LIBRARY UNiVERSlTY OF CALIFORNIA PAVl§ Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/correspondenceofOOnewtrich This "0-P Book" Is an Authorized Reprint of the Original Edition, Produced by Microfilm-Xerography by University Microfilms, Inc., Ann Arbor, Michigan, 1965 COERESPONDENCE ^ OF SIR ISAAC NEWTON AND PROFESSOR COTES, INCI.UD1NO LETTERS OF OTHER EMINENT MEN, NOW FIRST PUDLISIIED FROM THE ORIGINALS IN THE LIBRARY OF TRINITY COLLEGE, CAMBRIDGE; TOOETHER WITH AN APPENDIX, OONTAININO OTHER UNPUBLISHED LETTERS AND PAPERS BY NEWTON; ^\1TH NOTES, SYNOPTICAL VIEW OF THE PHEOSOPHEH'S LIFE, AND A VARIETY OF DETAILS ILLUSTRATIVE OP HIS HISTORY, BY J. EDLESTON, M.A., FELLOW OP THIKTTY OOLLEOE, CAMBRIDGE. LONDON: JOHN W. PARKER, WEST STRAND. CAMBRIDGE : JOHN DEIGHTON. M.DCCO.L. LIBRARY TTNTVERSITY OF CALIFORNIA ./ X 19r(nUti St ttt dntOtrsits Vrcss. CONTENTS. Words, or parts of words, enclosed wUhIn { } hare, with on© exception in p. 218, been added by the Editor for the purpose of illiutmtion or to supply omissions in the original MS. Preface Synoptical View op Newton's Lipr ..... N'oTEs xli PAOB ix xxi Newton's Dividends and Weeks of Rteidenco Exits and Redits . . . . Buttery Bills .... Lectures on Optics Algebra . Motion System of the World LETTEn I. Bentley to Cotes, May 21, 1700 11. Cotes to Newton, Aug. 18, ..... III. Newton to Cotes, Oct. 11, IV. Cctes to Newton, April 15, 1710 V. 30, VI. Newton to Cotes, May 1, VII. Cotes to Newton, 7, VIII. Newton to Cotes, 13, iX. Cotes to Newton, 17, X. 20, XI. Newton to Cotes, 80, XII. Cotes to Newton, June 1, ...... XIII. Newton to Cotes, 8, XIV. Cotes to Ne\vton, 11, XV. Newton to Cotes, 15, XVI. Cotes to Newton, 30, Ixxxii Ixxxv Ixxxvi xci xcii xcv ycviil 1 3 4 8 12 14 10 10 20 24 25 27 20 30 31 415479 IV CONTENTS. LfTTEn XVII. Nowton to Cotos, Juno 31,1710. XVIII. Cotes to Newton, Sept. 4, XIX. Newton to Cotes, 13, XX. Cotes to Newton, 21, XXI. Newton to Cotes, 30, XXII Oct. 27, XXIII Mar. 24, 1711. . XXIV Juno 7, XXV. Cotes to Newton, 9, . XXVI. Newton to Cotes, 10, XXVII. Cotes to Newton, 23, XXVIII July 19, XXIX. Newton to Cotea?^,. . . S8, . ...... XXX. Cotes to Newton, 30, XXXI. Sept. 4, XXXII. Newton to Cotes, Feb. 2, 1712 XXXIII. Cotes to Newton, 7. XXXI V. Newton to Cotes, 12, XXXV. Cotes to Newton, IG, XXXVI. Newton to Cotes, 19, XXXVII. Cotes to Newton, 23, XXX VI 1 1. Nowton to Cotes, 2G, XXXIX. Cotes to Newton. 20, XL. Cotes to Newton, Mar. 13, XLI. Newton to Cotes, 18, XLII April 3, XLIII 0, XLIV. Cotes to Newton, 14, XLV 16, XLVI. Newton to Cotes, 22, XLVI I. Cotes to Newton, 24, XLVIII 20, XLIX. Newton to Cotes, 24, (? 29) L. Cotes to Newton, May day, LI May 3, LII. Newton to Cotes, 10, Lin. Cotes to Newton, 13, LIV. 25, LV. Newton to Cotes, 27, CONTENTS. I.K1TI!R FAOI LVI. Cotes to Newton, July 20, 1712 118 LVII. Aug. 10 121 LVIII. Newton to Cotes, 12, 122 LIX. l^.aH) 12G LX. Cotes to Newton, 17, 127 LXI. Newton to Cotes, 2G, 120 LXII. Cotes to Newton, 20 132 LXIII. Newton to Cotes, Sept. 2, 134 LXIV. Cotes to Newton, 7, 13(5 LXV. Newton to Cotes, Sept. 13, 139 LXVI. Cotes to Newton, 15, ...... 140 LXVII. Newton to Cotes, 23, 141 LXVIII. , Oct. 14, LXIX. ; 21 143 LXX. Cotes to Newton, ..; .. 23, ... LXXI. Nov. 1 144 LXXII. 23, ...... 145 LXXIII. Newton to Cotes, Jan. G, 1713 ... LXXIV. Cotes to Newton, 13, 146 LXXV. Newton to Cotes, Mar. 2, 147 LXXVI. Newton and Bentlcy to Cotes, Mar. 5, 1713 148 LXXVII. Cotes to Newton, Mar. 8, 1713 ... LXXVIII. Cotes to Bentley, 10, • • • • 149 LXXIX. Bentley to Cotes, 12, 150 LXXX. Cotes to Newton, Feb. (? March) 18, 1713 151 LXXXI. Newton to Cotes, Mar. 28, 1713 154 LXXXII. 31 . 156 LXXXIII. Cotes to D' Clarke, June 25, 158 LXXXIV. Newton's Paper of Corrigenda and Addenda, Dec 1713 160 LXXXV. Cotes to Newton, Dec. 22, 1713 166 LXXXVI. Newton to Keill, (respecting a proposed answer to a Leibnizlan paper) Apr. 2, 1714 .... 169 LXXXVII. Newton to Keill, April 20, 1714 • 1 • • •170 LXXXVIII. May 11, 174 LXXXIX. 15, • . • • 176 • Since the sheet containing note " p. 171 was printed off, I have wen a paper by Brinkley on the origrin of the error in Newton's lat solution of the refllstance problem {Royal Irish Trantaetions, 1810, p. 45) in which the mistake is traced to its true source. VI CONTENTS. tKrrER Xa Cotes to Newtdi (after Apr. 25, 1716) XCL May 13, 1716 . XCII. Halley to KeiU, Oct. 3, XCIII. Newton to Keill, May 2, 1718 . XCIV. Newton to Arland, Oct. 22, 1722 . XCV. Cotes to his Uncle Smith, Doc. 31, 1698 XCVI. Smith to Cotes, Aug. 30, 1701 XCVII. Cotes to Smith, Sept. 9, XCVIII Feb. 10, 1708 XCIX Nov. 30, 1710 . C. Cotes to Ewer, Dec. 26 or 27, 1710 CI. Cotes to Ilallcy (exact dato uncertain) CII. Cotes to Jones, Feb. 16, 1711 cm. Jones to Cotes, Sept. 17, CIV. Cotes to Jones, 30, CV. Jones to Cotes, Oct. 26, CVI. Cotes to Jones, Nov. 11, CVII. Jones to Cotes, 16, CVIII. Cotes to Jones, 25, CIX. Jones to Cotes, Jan. 1, 1712 ClX.(bis) Newton to J. Smith, May 8, 1676 ex. Cotes to Jones, Jan. 1712 CXI. Jones to Cotes, Feb. 6, 1713 . CXII. Cotes to Jones, 13, CXIII. Jones to Cotes, Apr. 29, CXIV. Cotes to Jones, May 3, CXV. Jones to Cotes, July 11, CXVI. Cotes to Whiston, March, 1716 CXVII. Cotes to Lord Trevor, Jan. 10, 1716 CXVIII. Cotes to Dannye, March 15, 1716 CXIX. Brook Taylor to Prof. Smith, Nov. 27, 1718 CXX Dec. 11, .. Notice of three Letters from Taylor to Keill CXXI. Voltaire to Prof. Smith, Oct. 10 (N. S.), 1739 CXXII. Duke of Cumberland to Prof. Smith, July 3, 1740 CONTENTS. Til APPENDIX. I. Oldenburg to Newton, Jan. 2, 1072 . II. Newton to Oldenburg, Mar. 10, ..... Ill 19. IV 20. V ......30 VI. Apr. 13, VII Junell, VIII July30, VIII.(bis)01denburg to Newton, 10, . IX. Newton to Oldenburg, June 23, 1073 .... X. Examples of Algebraical Reduction given by Newton to Flamatced at Lecture, Midsummer, 1074 XI. Newton to Oldenburg, Nov. 13, 1076 . XII 30 XIII May 11, 1070 . XIV Aug.22, XV. Oct. 20, XVI Nov.14, XVII. Newton to D' Maddock. Feb. 7, 1670 . XVIII. Newton to Hooke, Dec. 3, 1080 . . . XIX. Newton to D' Briggs, June 20, 1682, on Vision XX Sept. 12, XXI. Apr. 26, 1085 XXI I. Directions given by Newton to Bentley respecting his mathe- matical reading, 1091 . XXII I. Newton to Locke, July 7, 1092 . XXIV. Newton to Leibniz, Oct. 10, 1093 XXV. Newton to Hawes, May 25, 1094, on Course of Mathematical Studies at Clmst's Hospital . . . . . XXVI 20, XXVII June 14, 1096 . XXVIII.Wallis to Newton, AprU 10 XXIX. Newton to Harington (a young Oxford student) May 30, 1098 XXX. Paper by Newton on Time of Vernal Equinox, Apr. 1700 a 2 Till CONTENTS. PAIIE XXXI. Nowton to hio coutiin. Sir John Newton, Apr. 1707 . . 307 XXXII. Critique by Newton on Leibniz's three Papers, 1712? XXXIII. Newton's Abstract of Paper on the different 'forms of the year in use among the nations of antiquity, Nov. 1713 . 314 XXXIV. Newton to Lord Townshend on a criminal under sentence ofdeath, Aug. 24, 1724 316 PREFACE. Newton's PhilosopMce Naturalis Principia Mathema^ tica, the most remarkable production of the human intel- lect that has yet been seen on the earth, whose mysterious path through space was first explained in its pages, was published about the middle of the year 1C87, a few weeks after his appearance before James's Ecclesiastical Commission, as the upholder of the rights of his Univer- sity and the laws of the realm, against the aggressions of arbitrary power. We are not informed how many copies of the work were printed, but the number probably was not large. If the extent of the impression had been rigorously limited to the number of persons likely to comprehend its contents, the volume would now have been one of excessive rarity. The work, however, seems to have found a readier sale than the abstruse nature of the subject and the engrossing interest of politics at that crisis of our history might have prepared us to expect ; and the sensation which it produced was long remem- bered, even by those who saw but darkly that tlie veil was now raised from the face of nature, which succes- sive generations of philosophers, from the first dawn of science, had vainly endeavoured to draw aside. It is true that, in a legal argument by Lord Mansfield, when Solicitor-General, the names of Locke and Newton are coupled with that of the author of Paradise Lost, as affording instances of the neglect shewn to works of X PREFACE. genius for a considerable time after their being given to the world. Dugald Stewart has assigned good reasons for doubting the correctness of the statement with re- spect to the Essay on the Human Understanding, and I believe the assertion to be equally unfounded as predi- cated of the Princi'pia, except so far as the slow recep- tion of the Newtonian doctrines, in some parts of the continent, may be considered as supplying ground for affirming the fact. Doubtless there were others besides Locke who tried to master the first principles, read the enunciations of the propositions, and accepted them either on the faith of the author's own word, or in re- liance upon the judgment of some known mathematician; nor was Bentley, we may rest assured, the only person in that inquisitive age who was struck with the wonder- ful truths developed by the new philosophy, and strove to attain to an intellectual appreciation of them. Locke's more popular book appeared in 1G90, and a second edi- tion was published in 1G04. The Principia seems to have been sold off with almost equal rapidity. In 1691 we hear of an improved edition of it as being in contem- plation. In 1C94 Newton renewed his attack on the lunar and planetary theories with a view to a new edi- tion of his book. And if Flamsteed, the Astronomer- Royal, had cordially co-operated with him in the humble capacity of an observer in the way that Newton pointed out and requested of him, (and for his almost unpardon- able omission to do so I know of no better apology that can be offered than that he did not understand the real nature and, consequently, the importance of the re- searches in which Newton was engaged, his purely empi- rical and tabular views never having been replaced in his PREFACE. xi mind by a clear conception of the Principle of Universal Gravitation,) the lunar theory would, if its creator did not overrate his own powers, have been completely in- vestigated, so far as he could do it, in the first few months of 1695, and a second edition of the Principia would probably have followed the execution of the task at no long interval. But science and the world were not destined to such good fortune. Flamsteed's infirmities of temper and bodily health conspired to thwart Newton's plans for the first half of the year just mentioned ; and the imperfect manner in which the Astronomer-Royal then met his wishes, leaves it uncertain whether we are to attribute the entire blame of the non-completion of the lunar theory in the latter half of the year to the circumstance of steps being at last taken by Newton's friends to provide for his material interests. His ap- pointment to the Wardenship of the Mint in March, 1696, was a bar to the further prosecution of his re- searches in physical astronomy. Henceforward his ofii- cial duties made it impossible for him to work continuously at his former pursuits ; his studies in mathematics and natural philosophy were by snatches and in the intervals of business. We shall accordingly find, when at length his consent to a new edition o^i\\Q Principia was wrung from him, that his necessary avocations seriously inter- fered with the progress of the work through the press. But his removal to a new sphere of labour did not abate his zeal for the promotion of science : the starving mathematician found in him a kind and liberal patron, and he was always ready with his purse and counsel to encourage any rational attempt to extract from nature more of her secrets. xii PREFACE. Probably as good an idea raay be formed of the actual feeling which prevailed with reference to the demand for a republication of the Principia, until the time when a new edition was finally determined upon, as would be conveyed by any description that I could give, if I cite a few notices referring to the subject, extracted from various contemporary letters and journals. 1691 Dec. 1 8. Fatio writing to Huygens from London says: " Mr. II est assez inutile de prier M"" Newton de faire une nouvelle edition de son livre. Je I'ai importune plusieurs fois sur ce sujet, sans I'avoir jamais pu flechir, Mais il n'est pas im- possible que j'entreprenne cette edition ; a qiioi je me sens d'autant plus portd, que je ne crois pas qu'il y ait personne qui entende a fond une si grande partie de ce livre que moi, graces aux peines que j'ai prises et au temps que j'ay employe pour en surmonter I'obscurite. D'ail- leurs je pourrois facilement aller faire un tour k Cambridge, et recevoir de Mr Newton meme Texplication de ce que je n'ai point entendu " Again, on Feb. 5, 1692 he writes : " Je n'ai encore ni aban- donne, ni embrasse absolutement la pensee de faire une seconde edition du livre de Mr New- ton." 1692 " Mr Newton is preparing a new System of Philo- sophy, which will be much larger and plainer than his Principia Philosophiaj Naturalis Phy- sico-Mathcmatica." (De la Croze's Works of the Learned for Jan. 169^. p. 269, under the head of " Cambridge.") "According to the best of our advices nothing considerable is doing new at Cambridge, but PREFACE. xiii Mr Newton's new System of PhUoPophy, and Mr Barnes's edition of Euripides." (Id. for March and April 1692, p. 398.) 1694 May 29. Huygcns, in a letter to Leibniz, speaks of **la nouvelle edition*' of the .Frincipia, "que doit procurer D. Grcgorius." Nov. 1. "I desire only such observations as tend to perfecting the theory of the planets, in order to a second edition of my book." Newton to Flamsteed (Baily, p. 138.) 1697 Dublin, Nov. 4. "I hear Mr Newton's Phil. Nat, Prin, Math, is out of press, and that he designs a 2nd Edition. Pray advise him to make it a little more plain to Readers not so well versed in Abstruse Mathematicks, a few Marginal Notes and references and Quotations would doe the business." (P. S. to a letter from W. Molyneux to Sloane. Orifj. Lett. Dk. Roy. Soc. M. i. 99.) 1699 July 15. J. Monroe, writing from Paris, says that Malebranche ** mightily commends ISIr Newton, adding at the same time that there were many things in his book that passed the bounds of his penetration, and that he would be very glad to see Dr Gregory's critick upon it." Orig. Lett. Bk. iloy. Soc. M. ii. 10. (Comp. Addison's ac- count of his visit to Malebranche at Paris, in the latter half of the year 1700. "His book is now reprinted with many additions, among which he shewed me a very pretty hypothesis of colours, which is diflcrent from that of Cartesius or ^Mr Newton, tho they may all three be true. He very much praised Mr Newton's mathematics, shook his head at the name of Hobbcs and told me he thought him a pauvre esprit.'* Letter to Bp. Hough from Lyons, Aikiii's Life, i. 91.) XIV PREFACE. 1700 Febr. ,|. " tTai appris aussi (je ne s^ai ou) qu'il donnera encore quelque chose sur le mouvement de la Lune; et on m'a dit aussi qu'il y aura une nouvelle edition de scs principes de la nature." (Leibniz to T. Burnet, Opp, Tom. vi. pars I. p. 2GG.) July 4. " The Royal Society have laboured to get his Theory of the Moon, Book of Colours &c. printed, but his excessive modesty has hitherto hindered him, but the Society will do what further they can ivith him." (Sloane to Leibniz, Orig, Lett. Bh, Roy. Soc. S. ii. 14.) 1701 In some MS. memoranda by David Gregory, dated Oxon. 21 May, of a variety of points upon which he wished to consult Newton we find the follow- ing : " To see if he has any design of reprinting his Principia Mathematica or any other thing." (Rigaud, Appendix to Essay, p. 80.) {1702, Monday} Nov. 30. "He owns there are a great many faults in his book, and has crossed it, and interleaved it, and writ in the margin of it, in a great many places. It is talked he designs tcr=Teprint itr though he would not own it. I asked him about his proof of a vacuum, and said that if there is such a matter as escapes through the pores of all sensible bodies, this could not be weighed I find he designs to alter that part, for he has writ in the margin. Materia sensibilis ; perceiving his reasons do not conclude in all matter what- soever." Bd. Greves to Lord Aston (Tixall Letters, n. 152), giving an account of a visit which he had paid to Newton the preceding Thursday in company with Sir E. Southcotc PREFACfc. XY at the request of Lord Aston, "a great lover of the mathematics, who would gladly be satis- fied in a difficulty or two of that science." 1704 Nov. 15. " The book {Newton's Optics} makes no noise in town, as the Principia did, which I hear he is preparing again for the press with necessary corrections." (Flamsteed to Pound, Greenwich MSS. xxxiii. 81.) The book had now become extremely scarce, and proportionately dear. Sir William Browne, who took his B.A. degree in 1711, states that when he was at Cambridge, he gave two guineas for a copy, " which was then esteemed a very cheap purchase." (Speech at Royal Society, Nov. 19, 1772, when he was eighty years of age, in Nichols's Literary Anecdotes, iii. 322.) Its original price seems to have been 10^. At last, in the beginning of 1709, Bentley's importunity prevailed over the scruples of the author, and induced him to entrust the superintendence of a new edition to the care of a promising young mathematician, Roger Cotes, Fellow of Trinity College, and recently appointed Professor of Astronomy and Experimental Philosophy. " Itaque cum Exemplaria prions Editionis rarissima admodum et im- mani pretio coemcnda suporcssent; suasit llle crebris efflagitationibus ct tantum non objurgando perpulit dcni- que Virum Praistantissimum, ncc modestia minus quara eruditione summa Insignem, ut novam hanc Operis Edi- tioncm, per omnia elimatam dcnuo et ogrcgiis insupcr acccssionibus ditatam, suis sumptibus et ausplciis prodire pateretur: Mihi vero, pro jure suo, pensum non ingra- tum demandavit, ut quam posset emendate id fieri cura- rcm." (Cotes, Pref. to 2nd ed.) In a letter to Professor XVI PREFACE. Sike, dated March 31, 1706 (the true date of which, I apprehend, from internal and external evidence, which it is not necessary to adduce here, to be 1709), Bentley says: **Pray tell Professor Cotes, that the book in your parcel, directed to him, is presented by Sir Isaac New- ton ; let him read it over with care, and I will tell him further of it in a particular letter. The bundle of wood cuts were found by Sir Isaac in his study, some of which he thinks may belong to the future sheets of his book. In the printed book are folded the MS. sheets that Sir Isaac has now finished." (Bentley's Correspondencey p. 231. Lond. 1842.) The book here alluded to was pro- bably a copy of the Pinncipia, containing Newton's MS. corrections and additions. This does not seem to have been the copy from which the second edition was printed, unless it was sent back to Newton for further modifica- tion. In May following, Cotes received intimation from Bentley that Newton would be glad to see him in town, and to put into his hands part of his revised copy of the Principia, The reader is now at the point where the Correspondence now offered to the public commences. This Correspondence, consisting of the letters which passed between Newton and Cotes relative to questions that arose connected with the new edition of the Prin- cipia, in the course of its passage through the press, is preserved, with some of the MS. sheets of Newton's in- terleaved copy of the first edition, and various mathe- matical papers in Cotes's handwriting, in the library of Trinity College. It was "collected from amongst the loose papers bequeathed " by Dr Robert Smith to the Rev. Edward Howkins, Fellow of Trinity College, who in 1779 demised the Collection, with a profile of New- PREFACE. XY at the request of Lord Aston, "a great lover of the mathematics, who would gladly be satis- fied in a difficulty or two of that science." 1704 Nov. 15. "The book {Newton's Optics} makes no noise in town, as the Principia did, which I hear he is preparing again for the press with necessary corrections." (Flamstced to Pound, Greenwich MSS. xxxiii. 81.) The book had now become extremely scarce, and proportionately dear. Sir AVilliam Browne, who took his B.A. degree in 1711, states that when he was at Cambridge, he gave two guineas for a copy, " which was then esteemed a very cheap purchase." (Speech at Royal Society, Nov. 19, 1772, when he was eighty years of age, in Nichols's Literary Anecdotes^ iii. 322.) Its original price seems to have been 10^. At last, in the beginning of 1709, Bentley's importunity prevailed over the scruples of the author, and induced hira to entrust the superintendence of a new edition to the care of a promising young mathematician, Roger Cotes, Fellow of Trinity College, and recently appointed Professor of Astronomy and Experimental Philosophy. " Itaque cum Exemplaria prioris Editionis rarissima admodum et im- niani prctio cocmcnda supcrcssent; suasit 111c crebris eflflagitationibus ct tantum non objurgando perpulit deni- que Virum Pra}stantissimum, nee modestia minus quam eruditione sunima Insiguem, ut novam banc Operis Edi- tioncm, per omnia elimatam dcnuo ct cgrcgiis insuper acccssionibus ditatam, suis sumptibus et ausplciis prodire pateretur: Jlihi vero, pro jure suo, pensum non ingra- tura demandavit, ut quam posset emendate id fieri cura- rcm." (Cotes, Pref. to 2nd ed.) In a letter to Professor XVI PREFACE. Sike, dated March 31, 1706 (the true date of which, I apprehend, from internal and external evidence, which it is not necessary to adduce here, to be 1709), Bentlcy says: **Pray tell Professor Cotes, that the book in your parcel, directed to him, is presented by Sir Isaac New- ton ; let him read it over with care, and I will tell him further of it in a particular letter. The bundle of wood cuts were found by Sir Isaac in his study, some of which he thinks may belong to the future sheets of his book. In the printed book are folded the MS. sheets that Sir Isaac has now finished." (Bentley's Correspondence^ p. 231. Lond. 1842.) The book here alluded to was pro- bably a copy of the Principia, containing Newton s MS. corrections and additions. This does not seem to have been the copy from which the second edition was printed, unless it was sent back to Newton for further modifica- tion. In May following, Cotes received intimation from Bentley that Newton would be glad to see him in town, and to put into his hands part of his revised copy of the Princ'Jpia. The reader is now at the point where the Correspondence now offered to the public commences. This Correspondence, consisting of the letters which passed between Newton and Cotes relative to questions that arose connected with the new edition of the Prin- cipia, in the course of its passage through the press, is preserved, with some of the MS. sheets of Newton's in- terleaved copy of the first edition, and various mathe- matical papers in Cotes's handwriting, in the library of Trinity College. It was "collected from amongst the loose papers bequeathed" by Dr Robert Smith to the Rev. Edward Ilowkins, Fellow of Trinity College, who in 1779 demised the Collection, with a profile of New- PREFACE. xvii ton, a lock of his hair, and other objects of interest, to the Society. The papers had come into Smith's posses- sion on the death of Cotes, who was his cousin. In their original state they contained among other things, which were afterwards lost, about twenty or thirty letters, written by Newton to Cotes " during the printing of the 2nd edition of the Principia" which were borrowed from Smith by Conduitt, who was collecting materials for a Life of Newton, and were never returned. They will, I suppose, be found among the papers which have descended with other property of Newton's from his niece, Catharine Barton (who married Conduitt), to the Earl of Portsmouth. Smith, in 1757, endeavoured, with the assistance of a friend, to obtain a clue to these letters which had belonged to him, and instituted inquiries, which were equally unsuccessful, respecting a common- place book of Newton's, "bound in green parchment," which he had formerly seen in the hands of William Jones, the father of the celebrated orientalist. Some correspondence which took place with reference to this subject is bound up with the Newtonian Letters and Papers. The late Mr Kidd, in 1796, saw in the possession of the Rev. Thomas Jones, Fellow of Trinity College, a copy of the Principia, " with an astonishing quantity of addi- tions and corrections" in Newton's hand. "Numerous loose papers of 4to form covered with diagrams and writing were placed between the leaves in different parts of the volume," which contained also "a loose copy of Ilalley's laudatory verses on the Principia, corrected throughout by the hand of D*" Bentley." Jones stated that this interesting volume was given to him by Mr xviii PREFACE. Davies, Senior Fellow of Trinity College, who received it from Smith, and he from Newton. All attempts that have been recently made to discover its existence have hitherto failed. I am inclined to think that it may be the identical volume alluded to in Bentley's letter to Sike, quoted above, in which case a link must be in- serted in the chain of its transmission between Newton and Smith. Of the other letters in the Trinity College Newtonian Collection which have been admitted into this publica- tion, those which were not written by or to Newton will bo foui\d, with few exceptions, to refer to him in some way or other, and to throw light upon the scien- tific history of the time. The Appendix contains various letters and papers, of more or less interest, from Newton's pen, collected principally from original sources. For details of these, and of other matter which is placed before the Corre- spondence, the reader is referred to the Table of Con- tents. The Portrait which accompanies this Work is taken, by the obliging permission of the Master and Fellows of Magdalen College, from an original drawing in Indian ink, which is preserved in the Pepysian Collection. It is uncertain when Pepys first became acquainted with Newton, but there is reason to think that their acquaint- ance began a short time previous to the Revolution, and they are known to have been on intimate terras in 1691 and 1693. The absence of Newton's name from the long list of persons who received at Pepys's funeral, in 1703, some token in memory of the deceased, may create PREFACE. xix a suspicion that their intimacy did not ripen into a friendship that continued unbroken to the last ; a cir- cumstance which need not excite much surprise when we reflect that neither the politics nor the morality of the Secretary of the Admiralty, under the two last Stuart kings, were at all congenial to Newton s taste. In assigning, therefore, the date of the portrait to the period of a few years on either side of 1G91, we shall not perhaps be very wide of the truth. If this supposi- tion be well-founded, this portrait may be considered as the most interesting of all the known portraits of our philosopher, as representing him at a time of his life the least remote from those memorable eighteen months which it cost him to produce the great work that has immortalized his name. The public is indebted to the liberality of the Master and Seniors of Trinity College for the appearance of the present volume. Trinity College, Cambridge, October IS50. SYNOPTICAL VIEW OF NEWTON'S LIFE. Quo fit ut omnis Votiv4 pateat veluti de&cripta tabellA 1642 Dec. 25. Isaac Newton born at TVoolsthorpe, near Grantham, Lincoln8liire(^). 1655 Sent to Grantham School. }656 Taken away from school and put to a;;ricultural employment. Reads matliematics while watching the sheep, and in consequence 1660 Sent back to school with the view of his going to College. 1661 Jun. 5. Admitted Subsizar at Trin. Coll. July 8. Matriculated Sizar f') (Quadrantarius). 1664 Feb. 19. Observations on two halos about the Moon(*). Thursday, Apr. 28. Elected Scholar (41< vacancies). 1665 Jan. Takes B.A. degree with 25 other Trinity men(*). May 20. Paper on fluxions ('), in which the notation of potntf is used. Nov. 13, "Discourse" on fluxions and their applications to tangents and curvature of curves ('). 1666 In the beginning of this year (the year beginning March 25) "applies himself to the grinding of Optic glasses of other figures than spherical," and "procures a triangular glass prism to try therewith the celebrated Phrenomena of Colours :" DISCOVERS THE UNEQUAL REFRANGIBILITY OP LIGHT (^), and abandoning in consequence the idea of improving the refract- ing telescope, leaves oft" his "glassworks," and turns his atten- tion to " Reflections," but while engaged thereon is " forced from Cambridge in {June} by the intervening plague (*), and it was more than two years before he proceeded further." May 16. Another paper on fluxions. First idea of gravity occurs to him from observing the fall of an apple (®) in the garden at Woolsthorpe ; proves (from Kepler's 3d law) that it must vary inversely as the square of the distance. b XXII SYNOPTICAL VIEW OF 1(}G6 Octob. Small tract on fluxions and fluents with their applica- tions to a variety of problems on tangents, curvature, areas, lengths, and centres of gravity of curves ("*). Nov. Small tract similar to the preceding, but apparently more comprehensive ("). (Notation by pointa in first and second fluxions. Basis of his larger tract of IO7I). 1667 Oct. 1. Elected minor fellow ("). Spiritual Chamber ("). 1668 March 16. Admitted major fellow. July 7. Created M.A.(^*) Makes a reflecting telescopc(") (probably towards the end of the year) : is interrupted until the autumn of 1671. 1669 Feb. 23. Describes his Reflecting Telescope in a letter to a friend. May 18. Letter of advice to his friend Francis Aston. July 31. His I)e Anali/si sent by Barrow to Collins. Oct. 29. Ap})oiiitcd Lucasian Professor ("). Dec. "Writes notes upon Kinkhuysen's Algebra sent by Collins through Barrow. 1670 Jan. 19. Letter to Collins ("). (Summation of harmonic series. Solution of equations by tables. Is writing notes at his leisure upon Kinkhuysen's Algebra). Feb. 6. Letter to Collins. (Solution of annuity problem, given all tho other quantities, find tho rate per cent. Kinkhuysen's Algebra not worth the pains of a formal comment). 18. Letter tO Collins ("). (Could give exacter solutions of tho annuity problem, but has no leisure for computations. Sees also a way of summing a harmonic series by logarithms). July 11. Letter to Collins (with his notes upon Kinkhuysen's Algebra)("'). — 16. Letter to Collins (proposing to make further additions to Kinkhuysen's Algebra, which is accordingly sent back for the purpose). Sept. 27. Letter to Collins (two mean proportionals cannot be found by trisecting an arc. General methods best adapted for instruction. Kinkhuysen's Algebra not so imperfect as ho had thought). 1671 July 20. Letter to Collins. (Prevented by a sudden fit of sick- ness from visiting him at the Duke of Buckingham's installa- tion as Chancellor. Will not, he fears, have time to return to discourse of infinite series before winter. Approximate sum of harmonic series). Autumn. ^Makes his 2nd Reflecting Telescope (in its essential parts like tho former) : it is sent up in December " for his Majesty's perusal ('^)." NEWTON'S LIFE. xxiii 1671 Dec. 21. Proposed candidate at the Royal Society by Dr Scth "NVard, Bishop of Salisbury. (Towards the end of the year) occupied in enlarging his method of infinite soric8("), and preparing 20 Opticall^ctures for the press. 1672 Jan. 6. Letter to OIdenburg("), "altering and enlarging the {Latin} descrij)tion (■') of his instrument which had been sent him for his review before it should go abroad" to lluygens at Paris. 11. Elected Fellow of the Royal Society. His telescope the subject of conversation at the meeting: the revised de- scription of it read ("). 10. Letter to Oldenburg on " a fit metalline matter" for the specula ; (announces his intention of sending to the Royal Society " an account of a philosophical discovery," " being the oddest, if not the most considerable detection, which hath hitherto been made in the operations of nature," viz. the con)- position of light). — 29. Letter to Oldenburg on the proportions of arsenic and bell-metal for specula. Feb. 6. Letter to Oldenburg communicating his discovery of the unequal refrangibility of the rays of light (read to the Soc. Feb. 8: printed in tlie Trans, for Feb. 19). — 10. Letter to Oldenburg, in acknowledgment of fho flat- tering reception of his letter of Feb. 6, and acceding to the wish of the Society that it should bo printed. — 20. Letter to Oldenburg, " promising an answer to Mr Ilooke's observations upon his new theory of light and colour," and acknowledging "the handsome and ingenious remarks'* in Huygens's letter on his telescope (read to the Soc. Feb. 22). March 16. Letter to 01denburg("). 19. Letter to Oldenburg, "containing several particu- lars relating to his new telescope (")," (read to the Soc ^Farch 21 : printed in Trans, for March 25). « 26. Letter to Oldenburg, " containing some more par- ticulars relating to his new telescope (")," (read to the Soc. March 28 : printed in Trans, for Apr. 22). — — 30. Letter to Oldenburg, " containing his answer to the difficulties objected by M. Auzout against his reflecting tele- scope ; as also the queries of M. Denys concerning it; together with his proposal of a way of using, instead of the little oval 62 XXIV SYNOPTICAL VIEW OP metal, a crystal figured like a A' prism (")." (Read to Soc. Apr. 4 : extract printed in Trans, for Apr. 22). 1672 Apr. 13. Latin letter to Oldenburg, in answer to the objections of Pardies (professor in the college of Clermont, in Paris) against his theory of light and colours (read to the Soc. Apr. 18: printed in Trans, for June 17). Same date. Letter to Oldenburg, " answering some experiments proposed by Sir Robert Moray, for the clearing of his theory of light and colours C-")" (read to the Soc. Apr. 18: extract printed in Trans, for May 20). May 4. Letter to Oldenburg, " containing his judgment of M, Cassegraine's telescope " (read to the Soc. May 8 : printed in Trans, for May 20). — 25. Letter to Collins (does not intend to publish his lectures) ('"'.). Juno 11. Letter to Oldenburg ("), accompanying (l) his 2nd answer to Pardies, who is satisfied by it, (printed in Trans. July ir>), and (2) his answer toHooke's "considerations upon his discourse on light and colours " (part of it read to the Soc. June 12: printed in the Trans. Nov. 18). — 19. Letter to Oldenburg from Woolsthorpe. July 0. Letter to Oldenburg from Stoke, in Northamptonshire, in answer to an inquiry concerning refraction, and containing 8 queries to test his theory of light and colours (partly printed in English and Latin in the Trans. July 15). — 8. Letter to Oldenburg from Stoke (containing remarks upon Huygens's letter of July 1, N. S.)("). — 13. Letter to Collins from Stoke. .. — . Oldenburg in which he repeats his inquiry about the 4 feet telescope, and desires to know the terms on which Cox will make one. ■ 30. Letter to Collins with a copy of his edition of Vare- nius's Geography ("). Letter to Oldenburg ("). Sept. 21. Letter to Oldenburg, in answer to one from Olden- burg of the 17th, inquiring whether the duplicate of July 16 had come to hand : (had drawn up some experiments adapted for determining the queries in his letter of July 6, and had intended from them to prove various propositions relating to colours by means of definitions and axioms, but prevented by other business from carrying out his design. But if the answer to Ilooko will conduce to the determination of any of the queries, it may be published). ^ NEWTON'S LIFE. XXV IG72 Doc 10. Letter to Collins, containing; (1 ) Rn account, requested by Collins in a letter received two days before, of his method of tangents ('*), and (2) "a long scribble" on James Gregorj-'s observations upon his paper on Cassegrain's telescope, (Very glad to have Barrow again, especially as Master). 1673 March 5. Joins in a protest against the claim of tho Heads of Houses to nominate for tho Public Oratorship. Votes for Isaac Craven of Trin. Coll. (not nominated) (**). ■ 8. Letter to Oldenburg (desires to withdraw from the Royal Society) C'). Apr. .3. Letter to Oldenburg, in answer to Iluygens's letter of Jan. 14 (read to the Soc. Apr. 9: printed in Trans. Oct. 6). 9. Letter to Collins (containing remarks upon Gregory's "candid reply"). May 20. Letter to Collins (»"). June 23. Letter to Oldenburg, thanking Ifuygens for tho pro- sent of his Horologium Oscillatoriuniy and replying to his remarks (in his letter of Jun. 10) upon Newton's letter of Apr. 3 (partly printed in Trans. July 21 )(="). Sept. 17. Letter to Collins: (postpones further discussion of telescope until Gregory pays his expected visit to Cambridge), 1674 June 20, Letter to Collins : (horizontal velocity of a bullet not uniform. Value of y in y' + a*y - i' = 0). Nov. 17. Letter to Collins: (mentions rules for solving incom- plete equations by logarithms). Dec. ■^.. Letter to Oldenburg : declines to take any notice of Linus's " conjecture :" however Oldenburg may direct him to the figure in the 2nd answer to Pardies, and signify " but not fiom me," that tlip experiment with tho prism was made on clear days, with the prism close to the hole and tho coloured image, not parallel but trans"'^r8e to the axis of the prism, (A letter was written by Oldenburg accordingly, and printed without Newton's knowledge in the Trans. Jan. 25, 1G75), 1675 Chemical pursuits ("). Jan. 22. Letter to IMichael Dary (length of an elliptic arc). — 28. Excused the weekly payments to tho Royal Society("), Feb. 18. Admitted F.R.S.(^'). Apr. 27. Obtains from the Crown a patent allowing tho Luca- ,sian Professor to hold a fellowship without being obliged to go into orders (*'). May 8. Letter to John Smith (construction of tables of square, cube, SiC. roots) (**). XXVI SYNOPTICAL VIEW OF 1675 July 24. ) A 27 I ^®*^®™ *^ ^^*® ^"*® ^" extraction of roots. Nov, 13. Letter to Oldenburg, with minute directions for Linus how to make the spectrum experiment : (communicated to the Soc. Nov. 18: principal part of it printed in Trans. Jan. 24, 167G) (**). Offers to send a paper on colours. — - 30. Letter to Oldenburg (is adding " an hypothesis " con- cerning light to his paper on colours. Description of ear- trumpet) ("). Dec. 1. Gives a copy of Irenteus (Paris, 1G75) to College Library. Decemb. Sends to the Roy. Soc. his papers, containing (1) his Hypothesis explaining the properties of light, (2) his explana- tion of the colours of thin plates, and of natural bodies ("). Dec. 14. Letter to Oldenburg (suggesting that the glass in the electrical experiment should be nearer the table than he had stated in his paper). — 21. Letter to Oldenburg, with (1) further directions respecting the electrical experiment (read to the Soc. Dec. 30(**), and the experiment ordered to be made at next meeting), and (2) remarks on Hooke's " insinuation." Communicates to Mercator his explanation of the Moon's li- hration (**). 1676 Jan. 10. Letter to Oldenburg, containing (l) suggestions re- specting the electrical experiment, (2) remarks upon Hooke's " insinuations," (3) further directions for Gascoines how to make the spectrum experiment. Oldenburg (Jan. 18) sends them to Gascoines, who requests Lucas (Linus's successor in the mathematical chair at Liege) to make the experiment. (Last part of the letter printed in Trans. Jan. 24, 1070). ' 13. At the meeting of the Royal Society, the electrical experiment being made according to Newton's "more par- ticular directions succeeded very well." " It was ordered that Mr Newton should have the thanks of the Society for giving himself the trouble of imparting to them such full instructions for making the experiment." 20. On the reading of the first 15 "observations" of Newton's discourse, the Society were " so well pleased " with them, that Oldenburg was ordered to desire him " to permit them to be published together with the rest." A passage was also read from his letter of Dec. 21, "stating the difference between his hypothesis and that of Mr Hooke," in NEWTON'S LIFE. xxvii allusion to what had fallen from Hooke at the meeting of Doc. 16. 167c Jan. 25. Letter to 01clenbur<», in acknowledgement of the favourable reception of his papers (•*), with alterations to be made in them. (Read to the Soo. Jan. 27). Feb. 3. On the reading of Newton's observations on colours, a discussion arose as to whether the difference of colour in the rays of light was not to bo attributed to the different velocities of the pulses rather than, as lie tliought, to a connate difference of refrangibility in the rays themselves. Ilooke expressed himself in favour of the former explanation. See Newton's Letter of Fob. 15. — 15. Letter to Oldenburg, answering the objection that had been raised at the meeting of Feb. 3. — 29. Letter to Oldenburg, occasioned by his having read in the Trans, for Jan. 24, Linus's letter of Feb. 25, 1075: it contains a particular answer to that letter, followed by expla- natory remarks for the behoof of Linus's friends. (Printed in Trans. March 25). Apr. 20. letter to Oldenburg, thanking him for " motioning to get " the spectrum experiment tried before the Royal So- ciety ("). Remarks upon Boyle's paper on the incalescence of gold and mercury. May 11. Letter to Oldenburg, thanking him for getting the experiment tried : during the summer may possibly work at his long-projected discourse about the prismatic colours("). Juno 8. At a meeting of the Soc. a letter from Lucas to Olden- burg (Liege, May 27) was read, containing partly an account of the success of the spectrum experiment, partly some new objections against Newton's theory of light and colours. A copy of the letter ordered to be sent to Newton immediately : printed in Trans, for Sept. 25. — — 13. Letter to Oldenburg, containing a general answer to Lucas with a promise of a particular one, and also "some communications of an algebraical nature for M. Leibniz, >vho by an express letter to Mr. Oldenburg had desiicd thorn." (read to the Soc. Juno 15: the part for Leibniz(") was sent to him at Paris, July 20). Aug. 22. Letter to Oldenburg (") (accompanied by another dated Aug. 18, the latter being an answer to Lucas, printed in Trans, for Sept. 25). Sept. 5. Letter to Collins. (Infinite Scrie3 of no great uso in xxviu SYNOPTICAL VIEW OP the nutnorical solution of equations. Tho University press cannot print Kinkhuysen's Algebra : the book is in the hands of a Cambrid/^e bookseller with a view to its being printed : shall add nothing to it. Will alter an expression or two in his paper about infinite series, if Collins thinks it should be printed). 1676 Oct. 24. Latin letter to Oldenburg (") for Leibniz, who desired explanation with reference to some points in tho letter of June 13. — 26. Letter to Oldenburg, with corrections for his letter of Oct. 24, &c.(") Nov. 8. Letter to Collins, thanking him for copies of the letters of Leibniz and Tschirnhaus, with remarks shewing that Leib- niz's method is net more general or easy than his own("). 14. Letter to Oldenburg (cider-fruit-trees : Lucas's 2nd letter: further alterations of his letter of Oct. 24)("). — 18. Letter to Oldenburg (answer to Lucas will not be ready so soon as he intended. Will never publish anything more on philosophy, after he has got clear of this dispute. Letter to Boy^C"). 28. Rejoinder to Lucasf"*). Subscribes £40 towards New Library. 1677 March 5. Letter of Collins to him (")• Sept. Death of Oldenburg. 1679 Feb. 7. Letter to Dr Maddock("). — H. Sir Thomas Exton, Master of Trin. Hall, and James Vernon, of Trinity, (the Duke of Monmouth's Secretary,) elected M.P. for the University. Newton plumps for the former("). ' 28. Letter to Boyle (physical qualities of bodies) (**). Nov. 8. Charles Montagu entered a feUow-commoner at Trinity College ("). December. Determines (in consequence of a letter from Hooke) the curve described by a body under tho action of a central force, and applies his theorem to the case of an ellipse ("). Gives copy of Huet's Demonstratio Evangelica to College Library. 1680 Jan. 21. Collins offers to print Newton's Algebra (along with Wallia's and Baker's), if the Society would take 60 copies, which the Council two years and a half afterwards agreed to do (July 12, 1682), bv^the design was carried out only with ^ respect to Baker and Wallis. NEWTON'S LIFE. xxix Lends the Collefije £lOO. for the New Library (sometime bo- twe;n Dec. 1079 and Michaelmas IGOO)^. 1680 Dec. 3. Letter tolloolcen. Gives copy of Grcw's Muswum Jlegalii Socictatlt to College Library. 1681 Jan. Promises to assist Adams (probably by advice and calcu- lations) in a survey of Enf!;land("). Feb. 28. Letter of Flamsteetl (through Crompton, Follow of Jes. Coll.) about the Cometh'). Apr. 16. Letter to Flamsteed about tho Comet ("). 1082 Apr. 3. Testimonial to Edw. Paget, Fellow of Trin. Coll., candidate for tho IMathematical Mastership of Christ's Hos- pital (^*)*. — Letter to Flamsteed (introducing Paget). ^ 1 lo r Letters to Briggs on Vision ("). 1683 Nov. 7. Votes for James Ilalman of Caius College, tho success- ful candidate for tho Registraryship. 10. Death of Collins. Dec. 2% Letter to Aubrey, who had offered some books for sale to Trinity College or the University ("). 1684 Jan. 19. Votes for James Manfoild of Trinity, the successful candidate for tho Librarianship. August. Halley on a visit to him, " learns the good news that he had brought the demonstration " of " tho laws of the ce- lestial motions to perfection." Newton cannot lay his hands upon his papers, but works them over again, and sends them in November by Paget to Halley in tho form of 4 theorems and 7 problems (^*). Halley "thereupon takes another jour- ney to Cambridge, on purpose to confer with him about them(")." 1685 Feb. 23. Letter to Aston (unsuccessful attempt to establish a philosophical society at Cambridge ("). Thanks for regis- tering at the Royal Society his " notions about motion"). Apr. 25. Letter to Briggs ("). DETERMINES THE ATTRACTIONS OP MASSES AND THUS COMPLETES THE DEMONSTRATION OP THE LAW OP UNIVFJISAL GRAVITATION. Summer. The 2nd book of the Principia finished. Sept. 10. Certificate of approval of Mabbot's Tables for renewal of leases ("). ■ 19. Letter to Flamsteed (is about to calculate the orbit of the comet of 1080 from 3 observations. Tides at solstices and equinoxes) ("). XXX SYNOPTICAL VIEW OF 1G85 Oct. 14. Letter to Flamstecd (acknowledging the receipt of Flamsteed's two letters in answer to the preceding). Dec. 30. Letter to Flamsteod (with thanks for information about comet of 1680 and Jupiter's satellites. Kepler makes Saturn's orbit too small. Requests the greatest elongations of any of Jupiter's satellites, and of Saturn's satellite) (*"). J 686 Jan. 13. Letter to Flamstecd (wishes to know the major axes of the orbits of Jupiter, Saturn and his satellite) ("). 22. Votes for John Laughton, of Trinity, the successful candidate for the Lihrarianship. Apr. 28. First Book op the Principia exhibited At the KoYAL Society ("*). May 19. At a meeting of the Society it was ordered " that M' Newton's PhilosopMw Naturalis Principia Mathematlca be printed forthwith in 4to. in a fair letter ; and that a letter be written to him to signify the Society's resolution, and to do- sire his opinion as to the print, volume, cuts, &c."("). June 2. Ilalley undertakes the publication of the Principia at his own expense ("). — 20. Letter to Ilalley (demolishing the claim set up by Uooke of having communicated to him the law of decrease of gravity according to the inverse square). — — 30. At a meeting of the Council of the Royal Society, the President was desired to license the Philosophias Natnralis Principia Mathematica. July 14. Letter to Ilalley (approves of the suggestion of having wood-cut-^. Conciliatory remarks respecting Ilooke). ■■■ 27- Letter to Ilalley (further remarks on ITooke's claim). Ar . 20. Letter to Halley (with Cor. 2 and 3 of Prop. xci. Lib. I. of Princip. on the attraction of a spheroid on a point in its axis produced, and on au internal point) (**). Sept. 3. Letter to Flamstecd ( sinian satellites. Cassini's observation of Jupiter's oblate: ' ). Autunm. Second Book of Princiwut made ready for the pres8("). Oct. 18. Letter to Ilalley (corrections of Scholium to Prop. 31. • Lib. I. : transformation of a trapezium into a parallelogram). 1687 Feb. 18. Letter to Ilalley (may have the second book of Prin- cipia when he pleases : has the sheets up to M : thanks him for putting forward the press again) ("). March 1. Tuesday. Letter to Ilalley, advising him that the 2nd book will arrive on Thursday night or Friday, by coach : obliged to him for pushing on the edition because of NEWTON'S LIFE. xxxi people's expectation^ tliougli otherwise he could bo as well satisfied to let it rest a year or two lonjifor (read to tlie Soc. 3Iarch 2). 1C87 Marcli 11. Deputed with Billers, tlie Public Orator, to carry to the Vico-Chancellur tlie opinions of the Non-Regent House respecting King James's second mandate, requiring the Uni- versity to confer upon Alban Francis, a Benedictine monk, tho degree of M.A. without the usual oaths(*''). Apr. 0. The 3rd book of the Principia "produced and presented" to the Royal Society (*"). 11. Appointed one of eight delegates to represent tho Senate, in conjunction with the Vice-Chancellor, before tho Ecclesiastical Commission ("'). Publication op the Principia (about Midsummer) ("). 1688 Spring. Cliarles IMontagu vacates his fellowship ("), Dec. 15. Votes for Archbishop Sancroft (for Chancellor of tho " Ufiiversity) who declines the office. 1689 Jan. 15. Elected one of the representatives of the University in tho Convention Parliament (""). First acquaintance with Locke. Furnishes him (March) with an easy proof of elliptic motion about a centre of force in ono ofthefociC*). June 12. Iluygens and Newton at tho Royal Society ("). Aug. 20. Parliament prorogued. ■ — Contemplated appointment to tho Provostship of King's College C^'^). Oct. 19. Meeting of Parliament ("). 1690 Jan. 27. Parliament prorogued. Feb. G. Parliament dissolved. — - 21. Sir Robert Sawyer, who had been expelled the IIouso of Commons, Jan. 20, for having been, as Attorney-General, one of the ])rosccutors of Sir Thomas Armstrong in 1084, re- elected M.P. for the University. Newton votes for him. Oct. 28. Letter to Locke : (will send, as desired, his " llisto* rical Account of two notable corruj)tion3 of scripture.** Ac- knowledgments to Lord and Lady Monmouth for their endeavours to procure him preferment) ("). Nov. 14. Letter to Locke, with the " Historical Account." 1691 Feb. 7« Letter to Locke (Daniel and Apocalypse). Juno 30. Letter to Locke. (Locke's good offices in trying to get him tho place of comptroller of the Mint Effijcts of look- ing at tho Sun's image in a mirror). xxxii SYNOPTICAL VIEW OP 1691 July (London). Testimonial to David Gregory, recommending him for tlie vacant chair of Astronomy at Oxford (•'). Directions to Bcntley about reading the Principia, (p. 273). Aug. 10. (London). Letter to Flamstecd (introducing David Gregory. Hopes Flamsteed will publish his catalogue of tho fixed stars before long. "Would willingly have his observa- tions of Jupiter and Saturn for next 4 or 5 years at least, or rather for tho next 12 or 15, before thinking further of their theory. Does the light of Jupiter's satellites, immediately before eclipse, incline either to red or blue, or become ruddier or paler than before ?) Dec. 13. Letter to Locke. (Declines "making a bustle" for the ^lastership of the Charter-House) (""). 1692 Jan. 26. Letter to Locke. (Charles Montagu a false friend. Desires to have his " Historical Account" returned.) Feb. 16. Letter to Locke. (Desires the translation and impres- sion of tho " Historical Account " to be stopped. Miracles). May 3. Letter to Locke (glad of his intended visit. Miracles). June. Observations on three halos about the Sun(^*^). July 7» Letter to Locke (Boyle's recipe for producing gold by means of red earth and mercury) (^°'). Aug. 2. Letter to Locke (Boyle's recipe. Discourages Locke from trying it). ^ \L' > Letters to Wallis, with illustrations of tho calculus Sept. 17. i of fluxions and fluents, sent at Wallis's request (***'). Nov. 21. Election of a Member for the University in the place of Sir Robt. Sawyer, deceased. Votes for the unsuccessful candidate, Dr Brookbank, of Trin. Hall("*). Dec. 10. First letter to Bentley. Paper on Acids (exact date uncertain) communicated to a friend this year(»**0- 1693 Jan. I7. Second letter to Bentley (^'»«). Feb. 11. Third letter to Bentley. ,^^ 25. Fourth letter to Bentley(*")- March 14. Letter to Fatio (proposing to make him such an allowance as might make his subsistence at Cambridge easy tohim)C"0- September. Bad state of health. 13. Letter to Samuel Pepj's (desiring to " withdraw from his acquaintance") C**0« ■ 16. Letter to Locke (beg^png his pardon for having had « hard thoughts" of him) ("0- NEWTON'S LIFE. xxxiii 1693 Oct. 15. Letter to Locke (explaining tlio circumstoncca under which the letter of Sept. 16 was written) ("')• 16. Letter to Leibniz, (p. 276). Nov. 23. ) D ' 16 I ^^^^^^"^ ^^ Pcpys on a problem in chances (""). 1694 May 7. Haunted house ('"). 1 1 . Charles Montagu, Chancellor of the Exchequer. — 25. Letter toITawes (explaining his views relative to the old and new schedules of niathcniatical studies at Clirist'a Hospital)("*). — 26. Letter to Hawes (supplementary to preceding). May. David Gregory at Cambridge ("*). July. Requested by the Royal Society to publish his optical and other treatises ("*). Sept. 1. Visits Flamstced at Greenwich, who shews him up- wards of 150 lunar observations, and a comparison of them with the places as calculated from tables ("^). Consequent correspondence between them, extending from Oct. until Sept. of the following year("*). Oct. 7. Letter to Flamsteed (describing what further observa- tions h(- will want, with which he believes he can " set right the moon's theory this winter"). — 24. Letter to Flamsteed (thanking him for his letter of Oct. 11, and particularly for the table of the difference of refractions of Sun and Venus. Parallactic Equation) ("*). Nov. 1. Letter to Flamsteed (errors in some of his observations. Lunar inequalities. Sun's menstrual parallax) ('-**). ■ 17. Letter to Flamsteed (will send back the two synopses of the Moon's places the next day, together with a table of refractions. His method of proceeding in determining the Moon's motions. Requests to have the JMoon's right ascen- sions and meridian altitudes just as they are observed without >, any correction : if Flamsteed will do him this favour, ho desires them as Flamsteed had observed them for tho last six months). Dec. 18. Letter to Flamsteed (Table of refractions not so accu- rate as it may be made : intends to correct it and send a new copy of it. Thanks Flamsteed for complying with his request of sending the Moon's right ascensions and meridian altitudes unreduced : begs her places on certain days which ho names : observations in this and next month or two of great importance). xxxiv SYNOPTICAL VIEW OF Deo. 20. Letter to Flamsteed (theoroni upon which his table of refraction is founded. Equations of the mean motions of Jupiter's satellites. " What you say about my having a mean opinion of you is a great mistake "). 1695 Jan. 15. Letter to Flamsteed (thinks he has discovered a new theorem in refractions, but intends to consider it a littlo further. Thanks Flamsteed for two lunar observations sent him, and as Flamsteed has calculated the ^loon's places in these and the other three observations of last month, will be glad to have a synopsis of the calculations. But for the rest of the observations, he merely wants the observed places ; at the same time is obliged to Flamsteed for offering to be at the pains of calculating them. Suggestions respecting the kind of time to be employed in taking the observations). — 20. Letter to Flamsteed (answer to Flamstced's childish question respecting a book which Flamsteed, two or three years before, had intended as a present to him. IMoon's hori- zontal parallax. Mas at last found out a new theorem in Refractions: is at present a little indisposed but hopes in a few days to be well enough again to finish the subject. The two observations mentioned in the last letter ("'). Promises to send a table of a small equation of Moon's parallax. If Flamsteed would rather have the observations perfectly his own in all respects, by calculating them himself, will stay his time). Feb. 16. Letter to Flamsteed (with thanks for the observations of Dec. and Ja-n. Has been engaged since ho wrote last upon making a new table of refractions, and has not yet finished it. Manly answer to Flamsteed's ungenerous suspicions of his observations having been communicated to Halley). March 15. Letter to Flamsteed (Candidates for mathematical mastership at Christ's Hospital. Encloses a copy of table of refractions now finished ("^). Will send the other tables ho promised in a few days). Apr. 23. Letter to Flamsteed (with the promised tables of ^loon's horizontal parallax, equations of apogee and eccentri- cities). 25(*"). Letter to Flamsteed (in reply to some remarks on the tables sent with his last letter). Jun. 14. Letter to Ilawcs (with new scheme of mathematical reading for Christ's ITo8pital)(^"). ^—- 29. Letter to Flamsteed (with thanks for solar tables. As Flamsteed's health and other business will not permit him to NEWTON'S LIFE. xxxy calculnto the Moon's places from olxjervations, he proposes once more that Flamstced slioiild ecnd tlio bare observntions and first of all thoso of 1G1)2. If not, let him propose some other way of supplying; the desired observationsi, or say plainly that he will not send any. Kecommends equestrian exercise). 1GD5 July 9. Letter to Flamstced (thankfully accepts the offer of the observations prior to 1690. Parallactic equation. Points out the kind of observations that he wants). — 20. Letter to Flamstced (has written to contradict tho report about llamsteed's not communicating his observations. Thanks for tho lunar observations. Has not yet compassed tho small equations, and bec^s him not to bo impatient for them. Forbears to take notice of some querulous expressions of Flamstecd's. " Pray take care of your health "). — — 27. Letter to Flamstced (is glad that all misunderstind- ings are composed. Describes the observations that he wants. Remuneration to Flamstecd's servant) ('"). Sept. 14. Letter to Flamstced (Ilalley's calculated orbit of tho comet of 1683 agrees with his own and Flamstecd's observa- tions to a minute. Is going on a journey and will not therefore have time to consider the lunar theory for a month or above. Hopes he gets ground of his distemper). Oct. 25. In the contest for the University plumps for tho Hon. II. Boyle. Nov. Rumour of his appointment to Mastership of Mint("*). 1696 Feb. 19. Votes for W. Ayloffo of Trin. successful candidate for the Public Oratorship. March 14. Letter to Ilalley (is not engaged upon the longi- tude. Not a candidate for any place in the Mint, nor would accept the Comptroller's place, if offered) ('"). — 19. Letter from Charles Montagu announcing his appoint- ment to Wardenship of ]\Iint. 1697 Jan. 30. Solution of John Bernoulli's two problems ("*): (read to the Soc. Feb. 24 : printed, without his name, in Trans, for Jan.). Feb. 11 . Letter to Ilalley : (has proposed Ilalley as a fit person to teach the mathematical principles of engineering)^"). End of June, or beginning of July. Examines boys at Christ's Hospital ("»). 1698 May 30. Letter to Ilarington, p. 302. July 25. Votes for Hon. II. Boyle (re-elected) ('»')• Dec. 4. Visit to Flamstced, in order to obtain 12 computed places of the Moon ('"). xxxvi SYN6HE1CAL VIEW OF Jan. 6. Letter to Flamsteed (explaining why he did not wish his name to bo meiitionod in the letter to Wallis, and stating, that there may be cases in which " friends should not be pub- lished without their leave ")("^). Feb. 11. Made Associe-Etranger ol" the French Academy ("*). Aug. 16. Exhibits at the Royal Society an improved form of his sextant (commonly called Iladlcy's) (*''*). r Nov. 30. Chosen member of Council of Royal Society (*"). This year the great re-coinage of silver was completed, having occupied the greater part of this and of the three preceding years (*"). Contributes towards the expenses of Lhuyd's Lithophylacii Britannici IcImographia("*). 1700 Apr. Paper on time of vernal equinox (p. 304). July 24. His opinion of the method proposed by an Italian mathematician for trisecting an angle, doubling the cube, and squaring the circle by means of a spiral line(""). 1701 Jan. 27. "NV^histon begins his Astronomical Lectures, as New- ton's deputy, receiving " the full profits of the place." May 28. His scale of heat read to the Society (""), (printed in the Trans, for ]\rarch- April). Nov. 26. Elected M.P. for the University("')- Dec. 10. Resigns his Professorship, and his Fellowship shortly aftcrC"). 1702 May 2.5. Parliament prorogued. (About June) his *'Lunie Theoria" published in Gregory's Astronomy (*"). July 2. Parliament dissolved. Autumn. On a visit to Locke at Gates ("*). 1703 J\Iay 15. Letter to Locke (giving his opinion of Locke's MS. papers on the Epistles to the Corinthians, and criticising his paraphrase on the 1st Ep. vii. 14). Nov. 30. Elected President of the Royal Society f"). 1704 Jan. 20. JVIentions to the Royal Society his burning-glass ("'). Feb. Publication of Optics ("0- Dec. 5. Note to Sloane (desiring him to be in readiness on the 7th, the day fixed for tlieir introduction to Prince George, for the purpose of having the honour of his signature in the Statute book of the Society, of which he was elected a mem- ber, Nov. 30). ■ 7' Waits on the Prince, and takes the opportunity of giving him a copy of Flamsteed's estimate of his Obser- vations. NEWTON'S LIFE. xxxvii 1704 Dec. 18. Letter to Flanisteed (inviting him to dinner to meet the gentlemen appointed by Prince George to inspect hift papers, and requesting liini to Lring his papers, or specimens of tluMn for the referees to examine). 20. Letter to Flamsteed (begging him to bring his papers for the referees to examine). 1705 Jan. 1. (^N. S.) Equivocal expressions in tho review of his tract, Do Quadratura Curvarum, in tho Leipsic Act8("*). (Origin of dispute on the priority of discovery of the new analysis). — 23. Report to Prince George recoinn»ending tho publica- tion of Flamstecd's Observations ("'). March 2. Letter to Flamsteed (earnestly desiring him to attend a meeting of the referees, in order to agree about an amanu- ensis, calculators^ and what else he has to propose for dispatch- ing the work). 7. Presents Royal Society with tho 1st Vol. of Rymer's Foedera, lately published (''"). Visit to Cambridge^'') Subscribes £60. towards the repairs of Trin. Coll. Chapel ('"). April Returns to London (about the 5th). 10. Knighted by Queen Anne at Trinity College. 24 or 25. Goes to Cambridge to contest the University. May 17. Defeated in tho contest for the University("'). June 8. Note to Flamstend (m .itlng him to meet the referees at dinner, 'Hhat wo may sot tho press a going as soon as possible "). Sept. 14. Noto to Sloane (begging him to get Ilauksbee to bring his air-pump some evening to his house. " I can then get some philosophical friends to see his experiments, who will otherwise be difficultly got together ")("*). — 17. Letter to Flamsteed (urging him to put his papers to press. " If you stick at anything, pray give Sir Chr. Wren and mo a meeting as soon as you can conveniently, that what you stick at may bo removed"). — Noto to Sloane (desiring llauksbeo's experiments to bo put off for a while, as Lord Halifax, Archbishop of Dublin, and Robartes are out of town). Nov. 14. Noto to Flamsteed (inviting him to meet tho referees at dinner, to finish the agreement and sign the articles about printing his book). 20. Signature to pedigree ('"). XXXViii SYNOPTICAL VIEW OP 1 700 J^tin edition of Optics ("•). Sept. 13. Note to Sloano (tliinks Bishop Wilkiiia's Legacy of £400 in 1072 should bo defended at any cost)C"). 1707 Jan. 14. Date of statutes of recently founded Plumian Pro- fessorship, drawn up partly under his eye("*). Apr. 9. Note to Flamsteed (requesting him to meet the referees, tliat all things may he now settled and adjusted, and to bring liis bill of disbursements). — Letter to Sir John Newton (recommending a poor kins- man as undertaker to conduct tho funeral of his cousin Coko)("''). 1709 Jan. 12. Gives tho Royal Society £20 ('""). Oct. 11. Commencement of his correspondence with Cotes relative to the 2nd ed. of tho Principia, extending from this date to March 31, 1713(''"). 1710 Sept. 13. Note to Sloane (glad that Sir Christopher and Mr Wren like tho house in Crane Court, proposed to bo pur- chased for the Royal Society, and hopes they will like the price also). Dec. 14. Promises to give £100 towards the easing of the debt for tho house, besides the £20 mentioned Jan. 12, 1709. 1713 Midsummer. Second edition of Principia("-). Nov. Paper on tho different kinds of years in use among tho nations of antiquity ('**). 1714 Apr. 2. Letter to Keill (respecting an answer to be made to Leibniz's "charta volans" as reprinted with remarks in tho Journal Literaire) Q^*). — — 20. Letter to Keill (on same subject). May 11. Letter to Keill (on same subject). Letter to Chamberlayne in reply to one from Leibniz of Apr. 28, (if it can be pointed out where he has wronged Leibniz, he will endeavour to make satisfaction, but he cannot retract what he knows to be true, and believes the Committee of tho Royal Society has not wronged Leibniz)(*"). — 15. Letter to Keill (in continuation of his letter of tho 11th). ^lay — June. One of Bishop Moore's Assessors at Bentley's trial ('«"). End of ^May or be- ) Evidence before a Committee of the House ginning of June. / of Commons, on the different methods of finding the longitude at sea(*"). Woodward's Classification of Fossils dedicated to him("*). NEWTON'S LIFE. xxxix lyjfi Feb. 26. Letter to Conti in answer to one from Ix>ibniz("*). May 18. Observations upon Leibniz's rej)!}' ("''). Juno .'5. Death of Cotcs('^')- J7I7 May 1(). Presents his portrait to the Royal Society^'*). Sept. 2L Report on the state of the Coin ('"). Nov. 2.3. Another Report on tlio CoinC'^. 1 718 Second edition of Optics('"). Jan. 2L At the IIouso of Lords with accounts relating to the coinC^"). ^lay 2. Letter to Keill (will John Benioulli's denial, in a jm- vate letter, of tlio autliorship of the Epistola jtro eminent^ MathematkOy satisfy him ('") ?) Oct. 22. Observations on the state of the Coin("'). Gift of £70 to the Royal Society ('^O- 1719 July 1.3. Present to Pound the Astronomer ("")• Letter to Monmort, enclosing one to Bernoulli C^")*. 1721 Third edition of Optics ("'). 1722 Attack of stone. Oct. 22. Letter to Arland the artist (thanking him for his pro- fessional services in tlic matter of a .j)luto in the Frencli translation of the Optics) (*''■). 1723 Jan. 17. Appoints (at a meeting of the Council of the Royal Society) IMartin Folkes liis Deputy or Vice- President. 1724 Apr. 27. Report on Wood's Halfpence and Farthings ("'). Jun. 25. Imprimatur for new edition of Ray's Sf/nopsig Plan- taritm Britannicarum, g^"^^* I Delisle in England C^^). Aug. 25. Letter to Lord Townshend (respecting a criminal under sentence of death for coining: thinks the law should take its course) ("*). Dec. 3. Letter to Halley (reque ting l)im to examine two of the calculated places in the elliptic orbit of the Coniet of 1080, and to calculate anotlier place, supposing the orbit a parabola) C^"). 1725 Jan. Violent cough and inflammation of tho lungs. Prevailed upon to take a house at Kensington. Feb. Fit of tho gout in both his feet (liad had a slight attack a few years before). Improved health after it. Letter to Mason, Rector of Colsterworth, notifying his subscrip- tion of £12. towards erecting a gallery in Colsterworth church C"). C2 xl SYNOPTICAL VIEW OF NEWTON'S LIFE. 1725 . March 7. Conversation with Conduitt on tho formation of the planetary bodies ("•). — 25. Grant of rents (£25) for four years of tho ancestral part of his estate at "Woolsthorpo to his god-son Isaac Warner. May 12. Letter to Mason (very glad to under8tanc\ that tho gallery in Colsterworth church is finished. The stirj/las in I^Iason's hands belonging to him to bo applied " to the use of the young people of tho parish that are learning to sing Psalms," according to Mason's desire). May 12. Letter to his tenant Percival of Woolsthorpe, agree- ing to a proposed distribution of tho commons there and at Colsterworth ("»). 27. Refuses his sanction to Freret's Translation of his Chronological Summary ('•*•). July 1. Visit of Abbe AlariC"). Date not given. Letter to Maclaurin (glad that he has a pros- pect of being joined to James Gregory in the Professorship of Mathematics at Edinburgh, and heartily wishes him good success) ("'^). Date not given. Letter to Lord Provost of Edinburgh (is ready to contribute £20 per ann. towards a provision for Maclaurin, if ho will act as assistant to Gregory). Towards the end of the year. Remarks upon Freret's observa- tions in his unauthorised translation of Newton's Chronologi- cal Summary ("=•). 1726 Third Edition of the Principia ("*)• May 10. Letter to Mason (with note for £3 for repair of the floor of Colsterworth church). 1727 Feb. 4. Letter to Mason (has procured assays to be made of the pieces of ore left with him by a Woolsthorpo friend of Mason's, but they contain no metal). Feb. 16. Writes Imprimatur for Hales's Vegetable Statics. March 2. Present for the last time at a meeting of the Royal Society, at which he calls attention to the fact of the Astrono- mer-Royal (Halley) having omitted to send to the Society a copy of his annual observations, as required by the late Queen's letterf"). — — 20. Monday, between 1 and 2 a.m. Dies(""). NOTES. (') *' Natus est Isaacus Neutonu8...1ioril primA veJ secunilli post mediam noctemt idque tempore ipso Plenilunii. Capillis eflloruit sensim in gummam canitiem veraw, Annum ajtatis inter trigcsimum & quadrngesimum.*' (Nicolas Fatio, in a printed copy of Latin Hexameters, entitled Neutonus EclogOt inserted in his copy of the 3rd ed. of the Principia which is preserved in the Bodleian Library.) For a description of liis person and liabits sec hi; nephew Conduitt*s account in Tumor's Grantham (pp. 103, 165), or Brewster's Newton^ pp. 340—342. According to Hamsteed he was short-sighted. " I happened once {during the year 1707} to visit the press while he was there, and took the opportunity to shew him how ill the compositor hud placed the types of the figures {in Flamstecd's Observations}... He put his head a little nearer to the paper, hut not near enough to see the fault, (for he is very near sighted,) and making- a slighting motion with his hand, said, • Mcthinks they are well enough.' " (Baily, p. 83.) (') This class of students were required to perform various menial services, which now seem to be considered degrading to a young man who is endeavouring by the force of his intellect to raise himself to his proper position in society. The following extract from the Conclusion Book of Trinity College, while it uftbrds an example of one of their duties, will also serve to illustrate tlie rampant buoyancy of the Academic youth at the period of the Restoration. "Jan. 16. 1660-1. Ordered also that no bachelor of what condition soever, nor any undergraduate, come into the upper butteries, save only a Sizar that is sent to see liis Tutor's quantum, and then to stay no longer than is requisite for that purpose, under penalty of 6cU for every time ; but if any shall leap over the hatch or strike a butler or his servant, upon this account of being hindered to come into the butteries, he shall undergo the censure of the Master and Seniors." (3) Optics, Bk. II. Part iv. Obs. 13. (*) The persons appointed (in conjunction with the Proctors, John Slader of Cath. Hall and Benj. Pulleyu of Trin. Newton's tutor) to examine the Questionlsts, were John Eachard^(the satirical author of The Grounds. ..of the contempt of the Clerf^y...) of Cath. Hall and Tho. Gipps of Trinity. I am sorry that I cannot gratify the curiosity of those who may expect to find here a notice of the Academical estimate formed of the acquirements of tlie most illustrious candidate that ever offered himself for a degree, as the " Ordo Senioritalis" of the Bachelors of Arts for this year is provokingly omitted in the Grace Jiook. (*) Shewing how to take the fluxion of (or to differentiate) an equation connecting any number of variables. It is referred to in a paper which peems to be part of a draught of his observations on Leibniz's letter of Apr. 9, 1716. (Kigaud's AppendiSf p. 23, compared with llaphson's llittunj of Fluxions, p. 116). (") lligaud and Raphson, u. s. C) The recipe described in the subjoined extiact is at least as worthy of being recorded as Tasso's malmsey, or Blackstone's port. " I have been credibly infonned that Sir Isaac Newton, when he applied himself to what is esteemed the greatest stretch xlii SYNOPTICAL VIEW OF NEWTON'S LIFE. of human invention and penetration (viz. the study, investigation and analysis of the theory of light and colours) to quicken his faculties and fix his attention, confined him- self to a small quantity of bread, during all the time, with a little sack and water, of which, without any regulation, he took as he found a craving or failure of spirit^" Cheyne*8 Natural Metfiod of curing diseases of the body and disorders of mind, 8^c. Lond. 1742, p. 81. (■) The College was ** dismissed" June 22 on the reappearance of the plague. The Fellows and Scholars were allowed their commons during their absence. Nev/ton received en this account 3s. 4d. weekly, for 13 weeks in the quarter ending Mich* W66. 12 Dec. 21 5 Lady Day, 1667. The College had been also dismissed the previous year, Aug. 8, on the breaking out of the plague, but Newton must have left Cambridge before that, as his name does not appear in the list of those who received extra coes for 6.^ weeks on the occasion. " Aug. 7, 1665. A month's commons (beginning Aug. 8) allowed to all Fellows and scholars which now go into the country upon occasion of the pestilence." (Conclusion Book). On the continuance of the scourge we find him, with others, receiving the allowance for commons for 12 weeks in the quarter ending Dec. 21, 1665, and for 13 weeks in that ending Lady-Day, 1666. (*) 'I'o the authorities for this anecdote (Biot, Journal des Savans, 1832, p. 265) may be added Green {Philosophy of Erpansive and Contractive Forces, p. 972), whose information on the point was derived from a very good source : ** quaj sententia...origi- nem ducit, uti omnis, ut fc-rtur, Cognitio nostra, a Porno, idquud accepi ab...amicissimo Martino Folkes." For the sentiment, compare the following from the meditations of a modern speculatist: " plcbiautcm vis gravitatis cognituplacuit... quia... corpora coelestia in orbes revolvi praisertim per tritissimam illam pomi coram Newtone delapsi historiam edocta securitatem adversus cocluni hausit, oblita scilicet, universaj generis humani, deinde Troja; mlseriaj prliicipiis poinum adfuisse, malum etiam scientiis philosophicis omen." Hegel's Dissertatio Fhibsophica de Orbitis Planetarum — an exercise written at the age of 31, jjro licentia docendi. Werke, Band 16, p. 18. Berlin, 1834. (''') In this tract his previous method of taking fluxions is extended to surds. The area of a curve, whoso ordinate is y, i» denoted by n !/• (Higaud's Append, p. 23.) (") Raphson, p. 116. Wilson's Appendix to Robins' Tracts (II. 351— 356). (") There were nine fellowships vacant; among them those of Duport, Thorn- dike, and Cowley (the last by er-fellow (Jitter to IldUey, July 27, 1686). But in the Junior Bursar's Book for the year ending at Michaelmas, 1673, we find the two entries " for selling M' Newton's chamber," *' for mending the slating. ..over M' Wickins," from which perhaps we may infer that one of them had changed his rooms in the interval between June and September*. In 1678 ho had a sizar living with him : " for mending over M"" Newton's sizar's chamber."|( JiiMim- Bursar's Hook.) The first notice of Newton's rooms which fixes their position, occurs in the Junior Bursar's Book for the year ending ut INlichaelmas 1603, and wo then find him inliabiting the rooms which well-informed tradition still points out to the stranger (the rooms on the first floor to the north of the Great Gateway) : "For mending the wall betwixt Mr Newton's garden and St John's" (probably a))Out the end of 1682). I am unable to determine satisfactorily the date of his taking these rooms, but the most probable supposition is that he went into tliein in the summer of 1679 1. Herbert Thorndikc preceded him in the occupation of them (with one or two removes): when Newton left Cambridge in 1696, they seem to have come into the possession of • If It wai Newton that change*!, we may find in th.it fact a foundation for the atntcmpnt made by a grandson of Wickins, who, in making mention of a woo<>en pint flagon given to his grandfather by Newton, says: " This with the whole furniture of the chambers devolved u|>on my ancestor upon Sir Isaac's leaving the college, and hath with some other articles remained in the family ever since." (Gent. Mag. Apr. H)<>2.) Wickins vacated his fellowship In ICa^ (eleven years before Newton left College), and had i-cascd to reside for several years. Yet, curiously enough, In Walker's account-book, quoted p. xnv, in the statement of the " income" of his rooms, there \% the following hem (date I/IG): " I'aid !)• Wickins a bill for rcpiiring what Mr Hanbury's brother UtoV. away, £\. 85." «'D» Wickins" was a son of Newton's fiiend, and had Just taken his bachelor's degree. Perhaps he had occupied part of the rooms jointly with llanbury, ♦ A view of Newton's rooms from the eai>t, with the garden attached, may be seen In Loggan's plate of the College. The following chronological notices, in conjunction with Loggan'i plates. xliv SYNOPTICAL VIEW OF NEWTON'S LIFE Daniel Hopkins, whom Bentley describes as " a Fellow of Trinity College and a very useful person in it, having the greatest number of pupils of any one amongst us " (CVr< retpondencey p. 185); Nat. Ilanbury (see p. 192) took them in 1704, and was succeeded in 1715 by *' Our hat" Walker, who continued in them until his death in 1764. Cum- berland, who came up a freshman in 1747, speaking of the kindness shewn to him by Walker, who was Vice-Master, says: *' He frequently invited me to his rooms, which 1 had 80 often visited as a child, and which had the further merit with me as having been the residence of Sir Isaac Newton, every relic of whose studies and experiments were respectfully preserved to the minutest particular, and pointed out to me by the good old Vice-Master with the most circumstantial precision. lie had many little anecdotes of my grandfather {Dentley}, which to me at least were interesting-, and an old servant Deborah, whom he made a kind of companion, and who was much in request for the many entertaininf; circumstances she could narrate of Sir Isaac Newton, when she waited upon him as his bedmaker, and also of D"" Bentley, with whom she lived for several years after Sir Isaac loft college, and at the death of my grandfatlier was passed over to D' Walker, in whose service she died." {Memoirs, p. 73.) What the '* relics " alluded to were I cannot exactly say. It happens that Walker's private account-book has been preserved. It contains a statement of what is called the "income" of his rooms, and an inventory of the furniture and movables in them and in the garden. In the list there appears a " thermometer," " a bureau bought of r)' Smith {the Mastjr}," a " violoncello (sold)," *' a picture of Vandyke," *' a barometer," and 10 pounds' worth of books, but there is nothing to indicate that any of these or the other articles ever belonged to Newton. In 1730 Walker made considerable alterations in the rooms. The same book contains his accounts with his bedmaker, Betty Baxter, and on her death, in Feb. 1744, with her sister " Deb." They seem to have been both women of thrift, and improved their capital by loans to their master. Deborah did not profit by her attendance upon Newton to learn the art of writing: in Walker's book, instead of her signature, slie appends, like our early kings, her mark. ('*) lie wjw 23rd on the list of 148 signed by the Sen. Proctor (Thomas Burnet, author of Theoria Telluris Sacra). (") It was 6 inches long, aperture something more than an inch, depth of plano- convex eye-pflass, one-sixth or one-seventh of au inch, magnifying power about 40. (Letter of Feb. 23, 1669 in INIacc. Corr. ii. 289. Comp. Brewster's Newton, p. 27.) (") The Lucasian statutes, dated Dec. 19, 1663 (they are printed in the Appendix to Whiston's Account of his Prosecution, ed. 1718-9) require the Professor to lecture at least once a week during term-time, on some portion "Geometriaj, Arithmeticje, Astro- nomic, Geographia;, Optica;, Staticaj aut alterius alicujus Mathematicae Disciplinae" ...•• per unius circiter horje spatium," and also two days in the week during- term-time (and during vacation one day, if the Professor is in residence) " per duas horas...omni- will enable the academical reader to plctiue to himself the College as it was when Newton walked to and fro within Its courts : 1670-1 Oerrard's Hostle rebuilt at the expense of Bishop Hacket and thence called Bishop's Hostle. 167(1 Peb. Foundation of new Library dug. 1678 Rooms over eight arches next the Library in north cloister finished, those next the library being built out of the subscriptions for the Library, those next to them to the east at the expense of Sir Thomas Sclatcr. 1681-S Rooms over right arches next the Library in south cloister built, those adjoining the library out of the library subscriptions, the others at the expense of D* Humfrey Babington. , 1G81 May J. Four statues on the top of the library by Cibber for which he received £80. 1685 Feb. New Library ceiled. UJ86 Library floor laid down. 16S7-8 Library paved. 1694 Ruinous part of King's Hostle pulleetwecn the Dutch and English at sea in 1672 was heard by the people who were out at work in the fields to the very centre of England : Mr Dcrham says it was heard 200 miles." The "observatory" in the passage quoted alwve is a prolepsLs for the "great gateway," which was not converted into an observatory until several years after Newton had left Cambridge. (8>) Appendix, No. VII. ("5) He also says, " I should be glad to hear whether Mr Cox hath finished the 4 feet telescope and what its efTects are... 1 Jut I know not whether I shall make any further trials myself, being desirous to prosecute some other studien." Mace. Corr. ii. 329. («') For a character of this work see Humboldt's Kmwm, Vol. i. The edition of 1681 seems to be almost a reprint of the preceding one, in spite of the ' ' auctior et emen- datior " of the title-page. (»*) Appendix, No. VIII. (8'^) This part of the letter is cited in the 3rd edition of the Principiot p. 246, instead of the letters to Leibniz referred to in the two first editions. Its contents were sent to Leibniz July 26, 1676, alonrr. ii. 405. See next note. («o) Lucas replied to Newton's letter of Aug. 18 in a letter of four pages closely written, dated Oct. 23, "containing further objections and experiments against Mr Newton's theory of light and colours with an examination of his experimentum crucis :" among other things he professes to prove that the red rays sulfer the same refraction as the blue ones. Newton sent an answer to this (Nov. 28), but with a determination that It should close the controversy. In a letter to Oldenburg (Nov. 18), he writes : *♦ 1 see I have made myself a slave to philosophy, but if 1 get free of Mr Linus's business, I will resolutely bid adieu to it eternally ; excepting what I do for my private satisfaction, or leave to come out after me ; for I see a man must either resolve to put out nothing new, or to become a slave to defend it." Mace. Cotr. ix. 405. His opponent, however, was not satisfied with the answer, and indited another letter (Feb. 2, 1677 N. S.), the sole value of which to us consists in its preserving for us a few words out of Newton's letter of Nov. 28. " In his last of Nov. 28," writes the Liege professor, " I still meet with new demurs.... He is pleased to quarrel with my examining his Experimentum Crucis, representing it ' a jostling out of the point in dispute by a new attempted digression,' or as he is pleased to term it * a running from one thing to another.* He tells us ' that he intends to take into consideration one or two of my experiments, w Inch I shall recommend for the best : and when there appears to be no weight in them, let others judge what there may be in the number of the rest'." Lucas closes his epistle with a desire that the whole of his previous letter of Oct. 23 may be printed, but the request was not attended to. The matter does not seem to have alto- gether dropt here, for in Oldenburg's letter to Leibniz of May 2, accompanying Newton's letter of Octob. 24 preceding, we read, " Ad alia nunc distrahitur Newtonus ab iis, qui Leodii, Francisco Lino succenturiati, novam ipsius de Lumine et Coloribus Theoriam vehementer insectantur: qua de re brevi plura accipies, ni raliones meas male subduxi," but our information extends no further. Goethe, in his "Geschichte der Farbenlehre" (Werke, Baud 55. Stuttg. 1833) gives an account of the reception of Newton's discovery of the composition of light, which does not indicate a very intimate acquaintance with the circumstances of the history. NOTES. liii For example, he does not know that the three penions whose suggestions or objections accompanied by Newton's answers are printed without their names in the Phibti^hieal Transactions were Moray, Hooke and Huygens. One of tliem, indeed, he conjectures riffhtly enough to be Hooke, the loss of whose pai>er of "considerations,** he says, is preatly to be regretted. It will, however, be found in a book which he himself quotes not many pnpres before, viz. Birch's History of the Royal Society, in. 10—15. In p. 56 he confounds John Gnscoines, Linus's pupil, with William Ga«coiffne, the inventor of telescopic sights, who fell at the age of 23 at Marston Moor fighting on the Royalist side. Again, Newton, in his answer to Lucas (Phil. Trans. Sept. 1076, p. 703) says that the principal experiments which Lucas had sent him were detailed in a ** tractate" which he had written upon light. Goethe, in quoting the passage, for the word *' tractate" writes " Optical Lectures,'* and adds that the statement " keineswegs der Wahrheit gemiiss ist" (p. 64). It is true tliat the treatise in question conrtisted in the main of the Optical Lectures, but it would not have been amiss to have ascertained the perfect identity of the two works before using language like that which has just been quoted. For Goethe's speculations on colours, see Whewell's Hist. hid. Sci. ii. Wilde's Geschichte der Optik, Theil. u. p. 153 sqq. (Berlin, 1843), and the works referred to by him. (*•) Printed in Wallis's Works, in. 646 (extracts from it in the Commercium Epist.). At tlie end of the letter Collins says: " Narrat mihi D. Loggan (Chalcographus) quod Effigiem tuam delineavit ille, in ordine ad Sculpturam ; Quae praifigenda sit libro tuo de Lumine, Coloribus, DioptriciSf &c. quem edecidum intendis. Qua de re desideramus esse certiores." Nothing further is known of the *' effigies" here spoken of. We may mention here Loggan's Dedication of his Plate of St Mary's Church. Its date is uncertain, as, though Loggan's Cantabrigia lllustrata was published in 1690, the dates of the separate plates range over a period of several years. " Clariss", Viro D". Isaaco Newton Matheseos apud Cuntubrigienses Profeisori l.ucasiano A'6'*. Trinitatis CoW^. ibidem, et Regice Societ'. Socio, Mathematieo, Phihsopho, Chymico consummatisso. Nee minus suavitate Morum et Candore Animi, Cum rerum Huma- narum Divinarumq: Periti^ spectabili, Hanc Tabulam Observantiaa ergo D. D. C. Q. Dav. Loggan." Loggan had the use of a room in Trinity College for his press. (") Appendix, No. XVII. (") In this and other instances where Newton is mentioned as voting at Univer- sity elections of Members of Parliament or Officers, our information is derived from the actual slips of paper on which each voter recorded his suffrage, and which are still preserved in the Registrary's office. A copy of Newton's voting paper on this occasion is given as a specimen. *♦ Isaacus Newton eligit Thomara Exton Militem in Burgensem hujus AcademisB in Regni Comitiis." (««) Boyle*s Life (by Birch) prefixed to his Works, p. 70. Mace. Corr. ii. 407. and elsewhere. («>) A very pretty story is told of him by his biogrrapher— how that in 1682 when his schoolfellow George Stepney was elected scholar from Westminster to Trinity Col- lege, Montagu, unable to l)ear the tlioughts of l)eii)g separated from his **dearist friend," went to College a year before the proper time — but, like many other pretty stories, it will not stand the test of dates. Montagu was matriculated Dec. 18, 1679, the "chamber" in which he "kept" in 1680 and following years is known, being the same, in fact, in which these lines are written, and on Oct. 6, 1681, he was made M.A. by Royal Mandate. (««) Newton seems to have been requested to give his opinion on a wild hypothesis of the heavens, which a Frenchman of the name of Mallemont had sent to the Royal Society. His judgment was given briefly, and with some reluctance, in a letter to Hooke, one of the Secretaries, (Nov. 28, read to the Soc. Dec. 4), in which, to make amends for the curtness of his answer, he suggested ** an experiment whereby to try whether the earth moves with a diurnal motion or not, viz. by the falling of a body from d liv SYNOPTICAL VIEW OP NEWTON'S LIFE. t considerable height, which, he alleged, must fall to the eastward of the perpendicular, if the earth moved. This proposal was highly approved of by the Society, & it wa» desired that it might be tried as soon as could be with convenience." At the meeting of the Soc. Dec. 1 1, Hooke read his answer to Newton's letter, in which he shewed that the path of the falling body would not be a spiral "as Mr. New- ton seemed to suppose," and that it would fall " not directly east, but to the south-east & more to the south than the east. It was desired that what was tryable in this expe- riment might be done with the first opportunity." At the meeting on Dec. 18, Hooke read his answer again, and also a reply to it from Newton, "containing his farther thoughts and examinations of what had been pro- pounded by Mr Hooke." He also gave an account of three trials that he had made of the experiment. At the meeting on Jan. 8, 1680, Hooke read another letter of his to Newton, giving *♦ a further account of his theory of circular motion 6c attraction, as also several obser- yations 6c deductions from it." Newton declined answering this letter. At the same meeting Hooke " was desired to make his trials " of Newton's experiment as soon an possible. («) Library Account Bk. for year from Dec. 22, 1679, to Dec. 22, 1680. The charge for the bond appears in the Sen. Bursar's Bk. for year ending Mich. 1G80. The money seems to have been repaid Nov. 12, 1688. Conclus. Bk. Feb. 5, 1C89. («9) Appendix, No. XVIII. («») Birch, IT, 65. A letter of his to his kinsman Sir John Newton, introducing Adams, is printed in Tumor's Grantham, p. 85, note. C") Gen, Diet, vii. 788. The originals of th'is and the other letters to Flamsleed down to 1698, are preserved in the Library of Corpus Christi College, Oxford. (]}) Gen. Diet, yii.lBX, (»)• " Cambridge April y« 3«» 1682 "These are to signify y* M' Ellis advising w**> me ab' a person fit to be intrusted w"» y« Charge of teaching Navigation toy* Boys of y* King's late foundation, I propounded M' Edw* Paget Master of Arts & ftellow of Trinity College in this University, as y« most promising person for this end I could think of; and that upon these considerations. He is of a temper very sober & industrious, as I am confident all that know him are ready to testify. He understands y« several parts of Mathematics, Arithmetic, Geometry, Algebra, Trigonometry, Geography, Astronomy, Navigation, & w** is y« surest character of a true Mathematical Genius, learned these of his own inclination & by his own Industry without a Teacher : And to make him y* readier in practicall Matters, hLs hand is very steady &c accurate, as well as his fancy & apprehension, good ; as may be •een by his writing & drawing w*'» his Pencil very well : Perfections w^h I conceive considerable for making y» Boys accurate & curious in their Draughts of Charts, Mapps & Prospects from Sea, wc>» joyn'd w*'» his knowledge in perspective and projections of ye Sphere will enable him to contrive & draw schemes after y* best manner for y« Boys apprehension, & perswades me y' he will not only be dexterous & nice in y« use of Instrum** but improve them : His long acquaintance also wi'» variety of Learning here, will help him to be methodical & clear in his teaching j w^*" much conduces to y«^ Boys ready & distinct apprehension of what they are taught. So y» tho it may be easy to find persons valuable for some of these Qualifications, yet considering him in all rcsixicts as I could not think of ony other person in this University so fit in my Opinion to be intrusted w''' a place of so great concerne as that of preparing Boys to make more skilful Navigators than formerly, so I believe it will be difficult to meet w'*> fitter persons abroad for that purpose. These things made me forward to propound him to y« Electors ; but to compare him w*** other Competitors 6c chuse y« best I leave wholly to their judgment. Is. Newton, Profess. IMath. Luc.** ( Pepysian MSS. 2612. p. 536). NOTES. Iv Newton also wrote to his friend Collins requesting him to use hi« interest in behalf of Paget. There is in the same MS. volume from which the above is taken^ a copy of the letter which Collins wrote in consequence, enumerating from Newton's letter to him Paget's qualifications, and dwelling upon the weight which the recommendation of the greatest mathematician of the age ought to have with the electors. (") Appendix, Nos. XIX. XX. (") **The charge of building" the College Library, "disables us from buyiog books at present We know not yet whether the University will purchase them, their chest being at present very low." Gentlcman''t Magazine^ LXl. 504. C*) The propositions here mentioned as sent to Ilalley, have been printed by Rigaud from the copy in the Register Uook of the Royal Society, vi. 218. (Appendix to Essay on Publication of Principiu, No. I.) It is to be observed, however, that the title which Rigaud g'ives to the Paper ( Newtoni Propositionet de Motu) is not to bo found in the MS. ('*) At the Meeting of the Royal Society, Dec. 10, " Mr Halley gave an account that he had lately seen M"" Newton at Cambridge, who had shcwcil him a curious treatise, De Motu { drawn up since August } ; which, upon Mr Ilalley's desire, was, ho said, promised to be sent to the Society, to bo entered upon their Register. Mr Ilalley was desired to put Mr Newton in mind of his promise for the securing his invention to himself till such time as he could be at leisure to publish it. Mr Paget was desired to join with Mr Ilalley." Birch, iv. 347. The treatise De MotUf mentioned here, was probably the same as that of which a copy is preserved in the University Library (Dd. IX. 46,) beginning "De motu cor- porum Liber primus, Definitiones," &c. consisting of the Lectures which he delivered as Lucasian Professor, (the first of them is dated Octob. 1684), and forming, to a certain extent, the first draught of the Principia. (See Letter CIV.). The paper which New- ton sent up to Haiiey, in Nov. 1684, was the germ of this treatise. It is probable that Halley produced the paper at the meeting on Dec. 10, though the fact is not recorded in the Journal Book. The treatise was never registered, but the paper was, apparently in February 1685, with the date Dec. 10, 1684. Rigaud*s idea that the paper which he has printed from the Register of the Royal Society (consisting of 4 theorems and 7 problems) is different from the paper which Newton sent to Halley, and that it was sent to the Society in Feb. 1685, is founded upon what I conceive to be a misapprehension of a passage in Newton's letter to Aston, (Feb. 23, 1685). The words are as follow : '• I thank you for entering in your Register my notions about motion. I designed them for you before now, but the examining several things has taken a greater part of my time than I expected, and a great deal of it to no purpose. And now I am to go into Lincolnshire for a montli or six weeks. After- wards I intend to finish it as soon as I can conveniently," &c. We possess only a part of the letter, and that in a copy. We cannot therefore be sure that the grammar is Newton's. It seems clear to me that what he "designed" for the Society " before now," was not yet finished and sent to the Society : that he was in fact working at his Treatise De Motu with a view to fulfil the promise whicli he had made to Halley, that he would *' send it to the Society to be entered upon their register." That the paper sent to Halley is identical with that which we find in the Register of the Royal Society, is evident from the whole tenor of our information on the subject : it is sufficient to refer to Halley'sown statement (Rigaud, Appendix to Kssay, p. 37), and a letter of his to Wallis, dated Dec. 11, 1686, in which he says: " JSIr Is. Newton about two years since gave me the inclosed propositions, touching the opposition of the medium to a direct impressed motion and to falling bodies, upon supposition that the opposition is as the velocity ; which tis possible is not true ; however, I thought any thing of his might not be unacceptable to you, and I beg your opinion thereupon, if it might not be (especially the 7th problem) somewhat better illustrated." (The original of this letter is in the collection of Dawson Turner, Ksq. Compare Birch, iv. 514. Rigaud, 77.) d2 hi SYNOPTICAL VIEW OF NEWTON'S LIFE. The probability is that Halley saw no immediate prospect of obtainiDg the treatise De Motu, and determined to secure the author's rights by at once reeristering the 11 propo- sitions which he bad received in November. (»«) Birch, IV. 370. Rigaud (Appendix to Essay, p. 24). Newton observes that •* that which chiefly dashed the businesa was tlie want of persons willing to try experi- ments, he whom we chiefly relied on refusing to concern himself in thatkind I should be very ready to concur with any persons for promoting such a design, so far as I can do it without engaging the loss of my own time in those things." (») Appendix, No. XXI. (»») Tables for renewing and purchasing of the leaset of Cathedral Churches and Colleges, H^e.y Cambridge, 1686. Newton's certificate prefixed to this work, the author of which was manciple {mancipium} or caterer of King's College, runs as follows: " Metho- dus hujus Libri recte se habet, nuroerique ut ex quibusdam ad calculum revocatis judico, satis exacte computantur. Is. Newton, Math. Prof. Luc." The later editions on the strength of this testimonial were published under the title of" Newton's Tables." In the treasury of Trinity College in a book labelled " Notitia E," which belonged to Humfrey Babington, as Bursar (1674 — 1678), containing "a true particular of the rents and leases belonging to Trin. Coll. 1674-5," there is a table and an explanation of it in Newton's handwriting, of the fines to be paid for renewing any number of years lapsed in a lease for 20 years. It is entitled Tabula redemptionalis ad reditus Collegii SS. Trinitatis accommodata. It is constructed on the hypotliesis that a lease for 20 years b worth 7 years' purchase, and that for the renewal of 7 years lapsed, one year's purchase must be paid, (This is equivalent to allowing the lessee between 12 and 13 per cent, for his money). This table vvrhich was apparently drawn up by Newton for Babington's oflicial use, continued to be employed by the College until 1700, when Bentley, on his appointment to the Mastership, introduced the 10 per cent, tables. The innovation however, according to Vice-lNIaster Walker, was unpalatable to the Seniors and Ofl^cers, whose "greediness for present sealing money" superadded to "quarrels in the Col- lege," compelled a return to the old system, and occasionally the granting of terms still more favourable lo the tenant. On Dr Robert Smith's succeeding to the IMas- tership in 1742, the 10 per cent, tables were introduced, and these were replaced in 1750, by 9 per cent, tables. (") Gen. Diet. vii. 793, where also the next four letters lo Flamsteed will be found. -=:-t-_:l,- (8") " You seem to insinuate as if Saturn had not yet any more satellites than one discovered by Hugeuius. I should be glad to know if it be so." If Flamsteed returned an answer to this question, it seems to have been still in the negative. Writing to him on Sept. 3, of the fallowing year, Newton says : " He [Mr Philips] tells me he apprehended by some of your discourses, tliat you had seen two of Cassini's Hew planets about Saturn. Ilugenius with a sixty foot glass could see none of them. Mr Halley (who was lately here) I find still suspicious of them, notwithstanding what Cassini has lately published of two more. I was glad to hear two of them confirmed by youT observation." IMr Philips' information does not appear to have been correct, for in a paper in Cotes's handwriting (Trin. Coll. Newtonian INISS. No. 382) which is apparently a memorandum of a conversation which he had had with Flamsteed some time between 1706 and 1716) it is stated "that he (Flamsteed) thought there were but 3 satellites of Saturn, himself had never seen above one." The first discovered satellite of Saturn (now the 6th, reckoning outwards) was ob- served by Huygens March 25, 1655. In 1671, 2, 3 Cassini discovered what is now the 8th, in 1672, 3 (while in pursuit of the last-mentioned one) the 5th, (see Phil. Trans, Iklarch25, 1673), and in 1684 the 3d and 4th : (an account of this last discovery, given in the Journal des Savons for April 1686, was mentioned at the Royal Society April 28, communicated at their next meeting, and printed in (he Transactions for May 25 : a letter from Cassini to Halley, dated Oct. 10, giving more correct elements of the then NOTES. Ivii known 5 satellites was read to the Society Nov. 3, and published in the Traniactiona for Apr.— June of the following year.) In the first edition of the Principia Newton mentioned only the Hug«nian tatellite, but in the second he introduced the others, availing himself of Cassini's paper in the Memoiret of the Academy for 1705, published in 170G (comp. p. 49 of this work). Pound (in 1718) was the first English astronomer who succeeded in observinj? the Cassinian satellites : this he did by means of corrected elements supplied by the younger Cassini, in the Mimoires for 1714 (published in 1717), and a telescope with an object-glass of 123 feet focal length, which Huygeus hod presented to tlic Royal Society in 1G91. (See Phil, Trans. Jan.— Apr. 1718. Delisle's " Second© Lcttre sur Ics Tables Astro- noraiques de M. Halley..." Journal des Savans, June, 1750). Flamsteed, however, was not convinced. (See his letter to A. Sharp, Sept. 13, 1718, Baily, p. 331). (**) The date is taken from the post-mark, which is Jan. 14, (82) f Dr Vincent, { Fellow of Clare Hall } presented to the Society a manuscript treatise intitled, Phibsophiie Naturalis principia mathematicaf and dedicated to the Society by ]Mr Isaac Newton, wliercin he gives a mathematical demonstration of the Copernican hypothesis as proposed by Kepler, and makes out all the phenomena of the celestial motions by the only supposition of a gravitation towards the center of sun decreasing as the squares of the distances therefrom reciprocally. It was ordered that a letter of thanks be written to Mr Newton ; and that the priming of his book be referred to the consideration of the Council : and that in tlie mean time the book be put into the hands of JNIr llalley, to make a reiwrt thereof to the council." Birch, iv. 479. For some account of Dr Vincent, see Whiston's MemoirSf who was his sizar. It may perhaps prevent further currency being given to the supposition of his being the husband of the lady to whom in early life Newton is said to have been attached, if I state that he was a Senior Fellow of Clare Hull at the time of his death (March 1722). (»=>) See Birch, iv. 484. (") At a meeting of the Council of the Royal Society "it was ordered that Mr Newton's book be printed, & that Mr llalley undertake the business of looking after it, & printing it at his own charge, which he engaged to do." Birch, iv. 406. (86) J\ly knowledge of this letter is derived from a memorandum by Halley, on the back of Newton's letter of July 14, mentioning among Newton's letters one of this date. The contents as stated above are purely conjectural, and founded upon a sen- tence in Newton's letter of Febr. 18, 1686-7, ('* I hope you received a letter with two corollaries 1 sent you in autumn,") coupled with the fact that the two corollaries above- mentioned are not found in Newton*8 MS. (*«) It had been finished in the summer of the preceding year. Writing to Halley June 20, 1686, he says that it " only wants transcribing and drawing the cuts fairly." (87) "I think I have the solution of your problem about the sun's parallax, but through other occasions shall scarce have time to think further on these tilings: and besides, I want something of observation." The "occasions" may refer to the anti- cipated effects of James's mandate, which had been received in Cambridge nine days before. See under March 11. (88) The first mandate was dated Febr. 7, received by the Vice-Chancellor on the 9th, and read to the Senate on the 21st, the second was dated Febr. 24, and read March 11. (89) " It contained the whole system of celestial motions, as well of the secondary as i>riraary planets, with tlie theory of comets, which is illustrated by the example of the great comet of 1680-1, proving that which appeared in the morning in Nov to have been the same that was observed in Dec. and Jan. in the evening." Birch, iv.530. The MS. sheets of the Principia (without the preface) have been bound up into a Volume which is preserved at the Royal Society. It is from no wish to detract from the Iviii SYNOPTICAL VIEW OP NEWTON'S LIFE. value of this treasure that I etate that I do not think the MS. to be in Newton*8 auto- lirraph. I believe it to be written by the same hand as the first draught of the Principia in the University Library. The author's own hand is easily recogrnised in both MSS, in additions and alterations. The Preface in the first edition has no date. The date " Dabam Cantabrigise e Collegio S. Trinitatis, Maii 8, 1686," first appeared in the second edition in 1713. See ndte to Febr. 1704. (•0) The following are the dates of the proceedings connected with this affair. Apr. 21. Vice-Chancellor and delegates appear before the Commissioners. Apr. 27. Give in their plea. May 7. Plea discussed. Vice-Chancellor sentenced to be deprived of his office, and suspended from his Mastership. May 12. The delegates reprimanded. Jeffreys wound up his address to them with the words : "Therefore I shall say to you what the scripture says, and rather because most of you are divines ; ' Go your way and sin no more, lest a worse thing come unto you."* See State Trials, or Cooper's Annals of Cambridge, Newton does not appear at all as a speaker during the proceedings. The Chancellor alludes twice to his having himself formerly been a member of the Univer- sity. Until some other College can establish a claim to him, Trinity College is liable to the suspicion of having had him for an alumnus, A " Georgius Jeffrys " was admitted pensioner there March 15, 1661-2, under Mr Hill, and he would therefore be a year junior to Newton. Under this date may be given the following entry in the College Account Book of the building of the New Library, which probably refers to our philosopher. " May 28, 1687. P^'-.-for erecting a scaffold for ]Mr Newton to measure the fret work of the stair- case : 4s. 6d," We may also notice under this year an elegant method given by him of finding (by infinitesimals) the volume of a segment of a parabolic conoid cutoff by a plane per- pendicular to the axis. "Construction and Demonstration as I received it from M. Isaac Newton, Prof, of the Mathematics, in Cambridge." Guager*s Magazine, by Wm. Hunt, Loud. 1687. (0^) Rigaud, 81, 82. Tlie copy which he gave to the College Library does not contain his autograph. In a copy in Emmanuel College Library is written, " Ex dono Authoris suirie docti lulii 13"°. 1687." The copy in Keill's catalogue of his books is priced at 10s., as also is a copy in Clare Ilall Library, given by Cornelius Crownfield to Cotes's friend Morgan, of which however the price at the time of the gift is put 5s. There is in the same Library a copy of the Theses printed at Edinburgh, in the first half of 1690, by James Gregory, of St Andrew's, containing a compend of the Principia^ alluded to in the Museum Criticum, ii. 518, note, and Brewster's Newton, p. 174, note. The following anecdote of Demoivre's first introduction to the Principia may not be altogether out of place here. The scene is probably to be laid in the year after its publication, when Newton is known to have been out of College. (See Table of Exit? and Redits). Demoivre, then about 21, was earning a livelihood in London by teaching mathematics, in which he thought himself a perfect master. " II en fut bientOt et bien singulierement desabus6. Le hasard le conduisit chez Mylord Devonshire dans le moment oil M. Newton venoit de laisser chez ce Seigneur un exemplaire de ses Principes. Le jeune Math««) " Besides a coach which I consider not, it is but 200^ per annum, with a confinement to the London air, & to such a way of living as I am not in love with." (w») Optics, Bk. 2. Part iv. Obs. la (loj) Appendix, No. XXIII. (»o») Wallis, 0/ip. II. 391. seqq. (>o*) Brookbank was originally of Trinity College. The successful candidate was the Hon. H. Boyle, " a near relation" of the Chancellor, (Duke of Somerset) who wrote a letter (Sept. 6) recommending him to the University. (Baker MSS. xxx. 355). (iM) It was read at a meeting of the Royal Society, Febr. 15, 1710, and ordered to be printed in the Transactions. It was printed in the Introduction to Vol. ii. of Harris's Lex. Techn. 1710. (io«) It may be doubted whether this letter is in Newton's handwriting. The conclusion " S' I am" &c., and the address, are evidently in his hand. (107) The four letters to Bentley were given to tlie College by Cumberland. They were printed in 1756, and reviewed by Johnson in the Literary Magazine. See Monk's Bentley, p. 33 j Brewster's Newton, p. 286. They first appeared in their correct order in Ikntley's Correspondence (Lond. 1842), the third and fourth having previously changed places. (108) '*i have now received the box of rulers, with your receipt of £14. I sent you that money because I thought it was just j & therefore you compliment me if you reckon it an obligation. The chamber next me is disposed of ; but that which I was contriving was ... to make you such an allowance, 6cc." Gentleman's Magazine, LXXXIV. 3. (io») Brewster's Life of Newtont p. 232. In this letter he says : '• I have neither ate nor slept well this twelvemonth, nor have my former consistency of mind." A fort- night ttfteryvards he apologized through a common friend for having written such •• a very odd letter," saying, " that it was in a distemper that much seized hb head, & that kept him awake for above five nights together.** lb. p. 234. I ("0) Dated "At the Bull, in Shoreditch." When he wrote this letter, he **had i not slept an hour a night for a fortnight together, & for five nights together not a wink." I See his letter of Oct. 15, in which he explains the cause of this state of his health. *• The \ last winter, by sleeping too often by my fire, I got an ill habit of sleeping ; & a dis- j temper, which this summer has been epidemical, put me farther out of order." Lord j King's Life of Locke, i. 420, Brewster's Life of Newton, p. 240, where the date is printed by mistake, Oct. 5. Intelligence of liis being out of health was conveyed in a very exaggerated form to Huygens in INIay of the following year by a Scotchman, of whom we know nothing whjitever except that his name was Colm, (M. Biot's Colin): this person's information as recorded in a sort of journal by Huygens, who was himself troubled at the time with symptoms which in little more than a year afterwards terminated fatally, and would drink in with a morbid sympathy the tale of the affliction of a kindred spirit, is in the following terras: "29 Muj. 1694. Narravit mihi D. Colm Scotus virum celeberriraum ac summum geometram Is. Neutonum in phrenesin incidisse, abhinc anno et 6 mcnsi- bus. An ex niraia studii assiduitate an dolore infortunii, quod incendio laboratorium chymicum et scripta quxdam amiserat? Cum ad Archiepiscopum Cantabrigiensera venissct, ea locutum, qua) alienationem mentis indicarent. Deinde ab amicis curam ejus susccptam, domoque clauso rcincdia volenti nolenti adhibita, quibus jam sanitalem ■•* recuperavit, ut jam rursus librum suum Principiorum Philosophiaj Mathcraaticorum NOTES. Ixi intelHgere incipiut." (Hugcnii Exereitationei..„Vy\eahioek, Fa«cic. ii. p, 171. Hag. Com. 1833). This extract was first published by M. liiot in the Dufgruphie Vnivtr$elte (art Newton, p. 168). Sir David Brewster hiis pointed out the improbability of the story and shewn the impossibility of reconciling it with known facts, {Life of JVeirfon, p. 230 foil.) but not to M. lliot*s sutisfaction. We will first quote at length an anecdote which has been brought to bear upon the question, which, however, I think an atten- tive perusal will prove to refer to a period some years antecedent to the ej.>och under consideration. It b found in a MS. diary written by a member of Su John's College, who, at the date of the entry about to be quoted, was in his second year of residence at Cambridge. He seems to have heard the anecdote in company, and immediately chronicled it in his journal. He does not tell us who waa his informant, and therefore we do not know the precise correction to be applied in this instance to an under- graduate's story. We shall not, however, probably err much in believin(; in the substantial truth of the narrative. It runs as follows : — " 1692. i'eb. 3d. What 1 heard to-day I must relate. There is one Mr Newton (whom I have very oft seen) Fellow of Trinity College, that is mighty famous for his learning, being a most excellent Mathematician, Philosopher, Divine, &c. He has been fellow of the Royal Society these many years, ic amongst other very learned Books &c Tracts he's written 6ne upon y« mathematical principles of Philosophy, which has got him a mighty name, he having received especially from Scotland abundance of congratulatory letters for the same : but of all the Books that he ever wrote there was one of colours &c light established upon thousands of Experiments which he had been 20 years of making, &t which had cost him many hunda-d of pounds. This Book which he valued so much, &c which was so much talked of, had the ill luck to perish, 6c be utterly lost just when the learned Author was almost at putting a conclusion at the same, after this manner : — In a winter's morning leaving it amongst his other Papers, on his Study table whilst he went to Chapel, the Candle which he had unfortunately left burning there too, catched hold by some means of other papers, & they fired the aforesaid Book, &c utterly consumed it, & several other valuable writings, & which is most wonderful did no further mischieC But when Mr Newton came from Chapel and had seen what was done, every one thought ho would have run mud, he was so troubled thereat that he was not himself for a Month after. A long account of this his system of light £c colours you may find in the Transactions of the Royal Society which he had sent up to them long before this sad mischance happened unto him." (Abraham do la Prymc's Dtary,in the possession of Prof. Pryme). The foregoing narrative is shewn by Sir David Brewster to be irreconcileable with Huygens's memorandum, on the supposition that they both refer to the same circum« stance. But, as I have stated, I believe De la Pryme's anecdote to refer to an earlier period not exactly known but admitting of being fixed within certain limits, as I will hereafter endeavour to point out. The discrepancy between the two statements is adverted to here solely for the purpose of noticing the singular hold which a traveller's gossip has acquired over M. Biot. '* Nous trouvons au contraire," observes that distinguished philosopher, " entre ces dates un parfait accord," and twits Sir David Brewster willi having overlooked the difference of calendar (^Journal de» Savant 1B32, p. 325). M. Biot tells us that in English documents, previous to the change of style in the middle of last century, we arc to add I to the year of our Lord for dates Iwitween January 1 and March 25, in order to find the year according to the present reckoning, and that therefore 1692 in the above extrac: is what would now be written 1693. It does not require a very extensive acquaintance with the literature of our diaries and correspondence to know that this rule is by no means a safe one to follow. In the case before us it is a matter of fact that the author of the diary commences the year in IxU SYNOPTICAL VIEW OP NEWTON'S LIFE. January*: (ex,gr. the death of Charles II. ia placed in Febr. 1685; under Jan. 1692, which follows 1691, the writer laments the loss of Robert Boyle who died Dec. 31) 1691 ; Dec, 1692 is followed by ** 1693 Jan. 1. This year begins very ill, &c." ; Dec. 1693 is followeil by " 1694 Jan. This month we sat for our degrees, &cc.") Sir David Brewster points to the fact that Newton wrote his four celebrated letters to Bentley during the time when Colm*8 gossiping statement represents him as having fallen into "phrenesis." Upon which M. Biot says, "nous admettrons volontiers roaintenant qu'il {the fire which consumed Newton's papers} est posterieur a la premiere lettre" (Journ. de SavanSt p. 332), and proposes to place the catastrophe between the 10th and 30th of Dec. 1692. " C*e8t d cela sans doute," he remarks, " que se rnpporte le passage Buivant des oeuvres de Wallis imprimOes en 1693....* Quara (metliodum) speraverim Neutonum ipsum aliquaudo fusius traditurum; ct quidem audio ilium hujusmodi aliquid prelo parutum habuisse anno 1671, sed quod (infortunio quodam) flammis periit.' Wallis, Tom. ii. p. 390. Le temps present du verbo audio, (-crit en 1693, ne peut s*appliquer qu'a un accident recent, tel que celui que les autres docu- mens nous attestent." Now the extract here quoted from Wallis is merely a trans- lation of what had originally appeared in English in his Algebra some years before. (Wallis's Algebra bears the booksellers' date of 1685. The bulk of the work was sent to London to be printed in 1676 or 7, but the printing was not proceeded with until about the beginning of Aug. 1683, some additions having been made to it in the mean time. The Preface is dated Nov. 20, 1684.) The passage alluded to is as follows: "But I here only give some specimen of what we hope Mr Newton will himself publish in due time. And it was, I hear, near ready for the press in 1671. But most of those papers have since (by a mischance) been unhappily burned" (p. 347). It is the more remarkable that M. Biot should have fallen into such an error, as nine lines below in tlie same page from which he has taken the above extract, Wallis goes on to say, " Atque ha;c sunt quaj, ex memoratis Newtoni Uteris excerpta, inserueram in editione Anglicana 1685." M. Biot makes another application of his chronological rule to Newton's fourth letter to Bentley, dated Febr. 11, 1693, which he affirms to mean our 1694, and that ** les propres expressions de celleci et sa relation avec les autres" shew that it was written a long time after the third, dated Febr. 25, 169^. Now the letters here called the third and fourth, though printed in that order until the appearance of Bentley's Correspondence in 1842, are wrongly placed. The four letters are endorsed by Bentley in the order in which they were received : on the back of the letter of Febr. 11, 1693 he has written "A 3«i Letter from M' Newton," and on that of Febr. 25, 169^ he has written " A 4''» Letter from Mr Newton." Besides, it can be sliewn, I think satisfac- torily, that Bentley's two last sermons were printed in 1693, and as Newton must have known that, his words in his letter of Febr. 11, **if this come not too late for your use" would have no meaning if they were written in 1694. By way of supplement to Sir David Brewster's refutation of the statement in Huygens*8 journal, it may bo observed that the words " Archiepiscopum Cantabri- giensem'* (probably a mistake for Cantuariensem) imply that the crisis of Newton's "phrenesis" took place in London. A glance at the Table in p. lxxxix. will shew that ho was not absent from College for more than a fortnight at a time in 1692 and 1693, and therefore if the calamity which IVI. Biot first made known to the world really occurred, Newton must have been brought down to Cambridge very soon. Now if this hud been the case, we should, almost to a certainty, have found Newton's name among the invalids in the Steward's Books, where a record is kept of the *' commons" allowed to sick fellows in their own rooms. I'or example, in the year in question, ending at • I am enabled, by the kindness of the family in whose possession the diary now it, to state tliii distinctly. NOTES. Ixiii Michaelmas 1693, we find one valetudinarian fellow allowed his commons in his rooms ("ex. CO,'*) for' 8 weeks, another for 1 : in 1(394 one for 6 weeks, another for 2; in 1692 one for 19 weeks, a second for 15 and a third for 20A ; in 1691 one for 9 weeks, another for 1^, three others for half a week eich, and another for 3. But probably the most elaborate and complete refutation will have less weight with the mj^jority of persons than the testimony of a trustworthy contfmi)orary witness. I will therefore lay before the reader an extract from a letter of Dr Wallis to Waller, the Secretary of the Royal Society, dated I^Iay 31, 1695, from which by the way it will be observed what "strength" Colm's story had " acquired'* in the course of its circulatioa to this country. Wullis had sent a copy of the second Volume of his Works as a present to Sturm a Professor at Altorf. Sturm wrote to thank Wallis for tlie present, and it is tins letter of thanks which Wallis alludes to in the beginning of the following extract : " I have, since, one from Sturmius, which signifies that ho had, some weeks before, received the Book I sent him. lie sends me word of a Rumor amongst them concerning Mr Newton as if his House & Books & all his Goods were Burnt, fie himself so disturbed in mind thereupon, as to be reduced to very ill circumstances. Which being all false, I thought fit presently to rectify that groundless mistake" { in a letter which he desires Waller to forward } . (Lett. Bk. Roy. Soc. W. 2. 50.) I may observe that I should not have devoted so large a space to so transparent a piece of exaggeration but for the remarkable fact of its adoption by M. l^iot, whose veneration for the creator of Natural Philosophy will not, I hope, suffer diminution by this exposure of an idle traveller's tale. (" Et si le sort eftt vdulu lo frapper aussi cruellement, quel sentiment dcvrait fuire naitro en nous son infortune, sinon do plaiudre et de v6uerer davantage cet autre Tiresias, dont I'intelligence se serait aiasi aveugld'C pour avoir vu de trop pres les secrets des dieux ? Toute autre pens^'e serait un sacrilege." Biot in Jour, des Sav, Apr. 1836, p. 216). A word may be added on the probable date of the fire in Newton's rooms. The notice which we have given above respecting the publication of Wallis's Algebra shews that the accident happened before Aug. 1683. The superior limit is the winter of 1677, 16T8 as Wallis believed copies of Leibniz's letters, the last of which was dated June 21, 1677, to have perished in the flames. (Letter to Lcibn. Dec. 1, 1696). One of the winters therefore from 1677 to 1682 (excluding perhaps that of 1680, 1681 during which we know a little more of Newton's movements than in the others) may bo fixed upon as the probable date of the occurrence. The version of the story in which " Diamond" is made to play a prominent part, and according to which the scene is laid in Newton's latter years, and consequently in London, may perhaps deserve a place here. "His temper was so mild and eqval, that scarce any accidents disturbed it. One instance in particular, which is authenti- cated by a person now living, [1780,] brings this assertion to a proof. Sir Isaac being called out of his study to a contiguous room, a little dog, called Diamond, the constant but incurious attendant of his master's researches, happened to be left among the papers, and by a fatality not to be retrieved, as it was in the latter part of Sir Isaac's days, threw down a lighted candle, which consumed the almost finished labours of some years. Sir Isaac returning too late, but to behold tlie tlreadful wreck, rebuke») See under Sept. 16. ("'') A Mr Smith " took a journey" to Cambridge for the purpose of consulting Newton on a problem in chances which had its origin in a lottery recently drawn, and brought with him a letter of introduction from Pepys. The Ist of Newton's letters is principally occupied with settling the meaning of the question (What are the chances of throwing 1 six with 6 dice, 2 sixes with .12 dice, and 3 sixes with 18 dice 1), The 2nd contains his "easy computation." See Pepys's Coirespondence, Ixiv SYNOPTICAL VIEW OF NEWTON'S LIFE. (u«) 4* On {the) Monday (night) likewise there being a great number of people at the door {of the haunted house,— it was a house opposite St. John's College in the occupation of Valentine Austin) there chanced to come by Mr Newton, fellow of Trinity College, a very learned man, and perceiving our fellows to have gone in {three fellows of St John's with a fellow -commoner of that college had rushed in armed with pistols) , and seeing several scholars about the door, Oh ye fools! says he, will you never have any wit 1 Know you not that all such things are mere cheats and impos- tures ? Fie ! fie ! go home for shame. And so he left them, scorning to go in." ( De la Pryme*(* MS. Diary, where there is a full account of the proceedings of the "spirit'* which the writer of the diary had received in a letter from Cambridge.) ("♦) Appendix, No. XXIV. (116) " Quoniam varii errores in Prop. 37 & 38 (Lib. ii.) irrepsere, illos omnes restitutos hie apponam, prout in autoris exemplari inveni, ineunte Maio 1694, dum Can- tabrigiae hsrerem, consulendi divini autoris gratik." MS. of Dav. Gregory (Rigaud. p. 100). ("«) «« July 4. Ordered that a letter be written to Mr Isaac Newton praying that he will please to communicate to the Society in order to be published his Treatise of light & colours & what other Mathematical or Physical Tnatises he has ready by him." Joum, Bk. (U7) ««Mr Newton coming to see me Sept. 1, 1694, and discoursing of the theory of the moon, to let him see wha: 1 had done in order to restore her motion, I produced and shewed him these 3 sheets { or synopses ) of her observed and calculated places compared.'* Flamsteed ap. Baily, p. 191. Shortly afterwards Flamsteed lent him copies of two of the synopses, of which Newton made transcripts at Cambridge. A copy of the 3d was sent Oct. 29. ("«) The whole of the known correspondence is printed in Baily's Flamsteed, pp. 133 — 160. Newton's letters are preserved in the library of Corpus Christi College, Oxford, to which Society they were givein in 1764 by S. Adee, M.D., formerly Scholar of the College. Mr Baily has attempted from this correspondence to shew, in opposition to a prevail- ing opinion, that Flamsteed manifested no unwillingness to furnish Newton with the observations necessary to enable him to complete the lunar theory, but, on the contrary, freely communicated every observation that Newton required. (Supplement to Flam- steed's History, pp. 708 — 720.) I regret that I cannot concur in Mr Baily's conclusion. Assuming, what is far from clear, that up to December, 1694, Flamsteed sent Newton all the observations that he asked for, I think that in the following month, and after- wards, we discover traces of a feeling which is scarcely compatible with Mr Baily's hypothesis. The following particulars are gleaned from Newton's letters, and Flam- steed's rough draughts or notes j additional light will be thrown upon the subject when the correspondence between them is made complete by the discovery of Flamsteed's actual letters, which it is hoped may be found among the Portsmouth papers : 1694 Dec. 6. Flamsteed promises to send Newton the observations that he wants after the Christfnas holidays, 1695 Jan. 15. Newton acknowledges the receipt of two observations uncalculated, and as Flamsteed had calculated these and the other three of last month, he desires a synopsis of the calculations, merely to save himself the trouble of doing what was already done. But as regards the rest of Flamsteed's observations, he repeats what he had said in his letter of Nov. 17, that he desires only the naked observations. — — 19. Flamsteed wrote back, " but no observations imparted... I have not time to send the synopsis now ; may do it hereafter : but would gladly see what places you have derived from the given Right Ascensions first. Shall give more hereafter." — — 26, Newton replies : " Since I perceive you have a mind to see whether we NOTES. Ixv can compute correctly, if you please to «enil me the latitude of Greenwich, I'll send you what you desire."..." I told you in autumn that it would be neces- sary to have about half of the ol)servations in your synopses set right by the correct places of the fixt stars. If you please to do it at your leisure, I'll send you a catalogrue of the observations." This request is agrain alluded to by Newton in his letters of Apr. 23 and July 9, but was never complied with. *• One thing," he continues, and we now come to an important part of the corre- spondence as affecting the question under discussion, " I did not consider. The olwervations being yours, perhaps you had rather have them perfectly your own in all respects, by dcterminingr the moon's longitude and latitude from them all yourself. If so (for that's what you have a very just right unto) I will stay your time. And when I have ^i a little further in the theory. ..I'll make a new table of the moon's eccentricities and equations of her apogee for finding her mean anomaly, and send you a copy of it Chuse you therefore whether you will compute the moon's places from the observbtions or leave that work to me." Tliis was answered in haste on the day on which it was received, but we do not know in what terms. Flamsteed sent a fuller answer, Feb. 7, with some lunar observations calculated and reduced, (among them the three mentioned by Newton Jan. 15, but not the two others.) In his draught of this answer he says: " I shall mind my business of the fixt stars and give him an account of my progress, whilst he is employed on the moon : and shall be very well pleased with an account of his success." Flamsteed accepted Newton's pro- posal with respect to the observations, hinting, at the same time, that he should devote himself to his catalogue of the fixt stars. At this point therefore New- ton's labours upon the lunar theory are suspended while he is " staying the time " of the Astronomer Royal. March 2. Flamsteed, in a draught of an answer to Newton's letter of Febr, 16, has these words : •' Vindication of myself for not imparting my observations, and an account of my northern correspondence." Apr. 23. Newton writes : ** When I have your materials, 1 reckon it { the moon's theory } will prove a work of about three or four montlis : ond when I have done it once I would have done with it for ever." June 29. Newton, who is still staying the Astronomer's time, thanks him for sending his solar tables (which Newton does not seem to have wanted): " IJut these, and almost nil your communications will be useless to me, unless you can propose some practicable way or other of supplying me with observations. For as your health and other business will not permit you to calculate the moon's places from your observations, so it was never my inclination to put you upon such a task, knowing that the tediousness of such a design will make me as weary with expectation as you with drudgery...! will therefore once more propose it to you { as he had done Nov. 17 and Jan. 15 } to send me your naked observations of the moon's right ascensions and meridional altitudes ; and leave it to me to get her places calculated from them. If you like thin propo- sal, then pray send me first your observations for the year 1692, and 1 will get them c:dculated, and send you a copy of the calculated places. IJut if you like it not, then I desire you would propose some otlier practicable method of sup- plying me with observations ; or else let me know plainly that I must be con- tent to lose all the time and pains I have hitherto taken about the moon's theory and about the table of refractions." July 2. Hamsteed, stung to the quick, offers not the mural arc observations of 1692, but the sextant observations from 1677 to 1690. It would also seem, from a statement written by Flamsteed on the back of Newton's Ittter, as if he had sent at the same tinjc the 30 obscrvaiions which he had made from Febr. 8 kvi SYNOPTICAL VIEW OF NEWTOJTS LIFE. to June 25 in the current year. But as Newton makea no mention of. having received them, merely saying, •♦ when you have computed your 30 observations, you will know no more of it { the parallactic equation } than at present," I suspect that there is some mistake in Flamsteed's memorandum. 1695 July 9. Newton writes: " After I had helped you where you stuck... { he par- ticularly mentions the table of refractions, which he says ) cost me above two months' hard labour which I should never have undertaken but upon your account, and which I told you I undertook that I might have something to return you for the observations you then gave me hopes of, and yet, when 1 had done, saw no prospect of obtaining them* or of getting your synopses rectified, I despaired of compassing the moon's theory, and had thoughts of giving it over as impracticable, and occasionally told a friend so who then made me a visit. But now you offer me those observations which you made before the year 1690, 1 thankfully accept of your oflTer, and will get as many of them com- puted as are sufficient for my puipose." >— — 13. Flamsteed sends his observations from Jan. to July 1677. — — 20. Newton says, "The report you mention { which was current in Lon- don about Flamsteed's not furnishing Newton with observations } was mucli against my mind, and I have written to put a stop to it. I thank you for.. .your lunar observations." — 27. Newton says, •' The other day I had an excuse sent me for what was said at London about your not communicating, and that the report should pro- ceed no furtlier. I am glad all misunderstandings are composed." He then specifies the further observations (out of the sextant stock ) that he wants. Sept. 14. Newton returned to Cambridge on Sept. 10, and went away again on the 14th : before leaving, he writes, " I have not yet got any time to think of the theory of the moon nor shall have leisure for it this month or above : which I thought fit to give you notice of, that you may not wonder at my silence." He however returned in a fortnight, but had sublunary matters to attend to, was named by rumour shortly afterwards as Master of tlie Mint, and in the March of next year was actually appointed Warden. ■»- 17. In Flamsteed's draught, written on Newton's letter, we read, *' My exercise will devour no small part of my time, and therefore I shall desire my friends to excuse me if I answer not their letters so fully nor readily as for- merly ; however, when you want more of my lunar observations { i.e. those made before 1690 with the sextant, not those which he had made or was making with the mural arc ] I shall cause them to be transcribed and it will be no trouble." Mr Bwly has printed the words "however no trouble" in italics ; the preceding part of the sentence is not however destitute of signi- ficance. Here the correspondence terminates. There are several allusions to it in Flamsteed's extant memoranda, two of which are produced here as evidence in the question we are examining : " { Mr Newton } ceased not to importune me (though he was informed of my illness) for more observations, and with that earnestness that looked as if he thought he had a right to command them, and had about 50 more imparted to him. But I did not think myself obliged to employ my pains to serve a person that was so inconsiderate as to presume he had a right to that which was only u courtesy. And I therefore went on with my business of the fixed stars ; leaving Mr Newton to examine the lunar obser- vations over again : which had he done, he had found that he needed not be so importu- nate for new, — the old would have been sufficient for the purpose and design for which • Flamsteed htu written on tlie letter " My sickness has hindercil." But wc shaU see by and by fh>m his own statement that that was not the sole cause. NOTES. Lxvii I had imparted them to him. I was therefore forced to leafo off my corre«pondence with him at that time." (Baily, p. 63.) Ajrain: ** I continued since fumi«hin(r him with lunar observations, as I gained them, until Midsummer 1G95, when bcin^ troubled with a distemper I was forced to intermit my correspondence with him." (lb. p. 191.) Upon the whole, I think, we may conclude that the combined action of Flamsteed's bad temper and bad health, for which great allowance muet be made, coupled with his professional jealousy of Halley* and his exaggerated opinion of the value of hit. own astronomical labours, has robbed us of the lunar theory in the form tliat its creator would have given it, and tiiat the following words contain more truth than is sometimes to be met with in epistolary statements : '* Flamstcedius suas de Luna obscrvalioncs Newtono negaverat. Inde factum aiunt quod hie qua;dam in motu Lunari adhuc indeterminata reliquit." (Leibniz to Roemer, Oct. 4, 1706. 0pp. Tom. iv. Pan ii. p. 126.) . . »,. ..... . , sin 0*« parallax ("») This inequality m the Moon's longitude is proportional to -7-^- , sin ])'» parallax its argument being ])'s mean anpulia.r distance from ©. •' On la consld6rer...avec raison comme unc des applications les plus delicates de I'analysc moderne.** (Hiot, Journ. des Sav. Apr. 1836, p. 218.) In his letter of July 9, 1695, Newton says that its maximum value scarce exceeds 2 or 3, or at most 4 minutes, Biirg ( Mtcan. Cel. Tom. III. p. 282) gives it 2', 2", 38. Compare Pont^coulant, iv. 605, who (ib. xiv. note) does not seem to be aware that this equation was known to Newton. M. Biot says that this equation is omitted in thesccond edition of the Priricij^ta, and suggests reasons to account for the omission. But see p. 120 of this work, where the " Variatio secunda" is de- scribed. ("0) This is now called the lunar equation of the Sun, "et Ton avail tout lieu de la considerer comme une des corrections les plus delicates des tables modernes." (Biot, Journ. des Sav. Apr. 1836, p. 220.) It = J^J!^ . dist. of^from Q ^ ^ ^.^ difference ti^'s mass dist. of Q) from tP of longitudes of]) and ©• The coefficient is given 8",83 in the M/lcan. Cel. Tom. iir. p. 108. Newton in the above letter says that he had not yet ascertained its magnitude, but that it may be assumed 16" or 20" until it be determined more exactly. Comp. Pontecoulan'., iv. 653. (1") Flamsteed's coquetry about his two observations draws from Newton a little playful irony— an indulgence extremely rare with him : " The places of the moon from your two observations I have not yet computed : for I thought it superfluous to do what you had done to my hands ; and desired a copy of your computations only to save myself that labour. But since I perceive you have a mind to see whether we can compute exactly, if you please to send me the latitude of Greenwich I'll send you what you desire." (Baiiy, p. 149.) (122) This is the table afterwards published by Ilalloy in the Phil. Trans. May— Aug. 1721, •• such as I long since received it from its Great Author." See Biot's third article on Baily*s Flamsteed in the Journal des Suvans for Nov. 1836, which he commences by observing that he is in arrcar with the article, "et pourtant, depuis environ ncuf mois que mon second article a paru, je n*ai pas 6t6 cccup6 d'autre chose que de sa con- tinuation. Ma'is, pendant tout co temps, je puis dire en v6rit6, comme Jacob, que j*ai lutt6 avec l'esprit." For the results of the struggle see that article, and his paper ** Analyse des Tables de refraction construites par Newton, avec I'indication des pro- cadi's num6riques par lesquels il a pu les calculer." ( lb. pp. 73S— 754.) * The torrents of vituperation poured by Flamsteed upon thia illutlrioui man are, I tielieve, to be explained on the principle alluded to: (Krpa/bi(v< Kepafitt..) At the meeting; of the Itnyal (kK'icty, June 1, 161)2, Hallcy read a paper vindicating his St Helena Observations " from some groundluu ex. ccptlons" of Flamsteed's. Ixviii SYNOPTICAL VIEW OP NEWTON'S LIFE. (»«•) Some delay occurred in sending this letter. Flamsteed did not receive it until May 6. (>««) Appendix, No. xxvi. ("») •' I shall order Will Martin...to pay him two guineas, if you please to let him call for them, or to pay it to his or your order in London if you please to let Bie know where." The words in this extract which follow '• pay him " are crossed out in the MS. and tiie word " guineas" altered into "shillings " apparently by Flam- steed. The words after "for them," to the end of the passag-e, are conjectural, tlie original writing bein& most skilfully blotted out. I believe however that it might be made out on a bright day, if it were thought worth the trouble. What motive Flam- •teed could have had for disguising any part of the above sentence I do not pretend to divine. It is curious that Mr Rigaud, who, at Mr Daily's request, examined the MS. with reference to this very point, should have overlooked the original " guineas." (Baily, p. 16.9, note.) (»3«) Wallis, writing to Ilalley from Oxford. Nov. 26, says: *• We are told here that he is made Master of the Mint, which if so, I doe congratulate to him and am his & your &c." Orig. Lett. Bk. Roy. Soc. W. 2. 66. See Appendix, p. 302. (»7) Mace. Co)T. II. 419. ("8) The original MS. with the address, " For the Right Honourable Charles Montagu Esq. Chancel lour of the Exchequer," is preserved at the Royal Society, Orig, Lett, Bk, N. l.eib. The problems are (1) To determine the brachistochron between two giv^n points not in the same vertical line : (2) APF' is a straight line passing through a fixed point A, and meeting a curve in P,P': to find the curve such that ilP" + i4P'" = constant. One of the two identical papers (a printed folio half- sheet) which were sent to Newton by Bernoulli, containing the problems, still exists in the Archives of the Royal Society, (Volume lettered "Arithmetic, Algebra," &c, 13). At the bottom, in Newton's hand, are the words " Chartam hanc ex Gallia missam accepi Jan. 29, 169^." (»2») Mace. Carr. n. 420. (•M) See Appendix, p. 299. ("1) " Isaac Newton chuseth the Hon**'* Henry Boyle Esq", Burghess of this University." The votes were given in English on account of the election occurring during the vacation. ("2) James Hodgson had calculated these 12 places for Newton by Flamsteed's orders, during the absence of the latter in Derbyshire, and sent them to him Sept, 8. Flamsteed on examining them Nov. 11, " found them all false," and computed them afresh. The results of these last calculations were communicated to Newton on his visit to Greenwich, Dec. 4. On December 29 Flamsteed sent him a correction of the time of one of the observa- tions, and after.vatds found that his results required further modification. " I acquainted him," he says, "there was a further fault in them, when I was last with him. He is reserved to me, contrary to his promise, I lie under no obligation to be open to him." (Baily, p. 166). Flamsteed was in London on Dec. 30 and 31, (Friday and Saturday), and the words '* when I was last with him," probably refer to one of those days. Newton was then aware of the liberty which Flamstceil accordC* toutc sa confidence. II alloit tous les soirs I'attendre dans un cafe (pro- bably Slaughter's Coffee House in St Martin's Lane } ou M. Moivro se rendoit des qu'il avoit fini ses lemons, et d'ou il I'emmenoit chez lui pour y passer la soiree dans des tete-a- t^te philosophique." ( Eloge, 1754). (167) •* I thank you for giving me timely notice of the caveat, and think we should stick at no charge for defending the legacy. What money shall be wanting for this purpose I'll advance till the Council shall be called. If you see Dr Harwood before me, pray desire him to have an eye upon this matter. I do not know the method of proceeding in these cases ; but he can tell us. I will take the first opportunity to inform myself of what is to be done." (Sloane MSS. Brit. Mus. 4054 ; printed without the date in Nichols's Illustrations of Lit. Hist. xiii. 59). The note in the same volume, dated Thursday night, (" Lady Betty Gayer being engaged for to- morrow, and at liberty on Monday or Tuesday, I beg the favour we may wait on you on either of those days at tliree o'clock, and that you will let us know which of those two days you are most at leisure,") is recommended to the attention of those who are versed in the "fashionable arrangements" of Anne's reign. (*") The trustees appointed under Plume's will (Covel, Bentley, Whiston, Fra. Thompson of Caius) were directed to frame statutes for the regulation of tlie Professor- ship, "with the advice of Sir John Ellis, (Master of Caius), Sir Isaae Newton and Dr { sic \ Flamsteed." Cotes, the first professor, was elected Oct. 16, 1707. Flam- steed wrote to Whiston Febr. 13, 1705-6, (compare Baily, p. 258,) recommending his assistant Mr John Witty for the Professorship. (Flamsteed's MSS. at Greenwich, XXXIII. 65). In Vol. lxix. of the same Collection, there is a long letter, dated Dec. 31, 1706,) from Ellis to Thomi)son, on the subject of the Professorship, in which Cotes is spoken of in very high terms, and in Vol. xxxin. p. 74, there is an answer to it, in which Flamsteed is reported as saying that "Trinity Gatehouse is not fit for" an ob- servatory, (see p. 200) " and that that of St John's is preferable, and that the Virtu tis Gotcway at Caius is better than either." Flamsteed wished a separate building to bo devoted to tlie purjjose. The substance of a note written by Prof. Smith on the fly-leaf of his copy of NOTES. Ixxv I!uyBCM*8 Cosmvtheoros (Hag, Com, 1698) and dated 1764, i* worth preserving. •• I have been well informed that Dr Plume, Archdeacon of llochest«r» was so pleased with this book, which the celebrated Mr Flamsteed had recommeudcd to him, as to leave by his will £1800 to found the Pluraian Professorship of Astronomy and £zperi- mental Philosophy, which 1 held many years after Mr Coles's decease." (»') Appendix, No. XXXI. (»«o) •' Instead of the like sum he intended after his death. It was ordered to b« put up by itself and to be subject to such end or benefaction as tite President shall direct." This no doubt is the foundation for Thomas Ileamc's scandal, " ho promUcd to become a benefactor to the Royal Society, but foiled." See under Doc. 14, of the following year. (ii^) It (ills pp. 4—157 of the present volume. ('") Mr Do Morgan, in his sketch of the life of Newton, says that in the 2nd etlitton Flamsteed's name was "erased in all the passages in which it appeared (we have verified, for this occasion, eight or nine places ourselves)." The name however will still be found in pages 441, 443, 455, 458, 4G5, 478 and 479 : the lost two references occur in some additional matter on comets, which was put into Cotes's hands in October 1712. (See p. 141 of this work.) I question very much whether the suppression of Flomsteetrs name in several places where it had appeared in tlie 1st edition was not such -as was necessary in the process of improving' the work. Newton's own experiments on the old echo in Trinity College cloister give way in the 2nd edition to more accurate researches, ("») The original of this paper is in the Britisli Museum, Add. MSS, 6489. fol. 67. ('♦ ex dono D°* Sharp"), It is printed in the Gentleman*t Mag. for Jan. 1755, pp.3 — 5, (Compare his Chronology, p. 71, sqq.) In the same MS. volume (fol. 69) is an abstract of the paper in Newton's hand, (printed in the Appendix to this Work, No. XXXIII.), which was embodied in a letter to Bishop Lloyd by an unknown writer, dated Nov. 7, 1713, of which the draught is preserved in the volume referred to (fol. G5, 66), beginniniif " I had the honor to receive and the pleasure to read the papers your Lordship directed to the Dean of Norwicli { Prideaux } : and before I sent them forward I communicatefl them to Sir Isaac Newton, according to your Lordship's order by Mr Archdeacon : when Sir Isaac brought them back, he told me that he found many excellent observa- tions in them about the ancient year, and at the same time acquainted me that ho had formerly discoursed with your Lord.ship about that year of 360 days, and represented" &c. (See Appendix, p. 314). Trininell, IVishop of Norwich, may possibly have been the writer of this letter, as, three years before, he was the organ of communication between Lloyd and Prideaux, conveying to the latter Lloyd's scheme of Daniel's 70 weeks. {PrUeuux's Life, p. 237). It would appear that Newton's abstract, and not the paper itself, was sent to Lloyd, but it docs not seem very clear why the abstract was drawn up at all. ("*) This and four other letters to Keill are printed in this volume, p. 169, foil. ('^^) John Chamberluyne was endeavouring to reconcile the two philosophers. He sent Newton's letter to Leibniz, who replied in a letter dated Vienna, Aug. 25, (Leibn. 0pp. III. 491) part of which was read by Chamberlayne ut the meeting of the Uoyal Society on Nov. 11. In it Leibniz " desires that some letters and papers of Mr Oldenburg and Mr Collins which he supposes to be in the custody of the Royal Society may bo communicated to him in order to his publishing a Commercium Kpistolicum in defence of himself ut his return from Vienna to Hanover. The Society was of opinion that Mr Leibniz ought either to make good his charge against Dr Keill or to ask pardon of the Society for suspecting their judgment and integrity in the Commerciura Kpistolicum alreiidy published by their order and approbation. But Mr Chamberlayne saying that Mr Leibniz designed in a sliort time to be in England, the farther consideration of this affair was referred to some other opportunity." Journ. Bk. There w in the BritisJi Museum (MSS. Birch, 4284) a copy in Newton's hand of Leibniz's letter of Aug. 25. ("«) The other assessors were Sir James Montagu, Dr Cannon, Prebendary of Kly, Ixxvi SYNOPTICAL VIEW OP NEWTON'S LIFE. Dr Samuel Clarke, Dr Henry Newton, Chancellor of the Diocese of London, and Dr Johnson, Chancellor of the Diocese of Ely. (Colbatch** MSS.) The trial after con- tinuing about B\x weeks, the Court holding its sittings two evenings in the week, ended on June 14. See Monk's Bentley, pp. 281—286. (»«») Commons* Journals, xvii. 677, 716. I do not consider M. Biot's abstract of the proceedings on this occasion (Biog, Univ, art. Newton, pp. 192, 193) as a model of accurate condensation t I will therefore exhibit Whiaton^s statement as. nearly as may be in his own words. In 1714 Whiston and Ditton commuLicated to Newton their method of discovering tlie longitude at sea by signals, aiid at his desire to HoIIey, as also to Sam. Clarke and Cotes, and soon had their approbation- ^o tar as to encourage them to apply to the House of Commons for a reward to such as should discover the Longitude. A Com- mittee was appointed to examine into the matter, and the four persons just mentioned were summoned to attend. " As soon as the Committee was set, which was a very large one, Newton, Halley, Clarke and Cotes appeared. A chair was placed for Sir I. Newton near the Chairman j ]Mr Clayton, M.P. for Liverpool } , and I stood at the back of it. What the rest had to say they delivered by word of mouth, but Sir I. Newton delivered what he had to say in a paper { referred to above } . Upon the reading of this paper, the Committee were at a loss, as not well understanding its contents: Sir I. Newton sitting still and saying nothing by way of explication. This gave the chairman an op- portunity which it was perceived he wanted of trying to drop the bill ; which he did by declaring his own opinion to be that ' Unless Sir I. Newton would say that the method now proposed was likely to be useful for the discovery of the Longitude, he was against making a bill in general for a reward for such a discovery* ; as Dr Clarke had particularly proposeknown saying attributed to Newton on the premature death of this promising mathematician is Robert Smith, who in his copy of the Uannonia Mensurarum, under Cotcs's epitaph, has written the words *' S' Isaac Newton, speaking of Mr Cotes, said, * If He had lived we might have known something '.'* In his Optics, (Vol. ii. art. 465, remarks^ p. 16) he gives the saying in exactly the same words, where in allusion to a theorem on the image of an object seen through a number of lenses he says: "That noble and beautiful theorem... was tlie last invention of that grreat Mathematician Mr Cotes, just before his death at the ago of 32: upon which occasion I am told Sir Isaac Newton said "...The autlior of Cotes's Life in the Biographie Universelte, who has been followetl by Delambre (p. 457), seems to have misunderstood this passage, taking Newton's remark to apply to the discovery of the optical tlieorem. Parne, who was six years junior to Smith, in his Collections fcr Hist, of Trill, Coll. p. 351, gives the saying with the single variation of "had" for "might have :" " On the death of Mr^otes Sir Isaac Newton b said to have expressed himself in these honourable and remarkable words....** (179J "The President in the chair. The President grave the Society his picture drawn by Mr Jervase for which he had their thanks." Joum. Bk, ("«) In pursuance of an Address to the king it was laid before the House of Lords on Jan. 21, 1718. In consequence of this Re])ort a Proclamation was issued in December 1717, reducing guineas from 2l5. 6d. to 2I5. ("*) This Report was accompanied with an Account of the Gold and Silver coine) A fourth edition was published in 1730 from a copy, it is suid, of the third cor- rected by the autlior's own hand, and left before his death with the bookseller. (>") p. 188. ("') Made jointly with E. Southwell and J. Scroope, Esqrs. Printed in Hibernian Patriot, ** being a Collection of the Drapier's Letters," ficc. 1730, p. 244. Comp. Scott's Swift, VI. 392, ed. 2. (iM) Xewton was then occupied with the 3rd edition of the Principia. Delisle tells us that Newton assured him that "si M. H alley avoit eu 6gard { in constructing his Lunar Tables } aux moindres Equations dont il a fait mention dans sa Theorie, et qu*il eut ajout6 une minute etdemie a la longitude de la Lune pour son ucc616ration physique dans notre teraps, il n*auroit trouv6 aucune difference sensible entre sea observations et le calcul." Journal det Savans, June 1750, p. 428. ( "») Appendix, No. XXXIV. (}**) Mace Corr, n. 435. Newton wanted the calculations for the 3rd ed. of the Principia. If Ualley re-examined the two calculations, the examiuation led to no new result, and if he performed the calculation for the place iu the parabolic orbit, no use was made of it in the 3rd ed. as had been intended. (18T) Gentleinan'i Mag. lix. 775, (with three other letters to the same person). It begins " A bad state of health makes me averse from minding business." (iM) Tumor's Grantham, p. 172. Brewster's Newton, p. 363. "Just after he was come out of a fit of the gout... ; he was better after it and his head clearer and his me- mory stronger than I had known them for some time." (189) Tumor's Grantham, p. 158. (»M) J»/ii/. TraM*. for 1725, p. 315. Brewster's Netutou, p. 262. The summary is entitled ** A Short Chronicle from tlie first memory of things in Europe to the conquest of Persia by Alexander the Great," and was afterwards published in his Chronology. It was drawn up in a few days at the request of the Princess of Wales. Conti, at her desire* was allowed to have a copji of it, from which when he went to France other transcripts were made. Newton's CAronoIo^i/, (Lond. 1728) appeared towards the end of 1727. Conduitt's Advertisement stales that it " was writ by the author many years since ; yet he lately revised it, and was actually preparing it for the press at the time of hisdeath." Martin Folkes writing to Morgan, Masterof Clare Hall, Jan. 6, 1727-8, says: '* I am glad you have been so well entertained with Sir Isaac's book, and at the same time to find my own opinion of it so entirely confirmed.. .but indeed I have had that satisfaction from several hands, and I even hear your Neighbour of the great College { Bentley } who spoke very slightingly of the performance before it appeared begins not to talk so magisterially as he did before, but \V. W. { VVhiston } continues in the same way, and declares he shall overturn it so easily that he shall not be able to extend the whole confutation to a sheet of paper." (»»i) " Pendant les deux mois que I'abb^ Alari passa a Londres { 1725 } , il visita I'universit^ de Cambridge, et le grand Newton, qui joulssait alors dans la capitale de I'Angleterre, de I'estirae g^n^rale de I'Europe et de cinquante mille livres de rente, en qualit6 d'intendant des monnaies. L'abb6 6tant all6 chez lui a neuf heures du matin, I'Anglais debuta par lui apprendre qu'il avait quatre-vingt-trois ana. On voyait daiis sa chambre le portrait du Lord Halifax, son protecteur et celui de l'abb6 Varignon dont il estimait les ouvrages de g6om6trie. Varignon et le ptre S^bastien carme, sont, dit-il, eeux qui ont le tnieux enlendu mon tyittme sur les couleurs. La conversation tomba ensuite sur I'histoire ancienne, dont Newton s'occupait alors. L'abb6, qui 6tait plein de la lecture des auteurs grecs et latins, I'ayant satisfait, il le pria a diner. Le repas fut detestable ; Newton 6tait avare, et il ne fit boire ii son convive que des vinsde Palme ou de Madere, qu'il rccevait en pr^sens. Apres le diner, il mena I'abb^ a la Societe NOTES. Ixxix royale de Londres, dont il ^tait president, et le fit aMeotr 4 la droits. La t^aiiM eora* mcnya et Newton s'endonnit. A la fm dc la seance, tout le nionde •iffna le reRiatre, et I'nbbe comino lus aulrea. Newton le ramena ensuile chcz lui, oil il le garda jusqu'^ neuf hcurcB du soir." ( Eisai H'utorique tur liolingbroktt^ compiled by General Grinioard) in LeUre$Historique$...de...Boliugbroke, i. 155, 1'aria. 1808). Alari was born in 1089 ; he was a friond, at leant for soroe time, of Dolinf^broke'i) and instructor of Louis XV. The intelligent roadur will make allowance for the wpico added to give pungtjncy to the story. The following is the simple record in the Journal Book of Alari 's visit, ' Mr Mildmay had leave to be prcsent| as also Mr Petre Joseph Alary, a French Gentleman." C*^) Life of Maclaurin, prefixed to bis Account of Sir Isaae Neu'(on*i PhUosophieal Discoveries. ( »") Printed in the Phil. Trans, for 1725, pp. 315—321. Comp. Brewster's New. ton, pp. 261— 2G5. The MS. written in a fine copper-plato hand is preserved in the Archives of the Royal Society, and is endorsed " read about the latter end of 1725,** In this paper he incidentally informs us that when he lived at Cambridge he used some- times to refresh himself with History and Chronology for a while, when he was weary with other studies. (»*) The Preface is dated Jan. 12, 1725-6. Twelve copies are stated to have been printed on large paper, (Rigaud's Bradley^ p. xi.), of which there is one in Trinity College Library, another in that of Queens' College, (a presentation copy from the author to his friend J. F. Fauquier,) and a third in the Library of the Royal Society, of which we find the following naive notice in the Journal Book. " March 31, 1726. Mr Folkes in the name uf the President gave the Society a Book richly bound in morocco leather aa a present for the Library, entitled Philosophiie Naturalis Principia Matht' miiticof printed at London 1726. The Society ordered thanks to be rendered to the President for this invaluable present." It is to be hoped that the correspondence which passed between Newton and his editor (Henry Pemberton, M.D.) during the progress of the work through the press will yet be discovered. See Rigaud's Essay, p. 107. PMlos. Mag. May 1836, p. 441. We may give here the anecdote quoted by Mr De Morgan firom Maty*s Memoirs of Demoivre {Phil. Trans, 1846, p. 109.) *' Comme tout ce qui regarde les grands hommes peut £tre interessant, on sera peut-etre bien aise de savoir que Newton a souvent dit h Mr. de Moivre que s'il avoit ^te moins vieux il auroit ^t6 tent6 de revoir sur les dernierea observations sa th^orie de la Lune, ou comme il s*exprimoit de Vattaquer de tiouveau {to have another pull at the moon). Je tiens ceci de Mr. de Moivre lui-rafime." ("*) Baily, Memoirs of Astron. Soe, viii. 188. (iM) u JMarch 23. The chair being vacant by the death of Sir Isaac Newton there was no meeting this day." {Journal Bk.) For the reflections which his death suggested tosome minds, see Boyer's Po/itica/ State of Great J3ritain (or Mzitch 1727, (Vol. xxxiii« pp. 327 — 330), In Mist's Weekly Journal for IMarch 25, the obituary opens with " Sir Isaac Newton, the greatest Mathematician that the World ever knew." Thomson's " Poem siicred to the IMemory of Sir Isaac Newton," (dedicated to Walpole) seems to have had a large circulation. I have a copy before me of the 5th edition, dated 1727. I wish that I had been able to contribute more local information respecting Sir Isaac Newton than it has been my fortune to meet with. But the age of "conversations with" and "reminiscences of had not yet arrived, and we do not know that any fellow of his College kept a diary. Thomas Parne, who took his B.A. degree in 1718, collected materials for the history of Trinity College, and had opportunities of conversing with men who had been contemporaries of Newton (for example, George Modd who was two years junior to Newton, and lived in College until his death in 1722). He has given us many particulars of more or less interest relating to Ray, Thorndike, Pearson, Barrow, Duport and other members of the College, but the only allusions to its chief pride and boast that I have found in his MSS. are the following': under the head of Ixxx SYNOPTICAL VIEW OP NEWTON'S LIFE. ** Writers** the name *' Newton" stands first in the list; the dates of his return as M.P. for the University and of his unsuccessful contest, (in the latter of which the majority agrainst him is erroneously stated) are given, and an anecdote is preserved of his absence of mind in these terms : " Newton hath come into the Hall without his Dand, and went towards St. Maries in his surplice ;" for which Tame quotes as his authority a " Mr Burwell," (perhaps Alexander Burrell, eleven years senior to Parne, who may have been a connexion of the Alexander Burrell who took his B. A. degree in 1670, and was chaplain of the College from Oct. 1673 to June 1681.) I do not know that I can find a more appropriate place for a similar anecdote which has already o appeared in English. It was told to the Swedish Professor Bjornstahl at Basle by John Bernoulli, son of the famous John, on Nov. 5, 1773 : " sagte uns, Newton sey eben- falls sehr zerstreut gewesen, und babe einmahl den Fing-er eineti Frauenzimmers genom- men, urn seine Tabakspfeife nachzustopfen." {Brief e auf...lUk.en. Leip«. und Ilostock. 1777—1783. v. 46). On Dec. 8, in the following year Bjornitahl paid a visit at Am- sterdam to the "gelehrten llerrn Fontein," an Anabaptist preacher and scholar of HemsterhuLs and Albert Schultens. " Ira Jahr 1738 hat er eine lleise nach Kiigluiid gemacht und mit dem grosseu Bentley Bekauntschaft unterhalteu. Zu Cambridge hat er verschiedne Auekdolen von Newton, welcher bervihmte Mann ncun oder zehn Jalir vorher gestorben war, gehtirt, unter andern : Newton habe geglaubt, dass Mahomed von Gott gesandt worden sey, um die Arabcr von der Finsterniss zuriick, und zum Glauben an einen Gott zu fiihren u. s. w. (Dies haben ihm wenigstens die Professoren oder Fellows zu Cambridge als eine besondre Merkwlirdigkeit aus Newtons Geschichte erziihlt ;) die im Koran und Mahomeds Leben vorkommendcn Fabelu und Wunder jcdochhabe dieser aufgeklarte Mann nicht geglaubt. £r sagte mir, Newton habe eine Abhandlung herausgegeben, um zu beweisen, die Stelle 1 Joliann. v. 7. sey nicht iicht, und der Text habe ohne diesea Vers einen weit bessern Zusammenhang.** ( lb, 462). The Professor was in England from April 1775 to March 1776. Writing from Oxford Oct. 24, 1775, after saying that he passes over many remarkable objects, such as the Marmora Oxoniensia, Cromwell's scull, Guy Faux's lantern, Blenheim, Stowe, 6cc, he proceeds : " Dagegen aber will ich einen Umstand mclden, der, wie ich mit Ueberzeugung weiss, bisher in keinem Buche vorkommt : diesen, dass wir unter andern in der hiesigen Nachbarschaft ausdrlicklich zudcm Ende eine Reise gethan haben, um die eigne Blicher- samralung des grossen und unsterblichen Ritters Newton zu sehen. Jetzt besitz sie Herr Doctor Musgrave... Rector zu Chinnor, achtzehn...Meilen von Oxford. Sie hat ihm un- gefehr vier hundert Pfund sterling gekostet. Hier findet man alle Ausgaben von Newton's Werken, und, welches das merkwUrdigste ist, am Rande mit seinen eigenhiindigen Anmerkungen angefUllt, und bisweilen mehrere Blatter am Schlusse der BUcber von ilim gauz vollgeschrieben. Ich zweifle nicht, dass ein Newtonianer hier nicht viel Vergniigen und manche ErlUuterung antreffen wUrde. Hier sah ich auch di\s seltne Buch von Herr Jones Vater, woven ich oben angemerkt habe, das der Sohn selbst es nichteinmahl besitze. Der Titel ist: Epitome of the Art of Practical Navigation... Lon- don, 1706. Noch ein sehr seltnes Buch von eben diesem Jones: (dies ist ganz ausser- ordentlich rar:) Synopsis Palmariorum Matheseos... London 1706...Uebngen8 sieht man, dass Newton eine vortrefllche Bibliothek gehabt hat. Alle griechirichen und latcin- ischen classischen SchriftstoUer finden sich daselbst. Sonst habe ich verschiedne eigenhiindige Briefe von Newton an Flamsteed gesehen, die in der Corpus-Christi-Bibli- othek zu Oxford aufbewahrt werdon. Zu Cambridge werden noch mehr Ilandschrifteu von ihm angetroflTen." (in. 288.) I have no means of confirming or impugning the accuracy of the account given by the simple-hearted Swede of the disposal of Newton's Library. A statement of its magnitude will be found at the end of the subjoined extract from Maude's Wemley- dale (p. 106.) NOTES. Ixxxi Newton'g nephew, Benjamin Smith, •• left a small ivory bust \ of hw uncle ] of ad- mirable workmanship by that celebrated artist, Marchand, which from its elef^nce, similitude and placid expression is truly valuable. It is said to have cost Sir Isaac 100 guineas and is specified in an authentic inventory of his effects, taken by virtuo of a commission of appraisement in April 1727, now in my possession. It appears that his personal estate amounted to jC31,82I 16s. \0d. which was distributed among eight relations, Sir Isaac dying- intestate:... as a proof of his benevolence. ..at his death there was owing him by one tenant £60 for 3 years rent, and by another for 2 ) years a smaller sum. ... { Ilis } wardrobe and cellar. ..in the valuation stand thns. Item, wearing apparel, woollen and linen, one silver hiltcd sword, and two canes, £8. 3j. Item, in the wine vault, a parcel of wine and cider in bottles, JC14. 16>. 6d. The furniture and luxuries of his house bearing nearly the like proportion, his library excepted, which consisted of 2000 volumes and 100 weight of pamphlets." It does not fall within the scope of our Chronological Synopsis of Newton's life to notice the great political events of his time, and I am therefore compelled to place here an extract from an in^renious French writer which lui^^ht otherwise have been given under a more convenient head. 1 leave it to future inquirers to ascertain the precise embarrassment alluded to in it, and to determine the probable extent to which we aro indebted for the story to tlie play of a lively imagination. '* Pour faire voir que runiversaliti* des talents est une chimSrc, je ne teux pas chercher mcs autorit^-s dans la classe commune des esprits ; roontons jusqu'i^ la sphere de ces g^nies rares qui, en faisant honneur h I'humanit^, humilient les hommes par la comparaison. Newton, qui a devin6 le systeme de Tunivers, du moins pour quelque temps, n'etoit pas regarde comme capable de tout par ceuz mdmes qui s'honoroient de I'avoir pour compatriote. Guillaume III, qui se connoissoit en hommes, £toit embarrass^ sur une affaire politique; on lui conscilla de consulter Newton; Newton, dit-il, n'est qu'un grand philosophe. Ce titre (;toit sans doute un ^loge rare ; mais enfin, dans cctte occasion- Ihf Newton n'etoit pas ce qu'il falloit, il en {'toit incapable, et n'6toit qu'un grand philosophe. II est vraisemblable, mais non pas d^'montr6, que, s'il eflt appliqu6 h la science du gouvernement Ics travaux qu'il avoit consacr^s k la connaissance de I'univers, le roi Guillaume n'edt pas d6daign6 ses conseils. Dans combien do circonstances, sur combien de questions le philosophe u'eOt-i'l pas rd'pondu h ceux qui lui auroient conseill6 de consulter le monarque : Guillaume n'est qu'un politique, qu'un grand roil" (Duclos's Considerations sur Us Moeurs, (Euvres, 1. 160. Paris, 1820.) Ixxxii SYNOPTICAL VIEW OF NEWTON'S LIFE. DIVIDENDS RECEIVED BY NEWTON, AND NUMBER OP WEEKS HE RESIDED EACU YEAR WHEN FELLOW OF TRINITY COLLEGK Tear emllng Weeks rcaiJed Year ending Weeks rcBldod Mlchaelimw. Hli DlWdend. by him. Blicliaclnuu. nia Dividend. by him. 1668* £15 AUin long ^IvacuUon.) 1685 £25 51 9 25 52 6 12.10 52 70 20 m 7 12. 10 45 1 16. IS*. 4i. 48 8 Nil. 45 2 16. IS*. 4d. 48 J Of Nil. 19 3 Nil. 49i 90 Nil. 294 4 25 51 1 12. 10 441 5 25 46 2 12. 10 49 6 25 50X 3 25 49i 7 12. 10*. 43| 4 Nil. 49 8 25 49 5t 34 50 9 25 38 6 34 274 80 25 36^ 7 Nil. 1 25 49 8 34 4 2 12. 10 46J 9 34 S 12. 10 46 1700 37 4 25 52 1§ 2 40 2 The dividend was voted at the annual audit in December, and paid by the Bursar " as money came to his hands," generally at the end of 6 or 12 months, but sometimes the payment was still further delayed. Newton's own receipts for his dividend, livery and stipend for the four years of Humfrey Babington's Bursarship are to be found in Babington's Day-Book, which is one of two or three that are still pre- • Steward's bill unpaid 19j. 7 Jd. As an illustration of the scrupulous exactness and regrularity which characterised Newton in all matters of business, it may be mentioned, that in two instances only was he in arrear with h's Steward's bill, viz. the one before us when he had just become Master of Arts, and pnibably did not know the proper mode of paying the bill until after the accounts for the year were made up, the other when he was absent in London as a member of the ('onvention Parliament. t Steward's bill unpaid i;&12«. X The augrmented dividend of this and subsequent years is in consequence of New- ton's increased standing in the College. $ lie must liave resigned hb fellowship before Dec. 21, 1701, otherwise the Bursar's Book would have contained a record of his receiving dividend for the quarter ending then. At the time of his resignation he stood 10th on the list: had he remained fellow until August of the year next but one following, he would have been elected a senior. DITIDENDS RECEIVED BY NEWTON. Ixxxiii eerved in the Muniment Room. Tho first of these rocoipta wo give at full length : the others are added for tho sake of tlie dates : — Oct 11, 1675. Rec'' then my wages as fellow for tho \ £a -to a whole year ending Mich, last ; My livery for the same year 1 IS 4 Pandoxator voted 1G73. 5 i dividend voted last audit 1674 12 10 In all £21 16 8 By me, Isaac Neavton. Nov. 20, 1675 R" tho later moiety of ^Ir Newton's div.... £l2 10 By me, John Battely. July 8, 1676 Wages for f year ending Midsummer £2 Livery for 1676 1 13 4 Pandoxator's div. granted at audit 1674.... 5 idiv last audit 1675.... 12 10 21 3 4 Nov. 16, 1676 Rec** the later i of div. granted 1675 £l2 10 "Wages for quarter ending IMich*. last 13 4 Dec. 13, 1677 Rec** wages for year ending Mich' £2 IS 4 Livery 1 13 4 Pandox. Div. Audit 1675 5 Do 1676 5 Firstjof div. Audit 1676 12 10 26 16 8 Nov. 22, 1678 Wages as fellow for year 1678 £2 13 4 Livery 1 IS 4 *• - Later J of div. granted Audit 1676. 12 10 Pandox. div 1677* 5 00 Later ^ of Mr Wickins's div. granted jr ^ 01 j»ir wicKinss aiv. grantea \ Audit 1676 / ^^ ^^ 34 6 8 Dec.30, 1678 Rec^div.granted Audit 1677 ^12 10 Also Mr Wickins's 12 10 Besides the dividend Newton was in receipt of the following emoluments from the College : 1. Pandoxator's dividend (from the profits of tho bakehouse and brewhouse) £2 10*. for year ending Mich'. 1668, and £5 annually afterwards except when ho did not reside tho major part of the year as in 1689, 1697, &c. Ixxxiv SYNOPTICAL VIEW OF NEWTON'S LIFE. 2. 3*. 4d. weekly during residence " pro pane et potu." (This sum represents 10 penny loaves, 10 quarts of sraall beer at Irf. a quart and 10 quarts of ale at 2d, a quart.) a 13*. 4d. for livery for year ending Mich'. 1668 and £1 13*. id, annually afterwards until Mich*. 1701. £2 3s, 4d. for stipend or wages for year ending Mich*. 1668, and I3^f. id, a quarter until the quarter ending Dec. 21, I7OI. If to these sources of income it be added that, as fellow, he had no* thing to pay for his dinners or room-rent, that his hereditary estate brought him in £80 and his professorship £100 a year, every reader can form his own opinion on the condition of Newton's worldly circum- stances while he was a resident fellow of Trinity College. NEWTON'S EXITS AND REDIT8, Ixxxv NEWTON'S EXITS AND REDIT8. [From tlio book in which the Fellows entered their names on going out of, or returning to, College. Tlio entries are generally in Newton's own hand, but sometiraea in that of North, tho Master, or of Lynnot when Vice-Master, and occasionally they seem to have been written by a servant. There is a 4to. book in tho Muniment Room containing tho Exits and Redits of the Bachelor Fellows and Scholars, commencing with Ootob. 1667. The first six names in it are those of tho six follows of Newton's year senior to him : tho second leaf of tho book, at the top of which Newton's nam© stood, with the dates of his Exits and Redits from Octob. 1667 until Midsummer of tho following year, has been cut out, tho lower portion of tho D belonging to the " Ds" prefixed to his name being the only part of tho entry relating to him that is left.] Year. Exit Rcdlt. Tear. Exit RtMltt. 1668 Sept. 29 1682 Feb. 21 Feb. 28 1669 Nov. 26 Dec. 8* Apr. 8 Apr. 29 J 1671 Apr. 17 May 11 May 10 1672 Jun. 18 Jul. 19 1683 ]\Iarch 27 Mays 1673 March 10 Apr. 1 May 21 1674 Aug. 28 Sept. 5 1685 ]\Iarch 27 Apr. 11 1675 Feb. 9 March I9 Jun. 11 Jun. 20 Oct. 14 Oct. 23 1687 March 25 1676 ]\Iay 27 Jun. 1 1688 March SO Apr. 25 1677 Feb. 20 March 3 Jun. 22 Jul. 17 March 26 1690 Feb. 4 Apr. 26 May 22 + March 10 Apr. 12 Jun. 8 Jun. 22 Jul. 2 1678 May 6 May 27 1691 Sept. 12 Sept. 19 1679 May 15 May 24 Dec. 31 II Jul. 19 1692 Jan. 21 Jul. 28 Nov. 27 1 169311 May SO Jun. 8. 168O March 11 1695 Sept. 10 Apr. 28 May 29 Sept. 14 Sept. 28 1681 March 15 March 26 1696 March 23 May 23 Apr. 20 * Newton was makings this entry under tlie Exits and had written more than half the first letter of his name when he found out his mistake. + Newton has also entered DrLynnet's Redit, who returned to College the same day. $ Two entries, one in Newton's hand, the otlier by North. Newton had been down in Lincolnshire, and a friend of hw availed himself of his return to Cambridge to employ him on a small commission, which it will be seen he lost no time in executing. •* Nov. 28, 1679. Retf* £11. 15s. 7d. by the hand of Mr Isaac Newton from Mr W. Walker, Rector of Grantham School." Dr Babington's Day-liook. Walker received the money from Mr Edw. Pawlet and he from James Thompson who owed it to Babington. § Newton had made nearly the whole of this entry under the year 1677, where on turning over the leaves of the book ample room offered itself, but when about to write the " 8 " he discovered the mistake. II He was in London in Jan. 1692. On the 9th of that mouth we find Pepys inviting Evelyn to his usual Saturday evening party to meet Dr Gale and Mr Newton, (3/«mojri, V. 181.2nded.) ^ Newton had entered an Exit for Apr. 15 ofthis year, but it was afterwards crossed out. Ixxxvi NEWTON'S WEEKLY BUTTERY BILLS. NEWTON'S WEEKLY BUTTERY BILLS, From October 1686, to February 1694, and from June 1698 to March 1702,/ar Breads Beert i^c. These relics of Newton's household expenditure are extracted from two mutilated Buttery Books in Trinity College Muniment Room. The Fellows' Buttery Books for the remainder of the period of his residence and all those of the Scholars during the time when he was undergraduate and bachelor, have, I fear, been destroyed by some per- son or persons, who, it is to be hoped, could not be supposed to know that books apparently so useless were indispensable for a correct history of the discovery of the new calculus and of the true theory of the world. Explanation op Abbreviations. no CO means not in commonst i. e. not in residence, or out of college. di or dimi means half the week, 0. P. or Com. P. means the fine for not delivering a Common Pktce in chapel after morning prayers. This Fine continued to be levied until 1830, when the system of compounding was introduced. M is supposed to stand for manf i. e. servant.* Ton. stands for tonsor (the College barber), ch. probably for chapel, Lett, for letters. * Newton mentions his servant twice in his correspondence with Flamsteed. (Baily, pp. 139, 157). " As for the places calculated from the tables, I will give you no trouble about them : my servant has lately learnt arithmetic, and, if I go on with this business of the moon, he shall learn astronomical calculations and examine them, and I will send you his corrections." (Letter of Nov. 17, 1694). *'I want not your calculations, but your observations only. For besides myself and my servant, S' Collins { of Catharine Hall } (whom 1 can employ for a little money, which I value not) tells me that he can calculate an eclipse, and work truly." (Letter of June 29, 1695). This may have been the John Perkins " Astrologus Cantabrigiensis," to whom Vincent Bourne addressed a copy of elegiacs, beginning Lusit, amabiliter lusit Fortuna jocosa, Et tunc, siquando, tunc oculata fuit; Cum tibi, Joannes, Newtoni sternere lectum; Cum tibi museum verrere diva dedit. And ending Nee melior lex est, ncc convenientior aequo, Quam siet astronomo scrvus ut astrologus. NEWTON'S WEEKLY BUTTERY BILLS. Ixxxvii •luUug 9. rf. 1686 Oct. 15 2 5i 22 7 2i 29 2 9i Nov . 5 3 H 12 3 oi 19 3 loi 26 2 9 Dec . 3 5 10 10 Oi 17 3 2 24 2 Hi 31 2 9i 1687 Jan. 7 2 11 14 7 21 10 6 28 3 11 Feb 4 3 5i 11 2 5i 18 3 H 25 3 Oi March 4 4 9 11 14 5J 18 5 8i 25 3 loi no CO. Apr. 1 8 no CO. 8 ... ... no CO. 15 ... ... di. no CO. 22 2 *i no CO. 29 • . . ... no CO. May 6 ... ... no CO. 13 ... ... di. no CO. 20 1 10 27 2 7 Juno 3 5 2i 10 10 17 3 7i 24 3 lOi July 1 4 5i 8 2 H 15 2 7i 22 2 11 euillug *• **• 1687 July 29 2 9i Aup^. 5 3 Oi 12 3 2i 19 3 1 26 4 24 Sept. 2 3 3i 9 16 lOj 16 13 6J 23 3 5i 30 4 4I Oct. 7 2 lOi 14 4 3 21 3 5J 28 3 4i Nov. 4 5 7i 11 6 14 18 3 25 2 5i Dec. 2 3 lOj 9 3 lOf 16 2 11 23 3 Oj 30 3 1 1688 Jan. 6 3 5J 13 3 5J 20 16 1 27 3 Oi Feb. 3 3 11 10 3 5J 17 3 Oi 24 15 7i March 2 8 Oi 9 10 7 16 3 1 23 3 4i 6.8. C. P. 30 19 2 no CO. Apr. 6 1 2i no CO. 13 2 20 di. no CO. 27 9i May 4 3 1 1688 no CO. no CO. no CO. di. no 1689 no CO. no CO. no CO. no CO. May 11 3 5J 18 4 2i 25 8 2 June 14 6 8 9 7i 15 12 7 22 2 lOi 8*.m. 29 11 July 6 13 CO. 20 1 1 27 3 2 Aug. 3 4 3 10 2 10} 17 3 8 24 10 5} 31 3 2 Sept. 7 18 8 14 10 7i 21 5 5J 28 4 Oct. 5 6 Ij 12 4 Hi 19 2 8 26 3 2 Nov. 2 3 1 9 3 1 16 2 lOi 23 4 lOi SO 4j^.6 Dec 7 2 8i 14 2 lOi 21 3 3i 28 6 8 Jan. 479 11 4 li 18 8 5 25 Feb. 1 8 15 /2 Ixxxviii NEWTON'S WEEKLY BUTTERY BILLS. WMk •odlug 1689 no CO. Feb. 22 no CO. March 1 no CO. Ton, . ^ 8 15 10 CO. Ton. \ 7#. Ch. 3*.j no. CO. no. CO. 22 no. CO. 29 no CO. Apr. 5 no CO 12 no CO. 19 no CO. 26 no CO. May 3 no CO. 10 no CO. 17 no CO, 24 no CO. 31 "^^^- I June 7 Ton.j no CO. 14 no CO. 21 no CO. 28 no CO. Jnly 5 no CO. 12 no CO. 19 no CO. 26 no CO. Aug. 2 no CO. 9 no CO. l6 no. CO. 23 no. CO. 30 no CO. Sept. 6 13 20 27 Oct. 4 11 di.no CO. 18 no CO. 25 no CO. Nov. 1 no CO. 8 I 6 10 1 6 5 Week MuUng 1689 no CO. Nov. 15 no CO. 22 no CO. 29 no CO. Dec. 6 no CO. 13 no CO. 20 no CO. 27 1690 no cou^ Jan. 3 *. rf. 1 6 10 17 24 31 Feb. 6 8 1 6 2 1 2 lOi 3 6i 3 1 2 9h 5 Sl no CO. no. CO. no CO. no CO. dimi. 1 no CO. ! 14 21 28 March 7 dimi. no co. 14 no CO. 21 no CO. 28 no CO, Apr. 4 no CO. 1 1 dimi. 18 25 May 2 9 16 dimi no co. 23 30 June 6 13 20 27 dimi. no CO . 11 18 25 1 9h 3 114 6 2J 5 sl 4 1 4 7i Wttk •udlng 1690 Aug. 1 8 15 22 29 Sept. 5 Ton. 10*. 12 no CO. 19 no CO. 26 Oct. 3 10 17 24 31 68. 8d. \ Com. P. f Nov 7 14 21 28 Dec. 5 12 19 26 3 5i 3 2 2 7J 3 2i 3 5i 2 llj 6 8 IS 3 10 lOj 8 6i 2 si \july 4 2 3i 19 6| 15 6^ 7 5i 1691 dimi. Jan. 2 dimi. no co. 9 16 23 30 Feb. 6 13 20 27 Ton. March 6 13 20 27 Apr. 3 10 17 24 *. d. 6 5i 11 11 3 3i 3 3J 3 lOi 5 14 2i 2 114 4 li 4 lOi 4 34 4 2I 9 2i 4 2i 3 04 4 114 3 lol 16 2I 3 11 3 8 4 5i 1 104 4 04 4 4 5 24 4 6 4 10 5 3 9 8 14 13 3 4 2 3 44 7 74 74 54 14 NEWTON'S WEEKLY BUTTERY BILLS. Ixxxix WMk •odlug s. d. Wr«k •adiug s. d. WMk •ndlnc *. d. 1691 May 1 5 1692 Feb. 5 7 Oi 1692 Nov. 18 4 8 5 12 4 25 5 Si 15 4 6i 19 5 Dec 2 6 2 22 5 li no CO. 26 1 11 9 S 7i 29 6 Oi March 4 7 6 16 7 9 To"-ljune 5 108. / 13 6i 11 3 8 23 5 Hi 18 5 4 SO 7 2i 12 3 lli 25 4 6i 1693 Jan. 6 4 10 19 S 9i Apr. 1 4 3i 13 8 4i 26 4 Si 8 10 7 20 6 1 July 3 3 7i 15 4 10 27 13 6i 10 7 22 4 Feb. 3 6 17 7 1 29 5 9 10 6 1 no CO. 24 6i May 6 4 8i 17 11 6i no CO. 31 13 8 Oi 24 7 no CO. Aug. 7 20 3 10 March 3 8 7i dimi.no co. 14 r"6i 27 3 6 10 5 6i 21 4 Juno 3 6 2 17 7 5i 28 3 10 5 li 24 6 Oi Sept. 4 4 2i 17 5 1 31 4 11 11 5 7i 24 3 7 Apr. 7 5 6i no CO. 18 1 4 July 1 4 6 14 4 6 25 3 5i 8 6 6 21 8 Oct. 2 4 5 15 4 8i 28 7 li 9 3 3i 22 5 li May 5 8 3 16 4 8i 29 4 6 12 4 8i 23 ^ 9k Aug. 5 5 19 5 30 4 4i 12 5 Oi 26 7 2i Nov. 6 S 9i 19 4 4i no CO I June 2 7 13 3 Oi 26 5 dimi. 20 2 74 Sept. 2 3 9i no CO . dimi. 9 2 4 27 3 101 9 5 8i 16 5 9i Dec. 4 6 7 16 4 23 5 5i 11 7 2i 23 4 8i no CO . dimi. 30 3 Si 18 25 5 3 9 30 Oct. 7 5 6 3 10 no CO dimi. • 1 July 7 3 7i 1692 Jan. 1 3 9 14 6 2i 14 5 3 no CO. 8 1 Oi 21 7 21 7 no CO. 15 28 5 8i 28 5 Oi 22 Nov. 4 5 6 Aug. 4 11 2 29 4 11 5 11 4 5i xo NEWTON'S WEEKLY BUTTERY BILLS. WMk Mdlug s. rf. Wwk - J ending '• ". JX *. d. 1693 Aug. 18 5 3i 1694 Jan. 26 5 7i 1699 25 4 10 Feb. 266 no CO. Sept. 8 16 Sept 1 6 962 no CO. Dec. 1 1 6 8 6 3 1700 noco.dim{. 15 4 1698 [In this and fol- no CO. Mar. 116 no CO. M. 22 6i lowing years wo have no CO. May 31 16 noco. dimi. 29 2 Hi copied only those no CO. Sept. 6)1 6 CP. J6 8 Oct. 6 5 dates where a charge 13 5 4 is put opposite his noco. Nov. 29 1 6 20 5 3i name. The sums con- 1701 27 4 5i sist principally of noco. Mar. 1 1 6 Nov. 3 4 3i quarterly payments.] no CO. May SO 1 6 10 5 111 no CO. June 3 1 6 noco. Sept. 12 1 6 17 8 H no CO. July 22 Nov. 21 6 4 24, 6 no CO. dimi. 29 3 6 Nov. 28 £1 10 Dec. 1 7 H no CO. Aug. 5 no CO. Dec. 5)1 6 Lett. J 5i 8 3 7 no CO. Sept. 9 1 6 15 5 7i no CO. Dec. 2 1 6 noco.C. P. 26 6 8 22 () 2 1^99 1702 29 9 no CO. Feb. 24 1 6 no CO. Mar. 6 1 6 1694 Jan. 5 5 no CO. June 3 16 May 8, name disappears 12 5 5] --}june23 6 8 from list of fellows. 19 5 6 TABLE OF NEWTON'S LECTURES AS LUCASIAN PROFESSOR. NEWTON'S LECTURES ON OPTICS (MS, Univ. Libr. Dd. 9. 67.) [The numbers on the right designate the pages in tho MS., those on the left tho pages in tho work as printed Lend. 1729.] Jan. I6J. 1— IS Optica? pars 1* — varia. Lect. 1(1—6) 13— 25 Ex codem— exi^ipt. Lect 2 ( 6— 12) 2f>— 34 Jam liquet — determinentur. Lect. 3(12—17) 35— 41 Sectio 2***— reflexes. Lect. 4(17—21) 42— 52 Cum eandem — attolluntur. Lect. 5(21—28) 5^— 62 Probleniatis— de aliis. Lect. 6(28—33) 62— 73 Ad eundcm — videar. Lect. 7(33—39) 74— 85 Sectio 3**" — proxime. Lect. 8(39—44) Octob. 1670. 85— 95 Prop. 12 — ajqualos. Lect. 9 (45—49) 95—105 TiCmma 5—~ixxv. q . e. d. Lect. 10 (49—54) 105—116 Prop. 17 — sufficiant. Lect. 11 (54—60) 116—125 Do radiorum— G^A"//. Lect. 12(60—64) 126—136 Sectio 4*" — possunt. Lect. 13 (64—69) 137—146 Prop. 32— dcfinitur. Lect. 14(69—75) 146—152 Prop. SQ — censeam. Lect. 15(74—77) [145]-153 Opticas pars 2*^ — disceptaturus. Lect. 1(1—5) 153—164 Prop. 1 — nequeant. Lect. 2( 5— 11) 164—171 Prop. 2 — censeam. Lect. 3(11—17) Octob. 1671. 171—181 Prop. 3 — commisceantur sibi. Lect 4(17—23) 182—189 Adhaec — judicaveris Lect. 5 (23—29) 189—197 Verum — manifestum est. Lect 6 (29—34) 197—207 Quinetiam — Prisma. Lect 7(34—41) 207—215 Ad hiec — cogantur. Lect 8 (41—46) 215—226 Prop. 5 — subjicient. Lect 9(46—54) 227—239 Sect. 2^*— emergcntis. Lect 10 (54—63) 239—247 Antequam — Hceat. Lect 11 (63—68) 248—260 2 De Plh-enomenis — posaint. Lect 12 (69—77) 261—269 3 De Phajnomonis^licere. Lect 13 (78—84) XCll TABLE OF NEWTON'S LECTURES. Octob. 1672. 269 — 277 4 De Phronomenis— patebunt. Lect 14 (84 — 90) 278 — ^285 Notiasimum — inforioris. Lect 15 (90 — 96) 285—291 Superest— decrevi. Lect. I6 (96— 101) The MS. does not seem to bo in Newton's hand, except some cor- rections hero and there, almost all the marginal notes, the diagrams and between 2 and 3 pages at the end. It was put into the hands of the Vice-Chancellor and delivered by him to Robert Peachy to be placed in the University Library, Octob. 21, l674. 1 — ^9 Computatio vel fit — in eadem ratione. 11—15 Do Additione — '-QOa^ Jaa-xx. 3 5* 1 5— '1 7 De Subductione ^^3 4- LECTURES ON ARITHMETIC AND ALGEBRA. (MS. Univ. Libr. Dd. 9. 68.) [The numbers on the left refer to the pages in the edition published by Whiston, Cantab. 1707.] Octob. 1673. Lect. 1(1—5) Lect. 2(5—8) Lect. 3 ( 8—10) Lect. 4 (10—12) Lect. 5 (13—15) Lect. 6 (15—18) Lect. 7 (18—21) Octob. 1674. Lect. 1 (21—22) Lect. 2 (22—24) Lect. 3 (25—27) Lect. 4 (27—28) Lect. 5 (28—29) Loot. 6 (30—32) Lect. 7 (32—33) Lect. 8 {33^35) Lect. 9 (35—37) Lect.lO (37—38) 18—21 De Multiplicatione— . 22 — 25 De Divisione — homogeneas. 25 — 30 Quod si quantitas — sufiicit. 31 — 34 De extractione Radicum — 279- 34—37 Extractionem radicis— observandum est. 37—40 E simplicibus — radicibus. 41,42,51,52* De Reductione ^hc. fit 53 — 55 Quod si divisor — — - . 55 — 57 De reductione Radicalium— et sic in aliis. 62 — 66 De forma iEquationis — = x*. 66 — 68 Reg. 4— docero. 69 — 72 De duabus — linquo. 72 — 74 Exterminatio — x df= 0. 74—76 Reg. 3 — asymmetria. • The part De Inventione Divisorum— totam quantitatem, pp. 42—51, is taken from the end of the MS. TABLE OP NEWTON'S LECTURES. xciu 76 — 79 Quomodo Qurestio—detorminandte. 79—81 Ciutcrum ut liujusinodi— . 156 — 159 Prob. 29 — proportionales. 159—161 Prob. 31— satisfaciet. 162—164 Prob. 33— = 0. 165—168 Prob. 35— aequales. 16*8 — 171 Idem brevius — altcrius ad A, 1 7 1 —1 72 Prob. 37— rectam FE, 172—174 Prob. 38— longitudinem DC. 174—176 Prob. 39— Q. E. F. 176 — 179 Prob. 40 — manifestura est. 1 79—1 82 Prob. 41 — oportuit. 1 82—1 86 Prob. 42— = VZ. 1 86 — 1 89 Analyseos — invcnicnda. 189— 192 J^rob. 43— j-j-^. Octob. 1675. Lect. I (39—41) Loct 2 (41—42) Lect. 3 (42—44) Loct. 4 (44 — 47) Lect. 5 (47 — 48) Lect. 6 (48—50) Loct. 7 (51—52) Lect. 8 (52—58) Lect. 9 (58—62) Loct. 10 (62—65) Octob. 1676. Lect. 1 (65—67) Lect. 2 (67—69) Loct. 3 (69—70) Lect. 4(70—71) Lect. 5 (71—72) Lect. 6 (72—74) Lect. 7 (74—76) Lect. 8 (76—78) Lect. 9 (78—79) Lcct.lO(80— 81) Octob. 1677. Lect. 1 (81—83) Lect. 2 (83—85) Lect. 3 (85—86) Lect. 4 (86—89) Lect. 5 (89—91) Lect. 6 (91—93) Lect. 7 (93—94) Lect 8 (94—96) Lect. 9 (96^99) Lect. 10 (99—100) Octob. 1678. Lect. 1 (100—101) Lect 2 (101—102) Lect. 3 (102—104) Lect. 4 (105—106) Lect. 5 (106—109) Lect. 6(109— 111) Octob. 1679. Lect. 1 (111—113) XCIV TABLE OP NEWTON'S LECTURES. 192—195 195—197 197—199 199—201 201—202 203—204 205—207 207—209 209—211 211—213 213— Sl5 215—218 218—221 221—224 224—227 227—230 231—234 234—240 240—243 243—247 247—250 250—253 253—257 257— 45— 47—50 50— 259—262 263—266 266—272 272—277 Prob. 44— cognosci potest. Prob. 45 — super basem. Prob. 47 — ^latera trianguli. Prob. 48 — ^triangiilum quaesitum. Prob. 49 aabb^O. Prob. 51 — ilia quresita. Prob. 52 — trajecit. Prob. 53—^. E. I. Prob. 54 — quaesitae. Prob. 55 — iequalis est Prob. 56— et EO. Hoc modo— Conica Sectio. Quod si quatuor — potest. Lect. 2 (114—115) Loot. 3 (116—117) Lect. 4(117— 118) Lect. 5(118— 119) Lect. 6(119— 120) Octob. 168O. Lect. 1 (120—121) Lect. 2 (121—124) Lect. 3 (124—125) Lect. 4 (125—127) Lect. 5 (127—128) Lect. 6 (128—130) Lect. 7 (130—132) Lect. 8 (132—134) Octob. 1681. Lect. 1 (134—137) Lect. 2 (137—140) Lect. 3 (140— 143) Lect. 4 (143—147) Lect. 5 (147—153) Lect. 6(153—157) Lect. 7 (157—161) ■limitibus cequationum. Lect 8 (161—164) Ex -^uationum— remotissimam - 5. Lect. 9 (l64 — 167) Si inter summam—coDsistunt. Lect.lO(l67 — 172) Prob. 58 21^^'^ ^^' Prob. 59 — invenire. Prob. 60— ^ar. Prob. 61— oportebit. Quomodo iEquationes — sit par. Sunt tamen radices — impossibiles dua?, Ubi termini duo — detegi possunt. Eadem operation ITorum vero — investigandoa sunt. Si jiuUus occurrit a: + 7 = 0. Si nuUus inveniri 66^ =0. Si quantitatis — kx-\-l (p. 258). Exempli gratia— J a a + cc. Hactenus — proponebatur. Si requatio— quadratico. Adjungere — potuorunt. ♦Octob. 1682. Lect 1 (172—176) Lect 2(176—180) Lect 3 (180—185) Lect 4(185—188) Lect 5 (188— 192) Lect 6 (192—196) Lect 7(197—203) Lect. 8 (203—207) • The matter in Lectures 1 — 4 is given in a modified form at the end of the MS., with a direction that it should be inserted at an earlier part of the vohime. For tluse four lec- tures, therefore, it is impossible to give exact references to the pages of the printed book. TABLE OP NEWTON'S LECTURES. xcv 277,58--6l 279—284 284—289 289—293 293—297 297—299 299—302 303—307 307—310 311—317 317—321 321—326 Ilactcnus 8?quationum — 1 1 + ^125. Loct. 9* (207 — 210) Ilactenus cequationum proprictatca— adhibctur. liCct. 10 (211— 215) Octob. I683t Lcct. 1 (216—219) Lect. 2 (219—222) Lcct. 3 (223—226) Lect. 4(226—228) Lect. 5(228—231) Lcct. 6(231—235) Lect. 7(235—237) Lect. 8 (238—243) Lect 9 (244—247) Si cui — immoror. Proponatur jam— conStruenda. Solvuntur etiam — dcmonstrantur. Construonda jam — demonstrandi, Proponatur jam — compinguntur. Ilto 8unt — Prop. XXXII. Quod si — nimius sum. Ilactenus — habeatur — r. Demonstratur— et BC. Scholium — proportionalium «. . Lect. 10 (247—251) At the end of the Volume are corrections and additions by Newton, and " De Inventione divisorura — nihil relinquit" (pp.42 — 51 of printed book.) LECTURES DE MOTU CORPORUM {MS. Univ, Libr. Dd. 9. 46.) [The numbers on the left denote the pages in the 1st ed. of tbo Principia : those on the right the leaves in the MS.] The title is " De motu corporum Liber primua." It forms the draught of tlie Ist book of the Principia, see p. 20^, note. Octob. 1684. 1 — 11 Dcfinitiones...Tractatuw pcquentem composui. Lect. 1(1—9) 12 — 20 Axiomata sive Leges Motus...et motus oorum inter so. Lcct. 2 (10—16) 20 — 29 Schol. Hactenus principia tradidi... ...in ultima rationc partis ad partem. Lect. 3 (16—20) • In the corrections at the end of tlie MS. part of this Lecture is ordftred to be trans- ferred to an earlier place in the Volume, and accordingly it appears in pp. 68 — 61 of the printed book. t The MS. in Lambeth Library, No. 592, (quoted by Rigaud, Etsay, p. 97, note) entitled "Trigonometriae Fundameuta a Viro 01. Iswico Newtou, Mathesoos Profes- sore, anao 1683 data/' contains merely rules for the solution of plane and sphoricjU triangles given to flenry Wharton probably at one of those private lensons mentioned in p. xlv. It consists of two folio leaves (i.e. of two pages and seven lines on the last page, the second being blank), forming part of a volume entitled '^Scripta Acaderaica &CC. annos inter 1682 et 1686, a me facta " &c. in Wharton's handwriting. xcvi TABLE OP NEWTON'S LECTURES. S9"~ S6 Lemma v. Similium figuraram... diminuendos sine limito. Lect. 4(20—25) 37— 49 Artio. 11. De Inventione Virium Centripetanim... ...in ccntrifugam versa. Lect. 5 (25 — 31) 50— 56 Artie, m. Do motu corporum in Conicis Soctionibus cxecn- trici8...majoribu8 axibus Ellipsodn. Lect. 6(31—36) 56— 63 Prop. XVI. Tlicorcina viii. lisdem positis... ...nimis obvias non adjungo*. Loot. 7 (36'^39) 67— Lemma xvt. A datia tribu8...pergo jam oxponero. Lect. 8(39-42) 115— Prop. XXI. Prob. XIII. Posi to quod vis.... Lect. 9(42— ) breaks off at fol. 44, in Prop. xxiv. with the words "gyrando, arcum Kk." Frinctp.}). 118. Octob. 1685. 79— 88 Prop. xxii. Prob. xiv....Problema impossibile est. Lect. 1 (58—63) 89 — 98 Prop. XXVI. Prob. xviii....8irailem et a?qualem. q. e. p. Lect. 2(63—68) 98 — 107 Lemma xxvn....irrationalem ut sequitur. Lect. 3(68—73) 107 — 114 Prop. XXXI. Prob. xxin....pergo jam exponere. Lect. 4(73—76) 115 — 125 Artie, vii. Do corporum a8censu...ad area,m DLme. Lect. 5(76—83) (whole of 7th section.) 125 — 131 Artie, vin. continens Inventionem orbium in quibus... ...adjiciamus pauca. Lect. 6(83 — 86) (whole of 8 th section.) 132 — IS** Artie, vni. {error for ix.}... oblique ascenderc. Lect. 7(86—89) * ** Demonstrationes hujus et pnecedentis ut nimis obvias non adjungo." In the Prtficipia the demonstrations of these two propositions (17th and 18th MS., 18th and I9th Frxnc'x-p.) are given complete. t This is Lemma xvi. of the ^rxncxif, p. 67. The Prop, which follows it in the MS. B Prop. XIX. Prob. xi., being Prop. xxi. Prob. xiii. of the Vr'xncip. pp. 68, 69. The reference to Lahire is not in the MS. having probably been suggested by Halley. (Newton to Halley, Octob. 18, 1686. Rigaud's Appendix, p. 47). Prop. XX. Prob. xii. in the MS. is Prop, xxx, Prob. xxii. in Princip. p. 104. Tlie difference in the numbers of the propositions arises from the circumstance of the 5th section which contains eight propositions having been afterwards inserted. After Prop. xx. comes a scholium containing the approximate solution of the same problem for the ellipse and hyperbola. Then follows the clause •' Hactenus... exponere " as in Princip, p. 114. TABLE OP NEWTON'S LECTURES. xcvii 137 — 144 Prop. xlv. Prob. xxxi....9ubindo dotonninamus. Lcct 8(90—95) 145 — 152 Artie. x. De Motibus Corporum... semper peragcnt. q. e. d. Lect. 9(95—99) 153 — Prop. Lii. Prob. xxxiv.... Lect 10 {9<) — ; breaks off in Prop. Liv. with tho words " qua vis altitudino CT per.** Princip, p. 159. Tho MS., it will be seen, is imperfect, ending abmptly at tho second page of fol. 102. Foil. 37—^4 aro rcpeate to Jones, lie mentions his having been last * I can discover no traces of this bill in any of the College Account Bookn. It may possibly have come into the Chapel Account, for which Cotes, as supcrin- lendcnt of tho repairs of the Chnpel, kept a separate book of rerripts and disburse- ments. Whether ths book is still in existence 1 am unable to say. 2 CORRESPONDENCE OF in town "about a year and a half ago") drawn thither, no doubt, by this noto of Bontlcy's, and expecting; to take with him down to Cambridge tho first instalment of Newton's corrected copy of the Principia. Newton however is still reluctant to part with it, having probably some further improvements to make, but promises to send the copy down in about a fortnight. So Cotes returns to Cambridge without tho "one part of tho book corrected for the press," which Bentley's letter had informed him was ready to be put into his hands some eight weeks before. Tlio copy docs not arrive in that fort- night, nor in tho next. The long vacation being nearly half over and no signs of tho promised copy appearing, tho young editor be- comes impatient. Ilenco his letter of Aug. 18, which however pro- duced no apparent effect, until his next-door neighbour Winston, one evening probably in September, newly arrived from London, (ho is known to have been in Cambridge on tho 29th of that montli) put into his hands " the greatest part of tho copy of tho Principia," end- ing at Prop. XXXI II Cor. 2 Lib. ii p. .320. That is followed some time afterwards by Newton's letter of Oct. 11, which apparently did not come through the post, being brought perhaps by some member of the University coming up on the beginning of term. Whiston, whose autobiography records so many other things certainly of not greater importance, makes no mention of his being employed as a messenger on this occasion: so absorbed was he in his Arian heresy and Apostolical Constitutions, with regard to which ho tells us "his best friends began to be greatly affrighted this summer at what they had heard he was going about." It is not likely that he found his old patron wanting in tho duties of friendship at this critical period of his life, and it is not impossible that Sir Isaac, in delivering to him a portion of a work containing so much close and profound reasoning, may have dropt a word of caution into his ear. The "election" referred to in this letter is probably that of a Head Master of Sir Joseph Williamson's Free ]\Iathematical School at Rochester, tho electors to which post are some 17 in number, consisting of tho Mayor, Recorder, eldest Resident Prebendary, &c. The Rev, John Colson was the first Head Master of this school, and was appointed June 1, 1709. He resigned the place March 1, I7I5, on being elected • Lucasian Professor. He was entered at Emmanuel April 23, 1728, and was one of the 71 persons in the King's list (William Warburton was another of the number) on whom the de- gree of M.A. was conferred at George II's visit to the University, April 25, 1728. On coming to reside as Lucasian Professor at Cam- bridge, ho was appointed Taylor Lecturer at Sidney College, where he was admitted "in convictum sociorum" 11 March I755, «tat. 60. NEWTON AND COTES. S It was for the purpose of boardin;^ with this same CoUon, and bcinf^ instructed by him "in ^lathematics and Philosophy and humane Icamin^^," tliat Garrick set. out from Lichfield on the morning of ^larch 2, 173} for London, accompanied by **one Mr Johnson," who was going "to try his fate with a tragedy, and to see to got him- self employed in some translation, either from the Latin or French," Christopher Ilussey was a senior Bachelor of Arts of Trinity Col- lege, and was elected Fellow the following October. On Winston's expulsion from the University (Oct. 30, 1710), he was appointed by him as his deputy in the Lucasian Chair, and " was ready to perform his duty, had not the heat of that time prevented him." Winston's Memoirs I. 312. lie was afterwards an unsuccessful candidate for the Professorship against "Blind" Saunderson (Nov. 20, 1711). Soo letter cviit, note. A slightly different date is assigned to this letter in the Bontley Correspondence (p. 370), and a widely different one is mentioned as being suggested by Bishop Monk (ib. p. 7^7). LETTER II. COTES TO NEWTON. S'. Cambridge August 18*^ 1709. The earnest desire I have to see a new Edition of y Princip. makes me somewhat impatient 'till we receive Y' Copy of it which You was pleased to promise me, about the middle of the last Month, You would send down in about a Fourtnights time. I hope You will pardon me for this uneasiness from which I cannot free my self & for giveing You this Trouble to let You know it. I have been so much obliged to You by Y"^ self & by Y"^ Book y* (I desire You to beleive me) I think my self bound in grati- tude to take all the Care I possibly can that it shall be correct. Some days ago I was examining the 2^ * Cor : of • In this Corollary is determined the Attraction of a Spheroid on a point in its axis produced, the attractive force of each particle varying inversely as the square of the distance. A paper by Cotes containing the investipfation is still preserved in the volume from which these letters are taken Nos. 24 and 25, 1—2 A CORRESPONDENCE OP Prop 91 Lib i and found it to be true by y* Quadratures of y« 1'* & 2** Curves of y« 8*^ Form of y* second Table in Y' Treatise De Quadrat, At the same time I went over y* whole Seventh & Eighth Forms which agreed with my Computation excepting y® First of y* Seventh & Fourth of y* Eighth which were as follows 4de -=- - 2rf/ SdeecT + 4>dfg8 Form : 7. 1. ^ — « t ^*jeg - vff Form: 8.4. +S6de/^ ^Sdegg -28dc/> -iGdeeg „ 8 a) XV XV V '\5dp "^dffg +}Odf' +I0deff _^ 24>rjeg^ - Grjffgg I take this Oportunity to return You my most hearty thanks for Y' many Favours & Civilitys to me who am Y' most Obliged humble Servant Far S' Isaac Newton at his House RoGfiR CoTES. in Jermin Street near St Jameses Church Westminster. LETTER III. NEWTON TO COTBS. S-" I sent you by M*" Whiston the greatest part of y* copy of my Principia in order to a new edition. I then forgot to correct an error in the first sheet pag 3 lin 20> 21, & to write plusquam duplo for quasi quadruplo & plusqtiam de- cuplo for quasi centuplo*. ■ These two corrections are not adopted literally in the 2d edition, the "quasi" for which Newton here substitutes "plusquam" being still retained in it. Perhaps Cotes had already altered the "quadruplo" and "centuplo" before receiving this letter, as so obvious an error could scarcely have escaped his attention. In the passage referred NEWTON AND COTES. 5 I forgot also to add the following Note to the end of Corol. 1 pag. 55 lin 6. Nam datis umbilico et puncto contactus & positione tangentis, describi potest Sectio conica qua) curvaturam datam ad punctum illud habcbit. Datur autem curvatura ex data vi centripeta: et Orbes duo se mutuo tangentes eadcm vi describi non possunt. I thank you for your Letter & the corrections of y* two Theorems in y* treatise de Quadratura. I would not have you be at the trouble of examining all the Demon- strations in the Principia. Its impossible to print the book w^'^out some faults & if you print by the copy sent you, correcting only such faults as occurr in reading over the sheets to correct them as they are printed off, you will have labour more then it's fit to give you. M'. Livebody is a composer (I mean M'^ Livebody who made the wooden cutts) & he thinks that he can sett the eutts better for printing off then other composers can, and offers to come down to Cambridge & assist in composing if it be thought fit. When you have printed off one or two sheets, if you please to send me a copy of them I will send you a further supply of wooden cutts. I am Yo' most humble & faithful servant London. Octob. 11. 1709. Is. Newton. For M' Cotes Professor of Astronomy in the University of Cambridt/e at his Chamber in Trinity College. Shortly after the date of tho above letter, Newton changed liis residence from Jermyn Street to Chelsea. Flamsteed, writing to Ab. to, Newton, speakiDgr of a ball shot horizontally with a eriven velocity from the top of a mountain to a distance of two miles before it reaches the ground, says, (as the words stand in the 2nd and 3rd editions) '* dupla cum velocitate quadi dupio longius per- geret, et decupla cum velocitate quasi decuplo longius." When he wrote "quadruplo " and *' centuplo," he was probably thinkingr of oblique projection. The passage in question occurs in 8omo additional remarks in illustration of J)cf. v., which were not given in tlie Ist edition. The MS. of them, unfortunately, does not appear in the Newtonian \'olunie. 6 CORRESPONDENCE OF Sharp Oct. 25, (Baily, p. 272) says : " He | Sir Isaac | is now re- moving to Chelsea, and has been lately much talked of ; but not much to his advantage. Our Society \ the Royal Society j is ruined by his close, politic, and cunning forecast; I fear past retrieving, for our Doctor's |Sloano| Transactions have been twice burlesqued publicly; and now we have had none published I think this four months." This burst of spleen would seem to bo in anticipation of the resolution passed by the Council of the Society Nov. 9, ordering Flamsteed's name " to be left out of the list of the Society for next year for not having com- plied with the order of Council made 12 Jan. }*JOl" relative to the payment of arrears. Here there is a break of 6 months in the correspondence until we come to Cotes's letter of April 15, 17^0, by which time nearly half of the whole work was printed off, the part then finished ending at p. 224, (2nd ed.) in the middle of the Lemma (II Lib. 2) in which the principles of fluxions are explained. A note by Mr Howkins states that there is wanting a letter of Cotes to Newton, dated Apr. 9, 1710, *"* de Cor. 1 and 6 Prop. ix. Lib. 2." No. 33 contains a draught in Cotes's writing of these two Corollaries, and two additional steps in the proof of the Proposition, but not (with the exception of the latter of the two steps) as they stand in the second edition. On the same paper Cotes has also written " dele Cor. 4 and 5, Prop, viii," which are accordingly omitted in the 2nd ed. It is probable, therefore, that if this missing letter of Apr. 9 referred to the Corollaries mentioned by Mr. Howkins, the proposed omission of Cor. 4 and 5, Prop. viii. and the introduction of the two steps into the reasoning of Prop. ix. in order to avoid a reference to the latter of the cancelled corollaries would also form a part of its contents. But besides this letter of Apr. 9 and Newton's answer to it, there is good reason for supposing that at least one other pair of letters passed between them during the interval from October to April. For (1) it seems probable that Cotes would return some answer to Newton's letter of Oct. 11, in explanation of his not adopting the precise language of the emendations contained in it ; and at all events he would attend to Newton's request to have one or two sheets sent to him, to say nothing of the presumption that ho would feel himself called upon to take some notice of Mr Livebody's offer of his services. (2) The 2nd method of finding the force to the centre of an ellipse given in p, 40 2nd Ed. is so much altered (in the opening part of it) from the form in which it stands in Newton's MS. (No. 9), that Cotes would scarcely have changed it without some communication from Newton on tho subject. At the head of this 2nd method Cotes has written ** vid. fol. NEWTON AND COTES. 7 scqu," but the leaf referred to is not to bo found. (3) From a mark in No. 11 it appears that the first word in p. 40 in tho proof sheet (H) was "corporis" which is now in the eightli and ninth lines lower down; so that some additional matter must have been introduced in a pro- cetling page after the proof of II was printed*, and tins almost neces- sarily implies tho receipt of instructions from Newton to that cft'ect, (perhaps at tho end of November or beginning of December, if we may judge from tho rate at which tho press was working). The loss of any letters in this interval is tho more to be regretted, because if ever the celebrated Scholium to Lemma II. Book 2 was touched upon in the correspondence between Newton and his Editor, the place for doing so would lie within this period. Tho missing letter of April 9, as has been said, may have contained remarks connected with Prop. VIII. which immediately follows that Scholium. The only alteration in the Scholium t made in the 2nd Edition, consists in tho addition of tho words " ct Idea generationis quantitatum" after ** nota- rum formulis." Tho "annis abhinc decem" referring to his second letter through Oldenburg to Leibniz, in Oct. 1076, is still retained, though 20 years intervened between tho publication of tho Ist and 2nd editions. In this interval, it may bo remarked, the quarrel between Bentley and the Seniors broke out, and we read of Cotes being present at two conferences at the lodge between tho conflicting parties, as a friend of the Master's. (See Monk's Bentley, p. 187.) On Jan. 18, 1710, 15entley cut INIiller's name out of tho boards. On Feb. 10, Miller presented tho petition, signed by thirty of tho fellows, to the Bishop of Ely. Great, however, as was tho delay which retarded the second *;dition of the Principia in its passage through the press, Cotes had * A comparison of Newton's MS. with sheet G of the 2nd Ed. shewH that the addi- tion must have been made in some sheet preceding that, but it is impossible to fix the exact place, as the part of the MS. which is preserved only begins with Prop. vi. Theor. v. t This Scholium was completely remodelled in the 3rd Ed. and Leibniz's name sup- pressed. The reader of these pages will smile at the followingr piece of information with which Montucla favours us (xii. 108): "On se demandera jicut etrc pourquoi cette suppression ne fut pas faite loi-s de I'edition des jtrincipes de 1713, puisquc alors la querelle £'toit encore dans toute sa chaleur ; en voici la raison, qui est une anecdote asst'Z i>eu connue et que je tiens de la mOme main que ce que j* ai dit ci-dessus {the * bonne main ' that had informed him that the notes on the Cominerciuvi EpistnUcum were written by Newton}. C'est que cette Edition fut faite u Cambridge, loin de Neuton et presque en cachette, par les soins do Cotes et de lientley, et que Ncuton en fut tres-m^content C est, en cffet, un proc6vn my ignorance to You y"^ to run y® hazard of leaving a blemish in a book I so OP much esteem*. Cor. 6. lin. ult. I would print thus — ut ~- , id est, ut secans anguli ejusdem, vel etiam reciproce ut Medii densitas. If I mistake not y® design of y® 8*** Co- rollary, I would alter it thus — Centro S intcrvallis continue proportionalibus 6M, SB, SC, &c. describe circulos quot- cunq: & statue tempus revolutionum omnium inter peri- metros duorum quorumvis ex his circulis, in Medio de quo egimus, esse ad tempus revolutionum omnium inter eosdem in Medio proposito, ut Medii propositi densitas mediocris inter hos circulos ad Medii de quo egimus densitatem mediocrem inter eosdem quam proximo ; sed & in eadem quoq : ratione esse secantem anguli quo Spiralis pra?finita in Medio de quo egimus secat radium ^15* ad secantem anguli quo Spiralis nova secat radium eundcm in Medio proposito: Atq; etiam ut sunt corundem angulo- rum tangentes ita esse numeruni revolutionum inter circu- • See the next and three following' Letters. 12 CORRESPONDENCE OP los eosdem duos quam proximo. Si ha3C passim — Prop. 16 must be altered for by my reckoning if y* centripetall force be as op<.+i ^^^® llesistance will be as ^^pnli * the Velocity as -—-r , & therefore y* Density as -^n'~^h • With y consent I would add this Corollary. Si vis cen- tripeta sit ut ■ ^^ , erit 1 - ^ n « 0, adeoq : Resistentia & Densitas Medii nulla erit ut in Prop ix Lib 1. Another Corollary might be added to shew in what cases y® Resist- ance is affirmative and in what cases negative. I beg of You to pardon the freedom of this Letter. Y"^ &c. LETTER V. COTES TO NEWTON. S' Apr. 30*'» 1710 I suppose M' Crownfield our Printer has delivered to You all y® Sheets that are already printed off. I desired him to wait upon You before he returned to Cambridge y* I might have Y"^ answer to my former Letter or at least to y* first part of it. The difficulty which I proposed to You concerning y* 4*** Corollary of Prop, xv I have since removed. Upon examination of y* Proposition I think I have observed another mistake in Cor. 3. which ballances y^ which I before mentioned * to You in y* Corollary. For if I be not deceived y*' force of Resistance is to y® Centri- petall force AS ^ Rr to TQ not as i?r to TQ. You will see my reasons in y® following alterations which I propose to You. Page 284. 1: 6 Ponantur qu89 in superiore Lem- viz. TQ being erroneously put = -— — in the Ist ed. instead of .5 J*^ . NEWTON AND COTES. 13 mate, & producatur »VQ ad V ut sit SV ajqualis SP» Tem- pore quovis, in Medio resistente, describat corpus arcum quam minimum FQ, & tempore duplo arcum quam minimum PR ; & decrementa horum arcuum ex resistentia oriunda, sive defectus ab arcubus qui in Medio non resistente iisdem temporibus deseriberentur, erunt ad invicem ut quadrata temporum in quibus generantur: est itaq: decrementum arcus PQ pars quarta decrementi arcus Pli. Postquam vero descriptus est arcus PQ in Medio resistente, si area) PSQ aaqualis capiatur area QSr, erit Qr arcus quem tem- pore reliquo corpus describet absq : ulteriore resistentia, arcuumq: QR, Qr differentia Rr dupla erit decrementi arcus PQ ; adeoq : vis resistentia) & vis centripeta sunt ad invicem ut lineolce ^Rr & TQ quas simul generant. Quoniam vis centripeta, qua corpus urgetur in P est — . Pag. 285. 1 : 5 — ^ FQ fit aiqualitatis. Quoniam decrementum arcus PQ, ex resistentia oriundum, sive hujus duplum Rr est ut re- sistentia & quadratum temporis conjunctim ; erit Eesisten- Rr tia ut -— ; . Erat autem PQ ad — Page 286 1 : 4. PQq X SP i FQ X PQ Nam vires illze sunt ut \Rr & TQ sive ut '^ - — & SQ hPQq , hoc est, ut i FQ & PQ seu \0S Si OP.— I satisfied SP my self more fully y* I am not mistaken in my reasoning after y" manner. If (as in Prop xvi) y* Centripetall force be as ■ , the force of Resistance will be to y* Centri- petall force as ^Rr to TQ ie as 1 - Jw, OS to OP. Put y® Centripetall force as — , & You will have « = O, & con- sequently 1 -iw, OS to OP as OS to OP, Therefore when y® Spiral coincides with y* line PS y® Resistance will be equall to y* Centripetall force & y* Body will descend "with an uniform Velocity as it ought to do, by Cor. 1 Prop 14 CORRESPONDENCE OP X V, & Cor 5. Prop iv Lib r. compared together, and also upon y* consideration y* y* velocity in y® Spiral of Prop XVI is as . , i e, as -— — . I have some things further to propose to You about y* remaining -part of Y"" copy, which I will not trouble You with till I have Y' answer to my former Letter Y"^ &c. LETTER VI. NEWTON TO COTES. S' Chelsea near London May 1'* 1710. I thank you for your letter with your remarks upon the papers now in the Press under your care. As soon as I could get some time to think on things of this kind, from V^* I have of late years disused myself, I examined them*, & all your corrections may stand till you come at page 287. In page 28G lin 4 for ^ OS read OS. In the same page let Corol. 4 stand thus. Corpus itaq : gyrari nequit in hac spirali nisi ubi vis resistentia) minor est quam vis centripeta. Fiat resistentia oBqualis vi centripetaD, et spi- ralis conveniet cum linea recta PS, et motus corporis ces- sabit. In page 287 & 288 the 8*^ Corollary may remain as in the Copy I sent you. In page 289 let the 16"* Pro- Rr position end thus et resistentia in P ut — — — — , sive ut PQ.SP^.SQ ' "^'"^^ "' OptsP^^* ^^"^ ""'' ("^ ^"- turn ^~ ] reciproce ut SP^^K Et propterea densi- tas in P est reciproce ut SP^If, * Newton does not seem to have worked the problem out himself, but to have taken Cotes's results (in Letter IV.) for granted. t The *' 5P- " is no doubt copied inadvertently from the Ist Ed. It should be SP. NEWTON AND COTES. 15 Corol. 1. Si vis centripcta sit reciproce ut SP^\ crit 1 - J n « 0, adeoq : resistentia et densitos Medii nulla erit ut in Propositione nona Libri prirai. Corol. 2. Si vis centripcta sit reciproce ut radii SP dignitas aliqua cujus index est major numcro 3, resistentia affirmativa in negativam mutabitur. When you sent me the sheets last printed off, I hap- pened to be from home, but a{t} night found them left at my house, and thank you for them. I am going to finish the next part of the copy I am to send you, & I hope to have it ready in due time if some experiments* succeed. I thank you once more for your corrections & for your care of the edition. I am S' Your most humble & most obedient servant Is. Newton. After the ^vriting of this Letter I received your second Letter dated Apr. 29. In the alterations you propose to be made in Prop xv you say. Postquam vero descriptus est arcus PQ in Medio resistente, si areso PSQ ajqualis capiatur area QSr, erit Qr arcus quem tempore reliquo corpus describet absq: ulteriore resistentia. And this would be true if the velocity of the body at Q were the same as when the arch PQ is described in the same time in Medio non resistente. But the velocity at Q being less in Medio resistente then in non resistente, the arch Qr will be less in the same proportion & thereby reduce Rr to half the bigness, & make the resistance to the cen- tripetal force as Rr to TQ. I hope therefore that what I have written on the other page of this Letter is right & • Probably experiments with glass balls dropt from the dome of St Paul's with a view to test his theory of the resistance of fluids. See Letter XXV. fin, and note. 16 CORRESPONDENCE OF that yo' difficulty will be removed by the words & motus corporis cessabit. I am Yo" ♦May 2^ I. N. For the R"*'t M' Roger Cotes Professor of MatJiematieh and Fellow of Triniti/ College in Cambridge. LETTER VII. COTES TO NEWTON. S', May 7. 1710 I received Y' Letter by y® last Post. I am not satis- fied that Y"^ words [et motus corporis cessabit] will remove y* difficulty proposed. They cannot in my opinion be reconciled with Cor. 1. I acknowledge Y' objection to be just against those words of mine [erit Qr arcus quem tem- pore reliquo corpus describet absq : ulteriore resistentia] I remember y* I inserted them into my Letter as I was hastily transcribing y* passage from another paper & was myself sensible of y® mistake soon after my Letter was gone from me. The alteration which I proposed, as it stood in y^ Paper, was thus J. [Ponantur qua3 in superiore Lemmate et producatur SQ ad V ut sit SV sequalis SP, Tempore quovis in Medio resistente, describat corpus ar- cum quam minimum PQ, & tempore duplo arcum quam minimum PR ; & decrementa horum arcuum ex resistentia oriunda, sive defectus ab arcubus qui in Medio non resis- tente iisdem temporibus describerentur, erunt ad invicem • The post mark is May 4. t Though addressed under thw title by Newton here, and in the remainder of the correspondence, Cotes was not ordained until three years afterwards, (deacon, May 29, 1713, priest the following day). t As may still be seen in tlie MS. of Letter V.(N6. 41), the words " Unde etiam erit decreraentum arcus PQ xquale dimidio lineolaa Kr," being crossed out and replaced by those which we have printed in p. 13, line 8, &c. " Postquam vero, &c." NEWTOX AND COTES. 17 ut quadrata temporum in quibus gencrantur. Est itaq : (lecrcmcntum arcus PQ pars quarta dccrcmenti arcus PR. Unde ctiam si area) PSQ roqualis capiatur area QSr, erit decrementum arcus PQ roquale dimidio lineola) Rr\ adcoq: vis resistentife & vis centripcta sunt ad inviccm ut lincoldo ^Rr Si TQ quas simul gencrant.] I am yet of opinion y* this alteration is just & that the resistance is to y* centri- petall force as }^Rr to TQ\ Your own objection does I think if You carefully consider it prove it to be so. To avoid further misunderstanding I m ill put down my demon- stration more at large thus P Q K U r L Tempore quovis in Medio resistente describat corpus arcum quam minimum PQ & tempore duplo f rcum quam minimum PR ; & decrementa horum arcuum ex resistentia oriunda sive defectus [QK, RL] ab arcubus [PK, PL] qui in ]\Iedio non resistente iisdem temporibus describercntur erunt ad invicem ut quadrata temporum in quibus generantur ; Est itaq: decrementum IQK] arcus PQ pars quarta dccrcmenti RL arcus PR. Unde etiam si area) PSQ a)qualis capiatur area QSr erit decrementum IQK] arcus PQ tequale dimidio lineola) Rr. [Nam ut SQ ad SP ita PK ad KL ita PQ ad Qr ita dividendo QK ad KL - Qr; ergo componendo PK ad PL ut QK ad {QK + KL - Qr sive) rL, unde rL^ZQKi sed erat RL = ^QK, itaq: Rr ^2 QK] adcoq: vis resistentia) & vis centripeta sunt ad invicem ut lineolao QK vel ^7?r & TQ quas simul gencrant. This I take for a direct demon- stration of the truth of what I proposed, & if You will be pleased to consider what I offered at y* end of my second Letter, You will {find J that also to amount to a demon- stratio per absurdum. I did there assume y* proportion of y® Resistance to y® Centripetall force to be as ^iJr to TQ & from y* assumption I deduced a consequence whose truth is very evident upon other considerations. But if You 2 18 CORRESPONDENCE OF take the proportion to be as Rr to TQ or any other way diflferent from y* of ^Rr to TQ, the consequence will be as evidently false; Therefore the proportion can be no other than y* of ^72r to TQ. You say in Y' Letter y* the 8*** Corollary may remain as in Y' copy, but in Y"^ copy there are no alterations of y® first Edition. That You may see the reason I had for the alteration I proposed, I will put JV for the number of Revolutions, T for y** Time of those Revolutions, D for y** Density of the Medium, t for y® tangent of y® Angle, a for y® secant of y® same. Now in Cor. 6 You put N as t, 3r as - or «, but in Cor 8 You put iV as -- or ^, T as a. The alteration which I proposed was to make y® 8^'' Corollary agree w*^ y® 6^\ for I am satisfied of y* truth of y® 6^\ In my first Letter I took notice of two mistakes in Prop xvi, You have consented y* one of 'em may be amended by putting l - ^w for ^n. The other You seem not to have observed which was y* y® Density is not reciprocally as SP' but reciprocally as SP : For the Resistance in P being as ^^p«+i ^"^ ^^ Velocity in P as -— ^^ , it follows y* y'' Density in P is as \^in, OS ^ 1 -Iw, OS ^, ^ ,, . . g op\ SP "*'* "^ OP. SP' ' '^^ °'"''*y ''"'"« "' y Resistance directly & y® square of y* Velocity inversly. If You consent to this correction as I do not doubt You will, I desire You to send me the words of y® Proposition as You would have them altered. It seems to me not im- proper to add somewhere in this xvi Prop, or in a Corollary to it That y® force of resistance is to y* eentripetall force as 1 -|w, OS to OP Y' &c. NEWTOX AND COTES. 19 LETTER VIII. NEWTON TO COTES. This letter is either misdated or was an unusually long time in arriving at its destination. It had not reached Cotes's hands when ho penned his short note of May 17. It lias no address, and was probably sent by a private hand, perliaps by Bentloy. M' Professor Chelsea. 13 May. 1710. I have reconsidered the 15*** Proposition with its Corol- laries & they may stand as you have put them in yo*" Let- ters. But in pag. 285 lin. 13 after the word coincident add the words, et angulus PS V* Jit rectus. Let the 16*^ Proposition stand thus Prop. XVI. Theor. xii. Si Medii densitas in locis singulis sit reeiproce ut dis- tantia locorum a centro immobili, sitq: vis centripeta reei- proce ut dignitas quajlibet ejusdem distantia): dico quod corpus gyrari potest in spirali quro radios omnes a centro illo ductos intersecat in angulo dato. Demonstratur eadem methodo cum Propositione supe- riore. Nam si vis centripeta in P sit reeiproce ut distan- tia9 SP dignitas qua3libet SP'"^^ cujus index est n + 1 ; col- ligetur ut supra, quod tempus quo corpus describit arcum quemvis PQ, erit ut PQ x SP"], et resistentia in P ut hoc est, ob datum ^-^ , reeiproce ut SP'^K Et propterea cum velocitas sit reeiproce ut SP^*, densitas in P erit reeiproce ut SP. Corol. 1. Resistentia est ad vim centripetam ut 1 - ^ n X 0*9 ad OP, • Cot€S has written PVQ, in the margin, t Cotes haa written SP^' in the margin. 2—2 20 CORRESPONDENCE OF Corel. 2. Si vis ccntripeta sit reciproce ut SP^"^^, erit 1 - ^ n - 0, adeoq : rcsistcntia et densitas Mcdii nulla erit, ut in Propositione nona Libri primi. Corol. 3. Si vis centripeta sit reciproce ut dignitas aliqua radii SP cujus index est major numero 3, rcsistcntia affirmativa in negativam mutabitur. Pag. 289, lin. 14. ffor data lege, read data velocitatis lege. Your most humble servant Is. Newton LETTER IX. COTES TO NEWTON. S'. Cambridge May l?*** 1710. After I had received Y' Letter I wrote to You again about a week ago, about some difficultys which still remain with me. The Compositor is now at a stand, & I dare not let him go on till You shall be pleased to send me Y"^ an- swer. Y'^ most Obedient and Faithfull Serv*. EooER Cotes. LETTER X. COTES TO NEWTON. S^ May 20. 1710 I thank You for Y*" Letter which came very season- ably. I now beg leave to propose to You some few alterations in the remaining part of Y' Copy. Page 293. 1:1 — secunda BFK (per Prop xix) pro mensura sua cequaliter premuntur. 1: 4 Hac pressione, pro mensura sua, NEWTON AND COTES. 21 & insuper — Page 303.1: G — nisi forte per particulos in- termedias virtute ilia auctas — I think these words were better left out ; for as I apprehend it, they alter y* case of y® Proposition. 1: 11 Ut si particula unaqusoq:— quadrato- cubi Densitatis. I think also y* this whole Period ought to be omitted, the two propositions containd in it seeming to me to be erroneous, fdess I mistake the sense of Y"^ words. Page 304. Coroll: 5 & G for [quadratum temporis directe] You have substituted in Y' copy [quadrato-quad- ratum temporis directe] I find >vritten in y* margin of Y' book by a different hand [quadr. quadratum temporis (credo)] This marginal note, not Y' own judgment, was I beleive y® occasion of Y*^ making the alteration. Page 308 1 : 10 I would omit y® words [si verbi gratia arcus alter sit altero duplo major]. With Y*^ leave I would begin the 311 page thus*. [Est itaq: incrementum vclocitatis ut V-RSi particula ilia temporis in qua factum est conjunctim: Sed & velocitas ipsa est ut incrementum contemporaneum spatii descripti directe & particula cadem temporis inverse. Unde cum resistentia (per Hypothesin) sit ut quadratum velocitatis, incrementum resistentia) erit (per Lem : ii) ut velocitas & incrementum velocitatis conjunctim, id est, ut momentum spatii & F— 72 conjunctim; atq: adeo si mo- mentum — In my Opinion this alteration is necessary to make the Demonstration accurate. When I first looked over this passage upon account of it I thought the whole construction erroneous. I therefore set my self, after the following manner, to examine how it ought to be, which I here put down for a further use I have of it. Taking .r, Xy V for quantitys analogous to the Force arising from y' gravity of y® Pendulous body, the force of resistance, & y* • In Prop. XXIX. Lib. 2. " Posito quod corpori in cycloide oscillanti resistitur in duplicata rationo vclocitatis : invenire rcsistentiam in locis singulis." This Proposition contains the geometrical construction of the expression ^ (2k$+ I - 2ka + 1 «-»*"-•), n being the first arc of descent. 22 CORRESPONDENCE OP velocity in />, tis evident y* the arch CD will also be as op, & the Fluxion of y* space BD already described will be as - 0?. If therefore t be put for y® moment of time in which the fluxion of y® space - w, the fluxion of y* velocity v, the fluxion of y* resistance x are generated ; You will have • V II —^t t> II d? - » X # But » II «v & therefore x || vv || — d7 X a? - « II «^ - aw. Assuming therefore the determi- nate quantity [a] of a just magnitude You will have this Equation ax ^ xoo ^ ooa. To construct this a3quation I introduced another indeterminate quantity \\j\ putting jr = j9 -f ^.r + ry & X ^ qoo ^- ry \ which values o^ x h x being substituted in y® former roquation I obtained this other aqoo + ary = px + qxx + ryx — cox. Then puttmg ay 9=1, p " «, I had the two following a3quations — « ar, jif«« + a? + ry & y® construction of these two [equations agreed intirely with Y' own Solution of y® Problem*. Being satisfied by this Analysis of y® truth of Y^ conclusion I easily saw y* my former difficulty lay in y® ambiguity of y® word [data] in line 1 & 5, & y® word [detur] in line 6. which I think may be remedied by the alteration which I propose. Page 312. 1: 21 I would leave out y® word [quamproxime]. Page 313. 1 : 29 f I would conclude the Demonstration thus — et ex aequo perturbate Fh seu MN * The analysis and construction of the problem will be found in Cotes*8 LogorMtria^ (Philos. Trans. Jan.— March, 1714, pp. 40 — 42. Harmonia Mensurarum, pp. 36 — 38.) t In Prop. XXX. Lib. 2. This Proposition contains the geometrical construction of jUie equation ~ (a' -6') = k i v'ds (h being the first arc of ascent), which is ob- tained by one integration from the equation of motion — — j— = -j . a — fcu", Cotes*8 sugr- gestion leads to further correspondence (see the next five letters). This and the preced- ing proposition may give us an idea of the trouble that Newton would take to exhibit hi« results in a synthetical form. NEWTON AND COTES. 23 ad Dd ut DK ad CF seu CM; Ideoq: summa omnium MNx CM, id est, ^CAq-- ^Caq seu Aa x^aB icqualis erit summj© omnium Dd x DK, id est, arero BKk VTa, quam rectangula omnia Dd x DK seu DKkd componunt. Q. E. D I was further satisfied y* there is no mistake in the Proposition or in this way of concluding it thus. Taking a: for CD & % for DK by y* abovementioned ooqua- tion ass wa «ai ^ xof it appeares y* ax + ^ir^r is equall to the Fhient of za, AVhence I conclude, if CL be taken on y* other side of y* point C equall to Ca & y* ordinate LQho erected, y* the indeterminate area DKVTa is equall to DK ■j-prx LQTa^-\CDq-\Caq & y* whole Area BKVTa is equall to ^CBq ^\Caq or Aax ^aB, Page 315. 1:7 I would read thus — & Ellipsis aBRVS, centro O, semiaxibus OB, OV — 1: 22 Thus. Nam cum Ellipsis vel Parabola aBRVS congruat — 1: 24 thus alterutram BRV vel VSa exccdit figuram — lin. penult, I would leave out [quam- proxime]. pag. 319. 1 : 13 You say [cum distantijo particu- larum Systematis unius sint ad distantias correspondentes particularum alterius, ut diameter particula) vel partis in Systemate, priore ad diametrum particula) vel partis cor* respondentis in altero.] The same thing is implied in the Demonstration of Prop. 32. I think it ought also to be expressed in y® words of y® 32 Proposition. Y' &c. 24 CORRESPONDENCE OF LETTER XI. NEWTON TO COTES. S' CholBca. May 30. 1710. The corrections w*** you have sent me in your Letter of May 20 are right. But I fear least that w*^^ relates to Prop. XXX may render the Demonstration thereof too obscure. And therefore I think that the Proposition with its Demonstration may stand, & in the end of it, after the words et sic eidem a3quabitur quam proximo, may be added these two sentences. Quinimo eidem ajquabitur accurate, ideoq: conclusiones pra3dicta9 sunt accuratoe. Nam si ad alteras partes puncti C capiatur CL jequalis ipsi Ca, et erigatur normaliter Z,Q ad Curvam aTVKB terminata, et pro Curvaa hujus area inde terminata a TVQL ad ordina- tam LQ applicata scribatur litera M ; area inde terminata a T VKD roqualis invenietur quantitati M, DK + \ CD'i - ^ Ca^t et area tota aTVKB quantitati ^ C5' - ^ Ca^, seu A a X \aB, The Scholium Generate w*^** in the former edition was printed in the end of the seventh Section, I would havr printed in the end of the sixt section next after Prop. XXXI. But it wants the following corrections Pag. 330. lin 21, 22, 23 &c read Scholium generale Ex his Propositionibus per oscillationes Pendulorum in Mediis quibuscunq:, invenire licet resistentiam Medio- rum. Aeris vero resistentiam investigavi per Expcrimenta sequentia. Globum ligneum pondere unciarum Roma- narum 57-^^, diamctro digitorum Londinensium 6^ fa- b{r}icatum, filo tenui &c. Pag. 340. lin 24, 25, blot out, omnino ut in Corollariis Propositionis xxxii demonstratum est. Pag 341 lin. 18 for resistentia read resistentioe. Pag 342 lin 21 blot out, Undo cum corpus tempore, & what follows to the end of the words, longitudincm duplam 30,556 digitorum. NEWTON AND COTES. 25 Pag. 343 lin 6 for pedum read digitorum. lb lin 8 read vi8 rcsistentico eodem tempore uniformiter continuata. lb lin 12 read posset. Pag 344 lin 13, 14 for prima, secunda, tertia read ter- tia quinta septima & for — read -^ , ^ ^ 193 193 Pag. 345 lin 7, 25 for dimidiata read subduplicata. lb. lin. 8 read Nam ratio 7^ - ^ ad 1 - ^ seu 10 J ad 1, non longe * Pag. 349 blot out the lines 17, 18, 19, 20, 21, 22, 23. 24, 25, 2G, 27 Pag. 350 lin. 32 blot out Quare eum globus aqueus in aere movendo & what follows to the end of the words, probe tamen cum prajcedcntibus congruebat. Pag 354 blot out the lines 11, 12, 13, 14, 16. In the beginning of Sect vii pag. 317 lin. 5 after the words similes sint, insert the words & proportionales. I am Your most humble servant. For the 11°^ M' CoTE», Professor Is. NewtoN of Astronomy t ^ Fellow of Trinity College in Cambridge. LETTER XII. COTES TO NEWTON. S' Juno. !•» 1710 I received Your Letter last night, by which You give Y"^ consent to the other alterations which I proposed, but seem to fear least y* which relates to Prop xxx may render the Demonstration thereof too obscure & therefore at the end of y® Corollary after the words [et sic eidem aequa- bitur quamproxime] You add [Quinimo eidem lequabitur • The words, " lb. lin. 8 non longe", are crossed out, apparently by Cotes, in pursuance of Newton's orders in letters X III . and XV. 26 CORRESPONDENCE OP accurate, &c] I beleive You designed those two sentences to be inserted pag. 314 Hn 18 after the words [erit etiam ajquale areiB BKTa quamproxime, & y* by some inadver- tency in y Letter You ordered them to be placed in page 315 1 : 25 after y* words [eidem ajquabitur quamproxime.] For though the Proposition it self & the first part of the Corollary ending w*^ the words [omnino ut in Propositione XXVIII demonstratum est] be accurate, yet as I understand it the remaining part of the Corollary is still but an Ap- proximation, the Ellipsis & Parabola mentioned in the latter part of y® Corollary not agreeing perfectly with the Figure BKVTa ; but by placing those two sentences as in Y*" Letter, even this latter part of the Corollary is declared to be accurate. I beg leave to express my self freely to You, I fear it will be look'd upon as a blemish in Y"^ book first to Demonstrate y* the Proposition is true & afterwards to assert it to be true accurate, I am of opinion y^ the alteration which I proposed pag. 313. 1 : 29 does make the Demonstration compleat to an intelligent Header. If You think good it may be put down more at large some such way as this which follows — et ex soquo perturbate {Fh seu) MN ad Z)d ut DK ad {CF seu) CM\ ideoq: summa om- nium MN X CM aaqualis erit summio omnium Dd y. DK, Ad punctum mobile M erigi semper intelligatur Ordinata rectangula asqualis indeterminata) CM^ qua? motu continuo ducatur in totam longitudinem Aa\ & trapezium ex illo motu descriptum sive huic a^quale rectangulum Aa x ^aB rsquabitur summge omnium MN x CM adeoq: summflD om- nium 7)d X DK, id est, arcfe BKkVTa. q.e.d. Or if You think the Demonstra- tion will even this way be too obscure, a new Scheme may be cut with y*' addition of y " lines here drawn & the A MN NEWTON AND COTES. 27 Demonstration may end thus — & ex roquo perturbate (FA sen) MN ad Dd ut DK ad {CF seu) CMx idcoq: MN X C3/ iequabitm* Z)d x DK, Erigantur normales AX, aZ sequales ipsis AC^ aC & jungatur XZ occurrens nor- malibus il/F, iW in F & /; & erit MY lequalis ipsi CM atq: adeo MN x YM ajquale />d x DK, & summa omnium MN X YM, id est, trapezium A a ZX sive huic rcquale rect- angulum Aa x ^aB aequabitur summoe omnium Dd x DK, hoc est, areoe BKkVTa q e.d. I think the first of these two way 3 sufficiently clear ; but will wait for Y' resolution Y' &c. LETTER XIII. NEWTON TO COTES. I have reconsidered your emendation of the xxx** Proposition w*^ the Demonstration & approve it after the manner you propose in the first of the two ways set do>vn in your Letter of June 1**. In my last letter, as I was sending it away, I crossed out four lines & should have struck out also these words relating to them [lb. lin. 8, read, Nam ratio 7^ - ^ ad 1 - ^ seu 10| ad 1, non longe] I am Yo' most humble Servant Chelsea Jun 8. * Is. Newton. 1710. I thank you for mending the Proposition Fw the Rot'"* M' Cotes Professor of Astronomy and fellow of Trinity College in Cambridge, This letter and the next must have crossed on tlio road. The poHt mark is Jun. 10* 28 CORRESPONDENCE OF LETTER XIV. COTES TO NEWTON. S' {Sunday} June U 1710. I received Y' Letter of May 30**». In that which 1 wrote to You by y® next Post instead of y® alteration in page 310. 1: 29 which You thought too obscure, I proposed the following — et ex asquo perturbate Fh seu MN ad Del ut Dk ad CF seu CM\ ideoq : summa omnium MN x CM jequalis erit summa) omnium Dd x. DK. Ad punctum mobile M erigi semper intelligatur Ordinata rectangula ajqualis indeterminatao CM^ qua) motu continuo ducatur in totam longitudinem Aa\ k trapezium ex illo motu descrip- tum sive huic a)quale rectangulum Aay.^aB cequabitur sum- ma9 omnium MN x CM adeoq : summae omnium Dd x DK^ id est, area9 BKkVTa. q.e.d. We are now at a stand expecting Y** resolution. You gave me orders in Y' Let- ter to print the Scholium Generale after y® sixth section & sent me Y"^ corrections of it. I have not had leasure since I received Y"* Letter to examine all the Calculations of y* Scholium, being at this time engaged in a Course of Experiments & having some other Buisness upon my Hands, but I have read it over & considered the reason- ing of it. Page 345. 1: 26 You say — {Si longitudo pen- duli....augeretur....arcuum descensu & subsequente as- censu descriptorum} differentia 0,4475 diminueretur in ratione velocitatis, adeoq: evaderet 0,4412. I do'nt see any reason for this diminution, but think it ought* to remain 0,4475 notwithstanding y' y® length of y® Pendulum is in- creased in the ratio of 126 to 1221, & thereby the time increased & y* velocity diminished in y® subduplicate ratio • This is also clear from the fact that the equation which connects the arcs of ilcsceut and ascent (the resistance varying as the square of the velocity) does not involve the length of the pendulum. NEWTON AND COTES. 29 of 126 to 122 J. You will see my reasons in what follows. QutB tradita sunt in Prop xxxi & ejus Corollariis obtinent ubi Oscillationes sunt Isochronae. At si oscillationum tern- pus quoq : mutctur, differentia inter areum desccnsu de- scriptum & areum subsequente ascensu deseriptum crit ut resistentia & quadratum temporis conjunetim : Nam totius retardationis particulne singula? ex quibus differentia ilia componitur sunt in hac ratione per Lem : x Libr. 1, Unde si detur longitudo arcus descripti & resistentia sit ut quadratum veloeitatis ; manebit differentia, utcunq : mutetur tempus atq: adeo veloeitas eorporis oscillantis. Nam ob datam longitudinem arcus descripti, tempus erit ut veloeitas inverse ; adeoq : differentia ilia, cum sit ut resistentia & quadratum temporis, erit ut quadratum velo- eitatis directe & quadratum veloeitatis inverse, ac proinde magnitudinem datam habebit. Idem aliter. (vide Fig Prop xxx) Manente longitudinc areus descripti aD augeatur longitudo Penduli. Si mutata longitudine Penduli mancret Resistentia, maneret quoq : ratio resistentise ad vim gravitatis atq : huic ajqualis ratio Ordinata) DK ad longitudinem Penduli ; adeoq : augenda esset Ordinata DK in ratione longitudinis Penduli. Verum ob auctam Penduli longitudinem augetur quoq : tempus in ratione ejus subduplicata adeoq: diminuitur veloeitas in eadem ratione subduplicata, & resistentia atq : huie propor- tionalis Ordinata DK diminuitur in ratione integra. Itaq : Ordinata DK diminuitur in eadem ratione qua prius auge- batur ac proinde manet ejusdem longitudinis, manetq : adeo magnitudo area) BKVTa atq : huic roquale rectangu- lum Aa X 1 ajB & differentia ilia Aa. If You admit of this reasoning, it will not only affect this place in page 345 but also pag. 348 1: 1 and Pag. 353. 1: 27 and page 341. 1: 10. In Page 346: 1: 23 You cite the Corollarys of Prop xl which are now to come after the Scholium ; there being no alteration of this place among the corrections You sent me, I do not know whether You took notice of it & 30 CORRESPONDENCE OF have therefore mentioned it to You. Page 348 1: 7 &c. You seem to confound the Differentia arcuum with y® Be- sistentia Glohi; the former is represented hy AV •{- CV^ & y* latter ought I think to be represented by ^-^K+fCF*. I desire Y' answer to tfiitXetter, when I receive it I will examine Sc alter y® Calculation, if there be occasion, ac- cording to Y' direction _ Y'-&c. LETTER XV. NEWTON TO COTES. I sent you a letter the last week in w*^^ I approved your correction of Prop xxx w*^ its demonstration according to the first of the two ways w'^^ you sent me in your Letter of June 1** & have now repeated in yours of June 11*^ w*^^ I received last tuesday morning {the 13th.} I thank you for that correction. In my last letter but one I crossed out four corrections w*^^ I had 'wrote down in it, & should have crossed out a fift w*^^ related to those four & was in these words. Pag. 345 lin. 8 lege, Nam ratio 7^ - -J- ad 1 - 1 seu lOf ad 1. The correction in the Scholium p. 345 lin 26, sent me in your last, is right, & I beg the favour that you would alter the calculations accordingly. In pag. 346 lin 23 strike out the words et propterea (per corollaria Prop xl Libri hujus) resistentia quam Globi majores & velociores in acre movendo sentiunt & so on to the end of the sentence In pag. 348 lin 7, 14, 15, 16 for yl & C put other letters* suppose F & G, writing, Designet jam FV -{- GV^ resistentiam Globi &c because AV + CV* was used before for the differentia arcuum. " Not adopted. Cotes altered this part of the Scholium in conformity with his remarks at the close of the preceding Letter. NEWTON AND COTES. 81 You need not give your self the trouble of examining all the calculations of the Scholium. Such errors as do not depend upon >vrong reasoning can be of no great consequence & may be corrected by the Reader. I am w*^ many thanks S"^ Your most humble servant Chelsea June 15*** 1710 Is. Newton. For tlie il"** M' Cotes Professor of Astronomy ^ Fellovj of Trinity College in Cambridge Cambridgeshire. LETTER XVI. COTES TO NEWTON. S"^ Juno 30 1710 We have now finished all Y' Copy & y* Scholium Ge- nerale. I received Y*" Letter of June 15*** in which You consent to y® alterations y* I proposed in y* Scholium. I have examined the whole Calculation & done it anew where I thought it necessary. The discourse it self is also a little altered in those places which I mentioned in my last, as You will perceive by y® 2 inclosed sheets {Oo & Pp}. They are not yet printed ofl*, but will stay for Your cor- rections if You shall think fit to make any, I could wish You would be pleased to look *em over, for I fear I may possibly have injured You. The Press being now at a stand I will take this oportunity to visit my Relations in Lincolnshire & Leicestershire. I hope I shall come back again to College in 5 or 6 weeks. \VTien I return I will Avrite to You to desire y* remaining part of Y' Copy. Y'&c. 32 CORRESPONDENCE OP LETTER XVII. NEWTON TO COTES. S' Chelsea Juno 31*. 1710. I received yours of June 30 this noon with the two inclosed proof sheets, & have perused them without ob- serving any faults except in the last page of the second sheet lin 28 where vires autem motrices should be vires autem acccleratrices. And in the preceding page (pag. 295) upon reconsidering the words of Prop, xxxiii, I think the words will be better understood if they run as in the former edition, viz* lisdem positis, dico quod Systematum partes majores rcsistuntur in ratione composita &c. The remaining part of the copy will be ready against your return from the visit you are going to make to your friends. I am w*^ my humble service to yo"" Master & many thanks to yo*^ self for your trouble in correcting this edition, S*" Yo'^ most humble servant For the R"* M' Cotes Professor of Is. Newton. Astronomi/ ^ Fellow of Trinity College in Cambridge. "Wanting— a Letter from M'. Cotes to S'. Is. Newton— dated ll'\ July 1710..." Note by Mr Howkins: who liero and elsewhere informs tis of the absence of letters, the dates of which wo should otherwise (from any thing that can be gathered from the correspondence itself) have been in ignorance of. Smith had probably made a list of all the letters, and Ilowkins on collecting and arranging them when they came into his possession, noted such as were missing. " This means July 1. Newton was not always exact in dating his letters. It may serve to make the lapius in this case less incredible, though most persons will be able to supply instances for themselves, if I mention that the letters which were delivered by the morning post at Cambridge, on July 1, 1847, were stamped June 31, and that one of them, written the previous day by a distinguished prelate, was dated April 30. NEWTON AND COTES. S3 LETTER XVIII. COTES TO NEWTON. S^ Monday Sopt 4*** 1710. I hope to be at Cambridge again oa Wednesday next. I have been somewhat longer in y* Country y" I at first in- tended, I hope You will exeuse me : For the future I shall, I hope, be ready without any further intermission to attend upon y® Edition of Y' Principia. I desire You to send the remaining part of Y"" Copy assoon as You ean. Y' most Humble Servant For S' Isaac Newton at his House Roger Cotes. near the College in Chelsea near London On his return to college Cotes would find that a slight change had just been introduced into the daily habits of the place, which, for the sake of those for whom the fact may possess an interest, may bo recorded here. " Sept. 4, at night Dr Smith the Senior Dean began the custom of standing at grace, chiefly upon my sollicitation, and all the HaJl readily complied with the alteration." Rud's Diary. LETTER XIX. NEWTON TO COTES. S' This Letter accompanies the next part* of the Prin- cipia. I am not certain that you have all y® cutts in wood, but if any be wanting pray send me a draught in paper of what is wanting & I'le get them cut {in} wood. I am S' Yo'^ most humble Servant Chelsea. Sept 13 1710. Is. Newton. For the R"** M' R. Cotes Professor of Mathematlcks ^ Fellow of Trinity CoU lege in y* University of Cambridge. • Beginning at p. 321, with part of Cor. 2, Prop, xxxiii. Lib. 2, and ending at |». 432, with Prop. xxiv. Lib. 3. 34 CORRESPONDENCE OP LETTER XX. COTES TO NEWTON. S'. Sept. 21"* 1710 I have received y* second part of Y' Copy, there arc wanting only two wooden cutts which I can get done at Cambridge. I have read over what relates to y® resistance of Fluids, I thank You for the satisfaction I have received in seeing y* Theory so perfectly compleated. I confess I was not a little surprized upon y® first reading of Prop. 36* ; but I now begin to be better reconciled to it. One of my greatest difficulties was an Experiment of Mons^ Marriotte which he says (page 245 Traite du Mouvment des Eauxf) he often repeated with great care. By his Ex- periment I concluded y* y* Velocity of y® effluent water was equall to y* gotten by an heavy body falling but from half y* Height of y*' Vessel. He tells us y* 14 Paris Pints of water were evacuated in a IMinute of time through a circular aperture of J Inch diameter, the altitude of y* Vessel being 13 feet. He describes the Paris pint to be y® 35*^ part of y* Cube of y* Paris foot. Therefore the water evacuated in a second was or Cubick inches. The Area of y® aperture 35 ^ 60 ^5 •^ ^ was ~ inches. Hence y* length of a Cylinder equall in magnitude to y® evacuated water & having y® above , . n .. T> . • 14 X l6 X 2 X 144. mentioned Aperture for its Basis is — ^ 11 X 25 Inches, and this length is y* space described in a second of time with y* uniform velocity of y® water as it passes • Making the velocity of efflux of a fluid through an orifice in the base of a cylindri- cal vessel to be that due to the height of the surface of the fluid above the orifice, a result first stated by Torricelli, and adopted by him as a principle, ( De motn Projecto- rum, Florcnt. 1644. p. 191.) In the Ist Ed. (Prop, xxxvii.) the velocity had been made that due to half the height. The MS. of tl»e Prop, which Cotes had before him when he wrote this Letter is wanting. t New Edit. Paris 1700. The Ist Ed. is dated 1686. NEWTON AND COTES. 35 through the aperture. The space described in a second of time with y* uniform Velocity acquired by any falling body in y* same time is (according to Tlio remainder of this letter is wanting: at the point where it breaks off Cotes is saying that, according to Iluygens's penduhim experiments, the velocity generated by gravity in l"07) = 30j Paris feet; and .'. the height duo to the velocity of efflux ■= \ — Tr—^i — ) • t- in feet, which lies between GJ and 6J. Some of the contents of the deficient part of the letter are men- tioned in Newton's letter of ^larch 24, 171?. The letter which was actually sent will probably bo found, with others that are wanting to complete this correspondence, in the Portsmouth Collection. The above-mentioned re:?ult of IMariotte's experiment had been brought before the notice of the Royal Society by Ilalloy at their meet- ings on March 18 and 25, 1C91. On the latter day an experiment (inaccurately described in the Journal Book) was made, in which the jet was found to rise " far above the middle of the height of the liquor, whence it is to bo noted that there is a mistake in the .37th Prop, of Mr Newton's 2nd Book, whereof it was ordered that Mr Newton should be certified." (It was probably in consequence of this sugges- tion that Newton revised the Proposition, and put it into the shape which surprised Cotes.) On Hallcy's recommendation, further experi- ments were made with a view to ascertain the cause of the discrepancy between the results derived from the observed height of the jet and the quantity of fluid discharged, but they only served to establish the fact, which remained unaccounted for until Newton (towards the end of 1710 and beginning of 1711), compelled by the statements of Cotes's letter of Oct. 5, I7IO, to investigate the subject afresh, found the true explanation in the difference between the velocities at the orifice, and at that part of the vein of issuing fluid where it ceases to contract. See Newton's letter of March 24, 171t. For an account of what has been done in this branch of Hydrodynamics, since Newton's time, see Rennie's Report to the British Association (meeting 1833) with the works there referred to, to which add Navier's Resume des Le9on8... sur r Application de la Mdcanique.-.Part. 2, 1838; and D'Aubuisson's Traite d'Hydraulique, 2nd Ed. 1840. 3—2 36 CORRESPONDENCE OF LETTER XXI. NEWTON TO COTES. S' London. Sept. 30. 1710 Since the receipt of your Letter I have been removing from Chelsea to London, w*^** has retarded my returning an answer to yo"^ last. I have not seen Mariots book con- cerning the motion of running water, but certainly there is something amiss in his experiment w*^^ you give me an ac- count of. ffor I have seen this experiment tried & it has been tried also before the Royal Society *, that a vessel a foot & an half or two foot high & six or eight inches wide with a hollow place in the side next the bottom & a small hole in the upper side of the hollow, being filled with water ; the water w*^^ spouted out of the small hole, rose right up in a small streame as high as the top of the water w*^^ stagnated in the vessel, abating only about half an inch by reason of the resistance of the air. The small hole was made in a thin plate of sheet tin and well polished, that the water might pass th{r}ough it with as little friction as possible. It was about the bigness of a hole made with an ordinary pin. The corrections you have made are very well & I thank you for them, & am glad that the Theory of the resistance of fluids does not displease you provided the xxxvi*^ Propo- sition be true, as I think it is. Direct your next Letters to me in S* Martins street neare Leicester fields. I am Yo*" most humble Servant For the R"** M' Cotes Professor of Astro- Is. New TON nomt/f ^ Fellow of Trinity College in Cambridge in Cambridgeshire. • An experiment of this kind attended with the same result was tried by Ilooke at a meeting of the Royal Society, April 1, 1691. The velocity of efflux was also the sub- ject of experiment or discussion at several other meetin(r<* in that year. See the Journal Book, March 18, 25. April 8, 22. NEWTON AND COTES. 37 *• Wanting, two letters from Mr Cotes to Sir Isaac Newton, dated 5th and 2()th Oct. 1710, concerning Prop, xxxvi. Lib. ii. I remember to have Been the wliolo of this Prop, as it ia now printed in the 2nd Edition, fairly written in Mr Cotcs's own hand ; but I fear it is lost, or inadvertently destroyed; as I cannot find it now. E. HOWKINS, 1770.** LETTER XXn. NEWTON TO COTES. I received both your Letters & am sensible that I must try three or four experiments before I can answer your former*. My time has been taken up partly with remov- ing to this house, partly with journeys about purchasing a house t for the Royal society & partly w*^ settling some matters in the Mint in order to go on w*^ y® coynagej that I have had no time to take these matters into considera- tion but hope w*4n a fortnight to try the experiments & * Of Oct. 5, containing probably, among other thini^, experiments Cotes had been tiiaking on fluids issuing from an orifice in a vessel, and which went to confirm Mariotte's. See letter of Newton in Macclesfield Corrtip. ii. 437. + In Crane Court. The Society met there for the first time on Nov. 8, having previously held tlieir meetings at Grcsham Colleg'e. The change, as is usual, was opposed by ?ome of the members. In 1782, Government assigned the Society apart- ments in Somerset House. See Weld's Hist, uf Royal Soc. i. 389, seqq. ; Kllis*s Let- ters of Eminent Literary Men, 346, (where C. Wren's letter should evidently be dated, 1711.) I The following table of gold and sUver coined yearly from Christmas, 1708, to Christmas, 1713, will shew approximately the times at which Newton's duties at the Mint would experience a pressure during the years over which this correspondence on the Principia extends. 1709 OOt.D. lbs. 2468 8II.VF.U. lbs. 25423 (in preceding year, 3751) 1710 3716 817 1711 9324 24768 1712 2855 1784 1713 13137 2333 Macclesfield Corre$p. u. 434. In the beginning of March 1711 the Royal Society changed their day of meeting to Thursday at 4, the President " being obliged to attend the Mint on Wednesdays.** 88 CORRESPONDENCE OP settle the matters in doubt & beg the favour that you will let the press stay till you hear from me again. 1 am Yo' most faithfuU friend Si humble Servant London. Octob 27*. 1710. Is. Newton For the R°* M' Cotes Professor of Astro- nomi/f at kU chamber in Trinity College in Cambridge. LETTER XXIII. NEWTON TO COTES. S* Martins street by Leicester ffiolds. Mar. 24*^ 171?. I send you at length the Paper for w^^ I have made you stay this half year. I beg your pardon for so long a delay. I hope you will find the difficulty cleared, but I know nott whether I have been able to express my self clearly enough upon this difficult subject, & leave it to you to mend any thing either in the expression or in the sense of what I send you. And if you meet w*^ any thing w^^ appears to you either erroneus or dubious, if you please to give me notice of it I will reconsider it. The emendations of Corol. 2 Prop 38 & Prop 40 are your own. You sent them to me in yours of Sept. 21, 1710, & I thank you for them. That you may have the clearer Idea of the experiments in the beginning of the inclosed paper, let ABCD represent a vessel full of water perforated in the side with a small hole EF made • Post Mark 28. t It 18 doubtful whether the " not" has not been added by another hand. If it be in Newton's handwriting, it is about the nearest approach to an instance of his crossing a * t', that I remember to have seen. NEWTON AND COTES. 39 in a very thin plate of sheet tin. And con- ceive that the water converges towards the hole from all parts of the vessel & passes it" through the hole with a converging motion & thereby grows into a smaller stream after it is past the hole then it was in the hole. In my trial the hole EF was 1^^^' of an* inch in diameter & about half an inch from' the hole the diameter of the stream US* was but ^ of an inch. And therefore the streame had the same velocity as if it had flowed directly out of a hole but JJ of an inch wide. And so in Marriotts experim* the stream had the same velocity as if it had flowed directly out of a hole but -^^ of an inch wide. In computing the velocity of the water w*^^ flows out we are not to take the diameter of the hole for the diameter of the streame, but to measure the diameter of the streame after it is come out of the hole & has formed itself into an caven & uniform stream. And the velocity thus found will be what a body would get in falling from y* top of the water : as is manifest also by the distance CG to which the stream will shoot it self, & also by the stream's ascending as high as the top of y* water stagnating in the vessel, if the motion be turned upwards. I am Your most humble & most obliged Servant For the R"*^ M' Roger Cotes Professor of Is. Newton Astronomy at his Cliamber in Trinity College in tJie University of Cambridge, • JRS b the diameter of the "sectio venae contractae," (a terra first used by Jurin, Philosoph. Transact. Sept.— Oct. 1722, p. 185; and afterwards by Dan. Bernoulli, Hydrodynam. p. 65. Jurin also uses ** vena contracta" to denote the same thing, and the expression b still retained in works on Hydrostatics, though differently defined by different writers, most of tliem describing it as that part of the issuing fluid between the orifice and the section whose diameter b RS.) 40 CORRESPONDENCE OF The " Paper" mentioned in the above letter seems to have consisted of four folio sheets, and to have included from Prop, xxxvi. to Prop. XXXIX. with part of Prop, xl., and a page of corrections (No. 111.) to be made in the conclusions of "the Experiments set down in the Scholium to the 40th Proposition sent you formerly." The first three leaves are wanting, the portion which is preserved beginning with the latter part of the 37th Prop. (No. 72). Tliero were several things in this ** Paper" which did not satisfy Cotes. (1) His " difficulty" about the SGth Prop, was not yet com- pletely removed. This probably led to his (missing) letter of March 31, which, if no other letter passed between them in the interval, brought him a satisfactory answer from Newton. This answer, sent apparently in a parcel from Bcntley, is also wanting. (2) Besides making other alterations of a minor character. Cotes has crossed out what is left of Prop, xxxvii., and written the Proposition out on another piece of paper (Nos;=?0^71) as it now stands in the 2nd Ed. with this note at the top : " Print this instead of wliat is blotted out in Prop, xxxvii." He has also modified part of the Scholium of this Prop, though not to the extent that Ilorsloy {Ncwtoni Op. ii. 412) attributes to him. He has drawn his pen through almost the wliole of Prop, xxxviii. and part of its 2nd Cor. and re- written the parts struck out as they now stand in the 2nd Ed. These were the materials of his letter of June 9. See introductory remarks to the fragment of that letter. "Wanting, a letter from Mr Cotes to Sir Isaac Newton, dated 31 st March 1711. Another dated 4th June 1 711." Note by Mr Howkins. LETTER XXIV. NEWTON TO COTES. Yo" of June 4*** I received the next day & thank you for it. I am glad you received what D*" Bentley sent you & that you think the difficulty removed, except what you mention about the manner of delivering y® 37*^ Pro- position, ffor clearing the sense of the first & second Paragraphs, these words may be added to the end of the second Paragraph after the word locatum. Circellus av^em NEWTON AND COTES. 41 sustinendo vim aquce defluentis mimiet ejua velocitatem, idq : in ratione qua minuit spatium per quod aqua Jam transit. Nam {per Cas, 5. Prop xxjcn^ ^ efus CoroL 6) aqua jam transihit per spatium annulare inter circellum fy latera canalis eadem velocitate qua prius transibat per canalis cavitatem totam^. And a little after where I have these words [augcatur velocitas circelli in eadem ratione et resistentia ejus auge- bitur in ratione duplicata] may be written these [augeatur veloeitas circelli in eadem ratione & resistentia ejus auge- bitur in eadem ratione bis, nempe semel ob auctam quan- titatcm aquas in quam circellus dato tempore agit & semel ob auctum motum quem circellus in singulas aqua3 partes imprimit. Nam partes fluidi similibus motibus agitabuntur atq: prius sed velocioribus et minore tempore*.] But since you are considering how to set this xxxvii^** Proposition in a cleare light I will suspend saying any thing more about it till I see your thoughts. I am Yo' humble servant London 7*^ June 1711 Is. Newton, For the R"** M' RooEit Cotes Pro/mor of Astronomy at hia Chamber in Trinity College in Cambridge LETTER XXV. COTES TO NEWTON. This is only the concluding part of a letter, which a note by Mr Howkins states to have been dated June 9th, I7II. In the words with which this fragment opens, Cotes is giving his emendation of Prop. XXX vni. as it stood in the " Paper" wliich Newton sent liim in liis letter of Marcli 24. The former part of the letter must have contained Prop, xxxvii. in the form in which Cotes had at last put it, and also Not adopted. 42 CORRESPONDENCE OP hia modification of the construction in the Scholium, whore Newton had made the latug rectum of the smaller parabolas S AB and tliat of the others 32 CD. Ilorsley saw some of Cotes's actual letters in the Portsmouth Col- lection, and this of the 9th of June among others. In a note on the 37th Prop, he says (ii. 404) : Iliec demonstratio a Cotesio tota est. Ilorsley also says tliat the Oth Cor. of Prop, xxxvii. is due to Cotes, and that in the letter in which it was sent (what the date of the letter was docs not appear certain) after explaining this Corollary he adds: "Hoc Corollarium lucem aliquam tuo in Corollario dccimo guantum sentio offundere possit." This 6th Corollary in Cotes's hand written on a slip of paper is still preserved (No. 07). )June 9. 1711.} • •*«**## {et propterea} Vis ilia qu89 tollere possit motum omnem Cylindri interea dum Cylindrus describat longi- tudinem quatuor diametrorum, Globi motum omnem toilet interea dum Globus describat duas tertias partes hujus longitudinis, id est, octo tertias partes diametri proprijo. Resistentia autem Cylindri est ad banc vim quamproxime ut densitas Fluidi ad densitatem Cylindri vel Globi, per Prop XXXVII, & resistentia Globi sequalis est resistentioB Cylindri per Lemm : v. vi. vii. I will remember to alter the 2** Corollary of this Proposition which You had forgotten to do in Your last Copy. I have computed y® Table pre- ceeding y® Scholium of Prop, xl & find some of the numbers to be amiss which I will take care to rectify ; as over against 0,9 G the space should be 0,7 1 96609 F; over against 3G the space should be 4^,6186570 F ; over against 4>G should be 6,6U3765F, I computed also all the Experiments & found my Calculations to agree nearly enough with Yours except in the 1** Experiment which I will alter throughout. Of the rest the greatest difference was in the 11 *^ in which y® result was 46 -g^ oscillations not 46 as You make it in Your corrections*, I took care to make a right allowance • Sent March 24. See p. 40. NEWTON AND COTES. 4S for y* narrowness of the Vessel. I desire You to send mo the Altitude from which the Globes fell in the 9*** Experi- ment. You had forgotten to mention it in Your Copy. The Six Experiments in y® Air * agree also very well with my computation, in the 5*^ the space should be 225^. 5**. Your most humble Servant B. C. LETTER XXVI. NEWTON TO COTES. I have read over & considered your alterations, & like them very well & return you my thanks. In y* end of Exper. 9, add, desanbentes altitudinem digitorum 182. I thank you also for correcting the numbers. I hope there will be no more occasion of stopping the press. After you have read the objection of Muysf taken from • These experiments were made by Hauksbee, June 9, 1710, with glass balU let fall, from the top of the Cupola of St Paul's, (nearly 220 feet). See Phitosoph. Trans. Oct. — Dec. 1710, p. 198. An account of them was read at a meeting of the lloyal Society, June 14, at which Newton presided. At the previous meetinpr, June 7, (the President then also in the chair) Hauksbee read a paper on some experiments of the same kind, which are described in the article in the Phil. Trans, just referred to. Newton assisted at similar experiments, made by Uesaguliers, April 27 and July 27, 1719, from tl»e upper gallery in the luntern on the top of the Cupola, a height of 272 feet. He with some other persons was below, and noted the difference in" the time of fall of the leaden and of the lighter bulls. See Phil, Trans. Sept._Oct. 1719, pp. 1071—1078. The ex- periments made on the latter day are introduced into the 3rd Ed. of the Principia, p.3o3. t In EUmenta Physices methodo Mathematlca Demonstrata, &c. Amstelod. 1711: a heavy quarto, reviewed in the Leipsic Acts for Sept. 171 1, and severely criticised by Leibniz and John Bernoulli in their Correspondence. In the ist Ed. of the Principia (p. 337), there is a Lemma which states that if a spherical or other vessel, filled with fluid f move rectilineurly with an accelerated velocity, the molecules of the contained Jluid participating equally in the motion of the ve^elwill remain at rest among themselves. Muys (p. 355), in opposition to this quotes a pitisago from the 4th Dialogue of Galileo's System. Cosmic, (p. 315, Lyons, 1641,) where Sal- viati, in attempting to explain the tides, takes the case of a vessel^ which contains water. 44 COllRESPONDENCE OF Galileo's experiment of the motion of a bucket full of water you will scarce expect very much from that author. I am S' Yo*" very humble servant SV Martins street London. June 18*'' 1711. Is. Newton For the R"** Af Cotes Profmor of Astro, notntff at his chamber in Trinity College in the University of Cambridge. LETTER XXVII. COTES TO NEWTON. S' Juno 23^ 1711 I received Your Letter & have delivered Y' Papers to the Printer. I hope we shall now go on without any fur- ther intermission. As for Muys, I have look'd over what relates to the resistance of Fluids. He acknowledges that what he offers upon y* subject at present is but crude & indigested & I am very willing to agree with him. His Objections as far as I can understand *em do not in any wise affect Your Book, much less the new Edition of it. One M"" Green of Clare-Hall has now in the Press a book* moving horizontallif^ and says that, If a farce be applied to retard the vessel^ the moleculet of the fluid will still retain their velocity, and the water mil rise at the anterior yart of the vessel. Ify on the contrary, the velocity of the vessel be increased^ the water will lag behind, and so will be higher at the hinder than at the fore part of the vessel. This fact the speaker proceeds still further to illustrate by referring^ to the boats used to convey fiesh water from Lizza Fusina to Venice. • The Principles of Natural Philosophy, in which is shewn the Insufficiency of the Present Systems^ Sec. &c. Camb. 1712. With a Latin Tract at the end, entitled, Geometria Solidorum^ &c. This eccentric writer also published A Demonstration of the Truth and Divinity of the Christian Religion, &c. Camb. 1711, and a large thick folio, (pp. 981) with the title of The Principles of the Philosophy of the Vxpansive and Con- tractive Forces, &c. Camb. 1727. In the Preface to this last work he says: *' Our Phi- losophy, as it is now received and embraced, is the product of Popish countries, NEWTON AND COTES. 45 of the like nature with Muys wherein I am informed he undertakes to overthrow the Principles of Your Philosophy. I do not expect very much from him, & I beleive You will not Your self when I have told You he is a Person who pretends to have solv'd y* grand Problem of y* Quadrature of the Circle. That the Press may not stop, I am now looking over Your Copy beforehand. I find nothing amiss till I come to Prop : 48. I will choose to make my Objec- tion against the Corollary, wherein You have these words [Nam lineola Physica 67, quamprimum ad locum suum primum EG redierit, quiescet;] This assertion cannot I think be reconciled with what You assert & prove in the Proposition [& propterea vis acceleratrix lincoloj Physicto 67 est ut ipsius distantia a medio vibrationis loco Q] I propose to alter the whole Proposition thus if You approve of it. [Propagentur pulsus in plagam BC a B versus C & designet BC intervallum corundem ab invicem. Sint E, Fy G puncta tria Physica Medii quiescentis in recta BC ad aaquales distantias sita \ ee, //*, gg spatia sc-qualia pcr- brevia per qua9 puncta ilia motu rcciproco singulis vibra- tionibus eunt & rcdeunt; 6, 0, 7 loca qua)vis intermedia eorundem punctorum ; & EF, FG lineola? Physica) seu Medii partes lineares punctis illis intcrjectio & successive imported to us from Italy and France All therefore which I design and intend, is to propose a Philosophy, which is truly EngUth, a Cuntnhriguni, and a Claren$ian one, as it was born, and educated, and studied in those places; And as my Name is not much worse in the Letters which belong to it, than those of Galileus or Des-Cartes, I shall venture to call the GIIKENIAN." Mr Green was not altoorcther a stranger to Newton when Cotes introduced a notice of him in this letter. On makin^f the discovery that the area of a circle is equal to four-fiftlis of the square of its diameter, shortly after taking his B. A. degree (1700), •' Dominum Newtonum accessi ut consulercm," says he, "orantem qui chartulas perlepreret, ipsis intactis, ne inspectis certe, rejecit, ag- gressus sum dein epistola, recusavit, (in the Preface to his Ceometria SoUdornm, his phrase is ' rescripsit nihil,') quid posthrcc arbitremini me putassel Saltern vel con- temptum me vel Problema." (lb. p. 940. 1st Lecture "ad Clarensem juvcntutcm.*'^ On the publication of Green's " Natural Philosophy" in 1712, where his quadrature of the circle was asserted, he tells us that Cotes was "so kind and obliging as to com- municate to me with great candour and friendship a demonstration aprainst it," which will be found lb. pp. 924-5. Cf. Letter CVL 46 CORRESPONDENCE OF translate in loca «0, (py & ef, fg. Recta? ......^ ee aoqualis ducatur recta PS, bisecetur eadem in O, centro O & intervallo OP describatur cir- culus SIP if & agatur diameter QR ad diame- triim PS perpendicularis. Per circuli hujus circumferentiam totam cum partibus suis expo- natur tempus totum vibrationis unius cum ip- sius partibus proportionalibus ; sic ut complete tempore quovis QII vel QESh, si demittatur ad PS perpendi- culum HL vel hi, & capiatur Ee lequalis OL vel 01, punc- tum Physicum JE reperiatur in €. Ilac lege punctum quodvis E eundo ab E per 6 ad e atq : inde redeundo, iisdem accelerationis ae retardationis gradib® vibrationes singulas peraget cum oscillante Pen- dulo. Probandum est quod singula Medii puncta Physica tali motu agitari debeant. Fingamus igitur Medium tali motu a causa quacunq : cieri, Sc videamus quid inde sequatur. In circumferentia PQSB capiantur cequales arcus ///, IK vel hi, ik cam habentes rationem ad circumferentiam totam quam habent ajquales rectaB EF, FG ad pulsuum intervallum totum BC* Et demissis perpendiculis IM, KN vel im, kn ; quoniam puncta jE, F, G motibus similibus successive agitantur & vibrationes suas integras ex itu & reditu compositas interea peragant dum pulsus transfertur a i? ad C, si QH vel QllSh sit tempus ab initio motus puncti E, erit QI vel QISi tempus ab initio motus puncti F, & QK vel QKSk tempus ab initio motus puncti G ; & propterea Ee, F(p, Gy erunt ipsis OL, OM, ON in itu punctorum, vel ipsis 01, Om, On in punctorum. reditu lequales respective. Unde €7 seu EG -f Gy ^ Ee in itu NEWTON AND COTES. 47 punctorum roqualis erit EG - LN, in reditu autcm aequalis EG + In, Sed ey latitude est seu expansio partis Medii EG in loco ey, Sc propterea expansio partis illius in itu, est ad ejus expansionem mcdiocrem ut EG - LN ad EG ; in reditu autcm ut EG + In seu EG + LN ad EG, Quare cum sit LN ad KIT ut JM ad radium O/, & KH ad JEG ut eircumferentia PQSBP ad i?C, id est, (si ponatur V pro radio circuli peripheriam habentis cequalem intervallo pul- suum BC) ut 0/ad F, et ex a?quo LN ad EG ut Llf ad V: erit expansio partis EG punctive Physici F in loco ey ad eo'pansionem mediocrem qvam habet in loco 8uo primo EG ut V- IM ad V in itu, utq : F + im ad V m reditu. Unde vis Elastica puncti F in loco €7 est ad vim ejus Elasticam mediocrem in loco EG ut ,^ — --- ad — in itu, in reditu K - IM V vero ut — — ; — ad — . Et eodem arffumento vires Elas- r+ im V tic£B punctorum Physicorum G & J^ in itu sunt ad vires mediocres ut --- — — - & r= — rrrr ad •- , & virium differen- V - KN V - HL V tia . sive excessus vis Elastica3 puncti y supra vim Elasti- cam puncti e est ad Medii vim Elasticam mediocrem ut KN-IIL 1 ^ u . . ad — , hoc est, ut VV-Vy.KN-Vy.HL'^KN y, HL F' vv ^^ v ^^^^ "** ^^ " ^^ ^^ ^* ®^ modo (ob angustos limites vibrationum) supponamus KN & ILL in- definite minores esse quantitate V, Quare cum quantitas V detur, excessus ille est ut KN - HL, hoc est (ob pro- portionals KN - HL ad HK & 031 ad 07, datasq : HK & OF) ut OM, id est, ut F(p, Et eodem argumento ex- cessus vis Elastic© puncti 7 supra vim Elasticam puncti e in reditu lineolae Physicso 6 7 est ut F(p. Sed excessus ille est vis qua hsBC lineola accelcratur ; & propterea vis acce- 48 CORRESPONDENCE OF leratrix lineolf© PhysicaD €y est ut ipsius distantia a medio vibrationis loco F, Proinde tempus (per Prop xxxviii Libr. 1) recte exponitur per arcum QI; & Medii pars linearis ey perget lege praoscripta moveri, id est, lege oscillantis Penduli : & par egt ratio partium omnium linea- rium ex quibus Medium totu componitur. q.e.d.] I was going to propose an alteration of the Corollary but I choose rather to leave it to Your self. It must be made to cor- respond mth what You have at the end of Page 372 where You cite it. I propose to alter Prop. 49 as follows, [p. 368. 1 : 28 — ad lineola) illius pondus ut IIK ^ A ad V X EG sive ut PO y^ A ad FF, nam HK erat ad EG ut PO ad F.] [1 : 32 — urgente vi ponderis in subduplicata ratione W ad PO x A atq : adeo — '] [1 : ult — in sub- duplicata ratione W ad PO ^ A & subduplicata ratione PO ad A conjunctim, id est, in ratione integra F ad A, Sed tempore vibrationis unius — .] [Ergo tempus — & reditu composita) ut V ad A, id est, ut BC ad circumferen- tiam circuli &c.] I propose to add the 2 following Corol- laries to Prop 49. Cor. 1. Velocitas pulsuum ea est quam acquirunt Gravia a)qualiter accelerato motu cadendo ct casu suo describendo dimidium altitudinis A, Nam tempore casus hiyus, cum velocitate cadendo acquisita, pulsus percurret spatium quod erit sequale toti altitudini A, adeoq : tempore oscillationis unius ex itu & reditu compositoB percurret spatium a3quale circumferentia) circuli radio A descripti ; est enim tempus casus ad tempus oscillationis ut radius circuli ad ejusdcm circumforentiam. Cor. 2. Unde cum altitudo ilia A sit ut Fluidi vis Elastica directe & dcnsitas ejusdcm inverse ; velocitas pul- suum erit in ratione composita ex subduplicata ratione densitatis inverse & subduplicata ratione vis Elasticro di- recte. I think the 47*^ Proposition is out of its place : for NEWTON AND COTES. 49 the Demonstration of it proceeds upon the supposition of the truth of the 48*^, & therefore it ought to follow the 48*^ & besides the 48*'' serves to form some Ideas which are necessary to the understanding of the 47*^*. If You agree that these Propositions should change places I would add the following words at y® end of the 47*^ which will then be the 48*^ [Hjcc Propositi© ultcrius patebit ex con- structione scquentis], I see nothing further in the 2** Book which I could wish might be altered. In the 3^ Book under Phoonom : 1, The Periodical times should be I**. 18''. 27'. 34>" 3**.13^13'. 42". 7**. s\^2\ 36'\ 1(3^16''. 32'. 9" and the Distantif© ex temporibus periodicis may be 5,667 9,017 14,384 25,299 I perceive You have made use of Cassini's Tables of Ju- piter's Satellits printed in 1693 in the Recudl d* ObservO' tions faites en plusieurs Voyages ^c. But Your numbers give the times of the Revolutions to Jupiters shadow, not to y® same point of y® Ecliptick. The Revolutions to the same point of the Ecliptick are (by those Tables) as I have set 'em down. Y"" time of the Revolution of Saturns outer- most Satellit differs from the time assigned by Hugenius in his Cosmotheoros & by Cassini in the Philosophical Transactions but I find it is y® time which was afterwards determin'd by Cassini in y* Memoires de VAcadem. 1705. You have made an addition to the 3** Proposition in which are these words [Hicc ratio obtinet in Orbe Luna) nostra). • The object of Prop, xlviii. is to shew that when pulses or undulations are pro*)a> gated in a fluid, the particles vibrate accordingr to the law of an oscillating pendulum. Prop. xLvii. shews how the velocity of propagation varies, and Prop. xlix. determines its quantity, the expression for which (\/^ . heighTof'homog. atmosph.) Laplace wai the first to prove, must (in the case of sound) be multiplied by A /spec, heat of air under a constan t pressure volume* Mtcan. Ctlest. < 121, 129. Poisson, Mtcan. ii. 716. VVhewell's lint. Ind. Sri ii. 4 50 CORRESPONDENCE OF In minore Orbe motus Aphelii minor esset in triplicata ratione minoris distantisD Lunze a Terra, & Fractio ^|^ diminui deberet in eadem ratione. Et propter banc dimi- nutionem vis qua Luna retinetur in Orbe suo est ad vim eandem in superficie Terra) ut 1 ad D-sa quamproxime, uti computum ineunti patebit] 1 sbould be glad to understand tbis place if it ^vill not be too great a trouble to make it out to me. I do not at present so mucb as understand what it is that You assert. I am S" y &c. LETTER XXVIII. COTES TO NEWTON. S'. July lO*** 1711 I wrote to You about a Month ago concerning the 48**^ Proposition of Y*^ second Book, & the last week I ordered the Printer to send You all the sheets which were printed off. If You have received these sheets You will perceive by *em that the Press is now at a stand. But having no Letter from You I fear the sheets have miscarried. The Compositor dunn's me every day, & I am forc'd to write to You again to beg ^^^nswer to my former Letter. I have received the last part* of Your Copy by D' Bently. I have now read over and examined all the calculations of the former part which ends in y® 432^ page. I will write to You concerning it assoon as 1 receive Your answer to my last Letter. I am S^ Y-" Sec, * Beginning at p. 433, with part of Prop, xxiv.. Lib. 3, and terminating^ at p. 510 with Prop. xLii. (end of 1st Ed.) Bentley returned to College on the 7th, (llud'fl Diary.) NE^VTON AND COTES. 51 LETTER XXIX. NEWTON TO COTES. S* Martins Street ia Leicester ffiolds London July 28*^ 1711. I received your Letters & the papers sent mc by the Printer But ever since I received yours of June 23 I have been so taken up with other aflairs that I have had no time to think of Mathematicks. But now being obliged to keep my chamber upon some indisposition w*^^ I hope will be over in a day or two* I have taken your letter into con- sideration. You think that in the Corollary to the 48*** Proposition these words [Nam lineola Physica ey quampri- mum ad locum suum primum redierit, quiescet] consist not w*^ what I assert & prove in the Proposition, viz* [& propterea vis acceleratrix lineolso Physic89 ey est ipsius distantia a medio vibrationis loco Q] But I suspect that you take the words [ad locum suum primum] in another sence then I might intend them, ffor when all the lineola; physical ey are returned to their first places or places in w*^^ they were before the vibrations began, the medium will be uniform as before & the vis acceleratrix of the lineola physica ey will cease, whether that lineola arrived to its first place in the beginning middle or end of the vibrations. For making the Corollary more intelligible, these words may be added to the end of it. Partes fluidi non quies- cent nisi in locis suis primis. Quamprimum in loca ilia motu retardato redierint, component Medium uniforme quietum quale erat ante vibrationes excitatas. In altering the 48*'* Proposition you have shortned the Demonstration. If you had proposed your alteration of the Corollary I should have been better able to compare the whole w*^ mine. • He was sufficiently recovered by the followinfir 'Jhursday, (Aug. 2,) to preside at a meeting of the Council of the Royal Society on that day. 4—2 52 CORRESPONDENCE OP Your emendations of Prop 49 are very well & the two Corollarys you propose may be added to it. And the 47*^ & 48*^ Propositions may change places, & at the end of the 47*^ these words may be added [Hsdc Propositio ulterius patebit ex constructione sequentis. I will write to you about {the} third book in my next. I am S' Your very humble servant For the Roror"'* M' Roger Cotes Professor Is. Newton. oj' Astronomy, at his Chamber in Trinity College in Cambridge. LETTER XXX. COTES TO NEWTON. S'. July 30*'' 1711. I have read Y"* Letter & find my self obliged to trouble You once more. I must beg leave to tell You I am not as yet satisfied as to the Inconsistency which I mention'd in my former Letter. You seem to say that when the Lineola Physica €7 is return'd to its first place, which You take to be the beginning of the Vibration, the Medium will be uniform as at first & consequently its Vis acceleratrix will cease. If upon the return of the Lineola to its first place it be granted that the Medium ^vill be uni- form 1 confess it must also be granted that the Vis Accele- ratrix will cease : but then if the Vis acceleratrix does cease in this place it must likewise be granted that its quantity is less than in places nearer the middle of y® Vibration where it does not cease, & of consequence its quantity will not be proportionable to the distance of the Lineola from ttie middle of the Vibration, for to be pro- portionable it ought not to cease in the beginning of the Vibration, but on the contrary it should be greater there than in any other place, & if it be greater there than in any other place the Medium will not then be uniform. NEWTON AND COTES. 53 This consideration was to me the occasion of altering the Proposition. By making the middle of the Vibration the loctts primus I saw this inconsistency might be avoided. But besides this, it appeares altogether reasonable upon other accounts that the loctis primus should be the middle of the Vibration. Suppose a Musical Chord to be put into motion ; tis certain its locus primus is the middle of its Vibration & consequently also y® locus primus of any lineola Pliysicd of Air which is contiguous to the Chord is in the middle of its own space of Vibration ; for the motion of this Lineola Physica follows & depends upon the motion of the contiguous Chord. And for the same reason, a second Lineola Physica not contiguous to the Chord but to the first Lineola will have its locus primus in the middle of its own Vibration, since its motion depends upon the first as the first did upon the Chord it self; & the same may be said of other Lineolce which are yet more remote from the Chord. Now assoon as the motion of the Chord ceases in its locus primus ie, in the middle of its Vibration, though it should perhaps be said y* the motion of the first Lineola would not cease of it self at the same time with it, yet tis evident it will be made to cease by the resistance of the Chord, for being contiguous to the Chord when it is arriv'd at its locus primus or the middle of its Vibration it can proceed no further towards the Chord whilst y* Chord maintains its rest, & it cannot return back again from the Chord as having no Vis Acceleratrix or acquired Impetus that way. And as this first Lineola ceases by y® resistance of y® Chord, so y® second ceases by y® resistance of y® first, & so on. By this You ^vill understand how I would alter the Corollary; but I chose rather to refer it to Your self, as fearing I could not express my thoughts with sufficient clearness & brevity & exactness at y® same time. What I have represented above is not so exact as it should be, for y® motions of the Lineolce must be supposed gradually to 64 CORRESPONDENCE OP cease with the motion of the Chord ; but I chose to ex- press my self as I have done that You might the more clearly understand me. In altering the Proposition I altered the 4*^ line of Page 366 by putting P/, Pw, Pn instead of Pn, Pm, PI; & in the 2^ line of Page 367 in- stead of [ob brevitatem pulsuum] I have put it [ob angus- tos limites vibrationum] for it would be truer & more to the purpose to say ob magnam pulsuum di8tantia\m} than to say oh brevitatem pulsuum. In Your Example taken from M' Sauveur the latitude of the Pulse is about 10 foot, when perhaps y® space of Vibration is not above y* 10^^ of an Inch at y® utmost. If You consent to my Alteration of the Proposition the Figure must be altered. I propose to have it cut like y® Figure I sent You, which docs better express the disproportion of y® breadths of y® Pulses & Vibrations than the former Figure. I am S^ Y*- &c. LETTER XXXI. COTES TO NEWTON. S"" Sept. 4*»» 1711 I received a Letter from you about a Month ago, & sent You an Answer to it the next day by y® Carrier, in which I gave You my reasons why I was not yet satisfied as to y' Inconsistency in the 48*^ Proposition & its Corol- lary which I formerly mention'd to You. I have not heard from You since y* time, & therefore I fear that either my Letter or Your Answer to it has miscarried. I shall be glad to know Your resolutions concerning this 48^^ Propo- sition assoon as You have leasure that the Press may go on. There were some things relating to the 3"^ Book in my former Letter, I hope You will not forget to let me know Your mind concerning them also. I am S-- Y^ &c. NEWTON XND COTES. 55 Newton's occupations at tho Mint (soo note J p. 37) coupled with liis duties as President of the Royal Society will probably bo sufficient to account for his not having had leisure to attend to tho two preceding letters until after the lapse of 5 or 6 months (Feb. 2, 1712, tho date of the next letter). The following dates will give ua a glimpse or two of him during some of these months. October 16, 1711: "The President |of tho Royal Society | ap- pointed a Council {a meeting of the Committee of Visitors of Greenwich Observatory} to be called on Friday come sevennight (the 26th) when Mr Hunt is ordered to desire Mr Flamsteed to meet the Council on that day at 11 o'clock, at their house in Crane Court in Fleet Street; to know of him if his instruments be in order, and fit to carry on tho necessary celestial observations." (Baily's Flamsteed, p. 96, 97 note). Three accounts of this meeting from Flamsteed's pen are extant, (Daily, p. 96, 228, 294), which bear painful marks of his unhappy temper soured by the mortification he felt at having a board of Vieitors " set over him." Jan. 31, 171 j}. Leibniz's 2nd letter to Dr Sloano (dated 29 Dec. 1711), complaining of " Keill's unfair dealing with him in his last letter, relating to the dispute between him and Sir Isaac Newton, was read : the letter was delivered to the President to consider tho contents thereof." (Journal Book of Royal See.) Tliis letter, in which Leib- niz, speaking of tho obnoxious passage in the Leipsic Acts for Jan. 1705, in tho review of Newton's tract De Quadratura, says " in illis circa banc rem quicquam cuiquam dctractum non rcperio, sed potius passim suum cuique tributum," led to the appointment of a Committee (March 6, 171 2)* to inspect the letters and papers relating to the sub* ject, who delivered in their Report, Apr. 24. — A great part of Cotes's correspondence with Jones falls within this interval (letters cm — ex) and may bo conveniently read hero as con- tributing towards filling up tho blank. — With the next letter the correspondence begins to be carried on with briskness. In a letter of Saunderson to Jones, March 16, 17^2, (Mac- clesfield Corrcsp. i. 264, where it is printed out of its chronological place,) a postscript adds that " Sir Is. Newton is much more intent upon his Principia than formerly, and writes almost every post about it, 80 that we are in great hopes to have it out in a very little time." 56 CORRESPONDENCE OP LETTER XXXII. NEWTON TO COTES. S' London 2^ Feb. ITli I have at length got some leasure to remove the diffi- culties w*^^ have stopt the press for some time, & I hope it will stop no more, ffor I think I shall now have time to remove the rest of your doubts concerning the third book if you please to send them. In reveiwing yo' letters I do not see but that y® xlviii*^ Proposition of the second Book with its Corollary may stand, ffor the particles of air go from their loca prima with a motion accelerated till they come to the middle of the pulses where the motion is swiftest. Then the motion retards till the particles come to the further end of the pulses. And therefore the loca prima are in the beginning of the pulses. There the force is greatest for putting y® particle into motion if any new pulses follow. But if no new pulse follows the force ceases & the particle continues in rest. In this Proposition pag. 366. lin. 12, this emenda- tion may be made. Quare cum sit LN ad KII ut IM ad radium OP, et KII ad EG ut circumferentia PHShP ad OP X BC BC; id est (si circumferentia dicatur % et dica- OP X BC tur F*,) ut OP ad seu OP ad V. Et ex a?quo LN ad EG ut IM ad V: erit expansio partis EG, punc- tive physici F, in loco €7, ad expansionem mediocrem quam pars ilia habet in loco suo prime EG ut V - IM ad Finitu, utq: V + im ad Fin reditu. Vnde vis elastica puncti F in loco ey est ad vim ejus elasticam mediocrem * Cotes did not adopt the part where t is brought in, but printed it as he proposed in his Letter of June 23, "(si ponatur V pro radio circuli circumferentiam habcntis xquulem intervallo pulsuum BC), &c." His suggestion of "ob angustos limites vi- brationum," (Letters June 23, July 30,) of which Newton takes no notice, is also introduced into the 2nd Ed. NEWTON AND COTES. 57 in loeo EG ut -- — j^^ ad — in itu, in reditu vero ut F— IM V 1 1 ad — . Et eodem argumento vires elastica) &c V + im V See lin 27. You stuck at a difficulty in the third Proposition of the third Book. I have revised it & the next Proposition & sent you them inclosed* as I think they may stand. What further Observations you have made upon the third Book or so many of them as you think fit if you please to send in yo*" next Letters, I will dispatch them out of hand. I shall be glad to have them all because I would have {the} third Book correct. I am Yo' most humble Servant For the R''^M.^CoTE3, Professor of Astro^ Is. Newton nomi/t at his chamber in Trinity College in Cambridge. LETTER XXXin. COTES TO NEWTON. I have received Your Letter & as to the buisness of sounds, I do intirely agree with You upon considering that matter over again. By Your alteration of y* 3** Prop : of y® 3^ Book, it is now very intelligible. What I have observed concerning the remaining part of Your Copy I will send You in the most convenient order I can. I begin with the 37*^* Proposition, in the 3** section of which You have these words [Eo autem tempore Luna distat a Sole \5^\qt, circitcr. Et Sol in hac distantia minus • A folio sheet, Nos. 127—129. To the 4th Proposition, a Scholium beginningf " Pkartut mensurando arcum, &c.*' is subjoined, which is a modification of what he had previously sent down in the second instalment of his copy of the Principia^ Sept. 13, 1710. He afterwards, (Letter XL!.), determined on omitting this Scholium, and placing it after Prop, xxxvii. Eventually, however, part of it was transferre«^ 44', 12«^'' C, l?^' 19' & 19^' 48' ad boream. Et excessus longitudinis Penduli auxerint. Observavit utiq : D. Picartus quod virga ferrea, qure tempore hyberno ubi gelabant frigora erat pedis unius longitudine, ad ignem calefacta evasit pedis unius cum quarta parte lineaj. Deinde D. de la Hire cum dua- bus tertiis partibus linesa. In priore casu calor major fuit quam in posteriore, in boc vero major fuit quam calor externarum partium corporis humani. Nam metalla ad solem sestivum valde incalescunt. At virga penduli quam hyberno, sed excessu quartam partem linese unius vix superante. Proinde differentia ilia prodiit baud minor quam 1^^- linea;, baud major quam 2^ linearum. Et inter bos limites quantitas mediocris est 2-^ linearum. Propter calores locorum in Zona torrida negligamus -^^ partes linea? et manebit differentia duarum linearum. Quare cum differentia ilia per Tabulam prtecedentem NEWTON AND COTES. 87 cx hypothesi quod Terra ex materia uniformiter clensa constat, sit tantum l yS^ij* lincaa: cxecssus altitudinisTerrje ad ajquatorem supra altitudincm ejus ad polos, qui erat milliarium 17 J, jam auotus in rationo difFerentiarum, fiet milliarium 31 ^f* ^am tarditas Penduli sub iEquatore defectum gravitatis arguit; et quo levior est materia eo major esse debet altitudo ejus ut pondere suo materiam sub Polis in ajquilibrio sustineat. Hinc figura umbne Terne per eclipses Lunas determi- nanda, non erit omnino circularis sed diameter ejus ab oriente in occidcntem ducta, major erit quam diameter ejus ab austro in boream ducta, excessu 55" eirciter. Et parallaxis maxima Luna? in Longitudinem paulo major erit quam ejus parallaxis maxima in Latitudinem. Ac Terra» semidiameter maxima erit pedum Parisiensium 197(54030, minima pedum 1J)()09860 & mediocris pedum 19686945 quam proximo. Cum gradus unus mensurante Picarto sit hexapedarum 57060, mensurante vero Cassino sit hexapedarum 57^^92 : Buspicantur aliqui scu parte duodecima diametri Lunto. Quibus omnibus experientia contrariatur. Certe Cassinus, definiendo gradum unum esse hexapedarum 57292, medium inter mensuras suas omnes, ex hypothesi de «;qualitate graduum assumpsit. Et quamvis Picartus in Gallioi limite boreali invenit gradum paulo minorem esse, tamen Nor- woodus noster in regionibus magis mensurando majus intervallum, invenit gradum paido majorem esse quam Cassinus invenerat. Et Cassinus ipse mensuram Picarti ob parvitatem intervalli mensurati non satis certam & exactam esse judicavit ubi mensuram gradus unius per intervallum longe majus definire aggressus. Dificrcntiro vero inter mensuras Cassini, Picarti & Norwoodi sunt propo • Newton liad written 92, but Cotes has altered it to 87. See Cotet'i next letter. t Cotes has druwn a line round the \ and written ,^ by the bide of it. 88 CORRESPONDENCE OP insensibiles & ab insensibilibus observationum errorlbus facile oriri potuere, ut nutationem axis Terraj pra)teream. Pag. 424 lin penult, read 229 ad 228. The rest of the Propositions to Prop, xxxvi, may continue as they are, w^^ y® corrections already sent you. I will speedily send you the corrections of y® xxxvi, xxxvii, & XXXIX Propositions. I am Yo' very humble Servant Isaac Newton. The following is in Cotea's hand. "Maxima 19767630 19688725 Minima 196O982O 197U886 Mediocris 19688725 304036lT Sem-. Sph : iEqu : I97U886 197018O5 Media Mcdianim." LETTER XLIII. NEWTON TO COTES. S' London Apr S*** 1712. I sent you by D"^ Bently my emendations of the 19*^ & 20*^ Propositions, & now send you those of the 36**^ & 37*^ When you have perused them I should be glad to have your thoughts upon them, & if any thing else want to be corrected before you come at y® 39*'' Proposition. In my next I intend to send you my emendations of that Propo sition. I am Yo*" most humble Servant For the R°** M' Cotes Professor of Astro- Is. Ne WTON. nomyy at his cTiamber in Trinity College in Cambridge. All that is preserved of the emendations of Prop, xxxvi. is contained in a small slip of paper (No. 192); it relates to the Corollary and is as follows : NEWTON AND COTES. 89 " In Prop. XXXVI. pag. 4G4 lin. 3, read 85820 ; & lin. 9 road, et digitorum undecim cum trieuto, Est enim hwc mcnsura ad mensuram pedum 85820 ut 1 ad 44038." Cotes, however, afterwards (letter of Apr. 20) altered the numbers in tlio Corollary otherwise, and the changes together with his other suggestions were approved of by Ne>vton (letter of May 10). The emended fonn of Prop, xxxvii. coincides with Newton's pre- vious copy (a transcript of which Cotes sent him Feb. 28), as far as the middle of the 2nd paragraph except that " carundem* appears in the right gender. It is not necessary therefore to print that common part again, but it will be sufficient to begin our transcript at the point where the first correction shews itself. (Nos. 195-198) ... "Si maxima harum altitudinum differentia sit pedum novem, erit L+Sad L-Sut 20? ad llf seu 41 ad 23. Qua) proportio satis congruit cum priore. Ob magnitudincm astus in Portu BUtoliw*^ observationibus Stunnii magis fidcndum esse videtur, ideoq : donee aliquid certius constiterit, proportionem 9 ad 5 usurpabimus. Cieterum ob aquarum reciprocos motus, a?8tus maximi non incidunt in ipsas Luminarium syzygia?, scd sunt tertii a syzygiis ut dictum fuit, seu proximo tertium Lunie post syzygias appulsum ad meridianum loci, vel potius tertium post tertiam circiter vel quartam a syzygiis horam appulsum ad meridianum loci. ^Estas ct hyems maximo vigent, non in ipsis solstitiis, scd ubi sol distat a novissimis solstitiis decima circiter vel undecima parte totius circuitus, seu gradibus plus minus 35. Et similiter maximus spstus maris oritur ab appulsu Luna? ad meridianum loci ubi Luna distat a Sole decima vel undecima parte motus totius ab a^stu ad lestum, seu gradibus plus minus septendecim cum dimidio. Et Sol in hac distantia minus augct vel diminuit motum maris a vi Lunte oriundum quam in ipsis syzygiis et quadraturis in rationo Radii ad sinum complemcuti distantia* hujus duplicatn? seu anguli graduum 3.5, hoc est, in ratione 1000000 ad 819152; ideoq: in analogia superioro pro ^ scribi debet 0,8191 52 >S'. Sed ct vis Luna3 in Quadraturis, ob Declinationem Luna? ab -^qua- tore, diminui debet. Nam Lima in Quadraturis vel potius in gradu 17j post Quadraturas, tempore /Equinoctiorum, in Doclinatione gra- duum plus minus 22 & 21' versatur. Et Luminaris ab Tlilquatoro Declinantis vis ad mare movcndum diminuitur in duplicata ratione sinus complcmenti Declinationis quamproxime. Et propterea vis Luna) in his Quadraturis est tantum 0,85539968 X. Est igitur Z/ + 0,81952*/S'adO,855399C8/;-0,81952*AS'ut9ad 5. Pra?terea diamctri Orbis in quo Luna abs^j : excentricitate movori 90 CORRESPONDENCE OF deberct sunt ad iiivicom ut 6y ad 70 (per Prop, xxviii,) ideoq : distantia Lun« a Terra in Syzygiis est ad distantiam ejus in Quadraturis ut 69 ad 70 ca»teri8 paribus. Et distantia ejus in gradu 17 j a 8yzyp;iis ubi a?8tu8 maximuB genoratur est ad distantium ejus in gradu 175 a Quad- raturis ubi cestus minimus generatur ut 83,8317 ad 84',8317, id est, ut 1 ad 1,0119286 vel 0,9882125 ad 1. Undefit l,0119286X + 0,819152/S' ad 0,9882125 x 0,85539908/.- 0,8 19152 >S' ut 9 ad 5. Et aS' ad Z ut 1 ad 45-. Corol. 1. Cum igitur aqua vi Solis agitata ascendat ad altitudinem pedis unius et digitorum undecim cum triento, eadem vi Lunae a«cendct ad altitudinem pedum octo et digitorum novcm. Tanta autcm vis &c. Corol. 2. Cum vis Lunte ad mare movcndum &c. Corol. 3. Quoniam vis Lunaj ad mare movendum est ad Solis vim consimilcm ut 4j ad 1, et vires ilia* (per Corol. 14 Prop. LxviLibr. i) sunt ut densitates corporum Lunto & Solis & cubi diametrorum appa- rentum conjunctim : erit densitas Lunrc ad densitatem Solis ut 4j ad 1 directo et cubus diametri Lunte ad cubum diametri Solis inverse, id est, (cum diametri mediocres apparentes Lunte et Solis sint 31'. 1 6" et 32' 12") ut 49112* ad 10000. Densitas autem Solis erat ad densitatem Terrse ut 100 ad 396 et propterea densitas Lunae est ad densitatem Terrae ut 491 12* ad 396OO sen 31 ad 25. Est igitur corpus Lunaj donsius et magis terrestre quam Terra nostra. Corol. 4. Et cum vera diameter Lunte (ex observationibus Astro- nomicis) sit ad voram diametrum Terras ut 100 ad 365, erit raassa Lunio ad massam Tome ut 1 ad 39i • Corol. 5. Et gravitas accelcratrix in superficio Luna3 orit triplo minor quam gravitas accelcratrix in superficio Terrae. Corol. 6. Et distantia centri Luna? a centre Terne erit ad distan- tiam centri Lunae a communi gravitatis centro Lunie ac Terrse ut 40 s ad 39\\ Corol. 7« Et distantia mediocris centrorum Lunje ac Terrae aequalis erit maximis Terra^ semidiametris 6*0j quam proximo. Nam Terrae semidiametor maxima fuit pedum Parisiensium 19764030. Et hujus- modi semidiametri 60 J- lequantur pedibus 1190782815. Et si htec sit distantia centrorum Solis et Luna?, eadem (per Corollarinm novissimu) erit ad distantiam centri Luna? a communi gravitatis centro Luna? ac Terra; ut 40^ ad 395-, quae proindo est pedum Il6ll6l352. Et cum Luna revolvatur respectu fixanim diebus 27 horis 7 & minutis primis 435-, sinus versus anguli quem Luna tempore minuti unius primi motu 8U0 medio circa commune gravitatis centrum Lunae ac Terrae describit • The last two figures are altered by Cotes to 51. The " n" in " sint" (lin. 18), the "2" ill lin. 37 & the •* 5" in lin. 6 (p. 91) seem also clue to him. NEWTON AND COTES. 91 est 1,275235, existonto Radio 100,000000,000000. Et nt iUdius est ad hunc einiim vcrsum ita sunt pedes II6II61 35 ad pedes 14,807536. Luna igitur vi ilia qua rctinetur in orbe, tempore niinuti unius priini cadendo describeret pedes 1 4,807536. Et lure vis (per Corol. Prop, iii est ad vim gravitatis nostra; in orbe Lunie ut 177S ad 178^; proindeq : corpus grave in orbe Lunoe ad distantiam pedum 1190782815 a centre Terra% vi gravitatis nostra) in Torram cadendo, tempore minuti unius primi describeret pedes 14,8J)08, & ad sexagctumam partem distantiw illius, id est ad distantiam pedum 1984638 a ceutro Terrtie, vi gravitatis in Tcrrara cadendo tempore minuti unius secundi describeret etiam pedes 14,8908, ot ad distantiam pedum 1 9694278 a ccntro Terne cadendo eodem tempore minuti unius secundi describeret pedes 15,1217 seu pedes 15, dig. 1, et lin. 55 . Et bac vi gravia cadunt in supcrficio Terra* in Latitudine urbis Lutetia) Parisiorum, ut supra ostensum est. Et distantia pedum 19694278 paulo major est quam Torrte semidiameter mediocris, et paulo minor quam scmidiameter globi cui Terra requalis est, suntq: diffcrentine insonsibiles ; ac proindo vis qua Luna rctinetur in Orbe suo ad distantiam prtedictam semidiame- trorum 60j-, si desccndatur in Terram, congruit cum vi gravitatis quam experimur in superficie Terrw. Corol. 8. Distantia mediocris centrorum Lunae ac Terras oequalis est mediocribus Terrce semidiametris 60j quamproxime. Nam tot semidiametri mediocres sunt pedum II9IO6OI72. Siquando mcnsuru) graduum in meridiano, longitudes* pendulorum isocbronorum in diverais parallelis Terruj, leges fluxus 8c refluxus marisi diametri apparentes SoHs et Luna', & Luniu parallaxis borizontalis ex pluenomenis accuratius dctcrminatuj fuerint : liccbit calculum huno omnem accuratius repetere." LETTER XLIV. COTES TO NEWTON. S^ Cambridge Aprill y* H*"* 1712 I have received Your Letter by D*" Bcntley & the other which You ^\Tote since. I have sent You two Proof Sheets t for Your re visa!, having made some alterations in them different from Your Copy. In Page 379 line 6 I have put [lin. 2 ■^'] instead of t Ccc,Ddd, pp. 377-392. 92 CORRESPONDENCE OF [lin. 2 J]. In line 10*** 1436,223 instead of 143,6223. In line 21■^ 2177,32 instead of 2177,52 In Page 382 I have put the proportion of 230 to 229 instead of 229 to 288* and altered the latter part of y*' Page accordingly & computed the Table anew in the next Page. The Latter Column supposes the measure of a degree at y* Latitude of 45^. 41 ' to be 57292 Toises as I think You put it in Your Table. The two extreme numbers are as the Cubes of 230 & 229, In y® rest the increment from y® ^'Equator is as the Versed Sine of y® doubled Latitude. In Page 38C lin: penult. 1 j^^^ for l -j-J-g^. Page 387 lin, 1 31 j3^ for 31 ^. Line 11"' I have put other numbers for y® semidiamcters of the Earth, wliich I desire You would examine, since there are different ways of coming at those numbers & I may not possibly have taken that which You like best. Line 21"* I put 95 Miles for 94. Line 27*^ 2'. 46" for 2'. 45, Line 32 Norivoodus noster in regionibus magis borealibus, the word borealibus or some- thing to y* effect was omitted in Your copy In Page 389 : line 20*'^ I have put 8**. 24 for 9**. 34. In the last Period of y® same xxiii** Proposition I have made an alteration which You will see. I think You have much improved the Method of the whole, but there seemes to be a mistake in y* Section of Prop xxxvii which begins with Prceterea diametri Orbis in quo Luna ^c. The Moons force in her Syzygies & Quad- ratures should be increased & diminished in the tripli- cate proportion of those distances to her mean distance reciprocally, Your correction is nearly according to y® duplicate proportion. I am streightned in time at present, & will explain myself more fully in my Next Your most humble Serv* R C • Slip of the pen for 228. NEWTON AND COTES. 93 LETTER XLV. COTES TO NEWTON. S**. Cambridge April IB*** 1712." I hope You have received the sheets sent You hy the Carrier for Your examination, with my Letter. I come now to the xxv*^ Proposition wliich I think were better to end thus ad dies 3G5. G^ 9'. id est, ut 1000 ad 178725 seu 1 ad 178 |{J. Unde ex proportione linearum TM, ML, datur etiam vis TM: & ha) sunt vires Solis quibus Lunso motus perturbantur. q.e.i. The two Periods which are left out may be removed to Prop: xxxvi for I think they are of no use till we come to that Proposition. If You remove them I suppose You will at the same time alter them, by putting in line 14*^ instead of y® proportion of 6ol* to 60 the proportion of 40 J- to 39-J-, if this be the Proportion which may at last stand in Corol. G^^ of Prop. xxxviA Now because the Proportion of 40 -J- to 39 } is made out in y® xxxvii*^ Proposition, the xxxvi & xxxvii*^ Propositions ought to change places, but this they cannot do because the xxxvii*^ does in other respects depend upon y® xxxvi^*^. Whence it appeares that there ought to be a further alteration in y® Form of these Propositions, that the former may not depend upon the latter. This may easily be done & I think the whole would be clearer & more !RIethodical if in y*" former Proposition the Problem were to find neither y® force of y® sun nor the force of y® moon, but only their proportion to each other, & in y® latter the Problem were to find the proportion of both forces to y* force of Gravity. And thus y^ 3^ 4}\ 5'\ G'\ r\ & 8*^ Corollarys of y® xxxvu*^ will belong to y® former, & the Corollary of y® xxxvi*^ together with the 1"* & 2^ corollarys of y* xxxvii*^ will belong to y® latter. There will be this • It should be ^. 94 CORRESPONDENCE OF further advantage in the change, That in y** 7^** Corollary of y" XXXVII*'' which will then be annex'd to y® former Proposition a good foundation may be laid for making out y* latter. In my Letter which I yesterday wrote to You I was somewhat in haste, I just mention'd a difficulty in Prop : XXXVII. Let iST be the Moons distance from y® Earth when she is 17°^ from her Syzygies & QThe her distance at 1?°^ from her Quadratures St MT her mean distance in y^ Octants. I think the force of y® Moon must be increased at S in the proportion of MT cub to *S'7' cub, & diminished at Q in the proportion of MT cub to QT cub. Your last corrections increase it at S in y® proportion of QT' to ST, which is nearly in the proportion of MTq to ST quad, & diminish it at Q in y® same proportion. I could wish when the whole is settled that the proportion of 4 J- to 1 may be retain'd for the sake of Proposition xxxix.* I think there is no Proposition in Your Book which does more deserve Your care. LETTER XLVI. NEWTON TO COTES. S"^ London Apr. 22. 1712. I have run my eye over the two proof sheets & approve yo' corrections. The sheets may be printed off. The xxv*^ Proposition may end thus. — ad dies 365, 6^' 9', id est ut 1000 ad 178725 seu 1 ad 178 |-g-. Invenimus autem in Pro- positione quarta quod, si Terra et Luna circa commune gravitatis centrum revolvantur, earum distantia mediocris • "Invenire Prnccessionem -i'quinoctiorum." NEWTON AND COTES. 95 ab invicem erit Co^ semidiametrorum mediocrium Tern© quamproxime. Et vis qua Luna in Orbe circa Tcrram quiescentem ad distantiam semidiametrorum 60 revolvi posset {est ad vim qua eodem tempore ad distantiam semidiametrorum 60 revolvi posset*,} ut 60^ ad 60 & hsoc vis ad vim gravitatis apud nos ut 1 ad 60 x 60. Ideoq: vis mediocris ML est ad vim gravitatis in superfieie TerroD ut 1 X 6iU^ ad 60 X 60 X 60 X 178 1 g, seu 1 ad 638092,6. Undo ex proportione linearum TM, ML, datur etiam vis TM, Et ha3 sunt vires Solis quibus motus Lunno perturbantur. q, e. i. I here referr the summ of y® forces uponf the Sun upon the earth & ^loon to the ]Moon alone & therefore consider the earth as resting & referr its motion to the Moon. I am satisfied that the force of the ^loon upon the Sea is in a triplicate ratio of her distance reciprocally & have altered the calculations accordingly, w^^ I send you in the inclosed paper together with the emendation of the SQ*** Proposition. T am Yo"^ most humble Servant For tlie IC'^W Cotes, Professor of Is. Newton. Astronomy, at his Chamber in Trinity College in Cambridge. The " inclosed paper" mentioned at the end of this letter is a folio sheet (Nos. 202,203,208), and contains Newton's further corrections of Prop. XXXVII. called for by the two preceding letters, and also those of the 39th Prop, which he had promised in his letters of Apr. 3d and 8th, (compare letters of March 18, Feb. 19, Feb. I6, and Feb. 12). It is not necessary to copy the whole of what relates to Prop, xxxvn. Every useful end will be answered by giving only those parts of it where it differs from the copy which Newton had recently sent (Apr. 8), leaving blanks to represent what is common to the two. The paper begins as follows : ** In Prop xxxvii read Ca'terum ob aquarum reci- • I have added the words between braces from the 1st Ed. The identity of temii< nation of the two clauses with " revolvi posset," combined with a little hurry in tran- scribing, will readily account for their omission. t This should be " of." 96 CORRESPONDENCE OF procos motu8...8eu proximo sequuntur tertium Luna;...vol potius (ut a Sturmio notatur) sunt tertii post diem noviliinii vel plonilunii, scu post lioram a novilunio vel plenllunio plus minus duodecimam, adcoq : incidunt in boram a novilunio vel plenilunio plus minus quadragcaimam tertiam. Incidunt voro in hoc portu in lioram septimam circitcr ab appulau Lunne ad meridianum loci, ideoq : proximo scquuntur appul- 8um Lunne ad meridianum ubi Luna distat a Sole vol oppositione Solis gradibus plus minus octodecim vol novcndccim in consequentia. iTlstas . . . Sol distat a solstitiis dccima circitcr parte totius circuitu)? seu gradibus plus minus 30 vel 37 ... a Solo dccima circitcr parte motus totius ab fpstu ad a}stum. Sit distantia ilia graduum plus minus 18z}. Et vis Solis in liac distantia Luute a syzygiis & quadraturis, minor erit ad augcndum et ad minuondum motum . . . scu anguli graduum 37, hoc est, in rationc 10000000 ad 798()3.'55. Idcoq : . . .debet 0,7980.355/9. ...in gradu 18^ post Quadraturas, in Declinatione graduum plus minus 22.13' vcrsatur est tantum 0,8570328//. Est igitur L + 0,7980355 S ad 0,8570328 L - 0,7980355 ^S* ut 9 ad 5 ut Q9 ad 70; ideoq: distantia.... ca^teris paribus. Et distantia) ejus in gradu 18| a syzygiis ... maximus generatur, & in gradu 18^ a qua- draturis ubi testus minimus generatur, sunt ad mediocrcm ejus distan- tiam ut 09,100082 & 09,899318 ad 6c)^^. Vires autem Luna^ ad mare movendum sunt in triplicata ratione distantiarum inverse, ideoq : vires in maxima et minima barum distantiarum sunt ad vim in medi{o}cri distantia ut O,98280l6 et 1,017342 ad 1. Undo fit 1,017342 Z + 0,7980355 aS' ad O,98280l0 x 0,8570328 Z - 0,7980355 /S' ut 9 ad 5. Et X = 4,4824 S. Corel. 1 & 2, as before. Corol.3. ... ut 4,4821. ad l...ut 4,4824 ad 1 ... sint 31'l0"i... ut 4892 ad 1000 ... ad densitatem Terrw ut 4892 ad 39OO seu 21 ad 17. Est igitur ... Corel. 4. ... ad massam Terra? ut 1 ad S^^SQS, Corel. 5. . . . erit quasi triple minor. . . Corel. 0. . . . ut 40,303 ad 30,^63, Corel. 7' Et mediocris distantia centrj Lunae a centro Terrae erit semidiametrorum maximarum Terrae 60J quam proximo. Nam semi- diameter maxima Terra) fuit pedum Parisiensium I97O7O3O, et medi- ocris distantia centrorum Terra) et Lunae ex hujusmodi semidiametris Go J constans, a?qualis est pcdlbus 1190999707. Et ha?c distantia (per Corollarium superius) est ad distantiam . . . centro Terrao et Lunas ut 40,303 ad 3%3G3, qua? proinde est pedum llOl 492740. Et cum Luna . . . centrum Terras et Luna? describit est 1275235, existente . . . pedes llOl 492740 ad pedes 14,811702. Luna ... in Orbe, cadendo in Terram, tempore minuti unius primi describet 1 Jes 14,811702. NEWTON AND COTES. 97 Et 81 ha?c vis augeatur in rationo 1775 ft^ 178S liabebitur vi« tota gravitatis in Orbe Luna? per Corol. Prop. iii. Et hac vi Lima cadendo, tempore minuti* iiniiia primi dcscribero debcrot pedes 1 4,89.') IS. Et ad sexagesimam partem hiijus distantiw, id est, ad distant iam pedum 19849995 a eentro Terra? corpus grave cadendo, tempore minuti unius secundi describere deberct etiam pedes 14,89513. Diminuatur ha»c distantia in subduplicata ratione pedum 14,89513 ad pedes 15,12028, et habebitur distantia pedum 19701651 a qua grave cadendo, codem tempore minuti unius secundi describet pedes 15,12028, id est \Hn\ca 15, dig 1, lin 5,32. Et hac vi ... urbis Lutethn Parisiorum^ lit supra ostensum est. Est autem distantia pedum 19701651 paulo minor quam semidiametor globi huic Terne wqualis et paulo major quam Terrae hujus semidiamctcr mediocris ut oportet. Sed differentia sunt insensibiles. Et propterea vis qua Luna ... ad distantiam maximanim Terras semidiametrorum 60 J, ea est quam vis gravitatis in superficio Terras requirit. Corol. 8 centrorum Terrae et Luna? est mediocrium Terr® semidiametrorum 60^ quam proximo. Nam semidiameter mediocris qucB erat pedum 19688725 est ad semidiametrum maximam pedum 19767630, ut 60^ ad 60^ quamproxime. In liis computationibus attractionem magneticam Terras non con- sideravimus, cujus utiq : quantitas perj)arva est et ignoratur. Siquando veroha?c attractio invcstigari potent, et mcnsura graduum in merldiano, ac longitudines . . . parallelis, legesq : motuum maris, & parallaxis Lunro cum diametris apparentibus Solis et Luna? ex pha?nomenis ..." The following are the corrections of the 39tli Prop. " In the xxxix'*" Proposition pag 470 lin 23 write r- id est (cum Terra) diameter minor PC vol aCsit ad diametrum majorem AC wi 229 ad 230,) ut 52441 ad 459; si annulus iste Terram secundum /Equa- torcm cingeret & utcrq: simul circa diametrum annuli revolveretur, motus annuli esset ad motum globi interioris (per hujus Lemma iri) ut 459 ad 52441 et 1000000 ad 925275 conjunctim, hoc est, ut 4590 ad 485223, ideoq: motus annuli esset ad summam motuum annuli ac globi ut 4590 ad 489813. Vndo si annulus globo adhaereat, & motum Buum quo ipsius Nodi seu puncta a^quinoctialia regrediuntur, cum globo communicet: motus qui restabit in annulo erit ad ipsius motum priorem ut 4590 ad 489813; et propterea motus punctorum aequinoctialium diminuctur in eadem ratione. Erit igitiir motus an- nuus punctorum ajquinoctialium corporis ex annulo et globo compositi • " minuti " here 6c " quam " p. 98. lia. 20 have been added by Coles, who has made a number of other alterations in the MS., the principal of wliich are mentioned in Letter XLVllI. 7 98 CORRESPONDENCE OF ad motum 20«* 11' 46", ut 1436 ad 39343 ot 4590 ad 489813 con- jiinctim, id est, ut 100 ad 292368. Vires autem quibus &c. Pag. 471 lin 19 write atq: adeo ad movenda puncta osquinoc- tialia ovaderet minor quam prius in rationo 2 ad 5. Ideoq : anniuis iEquinoctiorura rcgressus jam esset ad 20«' 11' 46" ut 10 ad 73092, ac proinde fieret 9" 56"' 50' ^ Cwterura hie motus ob inclinationom plani iEquatoris ad planum Ecliptica} mimicndu8 est, idq: in ratione sinus 91706 (qui sinus est complementi graduum 23 J) ad Radium 100000. Qua rationo motus iste jam fiet 9". 7"'. 20'*'. Ilajc est annua Prajcessio i^quinoctiorum a vi Solis oriunda. Vis autem Luna) ad mare movendum erat ad vim Solis ut 4,4824 ad 1 circiter. Et vis Lunss ad iEquinoctia movenda est ad vim Solis in eadem proportione. Indeq: prodit annua ^quinoctiorum Pracessio a vi Lunoe oriunda 40" 53"' 22", ac tota Proecessio annua a vi utraq: oriunda 50" . 00"' . 42' '. Et hie motus cum phaenomenis congruit. Nam Praecessio requinoctiorum ex Observationibus Astronomicis est minutorum secundorum plus minus quinquaginta Si altitudo Terrte ad iEquatorem superet altitudinem ejus ad Polos milliaribus plus quam I7J, materia ejus rarior erit ad circumferen- tiam quam ad centrum : et Prajcessio jEquinoctiorum ob altitudinem illam augeri, ob raritatem diminui debet. Descripsim usjam Systcma Solis, Terra?, Lunro, et Planetarum: superest ut do Cometis nonnulla adjiciantur." LETTER XL VII. COTES TO NEWTON. I have received Your last, but have not yet had time to try the Calculations of the inclosed sheet. I am satisfied as to the xxv*^ Proposition, upon reconsidering it. In Page 441, lin: 25, the first & last numbers are 368682 & 362046: they should be 368676 & 362047. The iEqua- tion* which results from hence will be 88487,19 - 12307251,44.r+75578,14.T?^-5082017,44cP''+ 42456,19** = 0, • The following is on a separate piece of paper, (No. 209) : iEquatio fit 88487,19 - 12307251,44 r + 75578,14ii - 5082017,44x« + 42456,19 jc* = 0. Inde X = 0.00719, CT = 1,00719, AT = 0,99281 adeoq : CT ad AT ut 70,041 ad 69,041, sive ut 70.i ad 69,-, vol 70.-3 a^ ^^h* Vera Radix ilerato examine est, 0071900057 ter exam : NEWTON AND COTES. 99 of which I find the Root to be 0,0071900057. If You approve of it I would alter the bottom of the Page thus [obtinetur w roqualis 0,007 19» & inde semidiameter CT fit 1,00719 & semidiameter AT 0,99281, qui numeri sunt ut 70^ & 69^ quam proximo. Est igitur distantia Luna) a Terra in Syzygiis ad ipsius distantiam in Quadraturis (seposita scilicet Eccentricitatis consideratione) ut C9^ ad 70^ vel numeris rotundis ut 69 ad 70] This will cause an alteration in the xxix*^ Proposition & in the xxxi"*, page 450. I have not computed the alterations for the xxix**"*, not knowing whether You will chuse the whole numbers 69 and 70 or the fractions 69^^ & 10-^. As for the other place in page 450*^ I took the numbers 69 & 70 that I might find what alteration would arise in the conclusion of y® xxxii** Proposition. The result of my computation is as follows. Pag: 450. lin: 18 [69 ad 70] Lin: 20, [si capiatur angulus 16". 2l'". s'\ 30*] Page 452**. lin: 5, [erat 32". 42'". 7''] Lin: 8, [illud est 17'". 43'". 11"] Lin: 10, [relinquit 16". 16"'. 37' ^ 42*'] Page 453, Lin: 22, [fit 39^ 38'. 7". 50'"] Lin: 23, [l9^ 49'. 3". 55""] Lin: ult: [seu S9fiS55'\ Page 454, Lin: 3, [id est, ut 9,0827646 ^ T^] Page 455, Lin: 4 [prodibit 0,ll88502]t Lin: 6, [est 1°. 29'. 58". 3'"] Lin: 7 [subductis relinquit 18°. 19'. 5". 52'"] Lin: 9 [relinquit 341°. 40'. 54". 8'"] Lin: 12 [qui propterea erit 19°. 18'. 1". 22'"] In finding the Number 0,1188502, 1 supposed y* ordinate e Z to bisect y** base NT by which meanes the series for y® Area TZeF converged quicker than the other for the Area Ne Z, so y* on account of this Latter I would not depend upon the last figure 2, I think the other are right. • Tliese alterations of Prop. xxix. form the subject of Letter L. X This correction, though approved by Newton, was subsequently modified (as also the four following corrections which depend upon it), 1 he result wliich w substituted for it in the 2nd Ed. leads to the value .1188496 for the area of the curve f^eVn : in the Ist Ed. it is .1188478. 7—2 100 CORRESPONDENCE OF In Line 14*^ You have 19°. 20'. Si", i'" from Flamsteeds Tables. By Your Theory in D' Gregory tis 19^ 2l'. 22". 3'"*. So in the following Proposition, page 45G. Lin 13 You have 9^ lO'. 40" ; by Your Theory tis 9°. n'. 3". There will need some other alterations in Prop, xxxiu** & its Corollary upon account of those in the preceding Proposition. You seem to depend too much upon Your Readers quickness when you say [ut rem perpendenti con- stabit] I hope when You review the whole You will make it easier to apprehend the agreement of the two Con- structions. I do not rightly understand line 12*^ of page 458 [Inclinationis autcm Variatio tantum augebitur per decre- mentum sinus IT, quantum diminuitur per decrementum motus Nodorum] I think I had observed nothing further before we come to y® XXXV i**^ Proposition. I am, S', Your most Humble Servant Trinity College Apr. 24**^ 1712 Eoger Cotes. For S' Isaac Newton at his House in St Martin's Street in Leicester Fields London. LETTER XLVIII. COTES TO NEWTON. S'. April 26*'* 1712. I have examin*d your last Emendations f of the xxxvii*** Proposition. I am very glad to see the whole so perfectly • Newton, in his next letter, adopts this correction and the followingf one. After- wards, however, (Letter LIT.) apparently forgetting that he had already given direc- tions about them, he orders 19*. 21'. 20". 45'" to be written in p. 455, and 9°. 11'. 3" in p. 456. Cotes, in his reply, (Letter LIII.) proposes to write 190.21'. 21". 50'" in p. 455, which Newton approves, (Letter LV.) Flamsteed's Tables here referred to, are printed at the end of his Doctrine of the Sphere^ London, 1680. t Sent in the Letter of Apr. 12. NEWTON AND COTES, 101 well settled & fairly stated, for without regard to the conclusion I think y* distance of 18^ degrees ought to be taken & is much better than 17^ or 15\ & the same may be said of y* other changes in y* principles from which the conclusion is inferr'd. In examining Your Numbers I found it necessary to alter most of them, I here send you others {instead of them \ for your approbation. Proeterea diametri Orbis in quo Luna...... sunt ad mediocrem ejus distantiam ut 69,098747 & 69,897345 ad 69^, Vires autcm Lunnc.ad vim in mediocri distan- tia ut 0,9830427 ct 1,017522 ad 1. Undo fit 1,017522 L + '0,7986355 S ad 0,9830427 x 0,85703271* -0,7986355*9 ut 9 ad 5. Et S ad L ut 1 ad 4,4815. Itaq: cum vis Solis sit ad vim gravitatis ut 1 ad 12868200 vis Lunoe erit ad vim gravitatis ut 1 ad 2871400. CoroL 1. Cum igitur* aqua vi Solis agitata asccndat ad altitudinem pedis unius & undecim digitorum cum octava parte digiti, eadem vi Lunio ascendet ad altitudi^ ncm octo pedum & digitorum octo. Tanta autem vis — Corol. 2. Cum vis Luna3 ad mare movendum sit ad vim gravitatis ut 1 ad 2871400 — Corol: 3. Quoniam vis Lunai ad mare movendum est ad Solis vim consimilem ut 4,4815 ad 1 et 32'. 12") ut 4891 ad 1000. Densitas autem Solis ad densitatem Terra) ut 4891 ad 3960 seu 21 ad 17. Est igitur Corol: 4 ad massam Terro9 ut 1 ad 39, 371. Corol: C ut 40,371 ad 39,371. • The word " igitur** is omitted in the 2nd Ed,, neither does it appear in Newton's first copy of the Prop, whicli k given at the end of Letter XXXIX. After the words " digitorum octo," the sentence is continued as follown in the 2nd Ed., "& vi utraque ad altitudinem pedum decem cum semisse, & ubi Luna est in Perigajo ad altitudinem pedum duodecim cum semisse & ultra, praisertim ubi ii-^stus ventis spirantibus adjuvatur. Tanta autem vis " corresponding: to Newton's copy just referred to. Cotes's omission of these words in this draught of his letter probably arose from the fact of Newton*s having omitted the passage in the emendations sent in his Letter of Apr. 8. 102 CORRESPONDENCE OF Corol: 7 ut 40,371 ad 39,371, quce proinde est pedum 1161498340 ita sunt pedes u6l4i.8340 ad pedes 14,811833 Et hac vi Luua eadendo, tempore minuti unius primi describere deberet pedes 14,89517 et ha- bebitur distantia pedum 19701 678 a qua grave eadendo, eodem tempore miimti unius secundi describet pedes 15,12028... In the XXXIX*'' Proposition. Vis autem Lunae ad mare movendum erat ad vim Solis ut 4,4815 ad 1 eirciter Prsccessio a vi Lunae oriunda 40". 52'". 52'^ ac tota Prse- cessio annua a vi utraq: oriunda 50". oo'". 12'*^. Et hie motus The xxxvi*'' Proposition depends upon the. latter part of the xxv*^, & must therefore stand as in the former Edition. I have altered the Corollary of it thus Corol. Cum vis ."SS^vim gravitatis lit 1 ad 289.,... mensura pedum Parisiensium 85820, vis Solaris de qua egimus, cum sit ad vim gravitatis ut 1 ad 12868200 atq: adeo ad vim illam eentrifugam ut 289 ad 12868200 seu l ad 44527, efficiet ut mensura tantum pedis unius Parisien- sis & digitorum undecini cum octava parte digiti. Est enim ha3C mensura ad mensuram pedum 85820 ut 1 ad 44527. I have altered the xxxviii*^ Proposition thus. Pag: 467. lin: 10 [id est, ut 39,371 ad 1 & 100 ad 365 conjunctim, seu 1079 ad 100. Unde cum mare nostrum vi Lunso attollatur ad pedes 8f , fluidum Lunare vi Terras attolli deberet ad pedes 93 J excessu pedum 187 Your very Humble Servt. R Cotes. NEWTON AND COTES. 103 LETTER XLIX. NEWTON TO COTES. The corrections made in yo' last of Apr. 24*** may all stand. In y* xxix*^ you may use either y* whole numbers 69 & 70 or the fractions 69^^ & 70^. In pag 455 lin 14 & pag 45G I have put the motion of the Nodes of Moon from y® Equinox & should have put it from y® fixt starrs. In y* first place therefore for I9^'20'3l" l'" ^vrite 19^21^22".3"' In y® second for 9°. lO' 40" write 9^ u'. s". In pag. 458 lin 11. write. [Et in eadem ratione minue- tur etiam Inclinationis Variatio.] And strike out the rest to the end of the Paragraph. In y® xxxiii** Proposition, pag 456, instead of y* words [ut rem perpendenti constabit] may be written [ut rem pcrpendenti & computationes instituenti constabit.] And the numbers in this Proposition are to be suited to y* alterations made in y® preceding Proposition as you men- tion. I am London Apr. 24**** Yo"^ most humble Servant 1712 Is. Newton For the R"** M' Roger Cotes Professor of Aatronomrff at his Chamber in Trinity College in Cambridge. LETTER L. COTES TO NEWTON. S' I have received Your last, & taking the whole numbers 69 & 70, the alteration in Pag: 442.t lin. penult, will be • The post mark is Ap. 29. t Prop. XXIX. Invenire Variationem Luni 104 COIIRESPONDENCE OF [68,6877 o-d numerum 69. Quo pacto tangens anguli CTP jam erit ad tangentem motu8 medii ut 68,6877 ad 70, & angulus CTP in Octantibus, ubi motus medius est 45^'. invenietur 44^^ 27'. 28": qui subductus de angulo motus medii 45° relinquit Variationem maximam 32'. 32". Hcec ita se habercnt si & Variatio maxima qua; sccus esset 32 . 32".* jam aucta in eadem ratione fit 35\ l0".t] You go on thus J. Hwc est ejus magnitudo in mediocri distantia Solis a Terra, neglectis differentiis quso a curva- tura Orbis magni majoriq: Solis actione in Lunara falcatam et novam quam in gibbosfi & plenam, oriri possint. In aliis distantiis Solis a Terra, Variatio maxima est in ratione qua? componitur ex duplicata ratione revolutionis Syno- dicae Lunaris (dato anni tempore) directe, et ratione anguli CTa directe, & triplicatii ratione distantia* Solis a Terra inverse ; id est, ex triplicata ratione revolutionis synodicae Lunaris directe et triplicata ratione distantiae Solis a Terra inverse. Ideoq: in Apogwo Solis Variatio maxima est S3'. 11" & in ejus Perigaeo 37'. 24", si modo eccentricitas Solis sit ad Orbis magni semidiametrum transversam ut 16J-5- ad 1000. Ilactenus Variationem investigavimus in Orbe non eccentrico in quo utiq: Luna in Octantibus suis semper est in mediocri sua distantia a Terra. Si Luna propter eccen- tricitatem suam, magis vel minus distat a Terra quam si locaretur in hoc Orbe, Variatio paulo major esse potest vel paulo minor quam pro Regula hie allata : sed excessum vel defectum ab Astronomis per Phaenomena determinan- dum relinquo. I was going to diminish § Your numbers 33'. 1 1", & • 32'. 34" in Newton'fl MS. t 35'. 12" in Newton's IMS. t N08. 149, 150. S In the margin of Newton's MS. (No. 149,) Cotes has actually made this diminu- tion, as he has done above, in the case of the numbers 32'. 34" and 35'. 12" at the end of the extract inclosed within biackets. NEWTON AND COTES. 105 37'. 24" by 2" which is nearly the diminution if those numbers are right, which I am forc'd to take upon trust not knowing how to state the proportion of the Moon's Periodical Revohitions nor consequently of her Synodical in the Apogee & Perigee of y* Sun. But I cannot fully satisfy my self about Your Rule. As I take it, the dupli- cate ratio of y® Synodical revolution of y* Moon & y* simple ratio of y'' angle CTa compose not the triplicate ratio of y* Synodical revolution alone but this triplicate ratio directly & y® simple ratio of y® periodical revolution inversly : the angle CTa being as y® Synodical revolution directly & y® Periodical revolution inversly. I have besides some scruple about introducing y* ratio of y* angle CTa, I have not throughly considered the thing, but I quajry whether it will not be sufficient to make the compounded ratio consist only of y® duplicate ratio of y® Synodical revolution directly & y® triplicate ratio of y® Sun's distance inversly according to y® 11*^ Corol: of Prop: lxvi*^ Lib. 1. I have transcribed y® whole y^ You may review it and order it as You think it should stand. Your &e. May day 1712. R C. In liis answer to this, (May 10,) Newton adheres to the statement that the Variation is proportional to (Moon's synodical period)^ dato anni tempore x / CTa (distance between Sun & Moon)' Cotes then (May 13) further explains his reasons for thinking that tlie / CTa should bo cancelled. Not receiving an answer, he writes again (May 25) to draw his attention to the point, and has the gratification of finding (see letter of May 27) that Sir Isaac has been convinced by liis arguments. 106 CORRESPONDENCE OF LETTER LI. COTES TO NEWTON. I fear I give You too much trouble with my Letters, but I think this will be my last till we come to the Theory of Comets. In the Corollary of the xxxiii'* Proposition I put 16". 19'". 27". instead of 16". 18"'. 41'" i. I am not cer- tain how You would compute that motion, & therefore I mention it to You, I found it by this Proportion: As 19'. 18'. 01". 23'" to 19". 21'. 22". 3'" so 16". 16'". 37". 42" to 16". 19'". 26'". 56'. In Your last letter You order page 458. lin 11. thus. [Et in eadem ratione minueter etiam Inclinationis Variatio] This will cause some alteration in the following Corollarys & in the xxxv*'' Proposition unless You design to consider the Moons Inclination only as moving in Orbe circulari. At the bottom of Page 461 You make use of 5^ 17'. 46" & 5° for the extream Inclinations ; In D"" Gregorys Astro- nomy You have 5\ 17'. 20" & 4^ 59'. 35". AVhich I suppose You find to be more agreeable to observations. In the first Paragraph of y® New Scholium* to Prop; XXXV*'' You have [ad 11'. 5o" circiter ascendit, & additur medio motui Lunaa ubi Terra pergit a Perihelio suo ad Aphelium & in opposita orbis parte suhducitur'] As I take it, the words additur & suhducitur should change places. You have not mentioned how to find this -Equation in every place. In the second Paragraph concerning the Annual Equa- tions of the Moon's Apogee & Node You have forgotten to mention when they must be added & when substracted. In the third Paragraph You say [Per Theoriam gravi- tatis constitit etiam quod actio Solis in Lunam paulo major • See the remarks which follow the Letter. NEWTON AND COTES. 107 sit ubi transversa diameter Orbis Limaris transit per Solem quam ubi eadem ad rectos est angulos cum linea Terram & Solem jungente & propterea Orbis Lunaris paulo minor est in priore casu quam in posteriore] I think it should be [paulo major est in priore] In the fourth Paragraph concerning y* ^Equation of y* Moon arising from y® position of her Nodes which You call Semestris secunday You have [additur vero medio motiu Lunje dum Nodi transeunt a Solis Syzygiis ad proximas Quadraturas & stthducitur in eorum transitu a Quadraturis ad Syzygias] As I apprehend it y® words additur & subduci- tur should change places. The sixth Paragraph I do not understand. The yEqua- tion which You there describe seems to be established not so much from Observations as from the Theory of gravity, but I cannot perceive how it answers Your design ex- press'd in these words. In Perihelio Terra? propter majo- rem vim Solis Apogajum Lunoe velocius movetur in epicy- clo circum centrum D (I suppose it should be centrum C) quam in Aphelio, idq: in triplicata ratione distantia) Terra) a Sole inverse. Ob ajquationem centri Solis in argumcnto annuo comprehensam Apogajum Lunaj velocius movebitur in epicyclo in duplicata ratione distantise Terrce a Solo inverse. Ut idem adhuc celerius moveatm* in ratione sim- plici distantia) inverse, sit &c.* Now the ^Equation which • We will add the remainder of the paragraph from Newton's MS. ( No. 170) : ' eit TD excentricitas primo sequata, et producatur TD ad E ut sit D£ ad TD ut daplutn excentricitatis Solis ad radium Orbis magni seu 33| ad 1000. Capiatur angulus EDF squalis argumento annuo, vel quod perinde est, agantur parallelae TS ac DF solem versus, et sit DF ipsi DK aqualis, et erit DTF xquatio annua apogxi Lunas fi FTS distantia Solis ab apogxo Luniu ter xquata, & TF excentricitas Lunx bis .Tquata in apogxum Luna; ter aquatuin tendens." 108 CORRESPONDENCE OF You describe in what follows does not in the least, as I see, depend upon the Sun's Anomaly but intirely upon y* Annual Argument of the Apogee. You will perhaps more easily perceive my difficulty if I tell You how I think the ^Equation should be stated to answer what was propos'd. Let CTD be y® JEquatio Semestris describ'd in y® preced- ing Paragraph ; produce CD to E, so y* DE may be to CD as 33^ to 1000; make the angle EDF equal to the Sun*s Anomaly, & the line DF equall to DE, & joyn TF: then will DTF be the second annual Equation of y® Apogee & TF be the Eccentricitas Lunse bis aquata in Apogaium Lunas ter roquatum tendens. The following Paragraph concludes thus*. Ducantur • The former part of this parayraph is as follows, (No. 170) : "Per eandem gravi- tatis Theoriam Sol fortius agit in Lunam annuatim ubi apogaeum Lunae et perigaeum Solis conjunguntur quam ubi oppoauntur. Etinde oriuntur aequationes duae periodicae, una medii motua Lunae, altera apogxiejus: quae quidetn acquationes nullae sunt ubi apogacura Lunae vel conjungitur cum perigso Solis vel eidem opponitur, et maximae in apogaeorum quadraturls. In aliis apogaeorum positionibus datam habent proportionem ad iuvicem, suntq : ut sinus distantiae apogaeorum ab invicem. /Equatio prior subduci- tur et posterior additur tibi apogaeum Lunae minus distat a perigaeo Solis in consequen- tia quam gradibus 180; prior voru additur & posterior subducitur ubi distantia ilia fit major, llarum a>quationum quantum sentio, /Equatio maxima apoga;i asccndit ad 15' vel 20' circiter, sed a>quatio maxima motus medii Lunae vix asceudit ad 30", et ob piirvitatem negligi potest donee quantitas ejus ex observationibus determinetur, Pro- ducatur excentricitas Lunaj bis acquata TF ad G ut sit FG sinus aiquationis maximae periodica) apogaii Lunae 15' vel 20' ad radium TF. Ducantur," &c. NEWTON AND COTES. 109 rcctre dusD parallelce TP, FII in Perigroum Solis tendcntcs, vel quod perinde est, capiatur angulus GFJI sequalis distan- tiai Perigaji Solis ab Apogrco Liinro, & sit FII ipsi FG ajqua- lis ; et angulus FT II crit ajquatio Periodica Apoga'i Luna;, & angulus PT// distantia Apogrci Luno) quarto aequati a Perigajo Solis et Til cccentricitas tertio wquata in apo- gseum quarto a*quatum tendens. Instead of which I pro- pose the following alteration, leaving out y® line TP in the Figure. Capiatur angulus GFII aiqualis distantio3 Apogaei Luna) a Perigico Solis in consequentia et sit FH ipsi FG aqualis, & angulus FTII erit yEquatio periodica Apogaii Luna} & Til cccentricitas tertio aquata in Apoga;um quarto a3quatum tendens. This Alteration will agree with what You lay down a little before in the same Paragraph, where speaking of this Periodical ^'Equation of y* Apogee You say additur ubi Apogyeum Lunje minus distat a Pe- rigaio Solis in consequentia quam gradibus 180 & subduei- tur ubi distantia ilia fit major, Which Rule I think is right but not agreable to the conclusion of the Paragraph which I therefore propose to alter. In the last Paragraph but one You say [pono medio- crem distantiam centri Luna; a centro Terra in Octantibus jpqualem esse 6o|^ semidiametris maximis Terrae] I desire to know whether You will here retain 6o^ or put instead of it 6o\ as in Corol 7*^ of Prop xxxvii**> Your &c. May. 3^ 1712 R C. The " New Scholium to Prop, xxxv.** whicli forms so largo a part of the subject of the preceding letter is a Scliolium on the Lunar Theory, containing a statement of the origin and quantity of various Limar Inequalities, and occupying the place of a short Scholium in th« Ist Ed. relative to the motion of the Moon's Apogee. It is written on three sides of a sheet of foolscap (Nos. 109 — 171) which seems to liave been doubled up and placed loosely between the pages of Newton's interleaved copy of the 1st Ed. It was probably sent to Cotes with the third and last division of Newton's copy^tho first week in July 110 CORRESPONDENCE OF 1711. (Letter xxviii and note). The reason why the Scholium appears on folio paper is, no doubt, that there was not room for it on tho quarto leaf in the interleaved copy : that quarto leaf is still preserved, and its first page (No. 190) is headed " Scholium" and is devoted to the opening words of it followed by an " &c." thus : ** llisce motuum Lunarium computationibus ostendero volui quod motus Lanares per Tlieoriam gravitatis &c." indicating that tho Scholium was to bo found written out on another paper. The second page of tho leaf contains some supplementary matter to be added to Prop, xxxvi. These minutim are mentioned for tho purpose of limiting the date of tho com- position of tho Scholium, as the circumstance of its being written on a folio sheet might have led one to suppose that it was sent down to Cambridge not as part of the copy, but as an emendation of copy pre- viously sent. Tho quarto leaves of Newton's handwriting in tho Newtonian Volume all formed part of his interleaved copy of tho Principia : those in folio were sent down in letters as corrections. The only exception to this remark that I have noticed is the sheet now referred to, which contains the Scholium on the Lunar Theory. A distinct idea of tho contents of this Scholium (or " first draught of tho Moon*s theory," as it is afterwards called), as it stood before undergoing the alterations which Newton made in it in consequence of the above letter from Cotes, may be obtained from tho following out- lino of it. It consists. of twelve paragraphs, which, for convenience of reference, I will number in the order in which they present themselves. 1. " Hisce motuum Lunarium computationibus . . . jequatio maxima erit 1 1' . 52"." (Annual Equation). 2. "Inveni etiam . . . tequatio maxima medii motus nodorum 9'. 27''." (Annual Equations of mean motion of apogee and nodes.) 3. " Per tlieoriam gravitatis. . .quadratura ad radium." (iEquatio semestris, the argument of which is = twice the distance of apogee from Sun, i. e. twice tho annual argument). 4. "Per eaudcm gravitatis tlieoriam ... ad 49" circiter ascendit." (iEquatio semestris sccunda, the argument of which is = twice the distance of node from Sun). Tho four preceding paragraphs stand as they are printed in tho 2nd Ed. with the exception of the modifications introduced in confor- mity with Cotcs's suggestions in the above letter. (See Letter lviii). In the 2nd the word " inverse" is also omitted after " si motus SoHs essct in triplicata ratione distantia\" 5. "Per eandcm gravitatis Theoriam apoga?um Luna?... in apo- gajum secundo rcquatum tendens". (The Equation of the centre and Evection combined, giving tho a?quatio semestris of the apogee and first correction of the eccentricity). NEWTON AND COTES. Ill Same as in 2nd Ed. with two exceptions : (1) the upper foeui of the moon's orbit (and not its centre according to Newton's subsequent correction in his paper of alterations, see letter Lvii.) is represented as describing the epicycle BDA ; by a similar inadvertence in paragraph 6 the apogee is made to move in that epicycle. (2) In the 2nd Ed. there is a clause " llabitis autem. . .per methodos notissimas" added at the end of the paragraph, which is in part transferred from paragrapli 9. Cotes has drawn two other lines in the figure (no doubt on receiving Newton's paper of alterations) viz. DE to the right, parallel to A /i, and DF making an acute angle with it (not an obtuse angle as in the figure in the 2nd Ed.) 6. "In porihelio terraa. . .in apoga?um Lunn? ter a?quatum tendens." (Third correction of the place of the apogee and second of the excen- tricity by an " annual equation" whoso argument « annual argument). This paragraph is given in the preceding lett(?r and note. It was completely remodelled in Newton's paper of alterations. Two para- graphs were substituted for it explanatory of what he says may bo called ** (equatio centri secunda" depending on the argument " dis- tance of moon from sun + dist. of moon's apogee from sun's apogee." The latter of them merely contains an approximation to its value. Newton's mode of determining the position of the centre of the moon's orbit in a secondary epicycle with centre D became the subject of an active correspondence between him and Cotes (letters lvii-lxvi). 7. " Per eandem gravitatis Theoriam Sol fortius agit ... in apo- ga?um quarto a3quatum tendens. (Fourth correction of the place of the apogee and third of the excentricity by a " periodical equation" whose argument = distance of apogee from Sun's perigee. ^lention is also made of a "periodical equation" of the Aloon's mean motion depending on the same argument having barely 30/' for its maximum : Damoiscau gives it 2", Plana O'' . 466, Pontecoulantl". 496-1". 108^ 0''.388, Burckhardt 0".l. See Pontdcoulant, torn. iv. pp. 451-465, 580, 604, 626 ; the two terms of which it is composed are of the fourth and fifth orders.) This paragraph will also be found in the preceding letter and proper note. a " Si tres anguli CTD, DTP & FTH ad singulos gradus angulorum BCD,EDFci Gi«V/ computentur & in Tabulas rcfcrantur. 112 CORRESPONDENCE OP et 81 logarithmi quoq: trium distantianim TD^ TF & Tffad radios TC TD et TF in partes 100000 divisos sinml compiitentur & in Tabulas reforantur : aggrcgatum trium angulorum sub signis suis + & - erit a?quatio tota apogwi, et aggregatum trium Logarithmorum erit Logarithmus excentricitatis verw." This and the preceding paragraph were not given in Newton's paper of alterations, whore another paragraph (" Si computatio accu- ratior...non multum errabitur") appeared relating to the "variatio secunda,'* which was omitted in the 3rd Ed. 9. " Ilabitis autem Lunce motu medio & apogaeo et excentricitate ultimum a^quatis, ut et Orbis diametro transversa partium 200000 ; ex his eruetur verus Lunae locus in orbe, et distantia ejus a Terra, idq : per methodos notissimas. Deinde per Variationem et Reductionem ad EcHpticam dabitur ejus loiigitudo et latitude vera." 10. " Diximus orbem Lume a viribus Solis per vices dilatari et contrahi & a^quationcs quasdam motuum Lunarium indo oriri. Indo etiam oritur variatio aliqua parallaxeos Lunae, sed quam inscnsibilem esse judico ; idcoq : in computationibus motuum Luna?, pro mcdiocri ejus distantia a centro Ternu semper usurj)o numcrum 100000, & pro Orbis diametro transversa numerum 200000, et ad parallaxim inves- tigandam pono mediocrem distantiam centri Luna? a centro Terroe in Octantibus iequalem esse 60j semidiametris maxirais Terra?. Semi- diametrum ejus maximam voco qua) a centjr|o ad cequatorem ducitur, minimam qua* a centro ad polos. Et hinc fit Lunoo parallaxis horizon- talis mcdiocris apparens in Octantibus 57' 5'\ in Syzygiis 57' 30" in quadraturis 56' 40". Luna? voro diameter mcdiocris apparens in Syzygiis 31 .30 in Quadraturis 31 .3 usurpari potest & Solis diameter mcdiocris 32 . 1 2." -=^^ 11. "Et cum atmosphajra Terra? ad usq : altitudinem milliarium 35 vel 40 refringat Lucem Solis et refringendo spargat eandem in umbram Terra?, & spargcndo lucem in confinio umbrte dilatet umbram: ad diametrum umbra) qua) per parallaxim prodit, addo minutum unum primum in eclipsibus Lunae, vel minutum unum cum triente." 12. " Theoria vero Luna? primo in Syzygiis, deinde etiam in qua- draturis" &c. as in the 2nd Ed. except (1) as regards the changes in some of the figures mentioned in Letter lxvi., and (2) the addition of the clause " & diffcrcntiam meridiauorum Observatorii hujus & Obsor- vatorii Rcgii Parisiensis 0''°'- Q*"'"- 20**"-." at the close of the paragraph, which does not appear here. NEWTOX AND COTES. 113 LETTER LII. NEWTON TO COTES. *"Prop. De Variatione Lunro p. 402." }2'* Ed.}. I have received three letters from you since my last. And the corrections w*^'' you send me in the two first of them may all stand. In the second of them dated May 1**, you cite my words. In aliis distantiis Solis a Terra Variatio maxima est in ratione qua) componitur ex duplicata ratione [temporis] rcvolutionis sy{n}odicfl3Lunaris (dato anni tempore directe, et ratione anguli CTa directe, et triplicata ratione distantiaQ Solis a Terra inverse. Ideoq: in Apogoco Solis Variatio maxima est S3\ \l" et in ejus Periga30 37' 24" si modo excentricitas Solis sit ad Orbis magni semidiametrum transversam ut 16-}-^- ad lOOO. Here S3 11 & 37 24 may be diminished by 2" & the word tem- poris may be inserted where you see it w*^in the brackets. The Variatio maxima is composed of the ratios of the time, the angle CTa, & the sun's force, as above ; because if any one of the three ratios be enlarged while the rest remain given, the variation will be enlarged. If the time alone be enlarged the Variation will be enlarged in a duplicate proportion, as may be gathered from the descent of falling bodies in a greater or less time. If the angle be enlarged the Variation w^*' is a proportional part of y* Angle will be inlarged in the same simple proportion, & the force also w*^^ is reciprocally as the cube of y* Suns distance enlarges the Variation in proportion to it self. In pag 445 write. Idem per Tabulas Astronomicas est 19. 21. 20. 45 f. Differentia minor est parte fere quadrin- gentesima motus totius, et ab Orbis &c. • Tn Cotes's hand. t This is the mean motion of the Moon*8 nodes in a Julian year. Dut it is the mean notion in a sidereal year that is required in the place referred to. See Cotcs*i annwer. 8 lU CORHESPONDEXOli: OF Pag 450 liii 13 write 9^'. ll'. 3". & lin 28 in Qiiadra- iuris aiitcm rcgrcdiuntur motu horario l6" 19'" 5\'\ I compute it thus. As JB to JD + Ali so is the mean horary motion of the Node to K)'. if/". 51'^ 1 am S*" Yo*" most humble Servant London lO'*^ May 1712 Is. Newton. At the bottom of pag 461 you may put the numbers 5^'. 17'. 20" & 4^"^ 59' 35" Pag 45G lin 1 instead of 38 ^^ write SS^^. The Lunar systeme must be altered " To M' Cotes Professor of Astronmny at hia Chamber in Trinity College in Cambridge LETTER LIII. COTES TO NEWTON. S'. May 13*** 1712 I have received Your last, but I am not yet clear that the ratio of y® angle CTa ought to be introduced in y® xxix*^ Proposition, though I do fully understand the reasons You give for it. As I apprehend it the duplicate ratio of y® Synodical time does itself account for the dilatation of the Angle, & therefore it ought not to be again ac- counted for. According to the reasoning of the 16*^ Corollary of Prop : lxvi. Lib. i, the Variatio maxima which is the angular Error of y® moon whilst she describes the half of y® Arch Cpa, is as the Square of y*' time imploy'd • This is all the notice that Newton at present takes of Cotes's remarks upon the Scholium on the Lunar Theory. The necessity of an "alteration" in "the Lunar Systeme" points to the 6th and 7th paragraphs of the Scholium, especially the former. About the end of June, we are to!d, he intended to send down his corrections " very soon," but even with tlie stimulus of a letter from Cotes (July 20), it is only a little before A\ig. 10 tlmt they are despatched to Cambridge, (Letters LVL, LVIL) NEWTON AND COTES. 115 in describing that half Arch directly & y* Cube of y" distance from y* Sun inversly : Or as the Square of y* Synodical time directly & y* Cube of y* distance invcrsly. Now I think the dilatation is accounted for by taking the angular Error which arises in the time of describing half y® arch Cpa, instead of y* Error which would arise in y* time of describing half y® arch CPA, The thing may be considered another way which perhaps will give more light to y® understanding of my difficulty The true Variatio maxima 35\ lo" arises from y® arch Cjoa, but the Variatio maxima 32'. 32" arises from the arch CPA, Now this latter by y® 16*^ Corollary of Prop lxvi Lib 1 must be altered with y® Square of y® Periodical time directly & the Cube of y® distance invcrsly, & so it will be more correct ; after it is thus corrected, the corrected true Variatio maxima will be dcduc'd from it, by enlarging or dilating it in y* proportion of y® Angle CTa to y® Angle CTJ Oi in the proportion of y® Synodical to y® Periodical time. There- fore the corrected true Variatio maxji}ma will be as the Square of y® Periodical time directly, the Cube of the distance invcrsly, the Synodical time directly & the Peri';- dical time inversly : that is, as the Periodical & Synodical times directly & the cube of y« distance inversly. In this latter way I scruple not to account for the dilatation, but in the former I think it is already accounted for by taking the Square of the Synodical time instead of the Square of y® Periodical. If You find the Objection to be of any moment, I desire you to send me otlier numbers instead of 33' . il". & 37'. 21.". If You choose to let the place stand, yet still there must be a further alteration of those numbers besides y® diminution by 2", for the Square of y' Synodical time compounded with y* ratio of y® angle CTa, makes not the triplicate ratio of y® Synodical time (upon which those numbers were computed but that triplicate 8—2 lit) CORRESPONDENCE OF ratio directly & y* ratio of y* Periodical time inversly as I observed in my former Letter. In Page 455 You direct me to write. Idem per Tabulas Astronomicas est 19°. 21'. 20". 45'". Differentia minor est parte fere quadringentesima motus totiiis &c. I would choose to put it thus. Idem per Tabulas Astronomicas est l9^ 21'. 21". 50"\ Differentia minor est parte trecen- tesima &c. For according to Flamsteed's Tables the motion of y® Nodes from y® Fix't stars in 20 Yeares or 7305 Days is l*"*^. 0*'». 27°. 6'. 53", and therefore in 365^ 6\ 9™ it is 19^ 21'. 21". 50'". The mean horary motion of y® Nodes by the same Tables is 7". 56"\ 56"" and as AB to AD + AB or as 373 to 766 so is 7". 56'". 56"' to 16". 19'". 26". Therefore in Pag: 456, lin: 28, I would write 16-". 19'". 26". Unless You find other reason for writing 16". 19'". 51'*^ as You put it in Your Letters. LETTER LIV. COTES TO NEWTON. S^ Trin: College May 26**» 1712 I have not yet received an answer to my last of May 13*** concerning the xxix*^ Proposition; I am therefore afraid it has miscarried. I sent You by D*^ Bentley a small Treatise* of my own • This was afterwards published in the Philosophical TransactionSi (Jan — March, 1714), and subsequently formed the first part of Cotes's Harmonia Mensurarurrif Cantab. 1722, edited by his cousin Rob. Smith. There is prefixed to it a short address to Ilalley as Secretary of the Royal Society, the first sentence of which is : " Mitto tibi, hortatu IlIustriBsimi Prnsidis Newtoni, quo; aliquot abhinc annis conscripseram de Rationibus dimetiendis." Cotes had succeeded in integratingr some general expressions, the inte- prrals of which involve logarithms. His Logometria contains the application of the results to the solution of a variety of problems. Compare Letter CX. fin. NEWTON AND COTES. J 17 concerning Logaritlims, of which the Title is, Elementa Logometrice together with the Figures belonging to it, I desire the favour of You to deliver 'em to M*" Livebody to be cut in Wood & to give him Your directions if he meets with any difficulty. 1 fear You are at this time taken up with other buisness, otherwise I would beg of You to pe- ruse the Treatise. You will find I am there proposing a new sort of Constructions in Geometry which appear to me very easy, simple & general. But I am fearfull of re- lying upon my own Judgment alone, which possibly in this matter may be too much byass'd. What I think to be right, may to others appear whimsical & of no use & I would not willingly give them the satisfaction of laughing at my Dreams. If You think I may venture to publish it, I shall be glad to know what may want to be corrected or altered either in the Matter or Expression. I have been forc'd to use some new Terms, as Modulus, Ratio modu- laris, &c. If others more proper occur to You upon reading the Papers, I shall be very willing to make any alteration. I hope You will pardon this Trouble I give You. I am Sir Your most Obliged & Humble Servant For S' Isaac Newton at his House RooER CoTES, m 8^ Martin*s Street in Leicester- Feilds London, LETTER LV. NEWTON TO COTES. S' I have reconsidered what you write about the Varia- tion & agree to it. You may leave out the words [et ratione anguli CTa dirccte] & instead of the numbers 118 CORRESPONDENCE OF 33' iT' & 37' 24" diminished by 2", write 33' 14" & S?' ll". fTor so I found them upon computing them anew. Also in pag 455 lin 14 you may write. Idem per Ta- bulas Astronomicas est Iff ^ 2l'. 2l". 50"'. DifFerentia minor est parte treeentcsima &c And pag 456 lin. 28 you may write 16". 19'". 2()". I received yo*" papers by D*" Bently & have run my eye over them. I intend to read them over again & get the cuts done for you as soon as I can find out M*^ Livebody. I am Yo** most humble Servant London May 27 1712 Is. Newton. For the U"*^ }ii* RooKU Cotes Professor of Astronomy at ?iis Chamber in Trinity College in Camhridge Brouglit probably by Bcntley. LETTER LVL COTES TO NEWTON. S'. Cambridge July 20*** 1712 It is now about three Weeks since D*" Bentley return'd from London. He told me, You then intended to send down Your Emendations of the Lunar Theory very soon. I have not received any thing from You since that time, & am therefore apprehensive of some miscarriage. He in- form'd me, You had thoughts of adding something further upon the Subject of Comets*, & besides a small Trea- tise concerning the Methods of Infinite Series & Fluxions. I hope You will go on with Your design : it were better that the publication of Your Book should be dcferr'd a little, than to have it depriv'd of those additions. I thank • This was (lone (sec Letter LXVIIT.), but the project with respect to scries and fluxions w«s ahandoned. NEWTON AND COTES. 119 You for tlTe Picture which I have received of him : 'tis rauch better done than tlie former; but I could have wish*d-it htid been taken from the first of M"" Thornhill's. I am Sir Your most Humble Servant For S' Isaac Newton at his //biw RonEU CoTES. in S* Mayt}n*s S(rM in Leicester Feilds London On the back of Cotes's draught of Apr. 20, there is the drauj;hi of a letter from him to Newton, wliich, from the alhision to the intended treatise on scries and fluxions, seems to liavo been written about the same time as the letter wo have just been reading. lie probably never sent it, but replaced it by the above, suppressing the suggestions and remarks which, upon second thoughts, he may have considered as out of place. We need not, liowever, withhold it here. It is as follows : "I am glad to understand by D' Bcntley that You have some tliouglits of adding to this Book a small Treatise of Infinite Series & the Method of Fluxions. I like the design very well, but I beg leave to make another proposal to You. "When this Book shall be finished I intended to have importun'd You to review Your Algebra for a better Edition of it & to have added to it those things which are published by M' Jones & what others You liave by You of the like nature. These together will make a Volume nearly of y* same size >vith Your Principia & may be printed in the same Character. Your Trcatiso of y" Cubick Curves should be reprinted, for I think the Enumeration is imperfect, there being five cases of ^Equations viz: xytf-¥ey-\ yy + gxxy -\xxy-^cy-\xy=\y-\ I should have acquainted You with this before M' Jones's book was published, if I had known any thing of the Printing of it, for I had observed it two or three yeares ago. I think there are some other things of less moment amiss in the same Treatise. I am S' Your most Humble Sorv' R. Cotes" 120 CORRESPONDENCE OF Here we miss two conniunicationB from NewtoD, one of which accompanied the MS. of the " Elementa Logometriae" on its return to its author, conveying his opinion of the tract in terms, the gist of which may be perceived, though more dimly than one could wish, through Cotes's litotes of " I am glad you are not displeased with it." (Next letter). The other coutained his corrections of the Scholium on the Lunar Theory, (see note on the postscript of Letter LI I). The nature of this lost paper may he easily collected from the correspond- ence that passed relative to parts of its contents. Newton overlooked Cotes's suggestions on the first four paragraphs of the Scholium, and commenced his paper of alterations with paragraph 5, probably with the words ^^ Horroxhis noster. . .Halleim suporiorem ElHpscos umbilicum," &c. The three last words are inadvertently copied from his first draught ; they ought to be "centrum Ellipseos," as Cotes points out in the next letter. The diagram belonging to this and two following paragraph)?, (the "new figure" mentioned in the next letter) seems to have been as represented in the annexed. Next came paragraph 6, (" In perihelio Terras" &c.) as it stands in the 2nd Ed. with the exception of the clerical errors rectified in the letter of Aug. 12, and the further correction (Aug. 20), in the mode of determining the z EDF. After that there was a new paragraph beginning "Computatio hujus motus dilBcilis est" &c. containing an approximation of the pre- ceding paragraph. (The "a}quatio centri secunda," whose argument is dist. of ]\Ioon from Sun + dist. of IMoon's apogee from Sim's apogee). This paragraph, in consequence of the difficulties which Cotes found in it, was alterwards rendered more perspicuous in the paper of Aug. 2G. Tlien followed another new paragraph describing the " Variatio pccunda," as it is printed in the 2nd Ed. except that " Aphelii" was twice written by mistake for "Apogtei." The Variatio secunda = - (2'1 - cosi'!^'+ 1') sin /), if PE= dist. of ])'8 apogee from ©'s peri- gee and D = dist. of )) from 0. And lastly, (omitting paragraphs 9, 10, 11, the first of which was j)artly removed to the end of paragraph 5) came the concluding para- graph " Thcoria vero Lunoa" &c. as printed in 2nd Ed. Compare the account of the first draught of tlie scholium which we have given after Cotes's letter of May 3. pp. 1 10 — 112. NEWTON AND COTES. 121 LETTER LVII. COTES TO NEWTON. S' Cambridge August 10''' 1712 I thank You for Your care of the Wooden Cutts which I received of the Carrier together with the Manuscript*. I am glad You are not displeased with it, & I wish You had signified what Emendations might be made in it. In my Letter of May the 3**, I mentioned some alter- ations in the former part of Your Lunar Theory. You have left me uncertain as to Your resolution about them, by taking no notice of them in Your Last in which Your correction of the latter part of the Theory is set down. I observe in the beginning of it, You have chang*d [et circulus BDA centro C intervallo CD descriptus erit Epi- cyclus ille in quo superior Ellipseos umbilicus locatur] for [Epicyclus ille in quo centrum Orbis Lunaris locatur]. I quaBry whether [Halleius superiorem Ellipseos umbilicum in Epicycle locavit] shoidd not be also chang'd into [Hal- leius centrum Ellipseos] I have not D"^ Halley's little Treatise by me concerning the Lunar Theory. I do not yet understand the Paragraph beginning with [In Perihelio Terra), propter majorem vim Solis &c.] As I apprehend it, the angle EDF in Your new Figure, should be equall to the excess of y® doubled annual argument of the Apogee above the Sun's mean Anomaly as I had suppos'd it in my Letter of May y® 3**. Your Rule concerning that angle is this ; [Et capiatur angulus EDF cequalis exccssui argumenti annui supra distantiam Aphelii Lunro ab Aplielio Solis.] I am uncertain how You understand the words [argumenti annui] ; they may signify either the Annual argument of y® moons apogee or the annual argument of the Sun, i. e, the Sun's mean Anomaly. I am also uncer- • Of the KUmenta Logometria: , 122 CORRESPONDENCE OF tain about y® words [Aphelii Lima3 ab Aphclio Solis] I suppose it should be wrote [Apogcoi I^una) ab Apogsoo Solis]. About the end of this Paragraph You say [Et concipe centrum orbis Lunro interca revolvi dum punc- tum D revolvitur circum centrum C] I do not perceive why it should be thus. The following Paragraph* is rather more obscure to me. I find I cannot form any conceptions of it, unless You will be pleased to give some further light to it. The ^Equation which You here call JEquatio centri sccunda is I perceive the same with that which in D"^ Gregories Astro- nomy You call zEquatio loci Lunce sexta I shall be very glad to learn from You more distinctly the reasoning by which it is established. I am S' Your oblcged Frcind & most Humble Servant LETTER LVIII. NEWTON TO COTES. Upon the receipt of yo" of Aug. 10*^ I have looked back upon yo*' of May 3'^ w^^' I had forgotten. In the first paragraph of y® new Scholium to Prop xxxv, where I have [ad iT 50" circitcr asccndit & additur medio motui Luna) ubi Terra pergit a Pcrihclio suo ad Aphelium et in oppo- sita Orbis parte suhducitur] the words additur & subducitur should change places, & after the word ascendit let these words be added [in aliis locis a)quationi centri solis propor- tionalis est,] In the end of the second Paragraph add these words. Additur vero ajquatio }«'ior & subducitur posterior ubi • Ikglnninej " Computatio hujus motup," &cc. NEWTON AND COTES. 123 Terra pergit a Perihclio suo ad Aphclium, & contrarium fit in opposita Orbis parte. In the third Paragraph the words [paulo minor est in priore casu] are in my copy [paulo major est in priorc casu] & should be so in yours. In the fourth Paragraph the words additur & subduci* tur should ehange places. In the beginning of the correction of the latter part of the Moons Theory you may write [Halleius centrum Ellip- seos in Epicyclo locavit.] In the next Paragraph beginning w**' the words [In Aphelio* Terroe &c] after the first sentence of the Para- graph the word Aphclium is written five times erroneously for the word Apoga)um. Write therefore [recta DE versus Apoga3um Luna) cxcessui Argumenti annui Apogroi Lunoo supra distantiam Apoga)i Luna) ab Apogfco Solis, vel forte oequalis cxcessui Argumenti annui & 360^ supra dis- tantiam Apog^ei Lunso ab Apogroo Solis Solis ab Apo- gioo Luna) Solis ab Apogajo proprio conjunctim. The Equation described in this Paragraph I had first from observations of Lunar Eclipses, & afterwards found that it answered the Theory of gravity in the manner here described. Its quantity when greatest came to about 2' lo"t by Eclipses. By y® Theory tis 2' 25". I suppose you understand that the force of y*^ Sun for disturbing the Moons motions is reciprocally as the cube of the distance of the earth from y® Sun. The motion of the center of the Moons Orb in y*' cycle BDAB arises from the force of the Sun, & as this force varies, the motion of the center of y^ jNIoons • Apparently a slip of the pen for " Periljelio." t This is the value given in the Luna: Theorut Newtoniana, in Gregory's Attronomif, In Mayer (modified by Lulande) it is 2' 9" ; Clairaut ijives it only -2()",8 ; Damoiscau -28",67; Plana -28",811 ; Pont6coulant -28" ,511 ; Burckhardt -27", 6. The terms which compose it arc of the 3rd and higher orders ( Ponti'coulant iv. pp. 577, 602),thc 15 first term Itcing - -r '" . <"'' = - 53" 174. Sec Letter LX V. 121 CORRESPONDENCE OF Orb should vary in this cycle both as to the length of the radius DC & as to y® velocity of the rotation of this radius about the center C, supposing the suns annual motion to be always equal & uniform, & that his distance from the earth only changed. But because the suns annual motion accelerates & retards in a duplicate proportion of the Suns distance reciprocally, & this acceleration & retardation is allowed for in the angle BCD so as to make the point D accelerate & retard in the same proportion in y* cycle J3DAB, here is a variation of the motion of the center of the Moons Orb in the cycle DDAB in a duplicate proportion of the suns distance reciprocally & this without altering the length of the radius CD. Had this variation been in a triplicate proportion there would have been no need of any further ajquation, but because it is only in a duplicate proportion, there wants a further allowance in a single proportion. And this allowance must be made w*^ respect to the Sun's motion & true place. If the suns true motion could be accelerated & retarded in this proportion, I would accelerate & retard the motion of the point D in y^ Epicy {c}le BDAB in the same proportion. But because this cannot be done, I make the allowance by the rotation of the line DF about y® center />, so that the center of the Moons orb may revolve about the center D in an Epicycle described by the point F, & about y® center C in a curvilinear Orb with a velocity reciprocally proportional to the cube of the distance of the earth from the Sun, or directly as the force of y® Sun w*^'' causeth this velocity ; or that the velo- city of the point F in the said curvilinear Orb be to the velocity of the point D in the Orb BDAB reciprocally as the distance of the earth from the Sun. And this "will come to pass quam pro{x}ime by determining y® length DF & the angle EDF as in the Theory. The next Paragraph beginning with the words [Com- putatio motus hujus diflicilis est] contcins only an approxi- NEWTON AND COTES. 125 mation of the former paragraph, by computing the angle at y" earth w*^** the line DF subtends at the Moon in her mean distance from the earth. For the translation of the center of the Moons Orb from D to F, creates the same translation of the whole orb of the INIoon & of the Moon in its Orb from the place in w^** they would otherwise be, & so makes an equation or angle at the Earth w*^** the line DF subtends at the Moon. If the Sun did not act upon the Moon tlie center of the Moons orb would be in the point C, By the action of Sun it is transferred from the center to the circumference of the Epicycle BDAD, If the earth moved uniformly in a concentric circle about the Sun so that y* action of the Sun upon the Moons Orb might be uniform, the center of her Orb would move uniformly in y* Epicy { c } le BDAB, 15y the inequality of the Suns action the center of the Moons orb is transferred from the center to the circumference of a secondary epicycle described with y® radius DC* about the point D. If the inequality of the Suns force or action on y® Moons orb arose only from the variation of the dis- tance of the earth from y® Sun & the angular motion of the earth about the Sun was uniform, the point D would move uniformly in the epicycle BDAB, the angle BCD w*^^ is double to the argumentum annuum increasing uni- formly & the center of the Moons orb would move uniform- ly about the point D in an Epicycle whose radius is 3DF, But the angular motion of the earth about the Sun not being uniform, the angular motion of the radius CD about the Center C is not uniform. If the angular motion of the earth about the Sun was as the cube of the distance of the earth from the Sun reciprocally, that is as the force of the Sun upon the Moons Orb, the angular velocity of the Radius CD about the center C would be in the same pro- ■ A slip of the pen for DF, 126 CORUESPONDENCK OF portion, & the center of the Moons orb being placed in the point D would have a velocity in the Orb BDAB propor- tional to the force of the Sun w^'' causeth it, & there would be no need of a secondary Epicycle about the center />. But because the angular motion of the earth about the Sun is but in a duplicate proportion of the distance of the Sun reciprocally, the motion of the point D in the epicycle BDA will {be} but in a duplicate propoi*tion & for making up this proportion a triplicate one, the center of the Moons Orb must be placed not in the point I) but in an Epicycle about the point Z), & the radius of the Epicycle must be but a third part of such a Radius as would make the epicycle alone answer to a triple proportion, so that the motion of the center of the Moons orb in this Epi- cycle & of the point D about the center C may together compound a motion in a triplicate proportion of the distance of the earth from the Sun reciprocally. In yo*" papers* 1 met w*^' nothing w^^' appeared to me to need correction. I am Yo' most humble Serv* London Aug. 12. 1712. Is. Newton. For the 11"'' M' Roger Cotes Pro/mov of Astroiiomy at his Cliamhcr in Trinity College in Canihrid(je, LETTER LIX. NEWTON TO COTES. S' London. 16 1 Aug. 1712. In the Letter I wrote to you two days ago, the words [Apoga)i Luna)] were interlined after the words [excessui " The Elemejita Logometria; t The post mark is Aug. 14. NEWTON AND COTES. 127 Argumcnti annui.]* Its better to strike out the interlined words, & at the end of the Paragraph to add this sen- tence. [Per Argumentum annuum intelligo excessum qui relinquitur subduccndo medium locum Apogaji Lunro semel ajquatum a vero loco Solis, vel a summa veri illius loci et 360^. Yo"" humble Servant For the R"'^ M' Cotks Professor Is. Nkwton of Astronomy at his chamber in Trinity College in Camhrllgc, The directions given in this billet were supcrsedetl l»y the coniinu- nication of Aug. 20. LETTER LX. COTES TO NEWTON. S" Cambridge August. 17*** 1712 I have received two Letters from You by the last Post & the foregoing. I thank You for the trouble You have given Your self to make the thing clearer to me, but am sorry to find You had mistaken my difficulty. I was very well satisfied as to the design of introducing a secondary Epicycle about y® point D: the motion which You had given the point F in that Epicycle was what I stuck at, & consequently Your manner also of determining the angle EDF, By making the angle BCD equal to the doubled annual argument of y® Moons Apogee the motion of the point D in the primary Epicycle DDAD was not yet enough accelerated in the Earths Perihelium nor enough retarded in the Earths Aphelium : the secondary Epicycle was therefore added that the velocity might be in a trip- licate instead of a duplicate proportion, & an increase of velocity be made in y® Earths Perihelium & a decrease be • All these five wouls are interlined. 128 CORRESPONDENCE OF made in its Aphelium. Hence it seem'd evident to nie, that the motion of y* point F in the secondary Epicycle ought to be such that it might arrive at y® place of its nearest distance from y® point C in y® enrths Perihclium & there by its motion conspiring with y® motion of the point D might render the compound of both the swiftest & again that it might arrive at y* place of its furthest dis- tance from the point C in y® earths Aphelium & there by its motion contrary to y® motion of y® point D might render the compound of both the slowest. Wherefore* if CD be produced to G so that DG ha equal to DF & on the other side between D & C, DH be also taken equal to DFx tis evident that in the Earths Aphelium DF will coincide with DG & in y® Earths Perihclium DF will coincide with DII so revolving about y® centre D y* the angle GDF may always be equal to the suns mean Anomaly. Hence the angle EDF or EDG - GDF or BCD - GDF will be equal to the excess of y® doubled Annual argument above y® suns mean Anomaly as I ob- serv'd in my last. This is the only way according to which I can apprehend the motion of y® point F in the secondary Epicycle to be regulated ; but I cannot perceive how it may be reconcil'd with Your way of determining the angle EDF] or with the time You Assign for its • Cotes does not give any figure : the annexed is added for the convenience of tlie reader. t t. e, by making it = annual argument - dist. of Moon's apogee from Sun's apogee, = twice annual argument - Sun*s anomaly. Cotes himself afterward!) (letter of Sept. 7) contends for this mode of determining the l EDFf taking the l GDF = Sun's true anomidy, not its mean, as he makes it in this and former letters. NEWTON AND COTES. 129 revolution by making it equal to y* time in which y* point Z> revolves about y* centre C, What I have here said will also affect the following Paragraph beginning with [Computatio motus hiyus dif- ficilis est &c.] But besides this there were two other difficultys containd in this Period [Et hncc recta [DF^ Rubtendit angulum ad Terram quern translatio ccntri Orbis Lunoo aloeoi^ ad locum Fgcncrat, & cujus duplum proj)- terea dici potest ^l^quatio ccntri secunda.] The angle at the Earth which DF subtends is y' angle DTF compre- hended by y^ lines TDy TF. I understood You thus, but I perceive by Your Letter that You do not mean the angle DTF, but an angle at y° Earth which is subtended by a line at the Moon equal & parallel to DF\ so y* I can now understand what follows [Et ha?c ajquatio est ut sinus anguli quem recta ilia DF cum recta a puncto F ad Lunam ducta continet quam proximo] which I could not before. However I am still at a loss to understand why You take the double of that angle for the JEquatio centri secunda. The following Paragraph describes the Variatio se- cunda. I suppose it was derived from Observations. In it the word Aphelium is twice used instead of Apogaoum. I am S*" Your &c. LETTER LXI. NEWTON TO COTES. S' London Aug 20. 1712. For removing the difficulties in the Theory of the Moon mentioned in yo" of Aug. 17 I have sent you the inclosed paper conteining some alterations in the descrip- tion of the latter part of that Theory. I had by mistake 9 130 CORRESPONDENCE OF writ [Aphelio Solia] & changed it to [Apogoeo Solis*] & should have changed it to [Perigajo Solis,] as I have done in this paper inclosed. By considering that the angle CDF is the complement of y** Suns Anomaly to a circle (as I have exprest it in the paper inclosed) you may perceive that whenever the Sun is in his Apoge the point F will fall be- tween the points D & C 8c so will be in its slowest motion in the Curve line w*^^ it describes about the center C If the line DF kept parallel to it self the points F 8c D would have equal motions : but by the revolving of the point F about the point D according to the order of the signes this motion of the point F is subducted from the motion of the point Z>, & the difference is the motion of the point F in the said curve line, w^^ motion is therefore the slowest that it can be. And on the contrary, in the Sun's Perige the line DF will lye in directum witli the line DC, & the motion of the point Ff in the said curve line will be at the swiftest being thef summ of the two motions. By the inclosed paper you will understand also why I took the double of the angle subtended by a line at the Moon equal & parallel to DFy £or the Equatio centri 8ecu\n]da. The line must be doubled at the superior focus of the Moon's Orb & carried thence to the Moon. I am Yo*" most humble Servant For the R"** M' Cotes Professor of Is. Newton. Astronomy at his Cliamher in Trinity College in Cambridge Paper inclosed in the above. Capiatur angulus BCD a3qualis duplo argumento annuo, seu dupla) distantia) veri loci Solis ab Apogteo Luna3 semel sequato, et erit CTD eequatio secundaj Apogiei In his letter of Aug. 12, adoptingr the conjecture thrown out by Cotes in his letter of Aug. 10. t The " F " and part of " the " are covered by the wax. \ lu the fair copy of the Scholium which Cotes made for the printer (No. 173), he NEWTON AND COTES. 131 Luna9 et TD excentricitas Orbis ejus. Habitis autcm Lunoo motu medio et Apogroo et cxcentricitatc, ut et Orbis axe majore partium 200000 ; ex his eruetur verus Luna} locus in Orbe et distantia ejus a Terra idq: per methodos notissimas. In perihelio Terrio, propter majorem vim Solis centrum Orbis Lunjo velocius movetur in epicyclo BDA circum centrum C quam in Aphclio, idq : in triplicata rationc dis- tantia? Terro9 a Sole inverse. Ob acquationem centri Solis in argumento annuo compreliensam, centrum Orbis Luna) velocius movetur in Epicyclo illo in duplicata rationc dis- tantiai Tcrrrc a Sole inverse. Vt idem adhuc velocius movcatur in ratione simplici distantiaj inverse; ab Orbis centro D agatur recta DE versus Apogaeum Lunne sou recta? TC parallela, et capiatur angulus EDF aqualis ex- cessui Argument! annul pra?dicti supra distantiam Apogaei Luna* a PerigaiO Solis in conscquentia ; vel quod perinde est, capiatur angulus CDF ajqualis complcmento Anomaliae verae Solis ad gradus s6o, Et sit DF ad DC ut dupla ex- centricitas Orbis magni ad distantiam mediocrem Solis a Terra et motus medius diurnus Solis ab Aphelio* Lunae ad motum medium diurnum Solis ab Apogaeo proprio con- junctim, id est, ut S3| ad 1000 et 52'. 27". 16'" ad 58'. 8". 10'" conjunctim, sive ut 3 ad 100. Et concipe centrum Orbis Lunae locari in puncto is et in Epicyclo cujus centrum est D et radius DF interea revolvi dum punctum D progredi- tur in circumferentia circuli DABD, Hac enim ratione velocitas qua centrum orbis Lunse circum centrum C in linea quadam curva movebitur, erit reciproce ut cubus dis- tantiae Solis a Terra quamproxime, ut oportet. Computatio motus hujus difficilis est, sed facilior red- has altered "secunda" into "semestris", and added the words " in Apogaium secundo icquatum tendens" after "Orbis ejus", in both instances returning to the phraseology of the first draught from wliich Newton had, probably without intending it, departed. • Altered by Cotes to Apoga'o. 9—2 132 CORRESPONDENCE OF detur per approximationem sequentem. Si distantia mc- diocris Lunoe a Terra sit partium 100000, et excentricitas TC sit partium 5505 ut supra : recta CB vel CD invenietur partium ll72j, et recta DF partium S5\, Et haec recta ad distantiam TC subtendit angulum ad Terram quem trans- latio centri Orbis a loco D ad locum F generat in motu centri hujus ; et eadem recta duplicata in situ parallelo ad distantiam superioris umbilici Orbis Luna) a Terra, sub- tendit eundem angulum, quem utiq: translatio ilia generat in motu umbilici, et ad distantiam Lunse a Terra subtendit angulum quem eadem translatio generat in motu Lunse, quiq: propterea roquatio centri secunda dici potest. Et ha»c aequatio in mediocri Lunaj distantia a Terra est ut sinus anguli quem recta ilia DF cum recta a puncto F ad Lunam ducta continet quamproxime, et ubi maxima est evadit 2' 25". Angulus autem quem recta DF et recta a puncto F ad Lunam ducta comprehendunt, invenitur &c. In the next Paragraph but one* write Ajyogoii twice for Aplielii, LETTER LXII. COTES TO NEWTON. S*" CamlMdgo Aug: 28*'* 1712 I received Yours with the inclosed paper, but cannot yet agree with You. In my former Letters I had suppos'd the point F to come the nearest to C in y*' Suns Perigee & to'be the furthest from C in the Suns Apogee : You on the contrary suppose it to be y® the nearest in y® Suns Apogee & the furthest in the Suns Perigee. According to your supposition the motion of y® point F in its curvilinear Orb " The words "but one" are added by mistake. They led Cotes to suspect that Newton's coi»y contuincd an ndditionul paragraph which was not in his. NEWTON AND COTES. 133 will then be the swiftest when that point is at its greatest distance from y* Centre C, & slowest at its least distance from the same, for we agree that tis the smftest in the Suns Perigee & slowest in his Apogee : whereas according to my supposition the smfltest motion accompanys the least distance & y® slowest the greatest, as I think it ought to do. By considering that the angle CBF is the complement of y® Suns Anomaly to a circle, You say, I may perceive that whenever the Sun is in his Apogee, the point F will fall between the points 7) & C, & so will be in its slowest motion in the Curve line which it describes about the cen- tre C. I do indeed perceive that y® point F will fall be- tween y® points D k C, but I think it will then be in its swiftest motion not its slowest. For since y* angle CDF is, by supposition, the complement of the suns Anomaly to a circle; it follows, that as that Anomaly is continually increasing its complement must be continually decreasing. Therefore the line DF does so revolve to the line DC as by its motion to diminish continually the angle CDFi Whence it appeares that in respect of y* line DC the line DF does revolve with a motion contrary to y* order of y* signcs 1 say in respect of y*' moveable line /)C, not in respect of y® Fixt Stars & it is in respect of y® line DC that its motion must be estimated in order to compound it with the motion of y® point D in the circle ADD, The motion then of y® point F in its passage over y** line DC or, by supposition, in the Suns Apogee does conspire with y*^ motion of y® point D & therefore the sum of y* two motions renders the motion of y*' point F in its Curvilinear Orb the swiftest in the Suns Apogee, which ought not to be. I think I apprehend Your meaning very well where You say, The line DF must be doubled at y*' superior Focus of the Moons Orb, & carried thence to the Moon : 134 CORRESPONDENCE OF but I cannot see any reason why y® doubled line at y* supe- rior Focus rather than the single line at y* centre, should be carried to the Moon, excepting that Observations may require it. Your &c. It. C. By Your Letter I suspect that in Your copy there is a Paragraph between that beginning with Computatio mottis htij'us dijficilis &c. & that beginning with Si computatio accu- ratior desideretur ; they immediately follow one the other in my Copy. LETTER LXIII. NEWTON TO COTES. The reason why the doubled line at the superior focus rather then the single one at the center should be carried to the Moon is this. The angles about the superior focus are (quamproxime) proportional to the times, those about y* Center are not. And therefore if the superior focus be translated, the line drawn from it to y® Moon will keep its parallelism, & by doing so will make the same translation in the Moon. As for your other difficulty, if the line DF kept parallel to it self, so as being produced to cut the line TB in a given angle the motion of the points D & F would be al- ways equal to one another. I do not speak of the angular motion of the lines CD and GF about the center C but of NEWTON AND COTES. 1S5 the local motion of the points D8cF in their curvilinear Orbs w*** in this case '\>'ill be two equal circles. Let the circle FMN* be described w*'* the center Cf & radius DF & be cut by the line CD in the point H Sc hy the line CD produced in the point J\L And if the line DF keep parallel to it self, the increase of the angle MDF >vill be equal to the increase of the angle BCD, I mcane that y* two angles will increase w*'* equal swiftness or have equal augmentations in equal times. And in this case the motions of the points D & F will be equal. But if the angle MDF increase but half so fast (w*^^ is the case of the Theory), the motion of the point F will be accelerated neare M & retarded nearc JV*. When the line DF keeps parallel to it self & has no angu- lar motion, its motion in it{s} orb will be equal to that of the point Z>. But if it has an angular motion according to the order of the letters FMIIF (as in the Theory) that angular motion will accelerate the point F neare M & retard it neare N*, You seem to consider the angular revolution of the line DF or CF in respect of the line DC, I consider not the relative angular motion of the line DF or CF but the absolute linear motion of the point F in its linear orb described about the point C in the unmoved plane of the Moons orb w^'^out any relation to the angular motion of the line CD. There is no Paragraph between that w*^^ begins w^** Conu' putatio motus hujus difficilis &c & that w*^*^ begins w*^ Si computatio accuratior desideretur &c If the words of the paper inclosed in my last are not right, pray correct them. After these two Paragraphs there is or should be a Para- graph concerning the refraction of the Atmosphere whereby the Diameter of the earths shadow is enlarged in Lunar " The ** N " should be " // " if wc follow the figure, as it w aUo in Coles'* figure, (letter LX.) It would miiurally drop from the pen after " M** •f- A slip of the pen for D. 13G CORRESPONDENCE OF Eclipses. That Paragraph was (I think) in the first draught I sent you of the Moons Theory*. I am Yo"^ most humble Servant London Sept 2*^ 1712. Is. Newton. For the R""* M' Roqeu Cotes Pro/mor of Astrimottii/ at his Chamber in Trinity College in Camhridije LETTER LXIV. COTES TO NEWTON. S' I received Your last, by which I do at length perceive, that You consider the absolute linear motion of the point F in its linear Orb described about the centre C, & not the angular revolution of the line CF about the same centre, which I had before supposed You to do. I am satisfied that this linear motion of the point jf^will be accelerated near M & retarded near N & therefore if it be the linear motion which ought to be considered in Your Theory & not the angular You do rightly in making the angle CDF equal to the complement of the Suns Anomaly to a Circle, or which is the same thing, in making the angle EDF equal to the excess of the Annual Argument above the distance of the Moons Apogee from the Suns Perigee. But I am of opinion that You ought rather to consider the angular motion of the point F than the linear. And if so, because the angular revolution of y^ line CF about the centre C in the unmoved plane of the Moons Orb, is accelerated near N & retarded near M\ the angle MDF nmst be taken equal to the suns Anomaly, or which is the same thing, the Angle EDF must {be} taken equal to the * It is parngrnph 11. Sec p. 112. NEWTON AND COTES. 137 excess of the Annual Argument above the distance of the Moons Apogee from the Sun's Apogee* I Avill not set down other reasons for considering the Angular motion rather than the linear, >yhieh may admit of dispute. What I ofTer is as follows. I suppose these words at y® end of the Paragraph answer to observations [ — subducendam si summa ilia sit minor semicirculo, ad- dendam si major. Sic habebitur — ^] But these words are not true by the Theory if the angle EDF be taken equal to the excess of the annual Argument above the distance of the Moons Apogee from the Suns Perigee, as it must be taken if the linear motion be considered. And they arc true by the Theory if the angle EDF be taken equal to the excess of y^ Annual Argument above y* distance of the Moons Apogee from the suns Apogee, as it must be taken if the angular motion be considered. Therefore tlie angu- lar motion ought to be considered rather than the linear, that the Theory may answer to the Observations. F Let DL be a line drawn from the point D to the Moon, then will the JEquatio ceiitri secimda be as the sine of the angle FDL. I suppose You agree with me that the ^Equation must be substracted whenever the angular dis- tance of y* line DL from the line DF taken according to the order of the signs is less than a semicircle & be added whenever y* distance is bigger, or in other words, that it • This 13 precisely the value which Newlon gave to the Z KDF by mistuke in Ww Letter of Aug. 12, (see his Letter of Aug. 26), and agalmt which Cotes argues iu \m Letter of Au^. 17, where he takes MDF = Sun's mean anomaly, not its true, as hire. 138 CORRESPONDENCE OF must be substracted whenever the excess of the Moons Anomaly above the angle EDF is less y" a semicircle & be added whenever that excess is bigger. If then the angle EDF be taken equal to y® excess of the Annual argument above the distance of the Moons Apogee from the Suns Perigee : the excess of the Moons Anomaly above the angle EDF will be equal to the sum of distances of the Moon from the Sun & of the Moons Apo- gee from the Suns Perigee, & therefore the ^Equation must be substracted when this sum is less y" a Semicircle & added when it is greater. Now this sum is less than a Semicircle when the sum of the distances of the Moon from the Sun & of the Moons Apogee from the Suns Apogee is greater than a Semicircle, and on the contrary the first sum is greater than a Semicircle when the second is less. Therefore the ^Equation must be substracted when the second sum is greater than a semicircle & added when it is less. But this Rule derived from the Theory is contrary to Your Eulc at the end of the Paragraph derived from Observation. From which contrariety I think it is evi- dent that the angle EDF ought not to be taken equal to y*' excess of the Annual Argument above the distance of the Moons Apogee from the Sun's Perigee & consequently the linear motion of the point F ought not to be considered but its angular motion. I am Your &c. Cambridge Sep*. T^ 1712 11 C NEWTON AND COTES. 139 LETTER LXV. NEWTON TO COTES. S' London Sopt. IS*** 1712. If it could be supposed that the force of the sun upon the Moon for disturbing her motions could be increased w%ut altering the periodical times of the sun & Moon, & that the Orb of the earth was concentric to the Sun : the line DF would vanish & the radius DC would be increased in proportion to the Sun's force without altering its angular motion about the center C. By the increase of the Suns force, the linear motion of the point D would bo increased by its moving in a larger orb, but its angular motion about the center C would remain the same as before. But the earths orb being excentric & the excentricity causing a variation of the Suns force upon the INIoon greater then in proportion to the variation of the Suns velocity, I compen- sate the excess or defect of the force by a secondary epi- cycle described w*^' the radius DF about the center D, so that the distance CF may increase or decrease accordingly as there is an excess or defect of the suns force & by in- creasing or decreasing cause the linear motion of the point F in the plane of the Moons Orb to be greater or less then the linear motion of the point D in the circle DDA in pro- portion to the said excess or defect of the suns force. I thank you for putting me upon examining the words [ — suhducendam si summa ilia ait minor semicirculo, addenda si major. Sic hahehitur ^-c] I have compared them with my calculations of the Moons place in Eclipses & fmd that they must be corrected & put [ — addendam si summa ilia sit minor semicirculoy suhducendam si major. Sic hahehitur Sfcl The Equation* I gathered from Observations many years ago & put it when greatest, to be 2' lo". The last Compare p. 123. 140 CORRESPONDENCE OF year I gathered its quan {ti} ty from observations to be 2' 25" wli.en greatest, but in describing it, committed the mistake w^** I have now corrected by reviewing my old calculations. I am S' Yo"" most humble Servant F(yr tlie R"** M' Roger Cotes Professor Is. Newton. o/ Astronomy J at his Chamber in Trinity College in Cambridge LETTER LXVI. COTES TO NEWTON. I have received Your last Letter. & am now sufficiently satisfied as to the JEquatio ceiitri secunda, I hope the de- scription of the Variatio secunda is accurate. The Para- graph concerning the refraction of the Atmosphere in Eclipses was in Your first draught, but was left out in Your Alteration* of it. There was also another Para- graph before it describing the dimensions of the Sun's & Moon's Diameters & Parallaxes which was also omitted in Your Paper of Alterations. I am uncertain whether You would have both of them inserted or that only concerning the Eifect of y® Atmosphere. They stood thusf. Diximus Orbem Luna? a viribus Solis &c. Et cum Atmosphtera Terrse &c. I suppose You would omit the first of these Paragraphs since the substance of it is in other parts of Your Book, excepting that You have 6oJ semidiameters in Corol. 7. Prop, xxxvii. Lib. iii instead of Go|-. Be pleased to send what You would have inserted. • Soe the introduction to Letter LVII. p. 120. t These form paragraphs 10 and II in the first draught of the Lunar Theory, and will \>e found in tlie account which we have given of it, (p. 112.) This being only the draujfht of his letter, Cotes has not copied them at full length. NEWTON AND COTES. 141 In the last Paragraph I suppose You have designedly altered Your first draught by putting yf 20®. 43'. 40" for V^* 20". 43'. 50", and ;sy 150. 20'. 00" for :c: 15^l9^50'^ and X 8". 20'. 00" for K 8®. 18'. 20''. Sept. 15. 1712 Your &c. II C. LETl'ER LXVir. NEWTON TO COTES. S' I beleive it will be suflleient to insert only the last of the two Paragraphs w*^^ you have copied in your last, viz* that w*^^' concerns the refraction of the Atmosphere. The alterations made in the last Paragraph of the Scholium were advisedly. The description of the Variatio secunda is derived only from phfcnomena & wants to be made more accurate by them that have Icasure & plenty of exact observations. The public must take it as it is. It brings the Moon nearer to the Sun in both the Quadra- tures. I am Yo' most humble Servant London. Sept. 23 1712. Is. Newton. For tJte R"^ M' Cotes Professor 0/ Astro- nomy in the University of Cambridge At his chamber in Trinity College. LETTER LXVIII. NEWTON TO COTES. S' I send you the conclusion* of the Theory of the Comets to be added at y® end of the book after the words • No3. 252 — ^255, beg-inning " Cieterum Cometarum revolventium, &c." and ending "primus omnium quod sciam deprehendit," (pp. 476 — 461 of 2nd Ed.) 142 CORRESPONDENCE OF [Dato autem Latere transverso datur etiam tempus periocli- cum Cometa) Q. J^. /.] There is an error* in the tenth Proposition of the second Book, Prob III, w*^^ will require the reprinting of about a sheet & an half. I was told of it since I wrote to you, & am correcting it. I will pay the charge of reprint- i ig it, & send it to you as soon as I can make it ready. With my service to D*" Bentley I remain Yo' most humble Servant London 14 Octob. 1712. Is. Newton. For the R"** M' Roger Cotes Professor of Astronomy at his Chamber in Trinity College in Cambridge * Tliis error in finding the value of the resistance to the motion of a projectile in the air (see Letter LXXIV.) was pointed out to Newton hy Nicolas Bernoulli (John's nephew), who was on a visit to England dtiring the months of September und October, 1712. *'Monente tandem D. Nic. Bernoulli quod error aliquis admissus fuissct in I'rop. x. Lib. It. constructioncm propositionis correxi ct correctam ei ostendi, et imprimi curavi non subdole sed eo cognoscente." Letter of Newton in Macclesfield Corr. ii. 437, Newton's result, when the curve described is a circle, had been previously shewn to be erroneous by John Bernoulli, in a Letter to Leibniz, in August, 1710, (see their Cor- respondence, II. 231), and in a communication made to the French Academy, in Jan. 1711, (see Memoires for 1711, pp.50 — 56, not published until 1714,) in an appendix to which his nephew corrects two others of Newton's examples, und professes to explain the origin of the mistake (en examinant avec soin sa solution gcaerale, j' en ay trouv6 Torigine). John afterwards resumed the inviting subject in the Leipsic Acts for Feb. and » March, 1713, (see Letters LXXXIL, LXXXVIL) It is remarkable that both of these I mathematicians mistook the source of the error. They imagined that Newton had / taken the coefficients of the successive powers of h in the expansion of (j + /i)" for the successive fluxions of a". This was one of the points upon which Keill was subsequently engaged in controversy with John Bernoulli or his partisans, who worked their crotchet with wearisome pertinacity in the Leipsic Acts. Keill informs us that Newton told Nicolas that the mistake did not arise from the use of series. Newton, through Nicolas, thanked the sturdy professor of Buide for the timely notification of the error, sent him a ; copy of his Analysis, &c., pubiishcd hy Jones in 1711, and nine days after the date of I this letter, proposed him as a member of the Royal Society, into wliich he was accord- ingly elected on the 1st of December following. NEWTON AND COTES. U3 LETTER LXIX. NEWTON TO COTES. I sent you last tucsday a sheet inclosed in a Letter. It concerned the* The Theory of Comets to be added to y® end of the book. I should be glad to hear that it came to your hands. I mentioned also an error that I was lately told of & yv^^ wants to be set right. I have heard nothing from you this month or above & should be glad of a line to know in what forwardness the Press is. I am Yo"^ most humble Servant London. Octob. 21. 1712. Is. Newton For the R*"* M' Roger Cotes Professor of Astronomy at his Chamber in Trinity College in Cambridge LETTER LXX. COTES TO NEWTON. S' October. 23. 1712. I received both Your last Letters, together with the Sheet to be added at the end of the Book, which was inclosed in the former. You mention'd an Error in the x*^ Proposition of the ii^ Book, which will require the reprinting of about a Sheet & an half. I have not re- vised that Proposition to see if I might find it out, but shall stay for Your corrections. The sheet which is now under the Press, ends in Page 492 of y® old Edition, and Page 456 of the new Edition. I have not observed any- thing of moment which may be altered in the Theory of Comets. In the new fourth Corollary f of Prop, xl I have t No. 245. 144 CORRESPONDENCE OF inserted after the first line [& quadratum radii illius ponatur esse partium lOOOOOOOO]. Pag. 490, lin. 5, I have put [in subduplicata ratione SQ ad St] instead of [in subduplicata ratione St ad SQ] In the last Page of the Book, lines 8 & 9, I design to i^ut 2G - 2C & 2T - 2S for G - C & T^S, unless You forbid it. I suppose the Astronomical compu- tations relating to the Comets are exact, having been exa- mined both by Your self & by D"" Halley. I should have given You notice sooner, that I had- re- ceived Your additional Sheet at the end of the Book, but that I expected D"* Bentley would have seen You before this time, for he once intended to have been at London a week sooner. I am S^ LETTER LXXI. COTES TO NEWTON. I here send You the Sheets as far as they are Printed off, that Your self or some freind may revise them, in order to see what Errata may be put in a Table. I know not whether You have got the Copper-plate of the Comet yet done. The Printer tells me there will be 750 requisite* The next week I shall be in the Countrey, when I return I suppose You will have the corrections ready which You mention'd for the Sheet to be reprinted I am Sir Your most Humble Serv* Nov. 1"*. 1712 Roger Cotes For S' Isaac Newton at His House in S* Martin' s-street Leicester feilds London NEWTON AND COTES. 145 LETTER LXXII. COTES TO NEWTON. I hope You have received the Sheets which I sent last, ending in Page 456 of the New Edition, We have since printed off 3 Sheets more, which take in the whole Book with the Additional Sheet, excepting about 20 lines. To fill up the following Sheet may be added a Table of the Contents of each Section, if You think fit. D"" Bentley was proposing to have subjoyned an Index to the whole, but particularly to the Third Book. If You approve of it, such an Index may soon be made. If Your alterations in the Second Book arc finished I desire You will be pleased to send 'em. I am Sir, Your most Humble Servant Cambridge Nov^^ 23**. 1712 Roger Cotes For S' Isaac Newton at his House in S* Martin's Street Leicester- Feilds London LETTER LXXIIL NEWTON TO COTES. S' I send you enclosed* the tenth Proposition of the Second book corrected. It will require the reprinting of a sheet & a quarter from pag 230 to pag. 240. There is wooden cut belonging to it w*^** I intend to send you by the next Carrier. I think this Proposition as it is now done will take up much the same space as before. If not, the • No«. 262-265. 10 146 CORRESPONDENCE OF space about the cuts may be made a little wider or a little narrower, or the number of lines in a page may be increased or diminished by a line. When this sheet & a quarter is printed oft' I liope your trouble of correcting will be at an end. As for making a Table to the book I leave it to you to do what you think. I beleive a short one will be sufficient. I shall send you in a few days a SchoIiu{m}* of about a quarter of a Sheet to be added to the {endj"*^ of the book: & some are perswading me to add an Appendix concerning the attraction of the small particles of bodies. It will take up about three quarters of a Sheet, but I am not yet resolved about it. I am Yo"* humble & obedient Servant London. Jan. 6. 171 §. Is. Newton For the R"** M' Cotes Professor of Astronomy at his Chamber in Trinity College in Cambridge. LETTER LXXIV. COTES TO NEWTON. S'. Cambridge Jan. IS*** 1713. I have considered Your alteration of Prop, x, Lib. ii. and am well satisfied with it. I observe that You have increased the Resistance in the proportion of 3 to 2, which is the only change in Your Conclusions, arising from hence (as I apprehend it) that in the new Figure LH is to NI as Roo to lloo ^ sSo^, whereas in y® former Figure kl was to FG as Roo to Roo + 2So^, Some things in Your Paper I have altered, they are not worth Your • These four letters within { } have disappeared with the wax. NEWTON AND COTES. 147 notice, being only faults in transcribing*. I have this day received the "Wooden Cut. I shall expect the Scholium at y® end of the Book & the Appendix at Your leasure. I am Sir Your Obliged Freind & Humble Servant Roger Cotes. For S'. Isaac Newton at his House in S* Martin*s Street Leicester. Feilds London LETTER LXXV. NEWTON TO COTES. The inclosed! is the Scholium w^^ I promised to send you, to be added to the end of the book. I intended to have said much more about the attraction of the small particles of bodies, but upon second thoughts I have chose rather to add but one short Paragraph about that part of Philosophy. This Scholium finishes the book. The cut for the Comet of 1G80 is going to be rolled off. I am Yo'^ most humble & obedient Servant London 2** March t 171f. Isaac Newton. For the Roy"** M' Roger Cotes Professor of Astronomy, at his Chamber in Trinity College in Cambridge. • Cotes, however, besides making the alterations alluded to here, has (perhaps from want of room) omitted a paragraph at the beginning of the Scholium of the Prop, (p. 269, Ed. 1, p. 240, Ed. 2.) in which Newton points out another mode of viewing the problem which is the subject of the Proposition. The parai^raph runs as follows : " Fingere liceret projectilia pergere in arcuum GH, HI, IK chordis & in Holis punctis Gf H, J, K per vim gravitatis £c vim resistcntite ngitari, perinde ut in Propositione prima Libri primi corpus per vim ccntripetam intermittentem agitabatur, deinde chordas in infinitum diminui ut vires reddantur continue. Et solutio Problematis hac ratione facillima cvadcrct." t Nos. 269, 270, 272. % The Post mark is March 3, (1 uesday.) 10—2 148 CORRESPONDENCE OP LETTER LXXVI. NEWTON AND BENTLEY TO COTES. S' I sent you by last tuesdays Post the last sheet of y* Prineipia, & told you that the cut for y® Comet of 1G80 was going to be rolled off. But we want the page where it is to be inserted in the book. I think y® page is 462 or 463. Pray send me w*^^ it is, that it may be graved upon the Plate for directing the Bookbinder where to insert it. I am Yo'^ most humble Servant London 5 March 171|. Is. Newton I have S"" Isaac's Leave to remind you of what You and I were talking of, An alphabetical Index, & a Preface in your own Name ; If you please to draw them up ready for y® press, to be printed after my Return to Cambridg, You will oblige Yours For the R"^ M' Roger Cotes Pro/mor of R Bentley. Astronomy^ at his Chamber in Trinity College in Cambridge LETTER LXXVIL COTES TO NEWTON. S'. I received both Your Letters with the last sheet of the Book inclosed in the former of them. The Paragraph beginning with Cceterum Trajectoriam quam Cometa descrip- sit &c., which is in the 497"' page of the former Edition, falls in the 465*^ page of the new Edition. This is the place to which I suppose You would refer the Cut for the Comet. I intend in a day or two to set about the Alphabetical Index. I will Avrite to D"" Bentley concerning the Preface by y* next Post. March. 8. 17^ I am S^ Your &c. NEWTON AND COTES. 149 LETTER LXXVIII. COTES TO BENTLEY. To D"" Bentley March. lO***. 171? S^ I received what You wrote to me in S' Isaac's Letter. I will set about the Index in a day or two. As to the Preface I should be glad to know from S' Isaac with what view he thinks proper to have it written. You know the book has been received abroad with some disadvantage, & the cause of it may easily be gues3*d at. The Commer- cium Epistolicum lately published by order of the R. So- ciety gives such indubitable proof of IVIr Leibnitz's want of candour that I shall not scruple in the least to speak out the full truth of the matter if it be thought convenient There are some peices of his looking this way which deserve a censure, as his Tentamen de Motuum Ccelestium causis*. If S** Isaac is willing that something of this nature may- be done, I should be glad if, whilst I am making the Index, he would be pleas'd to consider of it & put down a few notes of what he thinks most material to be insisted onT^ This I say upon supposition that I write the Preface my self. But I think it will be much more adviseable that You or He or both of You should Avrite it whilst You are in To>vn. You may depend upon it that I will own it & defend it as well as I can if hereafter there be occasion. I am S' &c. * Newtoo had himself drawn up some strictures upon this piece, which were made use of by the editors of the Commercium Epistolicum (p. 97). See the paper entitled " Ex Epistola cujusdam ad Amicum,*' printed in the Appendix to tliis work. J 50 CORRESPONDENCE OF LETTER LXXIX. BENTLEY TO COTES. Dear Sir, At S' Isaac Newton's March 12. I communicated your Letter to S'. Isaac, who happend to make me a visit this morning, & we appointed to meet this Evening at his House, & there to write you an Answer. For y® Close of your Letter, w*^^ proposes a Preface to be drawn up here, and to be fatherd by you, we will impute it to your Modesty ; but You must not press it further, but go about it your self. For y® subject of y® Preface, you know it must be to give an account, first of y® work it self, 2^*'^ of y® improvements of y® New Edition ; & then you have S^ Isaac's consent to add what you think proper about y° controversy of y® first Invention. You your self are full Master of it, & want no hints to be given you : However when it is drawn up, You shall have His & my Judgment, to suggest any thing y*. may improve it. Tis both our opinions, to spare y® Name of M. Leibnitz, and abstain from all words or Epithets of reproch ; for else, y* will be y® reply, (not that its untrue) but y* its rude & uncivil. S^ Isaac presents his service to you. I am Yours ForM*. Roger Cotes Professor of It. Bentley* Astrcyncymy at Trinity College in Cambridg. • The original of this Letter, which has been already printed in the Bentley Corre- spondence (p. 460), is in the possession of Dawson Turner, Esq., who has kindly fur- nished me with a new transcript of it. NEWTON AND COTES. 151 LETTER LXXX. COTES TO NEWTON. S' I have received D"^ Bentlys Letter in ansiyer to that which I wrote to him concerning the Preface. I am very well satisfied with the directions there given, & have accordingly been considering of the Matter. I think it will be proper besides the account of the Book & its im- provements, to add something more particularly concerning the manner of Philosophizing made use of & wherein it differs from that of Descartes and Others, I mean in first demonstrating the Principle it employs. This I would not only assert but make evident by a short deduction of the Principle of Gravity from the Phaenomena of Nature in a popular way that it may be understood by ordinary readers & may serve at y® same time as a specimen to them of the Method of y* whole Book. That You {may} y* better understand what I aim at I think to proceed in some such manner. [Tis one of y* primary Laws of Nature, that all bodys persevere in their state &c. Hence it follows that Bodys which are moved in curve-lines & continually hindred from going on along the tangents to those curve- lines must incessantly be acted upon by some force suffi- cient for that purpose. The Planets (tis matter of fact) revolve in Curve-lines, therefore. &c. [Again, tis Mathe- matically demonstrated that Corpus omne, quod movetur ^c. Prop, 2 Lib 1, & corpus omne, quod radio ^c, prop, 3 Lib 1, Now tis confess'd by all Astronomers that the Primary Planets about y® Sun & the Secondary about their re- spective primary doe describe areas proportional to the times. Therefore y*^ force by which they are continually diverted from the tangents of their Orbits is directed & tends towards their central Bodies; which force (from what cause soever it proceeds) may therefore not improperly be 152 CORRESPONDENCE OP call{ed} Centripetal in respect of the revolving Bodies & Attractive in respect of y* central ones. [Furthermore tis Mathematically demonstrated that. Cor. 6, Prop. 4. Lib. 1 & Cor. 1, Prop. 45, Lib. 1. But tis agreed upon by Astro- nomers that &c. or &c. Therefore the centripetal forces of the Primary Planets revolving about the Sun & of the Secondary Planets revolving about their Primary ones, are in a duplicate proportion &c. In this manner I would pro- ceed to the 4*^ Prop of Lib. iii & then to the 5*^. But in the first corollary of this 5*^ Proposition I meet with a difficulty*, it lyes in these words [Et cum attractio omnis mutua sit] I am persuaded they are then true when the Attraction may properly be so called, otherwise they may be false. You will understand my meaning by an Example. Suppose two Globes A & B placed at a distance from each other upon a Table, & that whilst y® Globe A remaines at rest the Globe B is moved towards it by an in- visible Hand; a by-stander who observes this motion but not the cause of it, will say that y® Globe B does certainly tend to the centre of y® Globe A^ & thereupon he may call the force of the invisible hand the centripetal force of B & the Attraction of A since the effect appeares the same as if it did truly proceed from a proper & real Attraction of A, But then I think he cannot by virtue of this Axiom £Attractio omnis mutua est] conclude contrary to his sense & Observation that the Globe A does also move towards the Globe B & will meet it at the common centre of Gravity of both bodies. This is what stops me in the train of * Tlie difRculty raised by Cotes here affords an instance of the temporary haze which may occasionally obscure tlie brightest intellects. Compare the story told of Lagrange by Biot (Journal des Savantty 1837, p. 84): "Lagrange tira un jour de sa poche un papier qu'il lut a I'Acad^mie, et qui contenait une d6monstration du fameux Pflstulatum d*Buclide, relatif a la th^orie des parallcles. Cette demonstration reposait 8ur un paralogisme Evident, qui parut tel u tout le monde ; et probablement Lagrange aussi le reconnut pour tel pendant sa lecture. Car, lorsqu^il eut fini, il remit son pa- pier dans sa poche, ct n'en parla plus. Un instant de silence univereel suivit, et Ton passa aus8it6t a d'autrcs objcts." NEWTON AND COTES. 153 reasoning by which I would make out as I said in a popular way Your 7*** Proposition of y* in** Book. I tihall be glad to have Your resolution of the difficulty, for such I take it to be. If it appeares so to You also, I think it should be obviated in the last Sheet of Your Book which is not yet printed off or by an Addendum to be printed with y* Errata Table. For till this objection be cleared I would not un- dertake to answer any one who should assert that You do Hypothesim fingerCt I think You seem tacitly to make this supposition that y® Attractive force resides in the Central Body After this Specimen I think it will be proper {to} add somethings by which your Book may be cleared from some prejudices which have been industriously laid against it. As that it deserts Mechanical causes, is built upon IMiracles, & recurrs to Occult qualitys. That You may not think it unnecessary to answer such Objections You may be pleased to consult a Weekly Paper called Memoires of Literature & sold by Ann Baldwin in Warwick -Lane. In the 18*** Number of y® second Volume of those Papers which was published May 5*^ 1712* You will find a very extraor- dinary Letter of Mr Leibnitz to Mr Hartsoekcr which will confirm what I have said. I do not propose to mention Mr Leibnitz's name, twere better to neglect him, but the Objections I think may very well be answered & even retorted upon the maintainers of Vortices. After I have spoke of Your Book it will come in my way to mention the Improvements of Geometry upon which Your Book is built, & there I must mention the time when those im- provements were first made & by whom they were made. I intend to say nothing of ]Mr Leibnitz, but desire You will give me leave to appeal to the Commercium Epis- • p. 137. Leibniz. 0pp. Torn. ii. Pare ii. p. 60. The letter is dated, Ilaoover, Feb. 10, 171 1. I^eibniz does not mention Newton's name. 154 CORRESPONDENCE OP tolicutn to vouch what I.shall say of Your self & to insert into my Preface the very words of the Judgment of the Society (page 120^^ Com. Ep) that foreigners may more generally be acquainted^ with the true state of the Case. Feb. ''^ 18. 171| The plan of the Preface sketched in the above letter was afterwards modified. The Indices compiled by Cotes supplied the place of "an account of the book", and the short preface wliicli Newton nent him in liis letter of March 31 made it unnecessary to enter into a detail of " its improve- ments." The intended notice of the method of fluxions and of the dispute relative to its discovery was abandoned, whether in consequence of Newton's declaration at the close of the letter just quoted that he " must not see it/' or from a feeling that it was better to leave the evidence in the Commercium Epistolicum to work its own way, wo have no precise information. Coten's Preface therefore is confined to an ex])osition of ** the manner of philosoj)liizing made use of" in the work, and to an examination of the objections of Leibniz (without mentioning his name) and of the system of Vortices. Leibniz in a letter (Apr. 9, I^Hj. N.S.) written under excitement, (it is his reply to Newton's raking fire of Feb. 26.) calls this Preface *'plcine d'aigreur," an expression which may be taken as a measure of that extraordinary man's sensitiveness at the time. LETTER LXXXI. NEWTON TO COTES. I had yo*"* of Feb W\ & the Difficulty you mention w^^ lies in these words [Et cum Attractio omnis mutua sit] is removed by considering that as in Geometry the word Hypothesis is not taken in so large a sense as to include the Axiomes & Postulates, so in Experimental Philosophy it is not to be taken in so large a sense as to include the * It is clear that this is a mistake for Marchf though Newton himself in his answer to this letter speaks of it as " yo""* of Feb. 18." NEWTON AND COTES. 155 first Principles or Axiomes w^^ I call the laws of motion. These Principles are deduced from Pha)nomena & made general by Induction : w*^^ is the highest evidence that a Proposition can have in this philosophy. And the word Hypothesis is here used by me to signify only such a Pro- position as is not a Phenomenon nor deduced from any Phajnomena but assumed or supposed w%ut any experi- mental proof. Now the mutual & mutually equal attrac- tion of bodies is a branch of the third Law of motion & how this branch is deduced from Phenomena you may see in the end of the Corollaries of y® Laws of ^lotion, pag. 22. If a body attracts another body contiguous to it & is not mutually attracted by the other: the attracted body will drive the other before it & both will go away together w*** an accelerated motion in infinitum, as it were by a self moving principle, cotrary to y® first law of motion, whereas there is no such phjcnomenon in all nature. At the end of the last Paragraph but two now ready to be printed off I desire you to add after the words [nihil aliud est quam ffatum et Natura.] these words : [Et hoic de Deo: de quo utiq: ex pha)nomenis disserere, ad Philo- sophiam experimentalem pertinct.] And for preventing exceptions against the use of the word Hypothesis I desire you to conclude the next Para- graph in this manner [Quicquid enim ex phrenomenis non deducitur Hypothesis vocanda est, et ejusmodi Hypotheses seu INIetaphysicae seu Physicaj seu Qualitatum occultarum seu Mechanic® in Philosophia experimentali locum non habent. In hac Philosophia Propositioncs dcducuntur ex phaenomenis & redduntur generales per Inductionem. Sic impenetrabilitas mobilitas & impetus corporum & leges motuum & gravitatis innotuere. Et satis est quod Gravi- ias corporu revera existat & agat secundum leges a nobis cxpositas & ad corporum coelcstium et maris nostri motus omnes sufficiat. 156 CORRESPONDENCE OP I have not time to finish this Letter but intend to write to you again on Tuesday. I am Yo' most humble Servant London. 28 March {Saturday} 1713. Is. Newton For the RoTorcnd M' Roger Cotks Professor of Astronomy/, at hia Cfiamber in Trinitij College in Cambridge. LETTER LXXXII. NEWTON TO COTES. S' London. 31 Mar. 1713. On Saturday last I wrote to you, representing that Experimental philosophy proceeds only upon Phenomena & deduces general Propositions from them only by Induc- tion. And such is the proof of mutual attraction. And the arguments for y* impenetrability, mobility & force of all bodies & for the laws of motion are no better. And he that in experimental Philosophy would except against any of these must draw his objection from some experi- ment or phaenomenon & not from a mere Hypothesis, if the Induction be of any force. In the same Letter, I sent you also an addition to the last Paragraph but two & an emendation to the last Para- graph but one in the paper now to be printed off in the end of the Book. I heare that M*" Bernoulli has sent a Paper* of 40 • Part of it appeared in the Number for Feb. 1713, pp. 77—95, the remainder in the March number, pp. 115—132. See Comm. Epistol. Leibn. and Bernoull. ii. 299. Bernoulli afterwards (Letter to Leibniz, Feb. i?, 1714), in consequence of his not re- ceiving a copy of the Commercium Epistolicum, and of the 2nd Ed. of the Principia, which Demoivre, in Newton's name, had promised more than a year before to send him, fancied that Newton was offended at his animadversions, and seems to have stated his suspicions to Demoivre ; but the tone of the article did not prevent the author of the Principia from expresbing his sense of the merits of Bernoulli's solution of his problem. NEWTON AND COTES. 157 pages to be published in the Acta Leipsica relating to what I have >vritten upon the curve Lines described by Projectiles in resisting Mediums. And therein he partly makes Observations upon what I have written & partly improves it. To prevent being blamed by him or others for any disingenuity in not acknowledging my oversights or slips in the first edition I believe it will not be amiss to print next after the old Proefatio ad Lectorem, the follow- ing Account of this new Edition. In hac secunda Principiorum Editione, multa sparsim emendantur & nonnulla adjiciuntur. In Libri primi Sect, ii, Inventio virium quibus corpora in Orbibus datis revolvi possint, facilior redditur et amplior. In Libri sccundi Sect. VII Thcoria resistentiro fluidorum accuratius invcsti- gatur & novis expcrimentis confirraatur. In Libro tertio Thcoria Luna) & Pra^cessio JEquinoctiorum ex Principiis suis plenius deducuntur, et Thcoria Cometarum pluribus et accuratius computatis Orbium exemplis confirmatur. 28 Mar. 1713. I. N. If you write any further Preface ft I must not see it J. for I find that I shall be examined about it. The cuts for y« Comet of 1680 & 1681 are printed off & will be sent to D' Bently this week by the Carrier. I am Yo' most humble Servant For tU R"** M' Cotes Professor of Astro- IsAAC Newton nomy in the University of Cambridge, At his Cluimber in Trinity College in Cambridge ** J'ai vu Mr. Neuuton, qui m*a dit, qu*il avoit lu avec beaucoup de plaiHir vdtre methodede resoudre le probleme de la resistance, il vous rend justice en Homme, qui n' est nullement offens^, il dit qu* elle est admirablement belle, 6c roeme qu* elle est commode pour des expressions finies." Extract from a Letter of Demoivre to Bernoulli in Leipsic Acts for July 1716, p. 309. t Newton seems to have particularly in his eye Cotefl*fl proposed allusion to the dispute about the invention of fluxions. :J: Compare Commerc. Epistol. 2nd Ed. ad Lectorem pag. penult "Quae nov» 158 CORRESPONDENCE OF This is tho last letter in the Trin. Coll. collection that passed be- tween Newton and his editor while the work was in the press. The proof-sheet however of the Scholium Generale must have been sent up to Newton, as there is a paper (No. 271) in his handwriting con- taining some alterations of the Scholium, in which the pages and lines are referred to as wo find them in the printed book. The Index was finished in April (letter cxiii), and tho Preface is dated May 12. In his letter of May 3 to Jones (letter cxiv), Cotes ** hopes the whole book may bo finished in a fortnight or 3 weeks :" ** it might have been done by this time" but for indisposition. It was not however until about June 18 that tho impression was finished. (See next letter). It was probably about this time that the Cambridge Aristarchus made his emendations of Ilalley's verses prefixed to tho Principia. See Rigaud's Essay, pp. 86, 87. LETTER LXXXIII. COTES TO D' SAM. CLARKE. S' Cambridge June 25*'* 1713. I received Your very kind Letter. I return You my thanks for Your corrections of the Preface, & particularly for Your advice in relation to that place where I seem'd to assert Gravity to be Essential to Bodies. I am fully of Your mind that it would have furnish'd matter for Cavil- ling, & therefore I struck it out immediately upon D' Cannon's mentioning Your Objection to me, & so it never was printed. The impression of the whole Book was finished about a week ago. INIy design in that passage was not to assert Gravity to be essential to Matter, but rather to assert that we are ignorant of the Essential propertys of Matter & that in re- Principionim editiuni praemissa sunt, Newtonus non vidit aDtequam Liber in lucem prodiit." Dalemhert's misstatement on this point ("preface faite sous les yeux de Tauteur," Encycloped. i. 854) is noticed by Wilson (Robins's Tracts, Appendix, II. 334). NEWTON AND COTES. 159 spect of our Knowledge Gravity might possibly lay as fair a claim to that Title as the other Propertys which I men- tion'd. For I understand by Essential propertys such pro- pertys without which no others belonging to the same substance can exist : and I would not undertake to prove that it were impossible for any of the other Properties of Bodies to exist without even Extension. Be pleased to present my humble Service to S' Isaac when You see him next, & let him know that the Book is finished* I am S^ Your much Obliged Freind & Humble Servant To D*^ Clark H C It appears from the above letter that a meaning has been given to expressions in Cotes's Preface wliich he did not intend them to convey. He has been understood to assert that gravity is an essential property of bodies: his words are *^ Inter primarias qualitates corporum univer- sonim vel Gravitas habobit locum ; vel Extensio, Mobilitas & Impeno- trabilltas non habebunt." His supposed views are controverted by D' Whewell (Philosophy of the Inductive Sciences, i. 249, or 258 2nd Ed.), and are quoted with approbation in a recent work (Le Cartesianisme ou la veritable renovation des sciences, par Bordas-Deraoulin, Paris 1843,— a work less remarkable for accuracy than for liveliness of declamation). Though Newton, says this last writer, had not the true idea of attrac- tion, "citte notion perce ct triomphe deju chez quelques-uns deses dis- ciples immediats, tcls que Roger C6tes." (i. 304). He also refers to Maupertuis and Lalande as holding the same opinion. " Pour moi, dit Lalande, je pense avec M. Maupertuis et la plupart des m^taphysicicns anglais, que I'attraction depend d'une propriete intrinseque do la ma- tiere." Astron. ed. 2. art. 3384." • On Monday July 27 Newton waited on the Queen with a copy of the new edition of his book. (Daily's Flamsteed, p. 98.) Jones's letter of thanks for a presentation copy (letter cxv) is dated July 11. Compare Bentley*a Correspondence, p. 465. Flumstceti gave \Ss. for a copy (Baily, p. 305). In Clare Hall Library are two copies of the book, one of which belonged to Cotes's friend Charles INIorgan " Kx dono Clariss'. Editoris Pr. P. 1*. C." and the other to Rob. Green " Pret. 15j." In a catalogue of Keill's library in his own hand-writing among the Lucasian papers the price of a copy is put down at £1. 160 CORRESPONDENCE OP ^ Newton was obliged on several occasions to protest against the doc- trine of innate gravity being ascribed to him. See letters to Bcntley, Jan. 17. Feb. 25. 1G9§. Advertisement to 2<* Ed. (in English) of his Optics, July 16, 1717 : " And to shew that I do not take Gravity for an essential Property of Bodies, I have added one question {the 21"} concerning its Cause, chusing to propose it by way of a Question, be- cause I am not yet satisfied about it for want of Experiments*." & his letter in Macclesfield Corrcsp. 11. 437. LETTER LXXXIV. This is not, properly spealcing, a letter, but a paper of Corrections and Additions sent by Newton to Cotes through Cornelius Crownfteld, the University Printer, six months after the publication of the book. See next letter. Corrigenda et Addenda in Lib. i. Pag 7. lin. 8, post veriore tempore adde mensurent. P. 10, 1. 6, post sed adde sunt P. 10. 1. 17, lege difficillimum est. P. 15 1 16 lege in plana, ut /)iVad pll. lb. 1. 20 veri- tatem ejus. P. 17 1. 20 pro communis lege corporis. P 31, 1. 38 AD et DB, P. 36, 1. 6 kge Cor. 5. P. 38 1 24 lege Corol. 2. 1. 26 lege Corol. 4. P. 41 1 5 lege P et Q. P. 42, 1. 8 post vol adde circu- lum concentrice tangit, id est. P 44 1 23 lege QR x RN + QN. P. 45, 1. 14 post hoc est adde (ob datam specie figuram illam) lb. 1. 21 post Spiralem adde concentrice. P. 46 1. 30 post intelligatur adde recta. lb. 1. ult. post PvxuV lege, Adde rectangulum uPv utrinq: et prodibit quadratum chorda? arcus PQ ajqnale rectangulo VPv. P. 47 1 4 post conica in P, lege • This declaration was probably drawn from him by the recent controversy between Leibniz and Clarke. NEWTON AND COTES. 161 adeoq: ex natura Sectionum Conicnrum, circuli hi\jiis chorda PT tequalis erit —^^ . P. 52, 1. 16 dele per. P. 54, 1. 4, jyoat area QT ^ SP adds quso dato tempore describitur. P. 57, 1. 25 post si ea adde sit. P. 59, 1. 7 7>o^« oxi principali ^qwtvq, adde id est axi in quo umbilici jacent. P. 61, 1. 12 lege ita ut sit GA ad AS et Ga ad a^S* ut est KB ad jBiS*, et axe A a, lb. 1. 15, 16 %e, et cum sit GA ad ^5* ut Ga ad aS, erit divisim Ga - GAy seu ^a ad aS-AS seu iS*// in eadem ratione. P. 86, 1. 7, ^o^f biseca adde in JT/ et N, P. 87 1. 7 lege per Prob. xiv. P. 89 & 90 in Figura jungatur FD, P. 92, in Figura jungantur FG et HL P. 101, 1. 6, 7, 8, %e. Nam centro O intervallo OA dcscribatur semicircu- lus AQB rectae LP si opus est products, occurrens in Q, junganturq: SQ^ OQ, quarura OQ producta occurrat arcui EFG in F, et in eandem OQ demittatur perpendiculum SE» lb. 1. 36 post qujo adde per punctum P transit et. P. 109, 1. 1 post Hyperbola adde rectangula. lb. in Sche' mate pro liter a O scrihatur liter a H, P. 117. 1 15 lege prio- ns in /. Et stantibus. P. 121 in Schemate e regione liter w p acHhatur litera K in Orhe VPK, P. 127, 1. 7, 9 graduum. P. 131 1. 17 lege^ m roqualis 1 et n, P. 136, 1. 2 pro Bp scribe BP. P. 137, 1. 16 post sinus versus adde est. P. 139, 1. 10 post adeoq: ad adde globi exterioris. 1. 12 post habet ad adde globi interioris. P. 148, 1. 4 post distantisB adde corporum. lb. 1. 7 pro tcrminos suos communi scribe terminum suum communem. P. 151, 1. 8, 21 scribe ad primum duorum. P. 151, 1. 18 scribe ut primum duorum. P. 156 1 31 scribe maximo. Nam. P 158 1 32 Post atq: adde ut, et post proportionalitate deh ut. lb. 1. 36 post non sit, adde reciproce. P. 166, 1. 9 dele quadratum temporis periodici et scribe 11 162 CORRESPONDENCE OP tempus periodicum. P. 169, 1. 26, 33, 34, & P. 170 1 3 pro C scribe O, et in schemate inter P ac T scribe literam O, P. 184 1 21 post area adde ABNA, P. 187 1 4 pro duplo eju8 scribe ejus duplo. P. 190, 1. 15 pro simUia scribe con- tinue proportionales SI, SB, SP, similia sunt. lb. 1. 19, post PE* adde, (ob proportionales IE ad PE ut IS ad SA) P. 191, 1. 7, %e corpus P erit ut ^„ . — - — . P 196 1. 25 post qua annidi acZtZe centro A intervallo AE in piano praidicto descripti. P. 197 1 24 pro diametro lege semidiametro. Corrigenda et addenda in Lib. ii. Pag. 213, lin. 10, 12 Pro EC et BD scribe BACH et BADE, lb. lin 14 post partes adde recta) AB, lb. 1. 24 jyro BC scribe BACH. lb. 1. 26 pro AH scribe BACH P. 214, 1. 33 post gravitatis qua adde corpus illud. P. 223 1. 20, 22 pro sesquialtera scribe sesquiplicata. P. 229, 1. 7, 8 lege omne ascendendi ad locum summum ut Sector Cir- culi, et tempus omne descendendi a loco summo ut Sector Hyperbolae. lb. 1. 13, 14, 15 post Circularis AtDui tem- pus lege omne ascendendi ad locum summum, & Sector Hyperbolicus ATD ut tempus omne descendendi a loco summo ; si modo Sectorum. lb. 1. 21, post ut lege qDp y^ ^2)quad. - — ^^j , id est, ob datam tD, ut. lb. 1. 26 post pjj mento adde velocitatis. 1. 30, post est ut adde tempus totum ascendendi ad locum summum. q.e.d. P. 233, lin. ult. pro ZQRo lege 2 QRo\ P. 240 1. 27 pro MX lege NX. P. 241, 1. 13 Parabola prjedicta?. P. 244, 1. 22 lege FG. Pag. 248, 1. 2 lege sit. n — 2 lb. 1. 10, pro omnis futuri lege totius lb. 1. 23, ^2 pro futuri lege totius. P. 249, 1. 20 post tempus adde totum P. 251, 1. 32 post et AB ut adde area. P. 255 1. 8 pro sit lege est. P. 285 1. 17 ^^05^ arcubus adde vel. P. 290 1. 31 pro aero NEWTON AND COTES. 163 scribe aqua. lb. 1. 34 pro aqua scribe acre. P. 300 1. 11 pro CD scribe AB, P. 301, 1. 7 post axis sui ndde unifomii- ter progrediendo. lb. 1 9 post diaraetri sua) adde uuiformi- ter progrediendo. lb. 1. 12 pro totum globi motum lege motum globi. lb. 1. lb ^ post diametri sua) adde uniforiniter progredicndo p. 317 I. penult, pro maximam G lege maximam //. Corrigenda ct Addenda in Lib. in. Pag. 358, 1. 3, 4 lege affirraatur. Corpora plura dura esse experimur ; oritur autem. P. 367, 1. 14 lege foret. P. 378 1. 28 pro circa annum lege anno. P. 379 1. 13, 23 pro centripetam lege centrifugam. P. 387 1. 22 lege quam. P. 396, 1. 17 pro erit Kk ad lege erit FK aqualis TK & Kk erit ad. lb 1 19 post FKkf adde erit. P. 399 1. 6 post Solem adde vel ab ea supcratur. P. 415 1. 12, 15 pro annua et annu«3 scribe scmcstris et semestri. P 422 1. 34 post hajc a)quatio adde maxima. P. 425 1. 23 dilatet. P 444 1. 33 dele formata est, et post inter se adde formata sunt. P 450 1. 16 lege ad ejus velo- citatem. P. 453 1 17 lege quorum AM, P 457 1. penult. & ult. post manentem dele parum diligenter definivit. Nam Cometa, ^ scribe ex observationibus definire neglexit. Cometa autem. P. 459. 1. 3 lege partium 100000. P 459 proxime post Tabulam lege Apparuit etiam hie Cometa mense Noverabri praicedende* in signis Virginis & Libra) ut Stella secundae vel tertiae magnitudinis, & Florentia? qui- dem ad horam octavam Italicam ea nocte quae mensis hujus diem vigcsimum & vigesimum primum intercessit, st. novo, id est, decimum & undccimum st. vet. visus fuit in signo Virginis sub stellis in sinistro pede [vel fcmure] Leonis cum Ascentione* recta graduum 165, referente Cas- sino. Erat igitur Cometa in mj 13^ circiter. Nam et 11—2 164 CORRESPONDENCE OP Hillus quidam hora quinta matutina die 12 vel potius 10 Novembris, Cantuaria) in Anglia distantiam ccepit* hi\ju3 Cometaj a Corde Lconis graduum scptcndccim in Orien- tem et a Cauda Lconis paulo plusquam graduum undecim in austrum. Unde Cometa tunc crat in -njj 12^ 24' cum latitudine borcali 2^ circitcr. Crassissiraoe fuerunt ha; obsen'ationes ; meliores'^glmt quae scquuntur. Pag. 459 lin 35 post Galletius etiam scribe Avenioni. lb. 1. 39 Cellius in =a. 13. so' lb. 1. 40 dele Roma;. P. 460 1. 33 post Au- strali l'^ 16' adde Cellius in ^ 28. lb. 1. 37 post, id est 2^ 2' vice linearum quinq: sequentium adde. Eodem die ad horam quintam matutinam Ballasora; in India Oricntali, capta est distantia Cometa; a Spica i^jt 7^ 35' in Orientem. In linea erat recta inter Spicam et Lancem australem, ideoq : versabatur in === 26^'. 58', cum Latitudine australi l'^'" ll' circiter ; et propterea post horas 5 & 40', ad horam scilicet quintam matutinam Londini erat in ^ 28^^ ll' cum Lati- tudine australi 1'^'' iG' circiter. Pag. 462 lin 30 post factoe videntur adde Die 22 ubi Cometa ex observatione Monte- nari crat in nt S*^*" 36' Venetiis, & propterea in ii\ 2^" 48' eadem hora matutina Londini : Hookius nostcr eundcm locavit in ir|^ 3 . 3o' ut supra. Montenarus in dcfectu Hookius in cxccssu errasse videntur. Nam et Ballasoraj eodem die ante ortum Solis, Cometa obscrvabatur in ni ^^^ 50', ideoq: eadem hora matutina Londini erat in nj, 3^"^ 5'. Die 24 ad horam quintam matutinam Ballasora; Cometa obscrvabatur in ii| ll^** 45', ideoq: ad horam quintam Lon- dini erat in m. 13^ circiter. Pag. 463 m Tabula priore pro ^ 27 . 52', m 2 56, rri 12 . 58, letje ^^ 28 . 0. m 3 . 5. rr^ 13 . 0. lb. initio secundoi Tabuloe addantur Novem. 9. 17 | 101551 | m, 12 . 25 . 50 1 . 43 . 30 Bor. Pag 472 lin 27 lege cadent. Pag 474 lin 23, inter Et et similis lege in Chronico Saxo- nico. lb. dele llOl vel. lb. lin. 26 post habet adde etiam. NEWTON AND COTES. 165 P. 478, 1. 25 pro prima lege sccunda. P. 482 1. 2, post spa- tiis adde ob defectum neris. lb. lin 18 lege ut se mutuo quam minime trahant. lb. 1. 29 lege non in corpus pro- prium (uti scntiunt quibus Dcus est anima mundi,) scd in servos. P. 483 1 36, post Fatum et Natura. adde, A neces- sitate Metaphysica, qua? utiq: eadem est semper et ubiqr, nulla oritur rerum variatio. Omnis ilia quae in mundo conspicitur pro locis ac temporibus diversitas a voluntate sola Entis necessario existcntis oriri potuit, Dicitur au- tem Deus per Allegoriam videre, audire, loqui, ridere, amare, odio habere, cupcre, dare, accipere, gaudere, irasci, pugnare, fabricare, condere, construere, & intclligentes (vitam infundendo) *<2:enerare. Nam sermo . ' c ^ . . . • Job 38 7 omnis de Deo a rebus humanis per similitu- , * „ * * * ^ Luc. 3. 38. dinem aliquam desumi solet. Et hajc de Deo; de quo utiq: ex pba?nomenis disserere ad Philoso- phiam cxperimentalem pertinet. The following notes are in Cotes's hand : tlioy are the elements of the next letter. p. 3. 1: 14 p. 41. 1: 3 p. 47 1: penult. p. 47. 1: 4 non emend. p. 109. in schem. non H pro O p. 148. 1. 7 n. p. 151. 1. 8. 18, 21 n p. 191. 1. 7 w. itjj .12«.25'.50" non lit p. 230, 1. penult, post incremento adde vclocitatis p. 460. p. 462 n intell. 166 CORRESPONDENCE OF LETTER LXXXV. COTES TO NEWTON. * S' I lately received from You by M' Crownfeild a Paper of Errata, Corrigenda & Addenda to be printed* & bound up with Your Principia. I take leave to send You some observations upon them. By comparing Your Catalogue with my Table of Corrigenda, I find you have omitted that of pag: 3. lin : 14. I think it convenient to make some such alteration, that You may not seem to assert what is false. You have also omitted that of pag, 47. lin. penult, which I think is requisite to determine Your meaning. "Whilst that Sheet was printing I remember I did not understand what it was that You there asserted, & not having then time to ex- amine the thing to the bottom, I was forc'd to let it go. Soon after I considered it, & found in what sense You{r} words could be true & accordingly made the Alteration. Since Your book has been published I have been ask'd the meaning of that place by one who told me he knew not what sense to put upon Y' words: I referr'd him to the Table of Corrigenda & then I perceiv'd he understood You. Your addition of pag. 47 lin. 4 should I think be omitted. For if that addition be made the 8 preceding lines are to no purpose & ought to be omitted. Tis very evident that PF is equal to _, by pag. 46 lin. ante- -» o penult. In pag 109 You direct to put H in the Figure instead of O. You mean instead of the lower O which bisects the transverse diameter of the Hyperbola. If this be • I am not aware that this table of Errata was ever printed. Cotes does not seem to Imvc been altogether pleased at the receipt of so formidable a list. NEWTON AND COTES. 167 done, then the Figure will not agree with the second line of this page, nor indeed with the whole Demonstration as it relates to the Hyperbola. In pag. 148 : lin, 7. I think the alteration should not be made. There arc three different distantice, & three different termini & one common angular motion. Pag. 151. You change prima the Fajminine into pri- mum y® Neutre. Tis my Opinion that this alteration is not necessary. I understand the printed text thus : prima duantm medie proportionalium quantitatum. If it were ad- viseable to make an alteration, I woiUd rather choose the Masculine & put it ; primus duorum medie proportionalium terminorum inter ^c, Pag. 191. lin. 7 I think wants no correction. I cannot understand by what reasoning You make one; You "will be pleas'd to reconsider it. If Your correction be true, it will be very necessary to explain it more fully. Page 463 in the beginning of the second Table I sup- pose You intended to put itK 12^ 25'. 50" not vri 12, 25. 50 as it is in Your written copy You order the 3 last lines of page 460, & the 2 first of page 461 to be struck out ; & in their room You place what follows. [Eodem die ad horam quintam matutinam Ballasora9 in India Orientali, capta est distantia Cometa; a Spica trR 7^. 35' Londini, erat in .^ 28^. ll' cum Latitudine australi 1^. l6 circiter.] I suppose You intended to make this addition at the end of the Paragraph which begins with Nov, 21. PonthoBtis 4-c. & would not have the 5 first lines of the following Paragraph struck out. I observe You have put down about 20 Errata besides those in my Table. I am glad to find they are not of any moment, such I mean as can give the reader any trouble. I had my self observed several of them, but I confess to You I was asham'd to put 'em in the Table, lest I should appear to be too diligent in trifles. Such Errata the 168 CORRSSrONDENCE OF NEWTON AND COTES. Reader expects to meet with, and they cannot well be avoided. After You have now Your self examined the Book & found these 20, I beleive You will not be surpriz'd if I tell You I can send You 20 more as considerable, which I have casually observ'd, & which seem to have escap'd You : & I am far from thinking these forty are all that may be found out, notwithstanding that I think the Edition to be very correct. I am sure it is much more so than the former, which was carefully enough printed ; for besides Your own corrections & those I acquainted You with whilst the Book was printing, I may venture to say I made some Hundreds, with which I never acquainted You I am S*- Your very Humble Serv* Dec. 22^ 1713. II. Cotes END OF CORRESPONDENCE ON THE PRINCIPIA. LETTERS OF NEWTON TO KEILL. LETTER LXXXVI*. NEWTON TO KEILIi. Yo' Letter of Feb. S*** I delayed to answer till tlie Journal Literaire for November and December should come out. It is just come from Holland & I desired M' Darby to send you a copy w*^'* I doubt he has not done because he sent one to me this morning w*^^ I reccon to be for you & I designe to send it to you the first opportunity by the Carrier. JM"" Leibnitz in August last, by one of his correspondents published a paperf in Germany conteining the judgment of a nameless Mathematician J in opj)osition to the judgment of the Committee of tlie Hoyal Society, with many reflexions annexed. This paper hath been sent to ]\r Johnson with remarks prefixed to it. And the whole is printed in the journal Literaire pag. 445. And * Letters LXXWI,, XCII., XClll. were formerly among the papers belonging to the Luodsian ]*rofessor. t A "cliarta volans," ilated 29 Jul. 1713, without name of place, printer, or author. 4: i.e. John Bernoulli nx the letter of June 7, 1713, to Leibniz. There are two circumstances connected with this letter — one of them affecting the writer of it, the other his correspondent— which are not calculated to add lustre to either of tliese great names. To mention the latter first: l^einoulli accompanied the letter with the request that in any use that might be made of it, his name mig-ht not be mixed up with the con- troversy. Leibniz observed his friend's injunction of secrecy at the time, but between two and three years afterwards, without Bernoulli's permission or knowledge, he quoted the letter with Bernoulli's name, in letters to Count Bothmar and — {qux legal ipia Ijycoris) — Madame lu Comtesse de Kilmansegg, He had shortly before intimated the fact in the plainest terms in his letter of April 0, 1716, to Conti for Newton. The other point alluded to wears a more serious aspect. Though Bernoulli was confessedly the writer of tl>e letter, (which accordingly appears in his Correspondence, published during his lifetime), he afterwards (1719), in a letter which he sent to Newton, dis- avowed the authorship of it. The following references will be sufficient to enable any reader to form his own judgment upon these two points. Leibn. ami Bernoull. Cimmefi. ii. 311, 323, 330, 334, 37}3. Leibniz. 0pp. iii. 459, 4()2. Macclesfield Cor- respomtenre, ii. 4'M). T)es Mai/.eaux to Conti, MSS. Birch, 42«1. fol. '222, Brit. Mur. 170 LETTERS OP now it is made so publick I think it requires an Answer. It is very reflecting upon the Committee of the Eoyal Society, & endeavours to derogate from the credit of some of the Letters published in the Commercium Epistolicum as if they were spurious. If you please when you have it, to consider of what Answer you think proper, I will within a Post or two send you my thoughts upon the Subject, that you may compare them w*^ your own sentiments & then draw up such an Answer as you think proper. You need not set your name to it. You may write either in English or in Latine & leave it to M*^ Johnson to get it translated into F{r}ench. M"" Darby will convey yo' An- swer to the Hague. I am Yo** most humble Servant London. 2 Apr. 1714. Is. Newton For D' JoiiN Keh^l, Professor of Astronomy, at his house in Ox^ ford. LETTER LXXXVII*. NEWTON TO KEILIi. S'^ I am glad you have read both the pieces concerning the Commercium inserted in the Journal Literaire & are of opinion that they must be immediately answered & are thinking of an Answer. As to what you want to know concerning things in the Principia contrary to the doctrine of fluxions or differences I take it to be this. In the Scholium of y* 10*^ Proposition of the second book of the Principia I have made use of y® method of Infinite Series for determining the Curves in w*^'' Projectiles will move in • This and the two following Letters were "the gift of Mr Watson, fellow of the College, 1771," (afterwards Bishop of LlandafT). They were formerly placed in a folio volume, which is now marked R. 4.G9. NEWTON TO KEILL. 171 a resisting Medium such as is air. John Bernoulli has published in the Acta Eruditorum for Febr. & March was a twelve month, a I*aper upon that Scholium, in w*** ho represents that the Method there used is the Method of fluxions, & that it appears thereby that I did not under- stand y* 2** ffluxions when I ^vrotc that Scholium because (as he thinks) I take the second terms of the series for the first fluxions, the third terms for the second fluxions & so on*. But he is mightily mistaken when he thinks that I there make use of the method of fluxions. Tis only a branch of y® method of converging series that I there make uses of. The Acta Eruditorum for the last year are but just come to London, & I find thereby that John Ber- noulli is the great Mathematician f who accuses me on this account. But I beleive it's better not to reflect upon hira for it nor so much as to name him any otherwise then by the general name of the great Mathematician. They are seeking to pick a quarrell with me & its better to lett them begin it still more openly without a provocation. There is another great Mathematician J to whom Leib- nitz referred the examination of the Commercium Episto- licum. He makes use of two arguments against me. One • See p. 142 note. An abortive attempt haa been made to revive this delusion by M. Jean Trembley (Berlin M6moires, 1798) in a paper which professes to overthrow Lagrange's explanation of the real source of the error in the expression for the resist- ance given in the 1st edition of the Principia. Lagrange has shewn (Th^orie dea Fonctions, Paris, 1813. pp. 339—349 : see also p. 6) that if powers of (the time of describing a small arc) above the square be neglected, we get Newton's first result, but that if we include terms involving 0^ we obtain the correct value. He has not, liowever, pointed out in what respect Newton*s ^omctrical expression is erroneous, or at what step of the demonstration the fallacy is introduced. The error consists in substitulinjc FG (which = /lo»+5o»= Jg0»-J^^ 03) for //r (which - Wu'+tJS.)"" a if 0"+i^ *^ e''). where r= resist, and u = vel. I am fully sensible of the danger of dissenting from that great geometer on a point of mathematics, but 1 think that a remarit to the effect just stated would have been less open to objection than his mode of arriving at the correct expression by substitution in an erroneous formula (p. 347. lines 15, 16, 17.) Lncroix (Calc. Uiff. et Int. torn. 3. p. 644. Parw. 1819) does not seem to have read the part of the Principia in question with mixh attention. t i.e. the "emincns quidam Mathematicus," quoted in the Charta Vohns. Sec next page, line 3. :J: John HernouUi. Sec preceding Letter and note. 172 LETTERS OP that I made no use of the prickt letters till of late, the other that when I wrote the Principia I understood not the second fluxions as a certain great Mathematician (Ber- noulli) has observed*. The Answer is that I use any nota- tion for fluents & any other notation for fluxions, & an unit for the fluxion of time or its exponent & the letter o for the moment of time or of its exponent, & the rect- angles of the fluxions & the moment o for the moments of other fluent quantities. That in the Analysis per cequatio- nes nuraero terminorum infinitas I represent fluents by the areas of figures, time by the Abscissa flowing uniformly, the fluxions of fluents by the Ordinates of curves, the moments of fluents by the rectangles under the Ordinates & the moment of the Abscissa : but do not confine my self to any certain symbols for the Ordinates or fluxions. That I do the same in the book of Quadratures & even to this day. That where I use prickt letters they signify not moments or differences w^^ are infinite little quantities but fluxions or the Ordinates of curves as the exponents of fluxions w^** are finite quantities, unless they be multi- plied by the symbol o (either exprest or understood) to make them infinitely little : but it is not necessary that the Ordinates of curves should be represented by prickt letters Such letters may be a convenient sort of notation but not necessary to the method. That prick letters are older symbols for fluxions then any used by M' Leibnitz : for he has no symbols for fluxions to this day. That the rect- angles under the Ordinates of curves & the moment O are older symbols for moments or differences then any used by ISV Leibnits they being used by me in my Analy- sis abovementioncd communicated by D' Barrow to M*" Collins in the year 16G0 & the symbols dx Si dy being not used by M"* Leibnitz before the year 1C77. And whereas IVl' Leibnits prnefixes the letter / to the Ordinate of a • "yueiiiadiuodum nb cininento quoj oKlcr, having been entered ut the mature age of 27. COTES TO NEWTON. 183 IIS it appeard to Me. The longer & brighter braneh of this Cross lay very nearly along the Eeliptick, the light of the shorter was so weak that I did not con- stantly see it. The colour of the Light of both was the same : I thought it was not so white as that of the Ring even in it's fainter parts, but verg'd a little towards the colour of very pale copper. You may observe, that in my Figure the branches of the Cross are represented as bounded by parallel lines, for so it was they appear'd to me. But there are others here, who saw a very differ- ent form. I have therefore sent You another Figure fy m^MMMkmi\4\mmA¥*iim'^ , 184 LETTER OF the mo8t remote of any I have met with from my own, This was dra^vn by a very ingenious Gentleman represent- ing the appearance as seen by himself. He differs also from me in this particular, viz' that he takes the Cross light to be only a continuation of the Ring whereas I make 'em to be intirely distinct from each other. I am Sir. May 13. 1715. Edmund Halley bom in London 1656, died 1742. . LETTER XCII. HALLEY TO KEILL. Dear S"^ London Octob 3° 1716 We have printed a French translation of y® account of the Commercium given in the Transactions*, in order to send it abroad : S"" Isaac is desirous it should be publisht in the Journal Literaire, and M"^ Gravesant has promised to gett it done, but cares not to do it as of his own head ; and therefore proposes~EIiat you would signifie to M*" John- son at the Hague, by a letter enclosed either to S'^ Isaac or me, that you are desirous that the said French paper be inserted in his Journal, as containing the whole state of y® controversy between you and M"^ Leibnitz. S' Isaac is unwilling to appear in it himself, for reasons I need not tell you, and therfore has ordered me to write to you about it, who have been his avowed Champion in this quarrell ; and he hopes you will gratifie him in this matter by the first opportunity f I have rec*^ Cloaks Lady days rent, but hear not one • For Jan. and Feb., 1715, pp. 173—224. "An Account of the Book entituled Commercium E}HStoUcum " t Keill, gladly enough, no doubt, complied witli the request. The French transla- tion of the "Account" or Abstract, alluded to, was inserted in the 7th Vol. of the Journal Literaire^ pp. 114—158, and 344—365. A Latin translation of the "Account" was prefixed to the 2nd Kd. of the Commercium Epiftolicum, (1722). IIALLEY TO KEILL. 185 word of Spctty; Pmy let mo know what I shall say to him about the Lease, and I will endeavour to make him pay the Years rent due at Lady day, or at least the best part of it, before I come down to you, which will not be long. I am Dear S' your most faithfull Serv* Edm: Halle y. LETTER XCIII. NEWTON TO KEILL. D' Keill I received about a month ago the inclosed Letter from M' Monmort*. It conteins some extracts of Letters to him from IVr Bernoulli & his son. The chief point is that W Bernoulli denies f that he is the author of y* Memoir entituled Epistola pro eminente &c that is inserted in the • Born 1678, died Oct. 7, (N.S.) 1719. Ho acted as a sort of messenger between tlio Codes of the Leibnizian bridge, as Fontenelle calls Demoulii, and some of the English mathematicians. See his Eloge by Fontenelle. We see him here-, and on another occasion (p. 187), in the amiable character of a peacemaker. The extracts from his letters, which were emulously published against each other after liis death, by the belligerent parties, shew that he could go considerable lengths in adapting his lan- guage to suit the different tastes of his correspondents. His pen has left us an im- passioned tribute to the beauty and accomplishments of Newton's niece, Miss Catha- rine Barton. Letter to Taylor, Apr. 1716, in Coutemp. Philos. p. 93. t In the Leipsic Acts for the following June, by way of Appendix to a paper on trajectories, Bernoulli's eldest son, Nicolas, then 23 years of age, took occasion to refer to the subject of the " Epistola pro eminente Mathematico," and to express his father's annoyance at the rumour which attributed it to him. He admits, says Nicolas, that at- the request of a friend, he put down in writing, *' sine ulla animi commotione," the main of the facts contained in the Letter, but his responsibility did not extend to the "modus scribendi" and form in which the Letter appeared. In confirmation of this, Nicolas, whose Latin, at this stage of his explanation, becomes somewhat obscure, points to the ludicrous oversight into which the toi-dhant writer falls towards the close of his diatribe, where the mask drops and Bernoulli is found speaking in his own person. " Examinentetiam considerentque, quam brevi via quamque diversa a New- toniana incesserit Bernoullius, {in the solution of the inverse problem of central forces}, dicantque postea, an alius quispiam praiter antagonUtam sibi persuadere possit, meam formulam ex Newtoniana esse dcsumtam." Leipsic Acts for July, 1716, p. 314. 186 LETTER OF Acts of Leipsic 1716. The Memoir it self lays it upon M*^ Bernoulli by the words meam solutianem, & if M*" Ber- noulli is injured thereby it is not you but the author of the Memoir who has injured him. The injury is public & in justice requires a public satisfaction, not from you but from him that has done the injury. The question is therefore whether you will take notice of M*^ Bernoulli's excusing himself in private or leave him to do it in publick. I have not yet returned any Answer to M' Monmort, be- cause I thought it best to stay till I had your sense upon this matter. I think to discourse also your friends D^ English* & D" Bower about it. I am Your faithful friend & humble Servant London. 2 May. 1718. {Friday}. Isaac Newton I pray return M*" Monmorts Letter by D"" Halley be- cause I am to answer it. For D' John Keill, Professor of Astronomy at Oxford. This letter, as has already been observed, p. 1 78. may have been the means of inducing Keill to suppress the answer whicli he had prepared to the '* Epistola pro cniinente Mathcmatico." Fragments of it, how- ever, may bo discerned in a Latin dross in the first few pages of a sub- sequent publication, the origin of which may claim a notice here. • Keiira cousin, John Inglb, M.D. Among the Lucasian MSS., (packet No. 3,) there are two short letters from him to Keill. In the first of them, (Dec. 19, 1717), after congratulating him on his marriage, the writer proceeds as follows : " Your papers have been in Sir Isaac's hands ever since they came into mine, and as yet I have heard nothing about them ; but as soon as 1 receive them, I shall endeavour to forward them to Holland by the first sure hand." These " papers *' were probably Keill 's answer to the Epistola pro eminente Malhematico. See antea, p. 178. The second letter, (Jan. 14, 171^), also relates to the aforesaid "papers." " 1 acquainted Sir Is. Newton that you was fully satisfyd with his corrections, and referr'd the whole to his judgement ; which he received very kindly, though he had been impatient to hear from you. But you have forgott to send me back his paper, as we had done to take a copy of it, and therefore you must send it me, to free Sir Is. of the trouble of going over it again Doctor Bower is yours." Bower was M.D. and Professor of Mathematics at Aber- deen. Ho and Inglis were Fellows of the Royal Society. NEWTON TO KEILL. 187 In the Journal Literaire for 1716 Keill had published an article* in defence of Newton against some remarks of John Bernoulli and his nephew relative to the inverse problem of central forces and the error in the 10th Prop. Book 2, of the 1st ed. of the Principia. An answer to this, framed under Bernoulli's eye by a pupil of the name of Crusius, appeared in the Leipsic Acts for October 1718, which had the effect of rousinn; Keill once more, lie drew up a reply to it in the shape of a Latin letter to Bernoulli, but while the brochure was passing through the press, Newton shewed him a letter which he had received (July 1719) from Bernoulli through Monmort, disavowing the authorship of the famous letter of June 7» 1713. Upon talking the matter over, Keill seems to have consented to proceed no further with the publica- tion of his pamphlet t. His pacific intentions, however, were scattered to the winds by the arrival of the May number of the Leipsic Acts (1719) coutaining a paper by Bernoulli^ in which that mathematician ushers * There is a MS. copy of this among the Lucasian papers, (packet No. 5) : it is en- titled ** Apjlogle pour le Chevalier Newton, dans laquelle on repond aux remarquet de Messieurs Jean et Nicolas Bernoully inserees dans les M^moires de I'Academie Hoyale des Sciences pour les ann^es, 1710 &c 1711, par J. Keill...** It appears tliat on Jan. 19, 1716, Halley wrote to Fontenelle with a view to this moreeau of Keill's being inserted in ihe Mimoires de l*Academie, where the papers against which it was directed had appeared. Monmort spoke in 'favour of the application, but the feeling of the majority of the members was adverse to it. (See Coutemplatio Phib$i*phicaf p. 85.) Fontenelle in his answer, (dated March 8,) a copy of which, ia Keill's hand, is extant in a folio book in the custody of the Lucasian Professor, says, *< Nous ne ccdons point ici aux Anglois meme en estime et en veneration pour M' Newton. Kt PAcademie voudroit fort qu' il fust possible " to insert Keill's paper in their Memoirs, but that it was their invariable rule to admit only articles written by members of their body. t Quantum sentio, a litibus in posterum abstinebit, (draught of a letter of Newton in Macclesfield Corres. u. 437. ) 1 assume that the letter, of which the draught is printed in the work referred to, without date or address, was addressed to Monmoit, (about the end of July, 1719,) though the editor (Preface, p. x) states that " it was found impossi* ble clearly to make out the date." '1 he point may be set at rest, if the letter to which thU is an answer, should turn up among tlic; Portsmouth Papers. X Joatinis Bernoulli Hespamio ad Non neminii Provocatumem, ejutque solutio quit$ti' onis ipai ab eodem propositu: de invenienda Linea curva quain describit projectile in medio resistente. Lt'ipsic ActSj May, 1719, p. 216. BernouU. 0/>/>. ii. 393. The tone and language of this piece are such, that even Ikrnoulli's friends, tlic conductors of the Acti, thought it necessary to apologize for inserting it without modification. Injustice to Keill, it ought to be observed, that the problem which led to this ex- plosion does not appear to have been sent as a challenge to Bernoulli, and still less to foreign mathematicians, as has been represented. It was mentioned incidentally in a pri^ate letter of his to Taylor, in which he expressed a wish that Bernoulli would apply his skill to questions of real utility (as, for instance, the one referred to, which Leibniz had attempted in vain), instead of wasting it upon such problems as that of Trajectories. An extract from this letter was (contrary to Keill's intention, and with- out his knowledge) sent by Taylor to Monmort, who forwarded it to Bernoulli. Keill seems to have intimated to Monmort, his dissatisfaction at the extract being communi- 188 LETTER OP in a construction which he gives of a generalization of Keill's projectile problem by a most violent attack upon its proposer. Forbearance was out of the question : Keill let loose his " Epistola ad Jo . . . Bernoulli," (London 1720) and gave further vent to his feelings in an ** Additamen- tum" appended to it, which he closed with some stinging extracts from Monmort's letters to Taylor who kindly supplied them for the pur- pose, — a species of weapon which enabled Bernoulli afterwards to take ample revenge by turning it upon Taylor (Leips. Act. May 1721, p. 207 seqq. BemouU. 0pp. ii. 493. seqq.). There are rough draughts of Keill's letter in English and Latin among the Lucasian papers, and part of it was read by Ilalloy (no doubt in the original English) at a meeting of the Royal Society May 28, 1719 at winch Newton presided. Before publishing it, Keill laid a complaint before the Royal Society agiiinst his adversary "for affronting him with scurrilous language," and called upon the Society to take steps " to shew their dislike of such foul proceedings." "The President ordered that the consideration of this complaint be deferred till Dr Halley (Secretary) comes to town, & that enquiry be made into precedents for the better information & direction of the Society." Journal Book, May 26, 1720. The Society does not seem to have moved any further in the matter. J. A. Arlaud or Arland, an eminent painter, bom at Geneva 1668, died 1764. "Newton fut son ami, et lui fit present de la version fran9aiso de son Optique ; il etait en correspondance avec lui." Biogr. Univ. At the age of 20 he went to live at Paris. LETTER XCIV. NEWTON TO ARLAND. Vir celeberrime, Gratias tibi debeo quam maxinias quod Schema expe- rimenti quo lux in colores primitivos & immutabiles sepa- ratur, emendasti, et longe elegantius reddidisti quam prius. Sed et me plurimum obligasti dum Schema illud in lamina cated to Bernoulli, for among the Lucasian papers, (packet No. 2) we find a very civil letter from Monmort to Keill, (it is not dated, but bears the London post mark, **Nov. 5," probably in 1718), in answer to one from Keill to him, (dated Sept. 3) in which he states that he thought that the extract was intended to be sent on to Bernoulli, and protests that if he had had any idea of the offence that he should give, he would never have sent it. NEWTON TO ARLAND. 189 senea incisum & inter imprimendum obtritum, refici curasti, ut impressio libri* elegantior redderetur. Gratias itaque reddo tibi quas possum amplissimas. Quod invcnta mca de natura lucis & colorum viris summis, D"° Cardinali Polignacf & D^^^ Abbati Bignon non displiceant, valde gau- deo. Utinam hajc vestratibus non minus placercnt quam elegantissimso vestrie & perfectissime delineatce picturro nostratibus placuerunt. Ut Deus te liberet a doloribus capitis & salvum conservet, ardentissimo precatur Servus tuus humillimus & obsequentissimus Dabam Londini 22 Oct. 1722. Isaacus Newton J. Celeberrimo Viro D°° Arland • Peter Coste's French translation of Newton's Optics^ Paris, 1722. t Born 1661, died 1741. Author of Anti- Lucretius (a posthumous Latin poem). It is said that he took great pains to have Newton's fundamental experiments on light properly performed in France, and had the honour of receiving a letter of thanks from our philosopher in consequence. X The original is in the Library at Geneva, to which institution Arlaud bequeathed several medals, paintings, &c. COTES'S CORRESPONDENCE WITH HIS UNCLE. LETTER XCV. COTES TO JOHN SMITH. Written while CotoB was at St Paul's School. S'. {London Dec. 31*. 1698} I am now very well recovered ; and am I thank God in as good health as ever. As for y* works of Kepler, and GalilsDO as far as I can learn they are dispersed in divers Volumes, put forth at different times. I have from severall choice Catalogues, as Draudius's Bibliotheca classica, A Catalogue of y* Mathematical! books in y® Savilian Library at Oxford, and y*t immense one of D"^ Francis Bernard's Library which is now under y® Auctioners Mallet at Lon- don and is Like to continue so for many Months, and severall others collected what I could find of those t{w}o Learned Authors. I send 'em you here in y® latest Edi- tions y* I could find there set down. You may from hence pitch upon those you most like of, & I shall be very glad to use my utmost endeavours to procure 'em for You— [Here follows in the MS a long list of Kepler's and Gali- leo's works, which it has not been considered necessary to print}. I suppose there might be added to each Catalogue especially to y' of Galilaeus. Perhaps this is more than You expected of theire Works. The first Tome of Gali- liBus's Works translated into English came out some Yeares ago Jin 1601} ; but y® Second is as yet unpublished • The day of the month is taken from the post mark. t " A Catalogue of the Library of the late learned D' Francis Bernard, Fellow of the CollcKe of Physicians, and Physician to S. Bartholomew's Hospital....which will be sold by Auction at the doctor's late Dwelling House in Little Britain : the Sale to l)egin on Tuesday, Octob. 4. 1698." COTES'S CORRESPONDENCE WITH HIS UNCLE. 191 and perhaps will never see light*. I have my self Galileo's Nuncius Sidereus put out at London in 8^° together with Kepler's Dioptricks and Gassendus's Astronomy ; if you please I ^nll send you 'em. You wrote of y* Quadrature of Curve's, as yet I cannot enquire of any Mathematician about 'em. S"" Edw : Sherbourn in his Appendix to his Translation of INIanilius's Astronom: {Lond. 1G75} tcU's us y* from M*" Isaac Newton is expected a New general Ana- lytical method by infinite Series for y* Quadrature of Cur- vilinear figures. I have D*" Wallis's Algebra {London 1685 j I think I bought it very cheape I am very well pleased w*^ y® Book. The D*"*'. Buisness therein is to shew y® Original, Progress & Advancement of Algebra from time to time, and by what steps it hath attained to y' height at which it now is he give{s} us a full Account of y® Methods used by Vieta Harriot Oughtred De-Chartes and Pell & others and of y® several methods of exhaustions, Indivisibles, Infinites, Approximations &c. amongst other things he speak's of squaring Curves and after other ways of approximations shewed he show's you this of M"^ New- tonf he determin's it impossible to do y® buisness exactly. In my mind there are many pretty things in y* book worth looking into. If you have a mind to see it, or have not seen it already I will send it w*^. Galila)o'9 Nuncius I thank you for your Directions about Instruments in your last letter dated December 21 You your self put me ofof y® Instrumentary way while I was with you but I meant In my Letter such Instrument's y* were not superseded by calculation or some more exact way ; as a Quadrant is ♦ ♦ ♦ bigg as y® * themselves • It was published in 1665, but nearly the whole impression was destmyed by the fire of T>ondon. See Macclesfield Coiresp, i. 120. t From the famous Letters of June 13 and Oct. 24, 1676, to Oldenburg, to be for- warded to Leifiniz. 192 OOTES'S CORRESPONDENCE sometimes be at a loss for But I will not be so bold as to ask my Grandfather for y* larger size. I wi{ll} * • • ♦ * little one in a concave case with y* Cir- cles only which will serve y^ end as well as y® largest size it will als{o} * * » ♦ pocket and ready upon all occasions. I am your very Obedient Servant and Nephew These For y* Reverend W Smith of Lea R. CoTES. nere Oainshorough IN Lincolnshire ^ Newark Bagg The lower part of the second leaf of the letter lias been torn off. LETTER XCVI. JOHN SMITH TO COTES. Dear Cos: Roger Aug: 30, 1701. I was very glad to hear of your welfare by your Father who befriended us w*^ his company about a fortnight ago ; he showed us your letter in w^^ you expressed a feeble inclination to come and see us in y® Country, we thank you for y*, and count it a favour y* you can spare us any share of your affection from your dear M" Mathesis ; I am glad to hear y* she so easily yields to your courtship, and has procured you such signal marks of favor from great men as D' Bently M' Hanbury*; I am sorry y* gentleman is so • Nathaniel Hanbury, elected from Westminster School to Trinity College, in 1677, admitted Minor Fellow, Sept. 17, 1683, (Charles Montagu was admitted Major Fellow on the following day). He published Horologia Scioterica Pralibata. ..L-ond. 1683 ; and Supplementum Analyticum ad JEquationes CartesianaSy Cantab. 1691. A paper by him on a mode of approximating to the value of 7r by the continual subdivision of an arc of 60", was produced at a meeting of the Royal Society, August 17, 1698. He filled various College offices, and we are told by Middleton, that Bentley " took oc- casion to convict him, in a solemn manner, by the testimony of all the College, of WITH HIS UNCLE. 193 overlookt as not to be Vice-pro {fe | ssor instead of M^ Whis- ton; for I believe he has far greater Mathemat: accom- plishments ; I hear he has a great respect for you ; con- sidering therefore y* favorable fair-promising circumstances you are under I cannot forbear presaging in your behalf, w* Ovid did to his friend, Scena manet dotes grandis Amice tuas. Divines you know are stiled prophets, as well as y* poets are, & I fimcy I shall be a true one in this; pro- vided you so moderate your studies as not to impair your health ; a journey into y* countrey once a year would do well for y* purpose ; what ? I warrant you, you have forgot- ten your old Ne quid nimis, & Interpone tuis &c. ♦ but I am resolved to remember you of em now & then ; I had writ to you before but expected ever & anon to have seen you here r there is in y® monthly accounts of y* works of y* learned, for y® year 1700, month December, a method for finding two middle proportional lines, w*^^ to me is false, there being a great error in y® demonstrat: pray look upon it a little ; I should be glad to hear of you, & of any new discovery ; I never saw yet what discoveries M*" Plally has made in his voyage, pray comunicate to me if there be any thing worth while ; & you will much oblige Your most affectionate friend & uncle J : Smith. My wife & son & daughter remember their kind love to you. For W Roger Cotes at Trinity Colkdg in Cambridg Deliver this in at Caxton to go to Camhridg being a common ncearer 8f habitual drunkard, aT;d without inflicting the least cen- sure upon him for all this, made him not long after |in 1712 & 1713} the Senii)r Dean.'* Miscellaneous Works, iii. 356. He was curate of St Michael** for many years. He died in Nov. 1715, and Colbatch was elected Senior in his place. • From that once popular school-book Dionysii Catonis Disticha dt Moribut ad Filium. Interpone tuis interdum gaudia curis, Ut possis animo quemvis sufferrc laborem. 13 194 COTES'S CORRESPONDENCE ITalloy was appointed (Aug. 19, 1698) to the command of the Paramore Pinky with orders to make a series of observations with a ■view to ascertain the law of the variation of the compass, " to call at his majesty's settlements in America & make such observations as are necessary for the better laying down the longitude & latitude of those places, & to attempt the discovery of what land lies to the south of the western ocean." lie set out on his expedition Nov. 29, and was carrying on his observations some degrees south of the line when the insubordination of his officers compelled him to return : he reached England at the end of June 1C99. In the following September he em- barked again in the Pink. In this second voyage after penetrating beyond the 52d degree of south latitude where ho was stopt by icebergs, he turned his course northwards, visiting among other places St Helena (a spot familiar to him by his sojourn there more than 20 years before), Pernambuco, Barbadoes, Bermudas and Newfoundland. After an absence of 12 months he arrived in the Thames in Sept. 1700. His observations were embodied in a General Chart which he published in 1701 "shewing at one view the variation of the Compass in all those seas where the English Navigators were acquainted." The Journals of his two voyages were published by A. Dalrymple (London 177»'>« 4to). These are the voyages to which Smith alludes, and about which the young Cambridge student could give him no information. But while Smith was writing this letter, the Captain was again afloat and en- gaged upon, if ho had not finished, another undertaking in which his activity and spirit of enterprise sought employment. At a meeting of the Royal Society, Juno 18, I7OI "the Vice-President (Sir John IIos- kyns) informed the Society that Mr Halley was gone on a new voyage, as he heard, having designed to make nice observations on the Tides & Currents in the Channel, for the Improvement of Navigation, that thereby by their different times, the going out of the Channel might bo more easy against contrary winds." And on July 30, a letter was read from Halley (Guernsey, 18 July) "giving an account that the weather having been fair for a Month past, he had made a great progress in the designs he had in making this voyage." The fruit of this voyage was a largo map of the British Channel published in 1702. WITH HIS UNCLE. J 95 LETTER XCVIL COTES TO JOHN SMIIH. Hon: S" ! Cambr: Sopt 9. 1701. I heartily thank You for Your kind Lett', & as heartily beg Your Pardon for suffering my self, by so long delay, to be as it were forc'd to returne You an Answ'. You are pleased to express a greate deale of Kindness to Me in Your Prophecies, as You call 'em, or, as I would rath' have it, Y' Wishes ; und' which Name, not y* other, I again thank You for it. I am sorry You should suspect me of forgetting my Ne quid nimia. I have learn't y*' lesson too p'fectly & 'twould be more adviseable (for y® Accomplish- ment of Y' Prophecies) to rememb' me of my Old Midta tulitfecitq: &c. The Mesolabe of y' wretched pretend', y' Quack Geomet' S* Julicn Potier, one cannot but admire for it*s grosness; & much more y* laborious confutation of it published in y* same paper some JMonths after. M' Halley's late discoveries I am wholly ignorant of. Surely You mistake Cambridg. Wee are situated in as dark a Corner of y® Land (in these Matters) as can well be de- sired. You have often mentioned to me y* Quadratures of Curves ; & particularly (which I now call to mind) You have wished to be satisfied in pag. 374 of Ne\vton. I per- suaded my self therefore y* something concerning this Matter might perhaps make amends for M' Halley's Story And y* I may be as short as is possible, I desire You to Consid' 2 Lem. 2 Lib. in which & it's Converse y® grounds of his Method of Fluxions are contained. To come to an Instance. Let AMD be any Curve; AMy AP, PM, any Chord, Abscist, Ordinate of this Cu{r}ve which w*** y® Arch AM are all unstable. Flowing, increasing or decreasing Quanti- tys ; and y^ too after a certain Law, in a certain proportion among themselves, according as y* Nature of y* Curve 13—2 196 OOTES'S CORRESPONDENCE requires. Let then for once AP stret*ch it self, & it's very first increase, it's primum nascem incrementumy it's Fluxion, it's moment be an infinitely little Pp ; w*^ it y* Ordinate, Chord, & Arch shall also change themselves into jt?m, Amy A Mm & their Moments or Fluxions will be Rm, Sm, Mm. The Area AMP will also have it's differentiola or Fluxion MPpm differing from y« D MPpR by y« A^® MRm infi- nitely little in respect of MP pR which is it self infinitely little in respect of AMP : Now quantities whose difference is infinitely little ought to be look'd upon as equall by 1 Lem. 1 Lib. Newt. For y® same reason y® Sectour AmM (which is y*' Fluxion of y® Bilinear Figure AM) may be account'd equall to y® A'® AMS. Now naming y® Abs. Ord. Ch. & Arch X, F, Uy Z. Pp, Rm, Smy Mm, will be Xy y, w, Zy according to y® second Lemma : or rath*" let us name y® Magnitudes themselves Xy y, w, z. & their Fluxions x-y.Uy z. 'Tis evident y* y® Fluxion of y^ Area will be « UMPpR^yx To particularize; let this Curve be y® Parabola, whose Area we know very well otherways. ax^yyy Sx^ ^yy orx^so ^yx ^ Fluxion of y® Area But y® Fluent of ahx^x (by Lem. 2 Lib 2 convers.) = |-a^a;5 « l^jy «= Area). In Newton's Hyperboloeid c? ^ x^y or (j?x~^ « y now in our case y® Fluxion of y® Abscist runs backward & is therefore = - a? and Fluxion of y® Area = - y^ « - c^x~^x whose Fluent y® Area «= d^x'^ or ~ X or xy is reciprocally as x. WITH HIS UNCLE. 197 This may p'haps serve as a Specimen of y* Method of Fluxions applied to y* buisness of Quadratures tho it's uses seem to be as inexhaustible as they are Natural! & Easy for by it y* great Geometers of our Age are enabled To draw Tangents, To rectifie, To find y" Evolutes, The Caus- ticks by reflection & refraction of all sorts of Curves, To measure y'' Surfaces generated by their rotation. The solids they comprehend, The Centers of Gravity, Oscillation & Percusn. of all these To resolve all sorts of Questions de Max & Min. To find y* Points of Inflection & Rebrous- sement (as y® French term it) in all Curves & y* Converse of all these & many more But what wonders docs it not do when applied to Nature ! where it Triumphs alone & admitts of no Partner But I transgress y* Bounds of a Letf Pray S"" pay my humble respects Y' very &c. to my Aunt ; and my Love to Cozz". R Cotes These to the Reverend W Smith Rector of Gate- Burton tiear Gainsborough by Newark '^ Caxton. This letter in which Cotes gives his old master an insight into the powers of the new Calculus was written in the long vacation between his 2(1 and 3d years. It is a very creditable performance for a junior soph. LETTER XCVIII. COTES TO JOHN SMITH. Cotes is now Plumian Professor. His appointment took place Oct 16. 1707. Honoured Unckle I have lately been at London ; I found Y' Letter at Cambridge upon my return. The occasion of my going up 198 OOTES'S -CORRESPONDENCE thither was partly to view a large Brass Sextant* of 5 foot Radius (y* had been makeing for us & is now finished) before it should be sent down. Whilst I was in Town S*" Isaac Newton gave orders for y® making of a Pendulum Clock which he designs as a present to our new Observa- tory. The Sextant will cost y* CoUedge 150^*^ & I bcleive S' Isaac's clock can cost him no less y" SO'**. We have another Instrument in hand for takeing y® Transits of Stars or y® Sun & Moon over y® Meridian & then we shall be pretty v^ell furnished for makeing Observations. All Alti- tudes You know may as well be taken by a Sextant as a Quadrant. We want another 200'^^ if we can procure it in y® University to raise up another Story over y® gate for Astronomical uses. I have lately hit upon a contrivance which I bcleive will be of very good use for observing Eclipses. You will easily understand it by this rude draught The Telescope a & is to be so directed as to look at y* Pole of y® World & thereby its axis will be parallel • December 10, 1707 : "The President in the chair. A draught of a Sextant made for the use of the Astronomical Professor in Cambridge was produced. Dr Harris & M' Halley reported that it was very exactly done by Mr Rowley." Jounutl Book of the Royal Society. A plate with a description of" this noble instrument'* is given in Harris's Lexicon Technicum, Vol. ii. Lend. 1710. WITH HIS UNCLE. 199 to y* Axis of y* Earth in which position it must be fixd by y* rings ef gh hl,cdis o. looking glass reflecting y* Object into the Telescope Tlien if y* Telescope revolve about its Axis within the rings with a motion correspondent to that of y' Earth about its Axis the Object will constantly be in y* Telescope for a whole day togather as You will easily understand by considering y* y* looking Glass parti- cipates of y* same uniform motion by being fixt to y* Tube. I have not described y* method of altering y* Inclination of y* glass according to y* different Declination of y* Object from y® ^Equator You will easily find out how y* may be done as also how a piece of Clock work if it be thought needfull may communicate to y* Telescope its due motion about its Axis*. I thank You for y* kind Judg- ment You made concerning my Paper about Projectiles. I have by me another such a Paper concerning y* motion of Pendulums which I drew up about y* same time with y*t. This or any thing else You know You have a right to command from mc haveing taught me all y' little which I understand in these matters. I am glad Coz Rob' has made so good progress in Mathematicks & y* he has a genius suited to those Study s as I all ways thought he had but I fear You are too diffident of his Abilitys. It will undoubtedly be more for his satisfaction & advantage to be admitted Pensioner y" Sizer, the other way if I can gett him a Poor Schollars place will be about lO''^ cheap"^ I wish You would resolve Y"^ self. I should be very sorry to have advised You amiss & I cannot now be certain of futuritys * It will be seen from this that Cotes anticipated 'a Gravc^ande in the principle of the lleliostAt, by upwards of thirty yenrs. lioth liowovcr had been forestallwl by Ilooke and Halley. llegist. Bk. Roy. Soc. ix. 23. For a description of that instrument see 's Uravesande's Physices V.lementa Mathematica, 3rd Ed. 1742, p. 715; Biot's TraitS de Phyaique, iti. 188. Compare Novi Coinmentarii Petropol. i. 291 ; Coi-40 B P op m /J IT n-5 5EF5 n-30 C pq 7 «-4 i^4 n-20 h m qr 1 n-3 II r m P C2 n-10 ' E rs € n-2 i^2 s m or m £1 n F St C To n-l >I n+10 t G m tv T n S ^/ V m V ir 100 n + 20 II vx e M + 1 li^ X m 4> 2C n + 30 I xy 1 n + 2 2F I in ^ 3C n + 40 y^ < n + 3 3F z in w S n + 50 L 5* \ n + 4 5r n + 5 5FG5 t]4> n + 6 G4 »;3 n+7 G3 »/2 « + 8 G2 ^1 n + g Gl n + 10 n G If] tv 100 M+ll IG 2»/ n + 12 2G In the First Table, Let n signify every 100*^ numb^ & F its root, wheth'. Square, Cube, or Sq. Square ; & ri - 50, n - 40, w - 30, &c. COTES AND JONES. 217 every 10***. numb"^; and J, J?, C, D, &c, their roots; and o, p, 7, r, &c, the differences of these roots ; & op, pq, qr, &e. their second differences, (that is op, the diff. of o & p, pq the diff. of p & q, &c.) and 7?* their third difference, that is, y®. common difference of >| P* ^^ signify the diff. of a, /3, 7, S, &c. And — the common diff. of o, tt, v, «, &c. 10 ^ ^ In the Second Table, Let w - 6, n - 5, « - 4, n - 3, &c signify y'. single numbers, 4>E, 5E or F5, F4>, Fs, &c. their Roots, 5 €y ^4, ^3, ^2, &c the diff. of those roots ; 8t — the common diff. of those differences for y*. ten 100 ^ numbers between w - 5 & n + 5, And so for y®. ten numbers between n + 5 & n + 15 ; let G5, G4, Gs, &c. signify y®. roots; 174, >;3, i;2, &c, their first differences, and - their second differences; and the 100 like for every denarie between n - 50 & w + 50. This explication of the Tables being p'mis'd, you may compute them thus ; 218 CORRESPONDENCE OF Out of «, (Square] extract I Cube jRoot, make ; F y\ ISq. Sq.) r loF 2n lOF 10ft) 20 ft) «/, S0«^ in am. 50s^ 3n 3n 3n lOF SOo) lOst ), ='8t, W. 4» 471 1 1 ^ i*^ ^ « 10 100 6000 + *a r, and — + e (T. ^ 10 1000 And these quantities Fy st, m, s, ^, & <7, being thus found, y\ rest are given by Addit". & Subduct. For 8t+m^r8y rS'\-mcsqr, &c. st'-m='tv, tv-m^vwt &c. Again fi + r««r, r + qr^q, &c. s-st'^ty t-tv^Vy &c. And F^a^Ey jE-r-A &c. F+t^Gy G + v^Hy&c. Further m mm *» « 10 '^ ^ 10 '^ 10 10 Lastly ^ + o- = e, e •fj'^^y &c, ^ - t « jy, »; - v «= ^, &C. These quantities being thus computed, in y®. first Table, to every lO***. number, tlie roots may be computed in y®. 2** Table to every numb^ by Addition and Subduction only ; For r+-ii = n, ti + — = r2,&c. 5» 100 ^ ^ 100 ^ y^fL^ir i^_*^ «2t, &c. ^ inn 2>' ^ inn V 100 100 Again F - ^ = /'I, /"I - ^1 « Fg, &c. /'+ 1^=17^, l/'+2^«2/', &c. • I have added the *qq^'^ I have also corrected some other errors of transcription. COTES AND JONES. 219 Thus you must proceed to five Figures on either hand, and then do the like in the next ten Figures, saying tv ^^ o n + ■» Jjl, tjl + ■• »72, &C. ' 100 ' ' 100 ' ' And the like for every Denarie between n - 50 & n + 50, In these Computations, Note, 1**. That they must be done every where to 10 or 11 decimal places, if you will have a Table of Roots exact to 8 of these places. gdiy If 5p Sc G5, the roots of n + 5 found two ways agree to 8 decimal places, it argues the whole works from which they were derived, to be true. And so of y*. roots of w + 15, w + 25, n - 5, &c. And also of y*. Terms J, * o, & a ; Z, ;» >K, & X, where two works meet. Let this there- fore be y®. Proof of y®. work. This S'. is w*. has occurr'd to me about your design, which I hope will do your business, the whole work being p'form'd by Addit. & Subduct : excepting y*. in y*. com- putation of every 100**^. number, there is required y*. Extraction of one root, & three divisions, to find F, a>, stf & m, S^ I am Your humble Scrv' Is. Newton. The person to whom this letter is written may be conjectured to be "John Smith, Philo-Accoinptant" author of Stereometrie^ Lond. 1673. (He must not be confounded with Cotes's uncle). In the Macclesfield Correspondence, ii. 370 — 374, tliere are two other letters on the ex- traction of roots from Newton to this same person (not to Collins, as tliere printed) dated July 24 and Aug. 27, 1075, in the fonner of which he refers to the metliod given in tlie foregoing letter. Mr J. Smith seems to have had a design of constructing Tables of Square, Cube and Biquadr. Roots, and consulted Newton as to the best mode of com- puting them. The Tables, if ever made, do not appear to have been published. The earliest I'ables of Roots are Briggs's IMS. Tables of the Square Roots of Numbers up to 1000 mentioned in Mayne's Merchant' $ Companion (London, 1674), p. 80. 220 CORRESPONDENCE OF LETTER ex. COTES TO JONES. Answer to Letter CIX, No date. I have received Your Letter with the inclosed Paper of S' Isaac Newton for which I return You my hearty thanks. His method seems to be excellently well suited to those particular purposes for which he designed it, & I do not doubt I shall find it very curious when I have lea- sure to examine it to y*^ bottom. What I intend to print will make but a small Volume, I cannot say it will be big- ger than that of S"^ Isaacs which You lately j)ublished. It ^vill contain the Lectures I have hitherto read in Publick, together with those which I shall read this Year, all of which amount to no more than Ten, for by the Statutes of my place I am obliged annually to make but two. I can- not indeed expect any profit from the Publication, twill be sufficient if y® expense of it can be defrayd. I have already put y® University to the charge of Types for some new characters which I have occasion to make use of & therefore for that reason as well as some others I cannot now draw back. What You mention that y® R: Society have chose me one of their Members is altogether a peice of news to me. If it be so, I shall be very sensible of the Honour they have done me. That Title may recommend my papers to y® Publick though they be printed at Cam- bridge. If You insist upon my Promise of sending those things to You before they are printed I shall be ready to make it good. What I have further concerning y® subject of differences consists of Ten Propositions whereof the Six first are particular & fitted for use & are sufficient for all cases that comonly hapjicn, the other four are general. You will be able to judge of my Method by y^ first Propo- COTES AND JONES. 221 sition which I here* send You. You may shew it to S' Isaac if You think it proper but I desire You would not shew it to others. I cannot so easily give You an Idea of my other peice concerning Logarithms but I find room enough in this Page to send Youf one thing out of it as a curiosity which may be understood independently of the rest. Rectificatio Logarithmica) — Oblata sit igitur Logarithmica &c. LETTER CXI. JONES TO COTES. Accompanying 4 copies of tho Commorcium Epistolicum, [Extract.] S' London Feb Q'\ 17}§ The R. Society having order'd one of their Books for you, & another for M"". Sanderson, also one for Trinity College Library, & one for the University Library ; I wou'd not miss the opportunity of paying you my respects by sending them : I need not tell the occasion & design of that Collection : you'l see readily that it affords such light concerning what it relates to, as cou*d not easily have bin discovered any other way : and also shews that your great Predecessor, whose illustrious Example, I don't doubt but you follow, never imploy*d his time about things ordinary. I have no ^Mathematical intelligence to send you ; M^ Keil * The tract of which Cotes sends a specimen to Jones will be found among his Opera Miscellanea, pp. 36 — 71. The title of it is •* Canonotitchnia sive Constructio Tabularum per Differentias." lie has not copied out the proposition in this draught of his letter, and therefore it will be sufficient to refer the curious reader to p. 36 of tha work just cited. t Here also Cotes has not taken the trouble to transcribe the proposition. It may be seen in his Logometria, {[larmonia Men$urarum^ pp. 23, 24.) 222 CORRESPONDENCE OF thinks he has discover d a very easy and Practical solution of the Keplerian Problem ♦ : the Problem of the Refrac- tion, or that concerning y®. description of the Curve de- scribed by a Ray of Light in passing thro the Atmosphere, is here done by two diflerent hands ; one of them endea- vours to apply it to Astronomical uses, \v^^. I suppose he has pretty well compassed. « • * • • « I am extremely pleas'd to find that S"^. Isaac's Book is so near being finish'd: his general Scholium I presume he'l soon send you, if 'tis not already done : and 'tis not less agreeable to me, to hear that your own Book is in such forwardness. P.S. I have sent to you four of the Comercium Epis- tolic. that is, one for your self, and y® other three as before mention'd which I desire you wou'd deliver, as from the Royal Society of London. LETTER CXII. COTES TO JONES. [Extract.] S'. Cambridge Feb', n^^ I have received Your obliging Letter together with the very agreeable gift of the Commercium Epistolicu. I have delivered one Copy to the University Library Keeper another to the Library-keeper of Our College and the third to M' Sanderson as from the Royal Society. You may be pleas'd to return our acknowledgments of the Favour. • Phil. Trans, for 1713, Vol. xxviii. pp. 1—10. COTES AND JONES. 223 I am very glad to see this Peice at length made pub- lick in which quicquam cuiquam detractum non reperio, sed potius passim suum cuique tributum*. LETTER CXIII. JONES TO COTES. S'. London Aprill 29*\ 1713. Ever since I received your very kind Letter, and Mouton's Book, I waited for an opportunity of sending you some old Manuscripts I had by me, and at last am oblig'd to Venture them by the Carrier ; They relate, in some measure, to the Method of Differences ; The folio one, I find, was "writ by one Nath. Torpcrleyf, a Shrop- shire man, who when young was Amanuensis to Vieta, but afterwards writ against him; he was contemporary with Briggs and Harriot, and intimately acquainted with them ; The Book, I think, can be of no other use to you, than in what relates to the History of that Method, and in having y® Satisfaction of seeing what has bin formerly done on that Subject. The other Small 4*° I\LS. is a piece of Mer- cator's about Differences, it seems to contain no great matter ; nor indeed, can I be satisfied, any thing that he has done, or any one else, so very considerable, as to deserve to accompany any piece of yours ; Therefore pray let us have your things entire, and as soon as conveniently you can. I am mightily pleas'd to see the end of the Principia, and return you many thanks for the very Instructive Index, • Commerc. Epistol. p. 119, (p. 239, 2ntl. Ed.) These are I^ibniz's words in his Letter to Sloane, Dec. 29, 1711, by which he unfortunately mode himself a party to the obnoxious languag-e of the Leipsic review of Newton's tract, "De Quadrature Cur- varum," Leips. Acts, Jan. 1705. t Compare Afacclesfield Corresp, ii. 5, note. 224 CORRESPONDENCE OF that you have taken the pains to add, and hope 'twill not be long before we shall see the Beginning of that Noble Book. I shall be in some pain till I hear that you have re- ceiv'd my old M:S. it being a favorite one, purely upon the account of some extravagancys in it, So very uncommon : But I shall think it safe when in your hands; I am S'. without reserve, your very affectionate friend and most humble Servant W: Jones. LETTER CXIV. COTES TO JONES. Dear S"^ I know not how to return You my thanks as I ought for Your readiness to assist me. The two Manuscripts of Torperly & Mercator are come very safe to my hands; I hope I shall return 'em to You without any damage. I have been lately, and am at present taken up with some College buisness, so that I have scarce yet had any time to look into 'em. If I find any thing in them of Moment, I believe I shall request You to let me print it .with my own, for I would not willingly have any one lose the Credit due to him. I am glad You can approve of the Index to the Prin- cipia. It was not design'd to be of any use to such Headers as Your self, but to those of ordinary capacity. I hope the whole Book may be finished in a fortnight or three Weeks. I have lately been out of Order, or it might have been done by this time I am S"" Your most Obliged Freind May 3**. 1713. and Servant B. Cotes. COTES AND JONES 225 LETTER CXV. JONES TO COTES. Dear S' juiy ii*>» ins Tis impossible to represent to you, with what pleasure I receiv'd your inestimable Present of the Principia, and am mucli concern'd to find my self so deeply charg'd with Obligations to you ; and such, I fear, as all my future en- deavours will never be able to requite. This Edition is indeed exceeding beautifull, and interspersed with great variety of admirable discoverys, so very natural to its great Author ; but is much more so, from the additional advan- tage of your excellent Preface prefix'd ; which I wish might be got publish'd in some of the foreign Journals; and since a better account of this Book cannot be given, I suppose it will not be difficult to get it done. Now this great Task being well over, I hope you'l think of publishing your own Papers, & not let such valu- able pieces lye by : As to w^ you mention*d in your last concerning my Old manuscripts, tho, for my part, I know of nothing worth your notice publickly in them, buit if you do find any, it the more answers the end of my sending it, and you know that you may do as you please ; S"- I am your most obedient humble Serv' W: Jones LETTER CXVL COTES TO {WHISTON.} Dear Sir {March 1715} I have lately seen two Schemes of the great Eclipse the one done by Your self, the other by D"" Halley. Yours being to be understood by those only who are acquainted 15 226 LETTER OF with Astronomy, has upon that account much the disad- vantage of the D"*" with most People. I take the Liberty to propose another Scheme to You, which I beleive would give a more general satisfaction than either of the other : I mean a Map of that part of the Heavens in Avhich the Sun will be at that time. If the sky be clear it will un- doubtedly be a great surprize to see the Stars, but twill be much more so to the Vulgar that You should be able to describe the Positions of 'em beforehand: this I am apt to think they will look upon as a greater peice of art, than to predict the Eclipse itself. By comparing the Ephemeris & Globe together I find there will be three Planets visible on the West of the Sun, Jupiter will be very near him, Venus will be about the Meridian, Mercury will lye between them. You have already spoken of the Moons Atmo- sphere, I think it would not be amiss if You desired Peo- ple to look if tlicy can observe the Suns also, I mean that light in the Heavens which D^ Gregory describes pretty largely in the Scholium to Prop. 8. Lib. 2 of his Astro- nomy. A representation of this may be inserted in the Map if You think fit, that it may be known beforehand how tis likely to appear. You may caution those who are desirous to see this faint light, that they prepare their eyes beforehand for it, by staying in some dark place for about a quarter of an hour before the Sun be totally obscur'd ; You know it requires about that time to bring our Eyes to the disposition they usually have in the night time for see- ing faint Lights. I would further advise, if You think fit to set about this Project, that You do it with exactness that Mathematicians may not dislike it, & that Your Ex- plications be written in a Popular way & as free as may be from Mathematical Terms that others may not dislike it. I suppose You have seen Cassini's Map & Reflections upon the Eclipse of 1699, printed in the Memoires of the Royal Academy of Sciences for that Year. If You have not yet COTES TO WIIISTON. 227 seen it, tis possible it may suggest something further to You. 1 shall not trouble You any longer upon this sub- ject. My Cozen Smith was chosen Fellow the last Elen. He takes his Master's Degree this next Commencement. Ho has already two Pupils & expects one or two more in a short time. He presents his humbl service to You ; both He & my self shall be obliged to You, if You can assist Bim by Your recommendation. I need not tell You, that as he is in all other respects well qualified for that Buiss- ness so he is very capable of instructing his Pupils in some parts of Knowledge which You & I esteem, & which very few Tutors in the University do at all pretend to. This letter was evidently written to AVhiston, who " a little before the famous total eclipse of tlie Sun, April 22, this year, 1715, pub- lished two schemes* of tliat eclipse," in the latter of which lie adopted Cotes's suj^gcstions, though ho makes no mention of his receiving any such assistance. "N.B. This most eminent eclipse, 1715, was exactly foretold by ]\r Flamstccd, D' Ilalley, & myself I myself by my lectures before ; by the sale of my schemes before & after ; by tho generous presents of my numerous & noble audience; wlio, at tho recommendation of my great friend, the lord Stanhope, then secretary of state, gave me a guinea apiece ; by the very uncommon present of twenty guineas from another of my great benefactors, tho duke of Newcastle; and of five guineas at night from tho lord Godolphin; gained in all about £120. by it." See Winston's Memoirs i. 204, 5. • The title of the first is "A Calculation of the prreat EcMpse of the Sun, Apr. 22. 1715, in y* morning, from Mr Flamsteed's Tables, as corrected according to S' Isaac Newton's Theory of y« Moon in y^ Astronomical Lectures " In the 2nd, which is larger and fuller than the 1st, the Eclipse is calculated " from S' I. Newton's last im- provements to his Theory of y«; INIoon." (It is dated, April 2, 1715). In the 1st Whistou had neglected to avail himself of the 2nd Ed. of the Principiuj a fact to which Cotes in the Letter of which we have here only the draught, may possibly have drawn his attention. Time of Eclipse at London. Whiston's Ist Scheme. His 2nd Halley. Flamstced. Observed Time. Beginning Q^ . 18' 8h. 7i' BV 7' 8h. 8' 8»>.6' Middle 9 . 24 9 . 14 9 . 13 9 . 13.i 9 . lO'. 45'' End 10 . 35 10 . 24i 10 .24 10. 24 10 > 20 15—2 228 LETTER OF LETTER CXVIL COTES TO LORD TREVOR. My Lord, Trinity College Cambr. Jan. 10*** 1716 When I waited upon Your Lordship with S"^ Isaac Newton, I remember my Lady Trevor was saying, that S' John Bernard was design'd for our College : I have since heard that He will come to us very soon. I have not been informed whether any Tutor is already provided for Him. If Your Lordship is not yet determin'd, I beg leave to propose one to You, His name is Smith, a Junior Fellow of the College, I have had the oportunity of an intimate knowledge of His Temper Behaviour & Learning, as He has been my Chamber-fellow for some yeares & as He is my Kinsman. I can therefore be bold to recommend Him to You as a person whom I think to be extraordinarily well qualified to satisfie Your expectation in all respects. If You desire to have S*" John instructed in the Mathematicks & the new Philosophy : I do assure Your Lordship, I know no one more capable of doing it with good success, both on account of His very great skill in those things & His easy way of teaching. Your Lordship was formerly pleas'd to desire me to assist M"* Trevor * that way : I was very sorry I might not do Your Lordship that service, for it was not my fault that I did not. The remembrance of it makes me beleive You have the same views for S*" John : I therefore thought it my duty as Avell to Your Lordship as to m}' Kinsman to write thus to You. If the appoint- ment of a Tutor shall be left to D"^ Bentlcy ; I know His opinion of M*" Smith is such, that He will think He cannot serve Your Lordship more, than by naming Him to You I am &c. ' E C • Lord Trevor's eldest son and successor in the title. He was entered a fellow- commoner at Trinity College, June 19, 1708, his tutor being Mr Nic. Clagett, Libra- rian of the Colleg-e, afterwards Dean of Rochester, and Bishop of St David's, from whence he was translated to Exeter. . COTES TO LORD TREVOR. 229 The application made in this letter was successful, but before tho formal result of it was realised, tlio wann heart that dictated it had ceased to beat, and tho grave had parted tho two chamber-fellows. Sir John Bernard was entered a Nobleman under Smith, July 6, \716, Cotes breathed his last on Juno 5. Lord Trevor was one of the twelve peers created by Queen Anno in order to turn tho balance in tho House of Lords in favour of tho peace of Utrecht. Ho was Chief Justice of tho Common Pleas in her reign, but shortly after tho acoesaion of George I. (Oct. 1714) ho was super- seded at^tho suggestion of Lord Chancellor Cowper, and tho appoint- ment was bestowed on Sir Peter King. See Lord Campbell's Chan- cellors IV. 349 note. 592. .593. Ho married for his second wife tho widow of Sir Robert Bernard, a brother of ^Irs Bcntley, and thus became step-father to the young baronet Sir John. On tho publication of the 2d Ed. of tho Principia, Bentley presented him with a copy of it. Bentloy's Correspondence, p. 4G5. LETl'ER CXVIIL COTES TO ROBERT DANNYE. Containing an account of the meteor of the (Jth of March 171 j . Tho following is an extract from tho Journal Book of the Royal Society. " March 7- ^m* Tho President in the Chair A letter of the lato M' Roger Cotes Math. Professor at Cambridge to the Reverend Mr Robert Dannye [dated March 15, 1716} was produced as communicated by M' Jurin of Trinity Coll. Cambridge. It contain'd some very remarkable circumstances seen by him in tho lato wonderful phoenomcnon seen about a twelve month since, as that about \ after seven there was a perfect Canopy of Rays ascending from all parts round tho Horizon, but no where reaching to it being about 10 or 15 degrees high on tho North Side & near forty on tho South, continuing in this state not above two minutes during w'*" interval several Colours appeared, some fainter & more permanent, others brighter but quickly vanishing, with several other curious remarks. This description being better circum- stanced than w* had before been communicated by most other obser>'ers, was thought worthy to bo preserv'd in the Transactions." It will bo found in tho Transactions for ^lay — August 1720. pp. 60-70, and in Smith's Optics (1738) Vol. i. pp. 07-70, and therefore it has not been thought necessary to reproduce it here. 230 CORRESPONDENCE OF COTES. This letter closes Cotos's corrospoiulcnco in the Trinity College Collection. Among the ]\Iacolesfiol(l Letters, however, there is one of a later date, addressed to his friend Jones only a month before his death, in answer to some inquiries respecting the progress of his tables of integrals upon which he was employed. At the beginning of the year he had returned to the subject of the integration of rational frac- tions, and in this letter ho refers exultingly to the success of his re- searches, animadverting upon a paper of Leibniz, (Lcips. Acts, 1702, p. 210) who was unable to integrate -j « . The letter is quoted by Smith (Harmon. Mcnsur. p. 113), and an extract from it is given by him in his account of that work printed in the Phil. Trans, for June — August 1722, pp. 140-148. Leips. Acts, April 1723, pp. 103, 104. One of the expressions which Cotes mentions in this letter as yielding to hb method f .— ,; -^^ , where q is some power of 2 j , Taylor sent to Monmort as a challenge from himself to the mathematicians of the continent, without dropping any allusion to the source to which he was indebted for the problem. Monmort transmitted the question to John Bernoulli and Ilermaim, tiie former of whom replied (Jan. 171^) by oflering to lay Taylor a wager of 50 guineas that he would produce a solution within a stipulated time, but upon condition that he should in his turn propose a problem to Taylor upon the same terms. Taylor at once declined tiie proposal in a lengthy reply, {Contempl, Philosoph. p. 109), but before it came to Bernoulli's hands, that mathematician apprehensive, ho says, lest his silence should be construed by some austere Englishmen (quidam ex severioribus Anglis) into an acknow- ledgment that the problem was beyond the strengtli of foreign analysts, had sent his solution, wliich ho had soon hit upon, for insertion in the Leipsic Acts (Leips. Acts, June 1710, p. 250. Bernoull. 0pp. ii. 402). Hermann's solution appeared in the Acts for August, p. 351. If an early death had not put an abrupt stop to his investigations. Cotes would no doubt have removed the restriction with respect to the value of q in the exjiression given above. His example, however, stimulated Demoivre to make the attempt, which was at last crowned with success. See Miscellanea Analt/tica^ Loud. 1730. Taylor says, (see Letter cxx, and Contempl. Philos. p. 113.) that ho himself could prove the possibility^ of the integration. END OF COTES'S CORRESPONDENCE LETTERS OF TAYLOR TO PROF. SMITH. Brook Taylor (bom 1685, died 1731) was entered a foUow-com- moner at St John's Collcf^e, Cambridge, in 1701, and took the degree of LL.B. in 1709, LL.D. in 1714. Treatises on the Differential Calculus have made his name familiar to many who can write out his Theorem without having any very precise idea of tlie personality of the dis- coverer of it. A life of him, prefixed to his tract Contcmplatio Philo' sop/iicay was printed in 1703 by liis grandson Sir W. Young. At the time when ho wrote the following letter lie was Secretary of the Royal Society, though, about a month before, ho had sent in his resignation of the office to his brother-secretary Halley {ContempL Philosoph. p. 103). On Dec. 1, 3Iachin was appointed to succeed him. Before the letter was sent off, it was read at the weekly meeting of the Society. " Nov, 27, 1718. Tlie President in the chair. D' Taylor read a letter he had drawn up for ]\r Smith, Professor of Astronomy ic Cambridge, re- questing him to communicate some curious discoveries in Geometry made by the late M' Cotes his predecessor & kinsman." Journal Book, LETTER CXIX. BROOK TAYLOR TO PROP. SMITH. Sir When I last saw your most excellent Predecessor M"^ Cotes I was so very much pleased with the account he gave me of some Mathematical Tracts he had thoughts of obliging the Publick with, particularly a Sett of Tables for the Squaring of Curves by the Measures of Ratio's & Angles, that I have not been able to forbear very fre- quently mentioning of them, and expressing my wishes that I might soon see them made publick. All Lovers of Mathematical Learning do heartily joyn with me in this, particularly the Royal Society is so sensible of the great usefulness of those Tables, that they have been pleased to order me to take this occasion to let you know that they shall think themselves very much obliged to you by the 232 LETTERS OP speedy publication of them, and shall be very glad to give you any assistance you may have occasion for in the doing of it. I myself, upon the memory of what M' Cotes sliew'd me, have made some Tables of the same nature, and am presst by some friends to publish them, as a thing they say will make amends for the injury you do the Publick and the memory of M' Cotes in so long suppressing his Papers. But I can by no means prevail upon myself to do this, being much more desireous to see M"" Cotes's own Tables publisht by you. And I shall be very glad in any manner to assist you in looking over the Papers them- selves, and in taking care of the Press, if the convenience of Types should make you think it proper to print them here, and your own affairs should make it inconvenient to you to attend this work wholly your self. I am Sir Your most humble Servant Norfolk Street Brook Taylor 27*^ Nov'; 1718 Seer P. S. If there be any other Papers of M*^ Cotes be- sides the Tables that are fit to be publisht and cannot be conveniently done so soon, the Tables, being a particular thing by themselves, may be printed seperate, leaving those other Papers to a more convenient opportunity. The purport of Smith'a answer may be gathered from the following extract from the Journal Book of the Royal Society. " Dec. 11. 171 8. There was read a letter from M' Smith, in answer to a letter of D' Taylor written to desire the hastening of the Edition of M' Cotes his Posthumous papers upon the Quadrature of Curves. M' Smith informs the Doctor that those papers are preparing with all convenient speed to be put in the press, & are designed to be printed by Subscription; that the Title of the Book is as follows: Ilarmonia Mensurarum, sivo Analysis et Synthesis per Kationcm et Angulorum raensuras promotce." BROOK TAYLOR TO PROF. SMITH. 233 LETTER CXX. BROOK TAYLOR TO PROP. SMITH. Sir I am very much obliged to you for the account you give me of your design to publish M' Cotes's Papers, and I am not only most ready myself, but all my acquaintance will do what is in their power to assist you in it. I have given your letter to D"" Halley, and I dont doubt but he will acquaint you with the thoughts of the Royal Society upon it. The great impatience I am in to see your Book publisht makes me a little concerned that it must depend upon a Subscription. For tho such a Book as this when publisht cannot want purchasers ; yet it will be very hard to find a sufficient number of Persons, who have knowledge enough in these studies to think it worth while to interest themselves in a Subscription that may turn to any account. And tho what you propose of having no money paid down, & the price being sett by the Vice chancellor, be very fair and easy to the Subscribers; yet there are a great many Persons who will not care to subscribe without knowing beforehand what will be the charge. In this I dont only \vrite my own sentiments, but also those of M' Jones, who is the best acquainted with affairs of this nature of any one I know, & whose character you can be no stranger to. He had a correspondance with M' Cotes upon this Subject, and would particularly be glad to do you any service in this matter. Upon account of what I have said I wish you could rather think of getting the Book publisht at the Charge of the University, or some other way. Perhaps the Royal Society would be inclined to do it. And it may be tried whether there may not be some encouragement got from the E. of Caernarvan. Wliat ever be your resolution I will do you all the Service 234 LETTERS OF I can in it. Particularly I will endeavor to get en- couragement from abroad by the Correspondance I have. Tho' I must be so just as tx) tell you that M' Cotes is but little known among the Foreigners. His Logometria is out of their Tast, (in short none of them have judgement enough to know how to esteem it,) & his Preface to the Principia is a prejudice to his disadvantage with them. Yet I dont doubt but the newness of the design will make them purchase the Book when it is out. I believe I can do all that M"^ Cotes has done in his Tables; for I can demonstrate that any Curve may be squared by Measures of Ratio's and Angles, whose Absciss beinff «, the Ordinate is in this form ■ ^ . - -^ , where t) is any index, & ^ & \ are any whole numbers affirma- tive or negative, & the denominator e-i-fz^+gz^'^+hz^^'SiC consists of any number of terms. You know very well that the irrational forms depend upon the rational ones. I have a different way from M' Cotes's*, and something more simple, of supplying the defect in Sir Is : Newton's 6"* form. I shall be very ready and glad to communicate to you any thing that I know in these matters that may render your Book the more compleat. I believed it might be some Service to the general design of it to have Tables of Natural Logarithms and Arcs answering to the Tangents, when the Radius is unite ; wherefore I have wrote to M^ Sharp at Little Horton near Bradford in Yorkshire, to know if he will undertake to make them. I desire you will direct to me in Norfolk Street, and • Given in his letter of May 5, 1716 to Jones, quoted p. 230 antea, which Taylor appears to have seen since writing the letter of Nov. 27, Smith having probably alluded to it in his answer. Newton's 6fh form (in his De Quadratura Curvanim) com- prises the integrals of two expressions equivalent to ^—r, r and ^ , in _ a + bx^ + cx* a + bx'+cx** the case where ft>2N/ur and a, 6, vn up a paper*, which I think soon to publish by itself." The 3rd (26 Aug. 1721) begins thus: "Tlio enclosed is just como to me from Abbe Conti, who desires me to convey it to you. He tells + me that he disputes continually with the French in favor of Sir Isaac Neuton and the English Mathematicians ; but that he can by no means make them sensible of the true nature of Sir Isaac's method, they not yet rightly understanding what ho means by first and last ratios of nascent and evanescent quantities I shall trouble you with no more at present, not knowing how unwelcome this little may be to you from me, upon account of what Bernoulli has publislit f out of my letters to Monmort in hopes to provoke your resentments against me." Taylor then enters into an elaborate explanation of tho offensive expression, in the course of which he lashes Monmort for " betraying so private a letter as that was," and Bernoulli for publishing it The apology seems to have come too late. The letter bears tho London post-mark of Aug. 28, and would therefore reach Oxford on the 29th, the day on which poor Keill died. The address is crossed. Fran9oi3-Mario Arouet (Voltaire) born 1694, died 1778. LETTER CXXI. VOLTAIRE TO PROF. SMITH. S' I have perus'd y' book of optics, I cannot be so mightily pleas'd with a book, without Loving the author, • Apologia D. Brook Taylor... contra... J, Bernonllium. (It is a reply to the charg-e of pla^arism brought against him in the "Epistola pro Eminente Mathematico" Leipfiic Acts, July 1716). Philosaph. Trans. March— May 1719, p. 955. Jo. Bernoulli Opp. II. 478. It was shewa by Jones to Newton before publication. See Taylor's letter to Jones, Mace. Corr. i. 279. Keill was already employed on his own account on his Epistola ad. ..Jo. Bernoulli. See p. 187, antea. t See Conti's letter to Taylor, (May 22, 1721), Contempl. Philos. p. 124. :J: In Jo. Burchardi... Epistola ad. ..Taylor (Leipsio Acts for May 1721, pp. 195 — ^228. Jo. Bernoulli Opp. ii, pp. 483 — 512), a reply to Taylor's Apologia. The words more especially referred to are as follows : " Entre nous, je suis un peu de I'avis de Mr. Bernoulli que Mr. Keill is better qualified for a Champion than for an Analyste." VOLTAIRE TO PROF. SMITH. 237 give me leave to submitt to y' judgement these little answer of mine, w^'ich I have writ against some ignorant ennemies of S' Isaae, Neiiton, whom you follow so closely in the path of truth and glory, I am S' Y' most humble obed {Hotel de Brie, rue Cloche- Servant Voltairb. Ferche] Farts the 10*** of October {1739} new stile, M^ Smith This letter was written during a short visit which Volta.iro made to Paris. Ho had run up from Bnissels in September, purposing to stay about a month in what he calls the worse than Cartesian tourUlloni of the French capital, but on tho day of his intended departure he had an attack of illness which detained him until the end of November, In a letter, written the day after the date of the one before us, he describes the plight ho was in between his two medical attendants ("on me saigne, on mo baigne"). Under these circumstances, added to long disuse of the language, we need not be surprised to find his English not quite so good as when he wrote a dozen years before during his resi- dence in this country. The " little answer " is his " Reponse aux objections principales qu* on a faites en France centre la philosophie de Newton," 8vo. Am- sterdam, 1739 (a defence against the attacks that had been made upon his EUmens de la Philosophie de Newton... 1738, and against miscon- ceptions on some points in the Newtonian philosophy). The following allusions to this tract occur in his Correspondence. Writing to Prince Frederic of Prussia, " the Solomon of the North/' in September, shortly after his arrival in Paris, he says, " II a fallu d'abord, en arrivant, re- pondre a beaucoup d'objections que j'ai trouveos repandues a Paris centre les d^couvertes de Newton. Mais ce petit devoir dont je me suis acquitte ne m'a point fait perdre de vue ce Mahomet {his tragedy} dont j'ai d6ja eu I'honneur d'envoyer les premices a votre altesso royale. Voici deux actes tl-la-fois." In a letter to Ilelvetius, dated a week previous to this letter to Smith, ho writes, *' Je ne sais comment je m*y prendrai pour envoyer une courte et modeste reponse que j'ai faite aux anti-newtoniens. Je suis I'enfant perdu d'un parti dont ]M. de Buffon est le chef, et je suis assez comme les soldats qui so battcnt de bon coeur sans trop entendre les interets de leur prince." Voltaire's " Elemens do la Philosophic de Newton, mis d la port^ 238 NOTE OF DUKE OF CUMBERLAND do tout le mondo " (the eight last words were added by the booksellers) issued from the press at Amsterdam in April, 1738, without his know- ledge. The impati«nco of the booksellers could not wait for his recovery from a fit of sickness, or for the alterations that he wished to make in the work, and they employed another hand to complete it by finishing the 23rd chapter, and writing two additional chapters (the 24th and 2oth). The book was reprinted at Paris (with a London title-page) the following July, accompanied with " eclaircissements" and a 2Gth chapter on the tides, supplied by Voltaire : these he also sent to the Dutch corsairs (as he denominates the booksellers) to be circulated with their edition. Before leaving Paris, in November, 1739, ho tells Fre- deric that a now edition was called for, and ho republished the work in an enlarged and otherwise altered form (1741)*, with flattering re- ferences to Smith's Optics (see, for example, the explanation of the sun or moon appearing larger on the horizon than on the meridian, Part 2, ch. viii. " le docteur Smith a la gloire d'avoir enfin trouve la solution complete d'un probleme sur lequel les plus grands g^nies avai- ent fait des systemes inutiles"). Journal des Savants y 1738. Billio- thi'que Franfaise, 1738, 1739. Voltaire's Corrcspondance. His Life in Blogr* Univ. (Beuchot's note). Bcuchot's Voltaire^ tom. 38. In a letter, written from Leyden in Feb. 1737, Voltaire says, " Je pars incessamment pour achever a Cambridge mon petit cours do new- tonisme :" (he had been studying the Newtonian philosophy for some weeks under 's Gravesande at Leyden, where he had taken shelter from the storm that burst upon him on the appearance of " Le Mondain "). But the announcement was intended only as a blind to his enemies. Ho in reality returned to his retreat at Ciroy, in Champagne. Some of his biographers state that his letters at this time were dated from Cambridge, but there are no letters so dated in his published Corre- spondence. William Augustus, son of George IL bom 1721, died 1705. LETTER CXXII. DUKE OF CUMBERLAND TO PROF. SMITH. {July 3. 1740}. Doctor Smith I desire you would lose no time in pro- viding a Sea Quadrant and Telescope for to fit my eye ; Lalande also mentions an edition in the followiniif year. TO PROF. SMITH. 239 my baggage goes at five this afternoon; I shall be ox- trcamly obliged to you. William Endorsed by D' Smith. " Tho Dulco of Cumberlanda Note to mo." This note was probably written by tho fnturo •'butcher,** wlien ho was on the point of setting out to join the squadron under Sir John Norris, which was supposed to be destined for an attack upon the Spanish fleet in Ferrol. " Friday, July 4. 1740. The Duke of Cum- berland who had been some time at his post in tho camp at Ilounslow {he was Colonel of the Coldstream Guards} left it on a sudden, and arrived at Portsmouth unexpected," where he " went aboard tho Victory Man of AVar as a Volunteer." — Gentleman* 8 MaffazinCy July 1740. The London Evening Post states that ho set out from St James's for Ports- mouth at 4 in the morning. Tho weather proving unfavourable, tho Admiral and the young Volunteer returned to London in September. The Duke was now turned 19. Smith had been in attendance upon him since June, 1739 (Conclusion Book, June 11). APPENDIX. Henry Oldenburg, born 1626 at Bremen, died 1677* Secretary of the Royal Society. He was a friend of Milton's. No. I. OLDENBURG TO NEWTON. Beginning of their Correspondence. Accompanying this letter were a figure and description in Latin of the reflecting telescope made by Newton the preceding autumn and sent up "for the King's perusal" in December. See Syn. View of Newton's Life under the year 167L S' Your Ingenuity is the occaon of this addresse by a hand unknowne to you. You have been so generous, as to impart to the Philosophers here, your Invention of contracting Telescopes. It having been considered, and examined here by some of y* most eminent in Opticall Science and practise, and applauded by them, they think it necessary to use some meanes to secure this Invention from y* Usurpaon of forreiners ; And therefore have taken care to represent by a scheme that first Specimen, sent hither by you, and to describe all y® parts of y® Instru- ment, together w*^ its effect, compared w*^ an ordinary, but much larger, Glasse; and to send this figure, and description by y® Secretary of y® B. Soc. (where you were lately by y* L^ B^. of Sarum proposed Candidat) in a solemn letter to Paris to M. Hugens*, thereby to prevent the arrogation of such strangers, as may perhaps have seen it here, or even w*^ you at Cambridge; it being too frequent, y* new Inventions and contrivances are snatched away from their true Authors by pretending bystanders ; • As OldeDburg had promised in a letter to JIuygens, Jan. 1. Letter Bk. Roy. Soc. v. 92. Append.] NEWTON AND OLDENBURG. 241 But yet it was not thought fit to send this away w*** out first giving you notice of it, and sending to you y* very figure and description, as it was here drawne up*; y* so you might adde, & alter, as you shall see cause ; w*** being done here w***, I shall desire your favour of returning it w*** all convenient speed, together w*^ such alterations, as you shall think fit to make therein. Though divers of y* most skillfull examiners agreed y* your Tube magnifyed, by measure, y* object here repre- sented by A-ft so much, as you see, above w* a much greater Telescope did; yet there were others, well versed also in Optic glasses, yS though they could not disprove that mensuraon, yet were positive to affirm, y* y* excesse of magnitude did not appeare such to their eye. Besides it was discoursed, y' by this way of yours it was longsome, & difficult to find y® Object: w^^ incon- venience yet they looked upon as possible to be remedied, I shall be glad, S^ to receive your speedy answer to these lines, and embrace all occasions to expresse my singular respects to your merit, as becomes S' Your humble Ser>'ant Jan. 2. 167^. Oldenburg J. Newton's answer, dated Jan. 6, will bo found in Mace. Corr. ii. 31 1, and (not complete) in Birch, ni. 2, Ilorsley, iv. 271. Comp. Syn. View under that date. No. II. NEWTON TO OLDENBURG. S' Cambridg March 16*'' 1671 }2 1 The book w*^^ my Carrier by forgetfulncsse disappointed me of the last week I have now received & thank you • Orig. Lett. Bk. Roy. Soc. N. 1. 37. Horsley iv. 270. t This is fig. 2. Tab. I. Phil. Tram. March 25, 1672. Or, see Horsley iv. fig. facing p. 280. ^ From a copy corrected by Oldenburg (Orig. Lett. Bk. Roy. Soc, 0. 2. 64). 16 242 CORRESPONDENCE OF [Append. for it. With the Telescope w*^^ I made I have sometimes seen remote objects & particularly the Moon very distinct in those p*' of it w*^^ were neare the sides of the visible angle. And at other times when it hath been otherwise put together it hath exhibited things not w^^out some con- fusion. W^^ difference I attributed chicfely to some imper- fection that might possibly be either in the figures of y* metalls or eye glasse, & once I found it caused by a little tarnishing of the Metall in 4 or 5 days of moist weather. One of the ffellows of o*^ College is making such another Telescope w*'* w*^^ last night I looked on Jupiter & he seemed as distinct & sharply defined as I have seen him in other Telescopes. When he hath finished it I will examin more strictly & send you an account of its per- formances, ffor it seemes to be something better then that ^ych J made. Yo' humble servant Tliese I. Newton* To Henry OLDENBUiia Esq : at his house abotit the middle of the old Pall-mail in Westminster. London "rec. March 18. 71" In Oldenburg's hand. No. III. NEWTON TO OLDENliURG. March 19. 1671 12|. AJler describing the performances of the instrument mentioned in the last letter he proceeds : This may be of some use to those that shall endeavour any thing in Reflexions ; for hereby they will in some measure be enabled to judge of the goodness of their Instruments. And for this end you may annex these observations made with this last instrument to the de- Orig. Lett. Bk. Roy. Soc. N. 1. 35. ArrEXD.] NEWTON AND OLDENBURG. 243 scription of it in the Transactions of this month. But my answer to M' Hooks observations will not be ready for them, because I intend to annex to that answer some further explications of the Theory which I shall not have leisure to do this week or fourtnight. I am in hast Yo' faithfull Serv« Endorsed by Oldenburg: I. New TON • " Rcc**. 20. Ans'*. 23 comiii{unicating} v" Comet and « sub cap. Cy^iii from Hevel." See Phil, Trans. March 25, 1072, p. 4017. No. IV. NEWTON TO OLDENBURG. S*" March 26*''. 1672 About 10 days since at night I saw a dull starr south west of Perseus, which I now take to have beene that Comet of which you give me information ; But it was very small & had not any visible tayle which made me regard it noe further, & I feare it will now bee difficult to find itf. Since my last letter I have further compared the two telescopes &c. {See Phil Trans, Apr. 22. 1672 p, 4032.) * * * # ♦ Thus much of these Telescopes, & at present I shiJI trouble you no further then to thanke you for your last intelligence, by which you have obliged Your faithfull servant I. Newton J. • Orig. Lett. JDk. Roy. Soc. N. 1. 36. For the first part of the letter see Phil Tram. March 25, 1672, p. 4009, where "considerable" is printed by mistake for " insensible.*' t Phil. Tram. March 25, 1672, p. 4018. X Lett. Bk. Roy. Soc. v. 187. Horsley iv. 275. 16—2 244 CORRESPONDENCE OF [Append. No. V. NEWTON TO OLDENBURG. ** Asserting the advantavill erect the object by performing the office of a double convex lens. The manner you have ex- pressed in the 3*^ scheme ; where suppose G to be the focus of the concave, and F -. : tip of the eye-glasse at \j!^ which the rays crosse tmce before their arri- vall at the eye. But it is convenient, that the first tryalls bee made * with Prisms whose sides are all of them plane. And thus much concerning Mons' Auzout*s considerations. To the queries of Mons' Denys I answer, 1. That a Tube of six inches is capable of bearing an aperture (limited next the eye) so large, that an obstacle of 1;^ or 1^ of an inch in breadth shall be requisite to intercept all the light com- ing from one point of the object towards the concave metall : But it is convenient, that the Tube bee a little wider than that aperture precisely requires, suppose 1^ or l§- of an inch, & not more; And the whole breadth of the metall should not bee lesse than two inches, because its figure to- -n— A O 246 CORRESPONDENCE OF [AlTEND. wards the edges will scarcely bee so true as to bee usefull. And by that meanes it may also bee conveniently fastened to the end of the Tube on the outside, so as at pleasure to bee taken off & layd up close from the Air, to preserve it from tarnishing. How the Diameter of the Tube is to bee enlarged ac- cording to its length, will appeare by the Table of Aperturs and charges which I sent you in my last letter of March the 2(i*^ Namely the Cube of its length should be propor- tionable to the square-square of its diameter or aperture at the me tall ; so that the advantage of augmenting the length of Tubes is by this way far greater than by refractions, where their length ought to bee proportionall to the square of the diameter of the aperture. 2. The breadth or shortest diameter of the little ovall- metall for a Tube of six inches should not bee greater than i, nor lesse than J of an inch ; And the longest Diameter should bee to the shortest as about 10 to 7. But you may more exactly determine these diameters for Tubes of all lengths after this manner ; In the 4*^ figure let AB repre- sent the ovall sett edg- wise ; DE the concave ; FG its axis; Gp the reflex of that axis; st the Diameter of the hole through which the light is transmitted to the eye ; & P the cen- ter of that hole. Pro- duce FG to TT, so that Gir may bee cquall to Gp; erect Tra- & ^t equall to ps &pt, & from a-Si T draw two feies, aD & tF, to 4;he utmost parts of the concave, w^^in the Tube intersecting AB in ^ & B; & AB shall bee the long djameter of the ovall ; which bisect in a?, & perpendicular to Fx erect xy & xz occurring ^vith vill in com- municating to me such observations as occurr concerning my Theories or Catadioptricall instruments, and I desire you to continue that favour to me. I shall immediately proceed to add what I promised to my answer to Mr, Hooks observations, & then send it you. Mons"* Hugens has very well observed the confusion of refractions near the edges of a lens, where its two superficies's are inclined much like • Lett. Sk, Roy. Soc. v. 193. This and some other letters have been printed by Horsley (Vol. IV.) from the MSS. at the Royal Society, but not so as altogether to supersede the necessity of their reappearance here in a more complete and accurate form. 248 CORRESPONDENCE OF [Append. the planes of a prisme whose refractions are in like manner confused. But it is not from the inclhiation of those super- ficies so much as from the heterogeneity of light that that confusion is caused: ffor by illuminating an object with homogeneall light, I have seen it far distincter through a Prism than I could by light that was heterogeneal. I suppose, the designe of S'^ Robt Moray's experiments is &c. {See Phil Tr. May 2^. 1672. p. 4060). « « « « « Thus far concerning S"" E,* Morays proposalls. I have nothing more at present unlesse to desire you, that in y® letter wherein I sent you the Table of apertures and charges you would change an expression concerning the six foot Tube where I intimated that it was none of the best in its kind, ffor least the friend, of whom it was borrowed, should thinke I depreciate it, I had rather that the expression should be a little intimated after this manner ; that I am not very well assured of its goodnesse, & therefore desire, that the other experiment of reading at 100 foot distances should rather be confided in. You will do me a favour to peruse the rest of that letter also before you commit it to the presse. ffor I writ it in so much hast, that I had no time to review it : And by rendring my expressions more perspicuous or lesse ambiguous you will still oblige Your faithfull Servant I. Newton*. No. VII. NEWTON TO OLDENBURG. S' June 11*** 1672. I have sent you my Answers to M' Hook & P. Pardies, w*^** I hope will bring with y™ y' satisfaction w*^^ I promised. Lett. Bk. Roy. Soc. v. 222. Append.] NEWTON AND OLDENBURG. 249 And as there is nothing in W, Hooks Considerations w*** w*^'' I am not well contented, so I presume there is as little in mine w^^ he ean exeep{t} against, since you will easily see that I have industriously avoyded y* intermixing of oblique & glancing expressions in my discourse. So y* I hope it will be needlesse to trouble the R. Society to adjust matters. However if there should possibly be any thing esteemed of y* kind, I desire it may be interpreted candidly & with respect to the contents of M'' Hooks Considerations, & I shall readily give way to y* mitigation of whatsoever y® Heads of y* 11. Society shall esteem personall. And concerning my former Answer to P. Pardies, I resigne to you y*' same liberty w*^^ he hath done for his Objections, of mollifying any expressions that may have a shew of harshnesse. Yo"^ Servant TJiese I. Newton •. To Henuy Oldenburg Esq : at his house abcut y" middle of y^ old Pall-maile in Westminlsjter London. No. vni. NEWTON TO OLDENBURG. Cambridg S' July 30*^ 1672 The last week I wrote to you that y* ^Metall w*^^ you sent me was well for closenesse & hardnesse but yet of a colour not very brisque & inclining to red. However if it be less apt to tarnish then any other mixture yet known, that will sufficiently recompense y* other imperfections. Yo" of July IG*** directed to Stoake is not yet come to my hands. I feare it is miscarried, and desire therefore you would favour me w^^ y* particulars w*^^ were in answer to Orig. Lett. Bk. Roy. Soc. N. 1. 39. 250 CORRESPONDENCE OF [Append. y' troublesome letter ^^ written last from Stoake, for w*^** I hegg yo' pardon. I send you by John Stiles 13' for the last quarter. Yo' humble Servant These Newton t To Henry Oldenburg Esq : at his house about the middle o/the old Pall-Maile in Westminster London ^9'^ 13*. "Reo. July 31. 72 Answ. codem. and repeated y® contents of my letter of July 16." Mem. by Oldenburg. No. VIII. (bis). OLDENBURG TO NEWTON. [Extract.] Sent in conformity with the wish expressed in the preceding letter. S' Lond. July 16. 1672. I have spoken with Mr Cock about the four foot Tube, which hath been ready a pretty while. He saith that the object-speculum (being a compound of copper, tin, tin- glasse, antimony and a little arsenick) is of about 6 inches diameter, wrought upon a tool of about 14 or 15 foot, and drawing 4 foot, more or less. He adds, that tis very good mettall, shewing the moon very well, but other objects faint ; perhaps for want of giving it its due charge. Tis lodged in a square box, with a lid at the end of it, for placing the speculum-plate, lodged in it, at such a dis- tance as shall be requisite. He offers to unpolish this plate again, and to send you this very Instrument for 5^^ ; and what alterations or emendations you shall direct to bee made herein upon triall, hee will make, without demanding • Dated July 13. It is printed in Gen. Diet. vir. 782. Mace. Corr. ir. 332. t Or'ig. Lett.Bk. Roy.Soc. N. 1. 41. Append.] NEWTON AND OLDENBURG. 25J any more money for that labour. I intend, god permitting, to send by the next conveniency of your Cambridge Carrier, J. Stiles, a piece of that very mettal, with the s** objeet-speeulum, w*^** the 4 foot Teleseope is com- pounded off. As to the steely Speculum, he saith, tis a pure Venice- Steel, forged with much care; not melted, nor com- pounded with any thing ; of 3 inches diameter, but bearing not so good a polish. And this he is not unwilling to send also to you to Cambridge for your examination, and further directions about it. Hee saith, that tis very hard & tedious to grind this steely matter true*. No. IX. NEWTON TO OLDENBURG. For the first part of the letter see Rlgaud^s Appendix to his Essay, No, VIII, pp. 42, 44, aiid the Phil, Trans, for Juli/ 21, 1C73, p. 6087. i|( « « « « Pray w*^ these Notes return my thanks to M. Hugens for his book. By a former letter of yo" I was a little dubious whether M. Slusius might not apprehend, by w* you wrote to him concerning me, y^ I pretended to his Method of drawing tangents ; untill I understood by M. Collins y* you signified to him y' you thought it here of a later date, ffor it seems to me that he was acquainted w*** it some yeares before he printed his Mesolabum & consequently before I understood it. But if it had been otherwise yet since he first imparted it to his friends & y* world, it ought deservedly to be accounted his. As for y® Methods they are y® same, though I beleive derived from different principles. But I know not whether his Principles afford • Orig. Lett. Bk. Hoy. Soc. 0. 2. 92. 252 ALGEBRAICAL PAPER [Append. it so general! as mine w^^ extend to Equations affected w*'* surd terms, w*^out reducing them to another form. But if you please let this pass. The incongruities you speak of, I pass by. But I must, as formerly, signify to you y* I intend to be no further soUicitous about matters of Philosophy. And there- fore I hope you will not take it ill if you find me cease from doing any thing more in y* kind, or rather y* you will favour me in my determination by preventing so far as you can conveniently any objections or other philoso- phical letters that may concern me. For your profer about my Quarterly payments I thank you. But I would not have you trouble yo'^self to get them excused if you have not done it already. And now being tired w*^ this long letter, I must in hast write myself Yo^ humble Servant Cambridg. June 23. 73. I. Newton*. No. X. Paper given by Newton to Flamsteed at lecture in 1674. It is printed here as exhibiting to us, perhaps in a more vivid manner than his actual lectures, the philosopher descending to the level of an elementary teacher. I. (a) + 6 = 0? per reductionem fit ao? + a6 - 6.^' wa ax '- wx seu wx = ox - ab, («) « w - c fit ^ ^ 2cy-cc a^ - abb a? ^ abb — (? '^yy - 3cy + cc seu c (y) — — o =a a? fit aa - ao? « xx. . ^ . X ^ ^ cxx a + b - X a^bb + aab^ — aabbx • Orig. Lett. Bk. Roy. Soc. N. 1. 47. The date is in Oldenburg's hand. The part of the letter which we have given here is crossed out in the MS. probably by Oldenburg. The whole of the letter is printed in Ilorsley iv. 342. ArPEPm.] GIVEN TO FLAMSTEED. 253 TT / \ oa-'^**? «. o + 2aa II. (a) — + a a or nt «rd7 -i .or a + 6 ^ + 6 + a6 y' - rt^y (i3) 7====f+a cv/afl-fefefity*- ofty + fl66 - 0. III. (a) \/«rt-fl.t? + «afl? fit aa^axmrnaw-^Qaof-haa seu tV = <7. (/3) v/s : aaa? + 2axx - a?^- o + a? = fit aax + Sa.f.r — ,r' « «' — i^oa.r + Saa'.r - .ir* sen .rar — 4rt.'«? - aa, (7) y = V «y + yy -a \/ay - yy primo fit y « \/ay - yy d{e}in2y = «. IV. (a) 2y = a fit y - ^a. (/3) ~ - a fit ,r - ^ . (7) aat-cx^ac fit u; « . (d) it? xat a " c — cc A- aac a?-a'cc = 0, tit .1' + a?j? — aaj7 — = 0.* + aacc 2ac — cc 2fls-c No. XI. NEWTON TO OLDENBURG. Not. 13. 1676. TJie principal part of the letter is printed in the Transaetiona for January 24, 1C76: the remainder is as follows : I have returnd you Mr Line's letter. It came to my hands but this week; the Gentleman by whom you sent it having not yet been at Cambridge but transmitting it to me from Oxford. • From the ori^nal paper in Newton'ti hand, pasted in at the beginning of Vol. 42 of Flamsteed's MSS. at Greenwich : at the l)ottom are the words •* Mr Newton's paper given at one of his lectures, Midsummer, 1674.'* Flanisteed wa? at Cambridge, from the end of May until July 13. He brought with him a Royal Mandate for tlie degree of M.A. which was conferred upon him on June 5. He had been admitted a pensioner nt Jesus College Dec. 21. 1670, durinjc^ a short stay he made at Cambridge on his return from London to Derby, when he also took the op|)ortunity of eiUing upon Barrow and Newton. Comp. Ibily, p. 29. I, III and IV (except y) will be found in the published Algebra Leeture$ {L^cU 6 and 7 ). Re^if. 3, 4, 5 pp. 65—67. 254 CORRESPONDENCE OF [Append. I had some thoughts of writing a further discours about colours to be read at one of yo' Assemblies, but find it yet against y* grain to put pen to paper any more on y* subject. But however I have one discourse by me of y* subject written when I sent my first letters to you about colours & of w*^** I then gave you notice. This you may command w" you think it will be convenient if y® custome of reading weekly discourses still continue*. In y* meane while I am S'" Yo"' humble Serv"* Is. Newton f. No. XII. NEWTON TO OLDENBURG. S' Cambr. Novemb 30 1675. I intended to have sent you y® papers this week but upon reviewing them it came into my mind to write ano- ther little scrible J to accompany them : You may expect *em y* next week. An ancient Gentleman I met at yo' Assemblies (whose name I cannot recollect,) being thick of hearing desired me to inquire after y® form of Mr Mace's Otocousticon a INIusitian here ; but he has not been in town since I came from London, but is somewhere in London about printing a book of Mu3iq:||. Yet y^ last week 1 had opportunity to inquire after it of his son & he ' ** Mr Oldenburg was ordered to thank him for this offer, and to desire him to send the said discourse ns soon as he pleased." Birch, iii. 232. t OrifT, leit, Jik. Roy. Soc. N. 1. 48. t "An Hypothesis explaininfr the properties of light, discoursed of in my several papers." liirch, iii. 2-18. II " Mustek's Monument," &c. &:c. Lond. 1676. Newton*8 name appears in the list of subscribers to the work. Thomas Mace was one of the ' Clerici' or Singing Men of Trinity College for more than 70 years (1635— 1706). Comp. Burney's Hist.of Muf. Vol. 3. Southey's Doctor, chapters 193-196. Cooper's Annals of Camb. under year 1690. Append.] NEWTON AND OLDENBURG. 255 tells 'me the form is this. A y* smal end to put into y* ear BC y"" length sup- pose two foot CD y^ >vide end suppose about eight inches over. Thetubei?Z>C tapers all y® way almost eavenly like a cone only at y* great Orifiee CD widens more, like y® end of a Trumpet. He has of several sizes. The biggest do y* best. If you can't recollect who y® Gentleman may be I suppose M' Hill can tell you, for I think M"" Hill was by when y" Gentleman spake to me, & y^ Gentleman dcsird me to write to either M*" Hill or you about it. For IIenry Oldenijuro Esq : at his house about y* middle of y' Old Pal-inel in Westminster London, Yo" in hast Is. Newton' No. XIII. NEWTON TO OLDENBURG. I hope M"* Linus's ffriends will acquiesce in y* late tryall of y'^ Exp* in debaitf, for y* procurement of w*^^ & for send- ing them notice of y® event, I return you my hearty thanks, as I have reason. I perceive I went upon a wrong suppo- • Orig. Lett. Dk, Roy. Soc. N. 1.49. t i.e. The Experiment on the Solar spectrum. •* Apr. 27. The Experiment of Mr Newtou which had been contested by Mr Linus and his fellows at Lienro, wan tried before the Society, according to Mr Newton's directions, and succeeded, as lie all along asserted it would do : and it was ordered, that Mr Oldenburg- should signify this success to those of Liege, who had formerly certified, { by a letter, Dec. 15, 1675 | that if the experiment were made before the Society, and succeeded according to Mr. Newton's assertions, they would acquiesce." Birch, in. 313. Linus had maintained that the sun's image was round, and the colours arranged parallel to the axis of the prism. 266 CORRESPONDENCE OF [Append. sition in what 1 wrote concerning M' Boyles Exp*. The Papers in yo*" hand I have no present need of: You may send them at yo"^ best leisure. . Sometime this Sommer it's possible I may make use of them, if I can but get some time to write y* other discourse about y® colours of y^ Prism w*^^ I have long intended. S*" I am Yo^ humble & obliged Servant Cambridge. May ll*^ 1G76. Is. Newton*. For Henry Oldenburo Esq : at his house about y' middle of y' old Pal-mall in Westminster London. Endorsed by Oldenburg : "RccM2May. Answ. by D' Sidnamt May l.'i. and sent by him liis Hypothesis explaining y* properties of light ; as also his discourse about y' various colors exhibited by transparent substances made very thin by being blown into bubles or otherwise form'd into plates, altho at a greater thicknes they appear very clear and colorlesse. In my letter accompanying these papers I imparted to M' Newton y* particulars contain'd in M. Leibniz his letter to me of May 12 1 67(3. from Paris st. n." In the letter just mentioned Leibniz desired information on the subject of the analytical discoveries recently made in England, and it was in compliance with this request that Newton, at the pressing solicitation of Collins and Oldenburg, drew up his celebrated letter of June 13. One of the questions in Leibniz's letter, of which an extract is printed in the Commercium Epistolicum., will probably sur- prise the modern student. The series (sin 6 =)0- jt+.... and its converse had been sent to him from this country, and he begs the favour of a demonstration of them. • Oris;. Lett. Bk. Roy.Soc. N. 1. 62. •f- Sydenham was going to Cambridge to take his M. D. degree. He was admitted at Pembroke, May 17 (from Alag-dalcn Hall, Oxford) and was made Doctor the follow- ing day. Append.] NEWTON AND OLDENBURO. 257 No. XIV. NEWTON TO OLDENBURO. Accompanying hia answer to Lucas (dated Aug. 18, and printed in the Trans, for Sept. 25). I have been stayed from writing to you longer then I intended by reason that I could not till of late meet w*^ a day clear enough at noon-time to try some of y* experi- ments herein set down. And now 1 have not sent you an answer so full as I intended at first but perhaps more to y* purpose considering who I have to deale w*^ whose buisiness it is to cavill. The other buisiness you wrote to me about viz: about stocking us w*^ fruit trees I shall be glad to pro- mote. Some inquiry I have made about it, & w^^'in a few days, when I have got some further information & dis- coursed it w*^ some that are most like to entertein y* pro- posall, I hope to give you a further account of it. In y* mean time I rest Yo'^ humble Servant Cambridge Aug: 22. 1676. Is. Newton* For Henry Oldenburg Esq : at his house about tlie middle of y^ old Pal-maill in Westmimter London. w*** care. No. XV. NEWTON TO OLDENBURO. S" Octob 26. 1676. Two days since, I sent you an answer to M. Leibnitz's excellent Letter. After it was gone, running my eyes over a transcript that I had made to be taken of it, I found some things w*^^ I could wish altered, & since I cannot now • Orig. TMt. m. Roy.Soc. N. 1. 54. 17 258 CORRESPONDENCE OF [Append. do it my self, I desire you would do it for me, before you send it away. In pag : 3. Sect : Pudet dicere.] ffor a Z>. Barrow tunc Matheseoa Professare write only per amicum Pag : 5. Sect : At quando.] After quibuscum potest com- parari ; write ad quod sufficit etiam hoc ipsum unicum jam descriptum Theorema si dehite concinnetur. Pro Trinomiis etiam et aliis quihusdam Regulas quasdem concinnavi &c. Pag : 6. Sect : Quamvis multa.] Where you find y® words Gregorianis ad Circulum et Hyperholam editis p^rsi- miles, for persimiles write ajffines Pag: 9 or 10. Sect: Theorema de.] ffor error erit — + — + &c. write error erit — h — + &c. 90 140 90 19^ Pag: 6 vel 7. Sect : Quamvis multa.] about y* end of y* section turn plenariam into plenam or rather blot y® word quite out. Pag : ult. vel penult. Sect : Ubi dixi]. write solutilia for solubilia. And if you observe any other such scapes pray do me y® favour to mend them. So in pag 5 or 6. Sect. Quamvis multa.] It may be perhaps more intellig { ib } le to "write cvOvvarei for euthunsi. Pag 8 or 9. Sect : Per seriem.] After y® words product ad multas figuras : you may if you please add these words, ut et ponendo summam terminorum 1 -■f + ^-iV'^'TV " ife "** '^ " :3V + ^V ^^ ^^^® ^^ totam seriem 1 - ;J^ + -J- - 1 + ^ - ^ + &c ut 1 + \/2 ad 2. Sed optimus ejus usus &c I feare I have been something too severe in taking notice of some oversights in M. Leibnitz letter considering y® goodnes & ingenuity of y® Author & y* it might have been my own fate in writing hastily to have committed y® like oversights. But yet they being I think real oversights I suppose he cannot be offended at it. If you think any thing be exprest too severely pray give me notice & I'le Atpe^nd.] NEWTON AND OLDENBURG. 259 endeavour to molliiy it, unless you will do it w*** a word or two of your own. I beleive INI. Leibnitz will not dislike y* Theorem towards y® beginning of my letter pag. 4 for squaring Curve lines Geometrically. Sometime when I have more leisure it's possible I may send him a fuller account of it : explaining how it is to be ordered for com- paring curvilinear figures w*^ one another, & how y* simplest figure is to be found w*^ w*^^ a propounded Curve may be compared. S"* I am Yo*" humble Servant Is. Newton*. Pray let none of my mathematical papers be printed w^^out my special licence. Some other things in M. Leibnitz letter I once thought to have touched upon, as y® resolution of affected aqua- tions, & y® impossibility of a geometric Quadrature of y* Circle in w*^^ M. Gregory seems to have tripped. But I shall add one thing here. That y® series of a?quations for y® sections of an angle by whole numbers, w*^^ M. Tschum- hause saith he can derive by an easy method one from an other, is conteined in y^ one aequation w*^* I put in y" 3** section of y® Problems in my former letter for cutting an angle in a given ratio, and in another aequation like that. Also y* coefficients of those equations may be all obteined n-Oxn-l n— 2x11—3 w-4xn— 5 by this progression 1 x ^^^_^ x __^ x -^— ^ X ^^ ^^^"-7 ^ ^^ rJ^Y^^ ^^g^ coefficient being l, y* 2^ 4xn-4 n-Oxn-1 ^ „rt ?i-Oxn-l ri-2xn-3 1 X . V® 3° 1 X X . &c. & n lxn-1 Ixw-l 2x71-2 being y® number by w*^^ y* angle is to be cut. as if n be 5. • MSS. Birch, Brit. Mus. 4294. The signature which waa cut out by some felonioua hand in 1833, has been recently r&stored. 17—2 260 CORRESPONDENCE OF [Append. 5 y^4t 3x2 1x0 then y* series is l x x — x that is 1 x 5 x 1 x ^ 1x4 2x3 8x2 & consequently y* coefficients 1.5.5. So if n be 6 y® 6x5 4x3 2x1 series is l x x — x - — x o that isl x6x|x|^xO 1x5 2x4 3x3 ^ & consequently y* coefficients 1.6.9.2. This scrible is not fit to be seen by any body nor scarce my other letter in y* blotted form I sent it, unless it be by a friend. For Henry Oldenburg Esq: at his house ahwit y* middle of y' old Pal-mall in Westminster London No. XVI. NEWTON TO OLDENBURG. S' I am desired to write to you about procuring a recom- mendation of us to M' Austin y® Oxonian planter. We hope yo' correspondent* will be pleased to do us y* favour as SLs{8ic} to recommend us to him, y* we may be furnished w*^ y* best sorts of Cider-fruit-trees. We desire only about 30 or 40 Graffs for y® first essay, & if those prove for o' pur- pose they will be desired in greater numbers. We desire graffs rather then sprags that we may y® sooner see what they will prove. They are not for M"" Blackley but some other persons about Cambridge, But M*" Austin need only direct his letters to me or to M"^ Bainbrigg ffellow of o' College. In y® mean time we return o^ thanks to you & your ffriend for y® good will you have already shewn us. M*" Lucas letter t I have received, & hope to send you an answer y** next Tuesday Post. I thank you for your care to prevent their prejudicing me in y® Society, as also • Dr John Deal, rector 5f Yeovil, wlio inherited a" zeal for the plantation of orchards for the making of cider." See liirch, iv. 235. t UateU Oct. 23. Append.] NEWTON AND OLDENBURG. 261 for giving me notice of y* things miswritten in my late letter. In pag 3 y* words you cite should run thus. Cujus 2x2x2 triplo adde Log. 0.8, siquidem sit - — - — - 10. But in 0.8 pag 8 y® signes of y® series i +i-^-4- + ^ + &c arc rightly put two + & two - after one another, it being a different series from y* of M. Leibnitz. But in y* next two or 3 lines, to prevent future mistake you may if you think fit, after y^ words res tardius ohtineretur per tangentem 45^, add these words jxixta seriem nobis commtmicatam, Seing y® letter is still in yo"" hands, you will do me y* favour to make these further amendments Pag. 3 Sect [Pudet dicere] cum D. Collinsio for ad D, Collinsium pag. 5. Exempl. 4 after y® words vel quibus libet dujiiU tatibus binomii cujuscunq : add licet non direct^ ubi index duj* nitatis est numerus integer. pag 6 or 7 in y® end of y* section quamvis multa I desire you would cross out y® words adeo ut in potestate habeam descriptionem omnium curvarum istius ordinis quce per 8 data ^'^ puncta determinantur. And in y® 2^ sentence of y® next section I could wish these words also numero infinite multaa were put out. pag 9f. Sect [Prceterea quae,'] for mihi quidem haud ita clara sunt put nondum percipio. And after a line or two where you see y® words et cert^ minor est labor, put out certe» By these alterations S*" you will oblige Yo' humble Servant {Tuesday} Nov. 14 1676. Is. Newton |. • "data" is written by mistake for "tantum." The words here ordered to be crossed out are inclosed within parentheses in the letter as printed in Wallis's 3rd Volume, and the Commercium Epistolicuni, wliere also leptem appears instead of octo. One of the points is supposed to be a double point. See Newton's Enumeralio Lin, Tert. Ord. t The place referred to is in p. 10. : MSS. Birch, Brit. Mus. 4294. 262 LETTER TO MADDOCK. [Append. • Just now I received Yo' packet conteining two books from M' Boyle. That for D"" Moor shall be conveyed to him. For the other I shall return my thanks to y® noble Author. For Henuy Oldenburg Esq : at his house about y* middle of y* old Palmail in Westminster Endorsed by Oldenburg: "Rec. Nov. 15. 76. written to D' Beall about part of y' contents of this letter. Nov. 16. 76. Anew. Nov. 25. 76." In another letter to Oldenburg written on the following Saturday, he saya : " I promised to send you an answer to Mr Lucas this next Tuesday, but I find I shall scarce finish what I have designed, so as to get a copy taken of it by that time, and therefore I beg your patience a week longer." Mace. Corr. ii. 405. The answer was accordingly sent on the 28th. All that is known respecting it is derived from Lucas's rejoinder. See Syn. View of Newton's Life, under Nov. 28, 1676, note. No. XVIL NEWTON TO DR JOSHUA MADDOCK. Maddock had sent Newton some specimens of a new branch of optics, devoted to the consideration of the properties of dark rays. Such a system would afibrd relief to those commentators who are em- barrassed by expressions like /ue\a/u0a€« €/)e/3ovrite to his Correspondent an Italian in London, to move that the Society would give him leave to dedicate • Allusion to Tiberius's peculiarity of vision. *' Cum praegrandibus oculis, et qui, quod mirum esset, noctu etiam et in tenebris viderent, sed ad breve." Sueton. Tib. 68. Comp. Plin. Nat. Hist. xi. 54. t Printed at the end of a Funeral Sermon on the death of Daniel Maddock by E. Latliam, M.D. Lond. 1745 : and Gentleman's Mag. Aug. 1782. 264 CORRESPONDENCE OF [Append. his Book to them. The said Italian being come from London hither before the Arrival of the Letters, upon the receipt of them applied himself to me and I promised him I would desire you to acquaint the Society with his Request. If you please to send their Answer to me, the Italian here will convey it into Italy. For the trials you made of an Experiment suggested by me about falling bodies*, I am indebted to you thanks which I thought to have returned by word of mouth, but not having yet the Opportunity must be content to do it by Letter &cf. Trinity College Decemb. 3'* 1680. William Briggs, bom about 1650, succeeded Tenison in his fellow- ship at Corpus Christi College, 16()8. A.M. 1670. M.D. 1677. See Mrtsters's Hist, of Corp. p. 249. No. XIX. NEWTON TO BRIGGS. S' I have perused yo' very ingenious Theory of Vision J in w*'** (to be free w*^ you as a friend should be) there seems to be some things more solid & satisfactory, others more disputable but yet plausibly suggested & well de- serving y® consideration of y® ingenious. The more satis- factory I take to be your asserting y* we see w*^ both eyes at once, yo*" speculation about y® use of y® musculua ohliquus inferior t yo' assigning every fibre in y® optick nerve of one eye to have its correspondent in y* of y® other, • See Synoptical View of Newton's Life under the year 1679 (note). •f- Roy. Soc. Letter Booky vni. 139. Hooke's Answer, dated Dec. 18, is given lb. 140. Compare Birch, IV. 61. J •• A New Theory of Vision " read at the meeting of the Royal Society, March 15, 1682, and printe«l in Ilooke's Philosophical Collections for tliat month. A paper in con- tinuation of it, *' with an examination of some late objections," appeared in the Phil. Tram, for May 1683. Append.] NEWTON AND BRIOGS. . 265 both w^** make all things appear to both eyes in one & y* same place & yo' solving hereby y* duplicity of y* object in distorted eyes & confuting y* childish opinion about y* splitting y* optick cone. The more disputable seems yo' notion about every pair of fellow fibres being unisons to one another, discords to y® rest, & this consonance making y* object seen w*^ two eyes appear but one for y* same reason that unison sounds, seem but one sound. I did think to have sent you what I fancy may be objected against this notion & so staid for time to write it down, but upon second thoughts I had rather reserve it for dis- course at o^ next meeting : and therefore shall add only my thanks for yo*" kind letter & p'sent. S' I am Yo' much obliged & humble servant Trin. Coll. Cambridge June 20*^ 1G82, Is. Newton*. For his honoured friend D' Wiluam BiiiGos at his hotise in Suffolk Street in London. No. XX. NEWTON TO BRIGOS. For his Hon'* ffriond D' W" Bnwos. S' Though I am of all men grown y* most shy of setting pen to paper about any thing that may lead into disputes yet yo' friendship overcomes me so far as y* I shall set down my suspicions about yo*" Theory, yet on this con- dition, that if I can Avrite but plain enough to make you understand me, I may leave all to yo'^ use w^^'out pressing it further on. For I designe not to confute or convince • From the original in the British Museum, Add. MSS. 4237. fol. 32. Part of thiu letter is lithographed in C. J. Smith's Hitt. and TJt. Cunoutie$, Lond. 1840. 266 CORRESPONDENCE OF [Append. you but only to present & submit my thoughts to yo' consideration & judgment. First then it seems not necessary that the bending of y* nerves in y* Thalamus opticus should cause a differing tension of y* ffibres. ffor those w*^^ have y® further way about, will be apt by nature to grow the longer. If y* arm of a tree be grown bent it follows not that the fibres on y* elbow are more stretcht then those on the concave side, but that they are longer. And if a straight arm of a tree be bent by force for some time, the fibres on y* elbow vi'''^ were at first on y* stretch will by degrees grow longer & longer till at length the arm stand of it's self in y® bended figure it was at first by force put into, that is till y* fibres on y® elbow be grown as much longer then y* rest as they go further about, & so have but the same degree of tension w*'* them. The observation is ordinary in twisted Codling hedges, fruit trees nailed up against a wall &c. And y® younger & more tender a tree is the sooner will it stand bent. How much more there- fore ought it to be so in that most tender substance of y® Optick nerves w*^^ grew bent from y* very beginning ? And whether if those nerves were carefully cut out of y® brain & outward coat & put into brine made as neare as could be of the same specific gravity w*^ y* nerves, they would unbend or exactly keep the same bent they had in y® brain may be worth considering, ffor though y® strength of a single fibre upon the stretch be inconsiderably little, yet all together ought to have as much strength to unbend y* nerve, as would suffice by outward application of y* hand to bend a straight nerve of y® same thickness, the dura Mater being taken off. M' Sheldrake* further suggests wittily that an object whether the axis opticus be directed above it, under it, or • A Fellow of Corpus, 7 years senior to Newton. Append.] NEWTON AND BRIOGS. 267 directly towards it, appears in all cases alike as to figure & colour excepting that in y* 3^ case tis distincter, w^^ pro- ceeds not from y* frame of y* nerves but from y* dis- tinctness of y* picture made in y* Retina in that case. But in y* first case where y' vision is made by y* fibres above & second where tis made by those below, the object appearing alike : he thinks it argues that the fibres above & below are of y** same constitution & tension, or at least if they be of a differing tension, that that tension has no efiect on y® mode of vision, but I understand you are already made acquainted w*^ his thoughts. It may be further considered that the cause of an objects appearing one to both eyes is not its appearing of y® same colour form & bigness to both, but in y* same situation or place. Distort one eye & you will see y* coincident images of y® object divide from one another & one of them remove from y® other upwards downwards or sideways to a greater or less distance according as y* distortion is; & when the eyes are let return to their natural posture the two images advance towards one another till they become coincident & by that coincidence appear but one. If we would then know why they appear but one, we must e{nj quire why they appear in one & y* same place & if we would know y® cause of that we must enquire why in other cases they appear in divers places variously situate & distant one from another, ffor that w*^^ can make their distance greater or less can make it none at all. Consider whats the cause of f heir being in y* same altitude when one is directly to y* right hand y* other to y® left & what of their being in y'* same coast or point of y® compas, when one is directly over y® other : these two causes joyned will make them in y® same altitude & coast at once that is in y* same place. The cause of situations is therefore to be enquired into. Now for finding out this y® analogy will stand between y* situations 268 CORRESPONDENCE OP [Append. of sounds & the situifCiDns of visible things, if we will compare these two senses. But the situations of sounds depend not on their tones. I can judge from whence an echo or other sound comes tho I see not y* sounding body, & this judgment depends not at all on y* tone. I judge it not from east because acute, from west because grave : but be y* tone what it will I judge it from hence or thence by some other principle. And by that principle I am apt to think a blind man may distinguish unisons one from another when y® one is on his right hand y® other on his left. And were our ears as good & accurate at distin- guishing y® coasts of audibles as our eyes are at distin- guishing y® coasts of visibles I conceive we shoidd judge no two sounds the same for being unisons unless they came so exactly from y® same coast as not to vary from one another a sensible point in situation to any side. Suppose then you had to do with one of so accurate an ear in distinguishing y® situation of sounds, how would you deale with him ? Would you tell him that you heard all unisons as but one sound ? He would tell you he had a better ear then so. He accounted no sounds y® same w*^^ differed in situation : & if your eyes were no better at y® situation of things then your ears, you would perhaps think all objects y® same, w^^ were of y® same colour. But for his part he found y* y*' like tension of strings & other sounding bodies did not make sounds one, but only of y® same tone : & therefore not allo>ving the supposition that it does make them one, the inference from thence that y® like tension of y® optick fibres made y® object to y® two eyes appeare one, he did not think himself obliged to admit. As he found y* tones depended on those tensions so perhaps might colours, but the situation of audibles depended not on those tensions, & therefore if the two senses hold analogy with one another, that of visibles does not, * & consequently the union of visibles as well as audibles which depends on the agreement AprEND.] NEWTON AND BRIOOS. 269 of situation as well as of colour or tone must have some other cause. But to leave this imaginary disputant, let us now consider what may be y* cause of y* various situations of things to y* eyes. If when we look but w*** one eye it be asked why objects appear thus & thus situated one to another the answer would be because they are really so situated among themselves & make their coloured pictures in y'' Retina so situated one to another as they are & those pictures transmit motional pictures into y* scnsorium in y® same situation & by the situation of those motional pictures one to another the soul judges of y* situation of things without. In like manner when we look with two eyes distorted so as to see y* same object double if it be asked why those objects appear e in this or that situation & distance one from another, the answer shoiUdbe because through y® two eyes are transmitted into y® sensorium two motional pictures by whose situation & distance then from one another the soule judges she sees two things so situate & distant. And if this be true then the reason why when the distortion ceases & y® eyes return to their natural posture the doubled object grows a single one is that the two motional pictures in y* sensorium come together & become coincident. But you will say, how is this coincidence made? I answer, what if I know not? Perhaps in y® sensorium, after some such way as y* Cartesians would have beleived or by some other way. Perhaps by y® mixing of y* marrow of y® nerves in their juncture before they enter the brain, the fibres on y* right side of each eye going to y* right side of y* head those on y® left side to y® left. If you mention y* experim* of y® nerve shrunck all y® way on one side y® head, that might be either by some unkind juyce abounding more on one side y® head y" on y* otlier, or by y® shrinking of y® coate of y® nene whose fibres & vessels 570 CORRESPONDENCE OP [Append. for nourishment perhaps do not cross in y* juncture as y* fibres of y* marrow may do. And its more probable y* y® stubborn coate being vitiated or wanting due nourishment shrank & made y® tender marrow yeild to its capacity, then that y® tender marrow by shrinking should make y* coate yeild. I know not whether you would have y® succus nutricius run along y® marrow. If you would, 'tis an opinion not yet proved & so not fit to ground an argument on. If you say y* in y® Camgelion & ffishcs y® nerves only touch one another without mixture & sometimes do not so much as touch ; *Tis true, but makes altogether against you. ffishcs looke one way with one eye y® other way with y® other: the Chamajlion looks up w*^ one eye, down w*^ t'other, to y® right hand w*^ this, to y® left w*^ y*, twisting his eyes severally this way or that way as he pleases. And in these Animals wliich do not look y® same way w**' both eyes what wonder if y® nerves do not joyn? To make them joyn would have been to no purpose & nature does nothing in vain. But then whilst in these animals where tis not necessary they are not joyned, in all others w*^^ look y® same way w*^ both eyes, so far as I can yet learn, they are joyned. Consider therefore for what reason they are joyned in y® one & not in the other, ffor God in y® frame of animals has done nothing w%ut reason. There is one thing more comes into my mind to object. Let y® circle D J represent the Retina, or if you will the end of y® optick nerve cut cross. A the end of a fibre above of most tension, C y® end of one below of least tension. D Sc G y^ ends of fibres above on either hand almost of as much tension as At F Si J the ends of others Append.] NEWTON AND BRIGGS. 271 below almost of as little tension as C E y* end of a fibre of less tension then ^ or G & of more then C or. J, And between A & Q G & J there will {be} fibres of equal tension w*^ E because between them there are in a continual series fibres of all degrees of tension between y* most tended at A Sc G 8c least tended at C & ./. And by the same argument that 3 fibres E, D k II of like tension are noted let the whole line of fibres of the same Degree of tension running from E to // be noted. Do you now say y* y^ reason why an object seen w*** two eyes appears but one is that y*' fibres in y® two eyes by w*^^ 'tis seen are unisons? then all objects seen by unison fibres must for y® same reason appear in one & y® same place that is all y' objects seen by the line of fibres E B II running from one side of the eye to y® other. fTor instance two stars one to y® right hand seen by y® fibres about //, the other to y® left seen by y® fibres about E ought to appear but one starr, & so of other objects, ffor if consonance unite objects seen w*^ the fibres of two eyes much more will it unite those seen w*^ those of y® same eye. And yet we find it much otherwise. AVhat soever it is that causes the two images of an object seen with both eyes to appear in y* same place so as to seem but one can make them upon distorting y® eyes separate one from y® other & go as readily & as far asunder to y® right hand & to y® left as upwards & downwards. You have now y® summ of what I can think of worth objecting set down in a tumultuary way as I could get time from my Sturbridge ffair friends. If I have any where exprest myself in a more peremptory way tlien becomes y® weaknes of y® argument pray look on that as done not in earnestness but for y® mode of discoursing. Whether any thing be so material as y* it may prove any way useful to you I cannot tell. But pray accept of it as written for that end. ffor having laid Philosophical 272 CORRESPONDENCE WITH BRIG OS. [Append. speculations aside nothing but y^ gratification of a friend would easily invite me to so large a scribble about things of this nature. S*" I am Yo' humble Servant Trin. Coll. Cambr. Sept. 12*^. 1682. Is. Newton*. No. XXI. NEWTON TO BRIGGS. Isaacus Newtonus Doctori Gulielmo Briggio. Vir Clarissime, Hisce tuis Tractatibusj duas magni nominis scientias uno opere promoves, Anatomiam dico & Opticam. Or- gani enim (in quo utraque versatur) artificio sumrao con- structi diligenter perscrutaris mysteria. In hujus dis- sectione peritiam & dexteritatem tuam non exiguo olim mihi oblectamento fuisse recordor. Musculis motoriis secundum situm suum naturalem eleganter a te expansis, caaterisque partibus coram expositis, sic ut singularum usus & ministcria non tarn intelligere liceret quam cernere, effecerat dudiim ut ex cultro tuo nihil non accuratum sperarem. Nee spem fallebat eximius ille Tractatus Ana- tomicus, quem postmoddm edidisti. Jam praxeos hujus uKpljieiav pcrgis ingeniosissima Theoria instruere & exor- nare. Et quis Theoriis condendis aptior extiterit, quam qui phienomenis accurate observandis navfirit operam? Nervos opticoa ex capillamentis varie tensis constare sup- ponis, eaque magis esse tensa qua) per iter longius porri- guntur; ex diversa autem tensione fieri ut objectorum partes singulaB non coincidant & confundantur inter se, sed • From the original in the British Museum, Add. MSS. 4237. lol. 34. t i. e. Opthalmographiaf Cantab. 1676 (2nd Ed. Lond. 1687) and his Theory of Viiion. Append.] DIRECTIONS TO BENTLEY. 278 pro situ 8110 natural! diversis in locis nppareant : & capilla- mentis amborum oculorum a;quali tcnsione factis coiiCordU bus, geminas objectorum species uniri. Sic ex tcnsione cliordarum, qud soni vel variantur vel concordant in Mu- sicji, colligere videris quid fieri debet in Opticii. Simplex etenim est Natura, & codem operandi tenore in immensA. eflectuum varietate sibi ipsa constare solet. Quanto vero magis in sensuum cognatorum causis ? Et quamvis aliam ctiam horum sensuum analogiam suspicari possim, ingcnio- sam tamen esse quam tute excogitasti, cert^ nemo non lubenter fatebitur. Nee inutilem censeo Dissertationem ultimam qua diluis objectiones. Inde Lector attentus & pleniCis intelliget Hypothesin totam, & in qua)stiones incidet vel tuis Meditationibus illustratas, vel novis experimentis & disquisitionibus posthac dirimendas. Id quod in usum cedet juventuti AcademicjB, & provectiores ad ulteriores in Philosophia progressus manuducet. Pergas itaque, vir ornatissime, scientias hasce pra)claris inventis, uti facis, excolere ; doceasque difficultatcs causarum naturalium tam facile solertia vinci posse, quiim solent conatibus vulgari- bus difficulter cedere. Daham Cantahrigice 7 Kal. Mail. Vale . 1686. No. XXII. Paper-of Difections given by Newton to Bentley, respecting the Books to bo read before entering upon the Principia. Date probably about July 1691. In 1691 the vigorous mind of Richard Bentley, who was then in his 30th year, was attracted to the revelations which the Principia had announced to the philosophical world some four years before, and with • This letter is prefixed to the Latin Version of Brigprs's Theory of Vhion (mnde at Newton's request) Load. 1685. 18 274 DIRECTIONS TO BENTLEY. [Append. the view of making himself acquainted with the "Great Charter of Modem Science/* as that immortal work is called by Dr Whewell, ho applied through his friend W. Wotton to John Craigo, a mathemati- cian of some eminence, for advice as to the course of reading to bo fol- lowed preparatory to the study of the volume itself. The appalling list of authors which Craige sent him (Juno 24, 1691 ; see Bentloy's Corresp. p. 736) probably induced him to repair to the fountain head, and the paper now before us was the result of that step. That Bcntley studied the Principia to some purpose was shewn by his two last sermons at the Boyle Lecture (founded by the "Will of Robert Boyle, who died Dec. 31, 1691) in November and December of the following year, which led him to consult our philosopher again upon some points that arose in tliem requiring elucidation. See New- ton's four Letters to Bentley in 1(592 and 1693. (Bcntley 's Corresp.) Next after Euclid's Elements the Elements of y* Conic sections are to be understood. And for this end you may read either the first part of y* Elementa Curvarum of John De Witt, or De la Hire's late treatise of y- conick sec- tions, or D' Barrow's epitome of Apollonius. For Algebra read first Barth{ol}in's introduction & then peruse such Problems as you will find scattered up & down in y* Commentaries on Cartes's Geometry & other Alegraical {sic] writings of Francis Scliooten. I do not mean y* you should read over all those Commentaries, but only y® solutions of such Problems as you will here & there meet with. You may meet with De AVitt's Elementa curvarum & Bartholin's introduction bound up together w*^ Carte's Geometry & Schooten's commentaries. For Astronomy read first y® short account of y® Copernican System in the end of Gassendus's Astronomy & then so much of Mercator's Astronomy as concerns y® same system & the new discoveries made in the heavens by Telescopes in the Appendix. These are sufficient for understanding my book : but if you can procure Ilugenius's Horologium oscillatorium, the perusal of that will make you much more ready. At y® first perusal of my Book it's enough if you under- Append.] LETTER TO LOCKE. 275 stand y" Propositions w*^ some of y* Demonstrations w*^** are easier then the rest. For when you understand y* easier they will afterwards give you light into y* harder. When you have read y* first GO pages, pass on to y* 3** Book & when you see the design of that you may turn back to such Propositions as you shall have a desire to know, or peruse the whole in order if you think fit*. Memorandum by Bcntlcy. "Directions from ^I' Newton by his own Hand" . No. XXIIi. NEWTON TO LOCKE. The first few lines of the letter are wanting. Locke had sent him some of Boyle's red earth, which tliat philosopher had a recipe for combining with mercury so as to " multiply" gold. In a letter >Nritten on the 2nd of tlie folloNving month, Newton " dissuadOs" Locke " from too hasty a trial of this recipe," whicli he states to bo " imperfect and useless." Lord King's Life of Lochy i. 412. as I can. You have sent much more earth then I ex- pected. For I desired only a specimen, having no inclination to prosecute the process. For in good earnest I have no opinion of it. But since you have a mind to prosecute it I should be glad to assist you all I can, having a liberty of communication allowed me by INIr B. in one case which reaches to you if it be done under y® same conditions in w^ I stand obliged to Mr B. ffor I presume you are already under the same obligations to him. But I feare I have lost y® first & third part out of my pockett. I thank you for • From the original, given, with Newton's four letters to Bentley, by Cumberland to Trinity College. 18—2 276 LETTER [Append. what you communicated to me out of yo' own notes about it. S' I am Yo' most humble Serv^ Cambridge Jul 7*** Is Newton 1692. When the hot weather is over I intend to try the begin- ning* tho y* success seems improbable f. For Zovfin Look, Esq. at M'. Paulen*s in Dorset Court in Channel Row in Westminster, No. XXIV. NEWTON TO LEIBNIZ. In answer to a letter of Leibniz dated ^ Mart. 1693, printed in Rapbson's Hist, of Fluxions, p. 119. Leibn, 0pp. in. 484. Ccleberrimo Viro Godefrido Gulielmo Leibnitio IsAAcus Newton S.P.D. Literse tuie, cilm. non statim acceptis responderem, e manibus elapsa? inter schedas meas diu latuere, nee in eas ante hesternum diem incidere potui. Id quod me moleste habuit, ctim amicitiam tuam maximi faciam, teq : inter summos hujus soeculi Geometras a multis retro annis habuerim, quemadmodum etiam data omni occasione tes- tatus sim Nam quamvis commercia philosophica & mathe- matica quammaxim^ fugiam, tamen metuebam ne amicitia nostra ex silentio decrementum acciperet, idq : maxime cum Wallisius noster Historiam Algebrae in lucem denuo • i. e. the first of tlic three parts of tho recipe, tlie effect of wliich, accordiner to Boyle, was the production of a mercury which would grow hot with gold, t From a transcript oblipringly made for me by I^ord Lovelace. Append.] TO LEIBNIZ. 277 missurus nova aliqua c litcris inscriiit qua8 olim per manus D°* Oldenburgi ad te conscripsi, & sic ansam mihi dedit ea etiam de re ad te scribendi. Postulavit enim ut methodum quandam duplicem aperirem quam litcris trans- positis ibi celaveram. Qiiocirca coactus sum qua potui brevitate cxponere methodum meam. fluxionum quam hac celaveram sententia. Data a;quatione quantitatea quotcunque fiueiites involvente invenire Jluxiones, Sf vice versa, Spero autem me nihil scripsisse quod tibi non placeat, et siquid sit quod reprehensione dignum censeas, ut litcris id mihi significes quoniam amicos pluris facio quam inventa ma- thematica. Eeductioncm quadraturarum ad Curvarum rectifica- tiones* quam desiderare videris, inveni talem. Sit Curvro cujusvis abscissa w, ordinata y, et area a«f, posito quod a sit data quantitas fiuat .2? uniformiter, sitque ejus fluxio ih « a, et ipsius y sit fluxio y. A dato puncto D in rectA positione data DE sumatur DD^x^ et agatur indcfinita DCG ea lege ut cosinus anguli DBG sit ad Radium ut fluxio yf ad fluxionem .r « a, et inveniatur curva FG quam recta BG perpetuo taugit. Id enim semper fieri potest • Twenty-six years later this problem, which Euler calls ** celebre illud prolhsma multutn inter Geomctras aKitatum," wos proposed by Hermann in the Leipsic Acts (Aug. 1719), and was solved by him in the number for Apr. 1723, and by J. Bernoulli in the number for Aug. 1724. The latter shews how to obtain a moie general solution. See also Newton's Geometria Anulytica (Horsley, i. 508), his letter to Oldenburg, Jun. 23, 1673, and Euler, Conunent. Petrop. Tom. v. p. 171, We find no allusion to Newton's solution in any of Leibniz's published letters or papers. In the fig"ure FD should be a straight line. The following may assist some readers in verifying Newton's construction. Let X, Y be the co-ordinates of the required curve, on the length of whose arc (S) the area of the proposed curve is to be made to depend. Then S =J'd Y Vl + P", (dX^PdY) , pYPdP = rv/i^i^«./^--. Assume r d P - Jx and -===-, = 2 ; and X will be found - ^i^lril^ ^ - ,, Vl+P' a a* dy * " dy t fluxio y.] This should be either •' fluxio x" or ito equal "y," 278 LETTER [Append. Geometrice ubi fluxionum .r & y relatio gcometrica est. Sit G punctum contactAs, et ubi ^ punctum B incidit in punctum D, incidat punctum G in punctum F, In tangente BG sumatur GO wqualis curva; GF^ et CII ajqualis rcctso FD, \ \ji et erit BH « ^r. Qua inventa habe- tur area qucesita ax. ^T b — E Quae vir summus Hugenius in mea notavit ingeniosa sunt*. Parallaxis soils niiagr videtur^uam ipse statueram, et motus sonorum forte magis rectilineus est, at cajlos materia aliqua subtili nimis implere videtur. Nam cum motus coolestes sint magls regulares quam si a vorticibus orirentur, et leges alias observent, adeo ut vortices non ad regendos sed ad perturbandos Planetarum et Cometarum motus conducant, cumque omnia caelorum et maris phae- nomena ex gravitate sola secundum leges a me descripttas agente accurate quantum scntio sequantur, et natura sim- plici-ssima sit, ipse causas alias omnes abdicandas judicavi et cajlos materia omni quantum fieri licet privandos, nc motus Planetarum et Cometarum impediantur aut reddan- tur irregulares. At interea si quis gravitatem una cum omnibus ejus legibus per actionem materioe alicujus subtilis explicuerit, et motus Planetarum et Cometarum ab liac materia non perturbatos \sic\ iri ostenderit, ego minime adversabor. Colorum phsonomena tam apparentium ut loquuntur quam fixorum rationes certissimas me invenisse puto, sed a libris edendis manum abstineo ne mihi lites ab imperitis intententur et controversiaa. Alius est New- • In an ** Addition" to his " Discours de la Cause de la Pesanteur." Leid. 1690. Nic. Fatio writing to Iluygens from London, Feb. 24, 1690, says: "Mr. Newton, Mr., recevra parfaitement bien tout ce que vous avez dit. Je I'ai trouv6 tantde fois pret a corrigcr son livre sur des choses que je lui disois, que je n'ai pil assez admirer sa facility, et particulierement sur les endroits que vous attaqtiez. II a quelquo peine n entendre le Francois, mais il s'en tire pourtant avec un dictionaire." Again, Apr. 11 : " Mr. Newton, Mr., m'a assure qu'il prenit en fort bonne part tout ce qui est dans le traitt^ de la cause de la pesanteur.** AppENn.] TO LEIBNIZ. 279 tonus*, cujus opera in Hbrorum cditorum indicibus tibi occurrunt. His contestari volui me tibi araicum intcger- rimiim esse & amicitiam tiiam maxinie faccre. Vale. Dabam Cantabrigiro, Octob. J J. IGOSf. Utinam rectificationeni Hyperbola) quam te invenisse dudum significasti in lucem cmitteres. No. XXV. NEWTON TO IIAWES. ^fr Edward Paget, Fellow of Trinity College, and Mathematical Master at Christ's Hospital, drew up in 1()94 a scheme of reading for • This refers to the following passage in I^ibnii's letter : ** In Hbrorum apud Anglo« ejitorum Indicibus occurrere mihi aliquoties libri Alathematici autore Neutono, scd dubitavi a Te cssent, quod velleni, an nb alio homonymo." The author in question was John Newton, D.D., a writer of mathematical school-books. Morhof was probably thinking of this same " Doctor," when he called our pliilosopher ** Mcdicus Anglus." (The passajje alluded to occurs in n posthumous part of the Volyhistor, but was written, apparently, not long after the publication of Newton's Analysis of Solar Light. The expression is retained in Fabricius's editions of fhc work 1732 and 1747. Morhof died in 1691). 'I'he title is retailed by Saxius Onomatt. v. 120: '♦ Isaacus Newtonus Wool- stropensis Anglus, Mcdicus, Mathematicus et Philosophus Ix)ndinensis..." Compare Report of Committee of House of Commons on abuses in the Mint (Apr. 8, 1697), in which, on the Moneyers alleffini? themselves to be a Corporation, it is stated that ** D' Newton, present Warden of the Mint, declared that he had never seen any such Grant or Patent to the Moneyers ; and believed they had no other Charter, but the general Charter of the Mint, which he had in his possession," 6ic. lluding*s Annals of the Coinage, iii. 536, 540. (London, 1817). D«" Henry Newton, Envoy Extraordinary (1704—1710) to the Grand Duke of Tuscany and Republic of Genoa, tells us that he occasionally received compliment* that were intended for his illustrious namesake: ** Et multa adhuc expectant [Itali] a Summo Mathematico ejusdem mecum Cognominis, (indc aliquoties contigit ex errore nominis, me quoque non mcis laudibus oruari) prxseitim vero Mundum qualem Deus, ipsi quoque Ilobbesio, il-Ucrnus Geometra, ab initio formaverat, atque sapientissimus Creator in mensura, & numero fie pondere disposuerat ; sed intellectu facilem, non solum Mathematicis, nee quidem illis ex plcbe, Icf^endum intucndumque; sicque tile demum optimu, sibi, Patria;, omni denique Posteritati consulat.*' Inciter from Florence, Oct. 1, 1705, in his Kpistolt£...L\xcait 1710. As a sort of compensation a Ictttr of thanks from Lord CowjMjr has been recently published as addressed to our pliiloso- pher, which 1 strongly suspect was intended for the author of the work just quoted. (Lord Campbell's Chancellors^ iv. 341.) f Partly from Crelle's Journal, Band xxxii. (where a portion of the letter is lithographed from the original in the Royal Library at Hanover), and partly from a copy in the British Museum, Add. MSS. 6399. fol. 56, which seems to have come to the Museum with Cole's Collections. The letter has been recently printed in the edition of Leibniz's works now in course of publication at Berlin. 280 CORRESPONDENCE OF [Append. the boys under hia care. At a meeting of the Committee of the Schools of the Hospital on the 9th of May, Mr IJawes, the Treasurer, was ** desired when he goes to Cambridge on Friday next to take with him a copy of the old and new schemes, and advise with the Professor and other Mathematicians in the University concerning them, and get their opinions in writing which of the two schemes they judge best." New- ton's opinion of their respective merits is conveyed in this letter, which was sent enclosed in another to Paget. ffiyr Nathanael Hawes, Esq. S' I now returne you the papers you left in my hands. The two Schemes of learning I have compared, and find that the old one wants methodizing & enlarging ; the want of method you may perceive by these instances. 1. Arithmetick is set down preposterously in the 12*^ Article after almost all the rest of Mathematicks. ffor a man may understand and teach Arithmetick w^^'out any other skill in Mathematicks, as writing Masters usually doe, but w*^out Arithmetick he can be skilled in noe other parte of Mathematicks, & therefore Arithmetick ought to have been set downe in the very first place as the ffounda- tion of all the rest. 2. The parts of Arithmetick are set downe in severall Articles preposterously, ffor Decimal Arithmetick and the Extraction of roots are enjoined in the 3** Article before the boyes have learnt Arithmetick in integers & vulgar fractions. Then in the 4*^ & S*'* Articles they are enjoined Logarithms. And after all this tliey are required in the 12*^ Article to learn Arithmetick in generall, as if they had learnt nothing of it before. 3. Geometry and Trigonometry are confounded to- gether in the first Article, and again in the 4***. Whereas Geometry ought to have made one Article and Trigono- metry another, ffor these are accounted distinct sciences. 4. The use of Logarithms w*^** is set downe in the 8*^ Article, ought to have preceded that of Artificial Sines & ArrEND.] NEWTON AND IIAWES. 281 Tangents vf^^ is in the 4*^ ffor how can any man under- stand the Logarithms of Sines and Tangents, before he understands the Logarithms of Numbers in generall. 5. The doctrine of the Globes is set down in the 11*** Article and the projection of the Sphere or globe and making of Maps is set down in the 10^**. whereas the doc- trine of the globes ought to precede the projection of the sphere & making of Maps. Ifor how can any man pro- ject the lines on a sphere or globe into Maps, before he is taught what those lines are ? 6. The 10^** Article is worded improperly. fFor instead of saying, The projection of the Sphere in circles or globe in a plain divers wai/e^, It should have been said The projec- tion of the Sphere or globe in circles on a plain divers wayes, ffor the projection of a spliere in circles and that of a Globe in a plain are neither equipollent phrases, nor branches of a distinction, & therefore cannot be put together disjunc- tively (as they are in this Article) w^^out an impropriety of speech. 7. The Rule for finding the Latitude by the Sun or Starrs in the sixth Article, and the questions of plane Sail- ing w*^ the use of the plane Sea Chart in the seventh, ought to have come after the Doctrine of the globes, & the making of Maps or Charts ; & yet these are set after the other in the 10^^ and 11*^ Articles. Soc alsoe in the second Article, the making of the Scale of hours, Rumbs and Longitude, is improperly joyned with the Rule of three, & more improperly set before the doctrine of y* Globes. And in generall the whole scheme is soe confused & imme- thodical, as mj es me think that they who drew it up, had noe regard to the order of the things, but set them downe by chance as they first thought upon them, w^^out giving themselves the trouble to digest and methodise the heap of things they had collected together ; w*^** makes me of ^opinion, that it will not be for the reputation of the 282 CORRESPONDENCE OP [Append. fibundation to continue this scheme any longer w%ut put- ting it at least into a new forme. But then for the things it conteins I account it but mean and of small extent. It seems to comprehend little more then the use of Instruments, and the bare practise of Seamen in their beaten road, \\^^ a child may easily learn by imitiition, as a Parrot does to speak, w%ut understand- ing in many cases the reason of what he does ; & w^^ an industrious blockhead, who <}an but remember what he has seen done, may attain to almost as soon as a child of parts, and he that knows it is not assisted thereby in inventing new things & practises, and correcting old ones, or in judging of what comes before him : Whereas the INIathematicall children, being the flower of the Hos- pitall, are capable of much better learning, & when well instructed and bound out to skilful! Masters, may in time furnish the Nation w^'' a more skilfuU sort of Sailors, Builders of Ships, Architects, Engineers and ]Mathematicall Artists of all sorts, both by Sea and Land, then llrance can at present boast of. The defects of the old scheme you may understand by these instances. 1. It conteins nothing more of Geometry than what Euclid has in the beginning of his first book, and in the 10*** and 12*^ propositions of his sixth booke. All which is next to nothing. 2. There is nothing at all of symbolical Arithmetick, w*^ tho' not requisite in the vulgar road of Seamen, yet to an inventive Artist may be of good use. 3. The taking of heights and distances, and measuring of planes and solids is alsoe wanting, tho of frequent use, 4. Nor is there any thing of spherical Trigonom ?*y, tho the foundacon of a great many usefull Problei- in Astronomy, Geography and Navigation. 5. Neither is there any thing of Sayling according to the severall Hypotheses, nor of Mercators Chart, nor of Append.] NEWTON AND HAWES; 283 computing the way of Ships tho things w*** a Sailor ought not to be ignorant of. 6. The finding the difference of Longitude, Amplitude, Azimuts and variation of the compass is alsoe omitted, tho these things are very useful! in long voyages, such as are those to the East Indies, and a INIariner who knows them not is an ignorant. 7. Nor is there one word of reasoning about force and motion, tho it be the very life and Soul of Mechanical skill and manual operations and there is nothing soe IMe- chanical as the frame & managem* of a ship. By these defects it's plain that the old scheme wants not onely methodizing, but alsoe an enlargem* of the learning, ffor some of the things here mentioned to be wanting, are requisite to make a Mariner skilfuU in the ordinary road, and the rest may be often found useful! to such as shall become eminent for skill & ingenuity, either in Sea affaires, or such other mechanicall offices and imploymcnts, as the King may have occasion in his Yards, Docks, fforts, and other places, to intrust them with. Now the imperfections of the whole scheme are pretty well supplyed in that new one w*^^' is proposed to be esta- blished, ffor this is methodical, short & comprehensive. It excells the old one beyond comparison ; I have returned it to you, w*^ some few alterations for making the affinity, coherence and good order of the several! parts of the learning, more cleare and conspicuous, & supplying 6ome defects. The alterations arc of noe very great moment, excepting the addition of the last Article, w*^^ conteins the science of Mechanicks. The rest is as perfect as I can make it without this Article, whether this should be added may be a question, but since you concur w*^ me in the affirmative, I'le set downe my reasons for the addition, ffor w^'^out the learning in this Article, a Man cannot be an able and Judicious Mechanick, & yet the contrivance & 284 CORRESPONDENCE OP [Append. managem* of Ships is almost wholly Mechanical. Tis true that by good naturall parts some men have a much better knack at Mechanical things then others, and on that acco* are sometimes reputed good Mechanicks, but yet w**'out the learning of this Article, they are soe ffarr from being soe, as a Man of a good Geometrical head who never learnt the Principles of Geometry, is from being a good Geometer, ffor whilst Mechanicks consist in the Doctrine of force and motion, and Geometry in that of magnitude and figure : he that can't reason about force and motion, is as far from being a true Mechanick, as he that can't reason about magnitude and figure from be^ng a Geometer. A Vulgar Mechanick can practice what he has been taught or seen done, but if he is in an error he knows not how to find it out and correct it, and if you put him out of his road, he is at a stand; Whereas he that is able to reason nimbly and judiciously about figure, force and motion, is never at rest till he gets over every rub. Expe- rience is necessary, but yet there is the same difference between a mere practical Mechanick and a rational one, as between a mere practical Surveyer or Guager and a good Geometer, or between an Empirick in Physick and a learned and a rational Physitian. Let it be therefore onely considered how Mechanical the frame of a Ship is, and on what a multitude of forces and motions the whole business and managem* of it depends. And then let it be further considered whether it be most for the advantage of Sea affaires that the ablest of our Marriners should be but mere Empiricks in Navigation, or that they should be alsoe able to reason well about those figures, forces, and motions they are hourly concerned in. And the same may be said in a great measure of divers others Mechanical employ- ments, as buildings of ships. Architecture, ffortification. Engineering, ffor of what consequence Mechanical skill is in such Mechanical employments may be known both by Append.] NEWTON AND HAWES. 285 the advantage it gave of old to Archimedes in defending his City against the Romans, by w*^** he made himself soe famous to all future ages, and by the advantage the ffrench at present have above all other Nations in the goodness of their Engineers, ffor it was by skill in this Article of learn- ing that Archimedes defended his City, And tho the ffrench Engineers are short of that great Mechanick, yet by coming nearer to him then our Artificers doe, we see how well they fortify and defend their owne Cities, and how readily they force and conquer those of their Enemies*. You may consider to what perfection that Nation by their Schooles for Sea-Officers had lately brought their Navall strength, even against all the disadvantages of nature, and yet your schoole is capable of out-doeing them, ffor their's are a mixture of all sorts of capacities, your's children of the best parts selected out of a great multitude. Their's are young men whose faculties for learning begin to be as stiff and inflexible as their bones, and whose minds are prepossest & diverted with other tilings, yours are children whose parts are Limber and pliable and free to receive all impressions. A great part of their schools are scarce capable of much better learning than that in your old scheme, your's have already shewn by experience that they are capable of all the learning in the new one, except the last Article, w*^^ has not yet been taught them, and yet after they have learnt the rest, will prove noo harder then that w*^^' they had learnt before. And as your children are a select Number for parts, and capable of all the learning here proposed, and it will be for the Honour & advantage of the Nation to introduce a new spirit of • The capture of Mons in 1691, that of Namur in 1692, and of Charleioi in 1693, were among- Vauban*8 recent triumphs. When Newton wrote the above remarks he probably little anticipated the example that wquld be set by " tlmt nation ' lo his own country in paying a tribute to hie genius. The " Newton " in the French sfeani navy is a corvette of 26 guns, 220 horse power. 286 CORRESPONDENCE OP [Append. useful! learning among the Seamen, soe it will give your children a higher reputation for preferrment. And 1 take it to be for the Honour of both King Charles his memory and of the foundation, that this School should be as learned for Sea affaires as you can well make it ; and probably it was his designe and will, it should be soe, tho all this learn- ing was not started when he founded it. If you admit this learning, your school will certainly grow into greater repu- tation, & may be thereby more apt to stir up new Bene- factors and set a Precedent of good learning to all future foundations of the same kind, and if you admit it not, your scheme of learning will be imperfect and leave roome for future foundations to outstrip yours, w*^^ I beleive would not be for it's honour. fFor the scheme of learning, as I now returne it to you is an entire thing w*^** cannot well want any of it's members, ffor 'tis nothing but a combina- tion of Arithmetick, Geometry, Perspective and Mecha- nicks, I mean Geometry as well in sphericall surfaces as in plane ones. Geometry is the foundation of Me- chanicks, & Mechanicks the accomplishm* & Crown of Geometry, & both are assisted by Arithmetick for com- puting and perspective for drawing figures : Soe that any part of this Systeme being taken away the rest remaines imperfect. These considerations have moved me to pro- pose this Article to you, but perhaps the Governors may see reasons against it of greater weight w*^'* I am not yet acq*^** with, & therefore I onely propose this business and leave it wholly to their prudence. The INIain difficulty that I can think of, is, that the learning of this Article may take up too much of the childrens time. And yet if for all the rest of their learn- ing they are allowed (as you tell me) but two yeares & halfe I question not but another halfe yeare would be abundantly sufficient for this addition, and then they would goe to sea w^*' a complete Systeme of Mathematical! learn- Append.] NEWTON AND IIAWES. 287 ing. And perhaps it may be more for their advantage to spend this halfe yeare at schoolc in an important part of learning w*^^ they cannot get at Sea, then at Sea in learn- ing what they will afterwards learn there more readily if well instructed at School, before they goe thither. If two yeares were not at first thought too much for the old scheme of learning w'^'* (before the addition of the Article of taking prospects) was very meane and narrow ; four or five yeares for this new scheme would be but a moderate allowance at that reckoning, & therefore tis very much if they can learn it in three. And yet perhaps they may run through all the parts of it in two yeares and an halfe ; but not soe well : And I Would advise that they should rather be allowed three full yeares, then be sent away smatterers in their learning. But whether they be allowed two yeares & an halfe or three yeares, 1 conceive the time of their examination ought to be stated, ffor the liberty w*^** the Masters of Ships have had of taking away the boys sometimes before they had gone through the whole course of their !Mathe- maticall learning, seems to me a mischief w*^** may deserve a reformation, ffor the sending abroad unripe boys can be neither a reputation to the School, nor advantage to the Nation ; Such boyes being not onely less knowing then others, but alsoe less able to make use of what they have learnt, & more apt to forget it, as smatterers in a Grainar school doe their Latine. Nor doe I see how the genius & method of the School in goeing through the whole course of the Mathematicall learning can be carried on soe evenly and advantagiously, as when y* Mathem" Master shall be at a certainty in the Number of Scholars, & in the time against which he is to make them fit. As the constitution now is you leave a bad Mathematicall Master a liberty of making excuses when ever he shall prove negligent, & discourage a good one 288 CORRESPONDENCE OF [ArrENi>. by the uncertainty of his business & method & of the satis- faction & reputation of bringing his Schollars to perfection, & alsoe by leaving him exposed to such humours as may desire by that meanes to take opportunity of hurting him in his business or reputation : whereas it's your interest to make the place as desirable as you can, that when it becomes void you may have the greater choice of such men as are fittest for it, & encourage them to goe on cheerfull}'' with their duty. And if it may be for the credit & interest of y® foundacon not onely that the boyes should be well learned, but alsoe that they should be placed abroad w*^ the best Masters, & the appointing two solemn times every yeare for examining five boyes at a time & binding them out apprentices may draw together a greater choice of good Masters then in the petty examinations at present. As a ffair draws together a greater Number of Chapmen than little markets doe: If the giving publick notice of those times may alsoe make the thing more solemn & more known to the Nation^ & thereby conduce to the honour of the foundation, & probably to the stirring up of new Benefactors : I should think the conjunction of soe many advantages may well deserve an establishment, unless there should be some great objection against it w^^ I am not yet aware of. ffor you have told me that when the boyes have run through their course of learning there will be noe danger of their not meeting with Masters at the next publick examination, and if any of them, should then happen to fail of Masters, they would at all times after that be at liberty to goe with such Masters as could be met with. And as for the Examinations I should think that the more publick they are, the more the School will be concerned for its reputation, & the greater will be the reputation w*^^ it may get by the good performance of the boyes. If there be any advantage in publick Examinations, the more publick they are the greater the advantage : if in Append.] NEWTON AND HAWES. 289 private ones the Governors may have it at their Visitations by able and diligent Examiners w*^ as much privacy and severity as they please : And if more such examinations shall upon any occasion be found requisite, yet I con« ceive they should be made onely by Examiners appointed by the Governors, & obliged, soe soon as the^ Examination is over to give an account to the Governors, & to noe body else \v*^out their permission, of what ever they find amiss. When the boyes are sent to Trinity house to be pub- lickly examined perhaps it would not be amiss that the Mathematical! Master send along w**^ them a larger & more particular draught of the things they have been taught, & are prepared to be examined in, then that scheme of leam^ ing w*^^ you establish, and that the draught of every Master with the alterations from time to time made in it and the Number of the boyes who at every examination answer well and readily to the things therein, be kept upon record in the school as a standard of the learning w*^^ the boyes are capable of w*4n the time allowed them. And when the boyes are put out apprentices, they may be exhorted or obliged by the Governors to communicate to the School (in gratitude to the place of their education) such accurate observations, curious discoveries and select draughts as they shall make abroad in their Voyages and ffactories for rectifying the longitudes and situation of places in the Maps, or otherwise improving Geography, Hydrography, Navigation, the building of Ships, Trade or any valuable knowledge of remote Nations or Regions. And these or other curiosities communicated by them may be kept together in a convenient place as an Ornament of the Schoole to be consulted upon occasions. I have hitherto considered onely the Kings ffoundation, and herein I have been free in comparing the old & new schemes of learning, and speaking my thoughts about them, because, as you told me, it was desired. I hope it will give noe 19 290 CORRESPONDENCE OF [Append. offence to any body, ffor at the first founding of the Schoole, the old scheme might serve very well for a tryall, till it was known what learning such young children might be capable of. And I presume that the Mathematicians who drew it up, intended for them nothing more then part of that learning which is taught to persons of riper age in the ffrench Schools, and thought it more advisable to leave the method of the things to the Mathematical Master, then to be accurate in what could not be made perfect. The conjunction of Mr Stones fToundation* with the Kings seems to be well designed : ffor as both the Honour and Interest of the Kings fToundation is consulted by making Mr. Stone's subservient & usefuU to it : Soe it is both for the Honour of Mr. Stone's fToundation to have this relation to the King's, and for the Interest of it, that his boyes may be preferred to the King's, where they will be bound out Apprentices w*^ a better allowance. But care should be taken that the Kings boyes be not retarded in their learning, by joyning w*^ them too great a Number of other boyes of inferior parts, soe as to hinder them from getting through their scheme of learning within the time limited. I like well the designe of establishing some Latin Authors to be read in the Schoole, because the best Mathematicall books are in that language, & by useing the boys to Mathematicall Latin, they will be enabled to understand them. The Synopsis Algehraica and Wards Trigonometry are well chosen and soe is Eticlides nova methodo in regard of the short time allowed the boyes. Yet Euclid himself (suppose in Barrow's edition) would doe them more good if it could be compassed within the time, and would be more usefull to them in reading other Authors afterwards. And therefore the Governors may • Henry Stone had, in 1693, bequeathed the bulk of his property to the Hospital, of which at least £50 a year was to be devoted to the improvement of the mathematical department of the school. Append.] NEWTON AND HAWES. 291 establish, if they think fit, that the Boyes read either Euclidea nova methodo or else at the discretion of the Ma- thematical! Master the first six books of Euclides Elements in Barrows edition for plane Geometry and the ll*** and 12*^ books thereof for Solids. fTor soe the Mathematical! Master will be at liberty to read the Elements themselves soe soon as he finds he can compass it and the rest of the scheme w*Mn the time limited. As for the Doctrine of the Sphere the first book of Mercator*8 Astronomy is brief and well adapted to the use of the Schoole and therefore may be appointed. And now I have told you my opinion in these things, I will give you Mr. Oughtred's, a Man whose judgment (if any man's) may be safely relyed upon, ffbr he in his book of the circles of proportion, in the end of what ho writes about Navigation (page 184) lias this exhortation to Seamen "And if, saith he, the Masters of Ships and Pilots will take the pains in the Journals of their Voyages diligently & faithfully to set down in several! columns, not oncly the Rumb they goe on and the measure of the Ships way in degrees, & the observation of Latitude and variation of their compass ; but alsoe their conjectures and reason of their correction they make of the aberrations they shall find, and the qualities & condition of their ship, and the diversities and seasons of the winds, and the secret motions or agitations of the Seas, when they begin, and how long they continue, how farr they extend & w^^ what inequality ; and what else they shall observe at Sea worthy consideration, & will be pleased freely to communicate the same with Artists, such as are indeed skilful! in the Mathematicks and lovers & enquirers of the truth : I doubt not but that there shall be in convenient time, brought to light many necessary precepts w*^^ may tend to y" perfecting of Navi- gation, and the help and safety of such whose Vocations doe inforce them to commit their lives and estates in the 19—2 292 CORRESPONDENCE OP [Append. vast Ocean to the providence of God." Thu8 farr that very good and judicious man Mr. Oughtred. I will add, that if instead of sending the Observations of Seamen to able Mathematicians at Land, the Land would send able Mathematicians to Sea, it would signify much more to the improvem* of Navigation and safety of Mens lives and estates on that element. I hope S*^ You will all interpret my freedome in this Letter candidly and pardon what you may therein think amiss, because I have written it with a good will to your ffoundation, and now I have spoke my thoughts I leave the whole business to the wisdome of your selfe and the Governors. I am, Hon'** S^ Your most humble & most obedient Servant, Cam^ idge May 25*^, 1694. Is. Newton. [Accompanying the above.] A 'Sew Scheme of Learning proposed for the Mathematical Boys in Christ's Hospital. {Paget's scheme with a few alterations by Newton who has also added the 10th article.} 1. Arithmetick in Integers, Vulgar fractions 8e Deci- mals, in Proportional numbers natural and Artificial, in Symbols of unknown Numbers & in Equations. 2. Geometry speculative and practical in planes and Solids. 3. The Application of Arithmetick to Geometry in determining and protracting Lines, Angles and figures by Numbers natural and Artificial, Symbols of Numbers and tables of Sines & Tangents. 4. The description and properties of figures in per- spective with the Arts of drawing and designing. Append.] NE^VTON AND HAWES. 293 5. The use of the best Instruments in working by proportionals taking Angles, heights and distances, and measuring planes and solids. 6. The Doctrine of the Globes and the Rudiments of Geography Hydrography and Astronomy. 7. The descriptions of the Globe in perspective com- monly called Projections and the Art of making Charts and Maps. 8. The Doctrine of Spherical Triangles w*^ their ap- plication in projecting and computing all the usefull Pro- blems in Geography, Astronomy and Navigation. 9. A full application of the learning aforesaid to Navigation particularly to the severall Hypotheses thereof, commonly called Plane, Great circle and Mercators sail- ing. As alsoe the use of Charts and Sea Instruments for observation and their application to the finding of the Latitude, difference of Longitude, Amplitudes, Azimuths and variation of the compass by the Sun or Starrs, w^** the knowledge of the Tides and Roman Calender, and the method of keeping Journals and of finding the difference of the Longitudes of Shores by the Eclipses of Jupiters Satellites. 10. The principles of reasoning about force & motion, particularly about the five mechanical powers, the stress of ropes and timber, the power of winds, tides, bullets and bombs, according to their velocity and direction against any plane, the line w*^^ a bullet describes, the force of weights and springs and the power of fluids to press against immersed bodies, and bear them up, and to resist their motions ; w*^ the application of this learning to Sea affaires, for contriving well and managing easily, speedily & dextrously. Levers, Pulleys, Skrews, Anchors, Pumps, Rudders, Guns, Sails and other Tackle, judging truly of the advantages & disadvantages of VesscUs, Havens, fforts, Engins and new Projects, & observing or discovering what 294 CORRESPONDENCE OF [Append. ever tends to make a Ship endure and Sail well, or other- wise to correct or improve Navigation. Is. Newton ♦, No. XXVI. NEWTON TO IIAWES. jfor Nathan". Ilawes, Esq. S' Yesterday I sent by the Carryer a Letter to you w*** the papers you left in my hands, inclosed in another to M^ Paget. In that I wrote to you, you mil find my thoughts set downe at large about the old and new schemes of learning. Looking this morning into S*" Jonas Moore''s Systcme of Mathematicks w*^^ he composed about 15 or 16 yeares agoe for the use of your schoole, I find by the title page and preface to that book, that the new Scheme was for the most part composed at that time by S"* Jonas, ifor there (as is mentioned in the preface) he proposes to teach in order these sciences. 1. Arithmetick vulgar, decimal and Logarithmical. 2. Practical Geometry. 3. Trigonometry plane and spherical, 4. Cosmography w^^ includes the Doctrine of the Globes with Astronomy and Geography. 5. Navigation with the making of Maps. After these and many Tables & Geographical Maps follow Algebra & speculative Geometry conteined in the first, 6*^ & 11*^ & 12*^ books of Euclid's Elements. The difference between this method and the new Scheme of learning now proposed lies in these things. 1. In the new scheme (as alsoe in the title page to S*" Jonas Moores book) Algebra is joyned w*^ Arithmetick, & speculative Geometry w**^ the practical ; w^^ certainly is • This and the two following letters are from ilie official copies in the Christ's Hospital Court Book. Append.] NEWTON AND IIAWES. 295 the best method for Schollars of good parts who are to learn both. But in the preface to S"^ Jonas Moores book Algebra & speculative Geometry are separated & taught apart after all the other Sciences ; w*^** is best for a mixturo of Schollars of all degrees of parts, some of w*** are not capable of learning the whole Scheme. 2. S^ Jonas joyns plane & spherical Trigonometry to- gether, but in the new scheme spherical Trigonometry ia set after the Doctrine of the Sphere w*^** is more proper for a learner. 3. S"^ Jonas omits perspective and Mechanicks & referrs the taking heights and distances & mensuration of planes & solids to the end of practical Geometry and plane Trigonometry : whereas in the new scheme perspective is inserted between them for delineating the heights, dis- tances and solids w*^^ are to be measured, & again after y*^ doctrine of the globes for the making of Maps. This I thought proper to signify to you, that the Governors of the Hospitall might have the judgment of S"" Jonas in this matter. fFor he follows not y® old scheme in any thing, but agrees well enough w*^ the new one, both in y* substance of the things he teaches, & in the order of them, if perspective & Mechanicks be inserted into his Systeme in their proper places. By S"^ Jonas his departing soe much from y® method of the whole scheme, and supply- ing some things w^^ were wanting in it & coming soe neare to the new one, you may gather that the old one in his judgm* wanted information, & that the new one is not much amiss. S"^ I am, Yo' most humble & most obed' Serv' Is. Newton. The new scheme with Newton's modifications was bent to "Wallia and David Gregory at Oxford, who gave their *' opinion and advice" respecting it in a joint paper, dated Juno 13, 1C94. "After a very large debate" on June 25, it was agreed to adopt the new scheme. The 296 CORRESPONDENCE OP [Append. Committee also stated it as their " opinion that the 10"" Art. in the now scheme about the 5 Mechanical powers cannot be taught under 6 months longer time than is allowed for their instruction in Mathema- tics. Also that the Court be desired to request Mr Newton to enlarge himself upon the aforesaid lO*^ Art. that so M' Paget may be the bettor qualified for their instruction therein, being very advantageous to the improvement of Navigation." It was at the same time ordered that "humble & hearty thanks be returned to Mr Newton, D" Wallis & Gregory for their extraordinary pains & kindness in this affair." A letter of thanks was accordingly sent Aug. 9, in which it is 'observed that " the plan requiring long & serious consideration, we have chosen a committee to consider tliereof, but being unwilling to defer our acknowledgments" &c. No. XXVII. NEWTON TO HAWES. S'. Cambridge June 14. 1696. I should have writ to you by Mr. Newton* but that I stay'd to consider further of y*' scheme of Mathematical learning before it be established, ffor the last Article seemed too indefinite to be subscribed, and in the forme it is there set downe, has noe books written of it„ & therefore I have changed it into the last A rticle of the scheme I now send you enclosed in this Letter, ffor this last Article eonteins as much of the other, as has been hitherto re- duced to a certain science and something more, and is definite, soe that the Master may know what he subscribes, and the Governors what the Master is obliged to by his sub- scription. It has alsoe books written upon every parte of it to make it more fit for the school. As for M^ Newton I never took him for a deep Mathematician, but recom- mended him as one who had Mathematicks enough for your business, w*^ such other qualifications as fitted him for • Mr Samuel Newton, who had been recently appointed to the IVIathcmatical Mastership at Christ's Hospital, vacant by Paget's resignation. Compare Newton's letters in Baily's Flamsteedf pp. 153, 154, 156. AprENn.] NEWTON AND IIAWES. 297 a Master in respect of temper and conduct as well as learning. But I reckon two yeares too short a time for this scheme of learning, and D'. Gregory who taught Mathematicks in Scotland w*** very good success, and was here last weeke, tells me that by the time he spent in teaching he should reckon three yeares little enough for this scheme. M'. Newton may try if he can compass it sufficiently in two yeares but if that be found too little, perhaps the wisdome of the Governors may soe order things as to allow him halfe a yeare more from the other schooles. ffor were it not for some Mathematicall bookes in Latine, I should think that language of soe little use to a Seaman as not to deserve four or five yeares of the chil- drens time, while Mathematicks are allowed but two ; I thank you for your concerne and pains in behalfe of M'. Newton, and am very glad to understand that he behaves himselfe so well, ffor tho* I was almost a stranger to him when I recommended him, yet since he was elected, I reckon myselfe concerned that he should answer my recommendation. The ill will you may have got by your acting for him I perceive is but of little extent and cannot hurt you. M*". CaswePs freinds at Oxford blame his freind* neere London, and some of them think the place would not have suited with his humour, soe that I am satisfyed you made the best choice. S*". Your most humble & most obedient Servant. Is: Newton. • Flamsteed, who had recommended Caswell as Paget's Buccessor. 298 CORRESPONDENCE OP [Append. [Enclosed in the above.] A Now Scheme of Learning proposed for the Mathematical boyes in Christ's Hospitall.* 1. Arithmetick in Integers, Vulgar fractions & Deci- mals, in Proportional numbers natural and Artificial, in Symbols of unknown Numbers & in Equations. 2. Geometry in Planes & Solids, with the Demon- strations thereof & y* practise by the Rule & Compass. 3. The application of Arithmetick to Geometry in determining & protracting lines, angles, and plane Trian- gles f, by numbers natural and artificial, Symbols of Num- bers, & Tables of Sines & Tangents. 4. The description & properties of ffigures (rectilinear & circular) in Perspective, w*^ the Art of Designing J & Drawing}. 5. The construction & use of the best Instruments in working by Proportionals, taking Angles, Heights & Dis- tances, & Surveying, Guaging, or otherwise measuring Planes & Solids. 6. Cosmo^Vi'p.^y, or the rudiments of Astronomy, Geo- graphy & Hydrography, with the Projections of the globe in Perspective, & the art of making Maps & Charts. 7. The doctrine of Spherical Triangles, with their application in projecting & computing all the useful Pro- blems in Astronomy, Geography & Navigation. 8. A full application of the Learning aforesaid to Navigation particularly to the several Hypotheses thereof commonly called Plane, Great circle. Parallel & Mercator's sailing. As also the use of Charts & Sea Instruments for Observation, & their application to the finding of the Lati- tude, difference of Longitude, Amplitudes, Azimuths & • There is a copy of this paper in Newton's handwriting in Trin. Coll. Library in a folio volume marked II. 5. 4. t In Newton's MS. it is " plane triangles & other figures." X These words change places in Newton's MS. Append.] NEWTON AND HAWES. 299 Variation of the Compas by y* Sun or stars, with the know- ledge of Tides, Currents & the Roman Calendar & the method of keeping Journals, & of finding the longitudes of shores by the Eclipses of Jupiters Satellites. 9. The mechanical Arts or Sciences of the five Powers, The laws of motion, Hydrostaticks, Gunnery & ffortifica- tion. A minute dated 19 July, 1695, states that **the consideration of the new sclieme... drawn up by Mr Newton... which was referred by the last Court to this Committee is for several reasons postponed until another time." The master seems to have found the scheme difficult to carry into practice, and a course of study formed by a fusion of the old and new schemes, and excluding Mechanics except " so much of gimncry as is necessary for sea service" was afterwards adopted. (Minutes of Apr. and June 10, 169(5.) A few notices of our philosopher, taken from the same source to which we are indebted for the three preccdinfl; letters, and exhibiting him in connexion with Christ's Hospital, may be given hero. " March 25, 1696. The Committee being informed that Mr Newton is in town {summoned by Charles Montagu's letter offering him the Wardenship of the ^lint} and will stay some days, desired the Treasurt^r to request him to examine and consider of the Library belonging to the Mathematical School.... and give his opinion what books are wanting that may be most useful and necessary. July 13, I6O7. The Committee did desire Mr Isaac Newton now present to deliver his opinion concerning the said jfive} boys, who was pleased to say that about 10 or 14 days since ho examined them and then found them perfected, except in a very few particulars, which by this time he don't question but they are masters of, and therefore is of opinion they are well qualified to be placed forth to sea as apprentices... And this Committee returned their unanimous thanks to Mr Professor Newton for his great kindness and pains taken herein." He is also mentioned as present at the Hospital meetings on Sept. 23, (visitation of all the schools in the hospital) and Dec. 16, 1697, on which latter day he was appointed one of a committee to consider how £100 might best be laid out for the improvement of the mathe- matical library. 300 LETTER OP WALLIS. [Aitend. No. XXVIII. WALLIS TO NEWTON. Sip, Oxford, Apr. 10, 1696. I was in hopes of seeing you in Oxford last summer ; which made me neglect sending you (by the Carrier) two Cuts which belonged, to the Volume you had before. They were not >\TOught oft at y® Rolling-Press when you had th« rest ; but are easy to be inserted in their proper places. I send them now, with the other Volume ; which I desire you to accept. I understand (from Mr Caswell) you have finished a Treatise about Light, Refraction and Colours; which I should be glad to see abroad. 'Tis pitty it was not out long since. If it be in English (as I hear it is) let it, how- ever, come out as it is ; & let those who desire to read it, learn English. I wish you would also print the two large Letters of June and August { October { 1676. I had intimation from Holland, as desired there by your friends, that somewhat of that kind were done; because your Notions (of Fluxions) pass there with great applause, by the name of Leibnitz's Calculus Differ entialis, I had this intimation when all but (part of) the Preface to this Volume was Printed-off'; so that I could onely insert (while the Press stay'd) that short intimation thereof which you there find. You are not so kind to your Reputation (& that of the Nation) as you might be, when you let things of worth ly by you so long, till others carry away the Reputation that is due to you. I have endeavoured to do you justice in that point ; and am now sorry that I did not print those two letters verbatim, I understand you are now about adjusting the Moons Motions ; and, amongst the rest, take notice of that of the Comon Center of Gravity of the Earth & Moon as a conjunct body : (a notion which, I think, was first started Append.] LETTER OP WALLIS. 301 by me, in my Discourse of the Flux and Reflux of the Sea.) And it must needs be of a like consideration in that of Jupiter with his Satellites, & of Saturn with his. (And I wonder we have not yet heard of any about ^loon.) But Saturn and Jupiter being so far off, the efTects thereof are less observable by us than that of the ^Moon. My advise upon the whole, is, that you would not be too slow in publishing what you do. I am S-^ Your very humble Servant, John Wallis*. For Mr Isaac Newton, Fellow of Trinity College, ^ Professor of Mathematick, in Cambridge. With a Book {the 1 at Vol. of Wallis's Works.} "NVallis was a strong advocate for the immediate publication of di»* coveries. In a letter to Waller (Sec. to Royal Soc), April 30, ho dwells upon the same topics, and speaks of Newton's Treatise as " finished & fairly transcribed some while since. I wisli he were called upon to print it without farther delay. Perhaps Mr Ilalley may prevail with him so to do, &c.'* Waller writes back May 15 " Mr Ilalley has promised to write to Mr Newton concerning those letters {to Leibniz} you mention. I hope they may be procured from him & thank you for the intimation thereof." Wallis writing to Ilalley Nov. 11, says : '* I have written several let- ters to Mr Newton about it {i. e. printing the two letters} pressing with some importunity the printing of them, and of his Treatise about Light and Colours (as being neither just to himself nor kind to the publick to delay it so long. As to the Letters I sent him a fair transcript ready for the press {Newton's copies of them may have perished in the fire which destroyed a mass of other papers, and, as Wallis supposed, Leibniz's answers among them ; see Wallis's Works III. 654 or CW- merc. Epistol. 110 or 211 ed. 2}, which if he would print, it might best be done here, (and I would take the care of it) But he did not seem forward for either As to that about laght & Colours • Orig. Lett. Bk, Roy. Soc. W. 2. 48. Part of it is printed in Raphson's Hist, of Fluxions, p. 120, 802 LETTER [Append. (for which I am more solicitous) your interest may possibly prevail with him better than mine to get it published." " In pursuance of" a letter from Ilalley dated Nov. 21, Wallis sent him copies of the two letters on the 20th, observing: "I am glad Mr Newton is inclinable to print some of the things he hath by him. So many as he hath on his hands at once do hinder one another. I am most fond of his Book of Light and Colours. His fear of disputes and cavils need not trouble him. It will be at his choice whether or not to answer them. His Hypothesis will defend itself. "We are told here that he is made Master of the Mint" &c. Orig, Lett. Bk. Roy. Soc, W. 2. 56. No. XXIX. NEWTON TO HAUINGTON. Mr John Harington (of the family of "Ariosto" Ilarington and ** Oceana" Harington), an undergraduate of Oxford, seems to have had some conversation with Newton upon a method which had occurred to him of representing musical intervals by the additions of the sides (3, 4, 5) of a right-angled triangle, and to have tlluded to the bearing of the subject upon the principles of architectural beauty. At Newton's request he sent the details of his method with remarks upon the appli- cation of harmonical ratios to architecture, in a letter dated Wadhain College May 22. 1698. The receipt of this letter Newton acknow- ledged in the following kind and encouraging terms. Sir, By the hands of your friend, Mr. Conset, I was favoured with your Demonstration of the Harmonic Ratios, from the Ordinances of the 47th of Euclid. I think it very explicit and more perfect than the Helicon of Ptolemy, as given by the learned Doctor Wallis. Your observations hereon are very just, and afford me some hints which, when time allows, I would pursue, and gladly assist you with any thing I can, to encourage your curiosity and labours in these matters. I see you have reduced, from this wonder- ful proposition, the inharmonics as well as the coincidences of agreement, all resulting from the given lines three, four, and five. You observe that the multiples hereof furnish those ratios that afford pleasure to the eye in architectural Append.] * TO IIARINOTON. 803 designs : I have, in former considerations, examined these things, and wish my other employments would permit my further noticing thereon, as it deserves much our strict scrutiny, and tends to exemplify the simplicity in all the works of the Creator ; however, I shall not cease to give my thoughts towards this subject at my leisure. I beg you to pursue these ingenious speculations, as your genius seems to incline you to mathematical researches. You remark that the ideas of beauty in surveying objects arises from their respective approximations to the simple con- structions, and that the pleasure is more or less, as the approaches are nearer to the harmonic ratios*. I believe you are right; portions of circles are more or less agreeable, as the segments give the idea of the perfect figure from whence they are derived. Your examinations of the sides of polygons with rectangles certainly quadrate with the harmonic ratios. I doubt some of them do not ; but then they are not such as give pleasure in the for- mation or use. These matters you must excuse my being exact in, during your inquiries, till more leisure gives me room to say with more certainty hereon. I presume you have consulted Kepler, !Mersenne, and other writers on the construction of figures. What you observe of the ancients not being acquainted with a division of the sesquialteral ratio is very right ; it is very strange that geniuses of their great talents, especially in such mathematical considera- tions, should not consider that, although the ratio of three to two was not divisible under that very denomination, yet its duple members six to four easily pointed out the ditone four to five, and the minor tierce six to five, which are the chief perfections of the diatonic system, and without which the ancient system was doubtless very imperfect. It • Comp. Kepler, Harmon. Munrfi, p. 126. In Architectonica quaecuiw^ue propor- tiones long^itudinis ad latitudinem vel crassitiem plurimum probantur, etiam a non Mathemnticis spectatoribus, os quum proximtc harmonicis inveniuntur. 304 REFORMATION OF CALENDAR. [Append. appears strange, that those whose nice scrutinies carried them so far as to produce the small limmas, should not have been more particular in examining the greater intervals, as they now appear so serviceable when thus divided. In fine, I am inclined to believe some general laws of the Creator prevailed with respect to the agreeable or unpleasing affections of all our senses ; at least the sup- position does not derogate from the wisdom or power of God, and seems highly consonant to the macrocosm in general. Whatever else your ingenious labours may pro- duce I shall attentively consider, but have such matters on my mind, that I am unable to give you more satisfaction at this time ; however, I beg your modesty will not be a means of preventing my hearing from you, as you proceed in these curious researches ; and be assured of the best services in the power of Your humble Servant, {Jermyn Street} May 30, 1698. Is. Newton*. No. XXX. The decree of the German Diet (Ratisbon, Sept. 23. 1699, see Mim^ tucla^ Hist, des Math. iv. 325,) refonning the Julian Calendar and order- ing (1) that the day after Febr. 18. 1700 should bo March 1, and (2) that Easter should bo determined by astronomical calculation (viz. of the exact time of the vernal equinox and the full moon following it), gave rise to considerable discussion among the theologians and scientific men of the Empire. In Leibniz's "Works (iv. pars ii. 115 — 137) will bo found the correspondence which he had with Roemcr upon the subject. Leibniz also consulted the French Academy (lb. 143) and the Royal Society on the second of the two Articles of the Ratisbon conclusum : his application to the latter body was laid before Newton, whose answer is contained in the paper now presented to the reader. • H. Harington's Nug84 Propositioncs princi pales earum quaB in Philosophia* Principiis Mathematicis hahentur cum Socictato Regia conimunicaro coopit, &c. 2. Ineunto anno 1(504 Ncwtonus Propositioncs cum Socictato Regia communicavit, &c. 3. Anno J 083 Ncwtonus Propositioncs 4. Anno 1083 ad finem vergento Nowtonus Propositioncs princi- pales carum...habontur Londinum misit eaodemque cum Socictato Regia mox communicata) sunt, &c. Newton first of all clearly wrote 1G84, then altered tho 4 to a 3, afterwards crossed all tho figures out and wrote distinctly 1683. I mention this the more particularly, because Mr Rigaud says (Essay, p. 20) that in tho MS. of tho latter of the two fragments which ho has published from th.e Macclesfield Collection, tho year 1683 was at first written, " tho last figure having been evidently altered to a 4." Newton therefore after endeavouring to recollect tlie exact year in which ho sent up tho fundamental propositions of tho Prindpia to London, antedated the event by a twelvemonth. Sec Syn. View of his Life, under date Nov. 1684. Ex Epistola cujusdam ad Amicum. Anno 1683 ad fmem vergente Newtonus Propositiones principales earum qurs in Philosophis9 Principiis Mathe- maticis habentur Londinum misit e^edemq : cum Socic- tato Regia mox communicatae sunt, annoq : 1686 Liber ille ad Societatem missus est ut imprimeretur, et pr» irao anno lucem vidit. Deinde anno 1688 epitome ejus in Actis Lipsicis impressa est, qua lecta D. Lcibnitius Epistolam de lineis opticis, Schediasma de res{is}tentia Medii & motu projcctilium gravium in Medio resistente, & Tentamen de motuum ccelcstium causis composuit & in Actis Lipsicis ineuntc^anno 1689 imprimi curavit, quasi Ipse quoque prtecipuas Newtoni de his rebus Proposi- tioncs invenissct idque methodo diversa ct Librum New- toni nondum vidissct.^ Qua licentia concessa Authorcs quilibet inventis suis facile privari possunt. Quam primum Append.] LEIBNIZ'S PAPERS. 309 Liber Newtoni lucem vidit exemplar ejus D. Nicolno Fatio datum est ut ad Leibnitium mitteretur. Vidcrat Leibnitius Epitomen ejus in Actis Lipsicis. Per commer- cium epistolicum quod cum viris doctis passim habcbat, cognoscere potuit Propositiones principalcs m libro illo contentas imo & librura ipsum procurarc. Sin Librum ipsura non vidisset videre tamon dcbuissct antequara sua de iisdem rebus cogitata publicaret, idq: ne festinando errarct in sub { j \ ecto novo ac difficili et Newtono injurius esset auferendo inventa ejus, et Lectori molestus repe- tendo qujB Xewtonus antea dixerat, & contentioncs de inventis excitaret, ut antea fecerat in causa ^Moutoni. Dicit enim in fine Scbediasmatis de resistentia Medii : Nobis nunc fundamenta Geometrica jecisse suffccerit in quibus* Qua3 de Lineis Opticis habet, primo intuitu ex New- tonianis consequuntur, positis sinubus incidentiaj et reflexi- onis a?qualibus. In scliediasmate de llesistentia Medii, Resistentiam cum Xcwtono duplicem facit, unam qua3 a Medii glutino- sitate et frictione oritur, alteram quae a Medii densitate. Priorem vocat absolutam, posteriorem relativam. Prio- rem facit velocitati proportionalem posteriorem cum Xcw- tono facit in duplicata ratione vclocitatis. Priorem tractat in tribus Articulis, eaq : sola tradit qua; Xewtonus in Libri secundi Propositionibus quatuor primis de hujusmodi resistentia prius dixerat. Posteriorem tractat in Articulo quarto quinto et sexto. Et qua) in articulo quarto habet Xewtoniana sunt. In quinto Propositiones quatuor (tertia quarta sexta et septima) sunt falsa) f. In sexto Propo- • There not being room for the remainder of the quotation in the MS., there is a mark after "quibus" apparently referring to anotlier paper which is lost. The whole of the i):u^sii!,'e will, however, be found quoted afterwards, p. 313, lin. 11. + Newton does not seem to have decided whetlier to write "non sunt veras" or "sunt falsaj." He first of all used the latter phrase, then crossed it out and wrote the former above it, but afterwards restored tiie old phrase underneath its oriffinal place. 310 STRICTURES ON [Append. sitiones sunt tantum diKOj, et iitraq : falsa est. Corpus enim, ubi resistentia est in duplicata ratione velocitatis, non fertur motu composito ea motibus duorum Articulorum praecedentium. Demonstret Leibnitius hasce sex Propo- sitiones si pro veris haberi velit. In tentamine de motuum ccelestium causis*, Leibnitius deducit circulationem harmonicam Planetarum a circu- latione harmonica Vorticum, & ascensum et descensum Planetarum ab eorum gravitate, dicitq : (in Propositione tertia) nihil referre quia sit motus rectilineus quo ad centrum acceditur vel ah ij^so receditur (quern motum vocat paracen- tricu) modo circulationes sunt harmoniccc. Imo multum refert. Nam si motus paracentricus si paulo velocior vel paulo tardier Apsides Planetarum non manebunt in locis suis, & propterea Sectiones oonicas non describentur. Co- nicas igitur Seetiones deseribi Leibnitius non demon- stravit. In sexta Tentaminis Propositione docet ex Phrenomenis Planetas motu harmonico ferri, in septima deducit inde motum liarmonicum vorticum. Qua? de Vorticibus dicuntur sunt mere hypothetica, & cum motu Cometarum conciliari non possunt, neque quadrant cum Planetarum temporibus periodieis qua3 sunt in ratione sesquiplicata distantiarum ab orbium centre communi. Hoc notavit Gregoriusf, et Pespondit Leibnitius Vortices non moveri motu harmonico nisi in singulis Planetarum orbibus seorsim spectatis ; in intervallis orbium vortices alia ratione moveri; id est, partes vorticum alternis vicibus harmonice et non harmo- nice per multa orbium intervalla revolvi. Miraculis plena est hsec hypothesis motumq : Cometarum adhue magis • Among the Lucasian MSS. there is a paper in Keill's handwriting entitled " Notae in Acta Erud...Anno 1689, Pag. 84 et scq." in which the errors of this essay of Leibniz's are briefly exposed, li seems to be the same as tliat mentioned by Wilson (Robins's Tractit II. 351) and apparently attributed by him to Newton. t Astron. Element, p. 102. Append.] LEIBNIZ'S PATERS; 311 perturbat & cum Vorticibiis Satellitum Planetarum minimc consistit. JMotiis Satellitum Jovis sunt summe regulares & Vorticem summe regularem circum Jovem requirunt : et hujusmodi Vortex impediet motum harmonicum Vorticis Solaris intra Orbem Jovis. Et pra3terea si Planetoe a Vorticibus deferuntur & gravitant etiam in Solem ut vult Leibnitius, ut ha) dua; vires seinvicem non perturbent, necesse est ut vis ilia qua Planetae deferuntur a vorticibus in Orbem & versus Solem incurvantur sit ipsa gravitas: cum tamen gravitas non minor sit ad polos Solis et Planetarum quam ad eorum ajquatores, vortices vero non agant ad polos, ad luce vis centripeta a motu harmonico oriunda debet esse reciproce non ut quadratum sed ut cubus distantia) Planeta) a Sole per Corol. 1 Prop. 4 Lib. 1 Principiorum Mathemat. Dcniq : Leibnitius nullam reddit causam motus harmonic! vorticum sed hunc motum sup- ponit tantum ut motibus Phmetarum a Kcplcro detectis consentaneum, ideoq: non demonstravit Planetas in Or- bibus EUipticis harmonice ferri. Et hoc non demonstrate nihil demonstravit quod alicujus sit momenti. Undecima Tentaminis Propositi© est ha;c. Conatus centrifugus exprimi potest per sinum versum anguli circula- tionis, Et vera quidem est hajc Propositi© ubi circulatio fit in circulo sine motu paracentrico. Sed ubi fit in Orbe excentrico Propositi© vera non est. Conatus centrifugus semper ajqualis est vi gravitatis & in contrarias partes dirigitur per tertiam motus Legem in Principiis IMathe- maticis Newtoni, et vis gravitatis exprimi non potest per sinum versum anguli circulationis, sed est reciproce ut quadratum Radii. Duodecima Tentaminis Propositi© ha;c est. Conatua centrifwji harmonice circulantis aunt in ratione radiorum reciproce triplicata. Pectins dixisset quod sunt in ratione radiorum reciproca diiplicata. Sunt qnim viribus gravitatis ajquales ut supra dictum est; et gravitas est in raticne radiorum reciproca duplicata. 312 STRICTURES ON [ArpoD. Decima quinta Tentaminis Propositio ha?c est. In omni circulatione harmonica elementum impetus paracentrici {hoc est incrementum aut decrementum velocitatis descendendi versus centrum vel ascendendi a centra) est differentia vel summa sollicitationis paracentricce, {hoc est impressionis a gravitate vel levitate aut causa simili factop) et dujili conatus centrifiuji ah ipsa circulatione harmonica orti. Summa quidem si levitas adsitt differentia si gravltas, Errorem eiiormem in liac Propositione Leibnitius postea correxit & pro cluplo conatu ccntrifugo conatum simplum scripsit (Vide Acta Lips. Anno 1706 pag. 447.) Sed Propositio tamen etiamnum falsa manet. Ob sollicitationem paracentricam & eonatiini cen- trifugum inter se aquales, differentia eorum nulla est, ideoq : elementum impetus paracentrici per banc Proposi- tionem semper debet esse nullum, et velocitas descendendi versus centrum vel ascendendi a centro semper debet esse uniformis. Quod verum esse non potest. Prajterca in De- monstratione hujus Propositionis error admittitur his verbis. Jam F2M cequ. {N2M seu) G^D^NP. Pro N2M Lie scribitur G^D quamvis G 2D sit major quam N2M excessu G^M, Tandem ex falsis hisce Propositionibus Leibnitius co- natur demonstrare, Quod Planeta? circa Solem in Ellipsi harmonice circulantis gravitas in Solem sit reciproce ut quadratum distantioe Planetae a Sole. Et ha3C est Leibnitii Propositio decima nona. Errat vero in Demonstratione citando duas falsas Propositiones duodecimam scilicet et decimam quintam quarum errores se mutuo corrigunt : Et errando Propositionem minime invenit minime demonstravit sed a Newtono invcntam et demonstratam conatus est aliter invenire et demonstrare ut suam faceret. Per duos errores se invicem corrigentes calculum aptare potuit ad eonclusionem propositam, veritatem invenire ac demon- strare non potuit. Propositio vigesima deducitur a decima nona ideoq ; non demonstratur. Append.] LEIBNIZ'S PAPERS. 3J3 Propositio vegesima [sic] prima et Propositio vigesima quiiita, niinorcm cxliibcnt vim ccntrifugam quam gravita- tem Plane toD in Solem ideoq : falsa) sunt. Motus Planetm in orbe non pendet ab cxcessu gravitatis supra vim centri- fugam (uti credit Leibnitius) sed Orbis incur vatur a gravi- tatis actionc sola, cui vis centrifuga (ut reactio vel resistentia) semper est ajqualis & contraria per motus Legem tertiara a Newtono positam. In fine Scliediasmatis dc resistentia Medii Leibnitius subjungit. Miilta ex his decluci possent praxi accommodata, sed nobis nimc fandameiita Geometrica jecisse stiff {ec]erit in quihus maxima consistcbat difficidtas, Et fortassis attente con* sideranti vias quasdam novas vel certe satis antea impeditas apendsse videhimur. Omnia autem respondent nostrce Analyst infinitorumf hoc est calculo summarum et differentiarum. Analysim banc per annos undecim vel duodecim Leib- nitius in differentiis primis jam exercuerat et notaverat differentias differentiarum per dd easq: ad inventionem puncti flexus contrarii applicuerat, sed problemata difficiliora per differentias differentiarum soluta nondum dederat. Jam vero per opus Newtonianum excitatus hroc aggreditur ac gloriatur se nunc fundamenta Geometrica jecisse in quibus maxima consistcbat difficultas et vias quasdam novas vel certe satis antea impeditas aperuisse & htec fecisse per Analysin suam infinitorum quam differentialcm vocat. Sed primo tamen conatu multipliciter erravit & per errorcs suos prodidit se methodum illam in difficilioribus hisce nondum probe intellexisse, prodidit se Propositioncs New- toni minime invenisse sed calculum tantum ad conclusioncs aptasse. Noverat methodum infinitorum Newtono prius cognitam fuisse ut ex ejus Epistolis manifestum est*. Nove- rat Propositioncs de resistentia mediorum deq: motibus • In the margin Newton has wrillen " vide pag," intending prohably to refer to Leibniz's letter to Wallis (I\Iuy 28, l()97)and his answer to Fatio, which are printed in the Comniercimn Epistolicum (pp. 104, 107). 21 314 ANCIENT FORMS [Appexd. corporum ccBlestium a Newtono primum inventas fuisse idq: per meth{od|um illam infinitorum, et omnia tamen sibi arrogat, & passim novis vcstit nominibus ne Newtonum sequi videatur. Quod prius feccrat cum Moutono hoc postea facere cum Newtono non dubitavit. Noverat etiam mcthodum serierum infinitarum a Newtono inventam fuisse et hujus methodi adminiculo Gregorium ineunte anno 1671 in sericm pro arcu ex tangente incidisse et tamen' banc seriem ut suam in Actis Lipsicis Anno 1682 magnifice in lucem edidit. • No. XXXIII. Seo Synoptical View of No-ytni's Life under date 1713 Nov. S"^ Isaac Newton represents that he did formerly dis- course w*^ your LordP about the ancient year of 360 days, & represented to yo'^ Lord^ that it was the Kalendar of the ancient Lunisolar year composed of the nearest round num- ber of Lunar months in a year & days in a Lunar month : that the ancients corrected this Kalendar monthly by the new moons & yearly by the returns of the four seasons, drop- ping a day or two when they found the Kalendar month of 30 days too long for the return of the Moon, and adding a month to the end of the year when they found the year of 12 Lunar months too shor£ for the return of the seasons & fruits of the earth : that Moses in describing the flood uses the Kalendar months not corrected by the course of the INIoon, the cloudy rainy weather not suffering her then to appear to Noah : that when Herodotus or any other author reccons 30 days to the months & 360 days to y® year, he understands the Kalendar month & year with- out correcting them by the courses of the Sun and Moon : that when Herodotus reccons by years of 12 & 13 months alternately for 70 years together, he understands the Diet- eris of the ancients continued 70 years without correcting Append.] OF THE YEAR. 315 it by the Luminaries : & that when we read of a week or a month or a year consisting of any other days then the natural, we are to reecon 7 days or 30 days or 3G0 days aecording to the Kalendar because where the days are not natural ones the Kalendar cannot be corrected by the courses of the Sun & Moon; and if the days be taken mystically for the years of any nation, we are to take these years in the vulgar sense for 7 or 30 or 3G0 practical years of that nation such as they commonly use in their civil affairs. S'^ Isaac saith further that he meets w*^ nothing in yo'^ LordP^ paper w*^'* in his opinion makes against what he tlreu represented to y^"" Lord*", that Suidas (in Sa^i) tells us that y® months of the Chaldees were Lunar, their cv dinary years composed of 12 Lunar months, and the Sarus composed of 18 such years & six months, w*^** months he takes to be intercalary (the end of all cycles of years being to know when to intercale the months of y* Luni- solar year for keeping the year to the seasons;) & that Censorinus mentions a Chaldean cycle of 12 years, & y* the Jews in returning from captivity called their own months by the names of the Chaldean, & that the feast Sacca* of the Babylonians was celebrated on y* 16*^ day of a Lunar month & kept to the same season of y® year, & that in all antiquity he meets w*^ no other sorts of years than the Luni-solar the Solar & the Lunar, & their Calendars & cycles f. • ■ AthenaEus xiv. 639. t From the original in Brit. Mus. Add. MSS. 6489. fol. < 316 LETTER TO LORD TOWNSHEND. [Append. No. XXXIV. NEWTON TO LORD TOWNSHEND. Containing an opinion of Bomo value in connexion with the subject of capital punishments. My Lord I know nothing of Edmund Metcalf convicted at Derby assizes of counterfeiting the coyne ; but since he is very evidently convicted, I am humbly of opinion that its better to let him suffer, than to venture his going on to counter- feit the coin & teach others to do so untill he can be con- victed again, ffor these people very seldom leave off. And its difficult to detect them. I say this with most humble submission to his Maj^ pleasure & remain My Lord your LordP'' most humble & obedient Servant Mint office Aug. 25, 1724. Is. Newton*- i**. Tawnshend {Secretary of State}, * From a copy communicated by P. O'Callaghan, Esq. The origiDal is in the possession of I\l. A. Donnadieu. THE END. Works published by John W. Parker, West Strand, London. Select List No. II. 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