To the Lovers of Truth and Students of Noble Sciences
Original
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.
Musean translation
Modern machine-assisted Musean edition by Mouseia, from Dee's 1570 Praeface. Latin quotations and titles are translated into English, using Dee's own English glosses where provided. Printer's type-size markup is normalized; marginal notes are kept as asides. Billingsley's separate address is omitted. Uncertain readings are marked.
To the unfeigned lovers of truth, and the constant students of the noble sciences, John Dee of London heartily wishes grace from heaven, and the most prosperous success in all their honorable attempts and exercises.
Divine Plato, that great master of many worthy philosophers, the steady affirmer and vigorous persuader of the One, the Good, and Being, was visited in his school and academy at various times — besides his ordinary scholars — by a certain kind of men, allured by the noble fame of Plato and the great commendation of his profound and profitable doctrine. But when such hearers, after long listening to him, perceived that the drift of his discourses came out to conclude that the One, the Good, and Being are spiritual, infinite, eternal, omnipotent, etc., nothing being alleged or expressed as to how worldly goods, how worldly dignity, how health, strength, or vigor of body might be attained, nor yet by what means a marvelous sensible and bodily bliss and felicity hereafter might be reached — at once the fancies of those hearers were dampened; their opinion of Plato was completely changed; indeed his doctrine was despised by them, and his school visited by them no more. Which thing his scholar Aristotle, narrowly considering, found the cause of it to be "that they had no forewarning and general information as to where his doctrine tended." For so they might have had occasion either to refrain from haunting his school — if they then disliked his aim and purpose — or to continue in it constantly, to their full satisfaction, if such his final scope and intent had been to their desire. For this reason Aristotle, ever after, used briefly to forewarn his own scholars and hearers "both of what matter, and also to what end, he undertook to speak or teach."
While I consider the diverse methods of these two excellent philosophers (and am most sure both that Plato could very well have taught otherwise, and that Aristotle might have dealt boldly with his hearers in the same way as Plato did), I am in no small pang of perplexity: because that which I dislike is easiest for me to perform (and to have Plato for my example), and that which I know to be most commendable, and (in this first bringing into common handling the mathematical arts) most necessary, is full of great difficulty and sundry dangers. Yet neither do I think it fitting, for so strange a matter as is now meant to be published, and to so strange an audience, to be bluntly at first put forth without a special preface; nor (imitating Aristotle) can I well hope that, according to the breadth and dignity of the mathematical discipline, I am able either plainly to set out the bounds of its subject matter, or precisely to express its chief purposes and most wonderful applications. And though I am sure that such as shrank from Plato's school, after they had perceived his final conclusion, would in these things have been his most diligent hearers — so infinitely might their desires in the end and at length be satisfied by our mathematical arts — yet by this my preface and forewarning, all such may, to their great benefit, be the sooner lured here; and also the Pythagorean and Platonic perfect scholar, and the constant profound philosopher, with more ease and speed may, like the bee, gather from this both wax and honey.
(The intent of this preface.)
Therefore, since I find great occasion — for the causes alleged, and further in respect of my art of mathematics in general — to use "a certain forewarning and preface, whose content shall be that mighty, most pleasant, and fruitful mathematical tree, with its main arms and second grafted branches: what each one of them is, and also what benefit in general is to be looked for, both from graft and from stock." And since this enterprise is so great that, to our time, it has never (to my knowledge) been achieved by anyone; and also it is most hard in these dreary days to win due and common credit for such rare and strange arts — nevertheless, if for my sincere endeavor to satisfy your honest expectation you will lend me your thankful mind a while, and apply your eye or ear attentively to such matter as my pen is able to deliver this time with speed, perhaps at once, and for the first saluting, you will find this preface a long enough lesson. And either you will by this be made much the apter for a second, or shortly become well able yourselves, from the lion's claw, to conjecture his royal symmetry and further property. Now then, gentle friends and countrymen, turn your eyes and bend your minds to that doctrine which for our present purpose my simple talent is able to yield you.
All things that are, and have being, are found under a triple general diversity. For either they are deemed supernatural, natural, or of a third being. Supernatural things are immaterial, simple, indivisible, incorruptible, and unchangeable. Natural things are material, compounded, divisible, corruptible, and changeable. Supernatural things are comprehended by the mind alone; natural things are able to be perceived by the outward sense. In natural things probability and conjecture have a place; but in supernatural things chief demonstration and most sure knowledge is to be had. By which properties and comparisons of these two, the state, condition, nature, and property of those things which we earlier called "of a third being" may more easily be described; which also, by a special name, are called Mathematical Things. For these, being in a manner middle between supernatural and natural things, are not so absolute and excellent as supernatural things, nor yet so base and gross as natural things; but they are immaterial, and nevertheless by material things able to be somewhat signified. And though their particular images by art are aggregable and divisible, yet the general forms notwithstanding remain constant, unchangeable, untransformable, and incorruptible. Neither can they at any time be perceived or judged by the sense. Nor yet for all that are they first conceived in the royal mind of man. But surmounting the imperfection of conjecture, supposition, and opinion, and falling short of high intellectual conception, they are the mercurial fruit of discursive reasoning, subsisting in perfect imagination. A marvelous neutrality have these mathematical things, and also a strange participation between supernatural, immortal, intellectual, simple and indivisible things, and natural, mortal, sensible, compounded and divisible things. Probability and sensible prose may well serve in natural things, and is commendable; in mathematical reasonings a probable argument is nothing regarded, nor yet the testimony of sense at all credited; but only a perfect demonstration of truths certain, necessary, and invincible, universally and necessarily concluded, is allowed as sufficient for "an argument exactly and purely mathematical."
(Note the word "unit" to express the Greek Monas, and not "unity," as we have all commonly used until now.)
Of mathematical things there are two principal kinds: namely, Number and Magnitude.
(Number.)
Number, we define, to be a certain mathematical sum of units. And a unit is that mathematical thing, indivisible, by sharing in some likeness of whose property any thing which in deed or in count is One may reasonably be called One. We account a unit a mathematical thing, though it is no number and also indivisible, because number materially consists of it, which principally is a mathematical thing.
(Magnitude.)
Magnitude is a mathematical thing, by sharing in some likeness of whose nature any thing is judged long, broad, or thick. "A thick magnitude we call a solid, or a body. What magnitude soever is solid or thick is also broad and long. A broad magnitude we call a surface or a plane. Every plane magnitude has also length. A long magnitude we call a line. A line is neither thick nor broad, but only long; every certain line has two ends:
(A point.)
The ends of a line are called points. A point is a mathematical thing, indivisible, which may have a certain determined position." If a point moves from a determined position, the way in which it moved is also a line, mathematically produced; after which, of the ancient mathematicians,
(A line.)
a line is called the race or course of a point. A point we define by the name of a mathematical thing, though it is no magnitude and is indivisible, because it is the proper end and bound of a line, which is a true magnitude.
(Magnitude.)
And magnitude we may define to be that mathematical thing which is divisible forever into divisible parts — long, broad, or thick. Therefore though a point is no magnitude, yet terminatively we reckon it a mathematical thing (as I said) by reason it is properly the end and bound of a line. Neither number nor magnitude has any materiality.
Plain English translation
Modern machine-assisted Plain English edition by Mouseia, from Dee's 1570 Praeface. Latin quotations and titles are translated into English, using Dee's own English glosses where provided. Printer's type-size markup is normalized; marginal notes are kept as asides. Billingsley's separate address is omitted. Uncertain readings are marked.
To all who love truth without pretense and who steadily study the noble sciences, John Dee of London warmly wishes grace from heaven and the greatest success in all their honorable work and exercises.
Divine Plato, the great teacher of many worthy philosophers, and the constant affirmer and powerful persuader of the One, the Good, and Being, was visited many times in his school and academy, besides his ordinary students, by a certain kind of men, allured by the noble fame of Plato and the great praise of his deep and profitable teaching. But when such hearers, after listening to him a long time, perceived that the point of his discourses came to this conclusion, that the One, the Good, and Being are spiritual, infinite, eternal, omnipotent, etc., with nothing alleged or explained about worldly goods, about worldly dignity, about health, strength or vigor of body, nor about the means by which a wonderful, sensible and bodily bliss and happiness hereafter might be attained—immediately those hearers were discouraged, their opinion of Plato completely changed, his doctrine was despised by them, and his school was visited by them no more. His student Aristotle, considering this closely, found the cause to be "that they had no general forewarning and information about where his teaching tended." For so they could have had reason either to stay away from his school (if they then disliked his scope and purpose) or to continue in it steadily, to their full satisfaction, if such a final scope and intent had been what they wanted. So Aristotle, from then on, always used to warn his own students and hearers briefly, "both of what subject matter, and also to what end, he took in hand to speak or teach." While I consider the different approaches of these two excellent philosophers (and I am most sure both that Plato could very well have taught otherwise, and that Aristotle could boldly have dealt with his hearers in the same way Plato did), I am in no small anguish of perplexity: because what I dislike is easiest for me to perform (and to have Plato as my example), while what I know to be most commendable, and (in this first bringing of the mathematical arts into common handling) most necessary, is full of great difficulty and various dangers. Yet I do not think it fitting, for so strange a matter as is now meant to be published, and to so strange an audience, to put it out bluntly at first with no special preface; nor, imitating Aristotle, can I well hope that, according to the fullness and dignity of the mathematical state, I am able either to set out plainly the material bounds, or to state precisely the chief purposes and most wonderful applications of it. And though I am sure that those who shrank from Plato's school, after they had perceived his final conclusion, would in these things have been his most diligent hearers (so infinitely might their desires be satisfied in the end and at length by our mathematical arts), yet by this preface and forewarning all such may be drawn here sooner, to their great benefit; and also the fully trained Pythagorean and Platonian student, and the steadfast profound philosopher, may with more ease and speed gather from it both wax and honey, like the bee.
The intent of this preface.
Seeing that I have good reason (for the reasons given, and further in respect of my general mathematical art) to use "a certain forewarning and preface, whose content shall be this: the mighty, most pleasant and fruitful Mathematical Tree, with its main arms and its second grafted branches—what each one of them is, and also what benefit in general may be looked for, from graft as much as from stock. And since this undertaking is so great that, up to our time, it has never (to my knowledge) been achieved by anyone; and since it is also very hard in these gloomy days to win proper and general credit for such rare and strange arts," still, if you will lend me your thankful mind a while, for my sincere effort to meet your honorable expectation, and apply your eye or ear attentively to whatever my pen can deliver quickly this time, perhaps at once, as a first greeting, you will find this preface a lesson long enough. And either you will be made much readier by it for a second one, or shortly become quite able yourselves, from the lion's claw, to judge its royal symmetry and further properties. So now, my good friends and countrymen, turn your eyes and bend your minds to that doctrine which, for our present purpose, my modest talent is able to yield you.
All things that are, and have being, fall under a general threefold difference. For either they are judged supernatural, natural, or of a third being. Supernatural things are immaterial, simple, indivisible, incorruptible, and unchangeable. Natural things are material, compound, divisible, corruptible, and changeable. Supernatural things are grasped by the mind alone; natural things can be perceived by the outward senses. In natural things probability and conjecture have a place; but in supernatural things the highest demonstration and the most certain knowledge is to be had. By these properties and comparisons of the two, the state, condition, nature and property of those things we earlier called of a third being can be more easily described, which by a special name are also called Mathematical Things. For these, being in a manner middle between supernatural and natural things, are not so complete and excellent as supernatural things, nor so base and coarse as natural things; but they are immaterial things, and yet can be signified to some degree by material things. And though their particular images can be assembled and divided by art, yet the general forms remain constant, unchangeable, untransformable, and incorruptible. They can never be perceived or judged by the senses. Nor yet for all that are they first conceived in the royal mind of man. But rising above the imperfection of conjecture, guesswork and opinion, and falling short of high intellectual understanding, they are the mediating fruit of Dianoetic reasoning, resting in perfect imagination. These mathematical things have a wonderful neutrality and a strange participation between supernatural, immortal, intellectual, simple and indivisible things, and natural, mortal, sensible, compound and divisible things. Probability and proof from the senses may well serve in natural things, and that is commendable; in mathematical reasoning a probable argument is not regarded at all, nor is the testimony of the senses credited at all; but only a perfect demonstration of certain, necessary and invincible truths, concluded universally and necessarily, is allowed as sufficient for "an argument exactly and purely mathematical."
Note that the word unit is used here to express the Greek monas, and not unity, as we have all commonly used until now.
Mathematical things fall into two main kinds: number and magnitude.
Number.
We define number as a certain mathematical sum of units. A unit is an indivisible mathematical thing; by partaking in some likeness of its property, anything that exists, or is counted as one, can reasonably be called one. We count a unit to be a mathematical thing, though it is no number and is also indivisible, because number consists of units materially, and number is itself chiefly a mathematical thing.
Magnitude.
Magnitude is a mathematical thing; by partaking in some likeness of its nature, anything is judged long, broad, or thick. "A thick magnitude we call a solid, or a body. Any magnitude that is solid or thick is also broad and long. A broad magnitude we call a surface or a plane. Every plane magnitude also has length. A long magnitude we call a line. A line is neither thick nor broad, but only long. Every definite line has two ends:
A point.
The ends of a line are called points. A point is an indivisible mathematical thing that may have a certain definite position." If a point moves from a definite position, the way in which it moved is also a line, produced mathematically; after which, according to the ancient mathematicians,
A line.
A line is called the race or course of a point. We define a point as a mathematical thing, though it is no magnitude and is indivisible, because it is the proper end and bound of a line, which is a true magnitude.
Magnitude.
And we may define magnitude as that mathematical thing which is forever divisible into divisible parts, long, broad or thick. So though a point is no magnitude, yet in a limiting sense we count it a mathematical thing (as I said), because it is properly the end and bound of a line. Neither number nor magnitude has any materiality.
About this section
To the Lovers of Truth and Students of Noble Sciences is part of The Mathematicall Praeface to the Elements of Geometrie by John Dee. This page sets the 16th-century English original beside 2 direct-source translations, so any line can be checked against the source. It is part 1 of 34 in the work, which is kept in the European tradition.
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