{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_22062/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_22062","id":"gutenberg_22062","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/22062","summary":null,"tags":["Category: Mathematics","Geometry -- Early works to 1800","Mathematics -- Early works to 1800","Philosophy -- Early works to 1800"],"text":"[Transcriber\u2019s Note:\n\nThis e-text includes characters that will only display in UTF-8\n(Unicode) text readers, including a few words of Greek:\n\n  \u03a4\u03b1\u03ba\u03c4\u03b9\u03ba\u1f74 [Taktik\u00ea]\n  \u00e3 \u1ebd \u00f5 \u0169 [overline or tilde to show following -n or -m]\n  \u2767 \u261e [leaf symbol; pointing-finger symbol]\n  \u2021 [double-ended dagger, used in size notations (below)]\n\nIf any of these characters do not display properly-- in particular, if\nthe diacritic does not appear directly above the letter-- or if the\nquotation marks in this paragraph appear as garbage, make sure your text\nreader\u2019s \u201ccharacter set\u201d or \u201cfile encoding\u201d is set to Unicode (UTF-8).\nYou may also need to change the default font. As a last resort, use the\nASCII version of this file instead.\n\nSome aspects of the original book had to be modified for all versions\nof this plain-text file.\n\nSuperscript letters are shown with ^: y^e, y^t.\n\nMarginal quotation marks are shown inline as \u201c and \u201d, approximating the\nbeginning and end of the marked passage. In the original text, no\nquotation marks were printed inline.\n\nParagraphs are broken up for sidenotes, with blank lines before and\nafter. Original paragraph breaks are shown as two blank lines. Brackets\nwithin the body text are in the original.\n\nAll sidenotes except the one beginning \u201cThis noble Earle\u201d were printed\nin italics; markup has been omitted to reduce visual clutter.\n\nAt least four sizes of text were used, often in combination with\n_italics_. The variants are shown here as:\n\n  +\u2021very large\u2021+\n  +larger+\n  =smaller=\n\nFurther errors and anomalies are listed at the end of the text, along\nwith those Euclid citations identified by number.]\n\n       *       *       *       *       *\n           *       *       *       *\n       *       *       *       *       *\n\n\n  THE ELEMENTS\n  OF GEOMETRIE\n\n  of the most auncient\n  Philosopher\n  _EVCLIDE_\n  of Megara.\n\n  _Faithfully (now first) translated\n  into the Englishe toung, by\n  _H. Billingsley_, Citizen of London_.\n\n  _Whereunto are annexed certaine\n  Scholies, Annotations, and Inuentions,\n  of the best Mathematiciens,\n  both of time past, and\n  in this our age._\n\n\n  _With a very fruitfull Pr\u00e6face made by _M. I. Dee_,\n  specifying the chiefe Mathematicall Sci\u1ebdces, what they are,\n  and wherunto commodious: where, also, are disclosed\n  certaine new Secrets Mathematicall and Mechanicall,\n  vntill these our daies, greatly missed._\n\n\n  Imprinted at London by _Iohn Daye_.\n\n\n\n\n\u2767 The Translator to the Reader.\n\n\n_There is (gentle Reader) nothing (the word of God onely set\napart) which so much beautifieth and adorneth the soule and\nminde of m\u00e3, as doth the knowledge of good artes and sciences:\nas the knowledge of naturall and morall Philosophie. The one\nsetteth before our eyes, the creatures of God, both in the\nheauens aboue, and in the earth beneath: in which as in a\nglasse, we beholde the exceding maiestie and wisedome of God,\nin adorning and beautifying them as we see: in geuing vnto them\nsuch wonderfull and manifolde proprieties, and naturall\nworkinges, and that so diuersly and in such varietie: farther in\nmaintaining and conseruing them continually, whereby to praise\nand adore him, as by S. Paule we are taught. The other teacheth\nvs rules and preceptes of vertue, how, in common life amongest\nmen, we ought to walke vprightly: what dueties pertaine to our\nselues, what pertaine to the gouernment or good order both of an\nhousholde, and also of a citie or common wealth. The reading\nlikewise of histories, conduceth not a litle, to the adorning of\nthe soule & minde of man, a studie of all men c\u00f5mended: by it\nare seene and knowen the artes and doinges of infinite wise men\ngone before vs. In histories are contained infinite examples of\nheroicall vertues to be of vs followed, and horrible examples of\nvices to be of vs eschewed. Many other artes also there are\nwhich beautifie the minde of man: but of all other none do more\ngarnishe & beautifie it, then those artes which are called\nMathematicall. Unto the knowledge of which no man can attaine,\nwithout the perfecte knowledge and instruction of the\nprinciples, groundes, and Elementes of Geometrie. But perfectly\nto be instructed in them, requireth diligent studie and reading\nof olde auncient authors. Amongest which, none for a beginner is\nto be preferred before the most auncient Philosopher _Euclide_\nof _Megara_. For of all others he hath in a true methode and\niuste order, gathered together whatsoeuer any before him had of\nthese Elementes written: inuenting also and adding many thinges\nof his owne: wherby he hath in due forme accomplished the arte:\nfirst geuing definitions, principles, & groundes, wherof he\ndeduceth his Propositions or conclusions, in such wonderfull\nwise, that that which goeth before, is of necessitie required to\nthe proufe of that which followeth. So that without the diligent\nstudie of _Euclides_ Elementes, it is impossible to attaine vnto\nthe perfecte knowledge of Geometrie, and consequently of any of\nthe other Mathematicall sciences. Wherefore considering the want\n& lacke of such good authors hitherto in our Englishe tounge,\nlamenting also the negligence, and lacke of zeale to their\ncountrey in those of our nation, to whom God hath geuen both\nknowledge, & also abilitie to translate into our tounge, and\nto publishe abroad such good authors, and bookes (the chiefe\ninstrumentes of all learninges): seing moreouer that many good\nwittes both of gentlemen and of others of all degrees, much\ndesirous and studious of these artes, and seeking for them as\nmuch as they can, sparing no paines, and yet frustrate of their\nintent, by no meanes attaining to that which they seeke: I haue\nfor their sakes, with some charge & great trauaile, faithfully\ntranslated into our vulgare to\u0169ge, & set abroad in Print, this\nbooke of _Euclide_. Whereunto I haue added easie and plaine\ndeclarations and examples by figures, of the definitions. In\nwhich booke also ye shall in due place finde manifolde\nadditions, Scholies, Annotations, and Inuentions: which I haue\ngathered out of many of the most famous & chiefe Mathematici\u1ebds,\nboth of old time, and in our age: as by diligent reading it in\ncourse, ye shall well perceaue. The fruite and gaine which I\nrequire for these my paines and trauaile, shall be nothing els,\nbut onely that thou gentle reader, will gratefully accept the\nsame: and that thou mayest thereby receaue some profite: and\nmoreouer to excite and stirre vp others learned, to do the like,\n& to take paines in that behalfe. By meanes wherof, our Englishe\ntounge shall no lesse be enriched with good Authors, then are\nother straunge tounges: as the Dutch, French, Italian, and\nSpanishe: in which are red all good authors in a maner, found\namongest the Grekes or Latines. Which is the chiefest cause,\nthat amongest th\u1ebd do florishe so many cunning and skilfull men,\nin the inuentions of straunge and wonderfull thinges, as in\nthese our daies we see there do. Which fruite and gaine if I\nattaine vnto, it shall encourage me hereafter, in such like sort\nto translate, and set abroad some other good authors, both\npertaining to religion (as partly I haue already done) and also\npertaining to the Mathematicall Artes. Thus gentle\nreader farewell._ (?\u00bf)\n\n\n    [Decoration]\n\n\n\n\n  \u2767 TO THE VNFAINED LOVERS\n  of truthe, and constant Studentes of Noble\n  _Sciences, _IOHN DEE_ of London, hartily_\n  wisheth grace from heauen, and most prosperous\n  _successe in all their honest attemptes and_\n  exercises.\n\n\nDiuine _Plato_, the great Master of many worthy Philosophers, and the\nconstant auoucher, and pithy perswader of _Vnum_, _Bonum_, and _Ens_: in\nhis Schole and Academie, sundry times (besides his ordinary Scholers)\nwas visited of a certaine kinde of men, allured by the noble fame of\n_Plato_, and the great commendation of hys profound and profitable\ndoctrine. But when such Hearers, after long harkening to him, perceaued,\nthat the drift of his discourses issued out, to conclude, this _Vnum_,\n_Bonum_, and _Ens_, to be Spirituall, Infinite, \u00c6ternall, Omnipotent,\n&c. Nothyng beyng alledged or expressed, How, worldly goods: how,\nworldly dignitie: how, health, Str\u1ebdgth or lustines of body: nor yet the\nmeanes, how a merueilous sensible and bodyly blysse and felicitie\nhereafter, might be atteyned: Straightway, the fantasies of those\nhearers, were dampt: their opinion of _Plato_, was clene chaunged: yea\nhis doctrine was by them despised: and his schole, no more of them\nvisited. Which thing, his Scholer, _Aristotle_, narrowly c\u00f5sidering,\nfounde the cause therof, to be, \u201cFor that they had no forwarnyng and\ninformation, in generall,\u201d whereto his doctrine tended. For, so, might\nthey haue had occasion, either to haue forborne his schole hauntyng: (if\nthey, then, had misliked his Scope and purpose) or constantly to haue\ncontinued therin: to their full satisfaction: if such his finall scope &\nintent, had ben to their desire. Wherfore, _Aristotle_, euer, after\nthat, vsed in brief, to forewarne his owne Scholers and hearers, \u201cboth\nof what matter, and also to what ende, he tooke in hand to speake, or\nteach.\u201d While I consider the diuerse trades of these two excellent\nPhilosophers (and am most sure, both, that _Plato_ right well, otherwise\ncould teach: and that _Aristotle_ mought boldely, with his hearers, haue\ndealt in like sorte as _Plato_ did) I am in no little pang of\nperplexitie: Bycause, that, which I mislike, is most easy for me to\nperforme (and to haue _Plato_ for my ex\u00e3ple.) And that, which I know to\nbe most commendable: and (in this first bringyng, into common handling,\nthe _Artes Mathematicall_) to be most necessary: is full of great\ndifficultie and sundry daungers. Yet, neither do I think it mete, for so\nstraunge matter (as now is ment to be published) and to so straunge an\naudience, to be bluntly, at first, put forth, without a peculiar\nPreface: Nor (Imitatyng _Aristotle_) well can I hope, that accordyng to\nthe amplenes and dignitie of the _State Mathematicall_, I am able,\neither playnly to prescribe the materiall boundes: or precisely to\nexpresse the chief purposes, and most wonderfull applications therof.\nAnd though I am sure, that such as did shrinke from _Plato_ his schole,\nafter they had perceiued his finall conclusion, would in these thinges\nhaue ben his most diligent hearers (so infinitely mought their desires,\nin fine and at length, by our _Artes Mathematicall_ be satisfied) yet,\nby this my Pr\u00e6face & forewarnyng, Aswell all such, may (to their great\nbehofe) the soner, hither be allured: as also the _Pythagoricall_, and\n_Platonicall_ perfect scholer, and the constant profound Philosopher,\nwith more ease and spede, may (like the Bee,) gather, hereby, both wax\nand hony.\n\n\n    [The intent of this Preface.]\n\nWherfore, seyng I finde great occasion (for the causes alleged, and\nfarder, in respect of my _Art Mathematike generall_) to vse \u201ca certaine\nforewarnyng and Pr\u00e6face, whose content shalbe, that mighty, most\nplesaunt, and frutefull _Mathematicall Tree_, with his chief armes and\nsecond (grifted) braunches: Both, what euery one is, and also, what\ncommodity, in generall, is to be looked for, aswell of griff as stocke:\nAnd forasmuch as this enterprise is so great, that, to this our tyme, it\nneuer was (to my knowledge) by any achieued: And also it is most hard,\nin these our drery dayes, to such rare and straunge Artes, to wyn due\nand common credit:\u201d Neuertheles, if, for my sincere endeuour to satisfie\nyour honest expectation, you will but lend me your th\u00e3kefull mynde a\nwhile: and, to such matter as, for this time, my penne (with spede) is\nhable to deliuer, apply your eye or eare attentifely: perchaunce, at\nonce, and for the first salutyng, this Preface you will finde a lesson\nlong enough. And either you will, for a second (by this) be made much\nthe apter: or shortly become, well hable your selues, of the lyons claw,\nto coniecture his royall symmetrie, and farder propertie. Now then,\ngentle, my frendes, and countrey men, Turne your eyes, and bend your\nmyndes to that doctrine, which for our present purpose, my simple talent\nis hable to yeld you.\n\n\nAll thinges which are, & haue beyng, are found vnder a triple diuersitie\ngenerall. For, either, they are demed Supernaturall, Naturall, or, of a\nthird being. Thinges Supernaturall, are immateriall, simple,\nindiuisible, incorruptible, & vnchangeable. Things Naturall, are\nmateriall, compounded, diuisible, corruptible, and chaungeable. Thinges\nSupernaturall, are, of the minde onely, comprehended: Things Naturall,\nof the sense exterior, ar hable to be perceiued. In thinges Naturall,\nprobabilitie and coniecture hath place: But in things Supernaturall,\nchief dem\u00f5stration, & most sure Science is to be had. By which\nproperties & comparasons of these two, more easily may be described, the\nstate, condition, nature and property of those thinges, which, we before\ntermed of a third being: which, by a peculier name also, are called\n_Thynges Mathematicall_. For, these, beyng (in a maner) middle, betwene\nthinges supernaturall and naturall: are not so absolute and excellent,\nas thinges supernatural: Nor yet so base and grosse, as things naturall:\nBut are thinges immateriall: and neuerthelesse, by materiall things\nhable somewhat to be signified. And though their particular Images, by\nArt, are aggregable and diuisible: yet the generall _Formes_,\nnotwithstandyng, are constant, vnchaungeable, vntr\u00e3sformable, and\nincorruptible. Neither of the sense, can they, at any tyme, be perceiued\nor iudged. Nor yet, for all that, in the royall mynde of man, first\nconceiued. But, surmountyng the imperfecti\u00f5 of coniecture, weenyng and\nopinion: and commyng short of high intellectuall c\u00f5cepti\u00f5, are the\nMercurial fruite of _Dian\u0153ticall_ discourse, in perfect imagination\nsubsistyng. A meruaylous newtralitie haue these thinges _Mathematicall_,\nand also a straunge participati\u00f5 betwene thinges supernaturall,\nimmortall, intellectual, simple and indiuisible: and thynges naturall,\nmortall, sensible, compounded and diuisible. Probabilitie and sensible\nprose, may well serue in thinges naturall: and is commendable: In\nMathematicall reasoninges, a probable Argument, is nothyng regarded: nor\nyet the testimony of sense, any whit credited: But onely a perfect\ndemonstration, of truthes certaine, necessary, and inuincible:\nvniuersally and necessaryly concluded: is allowed as sufficient for \u201can\nArgument exactly and purely Mathematical.\u201d\n\n\n    [Note the worde, Vnit, to expresse the Greke Monas,\n    & not Vnitie: as we haue all, commonly, till now, vsed.]\n\nOf _Mathematicall_ thinges, are two principall kindes: namely, _Number_,\nand _Magnitude_.\n\n    [Number.]\n\n_Number_, we define, to be, a certayne Mathematicall S\u0169me, of _Vnits_.\nAnd, an _Vnit_, is that thing Mathematicall, Indiuisible, by\nparticipation of some likenes of whose property, any thing, which is in\ndeede, or is counted One, may resonably be called One. We account an\n_Vnit_, a thing _Mathematicall_, though it be no Number, and also\nindiuisible: because, of it, materially, Number doth consist: which,\nprincipally, is a thing _Mathematicall_.\n\n    [Magnitude.]\n\n_Magnitude_ is a thing _Mathematicall_, by participation of some likenes\nof whose nature, any thing is iudged long, broade, or thicke. \u201cA thicke\n_Magnitude_ we call a _Solide_, or a _Body_. What _Magnitude_ so euer,\nis Solide or Thicke, is also broade, & long. A broade magnitude, we call\na _Superficies_ or a Plaine. Euery playne magnitude, hath also length.\nA long magnitude, we terme a _Line_. A _Line_ is neither thicke nor\nbroade, but onely long: Euery certayne Line, hath two endes:\n\n    [A point.]\n\nThe endes of a line, are _Pointes_ called. A _Point_, is a thing\n_Mathematicall_, indiuisible, which may haue a certayne determined\nsituation.\u201d If a Poynt moue from a determined situation, the way wherein\nit moued, is also a _Line_: mathematically produced, whereupon, of the\nauncient Mathematiciens,\n\n    [A Line.]\n\na _Line_ is called the race or course of a _Point_. A Poynt we define,\nby the name of a thing Mathematicall: though it be no Magnitude, and\nindiuisible: because it is the propre ende, and bound of a Line: which\nis a true _Magnitude_.\n\n    [Magnitude.]\n\nAnd _Magnitude_ we may define to be that thing _Mathematicall_, which is\ndiuisible for euer, in partes diuisible, long, broade or thicke.\nTherefore though a Poynt be no _Magnitude_, yet _Terminatiuely_, we\nrecken it a thing _Mathematicall_ (as I sayd) by reason it is properly\nthe end, and bound of a line. Neither _Number_, nor _Magnitude_, haue\nany Materialitie. First, we will consider of _Number_, and of the\nScience _Mathematicall_, to it appropriate, called _Arithmetike_: and\nafterward of _Magnitude_, and his Science, called _Geometrie_. But that\nname contenteth me not: whereof a word or two hereafter shall be sayd.\nHow Immateriall and free from all matter, _Number_ is, who doth not\nperceaue? yea, who doth not wonderfully w\u00f5der at it? For, neither pure\n_Element_, nor _Aristoteles, Quinta Essentia_, is hable to serue for\nNumber, as his propre matter. Nor yet the puritie and simplenes of\nSubstance Spirituall or Angelicall, will be found propre enough thereto.\nAnd therefore the great & godly Philosopher _Anitius Boetius_, sayd:\n_Omnia qu\u00e6cun[que] a prim\u00e6ua rerum natura constructa sunt, Numerorum\nvidentur ratione formata. Hoc enim fuit principale in animo Conditoris\nExemplar_. That is: +_All thinges (which from the very first originall\nbeing of thinges, haue bene framed and made) do appeare to be Formed by\nthe reason of Numbers. For this was the principall example or patterne\nin the minde of the Creator_.+ O comfortable allurement, O rauishing\nperswasion, to deale with a Science, whose Subiect, is so Auncient, so\npure, so excellent, so surmounting all creatures, so vsed of the\nAlmighty and incomprehensible wisdome of the Creator, in the distinct\ncreation of all creatures: in all their distinct partes, properties,\nnatures, and vertues, by order, and most absolute number, brought, from\n_Nothing_, to the _Formalitie_ of their being and state. By _Numbers_\npropertie therefore, of vs, by all possible meanes, (to the perfection\nof the Science) learned, we may both winde and draw our selues into the\ninward and deepe search and vew, of all creatures distinct vertues,\nnatures, properties, and _Formes_: And also, farder, arise, clime,\nascend, and mount vp (with Speculatiue winges) in spirit, to behold in\nthe Glas of Creation, the _Forme of Formes_, the _Exemplar Number_ of\nall thinges _Numerable_: both visible and inuisible, mortall and\nimmortall, Corporall and Spirituall. Part of this profound and diuine\nScience, had _Ioachim_ the Prophesier atteyned vnto: by _Numbers\nFormall, Naturall_, and _Rationall_, forseyng, concludyng, and\nforshewyng great particular euents, long before their comming. His\nbookes yet remainyng, hereof, are good profe: And the noble Earle of\n_Mirandula_, (besides that,) a sufficient witnesse: that _Ioachim, in\nhis prophesies, proceded by no other way, then by Numbers Formall_. And\nthis Earle hym selfe, in Rome,\n\n    [Ano. 1488.]\n\n* set vp 900. Conclusions, in all kinde of Sciences, openly to be\ndisputed of: and among the rest, in his Conclusions _Mathematicall_, (in\nthe eleuenth Conclusion) hath in Latin, this English sentence. _By\nNumbers, a way is had, to the searchyng out, and vnderstandyng of euery\nthyng, hable to be knowen. For the verifying of which Conclusion,\nI promise to aunswere to the 74. Questions, vnder written, by the way of\nNumbers_. Which C\u00f5clusions, I omit here to rehearse: aswell auoidyng\nsuperfluous prolixitie: as, bycause _Ioannes Picus, workes_, are\ncommonly had. But, in any case, I would wish that those Conclusions were\nred diligently, and perceiued of such, as are earnest Obseruers and\nConsiderers of the constant law of n\u0169bers: which is planted in thyngs\nNaturall and Supernaturall: and is prescribed to all Creatures,\ninuiolably to be kept. For, so, besides many other thinges, in those\nConclusions to be marked, it would apeare, how sincerely, & within my\nboundes, I disclose the wonderfull mysteries, by numbers, to be atteyned\nvnto.\n\n\nOf my former wordes, easy it is to be gathered, that _Number_ hath a\ntreble state: One, in the Creator: an other in euery Creature (in\nrespect of his complete constitution:) and the third, in Spirituall and\nAngelicall Myndes, and in the Soule of m\u00e3. In the first and third state,\n_Number_, is termed _Number Numbryng_. But in all Creatures, otherwise,\n_Number_, is termed _N\u0169ber Numbred_. And in our Soule, N\u0169ber beareth\nsuch a swaye, and hath such an affinitie therwith: that some of the old\n_Philosophers_ taught, _Mans Soule, to be a Number mouyng it selfe_. And\nin dede, in vs, though it be a very Accident: yet such an Accident it\nis, that before all Creatures it had perfect beyng, in the Creator,\nSempiternally. _Number Numbryng_ therfore, is the discretion discerning,\nand distincting of thinges. But in God the Creator, This discretion, in\nthe beginnyng, produced orderly and distinctly all thinges. For his\n_Numbryng_, then, was his Creatyng of all thinges. And his Continuall\n_Numbryng_, of all thinges, is the Conseruation of them in being: And,\nwhere and when he will lacke an _Vnit_: there and then, that particular\nthyng shalbe _Discreated_. Here I stay. But our Seuerallyng,\ndistinctyng, and _Numbryng_, createth nothyng: but of Multitude\nconsidered, maketh certaine and distinct determination. And albeit these\nthynges be waighty and truthes of great importance, yet (by the infinite\ngoodnes of the Almighty _Ternarie_,) Artificiall Methods and easy wayes\nare made, by which the zelous Philosopher, may wyn nere this Riuerish\n_Ida_, this Mountayne of Contemplation: and more then Contemplation. And\nalso, though _Number_, be a thyng so Immateriall, so diuine, and\n\u00e6ternall: yet by degrees, by litle and litle, stretchyng forth, and\napplying some likenes of it, as first, to thinges Spirituall: and then,\nbryngyng it lower, to thynges sensibly perceiued: as of a momentanye\nsounde iterated: then to the least thynges that may be seen, numerable:\nAnd at length, (most grossely,) to a multitude of any corporall thynges\nseen, or felt: and so, of these grosse and sensible thynges, we are\ntrayned to learne a certaine Image or likenes of numbers: and to vse\nArte in them to our pleasure and proffit. So grosse is our conuersation,\nand dull is our apprehension: while mortall Sense, in vs, ruleth the\ncommon wealth of our litle world. Hereby we say, Three Lyons, are three:\nor a _Ternarie_. Three Egles, are three, or a _Ternarie_.\n\n    [\u261e]\n\nWhich * _Ternaries_, are eche, the _Vnion_, _knot_, and _Vniformitie_,\nof three discrete and distinct _Vnits_. That is, we may in eche\n_Ternarie_, thrise, seuerally pointe, and shew a part, _One_, _One_, and\n_One_. Where, in Numbryng, we say One, two, Three. But how farre, these\nvisible Ones, do differre from our Indiuisible Vnits (in pure\n_Arithmetike_, principally considered) no man is ignorant. Yet from\nthese grosse and materiall thynges, may we be led vpward, by degrees,\nso, informyng our rude Imagination, toward the c\u00f5ceiuyng of _Numbers_,\nabsolutely (:Not supposing, nor admixtyng any thyng created, Corporall\nor Spirituall, to support, conteyne, or represent those _Numbers_\nimagined:) that at length, we may be hable, to finde the number of our\nowne name, gloriously exemplified and registred in the booke of the\n_Trinitie_ most blessed and \u00e6ternall.\n\n\nBut farder vnderstand, that vulgar Practisers, haue Numbers, otherwise,\nin sundry Considerations: and extend their name farder, then to Numbers,\nwhose least part is an _Vnit_. For the common Logist, Reckenmaster, or\nArithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lesse\npartes: and calleth them _Fractions_. As of an _Vnit_, he maketh an\nhalfe, and thus noteth it, \u00bd. and so of other, (infinitely diuerse)\npartes of an _Vnit_. Yea and farder, hath, _Fractions of Fractions. &c_.\nAnd, forasmuch, as, _Addition_, _Substraction_, _Multiplication_,\n_Diuision_ and _Extraction of Rotes_, are the chief, and sufficient\npartes of _Arithmetike_:\n\n    [Arithmetike.]\n\nwhich is, the _Science that demonstrateth the properties, of Numbers,\nand all operati\u00f5s, in numbers to be performed_:\n\n    [Note.]\n\n\u201cHow often, therfore, these fiue sundry sortes of Operations, do, for\nthe most part, of their execution, differre from the fiue operations of\nlike generall property and name, in our Whole numbers practisable, So\noften, (for a more distinct doctrine) we, vulgarly account and name it,\nan other kynde of _Arithmetike_.\u201d And by this reason:\n\n    [1.]\n\nthe Consideration, doctrine, and working, in whole numbers onely: where,\nof an _Vnit_, is no lesse part to be allowed: is named (as it were) an\n_Arithmetike_ by it selfe. And so of the _Arithmetike of Fractions_.\n\n    [2.]\n\nIn lyke sorte, the necessary, wonderfull and Secret doctrine of\nProportion, and proportionalytie hath purchased vnto it selfe a peculier\nmaner of handlyng and workyng: and so may seme an other forme of\n_Arithmetike_.\n\n    [3.]\n\nMoreouer, the _Astronomers_, for spede and more commodious calculation,\nhaue deuised a peculier maner of orderyng n\u0169bers, about theyr circular\nmotions, by Sexagenes, and Sexagesmes. By Signes, Degrees and Minutes\n&c. which commonly is called the _Arithmetike_ of _Astronomical_ or\n_Phisicall Fractions_. That, haue I briefly noted, by the name of\n_Arithmetike Circular_. Bycause it is also vsed in circles, not\n_Astronomicall. &c._\n\n    [4.]\n\nPractise hath led _Numbers_ farder, and hath framed them, to take vpon\nthem, the shew of _Magnitudes_ propertie: Which is _Incommensurabilitie_\nand _Irrationalitie_. (For in pure _Arithmetike_, an _Vnit_, is the\ncommon Measure of all Numbers.) And, here, N\u0169bers are become, as Lynes,\nPlaynes and Solides: some tymes _Rationall_, some tymes _Irrationall_.\nAnd haue propre and peculier characters, (as \u00b2\u221a. \u00b3\u221a. and so of other.\nWhich is to signifie _Rote Square, Rote Cubik: and so forth_:) & propre\nand peculier fashions in the fiue principall partes: Wherfore the\npractiser, estemeth this, a diuerse _Arithmetike_ from the other.\nPractise bryngeth in, here, diuerse compoundyng of Numbers: as some\ntyme, two, three, foure (or more) _Radicall_ n\u0169bers, diuersly knit, by\nsignes, of More & Lesse: as thus \u00b2\u221a12 + \u00b3\u221a15. Or thus \u2074\u221a19 + \u00b3\u221a12 - \u00b2\u221a2.\n&c. And some tyme with whole numbers, or fractions of whole Number, am\u00f5g\nthem: as 20 + \u00b2\u221a24. \u00b3\u221a16 + 33 - \u00b2\u221a10. \u2074\u221a44 + 12\u00bc + \u00b3\u221a9. And so,\ninfinitely, may hap the varietie. After this: Both the one and the other\nhath fractions incident: and so is this _Arithmetike_ greately enlarged,\nby diuerse exhibityng and vse of Compositions and mixtynges. Consider\nhow, I (beyng desirous to deliuer the student from error and\nCauillation) do giue to this _Practise_, the name of the _Arithmetike of\nRadicall numbers_: Not, of _Irrationall_ or _Surd Numbers_: which other\nwhile, are Rationall: though they haue the Signe of a Rote before them,\nwhich, _Arithmetike_ of whole Numbers most vsuall, would say they had no\nsuch Roote: and so account them _Surd Numbers_: which, generally spok\u1ebd,\nis vntrue: as _Euclides_ tenth booke may teach you. Therfore to call\nthem, generally, _Radicall Numbers_, (by reason of the signe \u221a.\nprefixed,) is a sure way: and a sufficient generall distinction from all\nother ordryng and vsing of Numbers: And yet (beside all this) Consider:\nthe infinite desire of knowledge, and incredible power of mans Search\nand Capacitye: how, they, ioyntly haue waded farder (by mixtyng of\nspeculation and practise) and haue found out, and atteyned to the very\nchief perfection (almost) of _Numbers_ Practicall vse. Which thing, is\nwell to be perceiued in that great Arithmeticall Arte of _\u00c6quation_:\ncommonly called the _Rule of Coss._ or _Algebra_. The Latines termed it,\n_Regulam Rei & Census_, that is, the +_Rule of the thyng and his\nvalue_+. With an apt name: comprehendyng the first and last pointes of\nthe worke. And the vulgar names, both in Italian, Frenche and Spanish,\ndepend (in namyng it,) vpon the signification of the Latin word, _Res_:\n+_A thing_+: vnleast they vse the name of _Algebra_. And therin\n(commonly) is a dubble error. The one, of them, which thinke it to be of\n_Geber_ his inuentyng: the other of such as call it _Algebra_. For,\nfirst, though _Geber_ for his great skill in Numbers, Geometry,\nAstronomy, and other maruailous Artes, mought haue semed hable to haue\nfirst deuised the sayd Rule: and also the name carryeth with it a very\nnere likenes of _Geber_ his name: yet true it is, that a _Greke_\nPhilosopher and Mathematicien, named _Diophantus_, before _Geber_ his\ntyme, wrote 13. bookes therof (of which, six are yet extant: and I had\nthem to *vse,\n\n    [* Anno. 1550.]\n\nof the famous Mathematicien, and my great frende, _Petrus Montaureus_:)\nAnd secondly, the very name, is _Algiebar_, and not _Algebra_: as by the\nArabien _Auicen_, may be proued: who hath these precise wordes in\nLatine, by _Andreas Alpagus_ (most perfect in the Arabik tung) so\ntranslated. _Scientia faciendi Algiebar & Almachabel. i. Scientia\ninueniendi numerum ignotum, per additionem Numeri, & diuisionem &\n\u00e6quationem_. Which is to say: +_The Science of workyng Algiebar and\nAlmachabel_+, that is, the +_Science of findyng an vnknowen number, by\nAddyng of a Number, & Diuision & \u00e6quation_+. Here haue you the name: and\nalso the principall partes of the Rule, touched. To name it, _The rule,\nor Art of \u00c6quation_, doth signifie the middle part and the State of the\nRule. This Rule, hath his peculier Characters:\n\n    [5.]\n\nand the principal partes of _Arithmetike_, to it appertayning, do\ndifferre from the other _Arithmeticall operations_. This _Arithmetike,\nhath N\u0169bers_ Simple, C\u00f5pound, Mixt: and Fractions, accordingly. This\nRule, and _Arithmetike of Algiebar_, is so profound, so generall and so\n(in maner) conteyneth the whole power of Numbers Application practicall:\nthat mans witt, can deale with nothyng, more proffitable about numbers:\nnor match, with a thyng, more mete for the diuine force of the Soule,\n(in humane Studies, affaires, or exercises) to be tryed in. Perchaunce\nyou looked for, (long ere now,) to haue had some particular profe, or\neuident testimony of the vse, proffit and Commodity of Arithmetike\nvulgar, in the Common lyfe and trade of men. Therto, then, I will now\nframe my selfe: But herein great care I haue, least length of sundry\nprofes, might make you deme, that either I did misdoute your zelous\nmynde to vertues schole: or els mistrust your hable witts, by some, to\ngesse much more. A profe then, foure, fiue, or six, such, will I bryng,\nas any reasonable man, therwith may be persuaded, to loue & honor, yea\nlearne and exercise the excellent Science of _Arithmetike_.\n\n\nAnd first: who, nerer at hand, can be a better witnesse of the frute\nreceiued by _Arithmetike_, then all kynde of Marchants? Though not all,\nalike, either nede it, or vse it. How could they forbeare the vse and\nhelpe of the Rule, called the Golden Rule? Simple and Compounde: both\nforward and backward? How might they misse _Arithmeticall_ helpe in the\nRules of Felowshyp: either without tyme, or with tyme? and betwene the\nMarchant & his Factor? The Rules of Bartering in wares onely: or part in\nwares, and part in money, would they gladly want? Our Marchant\nventurers, and Trauaylers ouer Sea, how could they order their doynges\niustly and without losse, vnleast certaine and generall Rules for\nExcha\u0169ge of money, and Rechaunge, were, for their vse, deuised? The Rule\nof Alligation, in how sundry cases, doth it conclude for them, such\nprecise verities, as neither by naturall witt, nor other experience,\nthey, were hable, els, to know? And (with the Marchant then to make an\nend) how ample & wonderfull is the Rule of False positions? especially\nas it is now, by two excellent Mathematiciens (of my familier\nacquayntance in their life time) enlarged? I meane _Gemma Frisius_, and\n_Simon Iacob_. Who can either in brief conclude, the generall and\nCapitall Rules? or who can Imagine the Myriades of sundry Cases, and\nparticular examples, in Act and earnest, continually wrought, tried and\nconcluded by the forenamed Rules, onely? How sundry other _Arithmeticall\npractises_, are commonly in Marchantes handes, and knowledge: They them\nselues, can, at large, testifie.\n\n\nThe Mintmaster, and Goldsmith, in their Mixture of Metals, either of\ndiuerse kindes, or diuerse values: how are they, or may they, exactly be\ndirected, and meruailously pleasured, if _Arithmetike_ be their guide?\nAnd the honorable Phisici\u00e3s, will gladly confesse them selues, much\nbeholding to the Science of _Arithmetike_, and that sundry wayes: But\nchiefly in their Art of Graduation, and compounde Medicines. And though\n_Galenus_, _Auerrois_, _Arnoldus_, _Lullus_, and other haue published\ntheir positions, aswell in the quantities of the Degrees aboue\nTemperament, as in the Rules, concluding the new _Forme_ resulting: yet\na more precise, commodious, and easy _Method_, is extant: by a\nCountreyman of ours\n\n    [R. B.]\n\n(aboue 200. yeares ago) inuented. And forasmuch as I am vncertaine, who\nhath the same: or when that litle Latin treatise, (as the Author writ\nit,) shall come to be Printed: (Both to declare the desire I haue to\npleasure my Countrey, wherin I may: and also, for very good profe of\nNumbers vse, in this most subtile and frutefull, Philosophicall\nConclusion,) I entend in the meane while, most briefly, and with my\nfarder helpe, to communicate the pith therof vnto you.\n\n\nFirst describe a circle: whose diameter let be an inch. Diuide the\nCircumference into foure equall partes. Fr\u00f5 the Center, by those 4.\nsections, extend 4. right lines: eche of 4. inches and a halfe long: or\nof as many as you liste, aboue 4. without the circumference of the\ncircle: So that they shall be of 4. inches long (at the least) without\nthe Circle. Make good euident markes, at euery inches end. If you list,\nyou may subdiuide the inches againe into 10. or 12. smaller partes,\nequall. At the endes of the lines, write the names of the 4. principall\nelementall Qualities. _Hote_ and _Colde_, one against the other. And\nlikewise _Moyst_ and _Dry_, one against the other. And in the Circle\nwrite _Temperate_. Which _Temperature_ hath a good Latitude: as\nappeareth by the Complexion of man. And therefore we haue allowed vnto\nit, the foresayd Circle: and not a point Mathematicall or Physicall.\n\n\n    [* Take some part of Lullus counsayle in his booke\n    de Q. Essentia.]\n\nNow, when you haue two thinges Miscible, whose degrees are * truely\nknowen: Of necessitie, either they are of one Quantitie and waight, or\nof diuerse. If they be of one Quantitie and waight: whether their\nformes, be Contrary Qualities, or of one kinde (but of diuerse\nintentions and degrees) or a _Temperate_, and a Contrary, _The forme\nresulting of their Mixture, is in the Middle betwene the degrees of the\nformes mixt_. As for example, let _A_, be _Moist_ in the first degree:\nand _B_, _Dry_ in the third degree. Adde 1. and 3. that maketh 4: the\nhalfe or middle of 4. is 2. This 2. is the middle, equally distant from\n_A_ and _B_\n\n    [* Note.]\n\n(for the * _Temperament_ is counted none. And for it, you must put a\nCiphre, if at any time, it be in mixture).\n\n                              HOTE\n                                +C\n                                |\n                                |\n                                +\n                                |\n                                |\n                                +\n                                |\n                                |\n                                +E\n                                |\n  MOIST                  A  TEMPERATE                B     DRYE\n    +------+------+------+------+------+------+------+------+\n                                |D\n                                |\n                                +\n                                |\n                                |\n                                +\n                                |\n                                |\n                                +\n                                |\n                                |\n                                +\n                              COLD\n\nCounting then from _B_, 2. degrees, toward _A_: you finde it to be _Dry_\nin the first degree: So is the _Forme resulting_ of the Mixture of _A_,\nand _B_, in our example. I will geue you an other example. Suppose, you\nhaue two thinges, as _C_, and _D_: and of _C_, the Heate to be in the 4.\ndegree: and of _D_, the Colde, to be remisse, euen vnto the\n_Temperament_. Now, for _C_, you take 4: and for _D_, you take a Ciphre:\nwhich, added vnto 4, yeldeth onely 4. The middle, or halfe, whereof, is\n2. Wherefore the _Forme resulting_ of _C_, and _D_, is Hote in the\nsecond degree: for, 2. degrees, accounted from _C_, toward _D_, ende\niuste in the 2. degree of heate. Of the third maner, I will geue also an\nexample: which let be this:\n\n    [Note.]\n\nI haue a liquid Medicine whose Qualitie of heate is in the 4. degree\nexalted: as was _C_, in the example foregoing: and an other liquid\nMedicine I haue: whose Qualitie, is heate, in the first degree. Of eche\nof these, I mixt a like quantitie: Subtract here, the lesse fr\u00f5 the\nmore: and the residue diuide into two equall partes: whereof, the one\npart, either added to the lesse, or subtracted from the higher degree,\ndoth produce the degree of the Forme resulting, by this mixture of _C_,\nand _E_. As, if from 4. ye abate 1. there resteth 3. the halfe of 3. is\n1\u00bd: Adde to 1. this 1\u00bd: you haue 2\u00bd. Or subtract from 4. this 1\u00bd: you\nhaue likewise 2\u00bd remayning. Which declareth, the _Forme resulting_, to\nbe _Heate_, in the middle of the third degree.\n\n\n    [The Second Rule.]\n\n\u201cBut if the Quantities of two thinges Commixt, be diuerse, and the\nIntensions (of their Formes Miscible) be in diuerse degrees, and\nheigthes. (Whether those Formes be of one kinde, or of Contrary kindes,\nor of a Temperate and a Contrary, _What proportion is of the lesse\nquantitie to the greater, the same shall be of the difference, which is\nbetwene the degree of the Forme resulting, and the degree of the greater\nquantitie of the thing miscible, to the difference, which is betwene the\nsame degree of the Forme resulting, and the degree of the lesse\nquantitie_. As for example. Let two pound of Liquor be geuen, hote in\nthe 4. degree: & one pound of Liquor be geuen, hote in the third\ndegree.\u201d I would gladly know the Forme resulting, in the Mixture of\nthese two Liquors. Set downe your n\u0169bers in order, thus.\n   ___________________________\n  |            |              |\n  |  {P}. _2._ |  _Hote. 4._  |\n  |            |              |\n  |  {P}. _1._ |  _Hote. 3._  |\n  |____________|______________|\n\nNow by the rule of Algiebar, haue I deuised a very easie, briefe, and\ngenerall maner of working in this case. Let vs first, suppose that\n_Middle Forme resulting_, to be 1{x}: as that Rule teacheth. And because\n(by our Rule, here geuen) as the waight of 1. is to 2: So is the\ndifference betwene 4. (the degree of the greater quantitie) and 1{x}: to\nthe difference betwene 1{x} and 3: (the degree of the thing, in lesse\nqu\u00e3titie. And with all, 1{x}, being alwayes in a certaine middell,\nbetwene the two heigthes or degrees). For the first difference, I set\n4-1{x}: and for the second, I set 1{x}-3. And, now againe, I say, as 1.\nis to 2. so is 4-1{x} to 1{x}-3. Wherfore, of these foure proportionall\nnumbers, the first and the fourth Multiplied, one by the other, do make\nas much, as the second and the third Multiplied the one by the other.\nLet these Multiplications be made accordingly. And of the first and the\nfourth, we haue 1{x}-3. and of the second & the third, 8-2{x}. Wherfore,\nour \u00c6quation is betwene 1{x}-3: and 8-2{x}. Which may be reduced,\naccording to the Arte of Algiebar: as, here, adding 3. to eche part,\ngeueth the \u00c6quation, thus, 1{x}=11-2{x}. And yet againe, contracting, or\nReducing it: Adde to eche part, 2{x}: Then haue you 3{x} \u00e6quall to 11:\nthus represented 3{x}=11. Wherefore, diuiding 11. by 3: the Quotient is\n3\u2154: the _Valew_ of our 1{x}, _Coss_, or _Thing_, first supposed. And\nthat is the heigth, or Intension of the _Forme resulting:_ which is,\n_Heate_, in two thirdes of the fourth degree: And here I set the shew of\nthe worke in conclusion, thus. The proufe hereof is easie: by\nsubtracting 3. from 3\u2154, resteth \u2154. Subtracte the same heigth of the\nForme resulting, (which is 3\u2154) fr\u00f5 4: then resteth \u2153: You see, that \u2154 is\ndouble to \u2153: as 2.{P}. is double to 1.{P}. So should it be: by the rule\nhere geuen. Note. As you added to eche part of the \u00c6quation, 3: so if ye\nfirst added to eche part 2{x}, it would stand, 3{x}-3=8. And now adding\nto eche part 3: you haue (as afore) 3{x}=11.\n    _________________________\n   |           |             | _\n   | {P}. _2._ |  _Hote. 4._ |   \u2153 _   _The forme_\n   |           |             |     _   _3\u2154 resulting._\n   | {P}. _1._ |  _Hote. 3._ | _ \u2154\n   |___________|_____________|\n\n\nAnd though I, here, speake onely of two thyngs Miscible: and most\ncommonly mo then three, foure, fiue or six, (&c.) are to be Mixed: (and\nin one Compound to be reduced: & the Forme resultyng of the same, to\nserue the turne) yet these Rules are sufficient: duely repeated and\niterated.\n\n    [Note.]\n\nIn procedyng first, with any two: and then, with the Forme Resulting,\nand an other: & so forth: For, the last worke, concludeth the Forme\nresultyng of them all: I nede nothing to speake, of the Mixture (here\nsupposed) what it is. Common Philosophie hath defined it, saying,\n_Mixtio est miscibilium, alteratorum, per minima coniunctorum, Vnio_.\nEuery word in the definition, is of great importance. I nede not also\nspend any time, to shew, how, the other manner of distributing of\ndegrees, doth agree to these Rules. Neither nede I of the farder vse\nbelonging to the Crosse of Graduation (before described) in this place\ndeclare, vnto such as are capable of that, which I haue all ready sayd.\nNeither yet with examples specifie the Manifold varieties, by the\nforesayd two generall Rules, to be ordered. The witty and Studious,\nhere, haue sufficient: And they which are not hable to atteine to this,\nwithout liuely teaching, and more in particular: would haue larger\ndiscoursing, then is mete in this place to be dealt withall: And other\n(perchaunce) with a proude snuffe will disdaine this litle: and would be\nvnthankefull for much more. I, therfore conclude: and wish such as haue\nmodest and earnest Philosophicall mindes, to laude God highly for this:\nand to Meruayle, that the profoundest and subtilest point, concerning\n_Mixture of Formes and Qualities Naturall_, is so Matcht and maryed with\nthe most simple, easie, and short way of the noble Rule of _Algiebar_.\nWho can remaine, therfore vnpersuaded, to loue, alow, and honor the\nexcellent Science of _Arithmetike_? For, here, you may perceiue that the\nlitle finger of _Arithmetike_, is of more might and contriuing, then a\nhunderd thousand mens wittes, of the middle sorte, are hable to\nperfourme, or truely to conclude, with out helpe thereof.\n\n\nNow will we farder, by the wise and valiant Capitaine, be certified,\nwhat helpe he hath, by the Rules of _Arithmetike_: in one of the Artes\nto him appertaining: And of the Grekes named\n\n    [\u03a4\u03b1\u03ba\u03c4\u03b9\u03ba\u1f74.]\n\n\u03a4\u03b1\u03ba\u03c4\u03b9\u03ba\u1f74. \u201cThat is, the Skill of Ordring Souldiers in Battell ray after\nthe best maner to all purposes.\u201d This Art so much dependeth vppon\nNumbers vse, and the Mathematicals, that _\u00c6lianus_ (the best writer\ntherof,) in his worke, to the _Emperour Hadrianus_, by his perfection,\nin the Mathematicals, (beyng greater, then other before him had,)\nthinketh his booke to passe all other the excellent workes, written of\nthat Art, vnto his dayes. For, of it, had written _\u00c6neas_: _Cyneas_ of\n_Thessaly_: _Pyrrhus Epirota_: and _Alexander_ his sonne: _Clearchus_:\n_Pausanias_: _Euangelus_: _Polybius_, familier frende to _Scipio_:\n_Eupolemus_: _Iphicrates_, _Possidonius_: and very many other worthy\nCapitaines, Philosophers and Princes of Immortall fame and memory: Whose\nfayrest floure of their garland (in this feat) was _Arithmetike_: and a\nlitle perceiuerance, in _Geometricall_ Figures. But in many other cases\ndoth _Arithmetike_ stand the Capitaine in great stede. As in\nproportionyng of vittayles, for the Army, either remaining at a stay: or\nsuddenly to be encreased with a certaine number of Souldiers: and for a\ncertain tyme. Or by good Art to diminish his company, to make the\nvictuals, longer to serue the remanent, & for a certaine determined\ntyme: if nede so require. And so in sundry his other accountes,\nReckeninges, Measurynges, and proportionynges, the wise, expert, and\nCircumspect Capitaine will affirme the Science of _Arithmetike_, to be\none of his chief Counsaylors, directers and aiders. Which thing (by good\nmeanes) was euident to the Noble, the Couragious, the loyall, and\nCurteous\n\n    [\u261e]\n\n_Iohn_, late Earle of Warwicke. Who was a yong Gentleman, throughly\nknowne to very few. Albeit his lusty valiantnes, force, and Skill in\nChiualrous feates and exercises: his humblenes, and frendelynes to all\nmen, were thinges, openly, of the world perceiued. But what rotes\n(otherwise,) vertue had fastened in his brest, what Rules of godly and\nhonorable life he had framed to him selfe: what vices, (in some then\nliuing) notable, he tooke great care to eschew: what manly vertues, in\nother noble men, (florishing before his eyes,) he Sythingly aspired\nafter: what prowesses he purposed and ment to achieue: with what feats\nand Artes, he began to furnish and fraught him selfe, for the better\nseruice of his Kyng and Countrey, both in peace & warre. These (I say)\nhis Heroicall Meditations, forecastinges and determinations, no twayne,\n(I thinke) beside my selfe, can so perfectly, and truely report. And\ntherfore, in Conscience, I count it my part, for the honor, preferment,\n& procuring of vertue (thus, briefly) to haue put his Name, in the\nRegister of _Fame Immortall_.\n\n\nTo our purpose. This _Iohn_, by one of his actes (besides many other:\nboth in England and Fraunce, by me, in him noted.) did disclose his\nharty loue to vertuous Sciences: and his noble intent, to excell in\nMartiall prowesse: When he, with humble request, and instant\nSolliciting: got the best Rules (either in time past by Greke or\nRomaine, or in our time vsed: and new Stratagemes therin deuised) for\nordring of all Companies, summes and Numbers of m\u1ebd, (Many, or few) with\none kinde of weapon, or mo, appointed: with Artillery, or without: on\nhorsebacke, or on fote: to giue, or take onset: to seem many, being few:\nto seem few, being many. To marche in battaile or Iornay: with many such\nfeates, to Foughten field, Skarmoush, or Ambushe appartaining:\n\n    [This noble Earle, dyed Anno. 1554. skarse of 24. yeares\n    of age: hauing no issue by his wife: Daughter to the Duke\n    of Somerset.]\n\nAnd of all these, liuely designementes (most curiously) to be in velame\nparchement described: with Notes & peculier markes, as the Arte\nrequireth: and all these Rules, and descriptions Arithmeticall, inclosed\nin a riche Case of Gold, he vsed to weare about his necke: as his Iuell\nmost precious, and Counsaylour most trusty. Thus, _Arithmetike_, of him,\nwas shryned in gold: Of _Numbers_ frute, he had good hope. Now, Numbers\ntherfore innumerable, in _Numbers_ prayse, his shryne shall finde.\n\n\nWhat nede I, (for farder profe to you) of the Scholemasters of Iustice,\nto require testimony: how nedefull, how frutefull, how skillfull a thing\n_Arithmetike_ is? I meane, the Lawyers of all sortes. Vndoubtedly, the\nCiuilians, can meruaylously declare: how, neither the Auncient Romaine\nlawes, without good knowledge of _Numbers art_, can be perceiued: Nor\n(Iustice in infinite Cases) without due proportion, (narrowly\nconsidered,) is hable to be executed. How Iustly, & with great knowledge\nof Arte, did _Papinianus_ institute a law of partition, and allowance,\nbetwene man and wife after a diuorce? But how _Accursius_, _Baldus_,\n_Bartolus_, _Iason_, _Alexander_, and finally _Alciatus_, (being\notherwise, notably well learned) do iumble, gesse, and erre, from the\n\u00e6quity, art and Intent of the lawmaker: _Arithmetike_ can detect, and\nconuince: and clerely, make the truth to shine. Good _Bartolus_, tyred\nin the examining & proportioning of the matter: and with _Accursius_\nGlosse, much cumbred: burst out, and sayd: _Nulla est in toto libro, hac\nglossa difficilior: Cuius computationem nec Scholastici nec Doctores\nintelligunt. &c._ That is: +_In the whole booke, there is no Glosse\nharder then this: Whose accoumpt or reckenyng, neither the Scholers, nor\nthe Doctours vnderstand. &c._+ What can they say of _Iulianus_ law, _Si\nita Scriptum. &c._ Of the Testators will iustly performing, betwene the\nwife, Sonne and daughter? How can they perceiue the \u00e6quitie of\n_Aphricanus_, _Arithmeticall_ Reckening, where he treateth of _Lex\nFalcidia_? How can they deliuer him, from his Reprouers: and their\nmaintainers: as _Ioannes_, _Accursius Hypolitus_ and _Alciatus_? How\nIustly and artificially, was _Africanus_ reckening made? Proportionating\nto the Sommes bequeathed, the Contributions of eche part? Namely, for\nthe hundred presently receiued, 17-1/7. And for the hundred, receiued\nafter ten monethes, 12-6/7: which make the 30: which were to be\nc\u00f5tributed by the legataries to the heire. For, what proportion, 100\nhath to 75: the same hath 17-1/7 to 12-6/7: Which is Sesquitertia: that\nis, as 4, to 3. which make 7. Wonderfull many places, in the Ciuile law,\nrequire an expert _Arithmeticien_, to vnderstand the deepe Iudgem\u1ebdt, &\nIust determinati\u00f5 of the Auncient Romaine Lawmakers. But much more\nexpert ought he to be, who should be hable, to decide with \u00e6quitie, the\ninfinite varietie of Cases, which do, or may happen, vnder euery one of\nthose lawes and ordinances Ciuile. Hereby, easely, ye may now\nconiecture: that in the Canon law: and in the lawes of the Realme (which\nwith vs, beare the chief Authoritie), Iustice and equity might be\ngreately preferred, and skilfully executed, through due skill of\nArithmetike, and proportions appertainyng. The worthy Philosophers, and\nprudent lawmakers (who haue written many bookes _De Republica:_ How the\nbest state of Common wealthes might be procured and mainteined,) haue\nvery well determined of Iustice: (which, not onely, is the Base and\nfoundacion of Common weales: but also the totall perfection of all our\nworkes, words, and thoughtes:) defining it,\n\n    [Iustice.]\n\n\u201cto be that vertue, by which, to euery one, is rendred, that to him\nappertaineth.\u201d God challengeth this at our handes, to be honored as God:\nto be loued, as a father: to be feared as a Lord & master. Our\nneighbours proporti\u00f5, is also prescribed of the Almighty lawmaker: which\nis, to do to other, euen as we would be done vnto. These proportions,\nare in Iustice necessary: in duety, commendable: and of Common wealthes,\nthe life, strength, stay and florishing. _Aristotle_ in his _Ethikes_\n(to fatch the sede of Iustice, and light of direction, to vse and\nexecute the same) was fayne to fly to the perfection, and power of\nNumbers: for proportions Arithmeticall and Geometricall. _Plato_ in his\nbooke called _Epinomis_ (which boke, is the Threasury of all his\ndoctrine) where, his purpose is, to seke a Science, which, when a man\nhad it, perfectly: he might seme, and so be, in dede, _Wise_. He,\nbriefly, of other Sciences discoursing, findeth them, not hable to bring\nit to passe: But of the Science of Numbers, he sayth. _Illa, qu\u00e6 numerum\nmortalium generi dedit, id profecto efficiet. Deum autem aliquem, magis\nquam fortunam, ad salutem nostram, hoc munus nobis arbitror contulisse.\n&c. Nam ipsum bonorum omnium Authorem, cur non maximi boni, Prudenti\u00e6\ndico, causam arbitramur? +That Science, verely, which hath taught\nmankynde number, shall be able to bryng it to passe. And, I thinke,\na certaine God, rather then fortune, to haue giuen vs this gift, for our\nblisse. For, why should we not Iudge him, who is the Author of all good\nthings, to be also the cause of the greatest good thyng, namely,\nWisedome?+_ There, at length, he proueth _Wisedome_ to be atteyned, by\ngood Skill of _Numbers_. With which great Testimony, and the manifold\nprofes, and reasons, before expressed, you may be sufficiently and fully\npersuaded: of the perfect Science of _Arithmetike_, to make this\naccounte: That\n\n    [\u261e]\n\nof all Sciences, next to _Theologie_, it is most diuine, most pure, most\nample and generall, most profounde, most subtile, most commodious and\nmost necessary. Whose next Sister, is the Absolute Science of\n_Magnitudes_: of which (by the Direction and aide of him, whose\n_Magnitude_ is Infinite, and of vs Incomprehensible) I now entend, so to\nwrite, that both with the _Multitude_, and also with the _Magnitude_ of\nMeruaylous and frutefull verities, you (my frendes and Countreymen) may\nbe stird vp, and awaked, to behold what certaine Artes and Sciences, (to\nour vnspeakable behofe) our heauenly father, hath for vs prepared, and\nreuealed, by sundry _Philosophers_ and _Mathematiciens_.\n\n\nBoth, _Number_ and _Magnitude_, haue a certaine Originall sede, (as it\nwere) of an incredible property: and of man, neuer hable, Fully, to be\ndeclared. Of _Number_, an Vnit, and of _Magnitude_, a Poynte, doo seeme\nto be much like Originall causes: But the diuersitie neuerthelesse, is\ngreat. We defined an _Vnit_, to be a thing Mathematicall Indiuisible:\nA Point, likewise, we sayd to be a Mathematicall thing Indiuisible. And\nfarder, that a Point may haue a certaine determined Situation: that is,\nthat we may assigne, and prescribe a Point, to be here, there, yonder.\n&c. Herein, (behold) our Vnit is free, and can abyde no bondage, or to\nbe tyed to any place, or seat: diuisible or indiuisible. Agayne, by\nreason, a Point may haue a Situation limited to him: a certaine motion,\ntherfore (to a place, and from a place) is to a Point incident and\nappertainyng. But an _Vnit_, can not be imagined to haue any motion.\nA Point, by his motion, produceth, Mathematically, a line: (as we sayd\nbefore) which is the first kinde of Magnitudes, and most simple: An\n_Vnit_, can not produce any number. A Line, though it be produced of a\nPoint moued, yet, it doth not consist of pointes: Number, though it be\nnot produced of an _Vnit_, yet doth it Consist of vnits, as a materiall\ncause. But formally,\n\n    [Number.]\n\nNumber, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting,\nis the workemanship of our minde: which, of distinct and discrete Vnits,\nmaketh a Number: by vniformitie, resulting of a certaine multitude of\nVnits. And so, euery number, may haue his least part, giuen: namely, an\nVnit: But not of a Magnitude, (no, not of a Lyne,) the least part can be\ngiu\u1ebd: by cause, infinitly, diuision therof, may be conceiued. All\nMagnitude, is either a Line, a Plaine, or a Solid. Which Line, Plaine,\nor Solid, of no Sense, can be perceiued, nor exactly by h\u00e3d (any way)\nrepresented: nor of Nature produced: But, as (by degrees) Number did\ncome to our perceiuerance: So, by visible formes, we are holpen to\nimagine, what our Line Mathematicall, is. What our Point, is. So\nprecise, are our Magnitudes, that one Line is no broader then an other:\nfor they haue no bredth: Nor our Plaines haue any thicknes. Nor yet our\nBodies, any weight: be they neuer so large of dimensi\u00f5. Our Bodyes, we\ncan haue Smaller, then either Arte or Nature can produce any: and\nGreater also, then all the world can comprehend. Our least Magnitudes,\ncan be diuided into so many partes, as the greatest. As, a Line of an\ninch long, (with vs) may be diuided into as many partes, as may the\ndiameter of the whole world, from East to West: or any way extended:\nWhat priuiledges, aboue all manual Arte, and Natures might, haue our two\nSciences Mathematicall? to exhibite, and to deale with thinges of such\npower, liberty, simplicity, puritie, and perfection? And in them, so\ncertainly, so orderly, so precisely to procede: as, excellent is that\nworkem\u00e3 Mechanicall Iudged, who nerest can approche to the representing\nof workes, Mathematically demonstrated?\n\n    [\u261e]\n\nAnd our two Sciences, remaining pure, and absolute, in their proper\ntermes, and in their owne Matter: to haue, and allowe, onely such\nDemonstrations, as are plaine, certaine, vniuersall, and of an \u00e6ternall\nveritye?\n\n    [Geometrie.]\n\nThis Science of _Magnitude_, his properties, conditions, and\nappertenances: commonly, now is, and from the beginnyng, hath of all\nPhilosophers, ben called _Geometrie_. But, veryly, with a name to base\nand scant, for a Science of such dignitie and amplenes. And, perchaunce,\nthat name, by c\u00f5mon and secret consent, of all wisemen, hitherto hath\nben suffred to remayne: that it might carry with it a perpetuall\nmemorye, of the first and notablest benefite, by that Science, to common\npeople shewed: Which was, when Boundes and meres of land and ground were\nlost, and confounded (as in _Egypt_, yearely, with the ouerflowyng of\n_Nilus_, the greatest and longest riuer in the world) or, that ground\nbequeathed, were to be assigned: or, ground sold, were to be layd out:\nor (when disorder preuailed) that Comm\u00f5s were distributed into\nseueralties. For, where, vpon these & such like occasi\u00f5s, Some by\nignor\u00e3ce, some by neglig\u1ebdce, Some by fraude, and some by violence, did\nwrongfully limite, measure, encroach, or challenge (by pretence of iust\ncontent, and measure) those landes and groundes: great losse,\ndisquietnes, murder, and warre did (full oft) ensue: Till, by Gods\nmercy, and mans Industrie, The perfect Science of Lines, Plaines, and\nSolides (like a diuine Iusticier,) gaue vnto euery man, his owne. The\npeople then, by this art pleasured, and greatly relieued, in their\nlandes iust measuring: & other Philosophers, writing Rules for land\nmeasuring: betwene them both, thus, confirmed the name of _Geometria_,\nthat is, (according to the very etimologie of the word) Land measuring.\nWherin, the people knew no farder, of Magnitudes vse, but in Plaines:\nand the Philosophers, of th\u1ebd, had no feet hearers, or Scholers: farder\nto disclose vnto, then of flat, plaine _Geometrie_. And though, these\nPhilosophers, knew of farder vse, and best vnderstode the etymologye of\nthe worde, yet this name _Geometria_, was of them applyed generally to\nall sortes of Magnitudes: vnleast, otherwhile, of _Plato_, and\n_Pythagoras_: When they would precisely declare their owne doctrine.\nThen, was\n\n    [* Plato. 7. de Rep.]\n\n* _Geometria_, with them, _Studium quod circa planum versatur_. But,\nwell you may perceiue by _Euclides Elementes_, that more ample is our\nScience, then to measure Plaines: and nothyng lesse therin is tought (of\npurpose) then how to measure Land. An other name, therfore, must nedes\nbe had, for our Mathematicall Science of Magnitudes: which regardeth\nneither clod, nor turff: neither hill, nor dale: neither earth nor\nheauen: but is absolute _Megethologia_: not creping on ground, and\ndasseling the eye, with pole perche, rod or lyne: but \u201cliftyng the hart\naboue the heauens, by inuisible lines, and\n\n    [\u261e]\n\nimmortall beames meteth with the reflexions, of the light\nincomprehensible: and so procureth Ioye, and perfection vnspeakable.\u201d Of\nwhich true vse of our _Megethica_, or _Megethologia_, _Diuine Plato_\nseemed to haue good taste, and iudgement: and (by the name of\n_Geometrie_) so noted it: and warned his Scholers therof: as, in hys\nseuenth _Dialog_, of the Common wealth, may euidently be sene. Where (in\nLatin) thus it is: right well translated: _Profecto, nobis hoc non\nnegabunt, Quicun[que] vel paululum quid Geometri\u00e6 gust\u00e2runt, quin h\u00e6c\nScientia, contr\u00e0, omnino se habeat, qu\u00e0m de ea loquuntur, qui in ipsa\nversantur._ In English, thus. +_Verely_+ (sayth _Plato_) +_whosoeuer\nhaue, (but euen very litle) tasted of Geometrie, will not denye vnto vs,\nthis: but that this Science, is of an other condicion, quite contrary to\nthat, which they that are exercised in it, do speake of it._+ And there\nit followeth, of our _Geometrie_, _Qu\u00f2d qu\u00e6ritur cognoscendi illius\ngratia, quod semper est, non & eius quod oritur quando[que] & interit.\nGeometria, eius quod est semper, Cognitio est. Attollet igitur\n(\u00f4 Generose vir) ad Veritatem, animum: at[que] ita, ad Philosophandum\npreparabit cogitationem, vt ad supera conuertamus: qu\u00e6, nunc, contra\nqu\u00e0m decet, ad inferiora deijcimus. &c. Qu\u00e0m maxim\u00e8 igitur pr\u00e6cipiendum\nest, vt qui pr\u00e6clarissimam hanc habit\u00e3t Civitatem, nullo modo,\nGeometriam spernant. Nam & qu\u00e6 pr\u00e6ter ipsius propositum, quodam modo\nesse videntur, haud exigua sunt. &c._ It must nedes be confessed (saith\n_Plato_) +_That =[Geometrie]= is learned, for the knowyng of that, which\nis euer: and not of that, which, in tyme, both is bred and is brought to\nan ende. &c. Geometrie is the knowledge of that which is euerlastyng. It\nwill lift vp therfore (O Gentle Syr) our mynde to the Veritie: and by\nthat meanes, it will prepare the Thought, to the Philosophicall loue of\nwisdome: that we may turne or conuert, toward heauenly thinges =[both\nmynde and thought]= which now, otherwise then becommeth vs, we cast down\non base or inferior things. &c. Chiefly, therfore, Commaundement must be\ngiuen, that such as do inhabit this most honorable Citie, by no meanes,\ndespise Geometrie. For euen those thinges =[done by it]= which, in\nmanner, seame to be, beside the purpose of Geometrie: are of no small\nimportance. &c._+ And besides the manifold vses of _Geometrie_, in\nmatters appertainyng to warre, he addeth more, of second vnpurposed\nfrute, and commoditye, arrising by _Geometrie_: saying: _Scimus quin\netiam, ad Disciplinas omnes facilius per discendas, interesse omnino,\nattigerit ne Geometriam aliquis, an non. &c. Hanc ergo Doctrinam,\nsecundo loco discendam Iuuenibus statuamus._ That is. +_But, also, we\nknow, that for the more easy learnyng of all Artes, it importeth much,\nwhether one haue any knowledge in Geometrie, or no. &c. Let vs therfore\nmake an ordinance or decree, that this Science, of young men shall be\nlearned in the second place._+ This was _Diuine Plato_ his Iudgement,\nboth of the purposed, chief, and perfect vse of _Geometrie_: and of his\nsecond, dependyng, deriuatiue commodities. And for vs, Christen men,\na thousand thousand mo occasions are, to haue nede of the helpe of *\n\n    [I. D.\n    * Herein, I would gladly shake of, the earthly name,\n    of Geometrie.]\n\n_Megethologicall_ Contemplations: wherby, to trayne our Imaginations and\nMyndes, by litle and litle, to forsake and abandon, the grosse and\ncorruptible Obiectes, of our vtward senses: and to apprehend, by sure\ndoctrine demonstratiue, Things Mathematicall. And by them, readily to be\nholpen and conducted to conceiue, discourse, and conclude of things\nIntellectual, Spirituall, \u00e6ternall, and such as concerne our Blisse\neuerlasting: which, otherwise (without Speciall priuiledge of\nIllumination, or Reuelation fr\u00f5 heauen) No mortall mans wyt (naturally)\nis hable to reach vnto, or to Compasse. And, veryly, by my small Talent\n(from aboue) I am hable to proue and testifie, that the litterall Text,\nand order of our diuine Law, Oracles, and Mysteries, require more skill\nin Numbers, and Magnitudes: then (commonly) the expositors haue vttered:\nbut rather onely (at the most) so warned: & shewed their own want\ntherin. (To name any, is nedeles: and to note the places, is, here, no\nplace: But if I be duely asked, my answere is ready.) And without the\nlitterall, Grammaticall, Mathematicall or Naturall verities of such\nplaces, by good and certaine Arte, perceiued, no Spirituall sense\n(propre to those places, by Absolute _Theologie_) will thereon depend.\n\n    [\u261e]\n\n\u201cNo man, therfore, can doute, but toward the atteyning of knowledge\nincomparable, and Heauenly Wisedome: Mathematicall Speculations, both of\nNumbers and Magnitudes: are meanes, aydes, and guides: ready, certaine,\nand necessary.\u201d From henceforth, in this my Preface, will I frame my\ntalke, to _Plato_ his fugitiue Scholers: or, rather, to such, who well\ncan, (and also wil,) vse their vtward senses, to the glory of God, the\nbenefite of their Countrey, and their owne secret contentation, or\nhonest preferment, on this earthly Scaffold. To them, I will orderly\nrecite, describe & declare a great Number of Artes, from our two\nMathematicall fountaines, deriued into the fieldes of _Nature_. Wherby,\nsuch Sedes, and Rotes, as lye depe hyd in the gro\u0169d of _Nature_, are\nrefreshed, quickened, and prouoked to grow, shote vp, floure, and giue\nfrute, infinite, and incredible. And these Artes, shalbe such, as vpon\nMagnitudes properties do depende, more, then vpon Number. And by good\nreason we may call them Artes, and Artes Mathematicall Deriuatiue: for\n(at this tyme) I Define\n\n    [An Arte.]\n\n+An Arte, to be a Methodicall c\u00f5plete Doctrine, hauing abundancy of\nsufficient, and peculier matter to deale with, by the allowance of the\nMetaphisicall Philosopher: the knowledge whereof, to humaine state is\nnecessarye.+ And that I account,\n\n    [Art Mathematicall Deriuatiue.]\n\n+An Art Mathematicall deriuatiue, which by Mathematicall demonstratiue\nMethod, in N\u0169bers, or Magnitudes, ordreth and confirmeth his doctrine,\nas much & as perfectly, as the matter subiect will admit.+ And for that,\nI entend to vse the name and propertie of a\n\n    [A Mechanitien.]\n\n_Mechanicien_, otherwise, then (hitherto) it hath ben vsed, I thinke it\ngood, (for distinction sake) to giue you also a brief description, what\nI meane therby. +A Mechanicien, or a Mechanicall workman is he, whose\nskill is, without knowledge of Mathematicall demonstration, perfectly to\nworke and finishe any sensible worke, by the Mathematicien principall or\nderiuatiue, demonstrated or demonstrable.+ Full well I know, that he\nwhich inuenteth, or maketh these demonstrations, is generally called _A\nspeculatiue Mechanicien_: which differreth nothyng from a _Mechanicall\nMathematicien_. So, in respect of diuerse actions, one man may haue the\nname of sundry artes: as, some tyme, of a Logicien, some tymes (in the\nsame matter otherwise handled) of a Rethoricien. Of these trifles,\nI make, (as now, in respect of my Preface,) small account: to fyle th\u1ebd\nfor the fine handlyng of subtile curious disputers. In other places,\nthey may commaunde me, to giue good reason: and yet, here, I will not be\nvnreasonable.\n\n\n    [+1.+]\n\nFirst, then, from the puritie, absolutenes, and Immaterialitie of\nPrincipall _Geometrie_, is that kinde of _Geometrie_ deriued, which\nvulgarly is counted _Geometrie_: and is the +Arte of Measuring sensible\nmagnitudes, their iust qu\u00e3tities and contentes.+\n\n    [Geometrie vulgar.]\n\nThis, teacheth to measure, either at hand: and the practiser, to be by\nthe thing Measured: and so, by due applying of Cumpase, Rule, Squire,\nYarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) to\nthe Length, Plaine, or Solide measured,\n\n    [1.]\n\n* to be certified, either of the length, perimetry, or distance lineall:\nand this is called, _Mecometrie_. Or\n\n    [2.]\n\n* to be certified of the content of any plaine Superficies: whether it\nbe in ground Surueyed, Borde, or Glasse measured, or such like thing:\nwhich measuring, is named _Embadometrie_.\n\n    [3.]\n\n* Or els to vnderstand the Soliditie, and content of any bodily thing:\nas of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels,\nsmall & great, of all fashions. Where, of Wine, Oyle, Beere, or Ale\nvessells, &c, the Measuring, commonly, hath a peculier name: and is\ncalled _Gaging_. And the generall name of these Solide measures, is\n_Stereometrie_.\n\n    [+2.+]\n\nOr els, this _vulgar Geometrie_, hath consideration to teach the\npractiser, how to measure things, with good distance betwene him and the\nthing measured: and to vnderstand thereby, either\n\n    [1.]\n\n* how Farre, a thing seene (on land or water) is from the measurer: and\nthis may be called _Apomecometrie_:\n\n    [2.]\n\nOr, how High or depe, aboue or vnder the leuel of the measurers st\u00e3ding,\nany thing is, which is sene on land or water, called _Hypsometrie_.\n\n    [3.]\n\n* Or, it informeth the measurer, how Broad any thing is, which is in the\nmeasurers vew: so it be on Land or Water, situated: and may be called\n_Platometrie_. Though I vse here to condition, the thing measured, to be\non Land, or Water Situated:\n\n    [Note.]\n\nyet, know for certaine, that the sundry heigthe of Cloudes, blasing\nStarres, and of the Mone, may (by these meanes) haue their distances\nfrom the earth: and, of the blasing Starres and Mone, the Soliditie\n(aswell as distances) to be measured: But because, neither these things\nare vulgarly taught: nor of a common practiser so ready to be executed:\nI, rather, let such measures be reckened incident to some of our other\nArtes, dealing with thinges on high, more purposely, then this vulgar\nLand measuring Geometrie doth: as in _Perspectiue_ and _Astronomie, &c._\n\n\nOf these Feates (farther applied) is Sprong the Feate of _Geodesie_, or\nLand Measuring: more cunningly to measure & Suruey Land, Woods, and\nWaters, a farre of. More cunningly, I say: But God knoweth (hitherto) in\nthese Realmes of England and Ireland (whether through ignorance or\nfraude, I can not tell, in euery particular)\n\n    [Note.]\n\nhow great wrong and iniurie hath (in my time) bene committed by vntrue\nmeasuring and surueying of Land or Woods, any way. And, this I am sure:\nthat the Value of the difference, betwene the truth and such Surueyes,\nwould haue bene hable to haue fo\u0169d (for euer) in eche of our two\nVniuersities, an excellent Mathematicall Reader: to eche, allowing\n(yearly) a hundred Markes of lawfull money of this realme: which, in\ndede, would seme requisit, here, to be had (though by other wayes\nprouided for) as well, as, the famous Vniuersitie of Paris, hath two\nMathematicall Readers: and eche, two hundreth French Crownes yearly, of\nthe French Kinges magnificent liberalitie onely. Now, againe, to our\npurpose returning: Moreouer, of the former knowledge Geometricall, are\ngrowen the Skills of _Geographie_, _Chorographie_, _Hydrographie_, and\n_Stratarithmetrie_.\n\n\n\u201c+\u2021Geographie\u2021+ teacheth wayes, by which, in s\u0169dry formes, (as\n_Sph\u00e6rike_, _Plaine_ or other), the Situation of Cities, Townes,\nVillages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, &\nsuch other things, vp\u00f5 the outface of the earthly Globe (either in the\nwhole, or in some principall m\u1ebdber and portion therof c\u00f5tayned) may be\ndescribed and designed, in c\u00f5mensurations Analogicall to Nature and\nveritie: and most aptly to our vew, may be represented.\u201d Of this Arte\nhow great pleasure, and how manifolde commodities do come vnto vs, daily\nand hourely: of most men, is perceaued. While, some, to beautifie their\nHalls, Parlers, Chambers, Galeries, Studies, or Libraries with: other\nsome, for thinges past, as battels fought, earthquakes, heauenly\nfyringes, & such occurentes, in histories mentioned: therby liuely, as\nit were, to vewe the place, the region adioyning, the distance from vs:\nand such other circumstances. Some other, presently to vewe the large\ndominion of the Turke: the wide Empire of the Moschouite: and the litle\nmorsell of ground, where Christendome (by profession) is certainly\nknowen. Litle, I say, in respecte of the rest. &c. Some, either for\ntheir owne iorneyes directing into farre landes: or to vnderstand of\nother mens trauailes. To conclude, some, for one purpose: and some, for\nan other, liketh, loueth, getteth, and vseth, Mappes, Chartes, &\nGeographicall Globes. Of whose vse, to speake sufficiently, would\nrequire a booke peculier.\n\n\n+\u2021Chorographie\u2021+ seemeth to be an vnderling, and a twig, of\n_Geographie_: and yet neuerthelesse, is in practise manifolde, and in\nvse very ample. \u201cThis teacheth Analogically to describe a small portion\nor circuite of ground, with the contentes: not regarding what\ncommensuration it hath to the whole, or any parcell, without it,\ncontained. But in the territory or parcell of ground which it taketh in\nhand to make description of, it leaueth out (or vndescribed) no notable,\nor odde thing, aboue the ground visible. Yea and sometimes, of thinges\nvnder ground, geueth some peculier marke: or warning: as of Mettall\nmines, Cole pittes, Stone quarries. &c.\u201d Thus, a Dukedome, a Shiere,\na Lordship, or lesse, may be described distinctly. But marueilous\npleasant, and profitable it is, in the exhibiting to our eye, and\ncommensuration, the plat of a Citie, Towne, Forte, or Pallace, in true\nSymmetry: not approching to any of them: and out of Gunne shot. &c.\nHereby, the _Architect_ may furnishe him selfe, with store of what\npatterns he liketh: to his great instruction: euen in those thinges\nwhich outwardly are proportioned: either simply in them selues: or\nrespectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also,\nterme this particular description of places, _Topographie_.\n\n\n\u201c+\u2021Hydrographie\u2021+, deliuereth to our knowledge, on Globe or in Plaine,\nthe perfect Analogicall description of the Ocean Sea coastes, through\nthe whole world: or in the chiefe and principall partes thereof:\u201d with\nthe Iles and chiefe particular places of daungers, conteyned within the\nboundes, and Sea coastes described: as, of Quicksandes, Bankes, Pittes,\nRockes, Races, Countertides, Whorlepooles. &c. This, dealeth with the\nElement of the water chiefly: as _Geographie_ did principally take the\nElement of the Earthes description (with his appertenances) to taske.\nAnd besides thys, _Hydrographie_, requireth a particular Register of\ncertaine Landmarkes (where markes may be had) from the sea, well hable\nto be skried, in what point of the Seacumpase they appeare, and what\napparent forme, Situation, and bignes they haue, in respecte of any\ndaungerous place in the sea, or nere vnto it, assigned: And in all\nCoastes, what Mone, maketh full Sea: and what way, the Tides and Ebbes,\ncome and go, the _Hydrographer_ ought to recorde. The Soundinges\nlikewise: and the Chanels wayes: their number, and depthes ordinarily,\nat ebbe and flud, ought the _Hydrographer_, by obseruation and diligence\nof _Measuring_, to haue certainly knowen. And many other pointes, are\nbelonging to perfecte _Hydrographie_, and for to make a _Rutter_, by: of\nwhich, I nede not here speake: as of the describing, in any place, vpon\nGlobe or Plaine, the 32. pointes of the Compase, truely: (wherof,\nscarsly foure, in England, haue right knowledge: bycause, the lines\ntherof, are no straight lines, nor Circles.) Of making due proiection of\na Sphere in plaine. Of the Variacion of the Compas, from true Northe:\nAnd such like matters (of great importance, all) I leaue to speake of,\nin this place: bycause, I may seame (al ready) to haue enlarged the\nboundes, and duety of an Hydographer, much more, then any man (to this\nday) hath noted, or prescribed. Yet am I well hable to proue, all these\nthinges, to appertaine, and also to be proper to the Hydrographer. The\nchief vse and ende of this Art, is the Art of Nauigation: but it hath\nother diuerse vses: euen by them to be enioyed, that neuer lacke sight\nof land.\n\n\n+\u2021Stratarithmetrie\u2021+, is the Skill, (appertainyng to the warre,) by\nwhich a man can set in figure, analogicall to any _Geometricall_ figure\nappointed, any certaine number or summe of men: of such a figure\ncapable: (by reason of the vsuall spaces betwene Souldiers allowed: and\nfor that, of men, can be made no Fractions. Yet, neuertheles, he can\norder the giuen summe of men, for the greatest such figure, that of\nthem, c\u00e3 be ordred) and certifie, of the ouerplus: (if any be) and of\nthe next certaine summe, which, with the ouerplus, will admit a figure\nexactly proportionall to the figure assigned. By which Skill, also, of\nany army or company of men: (the figure & sides of whose orderly\nstanding, or array, is knowen) he is able to expresse the iust number of\nmen, within that figure conteined: or (orderly) able to be conteined.\n\n    [* Note.]\n\n* And this figure, and sides therof, he is hable to know: either beyng\nby, and at hand: or a farre of. Thus farre, stretcheth the description\nand property of _Stratarithmetrie_: sufficient for this tyme and place.\n\n    [The difference betwene Stratarithmetrie and Tacticie.]\n\n\u201cIt differreth from the Feate _Tacticall_, _De aciebus instruendis._\nbycause, there, is necessary the wisedome and foresight, to what purpose\nhe so ordreth the men: and Skillfull hability, also, for any occasion,\nor purpose, to deuise and vse the aptest and most necessary order, array\nand figure of his Company and Summe of men.\u201d By figure, I meane: as,\neither of a _Perfect Square_, _Triangle_, _Circle_, _Ouale_, _long\nsquare_, (of the Grekes it is called _Eteromekes_) _Rhombe_, _Rhombo\u00efd_,\n_Lunular_, _Ryng_, _Serpentine_, and such other Geometricall figures:\nWhich, in warres, haue ben, and are to be vsed: for commodiousnes,\nnecessity, and auauntage &c. And no small skill ought he to haue, that\nshould make true report, or nere the truth, of the numbers and Summes,\nof footemen or horsemen, in the Enemyes ordring. A farre of, to make an\nestimate, betwene nere termes of More and Lesse, is not a thyng very\nrife, among those that gladly would do it.\n\n    [I. D.\n    Frende, you will finde it hard, to performe my description\n    of this Feate. But by Chorographie, you may helpe your selfe\n    some what: where the Figures knowne (in Sides and Angles)\n    are not Regular: And where, Resolution into Triangles can\n    serue. &c. And yet you will finde it strange to deale thus\n    generally with Arithmeticall figures: and, that for Battayle\n    ray. Their contentes, differ so much from like Geometricall\n    Figures.]\n\nGreat pollicy may be vsed of the Capitaines, (at tymes fete, and in\nplaces conuenient) as to vse Figures, which make greatest shew, of so\nmany as he hath: and vsing the aduauntage of the three kindes of vsuall\nspaces: (betwene footemen or horsemen) to take the largest: or when he\nwould seme to haue few, (beyng many:) contrarywise, in Figure, and\nspace. The Herald, Purseuant, Sergeant Royall, Capitaine, or who soeuer\nis carefull to come nere the truth herein, besides the Iudgement of his\nexpert eye, his skill of Ordering _Tacticall_, the helpe of his\nGeometricall instrument: Ring, or Staffe Astronomicall: (commodiously\nframed for cariage and vse) He may wonderfully helpe him selfe, by\nperspectiue Glasses. In which, (I trust) our posterity will proue more\nskillfull and expert, and to greater purposes, then in these dayes, can\n(almost) be credited to be possible.\n\n\nThus haue I lightly passed ouer the Artificiall Feates, chiefly\ndependyng vpon vulgar _Geometrie_: & commonly and generally reckened\nvnder the name of _Geometrie_. But there are other (very many)\n_Methodicall Artes_, which, declyning from the purity, simplicitie, and\nImmateriality, of our Principall Science of _Magnitudes_: do yet\nneuertheles vse the great ayde, direction, and Method of the sayd\nprincipall Science, and haue propre names, and distinct: both from the\nScience of _Geometrie_, (from which they are deriued) and one from the\nother. As +Perspectiue, Astronomie, Musike, Cosmographie, Astrologie,\nStatike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie,\nMenadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture,\nNauigation, Thaumaturgike+ and +Archemastrie+. I thinke it necessary,\norderly, of these to giue some peculier descriptions: and withall, to\ntouch some of their commodious vses, and so to make this Preface, to be\na little swete, pleasant Nosegaye for you: to comfort your Spirites,\nbeyng almost out of courage, and in despayre, (through brutish brute)\nWeenyng that _Geometrie_, had but serued for buildyng of an house, or a\ncurious bridge, or the roufe of Westminster hall, or some witty pretty\ndeuise, or engyn, appropriate to a Carpenter, or a Ioyner &c. That the\nthing is farre otherwise, then the world, (commonly) to this day, hath\ndemed, by worde and worke, good profe wilbe made.\n\n\nAmong these Artes, by good reason, +\u2021Perspectiue\u2021+ ought to be had, ere\nof _Astronomicall Apparences_, perfect knowledge can be atteyned. And\nbycause of the prerogatiue of _Light_, beyng the first of _Gods\nCreatures_: and the eye, the light of our body, and his Sense most\nmighty, and his organ most Artificiall and _Geometricall_: At\n_Perspectiue_, we will begyn therfore. +Perspectiue, is an Art\nMathematicall, which demonstrateth the maner, and properties, of all\nRadiations Direct, Broken, and Reflected.+ This Description, or\nNotation, is brief: but it reacheth so farre, as the world is wyde. It\nconcerneth all Creatures, all Actions, and passions, by Emanation of\nbeames perfourmed. Beames, or naturall lines, (here) I meane, not of\nlight onely, or of colour (though they, to eye, giue shew, witnes, and\nprofe, wherby to ground the Arte vpon) but also of other _Formes_, both\n_Substantiall_, and _Accidentall_, the certaine and determined actiue\nRadiall emanations. By this Art (omitting to speake of the highest\npointes) we may vse our eyes, and the light, with greater pleasure: and\nperfecter Iudgement: both of things, in light seen, & of other: which by\nlike order of Lightes Radiations, worke and produce their effectes. We\nmay be ashamed to be ignorant of the cause, why so sundry wayes our eye\nis deceiued, and abused: as, while the eye weeneth a ro\u0169d Globe or\nSphere (beyng farre of) to be a flat and plaine Circle, and so likewise\niudgeth a plaine Square, to be ro\u0169d: supposeth walles parallels, to\napproche, a farre of: rofe and floure parallels, the one to bend\ndownward, the other to rise vpward, at a little distance from you.\nAgaine, of thinges being in like swiftnes of mouing, to thinke the\nnerer, to moue faster: and the farder, much slower. Nay, of two thinges,\nwherof the one (incomparably) doth moue swifter then the other, to deme\nthe slower to moue very swift, & the other to stand: what an error is\nthis, of our eye? Of the Raynbow, both of his Colours, of the order of\nthe colours, of the bignes of it, the place and heith of it, (&c) to\nknow the causes demonstratiue, is it not pleasant, is it not necessary?\nof two or three Sonnes appearing: of Blasing Sterres: and such like\nthinges: by naturall causes, brought to passe, (and yet neuertheles, of\nfarder matter, Significatiue) is it not commodious for man to know the\nvery true cause, & occasion Naturall? Yea, rather, is it not, greatly,\nagainst the Souerainty of Mans nature, to be so ouershot and abused,\nwith thinges (at hand) before his eyes? as with a Pecockes tayle, and a\nDoues necke: or a whole ore, in water, holden, to seme broken. Thynges,\nfarre of, to seeme nere: and nere, to seme farre of. Small thinges, to\nseme great: and great, to seme small. One man, to seme an Army. Or a man\nto be curstly affrayed of his owne shaddow. Yea, so much, to feare,\nthat, if you, being (alone) nere a certaine glasse, and proffer, with\ndagger or sword, to foyne at the glasse, you shall suddenly be moued to\ngiue backe (in maner) by reason of an Image,\n\n    [\u261e A marueilous Glasse.]\n\nappearing in the ayre, betwene you & the glasse, with like hand, sword\nor dagger, & with like quicknes, foyning at your very eye, likewise as\nyou do at the Glasse. Straunge, this is, to heare of: but more\nmeruailous to behold, then these my wordes can signifie. And\nneuerthelesse by demonstration Opticall, the order and cause therof, is\ncertified: euen so, as the effect is consequent. Yea, thus much more,\ndare I take vpon me, toward the satisfying of the noble courrage, that\nlongeth ardently for the wisedome of Causes Naturall: as to let him\nvnderstand, that, in London, he may with his owne eyes, haue profe of\nthat, which I haue sayd herein. A Gentleman, (which, for his good\nseruice, done to his Countrey, is famous and honorable:\n\n    [S. W. P.]\n\nand for skill in the Mathematicall Sciences, and Languages, is the Od\nman of this land. &c.) euen he, is hable: and (I am sure) will, very\nwillingly, let the Glasse, and profe be sene: and so I (here) request\nhim: for the encrease of wisedome, in the honorable: and for the\nstopping of the mouthes malicious: and repressing the arrogancy of the\nignorant. Ye may easily gesse, what I meane. This Art of _Perspectiue_,\nis of that excellency, and may be led, to the certifying, and executing\nof such thinges, as no man would easily beleue: without Actuall profe\nperceiued. I speake nothing of _Naturall Philosophie_, which, without\n_Perspectiue_, can not be fully vnderstanded, nor perfectly atteined\nvnto. Nor, of _Astronomie_: which, without _Perspectiue_, can not well\nbe grounded: Nor _Astrologie_, naturally Verified, and auouched. That\npart hereof, which dealeth with Glasses (which name, Glasse, is a\ngenerall name, in this Arte, for any thing, from which, a Beame\nreboundeth) is called _Catoptrike_: and hath so many vses, both\nmerueilous, and proffitable: that, both, it would hold me to long, to\nnote therin the principall conclusions, all ready knowne: And also\n(perchaunce) some thinges, might lacke due credite with you: And I,\ntherby, to leese my labor: and you, to slip into light Iudgement,\n\n    [* \u261e]\n\nBefore you haue learned sufficiently the powre of Nature and Arte.\n\n\nNow, to procede: +\u2021Astronomie\u2021, is an Arte Mathematicall, which\ndemonstrateth the distance, magnitudes, and all naturall motions,\napparences, and passions propre to the Planets and fixed Sterres: for\nany time past, present and to come: in respect of a certaine Horizon, or\nwithout respect of any Horizon.+ By this Arte we are certified of the\ndistance of the Starry Skye, and of eche _Planete_ from the Centre of\nthe Earth: and of the greatnes of any Fixed starre sene, or _Planete_,\nin respect of the Earthes greatnes. As, we are sure (by this Arte) that\nthe Solidity, Massines and Body of the _Sonne_, conteineth the quantitie\nof the whole Earth and Sea, a hundred thre score and two times, lesse by\n\u215b one eight parte of the earth. But the Body of the whole earthly globe\nand Sea, is bigger then the body of the Mone, three and forty times\nlesse by \u215b of the Mone. Wherfore the _Sonne_ is bigger then the _Mone_,\n7000 times, lesse, by 59 39/64 that is, precisely 6940 25/64 bigger then\nthe _Mone_. And yet the vnskillfull man, would iudge them a like bigge.\nWherfore, of Necessity, the one is much farder from vs, then the other.\nThe _Sonne_, when he is fardest from the earth (which, now, in our age,\nis, when he is in the 8. degree, of Cancer) is, 1179 Semidiameters of\nthe Earth, distante. And the _Mone_ when she is fardest from the earth,\nis 68 Semidiameters of the earth and \u2153 The nerest, that the _Mone_\ncommeth to the earth, is Semidiameters 52\u00bc The distance of the Starry\nSkye is, fr\u00f5 vs, in Semidiameters of the earth 20081\u00bd Twenty thousand\nfourescore, one, and almost a halfe. Subtract from this, the _Mones_\nnerest distance, from the Earth: and therof remaineth Semidiameters of\nthe earth 20029\u00bc Twenty thousand nine and twenty and a quarter.\n\n    [Note.]\n\nSo thicke is the heauenly Palace, that the _Planetes_ haue all their\nexercise in, and most meruailously perfourme the Comma\u0169dement and Charge\nto them giuen by the omnipotent Maiestie of the king of kings. This is\nthat, which in _Genesis_ is called _Ha Rakia_. Consider it well. The\nSemidiameter of the earth, c\u00f5teineth of our common miles 3436 4/11 three\nthousand, foure hundred thirty six and foure eleuenth partes of one\nmyle: Such as the whole earth and Sea, round about, is 21600. One and\ntwenty thousand six hundred of our myles. Allowyng for euery degree of\nthe greatest circle, thre score myles. Now if you way well with your\nselfe but this litle parcell of frute _Astronomicall_, as concerning the\nbignesse, Distances of _Sonne_, _Mone_, _Sterry Sky_, and the huge\nmassines of _Ha Rakia_, will you not finde your Consciences moued, with\nthe kingly Prophet, to sing the confession of Gods Glory, and say, +_The\nHeauens declare the glory of God, and the Firmament =[Ha Rakia]= sheweth\nforth the workes of his handes_+. And so forth, for those fiue first\nstaues, of that kingly Psalme. Well, well, It is time for some to lay\nhold on wisedome, and to Iudge truly of thinges: and notso to expound\nthe Holy word, all by Allegories: as to Neglect the wisedome, powre and\nGoodnes of God, in, and by his Creatures, and Creation to be seen and\nlearned. By parables and Analogies of whose natures and properties, the\ncourse of the Holy Scripture, also, declareth to vs very many Mysteries.\nThe whole Frame of Gods Creatures, (which is the whole world,) is to vs,\na bright glasse: from which, by reflexion, reboundeth to our knowledge\nand perceiuerance, Beames, and Radiations: representing the Image of his\nInfinite goodnes, Omnipot\u1ebdcy, and wisedome. And we therby, are taught\nand persuaded to Glorifie our Creator, as God: and be thankefull\ntherfore. Could the Heathenistes finde these vses, of these most pure,\nbeawtifull, and Mighty Corporall Creatures: and shall we, after that the\ntrue _Sonne_ of rightwisenesse is risen aboue the _Horizon_, of our\ntemporall _Hemisph\u00e6rie_, and hath so abundantly streamed into our\nhartes, the direct beames of his goodnes, mercy, and grace: Whose heat\nAll Creatures feele: Spirituall and Corporall: Visible and Inuisible.\nShall we (I say) looke vpon the _Heauen_, _Sterres_, and _Planets_, as\nan Oxe and an Asse doth: no furder carefull or inquisitiue, what they\nare: why were they Created, How do they execute that they were Created\nfor? Seing, All Creatures, were for our sake created: and both we, and\nthey, Created, chiefly to glorifie the Almighty Creator: and that, by\nall meanes, to vs possible. _Nolite ignorare_ (saith _Plato in\nEpinomis_) _Astronomiam, Sapientissim\u0169 quiddam esse._ +_Be ye not\nignorant, Astronomie to be a thyng of excellent wisedome._+\n_Astronomie_, was to vs, from the beginning commended, and in maner\ncommaunded by God him selfe. In asmuch as he made the _Sonne_, _Mone_,\nand _Sterres_, to be to vs, for _Signes_, and knowledge of Seasons, and\nfor Distinctions of Dayes, and yeares. Many wordes nede not. But I wish,\neuery man should way this word, _Signes_. And besides that, conferre it\nalso with the tenth Chapter of _Hieremie_. And though Some thinke, that\nthere, they haue found a rod: Yet Modest Reason, will be indifferent\nIudge, who ought to be beaten therwith, in respect of our purpose.\nLeauing that: I pray you vnderstand this: that without great diligence\nof Obseruation, examination and Calculation, their periods and courses\n(wherby _Distinction_ of Seasons, yeares, and New Mones might precisely\nbe knowne) could not exactely be certified. Which thing to performe, is\nthat _Art_, which we here haue Defined to be _Astronomie_. Wherby, we\nmay haue the distinct Course of Times, dayes, yeares, and Ages: aswell\nfor Considerati\u00f5 of Sacred Prophesies, accomplished in due time,\nforetold: as for high Mysticall Solemnities holding: And for all other\nhumaine affaires, Conditions, and couenantes, vpon certaine time,\nbetwene man and man: with many other great vses: Wherin, (verely), would\nbe great incertainty, Confusion, vntruth, and brutish Barbarousnes:\nwithout the wonderfull diligence and skill of this Arte: continually\nlearning, and determining Times, and periodes of Time, by the Record of\nthe heauenly booke, wherin all times are written: and to be read with an\n_Astronomicall staffe_, in stede of a festue.\n\n\n+\u2021Musike\u2021+, of Motion, hath his Originall cause: Therfore, after the\nmotions most swift, and most Slow, which are in the Firmament, of Nature\nperformed: and vnder the _Astronomers Consideration_: now I will Speake\nof an other kinde of _Motion_, producing sound, audible, and of Man\nnumerable. _Musike_ I call here that _Science_, which of the Grekes is\ncalled _Harmonice_. Not medling with the Controuersie betwene the\nauncient _Harmonistes_, and _Canonistes_. +Musike is a Mathematicall\nScience, which teacheth, by sense and reason, perfectly to iudge, and\norder the diuersities of soundes, hye and low.+ _Astronomie_ and\n_Musike_ are Sisters, saith _Plato_. As, for _Astronomie_, the eyes: So,\nfor _Harmonious Motion_, the eares were made. But as _Astronomie_ hath a\nmore diuine Contemplation, and c\u00f5modity, then mortall eye can perceiue:\nSo, is _Musike_ to be considered,\n\n    [1.]\n\nthat the * Minde may be preferred, before the eare. And from audible\nsound, we ought to ascende, to the examination: which numbers are\n_Harmonious_, and which not. And why, either, the one are: or the other\nare not. I could at large,\n\n    [2.]\n\nin the heauenly * motions and distances, describe a meruailous Harmonie,\nof _Pythagoras_ Harpe\n\n    [3.]\n\nwith eight stringes. Also, somwhat might be sayd of _Mercurius_ * two\nHarpes,\n\n    [4.]\n\neche of foure Stringes Elementall. And very straunge matter, might be\nalledged of the _Harmonie_,\n\n    [5.]\n\nto our * Spirituall part appropriate. As in _Ptolomaus_ third boke, in\nthe fourth and sixth Chapters may appeare. *\n\n    [6.]\n\nAnd what is the cause of the apt bonde, and frendly felowship, of the\nIntellectuall and Mentall part of vs, with our grosse & corruptible\nbody: but a certaine Meane, and _Harmonious Spiritualitie, with both\nparticipatyng, & of both (in a maner) resultynge In\n\n    [7.]\n\nthe * Tune of Mans voyce, and also\n\n    [8.]\n\n* the sound of Instrument_, what might be sayd, of _Harmonie_: No common\nMusicien would lightly beleue.\n\n    [I. D.\n    Read in Aristotle his 8. booke of Politikes: the 5, 6, and 7.\n    chapters. Where you shall haue some occasion farder to thinke\n    of Musike, than commonly is thought.]\n\nBut of the sundry Mixture (as I may terme it) and concurse, diuerse\ncollation, and Application of these _Harmonies_: as of thre, foure,\nfiue, or mo: Maruailous haue the effectes ben: and yet may be founde,\nand produced the like: with some proportionall consideration for our\ntime, and being: in respect of the State, of the thinges then: in which,\nand by which, the wondrous effectes were wrought. _Democritus_ and\n_Theophrastus_ affirmed, that, by _Musike_, griefes and diseases of the\nMinde, and body might be cured, or inferred. And we finde in Recorde,\nthat _Terpander_, _Arion_, _Ismenias_, _Orpheus_, _Amphion_, _Dauid_,\n_Pythagoras_, _Empedocles_, _Asclepiades_ and _Timotheus_, by\n_Harmonicall_ Conson\u00e3cy, haue done, and brought to pas, thinges, more\nthen meruailous, to here of. Of them then, making no farder discourse,\nin this place: Sure I am, that Common _Musike_, commonly vsed, is found\nto the _Musiciens_ and Hearers, to be so Commodious and pleasant, That\nif I would say and dispute, but thus much: That it were to be otherwise\nvsed, then it is, I should finde more repreeuers, then I could finde\npriuy, or skilfull of my meaning. In thinges therfore euident, and\nbetter knowen, then I can expresse: and so allowed and liked of, (as I\nwould wish, some other thinges, had the like hap) I will spare to\nenlarge my lines any farder, but consequently follow my purpose.\n\n\n+\u2021Of Cosmographie\u2021+, I appointed briefly in this place, to geue you some\nintelligence. +Cosmographie, is the whole and perfect description of the\nheauenly, and also elementall parte of the world, and their homologall\napplication, and mutuall collation necessarie.+ This Art, requireth\n_Astronomie_, _Geographie_, _Hydrographie_ and _Musike_. Therfore, it is\nno small Arte, nor so simple, as in common practise, it is (slightly)\nconsidered. This matcheth Heauen, and the Earth, in one frame, and aptly\napplieth parts Corresp\u00f5dent: So, as, the Heauenly Globe, may (in\npractise) be duely described vpon the Geographicall, and Hydrographicall\nGlobe. And there, for vs to consider an _\u00c6quonoctiall Circle_, _an\nEcliptike line_, _Colures_, _Poles_, _Sterres_ in their true Longitudes,\nLatitudes, Declinations, and Verticalitie: also Climes, and Parallels:\nand by an _Horizon_ annexed, and reuolution of the earthly Globe (as the\nHeauen, is, by the _Primouant_, caried about in 24. \u00e6quall Houres) to\nlearne the Risinges and Settinges of Sterres (of _Virgill_ in his\n_Georgikes_: of _Hesiod_: of _Hippocrates_ in his _Medicinall Sph\u00e6re_,\nto Perdicca King of the Macedonians: of _Diocles_, to King _Antigonus_,\nand of other famous _Philosophers_ prescribed) a thing necessary, for\ndue manuring of the earth, for _Nauigation_, for the Alteration of mans\nbody: being, whole, Sicke, wounded, or brused. By the Reuolution, also,\nor mouing of the Globe Cosmographicall, the Rising and Setting of the\nSonne: the Lengthes, of dayes and nightes: the Houres and times (both\nnight and day) are knowne: with very many other pleasant and necessary\nvses: Wherof, some are knowne: but better remaine, for such to know and\nvse:\n\n    [\u261e]\n\nwho of a sparke of true fire, can make a wonderfull bonfire, by applying\nof due matter, duely.\n\n\n+\u2021Of Astrologie\u2021+, here I make an Arte, seuerall from _Astronomie_: not\nby new deuise, but by good reason and authoritie: for, +Astrologie, is\nan Arte Mathematicall, which reasonably demonstrateth the operations and\neffectes, of the naturall beames, of light, and secrete influence: of\nthe Sterres and Planets: in euery element and elementall body: at all\ntimes, in any Horizon assigned.+ This Arte is furnished with many other\ngreat Artes and experiences: As with perfecte _Perspectiue_,\n_Astronomie_, _Cosmographie_, _Naturall Philosophie_ of the 4.\nElementes, the Arte of Graduation, and some good vnderst\u00e3ding in\n_Musike_: and yet moreouer, with an other great Arte, hereafter\nfollowing, though I, here, set this before, for some considerations me\nmouing. Sufficient (you see) is the stuffe, to make this rare and\nsecrete Arte, of: and hard enough to frame to the Conclusion\nSyllogisticall. Yet both the manifolde and continuall trauailes of the\nmost auncient and wise Philosophers, for the atteyning of this Arte: and\nby examples of effectes, to confirme the same: hath left vnto vs\nsufficient proufe and witnesse: and we, also, daily may perceaue, That\nmans body, and all other Elementall bodies, are altered, disposed,\nordred, pleasured, and displeasured, by the Influentiall working of the\n_Sunne_, _Mone_, and the other Starres and Planets. And therfore, sayth\n_Aristotle_, in the first of his _Meteorologicall_ bookes, in the second\nChapter: _Est autem necessari\u00f2 Mundus iste, supernis lationibus fer\u00e8\ncontinuus. Vt, inde, vis eius vniuersa regatur. Ea siquidem Caus\u00e0 prima\nputanda omnibus est, vnde motus principium existit._ That is: +_This\n=[Elementall]= World is of necessitie, almost, next adioyning, to the\nheauenly motions: That, from thence, all his vertue or force may be\ngouerned. For, that is to be thought the first Cause vnto all: from\nwhich, the beginning of motion, is._+ And againe, in the tenth Chapter.\n_Oportet igitur & horum principia sumamus, & causas omnium similiter.\nPrincipium igitur vt mouens, pr\u00e6cipuum[que] & omnium primum, Circulus\nille est, in quo manifeste Solis latio, &c._ And so forth. His\n_Meteorologicall_ bookes, are full of argumentes, and effectuall\ndemonstrations, of the vertue, operation, and power of the heauenly\nbodies, in and vpon the fower Elementes, and other bodies, of them\n(either perfectly, or vnperfectly) composed. And in his second booke,\n_De Generatione & Corruptione_, in the tenth Chapter. _Quocirca & prima\nlatio, Ortus & Interitus causa non est: Sed obliqui Circuli latio: ea\nnam[que] & continua est, & duobus motibus fit:_ In Englishe, thus.\n+_Wherefore the vppermost motion, is not the cause of Generation and\nCorruption, but the motion of the Zodiake: for, that, both, is\ncontinuall, and is caused of two mouinges._+ And in his second booke,\nand second Chapter of hys _Physikes_. _Homo nam[que] generat hominem,\nat[que] Sol._ +_For Man (sayth he) and the Sonne, are cause of mans\ngeneration._+ Authorities may be brought, very many: both of 1000. 2000.\nyea and 3000. yeares Antiquitie: of great _Philosophers_, _Expert_,\n_Wise_, and godly men, for that Conclusion: which, daily and hourely, we\nmen, may discerne and perceaue by sense and reason: All beastes do\nfeele, and simply shew, by their actions and passions, outward and\ninward: All Plants, Herbes, Trees, Flowers, and Fruites. And finally,\nthe Elementes, and all thinges of the Elementes composed, do geue\nTestimonie (as _Aristotle_ sayd) that theyr +_Whole Dispositions,\nvertues, and naturall motions, depend of the Actiuitie of the heauenly\nmotions and Influences. Whereby, beside the specificall order and forme,\ndue to euery seede: and beside the Nature, propre to the Indiuiduall\nMatrix, of the thing produced: What shall be the heauenly Impression,\nthe perfect and circumspecte Astrologien hath to Conclude._+ Not onely\n(by _Apotelesmes_) \u03c4\u1f78 \u1f41\u03c4\u1f76]. but by Naturall and Mathematicall\ndemonstration \u03c4\u1f78 \u03b4\u03b9\u03cc\u03c4\u03b9. Whereunto, what Sciences are requisite (without\nexception) I partly haue here warned: And in my _Prop\u00e6deumes_ (besides\nother matter there disclosed) I haue Mathematically furnished vp the\nwhole Method: To this our age, not so carefully handled by any, that\neuer I saw, or heard of. I was,\n\n    [* Anno. 1548 and 1549. in Louayn.]\n\n(for * 21. yeares ago) by certaine earnest disputations, of the Learned\n_Gerardus Mercator_, and _Antonius Gogaua_, (and other,) therto so\nprouoked: and (by my constant and inuincible zeale to the veritie) in\nobseruations of Heauenly Influencies (to the Minute of time,) than, so\ndiligent: And chiefly by the Supernaturall influence, from the Starre of\nIacob, so directed: That any Modest and Sober Student, carefully and\ndiligently seking for the Truth, will both finde & c\u00f5fesse, therin, to\nbe the Veritie, of these my wordes: And also become a Reasonable\nReformer, of three Sortes of people: about these Influentiall\nOperations, greatly erring from the truth.\n\n    [Note.]\n\nWherof, the one, is +Light Beleuers+, the other, +Light Despisers+, and\nthe third +Light Practisers+. The first, & most c\u00f5mon Sort, thinke the\nHeauen and Sterres, to be answerable to any their doutes or desires:\n\n    [1.]\n\nwhich is not so: and, in dede, they, to much, ouer reache. The Second\nsorte thinke no Influentiall vertue (fr\u00f5 the heauenly bodies) to beare\nany Sway in Generation\n\n    [2.]\n\nand Corruption, in this Elementall world. And to the _Sunne_, _Mone_ and\n_Sterres_ (being so many, so pure, so bright, so wonderfull bigge, so\nfarre in distance, so manifold in their motions, so constant in their\nperiodes. &c.) they assigne a sleight, simple office or two, and so\nallow vnto th\u1ebd (according to their capacities) as much vertue, and power\nInfluentiall, as to the Signe of the _Sunne_, _Mone_, and seuen Sterres,\nhanged vp (for Signes) in London, for distinction of houses, & such\ngrosse helpes, in our worldly affaires: And they vnderstand not (or will\nnot vnderstand) of the other workinges, and vertues of the Heauenly\n_Sunne_, _Mone_, and _Sterres_: not so much, as the Mariner, or Husband\nman: no, not so much, as the _Elephant_ doth, as the _Cynocephalus_, as\nthe Porpentine doth: nor will allow these perfect, and incorruptible\nmighty bodies, so much vertuall Radiation, & Force, as they see in a\nlitle peece of a _Magnes stone_: which, at great distance, sheweth his\noperation. And perchaunce they thinke, the Sea & Riuers (as the Thames)\nto be some quicke thing, and so to ebbe, and flow, run in and out, of\nthem selues, at their owne fantasies. God helpe, God helpe. Surely,\nthese men, come to short: and either are to dull: or willfully blind:\nor, perhaps, to malicious. The third man, is the common and vulgare\n_Astrologien_, or Practiser: who, being not duely, artificially, and\nperfectly\n\n    [3.]\n\nfurnished: yet, either for vaine glory, or gayne: or like a simple dolt,\n& blinde Bayard, both in matter and maner, erreth: to the discredit of\nthe _Wary_, and modest _Astrologien_: and to the robbing of those most\nnoble corporall Creatures, of their Naturall Vertue: being most mighty:\nmost beneficiall to all elementall Generation, Corruption and the\nappartenances: and most Harmonious in their Monarchie: For which\nthinges, being knowen, and modestly vsed: we might highly, and\ncontinually glorifie God, with the princely Prophet, saying. +_The\nHeauens declare the Glorie of God: who made the Heau\u1ebds in his wisedome:\nwho made the Sonne, for to haue dominion of the day: the Mone and\nSterres to haue dominion of the nyght: whereby, Day to day vttereth\ntalke: and night, to night declareth knowledge. Prayse him, all ye\nSterres, and Light. Amen._+\n\n\nIn order, now foloweth, of +\u2021Statike\u2021+, somewhat to say, what we meane\nby that name: and what commodity, doth, on such Art, depend. +Statike,\nis an Arte Mathematicall, which demonstrateth the causes of heauynes,\nand lightnes of all thynges: and of motions and properties, to heauynes\nand lightnes, belonging.+ And for asmuch as, by the Bilanx, or Balance\n(as the chief sensible Instrument,) Experience of these demonstrations\nmay be had: we call this Art, _Statike:_ that is, _the Experimentes of\nthe Balance_. Oh, that men wist, what proffit, (all maner of wayes) by\nthis Arte might grow, to the hable examiner, and diligent practiser.\n\u201cThou onely, knowest all thinges precisely (O God) who hast made weight\nand Balance, thy Iudgement: who hast created all thinges in _Number,\nWaight, and Measure_: and hast wayed the mountaines and hils in a\nBalance: who hast peysed in thy hand, both Heauen and earth. We therfore\nwarned by the Sacred word, to Consider thy Creatures: and by that\nconsideration, to wynne a glyms (as it were,) or shaddow of\nperceiuerance, that thy wisedome, might, and goodnes is infinite, and\nvnspeakable, in thy Creatures declared: And being farder aduertised, by\nthy mercifull goodnes, that, three principall wayes, were, of the, vsed\nin Creation of all thy Creatures, namely, _Number_, _Waight_ and\n_Measure_, And for as much as, of _Number_ and _Measure_, the two Artes\n(auncient, famous, and to humaine vses most necessary,) are, all ready,\nsufficiently knowen and extant: This third key, we beseche thee (through\nthy accustomed goodnes,) that it may come to the nedefull and sufficient\nknowledge, of such thy Seruauntes, as in thy workemanship, would gladly\nfinde, thy true occasions (purposely of the vsed) whereby we should\nglorifie thy name, and shew forth (to the weaklinges in faith) thy\nwondrous wisedome and Goodnes. Amen.\u201d\n\n\nMeruaile nothing at this pang (godly frend, you Gentle and zelous\nStudent.) An other day, perchaunce, you will perceiue, what occasion\nmoued me. Here, as now, I will giue you some ground, and withall some\nshew, of certaine commodities, by this Arte arising. And bycause this\nArte is rare, my wordes and practises might be to darke: vnleast you had\nsome light, holden before the matter: and that, best will be, in giuing\nyou, out of _Archimedes_ demonstrations, a few principal Conclusions, as\nfoloweth.\n\n  +1.+\n\n  +The Superficies of euery Liquor, by it selfe consistyng, and in\n  quyet, is Sph\u00e6ricall: the centre whereof, is the same, which is the\n  centre of the Earth.+\n\n  +2.+\n\n  +If Solide Magnitudes, being of the same bignes, or qu\u00e3titie, that\n  any Liquor is, and hauyng also the same Waight: be let downe into\n  the same Liquor, they will settle downeward, so, that no parte of\n  them, shall be aboue the Superficies of the Liquor: and yet\n  neuertheles, they will not sinke vtterly downe, or drowne.+\n\n  +3.+\n\n  +If any Solide Magnitude beyng Lighter then a Liquor, be let downe\n  into the same Liquor, it will settle downe, so farre into the same\n  Liquor, that so great a quantitie of that Liquor, as is the parte of\n  the Solid Magnitude, settled downe into the same Liquor: is in\n  Waight, \u00e6quall, to the waight of the whole Solid Magnitude.+\n\n  +4.+\n\n  +Any Solide Magnitude, Lighter then a Liquor, forced downe into the\n  same Liquor, will moue vpward, with so great a power, by how much,\n  the Liquor hauyng \u00e6quall quantitie to the whole Magnitude, is\n  heauyer then the same Magnitude.+\n\n  +5.+\n\n  +Any Solid Magnitude, heauyer then a Liquor, beyng let downe into\n  the same Liquor, will sinke downe vtterly: And wilbe in that Liquor,\n  Lighter by so much, as is the waight or heauynes of the Liquor,\n  hauing bygnes or quantitie, \u00e6quall to the Solid Magnitude.+\n\n  +6.+\n\n      [I. D.\n      The Cutting of a Sph\u00e6re according to any proportion assigned\n      may by this proposition be done Mechanically by tempering\n      Liquor to a certayne waight in respect of the waight of the\n      Sph\u00e6re therein Swymming.]\n\n  +If any Solide Magnitude, Lighter then a Liquor, be let downe into\n  the same Liquor, the waight of the same Magnitude, will be, to the\n  Waight of the Liquor. (Which is \u00e6quall in quantitie to the whole\n  Magnitude,) in that proportion, that the parte, of the Magnitude\n  settled downe, is to the whole Magnitude.+\n\n\nBy these verities, great Errors may be reformed, in Opinion of the\nNaturall Motion of thinges, Light and Heauy. Which errors, are in\nNaturall Philosophie (almost) of all m\u1ebd allowed: to much trusting to\nAuthority: and false Suppositions. As, +Of any two bodyes, the heauyer,\nto moue downward faster then the lighter.+\n\n    [A common error, noted.]\n\nThis error, is not first by me, Noted: but by one _Iohn Baptist de\nBenedictis_. The chief of his propositions, is this: which seemeth a\nParadox.\n\n\n+If there be two bodyes of one forme, and of one kynde, \u00e6quall in\nquantitie or vn\u00e6quall,\n\n    [A paradox.]\n\nthey will moue by \u00e6quall space, in \u00e6quall tyme: So that both theyr\nmouynges be in ayre, or both in water: or in any one Middle.+\n\n\nHereupon, in the feate of +Gunnyng+,\n\n    [N. T.]\n\ncertaine good discourses (otherwise) may receiue great amendement, and\nfurderance.\n\n    [The wonderfull vse of these Propositions.]\n\nIn the entended purpose, also, allowing somwhat to the imperfection of\nNature: not aunswerable to the precisenes of demonstration. Moreouer, by\nthe foresaid propositions (wisely vsed.) The Ayre, the water, the Earth,\nthe Fire, may be nerely, knowen, how light or heauy they are (Naturally)\nin their assigned partes: or in the whole. And then, to thinges\nElementall, turning your practise: you may deale for the proportion of\nthe Elementes, in the thinges Compounded. Then, to the proportions of\nthe Humours in Man: their waightes: and the waight of his bones, and\nflesh. &c. Than, by waight, to haue consideration of the Force of man,\nany maner of way: in whole or in part. Then, may you, of Ships water\ndrawing, diuersly, in the Sea and in fresh water, haue pleasant\nconsideration: and of waying vp of any thing, sonken in Sea or in fresh\nwater &c. And (to lift vp your head a loft:) by waight, you may, as\nprecisely, as by any instrument els, measure the Diameters of _Sonne_\nand _Mone. &c._ Frende, I pray you, way these thinges, with the iust\nBalance of Reason. And you will finde Meruailes vpon Meruailes: And\nesteme one Drop of Truth (yea in Naturall Philosophie) more worth, then\nwhole Libraries of Opinions, vndemonstrated: or not aunswering to\nNatures Law, and your experience. Leauing these thinges, thus: I will\ngiue you two or three, light practises, to great purpose: and so finish\nmy Annotation _Staticall_. In Mathematicall matters, by the Mechaniciens\nayde, we will behold, here, the Commodity of waight.\n\n    [The practise Staticall, to know the proportion, betwene\n    the Cube, and the Sph\u00e6re.]\n\nMake a Cube, of any one Vniforme: and through like heauy stuffe: of the\nsame Stuffe, make a Sph\u00e6re or Globe, precisely, of a Diameter \u00e6quall to\nthe Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne,\nLead, Siluer. &c. (being, as I sayd, of like nature, condition, and like\nwaight throughout.) And you may, by Say Balance, haue prepared a great\nnumber of the smallest waightes: which, by those Balance can be\ndiscerned or tryed: and so, haue proceded to make you a perfect Pyle,\ncompany & Number of waightes: to the waight of six, eight, or twelue\npound waight: most diligently tryed, all. And of euery one, the Content\nknowen, in your least waight, that is wayable. [They that can not haue\nthese waightes of precisenes: may, by Sand, Vniforme, and well dusted,\nmake them a number of waightes, somewhat nere precisenes: by halfing\neuer the Sand: they shall, at length, come to a least common waight.\nTherein, I leaue the farder matter, to their discretion, whom nede shall\npinche.] The _Venetians_ consideration of waight, may seme precise\nenough: by eight descentes progressionall, * halfing, from a grayne.\n\n    [I. D.\n    * For, so, haue you .256. partes of a Graine.]\n\nYour Cube, Sph\u00e6re, apt Balance, and conuenient waightes, being ready:\nfall to worke.\u2749. First, way your Cube. Note the Number of the waight.\nWay, after that, your Sph\u00e6re. Note likewise, the N\u0169ber of the waight. If\nyou now find the waight of your Cube, to be to the waight of the Sph\u00e6re,\nas 21. is to 11: Then you see, how the Mechanicien and _Experimenter_,\nwithout Geometrie and Demonstration, are (as nerely in effect) tought\nthe proportion of the Cube to the Sphere: as I haue demonstrated it, in\nthe end of the twelfth boke of _Euclide_. Often, try with the same Cube\nand Sph\u00e6re. Then, chaunge, your Sph\u00e6re and Cube, to an other matter: or\nto an other bignes: till you haue made a perfect vniuersall Experience\nof it. Possible it is, that you shall wynne to nerer termes, in the\nproportion.\n\n\nWhen you haue found this one certaine Drop of Naturall veritie, procede\non, to Inferre, and duely to make assay, of matter depending. As,\nbycause it is well demonstrated, that a Cylinder, whose heith, and\nDiameter of his base, is \u00e6quall to the Diameter of the Sph\u00e6re, is\nSesquialter to the same Sph\u00e6re (that is, as 3. to 2:) To the number of\nthe waight of the Sph\u00e6re, adde halfe so much, as it is: and so haue you\nthe number of the waight of that Cylinder. Which is also Comprehended of\nour former Cube: So, that the base of that Cylinder, is a Circle\ndescribed in the Square, which is the base of our Cube. But the Cube and\nthe Cylinder, being both of one heith, haue their Bases in the same\nproportion, in the which, they are, one to an other, in their Massines\nor Soliditie. But, before, we haue two numbers, expressing their\nMassines, Solidities, and Quantities, by waight: wherfore,\n\n    [* =The proportion of the Square to the Circle inscribed.=]\n\nwe haue * the proportion of the Square, to the Circle, inscribed in the\nsame Square. And so are we fallen into the knowledge sensible, and\nExperimentall of _Archimedes_ great Secret: of him, by great trauaile of\nminde, sought and found. Wherfore, to any Circle giuen, you can giue a\nSquare \u00e6quall:\n\n    [* =The Squaring of the Circle, Mechanically.=]\n\n* as I haue taught, in my Annotation, vpon the first proposition of the\ntwelfth boke, And likewise, to any Square giuen, you may giue a Circle\n\u00e6quall:\n\n    [* =To any Square geuen, to geue a Circle, equall.=]\n\n* If you describe a Circle, which shall be in that proportion, to your\nCircle inscribed, as the Square is to the same Circle: This, you may do,\nby my Annotations, vpon the second proposition of the twelfth boke of\n_Euclide_, in my third Probleme there. Your diligence may come to a\nproportion, of the Square to the Circle inscribed, nerer the truth, then\nis the proportion of 14. to 11. And consider, that you may begyn at the\nCircle and Square, and so come to conclude of the Sph\u00e6re, & the Cube,\nwhat their proportion is: as now, you came from the Sph\u00e6re to the\nCircle. For, of Siluer, or Gold, or Latton Lamyns or plates (thorough\none hole draw\u1ebd, as the maner is) if you make a Square figure & way it:\nand then, describing theron, the Circle inscribed: & cut of, & file\naway, precisely (to the Circle) the ouerplus of the Square: you shall\nthen, waying your Circle, see, whether the waight of the Square, be to\nyour Circle, as 14. to 11. As I haue Noted, in the beginning of\n_Euclides_ twelfth boke. &c. after this resort to my last proposition,\nvpon the last of the twelfth. And there, helpe your selfe, to the end.\nAnd, here, Note this, by the way.\n\n    [Note Squaring of the Circle without knowledge of the\n    proportion betwene Circumference and Diameter.]\n\nThat we may Square the Circle, without hauing knowledge of the\nproportion, of the Circumference to the Diameter: as you haue here\nperceiued. And otherwayes also, I can demonstrate it. So that, many haue\ncumberd them selues superfluously, by trauailing in that point first,\nwhich was not of necessitie, first: and also very intricate. And easily,\nyou may, (and that diuersly) come to the knowledge of the Circumference:\nthe Circles Quantitie, being first knowen. Which thing, I leaue to your\nconsideration: making hast to despatch an other Magistrall Probleme: and\nto bring it, nerer to your knowledge, and readier dealing with, then the\nworld (before this day,) had it for you, that I can tell of. And that\nis, _A Mechanicall Dubblyng of the Cube: &c._ Which may, thus, be done:\n\n    [To Dubble the Cube redily: by Art Mechanicall: depending\n    vppon Demonstration Mathematicall.]\n\n+Make of Copper plates, or Tyn plates, a foursquare vpright Pyramis, or\na Cone: perfectly fashioned in the holow, within. Wherin, let great\ndiligence be vsed, to approche (as nere as may be) to the Mathematicall\nperfection of those figures. At their bases, let them be all open: euery\nwhere, els, most close, and iust to. From the vertex, to the\nCircumference of the base of the Cone: & to the sides of the base of the\nPyramis:+\n\n    [=I. D.=\n    =The 4. sides of this Pyramis must be 4. Isosceles\n    Triangles alike and \u00e6quall.=]\n\n+Let 4. straight lines be drawen, in the inside of the Cone and Pyramis:\nmakyng at their fall, on the perimeters of the bases, equall angles on\nboth sides them selues, with the sayd perimeters. These 4. lines (in the\nPyramis: and as many, in the Cone) diuide: one, in 12. \u00e6quall partes:\nand an other, in 24. an other, in 60, and an other, in 100. (reckenyng\nvp from the vertex.) Or vse other numbers of diuision, as experience\nshall teach you.+\n\n    [=I. D.=\n    =* In all workinges with this Pyramis or Cone, Let their\n    Situations be in all Pointes and Conditions, alike,\n    or all one: while you are about one Worke. Els you\n    will erre.=]\n\n+Then, * set your Cone or Pyramis, with the vertex downward,\nperpendicularly, in respect of the Base. (Though it be otherwayes, it\nhindreth nothyng.) So let th\u1ebd most stedily be stayed.+ Now, if there be\na Cube, which you wold haue Dubbled. Make you a prety Cube of Copper,\nSiluer, Lead, Tynne, Wood, Stone, or Bone. Or els make a hollow Cube, or\nCubik coffen, of Copper, Siluer, Tynne, or Wood &c. These, you may so\nproporti\u00f5 in respect of your Pyramis or Cone, that the Pyramis or Cone,\nwill be hable to conteine the waight of them, in water, 3. or 4. times:\nat the least: what stuff so euer they be made of. Let not your Solid\nangle, at the vertex, be to sharpe: but that the water may come with\nease, to the very vertex, of your hollow Cone or Pyramis. Put one of\nyour Solid Cubes in a Balance apt: take the waight therof exactly in\nwater. Powre that water, (without losse) into the hollow Pyramis or\nCone, quietly. Marke in your lines, what numbers the water Cutteth: Take\nthe waight of the same Cube againe: in the same kinde of water, which\nyou had before:\n\n    [=I. D.=\n    =* Consider well whan you must put your waters togyther:\n    and whan, you must empty your first water, out of your\n    Pyramis or Cone. Els you will erre.=]\n\nput that* also, into the Pyramis or Cone, where you did put the first.\nMarke now againe, in what number or place of the lines, the water\nCutteth them. Two wayes you may conclude your purpose: it is to wete,\neither by numbers or lines. By numbers: as, if you diuide the side of\nyour Fundamentall Cube into so many \u00e6quall partes, as it is capable of,\nconueniently, with your ease, and precisenes of the diuision. For, as\nthe number of your first and lesse line (in your hollow Pyramis or\nCone,) is to the second or greater (both being counted from the vertex)\nso shall the number of the side of your Fundamentall Cube, be to the\nn\u0169ber belonging to the Radicall side, of the Cube, dubble to your\nFundamentall Cube: Which being multiplied Cubik wise, will sone shew it\nselfe, whether it be dubble or no, to the Cubik number of your\nFundamentall Cube. By lines, thus: As your lesse and first line, (in\nyour hollow Pyramis or Cone,) is to the second or greater, so let the\nRadical side of your Fundam\u1ebdtall Cube, be to a fourth proportionall\nline, by the 12. proposition, of the sixth boke of _Euclide_. Which\nfourth line, shall be the Rote Cubik, or Radicall side of the Cube,\ndubble to your Fundamentall Cube: which is the thing we desired.\n\n    [\u261e God be thanked for this Inuention, & the fruite ensuing.]\n\nFor this, may I (with ioy) say, \u0395\u03a5\u03a1\u0397\u039a\u0391, \u0395\u03a5\u03a1\u0397\u039a\u0391, \u0395\u03a5\u03a1\u0397\u039a\u0391: thanking the\nholy and glorious Trinity: hauing greater cause therto, then\n\n    [* Vitruuius. Lib. 9. Cap. 3.]\n\n* _Archimedes_ had (for finding the fraude vsed in the Kinges Crowne, of\nGold): as all men may easily Iudge: by the diuersitie of the frute\nfollowing of the one, and the other. Where I spake before, of a hollow\nCubik Coffen: the like vse, is of it: and without waight. Thus. Fill it\nwith water, precisely full, and poure that water into your Pyramis or\nCone. And here note the lines cutting in your Pyramis or Cone. Againe,\nfill your coffen, like as you did before. Put that Water, also, to the\nfirst. Marke the second cutting of your lines. Now, as you proceded\nbefore, so must you here procede.\n\n    [* Note.]\n\n* And if the Cube, which you should Double, be neuer so great: you haue,\nthus, the proportion (in small) betwene your two litle Cubes: And then,\nthe side, of that great Cube (to be doubled) being the third, will haue\nthe fourth, found, to it proportionall: by the 12. of the sixth of\nEuclide.\n\n\n    [Note, as concerning the Sph\u00e6ricall Superficies of the Water.]\n\nNote, that all this while, I forget not my first Proposition Staticall,\nhere rehearsed: that, the Superficies of the water, is Sph\u00e6ricall.\nWherein, vse your discretion: to the first line, adding a small heare\nbreadth, more: and to the second, halfe a heare breadth more, to his\nlength. For, you will easily perceaue, that the difference can be no\ngreater, in any Pyramis or Cone, of you to be handled. Which you shall\nthus trye. _For finding the swelling of the water aboue leuell._\n\n    [\u261e]\n\n\u201cSquare the Semidiameter, from the Centre of the earth, to your first\nWaters Superficies. Square then, halfe the Subtendent of that watry\nSuperficies (which Subtendent must haue the equall partes of his\nmeasure, all one, with those of the Semidiameter of the earth to your\nwatry Superficies): Subtracte this square, from the first: Of the\nresidue, take the Rote Square. That Rote, Subtracte from your first\nSemidiameter of the earth to your watry Superficies: that, which\nremaineth, is the heith of the water, in the middle, aboue the leuell.\u201d\nWhich, you will finde, to be a thing insensible. And though it were\ngreatly sensible, *\n\n    [* Note.]\n\nyet, by helpe of my sixt Theoreme vpon the last Proposition of Euclides\ntwelfth booke, noted: you may reduce all, to a true Leuell. But, farther\ndiligence, of you is to be vsed, against accidentall causes of the\nwaters swelling: as by hauing (somwhat) with a moyst Sponge, before,\nmade moyst your hollow Pyramis or Cone, will preuent an accidentall\ncause of Swelling, &c. Experience will teach you abundantly: with great\nease, pleasure, and c\u00f5moditie.\n\n\nThus, may you Double the Cube Mechanically, Treble it, and so forth, in\nany proportion.\n\n    [Note this Abridgement of Dubbling the Cube. &c.]\n\nNow will I Abridge your paine, cost, and Care herein. Without all\npreparing of your Fundamentall Cubes: you may (alike) worke this\nConclusion. For, that, was rather a kinde of Experimentall dem\u00f5stration,\nthen the shortest way: and all, vpon one Mathematicall Demonstration\ndepending. \u201cTake water (as much as conueniently will serue your turne:\nas I warned before of your Fundamentall Cubes bignes) Way it precisely.\nPut that water, into your Pyramis or Cone. Of the same kinde of water,\nthen take againe, the same waight you had before: put that likewise into\nthe Pyramis or Cone. For, in eche time, your marking of the lines, how\nthe Water doth cut them, shall geue you the proportion betwen the\nRadicall sides, of any two Cubes, wherof the one is Double to the other:\nworking as before I haue taught you:\n\n    [* \u261e Note.]\n\n* sauing that for you Fundamentall Cube his Radicall side: here, you may\ntake a right line, at pleasure.\u201d\n\n\nYet farther proceding with our droppe of Naturall truth:\n\n    [To giue Cubes one to the other in any proportion,\n    Rationall or Irrationall.]\n\n+you may (now) geue Cubes, one to the other, in any proporti\u00f5 geu\u1ebd:\nRationall or Irrationall+: on this maner. Make a hollow Parallelipipedon\nof Copper or Tinne: with one Base w\u00e3ting, or open: as in our Cubike\nCoffen. Fr\u00f5 the bottome of that Parallelipipedon, raise vp, many\nperpendiculars, in euery of his fower sides. Now if any proportion be\nassigned you, in right lines: Cut one of your perpendiculars (or a line\nequall to it, or lesse then it) likewise: by the 10. of the sixth of\nEuclide. And those two partes, set in two sundry lines of those\nperpendiculars (or you may set them both, in one line) making their\nbeginninges, to be, at the base: and so their lengthes to extend vpward.\nNow, set your hollow Parallelipipedon, vpright, perpendicularly,\nsteadie. Poure in water, handsomly, to the heith of your shorter line.\nPoure that water, into the hollow Pyramis or Cone. Marke the place of\nthe rising. Settle your hollow Parallelipipedon againe. Poure water into\nit: vnto the heith of the second line, exactly.\n\n    [* Emptying the first.]\n\nPoure that water * duely into the hollow Pyramis or Cone: Marke now\nagaine, where the water cutteth the same line which you marked before.\nFor, there, as the first marked line, is to the second: So shall the two\nRadicall sides be, one to the other, of any two Cubes: which, in their\nSoliditie, shall haue the same proportion, which was at the first\nassigned: were it Rationall or Irrationall.\n\n\nThus, in sundry waies you may furnishe your selfe with such straunge and\nprofitable matter: which, long hath bene wished for. And though it be\nNaturally done and Mechanically: yet hath it a good Demonstration\nMathematicall.\n\n    [=The demonstrations of this Dubbling of the Cube, and of the\nrest.=]\n\nWhich is this: Alwaies, you haue two Like Pyramids: or two Like Cones,\nin the proportions assigned: and like Pyramids or Cones, are in\nproportion, one to the other, in the proportion of their Homologall\nsides (or lines) tripled. Wherefore, if to the first, and second lines,\nfound in your hollow Pyramis or Cone, you ioyne a third and a fourth, in\ncontinuall proportion: that fourth line, shall be to the first, as the\ngreater Pyramis or Cone, is to the lesse: by the 33. of the eleuenth of\nEuclide. If Pyramis to Pyramis, or Cone to Cone, be double,\n\n    [I. D.\n    = * Hereby, helpe your self to become a pr\u00e6cise practiser.\n    And so consider, how, nothing at all, you are hindred\n    (sensibly) by the Conuexitie of the water.=]\n\nthen shall * Line to Line, be also double, &c. But, as our first line,\nis to the second, so is the Radicall side of our Fundamentall Cube, to\nthe Radicall side of the Cube to be made, or to be doubled: and\ntherefore, to those twaine also, a third and a fourth line, in\ncontinuall proportion, ioyned: will geue the fourth line in that\nproportion to the first, as our fourth Pyramidall, or Conike line, was\nto his first: but that was double, or treble, &c. as the Pyramids or\nCones were, one to an other (as we haue proued) therfore, this fourth,\nshalbe also double or treble to the first, as the Pyramids or Cones were\none to an other: But our made Cube, is described of the second in\nproportion, of the fower proportionall lines:\n\n    [= * By the 33. of the eleuenth booke of Euclide.=]\n\ntherfore * as the fourth line, is to the first, so is that Cube, to the\nfirst Cube: and we haue proued the fourth line, to be to the first, as\nthe Pyramis or Cone, is to the Pyramis or Cone: Wherefore the Cube is to\nthe Cube, as Pyramis is to Pyramis, or Cone is to Cone.\n\n    [I. D.\n    = * And your diligence in practise, can so (in waight of\n    water) performe it: Therefore, now, you are able to geue\n    good reason of your whole doing.=]\n\nBut we * Suppose Pyramis to Pyramis, or Cone to Cone, to be double or\ntreble. &c. Therfore Cube, is to Cube, double, or treble, &c. Which was\nto be demonstrated. And of the Parallelipiped\u00f5, it is euid\u1ebdt, that the\nwater Solide Parallelipipedons, are one to the other, as their heithes\nare, seing they haue one base. Wherfore the Pyramids or Cones, made of\nthose water Parallelipipedons, are one to the other, as the lines are\n(one to the other) betwene which, our proportion was assigned. But the\nCubes made of lines, after the proporti\u00f5 of the Pyramidal or Conik\n_homologall_ lines, are one to the other, as the Pyramides or Cones are,\none to the other (as we before did proue) therfore, the Cubes made,\nshalbe one to the other, as the lines assigned, are one to the other:\nWhich was to be demonstrated. Note.\n\n    [* _Note this Corollary._]\n\n* This, my Demonstrati\u00f5 is more generall, then onely in Square Pyramis\nor Cone: Consider well. Thus, haue I, both Mathematically and\nMechanically, ben very long in wordes: yet (I trust) nothing tedious to\nthem, who, to these thinges, are well affected. And verily I am forced\n(auoiding prolixitie) to omit sundry such things, easie to be practised:\nwhich to the Mathematicien, would be a great Threasure: and to the\nMechanicien, no small gaine.\n\n    [* The great Commodities following of these new Inuentions.]\n\n* Now may you, +Betwene two lines giuen, finde two middle proportionals,\nin Continuall proportion: by the hollow Parallelipipedon, and the hollow\nPyramis, or Cone.+ Now, any Parallelipipedon rectangle being giuen: thre\nright lines may be found, proportionall in any proportion assigned, of\nwhich, shal be produced a Parallelipipedon, \u00e6quall to the\nParallelipipedon giuen. Hereof, I noted somwhat, vpon the 36.\nproposition, of the 11. boke of _Euclide_. Now, all those thinges, which\n_Vitruuius_ in his Architecture, specified hable to be done, by dubbling\nof the Cube: Or, by finding of two middle proportionall lines, betwene\ntwo lines giuen, may easely be performed. Now, that Probleme, which I\nnoted vnto you, in the end of my Addition, vpon the 34. of the 11. boke\nof _Euclide_, is proued possible. Now, may any regular body, be\nTransformed into an other, &c. Now, any regular body: any Sphere, yea\nany Mixt Solid: and (that more is) Irregular Solides, may be made (in\nany proporti\u00f5 assigned) like vnto the body, first giuen. Thus, of a\n_Manneken_, (as the _Dutch_ Painters terme it) in the same _Symmetrie_,\nmay a Giant be made: and that, with any gesture, by the Manneken vsed:\nand contrarywise. Now, may you, of any Mould, or Modell of a Ship, make\none, of the same Mould (in any assigned proportion) bigger or lesser.\n\n    [* \u261e]\n\nNow, may you, of any * Gunne, or little peece of ordina\u0169ce, make an\nother, with the same _Symmetrie_ (in all pointes) as great, and as\nlittle, as you will. Marke that: and thinke on it. Infinitely, +may you\napply this, so long sought for, and now so easily concluded: and\nwithall, so willingly and frankly communicated to such, as faithfully\ndeale with vertuous studies.+\n\n    [Such is the Fruite of the Mathematicall Sciences and Artes.]\n\nThus, can the Mathematicall minde, deale Speculatiuely in his own Arte:\nand by good meanes, Mount aboue the cloudes and sterres: And thirdly, he\ncan, by order, Descend, to frame Naturall thinges, to wonderfull vses:\nand when he list, retire home into his owne Centre: and there, prepare\nmore Meanes, to Ascend or Descend by: and, all, to the glory of God, and\nour honest delectation in earth.\n\n\nAlthough, the Printer, hath looked for this Pr\u00e6face, a day or two, yet\ncould I not bring my pen from the paper, before I had giuen you\ncomfortable warning, and brief instructions, of some of the Commodities,\nby _Statike_, hable to be reaped: In the rest, I will therfore, be as\nbrief, as it is possible: and with all, describing them, somwhat\naccordingly. And that, you shall perceiue, by this, which in order\ncommeth next. For, wheras, it is so ample and wonderfull, that, an whole\nyeare long, one might finde fruitfull matter therin, to speake of: and\nalso in practise, is a Threasure endeles: yet will I glanse ouer it,\nwith wordes very few.\n\n\nThis do I call +\u2021Anthropographie\u2021+. Which is an Art restored, and of my\npreferment to your Seruice. I pray you, thinke of it, as of one of the\nchief pointes, of Humane knowledge. Although it be, but now, first\nC\u00f5firmed, with this new name: yet the matter, hath from the beginning,\nben in consideration of all perfect Philosophers. +Anthropographie, is\nthe description of the Number, Measure, Waight, figure, Situation, and\ncolour of euery diuerse thing, conteyned in the perfect body of MAN:\nwith certain knowledge of the Symmetrie, figure, waight,\nCharacterization, and due locall motion, of any parcell of the sayd\nbody, assigned: and of N\u0169bers, to the sayd parcell appertainyng.+ This,\nis the one part of the Definition, mete for this place: Sufficient to\nnotifie, the particularitie, and excellency of the Arte: and why it is,\nhere, ascribed to the Mathematicals. Yf the description of the heauenly\npart of the world, had a peculier Art, called _Astronomie:_ If the\ndescription of the earthly Globe, hath his peculier arte, called\n_Geographie_. If the Matching of both, hath his peculier Arte, called\n_Cosmographie:_ Which is the Descripti\u00f5 of the whole, and vniuersall\nframe of the world: Why should not the description of\n\n    [MAN is the Lesse World.]\n\nhim, who is the Lesse world: and, fr\u00f5 the beginning, called\n_Microcosmus_ (that is. _The Lesse World._) And for whose sake, and\nseruice, all bodily creatures els, were created: Who, also,\nparticipateth with Spirites, and Angels: and is made to the Image and\nsimilitude of _God_: haue his peculier Art? and be called the _Arte of\nArtes_: rather, then, either to want a name, or to haue to base and\nimpropre a name? You must of sundry professions, borow or challenge\nhome, peculier partes hereof: and farder procede: as, God, Nature,\nReason and Experience shall informe you. The Anatomistes will restore to\nyou, some part: The Physiognomistes, some: The Chyromantistes some. The\nMetaposcopistes, some: The excellent, _Albert Durer_, a good part: the\nArte of Perspectiue, will somwhat, for the Eye, helpe forward:\n_Pythagoras_, _Hipocrates_, _Plato_, _Galenus_, _Meletius_, & many other\n(in certaine thinges) will be Contributaries. And farder, the Heauen,\nthe Earth, and all other Creatures, will eche shew, and offer their\nHarmonious seruice, to fill vp, that, which wanteth hereof: and with\nyour own Experience, concluding: you may Methodically register the\nwhole, for the posteritie: Whereby, good profe will be had, of our\nHarmonious, and\n\n    [Micro Cosmus.]\n\nMicrocosmicall constitution.\n\n    [* \u261e]\n\nThe outward Image, and vew hereof: to the Art of _Zographie_ and\nPainting, to Sculpture, and Architecture: (for Church, House, Fort, or\nShip) is most necessary and profitable: for that, it is the chiefe base\nand foundation of them.\n\n    [* Lib. 3. Cap. 1.]\n\nLooke in * _Vitruuius_, whether I deale sincerely for your behoufe, or\nno. Looke in _Albertus Durerus_, _De Symmetria humani Corporis_. Looke\nin the 27. and 28. Chapters, of the second booke, _De occulta\nPhilosophia_. Consider the _Arke_ of _Noe_. And by that, wade farther.\nRemember the _Delphicall Oracle NOSCE TEIPSVM_ +_(Knowe thy selfe)_+ so\nlong agoe pronounced: of so many a Philosopher repeated: and of the\n_Wisest_ attempted: And then, you will perceaue, how long agoe, you haue\nbene called to the Schole, where this Arte might be learned. Well. I am\nnothing affrayde, of the disdayne of some such, as thinke Sciences and\nArtes, to be but Seuen. Perhaps, those Such, may, with ignorance, and\nshame enough, come short of them Seuen also: and yet neuerthelesse they\ncan not prescribe a certaine number of Artes: and in eche, certaine\nvnpassable boundes, to God, Nature, and mans Industrie. New Artes, dayly\nrise vp: and there was no such order taken, that,\n\n    [\u261e]\n\nAll Artes, should in one age, or in one land, or of one man, be made\nknowen to the world. Let vs embrace the giftes of God, and wayes to\nwisedome, in this time of grace, from aboue, continually bestowed on\nthem, who thankefully will receiue them: _Et bonis Omnia Cooperabuntur\nin bonum._\n\n\n+\u2021Trochilike,\u2021 is that Art Mathematicall, which demonstrateth the\nproperties of all Circular motions, Simple and Compounde.+ And bycause\nthe frute hereof, vulgarly receiued, is in Wheles, it hath the name of\n_Trochilike:_ as a man would say, _Whele Art_. By this art, a Whele may\nbe geuen which shall moue ones about, in any tyme assigned. Two Wheles\nmay be giuen, whose turnynges about in one and the same tyme, (or equall\ntymes), shall haue, one to the other, any proportion appointed. By\nWheles, may a straight line be described: Likewise, a Spirall line in\nplaine, Conicall Section lines, and other Irregular lines, at pleasure,\nmay be drawen. These, and such like, are principall Conclusions of this\nArte: and helpe forward many pleasant and profitable Mechanicall workes:\n\n    [Saw Milles.]\n\nAs Milles, to Saw great and very long Deale bordes, no man being by.\nSuch haue I seene in Germany: and in the Citie of Prage: in the kingdome\nof Bohemia: Coyning Milles, Hand Milles for Corne grinding: And all\nmaner of Milles, and Whele worke: By Winde, Smoke, Water, Waight,\nSpring, Man or Beast, moued. Take in your hand, _Agricola De re\nMetallica:_ and then shall you (in all Mines) perceaue, how great nede\nis, of Whele worke. By Wheles, straunge workes and incredible, are done:\nas will, in other Artes hereafter, appeare. A wonderfull example of\nfarther possibilitie, and present commoditie, was sene in my time, in a\ncertaine Instrument: which by the Inuenter and Artificer (before) was\nsolde for xx. Talentes of Golde: and then had (by misfortune) receaued\nsome iniurie and hurt: And one _Ianellus_ of _Cremona_ did mend the\nsame, and presented it vnto the Emperour _Charles_ the fifth.\n_Hieronymus Cardanus_, can be my witnesse, that therein, was one Whele,\nwhich moued, and that, in such rate, that, in 7000. yeares onely, his\nowne periode should be finished. A thing almost incredible: But how\nfarre, I keepe me within my boundes: very many men (yet aliue) can tell.\n\n\n+\u2021Helicosophie\u2021+, is nere Sister to _Trochilike:_ and is, +An Arte\nMathematicall, which demonstrateth the designing of all Spirall lines in\nPlaine, on Cylinder, Cone, Sph\u00e6re, Conoid, and Sph\u00e6roid, and their\nproperties appertayning.+ The vse hereof, in _Architecture_, and diuerse\nInstrumentes and Engines, is most necessary. For, in many thinges, the\nSkrue worketh the feate, which, els, could not be performed. By helpe\nhereof,\n\n    [* Atheneus Lib. 5. cap. 8.]\n\nit is * recorded, that, where all the power of the Citie of Syracusa,\nwas not hable to moue a certaine Ship (being on ground) mightie\n_Archimedes_, setting to, his Skruish Engine, caused _Hiero_ the king,\nby him self, at ease, to remoue her, as he would.\n\n    [Proclus. Pag. 18.]\n\nWherat, the King wondring: \u0391\u03c0\u1f78 \u03c4\u03ac\u03c5\u03c4\u03b7\u03c2 \u03c4\u1fc6\u03c2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2, \u03c0\u03b5\u03c1\u1f76 \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2, \u0391\u03c1\u03c7\u03b9\u03bc\u03ae\u03b4\u03b5\u03b9\n\u03bb\u03ad\u03b3\u03bf\u03bd\u03c4\u03b9 \u03c0\u03b9\u03db\u03b5\u03c5\u03c4\u03b5\u03cc\u03bd. _From this day, forward_ (said the King) _Credit\nought to be giuen to Archimedes, what soeuer he sayth._\n\n\n+\u2021Pneumatithmie\u2021 demonstrateth by close hollow Geometricall Figures,\n(regular and irregular) the straunge properties (in motion or stay) of\nthe Water, Ayre, Smoke, and Fire, in theyr c\u00f5tinuitie, and as they are\nioyned to the Elementes next them.+ This Arte, to the Naturall\nPhilosopher, is very proffitable: to proue, that _Vacuum_, or _Emptines_\nis not in the world. And that, all Nature, abhorreth it so much: that,\ncontrary to ordinary law, the Elementes will moue or stand. As, Water to\nascend: rather then betwene him and Ayre, Space or place should be left,\nmore then (naturally) that qu\u00e3titie of Ayre requireth, or can fill.\nAgaine, Water to hang, and not descend: rather then by descending, to\nleaue Emptines at his backe. The like, is of Fire and Ayre: they will\ndescend: when, either, their C\u00f5tinuitie should be dissolued: or their\nnext Element forced from them. And as they will not be extended, to\ndiscontinuitie: So, will they not, nor yet of mans force, can be prest\nor pent, in space, not sufficient and aunswerable to their bodily\nsubstance. Great force and violence will they vse, to enioy their\nnaturall right and libertie.\n\n    [To go to the bottom of the Sea without daunger.]\n\nHereupon, two or three men together, by keping Ayre vnder a great\nCauldron, and forcyng the same downe, orderly, may without harme descend\nto the Sea bottome: and continue there a tyme &c. Where, Note, how the\nthicker Element (as the Water) giueth place to the thynner (as, is the\nayre:) and receiueth violence of the thinner, in maner. &c. Pumps and\nall maner of Bellowes, haue their ground of this Art: and many other\nstraunge deuises. As, _Hydraulica_, Organes goyng by water. &c. Of this\nFeat, (called commonly _Pneumatica_,) goodly workes are extant, both in\nGreke, and Latin. With old and learned Schole men, it is called\n_Scientia de pleno & vacuo._\n\n\n+\u2021Menadrie\u2021, is an Arte Mathematicall, which demonstrateth, how, aboue\nNatures vertue and power simple: Vertue and force may be multiplied: and\nso, to direct, to lift, to pull to, and to put or cast fro, any\nmultiplied or simple, determined Vertue, Waight or Force: naturally,\nnot, so, directible or moueable.+ Very much is this Art furdred by other\nArtes: as, in some pointes, by _Perspectiue_: in some, by _Statike_: in\nsome, by _Trochilike_: and in other, by _Helicosophie_: and\n_Pneumatithmie_. By this Art, all Cranes, Gybbettes, & Ingines to lift\nvp, or to force any thing, any maner way, are ordred: and the certaine\ncause of their force, is knowne: As, the force which one man hath with\nthe Duche waghen Racke: therwith, to set vp agayne, a mighty waghen\nladen, being ouerthrowne. The force of the Crossebow Racke, is\ncertainly, here, demonstrated. The reason, why one m\u00e3, doth with a\nleauer, lift that, which Sixe men, with their handes onely, could not,\nso easily do. By this Arte, in our common Cranes in London, where powre\nis to Crane vp, the waight of 2000. pound: by two Wheles more (by good\norder added) Arte concludeth, that there may be Craned vp 200000. pound\nwaight &c. So well knew _Archimedes_ this Arte: that he alone, with his\ndeuises and engynes, (twise or thrise) spoyled and discomfited the whole\nArmy and Hoste of the Romaines, besieging _Syracusa_,\n\n    [=Plutarchus in Marco Marcello.=]\n\n_Marcus Marcellus the Consul_, being their Generall Capitaine.\n\n    [=Synesius in Epistolis.=]\n\nSuch huge Stones, so many, with such force, and so farre, did he with\nhis engynes hayle among them, out of the Citie.\n\n    [=Polybius.=]\n\n    [=Plinius.=]\n\n    [=Quintilianus.=]\n\n    [=T. Liuius.=]\n\nAnd by Sea likewise: though their Ships might come to the walls of\n_Syracusa_, yet hee vtterly confounded the Romaine Nauye. What with his\nmighty Stones hurlyng:\n\n    [=* Athen\u00e6us.=]\n\nwhat with Pikes of * 18 fote long, made like shaftes: which he forced\nalmost a quarter of a myle. What, with his catchyng hold of their Shyps,\nand hoysing them vp aboue the water, and suddenly letting them fall into\nthe Sea againe:\n\n    [= * Galenus.=]\n\n    [=Anthemius.=]\n\nwhat with his * Burning Glasses: by which he fired their other Shippes a\nfar-of: what, with his other pollicies, deuises, and engines, he so\nmanfully acquit him selfe: that all the Force, courage, and pollicie of\nthe Romaines (for a great season) could nothing preuaile, for the\nwinning of Syracusa. Wherupon, the Romanes named _Archimedes_,\n_Briareus_, and _Centimanus_. _Zonaras_ maketh mention of one _Proclus_,\nwho so well had perceiued _Archimedes_ Arte of _Menadrie_, and had so\nwell inuented of his owne, that with his Burning Glasses,\n\n    [Burning Glasses.]\n\nbeing placed vpon the walles of Bysance, he multiplied so the heate of\nthe Sunne, and directed the beames of the same against his enemies Nauie\nwith such force, and so sodeinly (like lightening) that he burned and\ndestroyed both man and ship. And _Dion_specifieth of _Priscus_,\na _Geometricien_ in Bysance, who inuented and vsed sondry Engins, of\nForce multiplied: Which was cause, that the _Emperour Seuerus_ pardoned\nhim, his life, after he had wonne Bysance: Bycause he honored the Arte,\nwytt, and rare industrie of _Priscus_. But nothing inferior to the\ninuention of these engines of Force, was the inuention of Gunnes.\n\n    [Gunnes.]\n\nWhich, from an English man, had the occasion and order of first\ninuenting: though in an other land, and by other men, it was first\nexecuted. And they that should see the record, where the occasion and\norder generall, of Gunning, is first discoursed of, would thinke: that,\n\u201csmall thinges, slight, and c\u00f5mon: comming to wise mens consideration,\nand industrious mens handling, may grow to be of force incredible.\u201d\n\n\n+\u2021Hypogeiodie\u2021, is an Arte Mathematicall, demonstratyng, how, vnder the\nSph\u00e6ricall Superficies of the earth, at any depth, to any perpendicular\nline assigned (whose distance from the perpendicular of the entrance:\nand the Azimuth, likewise, in respect of the said entrance, is knowen)\ncertaine way may be pr\u00e6scribed and gone: And how, any way aboue the\nSuperficies of the earth designed, may vnder earth, at any depth\nlimited, be kept: goyng alwayes, perpendicularly, vnder the way, on\nearth designed: And, contrarywise, Any way, (straight or croked,) vnder\nthe earth, beyng giuen: vppon the vtface, or Superficies of the earth,\nto Lyne out the same: So, as, from the Centre of the earth,\nperpendiculars drawen to the Sph\u00e6ricall Superficies of the earth, shall\nprecisely fall in the Correspondent pointes of those two wayes. This,\nwith all other Cases and circumstances herein, and appertenances, this\nArte demonstrateth.+ This Arte, is very ample in varietie of\nConclusions: and very profitable sundry wayes to the Common Wealth. The\noccasion of my Inuenting this Arte, was at the request of two Gentlemen,\nwho had a certaine worke (of gaine) vnder ground: and their groundes did\nioyne ouer the worke: and by reason of the crokednes, diuers depthes,\nand heithes of the way vnder ground, they were in doubt, and at\ncontrouersie, vnder whose ground, as then, the worke was. The name onely\n(before this) was of me published, _De Itinere Subterraneo_: The rest,\nbe at Gods will. For Pioners, Miners, Diggers for Mettalls, Stone, Cole,\nand for secrete passages vnder ground, betwene place and place (as this\nland hath diuerse) and for other purposes, any man may easily perceaue,\nboth the great fruite of this Arte, and also in this Arte, the great\naide of Geometrie.\n\n\n+\u2021Hydragogie\u2021, demonstrateth the possible leading of Water, by Natures\nlawe, and by artificiall helpe, from any head (being a Spring, standing,\nor running Water) to any other place assigned.+ Long, hath this Arte\nbene in vse: and much thereof written: and very marueilous workes\ntherein, performed: as may yet appeare, in Italy: by the Ruynes\nremaining of the Aqueductes. In other places, of Riuers leading through\nthe Maine land, Nauigable many a Mile. And in other places, of the\nmarueilous forcinges of Water to Ascend. which all, declare the great\nskill, to be required of him, who should in this Arte be perfecte, for\nall occasions of waters possible leading. To speake of the allowance of\nthe Fall, for euery hundred foote: or of the Ventills (if the waters\nlabour be farre, and great) I neede not: Seing, at hand (about vs) many\nexpert men can sufficiently testifie, in effecte, the order: though the\nDemonstration of the Necessitie thereof, they know not: Nor yet, if they\nshould be led, vp and downe, and about Mountaines, from the head of the\nSpring: and then, a place being assigned: and of them, to be demaunded,\nhow low or high, that last place is, in respecte of the head, from which\n(so crokedly, and vp and downe) they be come: Perhaps, they would not,\nor could not, very redily, or nerely assoyle that question. _Geometrie_\ntherefore, is necessary to _Hydragogie_. Of the sundry wayes to force\nwater to ascend, eyther by _Tympane_, _Kettell mills_, _Skrue_,\n_Ctesibike_, or such like: in _Vitruuius_, _Agricola_, (and other,)\nfully, the maner may appeare. And so, thereby, also be most euident, how\nthe Artes, of _Pneumatithmie_, _Helicosophie_, _Statike_, _Trochilike_,\nand _Menadrie_, come to the furniture of this, in Speculation, and to\nthe Commoditie of the Common Wealth, in practise.\n\n\n+\u2021Horometrie\u2021, is an Arte Mathematicall, which dem\u00f5strateth, how, at all\ntimes appointed, the precise vsuall denominati\u00f5 of time, may be knowen,\nfor any place assigned.+ These wordes, are smoth and plaine easie\nEnglishe, but the reach of their meaning, is farther, then you woulde\nlightly imagine. Some part of this Arte, was called in olde time,\n_Gnomonice_: and of late, _Horologiographia_: and in Englishe, may be\ntermed, _Dialling_. Auncient is the vse, and more auncient, is the\nInuention. The vse, doth well appeare to haue bene (at the least) aboue\ntwo thousand and three hundred yeare agoe:\n\n    [4. Reg. 20.]\n\nin * King _Achaz_ Diall, then, by the Sunne, shewing the distinction of\ntime. By Sunne, Mone, and Sterres, this Dialling may be performed, and\nthe precise Time of day or night knowen. But the demonstratiue\ndelineation of these Dialls, of all sortes, requireth good skill, both\nof _Astronomie_, and _Geometrie_ Elementall, Sph\u00e6ricall, Ph\u00e6nomenall,\nand Conikall. Then, to vse the groundes of the Arte, for any regular\nSuperficies, in any place offred: and (in any possible apt position\ntherof) theron, to describe (all maner of wayes) how, vsuall howers, may\nbe (by the _Sunnes_ shadow) truely determined: will be found no sleight\nPainters worke. So to Paint, and prescribe the Sunnes Motion, to the\nbreadth of a heare. In this Feate (in my youth) I Inuented a way, +How\nin any Horizontall, Murall, or \u00c6quinoctiall Diall, &c. At all howers\n(the Sunne shining) the Signe and Degree ascendent, may be knowen.+\nWhich is a thing very necessary for the Rising of those fixed Sterres:\nwhose Operation in the Ayre, is of great might, euidently. I speake no\nfurther, of the vse hereof. Bur forasmuch as, Mans affaires require\nknowledge of Times & Momentes, when, neither Sunne, Mone, or Sterre, can\nbe sene: Therefore, by Industrie Mechanicall, was inuented, first, how,\nby Water, running orderly, the Time and howers might be knowen: whereof,\nthe famous _Ctesibius_, was Inuentor: a man, of _Vitruuius_, to the Skie\n(iustly) extolled. Then, after that, by Sand running, were howers\nmeasured: Then, by _Trochilike_ with waight: And of late time, by\n_Trochilike_ with Spring: without waight. All these, by Sunne or Sterres\ndirection (in certaine time) require ouersight and reformation,\naccording to the heauenly \u00c6quinoctiall Motion: besides the in\u00e6qualitie\nof their owne Operation. There remayneth (without parabolicall meaning\nherein) among the Philosophers,\n\n    [A perpetuall Motion.]\n\na more excellent, more commodious, and more marueilous way, then all\nthese: of hauing the motion of the Primouant (or first \u00e6quinoctiall\nmotion,) by Nature and Arte, Imitated: which you shall (by furder search\nin waightier studyes) hereafter, vnderstand more of. And so, it is tyme\nto finish this Annotation, of Tymes distinction, vsed in our common, and\npriuate affaires: The commoditie wherof, no man would want, that can\ntell, how to bestow his tyme.\n\n\n+\u2021Zographie\u2021, is an Arte Mathematicall, which teacheth and\ndemonstrateth, how, the Intersection of all visuall Pyramides, made by\nany playne assigned, (the Centre, distance, and lightes, beyng\ndetermined) may be, by lynes, and due propre colours, represented.+ A\nnotable Arte, is this: and would require a whole Volume, to declare the\nproperty thereof: and the Commodities ensuyng. Great skill of\n_Geometrie_, _Arithmetike_, _Perspectiue_, and _Anthropographie_, with\nmany other particular Artes, hath the _Zographer_, nede of, for his\nperfection. For, the most excellent Painter, (who is but the propre\nMechanicien, & Imitator sensible, of the Zographer) hath atteined to\nsuch perfection, that Sense of Man and beast, haue iudged thinges\npainted, to be things naturall, and not artificiall: aliue, and not\ndead. This Mechanicall Zographer (commonly called the Painter) is\nmeruailous in his skill: and seemeth to haue a certaine diuine power:\nAs, of frendes absent, to make a frendly, present comfort: yea, and of\nfrendes dead, to giue a continuall, silent presence: not onely with vs,\nbut with our posteritie, for many Ages. And so procedyng, Consider, How,\nin Winter, he can shew you, the liuely vew of Sommers Ioy, and riches:\nand in Sommer, exhibite the countenance of Winters dolefull State, and\nnakednes. Cities, Townes, Fortes, Woodes, Armyes, yea whole Kingdomes\n(be they neuer so farre, or greate) can he, with ease, bring with him,\nhome (to any mans Iudgement) as Paternes liuely, of the thinges\nrehearsed. In one little house, can he, enclose (with great pleasure of\nthe beholders,) the portrayture liuely, of all visible Creatures, either\non earth, or in the earth, liuing: or in the waters lying, Creping,\nslyding, or swimming: or of any foule, or fly, in the ayre flying. Nay,\nin respect of the Starres, the Skie, the Cloudes: yea, in the shew of\nthe very light it selfe (that Diuine Creature) can he match our eyes\nIudgement, most nerely. What a thing is this? thinges not yet being, he\ncan represent so, as, at their being, the Picture shall seame (in maner)\nto haue Created them. To what Artificer, is not Picture, a great\npleasure and Commoditie? Which of them all, will refuse the Direction\nand ayde of Picture? The Architect, the Goldsmith, and the Arras Weauer:\nof Picture, make great account. Our liuely Herbals, our portraitures of\nbirdes, beastes, and fishes: and our curious Anatomies, which way, are\nthey most perfectly made, or with most pleasure, of vs beholden? Is it\nnot, by Picture onely? And if Picture, by the Industry of the Painter,\nbe thus commodious and meruailous: what shall be thought of _Zographie_,\nthe Scholemaster of Picture, and chief gouernor? Though I mencion not\n_Sculpture_, in my Table of Artes Mathematicall: yet may all men\nperceiue, How, that _Picture_ and _Sculpture_, are Sisters germaine: and\nboth, right profitable, in a Comm\u00f5 wealth. and of _Sculpture_, aswell as\nof Picture, excellent Artificers haue written great bokes in\ncommendation. Witnesse I take, of _Georgio Vasari_, _Pittore Aretino_:\nof _Pomponius Gauricus_: and other. To these two Artes, (with other,) is\na certaine od Arte, called _Althalmasat_, much beholdyng: more, then the\ncommon _Sculptor_, _Entayler_, _Keruer_, _Cutter_, _Grauer_, _Founder_,\nor _Paynter (&c)_ know their Arte, to be commodious.\n\n\n    [An objection.]\n\n+\u2021Architecture\u2021+, to many may seme not worthy, or not mete, to be\nreckned among the _Artes Mathematicall_. To whom, I thinke good, to giue\nsome account of my so doyng. Not worthy, (will they say,) bycause it is\nbut for building, of a house, Pallace, Church, Forte, or such like,\ngrosse workes. And you, also, defined the _Artes Mathematicall_, to be\nsuch, as dealed with no Materiall or corruptible thing: and also did\ndemonstratiuely procede in their faculty, by Number or Magnitude. First,\n\n    [The Answer.]\n\nyou see, that I count, here, _Architecture_, among those _Artes\nMathematicall_, which are Deriued from the Principals: and you know,\nthat such, may deale with Naturall thinges, and sensible matter. Of\nwhich, \u201csome draw nerer, to the Simple and absolute Mathematicall\nSpeculation, then other do.\n\n    [\u261e]\n\nAnd though, the _Architect_ procureth, enformeth, & directeth, the\n_Mechanicien_, to handworke, & the building actuall, of house, Castell,\nor Pallace, and is chief Iudge of the same: yet, with him selfe (as\nchief _Master_ and _Architect_,) remaineth the Demonstratiue reason and\ncause, of the Mechaniciens worke: in Lyne, plaine, and Solid: by\n_Geometricall_, _Arithmeticall_, _Opticall_, _Musicall_,\n_Astronomicall_, _Cosmographicall_\u201d (& to be brief) by all the former\nDeriued _Artes Mathematicall_, and other Naturall Artes, hable to be\nconfirmed and stablished. If this be so: then, may you thinke, that\n_Architecture_, hath good and due allowance, in this honest Company of\n_Artes Mathematicall_ Deriuatiue. I will, herein, craue Iudgement of two\nmost perfect _Architectes_: the one, being _Vitruuius_, the Romaine: who\ndid write ten bookes thereof, to the Emperour _Augustus_ (in whose daies\nour Heauenly Archemaster, was borne): and the other, _Leo Baptista\nAlbertus_, a Florentine: who also published ten bookes therof.\n_Architectura_ (sayth _Vitruuius_) _est Scientia pluribus disciplinis &\nvarijs eruditionibus ornata: cuius Iudicio probantur omnia, qu\u00e6 ab\nc\u00e6teris Artificibus perficiuntur opera._ That is. +Architecture, is a\nScience garnished with many doctrines & diuerse instructions: by whose\nIudgement, all workes, by other workmen finished, are Iudged.+ It\nfolloweth. _Ea nascitur ex Fabrica, & Ratiocinatione. &c. Ratiocinatio\nautem est, qu\u00e6, res fabricatas, Solertia ac ratione proportionis,\ndemonstrare at[que] explicare potest. +Architecture, groweth of Framing,\nand Reasoning. &c. Reasoning, is that, which of thinges framed, with\nforecast, and proportion: can make demonstration, and manifest\ndeclaration.+_ Againe. _C\u00f9m, in omnibus enim rebus, t\u00f9m maxim\u00e8 etiam in\nArchitectura, h\u00e6c duo insunt: quod significatur, & quod significat.\nSignificatur proposita res, de qua dicitur: hanc autem Significat\nDemonstratio, rationibus doctrinarum explicata. +Forasmuch as, in all\nthinges: therefore chiefly in Architecture, these two thinges are: the\nthing signified: and that which signifieth. The thing propounded,\nwhereof we speake, is the thing Signified. But Demonstration, expressed\nwith the reasons of diuerse doctrines, doth signifie the same thing.+_\nAfter that. _Vt literatus sit, peritus Graphidos, eruditus Geometri\u00e6, &\nOptices non ignarus: instructus Arithmetica: historias complures\nnouerit, Philosophos diligenter audiuerit: Musicam sciuerit: Medicin\u00e6\nnon sit ignarus, responsa Iurisperitor\u0169 nouerit: Astrologiam, C\u00e6li[que]\nrationes cognitas habeat. +An Architect+_ (sayth he) +_ought to\nvnderstand Languages, to be skilfull of Painting, well instructed in\nGeometrie, not ignorant of Perspectiue, furnished with Arithmetike, haue\nknowledge of many histories, and diligently haue heard Philosophers,\nhaue skill of Musike, not ignorant of Physike, know the aunsweres of\nLawyers, and haue Astronomie, and the courses C\u00e6lestiall, in good\nknowledge._+ He geueth reason, orderly, wherefore all these Artes,\nDoctrines, and Instructions, are requisite in an excellent _Architect_.\nAnd (for breuitie) omitting the Latin text, thus he hath. +_Secondly, it\nis behofefull for an Architect to haue the knowledge of Painting: that\nhe may the more easilie fashion out, in patternes painted, the forme of\nwhat worke he liketh. And Geometrie, geueth to Architecture many helpes:\nand first teacheth the Vse of the Rule, and the Cumpasse: wherby\n(chiefly and easilie) the descriptions of Buildinges, are despatched in\nGroundplats: and the directions of Squires, Leuells, and Lines.\nLikewise, by Perspectiue, the Lightes of the heauen, are well led, in\nthe buildinges: from certaine quarters of the world. By Arithmetike, the\ncharges of Buildinges are summed together: the measures are expressed,\nand the hard questions of Symmetries, are by Geometricall Meanes and\nMethods discoursed on. &c. Besides this, of the Nature of thinges (which\nin Greke is called \u03c6\u03c5\u03c3\u03b9\u03bf\u03bb\u03bf\u03b3\u03af\u03b1) Philosophie doth make declaration. Which,\nit is necessary, for an Architect, with diligence to haue learned:\nbecause it hath many and diuers naturall questions: as specially, in\nAqueductes. For in their courses, leadinges about, in the leuell ground,\nand in the mountinges, the naturall Spirites or breathes are ingendred\ndiuers wayes: The hindrances, which they cause, no man can helpe, but\nhe, which out of Philosophie, hath learned the originall causes of\nthinges. Likewise, who soeuer shall read Ctesibius, or Archimedes\nbookes, (and of others, who haue written such Rules) can not thinke, as\nthey do: vnlesse he shall haue receaued of Philosophers, instructions in\nthese thinges. And Musike he must nedes know: that he may haue\nvnderstanding, both of Regular and Mathematicall Musike: that he may\ntemper well his Balistes, Catapultes, and Scorpions. &c. Moreouer, the\nBrasen Vessels, which in Theatres, are placed by Mathematicall order, in\nambries, vnder the steppes: and the diuersities of the soundes (which\ny^e Grecians call \u03b7\u03c7\u03b5\u1fd6\u03b1) are ordred according to Musicall Symphonies &\nHarmonies: being distributed in y^e Circuites, by Diatessaron, Diapente,\nand Diapason. That the conuenient voyce, of the players sound, wh\u1ebd it\ncame to these preparations, made in order, there being increased: with\ny^t increasing, might come more cleare & pleasant, to y^e eares of the\nlokers on. &c. And of Astronomie, is know\u1ebd y^e East, West, South, and\nNorth. The fashion of the heauen, the \u00c6quinox, the Solsticie, and the\ncourse of the sterres. Which thinges, vnleast one know: he can not\nperceiue, any thyng at all, the reason of Horologies. Seyng therfore\nthis ample Science, is garnished, beautified and stored, with so many\nand sundry skils and knowledges: I thinke, that none can iustly account\nthem selues Architectes, of the suddeyne. But they onely, who from their\nchildes yeares, ascendyng by these degrees of knowledges, beyng fostered\nvp with the atteynyng of many Languages and Artes, haue wonne to the\nhigh Tabernacle of Architecture. &c. And to whom Nature hath giuen such\nquicke Circumspection, sharpnes of witt, and Memorie, that they may be\nvery absolutely skillfull in Geometrie, Astronomie, Musike, and the rest\nof the Artes Mathematicall: Such, surmount and passe the callyng, and\nstate, of Architectes:\n\n    [A Mathematicien.]\n\nand are become Mathematiciens. &c. And they are found, seldome. As, in\ntymes past, was Aristarchus Samius: Philolaus, and Archytas, Tarentynes:\nApollonius Perg\u0119us: Eratosthenes Cyreneus: Archimedes, and Scopas,\nSyracusians. Who also, left to theyr posteritie, many Engines and\nGnomonicall workes: by numbers and naturall meanes, inuented and\ndeclared._+\n\n\nThus much, and the same wordes (in sense) in one onely Chapter of this\nInc\u00f5parable _Architect Vitruuius_, shall you finde. And if you should,\nbut take his boke in your hand, and slightly loke thorough it, you would\nsay straight way:\n\n    [Vitruuius.]\n\nThis is _Geometrie_, _Arithmetike_, _Astronomie_, _Musike_,\n_Anthropographie_, _Hydragogie_, _Horometrie_. _&c_. and (to c\u00f5clude)\nthe Storehouse of all workm\u00e3ship. Now, let vs listen to our other Iudge,\nour Florentine, _Leo Baptista_: and narrowly consider, how he doth\ndetermine of _Architecture_. _Sed ante[que] vltra progrediar. &c. +But\nbefore I procede any further +_(sayth he) +_I thinke, that I ought to\nexpresse, what man I would haue to bee allowed an Architect. For, I will\nnot bryng in place a Carpenter: as though you might Compare him to the\nChief Masters of other Artes. For the hand of the Carpenter, is the\nArchitectes Instrument._+\n\n    [VVho is an Architect.]\n\n+_But I will appoint the Architect to be \u201cthat man, who hath the skill,\n(by a certaine and meruailous meanes and way,) both in minde and\nImagination to determine and also in worke to finish: what workes so\neuer, by motion of waight, and cuppling and framyng together of bodyes,\nmay most aptly be Commodious for the worthiest Vses of Man.\u201d And that he\nmay be able to performe these thinges, he hath nede of atteynyng and\nknowledge of the best, and most worthy thynges. &c. The whole Feate of\nArchitecture in buildyng, consisteth in Lineamentes, and in Framyng. And\nthe whole power and skill of Lineamentes, tendeth to this: that the\nright and absolute way may be had, of Coaptyng and ioyning Lines and\nangles: by which, the face of the buildyng or frame, may be comprehended\nand concluded. And it is the property of Lineamentes, to prescribe vnto\nbuildynges, and euery part of them, an apt place, & certaine n\u0169ber:\na worthy maner, and a semely order: that, so, y^e whole forme and figure\nof the buildyng, may rest in the very Lineamentes. &c. And we may\nprescribe in mynde and imagination the whole formes, *\n\n    [* The Immaterialitie of perfect Architecture.]\n\nall material stuffe beyng secluded. Which point we shall atteyne, by\nNotyng and forepointyng the angles, and lines, by a sure and certaine\ndirection and connexion. Seyng then, these thinges, are thus:_+\n\n    [What, Lineament is.]\n\n+_Lineamente, shalbe the certaine and constant prescribyng, conceiued in\nmynde: made in lines and angles: and finished with a learned minde and\nwyt._+ \u201cWe thanke you Master _Baptist_, that you haue so aptly brought\nyour Arte, and phrase therof, to haue some Mathematicall perfection:\n\n    [Note.]\n\nby certaine order, n\u0169ber, forme, figure, and _Symmetrie_ mentall:\u201d all\nnaturall & sensible stuffe set a part. Now, then, it is euident, (Gentle\nreader) how aptely and worthely, I haue preferred _Architecture_, to be\nbred and fostered vp in the Dominion of the pereles _Princesse_,\n_Mathematica_: and to be a naturall Subiect of hers. And the name of\n_Architecture_, is of the principalitie, which this Science hath, aboue\nall other Artes. And _Plato_ affirmeth, the _Architect_ to be _Master_\nouer all, that make any worke. Wherupon, he is neither Smith, nor\nBuilder: nor, separately, any Artificer: but the Hed, the Prouost, the\nDirecter, and Iudge of all Artificiall workes, and all Artificers. For,\nthe true _Architect_, is hable to teach, Demonstrate, distribute,\ndescribe, and Iudge all workes wrought. And he, onely, searcheth out the\ncauses and reasons of all Artificiall thynges. Thus excellent, is\n_Architecture_: though few (in our dayes) atteyne thereto: yet may not\nthe Arte, be otherwise thought on, then in very dede it is worthy. Nor\nwe may not, of auncient Artes, make new and imperfect Definitions in our\ndayes: for scarsitie of Artificers: No more, than we may pynche in, the\nDefinitions of _Wisedome_, or _Honestie_, or of _Frendeshyp_ or of\n_Iustice_. No more will I consent, to Diminish any whit, of the\nperfection and dignitie, (by iust cause) allowed to absolute\n_Architecture_. Vnder the Direction of this Arte, are thre principall,\nnecessary _Mechanicall Artes_. Namely, _Howsing_, _Fortification_, and\n_Naupegie_. _Howsing_, I vnderstand, both for Diuine Seruice, and Mans\ncommon vsage: publike, and priuate. Of _Fortification_ and _Naupegie_,\nstraunge matter might be told you: But perchaunce, some will be tyred,\nwith this Bederoll, all ready rehearsed: and other some, will nycely nip\nmy grosse and homely discoursing with you: made in post hast: for feare\nyou should wante this true and frendly warnyng, and tast giuyng, of the\n_Power Mathematicall_. Lyfe is short, and vncertaine: Tymes are\nperilouse: &c. And still the Printer awayting, for my pen staying: All\nthese thinges, with farder matter of Ingratefulnes, giue me occasion to\npasse away, to the other Artes remainyng, with all spede possible.\n\n\n+The Arte of \u2021Nauigation\u2021, demonstrateth how, by the shortest good way,\nby the aptest Directi\u00f5, & in the shortest time, a sufficient Ship,\nbetwene any two places (in passage Nauigable,) assigned: may be\nc\u00f5ducted: and in all stormes, & naturall disturbances chauncyng, how, to\nvse the best possible meanes, whereby to recouer the place first\nassigned.+ What nede, the _Master Pilote_, hath of other Artes, here\nbefore recited, it is easie to know: as, of _Hydrographie_,\n_Astronomie_, _Astrologie_, and _Horometrie_. Presupposing continually,\nthe common Base, and foundacion of all: namely _Arithmetike_ and\n_Geometrie_. So that, he be hable to vnderstand, and Iudge his own\nnecessary Instrumentes, and furniture Necessary: Whether they be\nperfectly made or no: and also can, (if nede be) make them, hym selfe.\nAs Quadrantes, The Astronomers Ryng, The Astronomers staffe, The\nAstrolabe vniuersall. An Hydrographicall Globe. Charts Hydrographicall,\ntrue, (not with parallell Meridians). The Common Sea Compas: The Compas\nof variacion: The Proportionall, and Paradoxall Compasses\n\n    [Anno. 1559.]\n\n(of me Inuented, for our two Moscouy Master Pilotes, at the request of\nthe Company) Clockes with spryng: houre, halfe houre, and three houre\nSandglasses: & sundry other Instrum\u1ebdtes: And also, be hable, on Globe,\nor Playne to describe the Paradoxall Compasse: and duely to vse the\nsame, to all maner of purposes, whereto it was inuented. And also, be\nhable to Calculate the Planetes places for all tymes.\n\n\nMoreouer, with Sonne Mone or Sterre (or without) be hable to define the\nLongitude & Latitude of the place, which he is in: So that, the\nLongitude & Latitude of the place, from which he sayled, be giuen: or by\nhim, be knowne. whereto, appertayneth expert meanes, to be certified\neuer, of the Ships way. &c. And by foreseing the Rising, Settyng,\nNonestedyng, or Midnightyng of certaine tempestuous fixed Sterres: or\ntheir Coniunctions, and Anglynges with the Planetes, &c. he ought to\nhaue expert coniecture of Stormes, Tempestes, and Spoutes: and such lyke\nMeteorologicall effectes, daungerous on Sea. For (as _Plato_ sayth,)\n_Mutationes, opportunitates[que] temporum presentire, non minus rei\nmilitari, qu\u00e0m Agricultur\u00e6, Nauigationi[que] conuenit. +To foresee the\nalterations and opportunities of tymes, is conuenient, no lesse to the\nArt of Warre, then to Husbandry and Nauigation.+_ And besides such\ncunnyng meanes, more euident tokens in Sonne and Mone, ought of hym to\nbe knowen: such as (the Philosophicall Po\u00ebte) _Virgilius_ teacheth, in\nhys _Georgikes_. Where he sayth,\n\n    [Sidenote: Georgic. 1.]\n\n  _Sol quo[que] & exoriens & quum se condet in vndas,\n  Signa dabit, Solem certissima signa sequuntur. &c.\n    -------- Nam s\u00e6pe videmus,\n  Ipsius in vultu varios errare colores.\n  C\u00e6ruleus, pluuiam denunciat, igneus Euros.\n  Sin macul\u00e6 incipient rutilo immiscerier igni,\n  Omnia tum pariter vento, nimbis[que] videbis\n  Feruere: non illa quisquam me nocte per altum\n  Ire, ne[que] a terra moueat conuellere funem. &c.\n  Sol tibi signa dabit. Solem quis dicere falsum\n  Audeat? -------- &c._\n\nAnd so of Mone, Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, and\nBeastes, and of many thynges els, a certaine Sympathicall forewarnyng\nmay be had: sometymes to great pleasure and proffit, both on Sea and\nLand. Sufficiently, for my present purpose, it doth appeare, by the\npremisses, how _Mathematicall_, the _Arte_ of _Nauigation_, is: and how\nit nedeth and also vseth other _Mathematicall Artes_: And now, if I\nwould go about to speake of the manifold Commodities, commyng to this\nLand, and others, by Shypps and _Nauigation_, you might thinke, that I\ncatch at occasions, to vse many wordes, where no nede is.\n\n\nYet, this one thyng may I, (iustly) say. In _Nauigation_, none ought to\nhaue greater care, to be skillfull, then our English Pylotes. And\nperchaunce, Some, would more attempt: And other Some, more willingly\nwould be aydyng, it they wist certainely, What Priuiledge, God had\nendued this Iland with, by reason of Situation, most commodious for\n_Nauigation_, to Places most Famous & Riche. And though,\n\n    [* Anno. 1567 S. H. G.]\n\n(of * Late) a young Gentleman, a Courragious Capitaine, was in a great\nreadynes, with good hope, and great causes of persuasion, to haue\nventured, for a Discouerye, (either _Westerly_, by _Cape de Paramantia_:\nor _Esterly_, aboue _Noua Zemla_, and the _Cyremisses_) and was, at the\nvery nere tyme of Attemptyng, called and employed otherwise (both then,\nand since,) in great good seruice to his Countrey, as the Irish Rebels\nhaue * tasted:\n\n    [* Anno. 1569]\n\nYet, I say, (though the same Gentleman, doo not hereafter, deale\ntherewith) Some one, or other, should listen to the Matter: and by good\naduise, and discrete Circumspection, by little, and little, wynne to the\nsufficient knowledge of that +Trade+ and +Voyage+: Which, now, I would\nbe sory, (through Carelesnesse, want of Skill, and Courrage,) should\nremayne Vnknowne and vnheard of. Seyng, also, we are herein, halfe\nChallenged, by the learned, by halfe request, published. Therof, verely,\nmight grow Commoditye, to this Land chiefly, and to the rest of the\nChristen Common wealth, farre passing all riches and worldly Threasure.\n\n\n+\u2021Thaumaturgike\u2021, is that Art Mathematicall, which giueth certaine order\nto make straunge workes, of the sense to be perceiued, and of men\ngreatly to be wondred at.+ By sundry meanes, this _Wonder-worke_ is\nwrought. Some, by _Pneumatithmie_. As the workes of _Ctesibius_ and\n_Hero_, Some by waight. wherof _Tim\u00e6us_ speaketh. Some, by Stringes\nstrayned, or Springs, therwith Imitating liuely Motions. Some, by other\nmeanes, as the Images of Mercurie: and the brasen hed, made by _Albertus\nMagnus_, which dyd seme to speake. _Boethius_ was excellent in these\nfeates. To whom, _Cassiodorus_ writyng, sayth. +_Your purpose is to know\nprofound thynges: and to shew meruayles. By the disposition of your\nArte, Metals do low: Diomedes of brasse, doth blow a Trumpet loude:\na brasen Serpent hisseth: byrdes made, sing swetely. Small thynges we\nrehearse of you, who can Imitate the heauen. &c._+ Of the straunge\nSelfmouyng, which, at Saint Denys, by Paris,\n\n    [* Anno. 1551]\n\n* I saw, ones or twise (_Orontius_ beyng then with me, in Company) it\nwere to straunge to tell. But some haue written it. And yet, (I hope) it\nis there, of other to be sene. And by _Perspectiue_ also straunge\nthinges, are done. As partly (before) I gaue you to vnderstand in\n_Perspectiue_. As, to see in the Ayre, a loft, the lyuely Image of an\nother man, either walkyng to and fro: or standyng still. Likewise, to\ncome into an house, and there to see the liuely shew of Gold, Siluer or\nprecious stones: and commyng to take them in your hand, to finde nought\nbut Ayre. Hereby, haue some men (in all other matters counted wise)\nfouly ouershot th\u1ebd selues: misdeaming of the meanes. Therfore sayd\n_Claudius C\u00e6lestinus_.\n\n    [De his qu\u00e6 Mundo mirabiliter eueniunt. cap. 8.]\n\n_Hodie magn\u00e6 literatur\u00e6 viros & magna reputationis videmus, opera quedam\nquasi miranda, supra Natur\u00e3 putare: de quibus in Perspectiua doctus\ncausam faciliter reddidisset._ That is. +_Now a dayes, we see some men,\nyea of great learnyng and reputation, to Iudge certain workes as\nmeruaylous, aboue the power of Nature: Of which workes, one that were\nskillfull in Perspectiue might easely haue giuen the Cause._+ Of\n_Archimedes Sph\u00e6re_, _Cicero_ witnesseth.\n\n    [Tusc. 1.]\n\nWhich is very straunge to thinke on. +_For when Archimedes_+ (sayth he)\n+_did fasten in a Sph\u00e6re, the mouynges of the Sonne, Mone, and of the\nfiue other Planets, he did, as the God, which (in Tim\u00e6us of Plato) did\nmake the world. That, one turnyng, should rule motions most vnlike in\nslownes, and swiftnes._+ But a greater cause of meruayling we haue by\n_Claudianus_ report hereof. Who affirmeth this _Archimedes worke_, to\nhaue ben of Glasse. And discourseth of it more at large: which I omit.\nThe Doue of wood, which the _Mathematicien Archytas_ did make to flye,\nis by _Agellius_ spoken of. Of _D\u00e6dalus_ straunge Images, _Plato_\nreporteth. _Homere_ of _Vulcans Selfmouers_, (by secret wheles) leaueth\nin writyng. _Aristotle_, in hys _Politikes_, of both, maketh mention.\nMeruaylous was the workemanshyp, of late dayes, performed by good skill\nof _Trochilike. &c._ For in Noremberge, A flye of Iern, beyng let out of\nthe Artificers hand, did (as it were) fly about by the gestes, at the\ntable, and at length, as though it were weary, retourne to his masters\nhand agayne. Moreouer, an Artificiall Egle, was ordred, to fly out of\nthe same Towne, a mighty way, and that a loft in the Ayre, toward the\nEmperour comming thether: and followed hym, beyng come to the gate of\nthe towne. *\n\n    [* \u261e]\n\nThus, you see, what, Arte Mathematicall can performe, when Skill, will,\nIndustry, and Hability, are duely applyed to profe.\n\n\n    [A Digression.]\n\nAnd for these, and such like marueilous Actes and Feates, Naturally,\nMathematically, and Mechanically, wrought and contriued:\n\n    [Apologeticall.]\n\nought any honest Student, and Modest Christian Philosopher, be counted,\n& called a +Coniurer+? Shall the folly of Idiotes, and the Mallice of\nthe Scornfull, so much preuaile, that He, who seeketh no worldly gaine\nor glory at their handes: But onely, of God, the threasor of heauenly\nwisedome, & knowledge of pure veritie: Shall he (I say) in the meane\nspace, be robbed and spoiled of his honest name and fame? He that seketh\n(by S. Paules aduertisement) in the Creatures Properties, and wonderfull\nvertues, to finde iuste cause, to glorifie the \u00c6ternall, and Almightie\nCreator by: Shall that man, be (in hugger mugger) condemned, as a\nCompanion of the Helhoundes, and a Caller, and Coniurer of wicked and\ndamned Spirites? He that bewaileth his great want of time, sufficient\n(to his contentation) for learning of Godly wisdome, and Godly Verities\nin: and onely therin setteth all his delight: Will that m\u00e3 leese and\nabuse his time, in dealing with the Chiefe enemie of Christ our Redemer:\nthe deadly foe of all mankinde: the subtile and impudent peruerter of\nGodly Veritie: the Hypocriticall Crocodile: the Enuious Basiliske,\ncontinually desirous, in the twinke of an eye, to destroy all Mankinde,\nboth in Body and Soule, \u00e6ternally? Surely (for my part, somewhat to say\nherein) I haue not learned to make so brutish, and so wicked a Bargaine.\nShould I, for my xx. or xxv. yeares Studie: for two or three thousand\nMarkes spending: seuen or eight thousand Miles going and trauailing,\nonely for good learninges sake: And that, in all maner of wethers: in\nall maner of waies and passages: both early and late: in daunger of\nviolence by man: in daunger of destruction by wilde beastes: in hunger:\nin thirst: in perilous heates by day, with toyle on foote: in daungerous\ndampes of colde, by night, almost bereuing life: (as God knoweth): with\nlodginges, oft times, to small ease: and somtime to lesse securitie. And\nfor much more (then all this) done & suffred, for Learning and attaining\nof Wisedome: Should I (I pray you) for all this, no otherwise, nor more\nwarily: or (by Gods mercifulnes) no more luckily, haue fished, with so\nlarge, and costly, a Nette, so long time in drawing (and that with the\nhelpe and aduise of Lady Philosophie, & Queene Theologie): but at\nlength, to haue catched, and drawen vp, * a Frog?\n\n    [* A prouerb. Fayre fisht, and caught a Frog.]\n\nNay, a Deuill? For, so, doth the Common peuish Pratler Imagine and\nIangle: And, so, doth the Malicious skorner, secretly wishe, & brauely\nand boldly face down, behinde my backe. Ah, what a miserable thing, is\nthis kinde of Men? How great is the blindnes & boldnes, of the\nMultitude, in thinges aboue their Capacitie? What a Land: what a People:\nwhat Maners: what Times are these? Are they become Deuils, them selues:\nand, by false witnesse bearing against their Neighbour, would they also,\nbecome Murderers? Doth God, so long geue them respite, to reclaime them\nselues in, from this horrible slaundering of the giltlesse: contrary to\ntheir owne Consciences: and yet will they not cease? Doth the Innocent,\nforbeare the calling of them, Iuridically to aunswere him, according to\nthe rigour of the Lawes: and will they despise his Charitable pacience?\nAs they, against him, by name, do forge, fable, rage, and raise\nslaunder, by Worde & Print: Will they prouoke him, by worde and Print,\nlikewise, to Note their Names to the World: with their particular\ndeuises, fables, beastly Imaginations, and vnchristen-like slaunders?\nWell: Well. O (you such) my vnkinde Countrey men. O vnnaturall Countrey\nmen. O vnthankfull Countrey men. O Brainsicke, Rashe, Spitefull, and\nDisdainfull Countrey men. Why oppresse you me, thus violently, with your\nslaundering of me: Contrary to Veritie: and contrary to your owne\nConsciences? And I, to this hower, neither by worde, deede, or thought,\nhaue bene, any way, hurtfull, damageable, or iniurious to you, or yours?\nHaue I, so long, so dearly, so farre, so carefully, so painfully, so\ndaungerously sought & trauailed for the learning of Wisedome, &\natteyning of Vertue: And in the end (in your iudgem\u1ebdt) am I become,\nworse, then when I beg\u00e3? Worse, th\u1ebd a Mad man? A dangerous Member in the\nCommon Wealth: and no Member of the Church of Christ? Call you this, to\nbe Learned? Call you this, to be a Philosopher? and a louer of Wisedome?\nTo forsake the straight heauenly way: and to wallow in the broad way of\ndamnation? To forsake the light of heauenly Wisedome: and to lurke in\nthe dungeon of the Prince of darkenesse? To forsake the Veritie of God,\n& his Creatures: and to fawne vpon the Impudent, Craftie, Obstinate\nLier, and continuall disgracer of Gods Veritie, to the vttermost of his\npower? To forsake the Life & Blisse \u00c6ternall: and to cleaue vnto the\nAuthor of Death euerlasting? that Murderous Tyrant, most gredily\nawaiting the Pray of Mans Soule? Well: I thanke God and our Lorde Iesus\nChrist, for the Comfort which I haue by the Examples of other men,\nbefore my time: To whom, neither in godlines of life, nor in perfection\nof learning, I am worthy to be compared: and yet, they sustained the\nvery like Iniuries, that I do: or rather, greater. Pacient _Socrates_,\nhis _Apologie_ will testifie: _Apuleius_ his _Apologies_, will declare\nthe Brutishnesse of the Multitude. _Ioannes Picus_, Earle of Mirandula,\nhis _Apologie_ will teach you, of the Raging slaunder of the Malicious\nIgnorant against him. _Ioannes Trithemius_, his _Apologie_ will\nspecifie, how he had occasion to make publike Protestation: as well by\nreason of the Rude Simple: as also, in respect of such, as were counted\nto be of the wisest sort of men. \u201cMany could I recite: But I deferre the\nprecise and determined handling of this matter: being loth to detect the\nFolly & Mallice of my Natiue Countrey men. *\n\n    [* \u261e]\n\nWho, so hardly, can disgest or like any extraordinary course of\nPhilosophicall Studies: not falling within the Cumpasse of their\nCapacitie: or where they are not made priuie of the true and secrete\ncause, of such wonderfull Philosophicall Feates.\u201d These men, are of\nfower sortes, chiefly. The first, I may name, _Vaine pratling busie\nbodies_: The second, _Fond Frendes_: The third, _Imperfectly zelous_:\nand the fourth, _Malicious Ignorant_. To eche of these (briefly, and in\ncharitie) I will say a word or two, and so returne to my Pr\u00e6face.\n\n    [1.]\n\n_Vaine pratling busie bodies_, vse your idle assemblies, and\nconferences, otherwise, then in talke of matter, either aboue your\nCapacities, for hardnesse: or contrary to your Consciences, in Veritie.\n\n    [2.]\n\n_Fonde Frendes_, leaue of, so to commend your vnacquainted frend, vpon\nblinde affection: As, because he knoweth more, then the common Student:\nthat, therfore, he must needes be skilfull, and a doer, in such matter\nand maner, as you terme _Coniuring_. Weening, thereby, you aduaunce his\nfame: and that you make other men, great marueilers of your hap, to haue\nsuch a learned frend. Cease to ascribe Impietie, where you pretend\nAmitie. For, if your tounges were true, then were that your frend,\n_Vntrue_, both to God, and his Soueraigne. Such _Frendes_ and\n_Fondlinges_, I shake of, and renounce you: Shake you of, your Folly.\n\n    [3.]\n\n_Imperfectly zelous_, to you, do I say: that (perhaps) well, do you\nMeane: But farre you misse the Marke: If a Lambe you will kill, to feede\nthe flocke with his bloud. Sheepe, with Lambes bloud, haue no naturall\nsustenaunce: No more, is Christes flocke, with horrible slaunders, duely\n\u00e6dified. Nor your faire pretense, by such rashe ragged Rhetorike, any\nwhit, well graced. But such, as so vse me, will finde a fowle Cracke in\ntheir Credite. Speake that you know: And know, as you ought: Know not,\nby Heare say, when life lieth in daunger. Search to the quicke, & let\nCharitie be your guide.\n\n    [4.]\n\n_Malicious Ignorant_, what shall I say to thee? _Prohibe linguam tuam a\nmalo. A detractione parcite lingu\u00e6. +Cause thy toung to refraine fr\u00f5\neuill. Refraine your toung from slaunder.+_ Though your tounges be\nsharpned, Serpent like, & Adders poyson lye in your lippes:\n\n    [Psal. 140.]\n\nyet take heede, and thinke, betimes, with your selfe, _Vir linguosus non\nstabilietur in terra. Virum violentum venabitur malum, donec\npr\u00e6cipitetur._ For, sure I am, _Quia faciet Dominus Iudicium afflicti:\n& vindictam pauperum._\n\n\nThus, I require you, my assured frendes, and Countrey men (you\nMathematiciens, Mechaniciens, and Philosophers, Charitable and discrete)\nto deale in my behalf, with the light & vntrue tounged, my enuious\nAduersaries, or Fond frends. And farther, I would wishe, that at leysor,\nyou would consider, how _Basilius Magnus_, layeth _Moses_ and _Daniel_,\nbefore the eyes of those, which count all such Studies Philosophicall\n(as mine hath bene) to be vngodly, or vnprofitable. Waye well\n_S. Stephen_ his witnesse of _Moses_.\n\n    [Act. 7. C.]\n\n_Eruditus est Moses omni Sapientia \u00c6gyptior\u0169: & erat potens in verbis &\noperibus suis. +Moses was instructed in all maner of wisedome of the\n\u00c6gyptians: and he was of power both in his wordes, and workes.+_ You see\nthis Philosophicall Power & Wisedome, which _Moses_ had, to be nothing\nmisliked of the Holy Ghost. Yet _Plinius_ hath recorded, _Moses_ to be a\nwicked _Magicien_. And that (of force) must be, either for this\nPhilosophicall wisedome, learned, before his calling to the leading of\nthe Children of _Israel_: or for those his wonders, wrought before King\n_Pharao_, after he had the conducting of the _Israelites_. As concerning\nthe first, you perceaue, how _S. Stephen_, at his Martyrdome (being full\nof the Holy Ghost) in his Recapitulation of the olde Testament, hath\nmade mention of _Moses_ Philosophie: with good liking of it: And\n_Basilius Magnus_ also, auoucheth it, to haue bene to _Moses_ profitable\n(and therefore, I say, to the Church of God, necessary). But as\nc\u00f5cerning _Moses_ wonders, done before King _Pharao_: God, him selfe,\nsayd: _Vide vt omnia ostenta, qu\u00e6 posui in manu tua, facias coram\nPharaone. +See that thou do all those wonders before Pharao, which I\nhaue put in thy hand.+_ Thus, you euidently perceaue, how rashly,\n_Plinius_ hath slaundered _Moses_,\n\n    [Lib. 30. Cap. 1.]\n\nof vayne fraudulent _Magike_, saying: _Est & alia Magices Factio,\na Mose, Iamne, & Iotape, Iud\u00e6is pendens: sed multis millibus annorum\npost Zoroastrem. &c._\n\n    [1.]\n\nLet all such, therefore, who, in Iudgement and Skill of Philosophie, are\nfarre Inferior to _Plinie_, \u201ctake good heede, least they ouershoote them\nselues rashly,\u201d in\n\n    [\u261e]\n\nIudging of _Philosophers straunge Actes_: and the Meanes, how they are\ndone.\n\n    [2.]\n\nBut, much more, ought they to beware of forging, deuising, and imagining\nmonstrous feates, and wonderfull workes, when and where, no such were\ndone: no, not any sparke or likelihode, of such, as they, without all\nshame, do report.\n\n    [3.]\n\nAnd (to conclude) most of all, let them be ashamed of Man, and afraide\nof the dreadfull and Iuste Iudge: both Folishly or Maliciously to\ndeuise: and then, deuilishly to father their new fond Monsters on me:\nInnocent, in hand and hart: for trespacing either against the lawe of\nGod, or Man, in any my Studies or Exercises, Philosophicall, or\nMathematicall: As in due time, I hope, will be more manifest.\n\n\nNow end I, with +\u2021Archemastrie\u2021+. Which name, is not so new, as this\nArte is rare. For an other Arte, vnder this, a degree (for skill and\npower) hath bene indued with this English name before. And yet, this,\nmay serue for our purpose, sufficiently, at this present. +This Arte,\nteacheth to bryng to actuall experience sensible, all worthy conclusions\nby all the Artes Mathematicall purposed, & by true Naturall Philosophie\nconcluded: & both addeth to them a farder scope, in the termes of the\nsame Artes, & also by hys propre Method, and in peculier termes,\nprocedeth, with helpe of the foresayd Artes, to the performance of\ncomplet Experi\u1ebdces, which of no particular Art, are hable (Formally) to\nbe challenged.+ If you remember, how we considered _Architecture_, in\nrespect of all common handworkes: some light may you haue, therby, to\nvnderstand the Souerainty and propertie of this Science. _Science_ I may\ncall it, rather, then an Arte: for the excellency and Mastershyp it\nhath, ouer so many, and so mighty Artes and Sciences. And bycause it\nprocedeth by _Experiences_, and searcheth forth the causes of\nConclusions, by _Experiences_: and also putteth the Conclusions them\nselues, in _Experience_, it is named of some, _Scientia Experimentalis_.\nThe +_Experimentall Science_+. _Nicolaus Cusanus_ termeth it so, in hys\n_Experimentes Statikall_, And an other _Philosopher_,\n\n    [R. B.]\n\nof this land Natiue (the floure of whose worthy fame, can neuer dye nor\nwither) did write therof largely, at the request of _Clement the sixt_.\nThe Arte carrieth with it, a wonderfull Credit: By reason, it\ncertefieth, sensibly, fully, and completely to the vtmost power of\nNature, and Arte. This Arte, certifieth by _Experience_ complete and\nabsolute: and other Artes, with their Argumentes, and Demonstrations,\npersuade: and in wordes, proue very well their Conclusions. *\n\n    [\u261e]\n\nBut wordes, and Argumentes, are no sensible certifying: nor the full and\nfinall frute of Sciences practisable. And though some Artes, haue in\nthem, _Experiences_, yet they are not complete, and brought to the\nvttermost, they may be stretched vnto, and applyed sensibly. As for\nexample: the Naturall Philosopher disputeth and maketh goodly shew of\nreason: And the Astronomer, and the Opticall Mechanicien, put some\nthynges in _Experience_: but neither, all, that they may: nor yet\nsufficiently, and to the vtmost, those, which they do, There, then, the\n_Archemaster_ steppeth in, and leadeth forth on, the _Experiences_, by\norder of his doctrine _Experimentall_, to the chief and finall power of\nNaturall and Mathematicall Artes. Of two or three men, in whom, this\nDescription of _Archemastry_ was _Experimentally_, verified, I haue read\nand hard: and good record, is of their such perfection. So that, this\nArt, is no fantasticall Imagination: as some Sophister, might, _Cum suis\nInsolubilibus_, make a florish: and dassell your Imagination: and dash\nyour honest desire and Courage, from beleuing these thinges, so vnheard\nof, so meruaylous, & of such Importance. Well: as you will. I haue\nforewarned you. I haue done the part of a frende: I haue discharged my\nDuety toward God: for my small Talent, at hys most mercyfull handes\nreceiued. To this Science, doth the _Science Alnirangiat_, great\nSeruice. Muse nothyng of this name. I chaunge not the name, so vsed, and\nin Print published by other: beyng a name, propre to the Science. Vnder\nthis, commeth _Ars Sintrillia_, by _Artephius_, briefly written. But the\nchief Science, of the Archemaster, (in this world) as yet knowen, is an\nother (as it were) OPTICAL Science: wherof, the name shall be told (God\nwillyng) when I shall haue some, (more iust) occasion, therof, to\nDiscourse.\n\n\nHere, I must end, thus abruptly (Gentle frende, and vnfayned louer of\nhonest and necessary verities.) For, they, who haue (for your sake, and\nvertues cause) requested me, (an old forworne Mathematicien) to take pen\nin hand: (through the confidence they reposed in my long experience: and\ntryed sincerity) for the declaryng and reportyng somewhat, of the frute\nand commodity, by the +Artes Mathematicall, to be atteyned vnto+: euen\nthey, Sore agaynst their willes, are forced, for sundry causes, to\nsatisfie the workemans request, in endyng forthwith: He, so feareth\nthis, so new an attempt, & so costly: And in matter so slenderly\n(hetherto) among the common Sorte of Studentes, considered or estemed.\n\n\nAnd where I was willed, somewhat to alledge, why, in our vulgare Speche,\nthis part of the Principall Science of _Geometrie_, called _Euclides\nGeometricall Elementes_, is published, to your handlyng: being vnlatined\npeople, and not Vniuersitie Scholers: Verily, I thinke it nedelesse.\n\n\n    [1.]\n\nFor, the Honour, and Estimation of the +Vniuersities, and Graduates+,\nis, hereby, nothing diminished. Seing, from, and by their Nurse\nChildren, you receaue all this Benefite: how great soeuer it be.\n\n\n    [2.]\n\nNeither are their Studies, hereby, any whit hindred. No more, then the\nItalian _Vniuersities_, as _Academia Bononiensis_, _Ferrariensis_,\n_Florentina_, _Mediolanensis_, _Patauina_, _Papiensis_, _Perusina_,\n_Pisana_, _Romana_, _Senensis_, or any one of them, finde them selues,\nany deale, disgraced, or their Studies any thing hindred, by _Frater\nLucas de Burgo_, or by _Nicolaus Tartalea_, who in vulgar Italian\nlanguage, haue published, not onely _Euclides Geometrie_, but of\n_Archimedes_ somewhat: and in Arithmetike and Practicall Geometrie, very\nlarge volumes, all in their vulgar speche. Nor in Germany haue the\nfamous _Vniuersities_, any thing bene discontent with _Albertus\nDurerus_, his Geometricall Institutions in Dutch: or with _Gulielmus\nXylander_, his learned translation of the first sixe bookes of\n_Euclide_, out of the Greke into the high Dutch. Nor with _Gualterus H.\nRiffius_, his Geometricall Volume: very diligently translated into the\nhigh Dutch tounge, and published. Nor yet the _Vniuersities_ of Spaine,\nor Portugall, thinke their reputation to be decayed: or suppose any\ntheir Studies to be hindred by the Excellent _P. Nonnius_, his\nMathematicall workes, in vulgare speche by him put forth. Haue you not,\nlikewise, in the French tounge, the whole Mathematicall Quadriuie? and\nyet neither Paris, Orleance, or any of the other Vniuersities of\nFraunce, at any time, with the Translaters, or Publishers offended: or\nany mans Studie thereby hindred?\n\n\n    [3.]\n\nAnd surely, the Common and Vulgar Scholer (much more, the Gramarian)\nbefore his comming to the _Vniuersitie_, shall (or may) be, now\n(according to _Plato_ his Counsell) sufficiently instructed in\n_Arithmetike_ and _Geometrie_, for the better and easier learning of all\nmaner of _Philosophie_, _Academicall_, or _Peripateticall_. And by that\nmeanes, goe more cherefully, more skilfully, and spedily forwarde, in\nhis Studies, there to be learned. And, so, in lesse time, profite more,\nthen (otherwise) he should, or could do.\n\n\n    [4.]\n\nAlso many good and pregnant Englishe wittes, of young Gentlemen, and of\nother, who neuer intend to meddle with the profound search and Studie of\nPhilosophie (in the _Vniuersities_ to be learned) may neuerthelesse,\nnow, with more ease and libertie, haue good occasion, vertuously to\noccupie the sharpnesse of their wittes: where, els (perchance)\notherwise, they would in fond exercises, spend (or rather leese) their\ntime: neither seruing God: nor furdering the Weale, common or priuate.\n\n\n    [5.]\n\nAnd great Comfort, with good hope, may the _Vniuersities_ haue, by\nreason of this _Englishe_ +Geometrie, and Mathematicall Pr\u00e6face+, that\nthey (hereafter) shall be the more regarded, esteemed, and resorted\nvnto. For, when it shall be knowen and reported, that of the\n_Mathematicall Sciences_ onely, such great Commodities are ensuing (as I\nhaue specified): and that in dede, some of you vnlatined Studentes, can\nbe good witnesse, of such rare fruite by you enioyed (thereby): as\neither, before this, was not heard of: or els, not so fully credited:\n\u201cWell, may all men coniecture, that farre greater ayde, and better\nfurniture, to winne to the Perfection of all Philosophie,\n\n    [Vniuersities.]\n\nmay in the Vniuersities be had: being the Storehouses & Threasory of all\nSciences,\n\n    [\u261e]\n\nand all Artes, necessary for the best, and most noble State of Common\nWealthes.\u201d\n\n    [6.]\n\nBesides this, how many a Common Artificer, is there, in these Realmes of\nEngland and Ireland, that dealeth with Numbers, Rule, & Cumpasse: Who,\nwith their owne Skill and experience, already had, will be hable (by\nthese good helpes and informations) to finde out, and deuise, new\nworkes, straunge Engines, and Instrumentes: for sundry purposes in the\nCommon Wealth? or for priuate pleasure? and for the better maintayning\nof their owne estate? I will not (therefore) fight against myne owne\nshadowe. For, no man (I am sure) will open his mouth against this\nEnterprise. No m\u00e3 (I say) who either hath Charitie toward his brother\n(and would be glad of his furtherance in vertuous knowledge): or that\nhath any care & zeale for the bettering of the C\u00f5mon state of this\nRealme. Neither any, that make accompt, what the wiser sort of men (Sage\nand Stayed) do thinke of them. To none (therefore) will I make any\n_Apologie,_ for a vertuous acte doing: and for c\u00f5mending, or setting\nforth, Profitable Artes to English men, in the English toung. \u201cBut, vnto\nGod our Creator, let vs all be thankefull: for that, +_As he, of his\nGoodnes, by his Powre, and in his wisedome,\n\n    [\u261e]\n\nhath Created all thynges, in Number, Waight, and Measure_+: So, to vs,\nof hys great Mercy, he hath reuealed Meanes, whereby, to atteyne the\nsufficient and necessary knowledge of the foresayd hys three principall\nInstrumentes: Which Meanes, I haue abundantly proued vnto you, to be the\n_Sciences_ and _Artes Mathematicall_.\u201d\n\n\nAnd though I haue ben pinched with straightnes of tyme: that, no way,\nI could so pen downe the matter (in my Mynde) as I determined: hopyng of\nconuenient laysure: Yet. if vertuous zeale, and honest Intent prouoke\nand bryng you to the readyng and examinyng of this Compendious treatise,\nI do not doute, but, as the veritie therof (accordyng to our purpose)\nwill be euident vnto you: So the pith and force therof, will persuade\nyou: and the wonderfull frute therof, highly pleasure you. And that you\nmay the easier perceiue, and better remember, the principall pointes,\nwhereof my Preface treateth,\n\n    [The Ground platt of this Pr\u00e6face in a Table.]\n\nI will giue you the +Groundplatt+ of my whole discourse, in a Table\nannexed: from the first to the last, somewhat Methodically contriued.\n\n\nIf Hast, hath caused my poore pen, any where, to stumble: You will,\n  (I am sure) in part of recompence, (for my earnest and sincere\n    good will to pleasure you), Consider the rockish huge\n      mountaines, and the perilous vnbeaten wayes, which\n        (both night and day, for the while) it hath\n          toyled and labored through, to bryng you\n            this good Newes, and Comfortable\n              profe, of Vertues frute.\n\n    So, I Commit you vnto Gods Mercyfull direction, for the rest:\n      hartely besechyng hym, to prosper your Studyes, and\n        honest Intentes: to his Glory, & the Commodity\n          of our Countrey. _Amen_.\n\n\n                     _Written at my poore House\n                     At Mortlake._\n\n                      _Anno. 1570. February. 9._\n\n\n  [Decoration]\n\n\n\n\n  [Transcriber\u2019s Note:\n\n  The \u201cGroundplat\u201d was printed in the form of a stemma, or tree, on an\n  oversized fold-out page. The layout was impossible to reproduce for\n  this e-text, so the information has been rearranged in nested-list\n  form. Size markings (see note at beginning of e-text) are relative\n  within each paragraph.]\n\n\n      _J. DEE_\n\n  +\u2021Here haue you (according to my promisse) the Groundplat of\u2021+\n    +my MATHEMATICALL Pr\u00e6face: annexed to _Euclide_ (now first)+\n       published in our Englishe tounge. An. 1570. Febr. 3.\n\n\n+\u2021Sciences, and Artes Mathematicall,\u2021 are, either+\n\n  +\u2021Principall,\u2021 which are two, onely,+\n\n    +Arithmetike.+\n\n      +\u2021Simple\u2021+, Which dealeth with Numbers onely: and demonstrateth\n        all their properties and appertenances: where, an Vnit, is\n        Indiuisible.\n      +\u2021Mixt\u2021+, Which with aide of Geometrie principall, demonstrateth\n        some Arithmeticall Conclusion, or Purpose.\n\n    +Geometrie.+\n\n      +\u2021Simple\u2021+, Which dealeth with Magnitudes, onely: and\n        demonstrateth all their properties, passions, and\n        appertenances: whose Point, is Indiuisible.\n      +\u2021Mixt\u2021+, Which with aide of Arithmetike principall,\n        demonstrateth some Geometricall purpose, as +EVCLIDES\n        ELEMENTES+.\n\n    +\u2021The vse\u2021 whereof, is either,+\n\n      In thinges Supernaturall, \u00e6ternall, & Diuine: By Application,\n        _Ascending_.\n      In thinges Mathematicall: without farther Application.\n      In thinges Naturall: both Subst\u00e3tiall, & Accidentall, Visible,\n        & Inuisible. &c. By Application: _Descending_.\n\n    The like Vses and Applications are, (though in a degree lower) in\n    the +Artes Mathematicall Deriuatiue+.\n\n  +\u2021Deriuatiue\u2021 fr\u00f5 the Principalls: of which, some haue+\n\n    +\u2021The names of\u2021 the Principalls: as,+\n\n      +_Arithmetike_, vulgar: which considereth+\n\n        --Arithmetike of most vsuall whole numbers: And of Fractions to\n        them appertaining.\n        --Arithmetike of Proportions.\n        --Arithmetike Circular.\n        --Arithmetike of Radicall N\u0169bers: Simple, Compound, Mixt: And of\n        their Fractions.\n        --Arithmetike of Cossike N\u0169bers: with their Fractions: And the\n        great Arte of Algiebar.\n\n      +_Geometrie_, vulgar: which teacheth Measuring+\n\n        +\u2021At hand\u2021+\n\n          All Lengthes.--+Mecometrie.+\n          All Plaines: As, Land, Borde, Glasse, &c.--+Embadometrie.+\n          All Solids: As, Timber, Stone, Vessels, &c.--+Stereometrie.+\n\n        +\u2021With dist\u00e3ce\u2021+ from the thing Measured, as,\n\n          +\u2021How farre\u2021+, from the Measurer, any thing is: of him\n            sene, on Land or Water: called +Apomecometrie+.\n          +\u2021How high or deepe\u2021+, from the leuell of the Measurers\n            standing, any thing is: Seene of hym, on Land or Water:\n            called +Hypsometrie+.\n          +\u2021How broad\u2021+, a thing is, which is in the Measurers view:\n            so it be situated on Land or Water: called\n            +Platometrie+.\n\n        +\u2021Of which\u2021 are growen the Feates & Artes of+\n\n          +Geodesie+: more cunningly to Measure and Suruey Landes,\n            Woods, Waters. &c.\n          +Geographie.+\n          +Chorographie.+\n          +Hydrographie.+\n          +Stratarithmetrie.+\n\n    +\u2021Propre names\u2021 as+,\n\n      +Perspectiue,+--Which demonstrateth the maners and properties\n      of all Radiations: Directe, Broken, and Reflected.\n\n      +Astronomie,+--Which demonstrateth the Distances, Magnitudes,\n      and all Naturall motions, Apparences, and Passions, proper to\n      the Planets and fixed Starres: for any time, past, present, and\n      to come: in respecte of a certaine Horizon, or without respecte\n      of any Horizon.\n\n      +Musike,+--Which demonstrateth by reason, and teacheth by\n      sense, perfectly to iudge and order the diuersitie of Soundes,\n      hie or low.\n\n      +Cosmographie,+--Which, wholy and perfectly maketh description\n      of the Heauenlym and also Elementall part of the World: and of\n      these partes, maketh homologall application, and mutuall\n      collation necessary.\n\n      +Astrologie,+--Which reasonably demonstrateth the operations\n      and effectes of the naturall beames of light, and secrete\n      Influence of the Planets, and fixed Starres, in euery Element\n      and Elementall body: at all times, in any Horizon assigned.\n\n      +Statike,+--Which demonstrateth the causes of heauines and\n      lightnes of all thinges: and of the motions and properties to\n      heauines and lightnes belonging.\n\n      +Anthropographie,+ Which describeth the N\u0169ber, Measure, Waight,\n      Figure, Situation, and colour of euery diuers thing contained in\n      the perfecte body of MAN: and geueth certaine knowledge of the\n      Figure, Symmetrie, Waight, Characterization, & due Locall motion\n      of any percell of the said body assigned: and of numbers to the\n      said percell appertaining.\n\n      +Trochilike,+--Which demonstrateth the properties of all\n      Circular motions: Simple and Compound.\n\n      +Helicosophie,+--Which demonstrateth the designing of all\n      Spirall lines: in Plaine, on Cylinder, Cone, Sph\u00e6re, Cono\u00efd, and\n      Sph\u00e6roid: and their properties.\n\n      +Pneumatithmie,+--Which demonstrateth by close hollow\n      Geometricall figures (Regular and Irregular) the straunge\n      properties (in motion or stay) of the Water, Ayre, Smoke, and\n      Fire, in their Continuitie, and as they are ioyned to the\n      Elementes next them.\n\n      +Menadrie,+--Which demonstrateth, how, aboue Natures Vertue,\n      and power simple: Vertue and force, may be multiplied: and so\n      to directe, to lift, to pull to, and to put or cast fro, any\n      multiplied, or simple determined Vertue, Waight, or Force:\n      naturally, not, so, directible, or moueable.\n\n      +Hypogeiodie,+--Which demonstrateth, how, vnder the Sph\u00e6ricall\n      Superficies of the Earth, at any depth, to any perpendicular\n      line assigned (whose distance from the perpendicular of the\n      entrance: and the Azimuth likewise, in respecte of the sayd\n      entrance, is knowen) certaine way, may be prescribed and gone,\n      &c.\n\n      +Hydragogie,+--Which demonstrateth the possible leading of\n      water by Natures law, and by artificiall helpe, from any head\n      (being Spring, standing, or running water) to any other place\n      assigned.\n\n      +Horometrie,+--Which demonstrateth, how, at all times\n      appointed, the precise, vsuall denomination of time, may be\n      knowen, for any place assigned.\n\n      +Zographie,+--Which demonstrateth and teacheth, how, the\n      Intersection of all visuall Pyramids, made by any plaine\n      assigned (the Center, distance, and lightes being determined)\n      may be, by lines, and proper colours represented.\n\n      +Architecture,+--Which is a Science garnished with many\n      doctrines, and diuers Instructions: by whose iudgement, all\n      workes by other workmen finished, are iudged.\n\n      +Nauigation,+--Which demonstrateth, how, by the Shortest good\n      way, by the aptest direction, and in the shortest time:\n      a sufficient Shippe, betwene any two places (in passage\n      nauigable) assigned, may be conducted: and in all stormes and\n      naturall disturbances chauncing, how to vse the best possible\n      meanes, to recouer the place first assigned.\n\n      +Thaumaturgike,+--Which geueth certaine order to make straunge\n      workes, of the sense to be perceiued: and of men greatly to be\n      wondred at.\n\n      +Archemastrie,+--Which teacheth to bring to actuall experience\n      sensible, all worthy conclusions, by all the Artes Mathematicall\n      purposed: and by true Naturall philosophie, concluded: And both\n      addeth to them a farder Scope, in the termes of the same Artes:\n      and also, by his proper Method, and in peculiar termes,\n      procedeth, with helpe of the forsayd Artes, to the performance\n      of complete Experiences: which, of no particular Arte, are hable\n      (Formally) to be challenged.\n\n\n+\u00b6 Imprinted by _Iohn Day_.+\n\nAn. 1570. Feb. 25.\n\n       *       *       *       *       *\n           *       *       *       *\n       *       *       *       *       *\n\nErrors and Anomalies:\n\nUnless otherwise noted, spelling and punctuation are unchanged.\nErrors are listed below, with the original form, if changed, shown in\n[brackets]. Unusual words include \u201cfatch\u201d (probably used as a variant\nof \u201cfetch\u201d) and the mathematical terms \u201csexagene\u201d and \u201csexagesme\u201d.\n\n  How, worldly goods: how, worldly dignitie\n    [_\u201co\u201d in second \u201cworldly\u201d invisible_]\n  his most diligent hearers (so infinitely mought  [hearers) so]\n  the boundes, and duety of an Hydrographer  [Hydographer]\n  of the Grekes it is called _Eteromekes_\n    [_text unchanged: correct form is \u201cHeteromekes\u201d_]\n  \u03c4\u1f78 \u1f41\u03c4\u1f76  [_accent unchanged_]\n  in our worldly affaires  [wordly]\n  fall to worke.\u2749.\n    [_Some text readers may not display the oversized-asterisk symbol._]\n  _Emptying the first._  [Emptyting]\n  \u0391\u03c0\u1f78 \u03c4\u03ac\u03c5\u03c4\u03b7\u03c2 \u03c4\u1fc6\u03c2 \u1f21\u03bc\u03ad\u03c1\u03b1\u03c2, \u03c0\u03b5\u03c1\u1f76 \u03c0\u03b1\u03bd\u03c4\u1f78\u03c2, \u0391\u03c1\u03c7\u03b9\u03bc\u03ae\u03b4\u03b5\u03b9 \u03bb\u03ad\u03b3\u03bf\u03bd\u03c4\u03b9 \u03c0\u03b9\u03db\u03b5\u03c5\u03c4\u03b5\u03cc\u03bd\n    [\u1f21\u03bc\u03ae\u03c1\u03b1\u03c2 ... \u03c0\u03b9\u03db\u03b5\u03c5\u03c4\u03ad\u03bf\u03bc]\n  of the suddeyne  [snddeyne]\n  that the right and absolute way may be had  [he had]\n  Georgic I: [_The quoted segments, each ending in \u201c&c.\u201d, are\n    438-439; 451-457; 463-464._]\n\nAdditional Notes:\n\n  The Greek letter \u03b7 (eta) was consistently printed as if it were the\n    ou-ligature \u0223.\n  The Latin \u201c-que\u201d was written as an abbreviation resembling \u201c-q\u00b4;\u201d.\n    It is shown here as [que].\n\n  Mathematical symbols seen in the section accompanying the diagrams\n  could not be reproduced. The following substitutions were made:\n    --The curly \u201cP\u201d used for \u201cPounds\u201d is shown as {P}.\n    --The \u201cpotestas\u201d symbol, used to represent \u201cx\u201d (the unknown),\n    is shown as {x}.\n    --All roots were expressed as the \u201croot\u201d sign \u221a combined with\n    symbols for the power of 2 (doubled for power of 4, or fourth root)\n    and 3. They are shown as \u00b2\u221a \u00b3\u221a \u2074\u221a.\n\nEuclid:\n\nThe following Propositions were identified by number.\n\n6.12: (How) to find a fourth (line) proportional to three given straight\nlines.\n\n11.34: In equal parallelepipedal solids the bases are reciprocally\nproportional to the heights; and those parallelepipedal solids in which\nthe bases are reciprocally proportional to the heights are equal.\n\n11.36: If three straight lines are proportional, then the\nparallelepipedal solid formed out of the three equals the\nparallelepipedal solid on the mean which is equilateral, but equiangular\nwith the aforesaid solid.\n\n12.1: Similar polygons inscribed in circles are to one another as the\nsquares on their diameters.\n\n12.2: Circles are to one another as the squares on their diameters.\n\n12.18 (\u201clast\u201d): Spheres are to one another in triplicate ratio of their\nrespective diameters.","title":"The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara","tradition":"philosophy-ethics","translated_text":null,"word_count":33865}