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Ctltntfu uf lUrllrjiliM) I (TulliMU'. Jut Jllcmnnam •WXM^tfi-. CjUa<c JAMES S 1 I R L I N G OXFORD UNIVERSITY PRESS London Edinburgh Glasgow Copenhagen New York Toronto Melbourne Cape Town Bombay Calciitla Madras Shanghai HUMPHREY MILFORD Publisher to the University Uk irmMiM/ '^ocoJjk: iL-tn/^ ftpu^t .-kZ/cc ■^^'f^ ^;>u/C (^^iJaj tk cfn^^}^ 2^ m^WJU^. '>% ^aU htgyU^ <s^/e,- km^ /h^itc Jatic f'l*- if 'V/ >^n^ ^^^^ ^ l^hm^^ ^ ^ W^ JAMES STIRLING A SKETCH OF HIS Htfc auD KHorfts ALONG WITH HIS SCIENTIFIC CORRESPONDENCE BY CHARLES TWEEDIE M.A., B.Sc, F.K.S E. CAKNEGIE FELLOW, 1917-l'.t20; LECTURER IN PURE MATHEMATICS EDINBURGH UNIVERSITY OXFORD AT THE CLARENDON PRESS 1922 \ 80 4-2. 2^ 3 TO THE MEMORY OF JOHN STURGEON MACKAY. LL.D. TO WHOSP] 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 hy Dr. J. C. Mitchell, published in his work, Old Glcsgovj Essays (MacLehose, 1905). An interesting account of his life as manager of the Leadliills Mines is also given by Ramsay in liis f^cotland 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 contempoiary records, along with more accurate information regarding the part he played in the Tory interests, and the reason for his departure for Italy. Undoubtedl}", 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 that 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 ot" Professor E. T. Wliittaker's notes on Part I of Stirlino-'s MeUiodus Different lalit^, wliich he kindly put at my disposal. Stirlino-'s intluence as a niatlicniatician of pi'ofonml 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 Prol»al)ility 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 unwillingl}^ to make use of Stirling's Series, though its lack of convergence was aiiathema to him, IMore recentl}', Stirling has found disciples among Scandinavian mathema- ticians, and Stirling's theorems and investigations have been chosen Ijy 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 centur}-. I have little doubt that suitable research would add to their number. I have endeavoured to reproduce these as exactly as possible, and readers w^ill 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 7r/2 given by De Moivre's copy of Stirling's letter (taken from the Miscelkmea Amdytica) is not correct, being 1-5707963267U, 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 w^ith which the Garden letters were placed at my disposal by Mrs. Stirling, Gogar House, Stirling. 1 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 suitabh^ manuscript for the press, I had much valuable clerical assistance from my sister. Miss Jessie Tweedie. PREFACE ix Of the man}' friends who have helped to lighten my task I am particuUirly indebted to Dr. C. G. Knott, F.R.S., and to Professor E. T. Whittaker, F.R.S., of Edinburgli University; also to Professor George A. Gibson, of Glasgow Universitj% 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. i'"acsimile 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 tlianks. 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, MA., LL D., Headmaster of George Watson's College, Edinburgli. 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 L^niversity. 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 v^alue of £10, from A. G. Burgess, M.A., B.Sc., Rector of Rothesay Academy. Archibald Campbell, M.A., LL.B., Writer to the Si^-net, 36 Castle Street, Edinburoh. Jas. H. Craw, Estj., Secretar}^ 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, DSc, 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 Stiulixg to his Father, 1715 (pages 6-7) Frontisinece Facsimile of last page of Letter by Maclaurin to Stirling, 1728 (Letter No. 1) facing p. ^7 COAT OF AKMS OF TlIK STIKLINGS 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 horn at Garden in the comity 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 Stirling's 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 tlie 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. Earl}^ in the seventeenth century Sir Archibald Stirling of Keir Ijought 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. Tlic tsoii of John, Sir Arcliiltald Stirlint;, was a conspicuous Royalist in the Civil War, and was heavily fined hy 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 1G68. 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 1716, 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 16[):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.- YOUTH OF STIRLING Oxford Save for the account given by Ramsay of Ochtertyre (Scotia lul and Scotsmen, from the Ochtertyre MSS.),wh\ch is not trustworthy in dates at least, little is known of the early * Garden, pronounced (Jarden, or Gardenne. * Gan<r forward ; Scotlce for Allez en nvant. STIRLING AT OXFORD 3 years and education of Stirlint^^, prior to his journey to Oxford University in 1710. Ramsay, it is true, says that Stirhng studied for a time at Cdas<;o\v University. This would have been (|uitc 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 Ibum '. 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 sufficiently indicates the esteem in which he was held by his family, and their expectation of a promising futui-e for the youth. In one of these he narrates his experiences on the journey to London, and his endeavour to keep down expenses: ' I spent as little money on the road as I could. I could spend no less, seeing I went with such companj^, 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, l/jy, 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.' B 2 4 LIFE OF JAMES STIRLING ■ He thus ln'^an academic life at Oxford in n-ood s])ints, but us a nou-juiiiii; stiuleut. At tlus period Oxford University •svas not conspicuous for its intellectual activity. The Fellows f-eeui to have led lives of cond'ortal)le 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 loos-ely applied, and, naturally, complahit was made at any stringency later. At the time we speak of political ([Uestions 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-(iuarters of King Charles I during the Civil Wai', and his cavaliers were remendjered with regret when the town was occui)ied 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 (Hlstorij 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. Heariie, the anti(|uarian subdibrarian of the Bodleian. For Hearne all Tories were 'honest men', and nothing good was ever to be found in the ' \'ile 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 boiul of sympathy arose from their Tor}^ nay their Jacobite, principles, but it speaks well I'or 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 Liaeae Tertll Ordinis, a W'Ork which is still a recognized connnentary on Newton's Eauineration of Curves of I he 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 Balliol 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 Maj" 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! T/ie Duke of Ormond ! isic, 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 d. King George lieing informed of the proceedings of the cavaliers at Oxford, on Saturday and Sunday (May 28, 29), he is very angry, and by his order 'J'ownshend, 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 w^ere not less remiss on these occasions. The heads have had several meetings upon this atl'air, and they have draw^n up a programme, (for they are obliged to do something) to prevent the like hereaftei-; and this morning- very early, old Sherwin the yeoman bearlle w^as sent to London to represent the truth of the matter.' These mea.sures had a marked effect upon the celebration on June 10 of 'King James the Illd's' 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 Palliol College, and with honest j\Ir. John Leake, formerly of Hart Hall, and Rich. Clements (son to old Harry Clements the Ijookseller) he being a cavalier. We were very merry at Foxcombe, and came home between nine and ten,' etc. 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 5ire of age to go into orders : Rochester s^tands 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, tho IJishop is no enemy to our pi'inciples. In the meantime I've borrowed money of my friends till I'm ashamed to borrow an}'' more. I was resolved not to troul)le you while I could otherwise subsist ; but now I am forced to ask about 5 lib. or what in reason j^ou think fit to supply my present needs : STIRLING AT OXFORD 7 for ye little debts I have 1 can delay tlieni I hope till the i;ood 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 V>y the more prudent sort, and they are now rageing in Coventr}' 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"" 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 enough to impeach the rest the}' designed. I had a letter from Northside ^ 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 l)]*others sisters and all my relations I am Sir Your most dutifuU son Jas. Stirling. It was in the same year (1715) that Stirling first gave indications of his ability as a mathematician. In a letter - to Newton, of date Feb. 24, 1715, John Keill, of Oxford, mentions that the problem of orthogonal trajectories, which had been proposed by Leibidz, had recently been solved by ' Mr. Stirling, an undergraduate here ', as well as by others. The statement commonly made that Stirling was expelled ' James Stirling, son of the Laird of Northside (near Glasgow), is >;]>ecially mentioneil in the List of Persons concerned in the Rebellion of l7i5-6 (Scot. Hist. Soc.j. ~ Macclesfield, Citrrespondence of Scientijic Men, (5jT., vol. ii, ]). 421. 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"" Sterling, a Scotchman, of Balliol Coll. and M'" 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 lionest, 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/^ 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' Stirling of Balliol College, one of those turned out of their Scholarships upon account of the Oaths, hath the otter of a Professorship of Mathematicks in Italy, w^^ he hatli accepted of, and is about going thither. This Gentleman is printing a Book in the Mathematical way at the Theatre.^ ' We shall see presently that Stirling found himself compelled to refuse the proffered Chair. The circumstances in which he had this ofler are somewhat obscure ; and whether he ^ S.C.L. was a Degree (Student of Civil Law) i^arallel to that of B A., just as that of Bachelor of Civil Law (B.C.L.) is parallel to that of M.A. The degrer 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. '■^ 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 Abb^ 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 fiiendly welcome from Newton and the Fellows of the Royal Society. In a letter^ to Brook Taylor in 1721, Conti relates how ' M'' Newton me pria d'assembler a la Soci^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 Stiiling's ability and personal accomplishments, for Newton in a letter quoted by Brewster [Life of Neivtoa, ii, p. 308) querulously charges Conti with ' sending M"". Stirling to Italy, a person then unknown to me, to be ready to defend me there, if I would liave contributed to his maintenance '. The fact that Newton was a subscriber to Stirling's first venture, Lineae Tertii Ordinis Keutonicmae, sive Illustratio Tract aius D. Neutoni De Enumfieratione Linearuni 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 Iinpririiatur, 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 ^ Printed in the posthumous ContempJatio Philosophica 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 liis residence in Venice,^ Stirling is known in the I'.imily Histoiy of the Stirling's iisJumesSth^llng the Venetian. The invitation to Italy and the subsequent refuf-al are thus recorded in the Rawlinson MSS. in the Bodleian (materials collected by Dr. Richard Rawlinson for a continuation of Ward's Atheiuie Oxoniense^ up to 1750): * Jacobus Stirling, e coll. Baliol, exliibit. Scot, a Snell. jura- ment. R. G.^ recus. 1714, et in Italiam Nobilem virum Nicolaum Tron, Venetiarum Reipublicae ergo apud Anglos Legatum, secutus est, ubi religionis causa matheseos profes- sorium munus bibi 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 liad caused so mucli 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 ' I have endeavoured to ascertain the university to wh'ch Stirlinj^ was called. Professor G. Loria has informed nie that it was very probably Padua, Padua being the only i niversity in the Republic of Venice, the Quaiiier Latin of Venice according to Renan. It had been customary to select a foreigner for the chair of Mathematics. A foreigner (Hemnann) held it, and resigned it in 1713. It was then vacant until 1716, when Nicholas Bernoulli (afterwards Professor of Law at Bale) was appointed. Profe^^sor Favaro of Palua confirms the above, and adds that possibly some information nv'ght be gathered from the reports of the Venetian Ambassador, or from the records of the Reformatorcs Studii (the patrons of chidrs in a mediaeval university). To get this information it would be necessary to visit Venice. My chief dithculty 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 {nee also Evelyn's Diarij). C. T. "^ Kinj? George. AT VENICE 11 letter to Newton, expressino- his oi-atitiide, is here oivcii. It lias boon copied from Brewster's f/ife of Newton. Letter Venice 17 Aug. 1719. Sir I had the honour of 3'our letter about five weeks after the date. As j-our generosit}' is infinitely above my merite, so I reackon myself ever bound to serve 3'ou 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 a^k pardon for not returning my grateful acknow- ledgments before now. I wrote to M"" 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 3'ou having sent M"" Desaguliers a full account thereof. I beg leave to let you know that M"" 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 aho 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 oscillate 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. 1 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:! 3'ou 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. JVP Nicholas Bernoulli, as he hath been accused by D"" Keill of an illwill towards j'ou, wrote you a letter some time asTo to clear himself. But havino- in return desired me 12 LIFE OF JAMES STIRLING to assure 3'ou that what was printed in the Acta Paris. reUiting to 3'our 10 Pjop., lili. 2, was wrote before he had been in En^^hmd sent to his friends as his private opinion of the matter, and afterwards published witliout so much as Ins know led <(e. He is willing to make a full vindication of him- self as to that atlair 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'' 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 Eruditorinn where his uncle has lately wrote against D*" Keill he showed me that the theorems there about Quadratures are all corollarys from 3'our Quadratures ; and whereas M"" 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, nej)liew to one who is now become head of M'' 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"" Keill. How long he lived in Italy after his letter to Newton is not known; but life in the cultured atmosphere of Venice must liave been, otherwise, very congenial. It was a favourite haunt of the different members of the Bernoullian famil}'. The earliest letter to Stirling of a mathematical nature that has been preserved is one in 1719 from Nich. Pernoulli, 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 eonve3's the greetings of Poleni, Professor of Astronom}- at Padua At the same time Kiccati 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 Dljferentialis Neivioniana, published in the Ph'do- AT VENICE 13 sophical Tr<(vs<(ctions for 1719, witli the object of elucidating Newton's methods of Interpohition. London From 1719 to 1724 there is a o-ap in our information regarding Sterling. But a fragment of a letter Ijy him to his brother, Mr. John Stirling of Garden, shows that in July 17*24 he 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, 1725) 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 n)ay not be * quite idle ' he is preparing for the press an edition of . . .^ 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 fretjuent 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 .... Diiect 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 (Did. 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 ^ The name, unfortunately, is not legible. 14 LIFE OF .]AMES STIRLING connected with the Academy, and to this ad(b-ess most of the letters to him from contemporary niatliematicians, tliat luive been preserved, were directed. They form part of a hirger collection that was partly destroyed l)y lire, and early in the nineteenth century they were nearly lost altoj^ether throu<^h 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 fi-iends 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 17129 he could look forward with confidence to the future, for by that time he was able to wipe out his indel)tedness 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 l/.'iS, 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 liting up apartments for my former project still ly over my head.' Again on July 22, 1729, 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 1 find my business brings in about 200 L. a year, and is rather increasinof, 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 sunmier 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 ]\fcUioilus Differentia lis (17 30), 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 Roj^al Society in 1726, AT LONDON 15 a distinction that put him on an etjual footing with the .scientists that lived in, or fre(|iiented, 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 L^niversity. The o[)ening correspondence furnishes the best account we have of the unfortunate dispute between Maclaurin and Campbell regai'ding the priority of certain theorems in equations (vide MalJi. Gazette, January 1919). Maclaurin placed great reliance upon Stirling's judgment, and frequently consulted Stirling while engaged in writing his Treatlte of Flnxlonis. Their later letters are mainly concerned with their researches upon the Figure of the Earth and upon the Theory of Attrac- tion. In 1738, Stirling, at Maclaurin's special re<juest, 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, Profes&or of Mathematics at Genev^a, 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 to Stirling are preserved. A copy, kept by Stirling, of a letter to Cramer furnishes interest- ing information regarding his own views of his Methodus Differeiitialis, and also regarding the date at which the Supplement to De Moivre's Miacellanea 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 liad 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 JA1\IES STIRLING more. But there were two additional species, one of which was noted by Nicole in 1731. Murdocli in his Neiotoni Genesis Garvarutn per Urnhraa (1740) mentions that Cramer had told him of Bernoulli's discovery, hut without furnishinrr a date. Bernoulli's letter not only conhrnis Cramei''s state- ment, it also t;-ives undoubted precedence to Bernoulli over Stone's discovery of it in 1736. From 1730 onwards Stirling's life in London must have been one of considerable comfort, as his 'aflairs' 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 Polingbroke 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 liad 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 Cassirjis, 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 Todhunter's History of the Theory of Attraction and the Figure of lite Earth). Return to Scotland In 1735, a great change in his circumstances was occasioned by his appointment to the Managers!] ij) 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, w^ould suggest that he felt the change keenly. RETURN TO SCOTLAND 17 He was now well over forty years of age, l)ut, nothing daunted, he .set liiniself 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 tlie absorbing claims of his new duties is frequently brought to our notice. He appears to have discovered further important theorems regarding the Figui'e 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 accjuaintance 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 connnunicated 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 con-espondence 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, Gorr. 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.' 2447 C 18 LIFI^] OF JAMES STIRLING Euler, who was at tlie tiiiiu iusLalled in Putrograd, ili<l not then become a Fellow of the Iloyal Society. In 1741 he left Ivussia for Berlin, where, in 1744, he was made Director of the Mathematical Section of the Jjerlin Academy, and it is (juite possible that he had a share in conferring upon Stirling the honorary memVtership of tlie 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.^ He is also taken as one of the best types of the Scotsmen of his day by Ramsay in liis 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 i/rande manicre, without any trace of the pedantry one might have expected. Ramsay also narrates several anecdotes regarding Stirling's keen sense of humour.- The association between V^enice and the l^eadhills in Stirling's career is very remarkable. According to Ramsay, before Stirling left Venice, he had, at the reijuest 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, liis 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 acipiaintance, with the result that about 1735 the Scots IMining Company, which was controlled by a group ' 'Modern History of Leadliillh'. "^ I. c, vol ii. LEADHILLS 19 of Loudon merchants, associated witli the Sun Fire Office, selected him as manager of tlie Lcadhills 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, hut 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 ids predecessor's day were cleared off', and the mines became a source of profit to the shareholders. But his scientific pursuits had to be neglected. We find him, in his letters to Maclaurin, with whom he still frequently corresponded, complaining that he had no time to devote to their scientific i-esearches, and when writing to Euler 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. Li 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 j^ardens 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 rioht to the ground to another miner without fear of interference from the superior. In this way Stirling stimu- la'ed their industry, while at the eame time furnishing them with a healthy relaxation from their underground toil. The mmers 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^ 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 re([uireuients were well provided for by the Company, whose atfairs 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 carriaue. The Glasgow Kettle In the eighteenth century the rai)iilly ex])anding trade of Glasgow and the enterprise of her merchants made it highly desirable to have better water connnunication and to make the city a Port, and in 175'2 the Town Council opened a separate account to record the relative expenditure. The ' Of Stirling's private library two books have been preserved. One, on Geometry, was presented to hitii by Bernoulli in 1719. The other (now at Garden) is his co])y of Brook Taylor's Methodus Incremoitonim, which he boujifht in 1725. THE GLASGOW KETTLE 21 tii-st item in this account, which is headed ' Lock desi^n'd upon the River of Clyde ', runs thus : * Paid for a coniplinienfc 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 66^ oz at 8/ per oz £26 10 For chasinu- & Enaravinu" the Towns arms 1 14 4 £^^8 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 V «'is 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. ' Laing'^ Tomb is a prominent mural tablet (1620) on the right wall surrounding the churcb^-ard. 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 livino- 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 wliich any family might well be proud. WORKS PUBLISHED BY J. STIRLING (A) ENUMERATION OF CUBICS § 1. His first publication, Lineae Tertii Ordinis Neutonianae sive lilustratio Tradatas D. Neivtoni De Eimoneratlone Linear am Tertii Ordinis. Cui suhjungitur, Solutio Trium Prohlematum, was printed at the Sheldonian Theatre, at Oxford, in 1717. As the book^ 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 l;)y 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 ^ gave, in his Genesis Gurvarum i^er 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 ^ ■, ? » 7 ^ 2/" = «i^ + bx^ + CX + d. 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 " Edleston {CorresponfJene, &c., p. 2.3-5) 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 luivc stopped sliort \vith but hall' of the theory ? § 2. Newton stated that the algebraic ecpiation to a cubic can be reduced to one or other of the four forms (i) .ry'--\-cy, or (ii) xy, or (iii) y"^, or (iv) y, = ax'-^ + hx'^ + rx + d ; and he gave sufficient int'orniation 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 Serie><, 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 ditterential equations, though here he is not always very clear.) With some pride he gives on p. 32 the theorem ^ Let 2/ = ^ + ^^'' + ^^^'' + • • • ' then y may be expressed as . xy x~y x^''y' „ y =^ A-\ *^ + ^, H , ., + &c. ^ 1 .rx 1.2 r-x^ 1.2.3. r'lr^ applicaljle 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 tlieor^^ of asymp- totes. A rectilinear asymptote can cut the curve of degree ti in, at most, n—2 finite points. If two branches of tlie 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 oti' to infinity. A curve cainiot have more than u—l parallel asynq^totes, and if it has n—\, then it cannot cut these in any finite point. If the 2/-axis is parallel to an asymptote, the equation to the curve can have no term in ?/". From this follows the inq)()rtant corollary that the e(juation to a cubic curve ma}' alwa3s be found in the form , , „ ,. , , , . , {x + a)y'' = yf^(x}+f.,{x). ^ 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 sei-ies: but not always. Thus the quartie y ^ (ax^ + bx' + . . . + e)/ifx-' + (jx' + hx + k) has the asymptote ax hf— ag " = 7^1^' as found hy a series. The rest of the asymptotes are given by x = a, where a is any one of the roots of fx^-V(jx^-Vhx + k = 0. In the standard case of an ecjuation of degree n in x and y, if we assume tlie series y = Ax-vB+- + ^,-\-... and substitute in the given equation we find, in general, (1) an equation of degree n for A furnishing ii 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 tlie diagrams, furnish quite a good introduction to what we now call (jra'ph-tracing. He thus graphs the rational function y = f(x) / (p{x) with its asymptotes parallel to the y-i\\\^ found by ecjuating (p{x) to zero. The manner in which a curve approaches its asymptotes is explained by means of series. In the curves given by y — « + i^/ + . . . + A'*" there are only two infinite branches which are on the same, or opposite, sides of the ic-axis, according as n is even or odd. When x is large the lower terms in x may be neglected as compared with kx^\ Then follows the graphical discussion of quadratic, cubic, and quartie e(juations in x. The graph of 2/ = a;^ + "a? + h shows that the roots of the corresponding quadratic equation 26 WORKS PUBLISHED BY STIRLING in X arc real or imaginary according as the turning value ot" y is negative or positive. For the cuhic x''' + ax- + hx + c=Q he gives the excellent rule, which has recently been resuscitated, that the three roots are real and distinct only Avhen the graph of the corresponding function has two real turning values opposite in sign. A similar test is applied to discuss the reality of the roots of a quartic. These results arc required later in the enumeration of cubic curves. On p. G9 he gives the important theorem that a curve of degree n is determined by \n(n + 3) points on it.^ The demonstrations of Newton's general theorems in higher plane curves are then given in detail. An indication of some of these is interesting, and the modern geometer \vill note the entire absence of trigonometry. § 4. Diameter Theorem. Draw a line in a given direction to cut a curve in 1\,P^... P^^ ; and find on it such that '^OP = 0. As the line varies in position generates a straight line. Let the ecjuation to the curve be y'' + {ax + h)f'-'^ + ... = 0. (1) In the figure let AB = x, BC = y (so that A is what we call the origin). Take AF = —h/a\ and AE parallel to BC, and equal to —h/n. Let ED = z, DC =v; also let AB/ED = a. Then x = occ, y = DC—DB = V ■ 5 and substitution of these values in B leads to an ecjuation V'' + v«- 2/2 («) &c. = 0, in which the term in r"~' 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 ^07] . OP, = 0, ^OP, . OP,. OP, = 0, S:c. ' Also stated by Hermann (Phoronotnia). Fig. 1. ENUMEKATION OF CUBICS 27 Neuions Rect((^i<lh Theorem for a Conic, and generalization. The proof is made to dopciul on the theorem that il: 0(,, a.^, ... a, J are the roots of (pix) = .(•" + ax"-'^ + . . . + A- = 0, then (p(i)=(i-0(,)'i-o(.^-...ii-0(,). In the case of the cubic y^ + y- (ax + b) + y{cx^ + dx + e)+ fx^ + gx- + Itx + /• = 0. XPi Fm. 2. Let F^OPr^, QiOQ., he drawn in fixed directions through a point 0. Let i^i^ be the .r-axis, QiQ;. parallel to the y-axis, and let be the point (^, 0). Then 0Q,.0Q2.0(?3=/|'- + ryf + /.i + /.' OP, . OP, .OF, = j (ft + ge + hi + k\ so that the quotient OQ, . OQ, . OQJOP, . OP, . OP, = /(up to sign). But a change to parallel axes does not change/. .•. Arc. § 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^ = {hz^ + cz + d)v + e^ ^fz^ + gz + h ( 1 ) includes all lines of the third order, the r-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, BO or BD the corresponding ordinate v of the cubic. If F is the middle point of CD 7> ET 1 / ^ ^^^ + cz+d BF = \ (i\ + 1\,) = — J 28 WORKS PUBLISHED BY STIRLING so that the locus of F is the conic v= {h-J' + cz+(l)/2 z + a) ^vith real as3niiptotes GE •a\\^\ till. Fig. 3. Select these lines as new axes. Call GE X, and EG or ED y. Tlie cuhic o(jnation is of the same form as before, but EF must =K/2x, where 7v is constant, by the nature of the hyperbola. Therefore, the equation to the cubic is of the ^'oi'»i y^ - ey /x = ax^ + bx + c + d/x, or xy'^ — ey = ax^ + hx'^ + cx + d. (I) With a oood »"eal of inoenuity, the proof is indicated in the other cases. Prop. XVI (p 87). When (I is positive in (I) all three asymptotes are real. They are (i) X = 0, (ii) y = xVii +f>/ 2 -/(I, (iii) y= —xVa — b/2va. If I) = 0, the asymptotes are concuri'cnt. ENUMERATION OF CUBICS 29 If I) zfz 0, they form a triangle, inside which any oval of the cubic inuist lie, if there is an oval. The asymptotes (ii) and (iii) cut on the .r-axis, which is also a median of the asymptotic triangle. When c = 0, the point at infinity on the asjanptote (i) is a point of inflexion, and conveisely : 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 6"- — 4((c' = ±^ae\/a. Thus possible conditions for a diameter are c = 0. h" — \ac — iaeVa. h'^ — 4a('= —^aeVa. When any two of these are satisfied so is the third {a is positive and not zero). Tiius 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 these Stirling added 4 species, viz. species 11, p. 99, species 15, p. 1 00, and on p. 102, species 24 and 25. There still remained two species to be added (both arising from the standard form xy'''' — ax^-'thx + c). One of them was given by Nicole in 1731, and the other was communi- cated by N. Bernoulli,^ 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 liy a well- chosen example, especially in his later work. The publication of his commentary on Newton's Cables 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 l.erritor3^^ ^ See note to Letter. ^ 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 Lincae Terfii Ordinis was published in Paris in 1787. (' Isaaci Newtoni Enunieratio Lincarum Tertii Ordinis. Sequitur illustratio eiusdem tractatus lacobo Stirling.') 30 WORKS PUBLISHED BY STIRLING (B) JMETHODUS DIFFERENTIALIS. SIVE TRACTATUS DE SUMMATIONE ET INTERPOLATIONE SERIERUM INFINITARUM ■^ 7. The Mefhodus Differeidialis, as wc shall call it, is the most important product of Stirling's genius, by which lie is most generally known to mathematicians. The book is not, as the title may suggest, a treatise on the Dilierential Calculus, but is concerned with the Calculus of Finite Differences. It is divided into: (1) the Introduction (pp. 1-13); (2) the Suvi- mation of Series (pp. 14-84); (3) the Interpolation of Series (pp. 85-153;. In the Introduction he explains how the Series arc defined. Denote the terms l)}' T, T' , T", &c., and write s = T+r+r'+&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 0+1, 2"' at distance z+2, &c. ; where z is not necessarily an integer. For example, in Brouncker's Series (p. 26) 1 1 ' 1 + + — +... 1.23. 45. (3 any term is given by 1/45(5 +|) where 5 is, in succession, 2 ' ^ 2 ' "^2 ' ^^• A series may sometimes be specified by a relation connecting terms ; e.g. if T'=-.^-^T, then • y,.^5+;^+l^ ^^^^ s+1 Theorems of special interest arise when T can be ex- pressed as T^ A-^Bz + (Jz{z- \) + Bz{z-l}{z-2)+ c\:c., METHODUS DIFFEKENTIALIS 31 or as T = A+ ^ + + Ac, z 3(: + 1) the latter bein*;- useful when z is a lar^-e number. When T admits of either representation then after any transformation it should be reduced again to the same form. Thusif T ^ A+Bz + Gz{z~\)+..., then Tz = {A+B)z^{B + 2C)z{z-\) + (C+3i))s(:-l)(^-2) + ... To facilitate the reduction Stirling gives two formulae and two numerical tables. Let x{x+\){x + 2) ...{x + n-\) ^ 6V*" + ^'„' *'*-'+...+ (^V' ^ and l/x{.c+\)...{x^,i-\)= 2 (-ir^nV ■«'''', s=o then and ^=2 6V-"+Vs(.~+l)...(0 + r). The first table (p. 8) furnishes the values of P/ for the lower values of ii and s, and the second table (p. 11) the lower values of CJ. 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 (7,/ the Stirling Numhera of the First Species, and the numbers r„*' 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'^ + nz z{z+l) z{z+l){z+2) ' Nielsen, Ann. di Mat., 1904 ; or Theorie der Gammafunktion (Teubner, 1906). An account in English is given by me in the Proc. Edin. Math. Soc, 1918-19. Lagrange used them in his proof of Fermat's Theorem. 32 WORKS PUBLISHED liY STIRLING wliicli is L'tjuivalcnt to 1 _ 1 a <i((i + 1) x — a~ X X {.c + 1 ) x{x + 1 ) (.c + 2) ■ " ' when it is usually spoken of as ^tlrllivjs Series; but it had already been given before Stirling by Nicole and by Montmort. PARS I SUMMATIO SeKIEHUM. § 8. Stirling explains that he is not so much concerned with Scries the law of summation fur which is obvious or well known, as with the transformation of slowly converging series into scries tliat more rapidly converge, with their sum to any desired degree of accuracy. Let S = T ^ r+T"+...adoo, S'= T'+T" + ...ad^, S''= T"-\-...adrj,,&ii. Any ditiercnce-equaticHi connecting *S', /S", ..., T, T', ...z, may be transformed into another by writing for these, respec- tively , ,,/ ,,„ rnff . , 1 But when the number of terms in the series is finite, he takes T to be the last {S= .., + T" + r + T), so that >S"= S—T, and if S corresponds to s, >S" corresponds to :-l. On p. 16, he quotes a theorem of Newton,^ which furnishes a key to several of the theorems that fijllow later in the Mel hod as Dijfireidialls. In modern garb it may be thus stated, Z^>-\\-Z)'l-Kh =^'^^''-^^^^F{p + q, 1, p+\, Z). where F((i, b, c, z) denotes the hypergeometric series (6. h a{a+ 1) h{b ^ 1) ^ ^■^ iTc^"^ i.2.c(cTTy"^ ^•••- ' See also p. 113 of Methodus Diffeicntkilis. METH0DU8 DIFFERENTIALIS 33 When 5=1 we have, of cour.se, the Beta Function Jo Prop. I. §9. If T = A + Bz + Cz{z-l) + ... the sum of the first z terms is A:+^Jz+l)z+^{z+l)z{z-l) + ..., and Prop. II. If r=^i^+ ^ z{z+l) z.z+1 .z + 2 "" and s=T+T'+ . ad 00, .1 ., A B C , then ^ = — H -^ ■ +, &c., z 2Z.Z+1 Sz.z+l .z + 2 ' were both given previously by Nicole and Montmort, but Stirling carries their applications much further. E.g. To sum 1 1 1 \- 1 f- j2 ~ 2^ 3'-^ — This Stirling effects in the following characteristic fashion (pp 28, 29). rp 1 1 1! 2! 3! „ 52 z.z+l z.z+l .z + 2 z z + 3 &c. Hence o I 1! 2! „ ^S = ~ + + + , &C. z 2. Z.z+l 3. z.z+l. z + 2 ' Calculate >S' for 0=13. ■•• t|9 + iI6 + -- — -079,957,427. Add thereto T + i + ... + m= 1-564,976,638. The total is 1-644,934 065. Stirling did not probably know that this is equivalent to Itt^, until Euler sent him his well-known formulae for series of the kind. 2447 X) 34 WORKS PUBLISHED BY STIRLING Prop. III. If r = a.-4^ + -A^ +...[, ( : z.z+1 ) then the sum (to infinity) is ^z^n\ '' , ^^-^^ , G-2BX l^ l{l-x)z {l-x)z.z+l {1-X)3.Z+1 .Z + 2) '\ where A, B, C, ... denote tlie coefficients of the terms preceding those in which they occur. Thus A ^(' 71 h — Ax A = , B = 7- , &c. His well-chosen example gives the summation of the Series of Leibniz ^= 1-1+1-1 + .. .ad CO. 1, Here T = (- 1)--^ ^ .^^ fo^j^j i,^ writing i, If, 2^, &c. for z, so tliat 6=0, Sac. Calculate the sum for z = 12^ from the formula. It is -020,797,471,9. Add thereto l-^+.-.-is = -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 Leihnitius '. (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)8= (s - 1 ) 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 S and S' is S{z' + az'-' ^- ...) = mS' {z' + kz'~' + ...), METHODUS DIFFERENTIALIS 35 then the lust of the sums will be finite both ways only when m = 1 and k = a. In other words the infinite product 00 1 + a + ... 11 'li I n ,!= 1 e + + ... 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 Arltltvietira Iiijiiiitorum 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+{in-l)zr{= 0), , , „ m—l^ II A n+ 1 B n+2 G . then S = Th + + + <S:c. m z m + 1 m z+ 2 '>n ... 171 — 1 ,„ ^ . n A (A IS T, B IB , &c.). m z m If we take T = 1 it becomes f(.-«, 1, . ' ) = '-^^4«, 1, .,1). ^ 1 — m/ m V m/ [orF{a. l.y,^) = _Lf(y-a, 1, y,-^)] 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 hi I'rups. Vlli to Xil ' Stilling rctunis again ami again to the suuimation or transformation of the series delincd by ,^„^ z-m z-n ^, z z—n+1 Professor Wliittaker suggests that the relative theorems were doubtless invented to discuss the series 1 z — m 1 (z- m) (z — m+1) 1 „ -j- \. ^- '—^ : 1- etc. z—n z z—n + l z.z+1 z — n+2 which (up to a factor) represents the remainder after s— 1 terms in the series 1 \ —Hi \ 1 . m, . 2 — m 1 + -^—^ + ^^ o +•••. 1-rt 1 2 — n 1.2 3-/i 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 bis 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 Hdj^oc^'' when where A„, A,j j, ... are integral functions of n of degree r, and for which he finds a difierential equation (jiujional he calls it) af the rth order. He would have been clearer had he adopted the repre- sentation of integral functions as given by himself in the I iifroductioii. ' Cf. Andoyer, Bn/L Soc. Math, de France, 1905. METHODUS DIFFERENTIALIS 37 E.g. Suppose r = 2, and write the equation in the co- efficients as an((x + /3n + y n .??—!) + «„_j(a+ h .n—\ ^ c .n~\ .i\—2) + iV-c. = 0. Let y = HGj^x'", .'. y = ^'}irtjja;""\ y = Ii}i{n— l)rt„a:"~^ &c. Hence (cny + fSxy + yx"^})), + X {ay + hxy + cx'^y) + &c. = 0, or differs from zero by a function of x depending 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'^ y^ — r- x^ — x- y^ nulla alia radice explicabilem esse praeter duas exhibitas y = x- x^ / 6r^ + x^ / 120 r^ + ... y = Ax I -oj^-/2r^ + x^/24r'^+ ... quarum prior dat sinum, et posterior cosinum ex dato arcu x, et de qua posteriore dicis, quantitatem A quae aequalis est radio r ex aequatione tiuxionali non determinari. Ego non solum inveni seriem non posse habere banc foriiiam A : Bx'' + Cx^+... nisi fiat A = r, sed utramque a te exhibitam seriem compre- hendi sub alia generaliori, quae haec est : y = A-^Bx-^ Cx- + ... rr 1.2. rr 2.3. r? 1.2. rr B = -, F=-- D 38 WORKS PUBLISHED BY STIRLING in (ina coefficientes A, B, C, i^'c. hanc sequuntur relationem j^^_ rr-AA ,. -A ,. -B rr E= ^--, F = — , &c. 3 . 4 . r?' 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 exhibes pag. 84 pro radice aequationis y + a-ij-xij-x^y = sub aliis duabus generalioribus quae ex tuis particularibus compositae sunt comprelienduntur.' Bernoulli adds his solutions. (Vide Letter in question.) PARS SECUNDA DE Inteupolatione Serierum §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 ^ 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 1 + 1 + 2 ! + 3 ! + 4 ! + &c. in which the law is T\_^_^= zl\ (the law for the Gamma Function). If a is the term intermediate between 1 and 1, the corresponding intermediate terms are 2^^' 5 • 2"' ■2'2*'2^^' '^■C or, as Stirling puts it, h ■= |a, r = I A, (^'c. (Page 87) ' Cf. Wallenberg and Guldlierg, Thvorie dry linearen Differenzen- GleichiDigen (Teubner, 1911). METHODUS DIFFERENTIALIS 39 Prop. X\'IL Eveiy series admits of interpolation whose terms consist of factors admitting of interpolation. Thus, given the series 1, - A, , 7)', G, &c. ' p _/;+! ' 2J + 2 ' it will be sufficient to interpolate in 1 r r .r + \..., 1 'p p.p+l..., and divide. § 12. Prop. XVIII is of fundamental importance in many of the series discussed. In the two series r r + 1 , a, - a, 0, .... ' q q+1 ' ' if A and « 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 2^ — '>' from the beginning. The illustrations he gives can hardly furnish a proof, for /J — ?' 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 2/1 4 D G P which to meet the conditions must be written as 1^ 1+1 1+2 Suppose the term at distance ivi wanted. Here p — r = — ^. Write q — r — m or q = m+ \, and form the series a 2b 3c ' m+l' m + 2' m + s' Then the term wanted in the first series is that of this second series which precedes 1 by the interval — |. Tliis 40 WOTIKS PUBLISHED T.Y 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,z + A,^-^^,&<^., 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 Mefhndo I ncre mentor um, et postea Hermanns in Appendice ad PhoronomiaTn.' § 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 (i) = 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,+, = z T^ and T^= \, T.,= \. He interpolates between T^ and T-^^ to find jTnJ and then he has to divide by lOf, 9|, ... 1| to obtain T^. Since the numbers are rapidly increasing he uses their logarithms instead and actually calculates log Tnj from which he finds Tvl to be 11899423-08, so that Ti\ - -8862269251. He adds Tj = 1-7724538502, and this number, he says, is -/tt. (-/tt is actually 1-7724533509.) Also the corresponding entry among the numbers 1, 1, 4, 36 576, &c. is it} For inventive audacity Stirling's conclusion would be difficult to match, and its skilful application led him to ' Is it not possible that ho thus d.'t.'ct d that Tj = ^/tt ? METHODUS DIFFERENTIALS 41 results that aroused tlic admiration of his friend De Moivre. (Vide Miscellanea Analytlca.) In Prop. XXn, Ex. 1, it helps him in tlie interpolation of the term at intinit}' in the series 1 2 A 4 R en ^j T-^^' "5^' ■5'^^' • • ' 2 2.4 2.4.6 , or 1, -5 5 J (YC. 1 1.8 1.3.5 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 +xy to 2"^". Binet in his Memoir^ (pp. 319-20) proved that of the four solutions of the latter problem given by Stirling (1) and (3) are correct, wdiile (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, wanting h for the middle coefficient, gives the four formulae (1) (^y=7r^ii^(i I, n+\, 1). Of these (1) and (3) ^ are also the first and third of Stirling's ; while (2) and (4) replace the other tw^o given by Stirling, viz : 92 n 2 (^r (V) = li'^n+r V 12.32 „ - 1 h > &c. 2(2/1-3) 2.4(2/1-3) (2/1-5) * Binet, Me»K fnir Jrs Tnfe(/)riJes definies EitJeriinnes. "^ These are also the Kolutions ho gave in a letter to De Moivre to publish in the Miscellanea Analytka. (See pp. 46-48.) 42 WORKS PUBLISHED BY STIRLING nnl 2{2n-2) 2 . 4 (2 h- 2) (2u-4) J Clearly (4)' must be wrong since the factors 2 9i — 2, 2 7i - 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 Fia,l3,y, l)xF{~a,^,y-a, I) = 1. § 1 4. In Prop. XXIV the Beta Function is introduced (as an Integral) for the interpolation of r r(r+l) and the conclusion drawn (in modern notation) = ^^ (^J + 1 ) . . . (p + H - 1 ) / (yj + (/) (^j + (/ + 1 ) . . . ( /; + g + / 1 - 1 ) . Again, on p. 139, he solves the associated difference equation r + obtaining T = AF{—n, —z, r, 1); and Binet proves the interesting remark that had Stirling T -\- Z 1 1 added the solution of u'= u, where tt'= 7h>j u = jr.t r + z+ It 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, h, c, l)xF(-a, h, 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- volvino- laroe numbers are concerned. METHODUS DIFFERENTIALIS 43 STIRLING'S THEOREM When ii is a lar<i,c niiinber the product 1 .2.3 ...n = ii''W2ii7r e ^-", where < 6 < 1. Stirling actually gives the formula Log (1 . 2 . 3 ... .r) = i log (27r) + (x + i) log {x + ^) J 1 7 -(■»+2)-2.i2.(^,,+ i) + 8. 360 (a; + !)•'" "" with the law for the continuation of the series, De Moivre (Sup2). MlbC. Anal.) later expressed this result in the more convenient form log (1 . 2 ... a;) = I log (27r) + (« + 1) log x B. I 5o 1 T) 1 iM-l (2/t-lj=^« ur' Caueh}^ gave the remainder after the last term quoted as ^ ^ (-irO „^, 1 " (211 + 1) (2)1+ 2) a;-"+^ (5,, i?2, (^'C, denote the Bernoullian numbers.) More particularly the series ^ A_i_ J^l 1.2a; 3 . 4 a;'^ 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.^ Its relative accuracy is due to the fact ^ See Godefroy, Tlieorie des Series, or Bromwicli's Treatise on Series. 44 WORKS PUBLISHED BY STIRLING that the error coininitted at any sta(>;c, by neglecting i^„, 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. Legcndre has shown that if we write the !■ erics as Z (— l)""*"^ u„, then '^n+i/Un <(2u-l)2>i/47r^r2, and .-. < (n/3x)^. The terms therefore decrease so long as n does not exceed 3.r. When n = 3x the error is less in absolute value than •393409... xa-^e-''-'^. To later mathematicians, such as Gauss, who admitted onl}- the use of convergent series, Stirlino-'s Series was an insoluble riddle, but it now finds its place among the series defined as Asymjjtotic Series} To meet the objection to its divergence Binet (I.e., p. 22G) gave the convergent representation. log (a; — 1 ) ! = I log (2 tt) + (« — -1) log x — x 1 + 2 >S' + — s., + — s,+ ... 2 5 A •^4.5 ■* in which /S' denotes 1 1- ... ad oo. " {xi-iy {x + 2f' From this by the use of inverse factorials he deduces (p. 231) log(a;-l)!.= ilog(2 7r) + (^-|)log«-a^ 1 1 + 12(a;+ 1) 12 (a; +!)(.« + 2) 59 1 360 (a;+l)(n'+2)(.f + 3) 227 480(a)+l)...(« + 4) "^ ^ ^' § IG. The conclusion of Stirling's book is taken up witli various proltlems in intoi'polation, based partly on a papei- l)y him in the ndlosophical Transactions for 1719, and partly ' Vide Toincaie, Acta Math., 1886. STIRLING'S SERIES 45 on tlie 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 ii linear c(]uations in n variables, found ' per Aly;ebram vnli^arem '. A translation into English by Francis Holliday was published in 1749 ' with the autlior'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 Differentialis Newtoniana lllustrata Aathore lacolo 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, ^, 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-8 = 0, &c. The assumption of any one of these will furnish a linear equation in a, /3, &c., from which any one of these may be determined when all the others are known : e.g. to determine /dz / {\ -{■ z^), consider {l+z-)-\ {l+z^f, {l+z'^)\ &c. 46 WORKS PUBLISHED BY STIRLING The integrals of these oinittin<;- the first, are r, z-\rz'' j?,, &c. Take the latter as /:J, y. kc , so that a=/(/s/(l+s-^). The above equations give in succession tan-^s^:;; z-z''/^; s-sV^ + ^V^, &c. Other examples are easily constructed. Towards the end of his paper, while discussing a method of approximating to a slowly converging series, he furnishes what seems to be one of the earliest general tests for the convergence of a series. Consider the series of positive terms If, in the long run, 1111 > '^^n '^^n 1-1 '^n-V\ '^n + 2 the scries is convergent ; otherwise it is divergent. There are also the two papers on the Figure of tJte Earth, and on Stirling's Engine, to which reference has already been made. Li:tter fuom Stirling to De Moiyrk printed ix the Miscellanea A nalytica} (De Moivre was naturally much surprised by the intro- duction of TT into the calculation of the ratio of the coefficient of the middle term in (1 +.'•;" to the sum of all the coefficients. Cf. p. 172.) Quadrienium circiter abhinc, v'lr CI. cum significarem D. Alex. Cuming Problemata de Interpolatione & Sunnnatione Serierum aliaque cius generis (piae sub Analysi vulgo re- cepta non cadunt, solvi posse per Methodum Differentialem Newtoni ; respondit lUustrissimus vir se dubitare an Problema a te aliquot ante annos solutum de invenienda Uncia media in quavis dignitate Binomii solvi posset per Differentias. Ego dein curiositate inductus, k confidcns me viro de ]\Iathcsi bene merito gratum facturum, idem libentcr aggressus sum : ^ Miscellanen Anahjtica de Seriehus, pp. 170-2. PHILOSOPHICAL CONTRIBUTIONS 47 & f.iteor ortas esse difficultates quae impediere (|Uoniinus ad optatam conclusionem confestim pcrvenire potuerim, scd laboris hand piget, siquidem tandem asseentus sura solntionem adeo tibi probatam ut digneris eara propriis tnis scriptis inserere, Ea vero sic se habet. Si Index Dignitatis sit numerus par, appelletur ii ; vel si sit impar, vocetur n—l; eritque ut Uneia media ad summam onniinm eiusdem Dignitatis, ita unitas ad medium proportionale inter semi-circumferentiam Circuli & Seriem scquentem X + A + 9B + 25(7 49X> + 2X71 + 2 4xH + 4 6xn + 6 8x?i + 8 + SlE &c. lOx-^+lO Exempli gratia, si quaeratur ratio Unciae mediae, ad sum- mam omnium in Dignitate centesima vel nonagesima nona, erit n~ 100 (jui ductus in semiperipheriam Circuli 1 '5707963279 producit A primum terniinum Seriei ; dein erit B = ^^^A, C = ,f e 5, D = e¥^ G &c, 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 157-866984459 nonagesima nona. Problema etiam solvitur per reciprocam illius Seriei, etenim suunna omnium Unciarum est ad Unciam mediam in sub- duplicata ratione semiperipheriae Circuli ad Seriem 157-079032679 769998199 10658615 654820 37137 2734 246 26 3 1 A 9B 25C 49i) — — , + =^ + == + --— + =^ n+l 2x/t + 3 4X)i+5 6xrt+7 8X91 + 9 + 81^ .&c. 10 X 71 +11 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 •00€30316606304 3059789351 G5566915 2553229 143473 10470 934 98 12 1 Ut si sit li =100 ut Jintoa, eoiii})utuH L'l'it ut ill niarj^iiic vidcs, nbi suuiuia tcriui- iioruni prodit •00033144670787 cujus Rndix quadrata -079589^:373872 est ad uiiitatcm lit Uncia media ad suiiunam omnium in Dignitato centesiina vel nonagesiina noua. Sunt & aliae Series pro Solutione liujus Problematis aequo simplices ac eae liacte- nus allatae, sed paulo minus conver<4eiitcs. ubi Index Biuomii est numerus exi^uus. Caeterum in praxi non opus est recurrere ad Series; nam suffieit suinere mediam pro- -00633444670787 portionalem inter semicircumferentiam Circuli & n + ^; liaec enim semper approximabit propius quain duo priini Seriei termini, quorum etiam primus solus pleruinque sutiieit. Eadem vero Approximatio aliter i^ praxi accommodatior sic euunciatur. Pone 2a = cr= 1-2732395447352 ; eritqueut summa Unciarum ad mediam, ita unitas ad / quam proximo, existente errore in excessu circiter / • 16nn V 2 71 + 1 c St ii= 100, erit— — - = 006334525, ejusque radix quadrata •07958973 accurata est in sexta decimali, quae si dividatur per 16 nn, id est per 160000 dabit correctionem -00000050, & haec subducta de approximatione, relinquit numerum quaesitum •07958923 ju&tum in ultima figura. Similiter si sit n = 900, erit = -000706962545, cuius Radix quadrata -026588767 superat verum biuario 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 logaritlimos numerorum n + 2 & u— 2 dividatur per 16, & quotus adjiciatur dimidio logaritlimi Indicis n; liuic dein summa atljiciatur logaritlimus constans •0980599385151 hoc est dimidium logarithmi seiniperipheriae Circuli, & summa novissima est logaritlimus numeri (^ui est ad unitatem ut summa omnium Unciarum ad mediam. St n = 900 computus erit f log 900 1-4771212547 16)Dif. log 902 t^' 898( -0001206376 Lost constans ...... -0980599385 Summa 1-5753018308 PHILOSOPHICAL CONTRIBUTIONS 49 Et haec siinuna veruni supcrat biiiario in ultima tigura ; estque logaritliiiius numori 37-6098698 qui est ad unitatuiu ut Suuiina Unciaruiu ad uicdiam in dignitatc 900 vol 899. Et si vis illius nunieri reciprocuni, sumo complcmentum logarithmi, scilicet -2-4246i)81692, & numerus eidem corre- spondens inveuietur -0265887652. Et hae sunt Solutiones quae prodierunt per Method uni Differentialem Newtoni ; quarum demonstrationes jam non attingo. cum in animo sit l)revi publico impertire Tractatum quem de Interpolatione & Summatione serierum eonscripsi. Tid Stu(liosissir}ii 10 Jim. 1729 Jac. Stirling BIBLIOGRAPHY (1) Sir D. Brewster: Life of Ne" ton. (2) J. Brown: Epitaphs and Monumental In.'<criptions in Greyfriars Churchyard, Edinhitrah. 1867. (3) H. W. C. Davis: History of Balliol College, Oxford. (4) Edleston : Newton's Corr. spondence tcith Cotes, &,c. 1850. (5) W. Fraser: The Stirlin'js of Keir and their Private Papers. Privately printed, 1858. (6) Gentleman's Magazine for 1853: Modern Histnry of LeadhiUs. (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 Tnasure House in Scotland. 187t'). (12) J. Ramsay: Scotland and Scotsmen in the 18th Century. 1888. (13) S. P. Rigaud : Miscellaneous Works and Correspondence of the Rec. James Bradley, D.D. 1832. (14) B.Taylor: Contemplatio Philo^ophica. 1793. (15) W. W. R, Ball : Newton's Classification of Cubics, London Math. Soc. 1891. (16) Historical works of Cantor, Chasles (Aper9u), Montucla ; articles on Probability and Theory of Finite Differences in Encyclopedic des Sciences math^matiques ; modern text-books on Finite Differences by Markoff, Seliwanov, &c., and on Probability by Bertrand, Czuber, &c. (17) G. Cramer: Courhex algebriques. (18) P. H. Fuss : Corr. math, et 2>hysique de quelques celebres geometres du XVIir siecle. 1843. (19) M. Godefroi : Th^orie 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 Eeihen. 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 : Memoire sur les Integrales Euleriennes ; Jour. Ecole Poly. 1839. (ii) N.Nielsen: Tlieorie der Gammafunktion. Teubner, 1906. Also : Les Polynomes de Stirling. Copenhagen, 1820. (lii; G. Wallenberg und A. Guldberg: Tlieorie der lincaren Dijferenzen- gleichungen. Teubner, 1911. STIRLING'S SCIENTIFIC CORRESPONDENCE E 2 INTRODUCTION Much of the correfcpondence of James Stirling;- has been preserved at the family seat of Garden. In tlie collection are several letters from him to his iriends in Scotland, and numerous extracts from them are to be found in the Family History: — The Slirliags of Keir and their Private Papers, by W. 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 tl;e latter group of letters the earliest is one from Nicholas Bernoulli in 1719, and the last is one from M. Folkes, P.R.8. 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^ 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 diH'usion 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 ob.'^cure points in the history of Mathematics, as indicated in the notes. ]\laclaurin 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 Stirlino-, and those are chiefly copies made by Stirling himself. I had the good fortune to find four original letters from Stirling to Machiurin 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 Stirlino- 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 misht 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 ^erichus, and reference to a second letter in the Supplement to the same work. Ciamer's Letter III3 and the letter from Stirling to Castel V^, are reproduced in the Stirling Family Histoiy. CONTENTS PAGE I CORRESPONDENCE WITH MACLAURIN, 1728-1740. 57 11 Letters from Maclaurin to Stirling. 4 „ ,, Stirling to Maclaui in. 1 Letter „ Gray to Maclaurin relative to Stirling. The letters to Maclaurin have been obtained through the courtesy of Aberdeen University. Letter Ijo 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 fiom Cramer to Stirling. 1 Letter ,, Stirling to Cramer. IV N. BERNOULLI AND STIRLING, 1719-1733 . . 181 3 Letters from Bernoulli. 1 Letter ,, Stirling. L'ABBE CASTEL AND STIRLING, 1733 . . .151 1 Letter from Castel. 1 ,, ,, Stirling. VI CAMPAILLA AND STIRLING, 1738 .... 158 1 Letter from Campailla. 56 CONTENTS PAGE VI r J. BKADLEY 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 ])roblems. IX MACHIN 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, prol)aI)]3" 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 . . .198 ^:i/^^ i^ j-<^ ^f^K^y^^ r/ U£ ^ MZ'^ A^'ai^y 4^1^ I COLIN MACLAUKIN AND STIRLING (1) MacJaiirin to Stifling, 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 Roberts'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 oifer 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 piiblishinu- the rcnuiinder of my piece for wliieh I liave only the excuse of much teacliino- and my desi<;n 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 Manusci-ipts is ye following Article. I et x" - 2)x'^-'^ + qx" -^ _ rx"-'' &c. - be any equation proposed ; deduce from it an E(|uation for its Limits and from this last deduce an equation for its limits ; and by proceeding in this manner you will arrive at the (piadratick n X VI — 1 X x^ — 2 ( a - 1 ) ^xf + 2 7 = whose roots will be imposf^ible if 'jf- be less than q and therefor in that case at least two roots of ye proposed Equation will be impossible. Afterwards I shew that if 1 n — 1 „ r X (^ be less than 'pv 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 everyltody's hands here. The Observation is transform any Equation a? — i)X^ + qx - ?' = to another that shall have its roots less than the values of x by any ditference e : Let 2/ = x — e and 2/'' + 3e2/'- + 3C-2/ + C'' = where any Coefficient considered — 'p[j^—2pe'\) — 'pe^ as an Ecjuation gives for its roots + qij+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, tliird fluxions, &e. 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 tlio paper you have printed. I intended to have sent yoU one of my Theorems about the Collision of many Bodj^s 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 foi' his new Theory. I am with great Respect Sir Your most Obedient and Humble Servant Colin Maclaukin Edinburgh Dec' 7 1728. (2) Stirlwfj to MacJaurin, 1728 Sir A few days ago I received 3'our letter of the 7'^ 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 rioht way of niak[ing himself] known, but indeed I [quesjtion 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 othervvays. 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 ]\Ir 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' + ^a;* + Bx' + Cx^ + Bx + E = and Ez^ + Dz^ + Cz^ + Bz^ + ^0 + 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 erch 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 etjuations 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 e(|uation, 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 dinien[ ]ich his Rule discovers to have 6 impossible Roots, wliereas S'' Isaac's disc()[ ]ly two of the Six. [I] shal now make a remai-k on some of those Gentlemen who dispute for the new [njotion of Force to shew how * Inipossible roots (V). CORRESPONDENCE WITH MACLAURIN 61 mucli they depend one anotlicrs demonstrations which are to convince their Adversarys. Herman in his book page 113, I mean his Plioronomia, says In hac virium a^stimatione, prseeuntem habcmus Illustrissimum Leibnitium, qui eundeni non uno loco in Actis eruditorum Lei})>.ia3 indicavit qiiidem non tamen dcmonstravit, etsi apodictice demonstrari potest, ut forte alia id occasione ostendemus — He denj^s 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 liimself . 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 Woltius. And page 53 [ ] that perhaps some and those not the most scrupulous might doubt [ ] 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 Stirling 62 STIRLING'S SCIENTIFIC CORRESPONUENCE (3) Madaurin 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 liis 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 + Hx^-^ - /«;«-» + ira;"-iO- Ix^'^^ + Mx''-'^'^ &c = ' 1728 O.S. ; but 1729 N.S., cf. Letter !„. CORRESPONDENCE WITH MACLAURIN 63 and ^^^^ X D^ will always exceed EG - FB + GA - 11 n — \ n — 2 11 — 3 „ if m = ii X —— X -— - X -— — cVc. ^ »5 t: till you have as many factors as there are terms in the Equation proceeding 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"^ = P + 2Q. Now the number of those parts is m and therfor by the 4*"^ Lemma of the paper in the transactions for May 1726 (^i— 1)P must be greater than 2 Q and D^ {= P + 2Q) must be greater than Q or D^ greater than Q. Then I shew that 2m ^ ^ Q=EG-FB + GA-H and thence conclude that -— — B^ always exceeds 2 m "^ EG-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 r/i the number of Terms from E to H including both then shall EH = F + ^i^^lQ+'''^.'^^^R + '21±l.'I!l+l.'I^^S m+7 m+8 m + 9 m + 10 „ . + ^- -2- ^ — ^ *<=• where P expresses the squares of the parts of E multiplyed G4 STIRLING'S SCIENTIFIC CORRESPONDENCE by the (liH.siunlai' parts of C. (a term as far distant from the beginning of the E(|uation as H is from E) Q expresses the squares of tlic parts of the coefficient iunnediately preceding E viz. J) miiltiplycd by tlie dissimihir parts of the term next followini;- (' but one viz. in this case E itself. K expresses the s(|uarcs of tlie parts of the coefficient next precedin<^ E but one that is G multiplyed by the dissimihir 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 connnon Genesis of Equations produce it; and by disi^inn'lar parts those that involve not the same Quantitys. Tills general Theorem opens to me a vast variety of Tlieorems for comparing the products or S(juares of coefficients with one another of which those Intherto published are only particular Examples. Here I give you the theorem for comparing any two products of the same dimensions as EI and CL. Let s and m express the nundjer of terms that preceed C and / in the Equation then let 71/ - 1 n — s — 1 P = r X 7r~ X ^ 6' + 1 s + 2 n -s-2 &c. 6- + 3 X ■ii — m- -2 , n — m n — m—l and (7 = ; X — . . •^ m + 1 m + 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 alway exceed (JL when the roots are all real. Then I proceed to compare the ]3roducts of the Coefficients with the sums or ditterences of other products & one of the chief Theorems in that part is that mentioned above which Mr Cami)bell 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 Ecjuations when Sir Isaac's did not in proof of which I faithfully transcribe from my Manuscript the following Article. ' In the Equation x'-Ax^ + Bx-'-Cx'' + Dx-E= x^-\0x*-\-Z0x^-iix^ + 32x-d = CORRESPONDENCE WITH MACLAURIN 65 no impossible roots appear by Sir Isaac's rule. But i>'- x 2 III here is less than AC—D i'or n — 1 4 ,171—1 9 m — II X —~ — = 5 X - = 10 and == — 2 2 2m 20 now 2^0 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 tlie 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"" Isac's rule immediately follows. But I conclude from thence that he did not try to demonstrate compleatly Sir Isac's rule. If he 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 SCIKNTJFJC CORKEsrO]N])ENCE from tlie Limits is not so simple as tluit I luivc wliieli I may semi you a;4aiii. I now beg pardon lor this lung letter which I beg you would communicate to Mr Machin not by way ol' complaint against him fur whom 1 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 ?ome 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 Aflectionat Humble Servant Edinburgh CoLiN Maci.AUUIN 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 Ca, Gb, Ge, Cf &c. to determine ye direction of G itself after the stroke. CORRESPONDENCE WITH MACLAURIN 67 Suppose Da, Dh, De, Df &c. perpendicular to the directions CA, GB, GE, CF, &c. Imagine the Bodys G, A, B, E, F &c. to be placed in C, a, b, e, f &c. respectively ; find the centre of Gravity of all those Bodys so placed and let it be P. Draw DF and GG parallel to DF shall be ye direction of G after the stroke if the Bodys are perfectly hard. Adieu (4) Maclaur'ni to Stirlitig, 1729 J\Ir 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 E(j[uations. 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) lie 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 f3 68 STlHLlNCrS SCIENTIFIC CORRESPONDENCE that he had liit upon; and reflected that the generality of my TlieoreniR would satis- fy any judicious reader; I became less concerned. All I now desyre is to have my paper or at least the first part of it pnblisJied as soon as possible. I beg you may put Mr ]\rachin in mind of this. I doubt not but you and he will do what you can to liave 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 of 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 I ever have any opportunity of showing with how much Esteem it afiection I am Sir Your Most Obedient Humble Servant Colin MacLauiun Edinburgh May 1 1729 CORRESPONDENCE WITH MACLAURIN 69 (5) Maclaiirin to Sfirliiuj, 1720 Mr James Stirling at the Academy in little Tower Street London Sir Since I received your last I liave been mostly in the country. On my return I was surprised with a printed piece from ]\Ir Campbell ag-ainst me which the gentleman who franked tlie letter told me he sent you a copy oft". Tlie Gentleman indeed added he had not frank'd it it' he liad known tlie 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 M'ould 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 3'ou desyred him to write against mo. I am convinced this is false. Please to send me the letter I wrote to you in februar}^ if you have preserved it or a copy of it. I wish if it is not too nuich trouble you would send me a copy of all I said relating to IMr 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 whore 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 3'ou hear my repl3^ I have writ at large to Mr Folkes by this post who will show you my letter if 3'ou please. I assure you I am with great Esteem Sir Your ]\Iost Obedient Edinburgh Most Humble Servant nova's. 1729 CoLiN MacLauhin 70 STIRLING'S SCIENTIFIC CORRESPONDENCE (6) Stirling to Maclaurhi, 1720 To Mr Maclaiirin Professor of Mathematicks in the Universitj^ of Edenburgli Out of 3^our 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 puldished. but according to my taste not so elegant. Out of Letter of December 7, 1728 Let ic"— pa;"'^ + 9^.i'"~^ — raj''"^ &c. = 0, be txny Equation proposed, deduce from it an Equation for its Limits 9ia;""^ — 96- 1 X p.("-- + yt - 2 X g.c"-^ &c. = ; 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 d'c. A Cop3^ of 3' our Letter Fob 11, 172|. S^ Last Tuesday night I saw the philosophical Transactions for the month of October for the first time. At an}' rate in a few days 1 shall lie verj- easy about the whole matter. I am ^q.. S"" This is an exact cop}^ except the postscript which containing a Theoreme about the collision of Podys 1 presume is nothing to the present pui-pose. I am with all respect Sr Your most humble servant Ja: Stirling London 29 November 1 729 CORRESPONDENCE WITH MACLAURIN 71 (7) Maclanrin to Stirling Dear Sir I send you witli this letter m}^ answer to Mr George Campbell which I publish with regret being so far from deliofhtino- in such a difference that I have the greatest dislike at a publick dispute of this Nature. At the sfime time that I own this Aversion I can assure you it flows 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 delayed 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 ^ Written above the line. 72 STIRLING'S SCIENTIFIC CORRESPONDENCE end because it was like to have letjuired another hali'-sheet. I am sure I have given more tlian the subject deserves. I liave left out two or tliree paragraphs about his inconsistencys his stor}^ of some that visited me and found me so and so engaged Arc. This I answer in my manuscript letter sent to you, Nov. 5. I am indeed tyred wnth this affixir. I wished to have hoard from you what lie ol)jected 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 usage 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 Laurin 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 MacJaurhi, 17.1,2 London 2.3 Novom'" 1732 Dear Sir I had the favour of yours yesterday (S: inclosed a part of the abstract of your Supplement wilh a Letter to Mr Macliin, which, as you desired, 1 copyed & gave to him. He is of opinion that it will be iujproper 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 kiiul services to you. I am thinking that it will not be impro})er to move the Society at their first meetino- that Stirling be in Hodgson's room ; because he is nnich more capable of judging than him ; but in this I will follow Mr Machin's advice. I hope j^ou had m}' last, and am persuaded you will do in that affiiir what is fit. I have a great deal of business to do this evening. T will therefore only assure you that I am most faithfully Dear Sir Your most obedient i^' most humble Servant Jno Gray (9) Madaurin to Stirlh/g, 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 Countrj^ 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 k 13 degrees westerlj^ ; 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 Liid 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 ohli^eil to ask of you that I liad 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 lias 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 ma}^ seem to require here. When I was verj^ 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 toii'ethor with readino- some of the Antients more carefully than I had done in my younger years. I have !-ome 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 ever^diody 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 delaj^ 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. Peudjerton's book and the Doctor's delay diminish my influence in that very much. Looking over some letters I observed the other daj^^ 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 Ncnture to sett me down for seven Copys. Jjut I hop(! to gett moi"e if 1 had once fairl}' delivered Dr Smith's l>o(tk to the subscribeis. As to your Treatise of Series i got a copy sent uk; IVom 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 connnonly talk'd when I was in London in 1732. Please to give my humble ser\ice to Mr Machin and believe me to be ver}^ afFectionatly Sir Your Most Obedient Edinburgh Most Humble Servant Nov'-. IG. 1734. Colin MacLaurin I have taken tlie libert}^ to desyre Mr Warrender to take advice with you if any diffieultys 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^ To Mr James Stirling at Lead hills Dear Sir This is a copy of Maupertuis's letter which I thought it would be acceptable to } on to receive. I am told Mr Cassini would willingly find some fault with tlie Oliservation to save his father's doctrine, but is so much at a loss that he is obliged to suppose the instrument was twice disordered. H" I can be of any service to yon here in anything you may always command Dear Sir Your Most Obedient Humble Servant Ed^. feb.'" 4. 1737. Colin Mac L.\urin I forgot when yon was here to tell j^ou that last spring 1 1737 O.S. or 1738 N.S. 76 STIRLING'S SCIENTIFIC CORRESPONDENCE some Gentlemen liad formed a design of a philosophical society here wliicli tliey imagined might promote a spirit for natural knowledge in this country, that you was one of tlie meml)ers lirst thought of, and tliat Ld Hope & I were desyred to speak to you of it. I liopc and intreat j^ou will accept. The mnnhcr is limited to 45, of wliicli are L''^ Morton, Hope, El[)lnnston, St Clair, Lauderdale, Stormont, L'' president & Minto, S' Jolni Clark, D" Clark, Stevenson, St Clair, Pringle, Johnston, Simpson, Martin, Mess. Munroe, Craw, Short, Mr Will'" Carmichacl c^'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 nundjer. If j^ou would send us an^-thing it would lie most acceptable to them all iS: particularly to yours iS:c I had a letter from Mr De Moivre where he desyres to give his humble service to 3'OU. His book was to be out last week. Mcmpertms to Bradley A letter from Mons"" ]\Iaupertuis To Professor Bradley Dated at Paris Sepf 27^^ 1737 N.S. [Translated fi-om the French] Sir The Rank You hold among the Learned (k the great Discoveries with which you have enriched Astronomy, would oblige' me to give j^ou 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 }0u) by reason of the Share vou 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 3'our Assistance. Wherefore I have the honour to Accpiaint You Sir, That we are now returned from the Voycige we have made by Order of His Majesty to tlie Poler Circle. We have been so hajipy as CORRESPONDENCE WITH MACLAURIN 11 to be able, notvvithstaiKlino- the Severity ol* that Climate, to measure from Tornea northward a Distance o£ 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 tlie .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 Triancrles. We afterwards determined the Amplitude of this Arch by the Starr (5 Draco als, 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'' 173G. 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 Theoiy (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'-:50"-54 wliicli ditiers 3''i iVoiu tlio Amplitude (k'torniined liy S (Delta). Therefore takiu^j^ a Mean between these two amplitudes, Our Arch will be 57'-28"-72 which beino- couipared with the distance measured on the Earth, gives the Degree 57437'1 Toises; greater by 377-1 Toises than the J\li<ldle Degree of France. We looked upon the Verification which results from the Agreement between our two Amplitudes deduced from two ditterent (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 recpiisite for our operation to knov/^ 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 diff'ered 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 tiie two Degrees wliicli 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 E(|uator; 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 1 have one CORRESPONDENC^l^ WITH MACLAURIN 71) favour to Itei; of you, Sir, & hope you will not rcfus-c it lue; Which is, to let lue know if you have any immediate Observa- tions on the Aberration ot" our two Starrs 8 e^ a' DracoiitK; and if we have made proper (\)rrecti()ns for this AbeiTation. I shall have the honour, at Some Other time to eouumniicate to you our Experiments on Gravity, & the Whole detail of dur Operations, as soon as published. I have the honour to be with Sentiments of the highest Esteem Sir Yovu- Most humble iV' most Obedient Servant Maupiktuis I shall be much obliged to you if you will be pleased to Communicate . . , the Royal Soe . . . (11) Maclaarin to IStirJiiig, 1738 Mr. James Stirling at Lead Hills Dear Sir There is an ingenious young man here who I am very sure will please you for what you write about. I have promised him no more but that you will bear his charges in going & returning & give him gome small thing besides perhaps. I have not omitted to acquaint him that he will have opportunity to improve himself with you. He is a quiet modest industrious & accurat young man. I think I have mentioned him to you as one who seems to have a natural turn for making mathematical instruments, & deserves en- couragement. But his father is a poor minister who has ruined himself l)y lawsuits. If it will be time enough, it will be more convenient for him to go about the middle or end of May than just now. I have a part of a letter I writ for you some weeks ago in town, but some incidents hindered me from finishing it. 8;) SriRLTNG'S SCIENTIFIC CORRESPONDENCE I shall write soon by the post. This j^oes by a student who is to leave it for you at the lead hills. I aiu Dear Sir Your Most Obedient Dean near Ed'" Humble Servant April 1738. Colin MacLaurin JMr Do Moivre's Ijook is come but I have not had time to look much into it. I think you said you would send me Mr Machin's piece. I say a little of the centripetal foi'ces l)ut that part is now a printing off. Have you ever had occasion to enquire into the tiuent of such a (juantity as this X Va — XX Vh — XX Vc — X The common methods do not extend to it. My family is now come to this place, but I go every day to town to the coUeire. The removini^; & some incidents occasioned my delay in writing which I hope you will forgive. (1:2) Madimr'm to StirUng, 1738 To Mr. James Stirling at Leadhills Dear Sir This is to introduce Mr Maitland whom 1 have dispatched sooner than I intended because of your urging it in a letter I received on Monday last. I heartily thank you for Mr Machin's piece, and that you may not be deprived of the book bound in with it I shall send you my copy of it. I am persuaded many things are wanting in the inverse method of fluxions especially in what relates to fluents tliat are not reduced & perhaps are not reducible to the logarithms or circle. I give a chapter on these, distinguish them into various orders, and shew easy constructions of lines by whose CORRESPONDENCE WITH MACLAURIN 81 rectification they may be assigned, how to compare the more complex with the more simple & other things of tliis nature. But I suspect that some fluents (at least in some suppositions of the variable quantity) may be reduced to the circle or logaritlims that are not comprehended in the cases that have been considered by Cotes <% De Moivre. I could not hit upon a letter I had writ a great part of to you in our vacation week when I sought for it today. I shall mention sometliings of it as my memory serves. I easily found as you observed that the rigiit line AB attracts the particle P with the same force as the ark GED but I could make little use of this because when the figure revolves on the axis PE, the attractions of the circle generated by AE & of the spherical surface generated by CE are not equal. I found that what I had observed long ago of the attraction of spherical surfaces holds likewise of what is included betwixt two similar concentric spheroidical surfaces in- finitely near each other viz. That the attraction of the part convex towards the particle is equal to the attraction of the part concave towards it. This holds whether the particle be in the axis of the spheroid or not. Let EGKL be any solid, P the particle attracted, let PEK Fig. 5. Fig. 6. meet the solid in E &, K and any surface GHL in H, let NH be to EK in any invariable ratio, and the point N form a surface GXL. Then the attraction towards the solid GNLH shall be to the attraction of the solid EGKL in the same given ratio of NH to EK. Let ACE be a quadrant of a meridian, A tlie pole, E at the 2447 Q 8.2 STIRLING'S SCIENTIFIC CORRESPONDENCE equator, it' T^M be the divectiou of the onivity at L then CM sliall he to tlie ordinate LP in an invariable ratio. This ratio I cannot preciseh^ recollect unless I had my paper < which ai"e at the Dean. I remember it is compounded of two ratios but how I can- not suddenly recollect One of them I think is the ratio of the o'ravity at A to the force towards a sphere of the radius GA, the other is the ratio of the gravity at E to the force towards a sphere of the radius GE. I write this in a haste at the college because Mr Mait- land waits for it and I do not incline to detain him. On looking over the argument by which I had thought to have proved that the earth is a spheroid, I found that it supposed that in any right line GL from the center the gravity at L is to the centrifugal force as the gravity at / is to the centrifugal force. But this seems to need a proof. I have some more propositions, if they be worth your while I shall send them. Having no time to go home for the book I was to send I delay it till some carrier call to whom I shall give it. If you will send me your receipt for De iNToivre I shall cause one of the Booksellers get it down. In the mean time you may command my copy if you please. I am Dear Sir Your Most Obedient Ed-" May 12. Humble Servant 1738 Colin MacLaurin. (13) Stirling to JMucJanrbi, 1738 Leadhills 13 May 1738 Dear Sir I am obliged to you for dispatching IMr Maitland, for I am in a hast, ^ I hope he will do very well with smal assistance. CORRESPONDENCE WITH MACLAURIN 83 I shal be very gUul to see what you liave on tluents when your book comes out, particularly it' you can reduce to the area of a Conick Section, figured not comprehended in the Theorems of S'" Isaac, Cotes, or Be Moivre, I readily agree with you that great improvements may be in that piece of knowledge ; but the way to it is so rugged that I am afraid w^e arc not in the right path. Fig. 8. As to the attraction of an arch and its tangent being the same, on a particle placed in the center, it was of no use to me more than to you. What you say about the attraction of the concave and convex part of a spheroidical surface, being the same on a particle of matter, holds of any part of a spheroid comprehended l)etwixt two similar, concen- trical and similarly placed spheroidical surfaces, whether the distance betwixt them l)e infinitely smal or finite ; Suppose two such surfaces to l)e AEKB and GD^M, and a particle P placed any where ; through P and F the center of the spheroid, imagine a spherical surface to be described similar and similarly placed with AEKB; and that surface will cutt of the concave part from the convex part; and will divide the W'hole spheroid into two parts, whose attraction on P are equal ; which is true wdiether the particle P be w-ithout or with the spheroid. The reason of it is because the ellipsis passing through P and F, cutts all the lines AB and EK into equal parts, if they converge to P. And from the same principle follows what you say next in your letter, about the attraction of solids being in a given proportion : because the solides may be divided into cones wdiose vertex is the particle attracted. And what you say about LP being in an invariable ratio to CM is true ; but that ration cannot be assigned without G 2 84 STIRLING'S SCIENTIFIC CORRESPONDENCE the quadrature ol' the circle. And the Avhole problenic about tlie variation of (gravity on the Surface depends on it. When I firs solved that prol)lem, I supposed the attracted particle to be on the surface ; but now I am upon solving it, when the particle is placed with- out the spheroid on any distance, which I have not had time yet to do, altho I know I am master of it ; I have done it at the equator, I mean when the particle is in the plain of the equator produced ; Newton did it when it was in the axis produced. Suppose two ellipses similar described about the same center whose axes are EK and ek, and GL and gl the diameters of Fig. 9. P-= Fig. 10. their equators whose difference I suppose infinitely little : Let F the focus and C the center ; then if the elliptic ring revolve ai)out the axis EK and generat a solid ; and P be a particle in the axis produced, the gravitation of the particle P towards the solid comprehended betwixt the spheroidical surfaces will CORRESPONDENCE WITH MACLAURIN 85 be proportional to — jj-p^ — : that is in a ratio compounded of the direct ratio of a rectangle under the axes, and in the duplicate inverse ratio of the distance of the particle from either of the foci : whence it follows that the gravitation of the particle to the whole spheroid will be proportional to the bigness of the spheroid and the diflerence betwixt the ai'ch Or (described on the center C) and its tangent CF. Again if M be a particle in the plain of the equator pro- duced, it will gravitate to the part of the spheroid betwixt the two spheroidical surfaces with a foi'ce propoitional to — . And thence the gravitation of the particle FGVPC'-CF^ to the whole sphseroid will be found to depend on the quadra- ture of the circle, nay upon the forementioned difierence CF and Cr. I have gone no further ; but could accomplish what remains in a week or ?o if I had leisure. What I here send you are conclusions hastily drawn, and therefore I would not have them communicate because I have not yet examined them to my own satisfaction, and I write in such hast that I dont know if I have transcribed them right. I am in great liast DS. Your most obedient humble Servant James Siirling. (14) Maclatirin to Stirling, 1738 To Mr James Stirling at Leadhills Dear Sir I believe you will find Mr Maitland utefull & exact and am glad he has so good an opportunity of improving himself under your eye. I wish you had time to finish what you are doing relating to the figure of the earth. I am informed thst something is soon to be published on that subject at London by Celsius & others. 86 STIRLING'S SCIENTIFIC C0RRESP0NDENC1<: The account 1 gave you ot" some propositions had occurred to me on that subject was very imperfect. You may observe from what follows it, that when I spoke of concentric surfaces infinitely near I restricted it onl}'' that I mioht distinguish the parts more properly into such as were convex and concave towards the particle. I inquired into the ratio which I paid was invariable & obtained it in a simple enough series which I have not reduced to the quadrature of the circle, tho' I conclude from your more perfect solution that it must be reducible to it. I did not try the problem by the concentric surfaces but in a different manner. And tho' I think 3'our method must be better since an account of a different one may be agreable to you I shall describe the principal steps I took. Supposing PB the shorter axis, AC the transverse semi- axis. I first computed the fiuxion of the attraction of the solid generated by PMB while the figure revolves about the axis PB, and thence demonstrated what Mr Cotes says of the attraction of spheroids. By comparing what I had found in this with 3'our account of the attraction of P I drew immediately on reading j^our letter this consequence that seems worthy of notice. That if PM meet a circle described from the center P with the radius PC in N and NR be perpendicular to PB in R, &: PE be taken equal to CR, and EFG be a similar concentric semiellipse, then the attraction of P towards the solid generated by EFGE revolving about EG shall be equal to the attraction of P towards the solid generated by the segment PAM revolving about PB. This however I did not observe in the spheroid till I got your letter, in the sphei-e it is obvious. After I had made out Mr Cotes's theorems, I then proceeded to consider the attraction at the equator, and still sought the fluxion of the attraction of the solid which seemed then to me to be more easily obtained than that of the concentric surfaces in this case especially. I supposed therefore the solid to be projected orthogi'aphically on the plane of the meridian PA B D, the particle attracted I supposed to be directly over C, CORRESPONDENCE AVITH MACLAURIN 87 and to bo in the pt)lc of the meridian FABD, 2s CM k nCm to be any two infinitely near elHpses passing through the particle ; and then I computed the attrac- tion of the matter included betwixt these two ellipses, or the fluxion of the attraction of the solid represented by CPM. Thus I found that if CP = a, GA = h, CF(F being the focus of the generating ellipse) = c, then the attraction of a particle at the equator towards the spheroid is to the attraction towards a sphere of the radius 36-2 9c* CA as 7X1 + I) + &c : is to unit. 106- ' SG/y-* From this I computed the invariable ratio I mentioned in my last, wherein the difi'er- ence of the tangent OF & ark CZ entered by Mr Cotes' s theorem already spoke of. But by 3^our letter I perceive you have found the same invariable ratio without a series, by the quadrature of the circle only. From which I perceive that if the series I found be legitimate, as I cannot doubt but it is, it must be assignable by the circle. This perhaps would be easily found by examining it, but since you have done this already in effect I would willingly avoid the trouble. And only desyre you will let me know if the proportion given by this series agrees well enough with what you have found. I believe I might have computed your proportion from what you sent me^ but there are so many of my acquaintance in town this week & I have had so little time that I have not got it done. I have some suspicion from the fluxion that gave this series that it is reducible to the circle, or to the square of it, by a way I have sometimes made use of and I believe is not new, of transforming a fluxion by the negative logarithms, but I have not made the computation necessary to judge of this. You may be assured that I will communicate nothing of what you send me without your express alloiuance. I say something on this subject in my book, and would willingly add to it if you pleased, because since my book is grown to such a bulk I would willingly have as much new in it on the usefull problems as I can. I first proposed only to demon- 88 STIRLING'S SCIENTIFIC CORRESPONDENCE strate Mr Cotes's theorems in a brief manner enough after what Sir Isaac has on spheres, and so refer for the rest to your piece in the transactions; but I wouki think it more compleat to add this I have found since on the attraction at the equator &: either suVyoin that 3'ou had a more compleat solution which you would publish afterwards or mention, if you inclined that solution itself. In this I sliall do just as you pleaf-e. I have not as yet tryed if the method I took for the attrac- tion at the equator would succeed for computing the attraction at any other part of the spheroid, and hardly think it worth while to [ ] since you have a method that appears to be much better. All I have mentioned I did before I received your letter except the observation near [ j end of the first page of this letter, else I had not taken so much p[ains] about it. I was chiefly induced to try it, because I imagined the method to be different from your's, and sometimes by following a different method conclusions come out more simj)le ; but it has not proved &o in this instance as far as I can judge of your r[esult]. I told you there were some fluxions which I had ground to suspect depended on the circle & hyperbola besides those described already by authors but I did not say that I had reduced these fluxions That I sent you is one of them, in certain cases of the variable quantity. I resolve to try it, but it is my misfortune to get only starts for minding those things & to be often interrupted in the midst of a pursuit. The enquiry, as 3'ou say, is rugged and laborious. This is the first post as I am told to the lead hills since I got your letter, and I shall be obliged to you if you will let me know without delay whether the series I described agrees with your solution by the circle which I imagine you will see at a look. I am Dear Sir Your Most Obedient Dean May 20. 1738 Humble Servant Colin Maclaurin I have not the transaction by me where your paper is, else that perhaps would solve my question. CORRESPONDENCE WITH MACLAURIN 89 (15) Stirling to Maclanrin, 1738 To Mr Maclaiirin Professor of Mathematicks in Edenburo-li Leadhills 2G October 1738 D. S. I was sorry that when I was last in Edenbnrj^li I could not get time to wait on 3'ou. I got a letter this last summer from IVIr Machin wholly relating to the figure of the Earth and the new mensuration, he seems to think this a proper time for me to publish my proposition on that Subject when everybody is making a Noise about it : but I chuse rather to stay till the French arrive from the South ; which I hear will be veiy soon. And hitherto I have not been able to reconcile the measurement made in the north to the Theory : altho Dr Pound's and Mr Bradley's most accurate observations on the Diameters of Jupiter agree to two thirds of a second with m3^ computation. Mr Machin tells me you write to him that you had hit on a demonstration to prove the figure .of the earth to be a spheroid, on which I congratulate you, for my part hitherto I can only prove it by a compu- tation. I have lately had a letter from Mr Euler at Petersburg!!, who I am glad to find is under no uneasiness about your having fallen on the same Theorem with him, because both his and the demonstration were publickly read in the Academy about four years ago ; which makes me perfectly at quiet about it, for I was afraid of giving grounds of suspicion because I had long neglected to answere his first letter : his last one is full of a great many ingenious things, but it is long and I am not quite master of all the particulars. I have also heard lately from M. Clairaut, where he makes a great many apologies for having taken no notice of my paper about the figure of the earth when he sent his from Lapland to the Royal Society ; and he tells me he has carried the matter further since that time in a new paper which he has also sent 90 STIRLING'S SCIENTIFIC CORRESPONDENCE to tliu Royal Society : now lie says he has heard that I have been at some pains about that problemc and desires to have my opinion on his two papers. Tlie first I barely saw l)efore it was printed, and altho I had not time to read it thoroughly I soon saw that it was not of a low rank, as for the second I never saw it; and therefore I should be much ol)li<;ed to you if you could favour me with a sight of both, that I might be able to answere his letter. If you can, please send them to Mr Mait- land who will give them to Mr Charles Sherrif at Leith with w^hom I correspond weekly, and they shall be carefully and speedily returnetl. I haxe yet had no time to medle with that affair, and when I have, possibly I may not have inclina- tion ; but I shal be very glad to hear what you are doing & wdien we may expect to see your book Sir Your most obedient & most humble servant James Stirling. (IC) Maclaiirin to Sdrling, 1740 To M"" James Stirling At Leadhills Dear Sir I designed to have writ last Saturday, but having gone to tlie country that forenoon, I did not get homo that day. I am glad you are to send us a paper, and thank you for allowing M' Maitland to come here for some days to help me to forward the plates. I will acquaint him when I shall be ready for him, that I may make that my only business (besides my Colleges) while he is here. We have some daj^s of vacatioii about Christmas, if that time be not inconvenient for you I can find most leisure to apply to the figures then. I have so much drudgery in teaching, that I am commonly so fatigu'd at night I can do little business. M'' Short writes that an unlucky accident has happened to the frencli Mathematicians in Peru. It seems they were CORRESPONDENCE WITH MACLAURIN 91 shewing some iTeiieli guUantiy to the natives wives, wlio have murdered their sei'vants destroyed their Instruments & burn't their pajK'rs, the Gentlemen escaping narrowly themselves. What an ugly Article will this make; in a journal M"" Sliort saw the satellite of Venus Oct''. 23 for an hour in the morning, the phas is similar to that of Venus, but writes that he has never been able to sec it since. His account agrees with Ca&sini"s. It is a very shy planet it seems. M'' Graham has found that Brass has some influence on the magnetic needle, but I have not got a particular account of the experiments. I wish I had an opportunity to shew j^ou all that I have printed in my book relating to the attraction of spheroids and the figure of the Earth. In the mean time I shall give you some of the chief articles. 1. I begin with what I sent you two years ago, but the demonstration is somewhat difierent. 2. I give a general proposition concerning the attraction of a slice of a solid the figure of the section and position of the particle being given. 3. I apply this to spheres in a few words, and then to a spheroid. The attraction at the pole is measured by an area easily reduced to the circle. The attraction at the equator by the complement of this area to a certain rectangle. Here I take notice that you was the first that measured the attraction at the equator by a circle. 4. I easily reduce the al traction in the axis or equator produced to the attraction at the Pole and circumference of the equator, without any computation or new quadrature. 5. I apply this doctrine to the late observations & mensurations. G. The result of this leads me to shew that a density increasing towards the center accounts for a greater increase of gravitation from the ecpiator to the poles but not for a greater variation from the spherical figure ; and that it is the contrary, when the density decreases towards the center. I then compute both in several hypotheses of a varial)le density, and then propose it as a query whether D"" Halley's hypothesis may not best account for the increase of gravitation & of the degrees at the same time. I afterwards treat of Jupiter, and find that supposing his density to increa,se with the depth uniformly so as to be 4 times greater at the center than at the surface, the mean of D'' Pound's ratios will 92 STIRLlNcrs SCIENTIFIC CORUESPONDEXCF result. I find tlie variation i'rom Kepler's law in the periods of his satellites arising from the splieroidieal tigure of the primary cannot be sensible. I shall send you the proposition you mention and would have sent it today, but I have been somewhat out of order. It would be better to send j'ou the 2 or 3 sheets that relate to this subject if I could find a proper oi)portunit3\ 1 know not any particular reason for }tV Machin's printing that piece of late. M'" Short who engaged to send me the transactions has not as yet sent me M"" Clairaut's 2*^ paper. I have printed all my book, excepting the 3 last sheets. The printers are very slow in the algebraic part, and I have little time at this season of the year. This with the figures will retard the publication I believe to the spring. I am Dear Sir Your Most Obedient Humble Servant Colin MacL.vuiux. Edinburo-h: Dec'. 6. 1740 II SIR A. CUMING AND STIRLING Cuming to Stirling, 1728 Kensington July 4*'' 1728 These were transmitted me from Scotland this day by M"^ George Campbell. I am Dear M'" Stirling Your most obedient humble Servant Alex"". Cuming Let water run out of y^ circular hole NBRD whose radius BC = r. Let AC y® constant height of y° water above G y® center of y" hole be = a, and let Q = y"^ quantity of Water which wou'd be evacuated thro y^ same hole in any given time t ; providing y° water was to run out at all parts of y" hole with y" celerity at y° center C. Then y° quantity of water which will be evacuated in y° same time will be = B N( cHr D Fig. 13. 1 - 3- ^^^-2^i ^7^+4^6 ^ 13 5-7 Or'', + tX-x-- X — X X— T + IVC 4 G 8 10 12 a'' Let A DP be y^ elliptick Orbit which any of y" planets describes about y® Sun placed in one of y^ foci S, let i^ be 94- STIRLING'S SCIENTIFIC CORRESPONDENCE y" other I'ocus, C its center, ^1 y aphelion, P the perihehon, SMy^' mean distance of y" planet from y" sun, and let 7) be any place of y° planet. Let SM or CA he = r, ye lesser semi Axe CM = c, r — c = (l, the excentricity tiC = a, k let m represent y" de<^rees in an arch of a circle equal to y' radius or m = 57-29578. Let u be y*^ sine of y" angle AFl), and x the sine of its double y" radius being — r. Then y'' difference between y" angle AFD (which is y*' mean acquate anomaly) and y" mean anomaly Fig. 14. belonging to it, will be _ 2 md-^u^ 4 ma^u^ - 3 c3^:5 5 ~ + •nid 2r' c'r" X 1 + 6 ma'' lb' iVc 7 c:"!'^ 9c"-^f^ + 8c(:Z- + 2(P + 100c3cZ2+145c2cZ3+72c# + 12c?-^ li} 13c^ r* -c^c. From whence is deduced an easie method of determining y® true anomaly from y° mean anomaly being given. Let the angle Y be found which beaieth y° same proportion to an angle of 57-29578 degrees which half y® difference between y" semi axes bears to y'^ greater semi axe. Let also y® angle Z be found bearing y° same proportion to y® angle of § of 57-29578 degrees or .38-1971 degrees which y^ cube of y" eccentricity bears to y^ cube of ludf y" greater semi axe. Take an angle T proportional to y'^ time in which the Arch ^D is described or equal to y® mean anomaly. Then let y'' angle V be to y" angle Y as y'' sine of twice y'^ angle T is to y® radius, let also y ' angle X be to y*^ angle Z as y® cube of y® sine of T is to y° cube of y" radius, then y° mean acquat anomaly or AFD will be very near T+X+ F when T is less than 90, but T^X-V when T is more than 90° and less than 180=. Let z represent y^ ratio of y^ centripetal force at y^ acquator of any planet to y" power of gravity there, tlius in y" case of y° Earth z— gig- Then ye aequatorial diameter will be to the Polar, as 1 is to 1 — l-s 4 ■^%z- — -Mn%z'' kc. Ill G. CRAMER AND STIRLING (1) Cramer to Stirling^ 17 28 To Mr James Stirling F.R.S. in y° Academy in little Tower Street London Sir, Tis time to break ofFy^ silence vvich I kept so long, the' unwillingly. The wandering life of a traveller, and a long and tedious distemper, have been the only reason, why I did differ so long from giving you thanks for all the kindnesses and tokens of friendship you bestow'd upon me during my sojourn in London, and from making use of the permission you gave . me of writing to ye, and inquiring into the litteral news of your countr\', but chiefly into the news of your health wich is very dear to me. The very day of my departure I received a Letter from M"" Nicolas Bernoulli desiring me to present you his duties. In the same he demonstrates in an easy way, a General Principle whence it is not difficult to derive all y'' Propositions of M"" de Moivre about his Serus recur rentes. The principle is such. Let m + n + p + q, be the Index of y^ Series, and inquire into y'' Roots of y'^ Equation z'^ — mz^ — nz^—2)z — q= Let them be z, y, x, v: And make four Geometrical Series the Indices of whom be z, y, x, v. The Sum of y" respective Terms of these Geometrical Series is the respective Term of y^ Series recurrent four terms of wich may be given, because y® four first Terms of y'' Geometrical Serieses are taken ad 1)6 STIRLING'S SCIENTIFIC CORRESPONDENCE libitum he demonstrates also liis method for findin<^ the Com- ponent quantities of a Binomimu like 1 + :" hy y° Division of ye Circle I would fain know your opinion of this demonstration I found of M"" de Moivre's first Lemma in his Doctrine of Chances. The Lemma is such The number of chances for casting 7; + 1 points, with n Dices of / faces each is ^:) .p— 1 .^) — 2 ...p — 91 + 2 n q .q—\ ...q — n + 2 1.2. 3. ..91^1 1 1 . 2...n-l n{n—\)r.T—\ ...r — n-\-2 1-2 1.2 ...n-l {<l=P~f n.n—ln—2 s . s— 1 ... s — ?i + 2 „ . X = (VC V' = q — t 1.2.3 1.2. ..71-1 ^ . [s = r-f &c. The Series is abrupted when one Terra comes to be nought or negative. My demonstration is grounded upon that principle that the number of chances for casting ^? + 1 points with n Dices is equal to the number of chances for casting p and 2^—1 and p — 2 &c. to 2^~f+ 1 = 5'+ 1 points with n— 1 Dices. For it follows that y® number of chances for casting p points with one Dice is p^ — q^, wich is equal to nought if q is positive that is if p is bigger than/ and equal to one if |? = vel < /. Now the number of chances for casting p + l points with two Dices is equal to y® number of chances for casting 2^ with one Dice = 2^^ — <l^+ to y" number of chances for casting 2? — 1 with one Dice = p — 1 —q—1 &c &c &c &Q to y° number of chances for casting j>—/+ 1 with one Dice that is q + 1 — r+1 The Sum of y^ P* Col. p-q of the 2-^ Col. -q + r Total sum 2) — 2 (7 + r CORRESPONDENCE WITH CRAMER 97 I couVl proceed in the .same manner to the case of three Dices, then to four, and so forth, and if 1 will, demonstrate in general that if the Lennna holds for the case of n—1 Dices it holds too for n Dices. M'' 'S. Gravesandc, who is wholly employ 'd about y® Doctrine of forces, did comnuinicate me the following con- struction for the laws of percussion. Let A and B be two bodies Elastic or not Elastic. AL, BL their respective velocities before the shock. Let D be their Fig. 15. center of gravity, and DC be drawn perpendicular to AB of an indeterminate length. Draw AC, BG to be prolong'd if it needs. Now if the bodies are not Elastic, QC will be the common velocity after y® percussion. If they are Elastic, take Cs — GB and GT= CA and PT shall be the velocity of y« Body A, and PS the velocity of the Body B after y" Concussion. If they are imperfectly Elastic, take C'y to GS and Gt to GT as y® elasticity to the perfect elasticity and Gt, 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® Body B. But after y^ percussion CTM and GX are the forces of the bodies A and B, if they are elastic, and CQM GQN are these forces if they are not elastic, and AGB is the force lost in y® percussion M'' '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 2447 H 98 STIRLING'S SCIENTIFIC CORRESPONDENCE it cou'd be proved, witliout the new notion of forces, by this proposition. That y'' contemporaneous mutations of velocities of the two bodies are reciprocal to their masses wicli can be evinc'd in several manners, and very easily, if granted that the connnon center of gravity does not alter its velocity by the percussion. I am just ariived 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"" Machin publish his Treatise about y*" Theory of y*^ Moonl Is M'" de Moivre's Book ready to be published? Is there nothing under the press of S"" Isaac's remains? What are you about? Can we flatter ourselves of the hopes of seeing very soon your learned work about y'^ Series? All these and other news of that kind, if there are some, will be very acceptable to me ; and 111 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 fully persuaded, I am, with all esteem and consideration Sir Your most humble Most obedient Servant Paris, this i| X'^'° 1728 G. Cramer You can direct y° Answer A Messieurs Rilliet & Delavine, rue Grenier S* Lazare pour rendre ii M"^ Ci'amer a Paris. (2) Cramer to Stirling, 1729 To M-- James Stirling F R.S. at the Academy in little Tower Street London Here is, Dear Sir, a Letter from M"" Nich. Bernoulli in answer to yours, wich I received but t'other day. I send with it, CORRESPONDENCE WITH CRAMER 99 according to Ins Orders a Copy oi' his method of resolving y° quantity ^ ~ 27^ ii^ i^s component fractions the former part of wich he sent me to Paris, by M'" 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 l)y M"" Sinclair. I am with a great esteem Your most humble and obedient Servant Geneva the 6^^ January, 1729. N.S. G, Cramer. Methodus resolvendi quantitates l+gs" + s"^" in factores duarum Dimensionum, Auctore U*". Nicolao Bernoulli. Prob. I Resolvere quantitatem 1+^2-" + 5'-'* in factores duavum Dimensionum. Solut. Sit unus ex factoribus 1 —xz + zz & productum reliquorum l+az + bz- + cz\.. + )'z''-^ + s^"-- + tz''-'^ + S5« + rz''+\. . Ex comparatione terminorum homogeneorum product! horuni factorum cum terminis propo&itae quantitatis invenitur a = X, b = ax—l, c= bx — a & ita porrho usque ad t = sx — r, item ±q = 2s — 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 & eequentis. Jam vero si Chorda complement! BD alicujus arcus AD vocetur x & ladius AC = 1 Chordae arcuum multiplorum ejusdem arcus AD exprimentur respec- tive per eosdem terminos inventae Seriei recurrentis 1, a, b, c, &c. multi- plicatos per Chordam AD. Hinc .'i arcus AE i^\i ad arcum AD ut 11, ad 1, erit Chorda AE s.& Chordam AD ut / ad 1, id est AE — t x AD, & Chorda DfJ— s x AD. Ex natura vero quadrilateri ADEB h2 ]()() STIRLING'S SCIENTIFIC CORRESPONDENCE circiilo iiisciipti est AL' . l)B — AB . DE + AJ) . BK id est tx.AD ^ 2s. AD + AD. BE sive t.c = 2i> + BE = (quia + ry = 2s - tx) tx±'j+ HE, hinc BE =+q. Ex (|U0 sequitur (piod si arcus liabens pro Chortla complementi + () dividatur in u partes aeqnales quarum una sit arcus AD, hujus complementi Chorda futura sit x : vel si rem per Sinus conficere malinuis, dividendus est arcus habeus pro Cosinu + ^g in n partes aequales, qui si vocetur A, erit cosinus arcus — = -X Inv'ento valore ipsius x cognoscitur 1 —xz + zz unus ex factoribus (|uantitatis propositae l+(/c"+,:^". Sed & re- liqui factores liinc cognoscuntur. Si enim tota circumferentia vocetur C, habebunt onnies sequentes arcus A, C—A, C + A, 2 C- A, 2C+ A, 3 C— J., 3 6' + ^, &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 Coroll 2. Si capiatur arcus AH aetjualis alicui sequentium A G-A C + A 2C-A 20 + A il'C & fuerit arcuum n n 10 n a GG = z erit GH — radici quadratae factoris 1 —.xz + zz. Quia enim CF = \x erit GF = \x — z, ^ . FH= ^n-lx' & proinde GH — y 1 —xz+ zz. Coroll 3. Si g = 0, erit A = ^C, & reli(pii arcus dividendi ^C, |C, ^C, 1 6* &c. Hinc si dividatur tota cir- cumferentia in 4 II partes aequales AH, HI, IK, tV'C & ad singulos im- pares divisionis terminos H, K, j\f, iv'C. ex puncto G ducantur rectae GH, GK, &c erit horum onmium productum 1 +5^". CORRESPONDENCE WITH CRAMER 101 Probl. II Resolvere quantitateiu 1 +c2?t+i jj^ factorcs cluarimi Diuicn- sionum. Solut. Sit uniis ex factoribus l—xc + sz, & productum reliquorum 1 + (13 + bz^ + cz^... rz''-^ + s:"-- + f ;"-^ + tz"" + sc^+i + rs"+2_ _ ^ & invenitiir ut antea a = x, h = ax—l, c = hx — a, & ita porrlio usque ad t ~sx — r. Sed loco aequationis ±q= 2s — tx invenietur haec t = tx + s - id est, si ponatur arcus AD ad A V DP arcuin AE, ut 1 ad n, erit (quia t = . ^ & s = -r— , il' x = BD) ' AD AD AE- AE . BD + DE = 0. Sive DE =AE.BD- AE & aequatione in analogiam versa DE:AE= BD-IA = (facta DF = DC = AC = 1) BF: CB. Hinc tiianoula ADE, CFB, ob angulos ad ^ & i? aequales, erunt similia & angulus BCF = DAE. Ergo ang. BCF +ang CBF = ang DFG = ang DCF = ang DAE+ ang CBF Sed & ano- CDF = ang CBF. Hinc omnes tres anguli Trianguli CDF sunt aequales 2 ang. DAE +3 ang Ci?i^ ipsorum que mensura, id est, semieircumferentia = ^G= arc DE + ^ arc. AD = (quia arc DE = n-1 arc A D) — - — arc AD. C ideoque arcus AD := . Si imtur circumferentia Circuli ^ 2/t+l ^ dividatur in 2?^+ 1 partes aequales, (j[uarum una sit arcus AD, erit Chorda BD = x, vel si semieircumferentia in totidem partes aequales dividatur, erit cosinus unius partis -|a; unde cognoscetur factor 1 —xz + zz. Quia vero tot factores duarum dimensionum inveniendi sunt quot unitates i^unt in numero lu habebit totidem diversos valores qui erunt dupli cosinus 1, 3, 5, 7 &c partium semicircumferentiae in 2/1+1 partes aequales divisae : invenitur enim arcus ^D = singulis sequentibus C 36' 5(7 76' , . arcubus ? ? j 5 (fee, quia arcus Ah 2/1+1 2)i+l 2/t + l 2/t + l ' ^ 102 STIRLING'S SCIENTIFIC CORRESPONDENCE qui est ad arcuiii AB, ut n ad 1. potest iiitellioi auctus Integra Circuinferentia vel ejus multiplo, hoc niodo igitur resolvetur (|Uantitas proposita 1 + c^"''"^ in n factorcs duaruni diniensionum &^ unum factorcm 1 + - unius dinien- sionis. Coroll. Si fuerit CG = z, AC= CB= 1 & Circunit'erentia circuli dividatur in 4n + 2 partes aequales AH, HI, IK, i*(c ad si noulosini pares divisionis terniinos H, K, M, Ac ducantur rectae Gil, GK, GM, &c, erit horum omnium productum aequale Probl. Ill Re?olvere quantitntem l — -2»+i \y\ factores duarum Dimensionum. Solut. Sit unus ex i'actoribus \—xz + z: & productum reliquorum & invenietur s = t + 1x : roliqua vero se habcnt ut prius. Positis igitiir ut in Prob II arcu AJJBE = n arc A D, x = BD, AF I)F t = -^, s = ,~^^, erit DE= AE+AE. BD. AD AD Hinc DE:AE= BD+1 : 1 = (facta DF =^ DC = \)BF:CB Proinde triangula ADE, CFB habentia angulos ad E (l- B aequales erunt similia, & ano;BCF = ang DAE: quamobrem ano-: F = ang: DCF = ang 5Ci^- ano-: BCD = ang: DAE- ang: BCD. Hinc omnes trcs anguli triano-uli BGF sunt = ang: B + 2ang: D^i^"— ang: BCD: ipsorum que mensura I C = 1 arc :AD-\ arc : DBF- arc : BD = |arc :AD + arc : BE = n + ^ arc. AD-^ C. C 2G Hinc C = n + h- arc AD, & ai-c AD = 2 It T o (J ^ cuius dimidii, nempe , cosinus erit ^x. Si arcus ADBE ^ 2 71 + 1 CORRESPONDENCE WITH CRAMER 103 intelligatur auctiis inteij,n\ circumferenticl vel ejus multiplo invenieiitiu- reliijui valores ipsius -|.i' aequales cosiuibus arcuuin 2C 3C 46' , , , &c. Et 2*1+1 2 >i + 1 2ii + l sic resolvetur quantitas proposita 1— s-""*^ in n factores duarum Diuiensionum, & uniiin factorem I —: uiiius dimensionis. Coroll. Si in tio. Coroll. praeced. ad singulos pares terniinos /, L, N, &c. ducantur rectae GI, GL, GN, GO, &c. erit harum omnium productum — 1 — s-"+'. Probl IV Resolvere quantitatem 1 — c-" in factores duarum Dimensionum. Solut. Sit unus ex factoribus 1—xz + zz & productum reliquorum 1 + az -I- &5^ . . + rz''-^ + sz''-^ ± tz'"''^ - i-z'' - rs"+^ . . - bz^^''^ Hie quia terminus tz^^~'^ debet affici signo tam affirmative quam negativo, opportet esse t — 0, adeoque si ponatur arcus AE AD ad arcum AE ut 1 ad n, & per consequens t = ~xy\' erit AE — 0, k arcus huic Chordae respondens = vel C, vel 2C, vel 3C &c. Proinde arcus ^£' = alicui sequentium C 2C 30 , , , . ., C 2C 3(7 arcuum -5 — , — > ivc. iS: ia; = cosinibus arcuum -—} -— > —- > n U' a 2n 2 n 2 it &c qua ratione resolvitur quantitas 1— ,s-" in n—l factores duarum Dimensionum similes huic 1 —xz + zz,& alium factorem duarum dimensionum, nempe 1—zz. Coroll. Si in fig. Cor. 2 & 3, Probl I ad singulos pares terminos divisionis I. L, B, 0, Q, A, Ducantur reciae GI, GL, GB lI'c, erit harum omnium productum = 1 — j^". Coroll. geneiale. Si Circumferentia Circuli dividatiir in 2m partes aequales AH, HI, IK, &c, & ducantur rectae GH, GI, GK &c sive m sit numerus par, sive impar semper erit GHx GKxGM &c = l +z'", & GA x GIxGL &c = 1 -s'». Quod est Theorema Cotesii memoratum Act. Erud. Lips. 1723, pag. 170 et 171. 104 STIRLING'S SCIENTIFIC CORRESPONDENCE Supplemcntniu Eodein Auctore Probl. V Dividere fractionem :; — in fmctiones plures, 1 + qz^ + c-'* quarum denominatores ascendant tantum ad duas Dimen- sioncs. e — fz Solut. Sit una quaesitarum fractionum , = & sumnia ^ 1—ccz + sz oc + ^z + yzz + 8z'^ + ez'^ + &c ^^^^^^'^''^ l+az + bz- + cz^ + dz^ + &c Valor ipsius x determinatur in Problemate primo, & quan- titates 1, a, h, c, d, &c designant ut ibidem terminos Seriei recuiTcntis 1, x, xx—l, x^ — 2x, x* — 3xx+l, S:c. Valores autem ipsarum e & f post eliminationem ipsarum a, /?, y, 8 &c inveniuntur ut sequitur : neinpe si /? = 2, id est si 1 e—fz oc + ^z l+qz'^ + z'^ 1—xz + zz 1+xz + zz invenitur e = i & / = - x - Si 7i = 3, id est si 1 _ e-p Oi + ^z + yzz + Sz^ l+qz^' + z'' 1-xz + zz l+xz + xx-lzz + xz'^ + s^ 1 X invenitur e = 4, & f = -: si n = 4, id est si •^ * 3 XX — 1 1 e-fz l±qz + c'' 1 -XZ + ZZ a + Pz + yzz + Sz^ + es'^ + ^z"^ + 1+xz + xz-l zz + x-^-2xz-^ + xx—l z'^ + xz-' + 1 XX — 1 invenitur e -— ^ & /= - —, — -— : similiter si n = 5 invenitur e — | k f = -^ .' ~ ' , , ^ .neneraliter ob i-atio- 1 1 .s- neni prosfressionis jam satis nianifestam ent r = - ct/ = - » ubi s &, t significant duos postrenios tei-niinos Seriei rccurrentis 1, a, b, c, d, iVc. Hinc si in fi<;. Probl 1 .sit Chorda BJ'J = +q CORRESPONDENCE WITH CRAMER 105 & arciis AD = , erit s:t = DE.AE per ibi demonstrata, n & per consequens / = j, , ipsaque quaesita fractio 1 _ DE 6— /^ n~ n. AE'^ 1-xz + zz i-BJJz + zz Si porrho intelli(;atur arcus AE auctus Integra circumferentia vel ejus multiplo, ita ut inutentur valores ipsarum BD & DE, e — fz mutabitiir quoque valor fractionis -^ — — invenienturque successive omnes fractiones in quas proposita fractio l+n-n^^2n resolvi potest Q.E.F. Coroll. Si q = 0, DE= DB = x, AE = AB ^ 2, fractio I X ^ resolvitur in fractiones banc formam /«, 2n habentes. 1+s \—XZ + Z'^ Scbol 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 quani Moivraei qui invenit fractiones banc formam habentes I a — le II it — uii^ ubi a — \x — sinui f arcus BD, I — +\q = 9,mM\ 1—xz + zz ^ arcus BE, e = cosinui ^ arcus DE, potuisset enim adbibere 1 ez banc simpliciorcm expressionem n nV 1—u intelligendo per l — 2az + zz e non cosinum sed ipsum sinum i arcus DE Scbol II Non absimili methodo resolvi possunt fractiones vel I I -•^rt— 1 I ^^n Schol III Methodus praeced. supponit q minorem binario, quando autem a > 2, fractio :. ^j- resolvi potest ut 106 STIRLING'S SCIENTIFIC CORRESPONDENCE oc 8 ostendit Moivraeus, in duas lias + poiiendo 1 + X" 1+2/" £c" = s" X |(/ + V^qq- 1 & 2/" = -" X i 7 - ^iW - ^ > + — ^ ^ and /3 = ^ Schol. IV Sint a\ a^ a;", .r'", a;'^' &c valores omncs ipsius x sen radices hnjus aeqnationis, \/r/r/ — 4 =: |iz;+ V^xx—i'^ — \x— V\xx—i' vel potins hnjns ±q - ^X+ V'iu:;^;— l" + i«- V^XX-l^ in Coroll. Prob I inventae, et significent e, /, s, ^, idem qnod supra, per ea quae Peniberton non sine magno labore in\ enit in Epist. pag. 49. est t « x — x^ . « — a;" . a;— a;"i &c a; - *• . a^ — ^c" . a; — a;"' &c Denominator harum fractionum invenitur per Regulam Moivraei dividendo differentialem quantitatis ^x+ V^Xf:-l\ +^x— V^xx—1 per dx & liabebimus w *A/ • tA/ ~~" tA.' • t^ "^~ tX' tVO 11 ia' + V^xx —ir — n^x— y/ixx — 1 2yAa;a;-l = (per methodum Serierum recurrentium) nt. Hinc e = =: _, ^^- /■ — ut supra. /li yt lit Schol. V Ut Regula Moivraei quae i'acillimc deducitur ex art. 163 de I'Analyse des Infinim petits possit applicari, oportet aequationem esse dehite praeparatam, id est, ita comparatam ut nulla mutationc, multiplicatione vel divisione opus sit ad inveniendum terminum pure cognitum, qui prodit quando CORRESPONDENCK WITH CRAMER 107 Radix ah omiii vinculo liberatiir i^' tcrinimis altissiniac dioni- tatis iiuUo coefHcientc afficitur ut coutiugit in ista aequatione non aiitem in altera ^x+ \/i.r.r— 1 1 —^x— ^/^xx—1 I = Vqq — 4. (3) Cramer to Siirluuj, 1729 Viro Clarissimo, Doctissinio Jacobo Stirling L.A.M. & R.S. Socio Gabriel Cramer S.P.D. Dominum Klingnestierna Matheseos Professorem Vpsalien- sem aniicum meum intimuni eo digniorem e^^se familiaritate tua intelliges, quo tibi intimius innotescet. Is cum apud Germanos baud vulgaris Mathematici famam reportas&et & a Job. Bernoullio mibi magnopere commendatus mecum Parisiis degcret ; in Angliam profecturus est ut Matbe- maticorum tuique in primis consuetudine uteretur. Ubi tuum in me amorem intellexit, confidit his meis literis se apud te gratiosum fore quae ne spes cum fallat vehementer rogo te : Sed ut ad eam voluntatem quam tua sponte erga ipsum habiturus esses, tantus cumulus accedat commendatione mea, quanti me a te tieri intelligo. Hoc mibi gratius facere nihil potes. Vale. Dabam Genevae ad diem 20 Junii 1729. Mr James Stirling F.R.S. at y'' Academy in little Tower Street London. 108 STIRLING'S SCIENTIFIC CORRESPONDENCE (4) Cramer to Stirling, 1729 Mr James Stirling at the Academy in little Tower Street London Sir I received some days ago your dear letter, wicli in such a Town, and such a Time of Carnaval, I could not find any proper moment to answer sooner. I wrote this morning to Mr Nich. Bernoulli and presented him your compliments. I gave him advise too of your Mind of writing to him. As for his direction, if you will be so kind as to permit me to be the Mediator of that correspondence I'll be infinitely obliged to ye : and you ought l)ut to send me the Letter, wich shall arrive safe to him. I don't know whether he has thought upon that difficulty wich you made me advert to ; of finding any term whatsoever of a Series recurrens, when y" Divisor by wich it is produced being put equal to nought, has impossible roots : but I found an easy way of determining it by y'' help of Tables of Sines already calculated. For it is known that cich equation wich has impossible roots, has an even number of them and con- sequently may be reduced to as many (juadratick equations as many couples of impossible roots it has : therefore y'' fraction by y'' division of wich y" Series is produced may be reduced to as many fractions whose denominator shall be (jujidratick ; besides, perhaps, some others whose denominator is simple. Let the fraction whose denominator is quadratick be repre- sented by that ii:eneral expression where, in v" -^ ^ i 1 + mx + iixx '' case of two impossible roots n, is positive and mm less than 4 ;(,. Now in order to find any term whatsoever of the Series produced by that fraction for inst, y" term /*'' in order. Let V II be y'' Radius of a CircU^ and — l>e y'" Cosine of an Arch z of that Circle: take the Sine of y'' Arch c, multiply /-I it by H ^ , and divide it by y'' Sine of y'^ Arch z. 'Vhc quotient CORRESPONDENCE WITH CRAMER 109 will be y" Term reciiiired. The Deiuonstratioii follows easily from that Observation, that 1 being the first term ; and the sine of an Arch z y° second term of a Series recurreus, whose index is 2c — rr (<• being y*" cosine of y" arch z, and r y^ radius) each term I is etjual to ye Sine of y ' Arch Iz multiplied by y" l—\ power of Y radius. Where 'tis to be observed, that if m, be positive, you needs but to render all y*^ even Terms negatives. I am glad that M"" de Moivre's Lemma is by me demon- strated in a manner that pleases ye ; and since you have seen M'' De Moivre's own demonstration, I am anxious to know how far it agrees or differs from mine. I'll !-ee with a great pleasure M. Maclaurin's Book about vivid forces, but I fear it shall pass a long time before it comes into my hands, because English books come abroad very late : unless you wou'd be so good as to procure one to M'" Caille where I did lodge in Alderniary Churchyard, he shou'd pay for it, and find some way of sending it to me here in Paris. I'll be very obliged to ye for that trouble, and will be very glad to render ye any Services, when you'll judge fit to command. Shall M"" Bradley's account of y" newly observed motion of y fixt Stars appear in y*" Philosophical transactions, or by itself % If so, I desire you to take the same trouble about it; as about M^" Maclaurin's book. I long after seeing your book about Series, and intreat you not to put off y*^ printing of it, being sure that whatever set forth from yowx: hands is excellent, and will be very welcome in Publick. I desire you to be so kind as to give me advice, when M'' de Moivre's book shall be published, because M"" Caille has got a Subscription for me, and I'll be glad to peruse y" book as soon as it shall be publish'd. A learned friend of mine, M'' de Mairan, I should much oblige, if I cou'd by your help, give him an account of a Letter wich D"^ Halley wrote about twenty years ago, to M'' Maraldy, in answer to a Discourse, wich this printed in y® French Academy's Memoirs A° 1707. against y^ commonly received opinion of y" Successive propagation of Light : wherein he endeavours to argfue ao-ainst M'^' Roemer's and S'' Isaac Newton's 110 STIRLING'S SCIENTIFIC CORRESPONDENCE demonstration drawn iVoni y"" Observations of y'' Satellites of Jupiter'.s Emersions and Inniiersions. M'' de Mairan wishes to know, in what time exactly y" Letter was written, and its contents. If you cou'd help me to a copy of it, or, at least, to a short abstract of what is most material in it, I shou'd think myself infinitely oblidged t'ye. I am asliamed to trouble ye with so much business, but I hope your friendship will excuse me, and that in like cases, you will be not sparing of my trouble, wich I shall very willingly take, being with a great esteem and a sincere affection. Your most humble and obedient Servant G. Cramer Paris y W March 17 29. N.S. (5) Cramer to Stirling, 1729 To M' James Stirling, F.R.S. at the Academy in little Tower Street London Geneva, y' if May 1729. Sir The place whence I date this Letter, wall be, I hope, a sufficient excuse for having been so long in your Debt. I return you my humble thanks for all the trouble you took on my occasion, and shou'd think myself happy to find some opportunity of doing you any Service. I received, since y" last time, I wrote ye, a Letter from M' Nicolas Bernoulli who seems to be very glad of your correspondence and expects your Letters impatiently. My direction is now, A Moaslcur Cramer, Professeur en Mathematique a Geneve. You may spare y" trouble of freeing them, from London to Paris, if you'll wrap them in a sheet of Paper directed, A ]\[onsieur le Fevre Coinmis de la Poste, a Paris. I grant ye, my way of assigning a Term of a Recurring Series, wdien y^ Denominator of y ' Fraction hath impossible CORRESPONDENCE WITH CRAMER 111 Roots is not general enougli : for I thought not of y'' Case you make mention of: but I doubt veiy much of y*^ Possibility of a general Solution, for it seems to include a CJeneral Solution of any Equation. I have seen lately a Dissertation that M"" Daniel Bernoulh, IM"" John Bernoulli's son, did read in y'' Petersburg's Academy concerning the recurring Serieses. What seem'd to me most material and, I believe, new is that he deduces from this Serieses, an easy and elegant way of founiling by approxi- mation two Roots of any Equation, viz: the greatest and y^ smallest. The Method is such. Let the Equation be disposed after this form — 1 = ax + hx^ + ex"' + Sec, and make a recurring Series beginning by as much arbitrary Terms as dimensions The Equation has, and y" index of y® Series be « 4 6 + c + &c : and any Term divided by y'' subse- quent shall be equal or very near to y^ Smallest root. The greatest root is found in y'' same manner if this is y*' form of y® Equation and any Term of y'^ Series whose index is a + b + c be divided by y*^ precedent. The further you continue y^ Series y*^ better is y" Approximation. I think myself very oblidg'd t'ye for y'^ account you gave me of M"" Bradley's discovery, wich is indeed very noble, and pleased very much y" French Mathematicians, wich I com- municat'd it to. It seems wondrous now that those who made some attempts to determine y^ Parallax of y" fixt Stars, took no notice of y® successive propagation of y^ Light. This is very surprising too what he observed of the different variation of declination, of y'' Stars, greater for those wich are near y*^ Equinoxes, less for y^ Stars near y'' Solstices. It is plain, that the precession or change of Longitude being y° ?ame for two Stars, the one in or near y- Solstitial Colure, the t'other in or near y® Equinoxial Colure, the mutation of Declination of this shall be greater than y® mutation of Declination of y*" first. But, I suppose, M"" Bradley took into consideration this Difference, wich arises only from their 112 STIRLING'S SCIENTIFIC CORRESPONDENCE situation and found the true mutation of Declination more difi'erent than it shoud he if no extraordinary cause did inthie in it. I render you thanks too for y" account of D'' Halley'.s Letter to M"" Maraldy. INP de Mairan is very satisfied and ohligcd to ye. He hid me to offer ye liis Thanks and humhle respects. I long for receivin<;' news of your hook heing under y** press. My thirst of seeing it is rather increased, than quenched, hy the noble Theorem, you vouchsaf'd to comnuinicate me. I found indeeil a Demonstration of it, but as by chance, and, I think, not very general, and so your Method will give me a great pleasure. Here is my demonstration. It is known and easy to demonstrate that XX' X I — x'i is equal — 1-x'i '/+i i 1 m m + q + l x"" — m .m-l vi + q+1 .m + q m . m—1 m + q + 1 .m + q m+q—1 m.7)l—l .771—2 = x" &c. m + q+ 1 . VI + q . m + q — 1 . vi + q — 2 wich Series may be terminated to any Term, viz., to on .ni—l &c usque ad m — z + 2 ^m-z+ 1 m + q + 1 . m + q (kc uscjue ad m + q — z + 2 if you add this quantity m . m—1 i^c us(|ue ad in — z + I X 1-x'i 7)i + q + l . m + q &c us(jue ad m + q — z + 2 In the case of \—x= all the terms become ecjual to nought, but this last quantity, and it is icx"* X \ —x'l Til .m—\ &c usque ad m — z+\ m + (/ + 1 . m + q . . . m + q — z + 2 XX' •X l-x'J CORRESPONDENCE WITH CRAMER 113 Let m bo equal o + 7"— 1, and iit+q + l be r-r?.)-!, or q =j) — r—l, 3'ou'll havu xx'' + '~^ xl-xP- z + r—\ .z-\- r—2 ...z + r- z+'p-l .5 + /J-2...-; + |)- xx'''^x(l-'X)i'-''-^ Then XX-''''' ' X l—x>^' xx'''^ X (1 —a;) p r-l z + r—\ . z + r—2 ... z+r- ^^^^r.r+l.,.r + z-l^^^ z+2^-\ .z + p-2 ...z + 'p-z' ' ' p.'p+\...'p + z-\ I am with a oreat esteem and affection Sir Your most humble, most Obedient Servant G. Cramer (6) Cramer to StirUufj^ 1729 M"" 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"" Nichob 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'" Nicli. Bernoulli, since I sent him yonv 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"" John Bernoulli, who is always 114 STmLlNG'S SCIENTIFIC CORRESPONDENCE contriving again and again new Arguinunts lor liis Opinion about vivid forces. I don't know you liavc read what AU 'S Gravesande publisli'd in the Journal Lllteraire about that matter. 'Pis all metaphysical reasoning, in answer chiefly to the late D"" Clarke and M/' 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 pi'inciples have 1)rought ye to these Series. Tis nothing like 3'our Theorem for interpoling any Term in that Series A, A. B, 6' (^^c. : I sent all that to Mr Bernoulli. I render 3'e thanks for the account you gave M' Bernoulli of M"" Machin's Theorems. They peem indeed ver}^ well contrived for clearing S"" 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'' Machin, had thought of that Theorem. I wrote 3'ou in so few words oT M"" Dan. Bernoulli's Waj^ 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 haec forma 1 = ax + h.v^ + ca"^ + ex* + (^'c. Dein formetur Series incipiendo a tot terminis arbitrariis quot dimensiones habet Equntio, hac lege, ut si A, B, (J, D, E denotent terminos se invicem directo online consequentes, sit ubique "^ — aD-\-hG + cB-\-eA-\-&c sintcjue in hac Serie satis continuata duo termini proximi M & N, erit terminus antecedens M divisus per consequentcm N proximo 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 .1;'" = ua;'" ^ +^.^"'~- + ca;'"""-t-&c Formetur Series CORRESPONDENCE WITH CRAMER 115 incipiendo a tot tcrniiuis arbitrariis quot dimensionum est aeqiiatio, eaque talis, ut si A, B, G, D, E denotent tenninos directo ordine e Serie excerptos & contigiios, sit ul)i<iuc ^ = iiD + hC+cB + cA +&:c, sintque in liac Saric satis con- tinuata duo termini proximi M & N, erit terminus N divisus per praccedentem M proxime aecjualis radici maximae.' Tlie demonstration oi wich I conceive to bo tlius. Let the Roots of the Equation 1 = <i.v -\-hx" ■{■ ex" -\- &c. be -, -, - , Sec ^ X y and of tlie Equation x'^' = ((x"' ^ + hx^"~'^ + cx^'^~^ + (S:c be x, y, z, kc : and if the term M is in order I of the recurrent Series whose index is a -\- h -^ c + ^c this term M will be, for the values rt, h, c, c^'c of the lirst Equation -, + -i + -7 +&:g. and. ;>,.• y' c' for the values a, b, c, S:c in the second Ecpiation pJ + (jy^ + rz^ ; and the next term in order l+\, and called N shall be, for the first Equation -^^ + -^— + ^j—^ + (^^c and for the second X y z Ecjuation ^>a;' + ^ +q]/'^^ + 7-:^"^^ + &c. Now if x be the o-reatest and the smallest root the "•reater is /, or the further is that X term M from the beoinnino- of the Series, the oreater is - in coniparii-on with the other terms -^ + -j ^c, and -j^ in com- parison with -^— + -^^-f +^c. So that if I be infinite the terms -j + -^ &:c and -j^^ + t^j + (^'c are not to be considered but — , and -/xr make up the Terms M and N, the foi-mer of a;' «'■•"' ^ wich being divided by the latter gives you x. In the other Equation 'px^ and j9a;^+^ being infinitely greater than qyl + rz^ + Szc and qy^^^ +rz^'^^ +&:c make up the Terms M N _ px^^'^ + ii^c M ^»* + &c I am with a great respect Sir Your most humble and most Obedient Servant Geneva y^ 2G Decemb 1729 N.S. G. Cramer. I 2 and i\', and -^ = ' j — ^ — = « the greatest root. 116 STIRLING'S SCIENTIFIC CORRESPONDENCE As poon aa yours and IM"" de Moivre's books are printed, you'll oblige me very much to give notice of it to M'' 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 Coft'ee house over against y'' Royal Excliange. Cramer to Slirliiu/, 1730 M' James Stirling F.R.S. at the Academy in little Tower Street London Sir As there is no less than a j'ear, since I have no Letter from 3'e, 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'' Nich. Bernoulli in answer to yours, together with a Copy of his Method for finding y*" component quantities of a Binomium like this 1 +3" by the Division of the Circle. Extraordinar}^ businesses have, from that time hindred me always, from having the Pleasure of writing ye, and intjuiring after tlie Philosophical and Mathematical news of wich there is abundance in England in any time. I don't know whether your learned book about Serieses is published, but I wish and I hope it is. and y*^ Publick is not prived of your fine Inven- tions. I heard M'' de Moivre's book is out, but I have not seen it yet. You know without any doul)t, that M'" 'S Gravesande had made fome little improvement to your metliod, given in your book Enumeratio linearum 3" Ordinis (S:c for finding the difference of exponents Aritinnetically proportional in an infinite Series formed from a given equation : wich impro\e- ment he publishe<l at the end of his Mathescos universalis Elementa : but I found his Method wants yet a little correction, for it can induce into luTor, if the given equation, besides X and y contains their fiuxions. Let, for instance, the E(juation be Z|L +x'yy''-2x^yy \-u'y+ |^, = CORKliSPONDENCE WITH CRAMER 117 ami by S'' Is-tuic's Method ot! Parallolo^raiu, you'll tinu in the Series resulting {>/ = Ax^^ + Bx"^'' + &:c) n= 1, and sub- stitutin<;- ./; instead of y, and x instead of // : the indices shall be 9.4. 1 .4. 14. Whence, by D' Taylor's ]\lethod, r Ixjino- the common divisor is 1. By your methotl, the first term shall be AAx-2Ax+lx = or ^1^1 — 2^1 + 1=0, where ^1 has two equal valors, and therefore, by your method ?• = — = -• Mr 'S Gravcsande's Method "ives for r's value p 2 ^ 2^. But really r may be taken = 5, and the form of the Series is y = Ax + Bx^' + Cx^^ + &:c. This valor of r = 5, is deduced from this Rule, wich may be substituted to othei's. haviny; found, by the Parallelogram, the greatest terms of tho E([uation, and thereby the valor of ii ; see whether these terms give for y, or y, or y &c many eijual valors, and let 'p design the number of these equal valors of y, or y cV^c. Then substitute for y and y, y &c, x", a,"~', a;" - &c and write down the indexes of all the terms. Subtract them all from y" 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 tjivided, 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'y dy'^—2x''ydy + x^dy — 0, wieh divided by xUly, gives y(iy—2y + x = 0, where y has not many equal values, Theref. pj — I. 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 nundjer of Roots of an exponential E([uation, like this y-'' = I +x for the method you give in the 6 Coroll. of y" 2"^' Prop, of your book Enuineratio &g p. 18 does not succeed in this case. It is a thing pretty curious, that in the Curve represented by that Equation y'' = 1 +a;, or y — I +x' , the abscissa being = 0, the ordinate y is not 1, but of a very different value, tho' it seems at the first siirht, it must l)e 1, beino: 1 +o". I have happily conserved a Copy of M'' Bernoulli's Letter, 118 STIRLING'S SClb:NTIFIC COIUIESPONDENCE so iliat I can suiul it yc, il' you have not received y ' (jri^inal, wicli I pi'a}^ 1 may know I'roin ye, as soon as you can witliout any trouble at all. I am, with a jj;reat esteem and respect Sir Your most humble Geneva, the 22 X''"" 1730 N.S. most obedient Servant G. Cramer. (8) Slirliiifi io (■rauier, 1730 Copy of a Letter .scut to M'' Cramer at Geneva September 1730 Sir I Ijeg a thouf-and pardons for delaying so long to return you an ans^wer. I was designing it every day but unluckily hindi'ed 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° other for M"^ 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"" Machin pent me being an answer to what M'' Bernoulli write about his Small Book. The first part of my Book you see is about y" Suming of Series where I have made it my chief business to change them that conversfe 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 lind a tluxionary 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 Qua(irature perhaps, by wiiich means I take this matter to be carryed farther than it was before : this you will !-ec is the 15 Proposition and its Scholien I have taken an opportunity of clearing up a difiiculty about the extracting the Root of a fluxionary Equation, wliich is the only one that Sir Lsaac left to be done. This first part CORRESPONDENCE WITH CLIA]\1ER 119 lias l>i'en written 8 or 9 years ago, so that il" I were to write it again I should Scarce cliange anytliing in it; Ijut indeed that is nujre than I can say ior the Second part, because tliere was not above one halt' ot* 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 miglit have been better done. The 20 Prop: about y° Suming of Logarithms has been Considered by M"" Dc Moivre since y" publication of my Book, and he lias 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 ... c; whereof the last is z. ]\Iake /, : = Tabular log. of z, I, c=log. of 6-28318 which is the Circumference of a Circle whose Radius is unity, a — '43429 ... which is y- reciprocal of y'^ Hyperbolick Log of 10. and y" sum of y" Logarithms of the proposed numbers will be ^ whereas you will see that in my Series y'' Numerators are y^ alternate powers of 2, diminished by unity: the degree of convergency is y'' same in both, and indeed there is seldome occasion for above three Terms, reckoning — za the first : M"" De Moivre is to publish this with his manner of finding it out, which is (|uite different from mine, whicli is done by an old and well known principle, namely the taking of the differ- ence of the succes!-ive values of quantitys as you will see in y'^ 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. ^ The gap occurs in Stirling's copy of the letter. 120 STIRLING'S SCIEX'J'IFIC COKRESPONDENCE (9) Cramer to S/irUi/fj, 1731 31' Jiuues Stirling R.S.S. at the Academy in little Tower Street London Sir I guess Ity 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 tliat accident is the forgetfullness of a Merchant to whom M'" 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**^ of June. As soon as I received them, 1 sent M"" Bernoulli his Exemplary together with the Letter for him and the inclosed Letter of Mt Machin. And I resolved to write }ou 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 Docti'ine 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 Hatter 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 ]\I' N. Bernoulli has been this month elected Piofessor 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 his ,. . , ,. ,, ,, . T .r^-h.r+2h ... r + zh-h maimer ol intcnjonng tlie Series ; ^ -. j 2).p + o. /> + 2U ... p + zb — b 1 ii- -4 1 . r.r + b ...p-b by putiing it ('(lual to ; r-^- , or = CORRESPONDENCE WITH CRAMER 1*21 p + zb . . . zb + r — 1> , . , , 1 1 i- • i — 5 which cannot succeed l)Ut m some p . p + b ...O' — b few cases, wich have no difficulties. His Theorem sent to M'' 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 ... a+2J—lb n a + c. a -\- (■ -\- b . a -^ c + 2b . . . a + c +2> —'^b n n—1 + a + 2c.a + 2c + b...a + 2c+i:>-lb n 11 — 1 n — 2 1 >< -2- ^ ^^ a + 3c.a + 3c + b ...a + 3c+2:>—ib n n — 1 n — 2 n—3 T '^ ^ >< 3- "" "1- + kc. « + 4c.a + 4t-|-& ... a + ^c + 'p—lb (by putting ^-,5=^/1, 6' = ^^' B, D = -~— G, &c and Az + Bz' + a. ■■'> + Dz^ + etc = Hz" + is" + ^ + it s« + ^ + Xo" + '■' + etc) will be reduced into this p .f>+\ .p ^2 ...'[) ^n—\ jj_ p .p+l .p +2 . . .p + Uj a .a + b ... a +/> + n— lb a . a+b ... a +p + nb ^ P-P+^ ...y + n + l j^_ p.p+l .../) + n + 2 ^ ^^ a . a -^ b . . . ii + J) + n + 1 b a . a + b ... a +p + ii + 2 b or, (if you like rather to have but the sign + and not alternately + & —) into This p.p+l .p + 2 ... p + n-1 „ a + nc -{ p—\b . a + nc+p—2b ... a+ nc — n b 'p .p+1 .p + 2 ... p + n J + a + nc +p) —lb . . . a + nc — 11 + I b p.p+J\^p + 2...j, + n+l j^ ^ ^,^^ a + nc + p — lb ... a + nc — n+2h 1.2:2 STIRLING'S SCIENTIFIC CORRESPONDENCE where if c = h, A l.e'iMM- = //', ;ui<l B = C = X» = t^'c = all the Series is reduced to the first term (t . a + h .a + 2h ... (i + 'p+ ii — \b and, moreover, if you put again p = 1, you'll ha\ e M' Ber- noulli's Theorem I have also read over M"" Machin's Letter, but I cannot judge of their difference having not seen his Book. M"" Caille cou'd not find it. I am glad for what you say to M' 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 })rinted, for I shall read it with a great pleasure. I had willingly dehiyed this letter till 1 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 moi^t humble, mo.st obedient Geneva 18*^^ June 1731. and most faithfull Servant G. CllAMEH. (10) Cramer to StirUug, 1732 W James Stirling. K.S.S. at the Academy in little Tower Street London. Geneve, ce 22° Fevrier, 1732. Ne Soyes pas surpris, mon cher Mon.sieur, de recevoir si fcard la Keponse ji Voire chere Lcttre du Mois de May 1731, puisqu'il n'y a (jue tres pen de jours que Monsieur Bernoulli me I'a fait remettre. J'espere aus.si tpie vous me permeterds de vous t^crirc dans ma Langue nuiLernelle, puisque je sais que vous I'entendds fort bien. Et je crois vous eniuiyer moins en vous parlant une Langue qui vous est un peu dtrangcrc qu'en vous obligeant <\ lire un Anglois aussi barbare que celui que je pourrois vous ecrire. Je continue li vous rgndre mille graces pour le present (jue \'ous aves daigne me faire de votre CORRESPONDENCE WITH CRAMER 1:23 excellent Ouvrage, doiit jc vous ai accuse la reception dans unc Lettre ({ue vous dcvds avoir rt^u depuis I'envoy de la Votre. On ne peut ricn trouver dans le livre que d'excjuis pour ceux qui se plaisent aux Spc^culations dont vous a\'es enriclii les Mathematiques. Je n'en dirai davantage de peur (le paroitre vous flatter, 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"" Taylor pour trouver la forme d'une Serie iloit etre proposee, commc vous le remarqut^s sous une forme ditferente de celle qu'il a donnee, en ce que r doit etre, non le plus grand connnun diviseur des indices, mais bien celui des Differences des Indices. Mais pour qu'elle puisse s etendre a tous les cas possibles, M"" Gravesande dit qu'ayant substitue dans TEquatiou, yl,t" au lieu do y dkc 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 nmltiple de ce nombre II en donne I'exemple suivant. |ni + c^y - 2 .^'^2/^ + xy^ - ^5 = (|ue la substitution de Ax"- au lieu de y, change en -^ +Ax"+'-2A^x^''+^ + A^iiy''' + ' ^ = Done les indices sont 14, n+3, 2ii+2, 3)1+1, 9 ii. Par le Parallelogramme de M'' Newton on trouve pour la forme de la suite d'autant plus convergente que x est muindre, 11= 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 counnun diviseur est 5 ; Ain?i selon la Regie de M'' Taylor corrigee, la forme de la suite doit etre Ax + Bx''' + Cx^'^ + &c. Mais selon M"" 'S Gravesande si Ton veut determiner la valeur de A par le moyen des plus grands termes de I'equation (}ui sont Ax"+''^-2A\r^''+^ + A-^x-'" + \ ou Ax*-2A-x'^ + A'''x* dgales a zero et divises par a;* on trouve (ju'il a 2 valeurs egales. Done /' doit diviser les 2 differences 5 & 10, et entr'autres la plus petite par 2 ou 4, ou 6, &c. 1:24 STlllLING'8 SCIENTIFIC CORRESrONDENCE . \ « Ml H • Fig. 21. Ainsi r doit ctrc 2|, ct la forme dc la Seric sera Ax + Bar^ + Cx'' + I)x^^- + <S:c. Mais ccttc Re<;le de M' 'S. Gravesande iie paroit pas encore assess generale, car il peut aisement arriver dans les Ecpuitions tluxionelles que A ait plusieurs Valeurs e<^ales, sans (ju'ii y faille faire ancune attention. Ainsi quoi(iiie sa Regie donne toujours una Suite propre a determiner la Valeur de y, cependant elle ne donne pas toujours la plus simple. II faloit done (^tablir la Regie ainsi. Si les plus grands termes de I'l quation determines ])ar le Parallelogramme de M"" Newton, etant egalt^s a zero, font une Equation dans laquelle y ou quelcune de ses Fluxions ait plusieurs Valeurs ^gales, Divis^s la plus petite difference des Indices par le nombre de ces Valeurs ^gales, Et le plus grand commuu divifeur du Quotient et des autres Differences sera le nombre r clierche. Par exeniple, si I'Ecjuation cy-dessus avoit 6te -^ +x'y-2x'yy + x-y-y- ~^^ = on auroit trouve la nieme valeur de \i-= 1, les memes indices 14, 4, 4, 4, 9, les memes differences 5, 10, que cy-devant, k A auroit aussi deux Valeurs. Done selon la Regie de M'' 'S Gravesande ou auroit la meme forme de Serie, ^^ + i?a;'** + 6V + ("('c, Au lieu (|ue suivant la Regie (pie je viens de poser, les plus grands termes de TEcpiation x'^y—2x''yy-\-xSfy, (^gales a zero et divis^s par x' y donnent x—2y + yy=iQ (pii ne donne pas deux v^aleurs cgales de y ou y Ainsi il faudra simplement prendre pour r le plus grand conaiuni diviseur 5 des diff'erences 5, 10, Et la forme de la Serie est Ax + Bx^' + Cx^'^ +(%c. Ainsi si I'on calcule selon la forme de M"" 'S Gravesande, on trouve tons les Coefticiens des Termes pairs dgaux a zero. C'est h\ la Regie Generale. Mais il se rencontre (lueLpiefois des cas, ou il n'est pas si facile de I'appliquer. Les Termes places sur le Parallelogramme de M"" Newton peuvent se trouver sur une nieme ligne Verticale. Alors on ne peut en les CORRESPONDENCE WITH CRAMER 125 coinparant (letenniner la Valeiir <le Toxposant ii. Mais en supposant (pie le tenne le ])lu.s "^rand est celui (Hii a le plus yi'and oil le phis petit exposaut selon ([ii'oii vent (jue la Suite eonver<;'e, d'autant plus (pie x est plus petite ou plus oraiide : On determine par cette supposition la Valeur de // i^' la forme de rEt[uation. Mais la valeur du premier r et sonvcnt de (juel- ([ues autres coelficiens reste indctermiiK^e. Done si tons les termes places sur le Parallelof^ramme de ^P Newton se trouvent dans une meme Hone ol»li(jue, ou ce (pii revient au nic^'me, lorscpi'ayant substitue dans rp](]uation Ax" au lieu de y, & nAx'"~'^ au lieu de y, S:c les indices dcs termes resultans so peuvent tons rencontrer entre les Termes d'une Progression Arithmeti(|ue : alors re(|uation est a une ou plusieurs Paraboles, ou bien h une ou plusieiirs hyperboles, (|u'il est facile de determiner. Soit par exemple I'eriuation 2xx — 4xVay—15(iy = iH^ apres la substitution de Ax^^ au lieu de y, les indices seront 1, ^ a, n — \, qui sont en Progression Arithmetifpie. les supposant egaux on trouve n = 2. Soit done y — Ax^ et apres la Sub- stitution I'equation devient 2xx — 4xxVaA — ZQuAxx = ou, divisant par xx, 2—4 VaA — 30«^ = 0. Done les Racines sont 1—5 V<iA — Q, k 1+3 \/(7Z = 0. Dans ces Racines mettant ail lieu de A sa valeur "^ , elles se chanoent en 1 — 5 ~ — till . . . ^ i^' 1 + 3 / -^ = dont la multiplication produit XX — 2x Vay — 1 5 (< ?/ = qui est la fluente de la fluxion propos^e 2xx — ix ^<iy— IG'iy — Or cette equation designe deux demi Paraboles decrites sur le meme axe & du meme Sommet, les branches tirant d'un meme C(")te, dont la superieure a pour Paramelre 25a, ^ I'inferieure 9(/ ; L'abscisse commune est y, ^ I'ordonn^e de la premiere est x, celle de la seconde —x. Quant a I'Equation de la Courbe y-'^ = \ + x, voici la difficult^ qui m'avoit portt^ a vous demander si elle n'a qu'une ou deux brcUiches. C'est que quand x est un nombre pair, il semble que y doive avoir 2 Valeurs egales, I'une positive I'autre negative, puisque toute puissance paire a deux Racines. Par 126 STIRLING'S SCIENTIFIC CORRESPONDENCE Exeniplo qiiaud a; = 2, I'equjitioii dcvicnt y' = S, Done y = + V 3 & — v' 3. Mais quand x est impair, je ne trouve plus qu'une Valeur pour y. Car, par exemple, quand x = 3, I'cq nation 2/^=4 n'a qu'une racinc reelle, sea voir y = v^4 les deux autres Raeims y=-^V^6+ 7-1^10, & 7/ - -i ^10- v/-^^16 rtant iniaoinaires, II somble done qn'outre le Rameau on la l>ranche (pii est du Cote ou Ton prend les y positives, I'Ecjuation dcsio-no quelcpies points par-ci par-la du eute negatif, kK' non pas une Iti'anche entiere et continue ce (|ui est absurde. La difficulte est la nienic quand 1+x est ncoatif. Car a en juger par I'Ecjuation il .'•enibie ([uc y aura alterna- tivement des Valeui's rdelles et imaginaires, selon que x sera impair ou pair. La meme difticultc^ se preeente dans toutes les Courbes exponentielles sans en excepter la Logarithnn(iue. Je ne vois pas (jue personne ait donnd 1;\ dessus quelque (^elaircissement. Je soubaiterois que vous vous donnass-ies la peine de m'expli(|uer un peu pUis au lonij^ sur (|uel t'ondenient il vous paroit que y a deux valours egalcs mais avec des Signes contraires. En rcduisant en suite I'equation y^ = 1 + x je erois qu'on ne trouve qu'une seule suite, ce qui n'indiqueroit qu'une valeur. Mais le ( 'alcul est si lon<;', (pie je n'ai ni le courage ni le terns de I'entreprendre pour mioux m'assurer de ce soup^.on. Votre determination de la Valour do y (piand x est zero, est conforme h cello quo j'ai aussi trouvc'e par la memo nianiere et encore par ({uelques autres. Par Exemple. On pout ainsi construire la Courbe Sur I'Asymptoto CD soit decrite la Logaritbrnicjue, dont la Foutano-entc soit I'unite. Soit I'Oi-donnde AB eo-ale a la Soutano-ente ou j\ I'unite. Soit prise unc abscisse ([uelconque AAf—x. Pour trouver I'Ordonncc correspon- dante j\fP = y, jo trace la perpen- diculairo PMF rencontrant la Logaiitbrnique au point F. Par les points F k A je tire la Clioi'de ou secantc FAI, (pii I'oncontro I'Asymptotc en I. Je i)rens BK = BI ct elevant la perpendiculaiie K L je fais il/P = KL. Le point P est ii la Courbe l^Q cbercbeo. Car puisque CI B K D Fig. 22. CORRESPONDENCE WITH CRAMER 127 j\M=x, BM = x+l, Sc MF =Lx+l. Soit MP = LK = y tl' BK = BI = Lj/. Lcs Triangles Semblables AMF, ABI donncnt FM {L1,^\) : MA (x) : : BI (Ly) : BA (1 ) Done xLy = L.r + 1, ou y^' = x+\ Or (luand x = la Secante FAI devient la Tano-entc AC, iK' pieiiant BD = BG = \ (la i^ou- tano-ente) la Perpendiculairo J)E (i[\u est le nombre dont le Lo-aritlime est I'unite = 2-71 8281828450 i^x) sera r-ale a rOrdoiinee AQ. N.B. que cette Construction ne donne qu'une l)ranclie ] torn- la Courbe so. PQ. Mais ce qui i'ornie une nonvolle ditticulte, c'est (ju'en eher- chant la Soutano-ente an point Q il sernble (ju'il y ait deux ou 3 rameaux (|ui se coupent en ce point la. Car I'expression orenerale de la soutansjente est '- . Or cette x-1+x.ll+x expression devdent (en substituant au lieu de x la valeur= 0) • Done suivant I'art. 1G3 de TAnalyse des infininient petits, prennnt la Difterentielle ou iluxion du Numerateur et du 3 XX + 2 X Denoniinateur on trouve la soutan!>ente an point Q = — 7— : '^ ^ ^ —l{l+x} qui est encore ^. Done differentiant de nouveau, on trouve cette soutnngente = —Qxx — Sx-2= —2 (puisque x = 0). Or les Autheurs posent qu'on n'est oblige k ces differentiations que lorsque 2 ou plusieurs Rameaux de Courbe se coupent dans le point ou Ton cherche la soutangente Voyds Memoires de I'Academie de Paris. Annde 171G p. 75 & Ann(^e 1723 pag. 321. Edit, de Coll. Voyes aussi Fontenelle Elements de la Geometric de I'infini, p. 418 & 99. Votre Probleme du jet des Bombes est de la derniere im- portance par raport a cette branche de la Mechanique. Je serai infiniment curieux d'aprendre le re.sultat de vos Experiences & de Vos Calculs. J 'ai lu cet article de votre Lettre a plusieurs de mes Amis Oiliciers d'Artillerie, ches qui il a excite une merveilleuse curiosite. Ce que vous dites de la facilit(^ de votre solution ne pique pas moins la mienne, puisque la Solu- tion de M"- Jean Bernoulli (Acta Erud. 1719. p. 222, & 1721. p. 228) est si compliquee et inapliquable h la pratique. Je vous suplie, si vous aves compose quelque chose la dessus de daigner me la commiiniquer. 128 STIRLING'S SCIENTIFIC CORRESPONDENCE Jo voudrois bien en ecliaii(;u do votrc belle Lettre vous iiuli(iner aussi quel(juecliose digne de votre attention Mais il n'est pas donne a tout le nionde do Voler si luiuL. Je nxo rabaisse a do plus petits Sujets. Voici uu Problenie qui m'a oceupe ces jours passes, ct qui sera peut-dtre du oout de Mr de Moivre. Vous ne savi'^s peut-ttre pas ce que nous ap])ellons en Francois le jeu du Franc Carreau. Dans une chanibre pav^e de Carreaux, on jotte en I'air un Ecu. S'il retonibe sur un seul carreau, on dit (|u'il tombe franc, et celui qui I'a jettd gai^ne. S'il tonibe sur deux ou plusieurs Carreaux, c'est ti dire, s'il tombe sur la Raye qui separe deux Carreaux, celui qui I'a jette perd. C'est un Prob1i''nio ix resoudre d' (jui n'a point de difficult^. Trouver la Probabilite de gagncr ou de perdre, Les Carreaux & I'Ecu ^tant donn(^es, Mais si au lieu de jeter en I'air un Ecu qui est rond, on jettoit une Piece Quarrde, Le Probldnie m'a paru asses difficile, soit qu'il le soit naturellement, soit (jue la voye par laquelle je I'ai resolu ne soit pas la meilleure. Au reste j'ai re9u le Livre que Mr de Moivre m'a envo} 4 en present. J'ai pris la Libert*^ de lui en faire mes remercimens dans une Lettre dont j'ai charged un jeune liomme d'ici, qui est parti il y a quelques mois pour I'Angletei-re. Je ne scais s'il la lui aura remise n'en ayant eu dcpuis aucune nouvelle Je vous prie, quand vous le verrds de vouloir bien I'assurer de mes hund^les re-^pects, (S: de ma reconoissance. Temoign^s lui combien je suis sensible aux Marques publi(|ues qu'il m'a donn^es de son amitid. II ne sera pas trompt- dans sa Conjecture, quand il a cru que la 2^ Methode de M'' Nicolas Bernoulli est la meme que celle de Mr Stevens. II y a plus d'un an tiue je n'ai aucune nouvelle de ce dernier. Sa nouvelle Profession 1 oceupe entidrement. II a poui-tant rec;,n votre Livre avec vos Lettres, et vous aura sans doute repondu. Je suis avec une estime et une consideration toute parLiculiere Monsieur Votre tres lunnblo, iV' tres obdissant Serviteur G. Cram EH. CORRESPONDENCE WITH CRAMER 129 (11) Cramer to Stirliii;/, 1733 M' Jcuues Stirling. F.R.S. at the Academy in little Tower Street London Monsieur Voici uue Lettre que je viens de re^evoir pour vous de la part de M"" Nicol. Bernoulli. Elle est venue enfin aprds s'etre fait longtems attendre. Un nombre considerable d'occupations ni'empi^che d'avoir I'lionneur de vous ecrire plus au long. Voici seulement un Extrait de ce qu'il me marque touchant sa nouvelle Maniere de calculer les Numerateurs des fractions simples auxquelles se rciduit la fraction ^7- 1—~ . Soit suppose 1 e-fz + z-" + 2l-J'+l 1-JCZ+:: oc + ^z+yz- + 8z^'' +... + jUs"-- + J/o"-^ + . . . + CV^"-5 + Bz""-^ + Az^''-^^ 1 + az + 6s- + c;^ . . + rz"-'-" + sz'^-'' + tz'"-'^ + az" ... + az^''-"" + z^''-'^ et reduisant ces deux fractions au commun denominateur, en multipliant en Croix, & faisant oc+e — 1, ^^ les autres coeffi- ciens = on aura les Equations de la Tabl. I lesquelles apres avoir substitu^ pour x, ax, hx, ex, &:c respectivement a, 1+6, « + <?, b + d cV:c selon la nature de la suite r(^currente, I, a, h,c,d, &c se cliangeront en celles de la Tabl. II Tabl. I a+e:= 1 A-f^O 13-oiX + ae-f ^ B-Ax + e-af— y — ^x + a + be-af-O C—Bx + A + ae-bf=0 8-yx+l3 + ce-bf^ D-Cx + B + be-cf =^ i-Sx + y + de-cf = E—Dx + C + ce — df=0 ike &c 2447 K 130 STIRLING'S SCIENTIFIC CORRESPONDENCE Tabl. II 0^=1-6 A=f (3 = a-2<ie+f B = 2af-c y = b-3bc + 2af-e C= 3hf-2ae+f S = c-4ce + 3bf-2ae+f D = 4cf-3he + 2(if-e e = d - 5de + icf- 3bc + 2af- e E - 5df-ice + 3hf-2ae+f fi = s—n—lse + n — 2rf M = n—lsf—n — 2re — n—3qe + &c. +n—3qf-S:c. M—t — nte + n—lt<f fi = ntf— n —Ise — 70 — 2 re + (See. +n—2 rf— &c. Ces deux differentes valeurs de M ^galees ensemble donnent t — nte = 0, ou e = - & les deux valeurs de u donnent s — vlf n r- J OU f— -- , coninie i'ai trouve par induction dans la Solution ^ at *' '■ 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 publi(^ nouvellement en Angleterre en fait de Philoso- phie & de Mathematique. Soy^s persuade que je suis avec une extreme consideration & un Veritable attachement, Votre tres humble & tres ob^issant Serviteur G. Cramer. Geneve ce 10*^ Avril, 1733 IV N. BERNOULLI AND STIRLING (1) N. BeiiioulU to Stirling, 1719 D"° niihi plurimuni colende Peugratum milii fuit iiudius tertius accipere epistolam tuam, qua me ad mutuuiii epistolariim 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 difficultateni illam, quam de resistentia pendulorum movebam, ea hue redit. Posita gravi- tatis vi unifornii et resistentia proportionali veloeitati, non potest corpus grave oscillari in Oycloide ; hoc quidem inveni per calculum, sed quomodo ista impossibilitas a i^riori ex rationibus physicis demonstrari possit, adhucdum ignoro. Rogo igitur ut banc rem sedulo examines et quaeras construc- tionem Curvae, in qua abscissis denotantibus spatia oscillatione descripta (i.e. arcus Cycloidis interceptos inter punctum quietis et punctum quodvis ad quod mobile oscillando pertingit) apphcatae denotent resistentiam vel velocitatem mobihs in fine illorum spatiorum. D"^^ Newtonus pag. 282. dicit hanc Curvam 'proxl'nie esse Ellipsi Problema quod a D"" Taylor Geometris propositum mecum coiliunicavit D. Monmort, est sequens. Invenire per quadraturam circuli vel hyperbolae fluentem hujus quantitatis — -r- ^, ubi S significat numerum quemlibet integrum aftirmativum vel negativum, et X numerum aliquem Imjus progressionis 2, 4, 8, 16, 32 &c, petitur autem, ut hoc fiat sine ulla limitatione per radices k2 132 STIRLING'S SCIENTIFIC CORRESPONDENCE imaf:^inarias. Doniquc quod attinet ad Thcorcina Patrui mei pro conjiciendis Ciirvaruni areis in Scries convergciites, tuaiii que contra ejus generalitateni factam oppositionem, in ea re adliucduni tecum dissentio, et in mea opinione finnatus sum, post([uam nuper exemplum a te o))latum, et alia calculo subduxi ; deprehendi enim seriem, licet in infinitum abeat, tamen esse suulabilem, si area invenienda sit quadrabilis. De rebus aliis novis Matlicmaticis aut Philosophicis nihil, coiiiunicadum habeo, nisi quod Patruus meus miserit Lipsiam solutionem Proljlematis D' Taylori ((|Uod et ego jamdudum solvi) cum subjuncta appendice infra scripta. Quod superest Vale et fave. Dabam Patavii d. 29 Apr. 1719 Iipuus Polenus me enixe D"'^ Tuae rogavit ut suis verbis tibi Servo humillimo plurimam Salutem dicerem Nicolao Bernoulli Appendix Patrui Adjicere lubet quaedam milii inventa Theoremata, quae in reductionibus utilitatem suam liabent non exiguam. Demon- strationes eorum brevitatis gratia jam supprimo: Erunt inter Geometras qui facile invenient, quocirca illis eas relinquo. Definltio. Per q et I intelligo numeros qualescun(|ue in- tegros, fractos, attirmativos, negativos, rationales, irrationales. Per p intelligo tantum numerum integrum et atHrmativum, vel etiam cyphram. Sed per n et k intellectos volo numeros quoslibet integros affirmativos cxclusa cyphra. Theorema I + !•■ (IX : {e +fx'i)'' est algebraice cpiadrabilis. r . - +'!■+■■■ Theor. II Generalius, \x>"'(lx:{e+Jx'i)'' est algebraice quadi'aV»ilis. Theor. Ill -'U:: x!"i '^dx: (e +fx'i) '^ est algebraice quadra- bilis : Adeoque existente p = 0, erit etiam Ix^^-'i-'dxiie+fx'if''^" algebraice (piadrabilis. CORRESPONDENCE WITH BERNOULLI 133 Theor. V Theor. VI Theor. IV x^''dx:{e+fx'jy^ dependet a quadratnra luijiis dx : (e +fx'i). '«(/.<• : (c+/c'/)" dependet a quadratura ejusdem dx:{e+fx'i). I'P^+^dx ; (<> +fx'iy^ dependet a quadratura hujus j xhfx:(e+fx'i). Theor. VII Siimtis 8 et A in Casu Taylori erit '/-I dz:{e + fz'if quadrabilis per circuluni vel h^^perbolam. Corolloria (piae ex liisce Theorematibus deduci possent pulchra et miranda non minus quam utilia nunc omitto, sicut et plura alia ad quadraturaruni reductioneni spectantia, quae olim inveni ac passim cum Amicis coinunieavi. Ex. gr. Ex collatione Theorr. V et VI sequitur inveniri posse duos coeffi- cientes a et /3, ita ut algebraice quadrabilis. (a.r-p? + ^xf"i+'')dx : (e +fx'i)'"- sit (2) Bernoulli to Stirling^ 1729 Viro Clarissimo Jacobo Stirling S.P.D. Nic. Bernoulli. Pergrata fuit epistola, quam per coiliunem amieurn D. Cra- merum mihi baud pridem transmisisti et ad (|uam citius respondissem, si per varia impedimenta licuisset Gaudeo te valere et rem Mathematieam per impressionem libri de &uma- 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, (pii nunc apud nos versatur. Pauca quidem in eo intelligo, quia nullam adhuc operam 134 STIRLING'S SCIENTIFIC CORRESPONDENCE collocavi in lectione tertii libri Princi})iuiuiii J). Newtoni ; videris tamen mihi haud recte in cpistola tiia explicuisse (juid ipse vocat an E(juaiit. Verba sua sunt liaec : ' lie constructs a figure whose Sector CDF is proportional to the an^^'le ASB, and finds the point C which will make the fioure CD nearest to a Circle '. Existinio dicenduni fuisse ' ho constructs a fiourc, whose Sector CDF is equal to the area ASB, and finds the point 7^ which will make the fio-ure 67) nearest to a Circle.' Ceterum etiam si inveniatur punctum ali(jUod F ex quo xelocitas Planetae in utraque apside constituti eadem appareat ex hoc non sequitur ae(|uantem CD maxime accedere ad circulum, vel punctum i^ esse illud, ex (|U0 motus Planetae maxime uniformis appareat, ut D. Machin asserit pag. 41. Nam locus ex (pio Planeta in ^4 et P (fig. seq.) constitutus aeque velox apparet non est unicum punctum F sed Integra linea tertii ordinis FAffPf cu]us aequatio est Fig. 23. a — x . yy = a + l> — x .b — x.x positis AS — a, SF = h, Ag — x, flf = y- In hac igitui- linea et quidem in ejus ramo Pf datur fortassis punctum /. ex quo Fio. 24. Planeta apparet aeque velox in tribus punctis A, P, et D: adeoque ejus motus magis regularis vel uniformis ((uam ex puncto F. In ead. pag. 41. lin. IG omissa est vox reciprocally ; praeter hunc errorem in cadem pag. notavi, (juod Auctor videatur coihittere paralogismum, dum areas descriptas a corpore moto per arcum AR circa puncta S et F, item areas descriptas a lineis Fp et Fli dicit esse in duplicata ratione CORRESPONDENCE WITH BERNOULLI 135 perpendiculaiiuia in taugeiitoni (;ul punctuui R) deinissaium ex S et F; haec enim ratio obtinet tantum in barum arearuni fluxionibus, a qiiaruni proportioiialitate ad proportionalitatem ipsaruni areanun ar^iiinentari iion licet, ut scis ine olim (juoque ex alia oceasione monuisse ; nihilominus consecjuentia, quod area a linea Fp descripta aeqiialis sit areae a linea SR descriptae vera manet. Theorema illud, quod corpus ad duo fixa puncta attractum describat solida aequalia circa rectain conjungentcm ilia duo centra virium teniporilais aequalibus, verum esse deprehendo. Reli(]ua examinare non vacat. r r + I Theorema tuum pro interpolatione Seriei A, -A, B, /> p+ 1 C, D, &c per quadraturas Curvaruni deduci potest p+2 ' p+3 ' ^ ^ ex isto altero theoremate quod ante 19. annos cum D. de Monmort coiiiunicavi, 1 n n.n—l n .n—1 .11—2 a ~ a+b "^ 1 T2 .a + 2b ~ 1 . 2 . 3 . « + 3 6 a. ii-l.n-2 .n-3 , 1 . 2 . 3 . 4 . 5 ... /<6" + tvc = 1 . 2 . 3 . 4 . a + 4 6 a.a + b .a + 2h ...a + nb T Sed et sine quadraturis interpolatur facillime Series A, - A, r + 6 „ r + 2b r + 3b ^ . -, ^^ B, ; C, ; D, Arc ponendo p + b p + 2h /) + 3b ^ r .r + b .r + 2b ...r + zb — b r .r -\-b . r + 2b ... p — b 2) .p + b . p + 2b ... 2:> + zb — b r-\-zb.r + zh + b...zb+p — b , ,. p + zb .p + zb + b ... zb}-r — b vel etiam = ^ --. ; — , 'p .p + b .p) + ^'^ ... r — h prout p) major vel minor est quam r. Ex. gw Si s = 2^ erit terminus inter tertium 7 B et quartum yC medius = p-\-b ^ p+2b r . r + b.r+2b ...p-b , p + 2\b .p+3\b ... l^/> + r vel r+2|6.r+3i6 ... l^b+p 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 quail Jo z t'st imiiienis fractus, sed et quando differentia inter 2) et r est numerus magnus, quod ultimum, ut et valorem Seriei alicujus lente convergentis, ope Serierum quarundam infinitaruni pronitc convcrgcntium a te inveniri, ex littoris D"* Cramer intellexi, quaruin Serierum ddiionstrationein libenter videbo. Optarem spei tuae satisfacere tibi vieissim impertiendo nova fjuaedam inventa, sed dudum est quod Mathesis parum a me excolitur, ncc nisi in gratiam amicorum me subindo ad solu- tionem quorundam Problematuin accinxi, (juorum solutiones in Scbedis meis dispersae latent, et quoad maximam partem vix tanti sunt ut tecum coinunicari mcrcantur. D"™ Cramer rogavi, ut tibi transmittere velit Specimen method! meae (Pembertiana multo facilioris et cujus ipsnm participem feci) resolvendi fractioncm in i'ractiones luiius formae , ; 1 + qz"" + Z-" a + bz • I • \ +CZ + ZZ Dfis de Maupertuis Patruo meo nuper proposuit sequens Problema: A et B sunt duoignes quorum intensitates sunt ut p ad q, quaeritur per quam Curvain CD homo in dato loco G Fig. 25. constitutus recedere debeat, ut scntiat miniiinim ealorein, posito rationem cujusque ignis in objectum aH(juod esse in ratione reciproca duplicata distantiaium. Hujus Prol)lematis se(|urntem constructionem inveni. Centris ^ et i^ descriltantnr circuli acg, lulh aequalium CORRESPONDENCE WITH BERNOULLI 137 radioriiin Aa, Bh, juny.-mtiir AC, BC, secaiites circumferentiam horum circiiloruni in c, d et in eas demittantur perpendiculares ae, If, ex pnnctis c et (/ abscindantur arciis cff, dh, ea lege ut demissis perpendieularibiis a I, hi in radios Ag, Bh et perpen- dieularibus Ik, Im, in radios Aa, Bh et ductis en.fo, parallelis ad Aa, Bh sit eg— in ad dh — lo ut q ad p, erit, prodnctis radiis Ag, Bh, intersectio D punctiim Cnrvae quaesitae CD. En aliud Problcnia a Patruelo moo qui Potrol)urgi agit, mihi propositum. Circa punctuin A rectae positiono datae AE rotatur Curva ANOD, et ill quolibot Curvae ANO situ intelli- gatur punctuni maxime distans a recta AE, sitque A MO Curva quae transit per omnia puncta ; oportet invenire quaenam sint hae Curvae ita ut segmentum A MO A sit semper ad segmentum ANO A in ratione data in ad n. Ego iiiveni utramque Curvam OFse algebraicam. Idem proposuit sequens Problema cujus solutionem quoque inveni : Ex tribus altitudinibus stellae et duobus intervallis temporum invenire declinationem illius et elevationem poli. In Actis Lips, praeteriti anni pag. 523 : extat Problema, cujus solutionem talem dare possum, ut pro qualibet data Curva CBA possim invenire aliam Ahc, ita ut grave descendens ex quolibet Curvae CBA puncto B et descensu suo describens arcum BA, posteaque cum velocitate ac- quisita ascendens per Curvam Ahc integro suo ascensu describat arcum Ah aequalem arcui descensus BA ; motus autem fiat in medio resistente in ratione duplieata velocitatis. Sed invenire Curvam CBA talem, ut altera Ahc, in qua fit ascensus sit ips-a Curva descensus CBA ad alteram partem continuata, sive ut CBA et Ahc sint duo rami ejusdem Curvae videtur esse res altioris indaginis. Vale. D. Basileae d. 22. Xbris 1729. 138 STIRLING'S SCIENTIFIC CORRESPONDENCE (3) Sthihif) to BernotiUi, T/'PtO Cop3'' of a Letter sent to M"* Nicholas Bcrnouilli September 1730 Sir I was vcvy glad to hear of your welfare hy ^'our most obhgino- Lettei' ami luivc delayed answering it hitherto for no other reason but that I might he able at length to answer you in ever}^ particular : for seeing you desired the Demonstrations of the two Series which M' Ci'amer sent you, and these Demonstrations are such as could not be conveniently brought within the bounds of a Letter, I thought it was best to stay till ni}' book was read}' to be sent j'ou ; for you will find in it the principles explained by which I found these and such Series. Indeed I might have sent you my Book somewhat sooner, but unluckil}' I was taken up with an affair which obliged me far against my inclination to defer my answer till this time. As to M"" Machin's Treatise it was written in great hurry and designed only to shew wdiat mny be expected from his larger Treatise on that Subject & therefore it is no great wonder if you met with some difficulties in it, especially considering that not only his propositions but aho the prin- ciples from which most of them are deduced are new. I have prevailed on him to write an answer to that part of your Letter which relates to himself, which 1 now send you and hope it will satipfie you intirely till you shall see the Book he is now preparing for the press, which I am Confident will please j'ou extremely, as it clears up the Obscure parts of Newton's third Book of principles, and carrj^s the Theory of Gravity further than even Sir Laac himself did. And it is somewhat strange that altho the principles have been ]»ublished above 40 years, that no body has read further than the two first Books, altho they be barely Speculative and were written foi' no other reason but that the third might be understood. The Theoi-eme which M'' Cramer sent you for Interpoling by Quadratures may as you observe be deduced from one CORRESPONDENCE WITH BERNOULLI 139 which you sent to M"" Moninort 15 years ago, and so may it as easily be deduced from a more simple one which D' Wallis pulilished 75 years ago namely that -a;" is the Area of a Curve, whose Ordinate is a;""^ and I value it so much the more because the Demonstration of it is so very easy. But neither your Theoreme nor that of D"" Wallis is sufficient except in that case when the Series is so simple as to admit of Interpolation by a Binomial Curve, for if a Trino- mial or more Compound Curve be required we must liave recourse to the Comparing of Curves according to the 7 & 8 Propositions of Newton's Quadraturus, that being the generall principle for this kind of Interpolation. r r + h r + 2h I agree with you that the Series A, ~ A, , B, r C, *= -^ p p + b p + 2h &c. may be Interpoled without Quadratures, as you will see by many Examples in the 21, 22, 2G, &: 28 Propositions of my Book : but I am still at a loss to find out that it is to be done after the manner you propose by putting Indeed it is true that the Terms may be expressed by a Frac- tion, but to what purpose I know not ; for if the Term required be an Intermediate one, both the Numerator and Denominator of the Fraction will consist of an Infinite number of Factors, and therefore that is no Solution, for it is as Difficult, nay it is the very same Probleme, to find the Value of such a Fraction as to find the Value of the Term proposed. The fraction no more gives the value of a Term whose place is assigned, than the place of a Term being assigned gives the Fraction. Besides, that Method would not even give a primary Term which stands at a great distance from the begining of the Series : for the Number of Factors, tho not infinite, yet would be so great as to render the work altogether impracticable. But here I except the case where the difference betwixt 2^ tV r is not much greater than h, and at the same time is a multiple of it ; this is the only case when your Method will do, as far as I understand it ; but when this happens, the Series is interpoleable by the bare inspection of the Factors, even without the help of common algebra : and therefore 140 STIRLING'S SCIENTIFIC CORRESPONDENCE I hope 3'ou did not imagine tliat I designed to trouble a Gentleman of M"" Cramer's abilities with such a simple Ques- tion, or that I pretended to reduce it to Quadratures, altho perhaps I might take it for an Example of the general solution. I cannot but think that one of us has misunderstood the other, and therefore I should be glad to have your Method explained to me: for instance in the Series 1, ^A, ^B, |(7, |i), (^'c. which is the Simplest of all those which do not admit of an exact interpolation: how do 3'Ou find out that the Term which stands in the midle betwixt the first & second is equal to the ni)ml)er 1-570796 &c'l You know I find it to be such from the method of Quadi-aturcs, which demonstrates it to be double the area of a (.'ircle whose Diameter is Unity. And how doth your method give a Term remote from the l)egin- ing ; for instance the pi-oduct of a million of these Fractions I X f X I X f X -^ X ... f iggi^l which I can find in the quarter of an hour to be the number 1772-454 0724, as you may try by the Series which was sent you for finding the proportion which the midle Uncia in the Binomial has to the Sum of all the Unciae of the same Power. Altho you are pleased to say that you have not spent much time on Mathematicks of late, it would rather seem to be otherwise from the ingenious Problems which you mention; for my part, as their Solution depends not on new principles, and since I know not for what design they were proposed, I have not thought about them especialy since you say you have solved them ab-eady. M'' Klingcnstierna shewd me a Construction of the Probleme about two fires different from yours and Extremely Simple. He has also constructed the Probleme about a Curve revolving about a point, and whereas you have said without any limitation that you found both the Curves to be algcbi-aical, he observes that it is so only when the Areas mentioned in the Probleme are to one another as one number is to anotlici-. He has also solv'd the Piolileme about a Body falling down in a Curve, and afteiwards rising either in another or in the same continued; of which last 3'ou say vkletur esse res <iltioris indagiiiis: And as to the Probleme about finding the Latitude of the place (^- declination of a Star from having three altitudes of it, CORRESPONDENCE WITH BERNOULLI 141 and the times betwixt them, it is evident at first sight how it may be brought to an equation. M"" Klingenstierna had shewed me that part of your Demon- stration of Cotes's Theoreme which you liad ready when he left you ; and M"" Cramer sent me the same with the remaining part which j^ou sent to him about the begining of this Year : indeed I take it to be an elegant Demonstration and far Superior to that of the person you mentioned. But I suppose you know that M' De Moivre found out his Demonstration of the same Theoreme very soon after M"" Cotes's Book was published, which is now many years ago, and I am of opinion that it will please you, as it requires no Computation. And now I come to beg pardon for this long Letter and to assure you that I am with the greatest respect Sir Your most obedient most humble Servant James Stirling. (4) Bernoulli to Stirling, 1733 Viro Clarissimo Jacobo Stirling Nicolaus Bernoulli S.P.D. Epi.stolam tuam die 30 Scptembris 1730, seriptam una cum inclusa D'" Machin et cum eximio tuo (pro quo debitas ago gratias) Tractatu de SuilTatione et Interpolatione Serierum Infinitarum post annum fere accepi eo tempore, quo novae Stationi in nostra Academia Professioni nempe Juris admotus variisque occupationibus implicitus fui, quae me ex illo tem- pore a rerum Mathematicarum studio abduxerunt, et ab attenta et seria lectione Libri tui avocarunt. Est et alia dilatae responsionis causa. Perdideram epistolam tuam inter Schedas meas latentem, eamque multoties frustra quaesitam non nisi ante paucos dies inveni. Ignosce quaeso tam diuturnae morae. Alacrior quoque ad respondendum fuissem, si quae- dam a me dicta, quae tamen nunc sub silentio praetereo, paulo aequiori animo a te et a D"° Machin excepta fuissent. 142 STIRLING'S SCIENTIFIC CORRESPONDENCE Quae D""^ Machin rcgcssit contra objectioiiem mcam circa definitioneiii loci, ex quo Planetae uiotus maxime uniformis apparet verissima sunt. Fateor mo non attendissc ad motum medium aut ad motum retrogi'adum Planetae, sed studio id feci. Ego nunquam credidi Planetae motum apparere magis rcgularem aut magis uniformem eo ex loco, ex quo motus in tribus orbitae punctis aequalis apparet quam eo ex loco, ex quo motus in duobus tantum orbitae punctis aequalis apparet, id est, motum ex primo loco apparentem minus ditierre a motu medio, quam motus ex secundo loco apparens. Objectio mea erat tantum argumentum, ut vocant, ad hominem. Credebam Dnum Machin esti masse regularitatem vel uni- formitatem motus ex eo quod Planeta in utraque apside ex centro aequantis visus aeque velox appareat ; et ad hoc credendum me induxerunt haec verba pag. 42. ' The said center F will be the place about which the body will appear to have the most uniform motion. For in this case the point F will be in the middle of the figure LpD (which is the e(|uant for the motion about that point). So that the body will appear to move about the center F, as sivift ivhen it is iti its sloiuest motion in the remoter a2)sis A, as it does ivhen it is in its siuiftest motion in the nearest apsis P' quae verba sane alium sensum fundere videntur, quam sequentia quae habet in sua responsione : ' I did not conclude this to be the place of most uniform motion, because it is a place that reduces the velocity in two or three or more points to an equality, but because the motion throughout the revolution differs the least possible from the mean motion.' Obscuritatem verborum pag. 41. ubi Dnus Machin demon- strat acqualitatem arearum FjjL et SRA quae ansam praeluiit suspicandi paralogismum, non puto natain esse ex praeli crrato, sed ex festinatione, quam ipse Auctor se adhibuisse dicit ; si quidem non solum particula and cum in locum, quern dicit Auctor, transponenda est, sed delenda etiam particula sequens therefore, ego in meo exemplari locum sic correxi Pag. 40. lin. pen. pro areas scripsi fluxions of the areas LFp and AFR. Pag. 41. lin. 4. pro the areas scripsi and the fluxions of the areas ASR and AFR Pag. 41. lin. 8 delevi And therefore Ead. lin. post area adjunxi LFj). Ead. pag. lin. 10. pro that scripsi the area ASR. CORRESPONDENCE WITH BERNOULLI 143 Vehementer cnpio videre, qiioniodo theoreiiui tuuin pro interpolatione Seriei A, -A, B, G (Sec aeoue txcile ((]uod to per jocuin dixissc puto) dedvicatur ex theoremate Wallisii ante 75 annos publicato, quod neinpe - x^ sit Area Curvae en jus ordinata est cc""^ ac ex isto ineo theoremate quod me ante 15 annos ]\Ionmortio misisse scripseram, nimi- rum quod 1 . 2 . 3 . 4 . . . ii X 6" a .a-^b .a + 2b ... a-\- nb 1 n )i .n—\ a a+b 1 . 2 . ct + 2 6 n . n—l . n—2 . 1.2.3.«+36+*'^ Sane cum haec Series sit aequalis areae curvae cujvis ordinata est x"~'^ x 1 —xh^'' in casu x = I, sola substitutionc terminorum a te adliibitorum res immediate conHcitur Nam si lino-amus duasCurvas,unamcujus ordinata est a;'"' ^ x 1 —x^^~^~ > alteram cujus ordinata est x~'^''~^ x l—x''~^~ , faciendo b ~ 1, a =r at = z + r, n = p — r— 1 erunt istarum Curvarum Areae per theorema meuni 1 . 2 . 3 .4 ... 7>-?'-l , 1 . 2 . 3 . 4 . ... «-r- 1 ct r .r+\ .r+2 ... r+p — r — \ r-\- z .r + z+\ ... r-\-z +p - r — 1 adeoque prima ad secundam ut 1 ad r .r+\ ... r+ n~r—\ . .r . r + 1 ... ?' + s— 1 sive ad r-\-z.r + z+\...r+p — r—\ p) .p)-\-\ ... p + z — l id est, ut primus Seiiei interpolandae terminus ad alium cujus distantia a pi-imo = z, Dcmonstratio haec ubique supponit idipsum alterum theorema quod allegasti, nempe quod — a;'* = areae Curvae cujus ordinata est a^"^ (theorema melius notum ex methodo fluxionum quam ex Arithmetica Infinitorum Wallisii) quomodo enim potuissem dicere Seriem r + t^'C, esse aream curvae cuius ordinata et-t a a + b "^ x^'^xl - c^''" in casu .r = 1, nisi scivissem modum eruendi areas ex datis ordinatis? Sed hoc ipsum alterum theorema solum neutiquam sufficiens est etiam in istis Seriebus quae 144 STIRLING'S SCIENTIFIC CORRESPONDENCE adiuiitunt iiitci'[)()l;itioiicin per cnrvas binomialcs. Siinili iiiodo potiiisscs (licorc diiticillinia tlieoreiuata Nevvtoni et alioruiii do tiuadraturis ex dicto Wallisii facili dcduci posse. Quae dixisti de interpolationibus quae requirunt Curvas trino- iniales aut magis coinpositas, (]Uod nempe recurrenduui sit ad 7 et 8 Prop. Newt, de Quadraturis, ea non magis tangunt meuiu quain tuuni theoreina ; inihi animus non fuit tractatum scribere de interpolationibus, aut nieum theorema pro generali interpolationum remedio venditare, sed tantum tuuni a DiTo Cramero mihi missum theorema demonstrare. Quod attinet ad alterum uiodum interpolandi Seriem . r . r + h r + 2h . • ,-, • i A, -A, i B, , 6, ivc. (lui consistit m ponemlo r .r + b .r+2h ... r + sh — h__ r .r + h .r + 2h ...p — b p.^ + b .'p + 2h ...'p-\-zb — h r + zb.r + zb + b ...zb+2) — b , p + zb.p + zb + b ... zh + r — b . vel = — ; } i — ' lateor ilium non succedere J) . 2) + b . 'p + '^b ...r — o nisi iis in casibus, ubi differentia inter ^j et r est divisibilis per b, et sinuil nunierus non admodum magnus, quod ultimum in praecedentibus meis literis ipse jam agnovi. Fateor prae- terea sensum tlieorematis tui non recte intellexisse, credebam /j> T + 1 ■?■ + 2 enim in hac Serie A, ~A, ~ B, -G, &c. (luam Unus p p+1 p+2 Cramer tancjuam formulam generalem, non tanquam exem- plum alius generalioris mihi miserat, ^) et r significare numeros integros ; unde non capiebam cur hacc Series, utpote (juae accurate posset interpolari, ad quadraturas reduceretur. Sed his majora te praestitisse vidi cum voluptate in tuo libro, cujus Propositio 18 continet, ni i'allor, idipsum quod ego per modo dictum alterum interpolandi iiio(hnn monere volebam. In exemplo 1. Prop. 25. ubi tradis interpolationein unciarum binomii ad dignitatem indeteriuinatam elevati, inveni theorema non nmltum absimile praedicto meo theoremati. Si iractionis 1 .2.3.4... 71 X 6" ^ ,• -1 i 171 i. • „i; ;- -. r numerator dividatur per b^\ et sniguii a . a + b .a + 2b ...a + nb factores denoniinatoris cxcepto })rin)o per b, et ipsa fractio nmltiplicetur per prinunii I'actorem a, proveniet reciprocus it terminus unciae ordine n+ 1 in binomio ad dignitatem r + '"' CORRESPONDENCE WITH BERNOULLI 145 elevato ; liinc per theorenui incum, ut Area ordinatao x^'' 'x 1—x" ad -, ita imitas ad dictam unciaui. Ex. g-r. si a ^ ponatur a = 5, b = 1, it = 4 erit area ordinatae x'^xl—x , id est, i — l + f — f+l sive glo '^^1 | ut 1 ad 126 unciam termini quinti in dignitate noiia. Si a = 1, h = 2, it, = ^, erit area i ordinatae *" x l—xx , id est, quadrans circuli ciijus radius = 1, sive area circuli cujus diameter = 1, ad 1 sive ad quadratum circumscriptum, ut unitas ad terininum Wallisii Q inter- ponendem inter primum et secundum terminum Seriei 1, 2, 6, 20, 70, &:c quae continet uncias medias dionitatum parium, sive ad terminum qui consistit in medio inter duas uncias 1 et 1 in potestate simplici binomii ; sicut tu quoque invenisti in exemp. 2. dictae Prop. 25. Laboriosa quidem sed elegans est methodus per quam in- venisti ope Logarithmorum interpolationem Seriei 1, 1, 2, 6, 24, 120, &c in Ex 2. Prop. 21. Ceterum frustra quaesivi modum, quem dixisti in sequentibus monstrari, interpolandi hujiismodi Series absque Logarithmis, quod autem a te prae- stare posse nullus dubito. Termiimm qui consistit in medio inter duos primos 1 et 1 ope Theorematis mei sic eruo. Sit in dicto theor. a = n+1, b = 1 , eritque area ordinatae „ -- n 1.2.3.4 .11 1 .2.3.4...nx 1 .2.3...'M ^" X 1 X = = n + 1 . it + 2...2ii+l 1 .2.3 ... 2/H-l Fiat n=^ eritque area ordinatae Vx — xx i.e. area semicirculi, cujus diameter = 1, aequalis dimidio quadrato quaesiti termini. Hinc quo(|ue deducitur interpolatio terminorum intermediorum in hac Serie 1, 1, 3, 15, 105, 945, &c. Nam si liat a=l,b = 2, II 12 3 n X 2'* erit area ordinatae x*'xl—xx = ^ — '- — ~ ; sed in 1.3. 5 ... 1+2 II casu II = i praedicta area sit ae(|ualis areae circuli cujus diameter = 1, et numerator fractionis sit aequalis radici quad- ratae duplae istius arcae, per niodo ostensa, denominator autem fractionis sit aequalis termino qui consistit in medio inter secundum et tertium Seriei 1, 1, 3, 15, 105, 945, cVc proinde ut radix quadrata dimidiae areae circuli ad 1, ita unitas ad ternnnum ilium intermedium, qui per binarium divisus dabit medium inter duos primos 1 et 1 dictae Seriei. 2H7 L 146 STIRLING'S SCIENTIFIC CORRESPONDENCE Do iiKxlo invciiit^'iKli radiccin acquationis thixionalis per Seriem infinitarn, do ({110 a^is in Scliolio Piop. iilt. I 'art. I. ctiam e^o aliquoties cogitavi, at liac dc re scri])tuiii aliquoil comimicavi cuin Diio de Maupertuis ciun apiid 110s a<^eret, in quo sequentia observavi. Posse inveniri Series ^eneraliores quani quae inveniuntnr per parallelograiniini Newtoni ; non necesse esse ut indices dignitatiim in terniinis Seriei qiuiesitae aut aequationis transformatae cadant in eandem progressioneni aritlinietieain ; posse aliqnos indices esse irrationales ; et prop- terea tani Taylori reyulani in Prop 9 (juani tuani in Enunierat. Linear, tertii ordinis datam, pro deterniinanda forma Seriei fallere ; posse per terminos solitarios in ae(piatione trans- formata noiiunquani aliquid determinari, aljsque ut oinnes coefficientes 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 coefficientes in methodo Taylori evadentes = laborem calculi prolixiorem reddunt, quam paret. Sic pro Exemplo Taylori in Prop. 9. Method. Increm. pag. 31 1 +sx- z^xx — x = sequentes 4 Series inv^eni ; quarum tres priores sunt genera- liores illis quas Taylorus invenit. 40- 7 . 14 1 ^ -5+^165 , „ 288 , . 2^- «^=^2^"^-20^'+^' 4 +^^ 5¥87^'*' 3«. x= 2z^ + B-\z-'-rlBz-i-'-i^BBz-'' + -i^B'--i^z-i<kQ 4^ « = -z~'^~z-i — ^-iz-^-^%^z--^'i\c. Sic quoquc ol»servavi te non satis accurate rem examinasse, quando pag. 83 dicis, ae(|uationem r'^y- = 'rx^ — x-y'^ nulla alia radice explicabilem esse praeter duas exhibitas y = •^- 6-^ + 120,. - 5040^> + ^^'" "^ 2/ = ^lxl--.+ ^.-7-^^o+&c CORRESPONDENCE WITH BERNOULLI 14-7 quaiuiu })rior dat .siiiuui, et posterior cosiiiuiii ex dato arcu x ; et de (lua posteriore dicis, quantitatein A cjiiae aequalis est radio )• ex aequatione llnxioiiali nou deterininari. Ego 11011 solum iiiveni, Sericm iioii posse habere banc forniain A + Bx^ + Cx* + Dic" iS:c nisi fiat A = r, sed utranique a te exbibitam Seriem couiprebeiidi sub alia general iori, quae haec est : y = A-\- Bx + Cxx + Dx^' + Ex^ + &c in qua eoefficientes A, B, G, D i^'c banc sequuntur relationem BB="-^'\ C=-^—, D=-^-^, rr 1.2. rr 2.3. ?-/• c = A D — 1.2.r/ E= - C 3 . 4 . 7'r 4 . 5 . r?' Si fiat ^ = 0, babetur Series pro Sinu ; sin autem A fiat= r, babetur Series pro cosinu; sin vero A alium habeat valorem praeter bos duos, etiam alia Series praeter duas exbibitas erit radix aequationis fluxionalis propositae. Similiter Series illae quatuor, quas exbibes pag. 84. pro radice aequationis y + a^y — xy — x'^y = 0, sub aliis duabus generalioribus quae ex tuis particularibus compositae sunt, comprebenduntur. Duae nempe priores sub bac y = A+Bx + Cxx + Da;"' + Ex"^ + &c. in qua eoefficientes A et B babent valores arbitrarios, reliqui autem C, D, E, (kc sequentem ad priores babent relationem C'.^H^^, i)=L^^5, E^tz^c, F = ^-z:^D&c. 1.2' 2.3' 3.4' 4.5 Si £ = babetur tuarum Serierum prima, Si ^ = babetur secunda. Duae posteriores comprebenduntur sub bac generali forma y = Ax^ + Bx''' + C^"-^ + J)x-"-'^ + ^a;"-* + Fx-^-* + &c. ubi iterum A et B babent valores arbitrarios, g=-'''''-'a, e=-"-^-''-^c, 4 . « - 1 8 . (( - 2 12.«-3 ' ' 4.a+l ' ^^ a+2.a + 3 ^^a + 4.a4-5 8.a+2 ' 12.a + 3 ' l2 148 STIRLING'S SCIENTIFIC CORRESPONDENCE Si tiat B = exsuri;-it tua tertia Series, et si fiat x\ — exsur^it ([Uarta, in ([ua teniiini |)cr si^nuni + non per sinnuni — connccti debeiit. Jiicoinodum (juoiiiie est in tuis Si'riolnis, (|Uod Literae A, B, C, D iVrc mox pro coefficientilais terniinoruni, mox pro ipsis terniinis usurpentur. Hac (lata occasione describani hie ea quae ad (jiias lam tiias Series in Libro tuo de Enunieratione Linearuni tertii ordinis contentas notaverani eo tempore, (jiio hiinc Librum a l)no do Maupertuis comodatiim luibebam. Eum (|uidem nunc non habeo, sed in quadam mea Scheda liaec notata reperio. In Escemjylo 2 ikkj. 22. aequationis x'y + ayxJc -\- a-xJo — 2a^ Jo = radix y est = aA+(tiL aaA—d"^ a^A — a^ a^A—a^ . quando ^ = provenit Stirlingii solutio ; sed quando A -— a 2aa exsurgit y = a-\ • In Ex. 4. pag. 26. y-x-—3x'^xi/ + 2x-x' — a.i'>'r + (rx^ = radix y est = x + BxP— BBu.^ + - +B'''x''+ -aB—B*.x'^ 36 4 10 + iL aaB + ,— aB' + W . x"^ &c 324 180 2aa , 6rt'^ , 88a* ., „ item 1/ = 2a' + (:/— x ' + x~- x " &.C. '' 7 35 637 Pag. 28. aequationis y'-^ — (nj^ + o'-ij — a"' + x-y = radix y ^..2 ^.4 2 ^.G -,.8 non est := tt 4- r --^, (vc sed a H ^ H _ <xc. 2a 2tt' 2a \^a^ 32 a^ Pag 31. y = x+ —. „ ., + 7^—7—, — ^^^ a^ 2a" 7(('-' . . , ,„„ Pag. 34 Ex. 1. vSatisfacit etiam x' ic* 4*" Ax'^ „ 7/ = .1' + — ^- + — g- tVc. aa w^ a^ a" CORRESPONDENCE WITH BERNOULLI 149 In eadem Scheda iiotataiii reperio Speciein aliciuaui linearuni tertii ordinis a te et a Newtono oiiiiseani. Neinpe in Libri tui pag. 112, Sp. 58 ubi pro aeqaales et ejutdcin si(jnl legi debet aeqaales ajjirmativae: nam si radices sint aequales negativae, ligura non evadit cruciformis, sed habet crura ut in fig. 57. et praeterca piinctiiin conjiigatum in dianietio AB, ijuod reperitur faciendo abscif-sam = r . 2l> V.t igitur haec nova Species est di versa k Specie 53 Newtoni, apud quem in mentione Speciei 54 pro Imposibiles etiam legi debet aequales ajffirmaiivae. Probleniata de qui bus in fine epistolae meae nicntionem injeci, eum in finem subjunxi, ut petitioni tuae ali(|U0 modo obedirera inipertiendo nova quaedam Mathematica. Mos iste Probleniala proponendi et alios ad eoruni solutioncm amice iiivitandi, non est omnino culpandus, si is nempe scopus propo&itionis sit, ut coiiiunicatis invicem methodis solutionum Ars Analj'tica incrementum capiat. Dictorum Pi ol>lematum solutiones Patruus mens et ego cum Dnu Klingenstierna tum apud nos degente coraunicavimus ; hinc credo cons.tructionem quam hie tibi ostendit, Problematis de Curva recessus intra duos ignes, et quam tamquam valde simplicem laudas, non aliam esse quam Patrui mei, qui hoc Problema ope Trajec- loriarum Orthogonalium ingeniose quidem solvit, sed ipsius trajectoriae orthogonalis sive curvae (piaesitae constructionem non dedit, De problemate circa curvam circa datum punctum revolventem recte monitum est utramque curvam esse alge- braicam ; si areae de (piibus in Problemate sermo est, sint ut numerus ad numerum ; haec limitatio tanquam facile animavertenda a me studio omissa fuit. Vix est ut credam Problema in Act. Lips. 1728. pag. 523 propositum a d. Klin- genstierna solutum fuisse eo etiam in cas^u, de quo dixi, videri I'em esse altioris indaginis. Moivraei demonstratio Theorematis Cotesiani sive rosolutio fractionis ~ -, in fractiones simpliciores habentes z~>'-2lz" + l '■ denominatores duarum dimensionum milii perplacet, quamvis ob concisum sermonem explicatione (juadam opus habeat, Posteriorem partem demonstrationis meae, quam ex coinuni- catione Dm Crameri vidisti ab inductionis vitio liberavi 150 STIRLING'S SCIENTIFIC CORRESPONDENCE substitiita liquida et ricrida demonstratione, quam ad eundem D Crainerum Amicmn nostrum mitto in epistola cui banc ad te perferendam inehido. Vale. Dab. Basileae d. 1. Aprilis. 1733. P.S, Nescio quo fato acciderit ut nonien nieum in Cataloj^'o Socioruni R. S. omissum sit. Conjicio id factum esse bae ratione ; primum nomen meum mutatum fuisse in nomen Adgnati mei Nic. Bern. Professoris turn Bernensis, postea Petroburgensis ; dein ex catalogo expunctum post hujus obitum. Spero bunc errorem emendatum iri. V CASTEL AND STIRLING (1) Castel to Stirling^ 1733 Doctissime Vir Ltbenteh vidi quae de me in epistola ad clarissimum amicum D. de Ramsay seripsisti, et gratias pro benevolentia tua in me habeo quam plurimas. Jamdudum professus sum quanti sit apud me. Vidisti baud dubie quae in commentariis Trivoltiensibus seripsi circa opusculum tuum ultimum de seriebus infinitis tum summandis tum interpolandis. Quod nunc attinet ad aequabilitatem arearum Newtonianam, noUem mibi tribuisses errorem adeo crassum quasi lineam eandem duobus aliis non parallelis parallelam afficerem. Vel ipsa verba mea reclamant, licet verbis figura non satis re- spondet. Supposui enim statim cum Newtono lineolam Cc parallelam SB] et deinde distincte supposui lineolam aliam CR parallelam BT. Relegere potes liaec ipsissima verba mea pag. 539 (et tirant OR parallele a BT) quae si advertittas aliter profecto rem accepisses, neque demonstrationis meae errorem sed demonstrationis Newtonianae vitium deprehen- disses ; vitium dico non quidem geometricum sed pliysicum, quod plerisque summi illius geometrae demonstrationibus accidit, quae quidem geometrice verae sunt, a veritate physica autem omnino aberrant. Sensus itaque demonstrationis meae iste est. Suppono constructionem et demonstrationem Newtonianam circa puuctum >S'. En meum circa punctum T. Duco CR parallelam TB, et dico ATB = BTC, atque BTC = BTR. Ergo quod erat demonstrandum. Tam vera est haec demonstratio quam 152 STIRLING'S SCIENTIFIC CORRESPONDENCE Demonstratio Ncwtonii et si ([uithjuid circa illain dixi paginis totis 531, 32. 33. 34. 35. 3G. 37. 38 39. 40. 41, dignatus esses legere, sensisses noii in toto trium liiicai'uin iion parallclaniin parallelisiiio rem stare, sed in ipsa pi'aecipue curvariim geometricariim natura, qiiaruni latera infinitesimalia sunt omnino indeterminata ut hoc vel illo inodo physico resolvantur in determinationes laterales numero infinitas, Conclusio autem tua, non est mea, quam mihi affingis. Non sequitur ex mea demonstratione sectores D^ AED, DEB quos satis scio esse inaequales, esse aequales. A finito ad infinitum, ab infinitesimali ad finitum non valet eon- ^ JL — ^ — J sequentia. Diversa elemcnta, diversae y pq fluxiones dant fluentes omnino diver>as. In priori figura CB non est = RB, ncc lortasse TAB = SAB. Diversae sunt etiam vires centripetae Cc, GR. Vera est autem observatio Kepleri vera est Demonstratio Newtoni : sed non vere ista demonstratio Imic observationi applicatur: vel potius vera est utraque inclusiva non autem exclusiva, Punctum S centrum esse bine ita demonstratur ; ut centrum sit et T eodem modo, et quodvis punctum aliud, nullum enim est ad quod non dirigatur vis centripeta ut ipse adstruit Newtonus, varias versus vaiia puncta ciirvac dcfinicns vires centripetas. Excidit mihi superius plerasque Newtoni assertiones geo- metrice veras, physice falsas esse. Parce vir doctifcsime huic ingenuitati meae. admiror Newtonum nullum novi geometram illi anteferendum. Pbysicae vitiuni est : nimis geometrice tractari renuit, (juamvis tota sit geometrica, natura, ut ajunt, geometrizat senqier: sed geometria sesc infinitis acconnnodat CORRESPONDENCE WITH CASTEL 153 hypothesibus ; nee qnidiiuid o-eoiiietriciim est, coiitinuo pliy- sicum esse convincitur. geonietria circa absti'acta v^crsatur, circa possibilia, possibilia autein sunt nnmero infinita: unicuiu est in quolibet plienonieno naturae systema: nee a possibili ad actum valet consequentia. A quindecini circiter annis opusculum composui (juo pb}^- sicum Newtoni convellere totuni niihi videbar. Praelo paratuni erat opus ; suuinia mea Newtoni reverentia coliibuit ne publice illud juris facereni : nee faciam credo equidem tanta in animo meo insidet summi illius viri existimatio. Vale vir claris.>ime, meque tui observantissimuni, ^ervumque bumillimum habe. LuDOVic Casti L. Parisiis die 25 Martiis 1733 P.S. Status quaestionis est. vult Newtonus aequabilitatem arearum acquabili tempore descriptarum signum esse eertis- FiG. 30. simum, propriuni, unicum centri respectu cujus ea regnat aequabilitas. contend© ego signum illud esse omnino aequi- vocum. nee unam hac de re demonstrationem assero unicam impugnas clarissime vir. omnes sunt impugnandae si asser- tionem Newtonianam salvam velis. nam vel ea quae circa hie appositam figuram versatur totum systema Newtoni convellit: demonstrat enim 1", sine ulla vi centripeta, et sine ullo centre 154 STIRLING'S SCIENTIFIC CORRESPONDENCE aequabiles esse tanien areas circa pnnctuin E. 2 '. iiifiiiita esse puncta circa quae haec vigeat aequabilitas. at(|uc eiiim curvis eidi'iii olitinet iiuletenninato. ruit ergo propositio hace i'unda- mentalis Newtoniani systeinatis. (^^) Stirling io CasfcJ Reverendo Patri D° Ludovico Castel. Doctissiine Celeberrimeque Vir Gratias ago maximas propter epistolam quam nuper ad me scribere dignatus es, cui certe responsum antcliac dedissein, si per varia negotia licuisset. Commentaria trivoltiensia ad manus meas nondum pervenere, fateor tamen me pluribus noininibus tibi devinctum propter ea quae in aliis tuisoperibus de me scripta videram. Cur ego ad amicum commnnem D. Ramsay ea scripsi quae tibi paulo liberius videntur, in causa fuit tua erga me publico attestata benevolentia, quam certe credebam me satis renmnerari non posse, agnoscendo lil)rum tuuin de gravitate esse multiplici cruditione refertum si non libere etiam tecum communicarem objcctioncs quasdam mea opinione liaud male f undatas ; hoc enim ni fallor non minus quam illud munus est amici. Quantum ad aecjualitatem arearum circa centrum virium, ego in pagina 539 tui lil)ri credebam CR fuisse errorem praeli, Fig. 31. si quidem nulla istius modi linea extat in schemate ; et pro eadem legebam CV. Et procul dubio opportet CR et Cc esse unam atque eandem tam magnitudine (juam positione nisi fingas duas e.sse vires centripetas ut in tua epistola. Ibi suj^ponis demonstrationem Newtoni pro accjualitate arearum circa punctum ^, dein proi'ers propriam pro areis circa punctum CORRESPONDENCE WITH CASTEL 155 T, qiiam ;iis tain verain esse qnam earn Newtoni ; quod ego libeiiter concedo. Nam si existente >S' centro viriuin areae circa idem aecpiales sint per demonstratioiiem Newtoni ; annon per eandem demonstratioiiem areae erunt aequales circa aliud quodvis punctum T modo idem supponatur esse centrum virium ? Sed quid lioc ad nostram controversiam ago sane iioiidum percipio. Tuum est demonstrare areas esse aequales circa punctum quod non est centrum virium, alias inconcussa manebit Veritas propositionis Newtonianae. Inquis me si perlegerem paginas 531, 532 &c ' sensurum non in solo triura linearum parallelarum parallelismo rem stare, sed in ipsa praecipue eurvai-um geometricarum natura, quarum latera infinitesimalia sunt omiiino indeterminata ut hoc vel illo modo physico resolvantur in determinationes laterales numero infinitas '. Sed post lectas sedulo paginas mihi recom- mendatas, minime sentio rem stare in natura curvarum, etiainsi resolvi possint in latera infinitesimalia ad libitum. Et si CR et Gc supponantur non coincidere erunt duae vires centripetae, quo in casu nihil probari potest contra Newtonum. Ut autem coincidant est impossibile, quoniam aS'J. et TA non sunt parallelae. Revolvatur jam corpus in semicirculo ADB cujus centrum G, et E punctum quodvis in diametro AB, cui normalis sit GD. Dico impossibile esse areas circa puncta G 8c E descriptas esse temporibus proportionales. Sit enim si fieri potest. Itaqu« ex hypothesi erit ut tempus quo arcus AD describitur ad tem- pus quo arcus DB describitur ita quadrans AGD ad quadrantem DGB; et eadem de causa ut tempus quo describitur arcus ^D ad tempus quo describitur arcus DB ita area AED ad aream DEB; unde ex aequo ut quadrans ad quadrantem ita sector AED ad sectorem DEB, unde ob quadrantes ejusdem circuli sibi invicem aequales, erit area AED aequalis DEB. Quod est absurdum. nam prior excedit quadrantem, posterior vero ob eadem deficit triangulo GDE. Haec autem deducitur consequentia non arguendo a finite ad infinitum aut ab infinitesimali ad finitum, sed argumentando per aequalitatem rationis. Et in quacunque curva deferatur corpus, geometrice semper 156 STIRLING'S SCIENTIFIC CORRESPONDENCE deinonstrari potest, impossibile esse ut aieae eirca duo puncta descriptae sint teniporibus proportionales. A is verain esse observationcin Kepleri et veraiii esse deiiion- strationein Newtoni sed iion vore applicatam huic observationi quod ultimuiu veliin ostendes. Deinde ais ' punctum ^centrum esse ita deiiioustratnr ut centrum sit et T eodem iiiodo, et quodvis punetuni aliud, nullum enim est ad (juod non dirigitur vis centripeta, ut ipse adstruit Newtonus, varias versus varia puncta curvae definiens vires centripetas '. Newtonus ut demonstret vim qvia planetae retinentur in orbibus tendere ad centrum Solis, ostendit per prop. 2. lib. I corpus onnie quod movetur in ciirva, et radio ad punctum innnobile ducto describit areas Temporibus proportionales, urgeri a vi centripeta tendentc ad idem punctum ; quumque Keplerus observasset planetas describere areas circa solem temporibus proportionales, concludit vires quibus planetae retinentur in orbibus tendere ad centrum Solis. Et liaec est legitima argumentatio quoniam unicum tantum est punctum circa quod areae descriptae sunt temporibus proportionales. Unde constat nee punctum T nee aliud quodlibet probari posse centrum virium nisi prius ol)servetur areas circa idem descriptas esse temporibus proportionales. Newtonus definivit legem vis centripetae tendentis ad punc- tum (juodvis in genere, et exinde non sequitur eum adstruere vim centripetam tendere ad omnia puncta, e contra tota vis demonstrationis propositionis 1"'^° Lib I de aecjuabilitate arearum pendet ex hoc (juod vis centripeta dirigatur ad unicum punctum id(|ue immobile. Nam si dirigerentur ad punctum mobile, vel ad duo aut plura puncta propositio esset falsa. Et si vis centripeta tenderet ad duo puncta immobilia, turn triangulum confectum lineis jungentibus puncta ilia duo et centrum corporis moventis describeret solida proportionalia tempoi'ibus, ut paucis abliinc annis invenit D. IVIacliin. Lex autem pro pluribus centris (juam duobus nondum est I'eperta: aequalitas arearum ad unicum centrum pertinet. LKjuis pleras(|ue Newtoni assertiones esse geometrice veras, & physice i'alsas; banc distinctionem i'ateor me non capcrc. Nam secundum me assertio geometrice vera est propositio demonstrata; haec erit semper et ubi(|ue vera, nee falsa physice aut metaphysice, aut alio quovis modo. Fieri quidem CORRESPONDENCE WITH CASTEL 157 })ote.st propo^iitionem geometricam in rerimi natura locum noii habere propter ali(|uam suppositionein quae in natura non est, sed inde non seqnitur propositioneni esse falsani. Exempli <jjratia si nulla existat linea absolute recta in rerum natura, (um nullum cxstabit triangulum cujus tres anguli aequantur duobus rectis ; attamen est propositio vera non solum geo- metrice sed et in omnibus scientiis, quod tres anguli trianguli aequantur duobus angulis rectis modo latera ejus sint lineae rectae. Si tantum velis, non sequi conclusiones geometrice inventas existere nisi per cxperimenta vel observationes con- stiterit hypotheses quibus innituntur haec conclusiones existere, inficias non ibo. Si habes opusculum apud te quo physica Newtoni tota convelletur, oro te meo et omnium nostratum nomine ut eundem illico mandes praelo, neve patiare Newtoni reverentiam te cohibere a propaganda veritate; cujus amor apud nos ante- cellit reverentiam qua colimus mortalium quemvis. In conclusione dicis P"° sine ulla vi centripeta et sine ullo centre aequabiles esse tamen areas circa punctum E. In cujus contrarium aio demonstrationem Newtoni in eo fundari, quod sit vis centripeta continue agens, et quod vis ilia senq^er tendat ad unicum immobile centrum. Secundo dicis intinita esse puncta circa quae haec vigeat inaequalitas ; liujus autem impossibilitas geometrice demonstrari potest, de quo itaquo non est mihi disputandum. Adeoque post omnia quae ad me scripsisti, non percipio propositionem fundamentalem New- tonianae systematis mere ; ignoscas interim oro si tibi assentire nequeo, et obsecro ut tu legas hanc epistolam eodem animo quo ego eandem tcripseram. Quod supcrest valeas illustrissime Vir, meque tibi devinctissinmm et obhcquentissimum credas Jacob. Stuiling Londini Julii 1733 S.V. VI CAMPAILLA AND STIRLING (1) CamiKdlla to Stirling, 1738 Clarissiine, k Doctissime Domine QuAM priinum ad me successive pervenerunt quaedam Opera Insignis Scientiaruin Antistitis, & in Mathesi loiige praestantissimi Aequitis Angli Isaaci Newton votis annuente candido Amico nullani pati moram tanti Viri apud Vos illustre nonien, quiii oeius ea perlustrarem fecit, lit ut cximia nie tenuerit jucunditas, dum perlegerem mathematica Phjdoso- phiae Principia, nee minus dein Opticae libros, in nonnuUas incidi du])itationes, quas calamo inermi in binos includere Dialogos, lubuit. Praelo evulgare formidavi, neve mihi petu- lantis notam inureret, quam longe patet, Sapientum Respublica & indignationem apud Vestrates incurrerem ; (juod auderem censoria virga phylosophicam tangere hypothesim, (juam litera- rius Orbis eximio prosequitur lionore, raagnaq: reverentia colit. Tandem timorem ex animo prorsus excussit admodum Reverendus e Societate Jesu Pater Melchior Spedaleri, qui per Epistolam significavit, te mira, qua ornaris ingenuitate, ac candore ad Patreni Castel, hisce, quae subdo verbis scripsisse, ([uibus petisses, ut difficultates, quas adversus Newton haberet, typis statim mandaret, sicq: talia fando, eum adhortatus fuisti : ' Oro to, meo & omnium Nostrum nomine, ut illud praelo statim mandes: neve patiare reverentiam Newtoni plus apud te valere, (|uam amor Veritatis : nam certo apud Nos plus valet amor veritatis, quam reverentia, qua columus Mortalium quemvis'. Revocato igitur animo ab tui consilii heroica sinceritate, qui inter caeteros, quibus decoratur Societas Regia Londinensis Mathematicos & Phylosoplios, emicas cele- LETTER FROM CAMPAILLA 159 benimus, constitui nedum publice juris faceie, vcruin inodo Opuseuluin liot-ce qualecuuui: nieuin ad to traiisinittcrc. Unuin ab in<;eiiita Humanitatc tna euixo depraecor, Vir Clarissinie ne deiboncris ISapicntiae tuac dubia me cdocere; ab to uno enim solidain accipcre !-ententiani potcro ccrte ; eruiitq: inihi & dogmata, & oracula. Oalleo prorsus, ut rem porgratam, diu: praestes exoptatam hand valere famulatus mei offieia ; at reeorderis, oportet, (pios siiblimlori Sapientia ditavit natura, quaecumqiic agenda suscipinnt, virtute propria peragere, quae sibimet sola praemia dat. Vale interim, faelieissime vive, & dum to docentem habere obsecro, tuo noniini in omne aevum suscipe Motueae die sexta Mensis Maii 1738 Addictissimum & Obsequentissimum TlIOMAM CaMI'AILLA VII BRADLEY AND STIRLING (1) Stirling to Bradley, 1733 ' Tower-street, London, Nov. 24, 1733. Dear Sir, I was very sorry that I did not see you when last in town, because I wanted very much to have conversed about the experiment made in Jamaica, which I hear you have considered, as indeed I have also done. If the pendulum went slower there than here by 2' 16" in a sidereal day, and only 9" or 10" are to be allowed for the lengthening of it by heat, as Mr. Graham tells me, thence it would follow that the earth's diameters are as 189 to 190, or thereabouts, in which case the force of gravity at the equinoctial would be to the centrifugal force as 237^ is to unity; which is impossible, unless the diameter of the earth were above 9000 miles, and that differs so much from the measures of Norwood, Picart, and Cassini, that it cannot be admitted, nor consequently the experiment from whence it is deduced : and besides, I can prove from undoubted observations in astronomy, that Cas- sini's measure is very near the truth, for the diameter of the earth can be found surer by them than by any actual mensuration. If 29" could be allowed for the len;i-thenin<;: of tlie pendulum l)y heat, tliis experiment made at Jamaica would agree with other things, but Mr. Graham says that he cannot allow that by any means. I am very far from think- ing that the experiment was not exactly made, and indeed a greater absurdity would follow from llicher's experiment made in the island of Cayenna, which is the only one that can be depended on, which is mentioned in sir Isaac's Principia. ' Pp. 398-400 oi'Miscell. Worh>> S; Corrofp. of James Bradley. CORRESPONDENCE WITH BRADLEY 161 Althouy,h 1 have treated oi" the problem of the fi<;'ure of the earth in a manner which is new, yet I am still obliged to suppose the figuire of it to be an exact spheroid, and althoui^h I be sensible that this supposition is not sufficient to determine the number of vibrations to 8'' or 9" in a day, yet I know that the error cannot be so great as the Jamaica experiment makes it. If Mr. Graham be certain that not above lO'' can be allowed for the heat, it is as certain either that the mountains have a sensible effect on the pendulum, or some other thing, which will render the experiment entirely precarious. I find that sir Isaac in his 3d edit. Princip. mentions three observations of Dr. Pound, which )nake Jupiter's diameter about S?''; I want to know if that be the greatest diameter of Jupiter ; because if it be, then the lesser would be about 34'^ which would make too great an odds in the thing for which I want it. And I should be glad to know if you can help me to any observation which ascertains the moon's middle distance from the earth, which I could depend more on than the common ones ; if you could inform me of these things, I should be able quickly to make an end of what I shall say about the figure of the earth, which I would the more willingly do, because not only Mairan, but also Hugens, Herman, and Maupertuy, have all of them entirely mistaken the matter. I heartily wish j'ou all happiness, and the sooner I hear, the more you will oblige. Sir, your most humble servant, J. SlIRLINti. (2) Bradley to Stirling, 1733 To Mr James Stirling at the Academy in Tower Street London Dear Sir When I was last in London an unexpected accident obliged me to return hither sooner than I intended ; and hindred me from waiting on you, as I proposed to have done ; having been informed that you were then examining into the Dispute 2447 M 162 STIRLING'S SCIENTIFIC CORRESPONDENCE conccrniug the Figure of the Eartli. Not that I had much more to tell you, than what is contain'd in the Account of the Jamaica Experiment, which I left with M' Graham ; wherein I have stated the Facts as well as I could, and made such allowance for the lengthening of the Pendulum by Heat as former Observations and Experiments would warrant. The Result of all seem'd to be that tlie Clock went l'-58" p Diem slower in Jamaica than at London. I allowed only 8 2" on account of the different degrees of Heat, having no Authority from former experience to mal<e any greater Abate- ment ; so that I apprehend this Retardation of the Clock (so much greater than what is derived by a Computation founded on the Principles of Gravity and an uniform Density in ye several parts of the Earth) must be rather ascribed to an inequality in the Density of the parts of ye Earth near which the Clock is fix'd, than to the greater Heat. For the greatest part of the force of Gravity upon any particular Body arising from the parts of the Earth that are near it (the Action of ye remote parts being but small) does it not thence seem likely that a Body placed near a great Quantity of rarer Matter as Water &c: will not be attracted with so much Force as if it were in the midst of a large quantity of Denser Matter, as in a great Tract of Land (S:c ? and may it not thence follow that Clocks (tho' in the same Latitude) may yet not go alike, when placed on y*" Continent and on Islands or on larger and smaller Islands ? or may not the Mountains (as you observe) according as they contain Matter more or less Dense, contribute something towards such Inequalities. These considerations do at least suggest the necessity of a great variety of exact experiments made in difiercJit Places, situated in the same, as well as in different Latitudes, and I have (for this reason) proposed in the fore-mentioned Account, to have the Experiment repeated in several Places, in order to discover whether the Density of Different Regions be uniform or not ; for till that Point is settled, we may be at a loss for the true cause of this Difference between the Theory S: Experiment. As to the Diameters of Jupiter, 1 find from the Mean of several Observations which I made with the R. Society's Glass of 123 feet focus, that the greater Diameter is to the Lesser (when both were measured with a Micrometer) as 27 to 25. CORRESPONDENCE WITH BRADLEY 163 the greatest Diaiiietei- (at ll." mean Distance Ironi y'' ICarth or Sun) being just 39". This is the Case when ye Diameter was actually measured with the Micrometer; but by other observations of the Time of the Passage of some of the Satellites over ll^ Disk, compared with their greatest apparent Elonga- tions taken with a Micrometer, the Diameter of 1/ comes out only 37'' or 38", the ditterencc arising (as I conceive) from y° Dilatation of bight i^c. Having never made any Observations myself particularly with a view to determine the Moons mean Distance I can give you no information relating to that Point, but believe M"" Machin has examined that matter and lix'd it with all the accuracy that the best Observations we have, would enable him to do it. You would have had my Answer sooner, had I not been engaged in a Course iVc upon y'' conclusion of which I have taken the first opportunity of assuring you that I am with great Respect S' Your )nost obedient Oxford \ humble Serv* Dec. 2^^^ Ja: Bradley, 1733 M 2 VIII KLINGENSTIERNA AND STIRLING (1) Klingenstierna to StirUiifj, 1738 Viro Clarissimo, Doctissi moque Domino Jacobo Stirlingio Londinium at y- Academy in little Tower Street. Clariss, Viro Jacobo Stirlingio Sam. Klingenstierna S. p. d. Daplici nomine indulgentia Tua maximopere me egere sentio uno, quod multis singularis cujusdam benevolentiae documentis a te aftectus per tantum temporis spatium silucrim : altero quod nunc tandem silentium rumpens non dubitaverim negotiorum nonnullorum demandatione tibi esse molestus. Sed quemadmodum Te persuasissimum esse velim, me ofHcia it studia in me Tua, quae dum Londini agerem, multis modis expertus sum, prolixiori animi afi'ectu (|uam verborum apparatu agnoscere, seniper(j[ue agniturum esse: Ita spero ctiam te non aegre laturum, (juod Tibi amicorum optimo harumque rerum intelligentissimo ejusmodi negotia demandem, (juac ad comunium studiorum ([ualecunc^ue incrementum aliquid forte conferre poterunt. Constitui nimirum apparatum Instru- mentoruin Physicae Experimentali inservientium quam potcro perfectissimum niihi comparare. Eumque in fineni instrumenta (juae apud nos per })eritiam artificum fabricari possunt, confici LETTER FROM KLINGENSTIERNA 165 cuiavi. Ceteruni (jumii iiistniineiita optica millibi terrarum meliora quaiii Londini conficiantur, te etiaiu atque etiaiu oro, lit optima eoruni, quae se(|uens designatio exliibet, pro lue eliuas, A' Domino Claesson (cui curam numoruin pro iis solven- dorum, cV- transmittendorum Holmiam instrumentorum comisi) tradi facias. Certissimiis ero me bona habiturum instrumenta, si tu, harum rerum intellioentissimus Judex ea elet^eris & approbaveris. Si aliqua fuerint, quae apud artifices statim haberi non poterunt, ea mihi primum transmittas quae haberi nou poterunt, ea mihi primum transmittas quae haberi possunt, reliqua etiam missurus, quam primum parata fuerint. Optarim enim, ut ante hyemem, quam potero pluriina habeam. Si aliqua ratione heic locorum utilis tibi esse potero senties gratam animi vohmtatem mihi nunquam defuturam. Designatio Instrum. Vitra ad Tubum Astron. 16 pedd. circiter. Vitra ad Tub. Astron. 8 ped. Prismata et Lentes ad Newt. Theoriam Colorum demon - strandam. Laterna Magica cum figuris necessariis. Lens pro Camera ob&cura 4 ped. Specula Conica iVr Cylindrica cum picturis deformibus. Plana vitrea inter quae aqua ascendit in figura hyperbolica. Oculus artificialis. Tubus vitreus amplus pro electricitate vitri monstrar.da. Microscopium duplex cum apparatu necessario. Instrumenta pro Legibus Refractionis t*^ Reflexionis dete- gendis. Duo vitra concava pro Myopibus foe. unius pedis, Diaboli Cartesiani. Praeterea etiam libros nonnullos novos apud vos noviter editos libenter desideraverim, ut D°' Smith Systeme of Opticks: D"' MacLaurin Systema Algebrae, & si qui alii recens editi fuerint in Mathematicis, novi quid continentes, quales credo in Anglia, ingeniorum feracissima non deesse. ante alios aveo scire, utrum D°' Machin Theoria Gravitationis lucem viderit, vel quando videbit & quomodo valeant insignes viri IGG STIRLING'S SCIENTIFIC CORRESPONDENCE fautoresque inei lionoratissiivii D"' HallLyus, Moivreus, Machin, (jiiibus meis verbis salutem plurinuun iinpertias. Vale iiiteriia ife fave Tui Studiosissinio Holmiae d 19 S. Klingenstierna Septembris 1738. Problems of Klinoenstierna (1733?) Prohlema Sint in A & a duo ignes, quorum vires cale- faeiendi in distantiis aequalibus sint in data ratione AF ^i\iif, & creseentibus distantiis decrescant in ratione quadratorum distantiarum. Quaeritur per quam viam ab i<;nibus illis reeedere debeat viator in loco aliquo dato & eonstitutus, ut minimum sentiat calorem. Solutio Sit BD particula quam minima viae, qua viator a puncto quocunque B reeedere debet, ut ab ignibus A et a minimum calorem sentiat. Centro B intervallo BD describatur eircumferentia circuli DK, (S: erit intensitas caloris in D minor intensitate, ejus in quovis alio circumferentiae DK puncto. Quare si in eircumferentia ilia sumatur punctum (/ puncto D proximum, calor in d per naturam minimi aequalis eenseri potest calori in D. Sed calor in /) per h^^poth. est -rjr^ H j^^ AJJ Q/U . , . J AF of ^ AF at AF af & calor in d, -j— + —-, Ergo . y^ + -y- = -—■, + -^, Ad" ad^ ^ AD" aD- Ad- ad- „ , , AF AF af of & transponendo ^, - ^^, = ^^ - ^^, • Centris A k a intervallis AD k ad describantur arcus Dj) & dP, rectis Ad & aD occurrentes in ^) & 7\ k per principia methodi infinitesimalis erit -rr.., :r-r. = ..-,..> & ^ AD- Ad^ AD^ 1 1 2DP . . AF AF af af ,., — ,v, = ~i^ > a( ooque ae(|uatio-7-;— — -r-r = ' ^ — — — ad' uD' uD- ^ ^ AD- Ad- («P aW ^ ^ . , 2dp.AF 2DP.af „ ,. ., , mut'itur HI haiic, — ~ri\^r— = rrr^ ■> k dividendo per 2, ac AD' aD' ^ , ,, ,, ., 1 .„ n dp.AF DP.af pro AJ) aD, scribendo AB aB, ' „ = — . -•' . ^ AB' aB-" LETTER FROM KLINGENSTIERNA 1G7 Centris A & a intervallis AB, uB describantur arcus BE & Be rectis AD iSc aD occuiTentes in IiJ & e, & erit trian^uluin BBE simile triano-. Bdj), triangulum DBa simile triany;. dDP. Quare DB : Dd = BE: dp, & DB : Dd = Be : DP, adeo- que ex aequo BE -.dp = Be: DP Si itaque in aequatione ' p, — — ^~ pro d/p & DP substituantur earum propor- .. , DX.X o 1 1 . BE.AF Be. of tionales BE & Be, habetur — -r-r, — = — -^ • AB' aB-^ Centris A & a intervallis AF & af describantur eircum- ferentiae FQ & fq, rectis AB, AD, atque aB, aD occurrentes in /, L, & i, I, eritque ob similitudinem triangulorum ABE, n F' AW AIL, AB:BE=AI (id est AF): IL, unde * AB = IL. 168 STIRLING'S SCIENTIFIC CORRESPONDENCE Similiter ob similitudinein trianguloruin uBe, ail, erit (iB : Be = ai (id est af) : il, unde -rr = il. a B ^ .. . BE.AF Be.af BE.AF , Be.af Ergo SI in aequatione ^^,^ = -^^ pro ^^ ^v -^ IL il substituantur IL ^ il, transit ilia in liane : -t-tto = ^w * AB^ (iB- Ad reetara Aa demittantur normales LH, IG, BO, ig, Ih, ipsique Aa parallelae IN, in, rectis LH, Ih occurrentes in iY, n. Propter similitudinem triangulorum ABG, AIG, LIN, est AB:BC = AI (id est AF) : IG, & AB:BG= . . LI.IN; quare terminis ordinatim in se ductis AB^:BC^ = AFx LI: IGx IN, unde ^,= i^ti^, ' AB- AF.BC^ Similiter propter similitudinem triangulorum aBG, aig, tin, est aB : BG = ai (id est af) : ig aB : BG = . . li : in ; quare terminis ordinatim in se ductis aB^ : BG^ = . . af. li:ig. in ; unde -^, = ^^.'l^\.^ ah- af.BG^ ^ , . , , IL il IG.IN ig.in Sed mventum erat ^^ = ^,, ergo ^^^^ = -j-^^. & multiplicando per BG'-, ^. — = ^ • Est vero IG . IN elementum circuli IGHL, i\: ig.in elementum circuli igJil IGHL ighl , AF IGHL quare — ^^" = -^ ' adeoque — r; = . , , • ^ AF af ^ af ^gkl Sit >S' locus datus unde prodit viator. Jungantur AS, aS circumfercntiis FQ,fq occurrentes in II, r i^ demittantur RT, rt perpendiculares ad Aa. Et cum per jam dumonstrata, elementa IGHL, ighl ubique sint in data ratione AF ad af, erit etiam componendo, Summa IGHL, id est spatium RTHL, ad suiliam omnium ighl, id est spatium rthl, in eadem data ratione AF ad Af, unde scqucns prodit Gonstrudio. LETTER FROM KLINGENSTIERNA 169 Centris A S: a descriptis circulis FQ,fq, quorum radii AF, af sint proportionalos virilms calefacicndi io-nium A iS: a, jung-antur AS iV aS, circulis illis occurrentes in R iS: r, & demittantur Fig. 34. Rt, rt, normales ad Aa Rectis LH, Ih, itidem normalibus ad Aa, abscindantur Sj)atia TRLH, trih, quae sint in ratione AF ad af. Jungantur i^' producantur AL & al, donee conveniant in D, A' erit punctum D in curva quaesita 8D. Prohlema. In venire curvas AGBG i^ AHBl, quarum talis est ad se invicem relatio, ut curva prior AGBC rotata circa polum fixum A semper secetur ab altera AHBl in punctis summis 5, h, iV ut segmenta AGS A, AHBA semper sint in data ratione m ad n. Solutio. Rotetur curva AGBG circa punctum fixum A, donee perveniat in situm proxiuium AFDG, in quo situ secetur a curva AHBl in b. Centro A intervallo AB dcscribatur arcus BD curvae occurrens in D, ^ jungantur AD, Ah, quarum haec occurrat arcui BD in E. Et quia AGBA : AHBA - m : n, & AFhA : AHhA = rii : 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 : n. „ o^DE Dicatur AD, x\ Eb, dx; DE, dy; d' erit EB =^ iV BD = DE 4- = dy. m m m 170 STIRLING'S SCIENTIFIC CORRESPONDENCE Et quoniani per hyp. tautens ciirvae JUli(J in Ji paiallela est tan<;enti ejusdem in b, erit angulus rotationis BAD aequalis angulo (luem duae rectae ad curvaiu nonnales in punctis B & /; constituunt in centre circuli osculatoris. Ergo AD.DB = radius curvedinis in J) : ad elementuni curvae Dh, id est .,. , ,,, , , 97i + 9«, , xda'^dx -, (dicto Db — ds)x : (hi = -. — , — ^ ^ — ^-r '• ds, m dxdyds — xdydds dy . X ds'^ dx ds dx adcoque xds = ^ — j — -. vel 1 = dxdyds — xdydds dsdx — xdds' 11 nde ds dx — xdds =■ ds dx, m , , n -, -, . . n dx dds sen —xdds = — dsdx, nine = 9— > m ni x ds sumtisque lorarithmis — I- = I -r i iV' perficiendo quod restat ^ ^ m a ds ^ ^ reductionis : 11 =. = dy 2 n 2 n Centro A, intervallo a describatur circulus, eujus elemontum rectis AD, Ab comprehensum dicatur dz, eritque x -.dy = a : dt, xdz unde dy = ' — » & hoe valore substituto in aequatione modo inventa x'^'dx = dy, 1 2n in ^jd^-X^ transformatur ilia in banc m X " dx xdz / 2w 2m \a"' -X'" a seu ax"' dx 1 211 2 n = dz. LETTER FROM KLINGENSTIERNA 171 Ponatur x — a ~V m \- aequatio transibit in lianc; adv n y ^ck ua — vv quaosequentem snppeditat Pi'oblematis Constructioncm. Cen- tro A intervallo quovis AB describatur circuint'erentia circuli, in qua hinc inde a puncto quovis dato B sumantur arcus BC, BD Fig. 35. in ratione n ad m. Jungantur AG, AD & a puncto C demittatur CE normalis ad radium AB. In AC i^ AD sumantur AF CE " & AG aequales AB . -j^ 6^ erit punctum F in curva fixa AHBT, d' punctum G in curva rotatili AGBC. Coroll. Si fuerit m ad n ut numerus ad numeruin, utraque curvarum est A]oebraica,sive minus, earum constructio dependet a multisectione anguli tV rationis, seu quod idem est quadratura circuli & hj'perbolae. JX MACHIN AND STIRLING (1) MacMn to Stirling {1733?} To M"" Stirlino^ at the Academy in little Tower Street Dear Sir I intend to give you some short notes upon M'" Bernoulli's Letter, w*'' 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, iS: do intend withal to oblige him w*^ the solution of a Problem w"'' I now percieve he had proposed to himself but quitted rather than be at the pains to go through w*** 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 i^' one of the middle distances. And where it is that y*^ point \yes. 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*^'' may furnish me w^'' the oi:)portunit3' 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 iV especially the Problems since lie 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"". Your most faithful Friend & very humble Serv* Thursday morning J. Macuin. (2) Macliin to Stirling^ 1738 „ ^. Gresham College June 22. 1738 Dear Sn- ^ 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 ni}'' 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, appears, 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° world. There may I can assure you be as great a solitude from acquain- tance k conversation in a Town as in a Desert. But of this sufficient. Mons'^ Maupertuis has sent you a present of his book which I have deliverd to M'' Watts for you. It contains a complete account of the measurement in the North. M"" 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"^"^ I intended to have sent to you, but this book has rendered it unnecessary. 174 STIRLING'S SCIENTIFIC CORRESPONDENCE Tliurc 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" face of their Instrument turned contraryways. So that M'' Maupertuis was pdt to the necessity of procuring from England a certificate concerning the con- struction of M' Graham's Instrument, to show that it did not need that sort of verification. You will see that this measurement in y" North, if it l)e compared with y* in France, will serve to prove that y*^ figure is much more oblate than according to y*^ 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' De Lisle has published a Memoir read in tlie Academy at Petersburg w*^^ contains y'' scheme of a Grand Project of the Czarina for making a compleat Mapp of her whole Empire, and in w*^'' 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® meridian and many times more in the parallels. Your Proposition concerning y® 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'' 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 possil)le, but not without some investigation at least : unless you have hit upon a Demonstration w*'^ 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"" Macklaurin in a Letter to me dated in febry last, (and w*^^ 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'" figure of y ' Earth, and that CORRESPONDENCE WITH MACHIN 175 S' Is. had supposed but not demonstrated it to be a Splieioid, proceeds on in the following words, ' M' Stirling; if I reniendjer right told nie in April that none of those who have considered this subject have ghewed that it is accurately of that figuie. I hit upon a demonstration of this since he spoke to me w'"' 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 thonght everybody had known. But I shall take this opportunity to advise him to connnunicate his demonstration to you. And if he has found out a simple demonstration for it, I think it ought to l)e 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'^ Invention of M'" Euler's Series were I in your case I would not troul)le myself about it, but let it take its own course, if anything should arise your Letter to me w'"^ I shall keep will be a sufficient acquittal of yourself. M'' 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'' Watts goes out of Town, As to y'^ moon's Distance I have now materials to fix y*^ moon's Parallax, and chiefly by means of an Observation of the last Solar Eclipse at Edinburgh by M'" Macklaurin, and will take care as soon as I can make y° 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 Machin. X CLAIRAUT AND STIKLING (1) Clairmit to StirUng, 1738 Monsieur En cus qu'un Menioire sur la Figure de la Terre que j'envoyai de la Laponie a la Societe Royale, soit parv^enu juscjua vous et que vous I'ayes daigne lire, vous y aures reconnu plusieurs Theoremes dont vous avit^s donne auparavant les enoncds, parmi les belles decouvertes dont est rempli un morceau que vous av(^s insere dans les transact. Philosoph. de I'ann^e 1735 ou 1736. Vous aur^s ^t^ peut-etre etonne que traitant la meme matiere que vous je ne vous aye point eit^. Mais je vous supplie d'etre persuade que cela vient de ce que je ne connoissois point alors votre Memoire, et que si je I'eusse lu je me serois fait autant d'honneur de le eiter que j'ai ressenti de plaisir lorsque j'ai appris que je m't^tois rencontre^ avec vous. Depuis le terns o\x j'ai donne cette Piece j'ai pouss<^ mes recherches plus loin sur la nieme matiere, et j'envoye actuelle- ment mes nouvelles decouvertes a la Society Royale. Apr^s vous avoir fait ce recit Monsieur et vous avoir prid d'cxcuscr la liberti^ que j'ai pris de vous ecrire sans avoir I'lionneur d'etre connu de vous, oserois je vous demander une grace, c'est de vouloir bien jetter les ycux sur nion second Memoire que M"" Mortimer vous remcttra si vous le dai^'nes lire. Ce n'est pas seulement I'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 Gcometre anglois appelld M. Robbens que vous avies depuis pen travaille sur la memo matiere. LETTER FROM CLAIRAUT 177 Jc souluiitcrois done cxtroincment de scavoir si j'ui ete asses heureux encore pour m'etro rencontre avec voiis. Si au con- traire jc m'etois tronipe je vous serois infinimeiit obligd de me le dire i'rancliement afin ({ue je men corrigiasse. Qnoi (ju'il en soit si vous daign(^s me donner quelques momens, vous aurds bientut vii de quoy il est question et si mon memoire m'attire une reponse de vous je serai cliarmd de I'avoir fait parce qu'il y a deja longtems que je souhaite d'etre en liaison avec vous. Qufkpi'envie que j'en aye ne croyes pourtant pas Monsieur que je soye asses indiscret pour vous importuner sou vent par des lettres inutiles pleines de simples complimens. M'' Mortimer pourra vous dire ({uelle est ma conduite a son egard, J'en oserai de memo avec vous si vous me le permettes. En attendant j 'ai I'lionneur d'etre avec estime et respect Monsieur Votre tres humble et tres a Paris le 2 Octobre 1738 obeissant Serviteur Claikaut. P,S. En cas que vous veuillt^s me faire reponse il faudra avoir la bont(^ de remettre votre lettre a M. Mortimer. Si vous n'aimes a ecrire en francois, je decliifFre asses d'anglois pour entendre une lettre et quand ma science en cette langue ne suffiroit pas, j'aurois facilement du secours. XI EULEK AND STIRLING (1) Stirling to Eider, 1738^ Celeberrimo Doctissimocjue Viro Lconhardo Eiiler S.P.D Jacobus Stirling mihi Tantuin teinporis elapsuin est ex quo dignatus es (ad me) scribere, ut jam reseribere vix ausiin nisi tua humanitate fretiis. Per hosce duos annos plurimis negotiis implieitus sum, quae occasionem mihi dederunt frequenter eundi in Scotiam et dein Londinum redeundi. Et haec in causa fuerunt turn quod epistola tua sero ad manus meas pervenit, turn quod in liunc 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 sinnil (gratias) agendi gratias dudum debitas propter literas eximiis inventis refertas. Gratissimum mihi fuit Theorema tuum pro suinmandis Seriebus per aream Curvae et differcntias sive Huxiones Statim Terminorum quippe generale et praxi expcditum. (lllius) percepi item extendi ad phirima serierum genera, et (|Uod celerrime praecipuum et A plerunujue (celeriter) approximat. Forte non observasti theorema meum pro summandis Logarithmis Tui nihil aliu<l esse quam casum particularem tui Tlieorematis a * This is only StirUng's rough draft with all his corrections. Erasures are indicated by bracket!?. CORRESPONDENCE WITH EULER 179 eo generalis ; (quod ingenue fateor). Sed ct A gratius mild fuit quod (tuum) hunc invcntuni, (quoniam) de eodem (ego) quoque ego olim cogitaveram ; sed ultra prinium terminum non proeessi, approximav pro libitu et per euni solum (perveni satis expedit(i) ad valores Serierum satis expedite A scilicet per repetionem calculi, ut in resolutione aequa- tionum affectaruni ; cujusque specimen dedi (plurimis abliinc annis) in philosophicis nostris transactionibus : Quae liabes de inveniendis Logarithmis per Seriem Harmo- saltem nicam (non percipio, propter novi) obscura mihi a videntur, notationem quoniam a non recte intelligo (notationem.) Imprimis autem mihi placuit methodus tua summandi quasdam Series per potestates periferiae circuli, (quarura 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 ciedara hausisse facile (concedam) te idem (hausisse) ex novo fonte A ((et nuUus dubito to hactenus observasse, aut certe ex fundamento tuo facile percipies, alias series tuis tamen affines summari posse per potestates periferiae quarum indices sunt numeri impares. Verbi gratia, denotante ]) periferia, l^j = 1 — -3- + -I — f + 1 — &c ut vulgo notum 13,111 1 n 32^ 3- 5' 7^ 9-* 5 , 1 1 1 1 P 1536 3^ 5^ 7'' 95 c^-C.)) continentur in Series tuae (comprehenduntur sul^) forma generali 1 1 1 1 1 P ubi n est numerus par) eadem (tamen ad formulam scquen- n2 180 STIRLING'S SCIENTIFIC CORRESPONDENCE tein) nullo iiegotio reducitur, (scilicet) re<lucitur ad ronmdain sequcntem, 1111 1 o 1+ ■Sn+ 5"+ r'+ 9^+ 11" + ^^'' (iibi teniiiiii altcnii desiint, ct oniiies sub luic iorina compre- ot Ikuic sunimare liciisas suimiiarc) A doces A per potcstatem periferiae cujus index est n modo sit Ceterum si ((piando n est) imuicius par, (Si jam iiiutentur) signa tenninoruni alternoruin inutcntur ut cvadat Scries 1111 1 , 3" 5" 7" 9" 11" Haec inquain seii)})er suimiiari potest (at(|Ui! liaec Series, qiiaudo li est luunerus iiiipar suiliari potest) per dignitatem modo sit Humerus impar periferi (circiili) ciijiis index est h. (verbi gratia) uticpie si sit n = 1, (erit) ^p = 1 — -3 + i — t + 9 — tt + ^^ ut vulgo notum ' 32 ^ 3^ 5-^ 7-^ 9-^ ll'^ 5 . 11111., ' 153G ' 3 5^ 7^ 9' 11^^ &c. Et nuUus dul)ito te liactenus idem ubserxasse, aiit saltem facile observatiir ex fundamcnto tuo (jnod lil)enter videbo, quando (animus erit tibi idem impertire) ita tibi vi^nm fuerit. monendus es Mathcsoos Hie autem (ae(}uum est ut te moneam) D. Maclaurin a pro- fessorem (Matliescos) Edinburgi, post alicjuot tenq)us (brcvi) jam editurum lil)rum de fluxionibus cujus paginas ali(|uot (liactenus) imprcssas (niecum) mecum connnunicavit in (juibus duo liabet Theorcmata pro summandis seriebus per differentias termi- norum, (juorum alterum ipsissimum est (juod tu dudum mild (ad me) misisti, (et cujus ego cum illico certiorem feci). Et etiam si illc libenter promiserat se idem testaturum in sua praefationc, judicio tamcn tuo submitto annon velles (edcre) edere tuam epistolam A in nostris philosophicis trausactionibus. CORRESPONDENCE WITH EULER 181 Et si vis (luaedam illustrare vul demonstrare, (ant plura ut lucem videat adjicere, egoaiit) et cito inihi rescribere, ciirabo (tuam epistolaiii viseram lucem diu) Mntecjiiaiii ejus liber prodierit. Quod si animus erit hac (data) oecasione eligi unus ex Sociis nostrae Societatis (Academiae) Regiae, idem reli(iuis gratum (esse non) procul quando viderint praeclara tua inventa dubio gratum erit (postquam inventa tua viderint Et) mihi vero semper gratissinuim ut amicitiam (mihi licet immerenti) continuare difjneris Edinburgi IG Aprilis 1738 (2) Elder to Stirlinfj, 1738 lllustrissimo atque Celeberrimo Viro Jacobo Stirling S. P. D. Leonhard Euler Quo majore desiderio litteras a To Vir Celeb, expectavi, eo majorc gaudio me responsio Tua liumanissima afiecit, 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 mcditationes tam benevole accipere Tuumque do iis judicium mecum communicare volueris. Epistolam autem meam a Te dignam censeri, quae Transactionil)Us Vestris inseratur, id summae Tuae tribuo humanitati, atque in hunc tinem nonnullas amplificationes et dilucidationes superaddere visum est, quas pro arbitrio vel adjungere vel omittere poteris. Hac autem in re quicquam laudis Celeb. D. Maclauriu 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, quani de Suumiatione Serierum per potestates peri- \S2 STIRLING'S SCIENTIFIC CORRESPONDENCE phcriac circiili coinposui in nostris Coininentariis, (jui (jiiotiuinis prodeunt, brevi lucein publicam aspicict. In Connnontariis autom nostris jam editis ali(|Uot extant aliae nietliodi nicae Series sununandi quaruui (juaedani niultuni liabent Siniilitu- dinis cum Tuis in ej^regio Tuo opere traditis, sed (|uia tum temporis Tuum methodum diffcrentialem nondum videram, ejus quo(jue mcntionem facere non potui, uti debuissem. Misi etiam jam ante eomplures annos ad Illustris, Praesidem Vestrum D. Sloane schediasma quodpiam, in (pio <^eneralem constructioncm liujus aequationis y = yyx -\- ax^" X dcdi, quae acquatio ante multum erat agitata, at paucissimis tantum casibus cxponentis m constructa. Haee io^itur Diss-er- tatio, si etiamnum praesto esset, simul tanquam specimen produci posset, coram Societate vestra, quando me pro mendn-o recipere esset dignatura, quem quidem honorem Tibi Uni Vir Celeber, deberem. Sed vereor ut Incl3^tae Societati expediat me Socium eligere, qui ad Academiam nostram tam arete sum aliigatus, ut meditationes meas qualescunque hie primum pro- ducerc tenear, Ut autem ad Theorema, quo summa cujusque Seriei ex ejus termino dicto generali inveniri potest, rev'ertar, perspicuum est formulam datam eo majorem esse allaturam utilitatem, quo ejus plures termini habeantur, summa autem difficile esse videtui-, eam quousque lubuerit, continuare. Equidem ad plures quam duodecim terminos non pertetigi, quorum ultimos non ita pridem demum inveni ; hacc 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 sunnna hujus progressionis, puta ^ + i)'f C'+etc... +X ^ „ , X dX Xclx+ - -^ + 1.2 1 .2.3.2dx d^X df'X - + 1.2.3.4.5. Gt/u,'^ 1.2.3.4.5.6.7. iSdx' Zd'X bd'X + 1 .2.3 ...9. lOc/x'' 1.2.3 ... 11 .Grfic'-' CORRESPONDENCE WITH EULER ISJJ 1.2.3... 13.210(/a;i^ "*" 1 . 2.3 ... \5 .2dx^ ' 3617tZ'5X 43867(^^^Y 1 .2.3 ... 17.30(/.t;'^ 1 . 2 . 3 .. . 19 . 42(/a;'^ 1222277cZ^^X 1.2.3 ... 21.110(/a;i-* etc. ubi fluxio dx constans est posita. Haec aiitein expressio parumper luutata etiam ad summam seriei a tenuino A" in infinituin u&(|ue invcnienJam acconimodari potest. Hujus vero forinae praeter insigneni facilitatcm, quam siippoditat ad summas proxime inveniendas, oxiinius est usiis in veris sunnnis serieruni algebraicarum investigandis, quarum quidem sumniae absolute exhiberi possunt, ut si quaeratur summa hujus progressionis potestatum erit X = x'\ [Xdx = ^x''\ ^ = 12a;^\ J 13 dx iP X d^'^X —r- =10.11.12.0;", et ita porro, donee , ,., dx^ ' dx^' una cum sequentibus Terminis = Hinc igitur resultabit summa quaesita = x^" .1-12 „ lla-^ 22.Z' 33.r^ 5x^ 691 « f- 4-.l''^ 1 + J 13 2 07 10 3 2730 quam summam nescio, an ea per ullam aliam methodum tarn expeditam inveniri queat. Potest autem hac ratione aeque commode definiri summa hujus progressionis l+22i + 3-i + 4'-^i + ...+0'2', quod per alias vias labor insuperabilis videtur. Sin autem seriei propositae termini alternativi signis + et — fuerint aft'ecti, tum theorema istud minus commode adhiberi posset, quia ante binos terminos in unum eolligi oporteret. Pro hoc igitur serierum genere aliud investigavi Theorema priori quidem fere simile, quod ita se habet. Si quaeratur summa hujus seriei A-B + C-D+...+X, 184 STIRLING'S SCIENTIFIC CORRESPONDENCE iibi A' t>it tcniiiniis ciijiis exjJOiiL'iis sen index est ./', luiLetque sigimin vel + vel — proiii ,'■ nuinerus erit vel inipar vel par. Dico auteiu liujus progressionis siiuimam esse ^ , /X dX cPX = Const. ± (p + + 2 1.2.2dx 1 .2.3.4 .2da"'' 3d'X \7,rX 1.2,3... G.2dx^ 1 .2.. 3 ... 8.2(^.^7 155d'>X 2073d^^X + 1 .2.3 ... 10.2(/.^'-' 1 .2.3 ... 12.2(/rt;i' 33227cZ^3X ^ 1.2.3 ... 14. 2(/./,'^ Constantem autem ex uno casu, quo summa est eognita, determinari oportet. At si series sunimanda eonnexa sit cum Geomctriea pro- gressione hoc modo An + Bii^ + Cn" + . . . + Xn"" turn minus congruo utrumquc praecedentium tlieorematum adhibeietur. Summa enim coiinnodius invcniotur ex liac expressionc ^./nX (xdX ^d'^X Const. + n- (^_^_ J - J ^,^_ 1^,^/ ,. + 1 :2{n^rfdx'' y(F'X Sd'X 1.2.3. (vi - l)*(/,t^ '*' 1 .2.3.4(/t-l) ^-— i — etc. ) t-l)'(/.r'* / valores autem coefficientium a, /3, y, 8, etc sunt sc(|uentes oc = n ^ = n^ + n y — n' -^-^n- -\-n S ~ Qi* + 1 1 ,v- + 1 1 n~ + n € = lV'-\-2Qii' + GG/t'' + 2GH^ + li etc. cujus progressionis legem facile inspicies. En igitur tivia liujus generis TheorematM, (juac singula cortis easibus exiiiiiaiii liabe- bunt utilitatem ad sunmias serierum indacfandas. CORRESPONDENCE WITH EULER 185 Quod Jeiiule attinet ad suiiiiuatiout'S liujusmodi serierum, (juae contineutur in liac 1111 1 + ^. + ^. + ^7. + ^. + ^'tc. cxistente n nnmero pari eas duplici operatione sum consecutus, (juarum alteram uti recte conjectus Yir Celeb, dcduxi ex scrie 1+- h 1 + etc. altera vcro immediate mihi 3" 5" 7" illius summam praebuit. Priore modo utique summas etiam hujusmodi serierum I-^j+t ^i+^~ ^^^- existentc n numero impare detexi, invenique eas se habere, prorsus ac Tu indicas. Sunt autem summae tam pro paribus quam im- paribus exponentibus n sequentes ;? ^ 1 1 1 1 -^=1 1 \ etc. 4 3 5 7 \) p^ 1111, — = H H . + —. -\ — ^+ etc. ?/ 1111, — = 1 — — + -T, ; + —. — etc. 32 3' 5' 7- 9" ,/ 1111^ 5^/' 1111, — ^— = 1 r + —r ^+77^ — etc. 1536 3'^ 5^ 7' 9' «« 1111, 9^=^+3^+5«+r^+9^ + ^^^- 61// 1111, — = 1 ^H — ^ ?-l — ^ — etc. 194320 3^ 5' 7' 9^ 17p8 1 1 1 1 „ '-— r= 1 + — + ^- + — . H ; + tVC 161280 3^ 5-^ 7^ 9« etc. quae series omnes continentur in una hac generali : i+(-ir+(+ir+(-7r+(+if+etc. existente n numero integro. Si enim n est numerus par, turn 186 STIRLING'S SCIENTIFIC CORRESPONDENCE omnes terinini luibijltunt siyimm + ; sin auteni n sit iinpar, turn signa scsc altcrnatiiii insequcntur. Oinnes auteiii has suininas dcrivavi ex liae ae(|iiatione infinita ; ^ , s s^ s^ , 0=1 + + etc. I .a \ .2.3.(1 } .2. 3. 4. 5. a qua relatio inter arcum ,s ejusque siniim <i expriniitur in circulo cujus radius est 1. Quoniani igitur cideni sinui (i iimumerabiles areus s respondent, necesse est. Si s consideretur tan(juam radix istius aequationis, earn habituram esse infinites valores, eos(]ue oninos ex circuli indole cognitos. Sint ergo A, B, G, 1), etc. omnes illi arcns, (|uoruni idem est sinus a erit ex natura aequationum 1 + + etc. 1 . a 1 . 2 . 3 . a 1 . 2 . 3 . 4 . 5 . a = (>'1)('-b)C-5)*- Posita nunc ista fractionum serie ex omnibus illis arcubus formata —3 -7-, » -, » -^r etc. perspicuum est suillam banc f'rac- A B (J D ^ ^ tionum aequari coefficienti ipsius — s qui est = -; seu fore - = + T, + 7^, + 7; + etc. Simili modo summa factorii ex a A B U V binis fractionibus aequatur coefficienti ipsius a^ qui est = 0, unde erit 1/1 1 1 . \^ 1 / 1 1 1 . \ ' = 2 (Z + 5 + C + 'W - 2(^-^ + i^^ + t- + '^V' 1 ' 1 1,1,^ Porro summa factoru ex ternis fractionibus aecpialis esse 1 3 6(t debet coefficienti ipsius — s^, qui est = — r-' undo deducitur summa cultoru illarum fractionu, 1111,11 2. + 5^+(7^ + 2}3 + ^tc = ^--; CORRESPONDENCE WITH EULER 187 atque ita procedendo Miinniac reperientur oiuniuin serierum m hac <^enerali T7i + /jy, + 7*7^ + Jul + ^^^- compreliensaruin dummodo pro n siimatur numerus integer affirmativus. Si nunc pro sinu indelinito it ponatur sinus totus 1, illae ipsae oriuntur series quas Tecum cunununicavi. In istis autem summis notari meretur insignis afKnitas inter coefticientes numericos haruni suIITaru, atque terminos superioris progres- sionis, quani priniuui ad series quascun(|ue sumniantlas dedi, nempe liujus V 7 -Y (IX A ax + H etc. 1.2 1 .2.3.2(/a; Quo autem haec affinitas clarius pcrspiciatur, summas ipsas congruo modo expressas repraesentare visum est. 2M ., 1111^ I^:^2^^"=^+2^^+3^^+4^^+5^^+^^^- 1111. ^ ='+ 2^^+31 + ^^ + ^^ + ^t^- r , 1111. s , 1 1 1 1 ?^^-l+.7s+3-. + ^4^+p + etc. 1.2.3.'.:il.6 ^'" = ' + ^'" + ^" + ^ + o^> + ^^^• 2^691 ^, _ 1 1 J_ 1_ 1.2.3... 13.210^' " - 1 + ^. + gl^ + ^2 + 5!^ + etc. 21^35 ,, 1111^ -,l^* = 1 + ^4 + :7u + 7r4 + M4 + etc. 2-' .1 1 . .2.3. 2'\ 1 4. 5.6 1 .2.3 .4.5 2". .6. 3 7.G 1 . 2.3. 2^ .. 9 , 5 . 10 1 . 2 . 3 ... 1 5 . 2 2^^3017 ,, 1111, 1.2.3...17.30 ^^ =^+^'+i^+4l^' + ?^^ + ^^^- 2^". 43867 ,_ , 1 1 1 1 ^ /P''=^+^s+ ^+ in+ 77^ + etc. 1.2.3.... 19.42^ 2''\ 3222277 „„ , 1 1 1 1 ^ r7Yy^2>"= 1+^0+ 3.0 + ^.+ ^ + etc. 1.2.3. ... 2 etc 188 STIRLING'S SCIENTIFIC CORRESPONDENCE Hac scilicet convenientia aiiiiuadvcrsa mihi iiltcrius progredi licuit, quaiii si niethodo genuiiia inveniciidi coefficientes potestatu ipsiiis p, usus fuissein quippe qua labor niniis cvaderet operosus. Quamobrem non dul)ito, quin nexu hoc mirabili penitius cognito (mihi euim adluic sola constat obser- vatione) praeclara adjiimenta ad Analyseos proinotioneni sose sint proditura. Tu forte Vir Celeb, non difticultcr ncxum hunc ex ipsa rei natura derivabis. Dum haec scribo, accipio a Cel. Nicolao Bernoulli Prof. Juris Basiliensi et Membro Societatis Vestrae singularom deuionstrationem suniniae huius seriei \ ^ — o H — r, H — :, + etc. J 3- 5- 7^ (juam deducit ex suuiiua hujus notae 1— ^ + i — y+ etc. illam considerans tanquam hujus quadratum niinutum duplis factis binoru terminorum. Haec autem dupla facta seorsim con- templans multiferiam transforniat, tandenique ad seriem quandam regularem perducit, quam analytice ostendit pariter a Circuli quadratura pendere. Sed hac niethodo certe Viro Acutissimo non licuisset ad sunnuas altiorum potestatum pertingere. Eodem incommodo quoque laborat alia quaedam methodus mea, qua directe per solam analysin hujus seriei sunniiani 111,.. -. ,, XM- 1+ -r, A — r, + -S + etc. inveni, ex riua pariter nullam utili- tatem ad sequentes series suinmandas sum consecutus. Haec autem methodus ita se habet : Fluentem hujus fluxionis —z 5 qua arcus circuli cxprimitur cuius sinus est = ic V{l-a:x) ^ ^ •' existente sinu toto = 1, inultiplico per ipsam fluxionem ^-5 quo prodeat facti Hucns = ^ss, posito .s- pro arcu V{l—xx) illo cujus sinus est = x Si ergo post summationem peractam ponatur x = I, fiet s = -■> denotante p 'i'^ 1 ratifmem peri- pheriac ad diametrum ; ita ut hoc casu hal>eatur -^ • Fluens . . X . . autem ipsius — — , per seriem est ' V{\—xx)^ 1 , 1.3 . 1.3.5 „ = X + .» ■' H X' H X' + etc. 2.3 2.4.5 2.4.6.7 CORRESPONDENCE WITH EULER 189 Ducautur nunc sinmili termini in tluxionem -—- — — - efc " v(i — XX) suniantur tiuentes ita nt tiant — posito x — 0, turn vero = 1_ J{\-xx) = 1, ponatur x—\. Ita rcpcrietiir posito X — \. Siniili niodo erit - — ~ 'Jil-XX) 1 s'{\-xx) 3.3 1.3 r xJ'x 1 ''*'*1"'' 2. 4. 5 J ^{\~xx)~ 5.5 et ita porro, adoo ut tandem obtineatur ir 111, — = 1 + -", + - 7, + - ., + etc. 8 ^ 3^ 5- 7- Sed huic argumento jam ninnum sum innnoratus, ijuocirca Te ro<^o Vir Celeb, ut quae Ipse hac de re es meditatus, mecum benevole communicare veb's. Incidi alicpiando in banc expressionem notatu .satis (Hgnam : 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 nunieri primi naturaU ordine sese insequentes, denominatores vero sunt nuuieri pariter pares unitate distantes a numeratoribus. Hujus vero ex- pressionis valorem esse aream circuli cujus diameter est = 1, demonstrare possum. Quamobrem baec expressio aequalis erit huic Wallisianae 2.4.4.6.6.8.8. 10. 10 etc 3.3.5.5.7.7.9 . 9 .11 etc. Ut autem novi quiddam Tibi Vir (Jeleb. perscribam Tuoque acutissimo subjiciam judicio, communicabo quaedam proble- mata, quae inter Viros Celeberrimos Bernoullios et me al) alic{U0 tempore sunt versata. Proponebatur autem mihi inter alia problemata hoc, ut inter omnes curvas iisdem terminis con- tentas investigarem eam, in qua r"^s haberet valorem minimum, denotante s curvae arcum, et r radium curvaturae, quod problema ope consuetaru methodorum, quales Bernoulii, 100 STIRLING'S SCIENTIFIC CORRESPONDENCE Heniuiniius ct Taylorus Vcstcr dcdcrc, resolvi iioii potest, (jiiia in r fluxiones secundae ingrediuntur. Invcni autem jam ante incthodum universaleiii omnia huinsuiodi prohlomata solvendi, quae etiam ad fiuxioncs cujusqiie ordinis uxtunditur, cujus ope pro cnrva quaesita sequcntcm dedi acquationem c6"*./j + 6'"2/ = ("* + 1) >'"'« i" <i^^'^ •'■ *'t y coordinatas ortho- gonales hujus curvae denotant. Hinc autem sequitur casu, quo ';n = 1, cycloidem quacstioni satisfacere. Deinde etiam (^uaerebatur inter omnes tantum curvas cjusdem longitudinis, quae per duo data puncta duci possunt ea, in qua 7'"*s esset minimum ; hancque curvam deprehendi ista aequatione indicari a'"a; + />'"?/ + c'"s = (m+ 1) r'"b'. Praeterea quaerebantur etiam oscillationes sevi 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, ([uod aequalibus tem- poribus oscillationes suas absolvat ; hinc enim intelligitur (]Uot vibrationes data lamina dato tempore sit absolutui'a. Ego vero contra inter alia problema istud proposui, ut inveniantur super dato axe duae curvae algebraicae non rectificabiles, sed (juarum rectificatio a datae curvae quadratura pendeat, (|uac tamen arcuum eidem abscissae respondentium summam habeant ubique rectificabilem ; cujus problematis difKcillimi 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 Gt z ■ definiri debent, atque exprimat Vii illam quadraturam, a qua rectificatio utriusque curvae pcndere debet; sintque p et q quantitates (juaecunque algebraicae ex u et constantibus compositae. Quibus pro lubitu sumtis fiat V{\ +2^p) + V{l+qq) = r: V{1 ^-X)p) - V{\+ qq) = s, tum quaerantur scquentes valores P P P CORRESPONDENCE WITH EULER 191 item i> = -; E=^ cb F=^' A A b Ex liis quantitatibus porro forinentur istac V ^'''- n ^' f p ^ i = ^ U = - ■ et U = -r ' F D A Ex his deniqiic valoribus, (|ui omncs erunt algebraici sumta coininiini abscissa = — 3 j) fiat y=2^-R atque z = ^lAzi^ +Q; V V liacque ratione, cum p et q sint quantitatcs arbitrariae pro- blemati infinitis modis satisfieri poterit. Erunt enim ambae curvae algeljraicae, atque utriusque rectificatio pendcbit a fluente hujus fluxioiiis Fa. Summa vero amboru arcuum algebraice exprimi poterit. Est enim summa arcuum differentia vero eorum est = sx - OR -^EQ-FP+ f Fa . Detexi autem pro resolutione hujusmodi problematum pecu- liarem methodum, (juam Analysin infinitoru indeterminatam appellavi, atque jam maximam partem in singulari tractatu exposui. At tam longam epistolam scribendo vereor ne patientiam Tviam nimis fatigem : quamobrem rogo, ut pro- lixitati meae veniam des, eamque tribuas summae Tui existi- mationi, quam jamdudum concepi. Vale Vir Celebcrrime, meque uti coepisti amicitia Tua dignari perge. dabam Petropoli ad d. 27 Julii 1738. XII FOLKES AND STIRLING (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 t-orts. 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'" de Maupertuis the President and IVP' de Formey the Secretary. M"" Mitchell going your way I put it into his hands for you and congratulate you Sir upon this mark of the esteem of that Royal Academy upon their new establishment under their present President. Our old ffriend M"" IMontagu is well and we often talk of you together, and our old Master de Moivro 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. ]<"i'OLKks. Pr. R.S. member of the Royal Academies of Sciences of Paris and Berlin, and of the Society of M"^ Stirling Edinburgh NOTES UPON THE CORRESPONDENCE MACLAURIN (1698-1746), F.R.8. 1719 Colin Maclaiiiin was born at Kihnodan in Argyleshire, and attended Glasgow University. He became Professor ot" ^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 Geometrla Organica, 1720; Treatise of Fluxions, 1742; Treatise of Algebra, 1748, and an Accoant of Xewtoit's Fidlosophical Discoveries, 1748. His Treatise 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. T/ie Dispute betiveen 3Iaclaurin and Camphell. 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 GesvJiicJde der Matliematik. 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 Royal 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 conse(|uent 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 iniglit gain a livelihood in Loudon by teaching. Some of Campbell's papers were sent to London. One, at least, was read before the Royal Society, and, through the intluence 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 Folkes explaining his position, and o'ivinoj 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 Sii* A. Cuming are given in the list of subscribers to the MUcellauea Analyttca de Ser'iehus of De Moivre (1730). Xewtou's Theorem regarding the nature of the Root3 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 Avitliinetica U nlveraaH^. 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. Tran,s. 1864: Phil. Mag. 18GG.) Newton's Theoi'em may be stated thus (vide Todhunter's T/ieorg of E(inailont>). Consider the equation /{x) = a,x^' + ^^C\a,x^^-' + ... + ,^C,.a,.x"-'- + ... I a„ = 0. Form the two rows (^f (pianiities A^ A, A.,...A,^ NOTES UPON THE CORRESPONDENCE 195 where Call «,. «^.+i an associated couple of siiccet-sions. In such a couple tlic signs of «,. and (t,.+i may be alike and represent a Permanence, P ; or unlike, and represent a Variation V. Similarly for A^. and A^.^^. An associated couple may thus give rise to ( 1 ) a double Permanence, (2) a Permanence- Variation, (3) a Variation-Permanence, (4) a double Variation. Then we have New^ton's Rule : — The number of double Permanences in the series of couples is a superior limit of the number of negative roots ; and the number of ^'ariation-Permanences is an upper limit of the positive roots ; to that the number of Permanences in the Series A, A^...A^, is an upper limit to the number of the real roots of /(.i) = 0. Sylvester (v. Cdleded Works) was the first to fui'ni&h a demonstration of Newton's Theorem, and he gave the following generalization. Write /(a; + A) in the form and form the table f'o "i "n Jy Ay A|^ (where ^4^, ... A^^ are as before). Denote the numljer of double Permanences arit-ing therefrom by PP (A). Similarly denote by PP(n) the number corresponding to /(.r + /z). Then if /i > X, PPl/z)— PP(A) is either equal to the o2 196 NOTES UPON THE COREESPONDENCE number of ix'al roots ot" f(x) = butwet'ii jx ;ui(l A, or exceeds it by an even number. Letter I. 1. On p. 19 of his Defence (against Campbell) Maclaurin makes the statement : — 'In a Treall^e of Algebra, wliicli T composed in the Year 172G, and which, &ince that Time, lias been very publick in this Place, after giving the same Demonstration of the Doctrine of the LtinLtt<, as is now published in my second Letter, I add in Article 50 these Words, iVc' Maclaurin appears to be referring here to a course of lectures to his students. Maclaurin's Algehra did not appear until 174 8, 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 Miscellauea Aiudgtica, 1730. In 1738 appeared the second edition of his Doctrine of Chances, also referred to in the letters. Letter 1 3. This letter, dated by Maclaurin as Februar}' II"', 1728, should have been dated as February ll*'"', 172f. i.e. 1728 Old Style, or 1729 New Style. Stirling makes this correction in I. G, which consists of extracts from letters by Maclaurin. Lentil 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. C. Letter I. contains only extracts from letters of ]\iaelaurin, including one date<l October 22, 1 728, whicli is no longer in the Stirling collection. Ijetter 1. 7. In the spring of 1921 I had the good fortune to obtain a copy of Maclaurin's reply to Campbell. NO'l^ES T^PON THE CORRE>SPONDENCE 11)7 Jt is entitled: — ' A Defence ct" the Letter i)ultlislic(I in the Phihjsopliical Transactions for jMareh and April 1729, concerning the Impossible Roots of Equations : in a Letter from the Author to a Friend at London. Qui admonent aniiee, docemli sunt : <|ui inimice infeetantur, repellendi. Cicero ' The name of the ' Friend ' is not oiven. The ' Defence ' consists of twenty small quarto pao'es, 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 Autlior of the Remarks ' (on Maclaurin's Second Letter on impossil)le roots) : thougli also as ' the Remarker '• Maclaurin gives the extract from tlie letter of October 1728 (cf. I. 6), and adds: — ' See the 2d and 3d Examples of the Eighth Fro'positlon of tlio Lineae tertii Ordiuis Xewtonianae.' 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 Ma(hni, and we were both well satisfied that you had carried the IMatter to the greatest Height, as plainly appears b}^ what 3'ou 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 liavc the rest in such Readiness, I think 3'ou have it in j^our Power to do j^ourself Justice more than any Body else can. I mean by a speedy Publication of the remaining Part : For I am sure, if 3'OU 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 tliis 198 NOTES UPON THE CORRESPONDENCE kind A(lvic(\ resolved to send n]i my Second Paper as i-oon as possible." Maclaurin makes it clear that he had not intended his First Letter to Folkes to he published. It was printed without his knowledge. Had lie known in time, he would liave deferred its publication until he had more fully investigated additional theorems Avhich he had on the same subject; and he gives an extract from a letter from Folkes in corrolioration of his statement. Letter I. 8. Letter I. 8 is reproduced because of its reference to an office (in the Roj'al Society) for which Stirling had been thought fit. Letter I. 9. Letter I. 9 announces that Maelaurin has started to write his Treatise of Flnxi<ms. His conscientious reference to original authorities has been noted b}" Reitf {GeschicJde der UnendlicJien Relheii). The earlier proof-sheets of the Treatise, at least, passed through Stirling's hands. These facts bear interesting evidence regarding the Ealer- Maclaurin Summation Formidn, to which I have to return in connexion with the correspondence between Stirling and Euler in Letters XL Simp.son, referred to by Maclaurin, is doubtless his old teacher, Robert Simson, of Glasgow Univorsit}'. 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 Edinbui-gh, and is to be associated with the two letters of Maelaurin published in the Scots Magazine for June, 1804. The date of the letter of Maupertuis shows that Maclaurin should have given Feb. 4*', 173| as the date of his own. Maclaurin was more successful with Stirling than with R. Simson, who refused to become a member after IMaclaurin 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 THK CORRESPONDENCE 199 (Ki^aud). The original Fi-ench Jetter i.s preserved in one of the hotter hooks of the Royal Society of London. FuundatioR of the Philosophical Society of Edinhurgh. Letter L 10 confirms the date of foundation as 1737 (v. Forbes's Hidory 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 \vas not complete in numbers, for Stirling \Yas not yet a member. By 1739 the Society had outrun its original bounds, having forty-seven members M'hose 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, Piofessor 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 Easays and Observations, Physical and Literary (dated 1754, 17nG, 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 Feljruary. 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 printetl 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 Mawpertuis. 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 approximatel}?" that of an oblate spheroid, flatter at the poles than at the Equator. The m.) NOTES T^^OX THE COPIRESPOXDEXC'E Cassinis, arguing from iiR'asui'eiiK'iit.s oi' tliu arc uL' a Mcriilian in France, maintained that the figure was that of a prolate splieroid. There were thus two hostile camps, the X'cwtonians and the Cassinians. Pol- Pole A Y titwroK CASbTNI The French expedition to Lapland (173G-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 betW'Cen them refer mainly to their researches on Attraction and on the Figure of the Eartli. 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 alwa3^s correctly explained by Todhunter. Letter I. 11. The Dean, near Edinburgh, jMaclaurin's new address, now forms a residential suburb of Edinburgh. De Moivre's book is doubtless the second edition of tlie Doctrine of Chances (1738). letter T. 13. Tlie remark made by Stirling towards the conclusion that ' the gravitation of the particle to the wdiole spheroid will be found to depend on tlie quadrature of the circle ' seems to have given Maclaurin a good deal of trouble (cf. I. 14). NOTKS T^'OX THE (X:)RRESPONDENCE 201 ]\Ia('l;uiiiirs rct'crt'iKT to it in his Fluxions, § 647, as due to Stirling-, Mas iiicxplicaltlc to 'rodliunter, as Stii'lin<i^ never published his theorem. But Todliiniters conjecture {Hlttovij, vol. i, p. 139) that Maclaurin ma}^ 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 justitieation 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 Diskertatioiis (1 74.3). 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 J\Ioivre 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 controvers}^ regarding Fluxions. In his introduction to the Methodns Differential is, 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 CORRESPONDENT'E England with soiiii- oi" the duel's lie was iiistiuineiital in sirrano-inor n treaty for liis tribe. Later he fell into poverty, and was confined in the Fleet prison from 1737 to 1765, losing; his fellowship in the Pvo^'al Society for neglecting- to pay his annual fee. In 1766 he obtained admission to the Charterhouse and died tluM'e in 1 7 7.'. Ill CRAMER AND STIRLING Gabriel Cramer was born in 1 704 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 172 7 he started on a two years' tour, visiting Bale, where he studied under John Ber- noulli, and England, where he became acquainted with Stirling 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 d I' Analyse dcs lignes courhes algebriques. He also edited the works of James and John Bernoulli. Letter IIL 1. It is unfortunate for us that Cramer did not discover before 1732 that he wrote 'mi Anglois au&si barbare '. Regarding the history of the Probability Problem in III. 1, see Todhunter's History of the Theory of ProIxdnlUy (p. 84). De Moivre gives a much simpler solution in the Miscellanea Analytica (1730). Letter IIL 2. Compare Lettei' IV. 2 (Bernoulli). Letter III. 3. In this letter of introduction Cramer in the address describes Stirling as L.A.]\T. I do not know what these letters signify. Letter III. 8. Letter III. 8 contains valuable information regarding the manner in which Stirling wiote his Methodiis Diferentialis. The blank made for the formula given by De Moivre was never filled u[) : but the formula in (|uestion is of course easily NOTES UPON THE CORRESPONDENCE 203 obtained i'roni tlio Suj^plement to the M i scellanea Auahjtica of Do Moivre. We have also tlie important information that this Supplement appeared after tlie pnlilieation of Stirling's own Treatise. Letter III. 10. One will note Cramer's difficulties with the graph of 2/^"= 1 +.'-'; also his determination of (1 +a')^/-''' as x tends to zero. It is a pity there is no indication of Stirling's determination of this limit. Stirling' a Series and the claims to priority of De Moivre and Stirling. In the Bihlioteca Matliematica for 1904 (p. 207) Enestrom makes the following statement. ' Im Anschluss an den Bericht liber Stirling's Formel fiir 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 . . . ./•) = -i log 27r + {X + 4) log x .1 .1-2 , -x + A,,- + J., —^ + eVc, " X X"" zuerst von Moivre im Anhange an der Miec. analytica (17.30) angegeben und hergeleitet wurde. Moivre berichtet selbst dass Stirling ihm brieflich die Formel log (1 . 2 ... a-) = i log 273- + (a; + i) log (a; + I ) 2.12(a; + i) 8.360(« + i)=^ mitgeteilt liatte, und dass er selbst dadurch angeregt wurde die neue Formel auf eineni ganz anderen Wege auf zufinden.' Inasmuch as the only change effected by De Moivre is to give the expansion of log {x !j in descending powers of x instead of descending powers of x + ^, which has no special advantage when X is large, the priorit}^ of De Moivre to this important formula seems to me to rest on very slender foundations, unless we are to infer from Enestrom's reference to the 204 NOTES UPON THE CORRESPONDENCE Sujiidemciit Id tlu' Mli<rcflaiie(i Aiialijllcc thai l)c Moix to pulilisliL'd his result prioi- to Stirliui;-. Enestroins stateiiR-nt lias had considcraltle influence with subsequent writers (e.^-. Czuher and Le Roux, Cuhul des Probabllites; Selivano\- and Andoj'er, Calcul des Differences Finies, in the well-known Eacyc. des ^Sriences Math. ; Czuber, WaJir. Recliuvivi, 1903, s. 19), wlio refer for proof to tlie Siipi>. Misc. Anal, of De Moivre. Aeainst these we may put De Moivre's own statement in the third edition of the Doctrine of Chances (1756), oiven in tlie Appendix, p. 334, where, after giving a table of ^'alue.s for log {x !) for numerical \ alues 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 jMr James IStirliiig, published in his Supplement to the Ulitcellanea Analytica, and by Mr Stirling himself in his Methodus Different ialis, Prop. XXVIII. 'Let z — ^ be the last 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 7a 31 rt z ogz-az- 2 12s ^ 8 . 3603^ ~ 32 . 1260^5 127ft H _- — (vc 128. 1680-J added to 0-39908993 il 79, c^'C. which is half the Logarithm of a Circumference whose Radius is Unity, will be the Sum of the Logarithms of the given Series ; or the Logarithm of the Product 1x2x3x4x5... xz~\ &c.' There is thus no doubt in De Moivre's mind that the dis- covery of the theorem in question is not due to himself but to his friend Stirling. Date of ^u/rplement to the Miscellanea Aaahjtiai. At first sight the Supplement appears to bear the date Jan. 7, 17|§. Li such case it would almost certain]}' be anterior in [)ublication to Stirling's book. NOTES UPON THE CORRESPONDENX'E 205 Now this [supposition is quite erroneous. The Miscellanea Aaalytica, as originally published, bears the above date, and contains no supplement. (The first copy I consulted has no supplement.) An examination of a copy with the Supple'meiit shows two lists of Errata, the first after p. 250, and the f^econd after p. 22 of the Su^yjylemeiit, the latter list contain- ing Errata observed by De Moivre and his friends ' post editum libruni meum '. The letter III. 8 of Stirling puts it beyond a doubt that the Sa2}plemeiit had not appeared at the time he wrote (September 1730), so that its appearance was posterior to the publication of Stirling's Methodas Differe^dlalls. We have thus the following events in chronological order. De Moivre publishes the Mite. Anal, early in 1730. His friend Stirling points out to him the poor approximation he gives for log [x !) when x is large and sends him a formula of much greater accuracy. Stirling publishes his Meth. Diff. containing the famous Stirling Series. In the meantime De Moivre busies himself with Stirling's formula, and obtains it in a slightly different form but by an entirely different process: and he publi; lies his result as a Supijlement to his book and bound with it, but without changing the date of his book. He explains in his own garrulous way, which makes the reading of his works so attractive nowadays, how he had very nearly got at Stirling's Theorem before he had heard from Stirling. Will any scholar be bold enough to assert that the theorem is due to De Moivre in virtue of this latter statement, pub- lished after Stirling had given the theorem in all its generality in the Meth. Dijf. 1 You may speak of De Moivre's form of Stirling's Theorem if you please, but the merit of discover- ing a theorem of the kind eecms to rest indisputably with Stirling. IV N. BERNOULLI AND STIRLING Nicholas Bernoulli was born in 1687 at Bale in Switzerland, his father being a merchant in tliat town. His two uncles, James Bernoulli (1654-1705) and John Bernoulli (1607-1748), were both noted mathematicians. 20G NOTES UPON THE CORRESPONDENCE He .studied Hrst under the t\)nuei' at f!ale University, and then under the latter at Gri3nin»;"en, returning- with his uneh' John to Bale in 1 705. He devoted hiniselt' to the study of mathematics and law. He became F.R.S. in 171."}. (_)n the recommendation ot" Leibniz, he was in 1716 ap[)ointed Professor of Mathematics at Padua, resignini^,- in 171'.) and returning to Bale. In 1722 he was elected to the chair of Logic, and in 1731 to the chair of Law in Bale. He died in 1759. His cousins, the sons of Jolnij. Nicholas 1G95-1726; Daniel 1700-82; and John^ 1710-no were also noted mathematicians. Two of the three sons of Jolin^, viz. John, and James, also showed mathematical ability, so that we have here a remark- able instance of three generations of distinguished mathe- maticians in one family. Venice was a favourite resort of the Beruoullis about the time that Stirling resided there. Letter IV. 1. Letter IV. 1 is the earliest of the letters preserved in the mathematical correspondence of Stirling. When the actiuain- tance betwc'ii Bernoulli and Stirling began is unknown, but Bei'noulli in the course of his travels spent some time in Oxford in 1712, when Stirling was still an undergraduate. One is strongly tempted to suggest that it was at Oxford that they first met, for the disparity in their years was not very great, while the number of students of mathematical tastes cannot have been very large. The fact of Bernoulli's presence in Oxford I have discovered in the Corvespoudaace Math, et Physique, edited by N. Fuss, vol. ii, p. 183, where, in a letter to Daniel Bernoulli, Goldbach makes the remark : — 'Cum Oxonii agerem A. 1712, atque per unum alterumve diem communi diversorio uterer cum consobrino Tuo CI. Nicolao BernouUio, donavit me dissertatione (juadam Jacobi Bernoulli de seriebus intinitis Arc' (Lettre V Goldbach a D. BernouUi, 4 Nov. 1723) Licidentally we learn an interesting fact regarding Goldbach that has escaped the notice of M. Cantor, who, in the Vorwort to the second edition of his Gesdiivhte, gives 1718 as the earliest <late he has fonud in coiniexion with the traxels of Goldbach. NOTES UPON THE CORRESPONDENCE 207 Continuation as far as N. Bernoulli is concerned is found on p. 300 of vol. ii of Brewster's Life of Newton. He (i.e. Bernoulli) went to London in the summer^ of 1712, where he met with the kindest reception from Newton and Halley, a circumstance which he speaks of with much gratitude in a letter in wdiich he thanks Newton for a copy of the second edition of the Principia. (Letter dated Padua, May 31, 1717.) Query : Did Gold bach meet Newton 1 Taylor ti Problem. The problem &ent by Taylor to Montmort was a cliallenge to the continental mathematicians : — ' Problema analyticum omnibus geonietris non Anglis pro- positum : Invenire per quadraturam circuli vel hyperbolae Fluentem liujus quantitatis Taylor had obtained it in the posthumous papers of Cotes, who died in 1716, while his Harmoida Mensarum, in which the solution is given, w^as not published until 1722. The limitation on A was given by Taylor because cnly in such a case had Cotes etiected a solution. 'I'he challenge was really intended for John Bernoulli. John Bernoulli published a solution in May 1719 (Leip. Actis). Other solutions were given by Hermann, Professor of Mathe- matics at Padua,^ and by Ganfredi. (Montucla.) IV. 4. Letter IV. 4 is written in a typical Bernoullian spirit as a reply to Stirling's letter IV. 3. Bernoulli's letter, however, contains a number of valuable criticisms upon the tAvo pub- lished works of Stirlini; on Cubic Curves, and on Serieti, to which Stirlinc; would have had to a'ive careful attention had second editions of his w^orks ever been contemplated by him, and to which I may have to advert on another occasion. For the present I restrict my attention to the discovery Bernoulli makes known of a new variety of cubic omitted by ^ 'Visit to England duiing the months of September and October 1712.' (Edleston, note, p. U2.) - Formerly. 208 NOTES UPON THE CORRESPONDENCE l)otli Newton ;ui(l Stirling in tliuir uiiuiiiei-atiou of Cubic Curves. (Newton's error, whicli Bernoulli points out, is re- tained in tlie Horsley edition.) In the enumeration of the cubics oiven ])y the ecjuation only four of the six possible species are enunierateil Ijy Newton, and by Stirling; following Newton. Of the two missing species, Nicole in 1731 gave one (an oN'al and two infinite branches) corresponding to ay-=p-(x + (\-) {x + l3') or xy' = — [r {x — oi-) {x - /3-) . N. Bernoulli here announces (in 173?) the discovery of another, consisting of an acnode and two infinite branches as given by the equation XtJ- = + y- {X + Oi-f. Thus Bernoulli takes precedence of Stone 173G, Murdoch and De Gua 1740, to whom reference is made by \V. W. R. Ball, in his valuable memoir on Xeivtons Clasdjicatioii of Cuhic Carves (Trans. L.M.S. 1891). Murdoch {Neivtoni Genesis Gurvariim per Umbras, p. 87) has the remark : — ' Speciem hanc No Vlll Analogam apud Newtonum deside- rari animadverterat D. Nic. Bernoulli, quod me dim monuit I). Cramer, Phil, et Math, apud Genevenses Celebris Professor.' V CASTEL Louis Bertrand Castel (1688-1757), a Jesuit Father, was the autlior of Le vrai systcme de Newton. He became F.R.S. in 1730. Stirling's letter \ . 2 contains a clear exposition of what he understands by geometrical demonstration. VI CAMPAILLA Thomas Campailla was born at IModica in Sicily in 16G8. and died in 17-10. He studied in succession law, astrology'. NOTES UPON THE CORRESPONDENCE 209 and philosophy, and tinally devoted himself entirely to the Natural Sciences and Medicine. He was not a Fellow of the Royal Society. VII BRADLEY J. Bradley, 1692-1762, was a distinguished Astronomer. Like Stirling, he studied at Balliol College, Oxford. He became F.R.S. in 1718. In 1721 he was appointed to the chair of Astronomy in Oxford, in succession to Keill. He succeeded Halley as Astronomer Royal in 1 742. He discovered the aberration of the tixed stars and the nutation of the earth's axis. Both the letters here given are to be found in Rigaud's Bradley. Stirling's letter is taken from Rigaud ; and Bradley's reply is among the letters preserved at Garden. VIII KLINGENSTIERNA S. Klingenstierna was Professor of Mathematics at Upsala. It was through Cramer that he was introduced to Stirling (cf. Letter III. 3). In view of his researches in Optics, the letter here given is of some interest He became F.R.S. in 17.30. IX JOHN MACHIN John Machin, the astronomer, became F.R.S. in 1710 (the same year as Poleni, Professor of Astronomy at Padua, men- tioned in the postscript to IV. 1), and acted as Sec. R.S. from 1718 to 1747. He sat on the committee appointed in 1712 to investigate the dispute between Newton and Leibniz. In 1713 he became Professor of Astronomy at Gresham College. He died in 1751. Machin used the formula tt/ 4 = 4 arc tan -g- — arc tan gig to calculate tt to 100 places of decimals. His result is given (1706) in Jones's Syno2Jds Palmariorum Matheseos, in which the .symbol tt is first used for the number 3-14159 . . . :210 NOTES LIPOX THE COKRESPUNDENCE His 'Laws ot^ tlic IVIoon's Motion aceordiiin^ to Gravity' is appended to Motte's translation of the Prineipia. A greater work on Lunar Theory, begun in 1717, was never publislied : and relative manuscripts are in t\\c possession of the Royal Astronomical Society-. Letter IX. 1. Li connexion with this letter, which has no date, see the letters from Bernoulli to Stirling, IV. Letter IX. 2. Machin was keenly interested in the researches of Maclaurin and Stirling concerning the Figure of the Earth, though his name does not appear to find a place in Todhunter's Hidory of the subject. The book by Maupertuis is probably one on the Figure of the Earth mentioned by Todhunter (vol. i, p. 72 . Machin, in speaking of Stirling's Proposition concerning the Figure of the Earth, cannot refer to Stirling's Memoir entitled ' Of the Figure of the Earth and the Variation of Gravity on the Surface ', which appeared in the Phil. Trans. for 1735-6. Compare Stirling's letter to Maclaurin I. 15, in which he refers to his correspondence with Machin. I do not quite understand Machin in his reference to the invention of Eider's Series, though Stirling's letter, if it could be found, would explain. By 1738 Stirling had got definitely settled as Manager of the Lead Hills Mines in Scotland. He had apparently com- plained to Machin how he felt the isolation from his scientific friends and their researches in London. Machin's letter to him is written in the kindliest spirit of warm friendship. The book of De Moivre mentioned in the letter is doubtless the second edition of the Doctrine of Chanceti (1738j. X CLAIRAUT Born at Paris in 1713, Clairaut showed a wonderful pre- cocity for mathematics, and at eighteen years of age he NOTES UPON THE CORRESPONDENCE 211 piiblii-lied his celebrated ' Recherclics siir les Courbcs a double Courbure '. He took part in the expedition to Lapland under -Muupertuis to determine the length of the arc of the meridian. He made several contributions to the Theory of the Figure of the Earth, which he ultimately embodied in the classic work entitled Thcorie de la Figure de la Terre (^1743). His Theorle de la Lune appeared in 1765, shortly before his death. He was also the author of Elihnents de la Geometrie (1741), and of an Algebre (1746i. He became F.R.S. in 1737. He died in 17G5. ' Clairaut a eu pour el eve et pour amle la celebre Marquise de Chatelet, la docte et belle Emilie, qu'il a aid^e dans sa traduction du Livre des principes' (Marie, Hist. Math.), a state of affairs not over-pleasing to Voltaire. In the letter here given w^e find Clairaut introducing himself to Stirling. Cf. I. 15. Clairaut had frequent correspondence with Maclaurin, and several of the letters have been preserved. XI EULER Leonhard Euler (1707-83) was born at Bale in Switzerland. He studied Mathematics under John Bernoulli, having as fellow -students Nicholas and Daniel Bernoulli, the two sons of John Bernoulli. The two brothers were called to Petrograd in 1725, and Euler followed in 1727. In 1741, on the invita- tion of Frederick the Great, he went to Berlin, returning again in 1766 to Petrograd, where he died in 1783. For almost the whole of his second residence in Russia he was . totally blind, l)ut this misfortune had little effect on his wonderful production of mathematical memoirs. There is hardly a department of pure or mixed mathematics wdiich his genius has not enriched by memoirs of far-reaching impor- tance. A complete edition of his works has been undertaken by a Swiss commission. We are here only concerned with his relations with Stirling. Apparently Euler had opened the correspondence by a letter to Stilling, in which he announces, inter alia, the theorem known as the Euler-Maclaurin Theorem (Reiff', Geschichte der Unendlichen Reiheii). 1'his letter is not preserved, but copies p2 212 NOTES UPON THE CORRESPONDENCE of tlic Icttei's that passed between Euler and Stirling appear to have been in existence at Petrograd : and Professor Enestrom in his Vorldujiges VerzeicJniiK der Brlefe von und an L. Eider, 1726-41, furnishes the following dates : (1) Euler to Stirling, 9th June, 17.3G, (2) Stirling to Euler, April, 1738, (3) Euler to Stirling, 27th July, 1738. The letters preserved at Garden are doubtless (2) and (3). It remains to l)e seen whether the letters in Petrograd have survived the fury of the Revolution in Russia. Stirling's reply was much belated, for his time was now entirely devoted to the successful development of the Lead Hills Mines, of which he had been appointed manager a ye-AV or two before. The rounli draft of it is all that Stirling preserved, and is here given witli all his corrections and erasures. Stirling acknowledges the ini2:)ortance of Euler's Theorem, and remarks that his own theorem, ' Theorema meum ', for summing Logarithms is only a particular case. He informs Euler that Maclaurin has an identical theorem in the proof-sheets of a Treatise of Fluxions to appear shortly. At the same time he offers to communicate Euler's results to the Royal Society', and suggests that Euler should become a Fellow. With characteristic modesty and absolute freedom from jealousy, Euler in his reply waives his claim to priority over Maclaurin, and proposes that the Royal Society should publish a paper on the Equation of liiccati, which he had sent some 3"ears before to Sloane the President. There can be little doubt that luder and Maclaurin dis- covered the theorem independently, and llie suggestion made by Reiff to call it the Euler-Maclaurin Theorem seems fully justified. Maclaurin, by the way, does not refer to it in tlie intro- duction to liis Fluxions, but on p. C91 of his Treatise. Euler first gave his theorem without pi-oof in his Metliodus generalis summandi firoijressiones (Jomm. Pctrop. ad annos 1732, 1733: published 1738. The proof is given in Invodio ^ummae cujusque seriei ex dato termino (jcnerali (Jomm. Peirop., 173(1: published 1741. Compare Stirling's letter to Maclaurin I. 15. NOTES UPON THE CORRESPONDENCE 213 I cannot lierc further discuss Euler's letter, wliicli is almost encyclopaedic in its rany;e, save to say that Stirling had sliown in his Me(h. J^ijf. lunv to approximate with any desired 00 accuracy to ^ -^, > without hein^- aware ol" its expre SSI on n 1 as 77-/6. (See letters of Dan fJernoulli to Euler in Fuss, Corr. Math., t^'c.) As is well known, Euler became F.R.S. in 174G. XII M. FOLKES, P.R.S., TO STIRLING This is the letter of latest date in the correspondence. It conveys to Stirling the news that he had been made a member of the Royal Academy of Science at Berlin, an honour which has not hitherto been noted in any of the biographies of Stirling. May the Mr. Mitchell who brings the letter to Stirling not have been Maclaurin's friend, better known as Sir Andrew Mitchell, who afterwards became Ambassador at the court of Frederick the Great ? PRINTED IN ENGLAND AT THE OXFORD UNIVERSITY PRESS Date Due MAY 1 ; ?nnfi L. B. CAT. NO. 1 187 M-'Jffl QA2').S6H3 scni 3 5002 00228 1975 Stirling, James , , , , j James Stirling; a sketch of his life and Science QA 29 . S69 A3 Stirling , James !, 1692- ■1770. James St irling