PRINCIPIA. 



PRIXCIPIA. 



Ml 



U public. However, I remember thst Sir Christopher WM little 

 satisfied tbt he could do it ; and though Mr. Hooke then promised to 

 how U him, I do Dot find that in that particular ho has been 10 good 

 M hi wont. The August following, when I did myself the honour to 

 rUit you, I then leant the good news that you had brought this 

 demonstration to perfection : and you were pleated to promise me a 

 copy thereof, which the November following I received with a great 

 deal of satisfaction from Mr. Paget; and thereupon took nntli>r 

 journey to Cambridge, on purpose to confer with you about it, since 

 which time it has been entered upon the Register books of the Society. 

 As all this passed, Mr. Hooke was acquainted with it, and according to 

 the philosophically ambitious temper he is of, he would, had he been 

 master of a like demonstration, no longer have concealed it, the reason 

 he told Sir Christopher and me now ceasing. But now, he says, thi- i 

 bat one small part of an excellent system of natm v, whieh lie has 

 eonoeived, but has not yet completely mode out, so that he thinks not 

 fit to publish one part without the other. But I have plainly told lain, 

 that unless he produce another differing demonstration, and let the 

 world judge of it, neither I nor any one else can believe it. As to the 

 manner of Mr. Hooke's claiming the discovery, I fear it has been 

 represented in worse colours than it ought ; for he neither made 

 public application to the Society for justice, nor pretended you hod 

 all from him. The truth is this : Sir John Hoskyns, his particular 

 friend, being in the chair when Dr. Vincent presented your book, the 

 Doctor gave it its just encomium both as to the novelty and dignity 

 of the subject It was replied by another gentleman, that you had 

 carried the thing so far, that there was no more to be added. To 

 which the Vice-president replied, that it was so much the more to be 

 prized, for that it was both invented and perfected at the same time. 

 This gave Mr. Hooke offence, that Sir John did not, at that time, 

 make mention of what he hod, as he said, discovered to him ; upon 

 which they two, who till then were the most inseparable cronies, have 

 since scarce seen one another, and are utterly fallen out. After the 

 breaking up of that meeting, being adjourned to the coffee-house, 

 Mr. Hooke did there endeavour to gain belief, that ho had some such 

 thing by him, and that he gave you the first hint of this invention. 

 But I found, that they were all of opinion, that, nothing thereof 

 appearing in print, nor on the books of the Society, you ought to be 

 considered as the inventor. And if in truth he knew it before you, 

 he ought not to blame any but himself, for liaviug token no more 

 care to secure a discovery which he puts so much value on. What 

 application he has made in private, I know not ; but I am sure that 

 the Society have a very great satisfaction, in the honour you do them, 

 by the dedication of so worthy a treatise. Sir, I must now again beg 

 you, not to let your resentments run so high, as to deprive us of your 

 third book, wherein the application of your mathematical doctrine 

 to the theory of comets and several curious experiments, which, as I 

 guess by what you write, ought to compose it, will undoubtedly render 

 it acceptable to those who will call themselves Philosophers without 

 Mathematics, which are much the greater number. Now you approve 

 of the character and f^aper, I will push on the edition vigorously. I 

 have sometimes hod thoughts 'of having the cuts neatly done in wood, 

 so as to stand in the page with the demonstrations. It will be more 

 convenient, and not much more charge. If it please you to have it 

 so, I will try how well it con be done ; otherwise I will have them in 

 somewhat a larger size than those you have sent up. I am, Sir, 

 your most affectionate humble sen-ant, " . HALLEY." 



The authorities on this subject are 1, Rigaud, ' Historical Essay 

 on the First Publication of the Principia,' Oxford, 1838, 8vo. ; 2, Ed- 

 leston, ' Correspondence of Sir I. Newton and Prof. Cotes,' London, 

 1860, 8vo. ; 8, firewater, ' Memoirs of Sir I. Newton," London, 1855, 

 2 vuls. 8vo. Kigaud's research was of the most acute and accurate 

 character. Mr. Edleston has added a biography in the form of annals 

 with notes, the most convenient reference on Newton which exists : 

 and his researches, especially, those made in Trinity College, have been 

 laborious and successful. Sir D. Brewster had the celebrated Ports- 

 mouth papers in his possession, and has produced the most detailed 

 life of Newton which exists. An immense deal has been done for tin- 

 biography of Newton in the last twenty-five years : but the subject 

 will never be finally settled until the Portsmouth papers have been 

 allowed to undergo the most searching and leisurely scrutiny from men 

 of all opinions on disputed points. 



Newton had completed the first draught of the work by the cn<! i 

 1685, though none of it was sent till April, 1680. The third l>ook was 

 presented to the Society, April 6, 1687, in proof, probably, since the 

 whole was published (at ten or twelve shillings a copy) about Mid- 

 rummer, 1687. The order of the Council of the Society to license the 

 book was made on June 20, 1636, and the imprimatur of IVpys was 

 dated July 5. 



The part which Halley had in the matter would alone immortalise 

 bis name. He found out the ability of Newton to write such a work, 

 prevailed upon him to write it, took charge of the publication, pic- 

 vented the author from materially mutilating it in disgust, paiil the 

 expenses of printing, at a time when, owing to his father's death and 

 consequent litigation, he hod nothing to spare (which never happened 

 to him before or after), gave a copious explanation of it in tin- I 

 phical Transactions, and is generally admitted to have been for a long 



time the only person in Europe who showed that he thoroughly 

 ited the value of the work, and knew the place it must occupy 

 in the history of discovery. 



The interest attached to the second and third editions of the 

 Principia (snparinttndad by Cotes in 1713, and by Pembert.m in 1 7-H) 

 is considerable, with reference to the alterations made in them by 

 Newton. It would not however be worth while to specify these 

 alterations, which are numerous, some in correction of errors,' 

 in extension of views. With reference to the suppression of the 

 celebrated Scholium, see COMJIKHCMM EnsTOLlci'U and Kn \ 



The Principia of Newton contains the dedication to the Royal 

 Society, a short preface, verses by H.ilify in honour of N 

 nit ions, axioms, a first book on unresisted motion, a second <>u 

 resisted motion, and a third on the system of the universe. 11 

 verses were somewhat altered by Bentley in the second edition, l,nt 

 the original readings were very nearly restored in the third. N 

 wrote a short preface for each of the editions, and Cotes one < 

 adorable length for the second. The dates of the Newtonian prefaces 

 are. May 8, 1686; March 28,1713; January 12,1725-6. The follow-in;; 

 is the description of the contents of the third edition : 



The definitions comprise, 1. Quantity of rnnttvr measured by <1< 

 and volume jointly. 2. Quantity of motion [ MUMI vn M] liy\. 

 and quantity of matter jointly. 3. Vis insita, or 'vuim-iti.r. [INK.KTIA.] 

 4. Vis impressa, or external force. 5. Centripetal force. 6. Absolute 

 magnitude of centripetal force. 7. Accelerating force. 8. Moving 

 force. A scholium is added on time, space, and motion, the latter 

 considered absolutely and relatively. 



The axioms are the three laws of motion [MOTION, LAWS OF] niul 

 certain corollaries, namely : 1. The composition of velocitie 

 forces. 2. Their resolution, and deduction of the property of the 

 lever. 3. Momentum of a system in a given direction not changed by 

 the mutual action of the ]>arts. 4. Nor the motion of the cei 

 gravity. 5. Relative motion of bodies not altered by absolute i 

 of the space they move in. 6. Nor by equal and parallel acccli 

 forces applied to all A scholium is added, containing the oxpcrii 

 verifications of the third law in the cases of impact, attraction 

 operation of machines. 



THE FIRST BOOK * on unresisted motion, consists of fourteen sec- 

 tions, and ninety-eight propositions. The numbers in parentheses refer 

 to the propositions. 



Section 1 contains eleven lemmas and a scholium. This section is 

 explanatory of Newton's peculiar mode of reasoning, which subject, 

 with the contents of this section, will be treated under RATIOS, PKIXE 

 AND ULTIMATE. 



Section 2. On Centripetal Force. (1) Equal areas are descril 

 equal times. Six corollaries on the comparison of velocities and forces ; 

 the former inversely as the perpendiculars on the tangents, the latter 

 as the sagitbe of ores described in equal times. (2) If equal areas bo 

 described in equal times about a centre, fixed'or moving straightly and 

 uniformly, the force is centripetal. Two corollaries and scholium. 

 (3) In equiareol motion of a point about a moving centre, that point i . 

 acted on by a centripetal force, and by all the accelerating i 

 which act on the centre. Four corollaries and scholium. (4) In 

 different circles uniformly described, force varies as (vel) 3 -7- rad. Nine 

 corollaries and scholium* indicating the deduction in the case of (lie 

 planets. (5) Given the velocities in different parts of an orbit, to find 

 the centre of force. (6) Centripetal force in the middle of a small arc 

 is as sagitta -=- (time) 1 . Five corollaries; various ways of coin] 

 forces. (7) The orbit circular, centre anywhere within, to find tl 

 of force. 8 Cor. (8) Ditto, ditto, where the forces act in parallel 

 lines. Scholium ; same considerations apply to other conic sections. 

 (9) Law of force in equiangular spiral. Lemma 12 (the numbei 

 the lemmas runs on from the first section). Equality of parallelogram 

 about conjugates in conic sections. (10) Law of force in ellipse about 

 the centre. 2 Cor. and Schol. ; extension to the parabola. 



n 3. Motion in conic sections about the focus. (1 1 ) Law of force 

 in ellipse about focus. (12) Same for hyperbola. Lomma 1, 

 rectum in parabola always four times focal distance. Lemma 1 I 

 pendicular on tangent of parabola, mean between focal distances of 

 point of contact and vertex. 3 Cor. (13) Law of force in parabola 

 about focus. 2 Cor. (1-1) In conic Hcctions about same focal centre, 

 latent recta are in duplicate ratio of areas described. 1 Cor. (l;'n In 

 ellipses, periodic times are in sesquiplicate ratio of major axes. 1 Cor. 

 (16) And velocities are as perpendiculars oc tangents inversely, and 

 subduplicate ratio of latera recta directly. 9 Cor. ; comparison of 

 velocity in conic section and circle. (17) Given initial position and 

 velocity, required conic section described. 4 Cor. and .S-lml. 



a -I. ( In finding conic sections from a given focus, anil Fcctinn "i. 

 On finding conic sections of which no focus is given. These sections, 

 which carry on the lemmas from 15 to 27, both inclusive, and the pro- 

 positions from (18) to (29), both inclusive, are entirely geometrical 



This description is for reference, not for explanation, and in therefore very 

 briefly given. To gtre an account of every corollary and scholium would have 

 extended the article to a great length ; hence only eome of the more important 

 of them are described. But the rcfcrc nce to other articles In this work an- for 

 the elementary student. At the same time, u young student who can as much 

 a undi-rstand the meaning of the terms will U-arn more about the Principia 

 from this table of content! than from anything except th Principia itself. 



