The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara Part 1

The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara.

by John Dee.

TO THE VNFAINED LOVERS of truthe, and constant Studentes of n.o.ble _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 n.o.ble 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, aeternall, Omnipotent, &c. Nothyng beyng alledged or expressed, How, worldly goods: how, worldly dignitie: how, health, Str?gth or l.u.s.tines 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 csidering, 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 exple.) 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 Praeface & 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 Praeface, 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 thkefull 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 demstration, & 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, vntrsformable, 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 imperfecti of coniecture, weenyng and opinion: and commyng short of high intellectuall ccepti, are the Mercurial fruite of _Dianticall_ discourse, in perfect imagination subsistyng. A meruaylous newtralitie haue these thinges _Mathematicall_, and also a straunge partic.i.p.ati 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 princ.i.p.all kindes: namely, _Number_, and _Magnitude_.

[Number.]

_Number_, we define, to be, a certayne Mathematicall Sume, of _Vnits_.

And, an _Vnit_, is that thing Mathematicall, Indiuisible, by partic.i.p.ation 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, princ.i.p.ally, is a thing _Mathematicall_.

[Magnitude.]

_Magnitude_ is a thing _Mathematicall_, by partic.i.p.ation 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 wder 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 & G.o.dly Philosopher _Anitius Boetius_, sayd: _Omnia quaecun[que] a primaeua rerum natura constructa sunt, Numerorum videntur ratione formata. Hoc enim fuit princ.i.p.ale 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 princ.i.p.all 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 n.o.ble 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 Cclusions, I omit here to rehea.r.s.e: 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 nubers: 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 const.i.tution:) and the third, in Spirituall and Angelicall Myndes, and in the Soule of m. In the first and third state, _Number_, is termed _Number Numbryng_. But in all Creatures, otherwise, _Number_, is termed _Nuber Numbred_. And in our Soule, Nuber 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 G.o.d 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 Mult.i.tude 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 aeternall: 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 mult.i.tude 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_, princ.i.p.ally 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 cceiuyng 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 aeternall.

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, . 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 operatis, 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 nubers, about theyr circular motions, by s.e.xagenes, and s.e.xagesmes. 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, Nubers are become, as Lynes, Playnes and Solides: some tymes _Rationall_, some tymes _Irrationall_.

And haue propre and peculier characters, (as v. v. and so of other.

Which is to signifie _Rote Square, Rote Cubik: and so forth_:) & propre and peculier fashions in the fiue princ.i.p.all 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_ nubers, diuersly knit, by signes, of More & Lesse: as thus v12 + v15. Or thus 4v19 + v12 - v2.

&c. And some tyme with whole numbers, or fractions of whole Number, amg them: as 20 + v24. v16 + 33 - v10. 4v44 + 12 + v9. 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 spok?, is vntrue: as _Euclides_ tenth booke may teach you. Therfore to call them, generally, _Radicall Numbers_, (by reason of the signe v.

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 _aequation_: 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 & aequationem_. 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 & aequation_+. Here haue you the name: and also the princ.i.p.all partes of the Rule, touched. To name it, _The rule, or Art of aequation_, doth signifie the middle part and the State of the Rule. This Rule, hath his peculier Characters:

[5.]

and the princ.i.p.al partes of _Arithmetike_, to it appertayning, do differre from the other _Arithmeticall operations_. This _Arithmetike, hath Nubers_ Simple, Cpound, 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_.