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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-15 01:46:58 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-15 01:46:58 -0700 |
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+margin-right: -25%; width: 20%; line-height: normal; +font-style: italic;} +.sidenote.plain {font-style: normal; font-size: 88%;} + +span.sidequote {float: left; clear: left; font-size: 125%; +padding: 0em; margin-left: -3%; font-style: normal; +line-height: normal;} + +span.sidefinger {float: left; clear: left; padding: 0em; +margin-left: -8%;} +span.sidefinger:after {content: url("images/finger36.gif");} + + +/* text formatting */ + +.larger {font-size: 116%; line-height: normal;} +.largest {font-size: 133%; line-height: normal;} +.smaller {font-size: 83%;} +.leaf {font-size: 200%;} + +.xmath {font-family: sans-serif;} +.pmath {font-family: cursive;} + +/* my additions */ + +.greek {border-bottom: 1px dotted #666;} +ins.correction {text-decoration: none; border-bottom: thin dotted red;} + +.pagenum {position: absolute; left: 2%; font-size: 90%; +font-weight: bold; font-style: normal; text-align: left; +text-indent: 0em;} +.pagenum.verso {font-weight: normal;} + +div.mynote {background-color: #DDE; color: #000; padding: .5em 1em 1em; +margin: 1em 5%;} +div.mynote.end {background-color: #EEF; color: inherit; +border: 3px ridge #9AC; margin-top: 3em;} + +p.mynote {background-color: #DDE; color: #000; padding: 1em; +margin: 1em 5%; font-family: sans-serif; font-size: 90%;} +div.mynote p {font-family: sans-serif; font-size: 90%;} +div.mynote h5 {font-family: sans-serif; font-size: 95%; +font-weight: bold;} +div.mynote a {text-decoration: none;} + +</style> +</head> + +<body> + +<div style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara, by John Dee</div> +<div style='display:block; margin:1em 0'> +This eBook is for the use of anyone anywhere in the United States and +most other parts of the world at no cost and with almost no restrictions +whatsoever. You may copy it, give it away or re-use it under the terms +of the Project Gutenberg License included with this eBook or online +at <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you +are not located in the United States, you will have to check the laws of the +country where you are located before using this eBook. +</div> +<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara</div> +<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Author: John Dee</div> +<div style='display:block; margin:1em 0'>Release Date: July 13, 2007 [eBook #22062]<br /> +[Most recently updated: June 28, 2021]</div> +<div style='display:block; margin:1em 0'>Language: English</div> +<div style='display:block; margin:1em 0'>Character set encoding: UTF-8</div> +<div style='display:block; margin-left:2em; text-indent:-2em'>Produced by: Louise Hope, David Starner, Suzanne Lybarger and the Online Distributed Proofreading Team</div> +<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK THE MATHEMATICALL PRAEFACE ***</div> + +<div class="mynote"> + +<p>The variation in text sizes is as in the original.</p> + +<p>In the printed book, all sidenotes were shown in the outer margin. +For this e-text, they have been divided. Text sidenotes—including +single numerals—are in the right margin; symbols are on the left, +along with page numbers. The continuous quotation marks could not be +reproduced, so they are shown as oversized open quotes “ and close +quotes ” in the left margin.</p> + +<p>The original book did not have numbered pages. Instead it labeled all +recto (right-side) pages, grouped into six eight-page signatures. The +signatures themselves have an unusual sequence: <img src="images/finger20.gif" alt="pointer" /> (pointing finger); +* (asterisk); a; b; c; A. The verso (left-side) pages were +unmarked; they are identified here as ||. The <a href="#groundplat">final page</a> was an unnumbered foldout, about twice the +size of a normal page.</p> + +<p>Unless otherwise noted, spelling and punctuation are unchanged. +A few specific notes are given <a href="#textnotes">at the end of +the text</a>, along with those passages of <a href="#euclid">Euclid</a> identified by number.</p> + +</div> + +<p> </p> + +<!-- png 01 --> + +<p class="illustration"> +<img src="images/titlepage.png" width="417" height="641" +alt="text linked below" /> +<br /> +<br /> +<a href="#title_text">Title Page Text</a> +</p> + +<div class="main"> + +<!-- png 02 --> + +<hr class="spacer" /> + +<span class="pagenum"><img src="images/finger20.gif" alt="pointer" />.ii</span> +<!-- png 03 --> +<h5><span class="leaf">❧</span> The Translator to the Reader.</h5> + + +<p><span class="capletter"> +<img src="images/capT.png" width="151" height="153" +alt="T(There)" /></span> +<i>Here is (gentle Reader) nothing (the word of God onely set apart) +which so much beautifieth and adorneth the soule and minde of mã, as +doth the knowledge of good artes and sciences: as the knowledge of +naturall and morall Philosophie. The one setteth before our eyes, the +creatures of God, both in the heauens aboue, and in the earth beneath: +in which as in a glasse, we beholde the exceding maiestie and wisedome +of God, in adorning and beautifying them as we see: in geuing vnto them +such wonderfull and manifolde proprieties, and naturall workinges, and +that so diuersly and in such varietie: farther in maintaining and +conseruing them continually, whereby to praise and adore him, as by +S. Paule we are taught. The other teacheth vs rules and preceptes +of vertue, how, in common life amongest men, we ought to walke +vprightly: what dueties pertaine to our selues, what pertaine to the +gouernment or good order both of an housholde, and also of a citie or +common wealth. The reading likewise of histories, conduceth not a litle, +to the adorning of the soule & minde of man, a studie of all +men cõmended: by it are seene and knowen the artes and doinges of +infinite wise men gone before vs. In histories are contained infinite +examples of heroicall vertues to be of vs followed, and horrible +examples of vices to be of vs eschewed. Many other artes also there are +which beautifie the minde of man: but of all other none do more garnishe +& beautifie it, then those artes which are called Mathematicall. +Unto the knowledge of which no man can attaine, without the perfecte +knowledge and instruction of the principles, groundes, and Elementes of +Geometrie. But perfectly +<span class="pagenum verso">||</span> +<!-- png 04 --> +to be instructed in them, requireth diligent studie and reading of olde +auncient authors. Amongest which, none for a beginner is to be preferred +before the most auncient Philosopher </i>Euclide<i> of </i>Megara<i>. +For of all others he hath in a true methode and iuste order, gathered +together whatsoeuer any before him had of these Elementes written: +inuenting also and adding many thinges of his owne: wherby he hath in +due forme accomplished the arte: first geuing definitions, principles, +& groundes, wherof he deduceth his Propositions or conclusions, in +such wonderfull wise, that that which goeth before, is of necessitie +required to the proufe of that which followeth. So that without the +diligent studie of </i>Euclides<i> Elementes, it is impossible to +attaine vnto the perfecte knowledge of Geometrie, and consequently of +any of the other Mathematicall sciences. Wherefore considering the want +& lacke of such good authors hitherto in our Englishe tounge, +lamenting also the negligence, and lacke of zeale to their countrey in +those of our nation, to whom God hath geuen both knowledge, & also +abilitie to translate into our tounge, and to publishe abroad such good +authors, and bookes (the chiefe instrumentes of all learninges): seing +moreouer that many good wittes both of gentlemen and of others of all +degrees, much desirous and studious of these artes, and seeking for them +as much as they can, sparing no paines, and yet frustrate of their +intent, by no meanes attaining to that which they seeke: I haue for +their sakes, with some charge & great trauaile, faithfully +translated into our vulgare toũge, & set abroad in Print, this booke +of </i>Euclide<i>. Whereunto I haue added easie and plaine declarations +and examples by figures, of the definitions. In which booke also ye +shall in due place finde manifolde additions, Scholies, Annotations, and +Inuentions: which I haue gathered out of many of the most famous & +chiefe Mathematiciẽs, both of old time, and in our age: as by diligent +reading it in course, ye shall +<span class="pagenum"><img src="images/finger20.gif" alt="pointer" /><ins class="correction" title="text reads ‘.ii’">.iij</ins></span> +<!-- png 05 --> +well perceaue. The fruite and gaine which I require for these my paines +and trauaile, shall be nothing els, but onely that thou gentle reader, +will gratefully accept the same: and that thou mayest thereby receaue +some profite: and moreouer to excite and stirre vp others learned, to do +the like, & to take paines in that behalfe. By meanes wherof, our +Englishe tounge shall no lesse be enriched with good Authors, then are +other straunge tounges: as the Dutch, French, Italian, and Spanishe: in +which are red all good authors in a maner, found amongest the Grekes or +Latines. Which is the chiefest cause, that amongest thẽ do florishe so +many cunning and skilfull men, in the inuentions of straunge and +wonderfull thinges, as in these our daies we see there do. Which fruite +and gaine if I attaine vnto, it shall encourage me hereafter, in such +like sort to translate, and set abroad some other good authors, both +pertaining to religion (as partly I haue already done)<br /> +and also pertaining to the Mathematicall Artes.<br /> +Thus gentle reader farewell.</i></p> +<p>(?¿)</p> + + +<p class="illustration"> +<img src="images/dec_iii.png" width="132" height="136" +alt="decoration" /> +</p> + + +<!-- png 06 --> + +<hr class="spacer" /> + +<span class="pagenum verso">[<img src="images/finger20.gif" alt="pointer" />.iiij]</span> +<!-- png 07 --> + +<h5><span class="leaf">❧</span> +<span class="larger">TO THE VNFAINED LOVERS<br /> +of truthe, and constant Studentes of Noble<br /> +<i>Sciences, </i>IOHN DEE<i> of London, hartily</i></span><br /> +wisheth grace from heauen, and most prosperous<br /> +<i>successe in all their honest attemptes and</i><br /> +exercises.</h5> + + +<p><span class="capletter"> +<img src="images/capD.png" width="175" height="179" +alt="D(Divine)" /></span> +Iuine <i>Plato</i>, the great Master of many worthy Philosophers, and +the constant auoucher, and pithy perswader of <i>Vnum</i>, <i>Bonum</i>, +and <i>Ens</i>: in his Schole and Academie, sundry times (besides his +ordinary Scholers) was visited of a certaine kinde of men, allured by +the noble fame of <i>Plato</i>, and the great commendation of hys +profound and profitable doctrine. But when such Hearers, after long +harkening to him, perceaued, that the drift of his discourses issued +out, to conclude, this <i>Vnum</i>, <i>Bonum</i>, and <i>Ens</i>, to be +Spirituall, Infinite, Æternall, Omnipotent, &c. Nothyng beyng +alledged or expressed, How, worldly goods: how, <ins class="correction" title="‘o’ invisible">worldly</ins> dignitie: how, +health, Strẽgth or lustines of body: nor yet the meanes, how a +merueilous sensible and bodyly blysse and felicitie hereafter, might be +atteyned: Straightway, the fantasies of those hearers, were dampt: their +opinion of <i>Plato</i>, was clene chaunged: yea his doctrine was by +them despised: and his schole, no more of them visited. Which thing, his +Scholer, <i>Aristotle</i>, narrowly cõsidering, founde the cause therof, +to be, <span class="sidequote begin">“</span>For that they had no +forwarnyng and information, in generall,<span class="sidequote +end">”</span> whereto his doctrine tended. For, so, might they haue had +occasion, either to haue forborne his schole hauntyng: (if they, then, +had misliked his Scope and purpose) or constantly to haue continued +therin: to their full satisfaction: if such his finall scope & +intent, had ben to their desire. Wherfore, <i>Aristotle</i>, euer, after +that, vsed in brief, to forewarne his owne Scholers and hearers, <span +class="sidequote">“</span>both of what matter, and also to what ende, +he tooke in hand to speake, or teach.<span class="sidequote">”</span> +While I consider the diuerse trades of these two excellent Philosophers +(and am most sure, both, that <i>Plato</i> right well, otherwise could +teach: and that <i>Aristotle</i> mought boldely, with his hearers, haue +dealt in like sorte as <i>Plato</i> did) I am in no little pang of +perplexitie: Bycause, that, which I mislike, is most easy for me to +performe (and to haue <i>Plato</i> for my exãple.) And that, which I +know to be most commendable: and (in this first bringyng, into common +handling, the <i>Artes Mathematicall</i>) to be most necessary: is full +of great difficultie and sundry daungers. Yet, neither do I think it +mete, for so straunge matter (as now is ment to be published) and to so +straunge an audience, to be bluntly, at first, put forth, without a +peculiar Preface: Nor (Imitatyng <i>Aristotle</i>) well can I hope, that +accordyng to the amplenes and dignitie of the <i>State +Mathematicall</i>, I am able, either playnly to prescribe the +materiall boundes: or precisely to expresse the chief purposes, and most +wonderfull applications therof. And though I am sure, that such as did +shrinke from <i>Plato</i> his schole, after they had perceiued his +finall +<span class="pagenum verso">||</span> +<!-- png 08 --> +conclusion, would in these thinges haue ben his most diligent <ins class="correction" title="text has ‘hearers) so’">hearers (so</ins> +infinitely mought their desires, in fine and at length, by our <i>Artes +Mathematicall</i> be satisfied) yet, by this my Præface & +forewarnyng, Aswell all such, may (to their great behofe) the soner, +hither be allured: as also the <i>Pythagoricall</i>, and +<i>Platonicall</i> perfect scholer, and the constant profound +Philosopher, with more ease and spede, may (like the Bee,) gather, +hereby, both wax and hony.</p> + +<p>Wherfore, seyng I finde great occasion (for the causes alleged, and +farder, in respect of my <i>Art Mathematike generall</i>) to vse <span +class="sidequote">“</span>a certaine forewarnyng and Præface, whose +content shalbe, +<span class="sidenote"> +The intent of this Preface.</span> +that mighty, most plesaunt, and frutefull <i>Mathematicall Tree</i>, +with his chief armes and second (grifted) braunches: Both, what euery +one is, and also, what commodity, in generall, is to be looked for, +aswell of griff as stocke: And forasmuch as this enterprise is so great, +that, to this our tyme, it neuer was (to my knowledge) by any achieued: +And also it is most hard, in these our drery dayes, to such rare and +straunge Artes, to wyn due and common credit:<span class="sidequote">”</span> Neuertheles, if, for my sincere endeuour to +satisfie your honest expectation, you will but lend me your thãkefull +mynde a while: and, to such matter as, for this time, my penne (with +spede) is hable to deliuer, apply your eye or eare attentifely: +perchaunce, at once, and for the first salutyng, this Preface you will +finde a lesson long enough. And either you will, for a second (by this) +be made much the apter: or shortly become, well hable your selues, of +the lyons claw, to coniecture his royall symmetrie, and farder +propertie. Now then, gentle, my frendes, and countrey men, Turne your +eyes, and bend your myndes to that doctrine, which for our present +purpose, my simple talent is hable to yeld you.</p> + +<p>All thinges which are, & haue beyng, are found vnder a triple +diuersitie generall. For, either, they are demed Supernaturall, +Naturall, or, of a third being. Thinges Supernaturall, are immateriall, +simple, indiuisible, incorruptible, & vnchangeable. Things Naturall, +are materiall, compounded, diuisible, corruptible, and chaungeable. +Thinges Supernaturall, are, of the minde onely, comprehended: Things +Naturall, of the sense exterior, ar hable to be perceiued. In thinges +Naturall, probabilitie and coniecture hath place: But in things +Supernaturall, chief demõstration, & most sure Science is to be had. +By which properties & comparasons of these two, more easily may be +described, the state, condition, nature and property of those thinges, +which, we before termed of a third being: which, by a peculier name +also, are called <i>Thynges Mathematicall</i>. For, these, beyng (in a +maner) middle, betwene thinges supernaturall and naturall: are not so +absolute and excellent, as thinges supernatural: Nor yet so base and +grosse, as things naturall: But are thinges immateriall: and +neuerthelesse, by materiall things hable somewhat to be signified. And +though their particular Images, by Art, are aggregable and diuisible: +yet the generall <i>Formes</i>, notwithstandyng, are constant, +vnchaungeable, vntrãsformable, and incorruptible. Neither of the sense, +can they, at any tyme, be perceiued or iudged. Nor yet, for all that, in +the royall mynde of man, first conceiued. But, surmountyng the +imperfectiõ of coniecture, weenyng and opinion: and commyng short of +high intellectuall cõceptiõ, are the Mercurial fruite of +<i>Dianœticall</i> discourse, in perfect imagination subsistyng. +A meruaylous newtralitie haue these thinges <i>Mathematicall</i>, +and also a straunge participatiõ betwene thinges supernaturall, +immortall, intellectual, simple and indiuisible: and thynges naturall, +mortall, sensible, compounded and diuisible. Probabilitie and sensible +prose, may well serue in thinges naturall: and is commendable: In +Mathematicall reasoninges, a probable Argument, is nothyng +regarded: nor yet the testimony of sense, any whit credited: But onely a +perfect demonstration, of truthes certaine, necessary, and inuincible: +vniuersally and necessaryly concluded: +<span class="pagenum">*.i</span> +<!-- png 09 --> +is allowed as sufficient for <span class="sidequote">“</span>an +Argument exactly and purely Mathematical.<span class="sidequote">”</span></p> + +<p>Of <i>Mathematicall</i> thinges, are two principall kindes: namely, +<i>Number</i>, and <i>Magnitude</i>. +<span class="sidenote"> +Number.</span> +<i>Number</i>, we define, to be, a certayne Mathematicall Sũme, of +<i>Vnits</i>. +<span class="sidenote smaller"> +Note the worde, Vnit, to expresse the Greke Monas, & not Vnitie: as +we haue all, commonly, till now, vsed.</span> +And, an <i>Vnit</i>, is that thing Mathematicall, Indiuisible, by +participation of some likenes of whose property, any thing, which is in +deede, or is counted One, may resonably be called One. We account an +<i>Vnit</i>, a thing <i>Mathematicall</i>, though it be no Number, +and also indiuisible: because, of it, materially, Number doth consist: +which, principally, is a thing <i>Mathematicall</i>. +<span class="sidenote"> +Magnitude.</span> +<i>Magnitude</i> is a thing <i>Mathematicall</i>, by participation of +some likenes of whose nature, any thing is iudged long, broade, or +thicke. <span class="sidequote">“</span>A thicke <i>Magnitude</i> +we call a <i>Solide</i>, or a <i>Body</i>. What <i>Magnitude</i> so +euer, is Solide or Thicke, is also broade, & long. A broade +magnitude, we call a <i>Superficies</i> or a Plaine. Euery playne +magnitude, hath also length. A long magnitude, we terme a +<i>Line</i>. A <i>Line</i> is neither thicke nor broade, but onely +long: Euery certayne Line, hath two endes: +<span class="sidenote"> +A point.</span> +The endes of a line, are <i>Pointes</i> called. A&nbsp;<i>Point</i>, +is a thing <i>Mathematicall</i>, indiuisible, which may haue a certayne +determined situation.<span class="sidequote">”</span> If a Poynt moue +from a determined situation, the way wherein it moued, is also a +<i>Line</i>: mathematically produced, whereupon, of the auncient +Mathematiciens, +<span class="sidenote"> +A Line.</span> +a <i>Line</i> is called the race or course of a <i>Point</i>. +A Poynt we define, by the name of a thing Mathematicall: though it +be no Magnitude, and indiuisible: because it is the propre ende, and +bound of a Line: which is a true <i>Magnitude</i>. +<span class="sidenote"> +Magnitude.</span> +And <i>Magnitude</i> we may define to be that thing +<i>Mathematicall</i>, which is diuisible for euer, in partes diuisible, +long, broade or thicke. Therefore though a Poynt be no <i>Magnitude</i>, +yet <i>Terminatiuely</i>, we recken it a thing <i>Mathematicall</i> (as +I sayd) by reason it is properly the end, and bound of a line. Neither +<i>Number</i>, nor <i>Magnitude</i>, haue any Materialitie. First, we +will consider of <i>Number</i>, and of the Science <i>Mathematicall</i>, +to it appropriate, called <i>Arithmetike</i>: and afterward of +<i>Magnitude</i>, and his Science, called <i>Geometrie</i>. But that +name contenteth me not: whereof a word or two hereafter shall be sayd. +How Immateriall and free from all matter, <i>Number</i> is, who doth not +perceaue? yea, who doth not wonderfully wõder at it? For, neither +pure <i>Element</i>, nor <i>Aristoteles, Quinta Essentia</i>, is hable +to serue for Number, as his propre matter. Nor yet the puritie and +simplenes of Substance Spirituall or Angelicall, will be found propre +enough thereto. And therefore the great & godly Philosopher +<i>Anitius Boetius</i>, sayd: <i>Omnia quæcunq<sup>ue</sup> +a primæua rerum natura constructa sunt, Numerorum videntur ratione +formata. Hoc enim fuit principale in animo Conditoris Exemplar</i>. That +is: <span class="larger"><i>All thinges (which from the very first +originall being of thinges, haue bene framed and made) do appeare to be +Formed by the reason of Numbers. For this was the principall example or +patterne in the minde of the Creator</i>.</span> O comfortable +allurement, O rauishing perswasion, to deale with a Science, whose +Subiect, is so Auncient, so pure, so excellent, so surmounting all +creatures, so vsed of the Almighty and incomprehensible wisdome of the +Creator, in the distinct creation of all creatures: in all their +distinct partes, properties, natures, and vertues, by order, and most +absolute number, brought, from <i>Nothing</i>, to the <i>Formalitie</i> +of their being and state. By <i>Numbers</i> propertie therefore, of vs, +by all possible meanes, (to the perfection of the Science) learned, we +may both winde and draw our selues into the inward and deepe search and +vew, of all creatures distinct vertues, natures, properties, and +<i>Formes</i>: And also, farder, arise, clime, ascend, and mount vp +(with Speculatiue winges) in spirit, to behold in the Glas of Creation, +the <i>Forme of Formes</i>, the <i>Exemplar Number</i> of all thinges +<i>Numerable</i>: both visible and inuisible, mortall and +<span class="pagenum verso">||</span> +<!-- png 10 --> +immortall, Corporall and Spirituall. Part of this profound and diuine +Science, had <i>Ioachim</i> the Prophesier atteyned vnto: by <i>Numbers +Formall, Naturall</i>, and <i>Rationall</i>, forseyng, concludyng, and +forshewyng great particular euents, long before their comming. His +bookes yet remainyng, hereof, are good profe: And the noble Earle of +<i>Mirandula</i>, (besides that,) a sufficient witnesse: that +<i>Ioachim, in his prophesies, proceded by no other way, then by Numbers +Formall</i>. And this Earle hym selfe, in Rome, +<span class="sidenote"> +Ano. 1488.</span> +*set vp 900. Conclusions, in all kinde of Sciences, openly to be +disputed of: and among the rest, in his Conclusions +<i>Mathematicall</i>, (in the eleuenth Conclusion) hath in Latin, this +English sentence. <i>By Numbers, a way is had, to the searchyng +out, and vnderstandyng of euery thyng, hable to be knowen. For the +verifying of which Conclusion, I promise to aunswere to the 74. +Questions, vnder written, by the way of Numbers</i>. Which Cõclusions, +I omit here to rehearse: aswell auoidyng superfluous prolixitie: +as, bycause <i>Ioannes Picus, workes</i>, are commonly had. But, in any +case, I would wish that those Conclusions were red diligently, and +perceiued of such, as are earnest Obseruers and Considerers of the +constant law of nũbers: which is planted in thyngs Naturall and +Supernaturall: and is prescribed to all Creatures, inuiolably to be +kept. For, so, besides many other thinges, in those Conclusions to be +marked, it would apeare, how sincerely, & within my boundes, +I disclose the wonderfull mysteries, by numbers, to be atteyned +vnto.</p> + +<p>Of my former wordes, easy it is to be gathered, that <i>Number</i> +hath a treble state: One, in the Creator: an other in euery Creature (in +respect of his complete constitution:) and the third, in Spirituall and +Angelicall Myndes, and in the Soule of mã. In the first and third +state, <i>Number</i>, is termed <i>Number Numbryng</i>. But in all +Creatures, otherwise, <i>Number</i>, is termed <i>Nũber Numbred</i>. And +in our Soule, Nũber beareth such a swaye, and hath such an affinitie +therwith: that some of the old <i>Philosophers</i> taught, <i>Mans +Soule, to be a Number mouyng it selfe</i>. And in dede, in vs, though it +be a very Accident: yet such an Accident it is, that before all +Creatures it had perfect beyng, in the Creator, Sempiternally. <i>Number +Numbryng</i> therfore, is the discretion discerning, and distincting of +thinges. But in God the Creator, This discretion, in the beginnyng, +produced orderly and distinctly all thinges. For his <i>Numbryng</i>, +then, was his Creatyng of all thinges. And his Continuall +<i>Numbryng</i>, of all thinges, is the Conseruation of them in being: +And, where and when he will lacke an <i>Vnit</i>: there and then, that +particular thyng shalbe <i>Discreated</i>. Here I stay. But our +Seuerallyng, distinctyng, and <i>Numbryng</i>, createth nothyng: but of +Multitude considered, maketh certaine and distinct determination. And +albeit these thynges be waighty and truthes of great importance, yet (by +the infinite goodnes of the Almighty <i>Ternarie</i>,) Artificiall +Methods and easy wayes are made, by which the zelous Philosopher, may +wyn nere this Riuerish <i>Ida</i>, this Mountayne of Contemplation: and +more then Contemplation. And also, though <i>Number</i>, be a thyng so +Immateriall, so diuine, and æternall: yet by degrees, by litle and +litle, stretchyng forth, and applying some likenes of it, as first, to +thinges Spirituall: and then, bryngyng it lower, to thynges sensibly +perceiued: as of a momentanye sounde iterated: then to the least thynges +that may be seen, numerable: And at length, (most grossely,) to a +multitude of any corporall thynges seen, or felt: and so, of these +grosse and sensible thynges, we are trayned to learne a certaine Image +or likenes of numbers: and to vse Arte in them to our pleasure and +proffit. So grosse is our conuersation, and dull is our apprehension: +while mortall Sense, in vs, ruleth the common wealth of our litle world. +Hereby we say, Three Lyons, are three: or a <i>Ternarie</i>. Three +Egles, are three, or a <i>Ternarie</i>. +<span class="sidefinger"> + </span> +Which* <i>Ternaries</i>, are eche, the <i>Vnion</i>, <i>knot</i>, and +<i>Vniformitie</i>, of three discrete and distinct <i>Vnits</i>. That +is, we may in eche <i>Ternarie</i>, thrise, seuerally pointe, and shew a +part, <i>One</i>, <i>One</i>, and <i>One</i>. Where, in Numbryng, we say +One, two, +<span class="pagenum">*.ij </span> +<!-- png 11 --> +Three. But how farre, these visible Ones, do differre from our +Indiuisible Vnits (in pure <i>Arithmetike</i>, principally considered) +no man is ignorant. Yet from these grosse and materiall thynges, may we +be led vpward, by degrees, so, informyng our rude Imagination, toward +the cõceiuyng of <i>Numbers</i>, absolutely (:Not supposing, nor +admixtyng any thyng created, Corporall or Spirituall, to support, +conteyne, or represent those <i>Numbers</i> imagined:) that at length, +we may be hable, to finde the number of our owne name, gloriously +exemplified and registred in the booke of the <i>Trinitie</i> most +blessed and æternall.</p> + +<p>But farder vnderstand, that vulgar Practisers, haue Numbers, +otherwise, in sundry Considerations: and extend their name farder, then +to Numbers, whose least part is an <i>Vnit</i>. For the common Logist, +Reckenmaster, or Arithmeticien, in hys vsing of Numbers: of an Vnit, +imagineth lesse partes: and calleth them <i>Fractions</i>. As of an +<i>Vnit</i>, he maketh an halfe, and thus noteth it, ½. and so of other, +(infinitely diuerse) partes of an <i>Vnit</i>. Yea and farder, hath, +<i>Fractions of Fractions. &c</i>. And, forasmuch, as, +<i>Addition</i>, <i>Substraction</i>, <i>Multiplication</i>, +<i>Diuision</i> and <i>Extraction of Rotes</i>, are the chief, and +sufficient partes of <i>Arithmetike</i>: +<span class="sidenote"> +Arithmetike.</span> +which is, the <i>Science that demonstrateth the properties, of Numbers, +and all operatiõs, in numbers to be performed</i>: +<span class="sidenote"> +Note.</span> +<span class="sidequote">“</span>How often, therfore, these fiue sundry +sortes of Operations, do, for the most part, of their execution, +differre from the fiue operations of like generall property and name, in +our Whole numbers practisable, So often, (for a more distinct doctrine) +we, vulgarly account and name it, an other kynde of +<i>Arithmetike</i>.<span class="sidequote">”</span> And by this +reason: +<span class="sidenote"> +1.</span> +the Consideration, doctrine, and working, in whole numbers onely: where, +of an <i>Vnit</i>, is no lesse part to be allowed: is named (as it were) +an <i>Arithmetike</i> by it selfe. And so of the <i>Arithmetike of +Fractions</i>. +<span class="sidenote"> +2.</span> +In lyke sorte, the necessary, wonderfull and Secret doctrine of +Proportion, and proportionalytie hath purchased vnto it selfe a peculier +maner of handlyng and workyng: and so may seme an other forme of +<i>Arithmetike</i>. +<span class="sidenote"> +3.</span> +Moreouer, the <i>Astronomers</i>, for spede and more commodious +calculation, haue deuised a peculier maner of orderyng nũbers, about +theyr circular motions, by Sexagenes, and Sexagesmes. By Signes, Degrees +and Minutes &c. which commonly is called the <i>Arithmetike</i> of +<i>Astronomical</i> or <i>Phisicall Fractions</i>. That, haue I briefly +noted, by the name of <i>Arithmetike Circular</i>. Bycause it is also +vsed in circles, not <i>Astronomicall. &c.</i> +<span class="sidenote"> +4.</span> +Practise hath led <i>Numbers</i> farder, and hath framed them, to take +vpon them, the shew of <i>Magnitudes</i> propertie: Which is +<i>Incommensurabilitie</i> and <i>Irrationalitie</i>. (For in pure +<i>Arithmetike</i>, an <i>Vnit</i>, is the common Measure of all +Numbers.) And, here, Nũbers are become, as Lynes, Playnes and Solides: +some tymes <i>Rationall</i>, some tymes <i>Irrationall</i>. And haue +propre and peculier characters, (as <sup>2</sup>√. <sup>3</sup>√. and so +of other.<a class="tag" name="endtag1" id="endtag1" href="#endnote1">A</a> +Which is to signifie <i>Rote Square, Rote Cubik: and so forth</i>:) +& propre and peculier fashions in the fiue principall partes: +Wherfore the practiser, estemeth this, a diuerse <i>Arithmetike</i> +from the other. Practise bryngeth in, here, diuerse compoundyng of +Numbers: as some tyme, two, three, foure (or more) <i>Radicall</i> +nũbers, diuersly knit, by signes, of More & Lesse: as thus +<sup>2</sup>√12 + <sup>3</sup>√15. Or thus <sup>4</sup>√19 + +<sup>3</sup>√12 - <sup>2</sup>√2. &c. And some tyme with whole +numbers, or fractions of whole Number, amõg them: as 20 + +<sup>2</sup>√24. <sup>3</sup>√16 + 33 - <sup>2</sup>√10. <sup>4</sup>√44 ++ 12¼ + <sup>3</sup>√9. And so, infinitely, may hap the varietie. After +this: Both the one and the other hath fractions incident: and so is this +<i>Arithmetike</i> greately enlarged, by diuerse exhibityng and vse of +Compositions and mixtynges. Consider how, I (beyng desirous to +deliuer the student from error and Cauillation) do giue to this +<i>Practise</i>, the name of the <i>Arithmetike of Radicall numbers</i>: +Not, of <i>Irrationall</i> or <i>Surd Numbers</i>: which other while, +are Rationall: though they haue the Signe of a Rote before +<span class="pagenum verso">||</span> +<!-- png 12 --> +them, which, <i>Arithmetike</i> of whole Numbers most vsuall, would say +they had no such Roote: and so account them <i>Surd Numbers</i>: which, +generally spokẽ, is vntrue: as <i>Euclides</i> tenth booke may teach +you. Therfore to call them, generally, <i>Radicall Numbers</i>, (by +reason of the signe √. prefixed,) is a sure way: and a sufficient +generall distinction from all other ordryng and vsing of Numbers: And +yet (beside all this) Consider: the infinite desire of knowledge, and +incredible power of mans Search and Capacitye: how, they, ioyntly haue +waded farder (by mixtyng of speculation and practise) and haue found +out, and atteyned to the very chief perfection (almost) of +<i>Numbers</i> Practicall vse. Which thing, is well to be perceiued in +that great Arithmeticall Arte of <i>Æquation</i>: commonly called the +<i>Rule of Coss.</i> or <i>Algebra</i>. The Latines termed it, +<i>Regulam Rei & Census</i>, that is, the <span class="larger"><i>Rule of the thyng and his value</i></span>. With an apt +name: comprehendyng the first and last pointes of the worke. And the +vulgar names, both in Italian, Frenche and Spanish, depend (in namyng +it,) vpon the signification of the Latin word, <i>Res</i>: <span class="larger"><i>A thing</i></span>: vnleast they vse the name of +<i>Algebra</i>. And therin (commonly) is a dubble error. The one, of +them, which thinke it to be of <i>Geber</i> his inuentyng: the other of +such as call it <i>Algebra</i>. For, first, though <i>Geber</i> for his +great skill in Numbers, Geometry, Astronomy, and other maruailous Artes, +mought haue semed hable to haue first deuised the sayd Rule: and also +the name carryeth with it a very nere likenes of <i>Geber</i> his name: +yet true it is, that a <i>Greke</i> Philosopher and Mathematicien, named +<i>Diophantus</i>, before <i>Geber</i> his tyme, wrote 13. bookes therof +(of which, six are yet extant: and I had them to *vse, +<span class="sidenote"> +* Anno. 1550.</span> +of the famous Mathematicien, and my great frende, <i>Petrus +Montaureus</i>:) And secondly, the very name, is <i>Algiebar</i>, and +not <i>Algebra</i>: as by the Arabien <i>Auicen</i>, may be proued: who +hath these precise wordes in Latine, by <i>Andreas Alpagus</i> (most +perfect in the Arabik tung) so translated. <i>Scientia faciendi Algiebar +& Almachabel. i. Scientia inueniendi numerum ignotum, per additionem +Numeri, & diuisionem & æquationem</i>. Which is to say: <span +class="larger"><i>The Science of workyng Algiebar and +Almachabel</i></span>, that is, the <span class="larger"><i>Science of +findyng an vnknowen number, by Addyng of a Number, & Diuision & +æquation</i></span>. Here haue you the name: and also the principall +partes of the Rule, touched. To name it, <i>The rule, or Art of +Æquation</i>, doth signifie the middle part and the State of the Rule. +This Rule, hath his peculier Characters: +<span class="sidenote"> +5.</span> +and the principal partes of <i>Arithmetike</i>, to it appertayning, do +differre from the other <i>Arithmeticall operations</i>. This +<i>Arithmetike, hath Nũbers</i> Simple, Cõpound, Mixt: and Fractions, +accordingly. This Rule, and <i>Arithmetike of Algiebar</i>, is so +profound, so generall and so (in maner) conteyneth the whole power of +Numbers Application practicall: that mans witt, can deale with nothyng, +more proffitable about numbers: nor match, with a thyng, more mete for +the diuine force of the Soule, (in humane Studies, affaires, or +exercises) to be tryed in. Perchaunce you looked for, (long ere now,) to +haue had some particular profe, or euident testimony of the vse, proffit +and Commodity of Arithmetike vulgar, in the Common lyfe and trade of +men. Therto, then, I will now frame my selfe: But herein great care +I haue, least length of sundry profes, might make you deme, that either +I did misdoute your zelous mynde to vertues schole: or els mistrust your +hable witts, by some, to gesse much more. A profe then, foure, +fiue, or six, such, will I bryng, as any reasonable man, therwith may be +persuaded, to loue & honor, yea learne and exercise the excellent +Science of <i>Arithmetike</i>.</p> + +<p>And first: who, nerer at hand, can be a better witnesse of the frute +receiued by <i>Arithmetike</i>, then all kynde of Marchants? Though not +all, alike, either nede it, or vse it. How could they forbeare the vse +and helpe of the Rule, called the Golden +<span class="pagenum"> *.iij </span> +<!-- png 13 --> +Rule? Simple and Compounde: both forward and backward? How might they +misse <i>Arithmeticall</i> helpe in the Rules of Felowshyp: either +without tyme, or with tyme? and betwene the Marchant & his Factor? +The Rules of Bartering in wares onely: or part in wares, and part in +money, would they gladly want? Our Marchant venturers, and Trauaylers +ouer Sea, how could they order their doynges iustly and without losse, +vnleast certaine and generall Rules for Exchaũge of money, and +Rechaunge, were, for their vse, deuised? The Rule of Alligation, in how +sundry cases, doth it conclude for them, such precise verities, as +neither by naturall witt, nor other experience, they, were hable, els, +to know? And (with the Marchant then to make an end) how ample & +wonderfull is the Rule of False positions? especially as it is now, by +two excellent Mathematiciens (of my familier acquayntance in their life +time) enlarged? I meane <i>Gemma Frisius</i>, and <i>Simon +Iacob</i>. Who can either in brief conclude, the generall and Capitall +Rules? or who can Imagine the Myriades of sundry Cases, and particular +examples, in Act and earnest, continually wrought, tried and concluded +by the forenamed Rules, onely? How sundry other <i>Arithmeticall +practises</i>, are commonly in Marchantes handes, and knowledge: They +them selues, can, at large, testifie.</p> + +<p>The Mintmaster, and Goldsmith, in their Mixture of Metals, either of +diuerse kindes, or diuerse values: how are they, or may they, exactly be +directed, and meruailously pleasured, if <i>Arithmetike</i> be their +guide? And the honorable Phisiciãs, will gladly confesse them selues, +much beholding to the Science of <i>Arithmetike</i>, and that sundry +wayes: But chiefly in their Art of Graduation, and compounde Medicines. +And though <i>Galenus</i>, <i>Auerrois</i>, <i>Arnoldus</i>, +<i>Lullus</i>, and other haue published their positions, aswell in the +quantities of the Degrees aboue Temperament, as in the Rules, concluding +the new <i>Forme</i> resulting: yet a more precise, commodious, and easy +<i>Method</i>, is extant: by a Countreyman of ours +<span class="sidenote"> +R. B.</span> +(aboue 200. yeares ago) inuented. And forasmuch as I am vncertaine, who +hath the same: or when that litle Latin treatise, (as the Author writ +it,) shall come to be Printed: (Both to declare the desire I haue to +pleasure my Countrey, wherin I may: and also, for very good profe of +Numbers vse, in this most subtile and frutefull, Philosophicall +Conclusion,) I entend in the meane while, most briefly, and with my +farder helpe, to communicate the pith therof vnto you.</p> + +<p>First describe a circle: whose diameter let be an inch. Diuide the +Circumference into foure equall partes. Frõ the Center, by those 4. +sections, extend 4. right lines: eche of 4. inches and a halfe long: or +of as many as you liste, aboue 4. without the circumference of the +circle: So that they shall be of 4. inches long (at the least) without +the Circle. Make good euident markes, at euery inches end. If you list, +you may subdiuide the inches againe into 10. or 12. smaller partes, +equall. At the endes of the lines, write the names of the 4. principall +elementall Qualities. <i>Hote</i> and <i>Colde</i>, one against the +other. And likewise <i>Moyst</i> and <i>Dry</i>, one against the other. +And in the Circle write <i>Temperate</i>. Which <i>Temperature</i> hath +a good Latitude: as appeareth by the Complexion of man. And therefore we +haue allowed vnto it, the foresayd Circle: and not a point Mathematicall +or Physicall.<a class="tag" name="endtag2" id="endtag2" href="#endnote2">B</a></p> + +<p class="illustration"> +<img src="images/fourways.png" width="411" height="405" +alt="diagram: see end of text for alternative" /> +</p> + +<span class="sidenote"> +* <i>Take some part of Lullus counsayle in his booke de Q. +Essentia.</i></span> + +<p>Now, when you haue two thinges Miscible, whose degrees are * truely +knowen: Of necessitie, either they are of one Quantitie and waight, or +of diuerse. If they be of one Quantitie and waight: whether their +formes, be Contrary Qualities, or of one kinde (but of diuerse +intentions and degrees) or a <i>Temperate</i>, and a Contrary, <i>The +forme resulting of their Mixture, is in the Middle betwene the degrees +of +<span class="pagenum verso">||</span> +<!-- png 14 --> +the formes mixt</i>. As for example, let <i>A</i>, be <i>Moist</i> in +the first degree: and <i>B</i>, <i>Dry</i> in the third degree. Adde 1. +and 3. that maketh 4: the halfe or middle of 4. is 2. This 2. is the +middle, equally distant from <i>A</i> and <i>B</i> (for the +<span class="sidenote"> +* <i>Note.</i></span> +*<i>Temperament</i> is counted none. And for it, you must put a Ciphre, +if at any time, it be in mixture). Counting then from <i>B</i>, 2. +degrees, toward <i>A</i>: you finde it to be <i>Dry</i> in the first +degree: So is the <i>Forme resulting</i> of the Mixture of <i>A</i>, and +<i>B</i>, in our example. I will geue you an other example. +Suppose, you haue two thinges, as <i>C</i>, and <i>D</i>: and of +<i>C</i>, the Heate to be in the 4. degree: and of <i>D</i>, the Colde, +to be remisse, euen vnto the <i>Temperament</i>. Now, for <i>C</i>, you +take 4: and for <i>D</i>, you take a Ciphre: which, added vnto 4, +yeldeth onely 4. The middle, or halfe, whereof, is 2. Wherefore the +<i>Forme resulting</i> of <i>C</i>, and <i>D</i>, is Hote in the second +degree: for, 2. degrees, accounted from <i>C</i>, toward <i>D</i>, ende +iuste in the 2. degree of heate. Of the third maner, I will geue +also an example: which let be this: +<span class="sidenote"> +Note.</span> +I haue a liquid Medicine whose Qualitie of heate is in the 4. +degree exalted: as was <i>C</i>, in the example foregoing: and an other +liquid Medicine I haue: whose Qualitie, is heate, in the first degree. +Of eche of these, I mixt a like quantitie: Subtract here, the lesse +frõ the more: and the residue diuide into two equall partes: whereof, +the one part, either added to the lesse, or subtracted from the higher +degree, doth produce the degree of the +<span class="pagenum"> *.iiij </span> +<!-- png 15 --> +Forme resulting, by this mixture of <i>C</i>, and <i>E</i>. As, if from +4. ye abate 1. there resteth 3. the halfe of 3. is 1½: Adde to 1. this +1½: you haue 2½. Or subtract from 4. this 1½: you haue likewise 2½ +remayning. Which declareth, the <i>Forme resulting</i>, to be +<i>Heate</i>, in the middle of the third degree.</p> + +<p> +<span class="sidequote">“</span> +<span class="sidenote"> +The Second Rule.</span> +But if the Quantities of two thinges Commixt, be diuerse, and the +Intensions (of their Formes Miscible) be in diuerse degrees, and +heigthes. (Whether those Formes be of one kinde, or of Contrary kindes, +or of a Temperate and a Contrary, <i>What proportion is of the lesse +quantitie to the greater, the same shall be of the difference, which is +betwene the degree of the Forme resulting, and the degree of the greater +quantitie of the thing miscible, to the difference, which is betwene the +same degree of the Forme resulting, and the degree of the lesse +quantitie</i>. As for example. Let two pound of Liquor be geuen, hote in +the 4. degree: & one pound of Liquor be geuen, hote in the third +degree.<span class="sidequote">”</span> I would gladly know the +Forme resulting, in the Mixture of these two Liquors. Set downe your +nũbers in order, thus. +<span class="figfloat"> +<img src="images/diagram1.png" width="168" height="89" +alt="diagram: see end of text for alternative" /> +</span> +Now by the rule of Algiebar, haue I deuised a very easie, briefe, and +generall maner of working in this case. Let vs first, suppose that +<i>Middle Forme resulting</i>, to be 1<span class="xmath">X</span>: as +that Rule teacheth. And because (by our Rule, here geuen) as the waight +of 1. is to 2: So is the difference betwene 4. (the degree of the +greater quantitie) and 1<span class="xmath">X</span>: to the +difference betwene 1<span class="xmath">X</span> and 3: (the degree of +the thing, in lesse quãtitie. And with all, 1<span class="xmath">X</span>, being alwayes in a certaine middell, betwene the two +heigthes or degrees). For the first difference, I set 4-1<span +class="xmath">X</span>: and for the second, I set 1<span class="xmath">X</span>-3. And, now againe, I say, as 1. is to 2. so is +4-1<span class="xmath">X</span> to 1<span class="xmath">X</span>-3. +Wherfore, of these foure proportionall numbers, the first and the fourth +Multiplied, one by the other, do make as much, as the second and the +third Multiplied the one by the other. Let these Multiplications be made +accordingly. And of the first and the fourth, we haue 1<span class="xmath">X</span>-3. and of the second & the third, 8-2<span class="xmath">X</span>. Wherfore, our Æquation is betwene 1<span class="xmath">X</span>-3: and 8-2<span class="xmath">X</span>. Which may be +reduced, according to the Arte of Algiebar: as, here, adding 3. to eche +part, geueth the Æquation, thus, 1<span class="xmath">X</span>=11-2<span class="xmath">X</span>. And yet againe, +contracting, or Reducing it: Adde to eche part, 2<span class="xmath">X</span>: Then haue you 3<span class="xmath">X</span> æquall +to 11: thus represented 3<span class="xmath">X</span>=11. Wherefore, +diuiding 11. by 3: the Quotient is 3⅔: the <i>Valew</i> of our 1<span +class="xmath">X</span>, <i>Coss</i>, or <i>Thing</i>, first supposed. +And that is the heigth, or Intension of the <i>Forme resulting:</i> +which is, <i>Heate</i>, in two thirdes of the fourth degree: And here I +set the shew of the worke in conclusion, thus. The proufe hereof is +easie: by subtracting 3. from 3⅔, +<span class="figfloat"> +<img src="images/diagram2.png" width="437" height="130" +alt="diagram: see end of text for alternative" /> +</span> +resteth ⅔. Subtracte the same heigth of the Forme resulting, (which is +3⅔) frõ 4: then resteth ⅓: You see, that ⅔ is double to ⅓: as +2.<span class="pmath">P</span>. is double to 1.<span class="pmath">P</span>. So should it be: by the rule here geuen. Note. As you +added to eche part of the Æquation, 3: so if ye first added to eche part +2<span class="xmath">X</span>, it would stand, 3<span class="xmath">X</span>-3=8. And now adding to eche part 3: you haue (as afore) +3<span class="xmath">X</span>=11.</p> + +<p>And though I, here, speake onely of two thyngs Miscible: and most +commonly mo then three, foure, fiue or six, (&c.) are to be Mixed: +(and in one Compound +<span class="pagenum verso">||</span> +<!-- png 16 --> +to be reduced: & the Forme resultyng of the same, to serue the +turne) yet these Rules are sufficient: duely repeated and iterated. +<span class="sidenote"> +Note.</span> +In procedyng first, with any two: and then, with the Forme Resulting, +and an other: & so forth: For, the last worke, concludeth the Forme +resultyng of them all: I nede nothing to speake, of the Mixture +(here supposed) what it is. Common Philosophie hath defined it, saying, +<i>Mixtio est miscibilium, alteratorum, per minima coniunctorum, +Vnio</i>. Euery word in the definition, is of great importance. +I nede not also spend any time, to shew, how, the other manner of +distributing of degrees, doth agree to these Rules. Neither nede I of +the farder vse belonging to the Crosse of Graduation (before described) +in this place declare, vnto such as are capable of that, which I haue +all ready sayd. Neither yet with examples specifie the Manifold +varieties, by the foresayd two generall Rules, to be ordered. The witty +and Studious, here, haue sufficient: And they which are not hable to +atteine to this, without liuely teaching, and more in particular: would +haue larger discoursing, then is mete in this place to be dealt withall: +And other (perchaunce) with a proude snuffe will disdaine this litle: +and would be vnthankefull for much more. I, therfore conclude: and +wish such as haue modest and earnest Philosophicall mindes, to laude God +highly for this: and to Meruayle, that the profoundest and subtilest +point, concerning <i>Mixture of Formes and Qualities Naturall</i>, is so +Matcht and maryed with the most simple, easie, and short way of the +noble Rule of <i>Algiebar</i>. Who can remaine, therfore vnpersuaded, to +loue, alow, and honor the excellent Science of <i>Arithmetike</i>? For, +here, you may perceiue that the litle finger of <i>Arithmetike</i>, is +of more might and contriuing, then a hunderd thousand mens wittes, of +the middle sorte, are hable to perfourme, or truely to conclude, with +out helpe thereof.</p> + +<p>Now will we farder, by the wise and valiant Capitaine, be certified, +what helpe he hath, by the Rules of <i>Arithmetike</i>: in one of the +Artes to him appertaining: And of the Grekes named +<span class="sidenote"> +<span class="greek" title="Taktikê">Τακτικὴ</span>.</span> +<span class="greek" title="Taktikê">Τακτικὴ</span>. <span class="sidequote">“</span>That is, the Skill of Ordring Souldiers in Battell +ray after the best maner to all purposes.<span class="sidequote">”</span> This Art so much dependeth vppon Numbers vse, and +the Mathematicals, that <i>Ælianus</i> (the best writer therof,) in his +worke, to the <i>Emperour Hadrianus</i>, by his perfection, in the +Mathematicals, (beyng greater, then other before him had,) thinketh his +booke to passe all other the excellent workes, written of that Art, vnto +his dayes. For, of it, had written <i>Æneas</i>: <i>Cyneas</i> of +<i>Thessaly</i>: <i>Pyrrhus Epirota</i>: and <i>Alexander</i> his sonne: +<i>Clearchus</i>: <i>Pausanias</i>: <i>Euangelus</i>: <i>Polybius</i>, +familier frende to <i>Scipio</i>: <i>Eupolemus</i>: <i>Iphicrates</i>, +<i>Possidonius</i>: and very many other worthy Capitaines, Philosophers +and Princes of Immortall fame and memory: Whose fayrest floure of their +garland (in this feat) was <i>Arithmetike</i>: and a litle +perceiuerance, in <i>Geometricall</i> Figures. But in many other cases +doth <i>Arithmetike</i> stand the Capitaine in great stede. As in +proportionyng of vittayles, for the Army, either remaining at a stay: or +suddenly to be encreased with a certaine number of Souldiers: and for a +certain tyme. Or by good Art to diminish his company, to make the +victuals, longer to serue the remanent, & for a certaine determined +tyme: if nede so require. And so in sundry his other accountes, +Reckeninges, Measurynges, and proportionynges, the wise, expert, and +Circumspect Capitaine will affirme the Science of <i>Arithmetike</i>, to +be one of his chief Counsaylors, directers and aiders. Which thing (by +good meanes) was euident to the Noble, the Couragious, the loyall, and +Curteous +<span class="sidefinger"> + </span> +<i>Iohn</i>, late Earle of Warwicke. Who was a yong Gentleman, throughly +knowne to very few. Albeit his lusty valiantnes, force, and Skill in +Chiualrous feates and exercises: his humblenes, and frendelynes to all +men, were thinges, openly, of the world perceiued. But what rotes +(otherwise,) vertue had fastened in his brest, what Rules of godly and +honorable +<span class="pagenum">a.j </span> +<!-- png 17 --> +life he had framed to him selfe: what vices, (in some then liuing) +notable, he tooke great care to eschew: what manly vertues, in other +noble men, (florishing before his eyes,) he Sythingly aspired after: +what prowesses he purposed and ment to achieue: with what feats and +Artes, he began to furnish and fraught him selfe, for the better seruice +of his Kyng and Countrey, both in peace & warre. These (I say) +his Heroicall Meditations, forecastinges and determinations, no twayne, +(I thinke) beside my selfe, can so perfectly, and truely report. +And therfore, in Conscience, I count it my part, for the honor, +preferment, & procuring of vertue (thus, briefly) to haue put his +Name, in the Register of <i>Fame Immortall</i>.</p> + +<p>To our purpose. This <i>Iohn</i>, by one of his actes (besides many +other: both in England and Fraunce, by me, in him noted.) did disclose +his harty loue to vertuous Sciences: and his noble intent, to excell in +Martiall prowesse: When he, with humble request, and instant +Solliciting: got the best Rules (either in time past by Greke or +Romaine, or in our time vsed: and new Stratagemes therin deuised) for +ordring of all Companies, summes and Numbers of mẽ, (Many, or few) with +one kinde of weapon, or mo, appointed: with Artillery, or without: on +horsebacke, or on fote: to giue, or take onset: to seem many, being few: +to seem few, being many. To marche in battaile or Iornay: with many such +feates, to Foughten field, Skarmoush, or Ambushe appartaining: +<span class="sidenote plain"> +This noble Earle, dyed Anno. 1554. skarse of 24. yeares of age: hauing +no issue by his wife: Daughter to the Duke of Somerset.</span> +And of all these, liuely designementes (most curiously) to be in velame +parchement described: with Notes & peculier markes, as the Arte +requireth: and all these Rules, and descriptions Arithmeticall, inclosed +in a riche Case of Gold, he vsed to weare about his necke: as his Iuell +most precious, and Counsaylour most trusty. Thus, <i>Arithmetike</i>, of +him, was shryned in gold: Of <i>Numbers</i> frute, he had good hope. +Now, Numbers therfore innumerable, in <i>Numbers</i> prayse, his shryne +shall finde.</p> + +<p>What nede I, (for farder profe to you) of the Scholemasters of +Iustice, to require testimony: how nedefull, how frutefull, how +skillfull a thing <i>Arithmetike</i> is? I meane, the Lawyers of +all sortes. Vndoubtedly, the Ciuilians, can meruaylously declare: how, +neither the Auncient Romaine lawes, without good knowledge of <i>Numbers +art</i>, can be perceiued: Nor (Iustice in infinite Cases) without due +proportion, (narrowly considered,) is hable to be executed. How Iustly, +& with great knowledge of Arte, did <i>Papinianus</i> institute a +law of partition, and allowance, betwene man and wife after a diuorce? +But how <i>Accursius</i>, <i>Baldus</i>, <i>Bartolus</i>, <i>Iason</i>, +<i>Alexander</i>, and finally <i>Alciatus</i>, (being otherwise, notably +well learned) do iumble, gesse, and erre, from the æquity, art and +Intent of the lawmaker: <i>Arithmetike</i> can detect, and conuince: and +clerely, make the truth to shine. Good <i>Bartolus</i>, tyred in the +examining & proportioning of the matter: and with <i>Accursius</i> +Glosse, much cumbred: burst out, and sayd: <i>Nulla est in toto libro, +hac glossa difficilior: Cuius computationem nec Scholastici nec Doctores +intelligunt. &c.</i> That is: <span class="larger"><i>In the whole +booke, there is no Glosse harder then this: Whose accoumpt or reckenyng, +neither the Scholers, nor the Doctours vnderstand. &c.</i></span> +What can they say of <i>Iulianus</i> law, <i>Si ita Scriptum. +&c.</i> Of the Testators will iustly performing, betwene the wife, +Sonne and daughter? How can they perceiue the æquitie of +<i>Aphricanus</i>, <i>Arithmeticall</i> Reckening, where he treateth of +<i>Lex Falcidia</i>? How can they deliuer him, from his Reprouers: and +their maintainers: as <i>Ioannes</i>, <i>Accursius Hypolitus</i> and +<i>Alciatus</i>? How Iustly and artificially, was <i>Africanus</i> +reckening made? Proportionating to the Sommes bequeathed, the +Contributions of eche part? Namely, for the hundred presently receiued, +17 1/7. And for the hundred, receiued after ten monethes, +12 6/7: which make the 30: which were to be cõtributed by the +legataries to the heire. +<span class="pagenum verso">||</span> +<!-- png 18 --> +For, what proportion, 100 hath to 75: the same hath 17 1/7 to +12 6/7: Which is Sesquitertia: that is, as 4, to 3. which make 7. +Wonderfull many places, in the Ciuile law, require an expert +<i>Arithmeticien</i>, to vnderstand the deepe Iudgemẽt, & Iust +determinatiõ of the Auncient Romaine Lawmakers. But much more expert +ought he to be, who should be hable, to decide with æquitie, the +infinite varietie of Cases, which do, or may happen, vnder euery one of +those lawes and ordinances Ciuile. Hereby, easely, ye may now +coniecture: that in the Canon law: and in the lawes of the Realme (which +with vs, beare the chief Authoritie), Iustice and equity might be +greately preferred, and skilfully executed, through due skill of +Arithmetike, and proportions appertainyng. The worthy Philosophers, and +prudent lawmakers (who haue written many bookes <i>De Republica:</i> How +the best state of Common wealthes might be procured and mainteined,) +haue very well determined of Iustice: (which, not onely, is the Base and +foundacion of Common weales: but also the totall perfection of all our +workes, words, and thoughtes:) defining it, +<span class="sidenote"> +Iustice.</span> +<span class="sidequote">“</span>to be that vertue, by which, to euery +one, is rendred, that to him appertaineth.<span class="sidequote">”</span> God challengeth this at our handes, to be honored +as God: to be loued, as a father: to be feared as a Lord & master. +Our neighbours proportiõ, is also prescribed of the Almighty lawmaker: +which is, to do to other, euen as we would be done vnto. These +proportions, are in Iustice necessary: in duety, commendable: and of +Common wealthes, the life, strength, stay and florishing. +<i>Aristotle</i> in his <i>Ethikes</i> (to fatch the sede of Iustice, +and light of direction, to vse and execute the same) was fayne to fly to +the perfection, and power of Numbers: for proportions Arithmeticall and +Geometricall. <i>Plato</i> in his booke called <i>Epinomis</i> (which +boke, is the Threasury of all his doctrine) where, his purpose is, to +seke a Science, which, when a man had it, perfectly: he might seme, and +so be, in dede, <i>Wise</i>. He, briefly, of other Sciences discoursing, +findeth them, not hable to bring it to passe: But of the Science of +Numbers, he sayth. <i>Illa, quæ numerum mortalium generi dedit, id +profecto efficiet. Deum autem aliquem, magis quam fortunam, ad salutem +nostram, hoc munus nobis arbitror contulisse. &c. Nam ipsum bonorum +omnium Authorem, cur non maximi boni, Prudentiæ dico, causam arbitramur? +<span class="larger">That Science, verely, which hath taught mankynde +number, shall be able to bryng it to passe. And, I thinke, +a certaine God, rather then fortune, to haue giuen vs this gift, +for our blisse. For, why should we not Iudge him, who is the Author of +all good things, to be also the cause of the greatest good thyng, +namely, Wisedome?</span></i> There, at length, he proueth +<i>Wisedome</i> to be atteyned, by good Skill of <i>Numbers</i>. With +which great Testimony, and the manifold profes, and reasons, before +expressed, you may be sufficiently and fully persuaded: of the perfect +Science of <i>Arithmetike</i>, to make this accounte: That +<span class="sidefinger"> + </span> +of all Sciences, next to <i>Theologie</i>, it is most diuine, most pure, +most ample and generall, most profounde, most subtile, most commodious +and most necessary. Whose next Sister, is the Absolute Science of +<i>Magnitudes</i>: of which (by the Direction and aide of him, whose +<i>Magnitude</i> is Infinite, and of vs Incomprehensible) I now +entend, so to write, that both with the <i>Multitude</i>, and also with +the <i>Magnitude</i> of Meruaylous and frutefull verities, you (my +frendes and Countreymen) may be stird vp, and awaked, to behold what +certaine Artes and Sciences, (to our vnspeakable behofe) our heauenly +father, hath for vs prepared, and reuealed, by sundry +<i>Philosophers</i> and <i>Mathematiciens</i>.</p> + +<p>Both, <i>Number</i> and <i>Magnitude</i>, haue a certaine Originall +sede, (as it were) of an incredible property: and of man, neuer hable, +Fully, to be declared. Of <i>Number</i>, an Vnit, and of +<i>Magnitude</i>, a Poynte, doo seeme to be much like Originall +<span class="pagenum">a.ij </span> +<!-- png 19 --> +causes: But the diuersitie neuerthelesse, is great. We defined an +<i>Vnit</i>, to be a thing Mathematicall Indiuisible: A Point, +likewise, we sayd to be a Mathematicall thing Indiuisible. And farder, +that a Point may haue a certaine determined Situation: that is, that we +may assigne, and prescribe a Point, to be here, there, yonder. &c. +Herein, (behold) our Vnit is free, and can abyde no bondage, or to be +tyed to any place, or seat: diuisible or indiuisible. Agayne, by reason, +a Point may haue a Situation limited to him: a certaine +motion, therfore (to a place, and from a place) is to a Point incident +and appertainyng. But an <i>Vnit</i>, can not be imagined to haue any +motion. A Point, by his motion, produceth, Mathematically, +a line: (as we sayd before) which is the first kinde of Magnitudes, +and most simple: An <i>Vnit</i>, can not produce any number. +A Line, though it be produced of a Point moued, yet, it doth not +consist of pointes: Number, though it be not produced of an <i>Vnit</i>, +yet doth it Consist of vnits, as a materiall cause. But formally, +<span class="sidenote"> +Number.</span> +Number, is the Vnion, and Vnitie of Vnits. Which vnyting and knitting, +is the workemanship of our minde: which, of distinct and discrete Vnits, +maketh a Number: by vniformitie, resulting of a certaine multitude of +Vnits. And so, euery number, may haue his least part, giuen: namely, an +Vnit: But not of a Magnitude, (no, not of a Lyne,) the least part can be +giuẽ: by cause, infinitly, diuision therof, may be conceiued. All +Magnitude, is either a Line, a Plaine, or a Solid. Which Line, +Plaine, or Solid, of no Sense, can be perceiued, nor exactly by hãd (any +way) represented: nor of Nature produced: But, as (by degrees) Number +did come to our perceiuerance: So, by visible formes, we are holpen to +imagine, what our Line Mathematicall, is. What our Point, is. So +precise, are our Magnitudes, that one Line is no broader then an other: +for they haue no bredth: Nor our Plaines haue any thicknes. Nor yet our +Bodies, any weight: be they neuer so large of dimensiõ. Our Bodyes, we +can haue Smaller, then either Arte or Nature can produce any: and +Greater also, then all the world can comprehend. Our least Magnitudes, +can be diuided into so many partes, as the greatest. As, a Line of +an inch long, (with vs) may be diuided into as many partes, as may the +diameter of the whole world, from East to West: or any way extended: +What priuiledges, aboue all manual Arte, and Natures might, haue our two +Sciences Mathematicall? to exhibite, and to deale with thinges of such +power, liberty, simplicity, puritie, and perfection? And in them, so +certainly, so orderly, so precisely to procede: as, excellent is that +workemã Mechanicall Iudged, who nerest can approche to the representing +of workes, Mathematically demonstrated? +<span class="sidefinger"> + </span> +And our two Sciences, remaining pure, and absolute, in their proper +termes, and in their owne Matter: to haue, and allowe, onely such +Demonstrations, as are plaine, certaine, vniuersall, and of an æternall +veritye? +<span class="sidenote"> +Geometrie.</span> +This Science of <i>Magnitude</i>, his properties, conditions, and +appertenances: commonly, now is, and from the beginnyng, hath of all +Philosophers, ben called <i>Geometrie</i>. But, veryly, with a name to +base and scant, for a Science of such dignitie and amplenes. And, +perchaunce, that name, by cõmon and secret consent, of all wisemen, +hitherto hath ben suffred to remayne: that it might carry with it a +perpetuall memorye, of the first and notablest benefite, by that +Science, to common people shewed: Which was, when Boundes and meres of +land and ground were lost, and confounded (as in <i>Egypt</i>, yearely, +with the ouerflowyng of <i>Nilus</i>, the greatest and longest riuer in +the world) or, that ground bequeathed, were to be assigned: or, ground +sold, were to be layd out: or (when disorder preuailed) that Commõs were +distributed into seueralties. For, where, vpon these & such like +occasiõs, Some by ignorãce, some by negligẽce, Some by fraude, and some +by violence, did wrongfully limite, measure, encroach, or challenge (by +<span class="pagenum verso">||</span> +<!-- png 20 --> +pretence of iust content, and measure) those landes and groundes: great +losse, disquietnes, murder, and warre did (full oft) ensue: Till, by +Gods mercy, and mans Industrie, The perfect Science of Lines, Plaines, +and Solides (like a diuine Iusticier,) gaue vnto euery man, his owne. +The people then, by this art pleasured, and greatly relieued, in their +landes iust measuring: & other Philosophers, writing Rules for land +measuring: betwene them both, thus, confirmed the name of +<i>Geometria</i>, that is, (according to the very etimologie of the +word) Land measuring. Wherin, the people knew no farder, of Magnitudes +vse, but in Plaines: and the Philosophers, of thẽ, had no feet hearers, +or Scholers: farder to disclose vnto, then of flat, plaine +<i>Geometrie</i>. And though, these Philosophers, knew of farder vse, +and best vnderstode the etymologye of the worde, yet this name +<i>Geometria</i>, was of them applyed generally to all sortes of +Magnitudes: vnleast, otherwhile, of <i>Plato</i>, and <i>Pythagoras</i>: +When they would precisely declare their owne doctrine. Then, was +<span class="sidenote"> +* <i>Plato. 7. de Rep.</i></span> +*<i>Geometria</i>, with them, <i>Studium quod circa planum versatur</i>. +But, well you may perceiue by <i>Euclides Elementes</i>, that more ample +is our Science, then to measure Plaines: and nothyng lesse therin is +tought (of purpose) then how to measure Land. An other name, therfore, +must nedes be had, for our Mathematicall Science of Magnitudes: which +regardeth neither clod, nor turff: neither hill, nor dale: neither earth +nor heauen: but is absolute <i>Megethologia</i>: not creping on ground, +and dasseling the eye, with pole perche, rod or lyne: but <span class="sidequote">“</span>liftyng the hart aboue the heauens, by inuisible +lines, and +<span class="sidefinger"> + </span> +immortall beames meteth with the reflexions, of the light +incomprehensible: and so procureth Ioye, and perfection +vnspeakable.<span class="sidequote">”</span> Of which true vse of our +<i>Megethica</i>, or <i>Megethologia</i>, <i>Diuine Plato</i> seemed to +haue good taste, and iudgement: and (by the name of <i>Geometrie</i>) so +noted it: and warned his Scholers therof: as, in hys seuenth +<i>Dialog</i>, of the Common wealth, may euidently be sene. Where (in +Latin) thus it is: right well translated: <i>Profecto, nobis hoc non +negabunt, Quicunq<sup>ue</sup> vel paululum quid Geometriæ gustârunt, +quin hæc Scientia, contrà, omnino se habeat, quàm de ea loquuntur, qui +in ipsa versantur.</i> In English, thus. <span class="larger"><i>Verely</i></span> (sayth <i>Plato</i>) <span class="larger"><i>whosoeuer haue, (but euen very litle) tasted of Geometrie, +will not denye vnto vs, this: but that this Science, is of an other +condicion, quite contrary to that, which they that are exercised in it, +do speake of it.</i></span> And there it followeth, of our +<i>Geometrie</i>, <i>Quòd quæritur cognoscendi illius gratia, quod +semper est, non & eius quod oritur quandoq<sup>ue</sup> & +interit. Geometria, eius quod est semper, Cognitio est. Attollet igitur +(ô Generose vir) ad Veritatem, animum: atq<sup>ue</sup> ita, ad +Philosophandum preparabit cogitationem, vt ad supera conuertamus: quæ, +nunc, contra quàm decet, ad inferiora deijcimus. &c. Quàm maximè +igitur præcipiendum est, vt qui præclarissimam hanc habitãt Civitatem, +nullo modo, Geometriam spernant. Nam & quæ præter ipsius propositum, +quodam modo esse videntur, haud exigua sunt. &c.</i> It must nedes +be confessed (saith <i>Plato</i>) <span class="larger"><i>That</i></span> <span class="smaller">[<i>Geometrie</i>]</span> <span class="larger"><i>is +learned, for the knowyng of that, which is euer: and not of that, which, +in tyme, both is bred and is brought to an ende. &c. Geometrie is +the knowledge of that which is euerlastyng. It will lift vp therfore +(O Gentle Syr) our mynde to the Veritie: and by that meanes, it +will prepare the Thought, to the Philosophicall loue of wisdome: that we +may turne or conuert, toward heauenly thinges</i></span> <span class="smaller">[<i>both mynde and thought</i>]</span> <span class="larger"><i>which now, otherwise then becommeth vs, we cast down on base +or inferior things. &c. Chiefly, therfore, Commaundement must be +giuen, that such as do inhabit this most honorable Citie, by no meanes, +despise Geometrie. For euen those thinges</i></span> <span class="smaller">[<i>done by it</i>]</span> <span class="larger"><i>which, in +manner, seame to be, beside the purpose of Geometrie: are of</i></span> +<span class="pagenum">a.iij </span> +<!-- png 21 --> +<span class="larger"><i>no small importance. &c.</i></span> And +besides the manifold vses of <i>Geometrie</i>, in matters appertainyng +to warre, he addeth more, of second vnpurposed frute, and commoditye, +arrising by <i>Geometrie</i>: saying: <i>Scimus quin etiam, ad +Disciplinas omnes facilius per discendas, interesse omnino, attigerit ne +Geometriam aliquis, an non. &c. Hanc ergo Doctrinam, secundo loco +discendam Iuuenibus statuamus.</i> That is. <span class="larger"><i>But, also, we know, that for the more easy learnyng of all +Artes, it importeth much, whether one haue any knowledge in Geometrie, +or no. &c. Let vs therfore make an ordinance or decree, that this +Science, of young men shall be learned in the second place.</i></span> +This was <i>Diuine Plato</i> his Iudgement, both of the purposed, chief, +and perfect vse of <i>Geometrie</i>: and of his second, dependyng, +deriuatiue commodities. And for vs, Christen men, a thousand +thousand mo occasions are, to haue nede of the helpe of* +<span class="sidenote"> +I. D.<br /> +* Herein, I would gladly shake of, the earthly name, of +Geometrie.</span> +<i>Megethologicall</i> Contemplations: wherby, to trayne our +Imaginations and Myndes, by litle and litle, to forsake and abandon, the +grosse and corruptible Obiectes, of our vtward senses: and to apprehend, +by sure doctrine demonstratiue, Things Mathematicall. And by them, +readily to be holpen and conducted to conceiue, discourse, and conclude +of things Intellectual, Spirituall, æternall, and such as concerne our +Blisse euerlasting: which, otherwise (without Speciall priuiledge of +Illumination, or Reuelation frõ heauen) No mortall mans wyt (naturally) +is hable to reach vnto, or to Compasse. And, veryly, by my small Talent +(from aboue) I am hable to proue and testifie, that the litterall +Text, and order of our diuine Law, Oracles, and Mysteries, require more +skill in Numbers, and Magnitudes: then (commonly) the expositors haue +vttered: but rather onely (at the most) so warned: & shewed their +own want therin. (To name any, is nedeles: and to note the places, is, +here, no place: But if I be duely asked, my answere is ready.) And +without the litterall, Grammaticall, Mathematicall or Naturall verities +of such places, by good and certaine Arte, perceiued, no Spirituall +sense (propre to those places, by Absolute <i>Theologie</i>) will +thereon depend. +<span class="sidefinger"> + </span> +<span class="sidequote">“</span>No man, therfore, can doute, but +toward the atteyning of knowledge incomparable, and Heauenly Wisedome: +Mathematicall Speculations, both of Numbers and Magnitudes: are meanes, +aydes, and guides: ready, certaine, and necessary.<span class="sidequote">”</span> From henceforth, in this my Preface, will I frame +my talke, to <i>Plato</i> his fugitiue Scholers: or, rather, to such, +who well can, (and also wil,) vse their vtward senses, to the glory of +God, the benefite of their Countrey, and their owne secret contentation, +or honest preferment, on this earthly Scaffold. To them, I will +orderly recite, describe & declare a great Number of Artes, from our +two Mathematicall fountaines, deriued into the fieldes of <i>Nature</i>. +Wherby, such Sedes, and Rotes, as lye depe hyd in the groũd of +<i>Nature</i>, are refreshed, quickened, and prouoked to grow, shote vp, +floure, and giue frute, infinite, and incredible. And these Artes, +shalbe such, as vpon Magnitudes properties do depende, more, then vpon +Number. And by good reason we may call them Artes, and Artes +Mathematicall Deriuatiue: for (at this tyme) I Define +<span class="sidenote"> +An Arte.</span> +<span class="larger">An Arte, to be a Methodicall cõplete Doctrine, +hauing abundancy of sufficient, and peculier matter to deale with, by +the allowance of the Metaphisicall Philosopher: the knowledge whereof, +to humaine state is necessarye.</span> And that I account, +<span class="sidenote"> +Art Mathematicall Deriuatiue.</span> +<span class="larger">An Art Mathematicall deriuatiue, which by +Mathematicall demonstratiue Method, in Nũbers, or Magnitudes, ordreth +and confirmeth his doctrine, as much & as perfectly, as the matter +subiect will admit.</span> And for that, +<span class="pagenum verso">||</span> +<!-- png 22 --> +I entend to vse the name and propertie of a +<span class="sidenote"> +A Mechanitien.</span> +<i>Mechanicien</i>, otherwise, then (hitherto) it hath ben vsed, +I thinke it good, (for distinction sake) to giue you also a brief +description, what I meane therby. <span class="larger">A Mechanicien, +or a Mechanicall workman is he, whose skill is, without knowledge of +Mathematicall demonstration, perfectly to worke and finishe any sensible +worke, by the Mathematicien principall or deriuatiue, demonstrated or +demonstrable.</span> Full well I know, that he which inuenteth, or +maketh these demonstrations, is generally called <i>A speculatiue +Mechanicien</i>: which differreth nothyng from a <i>Mechanicall +Mathematicien</i>. So, in respect of diuerse actions, one man may haue +the name of sundry artes: as, some tyme, of a Logicien, some tymes (in +the same matter otherwise handled) of a Rethoricien. Of these trifles, +I make, (as now, in respect of my Preface,) small account: to fyle +thẽ for the fine handlyng of subtile curious disputers. In other places, +they may commaunde me, to giue good reason: and yet, here, I will +not be vnreasonable.</p> + +<p> +<span class="sidenote larger">1.</span> +First, then, from the puritie, absolutenes, and Immaterialitie of +Principall <i>Geometrie</i>, is that kinde of <i>Geometrie</i> deriued, +which vulgarly is counted <i>Geometrie</i>: and is the <span class="larger">Arte of Measuring sensible magnitudes, their iust quãtities and +contentes.</span> +<span class="sidenote"> +Geometrie vulgar.</span> +This, teacheth to measure, either at hand: and the practiser, to be by +the thing Measured: and so, by due applying of Cumpase, Rule, Squire, +Yarde, Ell, Perch, Pole, Line, Gaging rod, (or such like instrument) to +the Length, Plaine, or Solide measured, +<span class="sidenote"> +1.</span> +*to be certified, either of the length, perimetry, or distance lineall: +and this is called, <i>Mecometrie</i>. Or +<span class="sidenote"> +2.</span> +*to be certified of the content of any plaine Superficies: whether it be +in ground Surueyed, Borde, or Glasse measured, or such like thing: which +measuring, is named <i>Embadometrie</i>. +<span class="sidenote"> +3.</span> +*Or els to vnderstand the Soliditie, and content of any bodily thing: as +of Tymber and Stone, or the content of Pits, Pondes, Wells, Vessels, +small & great, of all fashions. Where, of Wine, Oyle, Beere, or Ale +vessells, &c, the Measuring, commonly, hath a peculier name: and is +called <i>Gaging</i>. And the generall name of these Solide measures, is +<i>Stereometrie</i>. +<span class="sidenote larger">2.</span> +Or els, this <i>vulgar Geometrie</i>, hath consideration to teach the +practiser, how to measure things, with good distance betwene him and the +thing measured: and to vnderstand thereby, either +<span class="sidenote"> +1.</span> +*how Farre, a thing seene (on land or water) is from the measurer: +and this may be called <i>Apomecometrie</i>: +<span class="sidenote"> +2.</span> +Or, how High or depe, aboue or vnder the leuel of the measurers stãding, +any thing is, which is sene on land or water, called <i>Hypsometrie</i>. +<span class="sidenote"> +3.</span> +*Or, it informeth the measurer, how Broad any thing is, which is in the +measurers vew: so it be on Land or Water, situated: and may be called +<i>Platometrie</i>. Though I vse here to condition, the thing measured, +to be on Land, or Water Situated: +<span class="sidenote"> +Note.</span> +yet, know for certaine, that the sundry heigthe of Cloudes, blasing +Starres, and of the Mone, may (by these meanes) haue their distances +from the earth: and, of the blasing Starres and Mone, the Soliditie +(aswell as distances) to be measured: But because, neither these things +are vulgarly taught: nor of a common practiser so ready to be executed: +I, rather, let such measures be reckened incident to some of our other +Artes, dealing with thinges on high, more purposely, then this vulgar +Land measuring Geometrie doth: as in <i>Perspectiue</i> and +<i>Astronomie, &c.</i></p> + +<p><span class="dropcap">O</span>F these Feates (farther applied) is +Sprong the Feate of <i>Geodesie</i>, or Land Measuring: more cunningly +to measure & Suruey Land, Woods, and Waters, a farre of. More +cunningly, I say: But God knoweth (hitherto) in these Realmes of +England and Ireland (whether through ignorance or fraude, I can not +tell, in euery particular) +<span class="sidenote"> +Note.</span> +how great wrong and iniurie hath (in my time) bene committed +<span class="pagenum">a.iij </span> +<!-- png 23 --> +by vntrue measuring and surueying of Land or Woods, any way. And, this I +am sure: that the Value of the difference, betwene the truth and such +Surueyes, would haue bene hable to haue foũd (for euer) in eche of our +two Vniuersities, an excellent Mathematicall Reader: to eche, allowing +(yearly) a hundred Markes of lawfull money of this realme: which, +in dede, would seme requisit, here, to be had (though by other wayes +prouided for) as well, as, the famous Vniuersitie of Paris, hath two +Mathematicall Readers: and eche, two hundreth French Crownes yearly, of +the French Kinges magnificent liberalitie onely. Now, againe, to our +purpose returning: Moreouer, of the former knowledge Geometricall, are +growen the Skills of <i>Geographie</i>, <i>Chorographie</i>, +<i>Hydrographie</i>, and <i>Stratarithmetrie</i>.</p> + +<p><span class="sidequote">“</span> +<span class="largest">Geographie</span> teacheth wayes, by which, in +sũdry formes, (as <i>Sphærike</i>, <i>Plaine</i> or other), the +Situation of Cities, Townes, Villages, Fortes, Castells, Mountaines, +Woods, Hauens, Riuers, Crekes, & such other things, vpõ the outface +of the earthly Globe (either in the whole, or in some principall mẽber +and portion therof cõtayned) may be described and designed, in +cõmensurations Analogicall to Nature and veritie: and most aptly to our +vew, may be represented.<span class="sidequote">”</span> Of this Arte +how great pleasure, and how manifolde commodities do come vnto vs, daily +and hourely: of most men, is perceaued. While, some, to beautifie their +Halls, Parlers, Chambers, Galeries, Studies, or Libraries with: other +some, for thinges past, as battels fought, earthquakes, heauenly +fyringes, & such occurentes, in histories mentioned: therby liuely, +as it were, to vewe the place, the region adioyning, the distance from +vs: and such other circumstances. Some other, presently to vewe the +large dominion of the Turke: the wide Empire of the Moschouite: and the +litle morsell of ground, where Christendome (by profession) is certainly +knowen. Litle, I say, in respecte of the rest. &c. Some, either +for their owne iorneyes directing into farre landes: or to vnderstand of +other mens trauailes. To conclude, some, for one purpose: and some, for +an other, liketh, loueth, getteth, and vseth, Mappes, Chartes, & +Geographicall Globes. Of whose vse, to speake sufficiently, would +require a booke peculier.</p> + +<p><span class="largest">Chorographie</span> seemeth to be an +vnderling, and a twig, of <i>Geographie</i>: and yet neuerthelesse, is +in practise manifolde, and in vse very ample. <span class="sidequote">“</span>This teacheth Analogically to describe a small +portion or circuite of ground, with the contentes: not regarding what +commensuration it hath to the whole, or any parcell, without it, +contained. But in the territory or parcell of ground which it taketh in +hand to make description of, it leaueth out (or vndescribed) no notable, +or odde thing, aboue the ground visible. Yea and sometimes, of thinges +vnder ground, geueth some peculier marke: or warning: as of Mettall +mines, Cole pittes, Stone quarries. &c.<span class="sidequote">”</span> Thus, a Dukedome, a Shiere, +a Lordship, or lesse, may be described distinctly. But marueilous +pleasant, and profitable it is, in the exhibiting to our eye, and +commensuration, the plat of a Citie, Towne, Forte, or Pallace, in true +Symmetry: not approching to any of them: and out of Gunne shot. &c. +Hereby, the <i>Architect</i> may furnishe him selfe, with store of what +patterns he liketh: to his great instruction: euen in those thinges +which outwardly are proportioned: either simply in them selues: or +respectiuely, to Hilles, Riuers, Hauens, and Woods adioyning. Some also, +terme this particular description of places, <i>Topographie</i>.</p> + +<p><span class="sidequote">“</span> +<span class="largest">Hydrographie</span>, deliuereth to our +knowledge, on Globe or in Plaine, the perfect Analogicall description of +the Ocean Sea coastes, through the whole world: or in the chiefe and +principall partes thereof:<span class="sidequote">”</span> with the +Iles and chiefe +<span class="pagenum verso">||</span> +<!-- png 24 --> +particular places of daungers, conteyned within the boundes, and Sea +coastes described: as, of Quicksandes, Bankes, Pittes, Rockes, Races, +Countertides, Whorlepooles. &c. This, dealeth with the Element of +the water chiefly: as <i>Geographie</i> did principally take the Element +of the Earthes description (with his appertenances) to taske. And +besides thys, <i>Hydrographie</i>, requireth a particular Register of +certaine Landmarkes (where markes may be had) from the sea, well hable +to be skried, in what point of the Seacumpase they appeare, and what +apparent forme, Situation, and bignes they haue, in respecte of any +daungerous place in the sea, or nere vnto it, assigned: And in all +Coastes, what Mone, maketh full Sea: and what way, the Tides and Ebbes, +come and go, the <i>Hydrographer</i> ought to recorde. The Soundinges +likewise: and the Chanels wayes: their number, and depthes ordinarily, +at ebbe and flud, ought the <i>Hydrographer</i>, by obseruation and +diligence of <i>Measuring</i>, to haue certainly knowen. And many other +pointes, are belonging to perfecte <i>Hydrographie</i>, and for to make +a <i>Rutter</i>, by: of which, I nede not here speake: as of the +describing, in any place, vpon Globe or Plaine, the 32. pointes of the +Compase, truely: (wherof, scarsly foure, in England, haue right +knowledge: bycause, the lines therof, are no straight lines, nor +Circles.) Of making due proiection of a Sphere in plaine. Of the +Variacion of the Compas, from true Northe: And such like matters (of +great importance, all) I leaue to speake of, in this place: +bycause, I may seame (al ready) to haue enlarged the boundes, and +duety of an <ins class="correction" title="text reads ‘Hydographer’">Hydrographer</ins>, much more, then any man (to this day) +hath noted, or prescribed. Yet am I well hable to proue, all these +thinges, to appertaine, and also to be proper to the Hydrographer. The +chief vse and ende of this Art, is the Art of Nauigation: but it hath +other diuerse vses: euen by them to be enioyed, that neuer lacke sight +of land.</p> + +<p><span class="largest">Stratarithmetrie</span>, is the Skill, +(appertainyng to the warre,) by which a man can set in figure, +analogicall to any <i>Geometricall</i> figure appointed, any certaine +number or summe of men: of such a figure capable: (by reason of the +vsuall spaces betwene Souldiers allowed: and for that, of men, can be +made no Fractions. Yet, neuertheles, he can order the giuen summe of +men, for the greatest such figure, that of them, cã be ordred) and +certifie, of the ouerplus: (if any be) and of the next certaine summe, +which, with the ouerplus, will admit a figure exactly proportionall to +the figure assigned. By which Skill, also, of any army or company of +men: (the figure & sides of whose orderly standing, or array, is +knowen) he is able to expresse the iust number of men, within that +figure conteined: or (orderly) able to be conteined. +<span class="sidenote"> +* <i>Note.</i></span> +*And this figure, and sides therof, he is hable to know: either beyng +by, and at hand: or a farre of. Thus farre, stretcheth the description +and property of <i>Stratarithmetrie</i>: sufficient for this tyme and +place. +<span class="sidenote"> +The difference betwene Stratarithmetrie and Tacticie.</span> +<span class="sidequote">“</span>It differreth from the Feate +<i>Tacticall</i>, <i>De aciebus instruendis.</i> bycause, there, is +necessary the wisedome and foresight, to what purpose he so ordreth the +men: and Skillfull hability, also, for any occasion, or purpose, to +deuise and vse the aptest and most necessary order, array and figure of +his Company and Summe of men.<span class="sidequote">”</span> By +figure, I meane: as, either of a <i>Perfect Square</i>, +<i>Triangle</i>, <i>Circle</i>, <i>Ouale</i>, <i>long square</i>, (of +the Grekes it is called <ins class="correction" title="text unchanged: correct form is ‘Heteromekes’"><i>Eteromekes</i></ins>) +<i>Rhombe</i>, <i>Rhomboïd</i>, <i>Lunular</i>, <i>Ryng</i>, +<i>Serpentine</i>, and such other Geometricall figures: Which, in +warres, haue ben, and are to be vsed: for commodiousnes, necessity, and +auauntage &c. And no small skill ought he to haue, that should make +true report, or nere the truth, of the numbers and Summes, of footemen +or horsemen, in the Enemyes ordring. A farre of, to make an +estimate, betwene nere termes of More and Lesse, is not a thyng very +rife, among those that gladly would +<span class="pagenum">b.j.</span> +<!-- png 25 --> +do it. +<span class="sidenote smaller"> +I. D.<br /> +Frende,<br /> +you will finde it hard, to performe my description of this Feate. But by +Chorographie, you may helpe your selfe some what: where the Figures +knowne (in Sides and Angles) are not Regular: And where, Resolution into +Triangles can serue. &c. And yet you will finde it strange to deale +thus generally with Arithmeticall figures: and, that for Battayle ray. +Their contentes, differ so much from like Geometricall Figures.</span> +Great pollicy may be vsed of the Capitaines, (at tymes fete, and in +places conuenient) as to vse Figures, which make greatest shew, of so +many as he hath: and vsing the aduauntage of the three kindes of vsuall +spaces: (betwene footemen or horsemen) to take the largest: or when he +would seme to haue few, (beyng many:) contrarywise, in Figure, and +space. The Herald, Purseuant, Sergeant Royall, Capitaine, or who soeuer +is carefull to come nere the truth herein, besides the Iudgement of his +expert eye, his skill of Ordering <i>Tacticall</i>, the helpe of his +Geometricall instrument: Ring, or Staffe Astronomicall: (commodiously +framed for cariage and vse) He may wonderfully helpe him selfe, by +perspectiue Glasses. In which, (I trust) our posterity will proue +more skillfull and expert, and to greater purposes, then in these dayes, +can (almost) be credited to be possible.</p> + +<p>Thus haue I lightly passed ouer the Artificiall Feates, chiefly +dependyng vpon vulgar <i>Geometrie</i>: & commonly and generally +reckened vnder the name of <i>Geometrie</i>. But there are other (very +many) <i>Methodicall Artes</i>, which, declyning from the purity, +simplicitie, and Immateriality, of our Principall Science of +<i>Magnitudes</i>: do yet neuertheles vse the great ayde, direction, and +Method of the sayd principall Science, and haue propre names, and +distinct: both from the Science of <i>Geometrie</i>, (from which they +are deriued) and one from the other. As <span class="larger">Perspectiue, Astronomie, Musike, Cosmographie, Astrologie, +Statike, Anthropographie, Trochilike, Helicosophie, Pneumatithmie, +Menadrie, Hypogeiodie, Hydragogie, Horometrie, Zographie, Architecture, +Nauigation, Thaumaturgike</span> and <span class="larger">Archemastrie</span>. I thinke it necessary, orderly, of +these to giue some peculier descriptions: and withall, to touch some of +their commodious vses, and so to make this Preface, to be a little +swete, pleasant Nosegaye for you: to comfort your Spirites, beyng almost +out of courage, and in despayre, (through brutish brute) Weenyng that +<i>Geometrie</i>, had but serued for buildyng of an house, or a curious +bridge, or the roufe of Westminster hall, or some witty pretty deuise, +or engyn, appropriate to a Carpenter, or a Ioyner &c. That the thing +is farre otherwise, then the world, (commonly) to this day, hath demed, +by worde and worke, good profe wilbe made.</p> + +<p>Among these Artes, by good reason, <span class="largest">Perspectiue</span> ought to be had, ere of <i>Astronomicall +Apparences</i>, perfect knowledge can be atteyned. And bycause of the +prerogatiue of <i>Light</i>, beyng the first of <i>Gods Creatures</i>: +and the eye, the light of our body, and his Sense most mighty, and his +organ most Artificiall and <i>Geometricall</i>: At <i>Perspectiue</i>, +we will begyn therfore. <span class="larger">Perspectiue, is an Art +Mathematicall, which demonstrateth the maner, and properties, of all +Radiations Direct, Broken, and Reflected.</span> This Description, or +Notation, is brief: but it reacheth so farre, as the world is wyde. It +concerneth all Creatures, all Actions, and passions, by Emanation of +beames perfourmed. Beames, or naturall lines, (here) I meane, not +of light onely, or of colour (though they, to eye, giue shew, witnes, +and profe, wherby to ground the Arte vpon) but also of other +<i>Formes</i>, both <i>Substantiall</i>, and <i>Accidentall</i>, the +certaine and determined actiue Radiall emanations. By this Art (omitting +to speake of the highest pointes) we may vse our eyes, and the light, +with greater pleasure: and perfecter Iudgement: both of things, in light +seen, & of other: which by like order of Lightes Radiations, worke +and produce their effectes. We may be ashamed to be ignorant of the +cause, why so sundry wayes our eye is deceiued, and abused: as, while +the eye weeneth a roũd Globe or Sphere (beyng farre of) to be a flat and +plaine Circle, and so likewise iudgeth +<span class="pagenum verso">||</span> +<!-- png 26 --> +a plaine Square, to be roũd: supposeth walles parallels, to approche, +a farre of: rofe and floure parallels, the one to bend downward, +the other to rise vpward, at a little distance from you. Againe, of +thinges being in like swiftnes of mouing, to thinke the nerer, to moue +faster: and the farder, much slower. Nay, of two thinges, wherof the one +(incomparably) doth moue swifter then the other, to deme the slower to +moue very swift, & the other to stand: what an error is this, of our +eye? Of the Raynbow, both of his Colours, of the order of the colours, +of the bignes of it, the place and heith of it, (&c) to know the +causes demonstratiue, is it not pleasant, is it not necessary? of two or +three Sonnes appearing: of Blasing Sterres: and such like thinges: by +naturall causes, brought to passe, (and yet neuertheles, of farder +matter, Significatiue) is it not commodious for man to know the very +true cause, & occasion Naturall? Yea, rather, is it not, greatly, +against the Souerainty of Mans nature, to be so ouershot and abused, +with thinges (at hand) before his eyes? as with a Pecockes tayle, and a +Doues necke: or a whole ore, in water, holden, to seme broken. Thynges, +farre of, to seeme nere: and nere, to seme farre of. Small thinges, to +seme great: and great, to seme small. One man, to seme an Army. Or a man +to be curstly affrayed of his owne shaddow. Yea, so much, to feare, +that, if you, being (alone) nere a certaine glasse, and proffer, with +dagger or sword, to foyne at the glasse, you shall suddenly be moued to +giue backe (in maner) by reason of an +<span class="sidenote"> +A marueilous Glasse.</span> +<span class="sidefinger"> </span> +Image, appearing in the ayre, betwene you & the glasse, with like +hand, sword or dagger, & with like quicknes, foyning at your very +eye, likewise as you do at the Glasse. Straunge, this is, to heare of: +but more meruailous to behold, then these my wordes can signifie. And +neuerthelesse by demonstration Opticall, the order and cause therof, is +certified: euen so, as the effect is consequent. Yea, thus much more, +dare I take vpon me, toward the satisfying of the noble courrage, that +longeth ardently for the wisedome of Causes Naturall: as to let him +vnderstand, that, in London, he may with his owne eyes, haue profe of +that, which I haue sayd herein. A Gentleman, (which, for his good +seruice, done to his Countrey, is famous and honorable: +<span class="sidenote"> +S. W. P.</span> +and for skill in the Mathematicall Sciences, and Languages, is the Od +man of this land. &c.) euen he, is hable: and (I am sure) will, +very willingly, let the Glasse, and profe be sene: and so I (here) +request him: for the encrease of wisedome, in the honorable: and for the +stopping of the mouthes malicious: and repressing the arrogancy of the +ignorant. Ye may easily gesse, what I meane. This Art of +<i>Perspectiue</i>, is of that excellency, and may be led, to the +certifying, and executing of such thinges, as no man would easily +beleue: without Actuall profe perceiued. I speake nothing of +<i>Naturall Philosophie</i>, which, without <i>Perspectiue</i>, can not +be fully vnderstanded, nor perfectly atteined vnto. Nor, of +<i>Astronomie</i>: which, without <i>Perspectiue</i>, can not well be +grounded: Nor <i>Astrologie</i>, naturally Verified, and auouched. That +part hereof, which dealeth with Glasses (which name, Glasse, is a +generall name, in this Arte, for any thing, from which, a Beame +reboundeth) is called <i>Catoptrike</i>: and hath so many vses, both +merueilous, and proffitable: that, both, it would hold me to long, to +note therin the principall conclusions, all ready knowne: And also +(perchaunce) some thinges, might lacke due credite with you: And I, +therby, to leese my labor: and you, to slip into light Iudgement*, +<span class="sidefinger"> + </span> +Before you haue learned sufficiently the powre of Nature and Arte.</p> + +<p>Now, to procede: <span class="largest">Astronomie</span>, <span +class="larger">is an Arte Mathematicall, which demonstrateth the +distance, magnitudes, and all naturall motions, apparences, and passions +propre to the Planets and fixed Sterres: for</span> +<span class="pagenum">b.ij </span> +<!-- png 27 --> +<span class="larger">any time past, present and to come: in respect of +a certaine Horizon, or without respect of any Horizon.</span> By this +Arte we are certified of the distance of the Starry Skye, and of eche +<i>Planete</i> from the Centre of the Earth: and of the greatnes of any +Fixed starre sene, or <i>Planete</i>, in respect of the Earthes +greatnes. As, we are sure (by this Arte) that the Solidity, Massines and +Body of the <i>Sonne</i>, conteineth the quantitie of the whole Earth +and Sea, a hundred thre score and two times, lesse by ⅛ one eight +parte of the earth. But the Body of the whole earthly globe and Sea, is +bigger then the body of the Mone, three and forty times lesse by ⅛ of +the Mone. Wherfore the <i>Sonne</i> is bigger then the <i>Mone</i>, 7000 +times, lesse, by 59 39/64 that is, precisely 6940 25/64 bigger then the +<i>Mone</i>. And yet the vnskillfull man, would iudge them a like bigge. +Wherfore, of Necessity, the one is much farder from vs, then the other. +The <i>Sonne</i>, when he is fardest from the earth (which, now, in our +age, is, when he is in the 8. degree, of Cancer) is, 1179 Semidiameters +of the Earth, distante. And the <i>Mone</i> when she is fardest from the +earth, is 68 Semidiameters of the earth and ⅓ The nerest, that the +<i>Mone</i> commeth to the earth, is Semidiameters 52¼ The distance of +the Starry Skye is, frõ vs, in Semidiameters of the earth 20081½ Twenty +thousand fourescore, one, and almost a halfe. Subtract from this, the +<i>Mones</i> nerest distance, from the Earth: and therof remaineth +Semidiameters of the earth 20029¼ Twenty thousand nine and twenty and a +quarter. +<span class="sidenote"> +Note.</span> +So thicke is the heauenly Palace, that the <i>Planetes</i> haue all +their exercise in, and most meruailously perfourme the Commaũdement and +Charge to them giuen by the omnipotent Maiestie of the king of kings. +This is that, which in <i>Genesis</i> is called <i>Ha Rakia</i>. +Consider it well. The Semidiameter of the earth, cõteineth of our common +miles 3436 4/11 three thousand, foure hundred thirty six and foure +eleuenth partes of one myle: Such as the whole earth and Sea, round +about, is 21600. One and twenty thousand six hundred of our myles. +Allowyng for euery degree of the greatest circle, thre score myles. Now +if you way well with your selfe but this litle parcell of frute +<i>Astronomicall</i>, as concerning the bignesse, Distances of +<i>Sonne</i>, <i>Mone</i>, <i>Sterry Sky</i>, and the huge massines of +<i>Ha Rakia</i>, will you not finde your Consciences moued, with the +kingly Prophet, to sing the confession of Gods Glory, and say, <span +class="larger"><i>The Heauens declare the glory of God, and the +Firmament</i></span> <span class="smaller">[<i>Ha Rakia</i>]</span> +<span class="larger"><i>sheweth forth the workes of his +handes</i></span>. And so forth, for those fiue first staues, of that +kingly Psalme. Well, well, It is time for some to lay hold on wisedome, +and to Iudge truly of thinges: and notso to expound the Holy word, all +by Allegories: as to Neglect the wisedome, powre and Goodnes of God, in, +and by his Creatures, and Creation to be seen and learned. By parables +and Analogies of whose natures and properties, the course of the Holy +Scripture, also, declareth to vs very many Mysteries. The whole Frame of +Gods Creatures, (which is the whole world,) is to vs, a bright +glasse: from which, by reflexion, reboundeth to our knowledge and +perceiuerance, Beames, and Radiations: representing the Image of his +Infinite goodnes, Omnipotẽcy, and wisedome. And we therby, are taught +and persuaded to Glorifie our Creator, as God: and be thankefull +therfore. Could the Heathenistes finde these vses, of these most pure, +beawtifull, and Mighty Corporall Creatures: and shall we, after that the +true <i>Sonne</i> of rightwisenesse is risen aboue the <i>Horizon</i>, +of our temporall <i>Hemisphærie</i>, and hath so abundantly streamed +into our hartes, the direct beames of his goodnes, mercy, and grace: +Whose heat All Creatures feele: Spirituall and Corporall: Visible and +<span class="pagenum verso">||</span> +<!-- png 28 --> +Inuisible. Shall we (I say) looke vpon the <i>Heauen</i>, +<i>Sterres</i>, and <i>Planets</i>, as an Oxe and an Asse doth: no +furder carefull or inquisitiue, what they are: why were they Created, +How do they execute that they were Created for? Seing, All Creatures, +were for our sake created: and both we, and they, Created, chiefly to +glorifie the Almighty Creator: and that, by all meanes, to vs possible. +<i>Nolite ignorare</i> (saith <i>Plato in Epinomis</i>) <i>Astronomiam, +Sapientissimũ quiddam esse.</i> <span class="larger"><i>Be ye not +ignorant, Astronomie to be a thyng of excellent wisedome.</i></span> +<i>Astronomie</i>, was to vs, from the beginning commended, and in maner +commaunded by God him selfe. In asmuch as he made the <i>Sonne</i>, +<i>Mone</i>, and <i>Sterres</i>, to be to vs, for <i>Signes</i>, and +knowledge of Seasons, and for Distinctions of Dayes, and yeares. Many +wordes nede not. But I wish, euery man should way this word, +<i>Signes</i>. And besides that, conferre it also with the tenth Chapter +of <i>Hieremie</i>. And though Some thinke, that there, they haue found +a rod: Yet Modest Reason, will be indifferent Iudge, who ought to be +beaten therwith, in respect of our purpose. Leauing that: I pray +you vnderstand this: that without great diligence of Obseruation, +examination and Calculation, their periods and courses (wherby +<i>Distinction</i> of Seasons, yeares, and New Mones might precisely be +knowne) could not exactely be certified. Which thing to performe, is +that <i>Art</i>, which we here haue Defined to be <i>Astronomie</i>. +Wherby, we may haue the distinct Course of Times, dayes, yeares, and +Ages: aswell for Consideratiõ of Sacred Prophesies, accomplished in due +time, foretold: as for high Mysticall Solemnities holding: And for all +other humaine affaires, Conditions, and couenantes, vpon certaine time, +betwene man and man: with many other great vses: Wherin, (verely), would +be great incertainty, Confusion, vntruth, and brutish Barbarousnes: +without the wonderfull diligence and skill of this Arte: continually +learning, and determining Times, and periodes of Time, by the Record of +the heauenly booke, wherin all times are written: and to be read with an +<i>Astronomicall staffe</i>, in stede of a festue.</p> + +<p><span class="largest">Musike</span>, of Motion, hath his Originall +cause: Therfore, after the motions most swift, and most Slow, which are +in the Firmament, of Nature performed: and vnder the <i>Astronomers +Consideration</i>: now I will Speake of an other kinde of <i>Motion</i>, +producing sound, audible, and of Man numerable. <i>Musike</i> I call +here that <i>Science</i>, which of the Grekes is called +<i>Harmonice</i>. Not medling with the Controuersie betwene the auncient +<i>Harmonistes</i>, and <i>Canonistes</i>. <span class="larger">Musike +is a Mathematicall Science, which teacheth, by sense and reason, +perfectly to iudge, and order the diuersities of soundes, hye and +low.</span> <i>Astronomie</i> and <i>Musike</i> are Sisters, saith +<i>Plato</i>. As, for <i>Astronomie</i>, the eyes: So, for <i>Harmonious +Motion</i>, the eares were made. But as <i>Astronomie</i> hath a more +diuine Contemplation, and cõmodity, then mortall eye can perceiue: So, +is <i>Musike</i> to be considered, that the +<span class="sidenote"> +1.</span> +*Minde may be preferred, before the eare. And from audible sound, we +ought to ascende, to the examination: which numbers are +<i>Harmonious</i>, and which not. And why, either, the one are: or the +other are not. I could at large, in the heauenly +<span class="sidenote"> +2.</span> +*motions and distances, describe a meruailous Harmonie, of +<i>Pythagoras</i> Harpe +<span class="sidenote"> +3.</span> +with eight stringes. Also, somwhat might be sayd of <i>Mercurius</i>* +<span class="sidenote"> +4.</span> +two Harpes, eche of foure Stringes Elementall. And very straunge matter, +might be alledged of the <i>Harmonie</i>, to our +<span class="sidenote"> +5.</span> +*Spirituall part appropriate. As in <i>Ptolomaus</i> third boke, in the +fourth and sixth Chapters may appeare.* +<span class="sidenote"> +6.</span> +And what is the cause of the apt bonde, and frendly felowship, of the +Intellectuall and Mentall part of vs, with our grosse & corruptible +body: but a certaine Meane, and <i>Harmonious Spiritualitie, with +<span class="pagenum">b.iii </span> +<!-- png 29 --> +both participatyng, & of both (in a maner) resultynge In the +<span class="sidenote"> +7.</span> +*Tune of Mans voyce, and also +<span class="sidenote"> +8.</span> +* the sound of Instrument</i>, what might be sayd, of <i>Harmonie</i>: +No common Musicien would lightly beleue. +<span class="sidenote smaller"> +I. D.<br /> +Read in Aristotle his 8. booke of Politikes: the 5, 6, and 7. +chapters. Where you shall haue some occasion farder to thinke of Musike, +than commonly is thought.</span> +But of the sundry Mixture (as I may terme it) and concurse, diuerse +collation, and Application of these <i>Harmonies</i>: as of thre, foure, +fiue, or mo: Maruailous haue the effectes ben: and yet may be founde, +and produced the like: with some proportionall consideration for our +time, and being: in respect of the State, of the thinges then: in which, +and by which, the wondrous effectes were wrought. <i>Democritus</i> and +<i>Theophrastus</i> affirmed, that, by <i>Musike</i>, griefes and +diseases of the Minde, and body might be cured, or inferred. And we +finde in Recorde, that <i>Terpander</i>, <i>Arion</i>, <i>Ismenias</i>, +<i>Orpheus</i>, <i>Amphion</i>, <i>Dauid</i>, <i>Pythagoras</i>, +<i>Empedocles</i>, <i>Asclepiades</i> and <i>Timotheus</i>, by +<i>Harmonicall</i> Consonãcy, haue done, and brought to pas, thinges, +more then meruailous, to here of. Of them then, making no farder +discourse, in this place: Sure I am, that Common <i>Musike</i>, commonly +vsed, is found to the <i>Musiciens</i> and Hearers, to be so Commodious +and pleasant, That if I would say and dispute, but thus much: That it +were to be otherwise vsed, then it is, I should finde more +repreeuers, then I could finde priuy, or skilfull of my meaning. In +thinges therfore euident, and better knowen, then I can expresse: and so +allowed and liked of, (as I would wish, some other thinges, had the like +hap) I will spare to enlarge my lines any farder, but consequently +follow my purpose.</p> + +<p><span class="largest">Of Cosmographie</span>, I appointed briefly +in this place, to geue you some intelligence. <span class="larger">Cosmographie, is the whole and perfect description of the +heauenly, and also elementall parte of the world, and their homologall +application, and mutuall collation necessarie.</span> This Art, +requireth <i>Astronomie</i>, <i>Geographie</i>, <i>Hydrographie</i> and +<i>Musike</i>. Therfore, it is no small Arte, nor so simple, as in +common practise, it is (slightly) considered. This matcheth Heauen, and +the Earth, in one frame, and aptly applieth parts Correspõdent: So, as, +the Heauenly Globe, may (in practise) be duely described vpon the +Geographicall, and Hydrographicall Globe. And there, for vs to consider +an <i>Æquonoctiall Circle</i>, <i>an Ecliptike line</i>, <i>Colures</i>, +<i>Poles</i>, <i>Sterres</i> in their true Longitudes, Latitudes, +Declinations, and Verticalitie: also Climes, and Parallels: and by an +<i>Horizon</i> annexed, and reuolution of the earthly Globe (as the +Heauen, is, by the <i>Primouant</i>, caried about in 24. æquall Houres) +to learne the Risinges and Settinges of Sterres (of <i>Virgill</i> in +his <i>Georgikes</i>: of <i>Hesiod</i>: of <i>Hippocrates</i> in his +<i>Medicinall Sphære</i>, to Perdicca King of the Macedonians: of +<i>Diocles</i>, to King <i>Antigonus</i>, and of other famous +<i>Philosophers</i> prescribed) a thing necessary, for due manuring +of the earth, for <i>Nauigation</i>, for the Alteration of mans body: +being, whole, Sicke, wounded, or brused. By the Reuolution, also, or +mouing of the Globe Cosmographicall, the Rising and Setting of the +Sonne: the Lengthes, of dayes and nightes: the Houres and times (both +night and day) are knowne: with very many other pleasant and necessary +vses: Wherof, some are knowne: but better remaine, for such to know and +vse: +<span class="sidefinger"> + </span> +who of a sparke of true fire, can make a wonderfull bonfire, by applying +of due matter, duely.</p> + +<p><span class="largest">Of Astrologie</span>, here I make an Arte, +seuerall from <i>Astronomie</i>: not by new deuise, but by good reason +and authoritie: for, <span class="larger">Astrologie, is an Arte +Mathematicall, which reasonably demonstrateth the operations and +effectes, of the naturall beames, of light, and secrete influence: of +the Sterres and Planets: in euery element and elementall body:</span> +<span class="pagenum verso">||</span> +<!-- png 30 --> +<span class="larger">at all times, in any Horizon assigned.</span> +This Arte is furnished with many other great Artes and experiences: As +with perfecte <i>Perspectiue</i>, <i>Astronomie</i>, +<i>Cosmographie</i>, <i>Naturall Philosophie</i> of the 4. Elementes, +the Arte of Graduation, and some good vnderstãding in <i>Musike</i>: and +yet moreouer, with an other great Arte, hereafter following, +though I, here, set this before, for some considerations me mouing. +Sufficient (you see) is the stuffe, to make this rare and secrete Arte, +of: and hard enough to frame to the Conclusion Syllogisticall. Yet both +the manifolde and continuall trauailes of the most auncient and wise +Philosophers, for the atteyning of this Arte: and by examples of +effectes, to confirme the same: hath left vnto vs sufficient proufe and +witnesse: and we, also, daily may perceaue, That mans body, and all +other Elementall bodies, are altered, disposed, ordred, pleasured, and +displeasured, by the Influentiall working of the <i>Sunne</i>, +<i>Mone</i>, and the other Starres and Planets. And therfore, sayth +<i>Aristotle</i>, in the first of his <i>Meteorologicall</i> bookes, in +the second Chapter: <i>Est autem necessariò Mundus iste, supernis +lationibus ferè continuus. Vt, inde, vis eius vniuersa regatur. Ea +siquidem Causà prima putanda omnibus est, vnde motus principium +existit.</i> That is: <span class="larger"><i>This</i></span> <span +class="smaller">[<i>Elementall</i>]</span> <span class="larger"><i>World is of necessitie, almost, next adioyning, to the +heauenly motions: That, from thence, all his vertue or force may be +gouerned. For, that is to be thought the first Cause vnto all: from +which, the beginning of motion, is.</i></span> And againe, in the tenth +Chapter. <i>Oportet igitur & horum principia sumamus, & causas +omnium similiter. Principium igitur vt mouens, præcipuumq<sup>ue</sup> +& omnium primum, Circulus ille est, in quo manifeste Solis latio, +&c.</i> And so forth. His <i>Meteorologicall</i> bookes, are full of +argumentes, and effectuall demonstrations, of the vertue, operation, and +power of the heauenly bodies, in and vpon the fower Elementes, and other +bodies, of them (either perfectly, or vnperfectly) composed. And in his +second booke, <i>De Generatione & Corruptione</i>, in the tenth +Chapter. <i>Quocirca & prima latio, Ortus & Interitus causa non +est: Sed obliqui Circuli latio: ea namq<sup>ue</sup> & continua est, +& duobus motibus fit:</i> In Englishe, thus. <span class="larger"><i>Wherefore the vppermost motion, is not the cause of +Generation and Corruption, but the motion of the Zodiake: for, that, +both, is continuall, and is caused of two mouinges.</i></span> And in +his second booke, and second Chapter of hys <i>Physikes</i>. <i>Homo +namq<sup>ue</sup> generat hominem, atq<sup>ue</sup> Sol.</i> <span class="larger"><i>For Man (sayth he) and the Sonne, are cause of mans +generation.</i></span> Authorities may be brought, very many: both of +1000. 2000. yea and 3000. yeares Antiquitie: of great +<i>Philosophers</i>, <i>Expert</i>, <i>Wise</i>, and godly men, for that +Conclusion: which, daily and hourely, we men, may discerne and perceaue +by sense and reason: All beastes do feele, and simply shew, by their +actions and passions, outward and inward: All Plants, Herbes, Trees, +Flowers, and Fruites. And finally, the Elementes, and all thinges of the +Elementes composed, do geue Testimonie (as <i>Aristotle</i> sayd) that +theyr <span class="larger"><i>Whole Dispositions, vertues, and +naturall motions, depend of the Actiuitie of the heauenly motions and +Influences. Whereby, beside the specificall order and forme, due to +euery seede: and beside the Nature, propre to the Indiuiduall Matrix, of +the thing produced: What shall be the heauenly Impression, the perfect +and circumspecte Astrologien hath to Conclude.</i></span> Not onely (by +<i>Apotelesmes</i>) <ins class="correction" title="accent unchanged"><span class="greek" title="to hoti">τὸ ὁτὶ</span></ins>. +but by Naturall and Mathematicall demonstration <span class="greek" +title="to dioti">τὸ διότι</span>. Whereunto, what Sciences are +requisite (without exception) I partly haue here warned: And in my +<i>Propædeumes</i> (besides other matter there disclosed) I haue +Mathematically furnished vp the whole Method: To this our age, not so +carefully handled by any, that +<span class="pagenum">b.iiij </span> +<!-- png 31 --> +euer I saw, or heard of. I was, +<span class="sidenote"> +* Anno. 1548 and 1549. in Louayn.</span> +(for *21. yeares ago) by certaine earnest disputations, of the Learned +<i>Gerardus Mercator</i>, and <i>Antonius Gogaua</i>, (and other,) +therto so prouoked: and (by my constant and inuincible zeale to the +veritie) in obseruations of Heauenly Influencies (to the Minute of +time,) than, so diligent: And chiefly by the Supernaturall influence, +from the Starre of Iacob, so directed: That any Modest and Sober +Student, carefully and diligently seking for the Truth, will both finde +& cõfesse, therin, to be the Veritie, of these my wordes: And also +become a Reasonable Reformer, of three Sortes of people: about these +Influentiall Operations, greatly erring from the truth. +<span class="sidenote"> +Note.</span> +Wherof, the one, is <span class="larger">Light Beleuers</span>, the +other, <span class="larger">Light Despisers</span>, and the third +<span class="larger">Light Practisers</span>. The first, & most +cõmon Sort, thinke the Heauen and Sterres, to be answerable to any their +doutes or desires: +<span class="sidenote"> +1.</span> +which is not so: and, in dede, they, to much, ouer reache. The Second +sorte thinke no Influentiall vertue (frõ the heauenly bodies) to beare +any Sway in Generation +<span class="sidenote"> +2.</span> +and Corruption, in this Elementall world. And to the <i>Sunne</i>, +<i>Mone</i> and <i>Sterres</i> (being so many, so pure, so bright, so +wonderfull bigge, so farre in distance, so manifold in their motions, so +constant in their periodes. &c.) they assigne a sleight, simple +office or two, and so allow vnto thẽ (according to their capacities) as +much vertue, and power Influentiall, as to the Signe of the +<i>Sunne</i>, <i>Mone</i>, and seuen Sterres, hanged vp (for Signes) in +London, for distinction of houses, & such grosse helpes, in our <ins +class="correction" title="text reads ‘wordly’">worldly</ins> +affaires: And they vnderstand not (or will not vnderstand) of the other +workinges, and vertues of the Heauenly <i>Sunne</i>, <i>Mone</i>, and +<i>Sterres</i>: not so much, as the Mariner, or Husband man: no, not so +much, as the <i>Elephant</i> doth, as the <i>Cynocephalus</i>, as the +Porpentine doth: nor will allow these perfect, and incorruptible mighty +bodies, so much vertuall Radiation, & Force, as they see in a litle +peece of a <i>Magnes stone</i>: which, at great distance, sheweth his +operation. And perchaunce they thinke, the Sea & Riuers (as the +Thames) to be some quicke thing, and so to ebbe, and flow, run in and +out, of them selues, at their owne fantasies. God helpe, God helpe. +Surely, these men, come to short: and either are to dull: or willfully +blind: or, perhaps, to malicious. The third man, is the common and +vulgare <i>Astrologien</i>, or Practiser: who, being not duely, +artificially, and perfectly +<span class="sidenote"> +3.</span> +furnished: yet, either for vaine glory, or gayne: or like a simple dolt, +& blinde Bayard, both in matter and maner, erreth: to the discredit +of the <i>Wary</i>, and modest <i>Astrologien</i>: and to the robbing of +those most noble corporall Creatures, of their Naturall Vertue: being +most mighty: most beneficiall to all elementall Generation, Corruption +and the appartenances: and most Harmonious in their Monarchie: For which +thinges, being knowen, and modestly vsed: we might highly, and +continually glorifie God, with the princely Prophet, saying. <span class="larger"><i>The Heauens declare the Glorie of God: who made the Heauẽs +in his wisedome: who made the Sonne, for to haue dominion of the day: +the Mone and Sterres to haue dominion of the nyght: whereby, Day to day +vttereth talke: and night, to night declareth knowledge. Prayse him, all +ye Sterres, and Light. Amen.</i></span></p> + +<p>In order, now foloweth, of <span class="largest">Statike</span>, +somewhat to say, what we meane by that name: and what commodity, doth, +on such Art, depend. <span class="larger">Statike, is an Arte +Mathematicall, which demonstrateth the causes of heauynes, and lightnes +of all thynges: and of motions and properties, to heauynes and lightnes, +belonging.</span> And for asmuch as, by the Bilanx, or Balance (as the +chief sensible Instrument,) Experience of these demonstrations may +<span class="pagenum verso">||</span> +<!-- png 32 --> +be had: we call this Art, <i>Statike:</i> that is, <i>the Experimentes +of the Balance</i>. Oh, that men wist, what proffit, (all maner of +wayes) by this Arte might grow, to the hable examiner, and diligent +practiser. <span class="sidequote">“</span>Thou onely, knowest all +thinges precisely (O God) who hast made weight and Balance, thy +Iudgement: who hast created all thinges in <i>Number, Waight, and +Measure</i>: and hast wayed the mountaines and hils in a Balance: who +hast peysed in thy hand, both Heauen and earth. We therfore warned by +the Sacred word, to Consider thy Creatures: and by that consideration, +to wynne a glyms (as it were,) or shaddow of perceiuerance, that thy +wisedome, might, and goodnes is infinite, and vnspeakable, in thy +Creatures declared: And being farder aduertised, by thy mercifull +goodnes, that, three principall wayes, were, of the, vsed in Creation of +all thy Creatures, namely, <i>Number</i>, <i>Waight</i> and +<i>Measure</i>, And for as much as, of <i>Number</i> and <i>Measure</i>, +the two Artes (auncient, famous, and to humaine vses most necessary,) +are, all ready, sufficiently knowen and extant: This third key, we +beseche thee (through thy accustomed goodnes,) that it may come to the +nedefull and sufficient knowledge, of such thy Seruauntes, as in thy +workemanship, would gladly finde, thy true occasions (purposely of the +vsed) whereby we should glorifie thy name, and shew forth (to the +weaklinges in faith) thy wondrous wisedome and Goodnes. Amen.<span class="sidequote">”</span></p> + +<p>Meruaile nothing at this pang (godly frend, you Gentle and zelous +Student.) An other day, perchaunce, you will perceiue, what occasion +moued me. Here, as now, I will giue you some ground, and withall +some shew, of certaine commodities, by this Arte arising. And bycause +this Arte is rare, my wordes and practises might be to darke: vnleast +you had some light, holden before the matter: and that, best will be, in +giuing you, out of <i>Archimedes</i> demonstrations, a few +principal Conclusions, as foloweth.</p> + +<h5>1.</h5> + +<p class="larger"> +The Superficies of euery Liquor, by it selfe consistyng, and in quyet, +is Sphæricall: the centre whereof, is the same, which is the centre of +the Earth.</p> + +<h5>2.</h5> + +<p class="larger"> +If Solide Magnitudes, being of the same bignes, or quãtitie, that any +Liquor is, and hauyng also the same Waight: be let downe into the same +Liquor, they will settle downeward, so, that no parte of them, shall be +aboue the Superficies of the Liquor: and yet neuertheles, they will not +sinke vtterly downe, or drowne.</p> + +<h5>3.</h5> + +<p class="larger"> +If any Solide Magnitude beyng Lighter then a Liquor, be let downe into +the same Liquor, it will settle downe, so farre into the same Liquor, +that so great a quantitie of that Liquor, as is the parte of the Solid +Magnitude, settled downe into the same Liquor: is in Waight, æquall, to +the waight of the whole Solid Magnitude.</p> + +<h5>4.</h5> + +<span class="pagenum">c.j.</span> +<p class="larger"> +Any Solide Magnitude, Lighter then a Liquor, forced downe +<!-- png 33 --> +into the same Liquor, will moue vpward, with so great a power, by how +much, the Liquor hauyng æquall quantitie to the whole Magnitude, is +heauyer then the same Magnitude.</p> + +<h5>5.</h5> + +<p class="larger"> +Any Solid Magnitude, heauyer then a Liquor, beyng let downe into the +same Liquor, will sinke downe vtterly: And wilbe in that Liquor, Lighter +by so much, as is the waight or heauynes of the Liquor, hauing bygnes or +quantitie, æquall to the Solid Magnitude.</p> + +<h5>6.</h5> + +<span class="sidenote smaller">I. D.<br /> +The Cutting of a Sphære according to any proportion assigned may by this +proposition be done Mechanically by tempering Liquor to a certayne +waight in respect of the waight of the Sphære therein Swymming.</span> + +<p class="larger"> +If any Solide Magnitude, Lighter then a Liquor, be let downe into the +same Liquor, the waight of the same Magnitude, will be, to the Waight of +the Liquor. (Which is æquall in quantitie to the whole Magnitude,) in +that proportion, that the parte, of the Magnitude settled downe, is to +the whole Magnitude.</p> + +<p><span class="dropcap">B</span>Y these verities, great Errors may be +reformed, in Opinion of the Naturall Motion of thinges, Light and Heauy. +Which errors, are in Naturall Philosophie (almost) of all mẽ allowed: to +much trusting to Authority: and false Suppositions. As, <span class="larger">Of any two bodyes, the heauyer, to moue downward faster then +the lighter.</span> +<span class="sidenote"> +A common error, noted.</span> +This error, is not first by me, Noted: but by one <i>Iohn Baptist de +Benedictis</i>. The chief of his propositions, is this: which seemeth a +Paradox.</p> + +<span class="sidenote"> +A paradox.</span> +<p><span class="larger">If there be two bodyes of one forme, and of +one kynde, æquall in quantitie or vnæquall, +they will moue by æquall space, in æquall tyme: So that both theyr +mouynges be in ayre, or both in water: or in any one Middle.</span></p> + +<p>Hereupon, in the feate of +<span class="sidenote"> +N. T.</span> +<span class="larger">Gunnyng</span>, certaine good discourses +(otherwise) may receiue great amendement, and furderance. +<span class="sidenote"> +The wonderfull vse of these Propositions.</span> +In the entended purpose, also, allowing somwhat to the imperfection of +Nature: not aunswerable to the precisenes of demonstration. Moreouer, by +the foresaid propositions (wisely vsed.) The Ayre, the water, the Earth, +the Fire, may be nerely, knowen, how light or heauy they are (Naturally) +in their assigned partes: or in the whole. And then, to thinges +Elementall, turning your practise: you may deale for the proportion of +the Elementes, in the thinges Compounded. Then, to the proportions of +the Humours in Man: their waightes: and the waight of his bones, and +flesh. &c. Than, by waight, to haue consideration of the Force of +man, any maner of way: in whole or in part. Then, may you, of Ships +water drawing, diuersly, in the Sea and in fresh water, haue pleasant +consideration: and of waying vp of any thing, sonken in Sea or in fresh +water &c. And (to lift vp your head a loft:) by waight, you may, as +precisely, as by any instrument els, measure the Diameters of +<i>Sonne</i> and <i>Mone. &c.</i> Frende, I pray you, way these +thinges, with the iust Balance of Reason. And you will finde Meruailes +vpon Meruailes: And esteme one Drop of Truth (yea in Naturall +Philosophie) more worth, then whole Libraries of Opinions, +vndemonstrated: or not aunswering to Natures Law, and your experience. +Leauing these +<span class="pagenum verso">||</span> +<!-- png 34 --> +thinges, thus: I will giue you two or three, light practises, to +great purpose: and so finish my Annotation <i>Staticall</i>. In +Mathematicall matters, by the Mechaniciens ayde, we will behold, here, +the Commodity of waight. +<span class="sidenote"> +The practise Staticall, to know the proportion, betwene the Cube, and +the Sphære.</span> +Make a Cube, of any one Vniforme: and through like heauy stuffe: of the +same Stuffe, make a Sphære or Globe, precisely, of a Diameter æquall to +the Radicall side of the Cube. Your stuffe, may be wood, Copper, Tinne, +Lead, Siluer. &c. (being, as I sayd, of like nature, condition, and +like waight throughout.) And you may, by Say Balance, haue prepared a +great number of the smallest waightes: which, by those Balance can be +discerned or tryed: and so, haue proceded to make you a perfect Pyle, +company & Number of waightes: to the waight of six, eight, or twelue +pound waight: most diligently tryed, all. And of euery one, the Content +knowen, in your least waight, that is wayable. [They that can not haue +these waightes of precisenes: may, by Sand, Vniforme, and well dusted, +make them a number of waightes, somewhat nere precisenes: by halfing +euer the Sand: they shall, at length, come to a least common waight. +Therein, I leaue the farder matter, to their discretion, whom nede +shall pinche.] The <i>Venetians</i> consideration of waight, may seme +precise enough: by eight descentes progressionall,* +<span class="sidenote"> +* I. D.<br /> +For, so, haue you .256. partes of a Graine.</span> +halfing, from a grayne. Your Cube, Sphære, apt Balance, and conuenient +waightes, being ready: fall to worke.❉. First, way your Cube. Note the +Number of the waight. Way, after that, your Sphære. Note likewise, the +Nũber of the waight. If you now find the waight of your Cube, to be to +the waight of the Sphære, as 21. is to 11: Then you see, how the +Mechanicien and <i>Experimenter</i>, without Geometrie and +Demonstration, are (as nerely in effect) tought the proportion of the +Cube to the Sphere: as I haue demonstrated it, in the end of the twelfth +boke of <i>Euclide</i>. Often, try with the same Cube and Sphære. Then, +chaunge, your Sphære and Cube, to an other matter: or to an other +bignes: till you haue made a perfect vniuersall Experience of it. +Possible it is, that you shall wynne to nerer termes, in the +proportion.</p> + +<p>When you haue found this one certaine Drop of Naturall veritie, +procede on, to Inferre, and duely to make assay, of matter depending. +As, bycause it is well demonstrated, that a Cylinder, whose heith, and +Diameter of his base, is æquall to the Diameter of the Sphære, is +Sesquialter to the same Sphære (that is, as 3. to 2:) To the number of +the waight of the Sphære, adde halfe so much, as it is: and so haue you +the number of the waight of that Cylinder. Which is also Comprehended of +our former Cube: So, that the base of that Cylinder, is a Circle +described in the Square, which is the base of our Cube. But the Cube and +the Cylinder, being both of one heith, haue their Bases in the same +proportion, in the which, they are, one to an other, in their Massines +or Soliditie. But, before, we haue two numbers, expressing their +Massines, Solidities, and Quantities, by waight: wherfore, we haue +<span class="sidenote smaller"> +* The proportion of the Square to the Circle inscribed.</span> +*the proportion of the Square, to the Circle, inscribed in the same +Square. And so are we fallen into the knowledge sensible, and +Experimentall of <i>Archimedes</i> great Secret: of him, by great +trauaile of minde, sought and found. Wherfore, to any Circle giuen, you +can giue a Square æquall: +<span class="sidenote smaller"> +* The Squaring of the Circle, Mechanically.</span> +*as I haue taught, in my Annotation, vpon the first proposition of the +twelfth boke, And likewise, to any Square giuen, you may giue a Circle +æquall: +<span class="sidenote smaller"> +* To any Square geuen, to geue a Circle, equall.</span> +*If you describe a Circle, which shall be in that proportion, to your +Circle inscribed, as the Square is to the same Circle: This, you may do, +by my Annotations, vpon the second proposition of the twelfth boke of +<i>Euclide</i>, in my third Probleme there. Your diligence may come to a +proportion, of the Square to the Circle inscribed, nerer the truth, then +is the proportion of 14. to 11. And consider, that you may begyn at the +Circle and Square, and so come to conclude of the Sphære, & the +Cube, what +<span class="pagenum">c.ij </span> +<!-- png 35 --> +their proportion is: as now, you came from the Sphære to the Circle. +For, of Siluer, or Gold, or Latton Lamyns or plates (thorough one hole +drawẽ, as the maner is) if you make a Square figure & way it: and +then, describing theron, the Circle inscribed: & cut of, & file +away, precisely (to the Circle) the ouerplus of the Square: you shall +then, waying your Circle, see, whether the waight of the Square, be to +your Circle, as 14. to 11. As I haue Noted, in the beginning of +<i>Euclides</i> twelfth boke. &c. after this resort to my last +proposition, vpon the last of the twelfth. And there, helpe your selfe, +to the end. And, here, Note this, by the way. +<span class="sidenote"> +Note Squaring of the Circle without knowledge of the proportion betwene +Circumference and Diameter.</span> +That we may Square the Circle, without hauing knowledge of the +proportion, of the Circumference to the Diameter: as you haue here +perceiued. And otherwayes also, I can demonstrate it. So that, many +haue cumberd them selues superfluously, by trauailing in that point +first, which was not of necessitie, first: and also very intricate. And +easily, you may, (and that diuersly) come to the knowledge of the +Circumference: the Circles Quantitie, being first knowen. Which thing, +I leaue to your consideration: making hast to despatch an other +Magistrall Probleme: and to bring it, nerer to your knowledge, and +readier dealing with, then the world (before this day,) had it for you, +that I can tell of. And that is, <i>A Mechanicall Dubblyng of the Cube: +&c.</i> Which may, thus, be done: +<span class="sidenote"> +To Dubble the Cube redily: by Art Mechanicall: depending vppon +Demonstration Mathematicall.</span> +<span class="larger">Make of Copper plates, or Tyn plates, +a foursquare vpright Pyramis, or a Cone: perfectly fashioned in the +holow, within. Wherin, let great diligence be vsed, to approche (as nere +as may be) to the Mathematicall perfection of those figures. At their +bases, let them be all open: euery where, els, most close, and iust to. +From the vertex, to the Circumference of the base of the Cone: & to +the sides of the base of the Pyramis:</span> +<span class="sidenote smaller">I. D.<br /> +The 4. sides of this Pyramis must be 4. Isosceles Triangles alike and +æquall.</span> +<span class="larger">Let 4. straight lines be drawen, in the inside of +the Cone and Pyramis: makyng at their fall, on the perimeters of the +bases, equall angles on both sides them selues, with the sayd +perimeters. These 4. lines (in the Pyramis: and as many, in the Cone) +diuide: one, in 12. æquall partes: and an other, in 24. an other, in 60, +and an other, in 100. (reckenyng vp from the vertex.) Or vse other +numbers of diuision, as experience shall teach you. Then,*</span> +<span class="sidenote smaller">I. D.<br /> +* In all workinges with this Pyramis or Cone, Let their Situations be in +all Pointes and Conditions, alike, or all one: while you are about one +Worke. Els you will erre.</span> +<span class="larger">set your Cone or Pyramis, with the vertex +downward, perpendicularly, in respect of the Base. (Though it be +otherwayes, it hindreth nothyng.) So let thẽ most stedily be +stayed.</span> Now, if there be a Cube, which you wold haue Dubbled. +Make you a prety Cube of Copper, Siluer, Lead, Tynne, Wood, Stone, or +Bone. Or els make a hollow Cube, or Cubik coffen, of Copper, Siluer, +Tynne, or Wood &c. These, you may so proportiõ in respect of your +Pyramis or Cone, that the Pyramis or Cone, will be hable to conteine the +waight of them, in water, 3. or 4. times: at the least: what stuff so +euer they be made of. Let not your Solid angle, at the vertex, be to +sharpe: but that the water may come with ease, to the very vertex, of +your hollow Cone or Pyramis. Put one of your Solid Cubes in a Balance +apt: take the waight therof exactly in water. Powre that water, (without +losse) into the hollow Pyramis or Cone, quietly. Marke in your lines, +what numbers the water Cutteth: Take the waight of the same Cube againe: +in the same kinde of water, which you had before: put that* also, +<span class="sidenote smaller">I. D.<br /> +* Consider well whan you must put your waters togyther: and whan, you +must empty your first water, out of your Pyramis or Cone. Els you will +erre.</span> +into the Pyramis or Cone, where you did put the first. Marke now againe, +in what number or place of the lines, the water Cutteth them. Two +<span class="pagenum verso">||</span> +<!-- png 36 --> +wayes you may conclude your purpose: it is to wete, either by numbers or +lines. By numbers: as, if you diuide the side of your Fundamentall Cube +into so many æquall partes, as it is capable of, conueniently, with your +ease, and precisenes of the diuision. For, as the number of your first +and lesse line (in your hollow Pyramis or Cone,) is to the second or +greater (both being counted from the vertex) so shall the number of the +side of your Fundamentall Cube, be to the nũber belonging to the +Radicall side, of the Cube, dubble to your Fundamentall Cube: Which +being multiplied Cubik wise, will sone shew it selfe, whether it be +dubble or no, to the Cubik number of your Fundamentall Cube. By lines, +thus: As your lesse and first line, (in your hollow Pyramis or Cone,) is +to the second or greater, so let the Radical side of your Fundamẽtall +Cube, be to a fourth proportionall line, by the 12. proposition, of the +sixth boke of <i>Euclide</i>. Which fourth line, shall be the Rote +Cubik, or Radicall side of the Cube, dubble to your Fundamentall Cube: +which is the thing we desired. +<span class="sidefinger"> + </span> +<span class="sidenote"> +God be thanked for this Inuention, & the fruite ensuing.</span> +For this, may I (with ioy) say, <span class="greek" title="EURÊKA, EURÊKA, EURÊKA">ΕΥΡΗΚΑ, ΕΥΡΗΚΑ, ΕΥΡΗΚΑ</span>: thanking the holy and +glorious Trinity: hauing greater cause therto, then +<span class="sidenote"> +* Vitruuius. Lib. 9. Cap. 3.</span> +*<i>Archimedes</i> had (for finding the fraude vsed in the Kinges +Crowne, of Gold): as all men may easily Iudge: by the diuersitie of the +frute following of the one, and the other. Where I spake before, of a +hollow Cubik Coffen: the like vse, is of it: and without waight. Thus. +Fill it with water, precisely full, and poure that water into your +Pyramis or Cone. And here note the lines cutting in your Pyramis or +Cone. Againe, fill your coffen, like as you did before. Put that Water, +also, to the first. Marke the second cutting of your lines. Now, as you +proceded before, so must you here procede. +<span class="sidenote"> +* Note.</span> +*And if the Cube, which you should Double, be neuer so great: you haue, +thus, the proportion (in small) betwene your two litle Cubes: And then, +the side, of that great Cube (to be doubled) being the third, will haue +the fourth, found, to it proportionall: by the 12. of the sixth of +Euclide.</p> + +<p> +<span class="sidenote"> +Note, as concerning the Sphæricall Superficies of the Water.</span> +Note, that all this while, I forget not my first Proposition +Staticall, here rehearsed: that, the Superficies of the water, is +Sphæricall. Wherein, vse your discretion: to the first line, adding a +small heare breadth, more: and to the second, halfe a heare breadth +more, to his length. For, you will easily perceaue, that the difference +can be no greater, in any Pyramis or Cone, of you to be handled. Which +you shall thus trye. <i>For finding the swelling of the water aboue +leuell.</i> +<span class="sidefinger"> + </span> +<span class="sidequote">“</span>Square the Semidiameter, from the +Centre of the earth, to your first Waters Superficies. Square then, +halfe the Subtendent of that watry Superficies (which Subtendent must +haue the equall partes of his measure, all one, with those of the +Semidiameter of the earth to your watry Superficies): Subtracte this +square, from the first: Of the residue, take the Rote Square. That Rote, +Subtracte from your first Semidiameter of the earth to your watry +Superficies: that, which remaineth, is the heith of the water, in the +middle, aboue the leuell.<span class="sidequote">”</span> Which, you +will finde, to be a thing insensible. And though it were greatly +sensible,* +<span class="sidenote"> +* Note.</span> +yet, by helpe of my sixt Theoreme vpon the last Proposition of Euclides +twelfth booke, noted: you may reduce all, to a true Leuell. But, farther +diligence, of you is to be vsed, against accidentall causes of the +waters swelling: as by hauing (somwhat) with a moyst Sponge, before, +made moyst your hollow Pyramis or Cone, will preuent an accidentall +cause of Swelling, &c. Experience will teach you abundantly: with +great ease, pleasure, and cõmoditie.</p> + +<p>Thus, may you Double the Cube Mechanically, Treble it, and so forth, +in any proportion. +<span class="sidenote"> +Note this Abridgement of Dubbling the Cube. &c.</span> +Now will I Abridge your paine, cost, and Care herein. Without all +preparing of your Fundamentall Cubes: you may (alike) worke this +Conclusion. For, that, was rather a kinde of Experimentall demõstration, +then the shortest way: +<span class="pagenum">c.iij </span> +<!-- png 37 --> +and all, vpon one Mathematicall Demonstration depending. <span class="sidequote">“</span>Take water (as much as conueniently will serue your +turne: as I warned before of your Fundamentall Cubes bignes) Way it +precisely. Put that water, into your Pyramis or Cone. Of the same kinde +of water, then take againe, the same waight you had before: put that +likewise into the Pyramis or Cone. For, in eche time, your marking of +the lines, how the Water doth cut them, shall geue you the proportion +betwen the Radicall sides, of any two Cubes, wherof the one is Double to +the other: working as before I haue taught you: +<span class="sidenote"> +* Note.</span> +<span class="sidefinger"> + </span> +*sauing that for you Fundamentall Cube his Radicall side: here, you may +take a right line, at pleasure.<span class="sidequote">”</span></p> + +<p>Yet farther proceding with our droppe of Naturall truth: +<span class="sidenote"> +To giue Cubes one to the other in any proportion, Rationall or +Irrationall.</span> +<span class="larger">you may (now) geue Cubes, one to the other, in +any proportiõ geuẽ: Rationall or Irrationall</span>: on this maner. Make +a hollow Parallelipipedon of Copper or Tinne: with one Base wãting, or +open: as in our Cubike Coffen. Frõ the bottome of that Parallelipipedon, +raise vp, many perpendiculars, in euery of his fower sides. Now if any +proportion be assigned you, in right lines: Cut one of your +perpendiculars (or a line equall to it, or lesse then it) likewise: by +the 10. of the sixth of Euclide. And those two partes, set in two sundry +lines of those perpendiculars (or you may set them both, in one line) +making their beginninges, to be, at the base: and so their lengthes to +extend vpward. Now, set your hollow Parallelipipedon, vpright, +perpendicularly, steadie. Poure in water, handsomly, to the heith of +your shorter line. Poure that water, into the hollow Pyramis or Cone. +Marke the place of the rising. Settle your hollow Parallelipipedon +againe. Poure water into it: vnto the heith of the second line, exactly. +Poure that water* +<span class="sidenote"> +* <ins class="correction" title="text reads ‘Emptyting’">Emptying</ins> the first.</span> +duely into the hollow Pyramis or Cone: Marke now againe, where the water +cutteth the same line which you marked before. For, there, as the first +marked line, is to the second: So shall the two Radicall sides be, one +to the other, of any two Cubes: which, in their Soliditie, shall haue +the same proportion, which was at the first assigned: were it Rationall +or Irrationall.</p> + +<p>Thus, in sundry waies you may furnishe your selfe with such straunge +and profitable matter: which, long hath bene wished for. And though it +be Naturally done and Mechanically: yet hath it a good Demonstration +Mathematicall. +<span class="sidenote smaller"> +The demonstrations of this Dubbling of the Cube, and of the rest.</span> +Which is this: Alwaies, you haue two Like Pyramids: or two Like Cones, +in the proportions assigned: and like Pyramids or Cones, are in +proportion, one to the other, in the proportion of their Homologall +sides (or lines) tripled. Wherefore, if to the first, and second lines, +found in your hollow Pyramis or Cone, you ioyne a third and a fourth, in +continuall proportion: that fourth line, shall be to the first, as the +greater Pyramis or Cone, is to the lesse: by the 33. of the eleuenth of +Euclide. If Pyramis to Pyramis, or Cone to Cone, be double, then shall* +<span class="sidenote smaller"> +I. D.<br /> +* Hereby, helpe your self to become a præcise practiser. And so +consider, how, nothing at all, you are hindred (sensibly) by the +Conuexitie of the water.</span> +Line to Line, be also double, &c. But, as our first line, is to the +second, so is the Radicall side of our Fundamentall Cube, to the +Radicall side of the Cube to be made, or to be doubled: and therefore, +to those twaine also, a third and a fourth line, in continuall +proportion, ioyned: will geue the fourth line in that proportion to the +first, as our fourth Pyramidall, or Conike line, was to his first: but +that was double, or treble, &c. as the Pyramids or Cones were, one +to an other (as we haue proued) therfore, this fourth, shalbe also +double or treble to the first, as the Pyramids or Cones were one to an +other: But our made Cube, is described of the second in proportion, of +the fower proportionall lines: therfore* +<span class="sidenote smaller"> +* By the 33. of the eleuenth booke of Euclide.</span> +as the fourth line, is to the first, so is that Cube, to the first Cube: +and we haue proued the fourth line, to be to the first, as the Pyramis +or Cone, is to the Pyramis or Cone: Wherefore the Cube is +<span class="pagenum verso">||</span> +<!-- png 38 --> +to the Cube, as Pyramis is to Pyramis, or Cone is to Cone. But we* +<span class="sidenote smaller"> +I. D.<br /> +* And your diligence in practise, can so (in waight of water) performe +it: Therefore, now, you are able to geue good reason of your whole +doing.</span> +Suppose Pyramis to Pyramis, or Cone to Cone, to be double or treble. +&c. Therfore Cube, is to Cube, double, or treble, &c. Which was +to be demonstrated. And of the Parallelipipedõ, it is euidẽt, that the +water Solide Parallelipipedons, are one to the other, as their heithes +are, seing they haue one base. Wherfore the Pyramids or Cones, made of +those water Parallelipipedons, are one to the other, as the lines are +(one to the other) betwene which, our proportion was assigned. But the +Cubes made of lines, after the proportiõ of the Pyramidal or Conik +<i>homologall</i> lines, are one to the other, as the Pyramides or Cones +are, one to the other (as we before did proue) therfore, the Cubes made, +shalbe one to the other, as the lines assigned, are one to the other: +Which was to be demonstrated. Note. +<span class="sidenote"> +* Note this Corollary.</span> +*This, my Demonstratiõ is more generall, then onely in Square Pyramis or +Cone: Consider well. Thus, haue I, both Mathematically and +Mechanically, ben very long in wordes: yet (I trust) nothing +tedious to them, who, to these thinges, are well affected. And verily I +am forced (auoiding prolixitie) to omit sundry such things, easie to be +practised: which to the Mathematicien, would be a great Threasure: and +to the Mechanicien, no small gaine. +<span class="sidenote"> +* The great Commodities following of these new Inuentions.</span> +*Now may you, <span class="larger">Betwene two lines giuen, finde two +middle proportionals, in Continuall proportion: by the hollow +Parallelipipedon, and the hollow Pyramis, or Cone.</span> Now, any +Parallelipipedon rectangle being giuen: thre right lines may be found, +proportionall in any proportion assigned, of which, shal be produced a +Parallelipipedon, æquall to the Parallelipipedon giuen. Hereof, +I noted somwhat, vpon the 36. proposition, of the 11. boke of +<i>Euclide</i>. Now, all those thinges, which <i>Vitruuius</i> in his +Architecture, specified hable to be done, by dubbling of the Cube: Or, +by finding of two middle proportionall lines, betwene two lines giuen, +may easely be performed. Now, that Probleme, which I noted vnto you, in +the end of my Addition, vpon the 34. of the 11. boke of <i>Euclide</i>, +is proued possible. Now, may any regular body, be Transformed into an +other, &c. Now, any regular body: any Sphere, yea any Mixt Solid: +and (that more is) Irregular Solides, may be made (in any proportiõ +assigned) like vnto the body, first giuen. Thus, of a <i>Manneken</i>, +(as the <i>Dutch</i> Painters terme it) in the same <i>Symmetrie</i>, +may a Giant be made: and that, with any gesture, by the Manneken vsed: +and contrarywise. Now, may you, of any Mould, or Modell of a Ship, make +one, of the same Mould (in any assigned proportion) bigger or lesser. +Now, may you, of any +<span class="sidefinger"> +* </span> +*Gunne, or little peece of ordinaũce, make an other, with the same +<i>Symmetrie</i> (in all pointes) as great, and as little, as you will. +Marke that: and thinke on it. Infinitely, <span class="larger">may you +apply this, so long sought for, and now so easily concluded: and +withall, so willingly and frankly communicated to such, as faithfully +deale with vertuous studies.</span> +<span class="sidenote"> +Such is the Fruite of the Mathematicall Sciences and Artes.</span> +Thus, can the Mathematicall minde, deale Speculatiuely in his own Arte: +and by good meanes, Mount aboue the cloudes and sterres: And thirdly, he +can, by order, Descend, to frame Naturall thinges, to wonderfull vses: +and when he list, retire home into his owne Centre: and there, prepare +more Meanes, to Ascend or Descend by: and, all, to the glory of God, and +our honest delectation in earth.</p> + +<p>Although, the Printer, hath looked for this Præface, a day or +two, yet could I not bring my pen from the paper, before I had giuen you +comfortable warning, and brief instructions, of some of the Commodities, +by <i>Statike</i>, hable to be reaped: In the rest, I will +therfore, be as brief, as it is possible: and with all, describing them, +somwhat accordingly. And that, you shall perceiue, by this, which in +order commeth +<span class="pagenum">c.iiij </span> +<!-- png 39 --> +next. For, wheras, it is so ample and wonderfull, that, an whole yeare +long, one might finde fruitfull matter therin, to speake of: and also in +practise, is a Threasure endeles: yet will I glanse ouer it, with wordes +very few.</p> + +<p><span class="dropcap">T</span>His do I call <span class="largest">Anthropographie</span>. Which is an Art restored, and of my +preferment to your Seruice. I pray you, thinke of it, as of one of +the chief pointes, of Humane knowledge. Although it be, but now, first +Cõfirmed, with this new name: yet the matter, hath from the beginning, +ben in consideration of all perfect Philosophers. <span class="larger">Anthropographie, is the description of the Number, Measure, +Waight, figure, Situation, and colour of euery diuerse thing, conteyned +in the perfect body of MAN: with certain knowledge of the Symmetrie, +figure, waight, Characterization, and due locall motion, of any parcell +of the sayd body, assigned: and of Nũbers, to the sayd parcell +appertainyng.</span> This, is the one part of the Definition, mete for +this place: Sufficient to notifie, the particularitie, and excellency of +the Arte: and why it is, here, ascribed to the Mathematicals. Yf the +description of the heauenly part of the world, had a peculier Art, +called <i>Astronomie:</i> If the description of the earthly Globe, hath +his peculier arte, called <i>Geographie</i>. If the Matching of both, +hath his peculier Arte, called <i>Cosmographie:</i> Which is the +Descriptiõ of the whole, and vniuersall frame of the world: Why should +not the description of +<span class="sidenote"> +MAN is the Lesse World.</span> +him, who is the Lesse world: and, frõ the beginning, called +<i>Microcosmus</i> (that is. <i>The Lesse World.</i>) And for whose +sake, and seruice, all bodily creatures els, were created: Who, also, +participateth with Spirites, and Angels: and is made to the Image and +similitude of <i>God</i>: haue his peculier Art? and be called the +<i>Arte of Artes</i>: rather, then, either to want a name, or to haue to +base and impropre a name? You must of sundry professions, borow or +challenge home, peculier partes hereof: and farder procede: as, God, +Nature, Reason and Experience shall informe you. The Anatomistes will +restore to you, some part: The Physiognomistes, some: The Chyromantistes +some. The Metaposcopistes, some: The excellent, <i>Albert Durer</i>, +a good part: the Arte of Perspectiue, will somwhat, for the Eye, +helpe forward: <i>Pythagoras</i>, <i>Hipocrates</i>, <i>Plato</i>, +<i>Galenus</i>, <i>Meletius</i>, & many other (in certaine thinges) +will be Contributaries. And farder, the Heauen, the Earth, and all other +Creatures, will eche shew, and offer their Harmonious seruice, to fill +vp, that, which wanteth hereof: and with your own Experience, +concluding: you may Methodically register the whole, for the posteritie: +Whereby, good profe will be had, of our Harmonious, and +<span class="sidenote"> +Micro Cosmus.</span> +Microcosmicall constitution. +<span class="sidefinger"> +* </span> +The outward Image, and vew hereof: to the Art of <i>Zographie</i> and +Painting, to Sculpture, and Architecture: (for Church, House, Fort, or +Ship) is most necessary and profitable: for that, it is the chiefe base +and foundation of them. Looke in +<span class="sidenote"> +* Lib. 3. Cap. 1.</span> +*<i>Vitruuius</i>, whether I deale sincerely for your behoufe, or no. +Looke in <i>Albertus Durerus</i>, <i>De Symmetria humani Corporis</i>. +Looke in the 27. and 28. Chapters, of the second booke, <i>De occulta +Philosophia</i>. Consider the <i>Arke</i> of <i>Noe</i>. And by that, +wade farther. Remember the <i>Delphicall Oracle NOSCE TEIPSVM</i> <span +class="larger"><i>(Knowe thy selfe)</i></span> so long agoe +pronounced: of so many a Philosopher repeated: and of the <i>Wisest</i> +attempted: And then, you will perceaue, how long agoe, you haue bene +called to the Schole, where this Arte might be learned. Well. I am +nothing affrayde, of the disdayne of some such, as thinke Sciences and +Artes, to be but Seuen. Perhaps, those Such, may, with ignorance, and +shame enough, come short of them Seuen also: and yet neuerthelesse +<span class="pagenum verso">||</span> +<!-- png 40 --> +they can not prescribe a certaine number of Artes: and in eche, certaine +vnpassable boundes, to God, Nature, and mans Industrie. New Artes, dayly +rise vp: and there was no such order taken, that, +<span class="sidefinger"> + </span> +All Artes, should in one age, or in one land, or of one man, be made +knowen to the world. Let vs embrace the giftes of God, and wayes to +wisedome, in this time of grace, from aboue, continually bestowed on +them, who thankefully will receiue them: <i>Et bonis Omnia Cooperabuntur +in bonum.</i></p> + +<p><span class="largest">Trochilike,</span> <span class="larger">is +that Art Mathematicall, which demonstrateth the properties of all +Circular motions, Simple and Compounde.</span> And bycause the frute +hereof, vulgarly receiued, is in Wheles, it hath the name of +<i>Trochilike:</i> as a man would say, <i>Whele Art</i>. By this art, +a Whele may be geuen which shall moue ones about, in any tyme +assigned. Two Wheles may be giuen, whose turnynges about in one and the +same tyme, (or equall tymes), shall haue, one to the other, any +proportion appointed. By Wheles, may a straight line be described: +Likewise, a Spirall line in plaine, Conicall Section lines, and +other Irregular lines, at pleasure, may be drawen. These, and such like, +are principall Conclusions of this Arte: and helpe forward many pleasant +and profitable Mechanicall workes: +<span class="sidenote"> +Saw Milles.</span> +As Milles, to Saw great and very long Deale bordes, no man being by. +Such haue I seene in Germany: and in the Citie of Prage: in the kingdome +of Bohemia: Coyning Milles, Hand Milles for Corne grinding: And all +maner of Milles, and Whele worke: By Winde, Smoke, Water, Waight, +Spring, Man or Beast, moued. Take in your hand, <i>Agricola De re +Metallica:</i> and then shall you (in all Mines) perceaue, how great +nede is, of Whele worke. By Wheles, straunge workes and incredible, are +done: as will, in other Artes hereafter, appeare. A wonderfull +example of farther possibilitie, and present commoditie, was sene in my +time, in a certaine Instrument: which by the Inuenter and Artificer +(before) was solde for xx. Talentes of Golde: and then had (by +misfortune) receaued some iniurie and hurt: And one <i>Ianellus</i> of +<i>Cremona</i> did mend the same, and presented it vnto the Emperour +<i>Charles</i> the fifth. <i>Hieronymus Cardanus</i>, can be my +witnesse, that therein, was one Whele, which moued, and that, in such +rate, that, in 7000. yeares onely, his owne periode should be finished. +A thing almost incredible: But how farre, I keepe me within my +boundes: very many men (yet aliue) can tell.</p> + +<p><span class="largest">Helicosophie</span>, is nere Sister to +<i>Trochilike:</i> and is, <span class="larger">An Arte Mathematicall, +which demonstrateth the designing of all Spirall lines in Plaine, on +Cylinder, Cone, Sphære, Conoid, and Sphæroid, and their properties +appertayning.</span> The vse hereof, in <i>Architecture</i>, and diuerse +Instrumentes and Engines, is most necessary. For, in many thinges, the +Skrue worketh the feate, which, els, could not be performed. By helpe +hereof, it is +<span class="sidenote"> +* Atheneus Lib. 5. cap. 8.</span> +*recorded, that, where all the power of the Citie of Syracusa, was not +hable to moue a certaine Ship (being on ground) mightie +<i>Archimedes</i>, setting to, his Skruish Engine, caused <i>Hiero</i> +the king, by him self, at ease, to remoue her, as he would. +<span class="sidenote"> +Proclus. Pag. 18.</span> +Wherat, the King wondring: <span class="greek" title="Apo tautês tês hêmêras [hêmeras], peri pantos, Archimêdei legonti pisteuteom [pisteuteon]">Απὸ τάυτης τῆς <ins class="correction" +title="error for ἡμέρας [hêmeras]">ἡμήρας</ins>, περὶ παντὸς, +Αρχιμήδει λέγοντι <ins class="correction" title="error for πιϛευτεόν [pisteuteon]">πιϛευτέομ</ins></span>. <i>From this +day, forward</i> (said the King) <i>Credit ought to be giuen to +Archimedes, what soeuer he sayth.</i></p> + +<p><span class="largest">Pneumatithmie</span> <span class="larger">demonstrateth by close hollow Geometricall Figures, (regular +and irregular) the straunge properties (in motion or stay) of the Water, +Ayre, Smoke, and Fire, in theyr cõtinuitie,</span> +<span class="pagenum">d.j </span> +<!-- png 41 --> +<span class="larger">and as they are ioyned to the Elementes next +them.</span> This Arte, to the Naturall Philosopher, is very +proffitable: to proue, that <i>Vacuum</i>, or <i>Emptines</i> is not in +the world. And that, all Nature, abhorreth it so much: that, contrary to +ordinary law, the Elementes will moue or stand. As, Water to ascend: +rather then betwene him and Ayre, Space or place should be left, more +then (naturally) that quãtitie of Ayre requireth, or can fill. Againe, +Water to hang, and not descend: rather then by descending, to leaue +Emptines at his backe. The like, is of Fire and Ayre: they will descend: +when, either, their Cõtinuitie should be dissolued: or their next +Element forced from them. And as they will not be extended, to +discontinuitie: So, will they not, nor yet of mans force, can be prest +or pent, in space, not sufficient and aunswerable to their bodily +substance. Great force and violence will they vse, to enioy their +naturall right and libertie. +<span class="sidenote"> +To go to the bottom of the Sea without daunger.</span> +Hereupon, two or three men together, by keping Ayre vnder a great +Cauldron, and forcyng the same downe, orderly, may without harme descend +to the Sea bottome: and continue there a tyme &c. Where, Note, how +the thicker Element (as the Water) giueth place to the thynner (as, is +the ayre:) and receiueth violence of the thinner, in maner. &c. +Pumps and all maner of Bellowes, haue their ground of this Art: and many +other straunge deuises. As, <i>Hydraulica</i>, Organes goyng by water. +&c. Of this Feat, (called commonly <i>Pneumatica</i>,) goodly workes +are extant, both in Greke, and Latin. With old and learned Schole men, +it is called <i>Scientia de pleno & vacuo.</i></p> + +<p><span class="largest">Menadrie</span><span class="larger">, is an +Arte Mathematicall, which demonstrateth, how, aboue Natures vertue and +power simple: Vertue and force may be multiplied: and so, to direct, to +lift, to pull to, and to put or cast fro, any multiplied or simple, +determined Vertue, Waight or Force: naturally, not, so, directible or +moueable.</span> Very much is this Art furdred by other Artes: as, in +some pointes, by <i>Perspectiue</i>: in some, by <i>Statike</i>: in +some, by <i>Trochilike</i>: and in other, by <i>Helicosophie</i>: and +<i>Pneumatithmie</i>. By this Art, all Cranes, Gybbettes, & Ingines +to lift vp, or to force any thing, any maner way, are ordred: and the +certaine cause of their force, is knowne: As, the force which one man +hath with the Duche waghen Racke: therwith, to set vp agayne, +a mighty waghen laden, being ouerthrowne. The force of the +Crossebow Racke, is certainly, here, demonstrated. The reason, why one +mã, doth with a leauer, lift that, which Sixe men, with their handes +onely, could not, so easily do. By this Arte, in our common Cranes in +London, where powre is to Crane vp, the waight of 2000. pound: by two +Wheles more (by good order added) Arte concludeth, that there may be +Craned vp 200000. pound waight &c. So well knew <i>Archimedes</i> +this Arte: that he alone, with his deuises and engynes, (twise or +thrise) spoyled and discomfited the whole Army and Hoste of the +Romaines, besieging <i>Syracusa</i>, +<span class="sidenote smaller"> +Plutarchus in Marco Marcello.</span> +<i>Marcus Marcellus the Consul</i>, being their Generall Capitaine. +<span class="sidenote smaller"> +Synesius in Epistolis.</span> +Such huge Stones, so many, with such force, and so farre, did he with +his engynes hayle among them, out of the Citie. +<span class="sidenote smaller"> +Polybius.</span> +<span class="sidenote smaller"> +Plinius.</span> +<span class="sidenote smaller"> +Quintilianus.</span> +<span class="sidenote smaller"> +T. Liuius.</span> +And by Sea likewise: though their Ships might come to the walls of +<i>Syracusa</i>, yet hee vtterly confounded the Romaine Nauye. What with +his mighty Stones hurlyng: what with Pikes of* +<span class="sidenote smaller"> +* Athenæus.</span> +18 fote long, made like shaftes: which he forced almost a quarter of a +myle. What, with his catchyng hold of their Shyps, and hoysing them vp +aboue the water, and suddenly letting them fall into the Sea againe: +what with his* +<span class="sidenote smaller"> +* Galenus.</span> +<span class="sidenote smaller"> +Anthemius.</span> +Burning Glasses: by which he fired their other Shippes a far-of: what, +with his other pollicies, deuises, and engines, he so manfully acquit +him selfe: that all the Force, courage, and pollicie of the Romaines +(for a great season) +<span class="pagenum verso">||</span> +<!-- png 42 --> +could nothing preuaile, for the winning of Syracusa. Wherupon, the +Romanes named <i>Archimedes</i>, <i>Briareus</i>, and <i>Centimanus</i>. +<i>Zonaras</i> maketh mention of one <i>Proclus</i>, who so well had +perceiued <i>Archimedes</i> Arte of <i>Menadrie</i>, and had so well +inuented of his owne, that with his Burning Glasses, +<span class="sidenote"> +Burning Glasses.</span> +being placed vpon the walles of Bysance, he multiplied so the heate of +the Sunne, and directed the beames of the same against his enemies Nauie +with such force, and so sodeinly (like lightening) that he burned and +destroyed both man and ship. And <i>Dion</i>specifieth of +<i>Priscus</i>, a <i>Geometricien</i> in Bysance, who inuented and +vsed sondry Engins, of Force multiplied: Which was cause, that the +<i>Emperour Seuerus</i> pardoned him, his life, after he had wonne +Bysance: Bycause he honored the Arte, wytt, and rare industrie of +<i>Priscus</i>. But nothing inferior to the inuention of these engines +of Force, was the inuention of Gunnes. +<span class="sidenote"> +Gunnes.</span> +Which, from an English man, had the occasion and order of first +inuenting: though in an other land, and by other men, it was first +executed. And they that should see the record, where the occasion and +order generall, of Gunning, is first discoursed of, would thinke: that, +<span class="sidequote">“</span>small thinges, slight, and cõmon: +comming to wise mens consideration, and industrious mens handling, may +grow to be of force incredible.<span class="sidequote">”</span></p> + +<p><span class="largest">Hypogeiodie</span><span class="larger">, is +an Arte Mathematicall, demonstratyng, how, vnder the Sphæricall +Superficies of the earth, at any depth, to any perpendicular line +assigned (whose distance from the perpendicular of the entrance: and the +Azimuth, likewise, in respect of the said entrance, is knowen) certaine +way may be præscribed and gone: And how, any way aboue the Superficies +of the earth designed, may vnder earth, at any depth limited, be kept: +goyng alwayes, perpendicularly, vnder the way, on earth designed: And, +contrarywise, Any way, (straight or croked,) vnder the earth, beyng +giuen: vppon the vtface, or Superficies of the earth, to Lyne out the +same: So, as, from the Centre of the earth, perpendiculars drawen to the +Sphæricall Superficies of the earth, shall precisely fall in the +Correspondent pointes of those two wayes. This, with all other Cases and +circumstances herein, and appertenances, this Arte demonstrateth.</span> +This Arte, is very ample in varietie of Conclusions: and very profitable +sundry wayes to the Common Wealth. The occasion of my Inuenting this +Arte, was at the request of two Gentlemen, who had a certaine worke (of +gaine) vnder ground: and their groundes did ioyne ouer the worke: and by +reason of the crokednes, diuers depthes, and heithes of the way vnder +ground, they were in doubt, and at controuersie, vnder whose ground, as +then, the worke was. The name onely (before this) was of me published, +<i>De Itinere Subterraneo</i>: The rest, be at Gods will. For Pioners, +Miners, Diggers for Mettalls, Stone, Cole, and for secrete passages +vnder ground, betwene place and place (as this land hath diuerse) and +for other purposes, any man may easily perceaue, both the great fruite +of this Arte, and also in this Arte, the great aide of Geometrie.</p> + +<p><span class="largest">Hydragogie</span><span class="larger">, +demonstrateth the possible leading of Water, by Natures lawe, and by +artificiall helpe, from any head (being a Spring, standing, or running +Water) to any other place assigned.</span> <span class="pagenum">d.ij +</span> +<!-- png 43 --> +Long, hath this Arte bene in vse: and much thereof written: and very +marueilous workes therein, performed: as may yet appeare, in Italy: by +the Ruynes remaining of the Aqueductes. In other places, of Riuers +leading through the Maine land, Nauigable many a Mile. And in other +places, of the marueilous forcinges of Water to Ascend. which all, +declare the great skill, to be required of him, who should in this Arte +be perfecte, for all occasions of waters possible leading. To speake of +the allowance of the Fall, for euery hundred foote: or of the Ventills +(if the waters labour be farre, and great) I neede not: Seing, at +hand (about vs) many expert men can sufficiently testifie, in effecte, +the order: though the Demonstration of the Necessitie thereof, they know +not: Nor yet, if they should be led, vp and downe, and about Mountaines, +from the head of the Spring: and then, a place being assigned: and +of them, to be demaunded, how low or high, that last place is, in +respecte of the head, from which (so crokedly, and vp and downe) they be +come: Perhaps, they would not, or could not, very redily, or nerely +assoyle that question. <i>Geometrie</i> therefore, is necessary to +<i>Hydragogie</i>. Of the sundry wayes to force water to ascend, eyther +by <i>Tympane</i>, <i>Kettell mills</i>, <i>Skrue</i>, <i>Ctesibike</i>, +or such like: in <i>Vitruuius</i>, <i>Agricola</i>, (and other,) fully, +the maner may appeare. And so, thereby, also be most euident, how the +Artes, of <i>Pneumatithmie</i>, <i>Helicosophie</i>, <i>Statike</i>, +<i>Trochilike</i>, and <i>Menadrie</i>, come to the furniture of this, +in Speculation, and to the Commoditie of the Common Wealth, in +practise.</p> + +<p><span class="largest">Horometrie</span><span class="larger">, is +an Arte Mathematicall, which demõstrateth, how, at all times appointed, +the precise vsuall denominatiõ of time, may be knowen, for any place +assigned.</span> These wordes, are smoth and plaine easie Englishe, but +the reach of their meaning, is farther, then you woulde lightly imagine. +Some part of this Arte, was called in olde time, <i>Gnomonice</i>: and +of late, <i>Horologiographia</i>: and in Englishe, may be termed, +<i>Dialling</i>. Auncient is the vse, and more auncient, is the +Inuention. The vse, doth well appeare to haue bene (at the least) aboue +two thousand and three hundred yeare agoe: in* +<span class="sidenote"> +4. Reg. 20.</span> +King <i>Achaz</i> Diall, then, by the Sunne, shewing the distinction of +time. By Sunne, Mone, and Sterres, this Dialling may be performed, and +the precise Time of day or night knowen. But the demonstratiue +delineation of these Dialls, of all sortes, requireth good skill, both +of <i>Astronomie</i>, and <i>Geometrie</i> Elementall, Sphæricall, +Phænomenall, and Conikall. Then, to vse the groundes of the Arte, for +any regular Superficies, in any place offred: and (in any possible apt +position therof) theron, to describe (all maner of wayes) how, vsuall +howers, may be (by the <i>Sunnes</i> shadow) truely determined: will be +found no sleight Painters worke. So to Paint, and prescribe the Sunnes +Motion, to the breadth of a heare. In this Feate (in my youth) +I Inuented a way, <span class="larger">How in any Horizontall, +Murall, or Æquinoctiall Diall, &c. At all howers (the Sunne shining) +the Signe and Degree ascendent, may be knowen.</span> Which is a thing +very necessary for the Rising of those fixed Sterres: whose Operation in +the Ayre, is of great might, euidently. I speake no further, of the +vse hereof. Bur forasmuch as, Mans affaires require knowledge of Times +& Momentes, when, neither Sunne, Mone, or Sterre, can be sene: +Therefore, by Industrie Mechanicall, was inuented, first, how, by Water, +running orderly, the Time and howers might be knowen: whereof, the +famous <i>Ctesibius</i>, was Inuentor: a man, of <i>Vitruuius</i>, +to the Skie (iustly) extolled. Then, after that, by Sand running, were +howers measured: Then, by <i>Trochilike</i> with waight: And of late +time, by <i>Trochilike</i> with Spring: without waight. All these, +<span class="pagenum verso">||</span> +<!-- png 44 --> +by Sunne or Sterres direction (in certaine time) require ouersight and +reformation, according to the heauenly Æquinoctiall Motion: besides the +inæqualitie of their owne Operation. There remayneth (without +parabolicall meaning herein) among the Philosophers, +<span class="sidenote"> +A perpetuall Motion.</span> +a more excellent, more commodious, and more marueilous way, then +all these: of hauing the motion of the Primouant (or first æquinoctiall +motion,) by Nature and Arte, Imitated: which you shall (by furder search +in waightier studyes) hereafter, vnderstand more of. And so, it is tyme +to finish this Annotation, of Tymes distinction, vsed in our common, and +priuate affaires: The commoditie wherof, no man would want, that can +tell, how to bestow his tyme.</p> + +<p><span class="largest">Zographie</span><span class="larger">, is +an Arte Mathematicall, which teacheth and demonstrateth, how, the +Intersection of all visuall Pyramides, made by any playne assigned, (the +Centre, distance, and lightes, beyng determined) may be, by lynes, and +due propre colours, represented.</span> A notable Arte, is this: and +would require a whole Volume, to declare the property thereof: and the +Commodities ensuyng. Great skill of <i>Geometrie</i>, +<i>Arithmetike</i>, <i>Perspectiue</i>, and <i>Anthropographie</i>, with +many other particular Artes, hath the <i>Zographer</i>, nede of, for his +perfection. For, the most excellent Painter, (who is but the propre +Mechanicien, & Imitator sensible, of the Zographer) hath atteined to +such perfection, that Sense of Man and beast, haue iudged thinges +painted, to be things naturall, and not artificiall: aliue, and not +dead. This Mechanicall Zographer (commonly called the Painter) is +meruailous in his skill: and seemeth to haue a certaine diuine power: +As, of frendes absent, to make a frendly, present comfort: yea, and of +frendes dead, to giue a continuall, silent presence: not onely with vs, +but with our posteritie, for many Ages. And so procedyng, Consider, How, +in Winter, he can shew you, the liuely vew of Sommers Ioy, and riches: +and in Sommer, exhibite the countenance of Winters dolefull State, and +nakednes. Cities, Townes, Fortes, Woodes, Armyes, yea whole Kingdomes +(be they neuer so farre, or greate) can he, with ease, bring with him, +home (to any mans Iudgement) as Paternes liuely, of the thinges +rehearsed. In one little house, can he, enclose (with great pleasure of +the beholders,) the portrayture liuely, of all visible Creatures, either +on earth, or in the earth, liuing: or in the waters lying, Creping, +slyding, or swimming: or of any foule, or fly, in the ayre flying. Nay, +in respect of the Starres, the Skie, the Cloudes: yea, in the shew of +the very light it selfe (that Diuine Creature) can he match our eyes +Iudgement, most nerely. What a thing is this? thinges not yet being, he +can represent so, as, at their being, the Picture shall seame (in maner) +to haue Created them. To what Artificer, is not Picture, a great +pleasure and Commoditie? Which of them all, will refuse the Direction +and ayde of Picture? The Architect, the Goldsmith, and the Arras Weauer: +of Picture, make great account. Our liuely Herbals, our portraitures of +birdes, beastes, and fishes: and our curious Anatomies, which way, are +they most perfectly made, or with most pleasure, of vs beholden? Is it +not, by Picture onely? And if Picture, by the Industry of the Painter, +be thus commodious and meruailous: what shall be thought of +<i>Zographie</i>, the Scholemaster of Picture, and chief gouernor? +Though I mencion not <i>Sculpture</i>, in my Table of Artes +Mathematicall: yet may all men perceiue, How, that <i>Picture</i> and +<i>Sculpture</i>, are Sisters germaine: and both, right profitable, in a +Commõ wealth. and of <i>Sculpture</i>, aswell as of Picture, excellent +Artificers haue written great bokes in commendation. Witnesse I take, of +<i>Georgio Vasari</i>, <i>Pittore Aretino</i>: of <i>Pomponius +Gauricus</i>: and other. To these two Artes, (with other,) is a certaine +od Arte, called <i>Althalmasat</i>, much beholdyng: more, then the +common <i>Sculptor</i>, <i>Entayler</i>, <i>Keruer</i>, <i>Cutter</i>, +<i>Grauer</i>, <i>Founder</i>, +<span class="pagenum">d.iij </span> +<!-- png 45 --> +or <i>Paynter (&c)</i> know their Arte, to be commodious.</p> + +<p><span class="sidenote"> +An objection.</span> +<span class="largest">Architecture</span>, to many may seme not +worthy, or not mete, to be reckned among the <i>Artes Mathematicall</i>. +To whom, I thinke good, to giue some account of my so doyng. Not +worthy, (will they say,) bycause it is but for building, of a house, +Pallace, Church, Forte, or such like, grosse workes. And you, also, +defined the <i>Artes Mathematicall</i>, to be such, as dealed with no +Materiall or corruptible thing: and also did demonstratiuely procede in +their faculty, by Number or Magnitude. First, +<span class="sidenote"> +The Answer.</span> +you see, that I count, here, <i>Architecture</i>, among those <i>Artes +Mathematicall</i>, which are Deriued from the Principals: and you know, +that such, may deale with Naturall thinges, and sensible matter. Of +which, <span class="sidequote">“</span>some draw nerer, to the Simple +and absolute Mathematicall Speculation, then other do. +<span class="sidefinger"> + </span> +And though, the <i>Architect</i> procureth, enformeth, & directeth, +the <i>Mechanicien</i>, to handworke, & the building actuall, of +house, Castell, or Pallace, and is chief Iudge of the same: yet, with +him selfe (as chief <i>Master</i> and <i>Architect</i>,) remaineth the +Demonstratiue reason and cause, of the Mechaniciens worke: in Lyne, +plaine, and Solid: by <i>Geometricall</i>, <i>Arithmeticall</i>, +<i>Opticall</i>, <i>Musicall</i>, <i>Astronomicall</i>, +<i>Cosmographicall</i><span class="sidequote">”</span> (& to +be brief) by all the former Deriued <i>Artes Mathematicall</i>, and +other Naturall Artes, hable to be confirmed and stablished. If this be +so: then, may you thinke, that <i>Architecture</i>, hath good and due +allowance, in this honest Company of <i>Artes Mathematicall</i> +Deriuatiue. I will, herein, craue Iudgement of two most perfect +<i>Architectes</i>: the one, being <i>Vitruuius</i>, the Romaine: who +did write ten bookes thereof, to the Emperour <i>Augustus</i> (in whose +daies our Heauenly Archemaster, was borne): and the other, <i>Leo +Baptista Albertus</i>, a Florentine: who also published ten bookes +therof. <i>Architectura</i> (sayth <i>Vitruuius</i>) <i>est Scientia +pluribus disciplinis & varijs eruditionibus ornata: cuius Iudicio +probantur omnia, quæ ab cæteris Artificibus perficiuntur opera.</i> That +is. <span class="larger">Architecture, is a Science garnished with +many doctrines & diuerse instructions: by whose Iudgement, all +workes, by other workmen finished, are Iudged.</span> It followeth. +<i>Ea nascitur ex Fabrica, & Ratiocinatione. &c. Ratiocinatio +autem est, quæ, res fabricatas, Solertia ac ratione proportionis, +demonstrare atq<sup>ue</sup> explicare potest. <span class="larger">Architecture, groweth of Framing, and Reasoning. &c. +Reasoning, is that, which of thinges framed, with forecast, and +proportion: can make demonstration, and manifest declaration.</span></i> +Againe. <i>Cùm, in omnibus enim rebus, tùm maximè etiam in Architectura, +hæc duo insunt: quod significatur, & quod significat. Significatur +proposita res, de qua dicitur: hanc autem Significat Demonstratio, +rationibus doctrinarum explicata. <span class="larger">Forasmuch as, +in all thinges: therefore chiefly in Architecture, these two thinges +are: the thing signified: and that which signifieth. The thing +propounded, whereof we speake, is the thing Signified. But +Demonstration, expressed with the reasons of diuerse doctrines, doth +signifie the same thing.</span></i> After that. <i>Vt literatus sit, +peritus Graphidos, eruditus Geometriæ, & Optices non ignarus: +instructus Arithmetica: historias complures nouerit, Philosophos +diligenter audiuerit: Musicam sciuerit: Medicinæ non sit ignarus, +responsa Iurisperitorũ nouerit: Astrologiam, Cæliq<sup>ue</sup> rationes +cognitas habeat. <span class="larger">An Architect</span></i> (sayth +he) <span class="larger"><i>ought to vnderstand Languages, to be +skilfull of Painting, well instructed in Geometrie, not ignorant of +Perspectiue, furnished with Arithmetike, haue knowledge of many +histories, and diligently haue heard Philosophers, haue skill of Musike, +not ignorant of Physike, know the aunsweres of Lawyers, and haue +Astronomie,</i></span> +<span class="pagenum verso">||</span> +<!-- png 46 --> +<span class="larger"><i>and the courses Cælestiall, in good +knowledge.</i></span> He geueth reason, orderly, wherefore all these +Artes, Doctrines, and Instructions, are requisite in an excellent +<i>Architect</i>. And (for breuitie) omitting the Latin text, thus he +hath. <span class="larger"><i>Secondly, it is behofefull for an +Architect to haue the knowledge of Painting: that he may the more +easilie fashion out, in patternes painted, the forme of what worke he +liketh. And Geometrie, geueth to Architecture many helpes: and first +teacheth the Vse of the Rule, and the Cumpasse: wherby (chiefly and +easilie) the descriptions of Buildinges, are despatched in Groundplats: +and the directions of Squires, Leuells, and Lines. Likewise, by +Perspectiue, the Lightes of the heauen, are well led, in the buildinges: +from certaine quarters of the world. By Arithmetike, the charges of +Buildinges are summed together: the measures are expressed, and the hard +questions of Symmetries, are by Geometricall Meanes and Methods +discoursed on. &c. Besides this, of the Nature of thinges (which in +Greke is called <span class="greek" title="phusiologia">φυσιολογία</span>) Philosophie doth make declaration. +Which, it is necessary, for an Architect, with diligence to haue +learned: because it hath many and diuers naturall questions: as +specially, in Aqueductes. For in their courses, leadinges about, in the +leuell ground, and in the mountinges, the naturall Spirites or breathes +are ingendred diuers wayes: The hindrances, which they cause, no man can +helpe, but he, which out of Philosophie, hath learned the originall +causes of thinges. Likewise, who soeuer shall read Ctesibius, or +Archimedes bookes, (and of others, who haue written such Rules) can not +thinke, as they do: vnlesse he shall haue receaued of Philosophers, +instructions in these thinges. And Musike he must nedes know: that he +may haue vnderstanding, both of Regular and Mathematicall Musike: that +he may temper well his Balistes, Catapultes, and Scorpions. &c. +Moreouer, the Brasen Vessels, which in Theatres, are placed by +Mathematicall order, in ambries, vnder the steppes: and the diuersities +of the soundes (which y<sup>e</sup> Grecians call <span class="greek" +title="êcheia">ηχεῖα</span>) are ordred according to Musicall +Symphonies & Harmonies: being distributed in y<sup>e</sup> +Circuites, by Diatessaron, Diapente, and Diapason. That the conuenient +voyce, of the players sound, whẽ it came to these preparations, made in +order, there being increased: with y<sup>t</sup> increasing, might come +more cleare & pleasant, to y<sup>e</sup> eares of the lokers on. +&c. And of Astronomie, is knowẽ y<sup>e</sup> East, West, South, and +North. The fashion of the heauen, the Æquinox, the Solsticie, and the +course of the sterres. Which thinges, vnleast one know: he can not +perceiue, any thyng at all, the reason of Horologies. Seyng therfore +this ample Science, is garnished, beautified and stored, with so many +and sundry skils and knowledges: I thinke, that none can iustly +account them selues Architectes, of the <ins class="correction" title="text reads ‘snddeyne’">suddeyne</ins>. But they onely, who from their +childes yeares, ascendyng by these degrees of knowledges, beyng fostered +vp with the atteynyng of many Languages and Artes, haue wonne to the +high Tabernacle of Architecture. &c. And to whom Nature hath giuen +such quicke Circumspection, sharpnes of witt, and Memorie, that they may +be very absolutely skillfull in Geometrie, Astronomie, Musike, and the +rest of the Artes Mathematicall:</i></span> +<span class="pagenum">d.iiij </span> +<!-- png 47 --> +<span class="larger"><i>Such, surmount and passe the callyng, and +state, of Architectes:</i></span> +<span class="sidenote"> +A Mathematicien.</span> +<span class="larger"><i>and are become Mathematiciens. &c. And +they are found, seldome. As, in tymes past, was Aristarchus Samius: +Philolaus, and Archytas, Tarentynes: Apollonius Pergęus: Eratosthenes +Cyreneus: Archimedes, and Scopas, Syracusians. Who also, left to theyr +posteritie, many Engines and Gnomonicall workes: by numbers and naturall +meanes, inuented and declared.</i></span></p> + +<p>Thus much, and the same wordes (in sense) in one onely Chapter of +this Incõparable <i>Architect Vitruuius</i>, shall you finde. And if you +should, but take his boke in your hand, and slightly loke thorough it, +you would say straight way: +<span class="sidenote"> +Vitruuius.</span> +This is <i>Geometrie</i>, <i>Arithmetike</i>, <i>Astronomie</i>, +<i>Musike</i>, <i>Anthropographie</i>, <i>Hydragogie</i>, +<i>Horometrie</i>. <i>&c</i>. and (to cõclude) the Storehouse of all +workmãship. Now, let vs listen to our other Iudge, our Florentine, +<i>Leo Baptista</i>: and narrowly consider, how he doth determine of +<i>Architecture</i>. <i>Sed anteq<sup>ue</sup> vltra progrediar. &c. +<span class="larger">But before I procede any further +</span></i>(sayth he) <span class="larger"><i>I thinke, that I ought +to expresse, what man I would haue to bee allowed an Architect. For, +I will not bryng in place a Carpenter: as though you might Compare +him to the Chief Masters of other Artes. For the hand of the Carpenter, +is the Architectes Instrument.</i></span> +<span class="sidenote"> +VVho is an Architect.</span> +<span class="larger"><i>But I will appoint the Architect to be <span +class="sidequote">“</span>that man, who hath the skill, (by a certaine +and meruailous meanes and way,) both in minde and Imagination to +determine and also in worke to finish: what workes so euer, by motion of +waight, and cuppling and framyng together of bodyes, may most aptly be +Commodious for the worthiest Vses of Man.<span class="sidequote">”</span> And that he may be able to performe these thinges, +he hath nede of atteynyng and knowledge of the best, and most worthy +thynges. &c. The whole Feate of Architecture in buildyng, consisteth +in Lineamentes, and in Framyng. And the whole power and skill of +Lineamentes, tendeth to this: that the right and absolute way may <ins +class="correction" title="text reads ‘he’">be</ins> had, of Coaptyng +and ioyning Lines and angles: by which, the face of the buildyng or +frame, may be comprehended and concluded. And it is the property of +Lineamentes, to prescribe vnto buildynges, and euery part of them, an +apt place, & certaine nũber: a worthy maner, and a semely +order: that, so, y<sup>e</sup> whole forme and figure of the buildyng, +may rest in the very Lineamentes. &c. And we may prescribe in mynde +and imagination the whole formes,* </i></span><span class="sidenote"> +* The Immaterialitie of perfect Architecture.</span> +<span class="larger"><i>all material stuffe beyng secluded. Which +point we shall atteyne, by Notyng and forepointyng the angles, and +lines, by a sure and certaine direction and connexion. Seyng then, these +thinges, are thus:</i></span> +<span class="sidenote"> +What, Lineament is.</span> +<span class="larger"><i>Lineamente, shalbe the certaine and constant +prescribyng, conceiued in mynde: made in lines and angles: and finished +with a learned minde and wyt.</i></span> <span class="sidequote">“</span>We thanke you Master <i>Baptist</i>, that you haue +so aptly brought your Arte, and phrase therof, to haue some +Mathematicall perfection: +<span class="sidenote"> +Note.</span> +by certaine order, nũber, forme, figure, and <i>Symmetrie</i> +mentall:<span class="sidequote">”</span> all naturall & sensible +stuffe set a part. Now, then, it is euident, (Gentle reader) how aptely +and worthely, I haue preferred <i>Architecture</i>, to be bred and +fostered vp in the Dominion of the pereles <i>Princesse</i>, +<i>Mathematica</i>: and to be a naturall Subiect of hers. And the name +of <i>Architecture</i>, is of the principalitie, which this Science +hath, aboue all other Artes. And <i>Plato</i> affirmeth, the +<i>Architect</i> to be <i>Master</i> ouer all, that make any worke. +Wherupon, he is neither Smith, nor Builder: nor, separately, any +Artificer: but the +<span class="pagenum verso">||</span> +<!-- png 48 --> +Hed, the Prouost, the Directer, and Iudge of all Artificiall workes, and +all Artificers. For, the true <i>Architect</i>, is hable to teach, +Demonstrate, distribute, describe, and Iudge all workes wrought. And he, +onely, searcheth out the causes and reasons of all Artificiall thynges. +Thus excellent, is <i>Architecture</i>: though few (in our dayes) +atteyne thereto: yet may not the Arte, be otherwise thought on, then in +very dede it is worthy. Nor we may not, of auncient Artes, make new and +imperfect Definitions in our dayes: for scarsitie of Artificers: No +more, than we may pynche in, the Definitions of <i>Wisedome</i>, or +<i>Honestie</i>, or of <i>Frendeshyp</i> or of <i>Iustice</i>. No more +will I consent, to Diminish any whit, of the perfection and dignitie, +(by iust cause) allowed to absolute <i>Architecture</i>. Vnder the +Direction of this Arte, are thre principall, necessary <i>Mechanicall +Artes</i>. Namely, <i>Howsing</i>, <i>Fortification</i>, and +<i>Naupegie</i>. <i>Howsing</i>, I vnderstand, both for Diuine +Seruice, and Mans common vsage: publike, and priuate. Of +<i>Fortification</i> and <i>Naupegie</i>, straunge matter might be told +you: But perchaunce, some will be tyred, with this Bederoll, all ready +rehearsed: and other some, will nycely nip my grosse and homely +discoursing with you: made in post hast: for feare you should wante this +true and frendly warnyng, and tast giuyng, of the <i>Power +Mathematicall</i>. Lyfe is short, and vncertaine: Tymes are perilouse: +&c. And still the Printer awayting, for my pen staying: All these +thinges, with farder matter of Ingratefulnes, giue me occasion to passe +away, to the other Artes remainyng, with all spede possible.</p> + +<p><span class="dropcap">T</span>He <span class="larger">Arte of +</span><span class="largest">Nauigation</span><span class="larger">, +demonstrateth how, by the shortest good way, by the aptest Directiõ, +& in the shortest time, a sufficient Ship, betwene any two +places (in passage Nauigable,) assigned: may be cõducted: and in all +stormes, & naturall disturbances chauncyng, how, to vse the best +possible meanes, whereby to recouer the place first assigned.</span> +What nede, the <i>Master Pilote</i>, hath of other Artes, here before +recited, it is easie to know: as, of <i>Hydrographie</i>, +<i>Astronomie</i>, <i>Astrologie</i>, and <i>Horometrie</i>. +Presupposing continually, the common Base, and foundacion of all: namely +<i>Arithmetike</i> and <i>Geometrie</i>. So that, he be hable to +vnderstand, and Iudge his own necessary Instrumentes, and furniture +Necessary: Whether they be perfectly made or no: and also can, (if nede +be) make them, hym selfe. As Quadrantes, The Astronomers Ryng, The +Astronomers staffe, The Astrolabe vniuersall. An Hydrographicall Globe. +Charts Hydrographicall, true, (not with parallell Meridians). The Common +Sea Compas: The Compas of variacion: The Proportionall, and Paradoxall +Compasses +<span class="sidenote"> +Anno. 1559.</span> +(of me Inuented, for our two Moscouy Master Pilotes, at the request of +the Company) Clockes with spryng: houre, halfe houre, and three houre +Sandglasses: & sundry other Instrumẽtes: And also, be hable, on +Globe, or Playne to describe the Paradoxall Compasse: and duely to vse +the same, to all maner of purposes, whereto it was inuented. And also, +be hable to Calculate the Planetes places for all tymes.</p> + +<p>Moreouer, with Sonne Mone or Sterre (or without) be hable to define +the Longitude & Latitude of the place, which he is in: So that, the +Longitude & Latitude of the place, from which he sayled, be giuen: +or by him, be knowne. whereto, appertayneth expert meanes, to be +certified euer, of the Ships way. &c. And by foreseing the Rising, +Settyng, Nonestedyng, or Midnightyng of certaine tempestuous fixed +Sterres: or their Coniunctions, and Anglynges with the Planetes, &c. +he ought to haue expert coniecture of Stormes, Tempestes, and Spoutes: +and such lyke Meteorologicall effectes, daungerous on Sea. For (as +<i>Plato</i> sayth,) <i>Mutationes, +<span class="pagenum">A.j </span> +<!-- png 49 --> +opportunitatesq<sup>ue</sup> temporum presentire, non minus rei +militari, quàm Agriculturæ, Nauigationiq<sup>ue</sup> conuenit. <span +class="larger">To foresee the alterations and opportunities of tymes, +is conuenient, no lesse to the Art of Warre, then to Husbandry and +Nauigation.</span></i> And besides such cunnyng meanes, more euident +tokens in Sonne and Mone, ought of hym to be knowen: such as (the +Philosophicall Poëte) <i>Virgilius</i> teacheth, in hys +<i>Georgikes</i>. Where he sayth,<a class="tag" name="endtag3" id="endtag3" href="#endnote3">C</a></p> + +<div class="poem"> +<p>Sol quoq<sup>ue</sup> & exoriens & quum se condet in +vndas,</p> +<p>Signa dabit, Solem certissima signa sequuntur. &c.</p> +<span class="sidenote"> +Georgic. 1.</span> +<p> ———— Nam sæpe videmus,</p> +<p>Ipsius in vultu varios errare colores.</p> +<p>Cæruleus, pluuiam denunciat, igneus Euros.</p> +<p>Sin maculæ incipient rutilo immiscerier igni,</p> +<p>Omnia tum pariter vento, nimbisq<sup>ue</sup> videbis</p> +<p>Feruere: non illa quisquam me nocte per altum</p> +<p>Ire, neq<sup>ue</sup> a terra moueat conuellere funem. &c.</p> +<p>Sol tibi signa dabit. Solem quis dicere falsum</p> +<p>Audeat? ———— &c.</p> +</div> + +<p>And so of Mone, Sterres, Water, Ayre, Fire, Wood, Stones, Birdes, and +Beastes, and of many thynges els, a certaine Sympathicall +forewarnyng may be had: sometymes to great pleasure and proffit, both on +Sea and Land. Sufficiently, for my present purpose, it doth appeare, by +the premisses, how <i>Mathematicall</i>, the <i>Arte</i> of +<i>Nauigation</i>, is: and how it nedeth and also vseth other +<i>Mathematicall Artes</i>: And now, if I would go about to speake of +the manifold Commodities, commyng to this Land, and others, by Shypps +and <i>Nauigation</i>, you might thinke, that I catch at occasions, to +vse many wordes, where no nede is.</p> + +<p>Yet, this one thyng may I, (iustly) say. In <i>Nauigation</i>, none +ought to haue greater care, to be skillfull, then our English Pylotes. +And perchaunce, Some, would more attempt: And other Some, more willingly +would be aydyng, it they wist certainely, What Priuiledge, God had +endued this Iland with, by reason of Situation, most commodious for +<i>Nauigation</i>, to Places most Famous & Riche. And though, +<span class="sidenote"> +* Anno. 1567 S. H. G.</span> +(of* Late) a young Gentleman, a Courragious Capitaine, was in +a great readynes, with good hope, and great causes of persuasion, to +haue ventured, for a Discouerye, (either <i>Westerly</i>, by <i>Cape de +Paramantia</i>: or <i>Esterly</i>, aboue <i>Noua Zemla</i>, and the +<i>Cyremisses</i>) and was, at the very nere tyme of Attemptyng, called +and employed otherwise (both then, and since,) in great good seruice to +his Countrey, as the Irish Rebels haue *tasted: +<span class="sidenote"> +* Anno. 1569</span> +Yet, I say, (though the same Gentleman, doo not hereafter, deale +therewith) Some one, or other, should listen to the Matter: and by good +aduise, and discrete Circumspection, by little, and little, wynne to the +sufficient knowledge of that <span class="larger">Trade</span> and +<span class="larger">Voyage</span>: Which, now, I would be sory, +(through Carelesnesse, want of Skill, and Courrage,) should remayne +Vnknowne and vnheard of. Seyng, also, we are herein, halfe Challenged, +by the learned, by halfe request, published. Therof, verely, might grow +Commoditye, to this Land chiefly, and to the rest of the Christen Common +wealth, farre passing all riches and worldly Threasure.</p> + +<p><span class="largest">Thaumaturgike</span><span class="larger">, +is that Art Mathematicall, which giueth certaine order to make straunge +workes, of the sense to be perceiued, and of men greatly to be wondred +at.</span> By sundry meanes, this <i>Wonder-worke</i> is wrought. Some, +by <i>Pneumatithmie</i>. As the workes of <i>Ctesibius</i> and +<i>Hero</i>, +<span class="pagenum verso">||</span> +<!-- png 50 --> +Some by waight. wherof <i>Timæus</i> speaketh. Some, by Stringes +strayned, or Springs, therwith Imitating liuely Motions. Some, by other +meanes, as the Images of Mercurie: and the brasen hed, made by +<i>Albertus Magnus</i>, which dyd seme to speake. <i>Boethius</i> was +excellent in these feates. To whom, <i>Cassiodorus</i> writyng, sayth. +<span class="larger"><i>Your purpose is to know profound thynges: and +to shew meruayles. By the disposition of your Arte, Metals do low: +Diomedes of brasse, doth blow a Trumpet loude: a brasen Serpent +hisseth: byrdes made, sing swetely. Small thynges we rehearse of you, +who can Imitate the heauen. &c.</i></span> Of the straunge +Selfmouyng, which, at Saint Denys, by Paris, +<span class="sidenote"> +* Anno. 1551</span> +*I saw, ones or twise (<i>Orontius</i> beyng then with me, in Company) +it were to straunge to tell. But some haue written it. And yet, +(I hope) it is there, of other to be sene. And by +<i>Perspectiue</i> also straunge thinges, are done. As partly (before) +I gaue you to vnderstand in <i>Perspectiue</i>. As, to see in the +Ayre, a loft, the lyuely Image of an other man, either walkyng to +and fro: or standyng still. Likewise, to come into an house, and there +to see the liuely shew of Gold, Siluer or precious stones: and commyng +to take them in your hand, to finde nought but Ayre. Hereby, haue some +men (in all other matters counted wise) fouly ouershot thẽ selues: +misdeaming of the meanes. Therfore sayd <i>Claudius Cælestinus</i>. +<span class="sidenote"> +De his quæ Mundo mirabiliter eueniunt. cap. 8.</span> +<i>Hodie magnæ literaturæ viros & magna reputationis videmus, opera +quedam quasi miranda, supra Naturã putare: de quibus in Perspectiua +doctus causam faciliter reddidisset.</i> That is. <span class="larger"><i>Now a dayes, we see some men, yea of great learnyng and +reputation, to Iudge certain workes as meruaylous, aboue the power of +Nature: Of which workes, one that were skillfull in Perspectiue might +easely haue giuen the Cause.</i></span> Of <i>Archimedes Sphære</i>, +<i>Cicero</i> witnesseth. +<span class="sidenote"> +Tusc. 1.</span> +Which is very straunge to thinke on. <span class="larger"><i>For when +Archimedes</i></span> (sayth he) <span class="larger"><i>did fasten in +a Sphære, the mouynges of the Sonne, Mone, and of the fiue other +Planets, he did, as the God, which (in Timæus of Plato) did make the +world. That, one turnyng, should rule motions most vnlike in slownes, +and swiftnes.</i></span> But a greater cause of meruayling we haue by +<i>Claudianus</i> report hereof. Who affirmeth this <i>Archimedes +worke</i>, to haue ben of Glasse. And discourseth of it more at large: +which I omit. The Doue of wood, which the <i>Mathematicien Archytas</i> +did make to flye, is by <i>Agellius</i> spoken of. Of <i>Dædalus</i> +straunge Images, <i>Plato</i> reporteth. <i>Homere</i> of <i>Vulcans +Selfmouers</i>, (by secret wheles) leaueth in writyng. <i>Aristotle</i>, +in hys <i>Politikes</i>, of both, maketh mention. Meruaylous was the +workemanshyp, of late dayes, performed by good skill of <i>Trochilike. +&c.</i> For in Noremberge, A flye of Iern, beyng let out of the +Artificers hand, did (as it were) fly about by the gestes, at the table, +and at length, as though it were weary, retourne to his masters hand +agayne. Moreouer, an Artificiall Egle, was ordred, to fly out of the +same Towne, a mighty way, and that a loft in the Ayre, toward the +Emperour comming thether: and followed hym, beyng come to the gate of +the towne.* +<span class="sidefinger"> +* </span> +Thus, you see, what, Arte Mathematicall can performe, when Skill, will, +Industry, and Hability, are duely applyed to profe.</p> + +<p> +<span class="sidenote"> +A Digression.</span> +And for these, and such like marueilous Actes and Feates, Naturally, +Mathematically, and Mechanically, wrought and contriued: +<span class="sidenote"> +Apologeticall.</span> +ought any honest Student, and Modest Christian Philosopher, be counted, +& called a <span class="larger">Coniurer</span>? Shall the folly +of Idiotes, and the Mallice of the Scornfull, so much preuaile, that He, +who seeketh no worldly gaine or glory at their handes: But onely, of +God, the threasor of heauenly wisedome, & knowledge of pure veritie: +Shall he (I say) in the meane +<span class="pagenum">A.ij </span> +<!-- png 51 --> +space, be robbed and spoiled of his honest name and fame? He that seketh +(by S. Paules aduertisement) in the Creatures Properties, and +wonderfull vertues, to finde iuste cause, to glorifie the Æternall, and +Almightie Creator by: Shall that man, be (in hugger mugger) condemned, +as a Companion of the Helhoundes, and a Caller, and Coniurer of wicked +and damned Spirites? He that bewaileth his great want of time, +sufficient (to his contentation) for learning of Godly wisdome, and +Godly Verities in: and onely therin setteth all his delight: Will that +mã leese and abuse his time, in dealing with the Chiefe enemie of Christ +our Redemer: the deadly foe of all mankinde: the subtile and impudent +peruerter of Godly Veritie: the Hypocriticall Crocodile: the Enuious +Basiliske, continually desirous, in the twinke of an eye, to destroy all +Mankinde, both in Body and Soule, æternally? Surely (for my part, +somewhat to say herein) I haue not learned to make so brutish, and +so wicked a Bargaine. Should I, for my xx. or xxv. yeares Studie: +for two or three thousand Markes spending: seuen or eight thousand Miles +going and trauailing, onely for good learninges sake: And that, in all +maner of wethers: in all maner of waies and passages: both early and +late: in daunger of violence by man: in daunger of destruction by wilde +beastes: in hunger: in thirst: in perilous heates by day, with toyle on +foote: in daungerous dampes of colde, by night, almost bereuing life: +(as God knoweth): with lodginges, oft times, to small ease: and somtime +to lesse securitie. And for much more (then all this) done & +suffred, for Learning and attaining of Wisedome: Should I (I pray +you) for all this, no otherwise, nor more warily: or (by Gods +mercifulnes) no more luckily, haue fished, with so large, and costly, +a Nette, so long time in drawing (and that with the helpe and +aduise of Lady Philosophie, & Queene Theologie): but at length, to +haue catched, and drawen vp,* +<span class="sidenote"> +* A prouerb. Fayre fisht, and caught a Frog.</span> +a Frog? Nay, a Deuill? For, so, doth the Common peuish Pratler +Imagine and Iangle: And, so, doth the Malicious skorner, secretly wishe, +& brauely and boldly face down, behinde my backe. Ah, what a +miserable thing, is this kinde of Men? How great is the blindnes & +boldnes, of the Multitude, in thinges aboue their Capacitie? What a +Land: what a People: what Maners: what Times are these? Are they become +Deuils, them selues: and, by false witnesse bearing against their +Neighbour, would they also, become Murderers? Doth God, so long geue +them respite, to reclaime them selues in, from this horrible slaundering +of the giltlesse: contrary to their owne Consciences: and yet will they +not cease? Doth the Innocent, forbeare the calling of them, Iuridically +to aunswere him, according to the rigour of the Lawes: and will they +despise his Charitable pacience? As they, against him, by name, do +forge, fable, rage, and raise slaunder, by Worde & Print: Will they +prouoke him, by worde and Print, likewise, to Note their Names to the +World: with their particular deuises, fables, beastly Imaginations, and +vnchristen-like slaunders? Well: Well. O (you such) my vnkinde +Countrey men. O vnnaturall Countrey men. O vnthankfull +Countrey men. O Brainsicke, Rashe, Spitefull, and Disdainfull +Countrey men. Why oppresse you me, thus violently, with your slaundering +of me: Contrary to Veritie: and contrary to your owne Consciences? +And I, to this hower, neither by worde, deede, or thought, haue +bene, any way, hurtfull, damageable, or iniurious to you, or yours? +Haue I, so long, so dearly, so farre, so carefully, so painfully, +so daungerously sought & trauailed for the learning of Wisedome, +& atteyning of Vertue: And in the end (in your iudgemẽt) am I +become, worse, then when I begã? Worse, thẽ a Mad man? A dangerous +Member in the Common Wealth: and no Member of the Church of Christ? Call +you this, to be Learned? Call you this, to be a Philosopher? and a louer +of Wisedome? To forsake the straight heauenly way: and to wallow in the +broad way of +<span class="pagenum verso">||</span> +<!-- png 52 --> +damnation? To forsake the light of heauenly Wisedome: and to lurke in +the dungeon of the Prince of darkenesse? To forsake the Veritie of God, +& his Creatures: and to fawne vpon the Impudent, Craftie, Obstinate +Lier, and continuall disgracer of Gods Veritie, to the vttermost of his +power? To forsake the Life & Blisse Æternall: and to cleaue vnto the +Author of Death euerlasting? that Murderous Tyrant, most gredily +awaiting the Pray of Mans Soule? Well: I thanke God and our Lorde +Iesus Christ, for the Comfort which I haue by the Examples of other men, +before my time: To whom, neither in godlines of life, nor in perfection +of learning, I am worthy to be compared: and yet, they sustained +the very like Iniuries, that I do: or rather, greater. Pacient +<i>Socrates</i>, his <i>Apologie</i> will testifie: <i>Apuleius</i> his +<i>Apologies</i>, will declare the Brutishnesse of the Multitude. +<i>Ioannes Picus</i>, Earle of Mirandula, his <i>Apologie</i> will teach +you, of the Raging slaunder of the Malicious Ignorant against him. +<i>Ioannes Trithemius</i>, his <i>Apologie</i> will specifie, how he had +occasion to make publike Protestation: as well by reason of the Rude +Simple: as also, in respect of such, as were counted to be of the wisest +sort of men. <span class="sidequote">“</span>Many could I recite: But +I deferre the precise and determined handling of this matter: being loth +to detect the Folly & Mallice of my Natiue Countrey men.* +<span class="sidefinger"> +* </span> +Who, so hardly, can disgest or like any extraordinary course of +Philosophicall Studies: not falling within the Cumpasse of their +Capacitie: or where they are not made priuie of the true and secrete +cause, of such wonderfull Philosophicall Feates.<span class="sidequote">”</span> These men, are of fower sortes, chiefly. The first, +I may name, <i>Vaine pratling busie bodies</i>: The second, <i>Fond +Frendes</i>: The third, <i>Imperfectly zelous</i>: and the fourth, +<i>Malicious Ignorant</i>. To eche of these (briefly, and in charitie) +I will say a word or two, and so returne to my Præface. +<span class="sidenote"> +1.</span> +<i>Vaine pratling busie bodies</i>, vse your idle assemblies, and +conferences, otherwise, then in talke of matter, either aboue your +Capacities, for hardnesse: or contrary to your Consciences, in Veritie. +<span class="sidenote"> +2.</span> +<i>Fonde Frendes</i>, leaue of, so to commend your vnacquainted frend, +vpon blinde affection: As, because he knoweth more, then the common +Student: that, therfore, he must needes be skilfull, and a doer, in such +matter and maner, as you terme <i>Coniuring</i>. Weening, thereby, you +aduaunce his fame: and that you make other men, great marueilers of your +hap, to haue such a learned frend. Cease to ascribe Impietie, where you +pretend Amitie. For, if your tounges were true, then were that your +frend, <i>Vntrue</i>, both to God, and his Soueraigne. Such +<i>Frendes</i> and <i>Fondlinges</i>, I shake of, and renounce you: +Shake you of, your Folly. +<span class="sidenote"> +3.</span> +<i>Imperfectly zelous</i>, to you, do I say: that (perhaps) well, do you +Meane: But farre you misse the Marke: If a Lambe you will kill, to feede +the flocke with his bloud. Sheepe, with Lambes bloud, haue no naturall +sustenaunce: No more, is Christes flocke, with horrible slaunders, duely +ædified. Nor your faire pretense, by such rashe ragged Rhetorike, any +whit, well graced. But such, as so vse me, will finde a fowle Cracke in +their Credite. Speake that you know: And know, as you ought: Know not, +by Heare say, when life lieth in daunger. Search to the quicke, & +let Charitie be your guide. +<span class="sidenote"> +4.</span> +<i>Malicious Ignorant</i>, what shall I say to thee? <i>Prohibe linguam +tuam a malo. A detractione parcite linguæ. <span class="larger">Cause thy toung to refraine frõ euill. Refraine your toung from +slaunder.</span></i> Though your tounges be sharpned, Serpent like, +& Adders poyson lye in your lippes: +<span class="sidenote"> +Psal. 140.</span> +yet take heede, and thinke, betimes, with your selfe, <i>Vir linguosus +non stabilietur in terra. Virum violentum venabitur malum, donec +præcipitetur.</i> For, sure I am, <i>Quia faciet Dominus Iudicium +afflicti: & vindictam pauperum.</i></p> + +<p>Thus, I require you, my assured frendes, and Countrey men (you +Mathematiciens, Mechaniciens, and Philosophers, Charitable and discrete) +to deale in my +<span class="pagenum">A.iij </span> +<!-- png 53 --> +behalf, with the light & vntrue tounged, my enuious Aduersaries, or +Fond frends. And farther, I would wishe, that at leysor, you would +consider, how <i>Basilius Magnus</i>, layeth <i>Moses</i> and +<i>Daniel</i>, before the eyes of those, which count all such Studies +Philosophicall (as mine hath bene) to be vngodly, or vnprofitable. Waye +well <i>S. Stephen</i> his witnesse of <i>Moses</i>. +<span class="sidenote"> +Act. 7. C.</span> +<i>Eruditus est Moses omni Sapientia Ægyptiorũ: & erat potens in +verbis & operibus suis. <span class="larger">Moses was instructed +in all maner of wisedome of the Ægyptians: and he was of power both in +his wordes, and workes.</span></i> You see this Philosophicall Power +& Wisedome, which <i>Moses</i> had, to be nothing misliked of the +Holy Ghost. Yet <i>Plinius</i> hath recorded, <i>Moses</i> to be a +wicked <i>Magicien</i>. And that (of force) must be, either for this +Philosophicall wisedome, learned, before his calling to the leading of +the Children of <i>Israel</i>: or for those his wonders, wrought before +King <i>Pharao</i>, after he had the conducting of the +<i>Israelites</i>. As concerning the first, you perceaue, how +<i>S. Stephen</i>, at his Martyrdome (being full of the Holy Ghost) +in his Recapitulation of the olde Testament, hath made mention of +<i>Moses</i> Philosophie: with good liking of it: And <i>Basilius +Magnus</i> also, auoucheth it, to haue bene to <i>Moses</i> profitable +(and therefore, I say, to the Church of God, necessary). But as +cõcerning <i>Moses</i> wonders, done before King <i>Pharao</i>: God, him +selfe, sayd: <i>Vide vt omnia ostenta, quæ posui in manu tua, facias +coram Pharaone. <span class="larger">See that thou do all those +wonders before Pharao, which I haue put in thy hand.</span></i> Thus, +you euidently perceaue, how rashly, <i>Plinius</i> hath slaundered +<i>Moses</i>, +<span class="sidenote"> +Lib. 30. Cap. 1.</span> +of vayne fraudulent <i>Magike</i>, saying: <i>Est & alia Magices +Factio, a Mose, Iamne, & Iotape, Iudæis pendens: sed multis +millibus annorum post Zoroastrem. &c.</i> +<span class="sidenote"> +1.</span> +Let all such, therefore, who, in Iudgement and Skill of Philosophie, are +farre Inferior to <i>Plinie</i>, <span class="sidequote">“</span>take +good heede, least they ouershoote them selues rashly,<span class="sidequote">”</span> in +<span class="sidefinger"> + </span> +Iudging of <i>Philosophers straunge Actes</i>: and the Meanes, how they +are done. +<span class="sidenote"> +2.</span> +But, much more, ought they to beware of forging, deuising, and imagining +monstrous feates, and wonderfull workes, when and where, no such were +done: no, not any sparke or likelihode, of such, as they, without all +shame, do report. +<span class="sidenote"> +3.</span> +And (to conclude) most of all, let them be ashamed of Man, and afraide +of the dreadfull and Iuste Iudge: both Folishly or Maliciously to +deuise: and then, deuilishly to father their new fond Monsters on me: +Innocent, in hand and hart: for trespacing either against the lawe of +God, or Man, in any my Studies or Exercises, Philosophicall, or +Mathematicall: As in due time, I hope, will be more manifest.</p> + +<p><span class="dropcap">N</span>Ow end I, with <span class="largest">Archemastrie</span>. Which name, is not so new, as this Arte +is rare. For an other Arte, vnder this, a degree (for skill and +power) hath bene indued with this English name before. And yet, this, +may serue for our purpose, sufficiently, at this present. <span class="larger">This Arte, teacheth to bryng to actuall experience sensible, +all worthy conclusions by all the Artes Mathematicall purposed, & by +true Naturall Philosophie concluded: & both addeth to them a farder +scope, in the termes of the same Artes, & also by hys propre Method, +and in peculier termes, procedeth, with helpe of the foresayd Artes, to +the performance of complet Experiẽces, which of no particular Art, are +hable (Formally) to be challenged.</span> If you remember, how we +considered <i>Architecture</i>, in respect of all common handworkes: +some light may you haue, therby, to vnderstand the Souerainty and +propertie of this Science. <i>Science</i> I may call it, rather, then an +Arte: for the excellency and Mastershyp it hath, ouer so many, and so +mighty Artes and +<span class="pagenum verso">||</span> +<!-- png 54 --> +Sciences. And bycause it procedeth by <i>Experiences</i>, and searcheth +forth the causes of Conclusions, by <i>Experiences</i>: and also putteth +the Conclusions them selues, in <i>Experience</i>, it is named of some, +<i>Scientia Experimentalis</i>. The <span class="larger"><i>Experimentall Science</i></span>. <i>Nicolaus Cusanus</i> +termeth it so, in hys <i>Experimentes Statikall</i>, And an other +<i>Philosopher</i>, +<span class="sidenote"> +R. B.</span> +of this land Natiue (the floure of whose worthy fame, can neuer dye nor +wither) did write therof largely, at the request of <i>Clement the +sixt</i>. The Arte carrieth with it, a wonderfull Credit: By +reason, it certefieth, sensibly, fully, and completely to the vtmost +power of Nature, and Arte. This Arte, certifieth by <i>Experience</i> +complete and absolute: and other Artes, with their Argumentes, and +Demonstrations, persuade: and in wordes, proue very well their +Conclusions.* But +<span class="sidefinger"> + </span> +wordes, and Argumentes, are no sensible certifying: nor the full and +finall frute of Sciences practisable. And though some Artes, haue in +them, <i>Experiences</i>, yet they are not complete, and brought to the +vttermost, they may be stretched vnto, and applyed sensibly. As for +example: the Naturall Philosopher disputeth and maketh goodly shew of +reason: And the Astronomer, and the Opticall Mechanicien, put some +thynges in <i>Experience</i>: but neither, all, that they may: nor yet +sufficiently, and to the vtmost, those, which they do, There, then, the +<i>Archemaster</i> steppeth in, and leadeth forth on, the +<i>Experiences</i>, by order of his doctrine <i>Experimentall</i>, to +the chief and finall power of Naturall and Mathematicall Artes. Of two +or three men, in whom, this Description of <i>Archemastry</i> was +<i>Experimentally</i>, verified, I haue read and hard: and good +record, is of their such perfection. So that, this Art, is no +fantasticall Imagination: as some Sophister, might, <i>Cum suis +Insolubilibus</i>, make a florish: and dassell your Imagination: and +dash your honest desire and Courage, from beleuing these thinges, so +vnheard of, so meruaylous, & of such Importance. Well: as you will. +I haue forewarned you. I haue done the part of a frende: +I haue discharged my Duety toward God: for my small Talent, at hys +most mercyfull handes receiued. To this Science, doth the <i>Science +Alnirangiat</i>, great Seruice. Muse nothyng of this name. +I chaunge not the name, so vsed, and in Print published by other: +beyng a name, propre to the Science. Vnder this, commeth <i>Ars +Sintrillia</i>, by <i>Artephius</i>, briefly written. But the chief +Science, of the Archemaster, (in this world) as yet knowen, is an other +(as it were) OPTICAL Science: wherof, the name shall be told (God +willyng) when I shall haue some, (more iust) occasion, therof, to +Discourse.</p> + +<p>Here, I must end, thus abruptly (Gentle frende, and vnfayned louer of +honest and necessary verities.) For, they, who haue (for your sake, and +vertues cause) requested me, (an old forworne Mathematicien) to take pen +in hand: (through the confidence they reposed in my long experience: and +tryed sincerity) for the declaryng and reportyng somewhat, of the frute +and commodity, by the <span class="larger">Artes Mathematicall, to be +atteyned vnto</span>: euen they, Sore agaynst their willes, are forced, +for sundry causes, to satisfie the workemans request, in endyng +forthwith: He, so feareth this, so new an attempt, & so costly: And +in matter so slenderly (hetherto) among the common Sorte of Studentes, +considered or estemed.</p> + +<p>And where I was willed, somewhat to alledge, why, in our vulgare +Speche, this part of the Principall Science of <i>Geometrie</i>, called +<i>Euclides Geometricall Elementes</i>, is published, to your handlyng: +being vnlatined people, and not Vniuersitie Scholers: Verily, +I thinke it nedelesse.</p> + +<p><span class="sidenote"> +1.</span> +For, the Honour, and Estimation of the <span class="larger">Vniuersities, and Graduates</span>, is, hereby, nothing +diminished. Seing, from, and by their Nurse Children, you receaue all +this Benefite: how great soeuer it be.</p> + +<span class="pagenum">A.iiij </span> +<!-- png 55 --> +<p><span class="sidenote"> +2.</span> +Neither are their Studies, hereby, any whit hindred. No more, then the +Italian <i>Vniuersities</i>, as <i>Academia Bononiensis</i>, +<i>Ferrariensis</i>, <i>Florentina</i>, <i>Mediolanensis</i>, +<i>Patauina</i>, <i>Papiensis</i>, <i>Perusina</i>, <i>Pisana</i>, +<i>Romana</i>, <i>Senensis</i>, or any one of them, finde them selues, +any deale, disgraced, or their Studies any thing hindred, by <i>Frater +Lucas de Burgo</i>, or by <i>Nicolaus Tartalea</i>, who in vulgar +Italian language, haue published, not onely <i>Euclides Geometrie</i>, +but of <i>Archimedes</i> somewhat: and in Arithmetike and Practicall +Geometrie, very large volumes, all in their vulgar speche. Nor in +Germany haue the famous <i>Vniuersities</i>, any thing bene discontent +with <i>Albertus Durerus</i>, his Geometricall Institutions in Dutch: or +with <i>Gulielmus Xylander</i>, his learned translation of the first +sixe bookes of <i>Euclide</i>, out of the Greke into the high Dutch. Nor +with <i>Gualterus H. Riffius</i>, his Geometricall Volume: very +diligently translated into the high Dutch tounge, and published. Nor yet +the <i>Vniuersities</i> of Spaine, or Portugall, thinke their reputation +to be decayed: or suppose any their Studies to be hindred by the +Excellent <i>P. Nonnius</i>, his Mathematicall workes, in vulgare +speche by him put forth. Haue you not, likewise, in the French tounge, +the whole Mathematicall Quadriuie? and yet neither Paris, Orleance, or +any of the other Vniuersities of Fraunce, at any time, with the +Translaters, or Publishers offended: or any mans Studie thereby +hindred?</p> + +<p><span class="sidenote"> +3.</span> +And surely, the Common and Vulgar Scholer (much more, the Gramarian) +before his comming to the <i>Vniuersitie</i>, shall (or may) be, now +(according to <i>Plato</i> his Counsell) sufficiently instructed in +<i>Arithmetike</i> and <i>Geometrie</i>, for the better and easier +learning of all maner of <i>Philosophie</i>, <i>Academicall</i>, or +<i>Peripateticall</i>. And by that meanes, goe more cherefully, more +skilfully, and spedily forwarde, in his Studies, there to be learned. +And, so, in lesse time, profite more, then (otherwise) he should, or +could do.</p> + +<p><span class="sidenote"> +4.</span> +Also many good and pregnant Englishe wittes, of young Gentlemen, and of +other, who neuer intend to meddle with the profound search and Studie of +Philosophie (in the <i>Vniuersities</i> to be learned) may +neuerthelesse, now, with more ease and libertie, haue good occasion, +vertuously to occupie the sharpnesse of their wittes: where, els +(perchance) otherwise, they would in fond exercises, spend (or rather +leese) their time: neither seruing God: nor furdering the Weale, common +or priuate.</p> + +<p><span class="sidenote"> +5.</span> +And great Comfort, with good hope, may the <i>Vniuersities</i> haue, by +reason of this <i>Englishe</i> <span class="larger">Geometrie, and +Mathematicall Præface</span>, that they (hereafter) shall be the more +regarded, esteemed, and resorted vnto. For, when it shall be knowen and +reported, that of the <i>Mathematicall Sciences</i> onely, such great +Commodities are ensuing (as I haue specified): and that in dede, some of +you vnlatined Studentes, can be good witnesse, of such rare fruite by +you enioyed (thereby): as either, before this, was not heard of: or els, +not so fully credited: <span class="sidequote">“</span>Well, may all +men coniecture, that farre greater ayde, and better furniture, to winne +to the Perfection of all Philosophie, +<span class="sidenote"> +Vniuersities.</span> +may in the Vniuersities be had: being the Storehouses & Threasory of +all Sciences, +<span class="sidefinger"> + </span> +and all Artes, necessary for the best, and most noble State of Common +Wealthes.<span class="sidequote">”</span></p> + +<p><span class="sidenote"> +6.</span> +Besides this, how many a Common Artificer, is there, in these Realmes of +England and Ireland, that dealeth with Numbers, Rule, & Cumpasse: +Who, with their owne Skill and experience, already had, will be hable +(by these good helpes and informations) to finde out, and deuise, new +workes, straunge Engines, and Instrumentes: for sundry purposes in the +Common Wealth? or for priuate pleasure? and for the better maintayning +of their owne estate? I will not (therefore) +<span class="pagenum verso">||</span> +<!-- png 56 --> +fight against myne owne shadowe. For, no man (I am sure) will open +his mouth against this Enterprise. No mã (I say) who either hath +Charitie toward his brother (and would be glad of his furtherance in +vertuous knowledge): or that hath any care & zeale for the bettering +of the Cõmon state of this Realme. Neither any, that make accompt, what +the wiser sort of men (Sage and Stayed) do thinke of them. To none +(therefore) will I make any <i>Apologie,</i> for a vertuous acte doing: +and for cõmending, or setting forth, Profitable Artes to English men, in +the English toung. <span class="sidequote">“</span>But, vnto God our +Creator, let vs all be thankefull: for that, <span class="larger"><i>As he, of his Goodnes, by his Powre, and in his wisedome, +<span class="sidefinger"> + </span> +hath Created all thynges, in Number, Waight, and Measure</i></span>: So, +to vs, of hys great Mercy, he hath reuealed Meanes, whereby, to atteyne +the sufficient and necessary knowledge of the foresayd hys three +principall Instrumentes: Which Meanes, I haue abundantly proued +vnto you, to be the <i>Sciences</i> and <i>Artes Mathematicall</i>.<span +class="sidequote">”</span></p> + +<p>And though I haue ben pinched with straightnes of tyme: that, no way, +I could so pen downe the matter (in my Mynde) as I determined: +hopyng of conuenient laysure: Yet. if vertuous zeale, and honest Intent +prouoke and bryng you to the readyng and examinyng of this Compendious +treatise, I do not doute, but, as the veritie therof (accordyng to +our purpose) will be euident vnto you: So the pith and force therof, +will persuade you: and the wonderfull frute therof, highly pleasure you. +And that you may the easier perceiue, and better remember, the +principall pointes, whereof my Preface treateth, +<span class="sidenote"> +The Ground platt of this Præface in a Table.</span> +I will giue you the <span class="larger">Groundplatt</span> of my +whole discourse, in a Table annexed: from the first to the last, +somewhat Methodically contriued.</p> + +<p class="center"> +If Hast, hath caused my poore pen, any where, to stumble: You will, +(I am<br /> +sure) in part of recompence, (for my earnest and sincere good will to +pleasure<br /> +you), Consider the rockish huge mountaines, and the perilous<br /> +vnbeaten wayes, which (both night and day, for the while) it<br /> +hath toyled and labored through, to bryng you this good<br /> +Newes, and Comfortable profe, of Vertues frute.</p> + +<p class="center"> +So, I Commit you vnto Gods Mercyfull direction, for the rest: +hartely<br /> +besechyng hym, to prosper your Studyes, and honest Intentes:<br /> +to his Glory, & the Commodity of our Countrey. <i>Amen</i>.</p> + +<p class="right"> +<i>Written at my poore House<br /> +At Mortlake.</i></p> + +<p class="right"> +<i>Anno. 1570. February. 9.</i></p> + +<p class="illustration"> +<img src="images/dec_end.png" width="119" height="121" +alt="decoration" /> +</p> + +</div> <!-- end div main--> + +<hr /> + +<p> +<a name="groundplat" id="groundplat"> </a> +</p> + +<p class="center smaller"><i>J. DEE</i></p> + +<p class="center larger">Here haue you (according to my promisse) the +Groundplat of</p> + +<p class="center">my MATHEMATICALL Præface: annexed to <i>Euclide</i> +(now first)</p> + +<p class="center smaller">published in our Englishe tounge. An. 1570. +Febr. 3.</p> + +<table class="groundplat" summary="impossible to summarize"> + +<!-- principal, 6 rows --> +<!-- row 1 of 6 --> + +<tr> +<td class="bracket righttop" rowspan="3"> + +</td> +<td class="bracket righttop" rowspan="2"> + +</td> +<td class="bracket righttop"> + +</td> +<td class="lower" colspan="4"> +<p><i><span class="larger">Simple,</span> <span class="small">Which +dealeth with Numbers onely: and demon­strateth all their properties +and apper­tenances: where, an Vnit, is +Indiui­sible.</span></i></p> +</td> +<td class="bracket lefttop"> + +</td> +<td class="bracket righttop"> + +</td> +<td rowspan="2"> +<p class="small"><i>In thinges Super­naturall, æternall, & +Diuine: By Appli­cation</i>, Ascending.</p></td> +<td class="bracket lefttop"> + +</td> +</tr> + +<!-- row 2 of 6 --> +<tr> +<!-- <td></td> <td></td> --> +<td class="bracket rightline"> +<p class="larger">Arithmetike.</p> +</td> +<td colspan="4" rowspan="2"> +<p><i><span class="larger">Mixt</span>, <span class="small">Which +with aide of Geometrie principall, demon­strateth some +Arith­meticall Conclusion, or Purpose.</span></i></p> +</td> +<td class="bracket leftline" rowspan="4"> + +</td> +<td class="bracket rightline" rowspan="4"> +<p><i><span class="larger">The vse</span><br /> +whereof, is either,</i></p> +</td> +<!-- <td></td> --> +<td class="bracket leftline" rowspan="4"> +<p><i><span class="small">The like Vses and Appli­cations are, +(though in a degree lower) in the</span></i> Artes Mathe­maticall +Deriuatiue.</p> +</td> +</tr> + +<!-- row 3 of 6 --> +<tr> +<!-- <td></td> --> +<td class="bracket rightline" rowspan="2"> +<p><i><span class="larger">Principall,</span> which are two, +onely,</i></p> +</td> +<td class="bracket rightbottom"> +<p> <br /> </p> +</td> +<!-- <td></td> <td></td> <td></td> <td></td> <td></td> --> +<td rowspan="2"> +<p class="small"><i>In thinges Mathe­maticall: without farther +Appli­cation.</i></p> +</td> +<!-- <td></td> --> +</tr> + +<!-- row 4 of 6 --> +<tr> +<td class="bracket rightline" rowspan="6"> +<p><span class="largest">Sciences, and Artes +Mathe­maticall,</span> <span class="larger">are, either</span></p> +</td> +<!-- <td></td> --> +<td class="bracket righttop"> +<p> <br /> </p> +</td> +<td colspan="4" rowspan="2"> +<p><i><span class="larger">Simple</span>, <span class="small">Which +dealeth with Magni­tudes, onely: and demon­strateth all their +properties, passions, and apper­tenances: whose Point, is +Indiui­sible.</span></i></p> +</td> +<!-- <td></td> <td></td> <td></td> <td></td> --> +</tr> + +<!-- row 5 of 6 --> +<tr> +<!-- <td></td> --> +<td class="bracket rightbottom"> +<p> <br /> </p> +</td> +<td class="bracket rightline"> +<p class="larger">Geometrie.</p> +</td> +<!-- <td></td> <td></td> <td></td> <td></td> <td></td> --> +<td rowspan="2"> +<p class="small"><i>In thinges Naturall: both Substã­tiall, & +Accidentall, Visible, & Inuisible. &c. By Appli­cation:</i> +Descending.</p> +</td> +<!-- <td></td> --> +</tr> + +<!-- row 6 of 6 --> +<tr> +<!-- <td></td> --> +<td> + +</td> +<td class="bracket rightbottom"> + +</td> +<td class="upper" colspan="4"> +<p><i><span class="larger">Mixt</span>, <span class="small">Which +with aide of Arith­metike principall, demon­strateth some +Geometricall purpose, as</span></i> EVCLIDES ELEMENTES.</p> +</td> +<td class="bracket leftbottom"> + +</td> +<td class="bracket rightbottom"> + +</td> +<!-- <td></td> --> +<td class="bracket leftbottom"> + +</td> +</tr> + +<!-- derivative, 5 rows --> +<!-- row 1 of 5 --> + +<tr> +<!-- <td></td> --> +<td class="bracket righttop" rowspan="2"> + +</td> +<td class="bracket righttop"> + +</td> +<td class="lower bracket single"> +<p><span class="larger">Arith­metike,</span> <span class="small"><i>vulgar: which consi­dereth</i></span></p> +</td> +<td colspan="7"> +<p>Arith­metike of most vsuall whole numbers: And of Fractions to +them apper­taining.</p> +<p>Arith­metike of Propor­tions.</p> +<p>Arith­metike Circular.</p> +<p>Arith­metike of Radicall Nũbers: Simple, Compound, Mixt: And of +their Fractions.</p> +<p>Arith­metike of Cossike Nũbers: with their Fractions: And the +great Arte of Algiebar.</p> +</td> +</tr> + +<!-- row 2 of 5 --> +<tr> +<!-- <td></td> <td></td> --> +<td class="lower bracket rightline"> +<p><i><span class="larger">The names of</span> the Princi­palls: +as,</i></p> +</td> +<td class="bracket righttop"> + +</td> +<td class="bracket single" colspan="2"> +<p class="larger"><i>At hand</i></p> +</td> +<td class="bracket single" colspan="3"> +<p>All Lengthes.—</p> +<p>All Plaines: As, Land, Borde, Glasse, &c.</p> +<p>All Solids: As, Timber, Stone, Vessels, &c.</p> +</td> +<td colspan="2"> +<p class="larger">Mecometrie.</p> +<p class="larger">Embadometrie.</p> +<p class="larger">Stereometrie.</p> +</td> +</tr> + +<!-- row 3 of 5 --> +<tr> +<!-- <td></td> --> +<td class="bracket rightline" rowspan="3"> +<p><i><span class="larger">Deriuatiue</span> frõ the Princi­palls: +of which, some haue</i></p> +</td> +<td class="bracket rightbottom"> + +</td> +<td class="upper bracket rightline"> +<p><span class="larger">Geometrie,</span> <span class="small"><i>vulgar: which teacheth Measuring</i></span></p> +</td> +<td class="bracket righttop" colspan="2"> + +</td> +<td class="lower"> +<p><i><span class="larger">How farre</span>, <span class="small">from the Measurer, any thing is: of him sene, on Land or Water: +called</span></i> <span class="larger">Apomecometrie</span>.</p> +</td> +<td class="bracket lefttop"> + +</td> +<td class="bracket righttop"> + +</td> +<td class="lower" colspan="2"> +<p><span class="larger">Geodesie:</span> <span class="small"><i>more +cunningly to Measure and Suruey Landes, Woods, Waters. +&c.</i></span></p> +</td> +</tr> + +<!-- row 4 of 5 --> +<tr> +<td class="bracket rightbottom" rowspan="2"> + +</td> +<!-- <td></td> --> +<td rowspan="2"> + +</td> +<td class="bracket rightbottom"> + +</td> +<td class="upper bracket rightline" colspan="2"> +<p><i><span class="larger">With distãce</span> <span class="small">from the thing Measured, as,</span></i></p> +</td> +<td> +<p><i><span class="larger">How high or deepe</span>, <span class="small">from the leuell of the Measurers standing, any thing is: Seene +of hym, on Land or Water: called</span></i> <span class="larger">Hypso­metrie</span>.</p> +</td> +<td class="bracket leftline"> + +</td> +<td class="bracket rightline"> +<p><i><span class="larger">Of which</span> are growen the Feates & +Artes of</i></p> +</td> +<td colspan="2"> +<p><span class="larger">Geographie.</span></p> +<p> </p> +<p><span class="larger">Choro­graphie.</span></p> +<p> </p> +<p><span class="larger">Hydro­graphie.</span></p> +</td> +</tr> + +<!-- row 5 of 5 --> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td> + +</td> +<td class="bracket rightbottom" colspan="2"> + +</td> +<td> +<p><i><span class="larger">How broad</span>, <span class="small">a thing is, which is in the Measurers view: so it be +situated on Land or Water: called</span></i> <span class="larger">Plato­metrie</span>.</p> +</td> +<td class="bracket leftbottom"> + +</td> +<td class="bracket rightbottom"> + +</td> +<td class="upper" colspan="2"> +<p><span class="larger">Strat­arith­metrie.</span></p> +</td> +</tr> + +<!-- proper names, 19 rows --> + +<tr> +<td rowspan="19"> </td> +<td class="bracket rightline" rowspan="8"> + +</td> +<td class="bracket righttop"> +<p> <br /> </p> +</td> +<td class="upper" colspan="2"> +<p class="larger">Perspectiue,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth the maners and properties +of all Radia­tions: Directe, Broken, and Reflected.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> --> +<td class="bracket rightline" rowspan="6"> + +</td> +<td class="upper" colspan="2"> +<p class="larger">Astro­nomie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth the Distances, +Magni­tudes, and all Naturall motions, Apparences, and Passions, +proper to the Planets and fixed Starres: for any time, past, present, +and to come: in respecte of a certaine Horizon, or without respecte of +any Horizon.</i></p> +</td> +</tr> + +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Musike,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth by reason, and teacheth +by sense, perfectly to iudge and order the diuer­sitie of Soundes, +hie or low.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Cosmo­graphie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which, wholy and perfectly maketh description of +the Heauenlym and also Elementall part of the World: and of these +partes, maketh homologall appli­cation, and mutuall collation +necessary.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Astro­logie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which reasonably demon­strateth the +opera­tions and effectes of the naturall beames of light, and +secrete Influence of the Planets, and fixed Starres, in euery Element +and Elementall body: at all times, in any Horizon assigned.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Statike,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth the causes of heauines +and lightnes of all thinges: and of the motions and properties to +heauines and lightnes belonging.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Anthropographie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which describeth the Nũber, Measure, Waight, +Figure, Situation, and colour of euery diuers thing contained in the +perfecte body of MAN: and geueth certaine knowledge of the Figure, +Symmetrie, Waight, Charac­terization, & due Locall motion of any +percell of the said body assigned: and of numbers to the said percell +apper­taining.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> --> +<td class="bracket rightline" rowspan="3"> +<p><i><span class="larger">Propre names</span><br /> +as,</i></p> +</td> +<td class="upper" colspan="2"> +<p class="larger">Trochilike,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth the properties of all +Circular motions: Simple and Compound.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> --> +<td class="bracket rightbottom" rowspan="2"> + +</td> +<!-- <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Helico­sophie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth the designing of all +Spirall lines: in Plaine, on Cylinder, Cone, Sphære, Conoïd, and +Sphæroid: and their properties.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Pneuma­tithmie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth by close hollow +Geometricall figures (Regular and Irregular) the straunge properties (in +motion or stay) of the Water, Ayre, Smoke, and Fire, in their +Conti­nuitie, and as they are ioyned to the Elementes next +them.</i></p> +</td> +</tr> + +<tr> +<!-- <td></td> --> +<td rowspan="9"> + +</td> +<td class="bracket rightline" rowspan="8"> + +</td> +<td class="upper" colspan="2"> +<p class="larger">Menadrie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth, how, aboue Natures +Vertue, and power simple: Vertue and force, may be multi­plied: and +so to directe, to lift, to pull to, and to put or cast fro, any +multi­plied, or simple deter­mined Vertue, Waight, or Force: +naturally, not, so, direc­tible, or moueable.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Hypogeiodie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth, how, vnder the +Sphæricall Super­ficies of the Earth, at any depth, to any +perpen­dicular line assigned (whose distance from the +perpen­dicular of the entrance: and the Azimuth likewise, in +respecte of the sayd entrance, is knowen) certaine way, may be +prescribed and gone, &c.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Hydra­gogie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth the possible leading of +water by Natures law, and by artificiall helpe, from any head (being +Spring, standing, or running water) to any other place assigned.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Horometrie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth, how, at all times +appointed, the precise, vsuall denomi­nation of time, may be knowen, +for any place assigned.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Zographie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth and teacheth, how, the +Inter­section of all visuall Pyramids, made by any plaine assigned +(the Center, distance, and lightes being deter­mined) may be, by +lines, and proper colours repre­sented.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Archi­tecture,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which is a Science garnished with many doctrines, +and diuers Instructions: by whose iudgement, all workes by other workmen +finished, are iudged.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Nauigation,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which demon­strateth, how, by the Shortest +good way, by the aptest direction, and in the shortest time: +a suffi­cient Shippe, betwene any two places (in passage +nauigable) assigned, may be conducted: and in all stormes and naturall +distur­bances chauncing, how to vse the best possible meanes, to +recouer the place first assigned.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> <td></td> --> +<td class="upper" colspan="2"> +<p class="larger">Thaumaturgike,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which geueth certaine order to make straunge +workes, of the sense to be perceiued: and of men greatly to be wondred +at.</i></p> +</td> +</tr> +<tr> +<!-- <td></td> <td></td> --> +<td class="bracket rightbottom"> + +</td> +<td class="upper" colspan="2"> +<p class="larger">Arche­mastrie,</p> +</td> +<td colspan="6"> +<p class="small"><i>Which teacheth to bring to actuall experience +sensible, all worthy conclu­sions, by all the Artes +Mathe­maticall purposed: and by true Naturall philo­sophie, +concluded: And both addeth to them a farder Scope, in the termes of the +same Artes: and also, by his proper Method, and in peculiar termes, +procedeth, with helpe of the forsayd Artes, to the perfor­mance of +complete Experi­ences: which, of no parti­cular Arte, are hable +(Formally) to be challenged.</i></p> +</td> +</tr> +</table> + + +<hr class="spacer" /> + +<h4><a name="title_text" id="title_text">Title Page Text</a></h4> + +<p class="illustration"> +<img src="images/titletext.png" width="394" height="543" +alt="full text below" /> +</p> + +<h5>THE ELEMENTS<br /> +<span class="smaller">OF GEOMETRIE</span><br /> +of the most auncient<br /> +Philosopher<br /> +<i>EVCLIDE</i><br /> +of Megara.</h5> + +<h5><i>Faithfully (now first) translated<br /> +into the Englishe toung, by<br /> +</i>H. Billingsley<i>, Citizen of London</i>.<br /> +<i>Whereunto are annexed certaine<br /> +<span class="smaller">Scholies, Annotations, and Inuentions,<br /> +of the best Mathematiciens,<br /> +both of time past, and<br /> +in this our age.</span></i></h5> + +<hr class="tiny" /> + +<h5><i>With a very fruitfull Præface made by </i>M. I. Dee<i>,<br /> +specifying the chiefe Mathematicall Sciẽces, what<br /> +<span class="smaller">they are, and wherunto commodious: where, also, +are disclosed certaine new Secrets Mathematicall and Mechanicall, vntill +these our daies, greatly missed.</span></i></h5> + +<hr class="tiny" /> + +<h5>Imprinted at London by <i>Iohn Daye</i>.</h5> + + +<div class="mynote end"> +<h5>Transcriber’s Footnotes</h5> + +<p><a name="endnote1" id="endnote1" href="#endtag1">A.</a> +<b>Mathematical Notation.</b> John Dee used the “root” sign √ in +combination with some less familiar symbols:</p> + +<table class="symbols" summary="list of mathematical symbols"> +<tr> +<td> +<img src="images/power_1.gif" width="38" height="35" +alt="‘power of 1’ symbol" /> +</td> +<td><p>“First power”, here used to express an unknown. <i>Shown in this +e-text as</i> <b>X</b> <i>(capitalized).</i></p> +</td> +</tr> +<tr> +<td> +<img src="images/root_square.gif" width="54" height="33" +alt="square root" /> +</td> +<td><p>Root sign combined with “second power” symbol = square root. +<i>Shown in this e-text as</i> <sup>2</sup>√.</p> +</td> +</tr> +<tr> +<td> +<img src="images/root_cubic.gif" width="60" height="33" +alt="cube root" /> +</td> +<td><p>Root sign combined with “third power” symbol = cube root. +<i>Shown in this e-text as</i> <sup>3</sup>√.</p> +</td> +</tr> +<tr> +<td> +<img src="images/root_fourth.gif" width="84" height="34" +alt="cube root" /> +</td> +<td><p>Doubled “second power” symbol = 4th power; with root sign = +fourth root. <i>Shown in this e-text as</i> <sup>4</sup>√.</p> +</td> +</tr> +</table> + +<p><a name="endnote2" id="endnote2" href="#endtag2">B.</a> +<b>Diagrams:</b> The symbol drawn as <i>P</i> (Pounds) is shown here as +<span class="pmath">P</span>. See above for <b>X</b> symbol.</p> + +<pre> + HOTE + +C + | + | + + + | + | + + + | + | + +E + | +MOIST A TEMPERATE B DRYE + +------+------+------+------+------+------+------+------+ + |D + | + + + | + | + + + | + | + + + | + | + + + COLD +</pre> + +<pre> + _____________________ +| | | +| {P}. 2. | Hote. 4. | +| | | +| {P}. 1. | Hote. 3. | +|_________|___________| +</pre> + +<pre> + _____________________ +| | | _ +| {P}. 2. | Hote. 4. | ⅓ _ The forme_ +| | | _ 3⅔ resulting. +| {P}. 1. | Hote. 3. | _ ⅔ +|_________|___________| +</pre> + +<p><a name="endnote3" id="endnote3" href="#endtag3">C.</a> +<b>“Vergilius teaches in his Georgikes.”</b> The quoted lines, with +breaks at each “&c.”, are 438-439; 451-457; 463-464.</p> + +<hr class="tiny" /> + +<h5><a name="euclid" id="euclid"> +Euclid citations</a></h5> + +<p>The following Propositions were identified by number.</p> + +<p><b>6.12</b>: (How) to find a fourth (line) proportional to three +given straight lines.</p> + +<p><b>11.34</b>: In equal parallelepipedal solids the bases are +reciprocally proportional to the heights; and those parallelepipedal +solids in which the bases are reciprocally proportional to the heights +are equal.</p> + +<p><b>11.36</b>: If three straight lines are proportional, then the +parallelepipedal solid formed out of the three equals the +parallelepipedal solid on the mean which is equilateral, but equiangular +with the aforesaid solid.</p> + +<p><b>12.1</b>: Similar polygons inscribed in circles are to one another +as the squares on their diameters.</p> + +<p><b>12.2</b>: Circles are to one another as the squares on their +diameters.</p> + +<p><b>12.18</b> (“last”): Spheres are to one another in triplicate ratio +of their respective diameters.</p> + +<hr class="tiny" /> + +<h5><a name="textnotes" id="textnotes"> +Notes on the text</a></h5> + +<p>The Greek letter η (eta) was consistently printed as if it were the +ou-ligature ȣ.</p> + +<p>The Latin <i>-que</i> was written as an abbreviation resembling +<b>-q´;</b>. It is shown here as q<sup>ue</sup>.</p> + +<p>Less common words include “fatch” (probably used as a variant of +“fetch”) and the mathematical terms “sexagene” and “sexagesme”.</p> + +</div> + +<div style='display:block; margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK THE MATHEMATICALL PRAEFACE ***</div> +<div style='text-align:left'> + +<div style='display:block; margin:1em 0'> +Updated editions will replace the previous one—the old editions will +be renamed. +</div> + +<div style='display:block; margin:1em 0'> +Creating the works from print editions not protected by U.S. copyright +law means that no one owns a United States copyright in these works, +so the Foundation (and you!) can copy and distribute it in the United +States without permission and without paying copyright +royalties. 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