Sunday, 14 June 2020

The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara John Dee

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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
D(Divine)Iuine Plato, the great Master of many worthy Philosophers, and the constant auoucher, and pithy perswader of VnumBonum, 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 VnumBonum, 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.
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, The intent of this Preface.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: *.iis allowed as sufficient for an Argument exactly and purely Mathematical.
Of Mathematicall thinges, are two principall kindes: namely, Number, and MagnitudeNumber.Number, we define, to be, a certayne Mathematicall Sũme, of VnitsNote the worde, Vnit, to expresse the Greke Monas, & not Vnitie: as we haue all, commonly, till now, vsed.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 MathematicallMagnitude.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.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 MagnitudeMagnitude.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 quæcunque 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.

THE WAY TO GEOMETRY. Being necessary and usefull, For Astronomers. Engineres. Geographers. Architecks. Land-meaters. Carpenters. Sea-men. Paynters. Carvers, &c.

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The Authors Preface.

Two things, I feare me, will here be objected against me: The one concerneth my selfe, directly: The other mine Author, and the worke I have taken in hand the translating of him. Concerning my selfe, I suppose, some will aske, Why I being a Divine; should meddle or busie my selfe with these prophane studies? Geometry may no way further Divinity, and therefore is no fit study for a Divine? This objection seemeth to smell of Brownisme, that is, of a ranke peevish humour overflowing the stomach of some, whereby they are caused to loath all manner of solid learning, yea of true Divinity it selfe, and therefore it doth not deserve an answer: And this we in our Title before signified. For we have not taken this paines for Turkes and others, who by the lawes of their profession are bound to abandon all manner of learning. But if any man shall propose it, as a question, with a desire of satisfaction, we are ready to answer him to the best of our abilitie. First, that Theologia vera est ars artium & scientia scientiarum, Divinity is the Art of Arts, and Science of Sciences; or Divinity is the Mistresse upon which all Arts and Sciences are to attend as servants and handmaides. And why then not Geometry? But in what place she should follow her, I dare not say: For I am no herald, and therefore I meddle not with precedencie: But if I were, she should be none of the hindermost of her traine.
The Oratour saith, and very truly doubtlesse, That, Omnes artes, quæ ad humanitatē pertinent, habent commune quoddam vinculum, & cognatione quadam inter se continentur. All Arts which pertaine unto humanity, they have a certaine common bond, and are knit together by a kinde of affinity. If then any Arts and Sciences may be thought necessary attendants upon this great Lady; Then surely Geometry amongst the rest must needes be one: For otherwise her traine will be but loose and shattered.
Plato saith τὸν θεὸν ἀκεὶ γεωμετρεῖνThat God doth alwayes worke by Geometry, that is, as the wiseman doth interprete it, Sap. XI. 21. Omnia in mensura & numero & pondere disponere. Dispose all things by measure, and number, and weight: Or, as the learned Plutarch speaketh; He adorneth and layeth out all the parts of the world according to rate, proportion, and similitude. Now who, I pray you, understandeth what these termes meane, but he which hath some meane skill in Geometry? Therefore none but such an one, may be able to declare and teach these things unto others.
How many things are there in holy Scripture which may not well be understood without some meane skill in Geometry? The Fabricke and bignesse of Noah's Arke: The Sciagraphy of the Temple set out by Ezechiel, Who may understand, but he that is skilfull in these Arts? I speake not of many and sundry words both in the New and Old Testaments, whose genuine and proper signification is merely Geometricall: And cannot well be conceived but of a Geometer.
And here, that I may speake it without offence, I would have it observed, how many men, much magnified for learning, not onely in their speeches, which alwayes are not premeditated, but even in their writings, exposed to the view and censure of all men, doe often paralogizein, speake much, and little to the purpose. This they could not so easily and often doe, if they had beene but meanely practised in these kinde of studies. Wherefore that Epigramme which was used to be written over their Philosophy Schoole doores, οὐδῆις ἀγεωμέτρητος εἴσιτωNo man ignorant of Geometry come within these doores: Now written over our Divinitie Schooles. And if any man shall thinke this an hard sentence, let him heare what Saint Augustine saith in the same case, Nemo ad divinarum humanarumq; rerum cognitionem accedat, nisi prius annumerandi artem addiscat: Let no man come neither within the Divinity nor Philosophy Schooles, except he have first learned Arithmeticke. Now that the one of these Arts cannot be learned without the other; Euclide our great Master, who made but one of both, hath sufficiently demonstrated.
If I should alledge the like practise of famous Divines, greatly admired for their great skill in this profession, as T. Peckham Arch-Bishop of Canterbury, Maurolycus Bishop of Messana in Sicilia, Cusanus Cardinall of Rome, and many others, before indifferent judges, I am sure I should not be condemned. Who doth not greatly magnifie the grave Seb. Munster, the nimble Ph. Melanchthon, and the noble Bernardino Baldo Abbot of Guastill, and the painefull Barth. Pitiscus of Grunberg, for their knowledge and paines in these Arts and Sciences? And thus much shall at this time suffice, to have spoken unto the first Question: If any shall require further satisfaction, those I referre unto the forenamed Authors, whose authority peradventure may more prevaile with them, then my reasons may.
The next is concerning mine Author, and the worke in hand Geometry, it must needs be confest we are beholden to Euclides Elements for: And he that would be rich in that profession, may have, if he be not covetous, his fill there, if he will labour hard, and take paines for it, it is true. But in what time thinke yau, may a man learne all Euclide, and so by him be made skilfull in this Art? By himselfe I know not whether ever or never: And with the helpe of another, although very expert, I will not promise him that hee shall attaine to perfection in many yeares.
Hippocrates the Prince of Physicians hath, as they say, in his workes laid out the whole Art of Physicke; but I marvell how long a man should study him alone, and read him over and over, before he should be a good Physician? I feare mee all the friends that he hath, and neighbours round about him, yea, and himselfe too, would all die before he should be able to hele them, or per adventure ere he should be able to know what they ail'd; and after 30, or 40. yeeres of such his study, I would be very loath to commit my selfe unto him. How much therefore are the students of this noble Science beholding unto those men, who by their industry, practise, and painefull travells, have shewed them a ready and certaine way through this wildernesse?
The Elements of Euclide they do containe generally the whole art of Geometry: But if you will offer to travell thorow them alone, you shall finde them, I will warrant you, Elements indeed: for there you may walke through the spacious Aire, and over the great and wide sea, and in and about the vaste and arid wildernesse many a day and night, before you shall know where you are. This Ramus, my Authour in reading him found to be true; and confesseth himselfe often to have beene at a stand: Often to have lost himselfe: Often to have hitte upon a rocke, when he had thought he had touch'd land.
Least therefore other men, in this journey doe not likewise loose themselves, for the benefit and safety, I meane, of others he hath prick'd them out a charde or chack'd out a way, which if thou shalt please to follow, it shall lead thee to thy wayes end, as directly, and in as short time, as conveniently may be. Yet in what time I cannot warrant thee: For all mens capacity, especially in these Arts, is not alike: All are not a like painefull, industrious, or diligent: All are not of the same ability of body, to be able to continue or sit at it: Or all not so free from other imployments or businesse calling them from their study, as some others are. For know this for certaine, Thou shalt here make no great progresse, except thou doe make it as it were a continued labour, Here you must observe that rule of the great Painter, Nulla dies sine linea, Let no day passe over your head, in which you draw not some diagram or figure or other.
One other thing let me also advise thee of, how capable soever thou art, refuse not, if thou maist have it, the helpe of a teacher; For except thou be another Hippocrates or Forcatelus, whō our Authour mentioneth, thou canst not in these Arts and Sciences attaine unto any great perfection without infinite patience and great losse of most precious time, For they are therefore called Μαθηματικόι, Mathematicks, that is, doctrinal or disciplinary Arts, because they are not to be attained unto by our owne information and industry; but by the helpe and instruction of others.
This Worke gentle Reader, was in part above 30. yeares since published by M. Thomas Hood, a learned man, and loving friend of mine, who teaching these Arts, in the Staplers Chappell in Leadenhall London, for the benefit of his Schollers and Auditory, did set out the Elements apart by themselves. The whole at large, with the Diagrammes, and Demonstrations, hee promised, as appeareth in the Preface to that his Worke, at his convenient leysure to send out shortly, after them. This for ought we know or can learne, is not by him or any other performed: And yet are those alone, without these of small use or none to a learner, where a teacher is not alwayes at hand. Wherefore we are bold being (encouraged thereunto by some private friends, and especially by the learned M. H. Brigges, professour of Geometry in the famous Vniversity of Oxford) to publish this of ours long since finished and ended.
The usuall termes, whether Latine or Greeke, commonly used by the Geometers, we have set downe and expressed in English, as well as we could, as others, writing of this argument in our language, have done before us. These termes, I doubt not, may by some in English otherwise be expressed, but how harsh those termes, may unto Mathematicall eares, at the first appeare, I will not say; and use in short time will make these familiar, and as pleasing to the eare as those possibly may be.
Our Authour, in the declaration of the Elements hath many passages, which in our judgement doe not make so much for the understanding of the matter in hand, as for the defence of the method here used, against Aristotle, Euclide, Proclus, and others, which we have therfore wholly omitted. Some other things, which in our opinion, might in some respect illustrate any particular in this businesse, we have here and there inserted. Out of the learned Finkius's Geometria Rotundi, Wee have added to the fifth Booke certaine Propositions with their Consectaries out of Ptolomi's Almagest. The painfull and diligent Rod. Snellius out of the Lectures and Annotations of B. Salignacus, I. Tho. Freigius, and others, hath illustrated and altered here and there some few things.




In issuing this volume of my Mathematical Puzzles, of which some have appeared in periodicals and others are given here for the first time, I must acknowledge the encouragement that I have received from many unknown correspondents, at home and abroad, who have expressed a desire to have the problems in a collected form, with some of the solutions given at greater length than is possible in magazines and newspapers. Though I have included a few old puzzles that have interested the world for generations, where I felt that there was something new to be said about them, the problems are in the main original. It is true that some of these have become widely known through the press, and it is possible that the reader may be glad to know their source.
On the question of Mathematical Puzzles in general there is, perhaps, little more to be said than I have written elsewhere. The history of the subject entails nothing short of the actual story of the beginnings and development of exact thinking in man. The historian must start from the time when man first succeeded in counting his ten fingers and in dividing an apple into two approximately equal parts. Every puzzle that is worthy of consideration can be referred to mathematics and logic. Every man, woman, and child who tries to "reason out" the answer to the simplest puzzle is working, though not of necessity consciously, on mathematical lines. Even those puzzles that we have no way of attacking except by haphazard attempts can be brought under a method of what has been called "glorified trial"—a system of shortening our labours by avoiding or eliminating what our reason tells us is useless. It is, in fact, not easy to say sometimes where the "empirical" begins and where it ends.
When a man says, "I have never solved a puzzle in my life," it is difficult to know exactly what he means, for every intelligent individual is doing it every day. The unfortunate inmates of our lunatic asylums are sent there expressly because they cannot solve puzzles—because they have lost their powers of reason. If there were no puzzles to solve, there would be no questions to ask; and if there were no questions to be asked, what a world it would be! We should all be equally omniscient, and conversation would be useless and idle.
It is possible that some few exceedingly sober-minded mathematicians, who are impatient of any terminology in their favourite science but the academic, and who object to the elusive x and y appearing under any other names, will have wished that various problems had been presented in a less popular dress and introduced with a less flippant phraseology. I can only refer them to the first word of my title and remind them that we are primarily out to be amused—not, it is true, without some hope of picking up morsels of knowledge by the way. If the manner is light, I can only say, in the words of Touchstone, that it is "an ill-favoured thing, sir, but my own; a poor humour of mine, sir."
As for the question of difficulty, some of the puzzles, especially in the Arithmetical and Algebraical category, are quite easy. Yet some of those examples that look the simplest should not be passed over without a little consideration, for now and again it will be found that there is some more or less subtle pitfall or trap into which the reader may be apt to fall. It is good exercise to cultivate the habit of being very wary over the exact wording of a puzzle. It teaches exactitude and caution. But some of the problems are very hard nuts indeed, and not unworthy of the attention of the advanced mathematician. Readers will doubtless select according to their individual tastes.
In many cases only the mere answers are given. This leaves the beginner something to do on his own behalf in working out the method of solution, and saves space that would be wasted from the point of view of the advanced student. On the other hand, in particular cases where it seemed likely to interest, I have given rather extensive solutions and treated problems in a general manner. It will often be found that the notes on one problem will serve to elucidate a good many others in the book; so that the reader's difficulties will sometimes be found cleared up as he advances. Where it is possible to say a thing in a manner that may be "understanded of the people" generally, I prefer to use this simple phraseology, and so engage the attention and interest of a larger public. The mathematician will in such cases have no difficulty in expressing the matter under consideration in terms of his familiar symbols.
I have taken the greatest care in reading the proofs, and trust that any errors that may have crept in are very few. If any such should occur, I can only plead, in the words of Horace, that "good Homer sometimes nods," or, as the bishop put it, "Not even the youngest curate in my diocese is infallible."
I have to express my thanks in particular to the proprietors of The Strand MagazineCassell's MagazineThe QueenTit-Bits, and The Weekly Dispatch for their courtesy in allowing me to reprint some of the puzzles that have appeared in their pages.
March 25, 1917

Philosophiae Naturalis Principia Mathematica by Isaac Newton





Cum Veteres Mechanicam (uti Author est Pappus) in verum Naturalium investigatione maximi fecerint, & recentiores, missis formis substantialibus & qualitatibus occultis, Phænomena Naturæ ad leges Mathematicas revocare aggressi sint: Visum est in hoc Tractatu Mathesin excolere quatenus ea ad Philosophiam spectat. Mechanicam vero duplicem Veteres constituerunt: Rationalem quæ per Demonstrationes accurate procedit, & Practicam. Ad practicam spectant Artes omnes Manuales, a quibus utiq; Mechanica nomen mutuata est. Cum autem Artifices parum accurate operari soleant, fit ut Mechanica omnis a Geometria ita distinguatur, ut quicquid accuratum sit ad Geometriam referatur, quicquid minus accuratum ad Mechanicam. Attamen errores non sunt Artis sed Artificum. Qui minus accurate operatur, imperfectior est Mechanicus, & si quis accuratissime operari posset, hic foret Mechanicus omnium perfectissimus. Nam & Linearum rectarum & Circulorum descriptiones in quibus Geometria fundatur, ad Mechanicam pertinent. Has lineas describere Geometria non docet sed postulat. Postulat enim ut Tyro easdem accurate describere prius didicerit quam limen attingat Geometriæ; dein, quomodo per has operationes Problemata solvantur, docet. Rectas & circulos describere Problemata sunt sed non Geometrica. Ex Mechanica postulatur horum solutio, in Geometria docetur solutorum usus. Ac gloriatur Geometria quod tam paucis principiis aliunde petitis tam multa præstet. Fundatur igitur Geometria in praxi Mechanica, & nihil aliud est quam Mechanicæ universalis pars illa quæ artem mensurandi accurate proponit ac demonstrat. Cum autem artes Manuales in corporibus movendis præcipue versentur, fit ut Geometria ad magnitudinem, Mechanica ad motum vulgo reseratur. Quo sensu Mechanica rationalis erit Scientia Motuum qui ex viribus quibuscunq; resultant, & virium quæ ad motus quoscunq; requiruntur, accurate proposita ac demonstrata. Pars hæc Mechanicæ a Veteribus in Potentiis quinque ad artes manuales spectantibus exculta fuit, qui Gravitatem (cum potentia manualis non sit) vix aliter quam in ponderibus per potentias illas movendis considerarunt. Nos autem non Artibus sed Philosophiæ consulentes, deq; potentiis non manualibus sed naturalibus scribentes, ea maxime tractamus quæ ad Gravitatem, levitatem, vim Elasticam, resistentiam Fluidorum & ejusmodi vires seu attractivas seu impulsivas spectant: Et ea propter hæc nostra tanquam Philosophiæ principia Mathematica proponimus. Omnis enim Philosophiæ difficultas in eo versari videtur, ut a Phænomenis motuum investigemus vires Naturæ, deinde ab his viribus demonstremus phænomena reliqua. Et hac spectant Propositiones generales quas Libro primo & secundo pertractavimus. In Libro autem tertio exemplum hujus rei proposuimus per explicationem Systematis mundani. Ibi enim, ex phænomenis cælestibus, per Propositiones in Libris prioribus Mathematice demonstratas, derivantur vires gravitatis quibus corpora ad Solem & Planetas singulos tendunt. Deinde ex his viribus per Propositiones etiam Mathematicas deducuntur motus Planetarum, Cometarum, Lunæ & Maris. Utinam cætera Naturæ phænomena ex principiis Mechanicis eodem argumentandi genere derivare liceret. Nam multa me movent ut nonnihil suspicer ea omnia ex viribus quibusdam pendere posse, quibus corporum particulæ per causas nondum cognitas vel in se mutuo impelluntur & secundum figuras regulares cohærent, vel ab invicem fugantur & recedunt: quibus viribus ignotis, Philosophi hactenus Naturam frustra tentarunt. Spero autem quod vel huic Philosophandi modo, vel veriori alicui, Principia hic posita lucem aliquam præbebunt.
In his edendis, Vir acutissimus & in omni literarum genere eruditissimus Edmundus Halleius operam navavit, nec solum Typothetarum Sphalmata correxit & Schemata incidi curavit, sed etiam Author fuit ut horum editionem aggrederer. Quippe cum demonstratam a me figuram Orbium cælestium impetraverat, rogare non destitit ut eadem cum Societate Regali communicarem, Quæ deinde hortatibus & benignis suis auspiciis effecit ut de eadem in lucem emittenda cogitare inciperem. At postquam Motuum Lunarium inæqualitates aggressus essem, deinde etiam alia tentare cæpissem quæ ad leges mensuras Gravitatis & aliarum virium, ad figuras a corporibus secundum datas quascunque leges attractis describendas, ad motus corporum plurium inter se, ad motus corporum in Mediis resistentibus, ad vires, densitates & motus Mediorum, ad Orbes Cometarum & similia spectant, editionem in aliud tempus differendam esse putavi, ut cætera rimarer & una in publicum darem. Quæ ad motus Lunares spectant, (imperfecta cum sint,) in Corollariis Propositionis LXVI. simul complexus sum, ne singula methodo prolixiore quam pro rei dignitate proponere, & sigillatim demonstrare tenerer, & seriem reliquarum Propositionum interrumpere. Nonnulla sero inventa locis minus idoneis inserere malui, quam numerum Propositionum & citationes mutare. Ut omnia candide legantur, & defectus, in materia tam difficili non tam reprehendantur, quam novis Lectorum conatibus investigentur, & benigne suppleantur, enixe rogo.






Sæculi Gentisque nostræ Decus egregium.
En tibi norma Poli, & divæ libramina Molis,
Computus atque Jovis; quas, dum primordia rerum
Pangeret, omniparens Leges violare Creator
Noluit, æternique operis fundamina fixit.
Intima panduntur victi penetralia cæli,
Nec latet extremos quæ Vis circumrotat Orbes.
Sol solio residens ad se jubet omnia prono
Tendere descensu, nec recto tramite currus
Sidereos patitur vastum per inane moveri;
Sed rapit immotis, se centro, singula Gyris.
Jam patet horrificis quæ sit via flexa Cometis;
Jam non miramur barbati Phænomena Astri.
Discimus hinc tandem qua causa argentea Phœbe
Passibus haud æquis graditur; cur subdita nulli
Hactenus Astronomo numerorum fræna recuset:
Cur remeant Nodi, curque Auges progrediuntur.
Discimus & quantis refluum vaga Cynthia Pontum
Viribus impellit, dum fractis fluctibus Ulvam
Deserit, ac Nautis suspectas nudat arenas;
Alternis vicibus suprema ad littora pulsans.
Quæ toties animos veterum torsere Sophorum,
Quæque Scholas frustra rauco certamine vexant
Obvia conspicimus nubem pellente Mathesi.
Jam dubios nulla caligine prægravat error
Queis Superum penetrare domos atque ardua Cœli
Scandere sublimis Genii concessit acumen.
Surgite Mortales, terrenas mittite curas
Atque hinc cœligenæ vires dignoscite Mentis
A pecudum vita longe lateque remotæ.
Qui scriptis jussit Tabulis compescere Cædes
Furta & Adulteria, & perjuræ crimina Fraudis;
Quive vagis populis circumdare mœnibus Urbes
Autor erat; Cererisve beavit munere gentes;
Vel qui curarum lenimen pressit ab Uva;
Vel qui Niliaca monstravit arundine pictos
Consociare sonos, oculisque exponere Voces;
Humanam sortem minus extulit; utpote pauca
Respiciens miseræ solummodo commoda vitæ.
Jam vero Superis convivæ admittimur, alti
Jura poli tractare licet, jamque abdita cœcæ
Claustra patent Terræ rerumque immobilis ordo,
Et quæ præteriti latuerunt sæcula mundi.
Talia monstrantem mecum celebrate Camænis,
Vos qui cœlesti gaudetis nectare vesci,
NEWTONVM clausi reserantem scrinia Veri,
NEWTONVM Musis charum, cui pectore puro
Phœbus adest, totoque incessit Numine mentem:
Nec fas est propius Mortali attingere Divos.







Def. I.

Quantitas Materiæ est mensura ejusdem orta ex illius Densitate & Magnitudine conjunctim.
Aer duplo densior in duplo spatio quadruplus est. Idem intellige de Nive et Pulveribus per compressionem vel liquefactionem condensatis. Et par est ratio corporum omnium, quæ per causas quascunq; diversimode condensantur. Medii interea, si quod fuerit, interstitia partium libere pervadentis, hic nullam rationem habeo. Hanc autem quantitatem sub nomine corporis vel Massæ in sequentibus passim intelligo. Innotescit ea per corporis cujusq; pondus. Nam ponderi proportionalem esse reperi per experimenta pendulorum accuratissime instituta, uti posthac docebitur.[2]

Def. II.

Quantitas motus est mensura ejusdem orta ex Velocitate et quantitate Materiæ conjunctim.
Motus totius est summa motuum in partibus singulis, adeoq; in corpore duplo majore æquali cum Velocitate duplus est, et dupla cum Velocitate quadruplus.

Def. III.

Materiæ vis insita est potentia resistendi, qua corpus unumquodq;, quantum in se est, perseverat in statu suo vel quiescendi vel movendi uniformiter in directum.
Hæc semper proportionalis est suo corpori, neq; differt quicquam ab inertia Massæ, nisi in modo concipiendi. Per inertiam materiæ fit ut corpus omne de statu suo vel quiescendi vel movendi difficulter deturbetur. Unde etiam vis insita nomine significantissimo vis inertiæ dici possit. Exercet vero corpus hanc vim solummodo in mutatione status sui per vim aliam in se impressam facta, estq; exercitium ejus sub diverso respectu et Resistentia et Impetus: Resistentia quatenus corpus ad conservandum statum suum reluctatur vi impressæ; Impetus quatenus corpus idem, vi resistentis obstaculi difficulter cedendo, conatur statum ejus mutare. Vulgus Resistentiam quiescentibus et Impetum moventibus tribuit; sed motus et quies, uti vulgo concipiuntur, respectu solo distinguuntur ab invicem, neq; semper vere quiescunt quæ vulgo tanquam quiescentia spectantur.

Def. IV.

Vis impressa est actio in corpus exercita, ad mutandum ejus statum vel quiescendi vel movendi uniformiter in directum.
Consistit hæc vis in actione sola, neq; post actionem permanet in corpore. Perseverat enim corpus in statu omni novo per solam [3]vim inertiæ. Est autem vis impressa diversarum originum, ut ex ictu, ex pressione, ex vi centripeta.

Def. V.

Vis centripeta est qua corpus versus punctum aliquod tanquam ad centrum trahitur, impellitur, vel utcunq; tendit.
Hujus generis est gravitas, qua corpus tendit ad centrum Terræ: Vis magnetica, qua ferrum petit centrum Magnetis, et vis illa, quæcunq; sit, qua Planetæ perpetuo retrahuntur a motibus rectilineis, et in lineis curvis revolvi coguntur. Est autem vis centripetæ quantitas trium generum, absoluta, acceleratrix et motrix.

Def. VI.

Vis centripetæ quantitas absoluta est mensura ejusdem major vel minor pro efficacia causæ eam propagantis a centro per regiones in circuitu.
Uti virtus Magnetica major in uno magnete, minor in alio.

Def. VII.

Vis centripetæ quantitas acceleratrix est ipsius mensura Velocitati proportionalis, quam dato tempore generat.
Uti Virtus Magnetis ejusdem major in minori Distantia, minor in majori: vel vis gravitans major in Vallibus, minor in cacuminibus præaltorum montium (ut experimento pendulorum constat) atq; adhuc minor (ut posthac patebit) in majoribus distantiis a Terra; in æqualibus autem distantiis eadem undiq; propterea quod corpora omnia cadentia (gravia an levia, magna an parva) sublata Aeris resistentia, æqualiter accelerat.

Def. VIII.

Vis centripetæ quantitas motrix est ipsius mensura proportionalis motui, quem dato tempore generat.
Uti pondus majus in majori corpore, minus in minore; inq; [4]corpore eodem majus prope terram, minus in cælis. Hæc vis est corporis totius centripetentia seu propensio in centrum & (ut ita dicam) pondus, & innotescit semper per vim ipsi contrariam & æqualem, qua descensus corporis impediri potest.
Hasce virium quantitates brevitatis gratia nominare licet vires absolutas, acceleratrices & motrices, & distinctionis gratia referre ad corpora, ad corporum loca, & ad centrum virium: Nimirum vim motricem ad corpus, tanquam conatum & propensionem totius in centrum, ex propensionibus omnium partium compositum; & vim acceleratricem ad locum corporis, tanquam efficaciam quandam, de centro per loca singula in circuitu diffusam, ad movenda corpora quæ in ipsis sunt; vim autem absolutam ad centrum, tanquam causa aliqua præditum, sine qua vires motrices non propagantur per regiones in circuitu; sive causa illa sit corpus aliquod centrale (quale est Magnes in centro vis Magneticæ vel Terra in centro vis gravitantis) sive alia aliqua quæ non apparet. Mathematicus saltem est hic conceptus. Nam virium causas & sedes physicas jam non expendo.
Est igitur vis acceleratrix ad vim motricem ut celeritas ad motum. Oritur enim quantitas motus ex celeritate ducta in quantitatem Materiæ, & vis motrix ex vi acceleratrice ducta in quantitatem ejusdem materiæ. Nam summa actionum vis acceleratricis in singulas corporis particulas est vis motrix totius. Unde juxta Superficiem Terræ, ubi gravitas acceleratrix seu vis gravitans in corporibus universis eadem est, gravitas motrix seu pondus est ut corpus: at si in regiones ascendatur ubi gravitas acceleratrix fit minor, pondus pariter minuetur, eritq; semper ut corpus in gravitatem acceleratricem ductum. Sic in regionibus ubi gravitas acceleratrix duplo minor est, pondus corporis duplo vel triplo minoris erit quadruplo vel sextuplo minus.
Porro attractiones et impulsus eodem sensu acceleratrices & motrices nomino. Voces autem attractionis, impulsus vel propensionis cujuscunq; in centrum, indifferenter et pro se mutuo promiscue usurpo, has vires non physice sed Mathematice tantum [5]considerando. Unde caveat lector ne per hujusmodi voces cogitet me speciem vel modum actionis causamve aut rationem physicam alicubi definire, vel centris (quæ sunt puncta Mathematica) vires vere et physice tribuere, si forte aut centra trahere, aut vires centrorum esse dixero.


Hactenus voces minus notas, quo in sensu in sequentibus accipiendæ sunt, explicare visum est. Nam tempus, spatium, locum et motum ut omnibus notissima non definio. Dicam tamen quod vulgus quantitates hasce non aliter quam ex relatione ad sensibilia concipit. Et inde oriuntur præjudicia quædam, quibus tollendis convenit easdem in absolutas & relativas, veras & apparentes, Mathematicas et vulgares distingui.
I. Tempus absolutum verum & Mathematicum, in se & natura sua absq; relatione ad externum quodvis, æquabiliter fluit, alioq; nomine dicitur Duratio; relativum apparens & vulgare est sensibilis & externa quævis Durationis per motum mensura, (seu accurata seu inæquabilis) qua vulgus vice veri temporis utitur; ut Hora, Dies, Mensis, Annus.
II. Spatium absolutum natura sua absq; relatione ad externum quodvis semper manet similare & immobile; relativum est spatii hujus mensura seu dimensio quælibet mobilis, quæ a sensibus nostris per situm suum ad corpora definitur, & a vulgo pro spatio immobili usurpatur: uti dimensio spatii subterranei, aerei vel cælestis definita per situm suum ad Terram. Idem sunt spatium absolutum & relativum, specie & magnitudine, sed non permanent idem semper numero. Nam si Terra, verbi gratia, movetur, spatium Aeris nostri quod relative & respectu Terræ semper manet idem, nunc erit una pars spatii absoluti in quam Aer transit, nunc alia pars ejus, & sic absolute mutabitur perpetuo.
III. Locus est pars spatii quam corpus occupat, estq; pro [6]ratione spatii vel absolutus vel relativus. Partem dico spatii, non situm corporis vel superficiem ambientem. Nam solidorum æqualium æquales semper sunt loci; Superficies autem ob dissimilitudinem figurarum ut plurimum inæquales sunt; situs vero proprie loquendo quantitatem non habent, neq; tam sunt loca quam affectiones locorum. Motus totius idem est cum summa motuum partium, hoc est, translatio totius de ipsius loco eadem cum summa translationum partium de locis suis, adeoq; locus totius idem cum summa locorum partium, & propterea internus & in corpore toto.
IV. Motus absolutus est translatio corporis de loco absoluto in locum absolutum, relativus de relativo in relativum. Sic in Navi quæ velis passis fertur, relativus corporis locus est navis regio illa in qua corpus versatur, seu cavitatis totius pars illa quam corpus implet, quæq; adeo movetur una cum Navi; & Quies relativa est permansio corporis in eadem illa navis regione vel parte cavitatis. At Quies vera est permansio corporis in eadem parte spatii illius immoti in qua Navis ipsa una cum cavitate sua & contentis universis movetur. Unde si Terra vere quiescit, corpus quod relative quiescit in Navi, movebitur vere et absolute ea cum Velocitate qua Navis movetur in Terra. Sin Terra etiam movetur, orietur verus et absolutus corporis motus partim ex Terræ motu vero in spatio immoto, partim ex Navis motu relativo in Terra; et si corpus etiam movetur relative in Navi, orietur verus ejus motus partim ex vero motu Terræ in spatio immoto, partim ex relativis motibus tum Navis in Terra, tum corporis in Navi, et ex his motibus relativis orietur corporis motus relativus in Terra. Ut si Terræ pars illa ubi Navis versatur moveatur vere in Orientem, cum Velocitate partium 10010, et velis ventoq; feratur Navis in Occidentem cum Velocitate partium decem, Nauta autem ambulet in Navi Orientem versus cum Velocitatis parte una, movebitur Nauta vere et absolute in spatio immoto cum Velocitatis partibus 10001 in Orientem, et relative in Terra Occidentem versus cum Velocitatis partibus novem.[7]
Tempus absolutum a relativo distinguitur in Astronomia per Æquationem Temporis vulgi. Inæquales enim sunt dies Naturales, qui vulgo tanquam æquales pro Mensura Temporis habentur. Hanc inæqualitatem corrigunt Astronomi ut ex veriore Tempore mensurent motus cælestes. Possibile est ut nullus sit motus æquabilis quo Tempus accurate mensuretur. Accelerari & retardari possunt motus omnes, sed fluxus Temporis absoluti mutari nequit. Eadem est duratio seu perseverantia existentiæ rerum, sive motus sint celeres, sive tardi, sive nulli; proinde hæc a mensuris suis sensibilibus merito distinguitur, & ex ijsdem colligitur per Æquationem Astronomicam. Hujus autem æquationis in determinandis Phænomenis necessitas, tum per experimentum Horologii oscillatorii, tum etiam per Eclipses Satellitum Jovis evincitur.
Ut partium Temporis ordo est immutabilis, sic etiam ordo partium Spatii. Moveantur hæ de locis suis, & movebuntur (ut ita dicam) de seipsis. Nam Tempora & Spatia sunt sui ipsorum & rerum omnium quasi loca. In Tempore quoad ordinem successionis; in Spatio quoad ordinem situs locantur universa. De illorum Essentia est ut sint loca, & loca primaria moveri absurdum est. Hæc sunt igitur absoluta loca, & solæ translationes de his locis sunt absoluti motus.
Verum quoniam hæ spatii partes videri nequeunt, & ab invicem per sensus nostros distingui, earum vice adhibemus mensuras sensibiles. Ex positionibus enim & distantiis rerum a corpore aliquo, quod spectamus ut immobile, definimus loca universa; deinde etiam & omnes motus æstimamus cum respectu ad prædicta loca, quatenus corpora ab iisdem transferri concipimus. Sic vice locorum & motuum absolutorum relativis utimur, nec incommode in rebus humanis: in Philosophicis autem abstrahendum est a sensibus. Fieri etenim potest ut nullum revera quiescat corpus, ad quod loca motusq; referantur.
Distinguuntur autem Quies & Motus absoluti & relativi ab invicem per eorum proprietates, causas & effectus. Quietis proprietas [8]est, quod corpora vere quiescentia quiescunt inter se. Ideoq; cum possibile sit ut corpus aliquod in regionibus fixarum, aut longe ultra, quiescat absolute; sciri autem non possit ex situ corporum ad invicem in regionibus nostris, utrum horum aliquod ad longinquum illud datam positionem servet, quies vera ex horum situ inter se definiri nequit.
Motus proprietas est, quod partes quæ datas servant positiones ad tota, participant motus eorundem totorum. Nam gyrantium partes omnes conantur recedere de axe motus, et progredientium impetus oritur ex conjuncto impetu partium singularum. Igitur motis corporibus ambientibus, moventur quæ in ambientibus relative quiescunt. Et propterea motus verus et absolutus definiri nequit per translationem e vicinia corporum, quæ tanquam quiescentia spectantur. Debent corpora externa non solum tanquam quiescentia spectari, sed etiam vere quiescere. Alioquin inclusa omnia, præter translationem e vicinia ambientium, participabunt etiam ambientium motus veros, et sublata illa translatione non vere quiescent, sed tanquam quiescentia solummodo spectabuntur; sunt enim ambientia ad inclusa ut totius pars exterior ad partem interiorem, vel ut cortex ad nucleum. Moto autem cortice, nucleus etiam, absq; translatione de vicinia corticis, ceu pars totius, movetur.
Præcedenti proprietati affinis est, quod moto loco movetur una locatum, adeoq; corpus, quod de loco moto movetur, participat etiam loci sui motum. Igitur motus omnes, qui de locis motis fiunt, sunt partes solummodo motuum integrorum et absolutorum, et motus omnis integer componitur ex motu corporis de loco suo primo, et motu loci hujus de loco suo, et sic deinceps, usq; dum perveniatur ad locum immotum, ut in exemplo Nautæ supra memorato. Unde motus integri et absoluti non nisi per loca immota definiri possunt, et propterea hos ad loca immota, relativos ad mobilia supra retuli: Loca autem immota non sunt, nisi quæ omnia ab infinito in infinitum datas [9]servant positiones ad invicem, atq; adeo semper manent immota, spatiumq; constituunt quod immobile appello.