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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 


assistance ot" Professor E. T. Wliittaker's notes on Part I of 
Stirlino-'s MeUiodus Different lalit^, wliich he kindly put at my 

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. 


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 



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, 

E. T. Whittaker, D.Sc, F.R.S., Professor of Mathematics, 

Edinburgh University. 


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. 







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 



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 

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 


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.- 



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. 


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 


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 


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 

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 


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. 

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 : 


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. 


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'. 


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. 


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 



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. 


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. 


letter to Newton, expressino- his oi-atitiide, is here oivcii. It 
lias boon copied from Brewster's f/ife of Newton. 


Venice 17 Aug. 1719. 


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 


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- 


sophical Tr<(vs<(ctions for 1719, witli the object of elucidating 
Newton's methods of Interpohition. 


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 

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. 


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, 


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 


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. 


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 


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.) 


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. 


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 


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. 


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. 


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. 



§ 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). 


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. 


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 



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 

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. 


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. 


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 



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. 


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.') 






■^ 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 

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^ ^^^^ 


Theorems of special interest arise when T can be ex- 
pressed as 

T^ A-^Bz + (Jz{z- \) + Bz{z-l}{z-2)+ c\:c., 


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. 

x{x+\){x + 2) ...{x + n-\) ^ 6V*" + ^'„' *'*-'+...+ (^V' ^ 

and l/x{.c+\)...{x^,i-\)= 2 (-ir^nV ■«'''', 


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. 


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 



§ 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" +^, 

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. 


When 5=1 we have, of, the Beta Function 

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. 


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) 


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. 

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'~' + ...), 


then the lust of the sums will be finite both ways only when 
m = 1 and k = a. 

In other words the infinite product 


1 + 


+ ... 




,!= 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 ~ ' 

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). 



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. 


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:"~^ 


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=-- 



in (ina coefficientes A, B, C, i^'c. hanc sequuntur relationem 

j^^_ rr-AA ,. -A ,. -B 


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.) 


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). 


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 


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 ? 


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 

Binet, wanting h for the middle coefficient, gives the four 

(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) = 


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.) 


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. 


§ 15. On pages 135-8 are given the formulae which have 
rendered Stirling's name familiar whenever calculations in- 
volvino- laroe numbers are concerned. 



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 


(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. 


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 


+ 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) 


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. 


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. 


§ 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 

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. 


The integrals of these oinittin<;- the first, are r, z-\rz'' j?,, &c. 
Take the latter as /:J, y. kc , so that 


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, 


'^^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 

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. 



& 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 + 








2X71 + 2 4xH + 4 6xn + 6 8x?i + 8 





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 










1 A 9B 25C 49i) 

— — , + =^ + == + --— + =^ 

n+l 2x/t + 3 4X)i+5 6xrt+7 8X91 + 9 




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. 












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 
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 


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 


(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. 


(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. 

(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. 


E 2 


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. 


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. 




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. 




G. CRAMER AND STIRLING, 1728-1733 ... 95 
10 Letters fiom Cramer to Stirling. 
1 Letter ,, Stirling to Cramer. 


N. BERNOULLI AND STIRLING, 1719-1733 . . 181 

3 Letters from Bernoulli. 
1 Letter ,, Stirling. 

1 Letter from Castel. 
1 ,, ,, Stirling. 


1 Letter from Campailla. 



VI r 

J. BKADLEY AND STIRLING, 1733 . . . .160 
1 Letter from Stirling. 
1 „ „ Bradley. 



1 Letter from Klingenstierna, also solutions of cer- 

tain ])roblems. 


MACHIN AND STIRLING, 1733 (?) and 1738 . . 172 

2 Letters from Machin. 


1 Letter from Clairaut. * 


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.) 


M. FOLKES, P.R.S., AND STIRLING, 1747 . . 192 
1 Letter from Folkes. 


^:i/^^ i^ j-<^ ^f^K^y^^ r/ U£ ^ MZ'^ A^'ai^y 4^1^ 



MacJaiirin to Stifling, 1728 

Mr James Stirling 
at the Academy in 
little Touer Street 


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 


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 

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 

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- 


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 


Your most Obedient and Humble Servant 

Colin Maclaukin 

Edinburgh Dec' 7 

Stirlwfj to MacJaurin, 1728 


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 


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). 


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 


Madaurin to Stirling, 1729^ 

Mr James Stirling 
at the Academy in 
little Tower Street 


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 !„. 


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 ^ ^ 


and thence conclude that -— — B^ always exceeds 

2 m "^ 


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 


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 




6- + 3 


■ii — m- 


, 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 

' In the Equation 

x'-Ax^ + Bx-'-Cx'' + Dx-E= 
x^-\0x*-\-Z0x^-iix^ + 32x-d = 


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 


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 

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. 


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. 


Maclaur'ni to Stirlitig, 1729 

J\Ir James Stirling 
at the Academy in 
little Tower Street 


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 



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 


Maclaiirin to Sfirliiuj, 1720 

Mr James Stirling 
at the Academy 

in little Tower Street 

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 


Your ]\Iost Obedient 
Edinburgh Most Humble Servant 

nova's. 1729 CoLiN MacLauhin 



Stirling to Maclaurhi, 1720 

To Mr Maclaiirin Professor of Mathematicks 
in the Universitj^ of 


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|. 

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 

Your most humble servant 

Ja: Stirling 
London 29 November 1 729 


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. 


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. 

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 


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 


Madaurin to Stirlh/g, 1734 


Mr James Stirling 
at Mr Watt Academy 
in little Tower Street 

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 


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. 


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 

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. 

Maclaurin to Stirling, 1738^ 


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 

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. 


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] 


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 


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 


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 

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 

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 


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 


Yovu- Most humble iV' most Obedient Servant 


I shall be much obliged to you if you will be pleased to 
Communicate . . , the Royal Soe . . . 

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. 


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. 


Madimr'm to StirUng, 1738 


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 



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 


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. 

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 



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 


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. 


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 


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 


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. 


Maclatirin to Stirling, 1738 


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. 


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, 


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 + 



&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- 


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. 



Stirling to Maclanrin, 1738 


Mr Maclaiirin Professor of Mathematicks 


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- 

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 


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 


Your most obedient & 
most humble servant 

James Stirling. 


Maclaiirin to Sdrling, 1740 


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 


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 


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 



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 

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 = 


N( cHr 


Fig. 13. 

1 - 


^^^-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 


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 ~ 




X 1 


6 ma'' lb' 


7 c:"!'^ 
9c"-^f^ + 8c(:Z- + 2(P 


100c3cZ2+145c2cZ3+72c# + 12c?-^ li} 
13c^ r* 


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. 




Cramer to Stirling^ 17 28 


Mr James Stirling F.R.S. in y° Academy 

in little Tower Street 



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 


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 


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 


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 



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 


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 


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. 


Cramer to Stirling, 1729 


M-- James Stirling F R.S. at the 
Academy in little Tower Street 

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, 


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 

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 



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 


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^". 


Probl. II 

Resolvere quantitateiu 1 +c2?t+i jj^ factorcs cluarimi Diuicn- 

Solut. Sit uniis ex factoribus l—xc + sz, & productum 

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 


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. 


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 ' ^ 


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- 

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 

& invenietur s = t + 1x : roliqua vero se habcnt ut prius. 

Positis igitiir ut in Prob II arcu AJJBE = n arc A D, x = BD, 


t = -^, s = ,~^^, erit DE= AE+AE. BD. 

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 = 


It T o 

(J ^ 

cuius dimidii, nempe , cosinus erit ^x. Si arcus ADBE 

^ 2 71 + 1 


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 

Solut. Sit unus ex factoribus 1—xz + zz & productum 

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 

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. 


Supplemcntniu Eodein Auctore 

Probl. V Dividere fractionem :; — in fmctiones plures, 

1 + qz^ + c-'* 

quarum denominatores ascendant tantum ad duas Dimen- 


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 


& arciis AD = , erit s:t = DE.AE per ibi demonstrata, 


& 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. 


\—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 


I I -•^rt— 1 I ^^n 

Schol III Methodus praeced. supponit q minorem binario, 
quando autem a > 2, fractio :. ^j- resolvi potest ut 


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 


ia' + V^xx —ir — n^x— y/ixx — 1 


= (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 


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. 

Cramer to Siirluuj, 1729 

Viro Clarissimo, Doctissinio 
Jacobo Stirling 
L.A.M. & R.S. Socio 
Gabriel Cramer 
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 



Cramer to Stirling, 1729 

Mr James Stirling at the Academy 

in little Tower Street 



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 

it by H ^ , and divide it by y'' Sine of y'^ Arch z. 'Vhc quotient 


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 

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 


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 

Your most humble and 

obedient Servant 

G. Cramer 
Paris y W March 17 29. N.S. 


Cramer to Stirling, 1729 

M' James Stirling, F.R.S. at the Academy 
in little Tower Street 

Geneva, y' if May 1729. 

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 


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 


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 


X I — x'i is equal — 


'/+i i 



m + q + l 

x"" — 



vi + q+1 .m + q 

m . m—1 

m + q + 1 .m + q m+q—1 

m.7)l—l .771—2 

= x" 


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 


•X l-x'J 



Let m bo equal o + 7"— 1, and iit+q + l be r-r?.)-!, or 
q =j) — r—l, 3'ou'll havu 

xx'' + '~^ 


z + r—\ .z-\- r—2 ...z + r- 

z+'p-l .5 + /J-2...-; + |)- 



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 


Your most humble, most 

Obedient Servant 

G. Cramer 

Cramer to StirUufj^ 1729 

M"" James Stirling at the Academy in 

little Tower Street 



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 


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 


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 


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 

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. 


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 


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+ |^, = 


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, 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, 


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 


Your most humble 

Geneva, the 22 X''"" 1730 N.S. most obedient Servant 

G. Cramer. 

Slirliiifi io (■rauier, 1730 

Copy of a Letter .scut to M'' Cramer at 
Geneva September 1730 


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 

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 


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 

D. Sir 

Your most Obedient & 

most humble Servant 
London September 1730 James Stirling. 

^ The gap occurs in Stirling's copy of the letter. 


Cramer to S/irUi/fj, 1731 

31' Jiuues Stirling R.S.S. at the Academy 
in little Tower Street 


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 = 


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 


a.a + b.a+2b ... a+2J—lb 


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- ^ ^^ 


+ 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 

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 


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 


Your moi^t humble, obedient 
Geneva 18*^^ June 1731. and most faithfull Servant 

G. CllAMEH. 

Cramer to StirUug, 1732 

W James Stirling. K.S.S. 

at the Academy in little Tower Street 


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 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 


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. 








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 


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 


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 

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 

Fig. 22. 


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 


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. 


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 

Votre tres lunnblo, iV' tres obdissant Serviteur 

G. Cram EH. 



Cramer to Stirliii;/, 1733 

M' Jcuues Stirling. F.R.S. 
at the Academy in little Tower Street 

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 


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 


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 
G. Cramer. 
Geneve ce 10*^ Avril, 1733 




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 



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 

Theor. Ill 


x!"i '^dx: (e +fx'i) '^ est algebraice quadra- 
bilis : Adeoque existente p = 0, erit etiam 

algebraice (piadrabilis. 


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 

I'P^+^dx ; (<> +fx'iy^ dependet a quadratura hujus 
j xhfx:(e+fx'i). 
Theor. VII Siimtis 8 et A in Casu Taylori erit 


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 


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 


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 


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 


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 


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 


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 


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, 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 

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 

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. 


Sthihif) to BernotiUi, T/'PtO 

Cop3'' of a Letter sent to M"* Nicholas 
Bcrnouilli September 1730 


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 


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 


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 

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, 


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 

Your most obedient 

most humble Servant 

James Stirling. 


Bernoulli to Stirling, 1733 

Viro Clarissimo Jacobo Stirling Nicolaus Bernoulli 

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. 


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. 


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 . 

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 


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 


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 


terminus unciae ordine n+ 1 in binomio ad dignitatem r + '"' 


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 


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 .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 


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. 


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 


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 = 


D — 


E= - 


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 ' 



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 

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" 


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 . 


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 


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. 



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 


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 



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 


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 


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 


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 


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 


}) 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. 


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- 


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 

Motueae die sexta Mensis Maii 1738 

Addictissimum & Obsequentissimum 




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. 


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 

Sir, your most humble servant, 

J. SlIRLINti. 

Bradley to Stirling, 1733 


Mr James Stirling 
at the Academy in Tower Street 
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 


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. 


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, 


M 2 


Klingenstierna to StirUiifj, 1738 

Viro Clarissimo, Doctissi 

moque Domino 

Jacobo Stirlingio 


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 


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 - 

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- 

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 


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^^ 


. , . 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 „ ,. ., , 

mut'itur HI haiic, — ~ri\^r— = rrr^ ■> k dividendo per 2, ac 

AD' aD' ^ 

, ,, ,, ., 1 .„ n dp.AF 
pro AJ) aD, scribendo AB aB, ' „ = — . -•' . 
^ AB' aB-" 



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 * 


= IL. 

Similiter ob similitudinein trianguloruin uBe, ail, erit 

(iB : Be = ai (id est af) : il, unde -rr = il. 

a B 

^ .. . 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^, ' 


Similiter propter similitudinem triangulorum aBG, aig, tin, 

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 

Sed mventum erat ^^ = ^,, ergo ^^^^ = -j-^^. 

& multiplicando per BG'-, ^. — = ^ • Est vero IG . IN 

elementum circuli IGHL, i\: 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. 



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 



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, 


, , 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 : 


=. = dy 

2 n 2 n 

Centro A, intervallo a describatur circulus, eujus elemontum 

rectis AD, Ab comprehensum dicatur dz, eritque x -.dy = a : dt, 

unde dy = ' — » & hoe valore substituto in aequatione modo 



= dy, 

1 2n in 



in banc 


X " dx 


/ 2w 2m 

\a"' -X'" 



ax"' dx 

1 211 2 n 

= dz. 



Ponatur x — a 



\- aequatio transibit in lianc; 

n y 


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. 




MacMn to Stirling {1733?} 


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 


herein behave according to the directions you are pleased 
to give. 

E"". Your most faithful 

Friend & very humble Serv* 
Thursday morning J. Macuin. 


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 

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. 


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 


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. 



Clairmit to StirUng, 1738 


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. 


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 

Votre tres humble et tres 
a Paris le 2 Octobre 1738 obeissant Serviteur 


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. 


Stirling to Eider, 1738^ 

Celeberrimo Doctissimocjue Viro 

Lconhardo Eiiler 


Jacobus Stirling 


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 

Terminorum quippe generale et praxi expcditum. (lllius) 

percepi item extendi ad phirima serierum genera, et (|Uod 

praecipuum et A plerunujue (celeriter) approximat. Forte 
non observasti theorema meum pro summandis Logarithmis 

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!?. 


generalis ; (quod ingenue fateor). Sed ct A gratius mild fuit 

(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- 

nicam (non percipio, propter novi) obscura mihi a videntur, 

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 


continentur in 
Series tuae (comprehenduntur sul^) forma generali 

1 1 1 1 1 P 

ubi n est numerus par) eadem (tamen ad formulam scquen- 



tein) nullo iiegotio reducitur, (scilicet) re<lucitur ad ronmdain 

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^^ 


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) 

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 

(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) 

tuam epistolam A in nostris philosophicis trausactionibus. 


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 

(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 

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- 


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 

- + Gt/u,'^ iSdx' 

Zd'X bd'X 


1 .2.3 ...9. lOc/x'' 1.2.3 ... 11 .Grfic'-' 


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;'^ 


1.2.3 ... 21.110(/a;i-* 


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- =;", 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, 


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' 
^ 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 


^./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 


cujus progressionis legem facile inspicies. En igitur tivia liujus 
generis TheorematM, (juac singula cortis easibus exiiiiiaiii liabe- 
bunt utilitatem ad sunmias serierum indacfandas. 


Quod Jeiiule attinet ad suiiiiuatiout'S liujusmodi serierum, 
(juae contineutur in liac 


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« 

quae series omnes continentur in una hac generali : 


existente n numero integro. Si enim n est numerus par, turn 


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 




debet coefficienti ipsius — s^, qui est = — r-' undo deducitur 

summa cultoru illarum fractionu, 


2. + 5^+(7^ + 2}3 + ^tc = ^--; 


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^ 


^ ='+ 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. 



1 . 

2'\ 1 








1 . 



.. 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 



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 


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 


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 


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 - — ~ 



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. . 37.41 
4.4.8. 12. 12. 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 10. 10 etc . 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, 


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 


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 

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. 



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 



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 

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 


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 

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,^ 



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 

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 



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 

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. 


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, 

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 

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 


kind A(lvic(\ resolved to send n]i my Second Paper as i-oon as 

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 

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 


(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 


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. 






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). 


]\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. 



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 

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 


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.'. 



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 


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 


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 


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 

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. 


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 


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. 


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. 


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. 


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.' 


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. 



Thomas Campailla was born at IModica in Sicily in 16G8. 
and died in 17-10. He studied in succession law, astrology'. 


and philosophy, and tinally devoted himself entirely to the 
Natural Sciences and Medicine. He was not a Fellow of 
the Royal Society. 



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. 



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. 


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 . . . 


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. 



Born at Paris in 1713, Clairaut showed a wonderful pre- 
cocity for mathematics, and at eighteen years of age he 


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. 


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 



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 

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. 


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 


accuracy to ^ -^, > without hein^- aware ol" its expre 

SSI on 



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. 


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 

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 ? 




MAY 1 

; ?nnfi 

L. B. CAT. NO. 1 





3 5002 00228 1975 

Stirling, James , , , , j 

James Stirling; a sketch of his life and 


QA 29 . 




, James 

!, 1692- 


James St