Chapter I: Part 1

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* * * * *
* * * *
* * * * *

THE ELEMENTS
OF GEOMETRIE

of the most auncient
Philosopher
_EVCLIDE_
of Megara.

_Faithfully (now first) translated
into the Englishe toung, by
_H. Billingsley_, Citizen of London_.

_Whereunto are annexed certaine
Scholies, Annotations, and Inuentions,
of the best Mathematiciens,
both of time past, and
in this our age._

_With a very fruitfull Præface made by _M. I. Dee_,
specifying the chiefe Mathematicall Sciẽces, what they are,
and wherunto commodious: where, also, are disclosed
certaine new Secrets Mathematicall and Mechanicall,
vntill these our daies, greatly missed._

Imprinted at London by _Iohn Daye_.

❧ The Translator to the Reader.

_There is (gentle Reader) nothing (the word of God onely set apart) which so much beautifieth and adorneth the soule and minde of mã, as doth the knowledge of good artes and sciences: as the knowledge of naturall and morall Philosophie. The one setteth before our eyes, the creatures of God, both in the heauens aboue, and in the earth beneath: in which as in a glasse, we beholde the exceding maiestie and wisedome of God, in adorning and beautifying them as we see: in geuing vnto them such wonderfull and manifolde proprieties, and naturall workinges, and that so diuersly and in such varietie: farther in maintaining and conseruing them continually, whereby to praise and adore him, as by S. Paule we are taught. The other teacheth vs rules and preceptes of vertue, how, in common life amongest men, we ought to walke vprightly: what dueties pertaine to our selues, what pertaine to the gouernment or good order both of an housholde, and also of a citie or common wealth. The reading likewise of histories, conduceth not a litle, to the adorning of the soule & minde of man, a studie of all men cõmended: by it are seene and knowen the artes and doinges of infinite wise men gone before vs. In histories are contained infinite examples of heroicall vertues to be of vs followed, and horrible examples of vices to be of vs eschewed. Many other artes also there are which beautifie the minde of man: but of all other none do more garnishe & beautifie it, then those artes which are called Mathematicall. Unto the knowledge of which no man can attaine, without the perfecte knowledge and instruction of the principles, groundes, and Elementes of Geometrie. But perfectly to be instructed in them, requireth diligent studie and reading of olde auncient authors. Amongest which, none for a beginner is to be preferred before the most auncient Philosopher _Euclide_ of _Megara_. For of all others he hath in a true methode and iuste order, gathered together whatsoeuer any before him had of these Elementes written: inuenting also and adding many thinges of his owne: wherby he hath in due forme accomplished the arte: first geuing definitions, principles, & groundes, wherof he deduceth his Propositions or conclusions, in such wonderfull wise, that that which goeth before, is of necessitie required to the proufe of that which followeth. So that without the diligent studie of _Euclides_ Elementes, it is impossible to attaine vnto the perfecte knowledge of Geometrie, and consequently of any of the other Mathematicall sciences. Wherefore considering the want & lacke of such good authors hitherto in our Englishe tounge, lamenting also the negligence, and lacke of zeale to their countrey in those of our nation, to whom God hath geuen both knowledge, & also abilitie to translate into our tounge, and to publishe abroad such good authors, and bookes (the chiefe instrumentes of all learninges): seing moreouer that many good wittes both of gentlemen and of others of all degrees, much desirous and studious of these artes, and seeking for them as much as they can, sparing no paines, and yet frustrate of their intent, by no meanes attaining to that which they seeke: I haue for their sakes, with some charge & great trauaile, faithfully translated into our vulgare toũge, & set abroad in Print, this booke of _Euclide_. Whereunto I haue added easie and plaine declarations and examples by figures, of the definitions. In which booke also ye shall in due place finde manifolde additions, Scholies, Annotations, and Inuentions: which I haue gathered out of many of the most famous & chiefe Mathematiciẽs, both of old time, and in our age: as by diligent reading it in course, ye shall well perceaue. The fruite and gaine which I require for these my paines and trauaile, shall be nothing els, but onely that thou gentle reader, will gratefully accept the same: and that thou mayest thereby receaue some profite: and moreouer to excite and stirre vp others learned, to do the like, & to take paines in that behalfe. By meanes wherof, our Englishe tounge shall no lesse be enriched with good Authors, then are other straunge tounges: as the Dutch, French, Italian, and Spanishe: in which are red all good authors in a maner, found amongest the Grekes or Latines. Which is the chiefest cause, that amongest thẽ do florishe so many cunning and skilfull men, in the inuentions of straunge and wonderfull thinges, as in these our daies we see there do. Which fruite and gaine if I attaine vnto, it shall encourage me hereafter, in such like sort to translate, and set abroad some other good authors, both pertaining to religion (as partly I haue already done) and also pertaining to the Mathematicall Artes. Thus gentle reader farewell._ (?¿)

[Decoration]

❧ TO THE VNFAINED LOVERS
of truthe, and constant Studentes of Noble
_Sciences, _IOHN DEE_ of London, hartily_
wisheth grace from heauen, and most prosperous
_successe in all their honest attemptes and_
exercises.

Diuine _Plato_, the great Master of many worthy Philosophers, and the constant auoucher, and pithy perswader of _Vnum_, _Bonum_, and _Ens_: in his Schole and Academie, sundry times (besides his ordinary Scholers) was visited of a certaine kinde of men, allured by the noble fame of _Plato_, and the great commendation of hys profound and profitable doctrine. But when such Hearers, after long harkening to him, perceaued, that the drift of his discourses issued out, to conclude, this _Vnum_, _Bonum_, and _Ens_, to be Spirituall, Infinite, Æternall, Omnipotent, &c. Nothyng beyng alledged or expressed, How, worldly goods: how, worldly dignitie: how, health, Strẽgth or lustines of body: nor yet the meanes, how a merueilous sensible and bodyly blysse and felicitie hereafter, might be atteyned: Straightway, the fantasies of those hearers, were dampt: their opinion of _Plato_, 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, that _Plato_ right well, otherwise could teach: and that _Aristotle_ mought boldely, with his hearers, haue dealt in like sorte as _Plato_ did) I am in no little pang of perplexitie: Bycause, that, which I mislike, is most easy for me to performe (and to haue _Plato_ for my exãple.) And that, which I know to be most commendable: and (in this first bringyng, into common handling, the _Artes 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 (Imitatyng _Aristotle_) well can I hope, that accordyng to the amplenes and dignitie of the _State 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 from _Plato_ his schole, after they had perceiued his finall conclusion, would in these thinges haue ben his most diligent hearers (so infinitely mought their desires, in fine and at length, by our _Artes Mathematicall_ be satisfied) yet, by this my Præface & forewarnyng, Aswell all such, may (to their great behofe) the soner, hither be allured: as also the _Pythagoricall_, and _Platonicall_ perfect scholer, and the constant profound Philosopher, with more ease and spede, may (like the Bee,) gather, hereby, both wax and hony.

[The intent of this Preface.]

Wherfore, seyng I finde great occasion (for the causes alleged, and farder, in respect of my _Art Mathematike generall_) to vse “a certaine forewarnyng and Præface, whose content shalbe, that mighty, most plesaunt, and frutefull _Mathematicall 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 called _Thynges 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 generall _Formes_, notwithstandyng, are constant, vnchaungeable, vntrãsformable, and incorruptible. Neither of the sense, can they, at any tyme, be perceiued or iudged. Nor yet, for all that, in the royall mynde of man, first conceiued. But, surmountyng the imperfectiõ of coniecture, weenyng and opinion: and commyng short of high intellectuall cõceptiõ, are the Mercurial fruite of _Dianœticall_ discourse, in perfect imagination subsistyng. A meruaylous newtralitie haue these thinges _Mathematicall_, 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.”

[Note the worde, Vnit, to expresse the Greke Monas,
& not Vnitie: as we haue all, commonly, till now, vsed.]

Of _Mathematicall_ thinges, are two principall kindes: namely, _Number_, and _Magnitude_.

[Number.]

_Number_, we define, to be, a certayne Mathematicall Sũme, of _Vnits_. And, an _Vnit_, is that thing Mathematicall, Indiuisible, by participation of some likenes of whose property, any thing, which is in deede, or is counted One, may resonably be called One. We account an _Vnit_, a thing _Mathematicall_, though it be no Number, and also indiuisible: because, of it, materially, Number doth consist: which, principally, is a thing _Mathematicall_.

[Magnitude.]

_Magnitude_ is a thing _Mathematicall_, by participation of some likenes of whose nature, any thing is iudged long, broade, or thicke. “A thicke _Magnitude_ we call a _Solide_, or a _Body_. What _Magnitude_ so euer, is Solide or Thicke, is also broade, & long. A broade magnitude, we call a _Superficies_ or a Plaine. Euery playne magnitude, hath also length. A long magnitude, we terme a _Line_. A _Line_ is neither thicke nor broade, but onely long: Euery certayne Line, hath two endes:

[A point.]

The endes of a line, are _Pointes_ called. A _Point_, is a thing _Mathematicall_, indiuisible, which may haue a certayne determined situation.” If a Poynt moue from a determined situation, the way wherein it moued, is also a _Line_: mathematically produced, whereupon, of the auncient Mathematiciens,

[A Line.]

a _Line_ is called the race or course of a _Point_. A Poynt we define, by the name of a thing Mathematicall: though it be no Magnitude, and indiuisible: because it is the propre ende, and bound of a Line: which is a true _Magnitude_.

[Magnitude.]

And _Magnitude_ we may define to be that thing _Mathematicall_, which is diuisible for euer, in partes diuisible, long, broade or thicke. Therefore though a Poynt be no _Magnitude_, yet _Terminatiuely_, we recken it a thing _Mathematicall_ (as I sayd) by reason it is properly the end, and bound of a line. Neither _Number_, nor _Magnitude_, haue any Materialitie. First, we will consider of _Number_, and of the Science _Mathematicall_, to it appropriate, called _Arithmetike_: and afterward of _Magnitude_, and his Science, called _Geometrie_. But that name contenteth me not: whereof a word or two hereafter shall be sayd. How Immateriall and free from all matter, _Number_ is, who doth not perceaue? yea, who doth not wonderfully wõder at it? For, neither pure _Element_, nor _Aristoteles, 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 Philosopher _Anitius Boetius_, sayd: _Omnia quæcun[que] a 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, from _Nothing_, to the _Formalitie_ of their being and state. By _Numbers_ propertie therefore, of vs, by all possible meanes, (to the perfection of the Science) learned, we may both winde and draw our selues into the inward and deepe search and vew, of all creatures distinct vertues, natures, properties, and _Formes_: And also, farder, arise, clime, ascend, and mount vp (with Speculatiue winges) in spirit, to behold in the Glas of Creation, the _Forme of Formes_, the _Exemplar Number_ of all thinges _Numerable_: both visible and inuisible, mortall and immortall, Corporall and Spirituall. Part of this profound and diuine Science, had _Ioachim_ the Prophesier atteyned vnto: by _Numbers Formall, Naturall_, and _Rationall_, forseyng, concludyng, and forshewyng great particular euents, long before their comming. His bookes yet remainyng, hereof, are good profe: And the noble Earle of _Mirandula_, (besides that,) a sufficient witnesse: that _Ioachim, 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 Conclusions _Mathematicall_, (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, bycause _Ioannes 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, that _Number_ hath a treble state: One, in the Creator: an other in euery Creature (in respect of his complete constitution:) and the third, in Spirituall and Angelicall Myndes, and in the Soule of mã. In the first and third state, _Number_, is termed _Number Numbryng_. But in all Creatures, otherwise, _Number_, is termed _Nũber Numbred_. And in our Soule, Nũber beareth such a swaye, and hath such an affinitie therwith: that some of the old _Philosophers_ taught, _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 Numbryng_ therfore, is the discretion discerning, and distincting of thinges. But in God the Creator, This discretion, in the beginnyng, produced orderly and distinctly all thinges. For his _Numbryng_, then, was his Creatyng of all thinges. And his Continuall _Numbryng_, of all thinges, is the Conseruation of them in being: And, where and when he will lacke an _Vnit_: there and then, that particular thyng shalbe _Discreated_. Here I stay. But our Seuerallyng, distinctyng, and _Numbryng_, createth nothyng: but of Multitude considered, maketh certaine and distinct determination. And albeit these thynges be waighty and truthes of great importance, yet (by the infinite goodnes of the Almighty _Ternarie_,) Artificiall Methods and easy wayes are made, by which the zelous Philosopher, may wyn nere this Riuerish _Ida_, this Mountayne of Contemplation: and more then Contemplation. And also, though _Number_, be a thyng so Immateriall, so diuine, and æternall: yet by degrees, by litle and litle, stretchyng forth, and applying some likenes of it, as first, to thinges Spirituall: and then, bryngyng it lower, to thynges sensibly perceiued: as of a momentanye sounde iterated: then to the least thynges that may be seen, numerable: And at length, (most grossely,) to a multitude of any corporall thynges seen, or felt: and so, of these grosse and sensible thynges, we are trayned to learne a certaine Image or likenes of numbers: and to vse Arte in them to our pleasure and proffit. So grosse is our conuersation, and dull is our apprehension: while mortall Sense, in vs, ruleth the common wealth of our litle world. Hereby we say, Three Lyons, are three: or a _Ternarie_. Three Egles, are three, or a _Ternarie_.

[☞]

Which * _Ternaries_, are eche, the _Vnion_, _knot_, and _Vniformitie_, of three discrete and distinct _Vnits_. That is, we may in eche _Ternarie_, thrise, seuerally pointe, and shew a part, _One_, _One_, and _One_. Where, in Numbryng, we say One, two, Three. But how farre, these visible Ones, do differre from our Indiuisible Vnits (in pure _Arithmetike_, principally considered) no man is ignorant. Yet from these grosse and materiall thynges, may we be led vpward, by degrees, so, informyng our rude Imagination, toward the cõceiuyng of _Numbers_, absolutely (:Not supposing, nor admixtyng any thyng created, Corporall or Spirituall, to support, conteyne, or represent those _Numbers_ imagined:) that at length, we may be hable, to finde the number of our owne name, gloriously exemplified and registred in the booke of the _Trinitie_ most 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 an _Vnit_. For the common Logist, Reckenmaster, or Arithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lesse partes: and calleth them _Fractions_. As of an _Vnit_, he maketh an halfe, and thus noteth it, ½. and so of other, (infinitely diuerse) partes of an _Vnit_. Yea and farder, hath, _Fractions of Fractions. &c_. And, forasmuch, as, _Addition_, _Substraction_, _Multiplication_, _Diuision_ and _Extraction of Rotes_, are the chief, and sufficient partes of _Arithmetike_:

[Arithmetike.]

which is, the _Science 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 of _Arithmetike_.” And by this reason:

[1.]

the Consideration, doctrine, and working, in whole numbers onely: where, of an _Vnit_, is no lesse part to be allowed: is named (as it were) an _Arithmetike_ by it selfe. And so of the _Arithmetike 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 of _Arithmetike_.

[3.]

Moreouer, the _Astronomers_, for spede and more commodious calculation, haue deuised a peculier maner of orderyng nũbers, about theyr circular motions, by Sexagenes, and Sexagesmes. By Signes, Degrees and Minutes &c. which commonly is called the _Arithmetike_ of _Astronomical_ or _Phisicall Fractions_. That, haue I briefly noted, by the name of _Arithmetike Circular_. Bycause it is also vsed in circles, not _Astronomicall. &c._

[4.]

Practise hath led _Numbers_ farder, and hath framed them, to take vpon them, the shew of _Magnitudes_ propertie: Which is _Incommensurabilitie_ and _Irrationalitie_. (For in pure _Arithmetike_, an _Vnit_, is the common Measure of all Numbers.) And, here, Nũbers are become, as Lynes, Playnes and Solides: some tymes _Rationall_, some tymes _Irrationall_. And haue propre and peculier characters, (as ²√. ³√. and so of other. Which is to signifie _Rote Square, Rote Cubik: and so forth_:) & propre and peculier fashions in the fiue principall partes: Wherfore the practiser, estemeth this, a diuerse _Arithmetike_ from the other. Practise bryngeth in, here, diuerse compoundyng of Numbers: as some tyme, two, three, foure (or more) _Radicall_ nũbers, diuersly knit, by signes, of More & Lesse: as thus ²√12 + ³√15. Or thus ⁴√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 this _Arithmetike_ greately enlarged, by diuerse exhibityng and vse of Compositions and mixtynges. Consider how, I (beyng desirous to deliuer the student from error and Cauillation) do giue to this _Practise_, the name of the _Arithmetike of Radicall numbers_: Not, of _Irrationall_ or _Surd Numbers_: which other while, are Rationall: though they haue the Signe of a Rote before them, which, _Arithmetike_ of whole Numbers most vsuall, would say they had no such Roote: and so account them _Surd Numbers_: which, generally spokẽ, is vntrue: as _Euclides_ tenth 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) of _Numbers_ Practicall vse. Which thing, is well to be perceiued in that great Arithmeticall Arte of _Æquation_: commonly called the _Rule of Coss._ or _Algebra_. The Latines termed it, _Regulam Rei & Census_, that is, the +_Rule 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 of _Algebra_. And therin (commonly) is a dubble error. The one, of them, which thinke it to be of _Geber_ his inuentyng: the other of such as call it _Algebra_. For, first, though _Geber_ for his great skill in Numbers, Geometry, Astronomy, and other maruailous Artes, mought haue semed hable to haue first deuised the sayd Rule: and also the name carryeth with it a very nere likenes of _Geber_ his name: yet true it is, that a _Greke_ Philosopher and Mathematicien, named _Diophantus_, before _Geber_ his 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, is _Algiebar_, and not _Algebra_: as by the Arabien _Auicen_, may be proued: who hath these precise wordes in Latine, by _Andreas 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, the +_Science 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 of _Arithmetike_, to it appertayning, do differre from the other _Arithmeticall operations_. This _Arithmetike, hath Nũbers_ Simple, Cõpound, Mixt: and Fractions, accordingly. This Rule, and _Arithmetike 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 of _Arithmetike_.

And first: who, nerer at hand, can be a better witnesse of the frute receiued by _Arithmetike_, then all kynde of Marchants? Though not all, alike, either nede it, or vse it. How could they forbeare the vse and helpe of the Rule, called the Golden Rule? Simple and Compounde: both forward and backward? How might they misse _Arithmeticall_ helpe in the Rules of Felowshyp: either without tyme, or with tyme? and betwene the Marchant & his Factor? The Rules of Bartering in wares onely: or part in wares, and part in money, would they gladly want? Our Marchant venturers, and Trauaylers ouer Sea, how could they order their doynges iustly and without losse, vnleast certaine and generall Rules for Exchaũge of money, and Rechaunge, were, for their vse, deuised? The Rule of Alligation, in how sundry cases, doth it conclude for them, such precise verities, as neither by naturall witt, nor other experience, they, were hable, els, to know? And (with the Marchant then to make an end) how ample & wonderfull is the Rule of False positions? especially as it is now, by two excellent Mathematiciens (of my familier acquayntance in their life time) enlarged? I meane _Gemma Frisius_, and _Simon 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 other _Arithmeticall 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, if _Arithmetike_ be their guide? And the honorable Phisiciãs, will gladly confesse them selues, much beholding to the Science of _Arithmetike_, and that sundry wayes: But chiefly in their Art of Graduation, and compounde Medicines. And though _Galenus_, _Auerrois_, _Arnoldus_, _Lullus_, and other haue published their positions, aswell in the quantities of the Degrees aboue Temperament, as in the Rules, concluding the new _Forme_ resulting: yet a more precise, commodious, and easy _Method_, is extant: by a Countreyman of ours

[R. 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. _Hote_ and _Colde_, one against the other. And likewise _Moyst_ and _Dry_, one against the other. And in the Circle write _Temperate_. Which _Temperature_ hath a good Latitude: as appeareth by the Complexion of man. And therefore we haue allowed vnto it, the foresayd Circle: and not a point Mathematicall or Physicall.

[* 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 a _Temperate_, and a Contrary, _The forme resulting of their Mixture, is in the Middle betwene the degrees of the formes mixt_. As for example, let _A_, be _Moist_ in the first degree: and _B_, _Dry_ in the third degree. Adde 1. and 3. that maketh 4: the halfe or middle of 4. is 2. This 2. is the middle, equally distant from _A_ and _B_

[* Note.]

(for the * _Temperament_ is counted none. And for it, you must put a Ciphre, if at any time, it be in mixture).

HOTE
+C
|
|
+
|
|
+
|
|
+E
|
MOIST A TEMPERATE B DRYE
+------+------+------+------+------+------+------+------+
|D
|
+
|
|
+
|
|
+
|
|
+
COLD

Counting then from _B_, 2. degrees, toward _A_: you finde it to be _Dry_ in the first degree: So is the _Forme resulting_ of the Mixture of _A_, and _B_, in our example. I will geue you an other example. Suppose, you haue two thinges, as _C_, and _D_: and of _C_, the Heate to be in the 4. degree: and of _D_, the Colde, to be remisse, euen vnto the _Temperament_. Now, for _C_, you take 4: and for _D_, you take a Ciphre: which, added vnto 4, yeldeth onely 4. The middle, or halfe, whereof, is 2. Wherefore the _Forme resulting_ of _C_, and _D_, is Hote in the second degree: for, 2. degrees, accounted from _C_, toward _D_, 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 was _C_, in the example foregoing: and an other liquid Medicine I haue: whose Qualitie, is heate, in the first degree. Of eche of these, I mixt a like quantitie: Subtract here, the lesse frõ the more: and the residue diuide into two equall partes: whereof, the one part, either added to the lesse, or subtracted from the higher degree, doth produce the degree of the Forme resulting, by this mixture of _C_, and _E_. As, if from 4. ye abate 1. there resteth 3. the halfe of 3. is 1½: Adde to 1. this 1½: you haue 2½. Or subtract from 4. this 1½: you haue likewise 2½ remayning. Which declareth, the _Forme resulting_, to be _Heate_, 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. ___________________________ | | | | {P}. _2._ | _Hote. 4._ | | | | | {P}. _1._ | _Hote. 3._ | |____________|______________|

Now by the rule of Algiebar, haue I deuised a very easie, briefe, and generall maner of working in this case. Let vs first, suppose that _Middle Forme resulting_, to be 1{x}: as that Rule teacheth. And because (by our Rule, here geuen) as the waight of 1. is to 2: So is the difference betwene 4. (the degree of the greater quantitie) and 1{x}: to the difference betwene 1{x} and 3: (the degree of the thing, in lesse quãtitie. And with all, 1{x}, being alwayes in a certaine middell, betwene the two heigthes or degrees). For the first difference, I set 4-1{x}: and for the second, I set 1{x}-3. And, now againe, I say, as 1. is to 2. so is 4-1{x} to 1{x}-3. Wherfore, of these foure proportionall numbers, the first and the fourth Multiplied, one by the other, do make as much, as the second and the third Multiplied the one by the other. Let these Multiplications be made accordingly. And of the first and the fourth, we haue 1{x}-3. and of the second & the third, 8-2{x}. Wherfore, our Æquation is betwene 1{x}-3: and 8-2{x}. Which may be reduced, according to the Arte of Algiebar: as, here, adding 3. to eche part, geueth the Æquation, thus, 1{x}=11-2{x}. And yet againe, contracting, or Reducing it: Adde to eche part, 2{x}: Then haue you 3{x} æquall to 11: thus represented 3{x}=11. Wherefore, diuiding 11. by 3: the Quotient is 3⅔: the _Valew_ of our 1{x}, _Coss_, or _Thing_, first supposed. And that is the heigth, or Intension of the _Forme 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, resteth ⅔. Subtracte the same heigth of the Forme resulting, (which is 3⅔) frõ 4: then resteth ⅓: You see, that ⅔ is double to ⅓: as 2.{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 2{x}, it would stand, 3{x}-3=8. And now adding to eche part 3: you haue (as afore) 3{x}=11. _________________________ | | | _ | {P}. _2._ | _Hote. 4._ | ⅓ _ _The forme_ | | | _ _3⅔ resulting._ | {P}. _1._ | _Hote. 3._ | _ ⅔ |___________|_____________|

And though I, here, speake onely of two thyngs Miscible: and most commonly mo then three, foure, fiue or six, (&c.) are to be Mixed: (and in one Compound to 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, concerning _Mixture of Formes and Qualities Naturall_, is so Matcht and maryed with the most simple, easie, and short way of the noble Rule of _Algiebar_. Who can remaine, therfore vnpersuaded, to loue, alow, and honor the excellent Science of _Arithmetike_? For, here, you may perceiue that the litle finger of _Arithmetike_, 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 of _Arithmetike_: in one of the Artes to him appertaining: And of the Grekes named

[Τακτικὴ.]

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, that _Ælianus_ (the best writer therof,) in his worke, to the _Emperour 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 written _Æneas_: _Cyneas_ of _Thessaly_: _Pyrrhus Epirota_: and _Alexander_ his sonne: _Clearchus_: _Pausanias_: _Euangelus_: _Polybius_, familier frende to _Scipio_: _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) was _Arithmetike_: and a litle perceiuerance, in _Geometricall_ Figures. But in many other cases doth _Arithmetike_ stand the Capitaine in great stede. As in proportionyng of vittayles, for the Army, either remaining at a stay: or suddenly to be encreased with a certaine number of Souldiers: and for a certain tyme. Or by good Art to diminish his company, to make the victuals, longer to serue the remanent, & for a certaine determined tyme: if nede so require. And so in sundry his other accountes, Reckeninges, Measurynges, and proportionynges, the wise, expert, and Circumspect Capitaine will affirme the Science of _Arithmetike_, to be one of his chief Counsaylors, directers and aiders. Which thing (by good meanes) was euident to the Noble, the Couragious, the loyall, and Curteous

[☞]

_Iohn_, late Earle of Warwicke. Who was a yong Gentleman, throughly knowne to very few. Albeit his lusty valiantnes, force, and Skill in Chiualrous feates and exercises: his humblenes, and frendelynes to all men, were thinges, openly, of the world perceiued. But what rotes (otherwise,) vertue had fastened in his brest, what Rules of godly and honorable life he had framed to him selfe: what vices, (in some then liuing) notable, he tooke great care to eschew: what manly vertues, in other noble men, (florishing before his eyes,) he Sythingly aspired after: what prowesses he purposed and ment to achieue: with what feats and Artes, he began to furnish and fraught him selfe, for the better seruice of his Kyng and Countrey, both in peace & warre. These (I say) his Heroicall Meditations, forecastinges and determinations, no twayne, (I thinke) beside my selfe, can so perfectly, and truely report. And therfore, in Conscience, I count it my part, for the honor, preferment, & procuring of vertue (thus, briefly) to haue put his Name, in the Register of _Fame Immortall_.

To our purpose. This _Iohn_, 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: Of _Numbers_ frute, he had good hope. Now, Numbers therfore innumerable, in _Numbers_ prayse, 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 thing _Arithmetike_ is? I meane, the Lawyers of all sortes. Vndoubtedly, the Ciuilians, can meruaylously declare: how, neither the Auncient Romaine lawes, without good knowledge of _Numbers 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, did _Papinianus_ institute a law of partition, and allowance, betwene man and wife after a diuorce? But how _Accursius_, _Baldus_, _Bartolus_, _Iason_, _Alexander_, and finally _Alciatus_, (being otherwise, notably well learned) do iumble, gesse, and erre, from the æquity, art and Intent of the lawmaker: _Arithmetike_ can detect, and conuince: and clerely, make the truth to shine. Good _Bartolus_, tyred in the examining & proportioning of the matter: and with _Accursius_ Glosse, 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 of _Iulianus_ law, _Si ita Scriptum. &c._ Of the Testators will iustly performing, betwene the wife, Sonne and daughter? How can they perceiue the æquitie of _Aphricanus_, _Arithmeticall_ Reckening, where he treateth of _Lex Falcidia_? How can they deliuer him, from his Reprouers: and their maintainers: as _Ioannes_, _Accursius Hypolitus_ and _Alciatus_? How Iustly and artificially, was _Africanus_ reckening made? Proportionating to the Sommes bequeathed, the Contributions of eche part? Namely, for the hundred presently receiued, 17-1/7. And for the hundred, receiued after ten monethes, 12-6/7: which make the 30: which were to be cõtributed by the legataries to the heire. For, what proportion, 100 hath to 75: the same hath 17-1/7 to 12-6/7: Which is Sesquitertia: that is, as 4, to 3. which make 7. Wonderfull many places, in the Ciuile law, require an expert _Arithmeticien_, to vnderstand the deepe Iudgemẽt, & Iust determinatiõ of the Auncient Romaine Lawmakers. But much more expert ought he to be, who should be hable, to decide with æquitie, the infinite varietie of Cases, which do, or may happen, vnder euery one of those lawes and ordinances Ciuile. Hereby, easely, ye may now coniecture: that in the Canon law: and in the lawes of the Realme (which with vs, beare the chief Authoritie), Iustice and equity might be greately preferred, and skilfully executed, through due skill of Arithmetike, and proportions appertainyng. The worthy Philosophers, and prudent lawmakers (who haue written many bookes _De 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. _Aristotle_ in his _Ethikes_ (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. _Plato_ in his booke called _Epinomis_ (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 proueth _Wisedome_ to be atteyned, by good Skill of _Numbers_. With which great Testimony, and the manifold profes, and reasons, before expressed, you may be sufficiently and fully persuaded: of the perfect Science of _Arithmetike_, to make this accounte: That

[☞]

of all Sciences, next to _Theologie_, it is most diuine, most pure, most ample and generall, most profounde, most subtile, most commodious and most necessary. Whose next Sister, is the Absolute Science of _Magnitudes_: of which (by the Direction and aide of him, whose _Magnitude_ is Infinite, and of vs Incomprehensible) I now entend, so to write, that both with the _Multitude_, and also with the _Magnitude_ of Meruaylous and frutefull verities, you (my frendes and Countreymen) may be stird vp, and awaked, to behold what certaine Artes and Sciences, (to our vnspeakable behofe) our heauenly father, hath for vs prepared, and reuealed, by sundry _Philosophers_ and _Mathematiciens_.

Both, _Number_ and _Magnitude_, haue a certaine Originall sede, (as it were) of an incredible property: and of man, neuer hable, Fully, to be declared. Of _Number_, an Vnit, and of _Magnitude_, a Poynte, doo seeme to be much like Originall causes: But the diuersitie neuerthelesse, is great. We defined an _Vnit_, to be a thing Mathematicall Indiuisible: A Point, likewise, we sayd to be a Mathematicall thing Indiuisible. And farder, that a Point may haue a certaine determined Situation: that is, that we may assigne, and prescribe a Point, to be here, there, yonder. &c. Herein, (behold) our Vnit is free, and can abyde no bondage, or to be tyed to any place, or seat: diuisible or indiuisible. Agayne, by reason, a Point may haue a Situation limited to him: a certaine motion, therfore (to a place, and from a place) is to a Point incident and appertainyng. But an _Vnit_, can not be imagined to haue any motion. A Point, by his motion, produceth, Mathematically, a line: (as we sayd before) which is the first kinde of Magnitudes, and most simple: An _Vnit_, can not produce any number. A Line, though it be produced of a Point moued, yet, it doth not consist of pointes: Number, though it be not produced of an _Vnit_, yet doth it Consist of vnits, as a materiall cause. But formally,

[Number.]