Part 1 (1/2)
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 rauis.h.i.+ng 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.