❧The Translator to the Reader.T(There)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 perfectlyto 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 PhilosopherEuclideofMegara. 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 ofEuclidesElementes, 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 ofEuclide. 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 shallwell 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)and also pertaining to the Mathematicall Artes.Thus gentle reader farewell.(?¿)❧TO THE VNFAINED LOVERSof truthe, and constant Studentes of NobleSciences,IOHN DEEof London, hartilywisheth grace from heauen, and most prosperoussuccesse in all their honest attemptes andexercises.D(Divine)IuinePlato, the great Master of many worthy Philosophers, and the constant auoucher, and pithy perswader ofVnum,Bonum, andEns: in his Schole and Academie, sundry times (besides his ordinary Scholers) was visited of a certaine kinde of men, allured by the noble fame ofPlato, 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, thisVnum,Bonum, andEns, to be Spirituall, Infinite, Æternall, Omnipotent, &c. Nothyng beyng alledged or expressed, How, worldly goods: how,worldlydignitie: 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 ofPlato, was clene chaunged: yea his doctrine was by them despised: and his schole, no more of them visited. Which thing, his Scholer,Aristotle, narrowly cõsidering, founde the cause therof, to be,“For that they had no forwarnyng and information, in generall,”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,Aristotle, euer, after that, vsed in brief, to forewarne his owne Scholers and hearers,“both of what matter, and also to what ende, he tooke in hand to speake, or teach.”While I consider the diuerse trades of these two excellent Philosophers (and am most sure, both, thatPlatoright well, otherwise could teach: and thatAristotlemought boldely, with his hearers, haue dealt in like sorte asPlatodid) I am in no little pang of perplexitie: Bycause, that, which I mislike, is most easy for me to performe (and to hauePlatofor my exãple.) And that, which I know to be most commendable: and (in this first bringyng, into common handling, theArtes Mathematicall) 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 (ImitatyngAristotle) well can I hope, that accordyng to the amplenes and dignitie of theState Mathematicall, 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 fromPlatohis schole, after they had perceiued his finallconclusion, would in these thinges haue ben his most diligenthearers (soinfinitely mought their desires, in fine and at length, by ourArtes Mathematicallbe satisfied) yet, by this my Præface & forewarnyng, Aswell all such, may (to their great behofe) the soner, hither be allured: as also thePythagoricall, andPlatonicallperfect scholer, and the constant profound Philosopher, with more ease and spede, may (like the Bee,) gather, hereby, both wax and hony.Wherfore, seyng I finde great occasion (for the causes alleged, and farder, in respect of myArt Mathematike generall) to vse“a certaine forewarnyng and Præface, whose content shalbe,The intent of this Preface.that mighty, most plesaunt, and frutefullMathematicall Tree, 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:”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.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 calledThynges Mathematicall. 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 generallFormes, 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 ofDianœticalldiscourse, in perfect imagination subsistyng. A meruaylous newtralitie haue these thingesMathematicall, 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:is allowed as sufficient for“an Argument exactly and purely Mathematical.”OfMathematicallthinges, are two principall kindes: namely,Number, andMagnitude.Number.Number, we define, to be, a certayne Mathematicall Sũme, ofVnits.Note the worde, Vnit, to expresse the Greke Monas, & not Vnitie: as we haue all, commonly, till now, vsed.And, anVnit, 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 anVnit, a thingMathematicall, though it be no Number, and also indiuisible: because, of it, materially, Number doth consist: which, principally, is a thingMathematicall.Magnitude.Magnitudeis a thingMathematicall, by participation of some likenes of whose nature, any thing is iudged long, broade, or thicke.“A thickeMagnitudewe call aSolide, or aBody. WhatMagnitudeso euer, is Solide or Thicke, is also broade, & long. A broade magnitude, we call aSuperficiesor a Plaine. Euery playne magnitude, hath also length. A long magnitude, we terme aLine. ALineis neither thicke nor broade, but onely long: Euery certayne Line, hath two endes:A point.The endes of a line, arePointescalled. A Point, is a thingMathematicall, indiuisible, which may haue a certayne determined situation.”If a Poynt moue from a determined situation, the way wherein it moued, is also aLine: mathematically produced, whereupon, of the auncient Mathematiciens,A Line.aLineis called the race or course of aPoint. 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 trueMagnitude.Magnitude.AndMagnitudewe may define to be that thingMathematicall, which is diuisible for euer, in partes diuisible, long, broade or thicke. Therefore though a Poynt be noMagnitude, yetTerminatiuely, we recken it a thingMathematicall(as I sayd) by reason it is properly the end, and bound of a line. NeitherNumber, norMagnitude, haue any Materialitie. First, we will consider ofNumber, and of the ScienceMathematicall, to it appropriate, calledArithmetike: and afterward ofMagnitude, and his Science, calledGeometrie. But that name contenteth me not: whereof a word or two hereafter shall be sayd. How Immateriall and free from all matter,Numberis, who doth not perceaue? yea, who doth not wonderfully wõder at it? For, neither pureElement, norAristoteles, Quinta Essentia, 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 PhilosopherAnitius Boetius, sayd:Omnia quæcunquea primæua rerum natura constructa sunt, Numerorum videntur ratione formata. Hoc enim fuit principale in animo Conditoris Exemplar. That is: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.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, fromNothing, to theFormalitieof their being and state. ByNumberspropertie 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, andFormes: And also, farder, arise, clime, ascend, and mount vp (with Speculatiue winges) in spirit, to behold in the Glas of Creation, theForme of Formes, theExemplar Numberof all thingesNumerable: both visible and inuisible, mortall andimmortall, Corporall and Spirituall. Part of this profound and diuine Science, hadIoachimthe Prophesier atteyned vnto: byNumbers Formall, Naturall, andRationall, forseyng, concludyng, and forshewyng great particular euents, long before their comming. His bookes yet remainyng, hereof, are good profe: And the noble Earle ofMirandula, (besides that,) a sufficient witnesse: thatIoachim, in his prophesies, proceded by no other way, then by Numbers Formall. And this Earle hym selfe, in Rome,Ano. 1488.*set vp 900. Conclusions, in all kinde of Sciences, openly to be disputed of: and among the rest, in his ConclusionsMathematicall, (in the eleuenth Conclusion) hath in Latin, this English sentence.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. Which Cõclusions, I omit here to rehearse: aswell auoidyng superfluous prolixitie: as, bycauseIoannes Picus, workes, 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.Of my former wordes, easy it is to be gathered, thatNumberhath 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,Number, is termedNumber Numbryng. But in all Creatures, otherwise,Number, is termedNũber Numbred. And in our Soule, Nũber beareth such a swaye, and hath such an affinitie therwith: that some of the oldPhilosopherstaught,Mans Soule, to be a Number mouyng it selfe. 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.Number Numbryngtherfore, 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 hisNumbryng, then, was his Creatyng of all thinges. And his ContinuallNumbryng, of all thinges, is the Conseruation of them in being: And, where and when he will lacke anVnit: there and then, that particular thyng shalbeDiscreated. Here I stay. But our Seuerallyng, distinctyng, andNumbryng, 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 AlmightyTernarie,) Artificiall Methods and easy wayes are made, by which the zelous Philosopher, may wyn nere this RiuerishIda, this Mountayne of Contemplation: and more then Contemplation. And also, thoughNumber, 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 aTernarie. Three Egles, are three, or aTernarie.Which*Ternaries, are eche, theVnion,knot, andVniformitie, of three discrete and distinctVnits. That is, we may in echeTernarie, thrise, seuerally pointe, and shew a part,One,One, andOne. Where, in Numbryng, we say One, two,Three. But how farre, these visible Ones, do differre from our Indiuisible Vnits (in pureArithmetike, 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 ofNumbers, absolutely (:Not supposing, nor admixtyng any thyng created, Corporall or Spirituall, to support, conteyne, or represent thoseNumbersimagined:) that at length, we may be hable, to finde the number of our owne name, gloriously exemplified and registred in the booke of theTrinitiemost blessed and æternall.But farder vnderstand, that vulgar Practisers, haue Numbers, otherwise, in sundry Considerations: and extend their name farder, then to Numbers, whose least part is anVnit. For the common Logist, Reckenmaster, or Arithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lesse partes: and calleth themFractions. As of anVnit, he maketh an halfe, and thus noteth it, ½. and so of other, (infinitely diuerse) partes of anVnit. Yea and farder, hath,Fractions of Fractions. &c. And, forasmuch, as,Addition,Substraction,Multiplication,DiuisionandExtraction of Rotes, are the chief, and sufficient partes ofArithmetike:Arithmetike.which is, theScience that demonstrateth the properties, of Numbers, and all operatiõs, in numbers to be performed:Note.“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 ofArithmetike.”And by this reason:1.the Consideration, doctrine, and working, in whole numbers onely: where, of anVnit, is no lesse part to be allowed: is named (as it were) anArithmetikeby it selfe. And so of theArithmetike of Fractions.2.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 ofArithmetike.3.Moreouer, theAstronomers, 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 theArithmetikeofAstronomicalorPhisicall Fractions. That, haue I briefly noted, by the name ofArithmetike Circular. Bycause it is also vsed in circles, notAstronomicall. &c.4.Practise hath ledNumbersfarder, and hath framed them, to take vpon them, the shew ofMagnitudespropertie: Which isIncommensurabilitieandIrrationalitie. (For in pureArithmetike, anVnit, is the common Measure of all Numbers.) And, here, Nũbers are become, as Lynes, Playnes and Solides: some tymesRationall, some tymesIrrationall. And haue propre and peculier characters, (as2√.3√. and so of other.AWhich is to signifieRote Square, Rote Cubik: and so forth:) & propre and peculier fashions in the fiue principall partes: Wherfore the practiser, estemeth this, a diuerseArithmetikefrom the other. Practise bryngeth in, here, diuerse compoundyng of Numbers: as some tyme, two, three, foure (or more)Radicallnũbers, diuersly knit, by signes, of More & Lesse: as thus2√12 +3√15. Or thus4√19 +3√12 -2√2. &c. And some tyme with whole numbers, or fractions of whole Number, amõg them: as 20 +2√24.3√16 + 33 -2√10.4√44 + 12¼ +3√9. And so, infinitely, may hap the varietie. After this: Both the one and the other hath fractions incident: and so is thisArithmetikegreately 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 thisPractise, the name of theArithmetike of Radicall numbers: Not, ofIrrationallorSurd Numbers: which other while, are Rationall: though they haue the Signe of a Rote beforethem, which,Arithmetikeof whole Numbers most vsuall, would say they had no such Roote: and so account themSurd Numbers: which, generally spokẽ, is vntrue: asEuclidestenth booke may teach you. Therfore to call them, generally,Radicall Numbers, (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) ofNumbersPracticall vse. Which thing, is well to be perceiued in that great Arithmeticall Arte ofÆquation: commonly called theRule of Coss.orAlgebra. The Latines termed it,Regulam Rei & Census, that is, theRule of the thyng and his value. 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,Res:A thing: vnleast they vse the name ofAlgebra. And therin (commonly) is a dubble error. The one, of them, which thinke it to be ofGeberhis inuentyng: the other of such as call itAlgebra. For, first, thoughGeberfor 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 ofGeberhis name: yet true it is, that aGrekePhilosopher and Mathematicien, namedDiophantus, beforeGeberhis tyme, wrote 13. bookes therof (of which, six are yet extant: and I had them to *vse,* Anno. 1550.of the famous Mathematicien, and my great frende,Petrus Montaureus:) And secondly, the very name, isAlgiebar, and notAlgebra: as by the ArabienAuicen, may be proued: who hath these precise wordes in Latine, byAndreas Alpagus(most perfect in the Arabik tung) so translated.Scientia faciendi Algiebar & Almachabel. i. Scientia inueniendi numerum ignotum, per additionem Numeri, & diuisionem & æquationem. Which is to say:The Science of workyng Algiebar and Almachabel, that is, theScience of findyng an vnknowen number, by Addyng of a Number, & Diuision & æquation. Here haue you the name: and also the principall partes of the Rule, touched. To name it,The rule, or Art of Æquation, doth signifie the middle part and the State of the Rule. This Rule, hath his peculier Characters:5.and the principal partes ofArithmetike, to it appertayning, do differre from the otherArithmeticall operations. ThisArithmetike, hath NũbersSimple, Cõpound, Mixt: and Fractions, accordingly. This Rule, andArithmetike of Algiebar, 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 ofArithmetike.And first: who, nerer at hand, can be a better witnesse of the frute receiued byArithmetike, 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 GoldenRule? Simple and Compounde: both forward and backward? How might they misseArithmeticallhelpe 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 meaneGemma Frisius, andSimon Iacob. 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 otherArithmeticall practises, are commonly in Marchantes handes, and knowledge: They them selues, can, at large, testifie.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, ifArithmetikebe their guide? And the honorable Phisiciãs, will gladly confesse them selues, much beholding to the Science ofArithmetike, and that sundry wayes: But chiefly in their Art of Graduation, and compounde Medicines. And thoughGalenus,Auerrois,Arnoldus,Lullus, and other haue published their positions, aswell in the quantities of the Degrees aboue Temperament, as in the Rules, concluding the newFormeresulting: yet a more precise, commodious, and easyMethod, is extant: by a Countreyman of oursR. B.(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.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.HoteandColde, one against the other. And likewiseMoystandDry, one against the other. And in the Circle writeTemperate. WhichTemperaturehath 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.Bdiagram: see end of text for alternative*Take some part of Lullus counsayle in his booke de Q. Essentia.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 aTemperate, and a Contrary,The forme resulting of their Mixture, is in the Middle betwene the degrees ofthe formes mixt. As for example, letA, beMoistin the first degree: andB,Dryin 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 fromAandB(for the*Note.*Temperamentis counted none. And for it, you must put a Ciphre, if at any time, it be in mixture). Counting then fromB, 2. degrees, towardA: you finde it to beDryin the first degree: So is theForme resultingof the Mixture ofA, andB, in our example. I will geue you an other example. Suppose, you haue two thinges, asC, andD: and ofC, the Heate to be in the 4. degree: and ofD, the Colde, to be remisse, euen vnto theTemperament. Now, forC, you take 4: and forD, you take a Ciphre: which, added vnto 4, yeldeth onely 4. The middle, or halfe, whereof, is 2. Wherefore theForme resultingofC, andD, is Hote in the second degree: for, 2. degrees, accounted fromC, towardD, ende iuste in the 2. degree of heate. Of the third maner, I will geue also an example: which let be this:Note.I haue a liquid Medicine whose Qualitie of heate is in the 4. degree exalted: as wasC, 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 theForme resulting, by this mixture ofC, andE. 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, theForme resulting, to beHeate, in the middle of the third degree.“The Second Rule.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,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. 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.”I would gladly know the Forme resulting, in the Mixture of these two Liquors. Set downe your nũbers in order, thus.diagram: see end of text for alternativeNow by the rule of Algiebar, haue I deuised a very easie, briefe, and generall maner of working in this case. Let vs first, suppose thatMiddle Forme resulting, to be 1X: 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 1X: to the difference betwene 1Xand 3: (the degree of the thing, in lesse quãtitie. And with all, 1X, being alwayes in a certaine middell, betwene the two heigthes or degrees). For the first difference, I set 4-1X: and for the second, I set 1X-3. And, now againe, I say, as 1. is to 2. so is 4-1Xto 1X-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 1X-3. and of the second & the third, 8-2X. Wherfore, our Æquation is betwene 1X-3: and 8-2X. Which may be reduced, according to the Arte of Algiebar: as, here, adding 3. to eche part, geueth the Æquation, thus, 1X=11-2X. And yet againe, contracting, or Reducing it: Adde to eche part, 2X: Then haue you 3Xæquall to 11: thus represented 3X=11. Wherefore, diuiding 11. by 3: the Quotient is 3⅔: theValewof our 1X,Coss, orThing, first supposed. And that is the heigth, or Intension of theForme resulting:which is,Heate, 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,diagram: see end of text for alternativeresteth 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.P. is double to 1.P. 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 2X, it would stand, 3X-3=8. And now adding to eche part 3: you haue (as afore) 3X=11.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 Compoundto be reduced: & the Forme resultyng of the same, to serue the turne) yet these Rules are sufficient: duely repeated and iterated.Note.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,Mixtio est miscibilium, alteratorum, per minima coniunctorum, Vnio. 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, concerningMixture of Formes and Qualities Naturall, is so Matcht and maryed with the most simple, easie, and short way of the noble Rule ofAlgiebar. Who can remaine, therfore vnpersuaded, to loue, alow, and honor the excellent Science ofArithmetike? For, here, you may perceiue that the litle finger ofArithmetike, 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.Now will we farder, by the wise and valiant Capitaine, be certified, what helpe he hath, by the Rules ofArithmetike: in one of the Artes to him appertaining: And of the Grekes namedΤακτικὴ.Τακτικὴ.“That is, the Skill of Ordring Souldiers in Battell ray after the best maner to all purposes.”This Art so much dependeth vppon Numbers vse, and the Mathematicals, thatÆlianus(the best writer therof,) in his worke, to theEmperour Hadrianus, 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 writtenÆneas:CyneasofThessaly:Pyrrhus Epirota: andAlexanderhis sonne:Clearchus:Pausanias:Euangelus:Polybius, familier frende toScipio:Eupolemus:Iphicrates,Possidonius: and very many other worthy Capitaines, Philosophers and Princes of Immortall fame and memory: Whose fayrest floure of their garland (in this feat) wasArithmetike: and a litle perceiuerance, inGeometricallFigures. But in many other cases dothArithmetikestand 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 ofArithmetike, 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 CurteousIohn, 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 honorablelife 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 ofFame Immortall.To our purpose. ThisIohn, 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:This noble Earle, dyed Anno. 1554. skarse of 24. yeares of age: hauing no issue by his wife: Daughter to the Duke of Somerset.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,Arithmetike, of him, was shryned in gold: OfNumbersfrute, he had good hope. Now, Numbers therfore innumerable, inNumbersprayse, his shryne shall finde.What nede I, (for farder profe to you) of the Scholemasters of Iustice, to require testimony: how nedefull, how frutefull, how skillfull a thingArithmetikeis? I meane, the Lawyers of all sortes. Vndoubtedly, the Ciuilians, can meruaylously declare: how, neither the Auncient Romaine lawes, without good knowledge ofNumbers art, 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, didPapinianusinstitute a law of partition, and allowance, betwene man and wife after a diuorce? But howAccursius,Baldus,Bartolus,Iason,Alexander, and finallyAlciatus, (being otherwise, notably well learned) do iumble, gesse, and erre, from the æquity, art and Intent of the lawmaker:Arithmetikecan detect, and conuince: and clerely, make the truth to shine. GoodBartolus, tyred in the examining & proportioning of the matter: and withAccursiusGlosse, much cumbred: burst out, and sayd:Nulla est in toto libro, hac glossa difficilior: Cuius computationem nec Scholastici nec Doctores intelligunt. &c.That is:In the whole booke, there is no Glosse harder then this: Whose accoumpt or reckenyng, neither the Scholers, nor the Doctours vnderstand. &c.What can they say ofIulianuslaw,Si ita Scriptum. &c.Of the Testators will iustly performing, betwene the wife, Sonne and daughter? How can they perceiue the æquitie ofAphricanus,ArithmeticallReckening, where he treateth ofLex Falcidia? How can they deliuer him, from his Reprouers: and their maintainers: asIoannes,Accursius HypolitusandAlciatus? How Iustly and artificially, wasAfricanusreckening 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.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 expertArithmeticien, 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 bookesDe Republica: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,Iustice.“to be that vertue, by which, to euery one, is rendred, that to him appertaineth.”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.Aristotlein hisEthikes(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.Platoin his booke calledEpinomis(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,Wise. He, briefly, of other Sciences discoursing, findeth them, not hable to bring it to passe: But of the Science of Numbers, he sayth.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?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?There, at length, he prouethWisedometo be atteyned, by good Skill ofNumbers. With which great Testimony, and the manifold profes, and reasons, before expressed, you may be sufficiently and fully persuaded: of the perfect Science ofArithmetike, to make this accounte: Thatof all Sciences, next toTheologie, 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 ofMagnitudes: of which (by the Direction and aide of him, whoseMagnitudeis Infinite, and of vs Incomprehensible) I now entend, so to write, that both with theMultitude, and also with theMagnitudeof 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 sundryPhilosophersandMathematiciens.Both,NumberandMagnitude, haue a certaine Originall sede, (as it were) of an incredible property: and of man, neuer hable, Fully, to be declared. OfNumber, an Vnit, and ofMagnitude, a Poynte, doo seeme to be much like Originallcauses: But the diuersitie neuerthelesse, is great. We defined anVnit, 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 anVnit, 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: AnVnit, 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 anVnit, yet doth it Consist of vnits, as a materiall cause. But formally,Number.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?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?Geometrie.This Science ofMagnitude, his properties, conditions, and appertenances: commonly, now is, and from the beginnyng, hath of all Philosophers, ben calledGeometrie. 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 inEgypt, yearely, with the ouerflowyng ofNilus, 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 (bypretence 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 ofGeometria, 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, plaineGeometrie. And though, these Philosophers, knew of farder vse, and best vnderstode the etymologye of the worde, yet this nameGeometria, was of them applyed generally to all sortes of Magnitudes: vnleast, otherwhile, ofPlato, andPythagoras: When they would precisely declare their owne doctrine. Then, was*Plato. 7. de Rep.*Geometria, with them,Studium quod circa planum versatur. But, well you may perceiue byEuclides Elementes, 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 absoluteMegethologia: not creping on ground, and dasseling the eye, with pole perche, rod or lyne: but“liftyng the hart aboue the heauens, by inuisible lines, andimmortall beames meteth with the reflexions, of the light incomprehensible: and so procureth Ioye, and perfection vnspeakable.”Of which true vse of ourMegethica, orMegethologia,Diuine Platoseemed to haue good taste, and iudgement: and (by the name ofGeometrie) so noted it: and warned his Scholers therof: as, in hys seuenthDialog, of the Common wealth, may euidently be sene. Where (in Latin) thus it is: right well translated:Profecto, nobis hoc non negabunt, Quicunquevel paululum quid Geometriæ gustârunt, quin hæc Scientia, contrà, omnino se habeat, quàm de ea loquuntur, qui in ipsa versantur.In English, thus.Verely(saythPlato)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.And there it followeth, of ourGeometrie,Quòd quæritur cognoscendi illius gratia, quod semper est, non & eius quod oritur quandoque& interit. Geometria, eius quod est semper, Cognitio est. Attollet igitur (ô Generose vir) ad Veritatem, animum: atqueita, 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.It must nedes be confessed (saithPlato)That[Geometrie]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[both mynde and thought]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[done by it]which, in manner, seame to be, beside the purpose of Geometrie: are ofno small importance. &c.And besides the manifold vses ofGeometrie, in matters appertainyng to warre, he addeth more, of second vnpurposed frute, and commoditye, arrising byGeometrie: saying: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.That is.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.This wasDiuine Platohis Iudgement, both of the purposed, chief, and perfect vse ofGeometrie: 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*I. D.* Herein, I would gladly shake of, the earthly name, of Geometrie.MegethologicallContemplations: 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 AbsoluteTheologie) will thereon depend.“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.”From henceforth, in this my Preface, will I frame my talke, toPlatohis 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 ofNature. Wherby, such Sedes, and Rotes, as lye depe hyd in the groũd ofNature, 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 DefineAn Arte.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.And that I account,Art Mathematicall Deriuatiue.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.And for that,I entend to vse the name and propertie of aA Mechanitien.Mechanicien, 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.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.Full well I know, that he which inuenteth, or maketh these demonstrations, is generally calledA speculatiue Mechanicien: which differreth nothyng from aMechanicall Mathematicien. 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.1.First, then, from the puritie, absolutenes, and Immaterialitie of PrincipallGeometrie, is that kinde ofGeometriederiued, which vulgarly is countedGeometrie: and is theArte of Measuring sensible magnitudes, their iust quãtities and contentes.Geometrie vulgar.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,1.*to be certified, either of the length, perimetry, or distance lineall: and this is called,Mecometrie. Or2.*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 namedEmbadometrie.3.*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 calledGaging. And the generall name of these Solide measures, isStereometrie.2.Or els, thisvulgar Geometrie, hath consideration to teach the practiser, how to measure things, with good distance betwene him and the thing measured: and to vnderstand thereby, either1.*how Farre, a thing seene (on land or water) is from the measurer: and this may be calledApomecometrie:2.Or, how High or depe, aboue or vnder the leuel of the measurers stãding, any thing is, which is sene on land or water, calledHypsometrie.3.*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 calledPlatometrie. Though I vse here to condition, the thing measured, to be on Land, or Water Situated:Note.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 inPerspectiueandAstronomie, &c.OF these Feates (farther applied) is Sprong the Feate ofGeodesie, 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)Note.how great wrong and iniurie hath (in my time) bene committedby 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 ofGeographie,Chorographie,Hydrographie, andStratarithmetrie.“Geographieteacheth wayes, by which, in sũdry formes, (asSphærike,Plaineor 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.”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.Chorographieseemeth to be an vnderling, and a twig, ofGeographie: and yet neuerthelesse, is in practise manifolde, and in vse very ample.“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.”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, theArchitectmay 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,Topographie.“Hydrographie, 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:”with the Iles and chiefeparticular 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: asGeographiedid principally take the Element of the Earthes description (with his appertenances) to taske. And besides thys,Hydrographie, 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, theHydrographerought to recorde. The Soundinges likewise: and the Chanels wayes: their number, and depthes ordinarily, at ebbe and flud, ought theHydrographer, by obseruation and diligence ofMeasuring, to haue certainly knowen. And many other pointes, are belonging to perfecteHydrographie, and for to make aRutter, 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 anHydrographer, 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.Stratarithmetrie, is the Skill, (appertainyng to the warre,) by which a man can set in figure, analogicall to anyGeometricallfigure 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.*Note.*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 ofStratarithmetrie: sufficient for this tyme and place.The difference betwene Stratarithmetrie and Tacticie.“It differreth from the FeateTacticall,De aciebus instruendis.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.”By figure, I meane: as, either of aPerfect Square,Triangle,Circle,Ouale,long square, (of the Grekes it is calledEteromekes)Rhombe,Rhomboïd,Lunular,Ryng,Serpentine, 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 woulddo it.I. D.Frende,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.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 OrderingTacticall, 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.
T(There)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 perfectlyto 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 PhilosopherEuclideofMegara. 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 ofEuclidesElementes, 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 ofEuclide. 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 shallwell 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)and also pertaining to the Mathematicall Artes.Thus gentle reader farewell.
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D(Divine)IuinePlato, the great Master of many worthy Philosophers, and the constant auoucher, and pithy perswader ofVnum,Bonum, andEns: in his Schole and Academie, sundry times (besides his ordinary Scholers) was visited of a certaine kinde of men, allured by the noble fame ofPlato, 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, thisVnum,Bonum, andEns, to be Spirituall, Infinite, Æternall, Omnipotent, &c. Nothyng beyng alledged or expressed, How, worldly goods: how,worldlydignitie: 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 ofPlato, was clene chaunged: yea his doctrine was by them despised: and his schole, no more of them visited. Which thing, his Scholer,Aristotle, narrowly cõsidering, founde the cause therof, to be,“For that they had no forwarnyng and information, in generall,”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,Aristotle, euer, after that, vsed in brief, to forewarne his owne Scholers and hearers,“both of what matter, and also to what ende, he tooke in hand to speake, or teach.”While I consider the diuerse trades of these two excellent Philosophers (and am most sure, both, thatPlatoright well, otherwise could teach: and thatAristotlemought boldely, with his hearers, haue dealt in like sorte asPlatodid) I am in no little pang of perplexitie: Bycause, that, which I mislike, is most easy for me to performe (and to hauePlatofor my exãple.) And that, which I know to be most commendable: and (in this first bringyng, into common handling, theArtes Mathematicall) 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 (ImitatyngAristotle) well can I hope, that accordyng to the amplenes and dignitie of theState Mathematicall, 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 fromPlatohis schole, after they had perceiued his finallconclusion, would in these thinges haue ben his most diligenthearers (soinfinitely mought their desires, in fine and at length, by ourArtes Mathematicallbe satisfied) yet, by this my Præface & forewarnyng, Aswell all such, may (to their great behofe) the soner, hither be allured: as also thePythagoricall, andPlatonicallperfect scholer, and the constant profound Philosopher, with more ease and spede, may (like the Bee,) gather, hereby, both wax and hony.
Wherfore, seyng I finde great occasion (for the causes alleged, and farder, in respect of myArt Mathematike generall) to vse“a certaine forewarnyng and Præface, whose content shalbe,The intent of this Preface.that mighty, most plesaunt, and frutefullMathematicall Tree, 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:”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.
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 calledThynges Mathematicall. 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 generallFormes, 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 ofDianœticalldiscourse, in perfect imagination subsistyng. A meruaylous newtralitie haue these thingesMathematicall, 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:is allowed as sufficient for“an Argument exactly and purely Mathematical.”
OfMathematicallthinges, are two principall kindes: namely,Number, andMagnitude.Number.Number, we define, to be, a certayne Mathematicall Sũme, ofVnits.Note the worde, Vnit, to expresse the Greke Monas, & not Vnitie: as we haue all, commonly, till now, vsed.And, anVnit, 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 anVnit, a thingMathematicall, though it be no Number, and also indiuisible: because, of it, materially, Number doth consist: which, principally, is a thingMathematicall.Magnitude.Magnitudeis a thingMathematicall, by participation of some likenes of whose nature, any thing is iudged long, broade, or thicke.“A thickeMagnitudewe call aSolide, or aBody. WhatMagnitudeso euer, is Solide or Thicke, is also broade, & long. A broade magnitude, we call aSuperficiesor a Plaine. Euery playne magnitude, hath also length. A long magnitude, we terme aLine. ALineis neither thicke nor broade, but onely long: Euery certayne Line, hath two endes:A point.The endes of a line, arePointescalled. A Point, is a thingMathematicall, indiuisible, which may haue a certayne determined situation.”If a Poynt moue from a determined situation, the way wherein it moued, is also aLine: mathematically produced, whereupon, of the auncient Mathematiciens,A Line.aLineis called the race or course of aPoint. 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 trueMagnitude.Magnitude.AndMagnitudewe may define to be that thingMathematicall, which is diuisible for euer, in partes diuisible, long, broade or thicke. Therefore though a Poynt be noMagnitude, yetTerminatiuely, we recken it a thingMathematicall(as I sayd) by reason it is properly the end, and bound of a line. NeitherNumber, norMagnitude, haue any Materialitie. First, we will consider ofNumber, and of the ScienceMathematicall, to it appropriate, calledArithmetike: and afterward ofMagnitude, and his Science, calledGeometrie. But that name contenteth me not: whereof a word or two hereafter shall be sayd. How Immateriall and free from all matter,Numberis, who doth not perceaue? yea, who doth not wonderfully wõder at it? For, neither pureElement, norAristoteles, Quinta Essentia, 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 PhilosopherAnitius Boetius, sayd:Omnia quæcunquea primæua rerum natura constructa sunt, Numerorum videntur ratione formata. Hoc enim fuit principale in animo Conditoris Exemplar. That is: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.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, fromNothing, to theFormalitieof their being and state. ByNumberspropertie 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, andFormes: And also, farder, arise, clime, ascend, and mount vp (with Speculatiue winges) in spirit, to behold in the Glas of Creation, theForme of Formes, theExemplar Numberof all thingesNumerable: both visible and inuisible, mortall andimmortall, Corporall and Spirituall. Part of this profound and diuine Science, hadIoachimthe Prophesier atteyned vnto: byNumbers Formall, Naturall, andRationall, forseyng, concludyng, and forshewyng great particular euents, long before their comming. His bookes yet remainyng, hereof, are good profe: And the noble Earle ofMirandula, (besides that,) a sufficient witnesse: thatIoachim, in his prophesies, proceded by no other way, then by Numbers Formall. And this Earle hym selfe, in Rome,Ano. 1488.*set vp 900. Conclusions, in all kinde of Sciences, openly to be disputed of: and among the rest, in his ConclusionsMathematicall, (in the eleuenth Conclusion) hath in Latin, this English sentence.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. Which Cõclusions, I omit here to rehearse: aswell auoidyng superfluous prolixitie: as, bycauseIoannes Picus, workes, 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.
Of my former wordes, easy it is to be gathered, thatNumberhath 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,Number, is termedNumber Numbryng. But in all Creatures, otherwise,Number, is termedNũber Numbred. And in our Soule, Nũber beareth such a swaye, and hath such an affinitie therwith: that some of the oldPhilosopherstaught,Mans Soule, to be a Number mouyng it selfe. 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.Number Numbryngtherfore, 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 hisNumbryng, then, was his Creatyng of all thinges. And his ContinuallNumbryng, of all thinges, is the Conseruation of them in being: And, where and when he will lacke anVnit: there and then, that particular thyng shalbeDiscreated. Here I stay. But our Seuerallyng, distinctyng, andNumbryng, 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 AlmightyTernarie,) Artificiall Methods and easy wayes are made, by which the zelous Philosopher, may wyn nere this RiuerishIda, this Mountayne of Contemplation: and more then Contemplation. And also, thoughNumber, 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 aTernarie. Three Egles, are three, or aTernarie.Which*Ternaries, are eche, theVnion,knot, andVniformitie, of three discrete and distinctVnits. That is, we may in echeTernarie, thrise, seuerally pointe, and shew a part,One,One, andOne. Where, in Numbryng, we say One, two,Three. But how farre, these visible Ones, do differre from our Indiuisible Vnits (in pureArithmetike, 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 ofNumbers, absolutely (:Not supposing, nor admixtyng any thyng created, Corporall or Spirituall, to support, conteyne, or represent thoseNumbersimagined:) that at length, we may be hable, to finde the number of our owne name, gloriously exemplified and registred in the booke of theTrinitiemost blessed and æternall.
But farder vnderstand, that vulgar Practisers, haue Numbers, otherwise, in sundry Considerations: and extend their name farder, then to Numbers, whose least part is anVnit. For the common Logist, Reckenmaster, or Arithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lesse partes: and calleth themFractions. As of anVnit, he maketh an halfe, and thus noteth it, ½. and so of other, (infinitely diuerse) partes of anVnit. Yea and farder, hath,Fractions of Fractions. &c. And, forasmuch, as,Addition,Substraction,Multiplication,DiuisionandExtraction of Rotes, are the chief, and sufficient partes ofArithmetike:Arithmetike.which is, theScience that demonstrateth the properties, of Numbers, and all operatiõs, in numbers to be performed:Note.“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 ofArithmetike.”And by this reason:1.the Consideration, doctrine, and working, in whole numbers onely: where, of anVnit, is no lesse part to be allowed: is named (as it were) anArithmetikeby it selfe. And so of theArithmetike of Fractions.2.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 ofArithmetike.3.Moreouer, theAstronomers, 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 theArithmetikeofAstronomicalorPhisicall Fractions. That, haue I briefly noted, by the name ofArithmetike Circular. Bycause it is also vsed in circles, notAstronomicall. &c.4.Practise hath ledNumbersfarder, and hath framed them, to take vpon them, the shew ofMagnitudespropertie: Which isIncommensurabilitieandIrrationalitie. (For in pureArithmetike, anVnit, is the common Measure of all Numbers.) And, here, Nũbers are become, as Lynes, Playnes and Solides: some tymesRationall, some tymesIrrationall. And haue propre and peculier characters, (as2√.3√. and so of other.AWhich is to signifieRote Square, Rote Cubik: and so forth:) & propre and peculier fashions in the fiue principall partes: Wherfore the practiser, estemeth this, a diuerseArithmetikefrom the other. Practise bryngeth in, here, diuerse compoundyng of Numbers: as some tyme, two, three, foure (or more)Radicallnũbers, diuersly knit, by signes, of More & Lesse: as thus2√12 +3√15. Or thus4√19 +3√12 -2√2. &c. And some tyme with whole numbers, or fractions of whole Number, amõg them: as 20 +2√24.3√16 + 33 -2√10.4√44 + 12¼ +3√9. And so, infinitely, may hap the varietie. After this: Both the one and the other hath fractions incident: and so is thisArithmetikegreately 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 thisPractise, the name of theArithmetike of Radicall numbers: Not, ofIrrationallorSurd Numbers: which other while, are Rationall: though they haue the Signe of a Rote beforethem, which,Arithmetikeof whole Numbers most vsuall, would say they had no such Roote: and so account themSurd Numbers: which, generally spokẽ, is vntrue: asEuclidestenth booke may teach you. Therfore to call them, generally,Radicall Numbers, (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) ofNumbersPracticall vse. Which thing, is well to be perceiued in that great Arithmeticall Arte ofÆquation: commonly called theRule of Coss.orAlgebra. The Latines termed it,Regulam Rei & Census, that is, theRule of the thyng and his value. 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,Res:A thing: vnleast they vse the name ofAlgebra. And therin (commonly) is a dubble error. The one, of them, which thinke it to be ofGeberhis inuentyng: the other of such as call itAlgebra. For, first, thoughGeberfor 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 ofGeberhis name: yet true it is, that aGrekePhilosopher and Mathematicien, namedDiophantus, beforeGeberhis tyme, wrote 13. bookes therof (of which, six are yet extant: and I had them to *vse,* Anno. 1550.of the famous Mathematicien, and my great frende,Petrus Montaureus:) And secondly, the very name, isAlgiebar, and notAlgebra: as by the ArabienAuicen, may be proued: who hath these precise wordes in Latine, byAndreas Alpagus(most perfect in the Arabik tung) so translated.Scientia faciendi Algiebar & Almachabel. i. Scientia inueniendi numerum ignotum, per additionem Numeri, & diuisionem & æquationem. Which is to say:The Science of workyng Algiebar and Almachabel, that is, theScience of findyng an vnknowen number, by Addyng of a Number, & Diuision & æquation. Here haue you the name: and also the principall partes of the Rule, touched. To name it,The rule, or Art of Æquation, doth signifie the middle part and the State of the Rule. This Rule, hath his peculier Characters:5.and the principal partes ofArithmetike, to it appertayning, do differre from the otherArithmeticall operations. ThisArithmetike, hath NũbersSimple, Cõpound, Mixt: and Fractions, accordingly. This Rule, andArithmetike of Algiebar, 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 ofArithmetike.
And first: who, nerer at hand, can be a better witnesse of the frute receiued byArithmetike, 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 GoldenRule? Simple and Compounde: both forward and backward? How might they misseArithmeticallhelpe 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 meaneGemma Frisius, andSimon Iacob. 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 otherArithmeticall practises, are commonly in Marchantes handes, and knowledge: They them selues, can, at large, testifie.
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, ifArithmetikebe their guide? And the honorable Phisiciãs, will gladly confesse them selues, much beholding to the Science ofArithmetike, and that sundry wayes: But chiefly in their Art of Graduation, and compounde Medicines. And thoughGalenus,Auerrois,Arnoldus,Lullus, and other haue published their positions, aswell in the quantities of the Degrees aboue Temperament, as in the Rules, concluding the newFormeresulting: yet a more precise, commodious, and easyMethod, is extant: by a Countreyman of oursR. B.(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.
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.HoteandColde, one against the other. And likewiseMoystandDry, one against the other. And in the Circle writeTemperate. WhichTemperaturehath 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.B
diagram: see end of text for alternative
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 aTemperate, and a Contrary,The forme resulting of their Mixture, is in the Middle betwene the degrees ofthe formes mixt. As for example, letA, beMoistin the first degree: andB,Dryin 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 fromAandB(for the*Note.*Temperamentis counted none. And for it, you must put a Ciphre, if at any time, it be in mixture). Counting then fromB, 2. degrees, towardA: you finde it to beDryin the first degree: So is theForme resultingof the Mixture ofA, andB, in our example. I will geue you an other example. Suppose, you haue two thinges, asC, andD: and ofC, the Heate to be in the 4. degree: and ofD, the Colde, to be remisse, euen vnto theTemperament. Now, forC, you take 4: and forD, you take a Ciphre: which, added vnto 4, yeldeth onely 4. The middle, or halfe, whereof, is 2. Wherefore theForme resultingofC, andD, is Hote in the second degree: for, 2. degrees, accounted fromC, towardD, ende iuste in the 2. degree of heate. Of the third maner, I will geue also an example: which let be this:Note.I haue a liquid Medicine whose Qualitie of heate is in the 4. degree exalted: as wasC, 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 theForme resulting, by this mixture ofC, andE. 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, theForme resulting, to beHeate, in the middle of the third degree.
“The Second Rule.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,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. 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.”I would gladly know the Forme resulting, in the Mixture of these two Liquors. Set downe your nũbers in order, thus.diagram: see end of text for alternativeNow by the rule of Algiebar, haue I deuised a very easie, briefe, and generall maner of working in this case. Let vs first, suppose thatMiddle Forme resulting, to be 1X: 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 1X: to the difference betwene 1Xand 3: (the degree of the thing, in lesse quãtitie. And with all, 1X, being alwayes in a certaine middell, betwene the two heigthes or degrees). For the first difference, I set 4-1X: and for the second, I set 1X-3. And, now againe, I say, as 1. is to 2. so is 4-1Xto 1X-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 1X-3. and of the second & the third, 8-2X. Wherfore, our Æquation is betwene 1X-3: and 8-2X. Which may be reduced, according to the Arte of Algiebar: as, here, adding 3. to eche part, geueth the Æquation, thus, 1X=11-2X. And yet againe, contracting, or Reducing it: Adde to eche part, 2X: Then haue you 3Xæquall to 11: thus represented 3X=11. Wherefore, diuiding 11. by 3: the Quotient is 3⅔: theValewof our 1X,Coss, orThing, first supposed. And that is the heigth, or Intension of theForme resulting:which is,Heate, 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,diagram: see end of text for alternativeresteth 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.P. is double to 1.P. 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 2X, it would stand, 3X-3=8. And now adding to eche part 3: you haue (as afore) 3X=11.
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 Compoundto be reduced: & the Forme resultyng of the same, to serue the turne) yet these Rules are sufficient: duely repeated and iterated.Note.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,Mixtio est miscibilium, alteratorum, per minima coniunctorum, Vnio. 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, concerningMixture of Formes and Qualities Naturall, is so Matcht and maryed with the most simple, easie, and short way of the noble Rule ofAlgiebar. Who can remaine, therfore vnpersuaded, to loue, alow, and honor the excellent Science ofArithmetike? For, here, you may perceiue that the litle finger ofArithmetike, 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.
Now will we farder, by the wise and valiant Capitaine, be certified, what helpe he hath, by the Rules ofArithmetike: in one of the Artes to him appertaining: And of the Grekes namedΤακτικὴ.Τακτικὴ.“That is, the Skill of Ordring Souldiers in Battell ray after the best maner to all purposes.”This Art so much dependeth vppon Numbers vse, and the Mathematicals, thatÆlianus(the best writer therof,) in his worke, to theEmperour Hadrianus, 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 writtenÆneas:CyneasofThessaly:Pyrrhus Epirota: andAlexanderhis sonne:Clearchus:Pausanias:Euangelus:Polybius, familier frende toScipio:Eupolemus:Iphicrates,Possidonius: and very many other worthy Capitaines, Philosophers and Princes of Immortall fame and memory: Whose fayrest floure of their garland (in this feat) wasArithmetike: and a litle perceiuerance, inGeometricallFigures. But in many other cases dothArithmetikestand 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 ofArithmetike, 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 CurteousIohn, 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 honorablelife 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 ofFame Immortall.
To our purpose. ThisIohn, 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:This noble Earle, dyed Anno. 1554. skarse of 24. yeares of age: hauing no issue by his wife: Daughter to the Duke of Somerset.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,Arithmetike, of him, was shryned in gold: OfNumbersfrute, he had good hope. Now, Numbers therfore innumerable, inNumbersprayse, his shryne shall finde.
What nede I, (for farder profe to you) of the Scholemasters of Iustice, to require testimony: how nedefull, how frutefull, how skillfull a thingArithmetikeis? I meane, the Lawyers of all sortes. Vndoubtedly, the Ciuilians, can meruaylously declare: how, neither the Auncient Romaine lawes, without good knowledge ofNumbers art, 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, didPapinianusinstitute a law of partition, and allowance, betwene man and wife after a diuorce? But howAccursius,Baldus,Bartolus,Iason,Alexander, and finallyAlciatus, (being otherwise, notably well learned) do iumble, gesse, and erre, from the æquity, art and Intent of the lawmaker:Arithmetikecan detect, and conuince: and clerely, make the truth to shine. GoodBartolus, tyred in the examining & proportioning of the matter: and withAccursiusGlosse, much cumbred: burst out, and sayd:Nulla est in toto libro, hac glossa difficilior: Cuius computationem nec Scholastici nec Doctores intelligunt. &c.That is:In the whole booke, there is no Glosse harder then this: Whose accoumpt or reckenyng, neither the Scholers, nor the Doctours vnderstand. &c.What can they say ofIulianuslaw,Si ita Scriptum. &c.Of the Testators will iustly performing, betwene the wife, Sonne and daughter? How can they perceiue the æquitie ofAphricanus,ArithmeticallReckening, where he treateth ofLex Falcidia? How can they deliuer him, from his Reprouers: and their maintainers: asIoannes,Accursius HypolitusandAlciatus? How Iustly and artificially, wasAfricanusreckening 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.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 expertArithmeticien, 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 bookesDe Republica: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,Iustice.“to be that vertue, by which, to euery one, is rendred, that to him appertaineth.”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.Aristotlein hisEthikes(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.Platoin his booke calledEpinomis(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,Wise. He, briefly, of other Sciences discoursing, findeth them, not hable to bring it to passe: But of the Science of Numbers, he sayth.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?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?There, at length, he prouethWisedometo be atteyned, by good Skill ofNumbers. With which great Testimony, and the manifold profes, and reasons, before expressed, you may be sufficiently and fully persuaded: of the perfect Science ofArithmetike, to make this accounte: Thatof all Sciences, next toTheologie, 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 ofMagnitudes: of which (by the Direction and aide of him, whoseMagnitudeis Infinite, and of vs Incomprehensible) I now entend, so to write, that both with theMultitude, and also with theMagnitudeof 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 sundryPhilosophersandMathematiciens.
Both,NumberandMagnitude, haue a certaine Originall sede, (as it were) of an incredible property: and of man, neuer hable, Fully, to be declared. OfNumber, an Vnit, and ofMagnitude, a Poynte, doo seeme to be much like Originallcauses: But the diuersitie neuerthelesse, is great. We defined anVnit, 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 anVnit, 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: AnVnit, 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 anVnit, yet doth it Consist of vnits, as a materiall cause. But formally,Number.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?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?Geometrie.This Science ofMagnitude, his properties, conditions, and appertenances: commonly, now is, and from the beginnyng, hath of all Philosophers, ben calledGeometrie. 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 inEgypt, yearely, with the ouerflowyng ofNilus, 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 (bypretence 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 ofGeometria, 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, plaineGeometrie. And though, these Philosophers, knew of farder vse, and best vnderstode the etymologye of the worde, yet this nameGeometria, was of them applyed generally to all sortes of Magnitudes: vnleast, otherwhile, ofPlato, andPythagoras: When they would precisely declare their owne doctrine. Then, was*Plato. 7. de Rep.*Geometria, with them,Studium quod circa planum versatur. But, well you may perceiue byEuclides Elementes, 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 absoluteMegethologia: not creping on ground, and dasseling the eye, with pole perche, rod or lyne: but“liftyng the hart aboue the heauens, by inuisible lines, andimmortall beames meteth with the reflexions, of the light incomprehensible: and so procureth Ioye, and perfection vnspeakable.”Of which true vse of ourMegethica, orMegethologia,Diuine Platoseemed to haue good taste, and iudgement: and (by the name ofGeometrie) so noted it: and warned his Scholers therof: as, in hys seuenthDialog, of the Common wealth, may euidently be sene. Where (in Latin) thus it is: right well translated:Profecto, nobis hoc non negabunt, Quicunquevel paululum quid Geometriæ gustârunt, quin hæc Scientia, contrà, omnino se habeat, quàm de ea loquuntur, qui in ipsa versantur.In English, thus.Verely(saythPlato)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.And there it followeth, of ourGeometrie,Quòd quæritur cognoscendi illius gratia, quod semper est, non & eius quod oritur quandoque& interit. Geometria, eius quod est semper, Cognitio est. Attollet igitur (ô Generose vir) ad Veritatem, animum: atqueita, 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.It must nedes be confessed (saithPlato)That[Geometrie]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[both mynde and thought]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[done by it]which, in manner, seame to be, beside the purpose of Geometrie: are ofno small importance. &c.And besides the manifold vses ofGeometrie, in matters appertainyng to warre, he addeth more, of second vnpurposed frute, and commoditye, arrising byGeometrie: saying: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.That is.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.This wasDiuine Platohis Iudgement, both of the purposed, chief, and perfect vse ofGeometrie: 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*I. D.* Herein, I would gladly shake of, the earthly name, of Geometrie.MegethologicallContemplations: 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 AbsoluteTheologie) will thereon depend.“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.”From henceforth, in this my Preface, will I frame my talke, toPlatohis 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 ofNature. Wherby, such Sedes, and Rotes, as lye depe hyd in the groũd ofNature, 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 DefineAn Arte.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.And that I account,Art Mathematicall Deriuatiue.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.And for that,I entend to vse the name and propertie of aA Mechanitien.Mechanicien, 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.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.Full well I know, that he which inuenteth, or maketh these demonstrations, is generally calledA speculatiue Mechanicien: which differreth nothyng from aMechanicall Mathematicien. 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.
1.First, then, from the puritie, absolutenes, and Immaterialitie of PrincipallGeometrie, is that kinde ofGeometriederiued, which vulgarly is countedGeometrie: and is theArte of Measuring sensible magnitudes, their iust quãtities and contentes.Geometrie vulgar.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,1.*to be certified, either of the length, perimetry, or distance lineall: and this is called,Mecometrie. Or2.*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 namedEmbadometrie.3.*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 calledGaging. And the generall name of these Solide measures, isStereometrie.2.Or els, thisvulgar Geometrie, hath consideration to teach the practiser, how to measure things, with good distance betwene him and the thing measured: and to vnderstand thereby, either1.*how Farre, a thing seene (on land or water) is from the measurer: and this may be calledApomecometrie:2.Or, how High or depe, aboue or vnder the leuel of the measurers stãding, any thing is, which is sene on land or water, calledHypsometrie.3.*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 calledPlatometrie. Though I vse here to condition, the thing measured, to be on Land, or Water Situated:Note.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 inPerspectiueandAstronomie, &c.
OF these Feates (farther applied) is Sprong the Feate ofGeodesie, 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)Note.how great wrong and iniurie hath (in my time) bene committedby 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 ofGeographie,Chorographie,Hydrographie, andStratarithmetrie.
“Geographieteacheth wayes, by which, in sũdry formes, (asSphærike,Plaineor 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.”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.
Chorographieseemeth to be an vnderling, and a twig, ofGeographie: and yet neuerthelesse, is in practise manifolde, and in vse very ample.“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.”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, theArchitectmay 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,Topographie.
“Hydrographie, 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:”with the Iles and chiefeparticular 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: asGeographiedid principally take the Element of the Earthes description (with his appertenances) to taske. And besides thys,Hydrographie, 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, theHydrographerought to recorde. The Soundinges likewise: and the Chanels wayes: their number, and depthes ordinarily, at ebbe and flud, ought theHydrographer, by obseruation and diligence ofMeasuring, to haue certainly knowen. And many other pointes, are belonging to perfecteHydrographie, and for to make aRutter, 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 anHydrographer, 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.
Stratarithmetrie, is the Skill, (appertainyng to the warre,) by which a man can set in figure, analogicall to anyGeometricallfigure 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.*Note.*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 ofStratarithmetrie: sufficient for this tyme and place.The difference betwene Stratarithmetrie and Tacticie.“It differreth from the FeateTacticall,De aciebus instruendis.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.”By figure, I meane: as, either of aPerfect Square,Triangle,Circle,Ouale,long square, (of the Grekes it is calledEteromekes)Rhombe,Rhomboïd,Lunular,Ryng,Serpentine, 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 woulddo it.I. D.Frende,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.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 OrderingTacticall, 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.