see time measured by\nclocks, dials, hour-glasses, and the like), have conceived sufficiently,\nthat there cannot be of time any other measure besides motion; and that\nthe most universal measure of motion, is a line described by some other\nmotion; which line being once exposed to sense, and the motion whereby\nit was described sufficiently explicated, will serve to measure all\nother motions and their time: for time and motion (time being but the\nmental image or remembrance of the motion) have but one and the same\ndimension, which is a line. But you, that would have me measure\n_swiftness_ and _slowness_ by longer and shorter motion, what do you\nmean by _longer_ and _shorter motion_? Is _longer_ and _shorter_ in the\nmotion, or in the duration of the motion, which is time? Or is the\nmotion, or the duration of the motion, that which is exposed, or\ndesigned by a line? Geometricians say often, _let the line A B be the\ntime_; but never say, _let the line A B be the motion_. There is no\nunlearned man that understandeth not what is time, and motion, and\nmeasure; and yet you, that undertake to teach it (most egregious\nprofessors) understand it not.\n\nAt the second article you bring another argument (which it seems in its\nproper place you had forgotten), to prove that a point is not quantity\nnot considered, but absolutely nothing; which is this, _That if a point\nbe not nothing, then the whole is greater than its two halves_. How does\nthat follow? Is it impossible when a line is divided into two halves,\nthat the middle point should be divided into two halves also, being\nquantity?\n\nAt the seventh article, I have sufficiently demonstrated, that all\nmotion is infinitely propagated, as far as space is filled with body.\nYou allege no fault in the demonstration, but object from sense, _that\nthe skipping of a flea is not propagated to the Indies_. If I ask you\nhow you know it, you may wonder perhaps, but answer you cannot. Are you\nphilosophers, or geometricians, or logicians, more than are the simplest\nof rural people? or are you not rather less, by as much as he that\nstandeth still in ignorance, is nearer to knowledge, than he that\nrunneth from it by erroneous learning?\n\nAnd, lastly, what an absurd objection is it which you make to the eighth\narticle, where I say that _when two bodies of equal magnitude fall upon\na third body, that which falls with greater velocity, imprints the\ngreater motion_? You object, _that not so much the magnitude is to be\nconsidered as the weight_; as if the weight made no difference in the\nvelocity, when notwithstanding weight is nothing else but motion\ndownward. Tell me, when a weighty body thrown upwards worketh on the\nbody it meeteth with, do you not then think it worketh the more for the\ngreatness, and the less for the weight.\n\n\n ==========\n\n\n OF THE FAULTS THAT OCCUR IN\n DEMONSTRATION.\n\n TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN\n THE UNIVERSITY OF OXFORD.\n\n\n LESSON IV.\n\nOf twenty articles which you say (of nineteen which I say) make the\nsixteenth chapter, you except but three, and confidently affirm the rest\nare false. On the contrary, except three or four faults, such as any\ngeometrician may see proceed not from ignorance of the subject, or from\nwant of the art of demonstration, (and such as any man might have mended\nof himself) but from security; I affirm that they are all true, and\ntruly demonstrated; and that all your objections proceed from mere\nignorance of the mathematics.\n\nThe first fault you find is this, that I express not (art. 1) what\n_impetus_ it is, which I would have to be multiplied into the time.\n\nThe last article of my thirteenth chapter was this, \u201c_If there be a\nnumber of quantities propounded, howsoever equal or unequal to one\nanother; and there be another quantity which so often taken as there be\nquantities propounded, is equal to their whole sum; that quantity I call\nthe mean arithmetical of them all_.\u201d Which definition I did there insert\nto serve me in the explication of those propositions of which the\nsixteenth chapter consisted, but did not use it here as I intended. My\nfirst proposition therefore as it standeth yet in the Latin, being this,\n\u201c_the velocity of any body moved during any time, is so much as is the\nproduct of the impetus in one point of time, multiplied into the whole\ntime_;\u201d to a man that hath not skill enough to supply what is wanting,\nis not intelligible. Therefore I have caused it in the English to go\nthus: \u201c_the velocity of any body in whatsoever time moved, hath its\nquantity determined by the sum of all the several_ (impetus)\n_quicknesses, which it hath in the several points of the time of the\nbody\u2019s motion_. And added, _that all the_ impetus _together taken\nthrough the whole time is the same thing with the mean_ impetus (which\nmean is defined (Chapter XIII. art. 29) _multiplied into the whole\ntime_.\u201d To this first article, as it is uncorrected in the Latin, you\nobject, _that meaning by_ impetus _some middle_ impetus, _and assigning\nnone, I determine nothing_. And it is true. But if you had been\ngeometricians sufficient to be professors, you would have shewed your\nskill much better, by making it appear that this middle _impetus_ could\nbe none but that, which being taken so often, as there be points in the\nline of time, would be equal to the sum of all the several _impetus_\ntaken in the points of time respectively; which you could not do.\n\nTo the _corollary_, you ask first how _impetus_ can be ordinately\napplied to a line; absurdly. For does not Archimedes sometimes say, and\nwith him many other excellent geometricians, _let such a line be the\ntime_? And do they not mean, that that line, or the motion over it, is\nthe measure of the time? And may not also a line serve to measure the\nswiftness of a motion? _You thought_, you say, _only lines ought to be\nsaid to be ordinately applied to lines_. Which I easily believe; for I\nsee you understand not that a line, though it be not the time itself,\nmay be the quantity of a time. You thought also, all you have said in\nyour _Elenchus_ , in your doctrine of the _angle of contact_, in your\n_Arithmetica Infinitorum_, and in your _Conics_ , is true; and yet it is\nalmost all proved false, and the rest nothing worth.\n\nSecondly, you object, that _I design a parallelogram by one only side_.\nIt was indeed a great oversight, and argueth somewhat against the man,\nbut nothing against his art. For he is not worthy to be thought a\ngeometrician that cannot supply such a fault as that, and correct his\nbook himself. Though you could not do it, yet another from beyond sea\ntook notice of the same fault in this manner, \u201c_He maketh a\nparallelogram of but one side_; it should be thus: _vel denique per\nparallelogrammum cujus unum latus est medium proportionale inter impetum\nmaximum (sive ultimo acquisitum) et impetus ejusdem maximi semissem;\nalterum vero latus, medium proportionale, inter totum tempus, et ejusdem\ntotius temporis semissem_.\u201d Which I therefore repeat, that you may learn\ngood manners; and know, that they who reprehend, ought also, when they\ncan, to add to their reprehension the correction.\n\nAt the second article, you are pleased to advise me, instead of _in omni\nmotu uniformi_, to put in _in omnibus motibus uniformibus_. You have a\nstrange opinion of your own judgment, to think you know to what end\nanother man useth any word, better than himself. My intention was only\nto consider motions uniform, and motions from rest uniformly, or\nregularly accelerated, that I might thereby compute the lengths of\ncrooked lines, such as are described by any of those motions. And\ntherefore it was enough to prove this theorem to be true in all uniform\nor uniformly accelerated _motion_, not _motions_; though it be true also\nin the plural. It seems you think a man must write all he knows, whether\nit conduce, or not, to his intended purpose. But that you may know that\nI was not (as you think), ignorant how far it might be extended, you may\nread it demonstrated at the same article in the English universally.\nAgainst the demonstration itself you run into another article, namely,\nthe thirteenth, which is this problem: \u201c_the length being given, which\nis passed over in a given time by uniform motion, to find the length\nwhich shall be passed over by motion uniformly accelerated in the same\ntime, so as that the_ impetus _last acquired be equal to the time_.\u201d\nWhich you recite imperfectly, thereby to make it seem that such a length\nis not determined. Whether you did this out of ignorance, or on purpose,\nthinking it a piece of wit, as your pretended mystery which goes\nimmediately before, I cannot tell, for in neither place can any wit be\nespied by any but yourselves. To imagine motions with their times and\nways, is a new business, and requires a steady brain, and a man that can\nconstantly read in his own thoughts, without being diverted by the noise\nof words. The want of this ability, made you stumble and fall\nunhandsomely in the very first place (that is in Chap. XIII. art. 13),\nwhere you venture to reckon both motion and time at once; and hath made\nyou in this chapter to stumble in the like manner at every step you go.\nAs, for example, when I say, _as the product of the time, and impetus,\nto the product of the time and impetus, so the space to the space when\nthe motion is uniform_; you come in with, _nay, rather as the time to\nthe time_; as if the parallelograms A I, and A H, were not also as the\ntimes A B, and A F. Thus it is, when men venture upon ways they never\nhad been in before, without a guide.\n\nIn the corollary, you are offended with the permutation of the\nproportion of times and lines, because you think, (you that have scarce\none right thought of the principles of geometry), that line and time are\nheterogeneous quantities. I know time and line are of divers natures;\nand more, that neither of them is _quantity_. Yet they may be both of\nthem _quanta_, that is, they may _have quantity_; but that their\nquantities are heterogeneous is false. For they are compared and\nmeasured both of them by straight lines. And to this there is nothing\ncontrary in the place cited by you out of Clavius; or if there were, it\nwere not to be valued. And to your question, what is the proportion of\nan _hour_ to an _ell_? I answer, it is the same proportion that _two\nhours_ have to _two ells_. You see your question is not so subtle as you\nthought it. By and bye you confess that in times and lines there is\n_quid homogeneum_ (this _quid_ is an infallible sign of not fully\nunderstanding what you say); which is false if you take it of the lines\nthemselves; though if you take it of their quantities, it is true\nwithout a _quid_. Lastly, you tell m\u201de how I might have expressed myself\nso as it might have been true. But because your expressions please me\nnot, I have not followed your advice.\n\nTo the third article, which is this: \u201c_In motu uniformiter a quiete\naccelerato_,\u201d _etc._ \u201c_In motion uniformly accelerated from rest, that\nis, when the impetus increaseth in proportion to the times, the length\nrun over in one time is to the length run over in another time, as the\nproduct of the impetus multiplied by the time, to the product of the\nimpetus multiplied by the time_;\u201d you object, \u201c_that the lengths run\nover are in that proportion which the impetus hath to the impetus; not\nthat which the impetus hath to the time, because impetus to time has no\nproportion, as being heterogeneous_.\u201d First, when you say the impetus,\ndo you mean some one impetus designed by some one of the unequal\nstraight lines parallel to the base B I? That is manifestly false. You\nmean the aggregate of all those unequal parallels. But that is the same\nthing with the time multiplied into the mean impetus. And so you say the\nsame that I do. Again, I ask, where it is that I say or dream that the\nlengths run over are in the proportion of the impetus to the times? Is\nit you or I that dream? And for your heterogeneity of the quantities of\ntime and of swiftness, I have already in divers places showed you your\nerror. Again, why do you make B I represent the lengths run over, which\nI make to be represented by D E, a line taken at pleasure; and you also\na few lines before make the same B I to design the greatest acquired\nimpetus? These are things which show that you are puzzled and entangled\nwith the unusual speculation of time and motion, and yet are thrust on\nwith pride and spite to adventure upon the examination of this chapter.\n\nSecondly, you grant the demonstration to be good, supposing I mean it,\nas I seem to speak, of one and the same motion. But why do I not mean it\nof one and the same motion, when I say not in _motions_, but in _motion_\nuniform? _Because_, say you, _in that which follows, I draw it also to\ndifferent motions_. You should have given at least one instance of it;\nbut there is no such matter. And yet the proposition is in that case\nalso true; though then it must not be demonstrated by the similitude of\ntriangles, as in the case present. And therefore the objections you make\nfrom different impetus acquired in the same time, and from other cases\nwhich you mention, are nothing worth.\n\nAt the fourth article, you allow the demonstration all the way (except\nthe faults of the third, which I have already proved to be none) till I\ncome to say, \u201c_that because the proportion of F K to B I is double to\nthe proportion of A F to A B, therefore the proportion of A B to A F is\ndouble to the proportion of B I to F K_.\u201d This you deny, and wonder at\nas strange, (for it is indeed strange to you), and in many places you\nexclaim against it as extreme ignorance in geometry. In this place you\nonly say, \u201c_no such matter; for though one proportion be double to\nanother, yet it does not follow that the converse is the double of the\nconverse_.\u201d So that this is the issue to which the question is reduced,\nwhether you have any or no geometry. I say, if there be three quantities\nin continual proportion, and the first be the least, the proportion of\nthe first to the second is double to the proportion of the first to the\nthird; and you deny it. The reason of our dissent consisteth in this,\nthat you think the doubling of a proportion to be the doubling of the\nquantity of the proportion, as well in proportions of defect, as in\nproportions of excess; and I think that the doubling of a proportion of\ndefect, is the doubling of the defect of the quantity of the same. As\nfor example in these three numbers, 1, 2, 4, which are in continual\nproportion, I say the quantity of the proportion of one to two, is\ndouble the quantity of the proportion of one to four. And the quantity\nof the proportion of one to four, is half the quantity of the proportion\nof one to two. And yet deny not but that the quantity of the defect in\nthe proportion of one to two is doubled in the proportion of one to\nfour. But because the doubling of defect makes greater defect, it maketh\nthe quantity of the proportion less. And as for the part which I hold in\nthis question, first, there is thus much demonstrated by Euclid, El. v.\nprop. 8; that the proportion of one to two, is greater than the\nproportion of one to four, though how much it is greater be not there\ndemonstrated. Secondly, I have demonstrated (Chap, XIII. art. 16); that\nit is twice as great, that is to say, (to a man that speaks English),\ndouble. The introducing of _duplicate_, _triplicate_, &c. instead of\n_double_, _triple_, &c. (though now they be words well understood by\nsuch as understand what proportion is), proceeded at first from such as\ndurst not for fear of absurdity, call the half of any thing double to\nthe whole, though it be manifest that the half of any defect is a double\nquantity to the whole defect; for want added to want maketh greater\nwant, that is, a less positive quantity. This difference between\n_double_ and _duplicate_, lighting upon weak understandings, has put men\nout of the way of true reasoning in very many questions of geometry.\nEuclid never used but one word both for _double_ and _duplicate_. It is\nthe same fault when men call half a quantity _subduplicate_, and a third\npart _subtriplicate_ of the whole, with intention (as in this case) to\nmake them pass for words of signification different from the _half_ and\nthe _third part_. Besides, from my definition of proportion (which is\nclear, and easy to be understood by all men, but such as have read the\ngeometry of others unluckily) I can demonstrate the same evidently and\nbriefly thus. My definition is this, _proportion is the quantity of one\nmagnitude taken comparatively to another_. Let there be therefore three\nquantities, 1, 2, 4, in continual proportion. Seeing therefore the\nquantity of four in respect of one, is twice as great as the quantity of\nthe same four in respect of 2, it followeth manifestly that the quantity\nof 1 in respect of 4, is twice as little as the quantity of the same 1\nin respect of 2; and consequently the quantity of 1 in respect of 2, is\ntwice as great as the quantity of the same 1 in respect of 4; which is\nthe thing I maintain in this question. Would not a man that employs his\ntime at bowls, choose rather to have the advantage given him of three in\nnine, than of one in nine? And why, but that three is a greater quantity\nin respect of nine, than is one? Which is as much as to say, three to\nnine hath a greater proportion than one to nine; as is demonstrated by\nEuclid, El. v. prop. 8. Is it not therefore (you that profess\nmathematics, and theology, and practise the depression of the truth in\nboth) well owled of you, to teach the contrary? But where you say \u201c_that\nthe point K_ (in the second figure of the table belonging to this\nsixteenth chapter) _is not in the parabolical line whose diameter is A\nB, and base B I, but in the parabolical line of the complement of my\nsemiparabola_ (_as I may learn from the twenty-third proposition of\nyour_ Arithmetica Infinitorum) _whose diameter is A C, and base I C_.\u201d\nWhat line is that? Is it the same line with that of my semiparabola, or\nnot the same? If the same, why find you fault? If not the same, you\nought to have made a semiparabola on the diameter A C, and base I C, and\nfollowing my construction made it appear that K is not in the line\nwherein I say it is; which you have not done, nor could do.\n\nThen again, running on in the same blindness of passion, you pretend I\nmake the proportion of B I to F K double to that of A B to A F, and then\nconfute it; when you knew I made the proportion of F K to B I, double to\nthat of F N, to B I, that is, of A F to A B; and this was it you should\nhave confuted. That which followeth is but a triumphing in your own\nignorance, wherein you also say, \u201c_that all that I afterwards build upon\nthis doctrine is false_.\u201d You see whether it be like to prove so or not.\nAs for your _Arithmetica Infinitorum_, I shall then read to you a piece\nof a lesson on it when I come to your objections against the next\nChapter. In the mean time let me tell you, it is not likely you should\nbe great geometricians, that know not what is quantity, nor measure, nor\nstraight, nor angle, nor homogeneous, nor heterogeneous, nor proportion,\nas I have already made appear in this and the former lessons.\n\nTo the first corollary of this fourth article your exception I confess\nis just, and (which I wonder at) without any incivility. But this argues\nnot ignorance, but security. For who is there that ever read any thing\nin the Conics, that knows not that the parts of a parabola cut off by\nlines parallel to the base, are in triplicate proportion to their bases?\nBut having hitherto designed the time by the diameter, and the impetus\nby the base; and in the next chapter (where I was to calculate the\nproportion of the parabola, to the parallelogram) intending to design\nthe time by the base, I mistook and put the diameter again for the time;\nwhich any man but you might as easily have corrected as reprehended.\n\nTo the second corollary, which is this, _that the lengths run over in\nequal times by motion so accelerated, as that the impetus increase in\ndouble proportion to their times, are as the differences of the cubic\nnumbers beginning at unity, that is, as seven, nineteen, thirty-seven,\n&c._ you say it is false. But why? \u201c_Because_\u201d say you \u201c_portions of the\nparabola of equal altitude, taken from the beginning, are not as those\nnumbers seven, nineteen, thirty-seven, &c._\u201d Does this, think you,\ncontradict any thing in this proposition of mine? Yes, because, you\nthink, the lengths gone over in equal times, are the same with the parts\nof the diameter cut off from the vertex, and proportional to the numbers\none, two, three, &c. Whereas the lengths run over, are as the aggregates\nof their velocities, that is, as the parts of the parabola itself, that\nis, as the cubes of their bases, that is, as the numbers one, eight,\ntwenty-seven, sixty-four, &c., and consequently the lengths run over in\nequal times, are as the differences of those cubic numbers, one, eight,\ntwenty-seven, sixty-four, whose differences are seven, nineteen,\nthirty-seven, &c. The cause of your mistake was, that you cannot yet,\nnor perhaps ever will, contemplate time and motion (which requireth a\nsteady brain) without confusion.\n\nThe third corollary you also say is false, \u201c_whether it be meant of\nmotion uniformly accelerated_ (as the words are) _or_ (_as perhaps_, you\nsay, _I meant it_) _of such motion as is accelerated in double\nproportion to the time_.\u201d You need not say perhaps I meant it. The words\nof the proposition are enough to make the meaning of the corollary\nunderstood. But so also you say it is false. Methinks you should have\noffered some little proof to make it seem so. You think your authority\nwill carry it. But on the contrary I believe rather that they that shall\nsee how your other objections hitherto have sped, will the rather think\nit true, because you think it false. The demonstration as it is, is\nevident enough; and therefore I saw no cause to change a word of it.\n\nTo the fifth article you object nothing, but that it dependeth on this\nproposition (Chap. XIII. art. 16): \u201c_That when three quantities are in\ncontinual proportion, and the first is the least, as in these numbers,\nfour, six, nine, the proportion of the first to the second, is double to\nthe proportion of the same first to the last_;\u201d which is there\ndemonstrated, and in the former lessons so amply explicated, as no man\ncan make any further doubt of the truth of it. And you will, I doubt\nnot, assent unto it. But in what estate of mind will you be then? A man\nof a tender forehead after so much insolence, and so much contumelious\nlanguage grounded upon arrogance and ignorance, would hardly endure to\noutlive it. In this vanity of yours, you ask me whether I be angry, or\nblush, or can endure to hear you. I have some reason to be angry; for\nwhat man can be so patient as not to be moved with so many injuries? And\nI have some reason to blush, considering the opinion men will have\nbeyond sea, (when they shall see this in Latin) of the geometry taught\nin Oxford. But to read the worst you can say against me, I can endure,\nas easily at least, as to read any thing you have written in your\ntreatises of the _Angle of Contact_, of the _Conic Sections_, or your\n_Arithmetica Infinitorum_.\n\nThe sixth, seventh, eighth articles, you say are sound. True. But never\nthe more to be thought so for your approbation, but the less; because\nyou are not fit, neither to reprehend, nor praise; and because all that\nyou have hitherto condemned as false, hath been proved true. Then you\nshow me how you could demonstrate the sixth and seventh articles a\nshorter way. But though there be your symbols, yet no man is obliged to\ntake them for demonstration. And though they be granted to be dumb\ndemonstrations, yet when they are taught to speak as they ought to do,\nthey will be longer demonstrations than these of mine.\n\nTo the ninth article, which is this, \u201c_If a body be moved by two movents\nat once, concurring in what angle soever, of which, one is moved\nuniformly, the other, with motion uniformly accelerated from rest, till\nit acquire an impetus equal to that of the uniform motion, the line in\nwhich the body is carried, shall be the crooked line of a\nsemiparabola_,\u201d you lift up your voice again, and ask, _what latitude?\nwhat diameter? what inclination_ of the diameter to the ordinate lines?\nIf your founder should see this, or the like objections of yours, he\nwould think his money ill bestowed. When I say, _in what angle soever_,\nyou ask, _in what angle?_ When I say _two movents, one uniform, the\nother uniformly accelerated, make the body describe a semiparabolical\nline_; you ask, _which is the diameter?_ as not knowing that the\naccelerated motion describes the diameter, and the other a parallel to\nthe base. And when I say _the two movents meet in a point, from which\npoint both the motions begin, and one of them from rest_, you ask me\n_what is the altitude?_ As if that point where the motion begins from\nrest were not the vertex; or that the vertex and base being given, you\nhad not wit enough to see that the altitude of the parabola is\ndetermined? When Galileo\u2019s proposition, which is the same with this of\nmine, supposed no more but a body moved by these two motions, to prove\nthe line described to be the crooked line of a semiparabola, I never\nthought of asking him what altitude, nor what diameter, nor what angle,\nnor what base, had his parabola. And when Archimedes said, let the line\nA B be the time, I should never have said to him, _do you think time to\nbe a line_, as you ask me whether I think impetus can be the base of a\nparabola. And why, but because I am not so egregious a mathematician, as\nyou are. In this giddiness of yours, caused by looking upon this\nintricate business of motion, and of time, and the concourse of motion\nuniform, and uniformly accelerated, you rave upon the numbers 1, 4, 9,\n16, &c. without reference to any thing that I had said; insomuch as any\none that had seen how much you have been deceived in them before, in\nyour scurvy book of _Arithmetica Infinitorum_, would presently conclude,\nthat this objection was nothing else but a fit of the same madness which\npossessed you there.\n\nMy tenth article is like my ninth; and your objections to it are the\nsame which are to the former. Therefore you must take for answer just\nthe same which I have given to your objection there.\n\nTo the eleventh, you say first, you have done it better at the\nsixty-fourth article of your _Arithmetica Infinitorum_. But what you\nhave done there, shall be examined when I come to the defence of my next\nchapter. And whereas I direct the reader for the finding of the\nproportions of the complements of those figures to the figures\nthemselves, to the table of art. 3, Chap, XVII., you say that if the\nincrease of the _spaces_, were to the increase of the times, as one to\ntwo, then the complement should be to the parallelogram as one to three,\nand say you find not (1)/(3) in the table. Did you not see that the\ntable is only of those figures which are described by the concourse of a\nmotion uniform with a motion accelerated? You had no reason therefore to\nlook for (1)/(3) in that table; for your case is of motion uniform\nconcurring with motion retarded, because you make not the proportions of\nthe spaces to the proportions of the times as two to one, but the\ncontrary; so that your objection ariseth from want of observing what you\nread. But I \u201c_may learn_\u201d you say, \u201c_these, and greater matters than\nthese, in your twenty-third and sixty-fourth propositions of your_\nArithmetica Infinitorum.\u201d This, which you say here is a great absurdity;\nbut if you mean I shall find greater there, I will not say against you.\nThis (1)/(3) you looked for, belongs to the complements of the figures\ncalculated in that table; which because you are not able to find out of\nyourselves, I will direct you to them. Your case is of (1)/(3) for the\ncomplement of a parabola. Take the denominator of the fraction which\nbelongs to the parabola, namely three, and for numerator take the\nnumerator of the fraction which belongs to the triangle, namely one, and\nyou have the fraction sought. And in like manner for the complement of\nany other figure. As, for example, of the second parabolaster, whose\nfraction hath for denominator five, take the numerator of the fraction\nof the same triangle which is one, and you have (1)/(3) for the fraction\nsought for; and so of the rest, taking always one for the numerator.\n\nThe twelfth article, which you say is miserably false, I have left\nstanding unaltered. For not comprehending the sense of the proposition,\nyou make a figure of your own, and fight against your own fancied\nmotions, different from mine. Other geometricians that understand the\nconstruction better, find no fault. And if you had in your own fifth\nfigure drawn a line through N parallel to A E, and upon that line\nsupposed your accelerated motion, you would quickly have seen that in\nthe time A E, the body moved from rest in A, would have fallen short of\nthe diagonal A D; and that all your extravagant pursuing of your own\nmistake had been absurd.\n\nMy thirteenth article you say is ridiculous. But why? \u201c_The impetus last\nacquired cannot_\u201d you say, \u201c_be equal to a time_.\u201d But the quantity of\nthe impetus may be equal to the quantity of a time, seeing they are both\nmeasured by line. And when they are measured by the same described line,\neach of their quantities is equal to that same line, and consequently to\none another. But when I meet with this kind of objection again, since I\nhave so often already shown it to be frivolous, and no less to be\nobjected against all the ancients that ever demonstrated any thing by\nmotion, than against me, I purpose to neglect it.\n\nSecondly, you object \u201c_that motion uniformly accelerated does no more\ndetermine swiftness, than motion uniform_.\u201d True; you needed not have\nused sixteen lines to set down that. But suppose I add, as I do, so as\nthe last acquired impetus be equal to the time. _But that_, you say, _is\nnot sense_; which is the objection I am to neglect. But, you say again,\nsupposing it sense, this limitation helps me nothing. Why? _Because_,\nyou say, _a parabola may be described upon a base given, and yet have\nany altitude, or any diameter one will_. Who doubts it? But how follows\nit from thence, that when a parabolical line is described by two\nmotions, one uniform, the other uniformly accelerated from rest, that\nthe determining of the base does not also determine the whole parabola?\nBut fifthly, you say, _that this equality of the impetus to the time\ndoes not determine the base_. Why not? _Because_, you say, _it is an\nerror proceeding from this, that I understand not what is_ ratio\nsubduplicata. I looked for this. I have shown and inculcated\nsufficiently before, but the error is on your side; and therefore must\ntell you, that this objection, and also a great part of the rest of your\nerrors in geometry, proceedeth from this, that you know not what\nproportion is. But see how wisely you argue about this duplication of\nproportion. For thus you say _verbatim_. \u201c_Stay a little. What\nproportion has duplicate proportion to single proportion? Is it always\nthe same? I think not for example, duplicate proportion_ (4)/(1) = (2 in\n2)/(1 in 1) _is double to the single_ (2)/(1). _Duplicate proportion_\n(9)/(1) = (3 in 3)/(1 in 1) _is triple to its single_ (3)/(1).\u201d Let any\nman, even of them that are most ready in your symbols, say in your\nbehalf (if he be not ashamed) that the proportion of nine to one is\ntriple to the proportion of three to one, as you do.\n\nIn the fourteenth, fifteenth, and sixteenth articles, you bid me repeat\nyour objections to the thirteenth. I have done it; and find that what\nyou have objected to the thirteenth, may as well be objected to these;\nand consequently, that my answer there will also serve me here.\nTherefore, if you can endure it, read the same answer over again.\n\nBut you have not yet done, you say, with these articles. Therefore\n(after you had for a while spoken perplextly, conjecturing, not without\njust cause, that I could not understand you) you say that to the end I\nmay the better perceive your meaning, I should take the example\nfollowing. \u201c_Let a movent (in the first figure of this chapter) be moved\nuniformly in the time A B, with the continual impetus A C, or B I, whose\nwhole velocity shall therefore be the parallelogram A C I B. And another\nmovent be uniformly accelerated, so as in the time A B it acquire the\nsame impetus B I. Now as the whole velocity, is to the whole velocity,\nso is the length run over, to the length run over._\u201d All this I\nacknowledge to be according to my sense, saving that your putting your\nword _movens_ instead of my word _mobile_ hath corrupted this article.\nFor in the first article, I meddle not with motion by concourse, wherein\nonly I have to do with two movents to make one motion; but in this I do,\nwherein my word is not _movens_ but _mobile_; by which it is easy to\nperceive you understand not this proposition. Then you proceed: \u201c_But\nthe length run over by that accelerated motion is greater than the\nlength run over by that uniform motion._\u201d Where do I say that? You\nanswer, \u201c_in the ninth and thirteenth article, in making A B (in the\nfifth figure) greater than A C; and A H (in the eighth figure) greater\nthan A B; and consequently, the triangle A B I, greater than the\nparallelogram A C I B_.\u201d That consequently is without consequence; for\nit importeth nothing at all in this demonstration, whether A B, or A C\nin the fifth figure be the greater. Besides I speak there of the\nconcourse of two movents, that describe the parabolical line A G D;\nwhere the increasing impetus (because it increaseth as the times) will\nbe designed by the ordinate lines in the parabola A G D B. And if both\nthe motions in A B and A C were uniform, the aggregate of the impetus\nwould be designed by the triangle A B D, which is less than the\nparallelogram A C D B. But you thought that the motion made by A C\nuniformly, is the same with the motion made uniformly in the same time\nby the motions in A B and A C concurring; so likewise, in the eighth\nfigure, there is nothing hinders A H from being greater than A B, unless\nI had said that A B had been described in the time A C with the whole\nimpetus A C maintained entire; of which there is nothing in the\nproposition, nor would at all have been pertinent to it. Therefore all\nthis new undertaking of the thirteenth, fourteenth, fifteenth, and\nsixteenth articles, is to as little purpose as your former objections.\nBut I perceive that these new and hard speculations, though they turn\nthe edge of your wit, turn not the edge of your malice.\n\nAt the seventeenth article, you show again the same confusion. Return to\nthe eighth figure: \u201c_if in a time given a body run over two lengths, one\nwith uniform, the other with accelerated motion_\u201d; as for example, if in\nthe same time A C, a body, run over the line A B with uniform motion,\nand the line A H with motion accelerated; \u201c_and again in a part of that\ntime it run over a part of the length A H, with uniform motion, and\nanother part of the same with motion accelerated_;\u201d as for example, in\nthe time A M it run over with uniform motion the line A I, and with\nmotion accelerated the line A B. _I say the excess of the whole A H\nabove the part A B, is to the excess of the whole A B above the part A\nI, as the whole A H to the whole A B._ But first you will say, that\nthese words _as the whole A H to the whole A B_, are left out in the\nproposition. But you acknowledge that it was my meaning; and you see it\nis expressed before I come to the demonstration. And therefore it was\nabsurdly done to reprehend it. Let us therefore pass to the\ndemonstration. Draw I K parallel to A C, and make up the parallelogram A\nI K M. And supposing first the acceleration to be uniform, divide I K in\nthe midst at N; and between I N, and I K, take a mean proportional I L.\n_And the straight line A L, drawn and produced, shall cut the line B D\nin F, and the line C G in G_ (which lines C G, and B D, as also H G and\nB F, are determined, though you could not carry it so long in memory, by\nthe demonstration of the thirteenth article). _For seeing A B is\ndescribed by motion uniformly accelerated, and A I by motion uniform in\nthe same time A M; and I L is a mean proportional between I N (the half\nof I K) and I K; therefore by the demonstration of the thirteenth\narticle, A I is a mean proportional between A B and the half of A B,\nnamely A O. Again, because A B is described by uniform motion, and A H\nby motion uniformly accelerated, both of them in the same time A C, B F\nis a mean proportional between B D and half B D, namely B E; therefore\nby the demonstration of the same thirteenth article, the straight line A\nL F produced will fall on G; and the line A H will be to the line A B,\nas the line A B to the line A I. And consequently as A H to A B, so H B\nto B I; which was to be demonstrated._ And by the like demonstration the\nsame may be proved, where the acceleration is in any other proportion\nthat can be assigned in numbers, saving that whereas this demonstration\ndependeth on the construction of the thirteenth article, if the motion\nhad been accelerated in double proportion to the times, it would have\ndepended on the fourteenth, where the lines are determined. Which\ndeterminations being not repeated, but declared before, in the\nthirteenth article, to which this diagram belongeth, you take no notice\nof, but go back to a figure belonging to another article, where there\nwas no use of these determinations. But because I see that the words of\nthe proposition, are as of four motions, and not of two motions made by\ntwice two movents, I must pardon them that have not rightly understood\nmy meaning; and I have now made the proposition according to the\ndemonstration. Which being done, all that you have said in very near two\nleaves of your _Elenchus_ comes to nothing; and the fault you find comes\nto no more than a too much trusting to the skill and diligence of the\nreader. And whereas after you had sufficiently troubled yourself upon\nthis occasion, you add, \u201c_that if Sir H. Savile had read my Geometry, he\nhad never given that censure of Joseph Scaliger, in his lecture upon\nEuclid, that he was the worst geometrician of all mortal men, not\nexceptioning so much as Orontius, but that praise should have been kept\nfor me_.\u201d You see by this time, at least others do, how little I ought\nto value that opinion; and that though I be the least of geometricians,\nyet my geometry is to yours as 1 to 0. I recite these words of yours, to\nlet the world see your indiscretion in mentioning so needlessly that\npassage of your founder. It is well known that Joseph Scaliger deserved\nas well of the state of learning, as any man before or since him; and\nthat though he failed in his ratiocination concerning the quadrature of\nthe circle, yet there appears in that very failing so much knowledge of\ngeometry, that Sir H. Savile could not but see that there were mortal\nmen very many that had less; and consequently he knew that that censure\nof his in a rigid sense (without the license of an hyperbole) was\nunjust. But who is there that will approve of such hyperboles to the\ndishonour of any but of unworthy persons, or think Joseph Scaliger\nunworthy of honour from learned men? Besides, it was not Sir H. Savile\nthat confuted that false quadrature, but Clavius. What honour was it\nthen for him to triumph in the victory of another? When a beast is slain\nby a lion, is it not easy for any of the fowls of the air to settle\nupon, and peck him? Lastly, though it were a great error in Scaliger,\nyet it was not so great a fault as the least sin; and I believe that a\npublic contumely done to any worthy person after his death, is not the\nleast of sins. Judge therefore whether you have not done indiscreetly,\nin reviving the only fault, perhaps that any man living can lay to your\nfounder\u2019s charge; and yet this error of Scaliger\u2019s was no greater than\none of your own of the like nature, in making the true spiral of\nArchimedes equal to half the circumference of the circle of the first\nrevolution; and then thinking to cover your fault by calling it\nafterwards an aggregate of arches of circles (which is no spiral at all\nof any kind) you do not repair but double the absurdity. What would Sir\nHenry Savile have said to this?\n\nThe eighteenth article is this, \u201c_in any parallelogram, if the two sides\nthat contain the angle be moved to their opposite sides, the one\nuniformly, the other uniformly accelerated; the side that is moved\nuniformly, by its concourse through all its longitude, hath the same\neffect which it would have if the other motion were also uniform, and\nthe line described were a mean proportional between the whole length,\nand the half of the same_.\u201d\n\nTo the proposition you object first, \u201c_that it is all one whether the\nother motion be uniform or not, because the effect of each of their\nmotions, is but to carry the body to the opposite side_.\u201d But do you\nthink that whatsoever be the motions, the body shall be carried by their\nconcourse always to the same point of the opposite side? If not, then\nthe effect is not all one when a motion is made by the concourse of two\nmotions uniform and accelerated, and when it is made by the concourse of\ntwo uniform or of two accelerated motions.\n\nSecondly, you say that these words, _and the line described were a mean\nproportional between the whole length, and the half of the same_, have\nno sense, or that you are deceived. True. For you are deceived; or\nrather you have not understanding enough distinctly to conceive variety\nof motions though distinctly expressed. For when a line is gone over\nwith motion uniformly accelerated, you cannot understand how a mean\nproportional can be taken between it and its half; or if you can, you\ncannot conceive that that mean can be gone over with uniform motion in\nthe same time that the whole line was run over by motion uniformly\naccelerated. Yet these are things conceivable, and your want of\nunderstanding must be made my fault.\n\nMy demonstration is this, _in the parallelogram A B C D, (Fig. 11). Let\nthe side A B be conceived to be moved uniformly till it lie in C D; and\nlet the time of that motion be A C, or B D. And in the same time let it\nbe conceived that A C is moved with uniform acceleration, till it lie in\nB D._ To which you object, _that then the acceleration last acquired\nmust be far greater than that wherewith A B is moved uniformly: else it\nshall never come to the place you would have it in the same time_. What\nproof bring you for this? None here. Where then? Nowhere that I\nremember. On the contrary I have proved (Art. 9 of the chapter) that the\nline described by the concourse of those two motions, namely, uniform\nfrom A B to C D, and uniformly accelerated from A C to B D, is the\ncrooked line of the semiparabola A H D. And though I had not, yet it is\nwell known that the same is demonstrated by Galileo. And seeing it is\nmanifest that in what proportion the motion is accelerated in the line A\nB, in the same proportion the impetus beginning from rest in A is\nincreased in the same times (which impetus is designed all the way by\nthe ordinate lines of the semiparabola), the greatest impetus acquired\nmust needs be the base of the semiparabola, namely B D, equal to A C,\nwhich designs the whole time. I cannot therefore imagine what should\nmake you say without proof, that the greatest acquired impetus is\ngreater than that which is designed by the base B D. Next you say, \u201cyou\nsee not to what end I divide A B in the middle at E.\u201d No wonder; for you\nhave seen nothing all the way. Others would see it is necessary for the\ndemonstration; as also that the point F is not to be taken arbitrarily;\nand likewise that the thirteenth article, which you admit not for proof,\nis sufficiently demonstrated, and your objections to it answered. By the\nway you advise me, where I say _percursam eodem motu uniformi, cum\nimpetu ubique_, &c. to blot out _cum_; because the _impetus_ is not a\n_companion_ in the way, but the _cause_. Pardon me in that I cannot take\nyour learned counsel; for the word _motu uniformi_ is the ablative of\nthe _cause_, and _impetu_ the ablative of the _manner_. But to come\nagain to your objections, you say, I make \u201c_a greater space run over in\nthe same time by the slower motion than by the swifter_.\u201d How does that\nappear? _because there is no doubt, but the swiftness is greater where\nthe greatest impetus is always maintained, than where it is attained to\nin the same time from rest_. True, but that is, when they are considered\nasunder without concourse, but not then when by the concourse they\ndebilitate one another, and describe a third line different from both\nthe lines, which they would describe singly. In this place I compare\ntheir effects as contributing to the description of the parabolical line\nA H D. What the effects of their several motions are, when they are\nconsidered asunder, is sufficiently shown before in the first article.\nYou should first have gotten into your minds the perfect and distinct\nideas of all the motions mentioned in this chapter, and then have\nventured upon the censure of them, but not before. And then you would\nhave seen that the body moved from A, describeth not the line A C, nor\nthe line A B, but a third, namely the semiparabolical line A H D.\n\nAgain, where I say, _Wherefore, if the whole A B be uniformly moved to C\nD, in the same time wherein A C is moved uniformly to F G_; you ask me\n\u201c_whether with the same impetus or not?_\u201d How is it possible that in the\nsame time two unequal lengths should be passed over the same impetus?\n\u201c_But why_,\u201d say you, \u201c_do you not tell us with what impetus A C comes\nto F G?_\u201d What need is there of that, when all men know that in uniform\nmotion and the same time, impetus is to impetus, as length to length?\nWhich to have expressed had not been pertinent to the demonstration.\nThat which follows in the demonstration, _rursus suppono quod latus A\nC_, &c. to these words, _ut ostensum est_, _Art. 12_, you confute with\nsaying you have proved that article to be false. But you may see now, if\nyou please, at the same place that I have proved your objection to be\nfrivolous.\n\nAfter this you run on without any argument against the rest of the\ndemonstration, showing nothing all the way, but that the variety and\nconcourse of motions, the speculations whereof you have not been used\nto, have made you giddy.\n\nTo the nineteenth article you apply the same objection which you made to\nthe eighteenth. Which having been answered, it appears that from the\nvery beginning of your Elenchus to this place all your objections\n(except such as are made to three or four mistakes of small importance\nin setting down my mind), are mere paralogisms, and such are less\npardonable than any paralogism in Orontius, both because the subject as\nless difficult is more easily mastered, and because the same faults are\nmost shamefully committed by a reprehender than by any other man.\n\nI had once added to these nineteen articles a twentieth, which was this:\n\u201c_If from a point in the circumference there be drawn a cord, and a\ntangent equal to it, the angle which they make shall be double to the\naggregate of all the angles made by the cords of all the equal arches\ninto which the arch given can possibly be divided_.\u201d Which proposition\nis true, and I did when I writ it think I might have use of it. But be\nit, or the demonstration of it true or false, seeing it was not\npublished by me, it is somewhat barbarous to charge me with the faults\nthereof. No doctor of humanity but would have thought it a poor and\nwretched malice, publicly to examine and censure papers of geometry\nnever published, by what means soever they came into his hands. I must\nconfess that in these words, _in such kind of progression arithmetical_\n(that is, which begins with 0) _the sum of all the numbers taken\ntogether, is equal to half the number that is made by multiplying the\ngreatest into the least_, there is a great error; for by this account\nthese numbers, 0, 1, 2, 3, 4, taken together, should be equal to\nnothing. I should have said they are equal to that number which is made\nby multiplying half the greatest into the number of the terms. There was\ntherefore, if those words were mine (for truly I have no copy of them,\nnor have had since the book was printed, and I have no great reason, as\nany man may see, to trust your faith) a great error in the writing, but\nnot an erroneous opinion in the writer. The demonstration so corrected\nis true. And the angles that have the proportions of the numbers 1, 2,\n3, 4, are in the table of your _Elenchus_ , fig. 12, the angles G A D, H\nD E, I E F, K F B. And if the divisions were infinite, so that the first\nwere not to be reckoned but as a cypher, the angle C A B would be double\nto them altogether. This mistake of mine, and the finding that I had\nmade no use of it in the whole book, was the cause why I thought fit to\nleave it quite out. But your professorships, could not forbear to take\noccasion thereby, to commend your zeal against _Leviathan_ to your\ndoctorships of divinity, by censuring it.\n\n\n ==========\n\n\n OF THE FAULTS THAT OCCUR IN\n DEMONSTRATION.\n\n TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN\n THE UNIVERSITY OF OXFORD.\n\n\n LESSON V.\n\n\nAt the seventeenth chapter, your first exception is to the definition of\nproportional proportions, which is this: \u201c_Four proportions are then\nproportional, when the first is to the second, as the third to the\nfourth_.\u201d The reader will hardly believe that your exception is in\nearnest. You say, I mean not by proportionality the \u201c_quantity of the\nproportions_.\u201d Yes I do. Therefore I say again, that _four proportions\nare then proportional, when the quantity of the first proportion, is to\nthe quantity of the second proportion, as the quantity of the third\nproportion, to the quantity of the fourth proportion_. Is not my meaning\nnow plainly enough expressed? Or is it not the same definition with the\nformer. But what do I mean, you will say, by the quantity of a\nproportion? I mean the determined greatness of it, that is, for example,\nin these numbers, the quantity of the proportion of two to three, is the\nsame with the quantity of the proportion of four to six, or six to nine;\nand again, the quantity of the proportion of six to four, is the same\nwith the quantity of the proportion of nine to six, or of three to two.\nBut now what do you mean by the quantity of a proportion? You mean that\ntwo and three, are the quantities of the proportion of two to three (for\nso Euclid calls them) and that six and four are the quantities of the\nproportion of six to four, which is the same with the proportion of\nthree to two. And by this rule, one and the same proportion shall have\nan infinite number of quantities; and consequently the quantity of a\nproportion can never be determined. I call one proportion double to\nanother, when one is equal to twice the other; as the proportion of four\nto one, is double to the proportion of two to one. You call that\nproportion double where one number, line, or quantity absolute, is\ndouble to the other; so that with you the proportion of two to one is a\ndouble proportion. It is easy to understand how the number two is double\nto one, but to what, I pray you, is double the proportion of two to one,\nor of one to two? Is not every double proportion double to some\nproportion? See whether this geometry of yours can be taken by any man\nof sound mind for sense. \u201c_But it is known_,\u201d you say, \u201c_that in\nproportions, double is one thing, and duplicate another_;\u201d so that it\nseems to you, that in talking of proportion men are allowed to speak\nsenselessly. \u201c_It is known_,\u201d you say. To whom? It is indeed in use at\nthis day to call _double duplicate_, and _triple triplicate_. And it is\nwell enough; for they are words that signify the same thing, but that\nthey differ (in what subject soever) I never heard till now. I am sure\nthat Euclid, whom you have undertaken to expound, maketh no such\ndifference. And even there where he putteth these numbers, one, two,\nfour, eight, &c. for numbers in _double_ proportion (which is the last\nproposition of the ninth element) he meaneth not that one to two, or two\nto one, is a _double_ proportion, but that every number in that\nprogression is _double_ to the number next before it; and yet he does\nnot call it _analogia dupla_, but _duplicate_. This distinction in\nproportions between _double_ and _duplicate_, proceeded long after from\nwant of knowledge that the proportion of one to two is _double_ to the\nproportion of one to four; and this from ignorance of the different\nnature of proportions of _excess_, and proportions of _defect_. And you\nthat have nothing but by tradition saw not the absurdities that did hang\nthereon.\n\nIn the second article I make E K, (fig. 1) the third part of L K, which\nyou say is false; and consequently the proposition undemonstrated. And\nthus you prove it false: \u201c_Let A C be to G C, or G K to G L, as eight to\none_ (_for seeing the point G is taken arbitrarily, we may place it\nwhere we will, &c._)\u201d and upon this placing of G arbitrarily, you prove\nwell enough that E K is not a third part of L K. But you did not then\nobserve, that I make _the altitude A G, less than any quantity given_,\nand by consequence E K to differ from a third part by a less difference\nthan any quantity that can be given. Therefore as yet the demonstration\nproceedeth well enough. But perceiving your oversight, you thought fit\n(though before, you thought this confutation sufficient) to endeavour to\nconfute it another way; but with much more evidence of ignorance. For\nwhen I come to say, _the proportion therefore between A C and G C is\ntriple, in arithmetical proportion, to the proportion between G K and G\nE, &c._ you say, \u201c_the proportion of A C to G C is the proportion of\nidentity, as also that of G K to G E.\u201d_ But why? Does my construction\nmake it so? Do not I make G C less than A C, though with less difference\nthan any quantity that can be assigned? And then where I say, _therefore\nE K is the third part of L K_, you come in, by parenthesis, with (_or a\nfourth, or a fifth, &c._). Upon what ground? Because you think it will\npass for current, without proof, that a point is nothing. Which if it\ndo, geometry also shall pass for nothing, as having no ground nor\nbeginning but in nothing. But I have already in a former lesson\nsufficiently showed you the consequence of that opinion. To which I may\nadd, that it destroys the method of _indivisibles_, invented by\nBonaventura; and upon which, not well understood, you have grounded all\nyour scurvy book of _Arithmetica Infinitorum_; where your indivisibles\nhave nothing to do, but as they are supposed to have quantity, that is\nto say, to be _divisibles_. You allow, it seems, your own nothings to be\nsomethings, and yet will not allow my somethings to be considered as\nnothing. The rest of your objections having no other ground than this,\n\u201c_that a point is nothing_,\u201d my whole demonstration standeth firm; and\nso do the demonstrations of all such geometricians, ancient and modern,\nas have inferred any thing in the manner following, viz. _If it be not\ngreater nor less, then it is equal. But it is neither greater nor less.\nTherefore, &c. If it be greater, say by how much. By so much. It is not\ngreater by so much. Therefore it is not greater. If it be less, say how\nmuch, &c._ Which being good demonstrations are together with mine\noverthrown by the nothingness of your _point_, or rather of your\nunderstanding; upon which you nevertheless have the vanity of advising\nme what to do, if I demonstrate the same again; meaning I should come to\nyour false, impossible, and absurd method of _Arithmetica Infinitorum_,\nworthy to be gilded, I do not mean with gold.\n\nAnd for your question, why I set the base of my figure upwards, you may\nbe sure it was not because I was afraid to say, that the proportions of\nthe ordinate lines beginning at the vertex were triplicate, or otherwise\nmultiplicate of the proportions of the intercepted parts of the\ndiameter. For I never doubted to call double duplicate, nor triple\ntriplicate, &c., or if I had, I should have avoided it afterwards at the\ntenth article of the same chapter. But because when I went about to\ncompare the proportions of the ordinate lines with those of their\ncontiguous diameters, the first thing I considered in them was in what\nmanner the base grew less and less till it vanished into a point. And\nthough the base had been placed below, it had not therefore required any\nchange in the demonstration. But I was the more apt to place the base\nuppermost, because the motion began at the base, and ended at the\nvertex. To proceed which way I pleased was in my own choice; and it is\nof grace that I give you any account of it at all.\n\nTo the third article, together with its table, you say, \u201c_it falls in\nthe ruin of the second; and that the same is to be understood of the\nsixth, seventh, eighth, and ninth_.\u201d For confutation whereof I need to\nsay no more, but that they all stand good by the confutation of your\nobjections to the second.\n\nTo the fourth article you say, \u201c_the description of those curvilineal\nfigures is easy_.\u201d True, to some men; and now that I have showed you the\nway, it is easy enough for you also. For the way you propound is wholly\ntranscribed out of the figure of the second article, which article you\nhad before rejected. For seeing the lines H F, G E, A B, &c. are equal\nto the lines C Q, C O, C D; and the lines Q F, O E, B D, equal to the\nlines C H, C G, C A; the proportion of D B to O E, will be triple (that\nis, triplicate) to the proportion of C O to G E; and the proportion of D\nB to Q F, triple to the proportion of C D to C Q; and consequently,\nbecause the complement B D C F E B is made by the decrease of A C in\ntriple proportion to that of the decrease of C D, it will be (by the\nsecond article) a third part of the figure A B E F C A. So that it comes\nall to one pass, whether we take triple proportion in decreasing to make\nthe complement, or triple proportion in increasing to make the figure;\nfor the proportion of H F to B A, is triple to the proportion of C H to\nC A. Wherefore you have done no more but what you have seen first done,\nsaving that from your construction you prove not the figure to be triple\nto the complement; perhaps because you have proved the contrary in your\n_Arithmetica Infinitorum_. But your way differs from mine, in that you\ncall the proportion subtriplicate, which I call triplicate; as if the\ndivers naming of the same thing made it differ from itself. You might as\nwell have said briefly, the proposition is true, but ill proved, because\nI call the proportion of one or two triple, or triplicate of that of one\nto eight; which you say is false, and hath infected the fourth, fifth,\nninth, tenth, eleventh, thirteenth, fourteenth, fifteenth, sixteenth,\nseventeenth, and nineteenth articles of the sixteenth chapter. But I\nsay, and you know now, that it is true; and that all those articles are\ndemonstrated.\n\nLastly you add, \u201c_Tu vero, in presente articulo, &c. id est, you bid\nfind as many mean proportionals as one will, between two given lines; as\nif that could not be done by the geometry of planes, &c._\u201d You might\nhave left out _Tu vero_ to seek an _Ego quidem_. But tell me, do you\nthink that you can find two mean proportionals (which is less than as\nmany as one will) by the geometry of planes? We shall see anon how you\ngo about it. I never said it was impossible, and if you look upon the\nplaces cited by you more attentively, you will find yourself mistaken.\nBut I say, the way to do it has not been yet found out, and therefore it\nmay prove a solid problem for anything you know.\n\nThe fifth article you reject, because it citeth the corollary of the\ntwenty-eighth article of the thirteenth chapter, where there is never a\nword to that purpose. But there is in the twenty-sixth article; which\nwas my own fault, though you knew not but it might have been the\nprinter\u2019s.\n\nTo the tenth you object for almost three leaves together, against these\nwords of mine, _because_, in the sixth figure, _B C is to B F in\ntriplicate proportion of C D to F E, therefore inverting, F E is to C D\nin triplicate proportion of B F to B C_. This you objected then. But now\nthat I have taught you so much geometry, as to know _that of three\nquantities, beginning at the least, if the third be to the first in\ntriplicate proportion of the second to the first, also by conversion the\nfirst to the second shall be in triplicate proportion of the first to\nthe third_; if it were to do again, you would not object it.\n\nMy eleventh article you would allow for demonstrated, if my second had\nbeen demonstrated, upon which it dependeth. Therefore seeing your\nobjections to that article are sufficiently answered, this article also\nis to be allowed.\n\nThe twelfth also is allowed upon the same reason. What falsities you\nshall find in such following propositions as depend upon the same second\narticle, we shall then see when I come to the places where you object\nagainst them.\n\nTo the thirteenth article you object, \u201c_that the same demonstration may\nbe as well applied to a portion of any conoeides, parabolical,\nhyperbolical, elliptical, or any other, as to the portion of a sphere_.\u201d\nBy the truth of this let any man judge of your and my geometry. Your\ncomparison of the sphere and conoeides, so far holds good, as to prove\nthat the superficies of the conoeides is greater than the superficies of\nthe cone described by the subtense of the parabolical, hyperbolical, or\nelliptical line. But when I come to say, that _the cause of the excess\nof the superficies of the portion of the sphere above the superficies of\nthe cone, consists in the angle D A B, and the cause of the excess of\nthe circle made upon the tangent A D, above the superficies of the same\ncone, consists in the magnitude of the same angle D A B_, how will you\napply this to your conoeides? For suppose that the crooked line A B (in\nthe seventh figure) were not an arch of a circle, do you think that the\nangles which it maketh with the subtense A B, at the points A and B,\nmust needs be equal? Or if they be not, does the excess of the\nsuperficies of the circle upon A D above the superficies of the cone, or\nthe excess of the superficies of the portion of the conoeides above the\nsuperficies of the same cone, consist in the angle D A B, or rather in\nthe magnitude of the two unequal angles D A B, and A B A? You should\nhave drawn some other crooked line, and made tangents to it through A\nand B, and you would presently have seen your error. See how you can\nanswer this; for if this demonstration of mine stand firm, I may be bold\nto say, though the same be well demonstrated by Archimedes, that this\nway of mine is more natural, as proceeding immediately from the natural\nefficient causes of the effect contained in the conclusion; and besides,\nmore brief and more easy to be followed by the fancy of the reader.\n\nTo the fourteenth article you say that I \u201c_commit a circle in that I\nrequire in the fourth article the finding of two mean proportionals, and\ncome not till now to show how it is to be done_.\u201d Nor now neither. But\nin the mean time you commit two mistakes in saying so. The place cited\nby you in the fourth article is, in the Latin, p. 215, line 26, in the\nEnglish, p. 255, line 24. Let any reader judge whether that be a\nrequiring it, or a supposing it to be done; this is your first mistake.\nThe second is, that in this place the proportion itself, which is, \u201c_If\nthese deficient figures could be described in a parallelogram\nexquisitely, there might be found thereby between any two lines given,\nas many mean proportionals as one would_,\u201d is a theorem, upon\nsupposition of these crooked lines exquisitely drawn; but you take it\nfor a problem.\n\nAnd proceeding in that error, you undertake the invention of two mean\nproportionals, using therein my first figure, which is of the same\nconstruction with the eighth that belongeth to this fourteenth article.\nYour construction is, \u201c_Let there be taken in the diameter C A, (fig. 1)\nthe two given lines, or two others proportional to them, as C H, C G,\nand their ordinate lines H F, G E (which by construction are in\nsubtriplicate proportion of the intercepted diameters). These lines will\nshow the proportions which those four proportionals are to have._\u201d But\nhow will you find the length of H F or G E, the ordinate lines? Will you\nnot do it by so drawing the crooked line C F E, as it may pass through\nboth the points F and E? You may make it pass through one of them, but\nto make it pass through the other, you must find two mean proportionals\nbetween G K and G L, or between H I and H P; which you cannot do, unless\nthe crooked line be exactly drawn; which it cannot be by the geometry of\nplanes. Go shew this demonstration of yours to Orontius, and see what he\nwill say to it.\n\nI am now come to an end of your objections to the seventeenth chapter,\nwhere you have an epiphonema not to be passed over in silence. But\nbecause you pretend to the demonstration of some of these propositions\nby another method in your _Arithmetica Infinitorum_, I shall first try\nwhether you be able to defend those demonstrations as well as I have\ndone these of mine by the method of motion.\n\nThe first proposition of your _Arithmetica Infinitorum_ is this lemma:\n\u201c_In a series, or row of quantities, arithmetically proportional,\nbeginning at a point or cypher, as 0, 1, 2, 3, 4, &c. to find the\nproportion of the aggregate of them all, to the aggregate of so many\ntimes the greatest, as there are terms_.\u201d This is to be done by\nmultiplying the greatest into half the number of the terms.\n\nThe demonstration is easy. But how do you demonstrate the same? \u201c_The\nmost simple way_,\u201d say you, \u201c_of finding this and some other problems,\nis to do the thing itself a little way, and to observe and compare the\nappearing proportions, and then by induction to conclude it\nuniversally_.\u201d Egregious logicians and geometricians, that think an\ninduction, without a numeration of all the particulars sufficient, to\ninfer a conclusion universal, and fit to be received for a geometrical\ndemonstration! But why do you limit it to the natural consecution of the\nnumbers, 0, 1, 2, 3, 4, &c? Is it not also true in these numbers, 0, 2,\n4, 6, &c. or in these, 0, 7, 14, 21, &c? Or in any numbers where the\ndifference of nothing and the first number is equal to the difference\nbetween the first and second, and between the second and third, &c.?\nAgain, are not these quantities, 1, 3, 5, 7, &c. in continual proportion\narithmetical? And if you put before them a cypher thus, 0, 1, 3, 5, 7,\ndo you think that the sum of them is equal to the half of five times\nseven? Therefore though your lemma be true, and by me (Chap. XIII. art.\n5) demonstrated; yet you did not know why it is true; which also appears\nmost evidently in the first proposition of your _Conic Sections_ , where\nfirst you have this, \u201c_that a parallelogram whose altitude is infinitely\nlittle, that is to say, none, is scarce anything else but a line_.\u201d Is\nthis the language of geometry? How do you determine this word _scarce_?\nThe least altitude, is somewhat or nothing. If somewhat, then the first\ncharacter of your arithmetical progression must not be a cypher; and\nconsequently the first eighteen propositions of this your _Arithmetica\nInfinitorum_ are all nought. If nothing, then your whole figure is\nwithout altitude, and consequently your understanding nought. Again, in\nthe same proposition, you say thus: \u201c_We will sometimes call those\nparallelograms rather by the name of lines than of parallelograms, at\nleast when there is no consideration of a determinate altitude; but\nwhere there is a consideration of a determinate altitude (which will\nhappen sometimes) there that little altitude shall be so far considered,\nas that being infinitely multiplied it may be equal to the altitude of\nthe whole figure._\u201d See here in what a confusion you are when you resist\nthe truth. When you consider no determinate altitude, that is no\nquantity of altitude, then you say your parallelogram shall be called a\nline. But when the altitude is determined, that is, when it is quantity,\nthen you will call it a parallelogram. Is not this the very same\ndoctrine which you so much wonder at and reprehend in me, in your\nobjections to my eighth chapter, and your word _considered_ used as I\nused it? It is very ugly in one that so bitterly reprehendeth a doctrine\nin another, to be driven upon the same himself by the force of truth\nwhen he thinks not on it. Again, seeing you admit in any case those\ninfinitely little altitudes to be quantity, what need you this\nlimitation of yours, \u201c_so far forth as that by multiplication they may\nbe made equal to the altitude of the whole figure_?\u201d May not the half,\nthe third, the fourth, or the fifth part, &c. be made equal to the whole\nby multiplication? Why could you not have said plainly, _so far forth as\nthat every one of those infinitely little altitudes be not only\nsomething but an aliquot part of the whole_? So you will have an\n_infinitely little_ altitude, that is to say, _a point to be both\nnothing and something and an aliquot part_. And all this proceeds from\nnot understanding the ground of your profession. Well, the lemma is\ntrue. Let us see the theorems you draw from it. The first is (p. 3)\n\u201c_that a triangle to a parallelogram of equal base and altitude is as\none to two_.\u201d The conclusion is true, but how know you that?\n\u201c_Because_,\u201d say you, \u201c_the triangle consists as it were_ [_as it were_,\nis no phrase of a geometrician] _of an infinite number of straight\nparallel lines_.\u201d Does it so? Then by your own doctrine, which is, that\n\u201c_lines have no breadth_,\u201d the altitude of your triangle consisteth of\nan infinite number of no altitudes, that is of an infinite number of\nnothings, and consequently the area of your triangle has no quantity. If\nyou say that by the parallels you mean infinitely little parallelograms,\nyou are never the better; for if infinitely little, either they are\nnothing, or if somewhat, yet seeing that no two sides of a triangle are\nparallel, those parallels cannot be parallelograms. I see they may be\ncounted for parallelograms by not considering the quantity of their\naltitudes in the demonstration. But you are barred of that plea, by your\nspiteful arguing against it in your _Elenchus_ . Therefore this third\nproposition, and with it the fourth, is undemonstrated.\n\nYour fifth proposition is, \u201c_the spiral line is equal to half the circle\nof the first revolution_.\u201d But what spiral line? We shall understand\nthat by your construction, which is this: \u201c_The straight line M A_ [in\nyour figure which I have placed at the end of the fifth lesson] _turned\nround (the point M remaining unmoved) is supposed to describe with its\npoint A the circle A O A, whilst some point, in the same M A, whilst it\ngoes about, is supposed to be moved uniformly from M to A, describing\nthe spiral line_.\u201d This therefore, is the spiral line of Archimedes; and\nyour proposition affirms it to be equal to the half of the circle A O A;\nwhich you perceived not long after to be false. But thinking it had been\ntrue, you go about to prove it, \u201c_by inscribing in the circle an\ninfinite multitude of equal angles, and consequently an infinite number\nof sectors, whose arches will therefore be in arithmetical proportion_;\u201d\nwhich is true. \u201c_And the aggregate of those arches equal to half the\ncircumference A O A_;\u201d which is true also. And thence you conclude\n\u201c_that the spiral line is equal to half the circumference of the circle\nA O A_;\u201d which is false. For the aggregate of that infinite number of\ninfinitely little arches, is not the spiral line made by your\nconstruction, seeing by your construction the line you make is\nmanifestly the spiral of Archimedes; whereas no number, though infinite,\nof arches of circles, how little soever, is any kind of spiral at all;\nand though you call it a spiral, that is but a patch to cover your\nfault, and deceiveth no man but yourself. Besides, you saw not how\nabsurd it was, for you that hold a point to be absolutely nothing, to\nmake an infinite number of equal angles (the radius increasing as the\nnumber of angles increaseth) and then to say, \u201c_that the arches of the\nsectors whose angles they are, are as_ 0, 1, 2, 3, 4, &c.\u201d For you make\nthe first angle 0, and all the rest equal to it; and so make 0, 0, 0, 0,\n0, &c. to be the same progression with 0, 1, 2, 3, 4, &c. The influence\nof this absurdity reacheth to the end of the eighteenth proposition. So\nmany are therefore false, or nothing worth. And you needed not to wonder\nthat the doctrine contained in them was omitted by Archimedes, who never\nwas so senseless as to think a spiral line was compounded of arches of\ncircles.\n\nYour nineteenth proposition is this other lemma: \u201c_In a series, or a\nrow, of quantities, beginning from a point, or cypher, and proceeding\naccording to the order of the square numbers, as_ 0, 1, 4, 9, 16, _&c.\nto find what proportion the whole series hath to so many times the\ngreatest_.\u201d And you conclude \u201c_the proportions to be that of 1 to 3_.\u201d\nWhich is false, as you shall presently see. First, let the series of\nsquares with the prefixed cypher, and under every one of them the\ngreatest 4 be (0 . 1 . 4)/(4 . 4 . 4). And you have for the sum of the\nsquares 5, and for thrice the greatest 12, the third part whereof is 4.\nBut 5 is greater than 4, by 1, that is, by one twelfth of 12; which\nquantity is somewhat, let it be called A. Again, let the row of squares\nbe lengthened one term further, and the greatepm divst set under every\none of them as (0 . 1 . 4 . 9)/(9 . 9 . 9 . 9). The sum of the squares\nis 14, and the sum of four times the greatest is 36, whereof the third\npart is 12. But 14 is greater than 12 by two unities, that is, by two\ntwelfths of 12, that is, by 2 A. The difference therefore between the\nsum of the squares, and the sum of so many times the greatest square, is\ngreater, when the cypher is followed by three squares, than when by but\ntwo. Again, let the row have five terms, as in these numbers (0 . 1\n. 4 . 9 . 16)/(16 . 16 . 16 . 16 . 16) with the greatest five times\ndescribed, and the sum of the squares will be 30, the sum of all the\ngreatest will be 80. The third part whereof is 26(2)/(3). But 30 is\ngreater than 26(2)/(3) by 3(1)/(3), that is, by three twelfths of\ntwelve, and (1)/(3) of a twelfth, that is, by 3(1)/(3) A. Likewise in\nthe series continued to six places with the greatest six times\nsubscribed, as ( 0 . 1 . 4 . 9 . 16 . 25)/(25 . 25 . 25 . 25 . 25\n. 25) the sum of the squares is 55, and the sum of the greatest six\ntimes taken is 150, the third part whereof is 50. But 55 is greater than\n50 by 5, that is, by five-twelfths of 12, that is by 5 A. And so\ncontinually as the row groweth longer, the excess also of the aggregate\nof the squares above the third part of the aggregate of so many times\nthe greatest square, growing greater. And consequently if the number of\nthe squares were infinite, their sum would be so far from being equal to\nthe third part of the aggregate of the greatest as often taken, as that\nit would be greater than it by a quantity greater than any that can be\ngiven or named.\n\nThat which deceived you was partly this, that you think, as you do in\nyour _Elenchus_ , that these fractions (1)/(12) (1)/(18) (1)/(24)\n(1)/(30) (1)/(36) &c. are proportions, as if (1)/(12) were the\nproportion of one to twelve, and consequently (2)/(12) double the\nproportion of one to twelve; which is as unintelligible as\nschool-divinity; and I assure you, far from the meaning of Mr. Ougthred\nin the sixth chapter of his _Clavis Mathematica_, where he says that\n4(3)/(7) is the proportion of 31 to 7; for his meaning is, that the\nproportion of 4(3)/(7) to one, is the proportion of 31 to 7; whereas if\nhe meant as you do, then 8(6)/(7) should be double the proportion of 31\nto 7. Partly also because you think (as in the end of the twentieth\nproposition) that if the proportion of the numerators of these fractions\n(1)/(12) (1)/(18) (1)/(24) (1)/(30) (1)/(36) to their denominators\ndecrease eternally, they shall so vanish at last as to leave the\nproportion of the sum of all the squares to the sum of the greatest so\noften taken, (that is, an infinite number of times), as one to three, or\nthe sum of the greatest to the sum of the increasing squares, as three\nto one; for which there is no more reason than for four to one, or five\nto one, or any other such proportion. For if the proportions come\neternally nearer and nearer to the subtriple, they must needs also come\nnearer and nearer to subquadruple; and you may as well conclude thence\nthat the upper quantities shall be to the lower quantities as one to\nfour, or as one to five, &c. as conclude they are as one to three. You\ncan see without admonition, what effect this false ground of yours will\nproduce in the whole structure of your _Arithmetica Infinitorum_; and\nhow it makes all that you have said unto the end of your thirty-eighth\nproposition, undemonstrated, and much of it false.\n\nThe thirty-ninth is this other lemma: \u201c_In a series of quantities\nbeginning with a point or cypher, and proceeding according to the series\nof the cubic numbers, as O. 1. 8. 27. 64, &c. to find the proportion of\nthe sum of the cubes to the sum of the greatest cube, so many times\ntaken as there be terms_.\u201d And you conclude that \u201c_they have a\nproportion of 1 to 4_;\u201d which is false.\n\nLet the first series be of three terms subscribed with the greatest (0.\n1. 8.)/(8. 8. 8.); the sum of the cubes is nine; the sum of all the\ngreatest is 24; a quarter whereof is 6. But 9 is greater than 6 by three\nunities. An unity is something. Let it be therefore A. Therefore the row\nof cubes is greater than a quarter of three times eight, by three A.\nAgain, let the series have four terms, as (0. 1. 8. 27)/(27. 27. 27.\n27); the sum of the cubes is 36; a quarter of the sum of all the\ngreatest is twenty-seven. But thirty-six is greater than twenty-seven by\nnine, that is, by 9 A. The excess therefore of the sum of the cubes\nabove the fourth part of the sum of all the greatest, is increased by\nthe increase of the number of terms. Again, let the terms be five, as\n(0. 1. 8. 27. 64)/(64. 64. 64. 64. 64), the sum of the cubes is one\nhundred; the sum of all the greatest three hundred and twenty; a quarter\nwhereof is eighty. But one hundred is greater than eighty by twenty,\nthat is, by 20 A. So you see that this lemma also is false. And yet\nthere is grounded upon it all that which you have of comparing parabolas\nand paraboloeides with the parallelograms wherein they are accommodated.\nAnd therefore though it be true, that the parabola is (2)/(3) and the\ncubical paraboloeides (3)/(4) of their parallelograms respectively, yet\nit is more than you were certain of when you referred me, for the\nlearning of geometry, to this book of yours. Besides, any man may\nperceive that without these two lemmas (which are mingled with all your\ncompounded series with their excesses) there is nothing demonstrated to\nthe end of your book: which to prosecute particularly, were but a vain\nexpense of time. Truly, were it not that I must defend my reputation, I\nshould not have showed the world how little there is of sound doctrine\nin any of your books. For when I think how dejected you will be for the\nfuture, and how the grief of so much time irrecoverably lost, together\nwith the conscience of taking so great a stipend, for mis-teaching the\nyoung men of the University, and the consideration of how much your\nfriends will be ashamed of you, will accompany you for the rest of your\nlife, I have more compassion for you than you have deserved. Your\ntreatise of the _Angle of Contact_ , I have before confuted in a very\nfew leaves. And for that of your _Conic Sections_ , it is so covered\nover with the scab of symbols, that I had not the patience to examine\nwhether it be well or ill demonstrated.\n\nYet I observed thus much, that you find a tangent to a point given in\nthe section by a diameter given; and in the next chapter after, you\nteach the finding of a diameter, which is not artificially done.\n\nI observe also, that you call the _parameter_ an imaginary line, as if\nthe place thereof were less determined than the diameter itself; and\nthen you take a mean proportional between the intercepted diameter, and\nits contiguous ordinate line, to find it. And it is true, you find it:\nbut the parameter has a determined quantity, to be found without taking\na mean proportional. For the diameter and half the section being given,\ndraw a tangent through the vertex, and dividing the angle in the midst\nwhich is made by the diameter and tangent, the line that so divideth the\nangle, will cut the crooked line. From the intersection draw a line (if\nit be a parabola) parallel to the diameter, and that line shall cut off\nin the tangent from the vertex the parameter sought. But if the section\nbe an ellipsis, or an hyperbole, you may use the same method, saving\nthat the line drawn from the intersection must not be parallel, but must\npass through the end of the transverse diameter, and then also it shall\ncut off a part of the tangent, which measured from the vertex is the\nparameter. So that there is no more reason to call the parameter an\nimaginary line than the diameter.\n\nLastly, I observe that in all this your new method of conics, you show\nnot how to find the _burning points_, which writers call the _foci_ and\n_umbilici_ of the section, which are of all other things belonging to\nthe conics most useful in philosophy. Why therefore were they not as\nworthy of your pains as the rest, for the rest also have already been\ndemonstrated by others? You know the focus of the parabola is in the\naxis distant from the vertex a quarter of the parameter. Know also that\nthe focus of an hyperbole, is in the axis, distant from the vertex, as\nmuch as the hypotenusal of a rectangled triangle, whose one side is half\nthe transverse axis, the other side half the mean proportional between\nthe whole transverse axis and the parameter, is greater than half the\ntransverse axis.\n\nThe cause why you have performed nothing in any of your books (saving\nthat in your _Elenchus_ you have spied a few negligences of mine, which\nI need not be ashamed of) is this, that you understood not what is\n_quantity_, _line_, _superficies_, _angle_, and _proportion_; without\nwhich you cannot have the science of any one proposition in geometry.\nFrom this one and first definition of Euclid, \u201c_a point is that whereof\nthere is no part_,\u201d understood by Sextus Empiricus, as you understand\nit, that is to say misunderstood, Sextus Empiricus had utterly destroyed\nmost of the rest, and demonstrated, that in geometry there is no\nscience, and by that means you have betrayed the most evident of the\nsciences to the sceptics. But as I understand it for _that whereof no\npart is reckoned_, his arguments have no force at all, and geometry is\nredeemed. If a line have no latitude, how shall a cylinder rolling on a\nplane, which it toucheth not but in a line, describe a superficies? How\ncan you affirm that any of those things can be without quantity, whereof\nthe one may be greater or less than the other? But in the common contact\nof divers circles the external circle maketh with the common tangent a\nless angle of contact than the internal. Why then is it not quantity? An\nangle is made by the concourse of two lines from several regions,\nconcurring, by their generation, in one and the same point. How then can\nyou say the angle of contact is no angle? One measure cannot be\napplicable at once to the angle of contact, and angle of conversion. How\nthen can you infer, if they be both angles, that they must be\nhomogeneous? Proportion is the relation of two quantities. How then can\na quotient or fraction, which is quantity absolute, be a proportion? But\nto come at last to your _Epiphonema_ , wherein, though I have perfectly\ndemonstrated all those propositions concerning the proportion of\nparabolasters to their parallelograms, and you have demonstrated none of\nthem (as you cannot now but plainly see), but committed most gross\nparalogisms, how could you be so transported with pride, as insolently\nto compare the setting of them forth as mine, to the act of him that\nsteals a horse, and comes to the gallows for it. You have read, I think,\nof the gallows set up by Haman. Remember therefore also who was hanged\nupon it.\n\nAfter your dejection I shall comfort you a little, a very little, with\nthis, that whereas this eighteenth chapter containeth two problems, one,\n\u201c_the finding of a straight line equal to the crooked line of a\nsemi-parabola_;\u201d the other, \u201c_the finding of straight lines equal to the\ncrooked lines of the parabolasters, in the table of the third article of\nthe seventeenth chapter_;\u201d you have truly demonstrated that they are\nboth false; and another hath also demonstrated the same another way.\nNevertheless, the fault was not in my method, but in a mistake of one\nline for another and such as was not hard to correct; and is now so\ncorrected in the English as you shall not be able (if you can\nsufficiently imagine motions) to reprehend. The fault was this, that in\nthe triangles which have the same base and altitude with the parabola\nand parabolaster, I take for designation of the mean uniform impetus, a\nmean proportional, in the first figure, between the whole diameter and\nits half, and, in the second figure, a mean proportional between the\nwhole diameter and its third part; which was manifestly false, and\ncontrary to what I had shown in the sixteenth chapter. Whereas I ought\nto have taken the half of the base, as now I have done, and thereby\nexhibited the straight lines equal to those crooked lines, as I\nundertook to do. Which error therefore proceeded not from want of skill,\nbut from want of care; and what I promised (as bold as you say the\npromise was), I have now performed.\n\nThe rest of your exceptions to this chapter, are to these words in the\nend: \u201c_There be some that say, that though there be equality between a\nstraight and crooked line, yet now, they say, after the fall of Adam, it\ncannot be found without the especial help of divine grace_.\u201d And you say\nyou think there be none that say so. I am not bound to tell you who they\nare. Nevertheless, that other men may see the spirit of an ambitious\npart of the clergy, I will tell you where I read it. It is in the\n_Prolegomena_ of Lalovera, a Jesuit, to his Quadrature of the Circle, p.\n13 and 14, in these words: \u201c_Quamvis circuli tetragonismus sit_ \u03c6\u03cd\u03c3\u03b5\u03b9\n_possibilis, an tamen etiam_ \u03c0\u03c1\u03cc\u03c2 \u1f21\u03bc\u1fb6\u03c2, _hoc est, post Ad\u00e6 lapsum homo\nejus scientiam absque speciali divin\u00e6 grati\u00e6 auxilio, possit comparare,\njure merito inquirunt theologi, pronunciantque; hanc veritatem tanta\nesse caligine involutam ut illam videre nemo possit, nisi ignoranti\u00e6 ex\nprimi parentis pr\u00e6varicatione propagatas tenebras indebitus divin\u00e6 lucis\nradius dissipet; quod verissimum esse sentio_.\u201d Wherein I observed that\nhe, supposing he had found that quadrature, would have us believe it was\nnot by the ordinary and natural help of God (whereby one man reasoneth,\njudgeth and remembereth better than another), but by a special (which\nmust be a supernatural) help of God, that he hath given to him of the\norder of Jesus above others that have attempted the same in vain.\nInsinuating thereby, as handsomely as he could, a special love of God\ntowards the Jesuits. But you taking no notice of the word _special_,\nwould have men think I held, that human sciences might be acquired\nwithout any help of God. And thereupon proceed in a great deal of ill\nlanguage to the end of your objections to this chapter. But I shall take\nnotice of your manners for altogether in my next lesson.\n\nAt the nineteenth chapter you see not, you say, the method. Like enough.\nIn this chapter I consider not the cause of reflection, which consisteth\nin the resistance of bodies natural; but I consider the consequences,\narising from the supposition of the equality of the angle of reflection,\nto that of incidence; leaving the causes both of reflection, and of\nrefraction, to be handled together in the twenty-fourth chapter. Which\nmethod, think what you will, I still think best.\n\nSecondly, you say I define not, here, but many chapters after, what an\nangle of incidence, and what an angle of reflection is. Had you not been\nmore hasty than diligent readers, you had found that those definitions\nof the angle of incidence, and of reflection, were here set down in the\nfirst article, and not deferred to the twenty-fourth. Let not therefore\nyour own oversight be any more brought in for an objection.\n\nThirdly, you say there is no great difficulty in the business of this\nchapter. It may be so, now it is down; but before it was done, I doubt\nnot but you that are a professor would have done the same, as well as\nyou have done that of the _Angle of Contact_ , or the business of your\n_Arithmetica Infinitorum_ . But what a novice in geometry would have\ndone I cannot tell.\n\nTo the third, fourth, and fifth article, you object a want of\ndetermination; and show it by instance, as to the third article. But\nwhat those determinations should be, you determine not, because you\ncould not. The words in the third article, are first these, _if there\nfall two straight lines parallel, &c._ which is too general. It should\nbe, _if there fall the same way two straight lines parallel, &c._ Next\nthese, _their reflected lines produced inwards shall make an angle, &c._\nThis also is too general. I should have said, _their reflected lines\nproduced inwards, if they meet within, shall make an angle, &c._ Which\ndone, both this article and the fourth and fifth are fully demonstrated.\nAnd without it, an intelligent reader had been satisfied, supplying the\nwant himself by the construction.\n\nTo the eighth, you object only the too great length and labour of it,\nbecause you can do it a shorter way. Perhaps so now, as being easy to\nshorten many of the demonstrations both of Euclid, and other the best\ngeometricians that are or have been. And this is all you had to say to\nmy nineteenth chapter. Before I proceed, I must put you in mind that\nthese words of yours, \u201c_adducis malleum, ut occidas muscam_,\u201d are not\ngood Latin, _malleum affers_, _malleum adhibes_, _malleo uteris_, are\ngood. When you speak of bringing bodies animate, _ducere_ and _adducere_\nare good, for there _to bring_, is _to guide or lead_. And of bodies\ninanimate, _adducere_ is good for _attrahere_, which is to draw to. But\nwhen you bring a hammer, will you say _adduco malleum_, _I lead a\nhammer_? A man may lead another man, and a ninny may be said to lead\nanother ninny, but not a hammer. Nevertheless, I should not have thought\nfit to reprehend this fault upon this occasion in an Englishman, nor to\ntake notice of it, but that I find you in some places nibbling, but\ncauselessly, at my Latin.\n\nConcerning the twentieth chapter, before I answer to the objections\nagainst the propositions themselves, I must answer to the exception you\nfirst take to these words of mine, \u201c_Qu\u00e6 de dimensione circuli et\nangulorum pronuntiata sunt tanquam exact\u00e8 inventa, accipiat lector\ntanquam dicta_ _problematic\u00e8._\u201d To which you say thus: \u201c_We are wont in\ngeometry to call some propositions theorems, others problems, &c. of\nwhich a theorem is that wherein some assertion is propounded to be\nproved; a problem that wherein something is commanded to be done_.\u201d Do\nyou mean _to be done_, and not proved? By your favour, a problem in all\nancient writers signifies no more but a proposition uttered, to the end\nto have it, by them to whom it is uttered, examined whether it be true\nor not true, faisable or not faisable; and differs not amongst\ngeometricians from a theorem but in the manner of propounding. For this\nproposition, _to make an equilateral triangle_, so propounded they call\na problem. But if propounded thus: _If upon the ends of a straight line\ngiven be described two circles, whose radius is the same straight line,\nand there be drawn from the intersection of the circles to their two\ncentres, two straight lines, there will be made an equilateral\ntriangle_, then they call it a theorem; and yet the proposition is the\nsame. Therefore these words, _accipiat lector tanquam dicta\nproblematic\u00e8_ signify plainly this, that I would have the reader, take\nfor propounded to him to examine, whether from my construction the\nquadrature of the circle can be truly inferred or not; and this is not\nto bid him, as you interpret it, to square the circle. And if you\nbelieve that _problematic\u00e8_ signifies probably, you have been very\nnegligent in observing the sense of the ancient Greek philosophers in\nthe word problem. Therefore your _solemus in geometria_, &c. is nothing\nto the purpose; nor had it been though you had spoken more properly, and\nsaid _solent_, leaving out yourselves.\n\n[Illustration:\n\n _Six Lessons._\n _Vol. VII. Eng. p.310_II. 325_\n]\n\nMy first article hath this title, \u201c_from a false supposition, a false\nquadrature of the circle_.\u201d Seeing therefore you were resolved to show\nwhere I erred, you should have proved either that the supposition was\ntrue, and the conclusion falsely inferred, or contrarily, that though\nthe supposition be false, yet the conclusion is true; for else you\nobject nothing to my geometry, but only to my judgment, in thinking fit\nto publish it; which nevertheless you cannot justly do, seeing it was\nlikely to give occasion to ingenious men (the practice of it being so\naccurate to sense) to inquire wherein the fallacy did consist. And for\nthe problem as it was first printed, but never published, and\nconsequently ought to have passed for a private paper stolen out of my\nstudy, your public objecting against it (in the opinion of all men that\nhave conversed so much with honest company as to know what belongs to\ncivil conversation), was sufficiently barbarous in divines. And seeing\nyou knew I had rejected that proposition, it was but a poor ambition to\ntake wing as you thought to do, like beetles from my egestions. But let\nthat be as it will, you will think strange now I should resume, and make\ngood, at least against your objection, that very same proposition. So\nmuch of the figure as is needful you will find noted with the same\nletters, and placed at the end of this fifth lesson. Wherein let B I, be\nan arch not greater than the radius of the circle, and divided into four\nequal parts, in L, N, O. Draw S N, the sine of the arch B N, and produce\nit to T, so as S T be double to S N, that is, equal to the chord B I.\nDraw likewise _a_ L, the sine of the arch B L, and produce it to _c_, so\nas _a c_ be quadruple to _a_ L, that is, equal to the two chords B N, N\nI. Upon the centre N with the radius N I, draw the arch I _d_, cutting B\nU the tangent in _d_. Then will B N produced cut the arch I _d_, in the\nmidst at _o_. In the line B S produced take S _b_, equal to B S; then\ndraw and produce _b_ N, and it will fall on the point _d_. And B _d_, S\nT, will be equal; and _d_ T joined and produced will fall upon _o_, the\nmidst of the arch I _d_. Join I T, and produce it to the tangent B U in\nU. I say, that the straight line I T U shall pass through _c_. For\nseeing B S, S _b_, are equal, and the angle at S a right angle, the\nstraight lines B N, and _b_ N, are also equal, and the triangles B N\n_b_, _d_ N _o_ like and equal; and the lines _d_ T, T _o_ equal. Draw _o\ni_ parallel to _d_ U, cutting I U in _i_; and the triangles _d_ T U, _o_\nT _i_ will also be like and equal. Produce S T to the arch _d o_ I in\n_e_, and produce it further to _f_, so that the line _e f_ be equal to T\n_e_; and then S _f_ will be equal to _a c_. Therefore _f c_ joined will\nbe parallel to B S. In _c f_ produced take _f g_ equal to _c f_; and\ndraw _g m_ parallel to _d_ U, cutting I U in _m_, and _d o_ in _n_; and\nlet the intersection of the two lines _a c_ and _d o_ be in _r_; which\nbeing done, the triangles _m n_ T, _r c_ T will be like and equal.\nTherefore _m n_ and _r c_ are equal; and consequently the straight line\nI _m_ T U shall pass through _c_. Dividing therefore _a c_ in the midst\nat _t_, and S N in the midst at _l_, and joining _t_ N, L _l_, the lines\nL _l_, _t_ N, and _c_ T produced, will all meet in one and the same\npoint of B S produced; suppose at _q_. Therefore the point _q_ being\ngiven by the two known points T and I, the lines drawn from _q_ through\nequal parts of the sine of the arch B I, (for example through the points\nP, Q, R, of the sine M I), shall cut off equal arches, as B L, L N, N O,\nO I. And this is enough to make good that problem, as to your objection.\n\nThe straight line therefore B U, for any thing you have said, is proved\nequal to the arch B I, and the division of any angle given into any\nproportion given, the quadrature of any sector, and the construction of\nany equilateral polygon is also given. And though in this also I should\nhave erred, yet it cannot be denied but that I have used a more natural,\na more geometrical, and a more perspicuous method in the search of this\nso difficult a problem, than you have done in your _Arithmetica\nInfinitorum_. For though it be true that the aggregate of all the mean\nproportionals between the radius, together with an infinitely little\npart of the same, and the radius wanting an infinitely little part of\nthe same; and again, between the radius, together with two infinitely\nlittle parts, and the radius wanting two infinitely little parts, and so\non eternally, will be equal to the quadrant (a thing which every mean\ngeometrician knew before); yet it was absurd to think those means could\nbe calculated in numbers by interpoling of a symbol; especially when you\nmake that symbol to stand for a number neither true nor surd; as if\nthere were a number that could neither be uttered in words, nor not be\nuttered in words. For what else is surd, but that which cannot be\nspoken?\n\nTo the fifth article, though your discourse be long, you object but two\nthings. One is, that \u201c_Whereas the spiral of Archimedes is made of two\nmotions, one straight, the other circular, both uniform, I taking the\nmotion compounded of them both for one of those that are compounded,\nconclude falsely, that the generation of the spiral is like to the\ngeneration of the parabola_.\u201d What heed you use to take in your\nreprehensions, appears most manifestly in this objection. For I say in\nthat demonstration of mine, that _the velocity of the point A in\ndescribing the spiral increaseth continually in proportion to the\ntimes_. For seeing it goes on uniformly in the semidiameter, it is\nimpossible it should not pass into greater and greater circles,\nproportionally to the times, and consequently it must have a swifter and\nswifter motion circular, to be compounded with the uniform motion in\nevery point of the radius as it turneth about. This objection therefore\nis nothing but an effect of a will, without cause, to contradict.\n\nThe other objection is, that \u201c_Granting all to be true hitherto, yet\nbecause it depends upon the finding of a straight line equal to a\nparabolical line in the eighteenth chapter, where I was deceived, I am\nalso deceived here_.\u201d True. But because in the eighteenth chapter of\nthis English edition I have found a straight line equal to the spiral\nline of Archimedes. I must here put you in mind that by these words in\nyour objections to the fifth article at your number two, _Quatenus verum\nest, etc._, _we have demonstrated prop. 10, 11, 13_, _Arithmetica\nInfinitorum_; you make it appear that you thought your spiral (made of\narches or circles) was the true spiral of Archimedes; which is fully as\nabsurd as the quadrature of Joseph Scaliger, whose geometry you so much\ndespise.\n\nTo the sixth article, which is a digression concerning the analytics of\ngeometricians, you deny _that the efficient cause of the construction\nought to be contained in the demonstration_. As if any problem could be\nknown to be truly done, otherwise than by knowing first how, that is to\nsay, by what efficient cause, and in what manner, it is to be done.\nWhatsoever is done without that knowledge, cannot be demonstrated to be\ndone; as you see in your computation of the parabola, and paraboloeides,\nin your _Arithmetica Infinitorum_.\n\nAnd whereas I said that _the ends of all straight lines drawn from a\nstraight line, and passing through one and the same point, if their\nparts be proportional, shall be in a straight line_; is true and\naccurate; as also, _if they begin in the circumference of a circle, they\nshall also be in the circumference of another circle_. And so is this:\n_if the proportion be duplicate, they shall be in a parabola_. All this\nI say is true and accurately spoken. But this was no place for the\ndemonstration of it. Others have done it. And I perceive by that you put\nin by parenthesis (\u201c_Intelligis credo inter duas peripherias\nconcentricas_\u201d) that you understand not what I mean.\n\nHitherto reach your objections to my geometry: for the rest of your\nbook, it containeth nothing but a collection of lies, wherewith you do\nwhat you can, to extenuate as vulgar, and disgrace as false, that which\nfolloweth, and to which you have made no special objection.\n\nI shall therefore only add in this place concerning your _Analytica per\nPotestates_, that it is no art. For the rule, both in Mr. Ougthred, and\nin Des Cartes, is this: \u201c_When a problem or question is propounded,\nsuppose the thing required done, and then using a fit ratiocination, put\nA or some other vowel for the magnitude sought_.\u201d How is a man the\nbetter for this rule without another rule, how to know when the\nratiocination is fit? There may therefore be in this kind of analysis\nmore or less natural prudence, according as the analyst is more or less\nwise, or as one man in choosing of the unknown quantity with which he\nwill begin, or in choosing the way of the consequences which he will\ndraw from the hypothesis, may have better luck than another. But this is\nnothing to art. A man may sometimes spend a whole day in deriving of\nconsequences in vain, and perhaps another time solve the same problem in\na few minutes.\n\nI shall also add, that symbols, though they shorten the writing, yet\nthey do not make the reader understand it sooner than if it were written\nin words. For the conception of the lines and figures (without which a\nman learneth nothing) must proceed from words either spoken or thought\nupon. So that there is a double labour of the mind, one to reduce your\nsymbols to words, which are also symbols, another to attend to the ideas\nwhich they signify. Besides, if you but consider how none of the\nancients ever used any of them in their published demonstrations of\ngeometry, nor in their books of arithmetic, more than for the roots and\npotestates themselves; and how bad success you have had yourself in the\nunskilful using of them, you will not, I think, for the future be so\nmuch in love with them as to demonstrate by them that first part you\npromise of your _Opera Mathematica_. In which, if you make not amends\nfor that which you have already published, you will much disgrace those\nmathematicians you address your epistles to, or otherwise have\ncommended; as also the Universities, as to this kind of learning, in the\nsight of learned men beyond sea. And thus having examined your pannier\nof Mathematics, and finding in it no knowledge, neither of quantity, nor\nof measure, nor of proportion, nor of time, nor of motion, nor of any\nthing, but only of certain characters, as if a hen had been scraping\nthere; I take out my hand again, to put it into your other pannier of\ntheology, and good manners. In the mean time I will trust the objections\nmade by you the astronomer (wherein there is neither close reasoning,\nnor good style, nor sharpness of wit, to impose upon any man) to the\ndiscretion of all sorts of readers.\n\n\n ==========\n\n\n OF MANNERS.\n\n TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN\n THE UNIVERSITY OF OXFORD.\n\n\n LESSON VI.\n\nHaving in the precedent lessons maintained the truth of my geometry, and\nsufficiently made appear that your objections against it are but so many\nerrors of your own, proceeding from misunderstanding of the propositions\nyou have read in Euclid, and other masters of geometry; I leave it to\nyour consideration to whom belong, according to your own sentence, the\nunhandsome attributes you so often give me upon supposition, that you\nyourselves are in the right, and I mistaken; and come now to purge\nmyself of those greater accusations which concern my manners. It cannot\nbe expected that there should be much science of any kind in a man that\nwanteth judgment; nor judgment in a man that knoweth not the manners due\nto a public disputation in writing; wherein the scope of either party\nought to be no other than the examination and manifestation of the\ntruth. For whatsoever is added of contumely, either directly or\n_scommatically_, is want of charity and uncivil, unless it be done by\nway of reddition from him that is first provoked to it. I say unless it\nbe by way of reddition; for so was the judgment given by the emperor\nVespasian in a quarrel between a senator and a knight of Rome which had\ngiven him ill language. For when the knight had proved that the first\nill language proceeded from the senator, the emperor acquitted him in\nthese words: \u201c_Maledici senatoribus non oportere; remaledicere, fas et\ncivile esse_.\u201d Nevertheless, now-a-days, uncivil words are commonly and\nbitterly used by all that write in matter of controversy, especially in\ndivinity, excepting now and then such writers as have been more than\nordinarily well bred, and have observed how heinous and hazardous a\nthing such contumely is amongst some sorts of men, whether that which is\nsaid in disgrace be true or false. For evil words by all men of\nunderstanding are taken for a defiance, and a challenge to open war. But\nthat you should have observed so much, who are yet in your mother\u2019s\nbelly, was not a thing to be much expected.\n\nThe faults in manners you lay to my charge are these: 1. _Self-conceit._\n2. _That I will be very angry with all men that do not presently submit\nto my dictates._ 3. _That I had my doctrine concerning Vision, out of\npapers which I had in my hands of Mr. Warner\u2019s._ 4. _That I have injured\nthe universities._ 5. _That I am an enemy to religion._ These are great\nfaults; but such as I cannot yet confess. And therefore I must, as well\nas I can, seek out the grounds upon which you build your accusation.\nWhich grounds (seeing you are not acquainted with my conversation) must\nbe either in my published writings, or reported to you by honest men,\nand without suspicion of interest in reporting it. As for my\nself-conceit and ostentation, you shall find no such matter in my\nwritings. That which you allege from thence is first, that in the\nepistle dedicatory I say of my book _De Corpore_, \u201c_though it be little,\nyet it is full; and if good may go for great, great enough._\u201d When a man\npresenting a gift great or small to his betters, adorneth it the best he\ncan to make it the more acceptable; he that thinks this to be\nostentation and self-conceit, is little versed in the common actions of\nhuman life. And in the same epistle, where I say of civil philosophy:\n\u201c_It is no ancienter than my book De Cive_;\u201d these words are added: \u201c_I\nsay it provoked, and that my detractors may see they lose their\nlabour_.\u201d But that which is truly said, and upon provocation, is not\nboasting, but defence. A short sum of that book of mine, now publicly in\nFrench, done by a gentleman I never saw, carrieth the title of _Ethics\nDemonstrated_ . The book itself translated into French, hath not only a\ngreat testimony from the translator Sorberius, but also from Gassendus,\nand Mersennus, who being both of the Roman religion had no cause to\npraise it, or the divines of England have no cause to find fault with\nit. Besides, you know that the doctrine therein contained is generally\nreceived by all but those of the clergy, who think their interest\nconcerned in being made subordinate to the civil power; whose\ntestimonies therefore are invalid. Why therefore, if I commend it also\nagainst them that dispraise it publicly, do you call it boasting? \u201c_You\nhave heard_,\u201d you say, \u201c_that I had promised the quadrature of the\ncircle, &c._\u201d You heard then that which was not true. I have been asked\nsometimes, by such as saw the figure before me, what I was doing, and I\nwas not afraid to say I was seeking for the solution of that problem;\nbut not that I had done it. And afterwards being asked of the success, I\nhave said, I thought it done. This is not boasting; and yet it was\nenough, when told again, to make a fool believe it was boasting. But\nyou, the astronomer, in the epistle before your philosophical essay, say\n\u201c_You had a great expectation of my philosophical and mathematical\nworks, before they were published_.\u201d It may be so. Is that my fault? Can\na man raise a great expectation of himself by boasting? If he could,\nneither of you would be long before you raised it of yourselves; saving\nthat what you have already published, has made it now too late. For I\nverily believe there was never seen worse reasoning than in that\nphilosophical essay; which any judicious reader would believe proceeded\nfrom a prevaricator, rather than from a man that believed himself; nor\nworse principles, than those in your books of Geometry. The expectation\nof that which should be written by me, was raised partly by the\n_Cogitata Physica-Mathematica_ of Mersennus, wherein I am often named\nwith honour; and partly by others with whom I then conversed in Paris,\nwithout any ostentation. That no man has a great expectation of any\nthing that shall proceed from either of you two, I am content to let it\nbe your praise.\n\nAnother argument of my self-conceit, you take from my contempt of the\nwriters of metaphysics and school-divinity. If that be a sign of\nself-conceit, I must confess I am guilty; and if your geometry had then\nbeen published, I had contemned that as much. But yet I cannot see the\nconsequence (unless you lend me your better logic) from despising\ninsignificant and absurd language, to self-conceit.\n\nAnd again, in your _Vindici\u00e6 Academiarum_, you put for boasting, that in\nmy _Leviathan_ , page 331, I would have _that book by entire sovereignty\nimposed upon the Universities_; and in my _Review_ , p. 713, that I say\nof my _Leviathan_ , \u201c_I think it may be profitably printed, and more\nprofitably taught in the University_.\u201d The cause of my writing that\nbook, was the consideration of what the ministers before, and in the\nbeginning of, the civil war, by their preaching and writing did\ncontribute thereunto. Which I saw not only to tend to the abatement of\nthe then civil power, but also to the gaining of as much thereof as they\ncould (as did afterwards more plainly appear) unto themselves. I saw\nalso that those ministers, and many other gentlemen who were of their\nopinion, brought their doctrines against the civil power from their\nstudies in the Universities. Seeing therefore that so much as could be\ncontributed to the peace of our country, and the settlement of sovereign\npower without any army, must proceed from teaching; I had reason to\nwish, that civil doctrine were truly taught in the Universities. And if\nI had not thought that mine was such, I had never written it. And having\nwritten it, if I had not recommended it to such as had the power to\ncause it to be taught, I had written it to no purpose. To me therefore\nthat never did write anything in philosophy to show my wit, but, as I\nthought at least, to benefit some part or other of mankind, it was very\nnecessary to commend my doctrine to such men as should have the power\nand right to regulate the Universities. I say my doctrine; I say not my\n_Leviathan_ . For wiser men may so digest the same doctrine as to fit it\nbetter for a public teaching. But as it is, I believe it hath framed the\nminds of a thousand gentlemen to a conscientious obedience to present\ngovernment, which otherwise would have wavered in that point. This\ntherefore was no vaunting, but a necessary part of the business I took\nin hand. You ought also to have considered, that this was said in the\nclose of that part of my book which concerneth policy merely civil.\nWhich part, if you, the astronomer, that now think the doctrine unworthy\nto be taught, were pleased once to honour with praises printed before\nit, you are not very constant nor ingenuous. But whether you did so or\nnot, I am not certain, though it was told me for certain. If it were not\nyou, it was somebody else whose judgment has as much weight at least as\nyours.\n\nAnd for anything you have to say from your own knowledge, I remember not\nthat I ever saw either of your faces. Yet you, the professor of\ngeometry, go about obliquely to make me believe that Vindex hath\ndiscoursed with me, once at least, though I remember it not. I suppose\nit therefore true; but this I am sure is false, that either he or any\nman living did ever hear me brag of my science, or praise myself, but\nwhen my defence required it. Perhaps some of our philosophers that were\nat Paris at the same time, and acquainted with the same learned men that\nI was acquainted with, might take for bragging the maintaining of my\nopinions, and the not yielding to the reasons alledged against them. If\nthat be ostentation, they tell you the truth. But you that are so wise\nshould have considered, that even such men as profess philosophy are\ncarried away with the passions of emulation and envy (the sole ground of\nthis your accusation) as well as other men, and instanced in yourselves.\nAnd this is sufficient to shake off your aspersions of ostentation and\nself-conceit. For if I added, that my acquaintance know that I am\nnaturally of modest rather than of boasting speech, you will not believe\nit; because you distinguish not between that which is said upon\nprovocation, and that which is said without provocation, from vain\nglory.\n\nThe next accusation is: \u201c_That I will be very angry with all men that do\nnot presently submit to my dictates; and that for advancing the\nreputation of my own skill, I care not what unworthy reflections I cast\non others_.\u201d This is in the epistle placed before the _Vindici\u00e6\nAcademiarum_, subscribed by N S, as the plain song for H D in the rest\nof the book to descant upon. I know well enough the authors\u2019 names; and\nam sorry that N S has lent his name to be abused to so ill a purpose.\nBut how does this appear? What argument, what witness is there of it?\nYou offer none; nor am I conscious of any. I begin to suspect since you,\nthe professor of geometry, have in your objections to the twentieth\nchapter these words concerning \u201c_Vindex, ocularis ille testis de quo hic\nagitur, erat, ni fallor, ille ipse_,\u201d--that Vindex himself, in other\ncompany, has bestowed a visit on me. Seeing you will have me believe it,\nlet it be so; and, as it is likely, not long after my return into\nEngland. At which time (for the reputation, it seems, I had gotten by my\nboasting) divers persons that professed to love philosophy and\nmathematics, came to see me; and some of them to let me see them, and\nhear and applaud what they applauded in themselves. I see now it hath\nhappened to me with Vindex, as it happened to Dr. Harvey with Moranus.\nMoranus, a jesuit, came out of Flanders hither, especially, as he says,\nto see what learned men in divinity, ethics, physics, and geometry, were\nhere yet alive, to the end that by discoursing with them in these\nsciences, he might correct either his own, or their errors. Amongst\nothers he was brought, he says, to that most civil and renowned old man\nDr. Harvey. That is very well. And in good earnest if he had made good\nuse of the time which was very patiently afforded him, he might have\nlearned of him (or of no man living) very much knowledge concerning the\ncirculation of the blood, the generation of living creatures, and many\nother difficult points of natural philosophy. And if he had had anything\nin him but common and childish learning, he could have showed it nowhere\nmore to his advantage, than before him that was so great a judge of such\nmatters. But what did he? That precious time (which was but little,\nbecause he was to depart again presently for Flanders) he bestowed\nwholly in venting his own childish opinions, not suffering the Doctor\nscarce to speak; losing thereby the benefit he came for, and discovering\nthat he came not to hear what others could say, but to show to others\nhow learned he was himself already. Why else did he take so little time,\nand so misspend it? Or why returned he not again? But when he had talked\naway his time, and found (though patiently and civilly heard) he was not\nmuch admired, he took occasion, writing against me, to be revenged of\nDr. Harvey, by slighting his learning publicly; and tells me that his\nlearning was only experiments; which he says I say have no more\ncertainty than civil histories. Which is false. My words are: \u201c_Ante hos\nnihil certi in physica erat pr\u00e6ter experimenta cuique sua, et historias\nnaturales, si tamen et h\u00e6 dicend\u00e6 cert\u00e6 sint, qu\u00e6 civilibus historiis\ncertiores non sunt_.\u201d Where I except expressly from uncertainty the\nexperiments that every man maketh to himself. But you see the near cut,\nby which vain glory joined with ignorance passeth quickly over to envy\nand contumely.\n\nThus it seems by your own confession I was used by Vindex. He comes with\nsome of my acquaintance in a visit. What he said I know not, but if he\ndiscoursed then, as in his philosophical essay he writeth, I will be\nbold to say of myself, I was so far from morosity, or, to use his\nphrase, from being tetrical, as I may very well have a good opinion of\nmy own patience. And if there passed between us the discourse you\nmention in your _Elenchus_, page 116, it was an incivility in him so\ngreat, that without great civility I could not have abstained from\nbidding him be gone. That which passed between us you say was this: \u201c_I\ncomplained that whereas I made sense, nothing but a perception of motion\nin the organ, nevertheless, the philosophy schools through all Europe,\nled by the text of Aristotle, teach another doctrine, namely, that\nsensation is performed by species_.\u201d This is a little mistaken. For I do\nglory, not complain, that whereas all the Universities of Europe hold\nsensation to proceed from species, I hold it to be a perception of\nmotion in the organ. The answer of Vindex, you say, was: \u201c_That the\nother hypothesis, whereby sense was explicated by the principles of\nmotion, was commonly admitted here before my book came out, as having\nbeen sufficiently delivered by Des Cartes, Gassendus, and Sir Kenelm\nDigby, before I had published anything in this kind_.\u201d This then, it\nseems, was it that made me angry. Truly I remember not an angry word\nthat ever I uttered in all my life to any man that came to see me,\nthough some of them have troubled me with very impertinent discourse;\nand with those that argued with me, how impertinently soever, I always\nthought it more civility to be somewhat earnest in the defence of my\nopinion, than by obstinate and affected silence to let them see I\ncontemned them, or hearkened not to what they said. If I were earnest in\nmaking good, that the manner of sensation by such motion as I had\nexplicated in my _Leviathan_ , is in none of the authors by him named,\nit was not anger, but a care of not offending him, with any sign of the\ncontempt which his discourse deserved. But it was incivility in him to\nmake use of a visit, which all men take for a profession of friendship,\nto tell me that that which I had already published for my own, was found\nbefore by Des Cartes, Gassendus, and Sir Kenelm Digby. But let any man\nread Des Cartes; he shall find, that he attributeth no motion at all to\nthe object of sense, but an inclination to action, which inclination no\nman can imagine what it meaneth. And for Gassendus, and Sir Kenelm\nDigby, it is manifest by their writings, that their opinions are not\ndifferent from that of Epicurus, which is very different from mine. Or\nif these two, or any of those I conversed with at Paris, had prevented\nme in publishing my own doctrine, yet since it was there known and\ndeclared for mine by Mersennus in the preface to his _Ballistica_ (of\nwhich the three first leaves are employed wholly in the setting forth of\nmy opinion concerning sense, and the rest of the faculties of the soul)\nthey ought not therefore to be said to have found it out before me. And\nconsequently this answer which you say was given me by Vindex was\nnothing else but untruth and envy; and, because it was done by way of\nvisit, incivility. But you have not alleged, nor can allege, any words\nof mine, from which can be drawn that I am so angry as you say I am with\nthose that submit not to my dictates. Though the discipline of the\nUniversity be never so good; yet certainly this behaviour of yours and\nhis are no good arguments to make it thought so. But you the professor\nof geometry, that out of my words spoken against Vindex in my twentieth\nchapter, argue my angry humour, do just as well, as when (in your\n_Arithmetica Infinitorum_) from the continual increase of the excess of\nthe row of squares above the third part of the aggregate of the\ngreatest, you conclude they shall at last be equal to it. For though you\nknew that Vindex had given me first the worst words that possibly can be\ngiven, yet you would have that return of mine to be a demonstration of\nan angry humour; not then knowing what I told you even now in the\nbeginning of this lesson, of the sentence given by Vespasian. But to\nthis point I shall speak again hereafter.\n\nYour third accusation is: \u201c_That I had my doctrine of vision, which I\npretended to be my own, out of papers which I had a long time in my\nhands of Mr. Warner\u2019s_.\u201d I never had sight of Mr. Warner\u2019s papers in all\nmy life, but that of _Vision by Refraction_ (which by his approbation I\ncarried with me to Park, and caused it to be printed under his own name,\nat the end of Mersennus his _Cogitata Physico-Mathematica_, which you\nmay have there seen, and another treatise of the proportions of alloy in\ngold and silver coin; which is nothing to the present purpose). In all\nmy conversation with him, I never heard him speak of anything he had\nwritten, or was writing, _De penicillo optico_. And it was from me that\nhe first heard it mentioned that light and colour were but fancy. Which\nhe embraced presently as a truth, and told me it would remove a rub he\nwas then come to in the discovery of the place of the image. If after my\ngoing hence he made any use of it (though he had it from me, and not I\nfrom him), it was well done. But wheresoever you find my principles,\nmake use of them, if you can, to demonstrate all the symptoms of vision;\nand I will do (or rather have done and mean to publish) the same; and\nlet it be judged by that, whether those principles be of mine, or other\nmen\u2019s invention. I give you time enough, and this advantage besides,\nthat much of my optics hath been privately read by others. For I never\nrefused to lend my papers to my friends, as knowing it to be a thing of\nno prejudice to the advancement of philosophy, though it be, as I have\nfound it since, some prejudice to the advancement of my own reputation\nin those sciences; which reputation I have always postposed to the\ncommon benefit of the studious.\n\nYou say further (you the geometrician) that I had the proposition of the\nspiral line equal to a parabolical line from Mr. Robervall: true. And if\nI had remembered it, I would have taken also his demonstration; though\nif I had published his, I would have suppressed mine. I was comparing in\nmy thoughts those two lines, spiral and parabolical, by the motions\nwherewith they were described; and considering those motions as uniform,\nand the lines from the centre to the circumference, not to be little\nparallelograms, but little sectors, I saw that to compound the true\nmotion of that point which described the spiral, I must have one line\nequal to half the perimeter, the other equal to half the diameter. But\nof all this I had not one word written. But being with Mersennus and Mr.\nRobervall in the cloister of the convent, I drew a figure on the wall,\nand Mr. Robervall perceiving the deduction I made, told me that since\nthe motions which make the parabolical line, are one uniform, the other\naccelerated, the motions that make the spiral must be so also; which I\npresently acknowledged; and he the next day, from this very method,\nbrought to Mersennus the demonstration of their equality. And this is\nthe story mentioned by Mersennus, prop. 25, corol. 2, of his\n_Hydraulica_; which I know not who hath most magnanimously interpreted\nto you in my disgrace.\n\nThe fourth accusation is: \u201c_That I have injured the Universities_.\u201d\nWherein? First, \u201c_In that I would have the doctrine of my Leviathan by\nentire sovereignty be imposed on them_.\u201d You often upbraid me with\nthinking well of my own doctrine; and grant by consequence, that I\nthought this doctrine good; I desired not therefore that anything should\nbe imposed upon them, but what (at least in my opinion) was good both\nfor the Commonwealth and them. Nay more, I would have the state make use\nof them to uphold the civil power, as the Pope did to uphold the\necclesiastical. Is it not absurdly done to call this an injury? But to\nquestion, you will say, whether the civil doctrine there taught be such\nas it ought to be, or not, is a disgrace to the Universities. If that be\ncertain, it is certain also that those sermons and books, which have\nbeen preached and published, both against the former and the present\ngovernment, directly or obliquely, were not made by such ministers and\nothers as had their breeding in the Universities; though all men know\nthe contrary. But the doctrine which I would have to be taught there,\nwhat is it? It is this: \u201c_That all men that live in a Commonwealth, and\nreceive protection of their lives and fortunes from the supreme governor\nthereof, are reciprocally bound, as far as they are able, and shall be\nrequired, to protect that governor_.\u201d Is it, think you, an unreasonable\nthing to impose the teaching of such doctrine upon the Universities? Or\nwill you say they taught it before, when you know that so many men which\ncame from the Universities to preach to the people, and so many others\nthat were not ministers, did stir the people up to resist the then\nsupreme civil power? And was it not truly therefore said, that the\nUniversities receiving their discipline from the authority of the pope,\nwere the shops and operatories of the clergy? Though the competition of\nthe papal and civil power be taken now away, yet the competition between\nthe ecclesiastical and the civil power hath manifestly enough appeared\nvery lately. But neither is this an upbraiding of an University (which\nis a corporation or body artificial), but of particular men, that desire\nto uphold the authority of a Church, as of a distinct thing from the\nCommonwealth. How would you have exclaimed, if, instead of recommending\nmy _Leviathan_ to be taught in the Universities, I had recommended the\nerecting of a new and lay-university, wherein lay-men should have the\nreading of physics, mathematics, moral philosophy, and politics, as the\nclergy have now the sole teaching of divinity? Yet the thing would be\nprofitable, and tend much to the polishing of man\u2019s nature, without much\npublic charge. There will need but one house, and the endowment of a few\nprofessions. And to make some learn the better, it would do very well\nthat none should come thither sent by their parents, as to a trade to\nget their living by, but that it should be a place for such ingenuous\nmen, as being free to dispose of their own time, love truth for itself.\nIn the mean time divinity may go on in Oxford and Cambridge to furnish\nthe pulpit with men to cry down the civil power, if they continue to do\nas they did. If I had, I say, made such a motion in my _Leviathan_ ,\nthough it would have offended the divines, yet it had been no injury.\nBut it is an injury, you will say, to deny in general the utility of the\nancient schools, and to deny that we have received from them our\ngeometry. True, if I had not spoken distinctly of the schools of\nphilosophy, and said expressly, that the geometricians passed not then\nunder the name of philosophers; and that in the school of Plato (the\nbest of the ancient philosophers) none were received that were not\nalready in some measure geometricians. Euclid taught geometry; but I\nnever heard of a sect of philosophers called Euclidians, or\nAlexandrians, or ranged with any of the other sects, as Peripatetics,\nStoics, Academics, Epicureans, Pyrrhonians, &c. But what is this to the\nUniversities of Christendom? Or why are we beholden for geometry to our\nuniversities, more than to Gresham College, or to private men in London,\nParis, and other places, which never taught or learned it in a public\nschool? For even those men that living in our Universities have most\nadvanced the mathematics, attained their knowledge by other means than\nthat of public lectures, where few auditors, and those of unequal\nproficiency, cannot make benefit by one and the same lesson. And the\ntrue use of public professors, especially in the mathematics, being to\nresolve the doubts, and problems, as far as they can, of such as come\nunto them with desire to be informed.\n\nThat the Universities now are not regulated by the Pope, but by the\ncivil power, is true, and well. But where say I the contrary? And thus\nmuch for the first injury.\n\nAnother, you say, is this, that in my _Leviathan_ , p. 670, I say: \u201c_The\nprincipal schools were ordained for the three professions of Roman\nreligion, Roman law, and the art of medicine_.\u201d Thirdly, that I say:\n\u201c_Philosophy had no otherwise place there than as a hand-maid to Roman\nreligion_.\u201d Fourthly: \u201c_Since the authority of Aristotle was received\nthere, that study is not properly philosophy, but Aristotelity_.\u201d\nFifthly: \u201c_That for geometry, till of late times it had no place there\nat all_.\u201d As for the second, it is too evident to be denied; the\nfellowships having been all ordained for those professions; and (saving\nthe change of religion) being so yet. Nor hath this any reflection upon\nthe Universities, either as they now are, or as they then were, seeing\nit was not in their own power to endow themselves, or to receive other\nlaws and discipline than their founder and the state was pleased to\nordain. For the third, it is also evident. For all men know that none\nbut the Roman religion had any stipend or preferment in any university,\nwhere that religion was established? No, nor for a great while, in their\ncommonwealths; but were everywhere persecuted as heretics. But you will\nsay, the words of my _Leviathan_ are not, philosophy \u201c_had no place_,\u201d\nbut \u201c_hath no place_.\u201d Are you not ashamed to lay to my charge a mistake\nof the word _hath_ for _had_? which was either a mistake of the printer,\nor if it were so in the copy, it could be no other than the mistake of a\nletter in the writing, unless you think you can make men believe that\nafter fifty years being acquainted with what was publicly professed and\npractised in Oxford and Cambridge, I knew not what religion they were\nof. This taking of advantage from the mistake of a word, or of a letter,\nI find also in the _Elenchus_, where for _pr\u00e6tendit se scire_, there is\n_pr\u00e6tendit scire_, which you the geometrician sufficiently mumble,\nmistaking it I think for an anglicism, not for a fault of the\nimpression.\n\nTo the fourth, you pretend, that men are not now so tied to Aristotle as\nnot to _enjoy a liberty of philosophising, though it were otherwise when\nI was conversant in Magdalen Hall_. Was it so then? Then am I absolved,\nunless you can show some public act of the university made since that\ntime to alter it. For it is not enough to name some few particular\ningenuous men that usurp that liberty in their private discourses, or,\nwith connivance, in their public disputations. And your doctrine, that\neven here you avow, of _abstracted essences_, _immaterial substances_,\nand of _nunc-stans_; and your improper language in using the word (not\nas mine, for I have it nowhere) _successive eternity_; as also your\ndoctrine of _condensation_, and your arguing from natural reason the\nincomprehensible mysteries of religion, and your malicious writing, are\nvery shrewd signs that you yourselves are none of those which you say do\n_freely philosophise_; but that both your philosophy and your language\nare under the servitude, not of the Roman religion, but of the ambition\nof some other doctors, that seek, as the Roman clergy did, to draw all\nhuman learning to the upholding of their power ecclesiastical. Hitherto\ntherefore there is no injury done to the universities. For the fifth,\nyou grant it, namely, \u201c_that till of late there was no allowance for the\nteaching of geometry_.\u201d But lest you should be thought to grant me\nanything, you say, you the astronomer, \u201c_geometry hath now so much place\nin the universities, that when Mr. Hobbes shall have published his\nphilosophical and geometrical pieces, you assure yourself you shall be\nable to find a greater number in the university who will understand as\nmuch, or more, of them than he desired they should_,\u201d &c. But though\nthis be true of the _now_, yet it maketh nothing against my _then_. I\nknow well enough that Sir Henry Savile\u2019s lectures were founded and\nendowed since. Did I deny _then_ that there were in Oxford many good\ngeometricians? But I deny _now_, that either of you is of the number.\nFor my philosophical and geometrical pieces are published, and you have\nunderstood only so much in them, as all men will easily see by your\nobjections to them, and by your own published geometry, that neither of\nyou understand anything either in philosophy or in geometry. And yet you\nwould have those books of yours to stand for an argument, and to be an\nindex of the philosophy and geometry to be found in the universities.\nWhich is a greater injury and disgrace to them, than any words of mine,\nthough interpreted by yourselves.\n\nYour last and greatest accusation, or rather railing (for an accusation\nshould contain, whether true or false, some particular fact, or certain\nwords, out of which it might seem at least to be inferred), is, that I\nam an enemy to religion. Your words are: \u201c_It is said that Mr. Hobbes is\nno otherwise an enemy to the Roman religion, saving only as it hath the\nname of religion_.\u201d This is said by Vindex. You, the geometrician, in\nyour epistle dedicatory, say thus: \u201c_With what pride and imperiousness\nhe tramples on all things both human and divine, uttering fearful and\nhorrible words of God_, (peace), _of sin, of the holy Scripture, of all\nincorporeal substances in general, of the immortal soul of man, and of\nthe rest of the weighty points of religion_ (down), _it is not so much\nto be doubted as lamented_.\u201d And at the end of your objections to the\neighteenth chapter, \u201c_Perhaps you take the whole history of the fall of\nAdam for a fable, which is no wonder, when you say the rules of\nhonouring and worshipping of God are to be taken from the laws_.\u201d Down,\nI say; you bark now at the supreme legislative power. Therefore it is\nnot I, but the laws which must rate you off. But do not many other men,\nas well as you, read my _Leviathan_ , and my other books? And yet they\nall find not such enmity in them against religion. Take heed of calling\nthem all atheists that have read and approved my _Leviathan_ . Do you\nthink I can be an atheist and not know it? Or knowing it, durst have\noffered my atheism to the press? Or do you think him an atheist, or a\ncontemner of the Holy Scripture, that sayeth nothing of the Deity but\nwhat he proveth by the Scripture? You that take so heinously that I\nwould have the rules of God\u2019s worship in a Christian commonwealth taken\nfrom the laws, tell me, from whom you would have them taken? From\nyourselves? Why so, more than from me? From the bishops? Right, if the\nsupreme power of the commonwealth will have it so; if not, why from them\nrather than from me? From a consistory of presbyters by themselves, or\njoined with lay-elders, whom they may sway as they please? Good, if the\nsupreme governor of the commonwealth will have it so; if not, why from\nthem, rather than from me, or from any man else? They are wiser and\nlearneder than I. It may be so; but it has not yet appeared. Howsoever,\nlet that be granted. Is there any man so very a fool as to subject\nhimself to the rules of other men in those things which so nearly\nconcern himself, for the title they assume of being wise and learned,\nunless they also have the sword which must protect them. But it seems\nyou understand the sword as comprehended. If so, do you not then receive\nthe rules of God\u2019s worship from the civil power? Yes, doubtless; and you\nwould expect, if your consistory had that sword, that no man should dare\nto exercise or teach any rules concerning God\u2019s worship which were not\nby you allowed. See therefore how much you have been transported by your\nmalice towards me, to injure the civil power by which you live. If you\nwere not despised, you would in some places and times, where and when\nthe laws are more severely executed, be shipped away for this your\nmadness to America, I would say, to Anticyra. What luck have I, when\nthis, of the laws being the rules of God\u2019s public worship, was by me\nsaid and applied to the vindication of the Church of England from the\npower of the Roman clergy, it should be followed with such a storm from\nthe ministers, presbyterian and episcopal, of the Church of England?\nAgain, for those other points, namely, that I approve not of incorporeal\nbodies, nor of other immortality of the soul, than that which the\nScripture calleth eternal life, I do but as the Scripture leads me. To\nthe texts whereof by me alleged, you should either have answered, or\nelse forborne to revile me for the conclusions I derived from them.\nLastly, what an absurd question is it to ask me whether it be in the\npower of the magistrate, whether the world be eternal or not? It were\nfit you knew it is in the power of the supreme magistrate to make a law\nfor the punishment of them that shall pronounce publicly of that\nquestion anything contrary to that which the law hath once pronounced.\nThe truth is, you are content that the papal power be cut off, and\ndeclaimed against as much as any man will; but the ecclesiastical power,\nwhich of late was aimed at by the clergy here, being a part thereof,\nevery violence done to the papal power is sensible to them yet; like\nthat which I have heard say of a man, whose leg being cut off for the\nprevention of a gangrene that began in his toe, would nevertheless\ncomplain of a pain in his toe, when his leg was cut off.\n\nThus much in my defence; which I believe if you had foreseen, this\naccusation of yours had been left out. I come now to examine (though it\nbe done in part already) what manners those are which I find everywhere\nin your writings.\n\nAnd first, how came it into your minds that a man can be an atheist, I\nmean an atheist in his conscience? I know that David confesseth of\nhimself, upon sight of the prosperity of the wicked, that his feet had\nalmost slipped, that is, that he had slipped into a short doubtfulness\nof the Divine Providence. And if anything else can cause a man to slip\nin the same kind, it is the seeing such as you (who though you write\nnothing but what is dictated to each of you by a doctor of divinity) do\nbreak the greatest of God\u2019s commandments, which is charity, in every\nline before his face. And though such forgettings of God be somewhat\nmore than short doubtings, and sudden transportations incident to human\npassion, yet I do not for that cause think you atheists and enemies of\nreligion, but only ignorant and imprudent Christians. But how, I say,\ncould you think me an atheist, unless it were because finding your\ndoubts of the Deity more frequent than other men do, you are thereby the\napter to fall upon that kind of reproach? Wherein you are like women of\npoor and evil education when they scold; amongst whom the readiest\ndisgraceful word is whore: why not thief, or any other ill name, but\nbecause, when they remember themselves, they think that reproach the\nlikeliest to be true?\n\nSecondly, tell me what crime it was which the Latins called by the name\nof _scelus_? You think not, unless you be Stoics, that all crimes are\nequal. _Scelus_ was never used but for a crime of greatest mischief, as\nthe taking away of life and honour; and besides, basely acted, as by\nsome clandestine way, or by such a way as might be covered with a lie.\nBut when you insinuate in a writing published that I am an atheist, you\nmake yourselves authors to the multitude, and do all you can to stir\nthem up to attempt upon my life; and if it succeed, then to sneak out of\nit by leaving the fault on them that are but actors. This is to\nendeavour great mischief basely, and therefore _scelus_. Again, to\ndeprive a man of the honour he hath merited, is no little wickedness;\nand this you endeavour to do by publishing falsely that I challenge as\nmy own the inventions of other men. This is therefore _scelus_ publicly\nto tell all the world that I will be angry with all men that do not\npresently submit to my dictates; to deprive me of the friendship of all\nthe world; great damage, and a lie, and yours. For to publish any\nuntruth of another man to his disgrace, on hearsay from his enemy, is\nthe same fault as if he published it on his own credit. If I should say\nI have heard that Dr. Wallis was esteemed at Oxford for a simple fellow,\nand much inferior to his fellow-professor Dr. Ward (as indeed I have\nheard, but do not believe it), though this be no great disgrace to Dr.\nWallis, yet he would think I did him injury. Therefore public accusation\nupon hearsay is _scelus_. And whosoever does any of these things does\n_scelerat\u00e8_. But you the professors of the mathematics at Oxford, by the\nadvice of two doctors of divinity have dealt thus with me. Therefore you\nhave done, I say not foolishly, though no wickedness be without folly,\nbut _scelerat\u00e8_, \u1f45\u03c0\u03b5\u03c1 \u1f14\u03b4\u03b5\u03b9 \u03b4\u03b5\u1fd6\u03be\u03b1\u03b9.\n\nThirdly, it is ill manners, in reprehending truth, to send a man in a\nboasting way to your own errors; as you the professor of geometry have\noften sent me to your two tractates of the _Angle of Contact_ and\n_Arithmetica Infinitorum_.\n\nFourthly, it is ill manners, to diminish the just reputation of worthy\nmen after they be dead, as you the professor of geometry have done in\nthe case of Joseph Scaliger.\n\nFifthly, when I had in my _Leviathan_ suffered the clergy of the Church\nof England to escape, you did imprudently in bringing any of them in\nagain. An Ulysses upon so light an occasion would not have ventured to\nreturn again into the cave of Polyphemus.\n\nLastly, how ill does such levity and scurrility, which both of you have\nshown so often in your writings, become the gravity and sanctity\nrequisite to the calling of the ministry? They are too many to be\nrepeated. Do but consider, you the geometrician, how unhandsome it is to\nplay upon my name, when both yours and mine are plebeian names; though\nfrom Willis by Wallis, you go from yours in Wallisius. The jest of using\nat every word _mi Hobbi_, is lost to them beyond sea. But this is not so\nill as some of the rest. I will write out one of them, as it is in the\nfourth page of your _Elenchus_: \u201c_Whence it appears that your Empusa was\nof the number of those fairies which you call in English hob-goblins.\nThe word is made of_ \u1f15\u03bd and \u03c0\u03bf\u03c5\u03c2; _and thence comes the children\u2019s play\ncalled the play of Empusa, Anglic\u00e8_ (hitherto in Latin all but\n_hob-goblins_, then follows in English) _fox, fox, come out of your\nhole_ (then in Latin again), _in which the boy that is called the fox,\nholds up one foot, and jumps with the other, which in English is to\nhop_.\u201d When a stranger shall read this, and hoping to find therein some\nwitty conceit, shall with much ado have gotten it interpreted and\nexplained to him, what will he think of our doctors of divinity at\nOxford, that will take so much pains as to go out of the language they\nset forth in, for so ridiculous a purpose? You will say it is a pretty\n_paranomasia_. How you call it there I know not, but it is commonly\ncalled here a _clinch_; and such a one as is too insipid for a boy of\ntwelve years old, and very unfit for the sanctity of a minister, and\ngravity of a doctor of divinity. But I pray you tell me where it was you\nread the word _empusa_ for the boy\u2019s play you speak of, or for any other\nplay amongst the Greeks? In this (as you have done throughout all your\nother writings) you presume too much upon your first cogitations. There\nbe a hundred other scoffing passages, and ill-favoured attributes given\nme in both your writings, which the reader will observe without my\npointing to them, as easily as you would have him; and which perhaps\nsome young students, finding them full of gall, will mistake for salt.\nTherefore to disabuse those young men, and to the end they may not\nadmire such kind of wit, I have here and there been a little sharper\nwith you than else I would have been. If you think I did not spare you,\nbut that I had not wit enough to give you as scornful names as you give\nme, are you content I should try? Yes (you the geometrician will say)\ngive me what names you please, so you call me not _Arithmetica\nInfinitorum_. I will not. Nor _Angle of Contact_ ; nor _Arch Spiral_ ;\nnor _Quotient_ . I will not. But I here dismiss you both together. So go\nyour ways, you _Uncivil Ecclesiastics, Inhuman Divines, Dedoctors of\nmorality, Unasinous Colleagues, Egregious pair of Issachars, most\nwretched Vindices and Indices Academiarum_; and remember Vespasian\u2019s\nlaw, that it is uncivil to give ill language first, but civil and lawful\nto return it. But much more remember the law of God, to obey your\nsovereigns in all things; and not only not to derogate from them, but\nalso to pray for them, and as far as you can to maintain their\nauthority, and therein your own protection. And, do you hear? take heed\nof speaking your mind so clearly in answering my _Leviathan_, as I have\ndone in writing it. You should do best not to meddle with it at all,\nbecause it is undertaken, and in part published already, and will be\nbetter performed, from term to term, by one Christopher Pike.\n\n\n\n\n \u03a3\u03a4\u0399\u0393\u039c\u0391\u0399\n\n \u0391\u03b3\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03af\u03b1\u03c2, \u0391\u03b3\u03c1\u03bf\u03b9\u03ba\u03af\u03b1\u03c2, \u0391\u03bd\u03c4\u03af\u03c0\u03bf\u03bb\u03b9\u03c4\u03b5\u03af\u03b1\u03c2, \u0391\u03bc\u03b1\u03b8\u03b5\u03af\u03b1\u03c2,\n\n OR\n\n MARKS\n\n OF THE\n\n ABSURD GEOMETRY, RURAL LANGUAGE, SCOTTISH\n CHURCH POLITICS, AND BARBARISMS\n\n OF\n\n JOHN WALLIS,\n PROFESSOR OF GEOMETRY AND DOCTOR OF DIVINITY.\n\n BY\n\n THOMAS HOBBES,\n\n OF MALMESBURY.\n\n TO THE RIGHT HONOURABLE\n\n HENRY, LORD PIERREPONT,\n\n VISCOUNT NEWARK, EARL OF KINGSTON, AND\n MARQUIS OF DORCHESTER.\n\n ==========\n\nMY MOST NOBLE LORD,\n\nI did not intend to trouble your Lordship twice with this contention\nbetween me and Dr. Wallis. But your Lordship sees how I am constrained\nto it; which, whatsoever reply the Doctor makes, I shall be constrained\nto no more. That which I have now said of his Geometry, Manners,\nDivinity, and Grammar, altogether is not much, though enough. As for\nthat which I here have written concerning his Geometry, which you will\nlook for first, is so clear, that not only your Lordship, and such as\nhave proceeded far in that science, but also any man else that doth but\nknow how to add and subtract proportions, (which is taught at the\ntwenty-third proposition of the sixth of Euclid), may see the Doctor is\nin the wrong. That which I say of his ill language and politics is yet\nshorter. The rest, which concerneth grammar, is almost all another\nman\u2019s, but so full of learning of that kind, as no man that taketh\ndelight in knowing the proprieties of the Greek and Latin tongues, will\nthink his time ill bestowed in the reading it. I give the Doctor no more\nill words, but am returned from his manners to my own. Your Lordship may\nperhaps say, my compliment in my title-page is somewhat coarse; and it\nis true. But, my Lord, it is since the writing of the title-page, that I\nam returned from the Doctor\u2019s manners to my own; which are such as I\nhope you will not be ashamed to own me, my Lord, for one of\n\n Your Lordship\u2019s most humble\n\n and obedient servants,\n\n THOMAS HOBBES.\n\n\n\n\n\n\n\n\n ==========\n\n\n\n\n TO\n\n DOCTOR WALLIS,\n\n IN ANSWER TO HIS\n\n SCHOOL DISCIPLINE\n\n ---\n\nSIR,\n\nWhen unprovoked you addressed unto me, in your _Elenchus_, your harsh\ncompliment with great security, wantonly to show your wit, I confess you\nmade me angry, and willing to put you into a better way of considering\nyour own forces, and to move you a little as you had moved me, which I\nperceive my lessons to you have in some measure done; but here you shall\nsee how easily I can bear your reproaches, now they proceed from anger,\nand how calmly I can argue with you about your geometry and other parts\nof learning.\n\nI shall in the first part confer with you about your _Arithmetica\nInfinitorum_, and afterwards compare our manner of elocution; then your\npolitics; and last of all your grammar and critics. Your spiral line is\ncondemned by him whose authority you use to prove me a plagiary, (that\nis, a man that stealeth other men\u2019s inventions, and arrogates them to\nhimself), whether it be Roberval or not that writ that paper, I am not\ncertain. But I think I shall be shortly; but whosoever it be, his\nauthority will serve no less to show that your doctrine of the spiral\nline, from the fifth to the eighteenth proposition of your _Arithmetica\nInfinitorum_, is all false; and that the principal fault therein (if all\nfaults be not principal in geometry, when they proceed from ignorance of\nthe science) is the same that I objected to you in my _Lessons_. And for\nthe author of that paper, when I am certain who it is, it will be then\ntime enough to vindicate myself concerning that name of plagiary. And\nwhereas he challenges the invention of your method delivered in your\n_Arithmetica Infinitorum_, to have been his before it was yours, I\nshall, I think, by and by say that which shall make him ashamed to own\nit; and those that writ those encomiastic epistles to you ashamed of the\nhonour they meant to you. I pass therefore to the nineteenth\nproposition, which in Latin is this: your geometry!\n\n\u201c_Si proponatur series quantitatum in duplicata ratione arithmetice\nproportionalium (sive juxta seriem numerorum quadraticorum) continue\ncrescentium, a puncto vel 0 inchoatarum, (puta ut 0. 1. 4. 9. 16. etc.),\npropositum sit, inquirere quam habeat illa rationem ad seriem totidem\nmaxim\u00e6 \u00e6qualium._\n\n\u201c_Fiat investigatio per modum inductionis ut_ (_in prop. 1_)\n\n_Eritque_,\n\n(0 + 1 = 1)/(1 + 1 = 2) = (1)/(3) + (1)/(6)\n\n(0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (1)/(3) + (1)/(12)\n\n(0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) = (1)/(3) + (1)/(18) _et sic\ndeinceps_.\n\n\u201c_Ratio proveniens est ubique major quam subtripla seu (1)/(3); excessus\nautem perpetuo decrescit prout numerus terminorum augetur (puta (1)/(6)\n(1)/(12) (1)/(18) (1)/(24) etc.) aucto nimirum fractionis denominatore\nsive consequente rationis in singulis locis numero senario (ut patet) ut\nsit rationis provenientis excessus supra subtriplam, ea quam habet\nunitas ad sextuplum numeri terminorum post 0; adeoque._\u201d\n\nThat is, if there be propounded a row of quantities in duplicate\nproportion of the quantities arithmetically proportional (or proceeding\nin the order of the square numbers) continually increasing; and\nbeginning at a point or 0; let it be propounded to find what proportion\nthe row hath; to as many quantities equal to the greatest;\n\nLet it be sought by induction (as in the first proposition).\n\nThe proportion arising is everywhere greater than subtriple, or (1)/(3),\nand the excess perpetually decreaseth as the number of terms is\naugmented, as here, (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c.\ndenominator of the fraction being in every place augmented by the number\nsix, as is manifest; so that the excess of the rising proportion above\nsubtriple is the same which unity hath to six times the number of terms\nafter 0; and so.\n\nSir, in these your characters I understand by the cross + that the\nquantities on each side of it are to be added together and make one\naggregate; and I understand by the two parallel lines = that the\nquantities between which they are placed are one to another equal; this\nis your meaning, or you should have told us what you meant else; I\nunderstand also, that in the first row 0 + 1 is equal to 1, and 1 + 1\nequal to 2; and that in the second row 0 + 1 + 4 is equal to 5; and 4 +\n4 + 4 equal to 12; but (which you are too apt to grant) I understand\nyour symbols no further; but must confer with yourself about the rest.\n\nAnd first I ask you (because fractions are commonly written in that\nmanner) whether in the uppermost row (which is (0 + 1 = 1)/(1 + 1 = 2) =\n(1)/(3) + (1)/(6))(0)/(1) be a fraction, (1)/(1) be a fraction, (1)/(2)\nbe a fraction, that is to say, a part of an unit, and if you will, for\nthe cypher\u2019s sake, whether (0)/(1), be an infinitely little part of 1;\nand whether (1)/(1) or 1 divided by 1 signify an unity? if that be your\nmeaning, then the fraction (0)/(1) added to the fraction (1)/(1) is\nequal to the fraction (1)/(2): But the fraction (0)/(1) is equal to O;\ntherefore the fraction (0)/(1) + (1)/(1) is equal to the fraction\n(1)/(1); and (1)/(1) equal to (1)/(2) which you will confess to be an\nabsurd conclusion, and cannot own that meaning.\n\nI ask you therefore again, if by (0)/(1) you mean the proportion of 0 to\n1; and consequently by (1)/(1) the proportion of 1 to 1, and by (1)/(2)\nthe proportion of 1 to 2: if so, then it will follow, that if the\nproportions of 0 to 1 and of 1 to 1 be compounded by addition, the\nproportion arising will be the proportion of 1 to 2. But the proportion\nof 0 to 1 is infinitely little, that is, none. Therefore the proposition\narising by composition will be that of 1 to 1, and equal (because of the\nsymbol =) to the proportion of 1 to 2, and so 1 = 2. This also is so\nabsurd that I dare say that you will not own it.\n\nThere may be another meaning yet: perhaps you mean that the uppermost\nquantity 0 + 1 is equal to the uppermost quantity 1; and the lowermost\nquantity 1 + 1 equal to the lowermost quantity 2: which is true. But how\nthen in this equation (1)/(2) = (1)/(3) + (1)/(6)? Is the uppermost\nquantity 1 equal to the uppermost quantity 1 + 1; or the lowermost\nquantity 2 equal to the lowermost quantity 3 + 6? Therefore neither can\nthis be your meaning. Unless you make your symbols more significant, you\nmust not blame me for want of understanding them.\n\nLet us now try what better success we shall have where the places are\nthree, as here:\n\n (0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (5)/(12) = (1)/(3) + (1)/(12):\n\nIf your symbols be fractions, the compound of them by addition is\n(5)/(4), for 0(1)/(4) and (4)/(4) make (5)/(4); and consequently\n(because of the symbol = ) (5)/(4) equal to (5)/(12), which is not to be\nallowed, and therefore that was not your meaning. If you meant that the\nproportions of 0 to 4 and of 1 to 4 and of 4 to 4 compounded, is equal\nto the proportion of 5 to 12, you will fall again into no less an\ninconvenience. For the proportion arising out of that composition will\nbe the proportion of 1 to 4. For the proportion of 0 to 4 is infinitely\nlittle. Then to compound the other two, set them in this order 1. 4. 4.\nand you have a proportion compounded of 1 to 4 and of 4 to 4, namely,\nthe proportion of the first to the last, which is of 1 to 4, which must\nbe equal, by this your meaning, to the proportion of 5 to 12, and\nconsequently as 5 to 12, so is 1 to 4, which you must not own. Lastly,\nif you mean that the uppermost quantities to the uppermost, and the\nlowermost to the lowermost in the first equation are equal, it is\ngranted, but then again in the second equation it is false. It concerns\nyour fame in the mathematics to look about how to justify these\nequations which are the premises to your conclusion following, namely,\nthat the proportion arising is every where greater than sub-triple, or a\nthird; and that the excess (that is, the excess above subtriple)\nperpetually decreaseth as the number of terms is augmented, as here\n(1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. which I will show you\nplainly is false.\n\nBut first I wonder why you were so angry with me for saying you made\nproportion to consist in the quotient, as to tell me it was abominably\nfalse, and to justify it, cite your own words _penes quotientem_; do not\nyou say here, the proportion is everywhere greater than subtriple, or\n(1)/(3)? And is not (1)/(3) the quotient of 1 divided by 3? You cannot\nsay in this place that _penes_ is understood; for if it were expressed\nyou would not be able to proceed.\n\nBut I return to your conclusion, that the excess of the proportion of\nthe increasing quantities above the third part of so many times the\ngreatest, decreaseth, as (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30),\n&c. For by this account in this row (0 + 1)/(1 + 1) = (1)/(2) where the\nquantity above exceeds the third part of the quantities below by\n(1)/(3), you make (1)/(3) equal to (1)/(6), which you do not mean. It\nmay be said your meaning is, that the proportion of 1 to the subtriple\nof 2 which is (2)/(3), exceedeth what? I cannot imagine what, nor\nproceed further where the terms be but two. Let us therefore take the\nsecond row, that is, (0 + 1 + 4)/(4 + 4 + 4) = (5)/(12). The sum above\nis 5, the sum below is 12, the third part whereof is 4; if you mean,\nthat the proportion of 5 to 4 exceeds the proportion of 4 to 12 (which\nis subtriple) by (1)/(12), you are out again. For 5 exceeds 4 by unity,\nwhich is (12)/(12). I do not think you will own such an equation as\n(12)/(12) = (1)/(12) Therefore I believe you mean (and your next\nproposition assures me of it), that the proportion of 5 to 4 exceeds\nsubtriple proportion by the proportion of 1 to 12; if you do so, you are\nyet deceived.\n\nFor if the proportion of 5 to 4 exceeds subtriple proportion by the\nproportion of 1 to 12, then subtriple proportion, that is, of 4 to 12\nadded to the proportion of 1 to 12 must make the proportion of 5 to 4.\nBut if you look on these quantities, 4, 12, 144, you will see, and must\nnot dissemble, that the proportion of 4 to 12 is subtriple, and the\nproportion of 12 to 144 is the same with that of 1 to 12. Therefore by\nyour assertion it must be as 5 to 4 so 4 to 144, which you must not own.\n\nAnd yet this is manifestly your meaning, as appeareth in these words:\n\u201c_Ut sit rationis provenientis excessus supra subtriplam ea quam habet\nunitas ad sextuplum numeri terminorum post 0, adeoque_,\u201d which cannot be\nrendered in English, nor need to be. For you express yourself in the\ntwentieth proposition very clearly; I noted it only that you may be more\nmerciful hereafter to the stumblings of a hasty pen. For _excessus ea\nquam_ does not well, nor is to be well excused by _subauditur ratio_.\nYour twentieth proposition is this:\n\n\u201c_Si proponatur series quantitatum in duplicata ratione arithmetice\nproportionalium (sive juxta seriem numerorum quadraticorum) continue\ncrescentium, a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem\ntotidem maxim\u00e6 \u00e6qualium subtriplam superabit; eritque excessus ea ratio\nquam habet unitas ad sextuplum numeri terminorum post 0, sive quam habet\nradix quadratica termini primi post 0 ad sextuplum radicis quadratic\u00e6\ntermini maximi._\u201d\n\nThat is, if there be propounded a row of quantities in duplicate\nproportion of arithmetically-proportionals (or according to the row of\nsquare numbers) continually increasing, and beginning with a point or O.\nThe proportion of that row to a row of so many equals to the greatest,\nshall be greater than subtriple proportion, and the excess shall be that\nproportion which unity hath to the sextuple of the number of terms after\n0, or the same which the square root of the first number after 0, hath\nto the sextuple of the square root of the greatest.\n\nFor proof whereof you have no more here than _patet ex pr\u00e6cedentibus_;\nand no more before but _adeoque_. You do not well to pass over such\ncurious propositions so slightly; none of the ancients did so, nor, that\nI remember, any man before yourself. The proposition is false, as you\nshall presently see.\n\nTake, for example, any one of your rows: as (0 + 1 + 4)/(4 + 4 + 4). By\nthis proportion of yours 1 + 4, which makes 5, is to 12 in more than\nsubtriple proportion; by the proportion of 1 to the sextuple of 2 which\nis 12. Put in order these three quantities 5, 4, 12, and you must see\nthe proportion of 5 to 12 is greater than the proportion of 4 to 12,\nthat is, subtriple proportion, by the proportion of 5 to 4. But by your\naccount the proportion of 5 to 4 is greater than that of 4 to 12 by the\nproportion of 1 to 12. Therefore, as 5 to 4 so is 1 to 12, which is a\nvery strange paradox.\n\nAfter this you bring in this consectary: \u201c_Cum autem crescente numero\nterminorum excessus ille supra rationem subtriplam continue minuatur, ut\ntandem quovis assignabili minor evadat (ut patet) si in infinitum\nproducatur, prorsus evaniturus est. Adeoque._\u201d\n\nThat is, seeing as the number of terms increaseth, that excess above\nsubtriple proportion continually decreaseth, so as at length it becomes\nless than any assignable (as is manifest) if it be produced infinitely,\nit shall utterly vanish, and so. And so what?\n\nSir, this consequence of yours is false. For two quantities being given,\nand the excess of the greater above the less, that excess may\ncontinually be decreased, and yet never quite vanish. Suppose any two\nunequal quantities differing by more than an unit, as 3 and 6, the\nexcess 3, let 3 be diminished, first by an unit, and the excess will be\n2, and the quantities will be 3 and 5; 5 is greater than 4, the excess\n1. Again, let 1 be diminished and made (1)/(2), the excess 4 and the\nquantities 3 and 4(1)/(2), 4(1)/(2) is yet greater than 4. Again\ndiminish the excess to (1)/(4), the quantities will be 3 and 4(1)/(4),\nyet still 4(1)/(4) is greater than 4. In the same manner you may proceed\nto (1)/(8) (1)/(16) (1)/(32), &c. infinitely; and yet you shall never\ncome within an unit (though your unit stand for 100 miles) of the lesser\nquantity propounded 3, if that 3 stands for 300 miles. The excesses\nabove subtriple proportion do not decrease in the manner you say it\ndoes, but in the manner which I now shall show you.\n\nIn the first row (0 + 1)/(1 + 1) a third of the quantities below is\n(2)/(3), set in order these three quantities 1 (2)/(9) (2)/(3). The\nfirst is 1, equal to the sum above, the last is (2)/(3), equal to the\nsubtriple of the sum below. The middlemost is (2)/(9) subtriple to the\nlast quantity (2)/(3). The excess of the proportion of 1 to (2)/(3)\nabove the subtriple proportion of (2)/(9) to (2)/(3) is the proportion\nof 1 to (2)/(9) that is of 9 to 2, that is, of 18 to 4.\n\nSecondly, in the second row, which is (0 + 1 + 4)/(4 + 4 + 4), a third\nof the sum below is 4, the sum above is 5. Set in order these\nquantities, 1, 5, 4, 12. There the proportion of 15 to 12 is the\nproportion of 5 to 4. The proportion of 4 to 12 is subtriple; the excess\nis the proportion of 15 to 4, which is less than the proportion of 18 to\n4, as it ought to be; but not less by the proportion of (1)/(6) to\n(1)/(12) as you would have it.\n\nThirdly, in the third row, which is (0 + 1 + 4 + 9)/(9 + 9 + 9 + 9). A\nthird of the sum below is 12, the sum above is 14. Set in order these\nquantities, 42, 4, 12. There the proportion of 42 to 12 is the same with\nthat of 14 to 4. And the proportion of 4 to 12 subtriple, less than the\nformer excess of 15 to 4. And so it goes on decreasing all the way in\nthis manner, 18 to 4, 15 to 4, 14 to 4, &c. which differs very much from\nyour 1 to 6, 1 to 12, 1 to 18, &c. and the cause of your mistake is\nthis: you call the twelfth part of twelve (1)/(12), and the eighteenth\npart of thirty-six you call (1)/(18), and so of the rest. But what need\nof all those equations in symbols, to show that the proportion\ndecreases; is there any man can doubt, but that the proportion of 1 to 2\nis greater than that of 5 to 12, or that of 5 to 12 greater than that of\n14 to 36, and so on continually forwards; or could you have fallen into\nthis error, unless you had taken, as you have done in very many places\nof your _Elenchus_, the fractions (1)/(6) and (1)/(12), &c. which are\nthe quotients of 1 divided by 6 and 12, for the very proportions of 1 to\n6 and 1 to 12. But notwithstanding the excess of the proportions of the\nincreasing quantities, to subtriple proportion decrease, still, as the\nnumber of terms increaseth, and that what proportions soever I shall\nassign, the decrement will in time (in time, I say, without proceeding\n_in infinitum_) produce a less, yet it does not follow that the row of\nincreasing quantities shall ever be equal to the third part of the row\nof so many equals to the last or greatest. For it is not, I hope, a\nparadox to you, that in two rows of quantities the proportion of the\nexcesses may decrease, and yet the excesses themselves increase, and do\nperpetually.\n\nFor in the second and third rows, which are (0 + 1 + 4 = 5)/(4 + 4 + 4\n= 12) and (0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) 5 exceeds the third\npart of 12 by a quarter of the square of 4, and 14 exceeds the third\npart of 36 by 2 quarters of the square of 4, and proceeding on, the sum\nof the increasing quantities where the terms are 5 (which sum is 30)\nexceedeth the third part of those below, (those below are 80, and their\nthird part 26(2)/(3)) by 3 quarters and (1)/(2) a quarter of the square\nof 4, and when the terms are 6, the quantities above will exceed the\nthird part of them below by 5 quarters of the square of 4. Would you\nhave men believe, that the further they go, the excess of the increasing\nquantities above the third part of those below shall be so much the\nless? And yet the proportions of those above, to the thirds of those\nbelow, shall decrease eternally; and therefore your twenty-first\nproposition is false, namely this:\n\n\u201c_Si proponatur series infinita quantitatum in duplicata ratione\narithmetice proportionalium (sive juxta seriem numerorum quadraticorum),\ncontinue crescentium a puncto sive 0 inchoatarum; erit illa ad seriem\ntotidem maxim\u00e6 \u00e6qualium, ut 1 ad 3._\u201d\n\nThat is, if an infinite row of quantities be propounded in duplicate\nproportion of arithmetically-proportionals (or according to the row of\nquadratic numbers), continually increasing and beginning from a point or\n0; that row shall be to the row of as many equals to the greatest, as 1\nto 3. This is false, _ut patet ex pr\u00e6cedentibus_; and, consequently, all\nthat you say in proof of the proportion of your _parabola_ to a\n_parallelogram_, or of the _spiral_ (the true _spiral_) to a _circle_ is\nin vain.\n\nBut your spiral puts me in mind of what you have under-written to the\ndiagram of your proposition 5. _The spiral, in both figures, was to be\ncontinued whole to the middle, but, by the carelessness of the graver,\nit is in one figure_ manca, _in the other_ intercisa.\n\nTruly, Sir, you will hardly make your reader believe that a graver could\ncommit those faults without the help of your own copy, nor that it had\nbeen in your copy, if you had known how to describe a spiral line then\nas now. This I had not said, though truth, but that you are pleased to\nsay, though not truth, that I attributed to the printer some faults of\nmine.\n\nI come now to the thirty-ninth proposition, which is this:\n\n\u201c_Si proponatur series quantitatum in triplicata ratione arithmetice\nproportionalium (sive juxta seriem numerorum cubicorum), continue\ncrescentium a puncto sive 0 inchoatarum (puta ut 0, 1, 8, 27, etc.),\npropositum sit inquirere quam habeat series illa rationem ad seriem\ntotidem maxim\u00e6 \u00e6qualium_:\n\n\u201c_Fiat investigatio per modum inductionis_ (_ut in prop. 1, et prop.\n19_):\n\n _Eritque_\n\n (0 + 1 = 1)/(1 + 1 = 2) = (2)/(4) = (1)/(4) + (1)/(4)\n\n (0 + 1 + 8 = 9)/(8 + 8 + 8 = 24) = (1)/(4) + (1)/(8)\n\n (0 + 1 + 8 + 27 = 36)/(27 + 27 + 27 + 27 = 108) = (4)/(12)\n = (1)/(4) + (1)/(12)\n\n _Et sic deinceps._\n\n\u201c_Ratio proveniens est ubique major quam subquadrupla, sive (1)/(4).\nExcessus autem perpetuo decrescit, pro ut numerus terminorum augetur,\nputa (1)/(4) (1)/(8) (1)/(12) (1)/(16) etc. Aucto nimirum fractionis\ndenominatore sive consequente rationis in singulis locis numero\nquaternatio, ut patet, ut sit rationis provenientis excessus supra\nsubquadruplam ea quam habet unitas ad quadruplum numeri terminorum post\n0 adeoque._\u201d\n\nThat is, if a row of quantities be propounded in triplicate proportion\nof arithmetically proportionals (or according to the row of cubic\nnumbers), continually increasing, and beginning from a point or 0, as 0,\n1, 8, 27, 64, &c., let it be propounded to inquire, what proportion that\nrow hath to a row of as many equals to the greatest.\n\nBe it sought by way of induction, as in proposition 1 and 19.\n\nThe proposition arising is everywhere greater than subquadruple, or\n(1)/(4), and the excess perpetually decreaseth as the number of terms\nincreaseth, as (1)/(4) (1)/(8) (1)/(12) (1)/(16) (1)/(20) &c. The\ndenominator of the fraction, or consequent of the proportion, being in\nevery place augmented by the number 4, as is manifest, so that the\nexcess of the arising proportion above subquadruple is the same with\nthat which an unit hath to the quadruple of the number of the terms\nafter 0, and so. Here are just the same faults which are in proposition\n19.\n\nFor, if (0)/(1) be a fraction, and (1)/(1) be a fraction, and (1)/(2) be\nanother fraction, then this equation (0 + 1 = 1)/(1 + 1 = 2) is false.\nFor this fraction (0)/(1) is equal to 0; and, therefore, we have (1)/(1)\n= (1)/(2), that is, the whole equal to half. But perhaps you do not mean\nthem fractions, but proportions; and, consequently, that the proportion\nof 0 to 1, and of 1 to 1, compounded by addition (I say by addition, not\nthat I, but that you think there is a composition of proportions by\nmultiplication, which I shall show you anon is false), must be equal to\nthe proportion of 1 to 2, which cannot be. For the proportion of 0 to 1\nis infinitely little, that is, none at all; and, consequently, the\nproportion of 1 to 1 is equal to the proportion of 1 to 2, which is\nagain absurd. There is no doubt but the whole number of 0 + 1 is equal\nto 1, and the whole number of 1 + 1 equal to 2. But, reckoning them as\nyou do, not for whole numbers, but for fractions or proportions, the\nequations are false.\n\nAgain, your second equation, (2)/(4) = (1)/(4) + (1)/(4), though meant\nof fractions, that is, of quotients, it be true, and serve nothing to\nyour purpose, yet, if it be meant of proportions, it is false. For the\nproportion of 1 to 4, and of 1 to 4 being compounded, are equal to the\nproportion of 1 to 16, and so you make the proportion of 2 to 4 equal to\nthe proportion of 1 to 16, where, as it is but subquaduplicate, as you\ncall it, or the quarter of it, as I call it. And, in the same manner,\nyou may demonstrate to yourself the same fault in all the other rows of\nhow many terms soever they consist. Therefore, you may give for lost\nthis thirty-ninth proposition, as well as all the other thirty-eight\nthat went before. As for the conclusion of it, which is, _that the\nexcess of the arising proportion_, &c. They are the words of your\nfortieth proposition, where you express yourself better, and make your\nerror more easy to be detected.\n\nThe proposition is this:\n\n\u201c_Si proponatur series quantitatum in triplicata ratione arithmetice\nproportionalium (sive juxta seriem numerorum cubicorum) continue\ncrescentium a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem\ntotidem maxim\u00e6 \u00e6qualium subquadruplam superabit; eritque excessus ea\nratio quam habet unitas ad quadruplum numeri terminorum post 0; sive\nquam habet radix cubica termini primi post 0 ad quadruplum radicis\ncubic\u00e6 termini maximi. Patet ex pr\u00e6cedente._\n\n\u201c_Quum autem crescente numero terminorum excessus ille supra rationem\nsubquadruplam ita continuo minuatur, ut tandem quolibet assignabili\nminor evadat, ut patet, si in infinitum procedatur, prorsus evaniturus\nest, adeoque._\n\n\u201c_Patet ex propositione_ _pr\u00e6cedente._\u201d\n\nThat is, if a row of quantities be propounded in triplicate proportion\nof arithmetically proportionals (or according to the row of cubic\nnumbers), continually increasing, and beginning at a point or 0; the\nproportion which that row hath to a row of as many equals to the\ngreatest, is greater than subquadruple proportion; and the excess is\nthat proportion which one unit hath to the quadruple of the number of\nterms after 0; or, which the cubic root of the first term after 0 hath\nto the quadruple of the root of the greatest term.\n\nIt is manifest by the precedent propositions.\n\nAnd, seeing the number of terms increasing, that excess above quadruple\nproportion doth so continually decrease, as that, at length, it becomes\nless than any proportion that can be assigned, as is manifest, if the\nproceeding be infinite, it shall quite vanish. And so\n\nThis conclusion was annexed to the end of your thirty-ninth proposition,\nas there proved. What cause you had to make a new proposition of it,\nwithout other proof than _patet ex pr\u00e6cedente_, I cannot imagine. But,\nhowsoever, the proposition is false.\n\nFor example, set forth any of your rows, as this of fewer terms:\n\n (0 + 1 + 8 + 27 = 36)/((27 + 27 + 27 + 27 =\n 108)\n\nThe row above is 36, the fourth part of the row below is 27. The\nquadruple of the number of terms after 0 is 12. Then, by your account,\nthe proportion of 36 to 108 is greater than subquadruple proportion by\nthe proportion of 1 to 12. For trial whereof, set in order these three\nquantities, 36, 27, 108. The proportion of 36 (the uppermost row) to 108\n(the lowermost row) is compounded by addition of the proportions 36 to\n27, and 27 to 108. And the proportion of 36 to 108, exceedeth the\nproportion of 27 to 108, by the proportion of 36 to 27. But the\nproportion of 27 to 108 is subquadruple proportion. Therefore, the\nproportion of 36 to 108 exceedeth subquadruple proportion, by the\nproportion of 36 to 27. And, by your account, by the proportion of 1 to\n12; and, consequently, as 36 to 27, so is 1 to 12. Did you think such\ndemonstrations as these should always pass?\n\nThen, for your inference from the decrease of the proportions of the\nexcess, to the vanishing of the excess itself, I have already showed it\nto be false; and by consequence that your next proposition, namely, the\nfortieth, is also false.\n\nThe proposition is this:\n\n\u201c_Si proponatur series infinita quantitatum in triplicata ratione\narithmetice proportionalium (sive juxta seriem numerorum cubicorum),\ncontinue crescentium a puncto sive 0 inchoatarum, erit illa ad seriem\ntotidem maxim\u00e6 \u00e6qualium, ut 1 ad 4, patet ex pr\u00e6cedente._\u201d\n\nThat is, if there be propounded an infinite row of quantities in\ntriplicate proportion of arithmetically proportionals (or according to\nthe row of cubic numbers), continually increasing, and beginning at a\npoint or 0; it shall be to the row of as many equals to the greatest as\n1 to 4. Manifest out of the precedent proposition.\n\nEven as manifest as that 36, 27, 1, 12, are proportionals. Seeing,\ntherefore, your doctrine of the spiral lines and the spaces is given by\nyourself for lost, and a vain attempt, your first forty-one propositions\nare undemonstrated, and the grounds of your demonstrations all false.\nThe cause whereof is partly your taking quotient for proportion, and a\npoint for 0, as you do in the first, sixteenth, and fortieth\npropositions, and in other places where you say, _beginning at a point\nor 0_, though now you deny you ever said either. There be very many\nplaces in your _Elenchus_, where you say both; and have no excuse for\nit, but that, in one of the places, you say the proportion is _penes\nquotientem_, which is to the same or no sense.\n\nYour forty-second proposition is grounded on the fortieth; and\ntherefore, though true, and demonstrated by others, is not demonstrated\nby you.\n\nYour forty-third is this:\n\n\u201c_Pari methodo invenietur ratio seriei infinit\u00e6 quantitatum arithmetice\nproportionalium in ratione quadruplicata, quintuplicata, sextuplicata,\netc., arithmetice proportionalium a puncto seu 0 inchoatarum, ad seriem\ntotidem maxim\u00e6 \u00e6qualium. Nempe in quadruplicata erit, ut 1 ad 5; in\nquintuplicata, ut 1 ad 6; in sextuplicata, ut 1 ad 7. Et sic deinceps._\u201d\n\nThat is, by the same method will be found, the proportion of an infinite\nrow of arithmetically proportionals, in proportion quadruplicate,\nquintuplicate, sextuplicate, &c., of arithmetically proportionals,\nbeginning at a point or 0, to the row of as many equals to the greatest;\nnamely, in quadruplicate, it shall be as 1 to 5; in quintuplicate, as 1\nto 6; in sextuplicate, as 1 to 7; and so forth.\n\nBut by the same method that I have demonstrated, that the propositions\n19, 20, 21, 39, 40, and 41, are false: any man else, that will examine\nthe forty-third may find it false also. And, because all the rest of the\npropositions of your _Arithmetica Infinitorum_ depend on these, they may\nsafely conclude, that there is nothing demonstrated in all that book,\nthough it consist of 194 propositions. The proportions of your\nparabolocides to their parallelograms are true, but the demonstrations\nfalse, and infer the contrary. Nor were they ever demonstrated (at least\nthe demonstrations are not extant) but by me; nor can they be\ndemonstrated, but upon the same grounds, concerning the nature of\nproportion, which I have clearly laid, and you not understood. For, if\nyou had, you could never have fallen into so gross an error as is this\nyour book of _Arithmetica Infinitorum_, or that of the angle of contact.\nYou may see by this, that your symbolic method is not only not at all\ninventive of new theorems, but also dangerous in expressing the old. If\nthe best masters of symbolics think for all this you are in the right,\nlet them declare it. I know how far the analysis by the powers of the\nlines extendeth, as well as the best of your half-learnt epistlers, that\napprove so easily of such analogisms as those, 5, 4, 1, 12, and 36, 27,\n1, 12, &c.\n\nIt is well for you that they who have the disposing of the professors\u2019\nplaces take not upon them to be judges of geometry. For, if they did,\nseeing you confess you have read these doctrines in your school, you had\nbeen in danger of being put out of your place.\n\nWhen the author of the paper wherein I am called Plagiary, and wherein\nthe honour is taken from you of being the first inventor of these fine\ntheorems, shall read this that I have here written, he will look to get\nno credit by it; especially if it be Roberval, which methinks it should\nnot be. For he understands what proportion is, better than to make 5 to\n4 the same with 1 to 12. Or to make, again, the proportion of 36 to 27\nthe same with that of 1 to 12; and innumerable _disproportionalites_\nthat may be inferred from the grounds you go on. But if it be Roberval\nindeed, that snatches this invention from you, when he shall see this\nburning coal hanging at it, he will let it fall again, for fear of\nspoiling his reputation.\n\nBut what shall I answer to the authority of the three great\nmathematicians that sent you those encomiastic letters. For the first,\nwhom you say I use to praise, I shall take better heed hereafter of\npraising any man for his learning whilst he is young, further than that\nhe is in a good way. But it seems he was in too ready a way of thinking\nvery well of himself, as you do of yourself. For the muddiness of my\nbrain I must confess it; but, Sir, ought not you to confess the same of\nyours? No, men of your tenets use not to do so. He wonders, say you, you\nthought it worth the while to foul your fingers about such a piece. It\nis well; every man abounds in his own sense. If you and I were to be\ncompared by the compliments that are given us in private letters, both\nyou and your complimentors would be out of countenance; which\ncompliments, besides that which has been printed and published in the\ncommendations of my writings, if it were put together, would make a\ngreater volume than either of your libels. And truly, Sir, I had never\nanswered your Elenchus as proceeding from Dr. Wallis, if I had not\nconsidered you also as the minister to execute the malice of that sort\nof people that are offended with my _Leviathan_.\n\nAs for the judgment of that public Professor that makes himself a\nwitness of the goodness of your geometry, a man may easily see by the\nletter itself that he is a dunce. And for the English person of quality\nwhom I know not, I can say no more yet than I can say of all three, that\nhe is so ill a geometrician, as not to detect those gross paralogisms as\ninfer that 5 to 4 and 1 to 12 are the same proportion. He came into the\ncry of those whom your title had deceived.\n\nAnd now I shall let you see that the composition of proportion by\nmultiplication, as it is in the fifth definition of the sixth element,\nis but another way of adding proportions one to another. Let the\nproportions be of 2 to 3, and of 4 to 5. Multiply 2 into 4 and 3 into 5,\nthe proportion arising is of 8 to 15. Put in order these three\nquantities, 8, 12, 15. The proportion therefore of 8 to 15, compounded\nof the proportions of 8 to 12, (that is, of 2 to 3) and of 12 to 15,\nthat is, of 4 to 5 by addition. Again, let the proportion be of 2 to 3,\nand of 4 to 5, multiply 2 into 5 and 3 into 4, the proportions arising\nis of 10 to 12. Put in order these three numbers, 10, 8, 12. The\nproportion 10 to 12 is compounded of the proportions of 10 to 8, that is\nof 5 to 4, and of 8 to 12, that is, of 2 to 3 by addition. I wonder you\nknow not this.\n\nI find not any more clamour against me for saying the proportion of 1 to\n2 is double to that of 1 to 4.\n\nYour book, you speak of, concerning proportion against _Meibomius_ is\nlike to be very useful when neither of you both do understand what\nproportion is.\n\nYou take exceptions, as that I say, that _Euclid_ has but one word for\n_double_ and _duplicate_; which nevertheless was said very truly, and\nthat word is sometimes \u03b4\u03b9\u03c0\u03bb\u03ac\u03c3\u03b9\u03bf\u03c2 and sometimes \u03b4\u03b9\u03c0\u03bb\u03ac\u03c3\u03b9\u03c9\u03bd. And you think\nyou have come off handsomely with asking me whether \u03b4\u03b9\u03c0\u03bb\u03ac\u03c3\u03b9\u03bf\u03c2 and\n\u03b4\u03b9\u03c0\u03bb\u03b1\u03c3\u03af\u03c9\u03bd be one word.\n\nNor are you now of the mind you were, that a point is not _quantity\nunconsidered_, but that in an infinite series it may be safely\nneglected. What is _neglected_ but unconsidered.\n\nNor do you any more stand to it, that the _quotient_ is the\n_proportion_. And yet were these the main grounds of your _Elenchus_.\n\nBut you will say, perhaps, I do answer to the defence you have now made\nin this your _School Discipline_: \u2019tis true. But \u2019tis not because you\nanswer never a word to my former objections against these propositions\n19, 89; but because you do so shift and wriggle, and throw out ink, that\nI cannot perceive which way you go, nor need I, especially in your\nvindication of your _Arithmetica Infinitorum_. Only I must take notice\nthat in the end of it, you have these words, \u201cWell, _Arithmetica\nInfinitorum_ _is come off clear_\u201d You see the contrary. For sprawling is\nno defence.\n\nIt is enough to me that I have clearly demonstrated both before\nsufficiently, and now again abundantly, that your book of _Arithmetica\nInfinitorum_ is all nought from the beginning to the end, and that\nthereby I have effected that your authority shall never hereafter be\ntaken for a prejudice. And, therefore, they that have a desire to know\nthe truth in the questions between us, will henceforth, if they be wise,\nexamine my geometry, by attentive reading me in my own writings, and\nthen examine, whether this writing of yours confute or enervate mine.\n\nThere is in my fifth lesson a proposition, with a diagram to it, to make\ngood, I dare say, at least against you, my twentieth chapter concerning\nthe dimension of a circle. If that demonstration be not shown to be\nfalse, your objections to that chapter, though by me rejected, come to\nnothing. I wonder why you pass it over in silence. But you are not, you\nsay, bound to answer it. True, nor yet to defend what you have written\nagainst me.\n\nBefore I give over the examination of your geometry, I must tell you\nthat your words, (p. 101 of your _School Discipline_), against the first\ncorollary are untrue.\n\nYour words are these: \u201c_you affirm that the proportion of the parabola A\nB I to the parabola A F K is triplicate to the proportion of the time A\nB to A F, as it is in the English_.\u201d This is not so. Let the reader turn\nto the place and judge. And going on you say, \u201c_or of the impetus B I to\nF K as it is in the Latin_.\u201d Nay, as it is in the English, and the other\nin the Latin. It is but your mistake; but a mistake is not easily\nexcused in a false accusation.\n\nYour exception to my saying, \u201c_that the differences of two quantities is\ntheir proportion_,\u201d (when they differ, as the no difference, when they\nbe equal), might have been put in amongst other marks of your not\nsufficiently understanding the Latin tongue. _Differre_ and\n_differentia_ differ no more than _vivere_ and _vita_, which is nothing\nat all, but as the other words require that go with them, which other\nwords you do not much use to consider. But _differre_ and _the quantity\nby which they differ_, are quite of another kind. _Differre_ (\u03c4\u1f78\n\u03b4\u03b9\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9\u03bd, \u03c4\u1f78 \u1f51\u03c0\u03b5\u03c1\u03ad\u03c7\u03b5\u03b9\u03bd) _differing_, _exceeding_, is not quantity, but\nrelation. But the quantity by which they differ is always a certain and\ndetermined quantity, yet the word _differentia_ serves for both, and is\nto be understood by the coherence with that which went before. But I had\nsaid before, and expressly to prevent cavil, that relation is nothing\nbut a comparison, and that proportion is nothing but relation of\nquantities, and so defined them, and therefore I did there use the word\n_differentia_ for _differing_, and not for the quantity which was left\nby subtraction. For a quantity is not a differing. This I thought the\nintelligent reader would of himself understand without putting me,\ninstead of _differentia_, to use (as some do, and I shall never do) the\nmongrel word \u03c4\u1f78 _differre_. And whereas in one only place for _differre\nternario_ I have writ _ternarius_, if you had understood what was\nclearly expressed before, you might have been sure it was not my\nmeaning, and therefore the excepting against it was either want of\nunderstanding, or want of candour, choose which you will.\n\nYou do not yet clear your doctrine of _condensation_ and _rarefaction_.\nBut I believe you will by degrees become satisfied that they who say the\nsame numerical body may be sometimes greater, sometimes less, speak\nabsurdly, and that _condensation_ and _rarefaction_ here, and\n_definitive_ and _circumscriptive_, and some other of your distinctions\nelsewhere are but snares, such as school divines have invented\n\n \u2014\u2014\u1fa5\u03c3\u03c0\u03b5\u03c1 \u03ac\u03c1\u03ac\u03c7\u03bd\u03b7\u03c2\n \u1f48\u03c5\u03bb\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03c7\u03ad\u03b6\u03b5\u03b9 \u1f00\u03bb\u03cd\u03c3\u03b5\u03b9\u03c2 \u03bc\u03c5\u03af\u03b1\u03b9\u03c2 \u1f00\u03b8\u03b1\u03c1\u03ad\u03c3\u03c3\u03b9,\n\nto entangle shallow wits.\n\nAnd that that distinction which you bring here, \u201c_that it is of the same\nquantity while it is in the same place, but it may be of a different\nquantity when it goes out of its place_,\u201d (as if the place added to, or\ntook any quantity from the body placed), is nothing but mere words. It\nis true that the body which swells changeth place, but it is not by\nbecoming itself a greater body, but by admixtion of air or other body,\nas when water riseth up in boiling, it taketh in some parts of air. But\nseeing the first place of the body is to the body equal, and the second\nplace equal to the same body, the places must also be equal to one\nanother, and consequently the dimensions of the body remain equal in\nboth places.\n\nSir, when I said that such doctrine was taught in the Universities, I\ndid not speak against the Universities, but against such as you. I have\ndone with your geometry, which is one \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74.\n\n RURAL LANGUAGE.\n\nAs for your eloquence, let the reader judge whether yours or mine be the\nmore _muddy_, though I in plain scolding should have outdone you, yet I\nhave this excuse which you have not, that I did but answer your\nchallenge at that weapon which you thought fit to choose. The catalogue\nof the hard language which you put in at pages 3 and 4 of your _School\nDiscipline_, I acknowledge to be mine, and would have been content you\nhad put in all. The titles you say I give you of _fools_, _beasts_, and\n_asses_, I do not give you, but drive back upon you, which is no more\nthan not to own them; for the rest of the catalogue, I like it so well\nas you could not have pleased me better than by setting those passages\ntogether to make them more conspicuous; that is all the defence I will\nmake to your accusations of that kind.\n\nAnd now I would have you to consider whether you will make the like\ndefence against the faults that I shall find in the language of your\n_School Discipline_.\n\nI observe, first, the facetiousness of your title-page, \u201c_Due correction\nfor Mr. Hobbes, or School Discipline, for not saying his Lessons\nright_.\u201d What a quibble is this upon the word lesson; besides, you know\nit has taken wind; for you vented it amongst your acquaintance at Oxford\nthen when my _Lessons_ were but upon the press. Do you think if you had\npretermitted that piece of wit, the opinion of your judgment would have\nbeen ere the less? But you were not content with this, but must make\nthis metaphor from the rod to take up a considerable part of your book,\nin which there is scarce anything that yourself can think wittily said\nbesides it. Consider also these words of yours: \u201c_It is to be hoped that\nin time you may come to learn the language, for you be come to great_ A\n_already_.\u201d And presently after, \u201c_were I great_ A, _before I would be\nwilling to be so used, I should wish myself little_ a _a hundred\ntimes_.\u201d Sir, you are a doctor of divinity and a professor of geometry,\nbut do not deceive yourself, this does not pass for wit in these parts,\nno, nor generally at Oxford; I have acquaintance there that will blush\nat the reading it.\n\nAgain, in another place you have these words: \u201c_Then you catechize us_,\n\u2018_what is your name? Are you geometricians? Who gave you that name_,\u2019\u201d\n&c. Besides in other places such abundance of the like insipid conceits,\nas would make men think, if they were no otherwise acquainted with the\nUniversity but by reading your books, that the dearth there of salt were\nvery great. If you have any passage more like to salt than these are\n(excepting _now and anon_) you may do well to show it to your\nacquaintance, lest they despise you; for, since the detection of your\ngeometry, you have nothing left you else to defend you from contempt.\nBut I pass over this kind of eloquence, and come to somewhat yet more\nrural.\n\nPage 27, line 1, you say I have given Euclid his _lurry_. And again,\npage 129, line 11, \u201c_and now he is left to learn his lurry_.\u201d I\nunderstand not the word _lurry_. I never read it before, nor heard it,\nas I remember, but once, and that was when a clown threatening another\nclown said he would give him _such a lurry come poop_, &c. Such words as\nthese do not become a learned mouth, much less are fit to be registered\nin the public writings of a doctor of divinity. In another place you\nhave these words, \u201c_just the same to a cow\u2019s thumb_,\u201d a pretty adage.\n\nPage 2, \u201c_But prithee tell me_.\u201d And again, page 95, \u201c_prithee tell me,\nwhy dost thou ask me such a question_,\u201d and the like in many other\nplaces.\n\nYou cannot but know how easy it is and was for me to have spoken to you\nin the same language. Why did I not? Because I thought that amongst men\nthat were civilly bred it would have redounded to my shame, as you have\ncause to fear that this will redound to yours. But what moved you to\nspeak in that manner? Were you angry? If I thought that the cause, I\ncould pardon it the sooner, but it must be very great anger that can put\na man, that professeth to teach good manners, so much out of his wits as\nto fall into such a language as this of yours. It was perhaps an\nimagination that you were talking to your inferior, which I will not\ngrant you, nor will the heralds, I believe, trouble themselves to decide\nthe question. But, howsoever, I do not find that civil men use to speak\nso to their inferiors. If you grant my learning but to be equal to\nyours, (which you may certainly do without very much disparaging of\nyourself abroad in the world), you may think it less insolence in me to\nspeak so to you in respect of my age, than for you to speak so to me in\nrespect of your young doctorship. You will find that for all your\ndoctorship, your elders, if otherwise of as good repute as you, will be\nrespected before you. But I am not sure that this language of yours\nproceeded from that cause; I am rather inclined to think you have not\nbeen enough in good company, and that there is still somewhat left in\nyour manners for which the honest youths of Hedington and Hincsey may\ncompare with you for good language, as great a doctor as you are.\n\nFor my verses of the Peak, though they be as ill in my opinion as I\nbelieve they are in yours, and made long since, yet they are not so\nobscene as that they ought to be blamed by Dr. Wallis. I pray you, sir,\nwhereas you have these words in your _School Discipline_, page 96,\n\u201c_unless you will say that one and the same motion may be now and anon\ntoo_.\u201d What was the reason you put these words, _now and anon too_, in a\ndifferent character, that makes them to be more taken notice of? Do you\nthink that the story of the minister that uttered his affection (if it\nbe not a slander) not unlawfully but unseasonably, is not known to\nothers as well as to you? What needed you then, when there was nothing\nthat I had said could give the occasion, to use those words; there is\nnothing in my verses that do _olere hircum_ so much as this of yours. I\nknow what good you can receive by ruminating on such ideas, or\ncherishing of such thoughts. But I go on to other words of mine by you\nreproached, \u201c_you may as well seek the focus of the parabola of Dives\nand Lazarus_,\u201d which you say is mocking the Scripture; to which I answer\nonly, that I intended not to mock the Scripture, but you, and that which\nwas not meant for mocking was none. And thus you have a second \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74.\n\n GRAMMAR AND CRITIQUES.\n\nI come now to the comparison of our Grammar and Critiques. You object\nfirst against the signification I give of \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74, and say thus: \u201c_What\nshould come into your cap_ (that, if you mark it, in a man that wears a\nsquare cap to one that wears a hat, is very witty) _to make you think\nthat_ \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 _signifies a mark or brand with a hot iron? I perceive\nwhere the business lies, it was_ \u03c3\u03c4\u03af\u03b3\u03bc\u03b1 _run in your mind when you\ntalked of_ \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74; _and because the words are somewhat alike you jumble\nthem both together_.\u201d Sir, I told you once before, you presume too much\nupon your first cogitations. Aristophanes, in _Ranis_, Act. V. Scen. 5,\n\n \u039a\u1f04\u03bd \u03bc\u1f74 \u03c4\u03b1\u03c7\u03ad\u03c9\u03c2 \u1f25\u03ba\u03c9\u03c3\u03b9\n \u039d\u1f74 \u03c4\u03b5\u03bd \u1f08\u03c0\u03cc\u03bb\u03bb\u03bf \u03c3\u03c4\u03af\u03be\u03b1\u03c2 \u1f00\u03c5\u03c4\u03bf\u1f7a\u03c2.\n\nThe old commentator upon the word \u03c3\u03c4\u03af\u03be\u03b1\u03c2 saith thus, \u03db\u03af\u03be\u03b1\u03c2 \u1f00\u03bd\u03c4\u1f76 \u03c4\u03bf\u1fe6\n\u03db\u03b9\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03b1\u03c2, \u1f20\u03bd \u03b3\u03ac\u03c1 \u03be\u03ad\u03bd\u03bf\u03c2. That is, \u03c3\u03c4\u03af\u03be\u03b1\u03c2 for \u03db\u03b9\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03b1\u03c2, for he\n(Adimantus) was not a citizen. I hope the commentator does not here mock\nAristophanes for jumbling \u03db\u03af\u03be\u03b1\u03c2 and \u03db\u03b9\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03b1\u03c2 together, for want of\nunderstanding Greek. No, \u03db\u03af\u03be\u03b1\u03c2 and \u03c3\u03c4\u03b9\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03b1\u03c2 signify the same, save\nthat for branding I seldom read \u03db\u03b9\u03b3\u03bc\u03b1\u03c4\u03af\u03c3\u03b1\u03c2 but \u03db\u03af\u03be\u03b1\u03c2. For \u03db\u03af\u03b3\u03bc\u03b1 does no\nmore signify a brand with a hot iron, than \u03db\u03b9\u03b3\u03bc\u1f74 a point made also with\na hot iron. They have both one common theme \u03db\u03af\u03b6\u03c9, which does not signify\n_pungo_, nor _interpungo_, nor _inuro_, for all your Lexicon, but _notam\nimprimere_, or _pungendo notare_, without any restriction to burning or\npunching. It is therefore no less proper to say that \u03db\u03b9\u03b3\u03bc\u1f74 is a mark\nwith a hot iron, than to say the same of \u03c3\u03c4\u03af\u03b3\u03bc\u03b1. The difference is only\nthis, that when they marked a slave, or a rascal, as you are not\nignorant is usually done here at the assizes in the hand or shoulder\nwith a hot iron, they called that \u03db\u03af\u03b3\u03bc\u03b1, not for the burning, but for\nthe mark. And as it would have been called \u03db\u03af\u03b3\u03bc\u03b1 that was imprinted on a\nslave, though made by staining or incision, so it is \u03db\u03b9\u03b3\u03bc\u1f74, though done\nwith a hot iron. And therefore there was no jumbling of those two words\ntogether, as for want of reading Greek authors, and by trusting too much\nto your dictionaries, which you say are proofs good enough for such a\nbusiness, you were made to imagine. The use I have made thereof was to\nshow that a point, both by the word \u03a3\u03b7\u03bc\u03b5\u1fd6\u03bf\u03bd in Euclid, and by the word\n\u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 in some others, was not _nothing_, but a _visible_ mark, the\nignorance whereof hath thrown you into so many paralogisms in geometry.\n\nBut do you think you can defend your _Adducis Malleum_ as well as I have\nnow defended my \u03db\u03b9\u03b3\u03bc\u1f74? You have brought, I confess, above a hundred\nplaces of authors, where there is the word _duco_, or some of its\ncompounds, but none of them will justify _Adducis Malleum_, and,\nexcepting two of those places, you yourself seem to condemn them all,\ncomparing yours with none of the rest but with these two only, both out\nof Plautus, by you not well understood. The first is in _Casina_, Act.\nV. Scen. 2, \u201c_Ubi intro hanc novam nuptam deduxi, via recta, clavem\nabduxi_;\u201d which you, presently presuming of your first thoughts, a\npeculiar fault to men of your principles, assure yourself is right. But\nif you look on the place as Scaliger reads it, cited by the commentator,\nyou will find it should be _obduxi_, and that _clavis_ is there used for\nthe bolt of the lock. Besides, he bolted it within. Whither then could\nhe carry away the key? The place is to be rendered thus, _when I had\nbrought in this new bride I presently locked the door_, and is this _as\nbad every whit_ as _Adducis Malleum_? The second place is in\n_Amphytruo_, Act. I. Scen. 1, \u201c_Eam_ (cirneam), _ut a matre fuerat\nnatum, plenam vini eduxi meri_,\u201d which you interpret _I brought out a\nflagon of wine_, unlearnedly. They are the words of Mercury transformed\ninto Sosia. And to try whether Mercury were Sosia or not, Sosia asked\nhim where he was and what he did during the battle; to which Mercury\nanswered, who knew where Sosia then was and what he did, _I was in the\ncellar, where I filled a cirnea, and brought it up full of wine, pure as\nit came from its mother_. By the mother of the wine meaning the vine,\nand alluding to the education of children, for _ebibi_ said _eduxi_, and\nwith an _emphasis_ in _meri_, because _cirnea_ (from \u039a\u03c6\u03bd\u03ac\u03c9, _misceo_)\nwas a vessel wherein they put water to temper to their wine. Intimating\nthat though the vessel was _cirnea_, yet the wine was _merum_. This is\nthe true sense of the place; but you will have _eduxi_ to be, _I brought\nout_, though he came not out himself. You see, sir, that neither this is\nso bad as _Adducis Malleum_.\n\nBut suppose out of some one place in some one blind author you had\nparalleled your _Adducis Malleum_, do you think it must therefore\npresently be held for good Latin? Why more than _learn his lurry_ must\nbe therefore thought good English a thousand years hence, because it\nwill be read in Dr. Wallis\u2019s long-lived works. But how do you construe\nthis passage (1 Tim. ii. 15) of the Greek Testament: \u03a3\u03c9\u03b8\u03ae\u03c3\u03b5\u03c4\u03b1\u03b9 \u03b4\u1f72 \u03b4\u03b9\u1f70\n\u03c4\u1fc6\u03c2 \u03c4\u03b5\u03ba\u03bd\u03bf\u03b3\u03bf\u03bd\u03af\u03b1\u03c2, \u1f10\u1f70\u03bd \u03bc\u03b5\u03af\u03bd\u03c9\u03c3\u03b9\u03bd \u1f10\u03bd \u03c0\u03af\u03c3\u03c4\u03b5\u03b9? You construe it thus: _she\nshall be saved notwithstanding child-bearing, if (the woman) remain in\nthe faith_. Is child-bearing any obstacle to the salvation of women? You\nmight as well have translated the first verse of the fifth of Romans in\nthis manner, _Being then justified by faith, we have peace with God\nnotwithstanding our Lord Jesus Christ_. I let pass your not finding in\n\u03c4\u03b5\u03ba\u03bd\u03bf\u03b3\u03bf\u03bd\u03af\u03b1\u03c2, as good a grammarian as you are, a nominative case to\n\u03bc\u03b5\u03af\u03bd\u03c9\u03c3\u03b9\u03bd. If you had remembered the place, 1 Pet. iii. 20, \u03b5\u03c3\u03ce\u03b8\u03b7\u03c3\u03b1\u03bd \u03b4\u1f76\n\u1f51\u03b4\u03b1\u03c4\u03bf\u03c2, that is, _they were saved in the waters_, you would have thought\nyour construction justified then very well; but you had been deceived,\nfor \u03b4\u03b9\u1f70 does not there signify _causam, ablationem impedimenti_, but\n_transitum_; not _cause or removing an impediment_, but _passage_. Being\ncome thus far, I found a friend that hath eased me of this dispute; for\nhe showed me a letter written to himself from a learned man, that hath\nout of very good authors collected enough to decide all the grammatical\nquestions between you and me, both Greek and Latin. He would not let me\nknow his name, nor anything of him but only this, that he had better\nornaments than to be willing to go clad abroad in the habit of a\ngrammarian. But he gave me leave to make use of so much of the letter as\nI thought fit in this dispute, which I have done, and have added it to\nthe end of this writing. But before I come to that, you must not take it\nill, though I have done with your _School Discipline_, if I examine a\nlittle some other of your printed writings as you have examined mine;\nfor neither you in geometry, nor such as you in church politics, cannot\nexpect to publish any unwholesome doctrine without some antidotes from\nme, as long as I can hold a pen. But why did you answer nothing to my\nsixth _Lesson_? Because, you say, it concerned your colleague only. No,\nsir, it concerned you also, and chiefly, for I have not heard that your\ncolleague holdeth those dangerous principles which I take notice of in\nyou, in my sixth _Lesson_, page 350, upon the occasion of these words,\nnot his but yours: \u201c_Perhaps you take the whole history of the fall of\nAdam for a fable, which is no wonder, seeing you say the rules of\nhonouring and worshipping of God are to be taken from the laws_.\u201d In\nanswer to which I said thus: \u201c_You that take so heinously, that I would\nhave the rule of God\u2019s worship in a Christian commonwealth to be taken\nfrom the laws, tell me from whom you would have them taken? From\nyourself? Why so, more than from me? From the bishops? Right, if the\nsupreme power of the commonwealth will have it so; if not, why from them\nrather than from me? From a consistory of presbyters themselves, or\njoined with lay elders, whom they may sway as they please? Good, if the\nsupreme governor of the commonwealth will have it so. If not, why from\nthem rather than from me, or from any man else? They are wiser and\nlearneder than I; it may be so, but it has not yet appeared. Howsoever,\nlet that be granted. Is there any man so very a fool as to subject\nhimself to the rules of other men in those things which do so nearly\nconcern himself, for the title they assume of being wise and learned,\nunless they also have the sword which must protect them? But it seems\nyou understand the sword as comprehended. If so, do not you then receive\nthe rules of God\u2019s worship from the civil power? Yes, doubtless; and you\nwould expect, if your consistory had that sword, that no man should dare\nto exercise or teach any rules concerning God\u2019s worship which were not\nby you allowed._\u201d\n\nThis will be thought strong arguing, if you do not answer it. But the\ntruth is, you could say nothing against it without too plainly\ndiscovering your disaffection to the government. And yet you have\ndiscovered it pretty well in your second _Thesis_, maintained in the Act\nat Oxford, 1654, and since by yourself published. This _Thesis_ I shall\nspeak briefly to.\n\n SCOTCH CHURCH POLITICS.\n\nYou define ministers of the Gospel to be those _to whom the preaching of\nthe Gospel by their office is enjoined by Christ_. Pray you, first, what\ndo you mean by saying preaching _ex officio is enjoined by Christ_? Are\nthey preachers _ex officio_, and afterwards enjoined to preach? _Ex\nofficio_ adds nothing to the definition; but a man may easily see your\npurpose to disjoin yourself from the state by inserting it.\n\nSecondly, I desire to know in what manner you will be able out of this\ndefinition to prove yourself a minister? Did Christ himself immediately\nenjoin you to preach, or give you orders? No. Who then, some bishop, or\nminister, or ministers? Yes; by what authority? Are you sure they had\nauthority immediately from Christ? No. How then are you sure but that\nthey might have none? At least, some of them through whom your authority\nis derived might have none. And therefore if you run back for your\nauthority towards the Apostles\u2019 times but a matter of sixscore years,\nyou will find your authority derived from the Pope, which words have a\nsound very unlike to the voice of the laws of England. And yet the Pope\nwill not own you. There is no man doubts but that you hold that your\noffice comes to you by successive imposition of hands from the time of\nthe Apostles; which opinion in those gentle terms passeth well enough;\nbut to say you derive your authority from thence, not through the\nauthority of the sovereign power civil, is too rude to be endured in a\nstate that would live in peace. In a word, you can never prove you are a\nminister, but by the supreme authority of the commonwealth. Why then do\nyou not put some such clause into your definition? As thus, _ministers\nof the Gospel are those to whom the preaching of the Gospel is enjoined\nby the sovereign power in the name of Christ_. What harm is there in\nthis definition, saving only it crosses the ambition of many men that\nhold your principles? Then you define the power of a minister thus:\n\u201c_The power of a minister is that which belongeth to a minister of the\nGospel in virtue of the office he holds, inasmuch as he holds a public\nstation, and is distinguished from private Christians. Such as is the\npower of preaching the Gospel, administering the sacrament, the use of\necclesiastical censures, and ordaining of ministers_,\u201d _&c._\n\nAgain, how will you prove out of this definition that you, or any man\nelse, hath the power of a minister, if it be not given him by him that\nis the sovereign of the commonwealth? For seeing, as I have now proved,\nit is from him that you must derive your ministry, you can have no other\npower than that which is limited in your orders, nor that neither longer\nthan he thinks fit. For if he give it you for the instruction of his\nsubjects in their duty, he may take it from you again whensoever he\nshall see you instruct them with undutiful and seditious principles. And\nif the sovereign power give me command, though without the ceremony of\nimposition of hands, to teach the doctrine of my _Leviathan_ in the\npulpit, why am not I, if my doctrine and life be as good as yours, a\nminister as well as you, and as public a person as you are? For _public\nperson_, primarily, is none but the civil sovereign, and so secondarily,\nall that are employed in the execution of any part of the public charge.\nFor all are his ministers, and therefore also Christ\u2019s ministers because\nhe is so; and other ministers are but his vicars, and ought not to do or\nsay anything to his people contrary to the intention of the sovereign in\ngiving them their commission.\n\nAgain, if you have in your commission a power to excommunicate, how can\nyou think that your sovereign who gave you that commission, intended it\nfor a commission to excommunicate himself? that is, as long as he stand\nexcommunicate, to deprive him of his kingdom. If all subjects were of\nyour mind, as I hope they will never be, they will have a very unquiet\nlife. And yet this has, as I have often heard, been practised in\nScotland, when ministers holding your principles had power enough,\nthough no right, to do it.\n\nAnd for administration of the sacraments, if by the supreme power of the\ncommonwealth it were committed to such of the laity as know how it ought\nto be done as well as you, they would _ipso facto_ be ministers as good\nas you. Likewise the right of ordination of ministers depends not now on\nthe imposition of hands of a minister or presbytery, but on the\nauthority of the Christian sovereign, Christ\u2019s immediate vicar and\nsupreme governor of all persons and judge of all causes, both spiritual\nand temporal, in his own dominions, which I believe you will not deny.\n\nThis being evident, what acts are those of yours which you call\n_authoritative_, and receive not from the authority of the civil power?\nA constable does the acts of a constable _authoritatively_ in that\nsense. Therefore you can no otherwise claim your power than a constable\nclaimeth his, who does not exercise his office in the constabulary of\nanother. But you forget that the Scribes and the Pharisees sit no more\nin Moses\u2019 chair.\n\nYou would have every minister to be a minister of the universal Church,\nand that it be lawful for you to preach your doctrine at Rome; if you\nwould be pleased to try, you would find the contrary. You bring no\nargument for it that looks like reason. Examples prove nothing, where\npersons, times, and other circumstances differ; as they differ very much\nnow when kings are Christians, from what they were then when kings\npersecuted Christians. It is easy to perceive what you aim at.\n\nYou would fain have market-day lectures set up by authority, (not by the\nauthority of the civil power, but by the authority of example of the\nApostles in the emission of preachers to the infidels), not knowing that\nany Christian may lawfully preach to the infidels; that is to say,\nproclaim unto them that Jesus is the Messiah, without need of being\notherways made a minister, as the deacons did in the Apostles\u2019 time; nor\nthat many teachers, unless they can agree better, do anything else but\nprepare men for faction, nay, rather you know it well enough, but it\nconduces to your end upon the market-days to dispose at once both town\nand country, under a false pretence of obedience to God, to a neglecting\nof the commandments of the civil sovereign, and make the subject to be\nwholly ruled by yourselves, wherein you have already found yourselves\ndeceived. You know how to trouble and sometimes undo a slack government,\nand had need to be warily looked to, but are not fit to hold the reins.\nAnd how should you, being men of so little judgment as not to see the\nnecessity of unity in the governor, and of absolute obedience in the\ngoverned, as is manifest out of the place of your _Elenchus_ above\nrecited. The doctrine of the duty of private men in a commonwealth is\nmuch more difficult, not only than the knowledge of your symbols, but\nalso than the knowledge of geometry itself. How then do you think, when\nyou err so grossly in a few equations, and in the use of most common\nwords, you should be fit to govern so great nations as England, Ireland,\nand Scotland, or so much as to teach them? For it is not reading but\njudgment that enables one man to teach another.\n\nI have one thing more to add, and that is the disaffection I am charged\nwithal to the universities. Concerning the Universities of Oxford and\nCambridge, I ever held them for the greatest and noblest means of\nadvancing learning of all kinds, where they should be therein employed,\nas being furnished with large endowments and other helps of study, and\nfrequented with abundance of young gentlemen of good families and good\nbreeding from their childhood. On the other side, in case the same means\nand the same wits should be employed in the advancing of the doctrines\nthat tend to the weakening of the public, and strengthening of the power\nof any private ambitious party, they would also be very effectual for\nthat; and consequently that if any doctrine tending to the diminishing\nof the civil power were taught there, not that the Universities were to\nblame, but only those men that in the universities, either in lectures,\nsermons, printed books, or theses, did teach such doctrine to their\nhearers or readers. Now you know very well that in the time of the Roman\nreligion, the power of the Pope in England was upheld principally by\nsuch teachers in the universities. You know also how much the divines\nthat held the same principles in Church government with you, have\ncontributed to our late troubles. Can I therefore be justly taxed with\ndisaffection to the universities for wishing this to be reformed? And it\nhath pleased God of late to reform it in a great measure, and indeed as\nI thought totally, when out comes this your _Thesis_ boldly maintained\nto show the contrary. Nor can I yet call this your doctrine the doctrine\nof the university; but surely it will not be unreasonable to think so,\nif by public act of the university it be not disavowed, which done, and\nthat as often as there shall be need, there can be no longer doubt but\nthat the universities of England are not only the noblest of all\nChristian universities, but also absolutely, and of the greatest benefit\nto this commonwealth that can be imagined, except that benefit of the\nhead itself that uniteth and ruleth all. I have not here particularized\nat length all the ill consequences that may be deduced from this\n_Thesis_ of yours, because I may, when further provoked, have somewhat\nto say that is new. So much for the third \u03db\u03b9\u03b3\u03bc\u1f74.\n\n AN EXTRACT OF A LETTER CONCERNING THE\n GRAMMATICAL PART OF THE CONTROVERSY\n BETWEEN MR. HOBBES AND DR. WALLIS.\n\n\nMr. Hobbes hath these words: \u201c_Longitudinem percursam motu uniformi, cum\nimpetu ubique ipsi B D \u00e6quali_.\u201d Dr. Wallis saith _cum_ were better out,\nunless you would have _impetus_ to be only a companion, not a _cause_.\nMr. Hobbes answered it was the _ablative case of the manner_. The truth\nis the ablative case of the _manner_ and _cause_ both, may be used with\nthe conjunction _cum_, as may be justified. Cicero in Lib. II. _De Nat.\nDeorum_: \u201c_Moliri aliquid cum labore operoso ac molesto_;\u201d and in his\noration for C\u00e6cina: \u201c_De se autem hoc pr\u00e6dicat, Antiocho Ebulii servo\nimperasse ut in C\u00e6cinam advenientem cum ferro invaderet_.\u201d Let us see\nthen what Dr. Wallis objects against Tully, where a casualty is\nimported, though we may use _with_ in English, yet not _cum_ in Latin;\nto kill with a sword, importing this to have an instrumental or causal\ninfluence, and not only that it hangs by the man\u2019s side whilst some\nother weapon is made use of, is not in Latin _occidere cum gladio_, but\n_gladio occidere_. This shows that the Doctor hath not forgot his\ngrammar, for the subsequent examples as well as this rule are borrowed\nthence. But yet he might have known that great personages have never\nconfined themselves to this pedantry, but have chosen to walk in a\ngreater latitude. Most of the elegancies and idioms of every language\nare exceptions to his grammar. But since Mr. Hobbes saith it is the\nablative case of the manner, there is no doubt it may be expressed with\n_cum_. The Doctor in the meantime knew no more than what Lilly had\ntaught him; Alvarez would have taught him more; and Vossius in his book,\n_De Constructione_, _cap._ XLVII. expressly teacheth, \u201c_Ablativos caus\u00e6,\ninstrumenti, vel modi, non a verbo regi sed a pr\u00e6positione omissa, a vel\nab, de, e vel ex, pr\u00e6, aut cum, ac pr\u00e6positiones eas quandoque exprimi\nnisi quod cum ablativis instrumenti haud temere invenias_;\u201d and\nafterwards he saith, \u201c_non timere imitandum_.\u201d If this be so, then did\nMr. Hobbes speak grammatically, and with Tully, but not _usually_. And\nmight not one retort upon the Doctor, that Vossius is as great a critic\nas he?\n\nHis next reflection is upon _pr\u00e6tendit scire_, this he saith is an\nAnglicism. If this be all his accusation, upon this score we shall lose\nmany expressions that are used by the best authors, which I take to be\ngood Latinisms, though they be also Anglicisms, the latter being but an\nimitation of the former. The Doctor therefore was too fierce to condemn\nupon so general an account, that which was not to have been censured for\nbeing an Anglicism, unless also it had been no Latinism. Mr. Hobbes\nreplies, that the printer had omitted _se_. He saith, this mends the\nmatter a little. It is very likely, for then it is just such another\nAnglicism as that of Quintilian: \u201c_Cum loricatus in foro ambularet,\npr\u00e6tendebat se id metu facere_.\u201d The Doctor certainly was very\nnegligent, or else he could not have missed this in Robert Stephen. Or\nhaply he was resolved to condemn Quintilian for this and that other\nAnglicism, \u201c_Ignorantia pr\u00e6tendi non potest_,\u201d as all those that have\nused _pr\u00e6tendo_, which are many and as good authors as Dr. Wallis, that\nmakes his own encomiasts (not an Englishman amongst them) to write\nAnglicisms.\n\nThen he blames \u201c_Tractatus hujus partis terti\u00e6, in qua motus et\nmagnitudo per se et abstracte consideravimus, terminum hic statuo_.\u201d\nHere I must confess the exception is colourable, yet I can parallel it\nwith the like objection made by Erasmus against Tully, out of whom\nErasmus quotes this passage: \u201c_Diutius commorans Athenis, quoniam venti\nnegabant solvendi facultatem, erat animus ad te scribere_;\u201d and excuses\nit thus, that Tully might have had at first in his thoughts _volebam_ or\n_statuebam_, which he afterwards relinquished for _erat animus_, and did\nnot remember what he had antecedently written, which did not vary from\nhis succeeding thoughts, but words. And this excuse may pass with any\nwho know that Mr. Hobbes values not the study of words, but as it serves\nto express his thoughts, which were the same whether he wrote _in qua\nmotus et magnitudo per se at abstracte considerati sunt_ or\n_consideravimus_. And if the Doctor will make this so capital, he must\nprove it _voluntary_, and show that it is greater than what is legible\nin the puny letter of his encomiast, whom he would have to be beyond\nexception.\n\nNow follows his ridiculous apology for _adducis malleum, ut occidas\nmuscam_. The cause why he did use that proverb, of his own phrasing, was\nthis. Mr. Hobbes had taken a great deal of pains to demonstrate what Dr.\nWallis thought he could have proved in short; upon this occasion he\nobjects, _adducis malleum ut occidas muscam_, which I shall suppose he\nintended to English thus, _you bring a beetle to kill a fly_. Mr. Hobbes\nretorted, that _adduco_ was not used in that sense. The Doctor\nvindicates himself thus: _duco_, _deduco_, _reduco_, _perduco_,\n_produco_, &c. signify the same thing, ergo, _adduco_ may be used in\nthat sense; which is a most ridiculous kind of arguing, where we are but\nto take up our language from others, and not to coin new phrases. It is\nnot the grammar that shall secure the Doctor, nor weak analogies, where\nelegance comes in contest. To justify his expression he must have showed\nit _usu tritum_, or alleged the authority of some author of great note\nfor it. I have not the leisure to examine his impertinent citations\nabout those other compounds, nor yet of that simple verb _duco_; nay, to\njustify his saying he hath not brought one parallel example. He talks\nindeed very high, that _duco_, with its compounds, is a word of a large\nsignification, and amongst the rest _to bring_, _fetch_, _carry_, &c. is\nso exceeding frequent in all authors, Plautus, Terence, Tully, C\u00e6sar,\nTacitus, Pliny, Seneca, Virgil, Horace, Ovid, Claudian, &c. that he must\nneeds be either maliciously blind, or a very stranger to the Latin\ntongue, that doth not know it, or can have the face to deny it. I read,\nwhat will be my doom for not allowing his Latin; yet I must profess I\ndare secure the Doctor for having read all authors, notwithstanding his\nassertion, and I hope he will do the like for me. And for those which he\nhath read, had he brought no better proofs than these, he had, I am\nsure, been whipped soundly in Westminster School, for his impudence as\nwell as ignorance, by the learned master thereof at present. But I dare\nfurther affirm, the Doctor hath not read in this point any, but only\nconsulted with Robert Stephen\u2019s _Thesaurus Lingu\u00e6 Latin\u00e6_, whence he\nhath borrowed his allegations in _adduco_; and for the other, I had not\nso much idle time as to compare them. And, lest the fact might be\ndiscovered, he hath sophisticated those authors whence Stephen cites the\nexpressions, and imposed upon them others. If it be not so, or that the\nDoctor could not write it right when the copy was right before him, let\nhim tell me where he did ever read in Plautus, _adducta res in\nfastidium_. I find the whole sentence in Pliny\u2019s preface to Vespasian\n(out of whom in the precedent paragraph he cites it) about the middle:\n_alia vero ita multis prodita, ut in factidium sint adducta_, which is\nthe very example Stephanus useth, although he doth premise his _adducta\nres in fastidium_. Let the Doctor tell where he ever did read in Horace,\n_Ova noctu\u00e6_, &c. _t\u00e6dium vini adducunt_. Did he, or any else, with the\ninterposition of an &c. make Trochaics? I say, and Stephanus says so,\ntoo, that it is in Pliny, lib. xiii. cap. 15, near the end; the whole\nsentence runs thus: _Ebriosis Ova noctu\u00e6 per triduum data in vino,\nt\u00e6dium ejus adducunt_. I doubt not but these are the places he aimed at,\nalthough he disguised and minced the quotations; if they be not, I\nshould be glad to augment my Stephanus with his additions.\n\nThese things premised, I come to consider the Doctor\u2019s proofs: _Res eo\nadducta est_: _adducta vita in extremum_: _adducta res in fastidium_:\n_rem ad mucrones et manus adducere_: _contractares et adducta in\naugustum_: _res ad concordiam adduci potest_: _in ordinem adducerem_:\n_adducere febres, sitim, tedium vini_ (all in Robert Stephen) betwixt\nwhich and _adducere malleum_, what a vast difference there is, I leave\nthem to umpire _qui terretes et religiosas nacti sunt aures_, who are\nthe competent judges of elegancy, and only cast in the verdict of one or\ntwo, who are in any place (where the purity of the Latin tongue\nflourisheth) of great esteem. Los\u00e6us, in his _Scop\u00e6 Lingu\u00e6 Latin\u00e6, ad\npurgandam Linguam a barbarie_, &c. (would any think that the Doctor\u2019s\nelegant expression, frequent in all authors, which none but the\nmalicious or ignorant can deny, should suffer so contumelious an\nexpurgation?) Los\u00e6us, I say, hath these words: _Adferre plerique minus\nattenti utuntur pro adducere. Quod Plautus, in Pseudolo, insigni exemplo\nnotat_.\n\n _CA._--Attuli hunc.\n _PS._--Quid attulisti?\n _CA._--Adduxi volui dicere.\n _PS._--Quis istic est?\n _CA._--Charinus.\n\n_Satis igitur admonet discriminis inter ducere, reducere, adducere, et\nabducere, qu\u00e6 de persona; et ferre, adferre_, &c. _qu\u00e6 de re dicuntur_.\nIdem, _Demetrium, quem ego novi, adduce: argentum non moror quin feras_.\n_Cavendum igitur est ne vulgi more_, (let the Doctor mark this, and know\nthat _this author is authentic amongst the Ciceronians_), _adferre de\npersona, dicamus, sed adducere; licet et hoc de certis quibusdam rebus\nnon inepte dicatur_. In this last clause he saith as much as Mr. Hobbes\nsaith, and what the Doctor proves; but, that ever the Doctor brought an\nexample which might resemble _adducis malleum_, is denied; for I have\nmentioned already his allegations, every one, of _adduco_. Another\nauthor, (a fit antagonist for the elegant Doctor), is the _Farrago\nsordidorum Verborum_, joined with the Epitome of L. Valla\u2019s\n_Elegancies_. He saith: _Accerse, adhuc Petrum, Latine dicitur, pro eo\nquod pueri dicunt, adfer Petrum_. And this may suffice to justify Mr.\nHobbes\u2019s exception who proceeded no further than this author to tell the\nDoctor that _adduco_ was used of animals. But the Doctor replies, _this\nsignification is true, but so may the other be also_. I say if it never\nhave been used so, it cannot be so, for we cannot coin new Latin words,\nno more than French or Spanish who are foreigners. Mr. Hobbes was upon\nthe negative, and not to disprove the contrary opinion. If the Doctor\nwould be believed, he must prove it by some example, (which is all the\nproof of elegancy), and till he do so, not to believe him, it is\nsufficient not to have cause. But, Doctor Wallis, _why not adduco for a\nhammer as well as a tree?_ I answer yes, equally for either, and yet for\nneither. Did ever anybody go about to mock his readers thus solemnly? I\ndo not find, to my best remembrance, any example of it in Stephen, and\nthe Doctor is not wiser than his book; if there be, it is strange the\nDoctor should omit the only pertinent example, and trouble us with such\nimpertinences for three or four pages. In Stephen there are _adducere\nhabenas_ and _adducere lorum_, but in a different sense. It is not\nimpossible I may guess at the Doctor\u2019s aim. In Tully _de Nat. Deor._ as\nI remember, there is this passage: _Quum autem ille respondisset, in\nagro ambulanti ramulum adductum ut remissus esset, in oculum suum\nrecidisse_, where it signifies nothing else but to be _bent_, _bowed_,\n_pulled back_, and in that sense, _the hammer of a clock_, or that of a\n_smith, when he fetcheth his stroke_, may be said _adduci_. And this, I\nconceive, the Doctor would have us in the close think to have been his\nmeaning; else, what doth he drive at in these words? \u201cWhen you have done\nthe best you can, you will not be able to find better words than\n_adducere malleum_ and _reducere_, to signify the two contrary motions\nof the _hammer_, the one when you strike with it (_excellently\ntrivial!_) the other when you take it back (_better and better_), _What\nto do?_ to fetch another stroke. If any can believe that this was his\nmeaning, I shall justify his Latin, but must leave it to him to prove\nits sense. If he intended no more, why did he go about to defend the\nother meaning, and never meddle with this? Which yet might have been\nproved by this one example of mine? May not, therefore, his own saying\nbe justly retorted upon him in this case, _Adducis malleum, ut occidas\nmuscam_?\n\nAnother exception is, _Fals\u00e6 sunt, et multa istiusmodi_\n(_propositiones_). I wish the Doctor could bring so good parallels, and\nso many, out of any author, for his _Adducis malleum_, as Tully affords\nin this case. Take one for all, out of the beginning of his _Paradoxes_:\n_Animadverti s\u00e6pe Catonem, cum in senatu sententiam diceret, Locos\ngraves ex Philosophia tractare, abhorrentes ab hoc usu forensi, et\npublico, sed dicendo consequi tamen, ut illa etiam populo probabilia\nviderentur_. This is but a _Sol\u00e6cophanes_, and hath many precedents\nmore, as in the second book of his _Academical Questions_, &c.\n\nI cannot now stay upon each particular passage; I do not see any\nnecessity of tracing the Doctor in all his vagaries. Now, he disallows\n_tanquam diceremus_, _as if we should say_. But why is that less\ntolerable than _tanquam feceris_, _as if you had done_? \u201cIt should be\n_quasi_, (forsooth!) or _ac si_, or _tanquam si_, which is Tully\u2019s own\nword.\u201d What is _tanquam si_ become but one word? _Tanquam si tua res\nagatur_, &c. Good Doctor, leave out Tully and all _Ciceronians_, or you\nwill for ever suffer for this, and your _Adducis malleum_. Is not this\nto put yourself on their verdict when you oppose Mr. Hobbes with Tully?\nBut the Doctor gives his reason. And though he hath had the luck in his\n_Adducis malleum_, to follow the first part of that saying, _Loquendum\ncum vulgo_, yet now it is, _sentiendum cum sapientibus_. For _tanquam_\nwithout _si_ signifies but _as_, not _as if_. It is pity the Doctor\ncould not argue in symbols too, that so we might not understand him; but\nsuppose all his papers to carry evidence with them, because they are\n_mathematically_ scratched. How does he construe this:--\n\n \u201cPlance tumes alto Drusorum sanguine, tanquam\n Feceris ipse aliquid, propter quod nobilis esses.\u201d\n\nSo C\u0153lius, one much esteemed by Cicero, who hath inserted his Epistles\ninto his works, saith, in his fifth Epistle (Tul. Epist. Fam. lib. viii.\nep. 5), _Omnia desiderantur ab eo tanquam nihil denegatum sit ei quo\nminus paratissimus esset qui publico negotio pr\u00e6positus est_. But it was\nnot possible the Doctor should know this, it not being in Stephen, where\nhis examples for _tanquam si_ are.\n\nBut, the Doctor having pitched upon this criticism, and penned it,\nsomebody, I believe, put him in mind of the absurdity thereof; and yet\nthe generous _Professor_, (who writes running hand and never transcribed\nhis papers, if I am not misinformed), presumed nobody else could be more\nintelligent than he, who had perused Stephen. He would not retract\nanything, but subjoins, \u201cThat he will allow it as passable, because\nother modern writers, and some of the ancients, have so used it, as Mr.\nHobbes hath done.\u201d I know not what authors the Doctor meant, for, if I\nam not much mistaken, I do not find any in Stephen. His citation of\nColumella is not right, (lib. v. cap. 5), nor can I deduce anything\nthence till I have read the passage, but, if he take Juvenal and C\u0153lius\nfor modern authors, I hope he will admit of Accius, N\u00e6vius, and\nCarmenta, for the only ancients. Let him think upon this criticism, and\nnever hope pardon for his _Adducis malleum_, which is not half so well\njustified, and yet none but _madmen_ or _fools_ reject it.\n\nBut certainly the Doctor should not have made it his business to object\n_Anglicisms_, in whose Elenchus I doubt not but there may be found such\nphrases as may serve to convince him that he is an Englishman, however\nScottified in his principles. If the Doctor doubt of it, or but desire a\ncatalogue, let him but signify his mind, and he shall be furnished with\na _Florilegium_. But I am now come to the main controversy about Empusa.\nThe Doctor saith nothing in defence of his _quibble_, nor gives any\nreason why he jumbled languages to make a silly clinch, which will not\npass for wit either at Oxford or at Cambridge; no, nor at Westminster.\n\nIt seems he had derived _Empusa_ from \u1f13\u03bd and \u03c0\u03bf\u1fe6\u03c2, and said it was a\nkind of _Hobgoblin_ that hopped upon one leg: and hence it was that the\nboys\u2019 play (_Fox come out of thy hole_) came to be called _Empusa_. I\nsuppose he means _Ludus Empus\u00e6_. This derivation he would have to be\ngood, and that we may know his reading, (though he hath scarce consulted\nany of the authors), he saith Mr. Hobbes did laugh at it, until somebody\ntold him that it was in the Scholiast of Aristophanes (as good a critic\nas Mr. Hobbes), Eustathius, Erasmus, C\u0153lius Rhodiginus, Stephanus,\nScapula, and Calepine. But sure he doth not think to scape so. To begin\nwith the last; Calepine doth indeed say, _uno incedit pede, unde et\nnomen_. But he is a _Modern_, and I do not see why his authority should\noutweigh mine if his author\u2019s reasons do not. He refers to Erasmus and\nRhodiginus. Erasmus in the adage, _Proteo mutabilior_ hath these words\nof Empusa: _Narrant autem uno videri pedi_--this is not to hop--_unde et\nnomen inditum putant_, \u1f1c\u03bc\u03c0\u03bf\u03c5\u03c3\u03b1\u03bd \u1f41\u03b9\u03bf\u03bd\u03b5\u1f76 \u1f11\u03bd\u03af\u03c0\u03bf\u03b4\u03b1. He doth not testify his\napprobation of the derivation at all, only lets you know what\netymologies some have given before him. And doth anybody think that Dr.\nHarmar was the first which began to show his wit, (or folly), in\netymologizing words? C\u0153lius Rhodiginus doth not own the derivation, only\nsaith, _Nominis ratio est, ut placet Eustathio, quia uno incedit\npede_;--is this to hop?--_sed nec desunt qui alterum interpretentur\nhabere \u00e6neum pedem, et inde appellatam Empusam; quod in Batrachis\nAristophanes expressit_. And then he recites the interpretation that\nAristophanes\u2019s Scholiast doth give upon the text, of which by and by. If\nany credit be to be attributed to this allegation, his last thoughts are\nopposite to Dr. Wallis; and _Empusa_ must be so called, not because she\nhopped upon one leg, but because she had but one, the other being brass.\nBut for the former derivation he refers to Eustathius.\n\nAs to Eustathius, I do easily conjecture that the reader doth believe\nthat Rhodiginus doth mean Eustathius upon Homer, for that is the book of\nmost repute and fame, his other piece being no way considerable for bulk\nor repute. But it is not that book, nor yet his History of Ismenias, but\nhis notes upon the 725th verse of Dionysius \u03a0\u03b5\u03c1\u1fd6\u03b7\u03b3\u03b7\u03c4\u03ae\u03c2. The poet had\nsaid of the stone _Jaspis_, that it was\n\n \u1f18\u03c7\u03b8\u03c1\u03b7\u1f76\u03bd \u1f18\u03bc\u03c0\u03bf\u03cd\u03c3\u03b7\u03c3\u03b9 \u03ba\u03b1\u1f76 \u1f04\u03bb\u03bb\u03bf\u03b9\u03c2 \u1f14\u03b9\u03b4\u03ce\u03bb\u03bf\u03b9\u03c3\u03b9\u03bd,\n\nUpon which Eustathius thus remarks: \u0394\u03bf\u03ba\u03b5\u1fd6 \u03b3\u03b1\u03c1 \u1f00\u03bb\u03b5\u03be\u03af\u03ba\u03b1\u03ba\u03bf\u03c2 \u03b5\u1f36\u03bd\u03b1\u03b9 \u1f21 \u03bb\u03af\u03b8\u03bf\u03c2\n\u1f05\u03c5\u03c4\u03b7, \u03ba\u03b1\u1f76 \u1f00\u03c0\u03bf\u03c4\u03c1\u03bf\u03c0\u03b9\u03b1\u03c2\u03c4\u03b9\u03ba\u1f74 \u03c6\u03b1\u03c3\u03bc\u1f00\u03c4\u03c9\u03bd, \u1f67\u03bd \u1f15\u03bd \u1f10\u03c2\u03c4\u03b9 \u03ba\u03b1\u1f76 \u1f21 \u1f1c\u03bc\u03c0\u03bf\u03c5\u03c3\u03b1, \u03b4\u03b1\u03b9\u03bc\u03bf\u03bd\u03b9\u03cc\u03bd\n\u03c4\u03b9 \u03c4\u03b5\u03c1\u03af \u03c4\u1f30\u1f7c \u1f19\u03ba\u03ac\u03c4\u03b7\u03bd, \u1f11\u03bd\u1f76 \u03c0\u03bf\u03b4\u1f76 \u03b4\u03bf\u03ba\u03bf\u1fe6\u03bd \u03b4\u1f30\u03ae\u03ba\u03b5\u03c3\u03b8\u03b1\u03b9\u00b7 (_fort\u00e8_ \u03b4\u03b9\u03b5\u03c1\u03b5\u03af\u03b4\u03b5\u03c3\u03b8\u03b1\u03b9\n_Steph._) \u1f44\u03b8\u03b5\u03bd \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c1\u03bf\u03bd\u03bf\u03bc\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9, \u1f61\u03c2 \u1f14\u03b9 \u03c4\u03b9\u03c2 \u1f14\u03b9\u03c0\u03b7 \u03bc\u03bf\u03bd\u03cc\u03c0\u03bf\u03c5\u03c2 \u03c0\u03bf\u03b4\u03b9 \u03b6\u03c9\u03bf\u1fe6\u00b7 \u1f61\u03c2\n\u03c4\u03bf\u1fe6 \u1f11\u03c4\u03ad\u03c1\u03bf\u03c5 \u03c0\u03bf\u03b4\u03bf\u03c2 \u03c7\u03b1\u03bb\u03ba\u03bf\u1fe6 \u1f44\u03bd\u03c4\u03bf\u03c2, \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f78\u03bd \u03bc\u1fe6\u03b8\u03bf\u03bd. This testimony doth not\nprove anything of _hopping_, and, as to the derivation, I cannot but say\nthat Eustathius had too much of the grammarian in him, and this is not\nthe first time, neither in this book, nor elsewhere, wherein he hath\ntrifled. It is observable out of the place, that there were more\n_Empusas_ than one, as, indeed, the name is applied by several men to\nany kind of frightful phantasm. And so it is used by several authors,\nand for as much as phantasms are various, according as the persons\naffrighted have been severally educated, &c. every man did impose this\nname upon his own apprehensions. This gave men occasion to fain _Empusa_\nas such--for who will believe that she was not apprehended as having\nfour legs, when she appeared in the form of a cow, dog, &c.--but, as\napprehended by _Bacchus_ and his man at that time. I do not find that\nshe appeared in any shape but such as made use of legs in going, whence\nI imagine that _Empus\u00e6_ might be opposite to the \u03b8\u03b5\u03bf\u1f76 \u03bd\u03b5\u03c0\u03bf\u03b4\u03b5\u03c2, which\nappellation was anciently fixed upon the gods, (_propitious_) upon a\ntwo-fold account; first, for that they were usually effigiated as having\nno feet, which is evident from ancient sculpture, and secondly, for that\nthey are all said not to walk, but rather swim, if I may so express that\n_non gradiuntur, sed fluunt_, which is the assertion of all the\ncommentators I have ever seen upon that verse of Virgil:--\n\n \u201cEt vera incessu patuit dea\u201d----\n\nThis whole discourse may be much illustrated from a passage in\nHeliodorus, \u00c6thiop. lib. iii. sec. 12, 13. Calasiris told Cnemon that\nthe Gods Apollo and Diana did appear unto him; Cnemon replied, \u1f08\u03bb\u03bb\u1f70 \u03c4\u03af\u03bd\u03b1\n\u03b4\u1f74 \u03c4\u03c1\u03cc\u03c0\u03bf\u03bd \u1f12\u03c6\u03b1\u03c3\u03ba\u03b5\u03c2 \u1f10\u03bd\u03b4\u03b5\u03b4\u03b5\u1fd6\u03c7\u03b8\u03b1\u1f31 \u03c3\u03bf\u03b9 \u03c4\u03bf\u1fe6\u03c2 \u03b8\u03b5\u03bf\u1fe6\u03c2 \u1f45\u03c4\u03b9 \u03bc\u1f74 \u1f10\u03bd\u03cd\u03c0\u03bd\u03b9\u03bf\u03bd \u1f26\u03bb\u03b8\u03bf\u03bd, \u1f00\u03bb\u03bb\u2019\n\u1f10\u03bd\u03b1\u03c1\u03b3\u1ff6\u03c2 \u1f10\u03c6\u1fb6\u03bd\u03b7\u03c3\u03b1\u03bd; upon this the old priest answered, that both gods and\ndemons, when they appear to men, may be discovered by the curious\nobserver, both in that they never shut their eyes, \u03ba\u03b1\u1f76 \u03c4\u1ff3 \u03b2\u03b1\u03b4\u03af\u03c3\u03bc\u03b1\u03c4\u03b9\n\u03c0\u03bb\u03ad\u03bf\u03bd, \u03bf\u1f50 \u03ba\u03b1\u03c4\u1f70 \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03c3\u03b9\u03bd \u03c4\u1ff6\u03bd \u03c0\u03bf\u03b4\u1ff6\u03bd \u03bf\u1f50\u03b4\u03ad \u03bc\u03b5\u03c4\u03ac\u03b8\u03b5\u03c3\u03b9\u03bd \u1f00\u03bd\u03c5\u03bf\u03bc\u03ad\u03bd\u03c9, \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u03c4\u1f70\n\u03c4\u03b9\u03bd\u03b1 \u03c1\u1f55\u03bc\u03b7\u03bd \u1f00\u03ad\u03c1\u03b9\u03bf\u03bd, \u03ba\u03b1\u1f76 \u1f41\u03c1\u03bc\u1f74\u03bd \u1f00\u03c0\u03b1\u03c1\u03b1\u03c0\u03cc\u03b4\u03b9\u03c3\u03c4\u03bf\u03bd, \u03c4\u03b5\u03bc\u03bd\u03cc\u03bd\u03c4\u03c9\u03bd \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u03c4\u1f78 \u03c0\u03b5\u03c1\u03b9\u03b5\u03c7\u03bf\u03bd\n\u1f22 \u03b4\u03b9\u03b1\u03c0\u03bf\u03c1\u03b5\u03c5\u03bf\u03bc\u03ad\u03bd\u03c9\u03bd. \u0394\u1f76\u03bf \u03b4\u1f74 \u03ba\u03b1\u1f76 \u03c4\u1f70 \u1f00\u03b3\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1 \u03c4\u1ff6\u03bd \u03b8\u03b5\u1ff6\u03bd \u1f08\u03b9\u03b3\u03cd\u03c0\u03c4\u03b9\u03bf\u03b9 \u03c4\u1f7c \u03c0\u1f78\u03b4\u03ad\n\u03b6\u03b5\u03c5\u03b3\u03bd\u03cd\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76 \u1f65\u03c3\u03c0\u03b5\u03c1 \u1f11\u03bd\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u1f35\u03c3\u03c4\u1fb6\u03c3\u03b9\u03bd. \u1f05 \u03b4\u1f74 \u03ba\u03b1\u1f76 \u1f4d\u03bc\u03b7\u03c1\u03bf\u03c2 \u1f10\u03b9\u03b4\u1ff6\u03c2, \u1f05\u03c4\u03b5\n\u1f08\u03b9\u03b3\u03cd\u03c0\u03c4\u03b9\u03bf\u03c2, \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f31\u03b5\u03c1\u1f70\u03bd \u03c0\u03ac\u03b9\u03b4\u03b5\u03c5\u03c3\u03b9\u03bd \u1f10\u03ba\u03b4\u03b9\u03b4\u03b1\u03c7\u03b8\u03b5\u03af\u03c2, \u03c3\u03c5\u03bc\u03b2\u03bf\u03bb\u03b9\u03ba\u1ff6\u03c2 \u03c4\u03bf\u1fd6\u03c2 \u1f14\u03c0\u03b5\u03c3\u03b9\u03bd\n\u1f10\u03bd\u03b1\u03c0\u03b5\u03b8\u03b5\u03c4\u03bf, \u03c4\u03bf\u1fd6\u03c2 \u03b4\u03c5\u03bd\u03b1\u03bc\u03ad\u03bd\u03bf\u03b9\u03c2 \u03c3\u03c5\u03bd\u03b9\u03ad\u03bd\u03b1\u03b9 \u03b3\u03bd\u03c9\u03c1\u03b9\u03b6\u03b5\u03b9\u03bd \u03ba\u03b1\u03c4\u03b1\u03bb\u03b9\u03c0\u03ce\u03bd, \u1f10\u03c0\u03b9 \u03c4\u03bf\u1fe6\n\u03c0\u03bf\u03c3\u03b5\u03b9\u03b4\u1ff6\u03bd\u03bf\u03c2, \u03c4\u03bf\n\n \u1f3c\u03c7\u03bd\u03b9\u03b1 \u03b3\u1f70\u03c1 \u03bc\u03b5\u03c4\u03cc\u03c0\u03b9\u03c3\u03b8\u03b5, \u03c0\u03bf\u03b4\u1ff6\u03bd \u1f20\u03b4\u03ad \u03ba\u03bd\u03b7\u03bc\u03ac\u03c9\u03bd\n \u1fec\u03b5\u1fd6 \u1f14\u03b3\u03bd\u03c9\u03bd \u1f00\u03c0\u03b9\u1f78\u03bd\u03c4\u03bf\u03c2.\n\n\u03bf\u1f34\u03bf\u03bd \u1fe5\u03ad\u03bf\u03bd\u03c4\u03bf\u03c2 \u1f10\u03bd \u03c4\u1fc6 \u03c0\u03bf\u03c1\u03b5\u03af\u03b1, \u03c4\u03bf\u1fe6\u03c4\u03bf \u03b3\u03ac\u03c1 \u03b5\u03c3\u03c4\u03b9 \u03c4\u03cc \u1fe5\u03b5\u1fd6 \u1f00\u03c0\u03b9\u03cc\u03bd\u03c4\u03bf\u03c2, \u03ba\u03b1\u1f76 \u03bf\u1fe6\u03c7 \u1f65\u03c2\n\u03c4\u03b9\u03bd\u03b5\u03c2 \u1f20\u03c0\u03ac\u03c4\u03b7\u03bd\u03c4\u03b1\u03b9, \u1fe5\u1fb3\u03b4\u03af\u03c9\u03c2 \u1f14\u03b3\u03bd\u03c9\u03bd \u1f51\u03c0\u03bf\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03bf\u03bd\u03c4\u03b5\u03c2. Farnaby, upon the place in\nVirgil, observes, that _Deorum incessus est continuus et \u00e6qualis, non\ndimotis pedibus, neque transpositis_, \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u03c4\u1f70 \u1fe5\u03cd\u03bc\u03b7\u03bd \u1f00\u03ad\u03c1\u03b9\u03bf\u03bd. Cornelius\nSchrevelius in the new Leyden notes saith, _Antiquissima qu\u00e6que Deorum\nsimulachra, quod observarunt viri magni, erant_ \u03c4\u03bf\u1fe6\u03c2 \u03c0\u03cc\u03b4\u03b1\u03c2 \u03c3\u03c5\u03bc\u03b2\u03b5\u03b2\u03b7\u03ba\u03cc\u03c4\u03b1,\n_diique ipsi non gradiuntur sed fluunt_. Their statues were said to\nstand rather upon columns than upon legs, for they seem to have been\nnothing but columns shaped out into this or that figure, the base\nwhereof carrying little of the representation of a foot. These things\nbeing premised, I suppose it easy for the intelligent reader to find out\nthe true etymology of _Empusa, quasi_ \u1f10\u03bd \u03c0\u03bf\u03c3\u03b9\u03bd \u03bf\u1fe6\u03c3\u03b1, or \u03b2\u03ac\u03b9\u03bd\u03bf\u03c5\u03c3\u03b1, from\ngoing on her feet, whereas the other _gods_ and _demons_ had a different\ngait. If any can dislike this deduction, and think her so named from\n\u1f11\u03bd\u03b9\u03c0\u03bf\u03c5\u03c2, whereas she always went upon two legs, (if her shape permitted\nit) though she might draw the one after her, as a man doth a wooden leg:\nI say, if any, notwithstanding what hath been said, can join issue with\nthe Doctor, my reply shall be \u03a3\u03bf\u1f76 \u03bc\u1f72\u03bd \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03b4\u03bf\u03ba\u03bf\u1fe6\u03bd\u03c4\u2019 \u1f10\u03c3\u03c4\u1f76\u03bd, \u1f10\u03bc\u1f78\u03b9 \u03b4\u1f72\n\u03c4\u03ac\u03b4\u03b5.\n\nNow, as to the words of Aristophanes upon which the Scholiast descants,\nthey are these:--speaking of an apparition strangely shaped, sometimes\nlike a camel, sometimes like an ox, a beautiful woman, a dog, &c.\nBacchus replies:\n\n \u1f1c\u03bc\u03c0\u03bf\u03c5\u03c3\u03b1 \u03c4\u03bf\u03b9\u03bd\u1f7a\u03bd \u03b3\u2019\u1f10\u03c3\u03c4\u03b9.\n \u039e\u0391. \u03c0\u03c5\u03c1\u1f76 \u03b3\u03bf\u1fe6\u03bd \u03bb\u03ac\u03bc\u03c0\u03b5\u03c4\u03b1\u03b9\n \u1f05\u03c0\u03b1\u03bd \u03c4\u03bf \u03c0\u03c1\u03bf\u03c3\u03c9\u03c0\u03bf\u03bd, \u03ba\u03b1\u1f76 \u03c3\u03ba\u03b5\u03bb\u03bf\u03c2 \u03c7\u03b1\u03bb\u03ba\u03bf\u1fe6\u03bd \u1f14\u03c7\u03b5\u03b9.\n \u0394\u0399. \u039d\u1f74 \u03c4\u1f78\u03bd \u03a0\u03bf\u03c3\u03b5\u03b9\u03b4\u1ff6, \u03ba\u03b1\u1f76 \u03b2\u03bf\u03bb\u03b9\u03c4\u03b9\u03bd\u03bf\u03bd \u03b8\u03ac\u03c4\u03b5\u03c1\u03bf\u03bd.\n \u039e\u0391. \u03a3\u1f01\u03c6\u2019 \u1f35\u03c3\u03b8\u03b9.\n\nThe Scholiast hereupon tells us that _Empusa_, was \u03a6\u03b1\u03bd\u03c4\u03b1\u03c3\u03bc\u03b1 \u03b4\u03b1\u03b9\u03bc\u03bf\u03bd\u03b9\u1ff6\u03b4\u03b5\u03c2\n\u1f51\u03c0\u1f78 \u1f19\u03ba\u03ac\u03c4\u03b7\u03c2 \u1f10\u03c0\u03b9\u03c0\u03b5\u03bc\u03c0\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c6\u03b1\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03bd \u03c4\u03bf\u1f34\u03c2 \u03b4\u03c5\u03c3\u03c4\u03c5\u03c7\u03bf\u1fe6\u03c3\u03b9\u03bd, \u1f45 \u03b4\u03bf\u03ba\u03b5\u1fd6 \u03c0\u03bf\u03bb\u03bb\u1f70\u03c2\n\u03bc\u03bf\u03c1\u03c6\u03b1\u03c2 \u03b1\u03bb\u03bb\u03b1\u03c3\u03c3\u03b5\u03c9 \u03ba\u03b1\u1f76 \u1f41\u03b9 \u03bc\u03b5\u03bd \u03c6\u03b1\u03c3\u03b9\u03bd \u1f00\u03c5\u03c4\u03b7\u03bd \u03bc\u03bf\u03bd\u03bf\u03c0\u03bf\u03b4\u03b1 \u03b5\u1fd6\u03bd\u03b1\u03b9, \u03ba\u03b1\u1f76\n\u1f10\u03c4\u03c5\u03bc\u03bf\u03bb\u03bf\u03b3\u03bf\u1fe6\u03c3\u03b9\u03bd\u2019 \u1f41\u03b9\u03bf\u03bd\u03b5\u1f76 \u1f11\u03bd\u03b9\u03c0\u03bf\u03b4\u03b1, \u03b4\u03b9\u1f70 \u03c4\u03bf \u1f11\u03bd\u1f76 \u03c0\u03bf\u03b4\u03b9 \u03ba\u03b5\u03c7\u03c1\u1fc6\u03c3\u03b8\u03b1\u03b9. And this is\nall that is material in the Scholiast, except that he adds by and by,\nthat \u03b2\u03bf\u03bb\u03b9\u03c4\u03b9\u03bd\u03bf\u03bd \u03c3\u03ba\u03b5\u03bb\u03bf\u03c2 is all one with the leg of an ass. And this very\ntext and Scholiast is that to which all the authors he names, and more,\ndo refer.\n\nI come now to Stephen, who, in his index, and in the word \u03c0\u03bf\u03b4\u03af\u03b6\u03c9, gives\nthe derivation of _Empusa_. \u03a0\u03bf\u03b4\u03b9\u03b6\u03c9, _gradior, incedo_, (not to hop) _sic\nSuidas_ \u1f1c\u03bc\u03c0\u03bf\u03c5\u03c3\u03b1\u03bd _dictam ait_ \u03c0\u03b1\u03c1\u1f70 \u03c4\u03bf \u1f11\u03bd\u1f76 \u03c0\u03bf\u03b4\u03b9\u03b6\u03b5\u03b9\u03bd. In the index thus:\n_sunt qui dictam putent_ \u03c0\u03b1\u03c1\u1f70 \u03c4\u1f78 \u1f11\u03bd\u1f76 \u03c0\u03bf\u03b4\u1f76\u03b6\u03b5\u03b9\u03bd, _quod uno incedat pedi,\nquasi_ \u1f1c\u03bc\u03c0\u03bf\u03c5\u03c3\u03b1\u03bd, _alterum enim pedem \u00e6neum habet_. But neither Stephen,\nnor any else, except _Suidas_, whom the hypercritical Doctor had not\nseen, no, not the Scholiast of Aristophanes (a better critic than Mr.\nHobbes) doth relate the etymology as their own. Nay, there is not one\nthat saith _Empusa_ hopped on one leg, which is to be proved out of\nthem. The great Etymological Dictionary deriveth it \u03c0\u03b1\u03c1\u1f70 \u03c4\u1f78 \u1f10\u03bc\u03c0\u03bf\u03b4\u03b9\u03b6\u03b5\u03b9\u03bd,\nto _hinder_, _let_, &c. its apparition being a token of ill luck. But,\nas to the Doctor\u2019s deduction, it saith, \u1f1c\u03bc\u03c0\u03bf\u03c5\u03c3\u03b1 \u03a8\u03b9\u03bb\u03bf\u1fe6\u03c0\u03b1\u03b9, \u03b5\u1f30 \u03ba\u03b1\u1f76 \u03b4\u03bf\u03ba\u03b5\u1fd6\n\u03c0\u03b1\u03c1\u1f70 \u03c4\u1f78 \u1f15\u03bd\u03b1 \u03c3\u03c5\u03b3\u03ba\u03b5\u1fd6\u03c3\u03b8\u03b1\u03b9. It doth only _seem_ so. And it is strange that\n\u1f11\u03bd should not alter only its _aspiration_, but change its \u03bd into \u03bc,\nwhich I can hardly believe admittable in Greek, least there should be no\ndifference betwixt its derivatives and those of \u1f10\u03bd. When I consider the\nseveral \u03bc\u03bf\u03c1\u03bc\u03cc\u03bd\u03b5\u03c2 which the Grecians had, some whereof did fly, some had\nno legs, &c., I can think that the origin of this name may have been\nthus: some amazed person saw a _spectrum_, and, giving another notice of\nit, his companion might answer, it is \u0392\u03c1\u03b9\u03bc\u1f7c, \u039c\u03bf\u03c1\u03bc\u1f7c \u1f29\u03ba\u1f70\u03c4\u03b7, but he,\nmeeting with a new phantasm, cries, \u1f10\u03bd \u03c0\u03bf\u03c3\u1f76 \u03b2\u03b1\u03af\u03bd\u03b5\u03b9 or \u03b2\u03b1\u03b4\u03af\u03b6\u03b5\u03b9, for which\napprehension of his, somebody coined this expression of \u1f1c\u03bc\u03c0\u03bf\u1fe6\u03c3\u03b1. It may\nalso be possibly deduced from \u1f18\u03bc\u03c0\u03bf\u03b4\u1f76\u03b6\u03c9, so that \u03c4\u03cd\u03c7\u03b7 \u1f10\u03bc\u03c0\u03bf\u03b4\u03b9\u03b6\u03bf\u03c5\u03c3\u03b1 might\nafterwards be reduced to the single term of _Empusa_. Nor do I much\ndoubt but that those who are conversant in languages, and know how that\nseveral expressions are often jumbled together to make up one word upon\nsuch like cases, will think this a probable origination. I believe,\nthen, that Mr. Hobbes\u2019s friend did never tell him it was in Eustathius,\nor that _Empusa_ was an _hopping phantasm_. It had two legs and went\nupon both, as a man may upon a wooden leg. \u1f1c\u03bc\u03c0\u03bf\u1fe6\u03c3\u03b1 is also a name for\nLamia, and such was that which Menippus might have married, which, I\nsuppose, did neither hop nor go upon one leg, for he might have\ndiscovered it. But Mr. Hobbes did not except against the derivation,\n(although he might justly, derivations made afterwards carrying more of\nfancy than of truth, and the Doctor is not excused for asserting what\nothers barely relate, none approve), but asked him where that is, in\nwhat authors _he read that boys\u2019 play to be so called_. To which\nquestion, the Doctor, to show his reading and the good authors he is\nconversant in, replies, _in Junius\u2019s Nomenclator, Rider and Thomas\u2019s\nDictionary, sufficient authors in such a business_, which, methinks, no\nman should say that were near to so copious a library. It is to be\nremembered that the trial now is in Westminster School, and amongst\nCiceronians, neither whereof will allow those to be sufficient authors\nof any Latin word. Alas, they are but _Vocabularies_; and, if they bring\nno author for their allegation, all that may be allowed them is, that,\nby way of allusion, our modern play may be called _Ludus Empus\u00e6_. But\nthat it is so called we must expect, till some author do give it the\nname. These are so good authors, that I have not either of them in my\nlibrary. But I have taken the pains to consult, first, Rider; I looked\nin him, (who was only author of the English Dictionary) and I could not\nfind any such thing. It is true, in the Latin Dictionary, which is\njoined with Rider, but made by Holyoke; (O that the Doctor would but\nmark!) in the index of obsolete words, there is _Ascoliasmus, Ludus\nEmpus\u00e6_, _Fox to thy hole_, for which word, not signification, he\nquoteth Junius. The same is in Thomasius, who refers to Junius in like\nmanner. But could the Doctor think the word obsolete, when the play is\nstill in fashion? Or, doth he think that this play is so ancient as to\nhave had a name so long ago, that it should now be grown obsolete? As\nfor Junius\u2019s interpretation of _Empusa_, it is this: _Empusa, spectrum,\nquod se infelicibus ingerit, uno pede ingrediens_. Had the Doctor ever\nread him, he would have quoted him for his derivation of _Empusa_, I\nsuppose. In Ascoliasmus, he saith, _Ascoliasmus, Empus\u00e6 Ludus, fit ubi,\naltero pede in aere librato, unico subsiliunt pede:_ \u1f00\u03c3\u03ba\u03bf\u03bb\u03b9\u03b1\u03c3\u03bc\u1f78\u03c2\n_Pollux; Almanic\u00e8, Hinckelen; Belgic\u00e8,_ _Op een been springhen;\nHinckepincken, Flandris_. But what is it in English he doth not tell,\nalthough he doth so in other places often. What the Doctor can pick out\nof the Dutch I know not; but, if that do not justify him, as I think it\ndoth not, he hath wronged Junius, and greatly imposed upon his readers.\n\nBut, to illustrate this controversy further, I cannot be persuaded the\nDoctor ever looked into Junius, for, if he had, I am confident,\naccording to his wonted accurateness, he would have cited Pollux\u2019s\n_Onomasticon_ into the bargain, for Junius refers to him, and I shall\nset down his words, that so the reader may see what _Ascoliasmus_ was,\nand all the Doctor\u2019s authors say _Ludus Empus\u00e6_ and _Ascoliasmus_ were\none and the same thing. Julius Pollux (lib. ix. cap. 7): \u1f49 \u03b4\u03b5\n\u1f08\u03c3\u03ba\u03bf\u03bb\u1f76\u03b1\u03c3\u03bc\u1f78\u03c2, (old editions read it, \u1f08\u2019\u03c3\u03ba\u03bf\u03bb\u03b9\u03b1\u03c3\u03bc\u1f78\u03c2 et \u03b1\u03c3\u03ba\u03bf\u03bb\u03b9\u03ac\u03b6\u03c9) \u03c4\u03bf\u1fe6\n\u1f11\u03c4\u03ad\u03c1\u03bf\u03c5 \u03c0\u03bf\u03b4\u1f78\u03c2 \u03b1\u1f30\u03c9\u03c1\u03bf\u03c5\u03bc\u03ad\u03bd\u03bf\u03c5, \u03ba\u03b1\u03c4\u1f70 \u03bc\u03cc\u03bd\u03bf\u03c5 \u03c4\u03bf\u1fe6 \u1f11\u03c4\u03ad\u03c1\u03bf\u03c5 \u03c0\u03b7\u03b4\u1fb6\u03bd \u1f14\u03c0\u03cc\u03b9\u03b5\u03b9; \u1f45\u03c0\u03b5\u03c1\n\u1f08\u03c3\u03ba\u03c9\u03bb\u03b9\u03ac\u03b6\u1f72\u03b9\u03bd \u1f60\u03bd\u03cc\u03bc\u03b1\u03b6\u03bf\u03bd\u00b7 \u1f24\u03c4\u03bf\u03b9 \u03b5\u1f30\u03c2 \u03bc\u1f22\u03ba\u03bf\u03c2 \u1f10\u03bd\u03ae\u03bb\u03bb\u03b1\u03bd\u03c4\u03bf, \u1f22 \u1f41 \u03bc\u1f72\u03bd \u1f10\u03b4\u03af\u03c9\u03ba\u03b5\u03bd \u03bf\u1f55\u03c4\u03c9\u03c2,\n\u03bf\u1f31 \u03b4\u1f72 \u1f51\u03c0\u03ad\u03c6\u03b5\u03c5\u03b3\u03bf\u03bd \u1f10\u03c0\u2019 \u1f00\u03bc\u03c6\u03bf\u1fd6\u03bd \u03b8\u1f72\u03bf\u03bd\u03c4\u03b5\u03c2, \u1f15\u03c9\u03c2 \u03c4\u03b9\u03bd\u1f78\u03c2 \u03c4\u1ff3 \u03c6\u03b5\u03c1\u03bf\u03bc\u03ad\u03bd\u1ff3 \u03c0\u03bf\u03b4\u1f76 \u1f41 \u03b4\u03b9\u1f7c\u03ba\u03c9\u03bd\n\u03b4\u03c5\u03bd\u03b7\u03b8\u1fc7 \u03c4\u03c5\u03c7\u03b5\u1fd6\u03bd\u00b7 \u1f24 \u03ba\u03b1\u1f76 \u03c3\u03c4\u03ac\u03bd\u03c4\u03b5\u03c2 \u1f10\u03c0\u03ae\u03b4\u03c9\u03bd, \u1f00\u03c1\u03b9\u03b8\u03bc\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u03c4\u1f70 \u03c0\u03b7\u03b4\u03ae\u03bc\u03b1\u03c4\u03b1\u00b7 \u03c0\u03c1\u03bf\u03c3\u03ad\u03ba\u03b5\u03b9\u03c4\u03bf\n\u03b3\u1f70\u03c1 \u03c4\u1ff7 \u03c0\u03bb\u03ae\u03b8\u03b5\u03b9 \u03c4\u1f78 \u03bd\u03b9\u03ba\u1fb6\u03bd. \u1f08\u03c3\u03ba\u03c9\u03bb\u03b9\u03ac\u03b6\u03b5\u03b9\u03bd \u03b4\u1f72 \u1f10\u03ba\u03b1\u03bb\u03b5\u1fd6\u03c4\u03bf \u03ba\u03b1\u1f76 \u03c4\u1f78 \u1f10\u03c0\u03b9\u03c0\u03b7\u03b4\u1fb6\u03bd \u1f00\u03c3\u03ba\u1ff7\n\u03ba\u03b5\u03bd\u1ff7 \u03ba\u03b1\u1f76 \u1f51\u03c0\u03bf\u03c0\u03bb\u03ad\u03c9 \u03c0\u03bd\u03b5\u03c5\u03bc\u03b1\u03c4\u03bf\u03c2, \u1f20\u03bb\u03b5\u03af\u03bc\u03bc\u03ad\u03bd\u03c9, \u1f35\u03bd\u03b1\u03c0\u03b5\u03c1 \u1f40\u03bb\u03b9\u03c3\u03b8\u03ac\u03bd\u03bf\u03b9\u03b5\u03bd \u03c0\u03b5\u03c1\u1f76 \u03c4\u1f74\u03bd\n\u1f00\u03bb\u03bf\u03b9\u03c6\u1f74\u03bd. \u201cSo that _Ascoliasmus_, and consequently, _Ludus Empus\u00e6_, was a\ncertain sport which consisted in hopping, whether it were by striving\nwho could hop furthest, or whether only one did pursue the rest hopping,\nand they fled before him on both legs, which game he was to continue\ntill he had caught one of his fellows, or whether it did consist in the\nboys\u2019 striving who could hop longest. Or, lastly, whether it did consist\nin hopping upon a certain bladder, which, being blown up and well oiled\nover, was placed upon the ground for them to hop upon, that so the\nunctuous bladder might slip from under them and give them a fall.\u201d And\nthis is all that Pollux holds forth. Now, of all these ways, there is\nnone that hath any resemblance with our _Fox to thy hole_; but the\nsecond: and yet, in its description, there is no mention of beating him\nwith gloves, as they do now-a-days, and wherein the play consists as\nwell as in hopping. It might, notwithstanding, be called _Ludus Empus\u00e6_,\nbut not in any sort our _Fox to thy hole_; so that the Doctor and his\nauthors are out, imposing that upon Junius and Pollux which they never\nsaid. And thus much may suffice as to this point. I shall only add out\nof Meursius\u2019s _Ludi Gr\u00e6ci_, that _Ascolia_ were not _Ludus Empus\u00e6_ but\n_Bacchisacra_, and he quotes Aristophanes\u2019s Scholiast in Plutus, \u1f08\u03c3\u03ba\u03ce\u03bb\u03b9\u03b1\n\u1f11\u03bf\u03c1\u03c4\u1f74 \u0394\u03b9\u03bf\u03bd\u03cd\u03c3\u03bf\u03c5 \u1f00\u03c3\u03ba\u1f78\u03bd \u03b3\u03b1\u03c1 \u03bf\u1f35\u03bd\u03bf\u03c5 \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2, \u1f11\u03bd\u1f76 \u03c0\u03bf\u03b4\u1f76 \u03c4\u03bf\u1fe6\u03c4\u03bf\u03bd \u1f10\u03c0\u03b5\u03c0\u03ae\u03b4\u03bf\u03bd, \u03ba\u03b1\u1f76\n\u1f41 \u03c0\u03b7\u03b4\u03ae\u03c3\u03b1\u03c2 \u1f06\u03b8\u03bb\u03bf\u03bd \u03b5\u1f36\u03c7\u03b5 \u03c4\u1f78\u03bd \u03bf\u1f35\u03bd\u03bf\u03c5. As also Hesychius, \u1f08\u03c3\u03ba\u03c9\u03bb\u03b9\u03ac\u03b6\u03b5\u03b9\u03bd, \u03ba\u03c5\u03c1\u03af\u03c9\u03c2\n\u03c4\u1f78 \u1f10\u03c0\u1f76 \u03c4\u03bf\u1fe6\u03c2 \u1f00\u03c3\u03ba\u03bf\u1f7a\u03c2 \u1f05\u03bb\u03bb\u03b5\u03c3\u03b8\u03b1\u03b9.\n\nBut I could have told the Doctor where he might have read of _Empusa_ as\nbeing the name of a certain sport or game, and that is, _in Turnebus\nAdversaria_, lib. xxvii. cap. 33. There he speaks of several games\nmentioned by Justinian in his _Code_, at the latter end of the third\nbook, one of which he takes to be named _Empusa_; adding withal, _that\nthe other are games, it is indisputable_, only _Empusa in lite et causa\nerit, quod nemo nobis facile assensurus sit Ludum esse, cum constet\nspectrum quoddam fuisse formas, varie mutans. Sed quid vetat eo nomine\nLudum fuisse? Certe ad vestigia vitiat\u00e6 Scriptur\u00e6 quam proximo accedit._\nYet he only is satisfied in this conjecture, till somebody else shall\nproduce a better. And now what shall I say? Was not Turnebus as good a\ncritic, and of as great reading as Dr. Wallis, who had read over Pollux,\nand yet is afraid that nobody will believe _Empusa_ to have been a game,\nand all he allegeth for it is, _quid vetat_? Truly, all I shall say, and\nso conclude this business, is, that he had read over an infinity of\nbooks, yet, had not had the happiness, which the Doctor had, to consult\nwith _Junius\u2019s Nomenclator, Thomasius and Rider\u2019s Dictionary, authors\nsufficient in such a case_.\n\nI now come to the Doctor\u2019s last and greatest triumph, at which I cannot\nbut stand in admiration, when I consider he hath not got the victory.\nHad the Doctor been pleased to have conversed with some of the fifth\nform in Westminster School, (for he needed not to have troubled the\nlearned master), he might have been better informed than to have exposed\nhimself thus.\n\nMr. Hobbes had said that \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 signified _a mark with a hot iron_; upon\nwhich saying the Doctor is pleased to play the droll thus: \u201cPrithee tell\nme, good Thomas, before we leave this point, (O the wit of a divinity\ndoctor!) who it was told thee that \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 was a mark with an hot iron,\nfor it is a notion I never heard till now, and do not believe it yet.\nNever believe him again that told thee that lie, for as sure as can be,\nhe did it to abuse thee; \u03db\u03b9\u03b3\u03bc\u1f74 signifies a distinctive point in writing,\nmade with a pen or quill, not a mark made with a _hot iron_, such as\nthey brand rogues withal; and, accordingly, \u03db\u03b9\u03b6\u03c9 \u03b4\u1fd6\u03b1\u03db\u03b9\u03b6\u03c9, _distinguo_,\n_interstinguo_, are often so used. It is also used of a _mathematical_\npoint, or somewhat else that is very small, \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 \u03c7\u03c1\u1f78\u03bd\u03bf\u03c5, a moment, or\nthe like. What should come in your cap, to make you think that \u03db\u1fd6\u03b3\u03bc\u1f74\nsignifies a mark or brand with a _hot iron_? I perceive where the\nbusiness lies; it was \u03db\u03af\u03b3\u03bc\u03b1 ran in your mind when you talked of \u03db\u03b9\u03b3\u03bc\u1f74,\nand, because the words are somewhat alike, you jumbled them both\ntogether, according to your usual care and accurateness, as if they had\nbeen the same.\u201d\n\nWhen I read this I cannot but be astonished at the Doctor\u2019s confidence,\nand applaud him who said, \u1f00\u03bc\u03ac\u03b8\u03b5\u03b9\u03b1 \u03b8\u03ac\u03c1\u03c3\u1f78\u03c2 \u03c6\u03ad\u03c1\u03b5\u03b9. That the Doctor should\nnever hear that \u03db\u03b9\u03b3\u03bc\u1f74 signifies _a mark with a hot iron_, is a manifest\nargument of his ignorance. But, that he should advise Mr. Hobbes not to\nbelieve his own readings, or any man\u2019s else that should tell him it did\nsignify any such thing, is a piece of notorious impudence. That \u03db\u03b9\u03b3\u03bc\u1f74\n_signifies a distinctive point in writing made with a pen or quill_, (is\na pen one thing and a quill another to write with?) nobody denies. But,\nit must be withal acknowledged it signifies many things else. I know the\nDoctor is a _good historian_, else he should not presume to object the\nwant of history to another; let him tell us how long ago it is since men\nhave made use of pens or quills in writing; for, if that invention be of\nno long standing, this signification must also be such, and so it could\nnot be that from any allusion thereunto the mathematicians used it for a\npoint. Another thing I would fain know of this great historian, how long\nago \u03db\u03af\u03b6\u03c9 and \u03b4\u03b9\u03b1\u03db\u03af\u03b6\u03c9 began to signify _interpungo_? For, if the\nmathematics were studied before the mystery of printing was found out,\n(as shall be proved whenever it shall please the Doctor, out of his no\nreading, to maintain the contrary), then the _mathematical_ use thereof\nshould have been named before the _grammatical_. And, if this word be\ntranslatitious, and that sciences were the effect of long contemplation,\nthe names used wherein are borrowed from talk, Mr. Hobbes did well to\nsay, that \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 precedaneously to that _indivisible_ signification\nwhich it afterwards had, did signify a _visible mark_ made by a hot\niron, or the like. And, in this procedure, he did no more than any man\nwould have done, who considers that all our knowledge proceeds from our\nsenses; as also that words do, _primarily_, signify things obvious to\n_sense_, and only _secondarily_, such as men call _incorporeal_. This\nleads me to a further consideration of this word. Hesychius, (of whom it\nis said that he is _Legendus non tanquam Lexicographus, sed tanquam\njustus author_), interprets \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74, \u03bd\u03c5\u03b3\u03bc\u03ae, which is a point of a greater\nor lesser size, made with any thing. So \u03db\u03af\u03b6\u03c9 signifies to prick or mark\nwith anything in any manner, and hath no impropriated signification in\nitself, but according to the writer that useth it. Thus, in a\n_grammarian_ \u03db\u03af\u03b6\u03c9 signifies to _distinguish_, by _pointing_ often;\nsometimes, even in them, it is the same with \u1f40\u03b2\u03b5\u03bb\u03af\u03b6\u03c9; sometimes it\nsignifies to set a mark that something is wanting in that place, which\nmarks were called \u03db\u03b9\u03b3\u03bc\u03b1\u03af. In matters of policy, \u03db\u03af\u03b6\u03c9 signifies to\n_disallow_, because they used to put a \u03db\u03b9\u03b3\u03bc\u1f74 (not \u03db\u03af\u03b3\u03bc\u03b1) before his name\nwho was either disapproved or to be mulcted. In punishment it signifies\nto _mark_ or _brand_, whereof I cannot at present remember any other\nways than that of an _hot iron_, which is most usual in authors, because\nmost practised by the ancients. But, that the mark which the _Turks_ and\nothers do imprint without burning may be said \u03db\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9, I do not doubt,\nno more than that Herodian did to give that term to the ancient Britons,\nof whom he says, \u03c4\u1f70 \u03c3\u03ce\u03bc\u03b1\u03c4\u03b1 \u1f10\u03db\u03af\u03b6\u03bf\u03bd\u03c4\u03bf \u03b3\u03c1\u03b1\u03c6\u03b1\u1fd6\u03c2 \u03c0\u03bf\u03b9\u03ba\u03af\u03bb\u03b1\u03b9\u03c2, \u03ba\u03b1\u1f76 \u03b6\u03ce\u03c9\u03bd\n\u03c0\u03b1\u03bd\u03c4\u03bf\u03b4\u03b1\u03c0\u1ff6\u03bd \u03b5\u1f30\u03ba\u03cc\u03c3\u03b9. Thus, horses that were branded with \u03ba\u03ac\u03c0\u03c0\u03b1 and \u03c3\u03b1\u03bd\n(\u03ba\u03bf\u03c0\u03c0\u1f00\u03c4\u03b9\u03b1\u03b9 and \u03c3\u03b1\u03bc\u03c6\u03bf\u03c1\u03b1\u03b9) were said \u03db\u03af\u03b6\u03b5\u03c3\u03b8\u03b1\u03b9. Thus, in its origin, \u03db\u03b9\u03b3\u03bc\u1f74\ndoth signify a _brand or mark with an hot iron_, or the like; and that\nmust be the proper signification of \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74, which is proper to \u03db\u03af\u03b6\u03c9,\nnone but such as Dr. Wallis can doubt. In its _descendants_ it is no\nless evident, for, from \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 comes _stigmosus_, which signifies to be\nbranded; _Vitelliana cicatrice stigmosus_, not _stigmatosus_. So Pliny\nin his Epistles, as Robert Stephen cites it. And \u03c3\u03c4\u03b9\u03b3\u03bc\u03b1\u03c4\u03b9\u03b1\u03c2 (the\nderivative of \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74, which signifies any mark, as well as a brand, even\nsuch as remain after stripes, being black and blue), was a nickname\nimposed upon the grammarian Nicanor, \u1f45\u03c4\u03b9 \u03c0\u03b5\u03c1\u1f76 \u03c3\u03c4\u03b9\u03b3\u03bc\u1ff6\u03bd \u1f10\u03c0\u03bf\u03bb\u03c5\u03bb\u03cc\u03b3\u03b7\u03c3\u03b5. And,\nthough we had not any examples of \u03c3\u03c4\u1fd6\u03b3\u03bc\u1f74 being used in this sense, yet,\nfrom thence, for any man to argue against it, (but he who knows no more\nthan Stephen tells him) is madness, unless he will deny that any word\nhath lost its right signification, and is used only, by the authors we\nhave, although neither the Doctor nor I have read all them, in its\nanalogical signification. I have always been of opinion, that \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74\nsignified a _single point_, big or little, it matters not; and \u03c3\u03c4\u03af\u03b3\u03bc\u03b1, a\n_composure of many_; as \u03b3\u03c1\u03b1\u03bc\u03bc\u1f74 signifies a _line_, and \u03b3\u03c1\u03ac\u03bc\u03bc\u03b1 a\n_letter_, made of several lines. For \u03c3\u03c4\u03af\u03b3\u03bc\u03b1 signified the _owl_, the\n_s\u00e6m\u00e6na_, the letter K, yea, _whole words, lines, epigrams_ engraven in\nmen\u2019s faces; and \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74, I doubt not, had signified _a single point_,\nhad such been used, and so it became translatitiously used by\ngrammarians and mathematicians. I could give grounds for this\nconjecture, and not be so impertinent as the Doctor in his sermon, where\nhe told men that \u03c3\u03bf\u03c6\u03cc\u03c2 was not in Homer; that from \u1f04\u03c6\u03c1\u03c9\u03bd came _ebrius_;\nthat _sobrietas_ was not bad Latin, and that _sobrius_ was once, as I\nremember, in Tully. Is this to speak suitably to the oracles of God, or\nrather to lash out into idle words? Hath the Doctor any ground to think\nthese are not impertinences? Or, are we, poor mortals, accountable for\nsuch _idle_ words as fall from us in private discourses, whilst these\nambassadors from heaven _droll_ in the pulpit without any danger of an\nafter-reckoning?\n\nBut I proceed to a further survey of the Doctor\u2019s intolerable ignorance.\nHis charge in the end of the _school-master\u2019s_ rant is, that he should\n_remember_ \u03c3\u03c4\u03af\u03b3\u03bc\u03b1 and \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 _are not all one_. I complained before that\nhe hath not cited Robert Stephen aright; now I must tell him he hath\nbeen negligent in the reading of Henry Stephen: for in him he might have\nfound that \u03c3\u03c4\u03af\u03b3\u03bc\u03b1 was sometimes all one with \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74, though there be no\nexample in him wherein \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 is used for \u03c3\u03c4\u03af\u03b3\u03bc\u03b1. Hath not Hesiod, (as\nStephen rightly citeth it), in his _Scutum_, 166-67.\n\n \u03a3\u03c4\u1f34\u03b3\u03bc\u03b1\u03c4\u03b1 \u03b4\u2019 \u1f65\u03c2 \u1f10\u03c0\u03ad\u03c6\u03b1\u03bd\u03c4\u03bf \u1f34\u03b4\u03b5\u1fd6\u03bd \u03b4\u03b5\u03b9\u03bd\u03bf\u1fd6\u03c3\u03b9 \u03b4\u03c1\u03ac\u03ba\u03bf\u03c5\u03c3\u03b9\n \u039a\u03c5\u03b1\u03bd\u03ad\u03b1 \u03ba\u03b1\u03c4\u1f70 \u03bd\u1ff6\u03c4\u03b1\n\n_ubi scholiastes_ \u1f65\u03c3\u03c0\u03b5\u03c1 \u03b4\u1f72 \u03c3\u03c4\u03b9\u03b3\u03bc\u03b1\u1f76 \u1f26\u03c3\u03b1\u03bd \u1f10\u03c0\u03ac\u03bd\u03c9, \u03c4\u1ff6\u03bd \u1fe5\u03ac\u03c7\u03b5\u03c9\u03bd \u03c4\u1ff6\u03bd \u03b4\u03c1\u03b1\u03ba\u03cc\u03bd\u03c4\u03c9\u03bd,\n\u03ba\u03b1\u03c4\u03ac\u03c3\u03c4\u03af\u03ba\u03c4\u03bf\u03b9 \u03b3\u1f70\u03c1 \u03ba\u03b1\u1f76 \u03c0\u03bf\u03b9\u03ba\u03af\u03bb\u03bf\u03b9 \u1f41\u03b9 \u1f44\u03c6\u03b5\u03b9\u03c2. So Johannes Diaconus upon the\nplace, a man who (if I may use the Doctor\u2019s phrase) was _as good a\ncritic as_ the Geometry Professor.\n\nThus much for the _Doctor_. To the understanding _reader_, I say that\n\u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 is used for burning with a hot iron: _2 Macchab._ ix. 11, where\nspeaking of Antiochus\u2019s lamentable death, his body putrefying and\nbreeding worms, he is said, \u1f10\u03b9\u03c2 \u03b5\u03c4\u03af\u03b3\u03bd\u03c9\u03c3\u03b9\u03bd \u03c4\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6 \u1f14\u03c1\u03c7\u03b5\u03b8\u03b1\u03b9 \u03b8\u03b5\u03af\u1fb3 \u03bc\u03ac\u03c3\u03c4\u03b9\u03b3\u03b9,\n\u03ba\u03b1\u03c4\u03b1 \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74\u03bd \u1f10\u03c0\u03b9\u03c4\u03b5\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03c4\u03b1\u1fd6\u03c2 \u1f00\u03bb\u03b3\u03b7\u03b4\u03cc\u03c3\u03b9; _being pained as if he had\nbeen pricked or burned with hot irons_. And that this is the meaning of\nthat elegant writer, shall be made good against the Doctor, when he\nshall please to defend the vulgar interpretation. Pausanias, in\n_B\u0153oticis_, speaking of Epaminondas, who had taken a town belonging to\nthe Sicyonians, called Ph\u0153bia (\u03a6\u03bf\u03c5\u03b2\u03af\u03b1) wherein were many B\u0153otian\nfugitives, who ought, by law, to have been put to death, saith he\ndismissed them under other names, giving them only a _brand_ or _mark_.\n\u03a0\u03cc\u03bb\u03b9\u03c3\u03bc\u03b1 \u1f11\u03bb\u1f7c\u03bd \u03a3\u03b9\u03ba\u03c5\u03c9\u03bd\u1f30\u03c9\u03bd \u03a6\u03bf\u03c5\u03b2\u03af\u03b1\u03bd, \u1f14\u03bd\u03b8\u1f70 \u1f26\u03c3\u03b1\u03bd \u03c4\u03bf \u03c0\u03bf\u03bb\u1f7a \u03bf\u1f31 \u0392\u03bf\u03b9\u03ce\u03c4\u03b9\u03bf\u03b9 \u03c6\u03c5\u03b3\u03ac\u03b4\u03b5\u03c2,\n\u03c3\u03c4\u03b9\u03b3\u03bc\u03ae\u03bd \u1f00\u03c6\u03af\u03b7\u03c3\u03b9 \u03c4\u03bf\u1fe6\u03c2 \u1f10\u03b3\u03ba\u03b1\u03c4\u03b1\u03bb\u03b7\u03c6\u03b8\u03ad\u03bd\u03c4\u03b1\u03c2 \u1f04\u03bb\u03bb\u03b7\u03bd \u03c3\u03c6\u03af\u03c3\u03b9\u03bd \u1f23\u03bd \u03b5\u03c4\u03c5\u03c7\u03b5 \u03c0\u03b1\u03c4\u03c1\u03af\u03b4\u03b1\n\u1f10\u03c0\u03bf\u03bd\u03bf\u03bc\u03ac\u03b6\u03c9\u03bd \u1f10\u03ba\u03ac\u03c3\u03c4\u03c9. It is true \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74\u03bd is here put _adverbially_, but\nthat doth not alter the case. Again, Zonaras, in the third tome of his\nHistory, in the life of the Emperor Theophilus, saith, that when\nTheophanes and another monk had reproved the said emperor for\ndemolishing images, he took and _stigmatized_ each of them with twelve\n_iambics_ in their faces: \u03b5\u1f36\u03c4\u03b1 \u03ba\u03b1\u1f76 \u03c4\u1f70\u03c2 \u1f44\u03c8\u03b5\u03b9\u03c2 \u1f00\u03c5\u03c4\u1ff6\u03bd \u03ba\u03ac\u03c4\u03b5\u03c3\u03c4\u03b9\u03be\u03b5 \u03ba\u03b1\u1f76 \u03c4\u03b1\u1fd6\u03c2\n\u03c3\u03c4\u03b9\u03b3\u03bc\u03b1\u1fd6\u03c2 \u03bc\u03ad\u03bb\u03b1\u03bd \u1f10\u03c0\u03ad\u03c7\u03b5\u03b5 \u03b3\u03c1\u03ac\u03bc\u03bc\u03b1\u03c4\u03b1 \u03b4\u1f72 \u1f10\u03c4\u03cd\u03c0\u03bf\u03c5\u03bd \u03c4\u1f70 \u03c3\u03c4\u03b9\u03b3\u03bc\u03b1\u03c4\u03b1, \u03c4\u1f70 \u03b4\u1f72 \u1f26\u03c3\u03b1\u03bd \u1f34\u03b1\u03bc\u03b2\u03bf\u03b9\n\u03bf\u1f57\u03c4\u03bf\u03b9. A place so evident, that I know not what the Doctor can reply.\nThis place is just parallel to what the same author saith in the life of\nIrene, \u03c4\u1f00\u03c2 \u1f44\u03c8\u03ad\u03b9\u03c2 \u03c3\u03c6\u03ce\u03bd \u03ba\u03b1\u03c4\u03b1\u03c3\u03c4\u03b9\u03be\u03b1\u03c2 \u1f10\u03bd \u03b3\u03c1\u03ac\u03bc\u03bc\u03b1\u03c3\u03b9, \u03bc\u03ad\u03bb\u03b1\u03bd\u03bf\u03c2 \u03b5\u03b3\u03c7\u03b5\u03bf\u03bc\u03ad\u03bd\u03bf\u03c5 \u03c4\u03bf\u1fd6\u03c2\n\u03c3\u03c4\u03af\u03b3\u03bc\u03b1\u03c3\u03b9. If the Doctor object that he is a modern author, he will never\nbe able to render him as inconsiderable as Adrianus Junius\u2019s\n_Nomenclator_, Thomasius and Rider. If any will deny that he writes good\nGreek, Hieronymus Wolfius will tell them, his only fault is\n\u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03bf\u03bb\u03bf\u03b3\u03b9\u03b1, _redundancy_ in words, and not the use of _bad_ ones.\n\nAnother example of \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 used in this sense, is in the collections out\nof Diodorus Siculus, lib. xxxiv. as they are to be found at the end of\nhis works, and as Photius hath transcribed them into his _Bibliotheca_.\nHe saith that the Romans did buy multitudes of servants and employ them\nin Sicily: \u039f\u1f37\u03c2, \u1f10\u03ba \u03c4\u1ff6\u03bd \u03c3\u03c9\u03bc\u03b1\u03c4\u03bf\u03c4\u03c1\u03bf\u03c6\u03b5\u03af\u03c9\u03bd \u1f00\u03b3\u03b5\u03bb\u03b7\u03b4\u1f78\u03bd \u03b1\u03c0\u03b1\u03c7\u03b8\u03b5\u1fd6\u03c3\u03b9\u03bd, \u1f10\u03c5\u03b8\u03cd\u03c2\n\u03c7\u03b1\u03c1\u03b1\u03ba\u03c4\u1fc6\u03c1\u03b1 \u1f10\u03c0\u03ad\u03b2\u03b1\u03bb\u03bb\u03bf\u03bd, \u03ba\u03b1\u1f76 \u03c3\u03c4\u03b9\u03b3\u03bc\u1f70\u03c2 \u03c4\u03bf\u1f34\u03c2 \u03c3\u03ce\u03bc\u03b1\u03c3\u03b9\u03bd. These are the words but\nof one author, but ought to pass for the judgment of two, seeing\nPhotius, by inserting them, hath made them his own.\n\nBesides, it is the judgment of a great _master_ of the Greek tongue,\nthat _stigmata non tam puncta ipsa quam punctis variatam superficiem\nGr\u00e6ci vocaverunt_. I need not, I suppose, name him, so great a critic as\nthe Doctor cannot be ignorant of him.\n\nNor, were \u03c3\u03c4\u03af\u03b3\u03bc\u03b1\u03c4\u03b1 commonly, but upon extraordinary occasions, imprinted\nwith an hot iron. The letters were first made by incision, then the\nblood _pressed_, and the place filled up with ink, the composition\nwhereof is to be seen in Aetius. And thus they did use to _matriculate_\nsoldiers also in the hand. Thus, did the Grecian emperor, in the\nprecedent example of Zonaras. And if the Doctor would more, let him\nrepair to Vinetus\u2019s comment upon the fifteenth Epigram of Ausonius.\n\nAnd now I conceive enough hath been said to vindicate Mr. Hobbes, and to\nshow the insufferable ignorance of the puny professor, and unlearned\ncritic. If any more shall be thought necessary, I shall take the pains\nto collect more examples and authorities, though I confess I had rather\nspend time otherwise, than in matter of so little moment. As for some\nother passages in his book, I am no competent judge of _symbolic\nstenography_. The Doctor (Sir Reverence) might have used a cleanlier\nexpression than that of a _shitten piece_, when he censures Mr. Hobbes\u2019s\nbook.\n\nHitherto the letter.[1] By which you may see _what came into my (not\nsquare) cap to call_ \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 _a mark with a hot iron, and that they who\ntold me_ that, did no more tell me a lie than they told you a lie that\nsaid the same of \u03c3\u03c4\u03af\u03b3\u03bc\u03b1; and, if \u03c3\u03c4\u03b9\u03b3\u03bc\u1f74 be not right as I use it now,\nthen call these notes not \u03c3\u03c4\u03b9\u03b3\u03c1\u03b1\u03c2, but \u03c3\u03c4\u03af\u03b3\u03bc\u03b1\u03c4\u03b1. I will not contend with\nyou for a trifle. For, howsoever you call them, you are like to be known\nby them. Sir, the calling of a divine hath justly taken from you some\ntime that might have been employed in geometry. The study of algebra\nhath taken from you another part, for algebra and geometry are not all\none; and you have cast away much time in practising and trusting to\nsymbolical writings; and for the authors of geometry you have read, you\nhave not examined their demonstrations to the bottom. Therefore, you\nperhaps may be, but are not yet, a geometrician, much less a good\ndivine. I would you had but so much ethics as to be civil. But you are a\nnotable critic; so fare you well, and consider what honour you do,\neither to the University where you are received for professor, or to the\nUniversity from whence you came thither, by your geometry; and what\nhonour you do to Emanuel College by your divinity; and what honour you\ndo to the degree of Doctor, with the manner of your language. And take\nthe counsel which you publish out of your encomiast his letter; think me\nno more worthy of your pains, you see how I have fouled your fingers.\n\n-----\n\nFootnote 1:\n\n Written by Henry Stubbe, M.A. of Christ Church, Oxford, who was,\n according to Anthony a Wood, \u201cthe most noted personage of his age that\n these late times have produced.\u201d\n\n-----\n\n\n\n\n THREE PAPERS\n\n PRESENTED TO THE ROYAL SOCIETY\n\n AGAINST DR. WALLIS.\n\n\n TOGETHER WITH\n\n CONSIDERATIONS\n\n ON DR. WALLIS\u2019S ANSWER TO THEM,\n\n BY\n\n THOMAS HOBBES,\n\n OF MALMESBURY.\n\n\n\n\n THREE PAPERS\n\n PRESENTED TO THE ROYAL SOCIETY.\n\n ==========\n\nTO THE RIGHT HONOURABLE AND OTHERS, THE LEARNED MEMBERS OF THE ROYAL\n SOCIETY, FOR THE ADVANCEMENT OF SCIENCES.\n\nPRESENTETH _to your consideration, your most humble servant, Thomas\nHobbes, (who hath spent much time upon the same subject), two\npropositions, whereof the one is lately published by Dr. Wallis, a\nmember of your Society, and Professor of Geometry; which if it should be\nfalse, and pass for truth, would be a great obstruction in the way to\nthe design you have undertaken. The other is a problem, which, if well\ndemonstrated, will be a considerable advancement of geometry; and though\nit should prove false, will in no wise be an impediment to the growth of\nany other part of philosophy._\n\n DR. WALLIS,\n DE MOTU, _Cap._ v. _Prop._ 1.\n\nIf there be understood an infinite row of quantities beginning with 0 or\n(1)/(0), and increasing continually according to the natural order of\nnumbers, 0, 1, 2, 3, &c. or according to the order of their squares, as,\n0, 1, 4, 9, &c. or according to the order of their cubes, as, 0, 1, 8,\n27, &c. whereof the last is given; the proportion of the whole, shall be\nto a row of as many, that are equal to the last, in the first case, as 1\nto 2; in the second case, as 1 to 3; in the third case, as 1 to 4, &c.\n\nThis proposition is the ground of all his doctrine concerning the\ncentres of gravity of all figures. Wherein may it please you to\nconsider:\n\nFirst, whether there can be understood an infinite row of quantities,\nwhereof the last can be given. Secondly, whether a finite quantity can\nbe divided into an infinite number of lesser quantities, or a finite\nquantity can consist of an infinite number of parts, which he buildeth\non as received from Cavallieri. Thirdly, whether (which in consequence\nhe maintaineth) there be any quantity greater than infinite. Fourthly,\nwhether there be, as he saith, any finite magnitude of which there is no\ncentre of gravity. Fifthly, whether there be any number infinite. For it\nis one thing to say, that a quantity may be divided perpetually without\nend, and another thing to say, that a quantity may be divided into an\ninfinite number of parts. Sixthly, if all this be false, whether that\nwhole book of _Arithmetica Infinitorum_, and that definition which he\nbuildeth on, and supposeth to be the doctrine of Cavallieri, be of any\nuse for the confirming or confuting of any propounded doctrine.\n\nHumbly praying you would be pleased to declare herein your judgment, the\nexamination thereof being so easy, that there needs no skill either in\ngeometry, or in the Latin tongue, or in the art of logic, but only of\nthe common understanding of mankind to guide your judgment by.\n\n THOMAS HOBBES,\n ROSET. _Prop._ v.\n\n _To find a straight line equal to two-fifths of the arc of a\n quadrant._\n\nI describe a square A B C D, and in it a quadrant D A C. Suppose D T be\ntwo-fifths of D C, then will the quadrantal arc T V be two-fifths of the\narc C A. Again let D R be a mean proportional between D C and D T; then\nwill the quadrantal arc R S be a mean proportional between the arc C A\nand the arc T V.\n\nSuppose further a right line were given equal to the arc C A, and a\nquadrantal arc therewith described; then will D C, C A, the arc on C A\nbe continually proportional. Set these proportionals in order by\nthemselves.\n\n D C, C A, arc on C A\u223a\n D R, R S, arc on R S\u223a\n D T, T V, arc on T V\u223a\n\nwhich are in continual proportion of the semi-diameter of the arc. And D\nC, D R, D T are in a continual proportion by construction, and therefore\nalso C A, R S, T V, and arc on C A, arc on R S, arc on T V, in continual\nproportion.\n\nTherefore as D C to R S, so is R S to the arc on T V. And D C, R S, the\narc on T V will be continually proportional. And because D C, C A, the\narc on C A are also continually proportional, and have the first\nantecedent D C common; the proportion of the arc on C A to the arc on T\nV is (by Eucl. xiv. 28) duplicate of the proportion of C A to R S, and\nthe arc on R S a mean proportional between the arc on C A and the arc on\nT V.\n\nNow if D C be greater than R S, also R S must be greater than the arc on\nT V; and the arc C A greater than the arc on R S. Therefore seeing D C,\nC A, arc on C A, are continually proportional; the arc on T V, the arc\non R S, the arc on C A cannot be continually proportional, which is\ncontrary to what has been demonstrated. Therefore D C is not greater\nthan R S. Suppose, then, R S to be greater than D C, then will the arc\non R S be a mean proportional between the arc on T V, and a greater arc\nthan that on C A; and so the inconvenience returneth. Therefore the\nsemidiameter D C is equal to the arc R S, and D R equal to T V, that is\nto say to two-fifths of the arc C A, which was to be demonstrated. Nor\nneedeth there much geometry for examining of this demonstration.\nTherefore I submit them both to your censure, as also the whole\n_Rosetum_, a copy whereof I have caused to be delivered to the secretary\nof your society.\n\n[Illustration]\n\n TO THE\n\n RIGHT HONOURABLE AND OTHERS,\n\n THE LEARNED MEMBERS\n\n OF\n\n THE ROYAL SOCIETY,\n\n FOR THE ADVANCEMENT OF THE SCIENCES.\n\n\n -------\n\n\nPresenteth to your consideration, your most humble servant Thomas\nHobbes, a confutation of a theorem which hath a long time passed for\ntruth; to the great hinderance of Geometry, and also of Natural\nPhilosophy, which thereon dependeth.\n\n THE THEOREM.\n\n_The four sides of a square being divided into any number of equal\nparts, for example into 10; and straight lines drawn through the\nopposite points, which will divide the square into 100 lesser squares;\nthe received opinion, and which Dr. Wallis commonly useth, is, that the\nroot of those 100, namely 10, is the side of the whole square._\n\n THE CONFUTATION.\n\n_The root 10 is a number of those squares, whereof the whole containeth\n100, whereof one square is an unity; therefore the root 10, is 10\nsquares: Therefore the root of 100 squares is 10 squares, and not the\nside of any square; because the side of a square is not a superficies,\nbut a line. For as the root of 100 unities is 10 unities, or of 100\nsoldiers 10 soldiers: so the root of 100 squares is 10 of those squares.\nTherefore the theorem is false; and more false, when the root is\naugmented by multiplying it by other greater numbers._\n\nHence it followeth, that no proposition can either be demonstrated or\nconfuted from this false theorem. Upon which, and upon the numeration of\ninfinites, is grounded all the geometry which Dr. Wallis hath hitherto\npublished.\n\nAnd your said servant humbly prayeth to have your judgment hereupon: and\nthat if you find it to be false, you will be pleased to correct the\nsame: and not to suffer so necessary a science as geometry to be\nstifled, to save the credit of a professor.\n\n TO THE\n\n RIGHT HONOURABLE AND OTHERS,\n\n THE LEARNED MEMBERS\n\n OF\n\n THE ROYAL SOCIETY,\n\n FOR THE ADVANCEMENT OF THE SCIENCES.\n\n ---\n\nYour most humble servant Thomas Hobbes presenteth, that the quantity of\na line calculated by extraction of roots is not to be truly found. And\nfurther presenteth to you the invention of a straight line equal to the\narc of a circle.\n\nA square root is a number which multiplied into itself produced a\nnumber.\n\n DEFINITION.\n\nAnd the number so produced is called a square number. For example:\nBecause 10 multiplied by 10 makes 100; the root is 10, and the square\nnumber 100.\n\n CONSEQUENT.\n\nIn the natural row of numbers, as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,\n13, 14, 15, 16, &c. every one is the square of some number in the same\nrow. But square numbers (beginning at 1) intermit first two numbers,\nthen four, then six, &c. So that none of the intermitted numbers is a\nsquare number, nor has any square root.\n\n PROP. I.\n\nA square root (speaking of quantity) is not a line, such as Euclid\ndefines, without latitude, but a rectangle.\n\n[Illustration]\n\nSuppose A B C D be the square, and A B, B C, C D, D A, be the sides, and\nevery side divided into 10 equal parts, and lines drawn through the\nopposite points of division; there will then be made 100 lesser squares,\nwhich taken altogether are equal to the square A B C D. Therefore the\nwhole square is 100, whereof one square is an unit; therefore 10 units,\nwhich is the root, is ten of the lesser squares, and consequently has\nlatitude; and therefore it cannot be the side of a square, which,\naccording to Euclid, is a line without latitude.\n\n CONSEQUENT.\n\nIt follows hence, that whosoever taketh for a principle, that a side of\na square is a mere line without latitude, and that the root of a square\nis such a line (as Dr. Wallis continually does) demonstrates nothing.\nBut if a line be divided into what number of equal parts soever, so the\nline have breadth allowed it (as all lines must, if they be drawn), and\nthe length be to the breadth as number to an unit; the side and the roof\nwill be all of one length.\n\n PROP. II.\n\n[Illustration]\n\nAny number given is produced by the greatest root multiplied into\nitself, and into the remaining fraction. Let the number given be two\nhundred squares, the greatest root is 14(4)/(14) squares. I say that 200\nis equal to the product of 14 into itself, together with 14 multiplied\ninto (4)/(14). For 14 multiplied into itself makes 196. And 14 into\n(4)/(14) makes (56)/(14) which is equal to 4. And 4 added to 196 maketh\n200; as was to be proved. Or take any other number 8, the greatest root\nis 2; which multiplied into itself is 4, and the remainder (2)/(4)\nmultiplied into 2, is 4, and both together 8.\n\n PROP. III.\n\nBut the same square calculated geometrically by the like parts,\nconsisteth (by Euclid II. 4) of the same numeral great square 196, and\nof the two rectangles under the greatest side 14, and the remainder of\nthe side, or (which is all one) of one rectangle under the greatest\nside, and double the remainder of the side; and further of the square of\nthe less segment; which altogether make 200, and moreover (1)/(49) of\nthose 200 squares, as by the operation itself appeareth thus:\n\n The side of the greater segment is 14(4)/(14)\n 14(4)/(14)\n Which multiplied into itself makes 200.\n\nThe product of 14, the greatest segment, into the two fractions\n(4)/(14), that is, into (4)/(14) (or into twice (2)/(14)) is (56)/(14)\n(that is 4); and that 4 added to 196 makes 200.\n\nLastly, the product of (2)/(14) into (2)/(14) or (1)/(7) into (1)/(7) is\n(1)/(49). And so the same square calculated by roots is less by (1)/(49)\nof one of those two hundred squares, than by the true and geometrical\ncalculation; as was to be demonstrated.\n\n CONSEQUENT.\n\nIt is hence manifest, that whosoever calculates the length of an arc or\nother line by the extraction of roots, must necessarily make it shorter\nthan the truth, unless the square have a true root.\n\n -------\n\n_The Radius of a Circle is a Mean Proportion between the Arc of a\n Quadrant and two-fifths of the same._\n\nDescribe a square A B C D, and in it a quadrant D C A. In the side D C\ntake D T two-fifths of D C, and between D C and D T a mean proportional\nD R, and describe the quadrantal arcs R S, T V. I say the arc R S is\nequal to the straight line D C. For seeing the proportion of D C to D T\nis duplicate of the proportion of D C to D R, it will be also duplicate\nof the proportion of the arc C A to the arc R S, and likewise duplicate\nof the proportion of the arc R S to the arc T V.\n\nSuppose some other arc, less or greater than the arc R S, to be equal to\nD C, as for example _r s_: then the proportion of the arc _r s_ to the\nstraight line D T will be duplicate of the proportion of R S to T V, or\nD R to D T. Which is absurd; because D _r_ is by construction greater or\nless than D R. Therefore the arc R S is equal to the side D C, which was\nto be demonstrated.\n\n COROL.\n\n[Illustration]\n\nHence it follows that D R is equal to two-fifths of the arc C A. For R\nS, T V, D T, being continually proportional, and the arc T V being\ndescribed by D T, the arc R S will be described by a straight line equal\nto T V. But R S is described by the straight line D R. Therefore D R is\nequal to T V, that is to two-fifths of C A.\n\nAnd your said servant most humbly prayeth you to consider, if the\ndemonstration be true and evident, whether the way of objecting against\nit by square root, used by Dr. Wallis; and whether all his geometry, as\nbeing built upon it, and upon his supposition of an infinite number, be\nnot false.\n\n\n\n\n CONSIDERATIONS\n\n UPON THE ANSWER OF DOCTOR WALLIS\n\n TO THE\n\n THREE PAPERS OF MR. HOBBES.\n\n\nDr. Wallis says, all that is affirmed, is but _if we_ SUPPOSE _that,\nthis will follow_.\n\nBut it seemeth to me, that if the supposition be impossible, then that\nwhich follows will either be false, or at least undemonstrated.\n\nFirst, this proposition being founded upon his _Arithmetica\nInfinitorum_, if there he affirm an absolute infiniteness, he must here\nalso be understood to affirm the same. But in his thirty-ninth\nproposition he saith thus: \u201c_Seeing that the number of terms increasing,\nthe excess above sub-quadruple is perpetually diminished, so at last it\nbecomes less than any proportion that can be assigned; if it proceed in\ninfinitum it must utterly vanish. And therefore if there be propounded\nan infinite row of quantities in triplicate proportion of quantities\narithmetically proportioned (that is, according to the row of cubical\nnumbers) beginning from a point or 0; that row shall be to a row of as\nmany, equal to the greater, as 1 to 4._\u201d It is therefore manifest that\nhe affirms, that in an infinite row of quantities the last is given; and\nhe knows well enough that this is but a shift.\n\nSecondly, he says, that usually in Euclid, and all after him, by\n_infinite_ is meant but, more than any assignable _finite_, or the\ngreatest possible. I am content it be so interpreted. But then from\nthence he must demonstrate those his conclusions, which he hath not yet\ndone. And when he shall have done it, not only the conclusions, but also\nthe demonstration, will be the same with mine in Cap. XIV. Art. 2, 3,\n&c. of my book _De Corpore_. And so he steals what he once condemned. A\nfine quality.\n\nThirdly, he says, (by Euclid\u2019s tenth proposition, but he tells not of\nwhat book), that a line may be bisected, and the halves of it may again\nbe bisected, and so onwards infinitely; and that upon such supposed\nsection infinitely continued, the parts must be supposed infinitely\nmany.\n\nI deny that; for Euclid, if he says a line may be divisible into parts\nperpetually divisible, he means that all the divisions, and all the\nparts arising from those divisions, are perpetually finite in number.\n\nFourthly, he says, that there may be supposed a row of quantities\ninfinitely many, and continually increasing, whereof the last is given.\n\nIt is true, a man may say, (if that be supposing) that white is black:\nbut, if _supposing_ be _thinking_, he cannot suppose an infinite row of\nquantities whereof the last is given. And if he say it, he can\ndemonstrate nothing from it.\n\nFifthly, he says (for one absurdity begets another) _that a superficies\nor solid may be supposed so constituted as to be_ infinitely long, _but_\nfinitely great, _(the breadth continually decreasing in greater\nproportion than the length increaseth), and so as to have no centre of\ngravity. Such is Toricellio\u2019s Solidum Hyperbolicum acutum, and others\ninnumerable, discovered by Dr. Wallis, Monsieur Fermat, and others. But,\nto determine this, requires more of geometry and logic, (whatsoever it\ndo of the Latin tongue), than Mr. Hobbes is master of._\n\nI do not remember this of Toricellio, and I doubt Dr. Wallis does him\nwrong and Monsieur Fermat too. For, to understand this for sense, it is\nnot required that a man should be a geometrician or a logician, but that\nhe should be mad.\n\nIn the next place, he puts to me a question as absurd as his answers are\nto mine. Let him ask himself, saith he, if he be still of opinion, _that\nthere is no argument in natural philosophy to prove that the world had a\nbeginning_. First, whether, in case it had no beginning, there must not\nhave passed an infinite number of years before Mr. Hobbes was born.\nSecondly, whether, at this time, there have not passed more, that is,\nmore than that infinite number. Thirdly, whether, in that infinite (or\nmore than infinite) number of years, there have not been a greater\nnumber of days and hours, and of which, hitherto, the last is given.\nFourthly, whether, if this be an absurdity, we have not then, (contrary\nto what Mr. Hobbes would persuade us), an argument in nature to prove\nthe world had a beginning.\n\nTo this I answer, not willingly, but in service to the truth, that, by\nthe same argument, he might as well prove that God had a beginning.\nThus, in case he had not, there must have passed an infinite length of\ntime before Mr. Hobbes was born; but there hath passed at this day more\nthan that infinite length, by eighty-four years. And this day, which is\nthe last, is given. If this be an absurdity, have we not then an\nargument in nature to prove that God had a beginning? Thus it is when\nmen entangle themselves in a dispute of that which they cannot\ncomprehend. But, perhaps, he looks for a solution of his argument to\nprove that there is somewhat greater than infinite; which I shall do so\nfar as to show it is not concluding. If from this day backwards to\neternity be more than infinite, and from Mr. Hobbes his birth backwards\nto the same eternity be infinite, then take away from this day backwards\nto the time of Adam, which is more than from this day to Mr. Hobbes his\nbirth, then that which remains backwards must be less than infinite. All\nthis arguing of infinites is but the ambition of school-boys.\n\n TO THE LATTER PART OF THE FIRST PAPER.\n\nThere is no doubt if we give what proportion we will of the radius to\nthe arc, but that the arc upon that arc will have the same proportion.\nBut that is nothing to my demonstration. He knows it, and wrongs the\nRoyal Society in presuming they cannot find the impertinence of it.\n\nMy proof is this: that if the arc on T V, and the arc R S, and the\nstraight line C D, be not equal, then the arc on T V, the arc on R S,\nand the arc on C A, cannot be proportional; which is manifest by\nsupposing in D C a less than the said D C, but equal to R S, and another\nstraight line, less than R S, equal to the arc on T V; and anybody may\nexamine it by himself.\n\nI have been asked by some that think themselves logicians, why I\nproceeded upon \u2156 rather than any other part of the radius. The reason I\nhad for it was, that, long ago, some Arabians had determined, that a\nstraight line, whose square is equal to 10 squares of half the radius,\nis equal to a quarter of the perimeter; but their demonstrations are\nlost. From that equality it follows, that the third proportional to the\nquadrant and radius, must be a mean proportional between the radius and\n\u2156 of the same. But, my answer to the logicians was, that, though I took\nany part of the radius to proceed on, and lighted on the truth by\nchance, the truth itself would appear by the absurdity arising from the\ndenial of it. And this is it that Aristotle means, where he\ndistinguishes between a direct demonstration and a demonstration leading\nto an absurdity. Hence it appears that Dr. Wallis\u2019s objections to my\n_Rosetum_ are invalid as built upon roots.\n\n TO THE SECOND PAPER.\n\nFirst, he says that it concerns him no more than other men, which is\ntrue. I meant it against the whole herd of them who apply their algebra\nto geometry. Secondly, he says that a bare number cannot be the side of\na square figure. I would know what he means by a bare number. Ten lines\nmay be the side of a square figure. Is there any number so bare, as by\nit we are not to conceive or consider anything numbered? Or, by 10\nnothings understands he bare 10? He struggles in vain, his conscience\npuzzles him. Thirdly, he says 10 squares is the root of 100 square\nsquares. To which I answer, first, that there is no such figure as a\nsquare square. Secondly, that it follows hence, that a root is a\nsuperficies, for such is 10 squares. Lastly, he says that, neither the\nnumber 10, nor 10 soldiers, is the root of 100 soldiers; because 100\nsoldiers is not the product of 10 soldiers into 10 soldiers. This last I\ngrant, because nothing but numbers can be multiplied into one another. A\nsoldier cannot be multiplied by a soldier. But no more can a square\nfigure by a square figure, though a square number may. Again, if a\ncaptain will place his 100 men in a square form, must he not take the\nroot of 100 to make a rank or file? And are not those 10 men?\n\n TO THE THIRD PAPER.\n\nHe objects nothing here, but that _the side of a square is not a\nsuperficies, but a line_, and that a _square root (speaking of quantity)\nis not a line, but a rectangle_, is a contradiction. The reader is to\njudge of that.\n\nTo his scoffings I say no more, but that they may be retorted in the\nsame words, and are therefore childish.\n\nAnd now I submit the whole to the Royal Society, with confidence that\nthey will never engage themselves in the maintenance of these\nunintelligible doctrines of Dr. Wallis, that tend to the suppression of\nthe sciences which they endeavour to advance.\n\n\n\n\n LETTERS\n AND OTHER PIECES.\n\n\n\n\n LETTERS AND OTHER PIECES.\n\n ==========\n\n\n I.\n A LETTER FROM MR. HOBBS TO MY MR.[2]\n\n HONORABLE SIR,\n\nThough I may goe whither and when I will for anie necessity you have of\nmy service, yet there is a necessity of good manners that obliges me as\nyo^r servant to lett you knowe att all times where to find me. Wee goe\nout of Paris 3 weekes hence, or sooner, towards Venice, but by what way\nI knowe not, because the ordinary high way through the territory of\nMilan is encumbered with the warre betweene the French and the\nSpaniards. Howsoever, wee have to be there in October next. If you\nrequire anie service that I can doe there, it may please you to convey\nyour command by Devonshire house. But if you command me nothing, I have\nforbidden my letters to look for answer: their busines being only to\ninforme and to lett you knowe that the image of your noblenes decayes\nnot in my memory, but abides fresh to keepe me eternally\n\n Your\n THO. HOBBS.\n\n-----\n\nFootnote 2:\n\n This letter is to be found in the British Museum, amongst the\n Lansdowne MSS. 238, entitled \u201ca collection of letters to and from\n persons of eminence in the reigns of Elizabeth, James I, and Charles\n I, made by some person in the service of Sir Gervas Clifton\u201d. It is\n without date: but the allusion to the war between France and Spain,\n and the passage in the VITA THO. HOBBES, \u201cAnno sequente qui erat\n Christi 1629, rogatus a nobilissimo viro domino Gervasio Clifton\u201d, &c.\n (p. xiv), show that it must have been written in either 1629 or 1630.\n\n-----\n\n\n II.\n TO A FRIEND IN ENGLAND.\n\n WORTHY SIR,\n\nI have been behind hand with you a long time for a letter I received of\nyours at Angers, that place affording nothing wherewith to pay a debt of\nthat kind, all matter of news being sooner known in England than here:\nand the news you writ me was of that kind, that none from England could\nbe more welcome, because it concerned the honour of Welbeck and Clifton,\ntwo houses in which I am very much obliged.\n\nMonsieur having given the slip to the Spaniards at Bruxelles, came to\nthe King about ten days ago at St. Germains, where he was received with\ngreat joy. The next day the Cardinal entertained him at Ruelle: and the\nday after that he went to Limours, where he is now, and from thence he\ngoes away shortly to Bloys, to stay there this winter. The Cardinal of\nLyons is going to Rome to treat about the annulling of Monsieur\u2019s\nmarriage, which is here by Parliament declared void, but yet they\nrequire the sentence of the Pope. There goes somebody thither on the\npart of his wife, to get the marriage approved: but who that is, I \u2019know\nnot. The Swedish party in Germany is in low estate, but the French\nprepare a great army for those parts, pretending to defend the places\nwhich the Swedes have put into the King of France his protection,\nwhereof Philipsbourgh is one; a place of importance for the Lower\nPalatinate. This is all the French news.\n\nFor your question, _why a man remembers less his own face, which he sees\noften in a glass, than the face of a friend that he has not seen of a\ngreat time_, my opinion in general is, that a man remembers best those\nfaces whereof he has had the greatest impressions, and that the\nimpressions are the greater for the oftener seeing them, and the longer\nstaying upon the sight of them. Now you know men look upon their own\nfaces but for short fits, but upon their friends\u2019 faces long time\ntogether, whilst they discourse or converse together; so that a man may\nreceive a greater impression from his friend\u2019s face in a day, than from\nhis own in a year; and according to this impression, the image will be\nfresher in his mind. Besides, the sight of one\u2019s friend\u2019s face two hours\ntogether, is of greater force to imprint the image of it, than the same\nquantity of time by intermissions. For the intermissions do easily\ndeface that which is but lightly imprinted. In general, I think that\nlasteth longer in the memory which hath been stronglier received by the\nsense.\n\nThis is my opinion of the question you propounded in your letter. Other\nnew truths I have none, at least they appear not new to me. Therefore if\nthis resolution of your first question seems probable, you may propound\nanother, wherein I will endeavour to satisfy you, as also in any thing\nof any other nature you shall command me, to my utmost power; taking it\nfor an honour to be esteemed by you, as I am in effect,\n\n Your humble and faithful servant\n THO. HOBBES.\n\n_Paris, Oct. 21/31, 1634._\n\nMy Lord Fielding and his Lady came to Paris on Saturday night last.\n\n\n III.\n TO MY WORTHY FRIEND MR. GLEN.[3]\n\n WORTHY SIR,\n\nI received here in Florence, two days since, a letter from you of the\n19th of January. It was long by the way; but when it came it did\nthoroughly recompence that delay. For it was worth all the pacquets I\nhad received a great while together. All that passeth in these parts is\nequally news, and therefore no news; else I would labour to requite your\nletter in that point, though in the handsome setting down of it, I\nshould still be your inferior.\n\nI long infinitely to see those books of the Sabbaoth[4], and am of your\nmind they will put such thoughts into the heads of vulgar people, as\nwill confer little to their good life. For when they see one of the ten\ncommandments to be _jus humanum_ merely, (as it must be if the Church\ncan alter it), they will hope also that the other nine may be so too.\nFor every man hitherto did believe that the ten commandments were the\nmoral, that is, the eternal law.\n\nI desire also to see Selden\u2019s _Mare Clausum_, having already a great\nopinion of it.\n\nYou may perhaps, by some that go to Paris, send me those of the\nSabbaoth, for the other being in Latin, I doubt not to find it in the\nRue St. Jaques.\n\nWe are now come hither from Rome, and hope to be in Paris by the end of\nJune. I thank you for your letter, and desire you to believe that I can\nnever grow strange to one, the goodness of whose acquaintance I have\nfound by so much experience. But I have to write to so many, that I\nwrite to you seldomer than I desire; which I pray pardon, and esteem me\n\n Your most affectionate friend\n and humble servant\n THO. HOBBES.\n\n_Florence, Apr. (6)/(16) 1636._\n\nMy Lord and Mr. Nicholls, and all our company commend them to you.\n\n-----\n\nFootnote 3:\n\n Probably George Glen, who was installed Prebend of Worcester in 1660,\n and died in 1669.\n\nFootnote 4:\n\n The History of the Sabbath. In two books. By Peter Heylyn. 4to. 1636.\n\n-----\n\n\n IV.\n LETTER TO SIR CHARLES CAVENDISH.[5]\n\n HONORABLE SIR,\n\n[Illustration]\n\nThe last weeke I had the honor to receave two letters from you at once,\none of the 30 of Dec., the other of the 7^{th} of Jan., w^{ch} I\nacknowledged, but could not answer in my last. In the first you begin\nwith a difficulty on the principle of Mons^r de Cartes, _that it is all\none to move a weight two spaces, or the double of that waight one\nspace_, and so on in other proportions: to w^{ch} you object the\ndifference of swiftnesse, w^{ch} is greater when a waight is moved two\nspaces than when double waight is moved one space. Certenly de Cartes\nhis meaning was by force the same that mine, namely, a multiplication of\nthe weight of a body in to the swiftnesse wherew^{th} it is moved. So\nthat when I move a pound two foote at the rate of a mile an howre, I do\nthe same as if of 2 poundes I moved one pound a foote at y^e rate of a\nmile an howre, the other pound another foote at the same rate, not in\ndirect\u016b, but parallell to the first pound. As if the wayt A B were moved\nto C D at the rate of a mile an howre, \u2019tis all one as if the waight A E\nwere moved to F H at the same rate. Here is all the difference: this\nswiftnesse or rate of a mile an howre is, in the first case, layd out in\nthe 2 spaces A G, G C, the latter, in the 2 spaces A G, E G. The first\ncase, as like as if a footman should run w^{th} double swiftnesse\nendwayes, w^{ch} is y^e doubling of swiftnesse in one man: in the other,\nit is as if you doubled the swiftnesse by doubling the man: for every\nman has his owne swiftnesse. And so A H is the swiftnesse A G doubled,\nas well as A D. For that, that Mons^r de Cartes will not have just twice\nthe force requisite to move the same weight twice as fast, I can say\nnothing. The papers I have of his touching that are in my trunk, w^{ch}\nhath bene taken by Dunkerkers, and taken againe from them by French, and\nat length recovered by frends I made: but I shall not have it yet this\nfortnight. In the meane time I am not in that opinion, but do assure\nmyself, the patient being the same, double force in the agent shall\nworke upon it double effect.\n\nIn the same letter you require a better explication of y^e proportion I\ngather betweene wayght and swiftnesse: wherein, because you have not my\nfigure, I imagine you have mistaken me very much. And first, you thinke,\nI suppose, D E equall to A B: w^{ch} I am sure is a mistake. For I put A\nB for any line you will to expresse a _minut\u016b secundum_. I will,\ntherefore, go over againe the demonstration I sent you before, and see\nif I can do it cleerer.\n\n[Illustration]\n\nLet A B stand for the time knowne wherein the waight D descendeth to E.\nAnd let there bee a cylinder of the same matter the waight D consisteth\nof, and let the altitude of that cylinder be D C: w^{ch} I shew before\nwas the swiftnesse wherew^{th} that cylinder _presseth_, not wherew^{th}\nit _falleth_. And wee are now to enquire how farre such matter as the\ncylinder is made of must descend from D, before it attayne a swiftnesse\nequall to this pressing swiftnesse D C. And I say it must fall to L. For\nin the time A B it is knowne that the waight in D will fall to E: and it\nis demonstrated by Gallileo, that when such waight comes to E, it shall\nbe able to go twice the space it hath fallen in the same time. Therefore\nthe waight D being in E, hath velocity to carry him the space D K\n(w^{ch} I put double to D E) in the same time A B. But I put B F equall\nto D K. Therefore, in the time A B, the waight\u2019s velocity acquired in E\nshall be such as to go from B to F without decrease of velocity by the\nway. Hence I go on to finde in what point the waight in D comes to where\nit getteth a velocity equall to C D. Therefore, I apply D C to G H,\nparallel to B F: and then it is, as the time A B to the time A G, so the\nvelocity acquired at the end of the time A B to the velocity acquired at\nthe end of the time A G. For the swiftnesse acquired from time to time\n(I say, not from place to place, but from time to time) are\nproportionable to the times wherein they are acquired: w^{ch} is the\npostulate on w^{ch} Galileo builds all his doctrine. And as A B to A G,\nso the line B F to the line G H. But, at the end of the time A B, the\nwaight D is by supposition in E, in that degree of velocity as to go B F\nor D K in the same time A B. The question therefore is, where the waight\nD shall be at the end of the time A G. For there it hath the velocity of\ngoing G H or D C in the same time, because the velocity G H is to the\nvelocity B F as the line G H to the line B F, or as the time of descent\nA G to the time A B. But, because the spaces of the descent are in\ndouble the proportion of the times of descent, make it as B F to G H,\nthat is, D K to D C, so D C to another, D L. The velocity, therefore,\nacquired in the point of descent E, namely the velocity D K or B F, is\nto the velocity acquired in the point L, namely, the velocity G H or D\nC, (w^{ch} is the velocity of the cylinder\u2019s waight), as D K to D L. And\ntherefore in L the waight D has acquired a velocity equal to the\nvelocity of the waight of the cylinder.\n\nIn the same letter you desire to knowe, how any medi\u016b, as water,\nretardeth the motion of a stone that falls into it. To w^{ch} I answer\nout of that you say afterwards, that nothing can hinder motion but\ncontrary motion: that the motion of the water, when a stone falls into\nit, is point blanke contrary to the motion of the stone. For the stone\nby descent causeth so much water to ascend as the bignesse of the stone\ncomes to. For imagine so much water taken out of the place w^{ch} the\nstone occupies, and layd upon the superficies of the water: it presseth\ndowneward as the stone does, and maketh the water that is below to rise\nupwards, and this rising upwards is contrary to the descent, and is no\nother operation than we see in scales, when of two equal bullets in\nmagnitude that w^{ch} is of heavier metal maketh the other to rise. And\nthus farre goes your letter of Dec. 30.\n\nFor the first qu\u00e6re in your second letter, concerning how we see in the\ntime the lucide body contracts itselfe, I have no other solution but\nthat w^{ch} your selfe hath given: w^{ch} is, that the reciprocation is\nso quicke, that the effect of the first motion lasteth till the next\ncomes, and longer. For by experience we observe that the end of a\nfirebrand swiftely moved about in circle, maketh a circle of fire:\nw^{ch} could not be, if the impression made at the beginning of the\ncirculation did not last till the end of it. For if the same firebrand\nbe moved slower, there will appear but a peece of a circle, bigger or\nlesser according to the swiftnesse or slownesse of y^e motion. For the\ncause of such reciprocation, it is hard to guess what it is. It may well\nbe the reaction of the medium. For though the medi\u016b yeld, yet it\nresisteth to: for there can be no passion w^{th}out reaction. And if a\nman could make an hypothesis to salve that contraction of y^e sun, yet\nsuch is the nature of naturall thinges, as a cause may be againe\ndemanded of such hypothesis: and never should one come to an end\nw^{th}out assigning the immediate hand of God. Whereas in mathematicall\nsciences wee come at last to a definition, w^{ch} is a beginning or\nprinciple, made true by pact and consent amongst ourselves. Further, you\nconceave a difficulty how the medium can be continually driven on, if\nthere be such an alternate contraction. To w^{ch}, first I answer, that\nthe motion forward is propagated to the utmost distance in an instant,\nand the first push is therefore enough, and in another instant is made\nthe returne back in y^e like manner. And though it were not done in an\ninstant, yet we see by experience in rivers, as in y^e Thames, that the\ntide goes upward towards London pushed by the water below, and yet at\nthe same instant the water below is going backe to the sea. For seeing\nit is high-water at Blackwall before \u2019tis so at Greenwich, the water\ngoes backe from Blackwall when it goes on at Greenwich. And so it would\nhappen, though Blackwall and Greenwich were nerer together then that any\nquantity given could come betweene.\n\n[Illustration]\n\nIn my letter from London, speaking of the refraction of a bullet, I\nthinke I delivered my opinion to be, that a bullet falling out of a\nthinner medium into a thicker, looseth in the entring nothing but motion\nperpendicular: but being entred, he looseth proportionably both of one\nand the other. For suppose a bullet, whose diameter is A B, be in the\nthiner medi\u016b, and enter at C into the thicker medium. The thicker\nmedium, at the first touch of B in the point C, worketh nothing upon the\nline A B. And when the diameter A B is entred, suppose halfe way, yet\nthe thicker medium operates laterally but on one halfe of it. So that in\nthe somme there is a losse of velocity perpendicular (to the quantity\nthat the diameter A B requires) without any offence to the motion\nlaterall, but so much of the diameter as is within the thicker medi\u016b is\nretarded both wayes, and looses of his absolute motion, w^{ch} is\ncompounded of perpendicular and laterall, and that proportionally.\nSuppose now that a bullet passe from A to D, and receave a peculiar\nlosse of his perpendicular motion by entring at D, so great that he\nproceed in the perpendicular but halfe so farre, as for example from D\nto I: and then being in, the thicknesse of the medium take away more of\nhis velocity both perpendicular and laterall, suppose halfe that w^{ch}\nwas left of the perpendicular motion and halfe of his first laterall\nmotion, so that the perpendicular motion is but D K, and the laterall\nmotion D E. Then will the line of refraction be D G. As for that\nargumentation of Des Cartes, it is, in my opinion, as I have heretofore\nendeavored to shew you, a mere paralogism.\n\nLastly, you make this qu\u00e6re, why light hath not at severall inclinations\nseverall swiftnesses as well as a bullet. The bullet itself passeth\nthrough the severall media: whereas in the motion of light, the body\nmoved, w^{ch} is the medi\u016b, entreth not into the other medium, but\nthrusteth it on: and so the parts of that medium thrust on one another,\nwhereby the laterall motion of the thicker medium hath nothing to worke\nupon, because nothing enters, but stoppes onely and retardes, in oblique\n_incidence_, that end w^{ch} comes first to it, and thereby causes a\nrefraction the contrary way to that of a bullet, in such manner as I set\nforth to you in one of my letters from hence concerning the cause of\nrefraction. And this is all I can say for the present to the qu\u00e6res of\ny^r two last letters.\n\nI have enquired concerning perspectives after the manner of De Cartes.\nMydorgius tells me there is none that goes about them, as a thing too\nhard to do. And I believe it. For here is one Mons^r de Bosne in towne,\nthat dwells at Bloys, an excellent workman, but by profession a lawier,\nand is counsellor of Bloys, and a better philosopher in my opinion then\nDe Cartes, and not inferior to him in the analytiques. I have his\nacquaintance by Pere Mersenne. He tells me he hath tryed De Cartes his\nway, but cannot do it: and now he workes upon a crooked line of his owne\ninvention. He sayes he shall have made one w^{th}in a moneth after he\nshall returne to Bloys: after that he will see what he can discover in\nthe heavens himselfe, and then if he discover any new thing he will let\nhis way be publique together w^{th} the effects. This is all the hope I\ncan give you yet. So w^{th} my prayers to God to keepe you in prosperity\nthis troublesome time, I rest\n\n Your most humble and obedient servant\n TH. HOBBES.\n\n_Paris, Feb. 8, stile no. 1641._\n\n To the Right Honorable\n Sr CHARLES CAVENDYSSHE\n present these\n\n /\n /\n /\n /\n at Wellinger.\n\n-----\n\nFootnote 5:\n\n Harleian MS. 6796.\n\n-----\n\n\n V.\n LETTER TO MR. BEALE.[6]\n\n SIR,\n\nThe young woman at Over-Haddon hath been visited by divers persons of\nthis house. My Lord himself hunting the hare one day at the Town\u2019s end,\nwith other gentlemen and some of his servants, went to see her on\npurpose: and they all agree with the relation you say was made to\nyourself. They further say on their own knowledge, that part of her\nBelly touches her Back-bone. She began (as her Mother says) to loose her\nappetite in December last, and had lost it quite in March following:\ninsomuch as that since that time she has not eaten nor drunk any thing\nat all, but only wetts her lips with a feather dipt in water. They were\ntold also that her gutts (she alwayes keeps her bed) lye out by her at\nher fundament shrunken. Some of the neighbouring ministers visit her\noften: others that see her for curiosity give her mony, sixpence or a\nshilling, which she refuseth, and her mother taketh. But it does not\nappear they gain by it so much as to breed a suspition of a cheat. The\nwoman is manifestly sick, and \u2019tis thought she cannot last much longer.\nHer talk (as the gentlewoman that went from this house told me) is most\nheavenly. To know the certainty, there bee many things necessary which\ncannot honestly be pryed into by a man. First, whether her gutts (as\n\u2019tis said) lye out. Secondly, whether any excrement pass that way, or\nnone at all. For if it pass, though in small quantity, yet it argues\nfood proportionable, which may, being little, bee given her secretly and\npass through the shrunken intestine, which may easily be kept clean.\nThirdly, whether no urine at all pass: for liquors also nourish as they\ngo. I think it were somewhat inhumane to examin these things too nearly,\nwhen it so little concerneth the commonwealth: nor do I know of any law\nthat authoriseth a Justice of peace, or other subject, to restrain the\nliberty of a sick person so farr as were needful for a discovery of this\nnature. I cannot therefore deliver any judgment in the case. The\nexamining whether such a thing as this bee a miracle, belongs I think to\nthe Church. Besides, I myself in a sickness have been without all manner\nof sustenance for more than six weeks together: which is enough to make\nmee think that six months would not have made it a miracle. Nor do I\nmuch wonder that a young woman of clear memory, hourely expecting death,\nshould bee more devout then at other times. \u2019Twas my own case. That\nwhich I wonder at most, is how her piety without instruction should bee\nso eloquent as \u2019tis reported.\n\n THO. HOBBES.\n\n_Chatsworth, Oct. 20. 68._\n\n-----\n\nFootnote 6:\n\n Amongst the MSS. of the Royal Society.\n\n-----\n\n\n VI.\n LETTER TO MR. OLDENBURG.[7]\n\n\n WORTHY S^R\n\nIn the last Transactions for September and October I find a letter\naddressed to you from D^r Wallis, in answer to my LUX MATHEMATICA. I\npray you tell me that are my old acquaintance, whether it be (his words\nand characters supposed to be interpreted) intelligible. I know very\nwell you understand sense both in Latine, Greeke, and many other\nlanguages. He shows you no ill consequence in any of my arguments.\nWhereas I say there is no proportion of _infinite_ to _finite_. He\nanswers, he meant _indefinite_; but derives not his conclusion from any\nother notion than simply _infinite_. I said the root of a square number\ncannot be the length of the side of a square figure, because a root is\npart of a square number, but length is no part of a square figure. To\nwhich he answers nothing. In like manner, he shuffles off all my other\nobjections, though he know well enough that whatsoever he has written in\nGeometry (except what he has taken from me and others) dependeth on the\ntruth of my objections. I perceive by many of his former writings that I\nhave reformed him somewhat as to the Principles of Geometry, though he\nthanke me not. He shuffles and struggles in vaine, he has the hooke in\nhis guills, I will give him line enough: for (which I pray you tell him)\nI will no more teach him by replying to any thing he shall hereafter\nwrite, whatsoever they shall say that are confident of his Geometry.\n_Qui volunt decipi, decipiuntur._ He tells you that I bring but _crambe\ns\u00e6pe cocta_. For which I have a just excuse, and all men do the same;\nthey repeat the same words often when they talk with them that cannot\nheare.\n\nI desire also this reasonable favour from you: that, if hereafter I\nshall send you any paper tending to the advancement of physiques or\nmathematiques, and not too long, you will cause it to be printed by him\nthat is printer to the Society, as you have done often for D^r Wallis:\nit will save me some charges.\n\n I am, S^r,\n\n Your affectionate frend and humble seruant\n\n THOMAS HOBBES.\n\n_November the 26th, 1672._\n\n ffor my worthy and much honoured\n frend M^r HENRY OLDENBURGH,\n Secretary to the Royal Society.\n\n-----\n\nFootnote 7:\n\n Amongst the MSS. of the Royal Society.\n\n-----\n\n\n VII.\n TO THE RIGHT HONOURABLE\n THE MARQUIS OF NEWCASTLE.[8]\n\nThe passions of man\u2019s mind, except onely one, may bee observed all in\nother living creatures. They have desires of all sorts, love, hatred,\nfeare, hope, anger, pitie, \u00e6mulation, and y^e like: onely of curiositie,\nwhich is y^e desire to know y^e causes of thinges, I never saw signe in\nany other living creature but in man. And where it is in man, I find\nalwaies a defalcation or abatement for it of another passion, which in\nbeastes is commonly predominant, namely, a ravenous qualitie, which in\nman is called _avarice_. The desire of knowledge and desire of needlesse\nriches are incompatible, and destructive one of another. And therefore\nas in the cognitive faculties reason, so in the motive curiositie, are\nthe markes that part y^e bounds of man\u2019s nature from that of beastes.\nWhich makes mee, when I heare a man, upon the discovery of any new and\ningenious knowledge or invention, aske gravely, that is to say,\nscornefully, _what \u2019tis good for_, meaning what monie it will bring in,\n(when he knows as little, to one that hath sufficient what that overplus\nof monie is good for), to esteeme that man not sufficiently removed bn\n484.png from brutalitie. Which I thought fit to say by way of\nanticipation to y^e grave scorners of philosophie, and that your\nlordship, after having performed so noble and honourable acts for\ndefence of your countrie, may thinke it no dishonour in this unfortunate\nleasure to have employed some thoughts in the speculation of the noblest\nof the senses, _vision_.\n\nThat which I have written of it is grounded especially upon that w^{ch}\nabout 16 yeares since I affirmed to your Lo^{PP} at Welbeck, that light\nis a fancy in the minde, caused by motion in the braine, which motion\nagaine is caused by the motion of y^e parts of such bodies as we call\n_lucid_: such as are the sunne and y^e fixed stars, and such as here on\nearth is fire. By putting you in mind hereof, I doe indeed call you to\nwitnesse of it: because, the same doctrine having since been published\nby another, I might bee challenged for building on another man\u2019s ground.\nYett philosophical ground I take to be of such a nature, that any man\nmay build upon it that will, especially if the owner himselfe will nott.\nBut upon this ground, with the helpe of some other speculations drawne\nfrom the nature of motion and action, I have, I thinke, derived y^e\nreason of all the ph\u00e6nomena I have mett with concerning light and\nvision, both solidly enough nott to be confuted, and withall easilie\nenough to be understood by such as can give that attention thereto which\nthe figures, whereby such motion as causeth vision is described, do\nrequire. All that I shall bee ever able to adde to it, is polishing:\nfor, being the first draught, it could nott bee so perfect as I hope\nhereafter to make it in Latin. Butt as it is, it will sufficiently give\nyour Lo^{PP} satisfaction in those _qu\u00e6res_ you were pleased to make\nconcerning this subject. I am content that it passe, in respect of some\ndrosse that yett cleaves to it, for ore: w^{ch} is much better than old\nends raked out of the kennell of sophisters\u2019 bookes. And for such I\ncommend it to your Lo^p, and myselfe to your accustomed good opinion:\nwhich hath beene hitherto so greate honour to mee, as I am nott known to\nthe world by any thing so much as by being,\n\n My most noble lord,\n\n Your Lo^{p\u2019s} most humble\n\n and most obliged servant\n\n THO. HOBBES.\n\n\nThe treatise ends with the following passage:--\n\n To conclude, I shall doe like those that build a new house where an\n old one stood before, that is to say, carry away the rubbish.\n\n And first, away goes the old opinion that the _shewes_ (which they\n call visible species) of all objects, are in all places, and all the\n babble _de extramittendo et intromittendo_. For their species are\n nothing else but fancie, made by the light proceeding directly or by\n repurcussion or refraction made from the object to y^e eye, and so\n moving the braine and other parts within.\n\n Secondly, the opinion which Vitellio takes for an axiome and\n foundation of his _Catoptricques_, that y^e place of y^e image by\n reflexion is in the perpendicular drawn from the object to the glasse.\n For it is false both in plaine glasses and in sph\u00e6ricall, whether\n convexe or concave.\n\n Thirdly, the opinion that light is engendred faster in hard bodies, as\n glasse, than in thin and fluid, as aire.\n\n Fourthly, that objects are seen by _penicilli_ that have their common\n base in the pupills: for y^e center of y^e eye is in their common\n base.\n\n Fifthly, the opinion that there bee other visuali lines by which wee\n see distinctly besides y^e optique axis.\n\n Sixthly, the opinion that perspective glasses and amplifying glasses\n are best made of hyperbolicall figures.\n\n Seventhly, the opinion that light is a bodie, or any other such thing\n than such light as wee have in dreames.\n\n Eighthly, that y^e object appeares greater and lesse in ye same\n proportion that y^e angles have under which they are seene.\n\n Lastly, is to be cast away the conceipt of millions of strings in ye\n optique nerve, by which the object playes upon the braine, and makes\n y^e soule listen unto it, and other innumerable such trash.\n\n How doe I feare that y^e attentive reader will find that which I have\n delivered concerning y^e _Optiques_ fitt to bee cast outt as rubbish\n among the rest. If hee doe, hee will recede from y^e authoritie of\n experience, which confirmeth all I have said. Butt if it bee found\n true doctrine, (though yett it wanteth polishing), I shall deserve the\n reputation of having beene y^e first to lay the grounds of two\n sciences; this of _Optiques_, y^e most curious, and y^t other of\n _Natural Justice_, which I have done in my booke DE CIVE, y^e most\n profitable of all other.\n\n-----\n\nFootnote 8:\n\n Harleian MS. 3360: a treatise on Optics, entitled \u201cA minute or first\n draught of the Optiques. In two parts. By Thomas Hobbes. At Paris,\n 1646.\u201d The second part, _On Vision_, we have in Latin, in the DE\n HOMINE: the first, _On Illumination_, was never published. The\n dedication to the Marquis of Newcastle, and the concluding paragraph,\n is all that is here given of the treatise.\n\n-----\n\n\n VIII.\n TO THE KING\u2019S MOST EXCELLENT MAJESTY\n THE HUMBLE PETITION OF THOMAS HOBBES[9]\n\nSheweth, that though your Majesty hath been pleased to take off the\nrestraint of late years laid upon the pensions payable out of your privy\npurse, yet your Majesty\u2019s Officers refuse to pay the pension of your\npetitioner without your Majesty\u2019s express command.\n\nAnd humbly beseaceth your Majesty, (considering his extreme age,\nperpetual infirmity, frequent and long sickness, and the aptness of his\nenemies to take any occasion to report that your petitioner by some ill\nbehaviour hath forfeited your wonted favour), that you would be pleased\nto renew your order for the payment of it in such manner as to his great\ncomfort he hath for many years enjoyed it.[10]\n\nAnd daily prayeth to God Almighty to bless your Majesty with long life,\nconstant health, and happinesse.\n\n-----\n\nFootnote 9:\n\n Additional MSS. 4292. Brit. Mus.\n\nFootnote 10:\n\n Deinde redux mihi Rex concessit habere quotannis\n Centum alias libras ipsius ex loculis:\n Dulce mihi donum.\n VITA _Carm. expres._ p. xcviii.\n\n-----\n\n\n\n\n\n\n\n\n END OF VOL. VII.\n\n\n\n\n\n\n\n\n RICHARDS, PRINTER, 100, ST. MARTIN\u2019S LANE.\n\n------------------------------------------------------------------------\n\n Transcriber\u2019s Note\n\nSome Greek passages employ the stigma ligature (\u2018st\u2019). The available\nUnicode character (\u03db) is nearly indistinguishable from the final form of\nsigma (\u03c2). Occasionally, the \u2018\u03bf\u03c5\u2019 ligature is employed. The only\navailable character (\u1d15) is a Latin, not a Greek character. It is\nrendered here as \u2018\u03bf\u03c5\u2019. There were also a number of instances of improper\nplacement of diacritical marks, particularly in cases where the\nbreathing mark and accents appear on the first rather than the second of\ntwo leading vowels, e.g. \u2018\u1f45\u03c5\u03c4\u03c9\u03c2\u2019 rather than \u2018\u03bf\u1f55\u03c4\u03c9\u03c2\u2019.\n\nAt 93.4, the response \u2018No sure\u2019 is obviously incorrect. It is most\nlikely that it should have read \u2018Not sure.\u2019, but it may also have been\nan unfinished line.\n\nOther errors deemed most likely to be the printer\u2019s have been corrected,\nand are noted here. Given the age of the text, any corrections were made\nsparingly. The references are to the page and line in the original.\n\n 8.35 with increasing [s]wiftness? Restored.\n 12.34 being so very heavy[?] Added.\n 54.8 too much or too[l i/ li]ttle Replaced.\n 62. was to be demonst[r]ated. Inserted.\n 65.33 which is 72[.] Added.\n 67.1 and multip[l]ying lines Inserted.\n 72.5 and subtil doc[t]rines. Inserted.\n 74.1 who begi[u/n]s his history Inverted.\n 91.16 the space be[t]ween. Inserted.\n 91.18 whose tap[-]hole is very little Inserted.\n 96.5 and th[a/e]n H will be east Replaced.\n 114.27 a spring[ ]upon the top Inserted.\n 158.19 the air which[ which] was Redundant.\n 163.11 8 degrees 30 minutes[.]? Removed.\n 192.19 (called by him [\u03b5/\u1f10]\u03c6\u03b1\u03c1\u03bc\u03cc\u03b6\u03bf\u03bd\u03c4\u03b1) Replaced.\n 200.5 may so precisely deter[ter]mine Removed.\n 206.1 lines whi[e/c]h shall never meet Replaced.\n 208.10 [\u1f45\u03c5/\u03bf\u1f55]\u03c4\u03c9\u03c2 \u1f14\u03c7\u03b5\u03b9 Replaced.\n 208.13 [\u1f45\u03c5/\u03bf\u1f55]\u03c4\u03c9\u03c2 \u1f14\u03c7\u03b5\u03b9 Replaced.\n 225.8 as by raref[r]action and condensation. Removed.\n 263.17 [\u201c]_The magnitude of an angle_ Added.\n 277.33 th[e/a]n of one in nine? Replaced.\n 297.26 At the seventee[n]th chapter Inserted.\n 300.22 _that of G K to G [E,/E.]_\u201d Replaced.\n 323.7 _tanquam dicta problematic\u00e8._[\u201d] Added.\n 338.31 and so mis[s]pend it? Inserted.\n 351.16 by which you live[,/.] Replaced.\n 376.4 _propositione pr\u00e6cedente._[\u201d] Added.\n 376.22 it shall quite vanish. [And so] Missing?\n 342.12 to the present purpose[)]. Added (likely).\n 382.12 whether \u03b4\u03b9\u03c0\u03bb\u03ac\u03c3\u03b9\u03bf\u03c2 and \u03b4\u03b9\u03c0\u03bb\u03b1\u03c3[\u03b9]\u03af\u03c9\u03bd be one Removed.\n 391.5 by the word \u03a3\u03b7\u03bc[~\u03b5/\u03b5\u1fd6]\u03bf\u03bd in Euclid Replaced.\n 412.17 \u03ba\u03b1[\u03b9/\u1f76] \u1f21 \u1f1c\u03bc\u03c0\u03bf\u03c5\u03c3\u03b1 Replaced.\n 413.26 \u03c4\u1ff6\u03bd \u03b8\u03b5\u1ff6\u03bd [\u0391/\u1f08]\u03b9\u03b3\u03cd\u03c0\u03c4\u03b9\u03bf\u03b9 Replaced.\n 413.27 \u03ba\u03b1[\u1f31/\u1f76] \u1f05\u03c4\u03b5 \u1f08\u03b9\u03b3\u03cd\u03c0\u03c4\u03b9\u03bf\u03c2, Replaced.\n 413.30 \u03c0\u03bf\u03b4\u1ff6\u03bd [\u1f74/\u1f20]\u03b4\u03ad \u03ba\u03bd\u03b7\u03bc\u03ac\u03c9\u03bd Replaced.\n 414.1 \u1fe5\u1fb3\u03b4\u03af\u03c9\u03c2[ ]\u1f14\u03b3\u03bd\u03c9\u03bd \u1f51\u03c0\u03bf\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03bf\u03bd\u03c4\u03b5\u03c2. Inserted.\n 414.4 \u1f00\u03bb\u03bb\u1f70 \u03ba\u03b1\u03c4\u1f70 \u1fe5\u03cd\u03bc\u03b7\u03bd [\u1f70/\u1f00]\u03ad\u03c1\u03b9\u03bf\u03bd Replaced.\n 415.23 deriveth it [\u03c4/\u03c0]\u03b1\u03c1\u1f70 \u03c4\u1f78 \u1f10\u03bc\u03c0\u03bf\u03b4\u03b9\u03b6\u03b5\u03b9\u03bd Replaced.\n 415.25 it saith, [\u1f1d/\u1f1c]\u03bc\u03c0\u03bf\u03c5\u03c3\u03b1 \u03a8\u03b9\u03bb\u03bf\u1fe6\u03c0\u03b1\u03b9 Replaced.\n 415.33 some had no legs, &c[.] Added.\n 416.7 \u03c4\u03cd\u03c7\u03b7 [\u1f72/\u1f10]\u03bc\u03c0\u03bf\u03b4\u03b9\u03b6\u03bf\u03c5\u03c3\u03b1 Replaced.\n 416.16 [\u1f1d/\u1f1c]\u03bc\u03c0\u03bf\u1fe6\u03c3\u03b1 is also a name for Lamia Replaced.\n 418.16 [\u1f41\u03b9/\u03bf\u1f31] \u03b4\u1f72 \u1f51\u03c0\u03ad\u03c6\u03b5\u03c5\u03b3\u03bf\u03bd Transposed.\n 418.20 \u03ba\u03b1\u1f76 \u03c4\u1f78 [\u1f72/\u1f10]\u03c0\u03b9\u03c0\u03b7\u03b4\u1fb6\u03bd Replaced.\n 419.13 \u03b3\u03b1\u03c1 [\u1f45\u03b9/\u03bf\u1f35]\u03bd\u03bf\u03c5 \u03c0\u03bb\u03b7\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2, Replaced.\n 419.14.1 \u03b5\u1f36\u03c7\u03b5 \u03c4\u1f78\u03bd [\u1f45\u03b9/\u03bf\u1f35]\u03bd\u03bf\u03c5. Replaced.\n 419.14.2 [\u0391/\u1f08]\u03c3\u03ba\u03c9\u03bb\u03b9\u03ac\u03b6\u03b5\u03b9\u03bd Replaced.\n 423.1 \u03ba\u03b1\u1f76 \u03b6\u03ce\u03c9\u03bd \u03c0\u03b1\u03bd\u03c4\u03bf\u03b4\u03b1\u03c0\u1ff6\u03bd [\u1f10\u03b9/\u03b5\u1f30]\u03ba\u03cc\u03c3\u03b9 Replaced.\n 455.27 two spaces th[e/a]n when double Replaced.\n 456.3 a pound two foote a[t] the rate of Added.\n 456.6 rate of a mile an how[er/re] Transposed.","title":"The English works of Thomas Hobbes of Malmesbury, Volume 07 (of 11)","tradition":"philosophy-ethics","translated_text":null,"word_count":132038}