(Sfontell Inineratta ffitbrarg Mtjam, Sfwn $ork BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE BMHESMttcg _ . Cornell University Library QA 29.S86A3 James Stirling; a sketch of his life and 3 1924 001 520 919 JAMES STIRLING OXFORD UNIVERSITY PRESS London Edinburgh Glasgow Copenhagen New York Toronto Melbourne Cape Town Bombay Calcutta Madras Shanghai HUMPHREY MILFORD Publisher to the University Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001520919 -&>**&* jj 'Ju/r tipc fir ^Add&TL. ^^^f^^*^«^>^^^ 'Tent/ hfyffe&tK,rfjmAcb , Al-it /uncr &*Mtb /m*> faty-fod&Q rfnup*4i&. My cio^k $04*4/ /Sufi nur aJe^ii^arMA^ <6ty p^n^-^Tti^nMunu. /£ If fat cm^ptfat/U, £&h#hc/ anb /cSivn/te >tdufyh theft TwtkcruP' csHWk* if 3r£&Uk, t **- i£*H£4*Jd/ Wfojcffixtof #uf yc£eJ(*h/ dvn- (fvr 6 " JAMES STIRLING A SKETCH OF HIS %\U and Oftorfts ALONG WITH HIS SCIENTIFIC CORRESPONDENCE BY CHARLES TWEEDIE M.A., B.Sc, F.R.S.E. CARNEGIE FELLOW, 1917-1920; LECTURER IN PURE MATHEMATICS EDINBURGH UNIVERSITY OXFORD AT THE CLARENDON PRESS 1922 5 TO THE MEMORY OF JOHN STURGEON MACKAY. LL.D. TO WHOSE INSPIRATION IS LARGELY DUE MY INTEREST IN THE HISTORY OF MATHEMATICS PREFACE The Life of Stirling has already formed the subject of a very readable article by Dr. J. C. Mitchell, published in his work, Old Glasgow Essays (MacLehose, 1905). An interesting account of his life as manager of the Leadhills Mines is also given by Ramsay in his Scotland and Scotsmen in the Eighteenth Century. The sketch I here present to readers furnishes further details regarding Stirling's student days at Balliol College, Oxford, as culled from contemporary records, along with more accurate information regarding the part he played in the Tory interests, and the reason for his departure for Italy. Undoubtedly, when at Oxford, he shared the strong Jacobite leanings of the rest of his family. Readers familiar with Graham's delightful Social Life in Scotland in the Eighteenth Century, and the scarcity of money among the Scottish landed gentry, will appreciate the tone of the letter to his father of June 1715, quoted in full in my sketch. Whether he ever attended the University of Glasgow is a moot point. Personally, I am inclined to think that he did, for it was then the fashion to enter the University at a much earlier age than now, and he was already about eighteen years of age when he proceeded to Oxford. Very little is known regarding his stay in Venice and the date of his return to Britain; but his private letters show tbat when he took up residence in London he was on intimate terms of friendship with Sir Isaac Newton and other dis- tinguished scholars in the capital. I have taken the opportunity here to add — what has hitherto not been attempted — a short account of Stirling's published works, and of their relation to current mathematical thought. In drawing up this account, I had the valuable viii PREFACE assistance of Professor E. T. Whittaker's notes on Part I of Stirling's Methodus Differ entialis, which he kindly put at my disposal. Stirling's influence as a mathematician of profound analytical skill has been a notable feature within the inner circle of mathematicians. Witness, for example, the tribute of praise rendered by Laplace in his papers on Probability and on the Laws of Functions of very large numbers. Binet, in a celebrated memoir on Definite Integrals, has shown Stirling's place as a pioneer of Gauss. Gauss himself had most unwillingly to make use of Stirling's Series, though its lack of convergence was anathema to him. More recently, Stirling has found disciples among Scandinavian mathema- ticians, and Stirling's theorems and investigations have been chosen by Professor Nielsen to lay the foundation of his Monograph on Gamma Functions. The Letters, forming the scientific correspondence of Stirling herewith published, make an interesting contribution to the history of mathematical science in the first half of the eighteenth century. I have little doubt that suitable research would add to their number. I have endeavoured to reproduce those as exactly as possible, and readers will please observe that errors which may be noted are not necessarily to be ascribed to negligence, either on my part or on that of the printer. For example, on page 47, the value of it/2 given by De Moivre's copy of Stirling's letter (taken from the Miscellanea Analytica) is not correct, being 1-57079632679, and not 1-5707963279 as there stated. A few notes on the letters have been added, but, in the main, the letters have been left to speak for themselves. I am deeply grateful for the readiness with which the Garden letters were placed at my disposal by Mrs. Stirling, Gogar House, Stirling. I am also indebted to the University of Aberdeen for permission to obtain copies of Stirling's letters to Maclaurin. In the troublesome process of preparing suitable manuscript for the press, I had much valuable clerical assistance from my sister, Miss Jessie Tweedie. PREFACE ix Of the many friends who have helped to lighten my task I am particularly indebted to Dr. C. G. Knott, F.R.S., and to Professor E. T. Whittaker, F.R.S., of Edinburgh University; also to Professor George A. Gibson, of Glasgow University, who gave me every encouragement to persevere in my research, and most willingly put at my disposal his mature criticism of the mathematicians contemporary with Stirling. Facsimile reproductions of letters by James Stirling and Colin Maclaurin have been inserted. These have never before appeared in published form, and will, it is hoped, be of interest to students of English or Scottish history, and to mathematical scholars generally. The heavy cost of printing during the past year would have made publication impossible but for the generous donations from the contributors mentioned in the subjoined list of subscribers, to whom I have to express my grateful thanks. CHARLES TWEEDIE. LIST OF SUBSCRIBERS The Trustees of the Carnegie Trust for Scotland (£50). Subscriptions, to the total value of £70, from Captain Archibald Stirling, of Kippen. General Archibald Stirling, of Keir. Sir John Maxwell Stirling Maxwell, Bart., of Pollok. John Alison, M A., LL D., Headmaster of George Watson's College, Edinburgh. George A. Gibson, M.A., LL.D.. Professor of Mathematics, Glasgow University. E. M. Horsburgh, M.A., D.Sc, A.M.I.C.E., Reader in Technical Mathematics. Edinburgh University. William Peddie, D.Sc, Professor of Physics, University College, Dundee. E. T. Whittaker, D.Sc, F.R.S., Professor of Mathematics, Edinburgh University. x LIST OF SUBSCRIBERS Subscriptions, to the total value of £10, from A. G. Burgess, M.A., B.Sc, Rector of Rothesay Academy. Archibald Campbell, MA., LL.B., "Writer to the Signet, 36 Castle Street, Edinburgh. Jas. H. Craw, Esq., Secretary of the Berwickshire Naturalists' Club, West Foulden, Berwick-on-Tweed. Alexander Morgan, M.A., D.Sc, Director of Studies, Edinburgh Provincial Training Centre. George Philip, D.Sc, Executive Officer, Ross and Cromarty Education Authority. Rev. A. Tweedie, M.A., B.D., Maryculter. Mrs. C. E. Walker, M.A., Villa Traquair, Stormont Road, Highgate, London. CONTENTS PAGE LIFE . . . 1 WORKS . . . .23 CORRESPONDENCE 51 FACSIMILES Facsimile of last page of Letter by Stirling to his Father, 1715 (pages 6-7) Frontispiece Facsimile of last page of Letter by Maclaurin to Stirling, 1728 (Letter No. 1) facing p. 57 COAT OP ARMS 01' THE STIIJLINGS OF GARDEN. LIFE OF JAMES STIRLING James Stirling, the celebrated mathematician, to whose name is attached the Theorem in Analysis known as Stirling's Theorem, was born at Garden in the county of Stirling, Scotland, in 1692. He was a member of the cadet branch of the Stirling family, usually described as the Stirling's of Garden. The Stirling family is one of the oldest of the landed families of Scotland. They appear as proprietors of land as early as the twelfth century. In 1 1 80, during the reign of William the Lion, a Stirling acquired the estate of Cawder (Cadder or Calder) in Lanarkshire, and it has been in the possession of the family ever since. Among the sixty-four different ways of spelling the name Stirling, a common one in those early days, was a variation of Striveling. In 1448, the estate of Keir in Perthshire was acquired by a Stirling. In 1534 or 1535 these two branches of the family were united by the marriage of James Striveling of Keir with Janet Striveling, the unfortunate heiress of Cawder. Since that time the main family has been, and remains, the Stirlings of Keir and Cawder. By his second wife, Jean Chisholm, James had a family, and of this family Elizabeth, the second daughter, married, in 1571-2, John Napier of Merchiston, the famous inventor of logarithms, whose lands in the Menteith marched with those of the Barony of Keir. This was not the first intermarriage between the Napiers and the Stirlings, for at the former Napier residence of Wright's Houses in Edinburgh (facing Gillespie Crescent), there is preserved a stone the armorial bearings on which record the marriage of a Napier to a Stirling in 1399. Early in the seventeenth century Sir Archibald Stirling of Keir bought the estate of Garden, in the parish of Kippen (Stirlingshire), and in 1613 he gave it to his son (Sir) John Stirling, when Garden for the first time became a separate 2 LIFE OF JAMES STIRLING estate of a Stirling. The son of John, Sir Archibald Stirling, was a conspicuous Royalist in the Civil War, and was heavily fined by Cromwell ; but his loyalty was rewarded at the Restoration, and he ascended the Scottish bench with the title of Lord Garden. Lord Garden, however, succeeded to the estate of Keir, and his younger son Archibald (1651-1715) became Laird of Garden in 1668. Archibald's eventful career is one long chapter of mis- fortunes. Like the rest of the Stirlings he adhered loyally to the Stuart cause. In 1708, he took part in the rising called the Gathering of the Brig of Turk. He was carried a prisoner to London, and then brought back to Edinburgh, where he was tried for high treason, but acquitted. He died in 1715, and thus escaped the penalty of forfeiture that weighed so heavily on his brother of Keir. He was twice married. By his first wife he had a son, Archibald, who succeeded him, and by his second marriage, with Anna, eldest daughter to Sir Alexander Hamilton of Haggs, near Linlithgow, he had a family of four sons and five daughters. James Stirling, the subject of this sketch, and born in 169;2, was the second surviving son of this marriage. (The sons were James, who died in infancy ; John, who acquired the Garden estate from his brother Archibald in 1717; James, the mathematician ; and Charles.) The Armorial Bearings of the Garden * branch of the Stirlings are : Shield: Argent on a Bend azure, three Buckles or : in chief, a crescent, gules. Crest : A Moor's Head in profile. Motto : Gang Forward. 2 YOUTH OF STIRLING Oxford Save for the account given by Ramsay of Ochtertyre (Scotland and Scotsmen, from the Ochtertyre MSS.), which is not trustworthy in dates at least, little is known of the early 1 Garden, pronounced Garden, or Gardenne. 2 Gang- forward ; Scotice for Allez en avant. STIRLING AT OXFORD 3 years and education of Stirling, prior to his journey to Oxford University in 1710. Ramsay, it is true, says that Stirling studied for a time at Glasgow University. This would have been quite in accordance with Stirling tradition, for those of the family who became students had invariably begun their career at Glasgow University; and the fact that Stirling was a Snell Exhibitioner at Oxford lends some colour to the statement. But there is no trace of his name in the University records. Addison, in his book on the Snell Exhibitioners, states that ' Stirling is said to have studied at the University of Glasgow, but his name does not appear in the Matriculation A lbum '. From the time that he proceeds on his journey to Oxford his career can be more definitely traced, though the accounts hitherto given of him require correction in several details. Some of the letters written by him to his parents during this period have fortunately been preserved. This fact alone s ufficiently indicates the esteem in which he was held by his family, and their expectation of a promising future for the youth. In one of these he narrates his experiences on the journey to London, and his endeavour to keep down expenses: 1 1 spent as little money on the road as I could. I could spend no less, seeing I went with such company, for they lived on the best meat and drink the road could afford. Non of them came so near the price of their horses as I did, altho' they kept them 14 days here, and payed every night 16 pence for the piece of them.' He reached Oxford towards the close of the year 1710. He was nominated Snell Exhibitioner on December 7, 1710, and he matriculated on January 18, J ?■££, paying £7 caution money. On the recommendation of the Earl of Mar he was nominated Warner Exhibitioner, and entered Balliol College on November 27, 1711. In a letter to his father of the same year (February 20, 1711) he gives some idea of his life at Oxford : ' Everything is very dear here. My shirts coast me 14 shillings Sterling a piece, and they are so course I can hardly wear them, and I had as fit hands for buy- ing them as I could.' . . . ' We have a very pleasant life as well as profiteable. We have very much to do, but there is nothing here like strickness. I was lately matriculate, and with the help of my tutor I escaped the oaths, but with much ado.' b2 4 LTFE OF JAMES STIELING He thus began academic life at Oxford in good spirits, but as a non-juring student. At this period Oxford University was not conspicuous for its intellectual activity. The Fellows seem to have led lives of comfortable ease, without paying much regard to the requirements of the students under their care. As we shall see in Stirling's case, the rules imposed upon Scholars were very loosely applied, and, naturally, complaint was made at any stringency later. At the time we speak of political questions were much in the thoughts of both students and college authorities. The University had always been faithful to the house of Stuart. It had received benefits from James I. For a time Oxford had been the head-quarters of King Charles I during the Civil War, and his cavaliers were remembered with regret when the town was occupied by the Parliamentary forces, and had to endure the impositions of Cromwell. At the time of Stirling's entry the reign of Queen Anne was drawing to a close. Partisan feeling between Whigs and Tories was strong, and of all the Colleges Balliol was most conspicuously Tory. According to Davis (History of Balliol College) Balliol 'was for the first half of the 18th century a stronghold of the most reactionary Toryism ', and county families, anxious to place their sons in a home of sound Tory principles, naturally turned to Balliol, despite the fact that Dr. Baron, the Master, was a stout Whig. It is, therefore, abundantly clear that Stirling had every reason to be content with his political surroundings at Balliol, with what results we shall see presently. Perhaps the best picture of the state of affairs is to be gathered from the pages of the invaluable Diary ofT. Hearne, the antiquarian sub-librarian of the Bodleian. For Hearne all Tories were ' honest men ', and nothing good was ever to be found in the ' vile Whigges '. His outspoken Tory sentiments led to his being deprived of his office, and almost of the privilege of consulting books in the Library, though he remained on familiar terms with most of the resident Dons. Luckily for us, James Stirling was one of his acquaintance, and mention of Stirling's name occurs frequently enough to enable us to form some idea of his career. Doubtless STIRLING AT OXFORD 5 their common bond of sympathy arose from their Tory, nay their Jacobite, principles, but it speaks well for the intellectual vigour of the younger man that he associated with a man of Hearne's scholarship. Moreover, Stirling must have been a diligent student, or he could never have acquired the scholarship that bore its fruit in 1717 in the production of his Lineae Tertii Ordinis, a work which is still a recognized commentary on Newton's Enumeration of Curves of the Third Order. But he was not the sort of man to be behindhand in the bold expression of his opinions, and he took a leading part among the Biilliol students in the disturbances of 1714-16. The accession of George of Hanover to the British throne was extremely unpopular in Oxford, and Hearne relates how on May 28, 1715, an attempt to celebrate the King's birthday was a stormy failure, while rioting on a large scale broke out next day. 'The people run up and down, crying, King James the Third! The True King, No 'usurper ! The Duke of Ormond ! &c, and healths were everywhere drunk suitable to the occasion, and every one at the same time drank to a new restauration, which I heartily wish may speedily happen.' . . . 'June 5. King George being informed of the proceedings of the cavaliers at Oxford, on Saturday and Sunday (May 28, 2D), he is very angry, and by his order Townshend, one of the Secretaries of State, hath sent rattling letters to Dr. Charlett, pro-vice-chancellor, and the Mayor. Dr. Charlett shewed me his this morning. This lord Townshend says his majesty (for so they will stile this silly usurper) hath been fully assured that the riots both nights were begun by scholars, and that scholars promoted them, and that he (Dr. Charlett) was so far from discountenancing them, that he did not endeavour in the least to suppress them. He likewise observed that his majesty was as well informed that the other magistrates were not less remiss on these occasions. The heads have had several meetings upon this affair, and they have drawn up a programme, (for they are obliged to do something) to prevent the like hereafter; and this morning very early, old Sherwin the yeoman beadle was sent to London to represent the truth of the matter.' These measures had a marked effect upon the celebration on June 10 of 'King James the IlldV birthday. Special 6 LIFE OF JAMES STIRLING precautions were taken to prevent a riotous outbreak. 'So that all honest men were obliged to drink King James's health, and to shew other tokens of loyalty, very privately in their own houses or else in their own chambers, or else out of town. For my own part I walked out of town to Fox- comb, with honest Will Fullerton, and Mr. Sterling, and Mr. Eccles, all three non-juring civilians of Ealliol College, and with honest Mr. John Leake, formerly of Hart Hall, and Rich. Clements (son to old Harry Clements the bookseller) he being a cavalier. We were very merry at Foxcombe, and came home between nine and ten,' &e. Several of the party were challenged on their return to Oxford, but no further mention is made of Stirling. On August 15 there was again rioting at Oxford, in which a prominent part was taken by scholars of Balliol. There can be little doubt that Stirling was implicated, though he seems to have displayed a commendable caution on June 10 by going out of town with a man so well known as Hearne. His own account of current events is given in the following letter to his father, which is the only trace of Jacobite corre- spondence with Scotland that has been preserved, if it can be so termed : — Oxon 23 July 1715. Sir, I wrote to you not long ago, but I have had no letter this pretty while. The Bishop of Rochester and our Master have renewed an old quarrell : the Bishop vents his wrath on my countrymen, and now is stopping the paying of our Exhibitions : it's true we ought to take Batchelours degrees by the foundation of these exhibitions, and quite them when we are of age to so into orders : Rochester stands on all those things, which his Predecessours use not to mind, and is resolved to keep every nicety to the rigor of the statute ; and accordingly he hath stoped our Exhibitions for a whole year, and so ows us 20 lib. apiece, he insists on knowing our ages, degrees, and wants security for our going into orders. 1 suppose those things may come to nought in a little while, the Bishop is no enemy to our principles. In the meantime I've borrowed money of my friends till I'm ashamed to borrow anjr more. I was resolved not to trouble you while I could otherwise subsist ; but now I am forced to ask about 5 lib. or what in reason you think fit to supply my present needs : STIRLING AT OXFORD 7 for ye little debts I have 1 can delay them I hope till the good humor shall take the Bishop. I doubt not to have the money one time or another, it's out of no ill will against us that he stops it, but he expects our wanting the money will make us solicite our Master to cringe to him, which is all he wants. No doubt you know what a generall change of the affec- tions of the people of England the late proceedings hath occasion : the mobbs begun on the 28 of May to pull down meeting houses and whiggs houses, and to this very day they continue doing the same, the mobb in Yorkshire and Lanca- shire amounted to severall thousands, and would have beat of the forces sent against them had they not been diswaded by the more prudent sort, and they are now rageing in Coventry and Baintry : so (as the court saith) the nation is just ripe for a rebellion. There were severall houses of late at London searched for the Chevalier, the D. of Berwick and M r Lesly. Oxford is impeached of high treason and high crimes and misdemanners and is now in the Touer, a little while ago both Whiggs and Tories wished him hanged, but he has gained some tories to stand his friends in opposition to the Whiggs. They cant make out eitough to impeach the rest they designed. I had a letter from Northside x lately. I shall delay an answere till I have the occasion of a frank. My cousin James sent me a letter the other day from Amster- dam, he is just come from the Canaries, and designs to return there without coming to Britain, he remembers himself very kindly to you and all friends with you. I give my humble duty to you and my mother and my kind respects to my brothers sisters and all my relations 1 am Sir Your most dutifull son Jan. Stirling. It was in the same year (1715) that Stirling first gave indications of his ability as a mathematician. In a letter 2 to Newton, of date Feb. 24, 1715, John Keill, of Oxford, mentions that the problem of orthogonal trajectories, which had been proposed by Leibniz, had recently been solved by ' Mr. Stirling, an undergraduate here ', as well as by others. The statement commonly made that Stirling was expelled 1 James Stirling, son of the Laird of Northside (near Glasgow), is specially mentioned in the List of Persons concerned in the Rebellion of 1745-6 (Scot. Hist. Soc). 2 Macclesfield, Correspondence of Scientific Men, §c, vol. n, p. 421. t 8 LIFE OF JAMES STIRLING from Oxford for his Jacobite leanings, and driven to take refuge in Venice, seems entirely devoid of foundation. Again Hearne's Diary comes to our aid, and indicates that Stirling was certainly under the observation of the government authorities :— '1715 Dec. 30 (Fri) On Wednesday Night last M r Sterling, a Scotchman, of Balliol Coll. and M r Gery, Gentleman Commoner of the same College, were taken up by the Guard of the Souldiers, now at Oxford, and not released till last night. They are both honest, non-juring Gentlemen of my acquaintance.' Also: '1716 July 21 (Sat.) One M> Sterling, a Non-juror of Bal. Coll. (and a Scotchman), having been prosecuted for cursing K. George (as they call the Duke of Brunswick), he was tryed this Assizes at Oxford, and the Jury brought him in not guilty.' The Records of Balliol bear witness to his tenure of the Snell and Warner Exhibitions down to September, 1716. (Also as S.C.L. 1 of one year's standing in September, 1715, and as S.C.L. in September, 1716.) There is no indication of his expulsion, though the last mention of him by Hearne informs us that he had lost his Scholarship for refusing to take 'the Oaths'. ' 1717. March 28 (Fri) M r Stirling of Balliol College, one of those turned out of their Scholarships upon account of the Oaths, hath the offer of a Professorship of Mathematicks in Italy, w oh he hath accepted of, and is about going thither. This Gentleman is printing a Book in the Mathematical way at the Theatre. 2 ' We shall see presently that Stirling found himself compelled to refuse the proffered Chair. The circumstances in which he had this offer are somewhat obscure ; and whether he 1 S.C.L. was a Degree (Student of Civil Law) parallel to that of B A., just as that of Bachelor of Civil Law (B.C.L.) is parallel to that of M.A. The degree has long been abolished, but its possession would suggest that Stirling had at one time the idea of adopting the profession of his grand- father, Lord Garden. 2 The Sheldonian Theatre, Oxford. STIRLING AT OXFORD 9 played any part in the Newton-Leibniz controversy is not certain. In the later stages of the controversy an inter- mediary between Leibniz and Newton was found in the Abbe - Conti, a noble Venetian, born at Padua in 1677, who, after spending nine years as a priest in Venice, gave up the Church, and went to reside in Paris, where he became a favourite in society. In 1715, accompanied by Montmort, he journeyed to London, and received a friendly welcome from Newton and the Fellows of the Royal Society. In a letter 1 to Brook Taylor in 1721, Conti relates how ' M r Newton me pria d'assembler a la Socie'te' les Ambassa- deurs et les autres strangers'. Conti and Nicholas Tron, the Venetian Ambassador at the English Court, became Fellows at the same time in 1715. How Conti came to meet Stirling is unknown to us ; but he must have formed a high opinion of Stirling's ability and personal accomplishments, for Newton in a letter quoted by Brewster {Life of Newton, ii, p. 308) querulously charges Conti with 'sending M r . Stirling to Italy, a person then unknown to me, to be ready to defend me there, if I would have contributed to his maintenance '. The fact that Newton- was a subscriber to Stirling's first venture, Lineae Tertii Ordinis Neutonianae, sive Illustratio Tractalus D. Neutoni De Enumeratione Linearum Tertii Ordinis, and doubtless the ' Book ' mentioned by Hearne, would suggest that Newton had met Stirling before the latter had left England. This little book is dedicated to Tron, and it was on Tron's invita- tion that Stirling accompanied him to Italy with a view to a chair in one of the Universities of the Republic. The long list of subscribers, the majority of whom were either Fellows or Students at Oxford, bears eloquent testimony to the repu- tation he had acquired locally at least as a good mathe- matician. The book was printed at the Sheldonian Theatre, and bears the Imprimatur, dated April 11, 1717, of John Baron, D.D., the Vice-Chancellor of the University, and Master of his own College of Balliol, who was also subscriber for six copies. Of the subscribers, forty-five are associated with Balliol. Richard Rawlinson, of St. John's, was also a 1 Printed in the posthumous Contemplatio Fhilosophica of Brook Taylor. 10 LIFE OF JAMES STIRLING subscriber, and W. Clements, the bookseller in London, took six copies. Thus Stirling left Oxford after publishing a mathematical work that was to earn him a reputation abroad as a scholar. Venice From his residence in Venice, 1 Stirling is known in the Family History of the Stirlings as James Stirling the Venetian. The invitation to Italy and the subsequent refusal are thus recorded in the Rawlinson MSS. in the Bodleian (materials collected by Dr. Richard Rawlinson for a continuation of Ward's Athenae Oxoniertses up to 1750) : ' Jacobus Stirling, e coll. Baliol, exhibit. Scot, a Snell. jura- ment. R. G* recus. 1714, et in Italian! Nobilem virum Nicolaum Tron, Venetiarum Reipublicae ergo apud Anglos Legatum, secutus est, ubi religionis causa matheseos profes- sorium munus sibi oblatum respuit.' The religious difficulty must have been a serious blow to Stirling's hopes, and placed him in great embarrassment, for his means were of the scantiest. But adherence to the Anglican Church was one of the most fundamental principles of the Tories, which had caused so much wavering in their ranks for the Catholic Chevalier, and there was no getting over the objection. We need not be surprised, therefore, that he got into serious difficulties, from which he was rescued in 1719 by the generosity of Newton, who had. henceforward at least, Stirling for one of his most devoted friends. Stirling's 1 I have endeavoured to ascertain the university to which Stirling was called. Professor G. Loria has informed me that it was very probably Padua, Padua being the only university in the Republic of Venice, the Quartier Latin of Venice according to Renan. It had been customary to- select a foreigner for the chnir of Mathematics. A foreigner (Hermann) held it, and resigned it in 1713. It was then vacant until 1716, when Nicholas Bernoulli (afterwards Professor of Law at Bale) was appointed. Professor Favaro of Padua confirms the above, and adds that possibly some information might be gathered from the reports of the Venetian Ambassador, or from the records of the Reformatores Studii (the patrons of chtiirs in a mediaeval university). To get this information it would be necessary to visit Venice. My chief difficulty here is to reconcile the date of Stirling's visit to Italy and the date of the vacancy. It may be added that a College for Scotch and English students still flourished at Padua at this time (see also Evelyn's Diary). C. T. 2 King George. AT VENICE 11 letter to Newton, expressing his gratitude, is here given. It has been copied from Brewster's Life of Newton. Letter Venice 17 Aug. 1719. Sir I had the honour of your letter about five weeks after the date. As your generosity is infinitely above my merite, so I reackon myself ever bound to serve you to the utmost : and, indeed, a present from a person of such worth is more valued by me than ten times the value from another. I humbly ask pardon for not returning my grateful acknow- ledgments before now. I wrote to M r Desaguliers to make my excuse while in the meantime I intended to send a supple- ment to the papers I sent, but now I'm willing they be printed as they are, being at present taken up with my own affair here wherewith I won't presume to trouble you having sent M r Desaguliers a full account thereof. I beg leave to let you know that M 1 ' Nicholas Bernoulli proposed to me to enquire into the curve which defines the resistances of a pendulum when the resistance is proportional to the velocity. I enquired into some of the most easy cases. and found that the pendulum, in the lowest point had no velocity, and consequently could perform but one half oscil- lation, and then rest. Bernoulli had found that before, as also one Count Ricato, which I understood after I communi- cated to Bernoulli what occurred to me. Then he asked me how in that hypothesis of resistance a pendulum could be said to oscill.tte since it only fell to the lowest point of the cycloid, and then rested. So I conjecture that his uncle sets him on to see what he can pick out of your writings that may any ways be cavilled against, for he has also been very busy in enquiring into some other parts of the Principles. I humbly beg pardon for this trouble, and pray God to prolong your daies, wishing that an opportunity should offer that I could demonstrate my gratefullness for the obligation:, you have been pleased to honour me with, I am with the greatest respect Sir Your most humble & most obedient serv' James Stirling. Venice 17 August 1719 n. st. P.S. M r Nicholas Bernoulli, as he hath been accused by D r Keill of an illwill towards you, wrote you a letter some time ago to clear himself. But having in return desired me 12 LIFE OF JAMES STIRLING to assure you that what was printed in the Acta Paris. relating to your 10 Prop., lib. 2, was wrote before he had been in England sent to his friends as his private opinion of the matter, and afterwards published without so much as his knowledge. He is willing to make a full vindication of him- self as to that affair whenever you'll please to desire it. He has laid the whole matter open to me, and if things are as he informs me D r Keill has been somewhat harsh in his case. For my part I can witness that I never hear him mention your name without respect and honour. When he showed me the Ada Eruditorivm where his uncle has lately wrote against D r Keill he showed me that the theorems there about Quadratures are all corollarys from your Quadratures ; and whereas M r John Bernoulli had said there, that it did not appear by your construction of the curve, Prop. 4, lib. 2, that the said construction could be reduced to Logarithms, he presently showed me Coroll. 2 of the said Proposition, where you show how it is reduced to logarithms, and he said he wondered at his uncle's oversight. I find more modesty in him as to your affairs than could be expected from a young man, nephew to one who is now become head of M r Leibnitz's party; and among the many conferences I've had with him I declare never to have heard a disrespectful word from him of any of our country but D r Keill. How long he lived in Italy after his letter to Newton is not known; but life in the cultured atmosphere of Venice must have been, otherwise, very congenial. It was a favourite haunt of the different members of the Bernoullian family. The earliest letter to Stirling of a mathematical nature that lias been preserved is one in 1719 from Nich. Bernoulli, F.R S., at that time Professor in the University of Padua. One is tempted to inquire whether Stirling did not meet Bernoulli and Goldbach on the occasion of their visit to Oxford in 1712. In the letter in question Bernoulli specially refers to their meeting in Venice, and also conveys the greetings of Poleni, Professor of Astronomy at Padua. At the same time Riccati was resident in Venice, which he refused to leave when offered a chair elsewhere. Ramsay says that Stirling made contributions to mathematics while resident in Italy, copies of which he brought home with him : but I have found no trace of them. The only paper of this period is his Methodus Differentialis Newtoniana, published in the Pldlo- AT VENICE 13 sophical Transactions for 1719, with the object of elucidating Newton's methods of Interpolation. London From 1719 to 1724 there is a gap in our information regarding Sterling. But a fragment of a letter by him to his brother, Mr. John Stirling of Garden, shows that in July 1724 lie was at Cader (Cawder or Calder, where the family of his uncle James, the dispossessed Laird of Keir, resided). Early in 1725 he was in London, as a letter to his brother John informs us (London, 5 June, 17 25) when he was making an effort towards ' getting into business '. ' It's not so easily done, all these things require patience and diligence at the beginning.' In the meantime, that he may not be ' quite idle ' he is preparing for the press an edition of . . . l Astronomy to which he is ' adding some things ' ; but for half a year the money will not come in, and he hopes his mother will provide towards his subsistence. ' So I cannot go to the country this summer but I have changed my lodgings and am now in a French house and frequent french Coffeehouses in order to attain the language which is absolutely necessary. So I have given over thoughts of making a living by teaching Mathematicks, but at present I am looking out sharp for any chub I can get to support me till I can do another way. S Isaac Newton lives a little way of in the country. I go frequently to see him, and find him extremely kind and serviceable in every thing I desire but he is much failed and not able to do as he has done .... Direct your letters to be left at Forrest's Coffee House near Charing Cross.' Thus in 1725, at 32 years of age, Stirling had not yet found a settled occupation which would furnish a competency. This project of ' getting into business ' was given up, for, some time after, he acquired an interest in Watt's Academy in Little Tower Street, where (Diet. Nat. Biog.) he taught Mechanics and Experimental Philosophy. It was the same Academy in which his countryman Thomson, the poet, taught for six months from May 1726, and where the latter composed portions of 'Summer'. For about ten years Stirling was 1 The name, unfortunately, is not legible. 14 LIFE OF JAMES STIRLING connected with the Academy, and to this address most of the letters to him from contemporary mathematicians, that have been preserved, were directed. They form part of a larger collection that was partly destroyed by fire, and early in the nineteenth century they were nearly lost altogether through the carelessness of Wallace and Leslie of Edinburgh Univer- sity, to whom' they had been sent on loan from Garden. There are also a few letters to his friends in Scotland from which one can gather a certain amount of information. In the earlier days of his struggle in London he may have had to seek assistance from them, but as his circumstances im- proved he showed as great a generosity in return. By 1729 he could look forward with confidence to the future, for by that time he was able to wipe out his indebtedness in con- nection with his installation in the Academy, as the following extracts from his letters show. In a letter to his brother, dated April 1728, he writes: ' I had 100 Lib. to pay down here when I came first to this Academy, and now have 70 Lib. more, all this for Instruments, and besides the expenses I was at in fiting up apartments for my former project still ly over my head.' Again on July 22, 1720, he writes: ' Besides with what money I am to pay next Michaelmas I shall have paid about 250 Lib. since I came to this house, for my share of the Instruments, after which time I shall be in away of saving, for I find my business brings in about 200 L. a year, and is rather increasing, and 60 or 70 L. serves me for cloaths and pocket money. I designed to have spent some time this summer among you, but on second thoughts I choose to publish some papers during my Leisure time, which have long lain by me. But I intend to execute my design is seeing you next summer if I find that my affairs will permit.' He had always a warm side for his friends in Scotland, and his letters to them are written in a bright and cheerful style. The reference to Newton is the only one he makes regarding his friends at the Royal Society, and the 'papers' he speaks of publishing are almost certainly his well-known Treatise the Methodus Differentitdis (1730), the first part of which he had drawn up some eight or nine years before (vide a letter to Cramer). He was admitted to the Royal Society in 1726, AT LONDON 15 a distinction that put him on an equal footing with the scientists that lived in, or frequented, London. It is most probable that his acquaintance with Maclaurin began at this time. They were both intimate friends of Newton, and fervent admirers of his genius, and both eagerly followed in his footsteps. Letters that passed between them are preserved at Garden and in Aberdeen University. The opening correspondence furnishes the best account we have of the unfortunate dispute between Maclaurin and Campbell regarding the priority of certain theorems in equations (vide Math. Gazette, January 1919). Maclaurin placed great reliance upon Stirling's judgment, and frequently consulted Stirling while engaged in writing his Treatise of Fluxions. Their later letters are mainly concerned with their researches upon the Figure of the Earth and upon the Theor}^ of Attrac- tion. In 1738, Stirling, at Maclaurin's special request, joined the Edinburgh Philosophical Society, in the foundation of which Maclaurin had taken so prominent a part in 1737. Maclaurin also begged for a contribution, but if Stirling gave a paper to the Society it has not been preserved or printed. In 1727 Gabriel Cramer, Professor of Mathematics at Geneva, received a welcome from the Royal Society on the occasion of his visit to London. He formed a warm friendship for Stirling, who was his senior by about twelve years, and several of his letters (o Stirling are preserved. A copy, kept by Stirling, of a letter to Cramer furnishes interest- ing information regarding his own views of his Methodus Differentialis, and also regarding the date at which the Supplement to De Moivre's Miscellanea Analytica was printed. Stirling had sent two copies of his treatise to Cramer, one of the copies being for Nich. Bernoulli, by this time Professor of Law at Bale. Cramer had requested to be the intermediary of the correspondence between Bernoulli and Stirling in order to have the advantage of their mathematical discussions. A few letters from Bernoulli are preserved, the last bearing the date 1733. In this letter Bernoulli pointed out several errata in the works of Stirling, and observed the omission, made by both Stirling and Newton, of a species in their enumeration of Cubic Curves. Newton gave seventy-two species, and Stirling in his little book of 1717 added four 16 LIFE OF JAMES STIRLING more. But there were two additional species, one of which was noted by Nicole in 1731. Murdoch in his Newtoni Genesis Carvarwm per Umbras (1740) mentions that Cramer had told him of Bernoulli's discovery, but without furnishing a date. Bernoulli's letter not only confirms Cramer's state- ment, it also gives undoubted precedence to Bernoulli over Stone's discovery of it iu 1736. From 1730 onwards Stirling's life in London must have been one of considerable comfort, as his ' affairs ' became prosperous, while he was a familiar figure at the Royal Society, where his opinions carried weight. According to Ramsay he was one of the brilliant group of philosophers that gathered round Eolingbroke on his return from exile. Of these Stirling most admired Berkeley. If he at all shared the opinions of the disillusioned politician, then he might still be a Tory, but it was improbable that he retained any loyalty to the Jacobite cause. When the Rebellion broke out in 1745 there is no trace of Stirling being implicated, though his uncle of Cawdor was imprisoned by the government and thus kept out of mischief. His studies were now directed towards the problem of the Figure of the Earth, the discussion of which had given rise to two rival theories, (i) that of Newton, who maintained that the Earth was flatter at the Poles than at the Equator, and (ii) that of the Cassims, who held exactly the opposite view. In 1735 Stirling contributed a short but important note on the subject which appeared in the Philosophical Transactions (vide Tod hunter's History of the Theory of Attraction and the Figure of the Earth). Return to Scotland In 1735, a great change in his circumstances was occasioned by his appointment to the Managership of the Leadhills Mines in Scotland. A more complete change from the busy social life of London to the monotonous and dreary moorland of Leadhills can hardly be imagined. At first he did not break entirely with London, but in a year or two he found it necessary to reside permanently in Scotland, and a letter from Machin to him in 1738, would suggest that he felt the change keenly. RETURN TO SCOTLAND 17 He was now well over forty years of age, but, nothing daunted, he set himself to the discharge of his new duties with all the energy and ability at his command. The letters he exchanged with Maclaurin and Machin show that his interest in scientific research remained unabated, though the want of time due to the absorbing claims of his new duties is frequently brought to our notice. He appears to have discovered further important theorems regarding the Figure of the Earth, which Machin urged him to print, but he never proceeded to publication. His reputation abroad, however, led the younger school of rising mathematicians to cultivate his acquaintance by correspondence, and to this we owe a letter from Clairaut, and also a long and interest- ing letter from Euler. Clairaut (1713-65), who had shown a remarkable precocity for mathematics, was a member of the French Commission under Maupertuis, sent out to Lapland to investigate the length of an arc of a meridian in northern latitudes, a result of which was to establish conclusively Newton's supposition as against the Cassinians. As Voltaire put it : Maupertuis ' avait aplati la Terre et les Cassinis.' While still in Lapland Clairaut sent to the Royal Society a paper, some of the conclusions in which had been already communicated by Stirling. An apology for his ignorance of Stirling's earlier publication furnished Clairaut with the ground for seeking the acquaintance of Stirling in 1738, and requesting his criticism of a second paper on the Figure of the Earth. The correspondence with Euler in 1736-8, in connection with the Euler-Maclaurin Theorem, has already been referred to by me in the Math. Gazette. Euler (1707-83) is the third member of the famous Swiss school of mathematicians with whom Stirling had correspondence. From his letters to Daniel Bernoulli (Fuss, Corr. Math.) it is quite clear that Euler was familiar with Stirling's earlier work. Stirling was so much impressed by Euler 's first letter that he suggested that Euler should allow his name to be put up for fellowship of the Royal Society. Euler's reply, which is fortunately preserved, is remarkable for its wonderful range of mathematical research ; so much so that Stirling wrote to Maclaurin that he was ' not yet fully master of it.' 18 LIFE OF JAMES STIRLING Euler, who was at the time installed in Petrograd, did not then become a Fellow of the Royal Society. In 1741 he left Russia for Berlin, where, in 1744, he was made Director of the Mathematical Section of the Berlin Academy, and it is quite possible that he had a share in conferring upon Stirling the honorary membership of the Academy in 1747. The informa- tion is contained in a letter of that date from Folkes, P.R.S., conveying the message to Stirling with the compliments of Maupertuis, the President, and the Secretary, De Formey. The letter furnishes the last glimpse we have of Stirling's connection with London. (He resigned his membership of the Royal Society in 1754.) Leadhills Regarding Stirling's residence in Scotland we are fortunately provided with much definite information. A detailed account of his skilful management f the mines is given in the Gentle- mans Magazine for 1853. 1 He is also taken as one of the best types of the Scotsmen of his day by Ramsay in his Scotland and Scotsmen. Ramsay, who always speaks of him as the Venetian, met him frequently on his visits to Keir and Garden, and had a profound regard for the courtly and genial society of the Venetian, who by his long residence abroad and in London had acquired to a marked degree la gram.de maniere, without any trace of the pedantry one might have expected. Ramsay also narrates several anecdotes regarding Stirling's keen sense of humour. 2 The association between Venice and the Leadhills in Stirling's career is very remarkable. According to Ramsay, before Stirling left Venice, he had, at the request of certain London merchants, acquired information regarding the manu- facture of plate glass. Indeed, it is asserted by some that owing to his discovery he had to flee from Venice, his life being in danger, though Ramsay makes no mention of this. Be that as it may, his return to London paved the way for further acquaintance, with the result that about 1735 the Scots Mining Company, which was controlled by a group 1 'Modern History of Leadhills'. 2 I. c., vol ii. LEADHILLS 19 of London merchants, associated with the Sun Fire Office, selected him as manager of the Leadhills mines. The company had been formed some twenty years previously with the object of developing the mining for metals, and had for managing director Sir John Erskine of Alva, a man of good ideas, but lacking in business capacity to put them into practice. Leases were taken in different parts of the country, but were all given up, with the exception of that of the Leadhills mines, the property of the Hopetoun family, which had already been worked for over a century. When Stirling was appointed the affairs of the Company were in a bad way. For the first year or more Stirling only resided at the mines for a few weeks, but about 1736 he took up definite residence, devoting his energies entirely to the interests of the Company. Gradually the debts that had accumulated in his predecessor's day were cleared off, and the mines became a source of profit to the shareholders. Eut his scientific pursuits had to be neglected. We find him, in his letters to Maelaurin, with whom he still frecmently corresponded, complaining that he had no time to devote to their scientific researches, and when writing to Eulcr he tells him that he is so much engrossed in business that he finds difficulty in concentrating his thoughts on mathematical subjects in the little time at his disposal. The village in which he and the miners lived was a bleak spot in bare moorland, nearly 1,300 feet above sea level. There was no road to it, and hardly even a track. Provisions and garden produce had to be sent from Edinburgh or Leith. In spite of these disadvantages Stirling has left indelible traces of his wise management, and many of his improvements have a wonderful smack of modernity. The miners were a rough, dissipated set of men, who had good wages but few of the comforts of life. Stirling's first care was to add to their comfort and to lead them by wise regulations to advance their own physical and mental welfare. In the first place he carefully graded the men, and worked them in shifts of six hours, so that with a six hours' day they had ample time at their disposal. To turn their leisure to profit they were encouraged to take up, free of charge, what we should now call ' allotments ', their size being restricted only C 2 20 LIFE OF JAMES STIRLING by the ability of the miners to cultivate. The gardens or crofts produced fair crops, and in time assumed a value in which the miner himself had a special claim, so that he could sell his right to the ground to another miner without fear of interference from the superior. In this way Stirling stimu- laed their industry, while at the game time furnishing them with a healthy relaxation from their underground toil. The miners were subject to a system of rules, drawn up for their guidance, by reference to which disputes could be amicably settled. They had also to make contributions for the main- tenance of their sick and aged. In 1740, doubtless with the aid of Allan Ramsay, the poet, who was a native of the place, a library was instituted, to the upkeep of which each miner had to make a small subscription. Stirling is thus an early precursor of Carnegie in the foundation of the free library. When Ramsay of Ochtertyre visited Leadhills in 1790 the library 1 contained several hundred volumes in the different departments of literature, and it still exists as a lasting memorial to Stirling's provision for the mental improvement of the miners. On the other hand, Stirling's own requirements were well provided for by the Company, whose affairs were so prosperous under his control. They saw to it that he was well housed. More than once they stocked his cellar with wines, while the salary they paid him enabled him to amass a considerable competency. "When, with the increase of years, he became (oo frail to move about with ease, they supplied him with a carriage. The Glasgow Kettle In the eighteenth century the rapidly expanding trade of Glasgow and the enterprise of her merchants made it highly desirable to have better water communication and to make the city a Port, and in 1752 the Town Council opened a separate account to record the relative expenditure. The 1 Of Stirling's private library two books have been preserved. One, on Geometry, was presented to him by Bernoulli in 1719. The other (now at Garden) is his copy of Brook Taylor's Methodus Incrementorum, which he bought in 1725. THE GLASGOW KETTLE 21 first item in this account, which is headed ' Lock design'd upon the River of Clyde ', runs thus : ' Paid for a compliment made by the Town to James Stirling, Mathematician for his service, pains, and trouble, in surveying the River towards deepening by locks, vizt For a Silver Tea Kettle and Lamp weighing 665 oz at 8/ per oz £^6 10 For chasing & Engraving the Towns arms 1 14 4 £28 4 4' Stirling had evidently performed his task gratuitously but with characteristic thoroughness ; and to this day, when the city holds festival, the Kettle is brought from Garden, where it reposes, in grateful memory of the services that occasioned the gift. To this period there belongs only one paper by Stirling, a very short article (Phil. Trans., 1745) entitled 'A Description of a machine to blow Fire by the Fall of Water'. The machine is known to engineers as Stirling's Engine, and furnishes an ingenious mechanical contrivance to create a current of air, due to falling water, sufficiently strong to blow a forge or to supply fresh air in a mine. Its invention is doubtless due to a practical difficulty in his experience as a mining manager. There is also preserved at Garden the manuscript of a treatise by Stirling on Weights and Measures. For thirty-five years Stirling held his managership. He died in 1770, at the ripe age of seventy-eight, when on a visit to Edinburgh to obtain medical treatment. Like Maclaurin and Matthew Stewart, he was buried in Greyfriars Churchyard, ' twa' corps lengths west of Laing's Tomb 7 as the Register Records grimly describe the locality. By his marriage with Barbara Watson, daughter of Mr. Watson of Thirtyacres, near Stirling, he had a daughter, Christian, who married her cousin, Archibald Stirling of Garden, his successor as manager of the mines; and their descendants retain possession of the estate of Garden. 1 LaingV Tomb is a prominent mural tablet (1620) on the rigbt wall surrounding- the churchyard. 22 LIFE OF JAMES STIRLING Thus closed a career filled with early romantic adventure and brilliant academic distinction, followed in later years by as marked success in the industrial field. As a mathematician Stirling is still a living power, and in recent years there has sprung up, more particularly in Scandinavian countries, quite a Stirling cult. His is a record of successful achievement of which any family might well be proud. WORKS PUBLISHED BY J. STIRLING (A) ENUMERATION OF CUBICS § 1. His first publication, Lineae Tertii Ordinis Neutonianae sive Illustratio Tractatus D. Newtoni Be Enumeratione Linearum Tertii Ordinis. Gui subjungitur, Solutio Trium Problematum, was printed at the Sheldonian Theatre, at Oxford, in 1717. As the book 1 is very scarce, I give a short account of its leading contents. By a transcendent effort of genius, Newton had, in the publication of his Enumeration of Cubic Curves, in 1704, made a great advance in the theory of higher plane curves, and brought order into the classification of cubics. He furnished no proofs of his statements in his tractate. Stirling was the first of three mathematicians from Scotland who earned for themselves a permanent reputation by their commentaries on Newton's work. Stirling proved all the theorems of Newton up to, and including, the enumeration of cubics. Maclaurin developed the organic description of curves (the basis for which is given by Newton), in his Geometria Organica (1720); and P. Murdoch 2 gave, in his Genesis Curvarum per Umbras (1740), a proof that all the curves of the third order can be obtained by suitable pro- jection from one of the five divergent parabolas given by the equation „ „ , „ , ^ y A = ax s + 6ar + ex + a. Stirling, in his explanatory book, follows precisely on the lines suggested by Newton's statements, though I doubt whether he had the assistance of Newton in so doing; for 1 Edleston (Correspondence, &c, p. 235) refers to a letter from Taylor to Keill, dated July 17, 1717, which gives a critique of Stirling's book. * Earlier proofs were given by Nicole and Clairaut in 1731 (Mem. de I Acad, des Sciences). 24 WORKS PUBLISHED BY STIRLING in that case why should he have stopped short with but half of the theory 1 § 2. Newton stated that the algebraic equation to a cubic can be reduced to one or other of the four forms (i) xy 2 + ey, or (ii) xy, or (iii) y 2 , or (iv) y, = ax 3 + bx 2 + cx + d; and he gave sufficient information as to the circumstances in which these happen. The demonstration of this statement forms the chief diffi- culty in the theory. Stirling finds it necessary to devote two-thirds of his little book of 128 pages to introductory matter. He bases the analytical discussion on Newton's doctrine of Series, and gives an adequate account of the use of the Parallelogram of Newton for expanding y in ascending or descending powers of x, x and y being connected by an algebraic equation. (He also applies his method to fluxional or differential equations, though here he is not always very clear.) With some pride he gives on p. 32 the theorem * Let y = A + Bx r +Cx 2r +..., then y may be expressed as J 1 . rx 1 . 2 r-x 2 1.2.3. r 6 x i applicable when x is very large if r is negative, or when x is very small if r is positive. As an example he establishes the Binomial Theorem of Newton (p. 36). Pages 41-58 are taken up with the general theory of asymp- totes. A rectilinear asymptote can cut the curve of degree n in, at most, n — 2 finite points. If two branches of the curve touch the same end of an asymptote, or opposite ends but on the same side of the asymptote, then three points of intersection go off' to infinity. A curve cannot have more than n — 1 parallel asymptotes, and if it has n — 1 , then it cannot cut these in any finite point. If the i/-axis is parallel to an asymptote, the equation to the curve can have no term in y n . From this follows the important corollary that the equation to a cubic curve may always be found in the form , , „ . , , , . . {x + a)y 2 = yf 2 (x)+f i (x). 1 Cf. Maclaurin's Theorem. ENUMERATION OF CUBICS 25 For all lines of .odd degree have real points at infinity. Asymptotes may be found by the doctrine of series: but not always. Thus the quartic y — (ax* + bx 3 + . . . + e)/(/.c' + gx 2 + lix + k) has the asymptote ax bf—ag as found by a series. The rest of the asymptotes are given by x = oc, where a. is any one of the roots of f,c 3 + gx 2 + hx + lc = 0. In the standard case of an equation of degree n in x and y, «m+«„-i+ ••■+%> = 0, if we assume the series C D y = Ax + B+ - + - +... x or and substitute in the given equation we find, in general, (1) an equation of degree n for A furnishing n values of A, (2) an equation involving A and B of the first degree in B, (3) an equation in A, B, and C, of the first degree in C, &c. So that in general we may expect n linear asymptotes y = Ax + B. § 3. Pages 58-69, with the diagrams, furnish quite a good introduction to what we now call graph-tracing. He thus graphs the rational function y =f(x)/(p(x) with its asymptotes parallel to the i/-axis found by equating n nr> CMXZ + b y = DC—DB = v > Fig. 1. and substitution of these values in D leads to an equation w» + ^- 2 /,(s) &c. = 0, in which the term in v 71 ^ is awanting. Let D coincide with and DC with OP. .: &c. Q. E. D. Stirling adds the extensions, not given by Newton, to a Diametral Conic, a Diametral Cubic, &c, corresponding to when 2 0P 1 . 0P 2 =0,2 0P 1 . 0P 2 . 0P 3 = 0, &c. 1 Also stated by Hermann (Phoivnomia). ENUMERATION OF CUBICS 27 Neivton's Rectangle Theorem for a Conic, and generalization. The proof is made to depend on the theorem that if cXj, a 2 , ... a,j are the roots of (x) = x n + ax 1 '- 1 + . . . + k — 0, then (£)= <£- (Xl )<£-oc 2 )-...(i-a n ). In the case of the cubic y s + y 2 (ax + b)+y (ex 2 + dx + e)+ fx z + gx 2 + hx + k = 0. Fig. 2. Let PflP^, Q 1 0Q 3 be drawn in fixed directions through a point 0. Let P^ be the a.'-axis, Q^ parallel to the y-nxis, and let be the point (£, 0). Then OQ^OQt.OQ^fF + gf + hi + k, 0P 1 .0P 2 .0P 3 = i(/f + <,f + A| + /c), so that the quotient 0& . 0Q 2 . 0Q 3 /0P l . 0P 2 . 0P 3 = /(up to sign). But a change to parallel axes does not change/. ,\ &c. § 5. After a brief enumeration of conies he proceeds to find in Prop. XV (p. 83) the reduction of the equation of a cubic to one or other of the four forms given by Newton. The equation (z + a)v 2 = (bz 2 + cz + d)v + ez 3 +fz 2 + gz + h (1) includes all lines of the third order, the i>-axis being parallel to an asymptote. First Case. Let all the terms be present in (1). Let A be the origin, AB any abscissa z, BG or BD the corresponding ordinate v of the cubic. If F is the middle point of CD bz 2 + cz+d BF=l(v 1+ v 2 ) = 2z+2a 28 WORKS PUBLISHED BY STIRLING so that the locus of F is the conic v = (bz 2 + cz + d)/2'z + a) with real asymptotes GE and GE. Fig. 3. Select these lines as new axes. Call GE x, and EG or ED y. The cubic equation is of the same form as before, but EF must = K/2x, where A" is constant, by the nature of the hyperbola. Therefore, the equation to the cubic is of the form yi_ e y/x = ax 2 + bx + c + d/x, or xy 2 — ey= ax 3 + bx 2 + cx + d. (I) With a good i"eal of ingenuity, the proof is indicated in the other cases. Prop. XVI (p 87). When a is positive in (I) all three asymptotes are real. They are (i) x = 0, (ii) y = x Va + b/ 2 Vu, (iii) y= —xVa — b/2Va. If b — 0, the asymptotes are concurrent. ENUMERATION OF CUBICS 29 If b =fc 0, they form a triangle, inside which any oval of the cubic must lie, if there is an oval. The asymptotes (ii) and (iii) cut on the ai-axis, which is also a median of the asymptotic triangle. When e = 0, the point at infinity on the asymptote (i) is a point of inflexion, and conversely : in that case the locus of F reduces to a straight line, which is a ' diameter ' of the curve. An inflexion at infinity and a diameter are always thus associated. The condition that (ii) or (iii) cuts the curve only at infinity is b 2 — iac — ±iueVa. Thus possible conditions for a diameter are e = 0. b 2 —±ac = AaeVa. Ir — 4uc= — 4«ev / tt. When any two of these are satisfied i-o is the third (a is positive and not zero). Thus a cubic may have no diameter, or one diameter, or three diameters. It cannot have two. § G. The enumeration of cubics is then proceeded with in the order given by Newton, to whose work the reader must go for the figures, which are not given by Stirling. Newton gave 72 species. To thefe Stirling added 4 species, viz. species 11, p. 99, species 15, p. 100, and on p. 102, species 24 and 25. There still remained two species to be added (both arising from the standard form xy 2 = ux 2 + bx + c). One of them was given by Nicole in 1731, and the other was communi- cated by N. Bernoulli, 1 in a letter to Stirling in 1733. While sufficiently lucid, Stirling's reasoning is admirably concise. He was never addicted to excess in the use of words, and often drove home the truth of a proposition by a well- chosen example, especially in his later work. The publication of his commentary on Newton's Cubics gave Stirling a place among mathematicians, and may have been the ground on which he was invited by Tron to accept a chair in Venetian territory. 2 1 See note to Letter. 2 In connection with both Newton and Stirling see W. W. Rouse Ball on 'Newton's Classification of Cubic Curves', London Math. Soc, 1891. Another edition of Stirling's Litieae Tertii Ordinis was published in Paris in 1787. (' Isaaci Newtoni Enumeratio Linearum Tertii Ordinis. Sequitur illustratio eiusdem tractatus Iacobo Stirling.') 30 WORKS PUBLISHED BY STIRLING (B) METHODUS DIFFERENTIALIS, SIVE TRACTATUS DE SUMMATIONE ET INTERPOLATIONE SERIERUM INFINITARUM § 7. The Mdhodus Differentialis, as we shall call it, is the most important product of Stirling's genius, by which he is most generally known to mathematicians. The book is not, as the title may suggest, a treatise on the Differential Calculus, but is concerned with the Calculus of Finite Differences. It is divided into: (1) the Introduction (pp. 1-13); (2) the Sum- mation of Series (pp. 14-84); (3) the Interpolation of Series (pp. 85-153). In the Introduction he explains how the Series are defined. Denote the terms hy T, T', T", &c, and write s = r+r+T"+&c. Suppose the terms arranged as ordinates to a line so that consecutive terms are always at the distance unity. Thus if T is at distance z from the origin, T' is at a distance z + 1 , T" at distance z + 2, &c. ; where z is not necessarily an integer. For example, in Brouncker's Series (p. 26) 11 1 172 + 3 74 + sTd +- " any term is given by l/4z(a+^) where z is, in succession, i,li,2f,&c. A series may sometimes be specified by a relation connecting terms ; e.g. it 1 1, then T"= z + n + 1 T , &c. 2+1 Theorems of special interest arise when T can be ex- pressed as T= A+Bz + Cz(z-l) + Dz(z-l)(z-2)+ &c, METHODUS DIFFERENTIALS 31 or as the latter being useful when z is a large number. When T admits of either representation, then after any transformation it should be reduced again to the same form. Thus if T=A + Bz + Cz{z-l) + ..., then Tz = (A+B)z + (B + 2C)z(z-l) + (C+3D)z(z-V)(z-2) + ... To facilitate the reduction Stirling gives two formulae and two numerical tables. Let x(x + 1) (x + 2) ... (x + n- 1) = C n u x" + C n ] x"- 1 + ...+ G H n ~ l x and l/ic(.t'+l) ...(x + n-l) = 2 (- 1)* r, t s /x n + s , then z n = r 2 n - i z + r^-*z(z-i) + ... + r\ +1 z(z-i) ...(z-n + i) and i= 2 C/-» +I /c(:+l)...(3 + r). The first table (p. 8) furnishes the values of P B S for the lower values of n and s, and the second table (p. 11) the lower values of Gj. Owing to the importance of these results, and the applica- tions which Stirling makes of them, it has been proposed by Professor Nielsen ' to call the numbers G n r the Stirling Numbers of the First Species, and the numbers T n s the Stirling Numbers of the Second Species. Nielsen has discussed their properties and indicated their affinities with the Bernoullian numbers. As an illustration Stirling deduces 1 1 1-n z 2 + nz z(z+l) ' z(z+l)(z+2) Ann. di Mat., 1904 ; or Theorie der Gammi Lccount in English is given by me in the Pn 1918-19. Lagrange used them in his proof of Fermat's Theorem. 1 Niels-en, Ann. di Mat., 1904 ; or Theorie der Gammafuhhtion (Teubner, 1906J. An account in English is given by me in the Proc. Edin. Math. Soc, 32 WORKS PUBLISHED BY STIRLING which is equivalent to 1 1 a a(a+l) = - + -7—7, + „,- .'*,. . „ i+-. a; — a a; a;(a:+l) x(x+l)(x + 2) when it is usually spoken of as Stirling's Series; but it had already been given before Stirling by Nicole and by Montmort. PARS I SlJMMATIO SeRIERUM. § 8. Stirling explains that he is not so much concerned with Series, the law of summation for which is obvious or well known, as with the transformation of slowly converging series into series that more rapidly converge, with their sum to any desired degree of accuracy. Let S = T \ T'+T"+..,adcc, 8'= T' + T"+...adao, S"= T"+...ad,&c. Any difference-equation connecting S, S', ..., T, T', ... z, may be transformed into another by writing for these, respec- tively ,-,, ,,// ,,,/ fn,l o , a , ... , i , l , ... , z + 1. But when the number of terms in the series is finite, he takes T to be the last {S= ... + T" + T' + T), so that S'= S—T, and if S corresponds to z, S' corresponds to :-l. On p. 16, he quotes a theorem of Newton, 1 which furnishes a key to several of the theorems that follow later in the Melhndus Differentialis. In modern garb it may be thus stated, ' z zP(l—z\1 z v -^{\~z)1-^dz = — -F(p + q, 1, p+1, z), o P where F(a, b, c, z) denotes the hypergeometric series a. b a(u+ 1) b(b +1) „ ■ 1+ lTc Z+ l.2.c( C+ l) Z+ -' 1 See also p. 113 of Methodus Differentialis. METHODUS DIFFERENTIALS 33 When z — 1 we have, of course, the Beta Function zP- l (l-z)l^dz. Jo Prop. I. §9. If T = A + Bz + Oz(z-l) + ... the sum of the first z terms is Az+-(z+l)z+~(z+l)z(z-l) + ..., and Prop. II. If T=-r^— + z(z+l) z .z+l .z + 2 and S=T+T' + adx, ,u ■ o A B ° then S — - + + + , &c, z 2z.z+l Zz. z+l .z + 2 were both given previously by Nicole and Montmort, but Stirling carries their applications much further. E.g. To sum 11 1 Ji + 2 2+ 3* + "" This Stirling effects in the following characteristic fashion (pp 28, 29). m 1 1 1! 2! 3! _ T = -„ = + + + , & c . z 1 z. z+l z. z+l. z + 2 z z+Z &c. Hence „ 1 1! 2! „ c = - H + V , &c. z 2.z.z+ 1 3 .z. z + l. z + 2 Calculate S for z — 13. ■•• lis +il* + - =-079,957,427. Add thereto f + i + ... + T ih = 1 '564,976,638. The total is 1-644,934,065. Stirling did not probably know that this is equivalent to |7r 2 , until Euler sent him his well-known formulae for series of the kind. !«47 1) 34 WORKS PUBLISHED BY STIRLING Prop. III. If T=afi+A- + — —+...], lz z.z+l ) then the sum (to infinity) is r+ „ ( a b — Ax G—2Bx , ) x z+n < + I- + &c > > 1(1 -x)z (\-x)z.z+\ (I -x) z.z+l .z + 2) ') where A, B, C, ... denote the coefficients of the terms preceding those in which they occur. Thus . a b—Ax „ A = - , B = , &c. 1—X (1— X) His well-chosen example gives the summation of the Series of Leibniz |=l-i + W+...adoo. i Here T= (- l) 2 -l^as found by writing f, li, 2|, &c. z for 0, so that 6 = 0, &c. Calculate the sum for z = 12± from the formula. It is -020,797,471,9. Add thereto 1-4+.. .-2V = -764,600,691.5, so that the sum of the total series is -785,398,163,4, a result which could never be attained by the simple addition of terms, ' id quod olim multum desiderabat Leibnitius '. (Stirling sums the same series by another process on p. 66.) This is an example of several numerical series, well known in his day, the summation of which had hitherto proved refractory, and which Stirling can sum to any desired degree of accuracy. Prop. IV is concerned with the problem of proceeding from an equation in S and S', say, to an equation in T and T'. E.g. From (z-n)S = (z-l)S\ he finds (z-n)T=zT'. Prop. V is taken up with applications of IV. § 9. Prop. VI gives an interesting theorem (pp. 37-8). If the equation connecting 8 and S' is 8(z° + az - 1 + ...) = mS' (z e + kz e ~ l + .. .), METHODUS DIFFERENTIALS 35 then the last of the sums will be finite both ways only when m = 1 and k = a. In other words the infinite product n 1+ - +... n 1=1 e+ - +... n is finite both ways only when e = 1 and a = f. This is one of the earliest general tests for the convergence of an Infinite Product of which Wallis (' Wallisius noster ' as Stirling calls him in his earlier book) furnished an illustra- tion, with rigorous proof, in the formula tt_ 2.2. 4. 4.6.6 ... 2 ~~ 1.3.3.5.5.7 ... ' published in his Arithmetica Infinitorum in 1655. Prop. VII gives a remarkable transformation of a series, in the discussion of which he has occasion to solve a Difference Equation by the method so universally employed nowadays of representing the solution by an Inverse Factorial Series. As stated by Stirling it runs thus : If the equation to a series is (z-n)T+{m-l)zT'(=0), m — 1 m n A n+ 1 B « + 2(7 . then S = T-\ + — + — + &c. m z m z + 1 m z + 2 *n . m— 1 n A s {A is T, Bis , &c). m :m If we take T = 1 it becomes V 1— m/ m \ m/ [<*F{*. l,y,x)= T L x F(y- a> l, y, J^j\ As Professor Whittaker has pointed out to me, the theorem in the latter form furnishes a remarkable anticipation of the well-known transformations of the Hypergeometric Series given by Kummer (Crelle, 15, 1836). D2 36 WORKS PUBLISHED BY STIRLING In Props. VIII to XII x Stirling returns again and again to the summation or transformation of the series denned by y,_ z—m z-n T z z — n+1 Professor Whittaker suggests that the relative theorems were doubtless invented to discuss the series 1 + z-m 1 (z-m)(z-m+l) 1 + && z — n z z — n+1 2.2+I z — n+2 which (up to a factor) represents the remainder after 2—1 terms in the series 1 1 — no 1 1 . m . 2 — m 1 + — i—n + r^ ^ +••■> 1— n 1 2 — n 1.2 3—n C 1 Jo x- n (l-x) m - 1 dx. After the work of Euler this integral was calculated by Gamma Functions. § 10. A number of theorems follow for summing a series ' accurate vel quam proxime ', all illustrated by well-chosen examples. Then, to show that his methods apply to series already well known, he takes up their application to the summation of Recurring Series, the invention of his friend De Moivre, the Huguenot refugee, who lived and died in London. He gives extensions to series when the terms at infinity are approximately of the recurrent type. Several examples are given of more complicated series such as Za n x n when K a n + K-i a n-l+ ■•■ — °> where X„, A,,-!, ... are integral functions of n of degree r, and for which he finds a differential equation (fluxional he calls it) of the ?'th order. He would have been clearer had he adopted the repre- sentation of integral functions as given by himself in the Introduction. 1 Cf. Andoyer, Bull. Soc. Math, de France, 1905. METHODUS DIFFERENTIALS 37 E.g. Suppose r = 2, and write the equation in the co- efficients as a n (ix + P n + yn. 11- l)+a n _ } (a+b.n-l 4 c.n-l.n — 2) + &c. = 0. Let y = 2a n x n , .: y = 2na n x n -\ y = 2n(n-l)a n x n - 2 , &c. Hence (ay + Pxy + yx 2 y), + x (ay + bxy + cx 2 y) + &c. = o, or differs from zero by a function of x dependiDg on the initial terms of the series, and easily calculated. The differential equation being obtained, its solution has next to be found when possible, and this he proceeds to do (pp. 79-84) by means of power series. Unfortunately, in the examples he takes he is not quite accurate in his conclusions. In the last letter from N. Bernoulli referred to above (1733) the latter remarks : ' Sic quoque observavi te non satis rem examinasse, quando pag. 83 dicis, aequationem r 2 y 2 = r 2 x 2 — x- y 2 nulla alia radice explicabilem esse praeter duas exhibitas y = x — x 3 /6r 2 + x !i /120r i + ... y = Ax l-x 2 /2r 2 + x i /24r i + ... quarum prior dat sinum, et posterior cosinum ex dato arcu a;, et de qua posteriore dicis, quantitatem A quae aequalis est radio r ex aequatione fluxionali non determinari. Ego non solum inveni seriem non posse habere hanc formam A i Bx 2 + Cx i +... nisi hat A = r, sed utramque a te exhibitam seriem compre- hendi sub alia generaliori, quae haec est : y = A + Bx + Cx 2 + . . . 38 WORKS PUBLISHED BY STIRLING in qua eoefficientes A, B, C, &c. hanc sequuntur relationem BB = rT ~ AA , Q= ~ A , D= - B rr 1 . 2 . rr 2 3 . rr JP C v D x E — , F = > &c. 3 . 4 . rr 4 . 5 . rr Si fiat A = 0, habetur series pro sinu, sin autem A fiat = r habetur series pro cosinu; sin vero A alium habeat valorem praeter hos duos, etiam alia series praeter duas exhibitas erit radix aequationis. Similiter series illae quatuor. quae e^hibes pag. 84 pro radice aequationis y + a 2 y — xy — x 2 y — sub aliis duabus generalioribus quae ex tuis particularibus compositae sunt comprehenduntur.' Bernoulli adds his solutions. (Vide Letter in question.) PARS SECUNDA de Intekpolatione Seiuerum § 11. The second part contains the solution of a number of problems in the treatment of which Stirling shows remarkable analytical skill. Again and again he solves Difference Equations by his method of Inverse Factorials. This is the method now adopted by modern writers 1 when large values of the variables are in question. In this short sketch I can only indicate very briefly a selection of some of his conclusions. A common principle applied is contained in the following : Being given a series of equidistant primary terms, and the law of their formation, intermediate terms follow the same law. Take for example the series l + l+2! + 3! + 4! + &c. in which the law is T' z+1 = zT z (the law for the Gamma Function). If a is the term intermediate between 1 and 1, the corresponding intermediate terms are 2 > "%•'%"> "2 • ~2~ • 2@/> (VC or, as Stirling puts it, b = fa, c = 1 6, &c. (Page 87) 1 Cf. Wallenberg and Guldberg, Theorie der linearen Differenzen- Gleichungen (Teubner, 1911). METHODUS DIFFERENTIALS 39 Prop. XVII. Every series admits of interpolation whose terms consist of factors admitting of interpolation. Thus, given the series 1, - -A, — — Ji, C, &c. p p+1 p + 2 ' it will be sufficient to interpolate in 1 r r.r + 1..., 1 p p.p+1..., and divide. § 12. Prop. XVIII is of fundamental importance in many of the series discussed. In the two series T T 4- 1 A, -A, L±±B,..., p p+1 r r+ 1 7 a, - a, b, ..., q q+1 if A and a are equal, then the term of the first series at the distance q — r from the beginning is equal to the term of the second series at the distance p — r from the beginning. The illustrations he gives can hardly furnish a proof, for p — r and q — r are not necessarily either integral or positive. (The proof may be put in a couple of lines by the use of Gamma Functions.) Example. Consider the series i 2 a 4 r en which to meet the conditions must be written as Suppose the term at distance m wanted. Here p — r=—%. Write q — r = m or q = m+l, and form the series a 26 3c 1> 7' 7.' ^' •••• m+1 m +2 m+3 Then the term wanted in the first series is that of this second series which precedes 1 by the interval — ^. This 40 WORKS PUBLISHED BY STIRLING artifice is often useful when m is a large number, provided the second series can be easily interpolated. He leaves these considerations to lay down the standard formulae of interpolation already established by Newton, viz. that known as Newton's Interpolation Formula f {z )=f(0)+A 1 z + A 2 Z ~^>,&o., and also the two formulae known as Stirling's Formulae, though they are really due to Newton. He also takes the opportunity to establish (p. 102) what is called Maclaurin's Series. ' Et hoc primus deprehendit D. Taylor in Methodo Incrementorum, et postea Hermanns in Appendice ad Phoronomiam.' § 13. In Prop. XXI he teaches by examples how to inter- polate near the beginning of a series. The second example (pp. 110-12) furnishes by pure calculation a most remarkable result, represented in modern rotation by the formula r a) = V*. About the same time Euler had obtained the same result by a different method (vide Fuss, Corresp. mathematique). Stirling proposes to find the term midway between 1 and 1 in the series 1, 1, 2, 6; 24, 120, &C The law here is T 2+1 = zT z and T x = 1, T 2 =l. He interpolates between T n and T 12 to find Znj and then he has to divide by 10 J, 9|, ... 1| to obtain T$. Since the numbers are rapidly increasing he uses their logarithms instead and actually calculates log T n % from which he finds Ti,| to be 11899423-08, so that Tii = -8862269251. He adds T% — 1-7724538502, and this number, he says, is Vtt. {Vtr is actually 1-7724538509.) Also the corresponding entry among the numbers 1, 1, 4, 36, 576, &c. is it. 1 For inventive audacity Stirling's conclusion would be difficult to match, and its skilful application led him to 1 Is it not possible that he thus detected that Tj = */n ? METHODUS DIFFERENTIALTS 41 results that aroused the admiration of his friend De Moivre. (Vide Miscellanea Analytica.) In Prop. XXII, Ex. 1, it helps him in the interpolation of the term at infinity in the series 1, I A, $B, iC, ..., or .2 2.4 2. 4. 6 a problem which faces him again in Prop XXIII, in which he gives a formula to find the ratio of the coefficient of the middle term in (1 +x) 2n to 2 2 ". Binet in his Memoir 1 (pp. 319-20) proved that of the four solutions of the latter problem given by Stirling (1) and (3) are correct, while (2) and (4) are wrong. As a matter of fact Stirling only proves (1) and (3) and leaves (2) and (4) to the reader. Binet, writing b for the middle coefficient, gives the four formulae ?2n. 2 0) (-j-) =rrnF(l f, n +l, 1). (2) Cir) = l {2n+l)F & -*' n + 1 > *>• < 3 > (M= ^hr/b *■*+*• v- Of these (1) and (3) 2 are also the first and third of Stirling's ; while (2) and (4) replace the other two given by Stirling, viz : o2n 2 < 2 )' (V) = \ (271+1) 1* 1 2 .3 2 . 1 1 ; . &C. 2(2)1-3) 2.4(2w-3)(2?i-5) 1 Binet, Minx, sur les Integrates diflnies EulMennes. 2 These are also the solufions he gave in a letter to De Moivre to publish in the Miscellanea Analytica. (See pp. 46-48.) 42 WORKS PUBLISHED BY STIRLING (*)' (i) Z = 1 nn 1 2 .3 2 . "I 2.4(2)1-2) (2n-i) J 2(2?i-2) 2.4(2n-2)(2n-4) Clearly (4)' must be wrong since the factors 2 n — 2, 2 n - 4, . . . include zero in their number. Binet remarks that the products of (1) and (4) and of (2) and (3) furnish the first examples known of Gauss's law F(a,p,y,l)xF(-a, ft y-a, 1) = 1. § 1 4. In Prop. XXIV the Beta Function is introduced (as an Integral) for the interpolation of v r (r + 1) P P(P+V and the conclusion drawn (in modern notation) B(2J + n, q)/B{ r+z+n r T he would have obtained A/T=F(n, -z, r + n, 1), i. e. he would have established the Gaussian formula F(a, b, c, l)*F( — a, b, c-a, 1) = 1. STIRLING'S SERIES §15. On pages 135-8 are given the formulae which have rendered Stirling's name familiar whenever calculations in- volving large numbers are concerned. METHODUS DIFFERENTIALS 43 STIRLING'S THEOREM When n is a large number the product -11+; 1.2.3 ...n = 7i n V2mre Un , where < 8 < 1. Stirling actually gives the formula Log (1 . 2 . 3 ... x) = J log(27r) + (a> + 4) log (aj + i) -(« + *)- ., 10 1 /^u + 2.12.(* + |) ' 8.360(a; + i) 3 with the law for the continuation of the series. De Moivre (Stipp. Mike. Anal.) later expressed this result in the more convenient form log (1.2 ...x) = |log(2n-) +(» + !) logic -* + r72.- ■B- 1 h 3.4 a; 3 ... + ( 1;" +1 - B» 1 10 n 1 l2» ™2»-l Cauchy gave the remainder after the last term quoted as H (271+1) (2 ft +2) x Zn+x (B t , B 2 , &c, denote the Bernoullian numbers.) More particularly the series 1 1.2* 3.4 a; 3 has been called the Series of Stirling. It is one of the most remarkable in the whole range of analysis to which quite a library of mathematical literature has been devoted. The series is divergent, and yet in spite of this fact, when n is very large and only a few of the initial terms are taken, the approximation to log n ! found by it is quite suitable for practical purposes. 1 Its relative accuracy is due to the fact 1 See Godefroy, Thiorie des Series, or Bromwich's Treatise on Series. 44 WORKS PUBLISHED BY STIRLING that the error committed at any stage, by neglecting JR n , is always less in absolute value than the first of the terms neglected , which suggests that the series should be discontinued when the minimum term is reached. Legendre has shown that if we write the i-eries as 2 (— 1)™ +1 u n , then U-n+i/Un < {2n-l)2n/47r 2 x 2 , and .-. < (to/ 3a?) 2 - The terms therefore decrease so long as n does not exceed 3 x. When n = 3x the error is less in absolute value than •393409... xx~i e~ ax . To later mathematicians, such as Gauss, who admitted only the use of convergent series, Stirling's Series was an insoluble riddle, but it now finds its place among the series defined as Asymptotic Series. 1 To meet the objection to its divergence Binet (1. c, p. 226) gave the convergent representation. log (x — 1) ! = -|log(27r) + (a; — ^) logo;— a; 1 „ 2 r , 3 1 + 2 2^2+374^3+ — ^+.. in which S„ denotes , + + ... ad co. " (sfl)" (x + 2) n From this by the use of inverse factorials he deduces (p. 231) log (a;- 1)! = | log (2 7r) + (as — |) log a; — a; 1 1 12(£B+ 1) 12(03+1) (03+ 2) 59 1 360 (a;+l)(a:+2)(a;+3) 227 480(03+1). ..(a; + 4) C ' § 16. The conclusion of Stirling's book is taken up with various problems in interpolation, based partly on a paper by him in the Philosophical Transactions for 1719, and partly 1 Vide Poincare, Acta Math., 1886. STIRLING'S SERIES 45 on the researches of Newton and Cotes. It may be noted that in Prop XXX he gives the expression of one of the roots of a system of n linear equations in n variables, found 'per Algebram vulgarem '. A translation into English by Francis Holliday was published in 1749 ' with the author's approbation ' There was also a second edition of the original treatise in 1764. (C) CONTRIBUTIONS TO THE PHILOSOPHICAL TRANSACTIONS § 1 7. Though Ramsay (loc. cit.) refers to writings by Stirling while in Italy, I am not acquainted with any such, save the first of his three papers printed in the Philosophical Transactions. It is entitled Methodus Different ialis Newtoniana Illustrata Aathore Iacobo Stirling, e Coll. Balliol. Oxon., and furnishes a useful commentary on Newton's Methodus Differentialis published in 1711. Stirling restricts his attention entirely to the case of equal increments and proves the three Inter- polation Formulae already referred to above (p 40). He deduces a number of special formulae, several of which are reproduced in his book of 1730. One of these may be noted on account of the uncanny accuracy of its approximations in certain cases. Let a, ft, y, 8, ... be a series of quantities, and write down the equations found by equating the differences to zero. a-/3=0, a-2/3 + y = 0, a-3/3 + 3y-4-^> ^ — 4T6""' atque perficiendo computum ut in margine, invenietursummaTerminorum 157-866984459, cuius Radix quadrata 125645129018 est ad unitatem ut summa omnium Unciarum ad mediam in Dignitate centesima, vel ut summa omnium ad alteram e mediis in Dignitate nonagenima nona. Problema etiam solvitur per reciprocam illius Seriei, etenim summa omnium Unciarum est ad Unciam mediam in sub- duplicata ratione semiperipheriae Circuli ad Seriem 157-079632679 769998199 16658615 654820 37137 2734 246 26 3 157-866984459 + A + 9B + 25 C + 49D n+1 2x11+3 4xii+5 6X71+7 Sxw + 9 + 81^ 10X71+11 _&c. vel quod eodem redit, ponatur a — -6366197723676, quoto scilicet qui prodit dividendo unitatem per semiperipheriam Circuli; & media proportionalis inter numerum a, & hanc Seriem, erit ad unitatem, ut Uncia media ad summam omnium. 48 WORKS PUBLISHED BY STIRLING •00630316606304 3059789351 65566915 2553229 143473 10470 934 98 12 1 Ut si sit in— 100 ut antea, computus erit ut in margine vides, ubi summa termi- norum prodit -00633444670787 cujus Radix quadrata -0795892373872 est ad unitatem ut Uncia media ad summam omnium in Dignitate centesima vel nonagesima nona. Sunt & aliae Series pro Solutione hujus Problematis aeque simplices ac eae hacte- nus allatae, sed paulo minus convergentes. ubi Index Binomii est numerus exiguus. Caeterum in praxi non opus est recurrere ad Series; nam sufficit sumere mediam pro- -00633444670787 portionalem inter semicircumferentiam Circuli & n + ^; haec enim semper approximabit propius quam duo primi Seriei termini, quorum etiam primus solus plerumque sufficit. Eadem vero Approximatio aliter & praxi accommodatior sic enunciatur. Pone 2it = c=l-2732395447352; eritque ut summa Unciarum ad mediam, ita unitas ad /- quam proxime, existente errore in excessu circiter / • 1 67174 V 271+1 n St n= 100, erit- =-006334525, ejusque radix quadrata •07958973 accurata est in sexta decimali, quae si dividatur per 167m, id est per 160000 dabit correctionem -00000050, & haec subducta de approximatione, relinquit numerum quaesitum •07958923 justum in ultima figura. Similiter si sit n = 900, erit = -000706962545, cuius Radix quadrata -026588767 superat verum binario in nona decimali, sin vero Correctio computetur ac subducatur de approximatione, habebitur numerus desideratus accuratus in decima tertia decimali. En autem approximationem aeque facilem & magis accura- tam, differentia inter logarithmos numerorum n + 2 & n — 2 dividatur per 16, & quotus adjiciatur dimidio logarithmi Indicis n; huie dein summa adjiciatur logarithmus constans •0980599385151 hoc est dimidium logarithmi semiperipheriae Circuli, & summa novissima est logarithmus numeri qui est ad unitatem ut summa omnium Unciarum ad mediam. St 7i = 900 computus erit £ log 900 1-4771212547 16)Dif. log 902 & 898 ( -0001206376 Log constans -0980599385 Summa 1-5753018308 PHILOSOPHICAL CONTRIBUTIONS 49 Et haec summa verum superat binario in ultima figura; estque logarithmus numeri 37-6098698 qui est ad unitatem ut Summa Unciarum ad mediam in dignitate 900 vel 899. Et si vis illius numeri reciprocum, sume complementum logarithmi, scilicet— 2-4246981692, & numerus eidem corre- spondens invenietur -0265887652. Et hae sunt Solutiones quae prodierunt per Methodum Differentialem Newtoni ; quarum demonstrationes jam non attingo, cum in animo sit brevi publico impertire Tractatum quem de Interpolatione & Summatione serierum conscripsi. Tui Studiosissimi 19 Jun. 1729 Jac. Stirling BIBLIOGRAPHY (1) Sir D. Brewster: Life of Newton. (2) J. Brown : Epitaphs and Monumental Inscriptions in Greyfriars Churchyard, Edinburgh. 1867. (3) H. W. C. Davis : History of Balliol College, Oxford. (4) Edleston: Newton's Correspondence ivith Cotes, $c. 1850. (5) W. Fraser: The Sterlings of Keir and their Private Papers. Privately printed, 1858. (6) Gentleman's Magazine for 1853: Modern History of Leadhills. (7) A. D. Godley's Oxford in the Eighteenth Century. 1908. (8) T. Hearne: Hearne's Diary, edited by Bliss, 1869; also by the Oxford Historical Society. (9) Macclesfield : Correspondence of Scientific Men. (10) G. 0. Mitchell : Old Glasgow Essays. 1905. (11) J. Moir Porteous: God's Treasure House in Scotland. 1876. (12) J. Ramsay : Scotland and Scotsmen in the 18th Century. 1888. (13) S. P. Rigaud : Miscellaneous Works and Correspondence of the Rev. James Bradley, D.D. 1832. (14) B.Taylor: Contemplatio Philosophica. 1793. (15) W. W. R. Ball : Newton's Classification of Cubics, London Math. Soc. 1891. (16) Historical works of Cantor, Chasles (Apercu), Montucla ; articles on Probability and Theory of Finite Differences in Encyclopidie des Sciences mathimatiques ; modern text-books on Finite Differences by Markoff, Seliwanov, &c, and on Probability by Bertrand, Czuber, &c. (17) G.Cramer: Courbes alge'briques. (18) P. H. Fuss : Corr. math, et physique de quelques cUebres giometres du XVIir siecle. 1843. (19) M. Godefroi ; Thiorie des Series. 1903. (20) C. Maclaurin : Treatise of Fluxions. 1742. (21) De Moivre : Doctrine of Chances. 1756. „ Miscellanea Analytica de Seriebus. 1730. (22) R. Reiff : Geschichte der Unendlichen Reihen. 1889. (23) I. Todhunter: History of Probability and History of Attraction and the Theory of the Figure of the Earth. (24) Any student wishing to study Stirling's methods cannot do better than read in the following order : (i) J. Binet : MSmoire sur les Intigrales EuUriennes ; Jour. Ecole Poly. 1839. (ii) N. Nielsen : Theorie der Gammafunktion. Teubner, 1906. Alscr: Les Polynomes de Stirling. Copenhagen, 1820. (lii) G. Wallenberg und A. Guldberg: Theorie der linearen Differenzen- gleichungen. Teubner, 1911. STIRLING'S SCIENTIFIC CORRESPONDENCE E 2 INTRODUCTION Much of the correspondence of James Stirling has been preserved at the family seat of Garden. In the collection are several letters from him to his friends in Scotland, and numerous extracts from them arc to be found in the Family History: — The Stirlings of Keir and their Private Papers, by VV. Fraser (Edinburgh, privately printed, 1858). Jn addition to these are letters of a scientific character which were with great courtesy placed at my disposal by Mrs. Stirling in 1917. Of the latter group of letters the earliest is one from Nicholas Bernoulli in 1719, and the last is one from M. Folkes, P.R.S. in 1747. Stirling enjoyed the acquaintance of most of the British mathematicians of his day, while his reputation and continental experience brought him into corre- spondence with continental scholars like Clairaut, Cramer, and Euler. It is interesting to note that all of his correspondents save Campailla were, or became Fellows of the Royal Society of London. (It is clear from letter XI X that Stirling suggested to Euler that he should become a Fellow.) The dates when they joined are indicated in the notes added to the letters. One learns from the letters how much depended on corre- spondence for the discussion of problems and the diffusion of new ideas, just as one would turn nowadays to the weekly and monthly journals of science. Several of the letters in the collection shed a good deal of light upon obscure points in the history of Mathematics, as indicated in the notes. Maclaurin appears to have been Stirling's chief correspondent and the letters between the two men are of particular interest to students of Scottish Mathematics. They were warm friends, though probably in opposite political camps, and Maclaurin had the benefit of Stirling's judgment when engaged upon his Treatise of Fluxions. 54 INTRODUCTION There are not many letters of Stirling, and those are chiefly copies made by Stirling himself. I had the good fortune to find four original letters from Stirling to Maclaurin in the Maclaurin MSS. preserved in Aberdeen, and they fit in admirably with the letters of the Garden collection. But I am convinced that other letters by Stirling are still to be found. Stirling is known to have had frequent correspondence with R. Simson, G. Cramer, and De Moivre, not to mention others, and the discovery of fresh letters might be the reward of careful search. Among letters of Stirling already published may be mentioned his letter to Newton in 1719 (Brewster's Newton), a letter to J. Bradley reproduced in the Works and Correspondence of Bradley, a letter to De Moivre in the Miscellanea Analytica de Seriebus, and reference to a second letter in the Supplement to the same work. Cramer's Letter III 3 and the letter from Stirling to Castel V 2 are reproduced in the Stirling Family History. CONTENTS PAGE CORRESPONDENCE WITH MACLAURIN, 1728-1740. 57 11 Letters from Maclaurin to Stirling. 4 ,, ,, Stilling to Maelaui in. 1 Letter „ Gray to Maclaurin relative to Stirling. The letters to Maclaurin have been obtained through the courtesy of Aberdeen University. Letter I 10 is a note attached to the translation of a letter from Maupertuis to Bradley. II LETTER FROM SIR A. CUMING TO STIRLING, 1728 . 93 III G. CRAMER AND STIRLING, 1728-1733 ... 95 10 Letters from Cramer to Stirling. 1 Letter ,, Stirling to Cramer. IV N. BERNOULLI AND STIRLING, 1719-1733 . . 131 3 Letters from Bernoulli. 1 Letter „ Stirling. V L'ABBE CASTEL AND STIRLING, 1733 . . .151 1 Letter from Castel. 1 „ „ Stirling. VI CAMPA1LLA AND STIRLING, 1738 .... 158 1 Letter from Campailla. 56 CONTENTS PAGE VII J. BEADLEY AND STIRLING, 1733 .... 160 1 Letter from Stirling. 1 „ „ Bradley. VIII S. KLINGENSTIERNA AND STIRLING, 1738 . . 164 1 Letter from Klingenstierna, also solutions of cer- tain problems. IX MACH1N AND STIRLING, 1733(?) and 1738 . . 172 2 Letters from Machin. X CLAIRAUT AND STIRLING, 1738 .... 176 1 Letter from Clairaut. XI EULER AND STIRLING, 1736-1738 .... 178 1 Letter from Stirling, 1738. 1 „ „ Euler, 1738. (Euler's first letter to Stirling, probably preserved at Petrograd, was written in 1736.) XII M. FOLKES, P.R.S., AND STIRLING, 1747 . . 192 1 Letter from Folkes. NOTES UPON THE CORRESPONDENCE . . .193 l I*. J fys#*lU&#> \fJ&Jy ~ffii »*> /W" u*u- ^ea, ^ /<*- / ^^^**k_^ j «/ J -f*t£ ff >tr2fo yfeJwj- >C»- J " H y«> M %«d& fa'o^ -fcfUT pt*c£\ f, C^iu^acL r nig ' J {gwXjtfa- •*/^\ ^rfait&QL/cuJt I COLIN MACLAURIN AND STIRLING (l) Madaurin to Stirling, 1728 Mr James Stirling at the Academy in little Touer Street London Sir Your last letter was very acceptable to me on several Accounts. I intend to set about publishing the piece on the Collision of Bodys very soon. I was obliged to delay it till now having been very busy taking up my Classes in the College. Your remarks on their experiments are certainly just. I intend if I can get a good opportunity by any of our members of parlia* to send you a copy of my remarks before I publish them. I have seen Koberts's paper since I came from Perthshire in August where I writ my remarks and find he has made some of the same observations as I had made ; nor could it well happen otherwise. I wish I had Mr Graham's Experiment at full length with Liberty to insert it. I design to write to him about this. I am much obliged to you for your kind offer and would accept of it if I was to publish this piece at London. I spoke to Col. Middleton and some others of influence here and find they have better hopes of success to . . . Mr Campbell in that Business than you have I think some of his performances deserved to be taken notice of. But as there is an imperfect piece of mine in the transactions for 1726 on the same subject I wish you had rather chose to publish some other of his pieces. I have been at pains to soften some prejudices and Jealousies that may possibly revive by it. It is true I have too long delayed 58 STIRLING'S SCIENTIFIC CORRESPONDENCE publishing the remainder of my piece for which I have only the excuse of much teaching and my design of giving a Treatise of Algebra where I was to treat that subject at large. I told you in my last I had the method of demonstrating that rule by the Limits. In one of my Manuscripts is ye following Article. I. et x n —px n ~ x + qx n - 2 — rx n ~ 3 &c. = be any equation proposed ; deduce from it an Equation for its Limits nx n ~ 1 — 'n - 1 xpx n ~ 2 + n - 2 x qx n ~ 3 &c. = and from this last deduce an equation for its limits ; and by proceeding in this manner you will arrive at the quadratick oi x n— 1 xx 2 — 2 (n— l)|xe+2g = whose roots will be impossible if — — p % be less than q U lb and therefor in that case at least two roots of ye proposed Equation will be impossible. Afterwards I shew that if -- x q 2 be less than pr two roots must be impossible by a quadratick equation deduced a little differently, and so of the other terms. But this matter is so easy I do not think it worth while to contend about it. I have some more concern about a remark I make in my Algebra on the transformation of Equations which has been of great use to me in demon- strating easily many rules in Algebra which I am afraid may be made use of in the paper you have printed because my dictates go through everybody's hands here. The Observation is transform any Equation x s —px 2 + qx - r = to another that shall have its roots less than the values of x by any difference e : Let y — x — e and y 3 + 3 ey 2 + 3 e 2 y + e 3 = where any Coefficient considered —py 2 — 2pey—pe 2 as an Equation gives for its roots + qy+qe the limits of the following — r Coefficient considered as an Equa- CORRESPONDENCE WITH MACLAURIN 59 tion. This holds in Equations of all sorts and from this I demonstrate many rules in a very easy manner. By it too I demonstrate a Theorem in y[our] (?) book where a Quantity is expressed by a series whose coefficients are first, second, third fluxions, &c. I shall be vexed a little if he has taken this from me. Pray let me know if there is any thing of this in the paper you have printed. I intended to have sent you one of my Theorems about the Collision of many Bodys striking one another in different directions in return for your admirable series. But I must leave that to another occasion. I expect to dispose of the six subscriptions I took for Mr De Moivre's Book. Please to give my humble service to Mr Machin and communicate what is above. I long for his new Theory. I am with great Respect Sir Your most Obedient and Humble Servant Colin Maclaurin Edinburgh Dee 1 7 1728. (2) Stirling to Maclaurin, 1728 Sir A few days ago I received your letter of the 7 th of this Moneth and am very glad that your Book is in so great a forwardness, but you have never yet told me in what language it is, altho at the same time I question not but it is in Latine. I should be very glad to see what you have done, and since you mention sending a Copy, you may send it under Cover to Mr Cuninghame of Balghane ; if I can do you any service as to getting Mr Grahams Experiment I wish you would let me know, I question not but that you may have liberty to print it, because probably it will be in our Transactions very soon. I am very glad that Coll. Midleton gives Mr Campbel encouragement to come to London, no doubt but bread might 60 STIRLING'S SCIENTIFIC CORRESPONDENCE be made by private teaching if a man had a right way of mak[ing himself] known, but indeed I [ques]tion'if Mr Campbel will not want a prompter in that p . I am apt to thi[nk that I ha]ve not given you a distinct account of his paper about in [ ] * because you se[em to thi]nk that I choose it out of a great many others to be printed [ ] which indeed would not have been so very candid before you had leasure to compleat your paper. But the Matter is quite otherways. For as soon as your paper was printed, Mr Campbel sent up his directly to Mr Machine, who at that time being very busy, delayed presenting it to the Society because the Correcting of Press would divert him from prosecuting his Theory of the Moon. Upon this delay Sir Alex. Cuming complained grieveously to Mr Machine that Mr Campbel was ill used, this made Mr Machine present it to the Society, upon which it was ordered to be printed, Mr Machine came to me and desired I would take the trouble of correcting it in the Press, which was all the Concern I had in it. And now I hope you are convinced that I did no more than yourself would have done had you been asked. Mr Campbels Method is grounded on the following observation. Let there be two equations x b + Ax*" + Bx 3 + Cx 2 + Dx + E — and Ez s + Dz i + Cz 3 + Bz i + Az + 1 = 0, where the reciprocals of the Roots of the one are the Roots of the other, then it is plain that the Roots in both are the same as to possibility and impossibility. He deduces from each of those a Quadratick Equation for the limits the common way, and on that founds his Demonstration. But he doth not use that property of equations which you have been pleased to communicate, indeed it is very simple and I can see at once what great use can be made of it, I had observed that the last Term but one gave the Fluxion of the equation, but never any further before you mentioned it. But Mr Campbell besides demonstrating Sir Isaac Rule [ ] one of his own more general, he exempli- fies it by an equation of 7 dimen[ ]ich his Rule discovers to have 6 impossible Roots, whereas S r Isaac's disco[ ]ly two of the Six. [I] shal now make a remark on some of those Gentlemen who dispute for the new [n]otion of Fox'ce to shew how 1 Impossible roots (?). CORRESPONDENCE WITH MACLAURIN 61 much they depend one anothers demonstrations which are to convince their Adversarys. Herman in his book page 113, I mean his Phoronomia, says In hac virium sestimatione, prseeuntem habemus Illustrissimum Leibnitium, qui eundem non uno loco in Actis eruditorum Leipsise indicavit quidem non tamen demonstravit, etsi apodictice demonstrari potest, ut forte alia id occasione ostendemus — He denys then that his friend Leibnitz ever did demonstrate it, but owns that it may be done and is in hope one time or other to do it himself. Poleni in his Book de Castellis page 49 tells us that Leibnitz demonstration was published ; and page 52 he mentions Bernoulli demonstration [ ] as published in Wolfius. And page 53 [ ] that perhaps some and those not the most scrupulous might doubt [ J Leibnitz's and Bernoullis demonstrations, and then page 61 he tells — is meaning in plain words, Demonstrationem inventam fuisse reor non tamen editam. So that it is very remarkable that a certain number of men should run into an opinion ; and all of them deny one another's proofs. For Herman denys Leibnitz demonstration, and Poleni denys all that ever were given, and declares further that he knows not possibly on what principles one should proceed in such a Demonstration, but at the same time, he resolves to be of the opinion : whether it be proved or not. But no doubt you have observed many more of their Absurdities as well as this. I have not seen Mr Machin since I got your letter, but shal carry him your complements, I am afraid it will be long before wee see his Theory, for Mr Hadly and he do not agree about some part of it. We expect in the first Transaction Mr Bradley's account of the new motion observed in the fixt Stars. I wish you good success, and hope to see your book soon, I am with all respect Sir London Your most obedient 31 December humble servant 1728 James Stirlinq 62 STIRLING'S SCIENTIFIC CORRESPONDENCE (3) Maclaurin to Stirling, 1729 ^ Mr James Stirling at the Academy in little Tower Street London. Sir Last tuesday night I saw the philosophical Transactions for the month of October for the first time. You may remember I wrote to you some time ago wishing some of Mr Campbell's papers might be taken notice of. I did not indeed then know that Mr Machin had any paper of his on the impossible roots. But even when I heard of it from you I was not much concerned because from a conversation with the Author on the street I concluded his method was from the equations for the Limits and never suspected that he had followed the very track which I had mark'd out in my paper in the transactions for May 1726 from the principle that the squares of the differences of Quantities are always positive as he has done in the latter part of this paper. As I never suspected that he had followed that Method I had no suspicion that he would prevent me in a Theorem that can be only obtained that way but cannot be overlooked in following that track. I cannot therefor but be a little concerned that after I had given the principles of my method and carried it some length and had it marked that my paper was to be continued another pursuing the very same thought should be published in the intervall ; at least I might have been acquainted that I might have sent the continuation of mine before the other was published. You would easily see that the latter part of Mr Campbell's paper after he has done with the limits is the very continuation of my theorems if you had the demonstrations. Let there be any Equation x n _ Ax 11 - 1 + Bx n ~ 2 - Gx n ~ 3 + Bx n ~ i -Ex n ' b + Fx n ~ 6 - 0x n ~'' + Hx n ~ 8 - Ix n - 9 + Kx n - 10 - /.x n ~ n + Mx n ~ 12 &c. = 1 1728 O.S. ; but 1729 N.S., cf. Letter L. CORRESPONDENCE WITH MACLAURIN 63 1)? — 1 and x D 2 will always exceed EG - FB + GA - H 2 m. J . „ tt-1 u-2 u-3 . it m = n x — - — x x &c. 2 3 4 till you have as many factors as there are terms in the Equation preceeding D. I have had this Theorem by me of a long time : and it easily arises from my Lemmata premised to my paper in the Trans- actions for May 1726. An abridgment of my demonstration as I have it in a book full of Calculs on these subjects is as follows. The square of the coefficient of D consists of the squares of its parts and of the double products of those parts multiplyed into each other. Call the sum of the first of these P the sum of the products Q and D 2 = P + 2Q. Now the number of those parts is m and therfor by the 4 th Lemma of the paper in the transactions for May 1726 (m— 1) P must be greater than 2 Q and D 2 ( = P + 2 Q) must be greater than Q 9? v ~~— 1 or — D 2 greater than Q. Then I shew that 2m 8 ^ Q=EC-FB + GA-H and thence conclude that — — D 2 always exceeds 2 m J EC-FB + GA-H when the roots of the equation are all real. I have a general Theoreme by which I am enabled to compare any products of coefficients with any other products of the same dimensions or with the Sums and Differences of any such products which to shew you how much I have considered this subject tho' I have been prevented when I thought myself very secure I now give you. Let E and H be any two coefficients and m the number of Terms from E to H including both then shall t?v d , — rr n , m + 2 ^ + 3^,^1+4 m+5 m + 6 EH=P + m+\Q + — — B + — — S m+7 m + 8 m + 9 m + 10 _ . + -j- — — — T &c. where P expresses the squares of the parts of E multiplyed 64 STIRLING'S SCIENTIFIC CORRESPONDENCE by the dissimilar parts of C. (a term as far distant from the beginning of the Equation as H is from E) Q expresses the squares of the parts of the coefficient immediately preceding E viz. I) multiplyed by the dissimilar parts of the term next following C but one viz. in this case E itself. M expresses the squares of the parts of the coefficient next preceding E but one that is G multiplyed by the dissimilar parts of the Term next following C but three viz. G ; and so on. Where I mean by the parts of a coefficient the terms that according to the common Genesis of Equations produce it ; and by dissimilar parts those that involve not the same Quantitys. This general Theorem opens to me a vast variety of Theorems for comparing the products or squares of coefficients with one another of which those hitherto published ave only particular Examples. Here I give you the theorem for comparing any two products of the same dimensions as EI and GL. Let s and m express the number of terms that preceed C and I in the Equation then let 71-1 71 — s — 1 11 — s — 2 „ p = x — - x &c. 1 8+1 s + 2 8 + 3 , 71 — 771 n—m—1 71 — m — 2 „ and q = x — x — &c. 1 m + 1 TTi + 2 m+3 continued in each till you have as many factors as there are terms from to E including one of them only; then shall - x EI alwav exceed CL when the roots are all real. P Then I proceed to compare the products of the Coefficients with the sums or differences of other products & one of the chief Theorems in that part is that mentioned above which Mr Campbell also found by the same method as is very apparent and could not miss in following the track I mark'd out in the transactions. I had observed that my rules gave often impossible roots in the Equations when Sir Isaac's did not in proof of which I faithfully transcribe from my Manuscript the following Article. ' In the Equation x s -Ax i + Bx 3 -Cx* + Dx-E = a) 5_ 10a ,4 +30a ,3_4 4a ,2 + 32 a ;_9 = o CORRESPONDENCE WITH MACLAURIN 65 no impossible roots appear by Sir Isaac's rule. But J5 2 x — here is less than AG — J) for it — 1 „ 4 1 m — 1 9 m = ;(, x — - — = 5 x - — 10 and = — 2 2 2m 20 now 2 9 o x 30 x 30 is less than 44 x 10 — 32 the first being 405 the latter 408 so that there must be impossible roots by our rule.' After that I give other Examples I believe you will easily allow I could not have invented these Theorems since tuesday last especially when at present by teaching six hours daily I have little relish left for such investigations. I showed too my theorems to some persons, who can witness for me. But I am afraid these things are not worthy your attention. Only as these things once cost me some pains I cannot but with some regret see myself prevented. However I think I can do myself sufficient justice by the length I have carried the subject beyond what it is in the transactions. I believe you will not find that Mr Campbell sent up his paper or at least the latter part of it so soon after I sent up mine which was in the beginning of 1726. One reason I have is that Mr Machin never mentioned it to me tho' I spent a whole day with him in September 1727 and talked to him on this subject and saw some other papers of Mr Campbell's in his hand at that time. So that I have ground to think that the paper of May 1726 led the Author into the latter part of his for October 1728. When I was with Mr Machin in September 1727 I then had not found a sufficient demonstration for the cases of Sir Isac's rule when there may be six or seven impossible roots arising by it. This part is entirely overlooked by this Author: for all he demonstrates amounts only to some pro- perties of Equations that have all their roots real ; from which he says indeed all S r Isac's rule immediately follows. But I conclude from thence that he did not try to demonstrate compleatly Sir Isac's rule. If be had tryed it new difficultys would have arisen which he has not thought of. The way he has taken to demonstrate Sir Isac's numbers 66 STIRLING'S SCIENTIFIC CORRESPONDENCE from the Limits is not so simple as that I have which I may send you again. I now beg pardon for this long letter which I beg you would communicate to Mr Machin not by way of complaint against him for whom I have more respect than for any Mathematician whatsoever ; but to do me justice in the matter of these impossible roots which I had thrown aside for some time and have now taken up with regret. I would have justice done me without disputing or displeasing anybody. At any [rate] in a few days I shall be very easy about the whole Matter. I am with the greatest Respect Sir Your Most Obedient Affectionat Humble Servant Edinburgh Colin Maclaurin febr. 11. 1728 Having room I send you here one of my Theorems about the Collision of Bodys. Let the Body G moving in the direction CD strike any number of Bodys of any magnitude A, B, E, F, &c. and make Fig. 4. them move in the lines Ga, Gb, Ce, Gf &c. to determine ye direction of C itself after the stroke. CORRESPONDENCE WITH MACLAURIN 67 Suppose Ba, Bb, Be, Bf &c. perpendicular to the directions CA, CB, CE, CF, &c. Imagine the Bodys G, A, B, E, F &c. to be placed in G, a, b, e, f &c. respectively ; find the centre of Gravity of all those Bodys so placed and let it be P. Draw BP and CG parallel to BP shall be ye direction of G after the stroke if the Bodys are perfectly hard. Adieu (4) Madaurin to Stirling, 1729 Mr James Stirling at the Academy in little Tower Street London Sir I delayed answering your last letter till I could tell you that now I have sent Mr Folkes the remainder of my paper concerning the impossible Roots of Equations. I sent him a part April 19 and the remainder last post. I thought to have finished it in our Vacation in March but a Gentleman compelled me to go to the Country with him all that time where we had nothing but diversions of one sort or other, so that I did not get looking into it once. However I am satisfyed that any person who will read this paper and compare it with Mr Campbell's will do me Justice. On comparing them further myself I (find) he has prevented me in one proposition only ; which I have stated without naming or citing him or his paper to be the least valuable. For I shew that some other rules I have deduced from my Theorems always discover impossible roots in an Equation when his rule discovers any, and often when his discovers none. I wish you could find time to read both the papers. I am sorry to find you so uneasy about what has happened in your last letter. It is over with me. When I found one of my Propositions in his paper I was at first a little in pain ; but when I found it was only one of a great many of mine f2 68 STIRLING'S SCIENTIFIC CORRESPONDENCE that he bad hit upon ; and reflected that the generality of my Theorems would satisfy any judicious reader; I became less concerned. All I now desyre is to have my paper or at least the first part of it published as soon as possible. I beg you may put Mr Machin in mind of this. I doubt not but you and he will do what you can to have this Justice done me. I could not but send the second part to Mr Folkes having sent him the first. I have at the end of my paper given some observations on Equations for the sake of those who may think the impossible roots may not deserve all this trouble. Mr Folkes will shew you the paper. I intend now to set about the Collisions of Bodys. The Proposition I sent you in my last letter is the foundation f all my Theorems about the impossible Roots. I have a little altered the form of it. It is the VI Proposition as I have sent them to Mr Folkes the first five having been given in 1726. I have made all my Theorems as I went over them last and transcribed them more simple than they were in my manuscripts ; and that occasioned this little delay : for your advice about sending up my paper soon perfectly pleased me. Abridgments and Additions that occurred as I transcribed it took up my time but it was about the third or fourth of April before I got beginning to it in earnest, and my teach- ing in the Colledge continuing still as before with other avocations ; you will allow I have not lost time. I have a particular sense of the Justice and kindness you have showed me in your last letter & will not forget it if 1 ever have any opportunity of showing with how much Esteem & affection I am Sir Your Most Obedient Humble Servant Colin MacLauuin Edinburgh May 1 1729 CORRESPONDENCE WITH MACLAURIN 69 (3) Maclaurin to Stirling, 1729 Mr James Stirling at the Academy in little Tower Street London Sir Since I received your last I have been mostly in the country. On my return I was surprised with a printed piece from Mr Campbell against me which the gentleman who franked the letter told me lie sent you a copy off. The Gentleman indeed added he had not frank'd it if he had known the nature of the paper ; and was ashamed of it. I wonder I had no message by a good hand from Mr Campbell before he printed these silly reports he diverts himself with. Good manners and prudence one would think ought to have led to another sort of conduct. He has misrepresented my paper much and found things in it I never asserted. I shall send you next post a fuller answer to it. His friends here give out that you desyred him to write against me. I am convinced this is false. Please to send me the letter I wrote to you in february if you have preserved it or a copy of it. I wish if it is not too much trouble you would send me a copy of all I said relating to Mr Campbell's taking the hint from my first paper in my letters to you. I wish you would allow me (if I print any defence) to publish your letter to me of the date of febr. 2 7 where you have expressed yourself very cautiously. But I will not do it without your permission. I hope the paper Mr Campbell has sent you will have little influence on you till you hear my reply. I have writ at large to Mr Folkes by this post who will show you my letter if you please. I assure you I am with great Esteem Sir Your Most Obedient Edinburgh Most Humble Servant nov r 6. 1729 Colin MacLaurin 70 STIRLING'S SCIENTIFIC CORRESPONDENCE (6) Stirling to Maclaurin, 1729 To Mr Maclaurin Professor of Mathematicks in the University of Edenburgh Out of your Letter of October 22, 1728 I have other ways of demonstrating the Rule about impossible roots & particularly one that was suggested to me from reading your book in 1718 drawn from the limits of Equations shorter than the one I have published, but according to my taste not so elegant. Out of Letter of December 7, 1728 Let x n — px n ~ 1 + qx n ~ 2 — rx n ' 3 &c. = 0, be any Equation proposed, deduce from it an Equation for its Limits nx n l —n — 1 xpx n ~ 2 + -it,— 2 x qx n ~' d &c. = 0; By it too I demonstrate a Theoreme in your book where a quantity is expresst by a Series whose coefficients are first, second, third fluxions &c. A Copy of your Letter Feb 11, 172|. S r Last Tuesday night I saw the philosophical Transactions for the month of October for the first time. At any rate in a few days I shall be very easy about the whole matter. I am &c. S r This is an exact copy except the postscript which containing a Theoreme about the collision of Bodys I presume is nothing to the present purpose. I am with all respect Sr Your most humble servant Ja: Stirling London 29 November 1729 CORRESPONDENCE WITH MACLAURIN 71 Maclaurin to Stirling Dear Sir I send you with this letter my answer to Mr George Campbell which I publish with regret being so far from delighting in such a difference that I have the greatest dislike at a publick dispute of this Nature. At the same time that I own this Aversion I can assure you it Hows not from any Consciousness of any other wrong I have done this Author than that I accepted of a settlement here that was proposed to me when some persons at Aberdeen were persecuting me and when a settlement here every way made me easy ; at the same time that he had some hopes tho' uncertain in a course of years of getting the same place. I was sensible however of this and therefor made it my great Concern to see him settled ever since I have been in this place, nay after my business had proceeded so well that it was indifferent to me whether he continued here or not in respect of Interest. However I have avoided everything that might seem writ in his strain and have left out many things lest they might look too strong, particularly in citing Mr Folkes's letter I left out his words that Mr Campbell's paper was writ with the greatest passion and partiality to himself, as you will see. I sent the first sheet in Manuscript to have been communicated to you above a fortnight ago by Mr Folkes that you might let me know if you desyred to have anything changed and have delaj'cd the publication till I thought there was time for an Answer to come to me. I have printed but a few Copys intending only to take of as much (without hurting him) * the Impression he endeavours to make as possible. It was to avoid little skirmishing that I have not followed him from page to page — but refuted the essentials of his piece, overlooking his Imaginations and Strictures upon them. I am at present in haste having several other letters to write on this subject. I avoid things together towards the 1 Written above the line. 72 STIRLING'S SCIENTIFIC CORRESPONDENCE end because it was like to have required another half-sheet. I am sure I have given more than the subject deserves. I have left out two or three paragraphs about his inconsistencys his story of some that visited me and found me so and so engaged &c. This I answer in my manuscript letter sent to you, Nov. 5. I am indeed tyred with this affair. I wished to have heard from you what he objected to the letter I wrote to you in the beginning of winter. I am truly sorry Mr Campbell has acted the part he has pleased to act. But my defence is in such terms after all his bad u?age of me as I believe to his own friends will shew I have no design to do him wrong and have been forced into this ungrateful part. It is true he speaks the same language ; with what ground let the most partial of his friends judge from what I have said in my defence. You may remember that my desyre of doing him service was what began our correspondence. I then could not have imagined what has happened. Please to forgive all the trouble I have given you on this Occasion and believe me to be Sir Your Most Obedient Humble Servant Colin Mac Lauein If you see Mr de Moivre soon, please to tell him I send him by this post a bill for six guineas and a letter directed to Slaughter's Coffee House. I did not know where else to direct for him. (8) Gray to Maclaurin, 1732 London 25 Novem r 1732 Dear Sir I had the favour of yours yesterday & inclosed a part of the abstract of your Supplement with a Letter to Mr Machin, which, as you desired, I copyed & gave to him. He is of opinion that it will be improper to put any part of your Abstract into our Abrigment, especially as matters stand. He will take care to do you all the justice he can and desires CORRESPONDENCE WITH MACLAURIN 73 his kind services to you. I am thinking that it will not be improper to move the Society at their first meeting that Stirling be in Hodgson's room ; because he is much more capable of judging than him ; but in this I will follow Mr Machin's advice. I hope you had my last, and am persuaded you will do in that affair what is fit. I have a great deal of business to do this evening. I will therefore only assure you that I am most faithfully Dear Sir Your most obedient & most humble Servant Jno Gray (9) Madaurin to Stirling, 1734 To Mr James Stirling at Mr Watt Academy in little Tower Street London Sir I was sorry on several accounts that I did not see you again before you left this Country. Being in the Country your letter about the Variation did not come to my hand till the time you said you had fix'd for your journey was so near that I thought a letter could not find you at Calder. I have observed it since I came to Town & found it betwixt 12 & 13 degrees westerly; the same had appeared in April last. But I am to take some more pains upon it which if necessary I shall communicate. Upon more consideration I did not think it best to write an answer to Dean Berkeley but to write a treatise of fluxions which might answer the purpose and be useful to my scholars. I intend that it shall be kid before you as soon as I shall send two or three sheets more of it to Mr Warrender that I may have your judgment of it with all openness & liberty. This 74 STIRLING'S SCIENTIFIC CORRESPONDENCE favour I am the rather obliged to ask of you that I had no body to examine it here before I sent it up on whose judgment I could perfectly depend. Robt. Simpson is lazy you know and perhaps has not considered that subject so much as some others. But I can entirely depend on your judgment. I am not at present inclined to put my name to it. Amongst other reasons there is one that in my writings in my younger years I have not perhaps come up to that accuracy which I may seem to require here. When I was very young I was an admirer too of infinites ; and it was Fontenelle's piece that gave me a disgust of them or at least confirmed it together with reading some of the Antients more carefully than I had done in my younger years. I have some thoughts in order to make this little treatise more compleat to endeavour to make some of Mr De Moivre's theorems more easy which I hope he will not take amiss as I intend to name everybody without naming myself. I have got some few promises as to Mr Machin's book and one of my correspondents writes me that he has got two subscriptions. I wonder at Dr Smith's obstinate delay which deprives me of the power of serving Mr Machin as yet so much as I desyre to do. It is from a certain number of hands that I get subscriptions of this kind. Pemberton's book and the Doctor's delay diminish my influence in that very much. Looking over some letters I observed the other day that you had once wrote to me you had got a copy from Mr Machin of the little piece he had printed on the Moon for me. If you can recollect to whom you sent it let me know ; for it never came to my hand ; and I know not how to get it here. Nor did the Copy of your treatise of Series come to my hand. You need not be uneasy at this : Only let me know what you can recollect about them. If Mr Machin's book happens to be published soon you may always venture to sett me down for seven Copys. But I hope to gett more if I had once fairly delivered Dr Smith's book to the subscribers. As to your Treatise of Series 1 got a copy sent me from one Stewart a Bookseller as a new book but about half a year after his son sent me a note of my being due half a guinea for it which I payed. But as I said I only mention these things in case you can recollect any thing further about them. CORRESPONDENCE WITH MACLAURIN 75 I observe in our newspapers that Dr Halley has found the longitude. I shall be glad to know if there is any more in this than what was commonly talk'd when I was in London in 1732. Please to give my humble service to Mr Machin and believe me to be very affectionatly Sir Your Most Obedient Edinburgh Most Humble Servant Nov. 16. 1734. Colin MacLaurin I have taken the liberty to desyre- Mr Warrender to take advice with you if any difficultys arise about the publishing the fluxions or the terms with a Bookseller. I would have given you more trouble perhaps but he was on some terms with me before you got to London. (10) Maclaurin to Stirling, 1738 l To Mr James Stirling at Leadhills Dear Sir This is a copy of Maupertuis's letter which I thought it would be acceptable to j ou to receive. I am told Mr Cassini would willingly find some fault with the Observation to save his father's doctrine, but is so much at a loss that he is obliged to suppose the instrument was twice disordered. If I can be of any service to you here in anything you may always command Dear Sir Your Most Obedient Humble Servant Ed r . feb. r 4. 1737. Colin Mac Laurin I forgot when you w r as here to tell you that last spring 1 1737 O.S. or 1738 N.S. 76 STIRLING'S SCIENTIFIC CORRESPONDENCE some Gentlemen had formed a design of a philosophical society here which they imagined might promote a spirit for natural knowledge in this country, that you was one of the members first thought of, and that Ld Hope & I were desyred to speak to you of it. I hope and intreat you will accept. The number is limited to 45, of which are L ds Morton, Hope, Elphinston, St Clair, Lauderdale, Stormont, L d president & Minto, S r John Clark, D ra Clark, Stevenson, St Clair, Pringle, Johnston, Simpson, Martin, Mess. Munroe, Craw, Short, Mr Will™ Carmichael &c. I shall write you a fuller account afterwards if you will allow me to tell them that you are willing to be of the number. If you would send us anything it would be most acceptable to them all & particularly to yours &c I had a letter from Mr De Moivre where he desyres to give his humble service to you. His book was to be out last week. Maupertuis to Bradley A letter from Mons r Maupertuis To Professor Bradley Dated at Paris Sepf 27^ h 1737 N.S. [Translated from the French] Sir The Rank You hold among the Learned & the great Discoveries with which you have enriched Astronomy, would oblige me to give you an Account of the Success of an Under- taking, which is of considerable consequence to Sciences (even tho' 1 were not moved to do it by my desire of having the honour to be known to you) by reason of the Share you have in the Work itself. Whereof a great part of the Exactitude is owing to an Instrument made on the Modell of yours, and towards the Construction of which I know you were pleased to lend your Assistance. Wherefore I have the honour to Acquaint You Sir, That we are now returned from the Voyage we have made by Order of His Majesty to the Poler Circle. We have been so happy as CORRESPONDENCE WITH MACLAURIN 77 to be able, notwithstanding the Severity of that Climate, to measure from Tornea northward a Distance of 55023-47 Toises on the Meridian. "We had this distance by a Basis the longest that ever has been made use of in this Sort of Work, & measured on the most level surface, that is, on the Ice, taken in the middle of eight Triangles. And the small number of these Triangles, together with the Situation of this great Basis in the Midst of them, Seem to promise us a great Degree of Exactness ; And leave us no room to apprehend any con- siderable Accumulation of Mistakes ; As it is to be feared in a Series of a greater Number of Triangles. We afterwards determined the Amplitude of this Arch by the Starr S Drucoids, Which we observed at each end with the Sector you are Acquainted with. This Starr was first observed over Kittis, one of the Ends, on the 4, 5, 6, 8. 10 of October 1736. And then we immediately carried our Sector by Water to Tornea, with all the precaution requisite its being any way put out of Order, And we observed the same Starr at Tornea the 1. 2, 3, 4 & 5, of Nov r 1730. By comparing these two Setts of Observations we found, That the Amplitude of our Arch (without making any other Correction than that which The procession of the Equinox requires) would be 57'-25"07. But upon making the necessary Correction according to your fine Theory (Parallax of Light) of the Aberration caused by the Motion of Light, This Amplitude by reason of the interval of Time between the Mean of the Observations, was greater by l"-83 : & consequently our Amplitude was 57'.27"-9. We were immediately Sensible that a Degree on the Meridian under the Polar Circle was much greater than that which had been formerly measured near Paris. In Spring of the ensuing Year we Recommenced this whole operation. At Tornea we observed Alpha Draconis on the 17, 18, & 19 of March 1737; and Afterwards set out for Kittis, Our Sector was this time drawn in a Sledge on the Snow, and went but a slow pace. We observed the Same Starr on the 4, 5 & 6 of Aprile 1737. By the Observations made at Tornea & Kittis we had 57'-25"-19; to Which Adding 5"-35 for the Aberration of this Starr during the time elapsed between the Middle of the Observations, we found for the Amplitude 78 STIRLING'S SCIENTIFIC CORRESPONDENCE of our Arch 57'-30"-54 which differs 3"§ from the Amplitude determined by S (Delta). Therefore taking a Mean between these two amplitudes, Our Arch will be 57'-28"-72 which being compared with the distance measured on the Earth, gives the Degree 57437-1 Toises; greater by 377-1 Toises than the Middle Degree of France. We looked upon the Verification which results from the Agreement between our two Amplitudes deduced from two different (Setts of) Operations (Joined to the precautions we had taken in the Carriage of the Sector) We looked (I say) upon this Verification to be more certain than any other that could be made ; and the more because our Instrument cannot from its Construction serve to be turned Contrary Ways. And that it was not requisite for our operation to know precisely the point of the Limb which answered to the Zenith. We verified the Arch of our Instrument to be 15°-§ by a Radius of 380 Toises, and a Tangent both measured on the Ice: and notwithstanding the great Opinion we had of Mr Graham's Abilities we were astonished to see, that upon taking the Mean of the Observations made by 5 Observers which agreed very well together ; The Arch of the Limb differed but 1" from what it ought to be According to the Construction. In fine, we Compard the degrees of the Limb with one Another, and were surprized to find that between the two Degrees which we had made use of, there is a Small Inequality, Which does not amount to l", & Which draws the two Amplitudes, we had found, Still nearer one Another. Thus, Sir, You See the Earth is Oblate, according to the Actual Measurements, as it has been already [found] by the Laws of Staticks : and this flatness appears even more considerable than Sir Isaac Newton thought it. I'm likewise of Opinion, both from the experiments we Made in the frigid Zone, & by those Which our Academicians sent us from their Expedition to the Equator ; that Gravity increaseth more towards the Pole, and diminishes more towards the Line, than Sir Isaac suppos'd it in his Table. And this is all conformable to the Remarks you made on Mr Campbell's Experiments at Jamaica. But I have one CORRESPONDENCE WITH MACLAURIN 79 favour to beg of you, Sir, & hope you will not refuse it me ; Which is, to let me know if you have any immediate Observa- tions on the Aberration of our two Starrs 8 ■ = CB and GT = GA and PT shall be the velocity of y u Body A, and PS the velocity of the Body B after y° Concussion. If they are imperfectly Elastic, take Cs to GS and Ct to GT as y 6 elasticity to the perfect elasticity and Ct, Gs shall be the velocitys of the Bodies A and B. In his opinion about the forces of the Bodies, this construction is very commodious, for before the percussion ALM represents the force of y° Body A, and BLN the force of y 6 Body B. But after y e percussion CTM and GX are the forces of the bodies A and B, if they are elastic, and CQM CQX are these forces if they are not elastic, and ACB is the force lost in y° percussion M r 'S Gravesande demonstrates it, by this proposition, That y° instantaneous mutations of forces in the two bodies, are proportional to their respective velocities. But I found that mi H 98 STIRLING'S SCIENTIFIC CORRESPONDENCE it cou'd be proved, without the new notion of forces, by this proposition. That y" contemporaneous mutations of velocities of the two bodies are reciprocal to their masses wich can be evinc'd in several manners, and very easily, if granted that the common center of gravity does not alter its velocity by the percussion. • I am just arrived at Paris, and so have no news from france to impart with ye. You'll oblige me very much, if you vouch- safe to answer to this, and inform me about your occupation and those of your Royal Society and its learned members. Did M r Machin publish his Treatise about y e Theory of y e Moon 1 Is M r de Moivre's Book ready to be published ? Is there nothing under the press of S r Isaac's remains'? What are you about? Can we flatter ourselves of the hopes of seeing very soon your learned work about y 9 Series'? All these and other news of that kind, if there are some, will be very acceptable to me ; and I'll neglect nothing for being able of returning you the like, as much as the sterility of the country I live in, and my own incapacity will allow. In the meanwhile, I desire you to be lofty persuaded, I am, with all esteem and consideration Tour most; humble Most obedient Servant Paris, tliis hh X bre 1723 G. Cramek You can direct y e Answer A Messieurs Rilliet iv Delavine, rue Grenier S l Lazare pour rendre a M r Cramer a Paris. Cramer to Stirling, 17:29 To M r James Stirling F R.S. at the Academy in little Tower Street London Here is, Dear Sir, a Letter from M r Nioh. Bernoulli in answer to yours, wich I received but t'other day. I send with it, CORRESPONDENCE WITH CRAMER 99 according to his Orders a Copy of his method of resolving y° quantity — in its component fractions the former 1 + qZ ' ^ part of wich he sent me to Paris, by M 1 ' Klingenstiern the supplement I had but in the same time with your Letter. I hope you have lately received from me an answer to your kind Letter brought by M r Sinclair. I am with a great esteem Your most humble and obedient Servant Geneva the 6 th January, 1729. N.S. G. Cramer. Methodus resolvendi quantitates 1 + qz H + z in in factores duarum Dimensionum, Auctore D°. Nicolao Bernoulli. Prob. I Resolvere quantitatem 1 ±qz 1l + z 2n in factores duarum Dimensionum. Solut. Sit unus ex factoribus 1 — xz + zz & productum reliquorum l+uz + bz 2 + cz\ . . + rz n ~ 3 + sz n ~ 2 + tz n ~ l + sz 11 + rz n+1 .. . + hz 211 '*- + az 2 n ~ 3 + s 2 " -2 . Ex comparatione terminorum homogeneorum producti horum factorum cum terminis propositae quantitatis invenitur a = x, b — ax—\, c— bx — a & ita porrho usque ad t = sx — r, item +q = 2a — tx, adeo ut quantitates 1, a, b, c, ... r, s, t con- stituant Seriem recurrentem in qua quilibet terminus per x multiplicatus est aequalis Summae praecedentis it sequentis. Jam vero si Chorda complementi BD alicujus arcus AD vocetur x & ladius AC = 1 Chordae arcuum multiplorum ejusdem arcus AD exprimentur respec- tive per eosdem terininos inventae Seriei recurrentis 1, a, b, c, &c. multi- plicatos per Choi'dam AD. Hinc si arcus AE sit ad arcum AD ut n ad 1, erit Chorda AE ad Chordam AD ut / ad 1, id est AE = t x AD, & Chorda DE— s x AD. Ex natura vero quadrilateri ADEB h2 100 STIRLING'S SCIENTIFIC CORRESPONDENCE circulo inscripti est AE . DB — AB . DE+AD . BE id est tx.AD = 2ts.AD + AD.BE sive tx = 2 s + BE = (quia + q = 2s- tx) tx±q + BE, hinc BE = +q. Ex quo sequitur quod si arcus habens pro Chorda complementi + q dividatur in n partes aequales quarum una sit arcus AD, hujus complementi Chorda futura sit x : vel si rem per Sinus conficere malimus, dividendus est arcus habens pro Cosinu + ^ q in n partes aequales, qui si vocetur A, erit cosinus arcus A 1 — = - x Invento valore ipsius x cognoscitur l—xz + zz unus n 2 ex factoribus quantitatis propositae 1 + qz n + z 2n . Sed & re- liqui factores hinc cognoscuntur. Si enim tota circumferentia vocetur C, habebunt omnes sequentes arcus A, C—A, C + A, 20- A, 2(7+ A, 3C—A, 3C+A, &c pro Cosinu +\q, quorum singuli in partes aequales divisi determinabunt totidem diversos valores ipsius x. Coroll. 1. Per methodum serierum recurrentium invenitur x = radici hujus aequationis Vqq-4: = (±x+ ■J\xx-\) n -(±x- J\xx-\) n Coroll 2. Si capiatur arcus AH aequalis alicui sequentium A C-A C+A 20- A 2C+ A arcuum &c & fuerit n n n n n CG = z erit GH — radici quadratae f actoris \—xz + zz. Quia enim OF '= \x erit GF — \x — z, proinde GH FH=J\-\j? & = Vl-xz + zz Coroll 3. Si q = 0, erit A = iC, & reliqui arcus dividendi |0, |(7, \0, %C &c. Hinc si dividatur tota cir- cumferentia in 4d partes aequales AH, HI, IK, &c & ad singulos im- pares divisionis terminos H, K, M, &c. ex puncto G ducantur rectae GH, GK, &c erit horum omnium productum \+z zn . CORRESPONDENCE WITH CRAMER 101 Probl. II Resolvere quantitatem 1 + z 2n+l in factores duarum Dimen- sionum. Solut. Sit unus ex factoribus 1—xz + zz, & productum reliquorum 1 + «z + bz 1 + cs 3 . . . rz n ' s + sz n ~ 2 + tz n ~ x + tz n + sz n+1 + rz n+2 . . . + bz 2n ~ 3 + as- n ~ i + z 2n ~K & invenitur ut antea a — x, b = ax—l, c=bx — a, & ifca porrho usque ad t — sx — r. Sed loco aequationis + q = 2 s - tx invenietur haec t = ix + s — 0. id est, si ponatur arcus AB ad A F T)F arcum AE, ut 1 ad n, erit (quia t = --= & s = ^-, & x = BD) A-D A.1) AE-AE.BB + BE=0. Sive BE = AE . BD-AE & aequatione in analogiam versa BE : A E = BB- 1 : 1 = (facta BF = BC = A C = 1) BF: CB. Hinc triangula ABE, CFB, ob angulos ad E & B aequales, erunt similia & angulus BCF = BAE. Ergo ang. BCF + wig GBF = ang BFG = ang BCF = ang BAE+ ang CBF Sed & ang GBF = ang CBi? 7 . Hinc omnes tres anguli Trianguli GBF sunt aequales 2 ang. BAE +3 ang CBi^ ipsorum que men&ura, id est, semicircumferentia = \C = arc DE+% arc. AB = (quia arc BE = n ~ 1 arc ^ D) — - — arc AB. G ideoque arcus AD — . Si imtur circumferentia Circuli H 2m +1 & dividatur in 2n+ 1 partes aequales, quarum una sit arcus AD, erit Chorda BD = x, vel si semicircumferentia in totidem partes aequales dividatur, erit cosinus unius partis \x uncle cognoscetur factor 1 — xz + zz. Quia vero tot factores duarum dimensionum inveniendi sunt quot imitates sunt in numero n, habebit totidem diversos valores qui erunt dupli cosinus 1, 3, 5, 7 iVc partium semicircumferentiae in 2n+ 1 partes aequales divisae : invenitur enim arcus AB = singulis sequentibus arcubus -j -■> - » -> Arc, quia arcus AE 2u+l 2ti+1 2h. + 1 2h+1 ,H 102 STIRLING'S SCIENTIFIC CORRESPONDENCE qui est ad arcum AD, ut n ad 1. potest intelligi auctus integra Circumferentia vel ejus multiplo, hoc modo igitur resolvetur quantitas proposita l+z in+1 in n J£ L factores duarum dimensionum & H unum factorem 1 + unius dimen- sionis. Coroll. Si fuerit CO = z, AC=CB=.l & Circumferentia circuli dividatur in 4/i+2 partes aequales AH, HI, IK, &c ad singulosimpares divisionis terminos H, K, M, &c ducantur rectae GH, OK, OM, &c, erit horum omnium productum aequale 1+Z 2 » +1 . Probl. Ill Refolvere quantitatem 1 — z 2n+1 in factores duarum Dimensionum. Solut. Sit unus ex factoribus 1— xz + zz & productum reliquorum l+az + bz 2 + ...+rz n - 3 + s: n -' i + tz n - 1 -tz n -tz n+l -r2 n+2 ... _] iz 2n-i_ az 2n-2_ e 2n-l & invenietur s = t + tx : reliqua vero se habent ut prius. Positis igitur ut in Prob II arcu ABBE = n arc AD, x = BD, t=-? s = ~, eritM= AE+AE. BD. AD AD Hinc DE:AE= BD+l-.l =( facta DF = DC = \)BF:GB Proinde triangula ADE, GFB habentia angulos ad E & B aequales erunt similia, & SLngBCF— ang DAE: quamobrem ang: F = ang: DCF = ang BCF- ang: BCD = ang: DAE- ang: BCD. Hinc omnes tres anguli trianguli BCF sunt = ang: B + 2 ang: DAE— ang: BCD: ipsorum que mensura | C = 1 arc :AD+ arc : DBE- arc : BD = -| arc : AD + arc : BE = « + f arc. AD-\C. C 2(7 Hinc C = n + i arc A D, & arc AD = ?l + i 2)1+1 c cuius dimidii, nempe , cosinus erit \x. Si arcus ADBE 2 n + 1 CORRESPONDENCE WITH CRAMER 103 intelligatur auctus integra circumferentia vel ejus multiplo invenientur reliqui valores ipsius \x aequales cosinibus arcuum 2(7 3C 46' D 27m' 2^n* 2TTP &c - Et ^ sic resolvetur quantitas proposita \—z' in+1 in n factores duarum Dimensionum, & unum factorem 1— s unius dimensionis. Coroll. Si in fig. Covoll. praeced. ad singulos pares terminos I, L, N, &c. ducantur rectae GI, GL, GN, GO, itc. erit harum omnium productum = 1 — z 2n+ \ Probl IV Resolvere quantitatem 1 - z 2n in factores duarum Dimensionum. Solut. Sit unus ex factoribus 1— xz + zz & productum reliquorum 1 + as + bz 2 . . . + rz n ~ 3 + sz n ~ 2 ± tz' 1 ' 1 - hz n - rz H+1 .. . - bz 2n ' i Hie quia terminus tz 71 ' 1 debet affici signo tarn affirmative) quam negativo, opportet esse t = 0, adeoque si ponatur arcus AE AD ad arcum..4i? ut 1 ad n, tv per consequens t= -j— j erit AE — 0, & arcus huic Chordae respondens = vel 0, vel 2 0, vel 3C &c. Proimle arcus _<4 i? = alicui sequentium 20 3t? „ „ , . .. C 20 30 arcuum -> — > — j Arc. & -J x = cosinibus arcuum — > — > — 3 n it, 11 * 2n 2 it, 2 n &c qua rati one resolvitur quantitas 1— z 2n in n—1 factores duarum Dimensionum similes huic 1— xz+zz, & alium factorem duarum dimensionum, nempe 1— zz. Coroll. Si in fig. Cur. 2 & 3, Probl I ad singulos pares terminos divisionis /, L, B, 0, Q, A, Ducantur rectae GI GL, GB &c, erit harum omnium productum = 1— z 2n . Coroll. geneiale. Si Circumferentia Circuli dividatnr in 2 m partes aequales AH, HI, IK, &c, & ducantur rectae GH, GI, GK &c sive m sit numerus par, sive impar semper erit GHxGKx QM &c = 1 + z m , Sc GAxGIx GL &c = 1 - z m . Quod est Theorema Cotesii memoratum Act. Erud. Lips. 1723, pag. 170 & 171. 104 STIKLING'S SCIENTIFIC CORRESPONDENCE Supplementum Eodem Auctore Probl. V Dividere fractionem 5- in f ractiones plures, 1 1 + qz^+ z 2 quarum denominatores ascendant tantum ad duas Dimen- siones. Solut. Sit una quaesitarum fractionum & summa 1—xz + zz (x + (3z + yzz + 8z 3 + ez 4 + &c rehquarum i+az + fe2 + ^ + ^ 4 + &e Valor ipsius x determinatur in Prdblemate primo, & quan- titates 1, a, b, c, d, &c designant ut ibidem terminos Seriei recurrentis 1, a;, xx— 1, x 3 —2x, x^ — Sxx+l, &c. Valores autem ipsarum e & / post eliminationem ipsarum ot, /?, y, S &c inveniuntur ut sequitur : nempe si v = 2, id est si 1 _ e-fz + a + (3z l±qz' i + z i 1—xz + zz 1+xz + zz invenitur e = i&/ , = ~x- Si n = 3, id est si 2 J 2 x 1 _ e-fz tx + /3z + yzz + Sz* l±qz* + z 6 ~~ 1-xz + zz \+xz + x~x~^lzz + xz' + z* 1 cc invenitur e = -|, &f=~ ~ -: si n = 4, id est si 1 e-fz 1 ± qz + z~~ 1 - xz + zz a + (3z + y zz + Sz 3 + eg 4 + f s 5 l+xz + xz — lzz + x' i —2xz" + xx—l z i + xz 6 + z 6 invenitur e - i & f= - — : similiter si n = 5 1 a;^— 2.r invenitur e = -| & / = - — . & eeneraliter ob ratio- J 5 x^-Zxx+l to nem progressionis jam satis manifestam erit e = -et/=--> ((. 71. f ubi s & t significant duos postremos terminos Seriei recurrentis 1, a, b, c, d, &c. Hinc si in fig. Probl I sit Chorda BE — +q CORRESPONDENCE WITH CRAMER 105 & arcus AB — — erit s : t = BE-.AE per ibi demonstrata, n if & per consequens /= — - , ipsaque quaesita fractio 1 _ BE e~fa n~n.AE Z 1-xz + zz l-BBz + zz Si porrho intelligatur arcus AE auctus integra circumferentia vel ejus multiplo, ita ut mutentur valores ipsarum BB & BE, e — fz mutabitur quoque valor fractionis — — invenienturque successive omnes fractiones in quas proposita fractio l ±q ~n + z *„ resol vi potest Q.E.F. Coroll. Si q = 0, BE = BB = x, AE = AB = 2, fractio 1 x resolvitur in fractiones hanc form am n 2n habentes. 1 + z 1 " l-xz + zz Schol I Solutio inventa congruit cum ea quam Pemberton ex calculo valde operoso deduxit in Epist. ad amicum pag. 48 & 49 & ejus appendice pag. 11, 12. Est quoque simplicior quam Moivraei qui invenit fractiones hanc formam habentes 1 «. — le n n—un" ubi a = \x = sinui -| arcus BB, I — +-|g = sinui 1—xz + zz -| arcus BE, e — cosinui \ arcus BE, potuisset enim adhibere 1 ez hanc simpliciorem expressionem n ^V 1—u intelligendo per 1 — 2 az + zz e non cosinum sed ipsum sinum \ arcus BE Schol II Non absimili methodo resolvi possunt fractiones 1 1 X vel: J ±2 2„-l 1 _.W Schol III Methodus praeced. supponit q minorem binario, quando autem q > 2, fractio -■ 2u resolvi potest ut 106 STIRLING'S SCIENTIFIC CORRESPONDENCE -l . 3 + ^-2,. .s+p-s" ' -p.^+L.^j + ^-l I am with a great esteem and affection Sir Your most humble, most Obedient Servant G. Cramer (6) Cramer to Stirling, 1729 M r James Stirling at the Academy in little Tower Street London Sir I received indeed in due time your last letter, with the inclosed for M r Nichol. Bernoulli which I sent him imme- diately; but several indispensable affairs, together with receiving no news from him, were the cause of my long delay in answering your most agreable Letter. I began to reproach myself my Laziness, when your worthy friend came with your dear Letter to awake me. I'll be very glad to find some opportunity to show him, by any Service I am able to do him, how much I am sensible of your kindnesses to me. I told you already I had no news from M r Nich. Bernoulli, since I sent him your learned Letter. I believe he is medi- tating you an answer : however I write to him to warn him it is high time to do it. I received in the meanwhile several .letters from his Uncle: D r John Bernoulli, who is always 114 STIRLING'S SCIENTIFIC CORRESPONDENCE contriving again and again new Arguments for his Opinion about vivid forces. I don't know you have read what M 1 ' 'S Gravesande publish'd in the Journal Litteraire about that matter. Tis all metaphysical reasoning, in answer chiefly to the late D r Clarke and M r MacLaurin. I read with a great pleasure your Elegant Series for finding the Middle Uncia of any Power of a Binomial, and for sum- ming a slow converging Series, but cannot imagine what principles have brought ye to these Series. Tis nothing like your Theorem for interpoling any Term in that Series P p+ 1 p + 2 I sent all that to Mr Bernoulli. I render ye thanks for the account j t ou gave M r Bernoulli of M r Machin's Theorems. They seem indeed very well contrived for clearing S r Isaac Newton's Theory of the Motion of the Moon and easily computing that Motion. I was mightily pleased with that Elegant improvement of Kepler's Proposition, of Areas described in Proportional Times, and the more pleased I was, that the Demonstration is so easy that I wondered no body, before M 1 ' Machin, had thought of that Theorem. I wrote you in so few words o" M r Dan. Bernoulli's Way of approximating to y° greatest and smallest root of any given Equation by the help of a recurrent Series, that I was almost unintelligible. Now here are his own words. ' Methodus inveniendae minimae radicis aequationis cujus- cumque tam numericae tarn algebraicae. Concilietur aequa- tioni propositae liaec forma 1 = ax + bx 2 + ex" + ex 1 + &c. Dein formetur Series incipiendo a tot terminis arbitrariis quot dimensiones habet Equatio, hac lege, ut si A, B, 0, D, E denotent terminos se invicem directo ordine consequentes, sit ubique 2 = uD + bG + cB + cA + &c sintquo in hac Serie satis continuata duo termini proximi M & N, erit terminus antecedens J\[ divisus per consequentem iV proxime aequalis Radici minimae quaesitae.' And after some cautions to be observed in several cases he goes on. ' Ut inveniatur Radix aequationis maxima, Proposita sit aequatio Catholica sic disposita x m = ax m - 1 + bx m ' 2 + e.r m - a + &c Formetur Series CORRESPONDENCE WITH CRAMER 115 incipiendo a tot terminis arbitrariis quot dimensionum est aequatio, eaque talis, ut si A, B, C, D, E denotent terminos directo ordine e Serie excerptos & contiguos, sit ubique 2 = aD + bC+cB + eA+&c, sintque in hac Ssrie satis con- tinuata duo termini proximi M & N, erit terminus N~ divisus per praeeedentem M proxime aequalis radici maximae.' The demonstration of wich I conceive to be thus. Let the Roots of the Equation 1 = ax + bx 2 + c* 3 + &c be - . - , - . &c ^ x y z' and of the Equation x m = ax m ^ 1 + bx iu ~- + cx m ~ 3 + &c be x, y, z, &c : and if the term M is in order I of the recurrent Series whose index is a + b + c + &c this term M will be, for the values a, b, c, &c of the first Equation -,- + —■ ■, + —, +&c. and, 1 x l y l z l for the values a, b, c, &c in the second Equation px l + qy l + r~) ; and the next term in order I + 1, and called N shall be, for the 1) Q V first Equation -j^ + -^ + ^j— j- + &c and for the second Equation px 1 * 1 + qy l+l + rz l+ x + &c. Now if x be the greatest and - the smallest root the greater is I, or the further is that X ■ V ■ term M from the beginning of the Series, the greater is — in o v iy comparison with the other terms -^ + -j &c, and - ?+ — in com- O T parison with -jVf + -^ + &c. So that if I be infinite the y z terms 4 + -* &c and -4r\ + -4t\ + &c are not to be considered but ^, and -£-r make up the Terms M and N, the former of x l x l+1 wich being divided by the latter gives you x. In the other Equation px l and px l+1 being infinitely greater than qy l + rz l + &c and qy Ul +r;' +1 + &c make up the Terms M A r to'^ + ^c and N, and Tf . = '- — p- 7— = » the greatest root. i\l px + 1\ c I am with a great respect Sir Your most humble and most Obedient Servant Geneva y e 26 Decemb 1729 N.S. G. Cramer. 1 2 116 STIRLING'S SCIENTIFIC CORRESPONDENCE As soon as yours and M r de Moivre's books are printed, you'll oblige me very much to give notice of it to M r Caille, that he may get them and send them to me. I believe he has changed his lodgings, but he uses to go to Bridge's Coffee house over against y° Royal Exchange. (?) Cramer to Stirling, 1730 M r James Stirling F.R.S. at the Academy in little Tower Street London Sir As there is no less than a year, since I have no Letter from ye, I don't know, whether I must not fear the Loss of a Letter wich I sent ye about that time, containing a Letter from M r Nich. Bernoulli in answer to yours, together with a Copy of his Method for finding y° component quantities of a Binomium like this 1 +z n by the Division of the Circle. Extraordinary businesses have, from that time hindred me always, from having the Pleasure of writing ye, and inquiring after the Philosophical and Mathematical news of wich there is abundance in England in any time. I don't know whether your learned book about Serieees is published, but I wish and I hope it is, and y 1 ' Publick is not prived of your fine Inven- tions. I heard M 1 ' de Moivre's book is out, but I have not seen it yet. You know without any doubt, that M 1 ' 'S Gravesande had made fome little improvement to your method, given in your book Enumeratio linearum 3" Ordinis &c for finding the difference of exponents Arithmetically proportional in an infinite Series formed from a given equation : wich improve- ment he published at the end of his Matheseos universalis Elementa : but I found his Method wants yet a little correction, for it can induce into Error, if the given equation, besides w and y contains their fluxions. Let, for instance, the Equation be » , „,3„.„\2 o,..3„,„-. , „.4„\ , X + a?yif-2x*yy \-x 4 y+ -^ - CORRESPONDENCE WITH CRAMER 117 and by S 1 ' Isaac's Method of Parallelogram, you'll find in the Series resulting (y = Ax" + Bx n f ' ' + &c) n—\, and sub- stituting x instead of y, and * instead of y : the indices shall be 9 . 4 . 4 . 4 . 1 4. Whence, by D r Taylor's Method, r being the common divisor is 1. By your method, the first term shall be AAx-2Ax+ lx = or AA — 2A + l=0, where A has two equal valors, and therefore, by your method r = — = -- Mr 'S Gravesande's Method gives for r's value p 2 b 2\. But really r may be taken = 5, and the form of the Series is j/ = Ax + Bx ti + Cx n + &c. This valor of r = 5, is deduced from this Rule, wich may be substituted to others, having found, by the Parallelogram, the greatest terms of the Equation, and thereby the valor of n ; see whether these terms give for y, or y, or y &c many equal valors, and let p design the number of these equal valors of y, or y &c. Then substitute for y and y, y &c, x", x H ~ l , x n ~ 2 &c and write down the indexes of all the terms. Subtract them all from y e greatest, or subtract the smallest from all the others ; accord- ing as the Parallelogram gave you the greatest or the least index. Divide the least of these differences by p, & of this so divided, and of all others, find the greatest common divisor. This shall be the valor of r. So in the Example cited, the Parallelogram gives for the greatest terms of y" Equation x"'ydy 2 —2x :i ydy + x i dy=0, wich divided by x :i dy, gives ydy—2y + x = 0, where y has not many equal values, Theref. p — 1. The indexes are 9.4.4.4.14. The difference 5 . 10. The common Divisor 5. Whence r — 5. I wou'd gladly know from ye, how one can find the number of Roots of an exponential Equation, like this y x = 1 +«; for the method you give in the C Coroll. of y" 2 nd Prop, of your book Enumeratio &c p. 18 does not succeed in this case. It is a thing pretty curious, that in the Curve represented by that Equation y x = 1 +x, or y = 1 -Yx', the abscissa being = 0, the ordinate y is not 1, but of a very different value, tho' i it seems at the first sight, it must be 1, being 1 +b". I have happily conserved a Copy of M r Bernoulli's Letter, 118 STIRLING'S SCIENTIFIC CORRESPONDENCE so that I can send it ye, if you have not received y e original, wich I pray, I may know from ye, as soon as you can without any trouble at all. I am, with a great esteem and respect Sir Your most humble Geneva, the 22 X bre 1 730 N.S. most obedient Servant G. Cramer. (8) Stirling to Cramer, 1730 Copy of a Letter sent to M r Cramer at Geneva September 1730 Sir I beg a thousand pardons for delaying so long to return you an answer. I was designing it every day but unluckily hindred by unexpected accidents. So that now I am quite ashamed to begin, and must intirely depend on your goodness. I send two Copies of my Book, one for yourself and y e other for M r Bernoulli which I hope you will transmit to him along with the letter directed to him. I have left it open for your perusal, and you will find a letter which M r Machin sent me being an answer to what M r Bernoulli write about his Small Book. The first part of my Book you see is about y 3 Suming of Series where I have made it my chief business to change them that converge slow into others that converge fast ; but that I might not seem quite to neglect the suming of those which are exactly sumable, I have shown how to find a Huxionary Equation which shall have any proposed Series for its root, by the Construction of which Equation the series will be sumed in the simplest manner possible, I mean either exactly or reduced to a Quadrature perhaps, by which means I take this matter to be carryed farther than it was before : this you will see is the 15 Proposition and its Scholien I have taken an opportunity of clearing up a difficulty about the extracting the Root of a fluxionary Equation, which is the only one that Sir Isaac left to be done. This first part CORRESPONDENCE WITH CRAMER 119 has been written 8 or 9 years ago, so that if I were to write it again I should Scarce change anything in it; But indeed that is more than I can say for the Second part, because there was not above one half of it finished when the begining of it was sent to the Printer. And altho' I am not conscious of any Errors in it but Typographical ones, yet I am sensible that it might have been better done. The 20 Prop: about y e Suming of Logarithms has been Considered by M r De Moivre since y° publication of my Book, and he has found a Series more simple than mine which is as follows. Let there be as many naturall numbers as you please 1, 2, 3, 4 ... z whereof the last is z. Make I, z = Tabular log. of z, I, i- = log. of 6-28318 which is the Circumference of a Circle whose Radius is unity, a = -43429 ... which is y° reciprocal of y e Hyperbolick Log of 10. and y e sum of y e Logarithms of the proposed numbers will be 1 whereas you will see that in my Series y u Numerators are y e alternate powers of 2, diminished by unity: the degree of convergency is y u same in both, and indeed there is seldome occasion for above three Terms, reckoning —za the first: M 1 ' De Moivre is to publish this with his manner of finding it out, which is quite different from mine, which is done by an old and well known principle, namely the taking of the differ- ence of the successive values of quantitys as you will see in y e Book, about which I shall be glad to have your opinion : and I hope you will write to me soon after this comes to hand, else 1 shall take it for granted that you have not forgiven me. I shall be always glad to hear of your wellfare, and to know your news of any kind whatsoever. I am with the greatest respect D. Sir Your most Obedient & most humble Servant London September 1730 James Stirling. 1 The gap occurs in Stirling's copy of the letter. 120 STIRLING'S SCIENTIFIC CORRESPONDENCE (0) Cramer to Stirling, 1731 M r James Stirling R.S.S. at the Academy in little Tower Street London Sir I guess by the date of your Letter you must be very angry with me, thinking, as you may well, my negligence in returning you an Answer quite unpardonable. But I beseech you to believe, I cou'd not be so ungrateful! as not to rendring you due thank for your fine present, wich I re- ceived but from five days. The chief reason of that accident is the forgetfullness of a Merchant to whom M 1 ' Caille gave the two Exemplarys of your Book for sending them to me, then his sickness, then the violence of the winter, than I know not what, so that, to my great misfortune, they came here but the 12 tb of June. As soon as I received them, I sent M r Bernoulli his Exemplary together with the Letter for him and the inclosed Letter of M r Machin. And I resolved to write you even before the perusing of your book that I coud justify myself of a so long and unexcusable delay. As far as I can see, by a superficial Lecture of the Titles of your Propositions, this Treatise is exceedingly curious, and carries far beyond what has been done heretofore a Doctrine of the utmost importance in the Analysis. I rejoice before- hand, for the advantages I shall reap from an attentive Lecture of it, and I flatter myself you shall be so kind as to permit me to improve this benefit by the correspondence you vouchsafe to keep with me. You shall know M r N. Bernoulli has been this month elected Professor of the Civil Law, in his own University, wich I fear will perhaps interrupt his Mathematical Studies. I have perused, as you permitted, your Letter to him, and, in my opinion you are in the right as to your objections against Ins c . , ,. ,i a ■ r.r + b.r+2b ... r + zb-b manner of Interpol) ns the faeries ; ; ; — = p.p + b. p + 2b ...p + zb — b by putting it equal to '—, ",' — ; J v & H r + zb...zb + p-b CORRESPONDENCE WITH CRAMER 121 rp -(- 2,}) , . . zh + t b or = —. — , which cannot succeed but in some p . p + b . . . r — b few cases, wich have no difficulties. His Theorem sent to M r Montmort seems to be usefull in many cases. I have found a demonstration of it very simple, and made it more general, in that manner. The Series 1 a.a + b.a+2b... u+p — lb a + c.a + c + b.a + c + 2b ... a + c +p —lb n M— 1 a + 2c.a + 2c + b...a + 2c + p—lb ib ii — \ n — 2 _ __ I x T- x HT a + 3c .a + 3c + b ... a + 3c+p>—lb n n—l n—2 a— 3 a + 4c .a + ic + b ... a + kc+p—lb ^ ,.• a t> c — b . „ c — 2b r , _. c — 3&^ (by putting ^4 = , B — A, C = 5, B — --- (7, ttec A <5 t: and J.z + &* + GV + X>3 4 + &c = #s» + iz" + x + Za»+ 2 + £;» + :i + &c) will be reduced into this p . p+l . p + 2 ... p + n — 1 „ p . p + 1 . p + 2 . . . p + n , + n + 1 b a .a + b ... a+2J + n + 2b or, (if you like rather to have but the sign + and not alternately + & — ) into This p -p + l -p+2 ... p+ it -I „ <( + nc4 p—lb. a + nc+p—2b ... a + nc — nb p .p+l .p+2 ...p + n . + a + nc +p —lb ... a + nc — n+lb -> .p+l .p+2 ...p + n+l a + nc+p — lb ... a + nc — n + 2b + p-p+J l -_p + z-p + ™+}_ K + &c 122 STIRLING'S SCIENTIFIC CORRESPONDENCE where if c = b, A being = b n , and B — G = D = &c = all the Series is reduced to the tirst term p.p+1 .p + 2 ... p + n—1 , n a . a + b . a + 2 b . . . a + p + n — 1 b and, moreover, if you put again p = 1, you'll have M 1 ' Ber- noulli's Theorem I have also read over M r Machin's Letter, but I cannot judge of their difference having not seen his Book. M r Caille cou'd not find it. I am glad for what you say to M r Bernoulli, he is preparing for the press a compleate Treatise about it. I conjure you to make me know as soon as it shall come forth, where it is printed, for I shall read it with a great pleasure. I had willingly delayed this letter till I had some news for ye, but I chusc rather to send this empty answer, than to put off any longer to tell ye I am with the greatest esteem and respect Sir Your most humble, most obedient Geneva 18 t;l June 1731. and most faithfull Servant G. Crameh. (10) Cramer to Stirling, 1732 M r James Stirling. R.S.S. at the Academy in little Tower Street London. Geneve, ce 22 e Fevrier, 1732. Ne Soye's pas surpris, mon cher Monsieur, de recevoir si tard la Reponse a Votre chere Lettre du Mois de May 1731, puisqu'il n'y a que tres peu de jours que Monsieur Bernoulli me l'a fait remettre. J'espere aussi que vous me permeteres de vous eerire dans ma Langue maternelle, puisque je sais que vous l'entendes fort bien. Et je crois vous ennuyer moins en vous parlant une Langue qui vous est un peu ^trangere qu'en vous obligeant a, lire un Anglois aussi barbare que celui que je pourrois vous ecrire. Je continue a vous rendre mille graces pour le present que vous aves daigne" me faire de v6tre CORKESPONDENCE WITH CRAMER 123 excellent Ouvrage, dont je vous ai accuse" la reception dans une Lettrc que vous deves avoir recu depuis l'envoy de la Votre. On ne peut rien trouver dans le livre que d'exquis pour ceux qui se plaisent aux Speculations dont vous are's enrichi les Mathematiques. Je n'en dirai da vantage de peur de paroitre vous natter, quoiqu'assurement ce que j 'en pourrois dire seroit fort au dessous de ce que j'en pense, et de ce que j'en devrois dire. La Regie de D r Taylor pour trouver la forme d'une Serie doit e"tre proposee, comme vous le remarque's sous une forme diti'erente de celle qu'il a donne"e en ce que r doit e.tre, non le plus grand commun diviseur des indices, niais bien celui des Differences des Indices. Mais pour qu'elle puisse s'etendre a tous les cas possibles, M r Gravcsande dit qu'ayant substitue" dans l'Equation, Ax n au lieu de y ttc il faut chercher la Valeur de A & s'il se trouve qu'il ait plusieurs valeurs egales, il faut prendre pour r le plus grand commun diviseur des Differences, mais tel qu'il mesure la plus petite par le nombre des valeurs egales de A ou par un multiple de ce nombre II en donne l'exemple suivant. ^ + a?y-2j;Hf + xif - J = que la substitution de Ax n au lieu de y, change en ±- +Ax n+3 -2A i a?» + * + A 3 aB in+ i-'=^= r - = b w a 6 Done les indices sont 14, n + S, 2n+2, 3n+l, 9». Par le Parallelograinme de M r Newton on trouve pour la forme de la suite d'autant plus convergente que if est moindre, n = 1, ce qui change les indices en 14, 4, 4, 4, 9. Otant le plus petit des autres, les differences sont 5, 10. Le plus grand commun diviseur est 5 ; Ainsi selon la Regie de M r Taylor corrigde, la forme de la suite doit &re Ax + Bx r ' + Cx n +lkc. Mais selon M' 'S Gravesande si Ton veut determiner la valeur de A par le moyen des plus grands termes de l'equation qui sont Ax n+ ' i -2A 2 x- n+2 + A !i x'' n + \ ou Ax i -2A 2 x i + A 3 x i e'gale's a zero et divise's par a; 4 on trouve qu'il a 2 valeurs egales. Done r doit diviser les 2 differences 5 & 10, et entr'autres la plus petite par 2 ou 4, ou 6, &c. 124 STIRLING'S SCIENTIFIC CORRESPONDENCE Ainsi r doit etre 2|, et la forme de la Serie sera Ax + Bx* + Gx 6 + Dx^ + &c. Mais eette Regie de M r 'S. Gravesande ne paroit pas encore asses generale, car il peut aisement arriver dans les Equations fluxionelles que A ait plusieurs Valeurs egales, sans qu'il y faille faire aucune attention. Ainsi quoique sa Regie donne tou jours une Suite propre a determiner la Valeur de y, cependant elle ne donne pas toujoUrs la plus simple. II faloit done e"tablir la Regie ainsi. Si les plus grands termes de l'i quation determines par le Parallelogramme de M r Newton, etant egales a zero, font une Equation dans laquelle y ou quelcune de ses Fluxions ait plusieurs Valeurs egales, Divises la plus petite difference des Indices par le nombre de ces Valeurs egales, Et le plus grand commun diviseur du Quotient et des autres Differences sera le nombre r cherchd Par exemple, si l'Equation cy-dessus avoit 6t6 . \ * Hi * \ ■ Fig. 21. JOOC w ... ... , .., xy' + x i y-2x\jy + x s yhj- -^ Lb on auroit trouve" la meme valeur de u= 1, les memes indices 14, 4, 4, 4, 9, les meraes differences 5, 1 0, que cy-devant, & A auroit aussi deux Valeurs, Done selon la Regie de M r 'S Gravesande ou auroit la meme forme de Serie, Ax + £x :i - + Cx e + &c, Au lieu que suivant la Regie que je viens de poser, les plus grands termes de l'Equation x i y — 2x )i yy + x*y 2 y, egales a zero et divise's par x"y donnent x—2y+yy = qui ne donne pas deux valeurs egales de y ou y Ainsi il faudra simplement prendre pour r le plus grand commun diviseur 5 des differences 5, 10, Et la forme de la Serie est Ax + Bx'' + Cx n +&c. Ainsi si Ton calcule selon la forme de M r 'S Gravesande, on trouve tous les Coefficiens des Termes pairs egaux a zero. C'est la la Regie Generale. Mais il se rencontre quelquefois des cas, ou il n'est pas si facile de l'appliquer. Les Termes place's sur le Parallelogramme de M r Newton peuvent se trouver sur une meme ligne Verticale. Alors on ne peut en les CORRESPONDENCE WITH CRAMER 125 comparant determiner la Valeur de l'exposant u. Mais en supposant que le terme le plus grand est celni qui a le plus grand ou le plus petit exposant selon qu'on veut que la Suite converge, d'autant plus que x est plus petite ou plus grande : On determine par cette supposition la Valeur de -it & la forme de l'Equation. Mais la valeur du premier r et souvent de quel- ques autres coeffieiens reste indeterminde. Done si tous les termes places sur le Parallelogramme de M r Newton se trouvent dans une meme ligne oblique, ou ce qui revient au meme, lorsqn'ayant substitue' clans l'Equation Ax n au lieu de y, & nAx 1l ~ l au lieu de y, &c les indices des termes resultans se peuvent tous rencontrer entre les Termes d'une Progression Arithmetique: alors l'equation est a une ou plusieurs Paraboles, ou bien a une ou plusieurs hyperboles, qu'il est facile de determiner. Soit par exemple l'equation 2xx — ixVay — 15 ay — & apres la substitution de Ax n au lieu de y, les indices seront 1, \ ib, Ti — 1, qui sont en Progression Arithmetique. les supposant egaux on trouve n = 2. Soit done y = Ax 2 et apres la Sub- stitution l'equation devient 2xx — AxxVaA — 30aAxx = ou, divisant par x.v, 2—4 VuA — 30aJ. = 0. Done les Racines sont 1 — 5 VaA = 0, & 1+3 -/aA = 0. Dans ces Racines mettant au lieu de A sa valeur ^ , elles se changent en 1 — 5 /— = lav iv 1 + 3 I -~ = dont la multiplication produit xx — 2xVay—lBuy = qui est la tiuentc de la fluxion proposee 2x.r,— 4x ^ente au point Q — — ; ; ° v ^ -l(\ + x) qui est encore £. Done differentiant de nouveau, on trouve cette soutangente= — Qxx — 8x — 2 = — 2 (puisque x = 0). Or les Autheurs poscnt qu'on n'est oblige* a ces differentiations que lorsque 2 ou plusieurs Rameaux de Courbe se coupent dans le point ou Ton cherche la soutangente Voyes Memoires de l'Academie de Paris. Anne"e 171G p. 75 & Annee 1723 pag. 321. Edit, de Coll. Voye's aussi Fontenelle Elements de la Geometric de l'infini, p. 418 & 99. Votre Probleme du jet des Bombes est de la dernicre im- portance par raport a cette branche de la Mechanique. Je serai infiniment curieux d'aprendrc le resultat de vos Experiences & de Vos Calculs. J 'ai lu cet article de votre Lettre a plusieurs de mes Amis OrHcicrs d'Artillerie, ches qui il a excite une merveilleuse curiosity. Ce que vous dites de la facilite - de votre solution ne pique pas moms la mienne, puisque la Solu- tion de M r Jean Bernoulli (Acta Enid. 1719. p. 222, cv 1721. p. 228) est si complique'e et inapliquable a la pratique. Je vous suplie, si vous av^s compose - quelque chose la dessus de daigner me la communiquer. 128 STIRLING'S SCIENTIFIC CORRESPONDENCE Je voudrois bien en ^change de votre belle Lettre vous indiquer aussi quelquechose digne de votre attention Mais il n'est pas donne a tout le monde de Voler si haut. Je me rabaisse a de plus petits Sujets. Voiei un Probleme qui m'a occupe ces jours passes, et qui sera peut-dtre du gout de Mr de Moivre. Vous ne save's peut-^tre pas ce que nous appellons en Francois le jeu du Franc Carreau. Dans une chambre pavee de Carreaux, on jette en l'air un Ecu. S'il retombe sur un seul carreau, on dit qu'il tombe franc, et celui qui l'a jette" gagne. S'il tombe sur deux ou plusieurs Carreaux, c'est a dire, s'il tombe sur la Raye qui separe deux Carreaux, celui qui l'a jette perd. C'est un Probleme a resoudre & qui n'a point de difficulty. Trouver la Probability de gagner ou de perdre, Les Carreaux & l'Ecu elant donn^es. Mais si au lieu de jeter en l'air un Ecu qui est rond, on jettoit une Piece Quarree, Le Probleme m'a paru asses difficile, soit qu'il le soit naturellement, soit que la voye par la quelle je l'ai resolu ne soit pas la meilleuru. Au reste j'ai recu le Livre que Mr de Moivre m'a envo} e" en present. J'ai pris la Liberte" de lui en faire mes remercimens dans une Lettre dont j'ai charge" un jeune homme d'ici, qui est parti il y a quelques mois pour l'Angleterre. Je ne scais s'il la lui aura remise n'en ayant eu depuis aucune nouvelle Je vous prie, quand vous le verres de vouloir bien l'assurer de mes humbles respects, & de ma reconoissance. Temoignc's lui combien je suis sensible aux Marques publiques qu'il m'a donne'es de son auntie". II ne sera pas trompe" dans sa Conjecture, quand il a cru que la 2 e Methode de M 1 ' Nicolas Bernoulli est la meme que celle de Mr Stevens. II y a plus d'un an que je n'ai aucune nouvelle de ce dernier. Sa nouvelle Profession 1 occupe entieYement. II a pourtant recu votre Livre avec vos Lettres, et vous aura sans doute repondu. Je suis avec une estime et une consideration toute parliculiere Monsieur Votre tre"s humble, & tres obe"issant Serviteur G. Cramer. CORRESPONDENCE WITH CRAMER 129 (11) Cramer to Stirling, 1733 M' James Stirling. F.R.S. at the Academy in little Tower Street London Monsieur Voici une Lettre que je viens de recevoir pour vous de la part de M r Nicol. Bernoulli. Elle est venue enfin apre's s'etre fait longtems attendre. Un nombre considerable d'occupations m'empe'che d'avoir l'honneur de vous ecrire plus au long. Voici seulement un Extrait de ce qu'il me marque touchant sa nouvelle Manie're de calculer les Numerateurs des fractions simples auxquelles se reYluit la fraction -„ n j-^ — i ■ ^°^ Z" t ^ LZ -\- 1 suppose" z* n +2lz n +l 1-xz + zz com me j'ai trouve" par induction dans la Solution de mon Probl. 5. Je vous soupplie, Monsieur, de vouloir bien me faire la grace de me donner de Vos nouvelles, & de m'informer de ce qui s'est publie nouvellement en Angleterre en fait de PhiloEO- phie & de Mathematique. Soyes persuade que je suis avec une extreme consideration & un Veritable attachement, Votre tres humble & tres obeissant Serviteur G. Cramer. Geneve ce 10 e Avril, 1733 IV N. BERNOULLI AND STIRLING (l) N. Bernoulli to Stirling, 1719 D ue mihi plurimum colunde Peugkatum mihi fuit nudius tertius aecipere epistolam tuam, qua me ad mutuum epistolarum coiiiercium invitare voluisti, gaudeoque quod ea, de quibus ante hac Venetiis egimus, consideratione tua digna esse judices, quia igitur ea tibi in memoriam revocari cupis petitioni tuae libenter morem geram, quod attinet primo ad difficultatem illam, quam de resistentia pendulorum movebam, ea hue redit. Posita gravi- tatis vi uniformi et resistentia proportionali velocitati, non potest corpus grave oscillari in Cyeloide ; hoc quidem inveni per calculum, sed quomodo ista impossibilitas a priori ex rationibus physicis demonstrari possit, adhucdum ignoro. Rogo igitur ut hanc rem sedulo examines et quaeras construc- tioneni Curvae, in qua abscissis denotantibus spatia oscillatione descripta (i.e. arcus Cycloidis interceptos inter punctum quietis et punctum quodvis ad quod mobile oscillando pertingit) applicatae denotent resistentiam vel velocitatem mobilis in fine illorum spatiorum. D n3 Newtonus pag. 282. dicit hanc Curvam proxime esse Ellijisi. Problema quod a D"° Taylor Geometris propositum mecum comunicavit D. Monmort, est sequens. Invenire per quadraturam ch-culi vel hyperbolae fluentem hujus quantitatis . — ^ t , ubi S significat numerum quemlibet integrum attirmativum vel negativum, et X numerum aliquem hujus progressionis 2, 4, 8, 16, 32 &c, petitur autem, ut hoc fiat sine ulla limitatioue per radices k2 132 STIRLING'S SCIENTIFIC CORRESPONDENCE imaginarias. Denique quod attinet ad Theorema Patrui mei pro conjiciendis Curvarum areis in Series convergentes, tuam que contra ejus generalitatem factam oppositionem, in ea re adhucdum tecum dissentio, et in mea opinione firmatus sum, postquam nuper exemplum a te oblatum, et alia calculo subduxi ; deprehendi enim seriem, licet in infinitum abeat, tamen esse sumabilem, si area invenienda sit quadrabilis. De rebus aliis novis Mathematicis aut Philosophicis nihil, comunicadum habeo, nisi quod Patruus meus miserit Lipsiam solutionem Problematis D 1 Taylori (quod et ego jamdudum solvi) cum subjuncta appendice infra scripta. Quod superest Vale et fave. Dabam Patavii d. 29 Apr. 1719 Ill mu3 Polenus me enixe D" is Tuae rogavit ut suis verbis tibi Servo humillimo plurimam Salutem dicerem Nicolao Bernoulli Appendix Patrui Adjicere lubet quaedam mihi inventa Theoremata, quae in reductionibus utilitatem suam habent non exiguam. Demon- strationes eorum brevitatis gratia jam supprimo : Erunt inter Geometras qui facile invenient, quocirca illis eas relinquo. Definttio. Per q et I intelligo numeros qualescunque in- tegros, fractos, attirmativos, negativos, rationales, irrationales. Per p intelligo tantum numerum integrum et affirmativum, vel etiam cyphram. Sed per n et k intellectos volo numeros quoslibet integros affirmativos exclusa cyphra. Theorema I ax : (e +fx'iy est algebraice quadrabilis. Theor. II Generalius, quadrabilis. x p, 'dx : (e +fxi)~ q est algebraice _i x k l~ 1 dx:(e+fxl) i est algebraice quadra- Theor. Ill bilis : Adeoque existente p = 0, erit etiam i x k l- l dx:(e+fxl) i algebraice quadrabilis. ,- J -+* CORRESPONDENCE WITH BERNOULLI 133 Theor. V Theor. VI Theor. IV x*">dx:(e+fx1) n dependet a quadratura hujus dx : (e +fx«). x-fidx : (e +fafl) n dependet a quadratura ejusdem dx: (e+fx'l). xw+'dx ; (c +fx r i) n dependet a quadratura hujus x l dx : (<• +/("?). Theor. VII Sumti.s S et X in Casu Taylori erit ;?-' h:(e + fziy i quadrabilis per circulum vel hyperbolam. Corolloria quae ex hisce Theorematibus deduci possent pulchra et miranda non minus quam utilia nunc omitto, sicut et plura alia ad quadraturarum reductioneni spectantia, quae olim inveni ac passim cum Amicis comunicavi. Ex. gr. Ex collatione Tkeorr. V et VI sequitur inveniri posse duos coeffi- cientes a et /?, ita ut algebraice quadrabilis. (a.r-c? + (3x<"> + '<)da' : (e +fx r i) n sit (2) Bernoulli to Stirling, 1729 Viro Clarissimo Jacobo Stirling S.P.D. Nic. Bernoulli. Pergrata fuit epistola, quam per coiliunem amicum D. Cra- merum mihi haud pridem transmisisti et ad quam citius respondissem, si per x&y'va impedimenta licuisset Gaudeo te valere et rem MathcmatiGam per impressionem libri de suTfia- tione et interpolatione Serierum novis inventis locupletare. Gratias tibi ago pro illis quae prolixe narrasti de nova theoria Lunae a D. Machin inventa, cujus hac de re libellum nuperrime mihi donavit D. de Maupertuis, qui nunc apud nos versatur. Pauca quidem in eo intelligo, quia nullam adhuc operam 134 STIRLING'S SCIENTIFIC CORRESPONDENCE eollocavi in lectione tertii libri Principiorum D. Newtoni ; videris tamen mihi haud recte in epistola tua explicuisse quid ipse vocat an, Equant. Verba sua sunt haec : 'he constructs a figure whose Sector CDF is proportional to the angle ASB, and finds the point C which will make the figure CD nearest to a Circle'. Existiino dicendum fuisse 'he constructs a figure, whose Sector CDF is equal to the area ASB, and finds the point F which will make the figure CD nearest to a Circle.' Ceterum etiam si inveniatur punctum aliquod F ex quo velocitas Planetae in utraque apside constituti eadem appareat ex hoc non sequitur aequantem CD maxime accedere ad circulum, vel punctum F esse illud, ex quo motus Planetae maxime uniformis appareat, ut D. Machin asserit pag. 41. Nam locus ex quo Planeta in A et P (fig. seq.) constitutus aeque velox apparet non est unicum punctum F sed integra linea tertii ordinis FAffPf cujus aequatio est Fro. 23. a — x . yy = a + b — x . b — x . x positis AS = a, SP = b, Ag — x, gf = y. In hac igitur linea et quidem in ejus ramo Pf datur fortassis punctum /, ex quo Fig. 24. Planeta apparet aeque velox in tribus punctis A, P, et D: adeoque ejus motus magis regularis vel uniformis quam ex puncto F. In ead. pag. 41.1in. 16 omissa est vox reciprocally; praeter hunc errorem in cadem pag. notavi, quod Auctor videatur comittere paralogismum, dum areas descriptas a corpore moto per arcum Alt circa puncta S et F, item areas descriptas a lineis Fp et FR dicit esse in duplicata ratione CORRESPONDENCE WITH BERNOULLI 135 perpendicularium in tangentem (ad punctum R) demissarum ex S et F; haec enim ratio obtinet tantum in harum arearum fluxionibus, a quarum pvoportionalitate ad proportionalitatem ipsarum arearum argumentari non licet, ut scis me olim quoque ex alia occasione monuisse; nihilominus consequentia, quod area a linea Fp descripta aequalis sit areae a linea MR descriptae vera manet. Theorema illud, quod corpus ad duo fixa puncta attractum describat solida aequalia circa rectam conjungentem ilia duo centra virium temporibus aequalibus, verum esse deprehendo. Reliqua examinare non vacat. Theorema tuum pro interpolatione Seriei A, -A, —— B, r + 2r+3_. , /> P+ 1 —tt, ^> T q ^' ^ C P er quadraturas Curvarum deduci potest P + & J) + O ex isto altero theoremate quod ante 19. annos cum D. de Monmort coinunicavi, 1 n n.n—1 a . n— 1 . n— 2 a a+b I .2 .a+2b 1 . 2 . 3 . a + 3 b n . n— 1 . it — 2 . /(,— 3 „ 1 . 2 . 3 .4 . 5 ... nb 11 + — ; — ^— 7. — : ; ^c = 1 . 2 . 3 . 4 . a + ib a . a + b .a + 2b ... a + nb r Sed et sine quadraturis interpolatur facillime Series A, - A, r + b r + 2b r + 36 p j B, —-. V, ; D, ivc ponendo p + b p + 2b p + 3b r r .r + b .r + 2b ...r + zb — b r .r + b . r+2b ... p — b p .p + b .jj+2b ... 2^ + zb — b r + zb . r + zb + b . . . zb + p — b , .. p + zb . p + zb + b ... zb + r — b vel etiam = 2 *-= ; ; — , p .p + b .p+2b ... r—b prout p major vel minor est quam r. Ex. gr. Si z = 2\ erit terminus inter tertium r + b r + 2b .. r B et quartum r U medius = p + b ^ p+2b r . r + b.r+2b ...p-b p+2\b.p + 3\b ... l±b+r r+2^b.r+3±b ... l±b+p p./) + b.p + 2b ... r-b Aliud vero est interpolare ejusmodi Series quando valor ipsius z non est numerus integer, aliud invenire per approxi- mationem aliquam earundem Serierum terminos non tantum 136 STIRLING'S SCIENTIFIC CORRESPONDENCE quando z est numerus fractus, sed et quando differentia inter 'p et r est numerus magnus, quod ultimum, ut et valorem Seriei alicujus lente convergentis, ope Serierum quarundam infmitarum promte convergentium a te inveniri, ex litteris D ni Cramer intellexi, quarum Serierum demonstrationem libenter videbo. Optarem spei tuae satisfacere tibi vieissim impertiendo nova quaedam inventa, sed dudum est quod Mathesis parum a me excolitur, nee nisi in gratiam amicorum me subinde ad solu- tionem quorundam Problematum accinxi, quorum solutiones in Sehedis meis dispersae latent, et quoad maximam partem vix tanti sunt ut tecum comunicari mereantur. D um Cramer rogavi, ut tibi transmittere velit Specimen methodi meae (Pembertiana multo facilioris et cujus ipsum participem feci) resolvendi fractionem in fractiones hujus formae , , \+qz n + z An J a + bz < i ' 1 ±cz + zz Dus de Maupertuis Patruo meo nuper proposuit sequens Problema: A et B sunt duo ignes quorum intensitates sunt ut p ad q, quaeritur per quam Curvam CD homo in dato loco C Fig. 25. constitutus recedere debeat, ut sentiat minimum caloi-em, posito rationem cujusque ignis in objecturn aliquod esse in ratione reciproca duplicata distantiarum. Hujus Problematis sequentem constructionem inveni. Centris A et B describantur circuli acg, bdh aequalium CORRESPONDENCE WITH BERNOULLI 137 radiorum Aa, Bb, jungantur AC, BO, secantes circumfereutiam horum circuloruni in c,:. ± e: j-Vnidcare numeros integro.- : unde non eapiebani e.ir h&ec >rri-s urpote quae accurate posset interpolari. ad quadratur&s redueeretur. Sed his majora te praestitisse \ i-li cum voluptace in tuo libro. cujus Propositio IS continet. ni fallor, i -.liv.su :n qnou ep? per modo dictum alterum interpolandi modum tuonere Tolebam. In exemplo 1. Prop. 25. ubi trails interpolationem unciarum binomii ad dignitatem iudeterminatam elevati. inveni theorema non multum absimile praedicto meo theoivuuui. Si fraetionis ] .2.3.4... nx&» .. ., i. » ■ r ; -. r numerator dividatur per b M , et sin^uli ti . u — b . , ( 4. 2 o . . . a + no faetores denominatoris excepto primo per b. e: :psa fraetio multiphcetur per primum factorem a, proveniet reerproeus terminus uneiae ordine n + 1 in binomio ad dignitatem t + ** CORRESPONDENCE WITH BERNOULLI 145 elevato; hinc per theorcma meum, ut Area ordinatae — — n 1 x a x 1—%I> ad-,ita unitas ad dictain unciam. Ex. gr. si ponatur a = 5, b = 1, n = 4 erit area ordinatae a 4 x \-x, id est, i-f + f-f + l sive ^h »<•! J ut 1 ad 126 unciam termini quinti in dignitate noria. Si a- I, b = 2, n = a, erit area ordinatae a; x 1 — a-ai , id est, quadrans circuli cujus radius = 1, sive area circuli cujus diameter = 1, ad 1 sive ad quadratum circumscriptum, ut unitas ad terminum Wallisii D inter- ponendem inter primum et secundum terminum Seriei 1, 2, 6, 20, 70, &c quae continet uncias medias dignitatum parium, sive ad terminum qui consistit in medio inter duas uncias 1 et 1 in potestate simplici binomii; sicut tu quoquo invenisti in exump. 2. dictae Prop. 25. Lalioiiusa quidem sed elegans est methodus per quam in- venisti opu Logarithmorum interpolationem Seriei 1, 1, 2, 6, 24, 120, itc in Ex 2. Prop. 21. Ceterum frustra quaesivi modum, quern dixisti in sequentibus monstrari, interpolandi hujiii-modi Series absque Logarithmis, quod autem a te prae- stare posse nullus dubito. Terminum qui consistit in medio inter duos primos 1 et 1 ope Theorematis mei sic eruo. Sit in dicto theor. a=H+l,6=l, eritque area ordinatae „ « 1.2.3.4 . ii 1 . 2. 3.4... to x 1 .2. 3. ..% n + l . n + 2...2n+l 1 . 2 . 3 ... 2 n+ 1 Fiat to = \ eritque area ordinatae Vx — xx i. e. area semicirculi, cujus diameter = 1, aequalis dimidio (juadrato quaesiti termini. Hinc quoque deducitur interpolatio terminorum intermediorum in hac Seiie 1, 1, 3, 15, 105, 945, tvc. Nam si h'at a = 1, b = 2, ., ,. „ „ n 1.2.3 ... 91X2" , . erit area ordinatae a; u xl— a;a; =- ; sed in 1 . 3. 5 ... 1 +2/4 casu n = -| praodicta area sit aequalis areae circuli cujus diameter = 1 , et numerator £ ractionis sit aequalis radici quad- ratae duplae istius arcae, per modo ostensa, denominator autem fractionis sit aequalis termino qui consistit in medio inter secundum et tertium Seriei 1, 1, 3, 15, 105, 945, &c proinde ut radix quadrata dimidiae areae circuli ad 1, ita unitas ad terminum ilium intermedium, qui per binarium divisus dabit medium inter duos primos 1 et 1 dictae Seriei. 146 STIRLING'S SCIENTIFIC CORRESPONDENCE De modo inveniendi radicem aequationis fluxionalis per Seriem infinitam, de quo agis in Scholio Prop. ult. Part. I. etiam ego aliquoties cogitavi, efhac de re scriptum aliquod comunicavi cum Dno de Maupertuis cum apud nos ageret, in quo sequentia observavi. Posse inveniri Series generaliores quam quae inveniuntur per parallelogramum Newtoni ; non necesse esse ut indices dignitatum in terminis Seriei quaesitae aut aequationis transformatae cadant in eandem progressionem arithmeticam ; posse aliquos indices esse irrationales ; et prop- terea tarn Taylori regulam in Prop 9 quam tuam in PTnumerat. Linear, tertii ordinis datam, pro determinanda forma Seriei fallere; posse per terminos solitarios in aequatiotie trans- formata nonunquam aliquid determinari, absque ut omnes coefBcientes fiant aequales nihilo ; non necesse esse, ut Serierum in aequatione transformata provenientium ad minimum duorum terminorum primorum indices inter se aequentur, ut deter- minetur coefficiens primus A, quia hie nonunquam potest ad arbitrium assumi ; posse evitari terminos superfluos, quorum coefBcientes in methodo Taylori evadentes = laborem calculi prolixiorem reddunt, quam paret. Sic pro Exemplo Taylori in Prop. 9. Method. Increm. pag. 31 1 + zx- z%xx — x — sequentes 4 Series inveni ; quarum tres priores sunt genera- liores i His quas Taylorus invenit. l a . x=A + Bz + $z 2 + ±Az 3 - 1 t- s ABzi + Y \ s Bz i --£- s AA + BBz3 7 , 14 ! _ -5+^165 , A „ 288 ,. 2 ' *= 2^-29 Z " +GZ 4 +4 *-588^ 2&C 3*. x = 2zi + B-±Z-' 1 +i^-i- T 1 g£50- 2 + ^o-B 3 -2 3 o^ 5 * c 4 a . x = ~z- x — z'^—^iz~ i —-%-z"^~&c. Sic quoque observavi te non satis accurate rem examinasse, quando pag. 8 3 dicis, aequationem r 2 y 2 = r 2 x 2 — x 2 y 2 nulla alia radice explicabilem esse praeter duas exhibitas V = *-«?+ 120T'-5040^ + * Cet X 1 X* X s CORRESPONDENCE WITH BERNOULLI 147 quarum prior dat sinum, et posterior cosinum ex dato arcu .r ; et de qua posteriore dieis, quantitatem A quae aequalis est radio r ex aequatione nuxionali non determinari. Ego non solum inveni, Seriem non posse habere hanc formam A + Bx 2 + Gx i + Dx a itc nisi fiat A = r, sed utramque a te exhibitam Seriem comprehendi sub alia generaliori, quae haec est : y = A + Bx + Cxx + Dx* + Ex i + &c in qua coefficientes A, B, G, D &c hanc sequuntur relationem rr 1 . 2 . ?•/• 2 . 3 . rr E= ( ±—, F = ^— &c 3 . 4 . rr 4 .5 .rr Si fiat A = 0, habetur Series pro Sinu ; sin autem A fiat = 7", habetur Series pro cosinu ; sin vero A alium habeat valorem praeter hos duos, etiam alia Series praeter duas exhibitas erit radix aequationis fluxionalis propositae. Similiter Series illae quatuor, quas exhibes pag. 84. pro radice aequationis y + a 2 y — xy — x 2 y = 0, sub aliis duabus generalioribus quae ex tuis particularibus compositae sunt, comprehenduntur. Duae nempe priores sub hac y = A + Bx + Give + Dx^ + Ex* + &c. in qua coefficientes A et B habent valores arbitrarios, reliqui autem C, D, E, &c sequentem ad priores habent relationem 0-oa D== lz^.B, E=tZ^C, F = ^^D&e. 1.2 2.3 3.4 4.5 Si B = habetur tuarum Serierum prima, Si .4 = habetur secunda. Duae posteriores comprehenduntur sub hac generali forma y = Ax a + Bx' a + Cx a - 2 + J)x-"- 2 + Ex a - i + Fx^^ + iScc. ubi iterum A et B habent valores arbitrarios, C=-^A, E=-^^C, 4 . a — 1 a .a— & ., « — 4.a — 5 n a.a+l n g= - 12.a-3 E > &C > D= 47a+l B > u+2.a + 3 a + i.a + B 8.a+2 12.a + 3 L2 148 STIRLING'S SCIENTIFIC CORRESPONDENCE Si fiat B = exsurgit tua tertia Series, et si fiat ^4 = exsurgit quarta, in qua termini per signum + non per signum — connecti clebent. Jncomodum quoque est in tuis Seriebus, quod Literae A, B, C, D itc mox pro coefficientibus terminorum, mox pro ipsis terminis usurpentur. Hac data occasione describam liic ea quae ad quasilam tuas Series in Libro tuo de Enumerationc Linearum tertii ordinis contentas notaveram eo tempore, quo hunc Librum a Dno de Maupertuis comodatum habebam. Eum quidem nunc non liabeo, sed in quad am mea Scheda haec notata reperio. In Exempli) 2 pay. 22. aequationis x^ij + ayxx + a 2 xx — 2a ? 'x = radix y est = . (lA+mi aaA—a 3 a ?J A — a i a i A—u 5 A + —^ + ~n^ - + t-t^ + 2^^? + &c quando A = provenit Stirlingii solutio ; sed quando A — a 2«« exsurgit y = a + • In Ex. 4. pag. 26. y-j; 2 —3x 2 xy + 2x-x-—a,iy 2 + & dividendo per 2, ac AB 6 aD d ■ r a 7> n -u j ad n dp.AF BP.af pro AB aB, Kcnbendo AB aB, ' = — ^ . AB J aB d LETTER FROM KLINGENSTIERNA 167 Centris A & a intervallis AB, aB describantur arcus BE & Be rectis AD & aD occurrentes in E & e, & erit triangulum BBE simile triang. Ddp, triangulum DBa simile triang. dDP. Quare DB : Dd = BE: dp, & DB : Dd = JBe : DP, adeo- que ex aequo BE : dp = Be : DP Si itaque in aequatione -±— = -l^i pro dp & DP substituantur earum propor- AB* aB* r 1 BE.AF Be.af aB* tionales BE & Be, habetur AB' aB" Centris A & a intervallis AF & uf describantur circum- ferentiae FQ & fq, rectis AB, AD, atque aB, aD occurrentes in I, L, & •/., I, eritque ob similitudinem triangulorum ABE, BE AF AIL AB:BE=AI (id est AF): IL, unde A£ = IL. 168 STIRLING'S SCIENTIFIC CORRESPONDENCE Similiter ob similitudinem triangulorum aBe, ail, erit . , 7 . Be . af ., aB:Be= ai (id est af) : il, unde — -jf- = %l. BE.AF Be.af BE.AF , Be.af Ergo si in aequatione AR , - -^- pro A£ & -^- substituantur IL & il, transit ilia in hanc : -j-™ = ~R2 " Ad rectam Aa demittantur normal es LH, IG, BC, ig, Ih, ipsique Aa parallelae IN, in, rectis LH, Ih occurrentes in N, n. Propter similitudinem triangulorum ABC, AIG, LIN, est AB:BG = AI (id est AF) : IG, & AB:BC= . . LI: IN; quare terminis ordinatim in se ductis AB 2 :BC 2 = AFxLI:IGxIN, unde ^= j^, . Similiter propter similitudinem triangulorum aBG, aig, lin, est aB : BC = ai (id est af) : ig aB : BC = . . li:in; quare terminis ordinatim in se ductis aB 2 :BC*= . . af.li:ig .in; unde ^ = ~^ a , . IL U IG-.IN iq.in Sed inventum erat -^ = ^, ergo ^^-^ =-. _^., & multiplicando per BC 2 , — ^= — = ' ■ Est vero IG . IN elementum circuli IGHL, it igr . in elementum circuli iglil IGHL ighl , AF IGHL quare __ = _ , ade0 que ^ = — w • Sit 5 locus datus unde prodit viator. Jungantur AS, aS circumferentiis FQ,fq occurrentes in R, r & demittantur RT, rt perpendiculares ad Aa. Et cum per jam demonstrata, elementa IGHL, ighl ubique sint in data ratione AF ad af, erit etiam componendo, Summa IGHL, id est spatium RTHL, ad suiiiam omnium ighl, id est spatium rthl, in eadem data ratione AF ad Af, unde sequens prodit Constructio. LETTER FROM KLINGENSTIERNA 169 Centris A & a descriptis circulis FQ.fq, quorum radii AF, af sint proportionates viribus calefaciendi ignium A & a, jungantur AS & aS, circulis illis occurrentes in R & r, & demittantur Fig. 34. Rt, rt, normales ad Aa Rectis LH, Ih, itidem normalibus ad Aa, abscindantur Spatia TRLH, trlh, quae sint in ratione AF ad af. Jungantur & producantur AL & al, donee conveniant in D, & erit punctum D in curva quaesita 8D. Problema. Invenire curvas AGBC & AHB1, quarum talis est ad se invicem relatio, ut curva prior AGBC rotata circa polum fixum A semper secetur ab altera AHB1 in punctis summis B, b, & ut segmenta AGBA, AHBA semper sint in data ratione m ad ??. Solutio. Rotetur curva AGBC circa punctum fixum A, donee perveniat in situm proximum AFDG, in quo situ secetur a curva AHBI in b. Centro A intervallo AB describatur arcus BD curvae occurrens in D, & jungantur AD, Ab, quarum haec occurrat arcui BD in E. Et quia A GBA : AHBA = m:n, \ r AFbA : AHbA = m : n, erit etiam dividendo A FbA - A GBA : AHbA - AHBA = m : n, id est, Triangulum ADb : triang. ABb = m:n, unde ob basin communem Bb, erit DE-.BE = m : h. . „„ 11 DE Dicatur AD, a:; Eb, dx; DE, dy; & erit EB = m (V BD = DE + ix DE m + n 'in dy. 170 STIRLING'S SCIENTIFIC CORRESPONDENCE Et quoniam per hyp. tangens curvae 1GBC in B parallela est tangenti ejusdem in b, erit angulus rotationis BAD aequalis angulo quern duae rectae ad curvam normales in punctis D & b constituunt in centro cireuli osculatoris. Ergo AD:DB = radius curvedinis in D : ad elementum curvae Db, id est , ,. , _,, , , m + n , xds 2 dx (dicto Db = ds)x : dy = m dxdyds — xdydds' m + n . m + n 7 7 ay .xds^dx dsdx j 7 m , „ m adeoque xds = vel 1 dxdyds — xdydds dsdx — xdds' nnde dsdx — xdds = dsdx, m 7 7 n 7 7 , . n dx dds seu —xdds = — dsdx, nine = =— > m m x ds sumtisque logarithmis — l- = l-^, & perficiendo quod restat reductionis : n x m dx , = = dy \ja m —x m Centro A, intervallo a describatur circulus, cujus elementum rectis AD, Ab comprehensum dicatur dz, eritque x:dy = a:dz, unde dy = > & hoc valore substituto in aequatione modo inventa * x m dx = dy, 4 *2 it 2 n transformatur ilia in hanc m a. n dx xdz j 'In tn a \a m —x m ax m dx 7 seu — _______ — dz. j -in a" 1 — „"' LETTER FROM KLLNGENSTIERNA 171 Ponatur x V a — a m iv aequatio transibit in hanc ; adv n Vaa — = dz vv quae sequentem suppeditat Problematis Constructionem. Cen- tro A intervallo quo vis AB describatur circumt'erentia circuli, in qua hinc inde a puncto quovis dato B sumantur arcus BC, BD Fig. 35. in ratione n ad m. Jungantur AC, AD & a puncto (7demittatur CE normalis ad radium AB. In AC & AD sumantur AF CE & AG aequales AB . -jt. & erit punctum F in eurva fixa AHBT, A punctum G in curva rotatili AGBC. Coroll. Si fuerit m ad n ut numerus ad numerum, utraque curvarum est Algebraica,sive minus, earum constructio dependet a multisectione anguli it rationis. seu quod idem est quadratura circuli & hyperbolae. JX MACHIN AND STIRLING (l) Machin to Stirling (1733?) To M r Stirling at the Academy in little Tower Street Dear Sir I intend to give you some short notes upon M r Bernoulli's Letter, w ch if you approve of it shall be addrest in a Letter to yourself. It shall be ready against the beginning of next week, unless anything material happen to hinder it. I have reason to believe that if he be a man of any candour, I shall be able to give him entire satisfaction as to every objection that he makes, & do intend withal to oblige him w t!l the solution of a Problem w oh I now percieve he had proposed to himself but quitted rather than be at the pains to go through w th it. And that is whether there be a point in his locus from whence the Planet will appear to move equally swift in the Apsides cv one of the middle distances. And where it is that y e point lyes. As I apprehend he may have communi- cated some of his remarks to others as well as yourself or may have hinted that he has made some; I should be glad to a word or line know by the bearer, whether you will give me leave to shew this Letter to the Society upon the foot of there being some new Problems in it, w ch may furnish me w th the opportunity of saying that his Objections are to be answered. I do not mean to have the Letter read, but only to have the Contents of it mentioned & especially the Problems since he seems to have sent those on purpose to be proposed to others. I shall CORRESPONDENCE WITH MACHIN 173 herein behave according to the directions you are pleased to give. E r . Your most faithful Friend & very humble Serv 4 Thursday morning J. Maciiin. (2) Maciiin to Stirling, 1738 ^ . Gresham College June 22. 1738 Dear Sir 6 The date of your obliging Letter when I cast my eye upon it gives me great concern. I was ashamed when I received a Letter from you to think you had prevented me in paying my respects to you first, but am now confounded in the reflection of having slipt so long a time without return- ing an answer to it. Sure I am in the case of Endymion! But every day has brought its business and its impertinence to engage me and to interrupt me. Were there time I could plead perhaps more things in my excuse than you may be apt to imagine. This long vacation which begins today, appeai-s, if it deceive me not in my expectation, as one of y° greatest blessings I have long since enjoyed. If I am tardy after this, then believe (what would grieve me if you should believe) that you are one that are not in my thoughts. Think not that you are singular in your retirement from y e world. There may I can assure you be as great a solitude from acquain- tance & conversation in a Town as in a Desert. But of this sufficient. Mons r Maupertuis has sent you a present of his book which I have deliverd to M r Watts for you. It contains a complete account of the measurement in the North. M r Celsius likewise published two or 3 sheets on y° same subject chiefly to shew that Cassini's measurement was far inferior to this in point of exactness, and which I suppose you will need no argument to prove when you have read over M. Maupertuis's book. We have also had from time to time scraps of accounts communicated to us, still in expectation of something more perfect, w ch I intended to have sent to you, but this book has rendered it unnecessary. 174 STIRLING'S SCIENTIFIC CORRESPONDENCE There have been great wrangles and disputes in France about this measurement. Cassini has endeavoured to bring the exactness of it into Question. Because the Gentlemen did not verify the truth of their astronomical observations, by double observation with y b face of their Instrument turned contrary ways. So that M 1 ' Maupertuis was pat to the necessity of procuring from England a certificate concerning the con- struction of M 1 ' Graham's Instrument, to show that it did not need that sort of verification. You will see that this measurement in y e North, if it be compared with y* in France, will serve to prove that y e figure is much more oblate than according to y e rule. But perhaps it will be safer to wait for the account from Peru before any conclusion be drawn. These Gentlemen have also compleated their work and are returning home where they are expected in a short time. Mons r De Lisle has published a Memoir read in the Academy at Petersburg w ch contains y e scheme of a Grand Project of the Czarina for making a compleat Mapp of her whole Empire, and in w ch there is a design of making such a measure- ment not only from North to South but from East to West also as will far surpass any thing that was ever yet thought of ; it being to contain above 20 degrees of y e meridian and many times more in the parallels. Your Proposition concerning y e figure (wherein all my friends can witness how much I envy you) could never find a time to appear in the world with a better grace than at present, Now when y° great Princes of y e Earth seem to have their minds so fix't upon it. But for other reasons I should be glad if your Proposition could be published in some manner or other as soon as possible, but not without some investigation at least : unless you have hit upon a Demonstration w ch would be better, because I find several people are concerning themselves upon that subject. I have kept your paper safe in my own custody, nor has any one had the perusal of it. Nor shall I believe that any one will find it out till I see it. But M r Macklaurin in a Letter to me dated in febiy last, (and w ch was not deliver'd to me but about a month ago, the Gentleman being ill to whose care it was entrusted) taking occasion to speak of y e figure of y 3 Earth, and that CORRESPONDENCE WITH MACHIN 175 S r Is. had supposed but not demonstrated it to be a Spheroid, proceeds on in the following words, ' M r Stirling if I remember right told me in April that none of those who have considered this subject have shewed that it is accurately of that figure. I hit upon a demonstration of this since he spoke to me w cl! seems to be pretty simple.' I have given you his own words for fear of a mistake, because I am surprised you did not take that opportunity to inform him, that you had found it to be of that figure. For that nobody has yet shewn it to be so is what I thought everybody had known. But I shall take this opportunity to advise him to communicate his demonstration to you. And if he has found out a simple demonstration for it, I think it ought to be highly valued, for it does not seem easy to come at it. I own I have not had time to pursue a thought I had upon it, and which I apprehended and do still apprehend might lead to the demonstration and shall be very glad if he or any one else by doing it before shall save me that trouble. As to y e Invention of M r Euler's Series were I in your case I would not trouble myself about it, but let it take its own course, if anything should arise your Letter to me w cl1 I shall keep will be a sufficient acquittal of yourself. M r Moivre's Book is now published but I have not got it yet nor have I been able to see him but once since I reced your Letter and as to this conveyance I was but just now apprized of it and have but just time to get this ready before M r Watts goes out of Town. As to y c moon's Distance I have now materials to fix y e moon's Parallax, and chiefly by means of an Observation of the last Solar Eclipse at Edinburgh by M r Macklaurin, and will take care as soon as I can make y e calculation to send it to you. There are some other matters whereto I should speak which I must now defer to another opportunity, and only say now that I am with affectionate regard Your most faithful friend & very humble Servant John Maohin. X CLAIRAUT AND STIRLING (1) Clairaid to Stirling, 1738 Monsieur En cas qu'un Memoire sur la Figure de la Terre que j'envoyai cle la Laponie a la Socie'te Royale, soit parvenu jusqu'a vous efc que vous l'ayt^s daigne lire, vous y aur^s reconnu plusieurs Theoremes dont vous avie's donne" auparavant les enonce's, parrni les belles decouvertes dont est rempli un morceau que vous ave"s insere dans les transact. Philosoph. de l'annde 1735 ou 1736. Vous aures 6t6 peut-etre etonne' que traitant la meme matiere que vous je ne vous aye point cite - . Mais je vous supplie d'etre persuade - que cela vient de ce que je ne connoissois point alors votre Memoire, et que si je l'eusse Id je me serois fait autant d'honneur de le citer que j'ai ressenti de plaisir lorsque j'ai appris que je m'dtois rencontre" avec vous. Depuis le terns ou j'ai donne cette Piece j'ai pousse" mes recherches plus loin sur la meme matiere, et j'envoye actuelle- ment mes nouvelles decouvertes a la Society Royale. Apre*s vous avoir fait ce recit Monsieur et vous avoir prie" d'excuser la liberty que j'ai pris de vous ecrire sans avoir l'honneur d'etre connu de vous, oserois je vous demander une grace, c'est de vouloir bien jetter les yeux sur mon second Memoire que M r Mortimer vous remettra si vous le daigne's lire. Ce n'est pas seulement l'envie d'etre connu de vous qui m'engage a vous prier de me faire cette grace, Mais c'est que j'ai appris par un ami qui a vu a Paris un Geometre anglois appelle - M. Robbens que vous avie's depuis peu travaille' sur la meme matiere. LETTER FROM CLAIRAUT 177 Je souhaiterois done extremement de scavoir si j'ai &t£ asses heureux encore pour m'etre rencontre avec vous. Si au con- fcraire je m'etois trompe je vous serois infinimeiifc oblige 1 de me le dire franchement afin que je m'en corrigiasse. Quoi qu'il en soit si vous daigne"s me donner quelques momens, vous aure*s bientot vu. de quoy il est question et si mon memoire m'attire une reponse de vous je serai charme" de l'avoir fait parce qu'il y a deja longtems que je souhaite d'etre en liaison avec vous. Quelqu'envie que j'en aye ne croy^s pourtant pas Monsieur que je soye asses indiscret pour vous impor tuner sou vent par des lettres inutiles pleines de simples complimens. M 1 ' Mortimer pourra vous dire quelle est ma conduite a son egard, J'en oserai de meme avec vous si vous me le perinett^s. En attendant j'ai l'honneur d'etre avec estime et respect Monsieur Votre tre"s humble et tre\s a Paris le 2 Octobre 1738 obeissant Serviteur Ciaieaut. P.S. En cas que vous veuille's me faire reponse il faudra avoir la bonte - de remettre votre lettre a M. Mortimer. Si vous n'aimes a ecrire en francois, je dechiffre ass^s d'anglois pour entendre une lettre et quand ma science en cette langue ne suffiroit pas, j'aurois facilement du secours. XI EULER AND STIRLING (l) Stirling to Euler, 1738 x Celeberrimo Doctissimoque Viro Leonhardo Euler S.P.D Jacobus Stirling mihi Tantum temporis elapsum est ex quo dignatus es (ad me) scribere, ut jam rescribere vix ausim nisi tua humanitate fretus. Per hosce duos annos plurimis negotiis implicitus sum, quae occasionem mihi dederunt frequenter eundi in Scotiam et dein Londinum redeundi. Et haee in causa fuerunt turn quod epistola tua sero ad manus meas pervenit, turn quod in hunc usque diem vix suppeterat tempus eundem perlegendi ea qua meretur attentione. Nam postquam speculationes sunt diu interruptae, ne dicam obsoletae, patientia opus est ante- quam induci possit animus iterum de iisdem cogitare. Hanc igitur primam corripio occasionem testandi meam in te Obser- vantiam et simul (gratias) agendi gratias dudum debitas propter literas eximiis inventis refertas. Gratissimum mihi fuit Theorema tuum pro summandis Seriebus per aream Curvae et differentias sive fluxiones Statim Terminorum quippe generale et praxi expeditum. (Illius) percepi item extendi ad plurima serierum genera, et quod celerrime praecipuum et a plerumque (celeriter) approximat. Forte non observasti theorema meum pro summandis Logarithmis Tui nihil aliud esse quam casum particularem tui Theorematis a 1 This is only Stirling's rough draft with all his corrections. Erasures are indicated by brackets. CORRESPONDENCE WITH EULER 179 eo generalis ; (quod ingenue fateor). Sed et A gratius milri fuit quod (tuum) hunc inventum, (quoniam) de eodera (ego) quoque ego olim cogitaveram ; sed ultra primum terminum non processi, appioximav pro libitu et per eum solum (perveni satis expedite) ad valores Serierum satis expedite A scilicet per repetionem calculi, ut in resolutione aequa- tionum affectarum ; cujusque specimen dedi (plurimis abhinc annis) in philosophicis nostris transactionibus : Quae babes de inveniendis Logaritbmis per Seriem Harmo- saltem nicam (non percipio, propter novi) obscura mini A videntur, notationem quoniam A non recte intelligo (notationem.) Imprimis autem mihi placuit methodus tua summandi quasdam Series per potestates periferiae circuli, (quarum indices sunt numeri pares). Hoc fateor (omnino novum et) et omnino novum habeat admodum ingeniosum a nee video quod A quicquid commune methodis receptis (affin habeat) cum (iis quae hactenus publicantur,) adeo ut ciedam hausisso facile (concedam) te idem (hausisse) ex novo fonte A ((et nullus dubito to hactenus observasse, aut certe ex lundamento tuo facile percipies, alias series tuis tamen affines summari posse per potestates periferiae quarum indices sunt numeri impares. Verbi gratia, denotante p periferia, \p—\— -| + 3 — t + |— &c ut vulgo notum 5 1111, 1536 3 5 5° 7° 9 5 *c.)) continentur in Series tuae (comprehenduntur sub) forma generali 1 1 1 11. 1 4 _ 1 _i_ _l_ 4- J_ Ato "•" 2" 3™ 4™ 5" 6™ ubi n est numerus par) eadem (tamen ad formulam sequen- n2 180 STIRLING'S SCIENTIFIC CORRESPONDENCE tem) nullo negotio reducitur, (scilicet) reducitur ad formulam sequentem, 1111 1 „ 1 + T* + 5* + r« + ¥- + n» + &c (ubi termini alterni desunt, et omnes sub hac forma compre- ct hanc summave hensas summare) A doces A per potestatem periferiae cuj us index est n modo sit Ceterum si (quando n est) numerus par. (Si jam mutentur) signa terminorum alternorum mutentur ut evadat Series 1 1 1 1 1 — r, H r - _l + &c. 3" 5" 7" 9" 11" Haec inquam semper summari potest (atque haec Series, quando n est numerus impar sumari potest) per dignitatem modo sit numerus impav periferi (circuli) cujus index est n. (verbi gratia) utique si sit n = 1 , (erit) \p = 1 — A + § — \ + I — T X T + &c ut vulgo notum n = 3, 3 2 P 3 = 1- F +^- 1 1 Y 6 + 9* ~~ 1 r 5 11 Tr — 1 1 1 1 ~ 7 1 + ¥ -T^+*e 1536 ' 3 5* yi = 5 ' 1536 l 3 Et nullus dubito te hactenus idem observasse, ant saltern facile observatur ex fundamento tuo quod libenter videbo, quando (animus erit tibi idem impertire) ita tibi vi»um fuerit. monendus es Matheseos Hie autem (aequum est ut te moneam) D. Maclaurin a pro- fessorem (Matheseos) Edinburgi, post aliquot tempus (brevi) jam editurum librum de fluxionibus cujus paginas aliquot (hactenus) impressas (mecum) mecum communicavit in quibus duo habet Theoremata pro summandis seriebus per differentiae termi- norum, quorum alterum ipsissimum est quod tu dudum mihi (ad me) misisti, (et cujus ego eum illico certiorem feci). Et etiam si ille libenter promiserat se idem testaturum in sua praefatione, judicio tamen tuo submitto annon velles (edere) edere tuam epistolam a in nostris philosophicis transactionibus. CORRESPONDENCE WITH EULER 181 Et si vis quaedam illustrare vel demonstrare, (aut plura n.. ut lucein videat aclpcere, ego aut) et cito mihi reseribere, curabo (tuam epistolam viseram lucem diu) nntequam ejus liber prodierit. Quod si animus erit hac (data) occasione eligi unus ex Sociis nostrae Societatis (Academiae) Regiae, idem reliquis gratum (esse non) proeul quando viderint pracclara tua inventa dubio gratum erit (postquam inventa tua viderint Et) mihi vero semper gratissimum ut amicitiam (mihi licet immerenti) continuare digneris Edinburgi 10 Aprilis 1738 (2) Euler to Stirling, 1738 Illustrissimo atque Celeberrimo Viro Jacobo Stirling S. R D. Leonhard Euler Quo majore desiderio litteras a Te Vir Celeb, expectavi, eo majore gaudio me responsio Tua humanissima affeeit, qua, eo magis sum delectatus, quod non solum litteras meas Tibi non ingratas fuisse video, sed Temet etiam ad commercium hoc inceptum continuandum invitare. Gratias igitur Tibi habeo maximas, quod tenues meas meditationes tarn benevole accipere Tuumque de iis judicium mecum communicare volueris. Epistolam autem meam a Te dignam censeri, quae Transactionibus Vestris inseratur, id summae Tuae tribuo humanitati, atque in hunc finem nonnullas amplificationes et dilucidationes superaddere visum est, quas pro arbitrio vel adjungere vel omittere poteris. Hac autem in re quicquam laudis Celeb. D. Maclaurin derogari minime vellem, cum is forte ante me in idem Theorema seriebus summandis oserviens incident, et idcirco primus ejus Inventor nominari mereatur. Ego enim circiter ante quadriennium istud Theorema inveni, quo tempore etiam ejus demonstrationem et usum coram Academia nostra fusius exposui, quae dissertatio mea pariter ac ilia, quam de Summatione Serierum per potestates peri- 182 STIRLING'S SCIENTIFIC CORRESPONDENCE pheriae circuli composui in nostris Commentariis, qui quotannis prodeunt, brevi lucem publicam aspiciet. In Commentariis autem nostris jam editis aliquot extant aliae methodi meae Series summandi quarum quaedam multum habent Similitu- dinis cum Tuis in egregio Tuo opere traditis, sed quia turn teinporis Tuum methodum differentialem nondum videram, ejus quoque mentionem facere non.potui, uti debuissem. Misi etiam jam ante complures annos ad Illustris. Praesidem Vestrum D. Sloane schediasma quodpiam, in quo generalem constructionem hujus aequationis y = yyx + ax m x dedi, quae aequatio ante multum erat agitata, at paucissimis tantum casibus exponentis m constructa. Haec igitur Dister- tatio, si etiamnum praesto esset, simul tanquam specimen produci posset, coram Societate vestra, quando me pro membro recipere esset dignatura, quern quidem honorem Tibi Uni Vir Celeber. deberem. Sed vereor ut Inclytae Societati expediat me Socium eligere, qui ad Academiam nostram tarn arete sum alligatus, ut meditationes meas qualescunque hie primum pro- ducere tenear. Ut autem ad Theorema, quo summa cujusque Seriei ex ejus termino dicto generali inveniri potest, revertar, perspicuum est formulam datam eo majorem esse allaturam utilitatem, quo ejus plures termini habeantur, summa autem difficile esse videtur, earn quousque lubuerit, continuare. Equidem ad plures quam duodecim terminos non pertetigi, quorum ultimos non ita pridem demum inveni ; haec autem expressio se habet ut sequitur. Si Seriei cujuscunque terminus primus fuerit A, secundus B, tertius C, etc. isque cujus index est x sit = X : erit summa hujus progressions, puta A + £ + C+etc... + X = \xdx + ^+ '' X 1.2 1 . 2.3.2ote d s X d'X + 1.2.3.4.5. ticte 3 1.2.3.4.5.6.7. Gduf 3d'X 5d 9 X + 1.2.3 ...9. lOdx 1 1.2.3 ... 11 .&dx 9 CORRESPONDENCE WITH EULER 183 691d u Z 35d 13 Z + 1.2.3 ... 13. 210c/ce u 1 . 2 . 3 ... 15 . 2 da; 1 ' 3617d 15 Z 43867d 17 X 1 .2.3 ... 17.30d^' 5 1 .2.3 ... 19.42da=' 7 1222277d 19 X 1.2.3... 21.110da: 1! etc. ubi fluxio dx constans est posita. Haec autem expressio parumper mutata etiam ad summam seriei a termino X in infinitum usque inveniendam accommodari potest. Hujus vero formae praeter insignem facilitatem, quam suppeditat ad summas proxime inveniendas, eximius est usus in veris summis serierum algebraicarum investigandis, quarum quidem summae absolute exhiberi possunt, ut si quaeratur summa hujus progressionis potestatum l + 2 12 +3 12 +4 12 +5 ,2 +,.. + a; ]2 , erit X = x™, [Xdx = -^a: 13 , ~= 12a; 11 , J 13 da; d 3 X d 13 X —r^r = 10 . 1 1 . 1 2 . a; 9 , et ita porro, donee , ... da; 3 • v dx 16 una cum sequentibus Terminis = Hinc igitur resultabit summa quaesita = x 13 x 12 u Ha; 9 22,r 7 33a; 5 5a? 3 _ 691a; _ + _ +x — + — - To" + 3 ~ 2730' quam summam nescio, an ea per ullam aliam methodum tam expeditam inveniri queat. Potest autem hac ratione aeque commode definiri summa hujus progressionis l + 2 21 + 3 21 + 4 21 + ...+a! 21 , quod per alias vias labor insuperabilis videtur. Sin autem seriei propositae termini alternativi signis + et - fuerint afi'ecti, turn theorema istud minus commode adhiberi posset, quia ante binos terminos in unum colligi oporteret. Pro hoc igitur serierum genere aliud investigavi Theorema priori quidem fere simile, quod ita se habet. Si quaeratur summa hujus seriei A-B + G-D+...±X, 184 STIRLING'S SCIENTIFIC CORRESPONDENCE ubi X sit terminus cujus exponens seu index est a; habetque signum vel + vel — prout x numerus erit vel impar vel par. Dieo autem hujus progressionis summam esse { X . clX d 3 X = Const. + ( ^ + + - V12 1.2.2dx 1.2.3.4.2cfar 3d a X 17u 7 X 1 .2.3 ... 6.2cte 5 1 .2.3 ... 8.2cte 7 155d"X 2073d n X + 1.2.3 ... 10.2t^' J 1 .2.3 ... 12.2dx 11 33227d 13 X _L Sirn 1.2.3 ... 14.2cte 13 Constantem autem ex uno casu, quo summa est eognita, deterrninari oportet. At si series summanda connexa sit cum Geometrica pro- gressione hoc modo An + Bn 2 + Cn* + ...+ Xn x turn minus congrue utrumque praecedentium theorematum adhiberetur. Summa enim commodius invenietur ex hac expressione ,pl adX (3d 2 X Const. +n{ n _ l - 1( ^ n _ 1} , dx + U2 (n- \fdx 2 yd z X 8d*X 1.2.3. {n-l^dx 6 + 1.2.3.4 (n- \fdx" — etc. ) valores autem coefficientium a, /3, y, S, etc sunt sequentes a = n /J = n 2 + n y = n 3 + 4n 2 + n S = n* + 11 n* + 11 n 2 + n e = n 5 + 26n' + 66n z + 26n 2 + n etc. cujus progressionis legem facile inspicies. En igitur tria hujuS generis Theoremata, quae singula certis easibus eximiam habe- bunt utilitatem ad summas serierum indagandas. CORRESPONDENCE WITH EULER 185 Quod deinde attinet ad summationes hujusmodi serierum, quae continentur in hac 1111 1+27.+ 3S+4S+5S+ etC - existente n numero pari eas duplici operatione sum consecutus, quarum alteram uti recte conjectus Vir Celeb, deduxi ex serie 1+ h 1 — + etc. altera vero immediate mihi 3 n 5" 7" illius summam praebuit. Priore modo utique summas etiam hujusmodi serierum 1 - — + — - — + ^ - etc. existente n numero impare detexi, invenique eas se habere, pror&us ac Tu indicas. Sunt autem summae tam pro paribus quam im- paribus exponentibus n sequentes p 1111 t. = l 1 1 etc. 4 3 5 7 9 « 2 1111, — = 1-1 1 i H ; H ; + etc. 8 + 3- 5 2 7* 9 2 p s 1111, -c_— i i f- — — etc. 32 3 3 5 3 7 s 9 J V* 1 1 I l l4 . i— = 1A 1 1 — H — t+ etc. 96 3 4+ 5> ^ 7 1 T 9 4 5w> 111,1, i — 1 1 h — — etc. 1536 _ 3 5 + 5 s 7> 9° «« I 1 1 1 , , £__ _ i j i 1 — - + -= + etc. 960 _ + 3 U + 5« + 7° + 9" 61P T _ , 1 A 1 _ I + i _i u 1 — - — etc. ~ * 3 7 5 7 7 7 ^ 9 7 194320 3 17/> 8 1 1 1 l x Til280 ~~ + 3 8 + 5* + 7 S ^ 9 8 etc. quae series omnes continentur in una hac generah : existente n numero integro. Si enim n est numerus par, turn 186 STIRLING'S SCIENTIFIC CORRESPONDENCE omnes termini habebunt signum + ; sin autem n sit impar, turn sign a sese alternating insequentur. Omnes autem has summas derivavi ex hac aequatione infinita ; s s 3 s 5 = 1 h h etc. 1 .a 1 .2. 3. a 1.2. 3. 4. 5. a qua relatio inter arcum s ejusque sinum a exprimitur in circulo cujus radius est 1. Quoniam igitur eidem sinui a innumerabiles arcus s respondent, necesse est. Si s consideretur tanquam radix istius aequationis, earn habit uram esse infinites valores, eosque omnes ex circuli indole cognitos. Sint ergo A, B, C, D, etc. omnes illi arcus, quorum idem est sinus a erit ex natura aequationum S 6 3 S 5 1 h \- etc. 1 . a 1 . 2 . 3 . a 1.2. 3. 4. 5. a = (•-2) O-Dc ->- Posita nunc ista fractionum serie ex omnibus illis arcubus formata ~r> -^5 ^. =r etc. perspicuum est sumam hanc frac- A B C D r r tionum aequari coefficienti ipsius — s qui est = -; seu fore - = +_+-+-+ etc. Simili modo summa factoru e^ a A B V B binis f'ractionibus aequatur coefficienti ipsius s 2 qui est = 0. unde erit 1/1 1 1 x \ 2 ! / ! 1 , 1 , , \ ° = 2 (l + B + G + 6tC ) ~ 2 U» + W + G* + ei V' 1 1 1 1 ! , l Porro summa factoru ex ternis fractionibus aequalis esse debet coefficienti ipsius — s 3 , qui est = — —» unde deducitur summa cuboru illarum fractionu, 1 1 1 _1_ t _1 J_ Z 3 + 5» + C 3 " + B 3 + ~ a 3 2a' CORRESPONDENCE WITH EULER 187 atque ita procedendo summae reperientur omnium serierum in hac generali —+ — ++_+ e tc. comprehensarum dummodo pro n sumatur numerus integer affirmativus. Si nunc pro sinu indefinito « ponatur sinus totus 1, illae ipsae oriuntur series quas Tecum communicavi. In istis autem summis notari meretur insignis affinitas inter coefficientes numericos harum sumaru, atque terminos superioris progres- sionis, quam primurn ad series quascunque summandas dedi, nempe hujus J v 7 A' F« + 3" + 4* + 6* + ^ 2". 43867 lg 1 1 , J_, J_, etc TTJ^l9T* p = + ^ ^ ^ 5 ' B 2 19 . 3222277 20 , , 1 , 1 , * , J_ + etc r^sTT^TTTIo^ = 1+ 2* + 3^0 + i* + 6* + etc - etc. 188 STIRLING'S SCIENTIFIC CORRESPONDENCE Hac scilicet convenientia animad versa mihi ulterius progrecli licuit, quam si rnethodo genuina inveniendi coefficientes potestatu ipsius p, usus fuissem quippe qua labor nimis evaderet operosus. Quamobrem non dubito, quin nexu hoe mirabili penitius cognito (mihi enim adhuc sola constat obser- vatione) praeclara adjumenta ad Analyseos promotionem sese sint proditura. Tu forte Vir Celeb, non difficulter nexum hunc ex ipsa rei natura derivabis. Dura haec scribo, accipio a Cel. Nicolao Bernoulli Prof. Juris Basiliensi et Membro Societatis Yestrae singularem demonstrationeru summae hums seriei 1 H — , H ; H , + etc. J 3 2 5 2 7'-' quam deducit ex summa hujus notae 1— J + i - \+ etc. illam considerans tanquam hujus quadratum minutum duplis factis binorii terminorum. Haec autem dupla facta seorsim con- templans multiferiam transformat, tandemque ad seriem quandam regularein perducit, quam analytice ostendit pariter a Circuli quadratura pendere. Sed hac rnethodo certe Viro Acutissimo non licuisset ad summas altiorum potestatum pertingere. Eodem incommodo quoque laborat alia quaedam methodus mea, qua directe per solam analysin hujus seriei summam 1+ — + — h — - + etc. inveni, ex qua pariter nullam utili- tatem ad sequentes series summandas sum consecutus. Haec autem methodus ita se habet: Fluentem hujus fluxionis > qua arcus circuli exprimitur cuius sinus est = x V(l-.rx) l existente sinu toto = 1, multiplico per ipsam fluxionem quo prodeat facti fluens = %ss, posito s pro arcu V(\— xx) illo cujus sinus est = x Si ergo post summationem peractam ponatur x = 1, fiet s =^> denotante p ad 1 rationem peri- pheriae ad diametrum ; ita ut hoc casu habeatur — • Fluens autem ipsius -— — per seriem est 1 1.3 . 1.3.5 . , — x -\ x 3 A x" -\ x' + etc. + 2.3 + 2.4.5 ^2.4.6.7 CORRESPONDENCE WITH EULER 189 Ducantur nunc singuli termini in fluxionem et v (1— XX) sumantur fluentes ita ufc fiant = posito x = 0, turn vero = l- V(\-xx) = 1, ponatur x — 1 Ita reper' ^ 1 "' ] posito x = 1. Simili modo erit r u X V ( 1 — xx) ! i 1 2.3^ , 1.3 atnue '(l-.w) _ 3.3' X s ' X 1 1 2.4. V {I -xx) 5.5 et ita porro, adeo ut tandem obtiueatur ]>* , 1 1 1 g - ! + 15 + 53 + ^ + etC ' Sed huic argumento jam nimium sum immoratus, quocirca Te rogo Vir Celeb, ut quae Ipse hac de re es meditatus, mecum benevole communicare velis. Incidi aliquando in hanc expressionem notatu satis dign c am : 3.5. 7. 11 . 13. 17. 19.23.29.31 .37.41 4.4.8. 12. 12. 16.20.24.28.32.36. 40 cujus numeratores sunt omnes numeri primi nafcurali ordine sese insequentes, denominatores vero sunt numeri pariter pares unitate distantes a numeratoribus. Hujus vero ex- pressionis valorem esse aream circuli cujus diameter est = 1, demonstrare possum. Quamobrem haec expressio aeqnalis erit huic Wallisianae 2.4.4.6.6.8.8.10. 10 etc 3.3.5.5.7.7.9 . !l .11 etc. Ut autem novi quiddam Tibi Vir Celeb, perscribam Tuoque acutissimo subjiciam judicio, communicabo quaedam proble- mata, quae inter Viros Celeberrimos Bernoullios et me ab aliquo tempore sunt versata. Proponebatur autem mihi inter alia problemata hoc, ut inter omnes curvas iisdem terminis con- tentas investigarem earn, in qua r m ii haberet valorem minimum, denotante s curvae arcum, et r radium curvaturae, quod problema ope consuetaru methodorum, quales Bernoulii, 190 STIRLING'S SCIENTIFIC CORRESPONDENCE Hermannus et Taylorus Vester dedere, resolvi non potest, quia in r fluxiones secundae ingrediuutur. Inveni autem jam ante methodum universalem omnia hujusmodi problemata solvendi, quae etiam ad fluxiones cujusque ordinis extenditur, cujus ope pro curva quaesita sequentem dedi aequationem a m x + b m y = (m+ 1) r m k in qua x et y coordinatas ortho- gonales hujus curvae denotant. Hinc autem sequitur casu, quo to = 1, cycloidem quaestioni satisfacere. Deinde etiam quaerebatur inter omnes tantum curvas ejusdem longitudinis, quae per duo data puncta duci possunt ea, in qua r m s esset minimum; hancque curvam deprehendi ista aequatione indicari a m x + b m y + c m s = (to+ 1) r m s. Praeterea quaerebantur etiam oscillationes seu vibrationes laminae elasticae parieti firmo altero termino infixae, cui quaestioni ita satisfeci, ut primo curvam, quam lamina inter vibrandum induit, determinarem, atque secundo longitudinem penduli simplicis isochroni definirem, quod aequalibus teni- poribus oscillationes suas absolvat ; hinc enim intelligitur quot vibrationes data lamina dato tempore sit absolutura. Ego vero contra inter alia problema istud proposui, ut inveniantur super dato axe duae curvae algebraicae non rectificabiles, sed quarum rectificatio a datae curvae quadratura pendeat, quae tamen arcuum eidem abscissae respondentium summam habeant ubique reetificabilem ; cujus problematis dimcillimi visi, neque a Bernoullio soluti, sequentem adeptus sum solutionem. Posita abscissa utrique curvae communi = x ; sit alterius curvae applicata = y ; alterius vero = z. Assumatur nova variabilis u ex qua et constantibus variabiles x, y et z definiri debent, atque exprimat rectificatio utriusque curvae pendere debet; sintque p et q quantitates quaecunque algebraicae ex u et constantibus compositae. Quibus pro lubitu sumtis fiat V(l +pp) + V(l +qq) = r: V(l +pp) - -/(l + qq) = s, turn quaerantur sequentes valores A = h B = t; = 1 p p p Vw illam quadraturam, a qua CORRESPONDENCE WITH EULER 191 item £ = -; E= l i et F = *L. A A i> Ex his quantitatibus porro formentur istae 1- T , Q = 3 , et iJ = 3 . Ex his denique valoribus, qui omnes erunt algebraici sumta i> communi abscissa = _ , P fiat y = P^-R a t q ue z = qA-Bq +Q . P P hacque ratione, cum p et q sint quantitates arbitrariae pro- blemati infinitis modis satisfied poterit. Erunt enim ambae curvae algebraicae, atque utriusque rectificatio pendebit a fluente hujus fluxionis Vu. Summa vero amboru arcuum algebraice exprimi poterit. Est enim summa arcuum = rx-BR + DQ-P differentia vero eorum est = sx-GR + EQ-FP + Vu. Detexi autem pro resolutione hujusmodi problematum pecu- liarem methodum, quam Analysin infinitoru indeterminatam appellavi, atque jam maximam partem in singulari tractatu exposui. At tarn longam epistolam scribendo vereor ne patientiam Tuam nimis i'atigem: quamobrem rogo, ut pro- lixitati meae veniam des, eamque tribuas summae Tui existi- mationi, quam jamdudum concepi. Vale Vir Celeberrime, meque uti coepisti amicitia Tua dignari perge. dabam Petropoli ad d. 27 Julii 1738. XII FOLKES AND STIELING (1) Folkes to Stirling, 1747 Dear Sir After so many years absence I am proud of an oppor- tunity of assuring you of my most sincere respect and good wishes for your prosperity and happiness of all sorts. I received the day before yesterday of a Gentleman just arrived from Berlin, the enclosed Diploma which I am desired to convey to you with the best respects of the Royal Academy of Sciences of Prussia, and more particularly of M r de Maupertuis the President and M r de Formey the Secretary. M r Mitchell going your way I put it into his hands for you and congratulate you Sir upon this mai-k of the esteem of that Royal Academy upon their new establishment under their present President. Our old ffriend M r Montagu is well and we often talk of you together, and our old Master de Moivre whom we dined with the other day on the occasion of his compleating his eightieth year. I remain with the truest esteem and affection Dear Sir Your most obedient humble servant London June 10. 1747 M. Ffolkes. Pr. R.S. member of the Royal Academies of Sciences of Paris and Berlin, and of the Society of Edinburgh M 1 Stirling NOTES UPON THE CORRESPONDENCE MACLAURIN (1698-1746), F.R.S. 1710 Colin Maclaurin was bom at Kilmodan in Argyleshire, and attended Glasgow University. He became Professor of Mathematics at Aberdeen in 1717, and in 1725 was appointed to the chair of Mathematics in Edinburgh University. He died in 1746. His published works are Geometria Orr/auica, 1720; Treatise of Fluxions, 1742; Treatise of Algebra, 1748, and an Account of Newton's Philosophical Discoveries, 1748. His Treat ue of Fluxions, which made a suitable reply to the attack by Berkeley, also gives an account of his own important researches in the Theory of Attraction. The Dispute between Maclaurin and Campbell. Letters I. 1 to I. 7 are mainly concerned with a dispute between Colin Maclaurin and George Campbell, a pretty full account of which is given in Cantor's Geschichte tier Mathematik. But the correspondence before us gives a good deal of fresh information, as well as practically the only details known regarding George Campbell, about whom the Histories of the Campbell Clan are silent, in spite of the fact that he was a Fellow of the Roj^al Society, being elected in 1730. From Letter I. 1, it would appear that when Maclaurin, glad to leave Aberdeen University owing to the friction arising from his absence in France, and consequent neglect of his professorial duties, accepted the succession to Professor Gregory in the Chair at Edinburgh, he had in a sense stood in the way of Campbell for promotion to the same office. Feeling this, he had done his best to advance Campbell's interests otherwise and had corresponded to this intent with Stirling, who 194 NOTES UPON THE CORRESPONDENCE suggested that Campbell might gain a livelihood in London by teaching. Some of Campbell's papers were sent to London. One, at least, was read before the Royal Society, and, through the influence of that erratic genius, Sir A. Cuming, ordered to be printed in the Transactions. Stirling himself read the paper in proof for the Society. When the paper appeared Maclaurin was much perturbed to find that it contained some theorems he had himself under discussion as a continuation of his own on the Impossible Roots of an Equation. He wrote letters to Folkcs explaining his position, and giving fresh additional theorems. But the matter did not end here. For Campbell in a jealous mood wrote and published an attack upon Maclaurin, who found himself compelled to make a similar public defence. An attempt was also made to embroil Stirling with Maclaurin, fortunately without success. Practically nothing further is known regarding George Campbell (who is not to be confused with Colin Campbell, F.R.S., of the Jamaica Experiment, mentioned in Letter I. 10). The names of G. Campbell and Sir A. Cuming are given in the list of subscribers to the Miscellanea Analytica cle Seriebus of De Moivre (1730). Xeiuton's Theorem regarding the nature of the Roots of an Algebraic Equation. Neither Campbell nor Maclaurin attained the object aimed at, — to furnish a demonstration of Newton's Theorem, stated without proof in the Arithmetica Universalis. Other as eminent mathematicians were to try and fail, and it was not until the middle of the nineteenth century that a solution was furnished by Sylvester, who also gave a generali- zation. (Phil. Trans. 1864: Phil. Mag. 1866.) Newton's Theorem may be stated thus (vide Todhunter's Theory of Equations). Consider the equation / (x) = a x n + „ A, PP(/i)-PP(\) is either equal to the 02 196 NOTES UPON THE CORRESPONDENCE number of real roots of f(x) - between fi and X, or exceeds it by an even number. Letter I. 1. On p. 19 of his Defence (against Campbell) Maclaurin makes the statement : — ' In a Treatise of Algebra, which I composed in the Year 1726, and which, since that Time, has been very publick in this Place, after giving the same Demonstration of the Doctrine of the Limits, as is now published in my second Letter, I add in Article 50 these Words, &c.' Maclaurin appears to be referring here to a course of lectures to his students. Maclaurin's Algebra did not appear until 1748, after his death. It was in English, but contained an important appendix in Latin on the Properties of Curves. De Moivre's book referred to is his Miscellanea Analytica, 1730. In 1738 appeared the second edition of his Doctrine of Chances, also referred to in the letters. Letter I 3. This letter, dated by Maclaurin as February 1 1 th , 1728, should have been dated as February 11 th , 172|, i.e. 1728 Old Style, or 1729 New Style. Stirling makes this correction in I. 6, which consists of extracts from letters by Maclaurin. Until this had been noted, the first three letters seemed hopelessly confused. Maclaurin shows the same slovenliness in the important note of his, I. 10, attached to the letter from Maupertuis to Bradley. Letter I. 6. Letter I. 6 contains only extracts from letters of Maclaurin, including one dated October 22. 1728, which is no longer in the Stirling collection. Letter I. 7. In the spring of 1921 I had the good fortune to obtain a copy of Maclaurin's reply to Campbell. NOTES UPON THE CORRESPONDENCE 197 It is entitled : — ' A Defence of the Letter published in the Philosophical Transactions for March and April 1729, concerning the Impossible Roots of Equations; in a Letter from the Author to a Friend at London. Qui admonent amice, docendi sunt : qui inimice infectantur, repellendi. Cicero ' The name of the 'Friend' is not given. The 'Defence' consists of twenty small quarto pages, and contains numerous extracts from the letters to Stirling; and towards the end Campbell's statements regarding Maclaurin's theorems are refuted. Campbell is generally referred to as ' the Author of the Remarks ' (on Maclaurin's Second Letter on impossible roots) ; though also as ' the Remarker '. Maclaurin gives the extract from the letter of October 17 28 (cf. I. 6), and adds: — ' See the 2d and 3d Examples of the Eighth Proposition of the Lineae terlii Ordiids Netiotonianae.' There is also the following passage containing an extract from a letter by Stirling, not otherwise known : — ' I had an Answer from this Gentleman in March, from which, with his Leave, I have transcribed the following Article : '' I shewed your Letter (says he) to Mr Machin, and we were both well satisfied that you had carried the Matter to the greatest Height, as plainly appears by what you have said in your Letter. But it is indeed a Misfortune, that you was so long in giving us the Second Part, after you "had delivered some of your Principles in the First:— Since you have published Part of your Paper before Mr C 11, and now have the rest in such Readiness, I think you have it in your Power to do yourself Justice more than any Body else can. I mean by a speedy Publication of the remaining Part : For I am sure, if you do that, there is no Mathematician, but who must needs see, That it is your own Invention, after the Result of a great Deal of Study that way." I received this Letter in March, and, in consequence of this 198 NOTES UPON THE CORRESPONDENCE kind Advice, resolved to send up my Second Paper as soon as possible.' Maclaurin makes it clear that he had not intended his First Letter to Folkes to be published. It was printed without his knowledge. Had he known in time, he would have deferred its publication until he had more fully investigated additional theorems which he had on the same subject ; and he gives an extract from a letter from Folkes in corroboration of his statement. Letter I. 8. Letter I. 8 is reproduced because of its reference to an office (in the Royal Society) for which Stirling had been thought fit. Letter I. 9. Letter I. 9 announces that Maclaurin has started to write his Treatise of Fluxions. His conscientious reference to original authorities has been noted by Reiff (Geschiohte der U nendlichen Reihen). The earlier proof-sheets of the Treatise, at least, passed through Stirling's hands. These facts bear interesting evidence regarding the Euler- M^aclaurin Summation Formula, to which I have to return in connexion with the correspondence between Stirling and Euler in Letters XI. Simpson, referred to by Maclaurin, is doubtless his old teacher, Robert Simson, of Glasgow University. Letter I. 10. Letter I. 10, which is a mere scrawl written on the outside of the copy of the letter from Maupertuis to Bradley, is of interest in the history of the Royal Society of Edinburgh, and is to be associated with the two letters of Maclaurin published in the Scots Magazine for June, 1804. The date of the letter of Maupertuis shows that Maclaurin should have given Feb. 4 th , 173| as the date of his own. Maclaurin was more successful with Stirling than with R. Simson, who refused to become a member after Maclaurin had got him nominated. (Scots Mag.) Bradley's translation of the letter of Maupertuis is repro- duced in the Works and Correspondence of Bradley, 1832 NOTES UPON THE CORRESPONDENCE 199 (Rigaud). The original French letter is preserved in one of the letter books of the Royal Society of London. Foundation of the Philosophical Society of Edinburgh. Letter I. 10 confirms the date of foundation as 1737 (v. Forties's Hittory of the Royal Society of Edinburgh, in General Index Trans. R.S.E. published 1890). But at the date of this letter I. 10 the Society was not complete in numbers, for Stirling was not yet a member. By 1739 the Society had outrun its original bounds, having forty-seven members whose names are given (p. 26 of Gen. Index Trans. R.S.E.). More or less informal meetings were held in 1 737. Maclaurin and Dr. Plummer, Professor of Chemistry in the University, were the Secretaries. The Rebellion of 1745 seriously affected the activity of the Society, and Maclaurin's death in 1746 was also a severe blow. The papers read before the Society had been in Maclaurin's hands, but only some of these were found. Three volumes of Essays and Observations, Physical and Literary (dated 1754, 1756, 1771), were published. The papers in Vol. I are not in chronological order, but those by Plummer are fortunately dated, the first bearing the date January 3, 1738. Dr. Pringle, afterwards President of the Royal Society of London, followed in February. Then it was Maclaurin's turn in March, when he gave two papers, one being on the Figure of the Earth (Scots Magazine). These two papers are not printed in the Essays, &c. But among the Maclaurin MSS. preserved in Aberdeen University there is one entitled ' An Essay on the Figure of the Earth '. On the foundation of the Royal Society of Edinburgh in 1783 the members of the Philosophical Society were assumed as Fellows. Maclaurin's son John (Lord Dreghorn) is one of those mentioned in the original charter of the Royal Society. Lttter of Maupertnis. The letter of Maupertuis must have given lively satisfaction to Maclaurin and Stirling. Newton had assumed as a postulate that the figure of the Earth is approximately that of an oblate spheroid, flatter at the poles than at the Equator. The 20 J NOTES UPON THE CORRESPONDENCE Cassinis, arguing from measurements of the arc of a Meridian in France, maintained that the figure was that of a prolate spheroid. There were thus two hostile camps, the Newtonians and the Cassinians. Pole Pole Pole h£wtoh CASSINI The French expedition to Lapland (1736-7) with Mauper- tuis as leader, and Clairaut as one of the party, conclusively established the accuracy of Newton's hypothesis. In the words of Voltaire, Maupertuis had 'aplati les Poles et les Cassinis '. Both Stirling and Maclaurin made important contributions to the subject, and the rest of the letters preserved as passing between them refer mainly to their researches on Attraction and on the Figure of the Earth. Readers who are interested cannot do better than consult Todhunter's History of the Theory of Attraction and of the Figure of the Earth, for full details. The letters, however, clear up some difficulties that were not always correctly explained by Todhunter. Letter I. 11. The Dean, near Edinburgh, Maclaurin's new address, now forms a residential suburb of Edinburgh. De Moivre's book is doubtless the second edition of the Doctrine of Chances (1 738). Letter I. 13. The remark made by Stirling towards the conclusion that ' the gravitation of the particle to the whole spheroid will be found to depend on the quadrature of the circle' seems to have given Maclaurin a good deal of trouble (cf. I. 14). NOTES UPON THE CORRESPONDENCE 201 Maclaurin's reference to it in his Fluxions, § 647, as due to Stirling, was inexplicable to Todhunter, as Stirling never published his theorem. But Todhunter's conjecture {History, vol. i, p. 139) that Maclaurin may have inadvertently written Stirling for Simpson is of course quite a mistake. Letter I. 15. Compare the correspondence with Machin IX, Clairaut X, and Euler XL Letter I. 16. This letter, dated 1740, furnishes ample justification of Todhunter's contention that the researches of Maclaurin, ' the creator of the theory of the attraction of ellipsoids ', are quite independent of those given by T. Simpson in his Mathematical Dissertations (1743). Simpson lays claim to priority in certain theorems of the Fluxions on the ground that these given by himself were read before the Royal Society in 1741. The Treatise of Fluxions so near completion in 1740 was not published until 1742. II CUMING Sir A. Cuming (1690 ?— 17 75) was the only son of Sir Alex- ander Cuming, M.P., the first baronet of Culter, Aberdeen. Cuming went to the Scotch bar, but gave up his profession on receiving a pension. In 1720 he became a Fellow of the Royal Society. Though no mathematical writings of his are known, he seems to have been possessed of mathematical ability. He was on friendly terms with De Moivre and Stirling, both of whom acknowledge their indebtedness to him for valuable suggestions. At Aberdeen there is preserved a short letter (Nov. 3, 1744) from him to Maclaurin, in which he shows his interest in the controversy regarding Fluxions. In his introduction to the Methodus Differential, Stirling speaks of him as ■ Spectatissimus Vir'. Being a friend of Campbell he had a share in the dispute between Maclaurin and Campbell. In 1729-30 he was in the American Colonies, visited the Cherokees, and became one of their chiefs. On his return to 202 NOTES UPON THE CORRESPONDENCE England with some of the chiefs he was instrumental in ai'ranging a treat}* for his tribe. Later he fell into poverty, and was confined in the Fleet prison from 1737 to 1765, losing his fellowship in the Royal Society for neglecting to pay his annual fee. In 1766 he obtained admission to the Charterhouse and died there in 1775. Ill CRAMER AND STIRLING Gabriel Cramer was born in 1704 in Geneva, where his father practised medicine. In 1724 he was, conjointly with Calandrini, entrusted with the instruction in Mathematics at the University of Geneva. In 1727 he started on a two years' tour, visiting Bale, where he studied under John Ber- noulli, and England, where he became acquainted with Stilling and De Moivre, and returning by Paris. He became F.R.S. in 1748. He died in 1752. He is best known through his Introduction a V Analyse des lignes courbes algebriqn.es. He also edited the works of James and John Bernoulli. Letter III. 1. It is unfortunate for us that Cramer did not discover before 1732 that he wrote 'un Anglois aussi barbare'. Regarding the history of the Probability Problem in III. 1, see Todhunter s History of the Theory of Probability (p. 84). De Moivre gives a much simpler solution in the Miscellanea Analytica (1730). Letter III. 2. Compare Letter IV. 2 i Bernoulli). Letter III. 3. In this letter of introduction Cramer in the address describes Stirling as LA.M. I do not know what these letters signify. Letter III. 8. Letter III. 8 contains valuable information regarding the manner in which Stirling wrote his Nethodus Liferent talis. The blank made for the formula given by De Moivre was never filled up : but the formula in question i^ of course easilj* NOTES UPON THE CORRESPONDENCE 203 obtained from the Supplement to the Miscellanea Analytica of De Moivre. We have also the important information that this Supplement appeared after the publication of Stirling's own Treatise. Letter III. 10. One will note Cramer's difficulties with the graph of y x — 1+x; also his determination of (l+x) l/x as x tends to zero. It is a pity there is no indication of Stirling's determination of this limit. Stirling's Series and the claims 1o priority of De Moivre and Stirling. In the Biblioteca Mathematica for 1904 (p. 207) Enestrom makes the following statement. 'Im Anschluss an den Bericht iiber Stirling's Formel fur die Summe einer Anzahl von Logarithmen ware es angezeigt mitzuteilen dass die bekannte Formel dieser Art die man jetzt ziemlich allgemein gewohnt ist als die Stirlingsche Formel zu bezeichnen, namlich log (1 . 2 . 3 . . . ,r) = \ log 2tt + (x + •§) log x -x + A,,- + A. —^- +&c, * X X 6 zuerst von Moivre im Anhange an der Misc. analytica (1730) angegeben und hergeleitet wurde. Moivre berichtet selbst dass Stirling ihm brieflich die Formel log(1.2...a;) = $log2»r + (a; + 4)log(a;+!) -(* + $)- o 10 L. i, + 2.12(33 + ^) 8.360(« + -i) 3 mitgeteilt hatte, und dass er selbst dadurch angeregt wurde die neue Formel auf einem ganz anderen Wege aufzufinden.' Inasmuch as the only change effected by De Moivre is to give the expansion of log (x I) in descending powers of x instead of descending powers of x + \, which has no special advantage when x is large, the priority of De Moivre to this important formula seems to me to rest on very slender foundations, unless we are to infer from Enestrdm's reference to the 204 NOTES UPON THE CORRESPONDENCE Supplement to the Miscellanea Analytic a that De Moivre published his result prior to Stirling. Enestrom's statement has had considerable influence with subsequent writers (e.g. C'zuber and Le Roux, Calcul des Probahilites: Selivanov and Andoyer, Calcul des Differences Finiev, in the well-known Encyc. des Sciences Math. ; Czuber, Wahr. Rechnung, 1903, s. 19), who refer for proof to the Supp. Misc. Anal, of De Moivre. Against these we may put De lloivre's own statement in the third edition of the Doctrine of Chances (1756), given in the Appendix, p. 334, where, after giving a table of values for log (x !) for numerical values of x he goes on to add : — ' If we would examine these numbers, or continue the Table farther on, we have that excellent Rule communicated to the Author by Mr James Stirling, published in his Supplement to the Mi.-cellanea Analytica, and by Mr Stirling himself in his Methodus Different hills, Prop XXVIII. 'Let z — \ be the la^t term of any Series of the natural Numbers 1, 2, 3, 4, 5, ...:-|; «= -43429448190325 the reciprocal of Neper's Logarithm of 10: Then three or four terms of this Series a la 31a z Logz-az - Y^2~z + 8.360s 3 ~~ 32 . 1260l i 127a „ 128.1680'- 7 added to 0-399089934179,