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JAMES S 1 I R L I N G
OXFORD UNIVERSITY PRESS
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HUMPHREY MILFORD
Publisher to the University
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JAMES STIRLING
A SKETCH OF HIS
Htfc auD KHorfts
ALONG WITH HIS
SCIENTIFIC CORRESPONDENCE
BY
CHARLES TWEEDIE
M.A., B.Sc, F.K.S E.
CAKNEGIE FELLOW, 1917-l'.t20; LECTURER IN PURE MATHEMATICS
EDINBURGH UNIVERSITY
OXFORD
AT THE CLARENDON PRESS
1922
\ 80 4-2.
2^
3
TO THE MEMORY OF
JOHN STURGEON MACKAY. LL.D.
TO WHOSP] INSPIRATION IS LARGELY DUE
MY INTEREST IN THE
HISTORY OF MATHEMATICS
PREFACE
The Life of Stirling has already formed the subject of
a very readable article hy Dr. J. C. Mitchell, published in his
work, Old Glcsgovj Essays (MacLehose, 1905). An interesting
account of his life as manager of the Leadliills Mines is also
given by Ramsay in liis f^cotland and Scotsmen in the
Eighteenth Century .
The sketch I here present to readers furnishes further
details regarding Stirling's student days at Balliol College,
Oxford, as culled from contempoiary records, along with more
accurate information regarding the part he played in the
Tory interests, and the reason for his departure for Italy.
Undoubtedl}", when at Oxford, he shared the strong Jacobite
leanings of the rest of his family. Readers familiar with
Graham's delightful Social Life in Scotland in the Eighteenth
Century, and the scarcity of money among the Scottish
landed gentry, will appreciate the tone of the letter to his
father of June 1715, quoted in full in my sketch.
Whether he ever attended the University of Glasgow is
a moot point. Personally, I am inclined to think that he did,
for it was then the fashion to enter the University at a much
earlier age than now, and he was already about eighteen
years of age when he proceeded to Oxford.
Very little is known regarding his stay in Venice and the
date of his return to Britain ; but his private letters show
that when he took up residence in London he was on intimate
terms of friendship with Sir Isaac Newton and other dis-
tinguished scholars in the capital.
I have taken the opportunity here to add — what has
hitherto not been attempted — a short account of Stirling's
published works, and of their relation to current mathematical
thought. In drawing up this account, I had the valuable
viii PREFACE
assistance ot" Professor E. T. Wliittaker's notes on Part I of
Stirlino-'s MeUiodus Different lalit^, wliich he kindly put at my
disposal.
Stirlino-'s intluence as a niatlicniatician of pi'ofonml
analytical skill has been a notable feature within the inner
circle of mathematicians. Witness, for example, the tribute
of praise rendered by Laplace in his papers on Prol»al)ility
and on the Laws of Functions of very large numbers. Binet,
in a celebrated memoir on Definite Integrals, has shown
Stirling's place as a pioneer of Gauss. Gauss himself had
most unwillingl}^ to make use of Stirling's Series, though its
lack of convergence was aiiathema to him, IMore recentl}',
Stirling has found disciples among Scandinavian mathema-
ticians, and Stirling's theorems and investigations have been
chosen Ijy Professor Nielsen to lay the foundation of his
Monograph on Gamma Functions.
The Letters, forming the scientific correspondence of Stirling
herewith published, make an interesting contribution to
the history of mathematical science in the first half of the
eighteenth centur}-. I have little doubt that suitable research
would add to their number. I have endeavoured to reproduce
these as exactly as possible, and readers w^ill please observe
that errors which may be noted are not necessarily to be
ascribed to negligence, either on my part or on that of the
printer. For example, on page 47, the value of 7r/2 given
by De Moivre's copy of Stirling's letter (taken from the
Miscelkmea Amdytica) is not correct, being 1-5707963267U,
and not 1-5707963279 as there stated.
A few notes on the letters have been added, but, in the
main, the letters have been left to speak for themselves.
I am deeply grateful for the readiness w^ith which the
Garden letters were placed at my disposal by Mrs. Stirling,
Gogar House, Stirling. 1 am also indebted to the University
of Aberdeen for permission to obtain copies of Stirling's
letters to Maclaurin.
In the troublesome process of preparing suitabh^ manuscript
for the press, I had much valuable clerical assistance from
my sister. Miss Jessie Tweedie.
PREFACE ix
Of the man}' friends who have helped to lighten my task
I am particuUirly indebted to Dr. C. G. Knott, F.R.S., and to
Professor E. T. Whittaker, F.R.S., of Edinburgli University;
also to Professor George A. Gibson, of Glasgow Universitj%
who gave me every encouragement to persevere in my
research, and most willingly put at my disposal his mature
criticism of the mathematicians contemporary with Stirling.
i'"acsimile reproductions of letters by James Stirling and
Colin Maclaurin have been inserted. These have never before
appeared in published form, and will, it is hoped, be of interest
to students of English or Scottish history, and to mathematical
scholars generally.
The heavy cost of printing during the past year would
have made publication impossible but for the generous
donations from the contributors mentioned in the subjoined
list of subscribers, to whom I have to express my grateful
tlianks.
CHARLES TWEEDIE.
LIST OF SUBSCRIBERS
The Trustees of the Carnegie Trust for Scotland (£50).
Subscriptions, to the total value of £70, from
Captain Archibald Stirling, of Kippen.
General Archibald Stirling, of Keir.
Sir John Maxwell Stirling- Maxwell, Bart., of Pollok.
John Alison, MA., LL D., Headmaster of George Watson's
College, Edinburgli.
George A. Gibson, M.A., LL.D.. Professor of Mathematics,
Glasgow University.
E. M. Horsburgh, M.A., D.Sc, A.M.I.C.E., Reader in Technical
Mathematics. Edinburgh L^niversity.
William Peddie, D.Sc, Professor of Physics, University College,
Dundee.
E. T. Whittaker, D.Sc, F.R.S., Professor of Mathematics,
Edinburgh University.
X LIST OF SUBSCRIBERS
Subscriptions, to the total v^alue of £10, from
A. G. Burgess, M.A., B.Sc., Rector of Rothesay Academy.
Archibald Campbell, M.A., LL.B., Writer to the Si^-net,
36 Castle Street, Edinburoh.
Jas. H. Craw, Estj., Secretar}^ of the Berwickshire Naturalists'
Club, West Foulden, Berwick-on-Tweed.
Alexander Morgan, M.A., D.Sc, Director of Studies, Edinburgh
Provincial Training Centre.
George Philip, DSc, Executive Officer, Ross and Cromarty
Education Authority.
Rev. A. Tweedie, M.A., B.D., Maryculter.
Mrs. C. E. Walker, M.A., Villa Traquair. Stormont Road,
Highgate, London.
CONTENTS
PAGE
LIFE 1
WORKS 23
CORRESPONDENCE 51
FACSIMILES
Facsimile of last page of Letter by
Stiulixg to his Father, 1715 (pages 6-7) Frontisinece
Facsimile of last page of Letter by
Maclaurin to Stirling, 1728 (Letter No. 1)
facing p. ^7
COAT OF AKMS OF TlIK STIKLINGS OF GARDEN,
LIFE OF JAMES STIRLING
James Stirling, the celebrated mathematician, to whose
name is attached the Theorem in Analysis known as Stirling's
Theorem, was horn at Garden in the comity of Stirling,
Scotland, in 1692. He was a member of the cadet branch
of the Stirling family, usually described as the Stirling's of
Garden.
The Stirling family is one of the oldest of the landed
families of Scotland. They appear as proprietors of land
as early as the twelfth century. In 1 1 80, during the reign
of William the Lion, a Stirling acquired the estate of Cawder
(Cadder or Calder) in Lanarkshire, and it has been in the
possession of the family ever since. Among the sixty-four
different ways of spelling the name Stirling, a common one
in those early days, was a variation of Striveling.
In 1448, the estate of Keir in Perthshire was acquired by
a Stirling. In 1534 or 1535 these two branches of the family
were united by the marriage of James Striveling of Keir
with Janet Striveling, the unfortunate heiress of Cawder.
Since that time the main family has been, and remains,
the Stirling's of Keir and Cawder. By his second wife, Jean
Chisholm, James had a family, and of this family Elizabeth,
the second daughter, married, in 1571-2, John Napier of
Merchiston, the famous inventor of logarithms, whose lands
in the Menteith marched with those of the Barony of Keir.
This was not the first intermarriage between the Napiers
and the Stirlings, for at tlie former Napier residence of
Wright's Houses in Edinburgh (facing Gillespie Crescent),
there is preserved a stone the armorial bearings on which
record the marriage of a Napier to a Stirling in 1399.
Earl}^ in the seventeenth century Sir Archibald Stirling
of Keir Ijought the estate of Garden, in the parish of Kippen
(Stirlingshire), and in 1613 he gave it to his son (Sir) John
Stirling, when Garden for the first time became a separate
2 LIFE OF JAMES STIRLING
estate of a Stirling. Tlic tsoii of John, Sir Arcliiltald Stirlint;,
was a conspicuous Royalist in the Civil War, and was heavily
fined hy Cromwell; but his loyalty was rewarded at the
Restoration, and he ascended the Scottish bench with the title
of Lord Garden. Lord Garden, however, succeeded to the
estate of Keir, and his younger son Archibald (1651-1715)
became Laird of Garden in 1G68.
Archibald's eventful career is one long chapter of mis-
fortunes. Like the rest of the Stirlings he adhered loyally
to the Stuart cause. In 1708, he took part in the rising
called the Gathering of the Brig of Turk. He was carried
a prisoner to London, and then brought back to Edinburgh,
where he was tried for high treason, but acquitted. He died
in 1716, and thus escaped the penalty of forfeiture that
weighed so heavily on his brother of Keir. He was twice
married. By his first wife he had a son, Archibald, who
succeeded him, and by his second marriage, with Anna,
eldest daughter to Sir Alexander Hamilton of Haggs, near
Linlithgow, he had a family of four sons and five daughters.
James Stirling, the subject of this sketch, and born in 16[):2,
was the second surviving son of this marriage. (The sons
were James, who died in infancy ; John, who acquired the
Garden estate from his brother Archibald in 1717; James,
the mathematician ; and Charles.)
The Armorial Bearings of the Garden ^ branch of the
Stirlings are :
Shield: Argent on a Bend azure, three Buckles or : in chief,
a crescent, gules.
Crest : A Moor's Head in profile.
Motto : Gang Forward.-
YOUTH OF STIRLING
Oxford
Save for the account given by Ramsay of Ochtertyre
(Scotia lul and Scotsmen, from the Ochtertyre MSS.),wh\ch is not
trustworthy in dates at least, little is known of the early
* Garden, pronounced (Jarden, or Gardenne.
* Gan<r forward ; Scotlce for Allez en nvant.
STIRLING AT OXFORD 3
years and education of Stirlint^^, prior to his journey to Oxford
University in 1710.
Ramsay, it is true, says that Stirhng studied for a time
at Cdas<;o\v University. This would have been (|uitc in
accordance with Stirling tradition, for those of the family
who became students had invariably begun their career at
Glasgow University ; and the fact that Stirling was a Snell
Exhibitioner at Oxford lends some colour to the statement.
But there is no trace of his name in the University records.
Addison, in his book on the Snell Exhibitioners, states that
' Stirling is said to have studied at the University of Glasgow,
but his name does not appear in the Matriculation A Ibum '.
From the time that he proceeds on his journey to Oxford
his career can be more definitely traced, though the accounts
hitherto given of him require correction in several details.
Some of the letters written by him to his parents during this
period have fortunately been preserved. This fact alone
sufficiently indicates the esteem in which he was held by his
family, and their expectation of a promising futui-e for the
youth. In one of these he narrates his experiences on the
journey to London, and his endeavour to keep down expenses:
' I spent as little money on the road as I could. I could spend
no less, seeing I went with such companj^, for they lived on
the best meat and drink the road could afford. Non of them
came so near the price of their horses as I did, altho' they
kept them 14 days here, and payed every night 16 pence
for the piece of them.' He reached Oxford towards the close
of the year 1710. He was nominated Snell Exhibitioner on
December 7, 1710, and he matriculated on January 18, l/jy,
paying £7 caution money. On the recommendation of the
Earl of Mar he was nominated Warner Exhibitioner, and
entered Balliol College on November 27, 1711. In a letter to
his father of the same year (February 20, 1711) he gives some
idea of his life at Oxford : ' Everything is very dear here. My
shirts coast me 14 shillings Sterling a piece, and they are so
course I can hardly wear them, and I had as fit hands for buy-
ing them as I could.' . . . ' We have a very pleasant life as well
as profiteable. We have very much to do, but there is nothing
here like strickness. I was lately matriculate, and with the
help of my tutor I escaped the oaths, but with much ado.'
B 2
4 LIFE OF JAMES STIRLING ■
He thus ln'^an academic life at Oxford in n-ood s])ints, but
us a nou-juiiiii; stiuleut. At tlus period Oxford University
•svas not conspicuous for its intellectual activity. The Fellows
f-eeui to have led lives of cond'ortal)le ease, without paying
much regard to the requirements of the students under
their care.
As we shall see in Stirling's case, the rules imposed upon
Scholars were very loos-ely applied, and, naturally, complahit
was made at any stringency later. At the time we speak of
political ([Uestions were much in the thoughts of both students
and college authorities. The University had always been
faithful to the house of Stuart. It had received benefits from
James I.
For a time Oxford had been the head-(iuarters of King
Charles I during the Civil Wai', and his cavaliers were
remendjered with regret when the town was occui)ied by the
Parliamentary forces, and had to endure the impositions of
Cromwell. At the time of Stirling's entry the reign of Queen
Anne was drawing to a close. Partisan feeling between
Whigs and Tories was strong, and of all the Colleges Balliol
was most conspicuously Tory. According to Davis (Hlstorij
of Balliol College) Balliol ' was for the first half of the
18th century a stronghold of the most reactionary Toryism',
and county families, anxious to place their sons in a home
of sound Tory principles, naturally turned to Balliol, despite
the fact that Dr. Baron, the Master, was a stout Whig. It is,
therefore, abundantly clear that Stirling had every reason to
be content with his political surroundings at Balliol, with
what results we shall see presently. Perhaps the best picture
of the state of affairs is to be gathered from the pages of the
invaluable Diary ofT. Heariie, the anti(|uarian subdibrarian of
the Bodleian. For Hearne all Tories were 'honest men', and
nothing good was ever to be found in the ' \'ile Whigges'. His
outspoken Tory sentiments led to his being deprived of his
office, and almost of the privilege of consulting books in the
Library, though he remained on familiar terms with most of
the resident Dons.
Luckily for us, James Stirling was one of his acquaintance,
and mention of Stirling's name occurs frequently enough
to enable us to form some idea of his career. Doubtless
STIRLING AT OXFORD 5
their common boiul of sympathy arose from their Tor}^ nay
their Jacobite, principles, but it speaks well I'or the intellectual
vigour of the younger man that he associated with a man of
Hearne's scholarship. Moreover, Stirling must have been
a diligent student, or he could never have acquired the
scholarship that bore its fruit in 1717 in the production of
his Liaeae Tertll Ordinis, a W'Ork which is still a recognized
connnentary on Newton's Eauineration of Curves of I he Third
Order. But he was not the sort of man to be behindhand
in the bold expression of his opinions, and he took a leading
part among the Balliol students in the disturbances of
1714-16.
The accession of George of Hanover to the British throne
was extremely unpopular in Oxford, and Hearne relates how
on Maj" 28, 1715, an attempt to celebrate the King's birthday
was a stormy failure, while rioting on a large scale broke out
next day.
' The people run up and down, crying. King James the
Third! The True King, No 'usurper! T/ie Duke of Ormond !
isic, and healths were everywhere drunk suitable to the
occasion, and every one at the same time drank to a new
restauration, which I heartily wish may speedily happen.' . . .
'June d. King George lieing informed of the proceedings
of the cavaliers at Oxford, on Saturday and Sunday (May 28,
29), he is very angry, and by his order 'J'ownshend, one of
the Secretaries of State, hath sent rattling letters to
Dr. Charlett, pro- vice-chancellor, and the Mayor. Dr. Charlett
shewed me his this morning. This lord Townshend says his
majesty (for so they will stile this silly usurper) hath been
fully assured that the riots both nights were begun by scholars,
and that scholars promoted them, and that he (Dr. Charlett)
was so far from discountenancing them, that he did not
endeavour in the least to suppress them. He likewise
observed that his majesty was as well informed that the
other magistrates w^ere not less remiss on these occasions.
The heads have had several meetings upon this atl'air, and
they have draw^n up a programme, (for they are obliged to do
something) to prevent the like hereaftei-; and this morning-
very early, old Sherwin the yeoman bearlle w^as sent to
London to represent the truth of the matter.'
These mea.sures had a marked effect upon the celebration
on June 10 of 'King James the Illd's' birthday. Special
6 LIFE OF JAMES STIRLING
precautions were taken to prevent a riotous outbreak. ' So
that all honest men were obliged to drink King James's
health, and to shew other tokens of loyalty, very privately in
their own houses or else in their own chambers, or else out
of town. For my own part I walked out of town to Fox-
comb, with honest Will Fullerton, and Mr. Sterling, and
Mr. Eccles, all three non-juring civilians of Palliol College,
and with honest j\Ir. John Leake, formerly of Hart Hall, and
Rich. Clements (son to old Harry Clements the Ijookseller)
he being a cavalier. We were very merry at Foxcombe, and
came home between nine and ten,' etc. Several of the party
were challenged on their return to Oxford, but no further
mention is made of Stirling.
On August 15 there was again rioting at Oxford, in which
a prominent part was taken by scholars of Balliol. There
can be little doubt that Stirling was implicated, though he
seems to have displayed a commendable caution on June 10
by going out of town with a man so well known as Hearne.
His own account of current events is given in the following
letter to his father, which is the only trace of Jacobite corre-
spondence with Scotland that has been preserved, if it can be
so termed : —
Oxon 23 July 1715.
Sir,
I wrote to you not long ago, but I have had no letter
this pretty while. The Bishop of Rochester and our Master
have renewed an old quarrell : the Bishop vents his wrath
on my countrymen, and now is' stopping the paying of our
Exhibitions: it's true we ought to take Batchelours degrees
by the foundation of these exhibitions, and quite them when
we 5ire of age to go into orders : Rochester s^tands on all those
things, which his Predecessours use not to mind, and is
resolved to keep every nicety to the rigor of the statute ;
and accordingly he hath stoped our Exhibitions for a whole
year, and so ows us 20 lib. apiece, he insists on knowing our
ages, degrees, and wants security for our going into orders.
1 suppose those things may come to nought in a little while,
tho IJishop is no enemy to our pi'inciples. In the meantime
I've borrowed money of my friends till I'm ashamed to borrow
an}'' more. I was resolved not to troul)le you while I could
otherwise subsist ; but now I am forced to ask about 5 lib. or
what in reason j^ou think fit to supply my present needs :
STIRLING AT OXFORD 7
for ye little debts I have 1 can delay tlieni I hope till the
i;ood humor shall take the Bishop. I doubt not to have the
money one time or another, it's out of no ill will against us that
he stops it, but he expects our wanting the money will make us
solicite our Master to cringe to him, which is all he wants.
No doubt you know what a generall change of the affec-
tions of the people of England the late proceedings hath
occasion : the mobbs begun on the 28 of May to pull down
meeting houses and whiggs houses, and to this very day they
continue doing the same, the mobb in Yorkshire and Lanca-
shire amounted to severall thousands, and would have beat
of the forces sent against them had they not been diswaded
V>y the more prudent sort, and they are now rageing in
Coventr}' and Baintry : so (as the court saith) the nation is
just ripe for a rebellion. There were severall houses of late
at London searched for the Chevalier, the D. of Berwick and
M"" Lesly. Oxford is impeached of high treason and high
crimes and misdemanners and is now in the Touer, a little
while ago both Whiggs and Tories wished him hanged, but he
has gained some tories to stand his friends in opposition
to the Whiggs. They cant make out enough to impeach the
rest the}' designed. I had a letter from Northside ^ lately.
I shall delay an answere till I have the occasion of a frank.
My cousin James sent me a letter the other day from Amster-
dam, he is just come from the Canaries, and designs to return
there without coming to Britain, he remembers himself very
kindly to you and all friends with you. I give my humble
duty to you and my mother and my kind respects to my
l)]*others sisters and all my relations
I am Sir
Your most dutifuU son
Jas. Stirling.
It was in the same year (1715) that Stirling first gave
indications of his ability as a mathematician. In a letter - to
Newton, of date Feb. 24, 1715, John Keill, of Oxford, mentions
that the problem of orthogonal trajectories, which had been
proposed by Leibidz, had recently been solved by ' Mr. Stirling,
an undergraduate here ', as well as by others.
The statement commonly made that Stirling was expelled
' James Stirling, son of the Laird of Northside (near Glasgow), is
>;]>ecially mentioneil in the List of Persons concerned in the Rebellion of
l7i5-6 (Scot. Hist. Soc.j.
~ Macclesfield, Citrrespondence of Scientijic Men, (5jT., vol. ii, ]). 421.
8 LIFE OF JAMES STIRLING
from Oxford for his Jacobite leanings, and driven to take
refuge in Venice, seems entirely devoid of foundation. Again
Hearne's Diary comes to our aid, and indicates that Stirling-
was certainly under the observation of the government
authorities : —
'1715 Dec. 30 (Fri)
On Wednesday Night last M"" Sterling, a Scotchman, of
Balliol Coll. and M'" Gery, Gentleman Commoner of the same
College, were taken up by the Guard of the Souldiers, now
at Oxford, and not released till last night. They are both
lionest, non-juring Gentlemen of my acquaintance.'
Also :
' 1716
July 21 (Sat.) One M' Sterling, a Non-juror of
Bal. Coll. (and a Scotchman), having been prosecuted for
cursing K. George (as they call the Duke of Brunswick), he
was tryed this Assizes at Oxford, and the Jury brought him
in not guilty.'
The Records of Balliol bear witness to his tenure of the
Snell and Warner Exhibitions down to September, 1716.
(Also as S.C.L/^ of one year's standing in September, 1715.
and as S.C.L. in September, 1716.) There is no indication
of his expulsion, though the last mention of him by Hearne
informs us that he had lost his Scholarship for refusing to
take "the Oaths'.
'1717.
March 28 (Fri)
M' Stirling of Balliol College, one of those turned out of
their Scholarships upon account of the Oaths, hath the otter
of a Professorship of Mathematicks in Italy, w^^ he hatli
accepted of, and is about going thither. This Gentleman is
printing a Book in the Mathematical way at the Theatre.^ '
We shall see presently that Stirling found himself compelled
to refuse the proffered Chair. The circumstances in which
he had this ofler are somewhat obscure ; and whether he
^ S.C.L. was a Degree (Student of Civil Law) i^arallel to that of B A.,
just as that of Bachelor of Civil Law (B.C.L.) is parallel to that of M.A.
The degrer has long been abolished, but its possession would suggest that
Stirling had at one time the idea of adopting the profession of his grand-
father. Lord Garden.
'■^ The Sheldonian Theatre, Oxford.
STIRLING AT OXFORD 9
played any part in the Newton-Leibniz controversy is not
certain. In the later stages of the controversy an inter-
mediary between Leibniz and Newton was found in the
Abb^ Conti, a noble Venetian, born at Padua in 1677, who,
after spending nine years as a priest in Venice, gave up the
Church, and went to reside in Paris, where he became a
favourite in society. In 1715, accompanied by Montmort,
he journeyed to London, and received a fiiendly welcome
from Newton and the Fellows of the Royal Society. In
a letter^ to Brook Taylor in 1721, Conti relates how
' M'' Newton me pria d'assembler a la Soci^te les Ambassa-
deurs et les autres strangers'. Conti and Nicholas Tron,
the Venetian Ambassador at the English Court, became
Fellows at the same time in 1715.
How Conti came to meet Stirling is unknown to us ; but
he must have formed a high opinion of Stiiling's ability and
personal accomplishments, for Newton in a letter quoted by
Brewster [Life of Neivtoa, ii, p. 308) querulously charges
Conti with ' sending M"". Stirling to Italy, a person then
unknown to me, to be ready to defend me there, if I would
liave contributed to his maintenance '. The fact that Newton
was a subscriber to Stirling's first venture, Lineae Tertii
Ordinis Keutonicmae, sive Illustratio Tract aius D. Neutoni
De Enumfieratione Linearuni Tertii Ordinis, and doubtless
the ' Book ' mentioned by Hearne, would suggest that Newton
had met Stirling before the latter had left England. This
little book is dedicated to Tron, and it was on Tron's invita-
tion that Stirling accompanied him to Italy with a view to
a chair in one of the Universities of the Republic. The long
list of subscribers, the majority of whom were either Fellows
or Students at Oxford, bears eloquent testimony to the repu-
tation he had acquired locally at least as a good mathe-
matician. The book was printed at the Sheldonian Theatre,
and bears the Iinpririiatur, dated April 11, 1717, of
John Baron, D.D., the Vice-Chancellor of the University, and
Master of his own College of Balliol, who was also subscriber
for six copies. Of the subscribers, forty-five are associated
with Balliol. Richard Rawlinson, of St. John's, was also a
^ Printed in the posthumous ContempJatio Philosophica of Brook
Taylor.
10 LIFE OF JAMES STIRLING
subscriber, and W. Clements, the bookseller in London, took
six copies. Thus Stirling left Oxford after publishing a
mathematical work that was to earn him a reputation abroad
as a scholar.
Venice
From liis residence in Venice,^ Stirling is known in the
I'.imily Histoiy of the Stirling's iisJumesSth^llng the Venetian.
The invitation to Italy and the subsequent refuf-al are thus
recorded in the Rawlinson MSS. in the Bodleian (materials
collected by Dr. Richard Rawlinson for a continuation of
Ward's Atheiuie Oxoniense^ up to 1750):
* Jacobus Stirling, e coll. Baliol, exliibit. Scot, a Snell. jura-
ment. R. G.^ recus. 1714, et in Italiam Nobilem virum
Nicolaum Tron, Venetiarum Reipublicae ergo apud Anglos
Legatum, secutus est, ubi religionis causa matheseos profes-
sorium munus bibi oblatum respuit.'
The religious difficulty must have been a serious blow to
Stirling's hopes, and placed him in great embarrassment, for
his means were of the scantiest. But adherence to the
Anglican Church was one of the most fundamental principles
of the Tories, which liad caused so mucli wavering in their
ranks for the Catholic Chevalier, and there was no getting
over the objection. We need not be surprised, therefore, that
he got into serious difficulties, from which he was rescued in
1719 by the generosity of Newton, who had. henceforward at
least, Stirling for one of his most devoted friends. Stirling's
' I have endeavoured to ascertain the university to wh'ch Stirlinj^
was called. Professor G. Loria has informed nie that it was very probably
Padua, Padua being the only i niversity in the Republic of Venice, the
Quaiiier Latin of Venice according to Renan. It had been customary to
select a foreigner for the chair of Mathematics. A foreigner (Hemnann)
held it, and resigned it in 1713. It was then vacant until 1716, when
Nicholas Bernoulli (afterwards Professor of Law at Bale) was appointed.
Profe^^sor Favaro of Palua confirms the above, and adds that possibly
some information nv'ght be gathered from the reports of the Venetian
Ambassador, or from the records of the Reformatorcs Studii (the patrons
of chidrs in a mediaeval university). To get this information it would be
necessary to visit Venice. My chief dithculty here is to reconcile the
date of Stirling's visit to Italy and the date of the vacancy. It may be
added that a College for Scotch and English students still flourished at
Padua at this time {nee also Evelyn's Diarij). C. T.
"^ Kinj? George.
AT VENICE 11
letter to Newton, expressino- his oi-atitiide, is here oivcii. It
lias boon copied from Brewster's f/ife of Newton.
Letter
Venice 17 Aug. 1719.
Sir
I had the honour of 3'our letter about five weeks after
the date. As j-our generosit}' is infinitely above my merite,
so I reackon myself ever bound to serve 3'ou to the utmost :
and, indeed, a present from a person of such worth is more
valued by me than ten times the value from another. I
humbly a^k pardon for not returning my grateful acknow-
ledgments before now. I wrote to M"" Desaguliers to make
my excuse while in the meantime I intended to send a supple-
ment to the papers I sent, but now I'm willing they be
printed as they are. being at present taken up with my own
affair here wherewith I won't presume to trouble 3'ou having
sent M"" Desaguliers a full account thereof.
I beg leave to let you know that M"" Nicholas Bernoulli
proposed to me to enquire into the curve which defines the
resistances of a pendulum when the resistance is proportional
to the velocity. I enquired into some of the most easy cases.
and found that the pendulum, in the lowest point had no
velocity, and consequently could perform but one half oscil-
lation, and then rest. Bernoulli had found that before, as
aho one Count Ricato, which I understood after I communi-
cated to Bernoulli what occurred to me. Then he asked me
how in that hypothesis of resistance a pendulum could be said
to oscillate since it only fell to the lowest point of the cycloid,
and then rested. So I conjecture that his uncle sets him on
to see what he can pick out of your writings that may any
ways be cavilled against, for he has also been very busy in
enquiring into some other parts of the Principles.
1 humbly beg pardon for this trouble, and pray God to
prolong your daies, wishing that an opportunity should offer
that I could demonstrate my gratefullness for the obligation:!
3'ou have been pleased to honour me with,
I am with the greatest respect Sir
Your most humble & most obedient serv'
James Stirling.
Venice 17 August 1719 n. st.
P.S. JVP Nicholas Bernoulli, as he hath been accused by
D"" Keill of an illwill towards j'ou, wrote you a letter some
time asTo to clear himself. But havino- in return desired me
12 LIFE OF JAMES STIRLING
to assure 3'ou that what was printed in the Acta Paris.
reUiting to 3'our 10 Pjop., lili. 2, was wrote before he had
been in En^^hmd sent to his friends as his private opinion of
the matter, and afterwards published witliout so much as Ins
know led <(e. He is willing to make a full vindication of him-
self as to that atlair whenever you'll please to desire it. He
has laid the whole matter open to me, and if things are as he
informs me D'' Keill has been somewhat harsh in his case.
For my part I can witness that I never hear him mention
your name without respect and honour. When he showed
me the Ada Eruditorinn where his uncle has lately wrote
against D*" Keill he showed me that the theorems there about
Quadratures are all corollarys from 3'our Quadratures ; and
whereas M"" John Bernoulli had said there, that it did not
appear by your construction of the curve, Prop. 4, lib. 2, that
the said construction could be reduced to Logarithms, he
presently showed me Coroll. 2 of the said Proposition, where
you show how it is reduced to logarithms, and he said he
wondered at his uncle's oversight. I find more modesty in
him as to your affairs than could be expected from a young
man, nej)liew to one who is now become head of M'' Leibnitz's
party ; and among the many conferences I've had with him
I declare never to have heard a disrespectful word from him
of any of our country but D"" Keill.
How long he lived in Italy after his letter to Newton is not
known; but life in the cultured atmosphere of Venice must
liave been, otherwise, very congenial. It was a favourite
haunt of the different members of the Bernoullian famil}'.
The earliest letter to Stirling of a mathematical nature that
has been preserved is one in 1719 from Nich. Pernoulli,
F.R S., at that time Professor in the University of Padua.
One is tempted to inquire whether Stirling did not meet
Bernoulli and Goldbach on the occasion of their visit to
Oxford in 1712. In the letter in question Bernoulli specially
refers to their meeting in Venice, and also eonve3's the
greetings of Poleni, Professor of Astronom}- at Padua At the
same time Kiccati was resident in Venice, which he refused
to leave when offered a chair elsewhere. Ramsay says that
Stirling made contributions to mathematics while resident in
Italy, copies of which he brought home with him : but I have
found no trace of them. The only paper of this period is his
Methodus Dljferentialis Neivioniana, published in the Ph'do-
AT VENICE 13
sophical Tr<(vs<(ctions for 1719, witli the object of elucidating
Newton's methods of Interpohition.
London
From 1719 to 1724 there is a o-ap in our information
regarding Sterling. But a fragment of a letter Ijy him to
his brother, Mr. John Stirling of Garden, shows that in July
17*24 he was at Cader (Cawder or Calder, where the family
of his uncle James, the dispossessed Laird of Keir, resided).
Early in 1725 he was in London, as a letter to his brother
John informs us (London, 5 June, 1725) when he was making
an effort towards 'getting into business'. 'It's not so easily
done, all these things require patience and diligence at the
beginning.' In the meantime, that he n)ay not be * quite idle '
he is preparing for the press an edition of . . .^ Astronomy
to which he is 'adding some things'; but for half a year the
money will not come in, and he hopes his mother will provide
towards his subsistence.
' So I cannot go to the country this summer but I have
changed my lodgings and am now in a French house and
fretjuent french Coffeehouses in order to attain the language
which is absolutely necessary. So I have given over thoughts
of making a living by teaching Mathematicks, but at present
I am looking out sharp for any chub I can get to support me
till I can do another way. S Isaac Newton lives a little way
of in the country. I go frequently to see him, and find him
extremely kind and serviceable in every thing I desire but he
is much failed and not able to do as he has done .... Diiect
your letters to be left at Forrest's Coffee House near Charing
Cross.'
Thus in 1725, at 32 years of age, Stirling had not yet
found a settled occupation which would furnish a competency.
This project of ' getting into business ' was given up, for, some
time after, he acquired an interest in Watt's Academy in
Little Tower Street, where (Did. Nat. Biog.) he taught
Mechanics and Experimental Philosophy. It was the same
Academy in which his countryman Thomson, the poet, taught
for six months from May 1726, and where the latter composed
portions of ' Summer '. For about ten years Stirling was
^ The name, unfortunately, is not legible.
14 LIFE OF .]AMES STIRLING
connected with the Academy, and to this ad(b-ess most of the
letters to him from contemporary niatliematicians, tliat luive
been preserved, were directed. They form part of a hirger
collection that was partly destroyed l)y lire, and early in the
nineteenth century they were nearly lost altoj^ether throu<^h
the carelessness of Wallace and Leslie of Edinburgh Univer-
sity, to whom' they had been sent on loan from Garden,
There are also a few letters to his fi-iends in Scotland from
which one can gather a certain amount of information. In
the earlier days of his struggle in London he may have had
to seek assistance from them, but as his circumstances im-
proved he showed as great a generosity in return. By 17129
he could look forward with confidence to the future, for by
that time he was able to wipe out his indel)tedness in con-
nection with his installation in the Academy, as the following
extracts from his letters show.
In a letter to his brother, dated April l/.'iS, he writes:
' I had 100 Lib. to pay down here when I came first to this
Academy, and now have 70 Lib. more, all this for Instruments,
and besides the expenses I was at in liting up apartments for
my former project still ly over my head.'
Again on July 22, 1729, he writes:
'Besides with what money I am to pay next Michaelmas I shall
have paid about 250 Lib. since I came to this house, for my
share of the Instruments, after which time I shall be in away
of saving, for 1 find my business brings in about 200 L. a year,
and is rather increasinof, and 60 or 70 L. serves me for cloaths
and pocket money. I designed to have spent some time this
summer among you, but on .second thoughts I choose to publish
some papers during my Leisure time, which have long lain by
me. But I intend to execute my design is seeing you next
sunmier if I find that my affairs will permit.'
He had always a warm .side for his friends in Scotland, and
his letters to them are written in a bright and cheerful style.
The reference to Newton is the only one he makes regarding
his friends at the Royal Society, and the 'papers' he speaks
of publishing are almost certainly his well-known Treatise
the ]\fcUioilus Differentia lis (17 30), the first part of which he
had drawn up some eight or nine years before (vide a letter to
Cramer). He was admitted to the Roj^al Society in 1726,
AT LONDON 15
a distinction that put him on an etjual footing with the
.scientists that lived in, or fre(|iiented, London. It is most
probable that his acquaintance with Maclaurin began at this
time. They were both intimate friends of Newton, and
fervent admirers of his genius, and both eagerly followed
in his footsteps. Letters that passed between them are
preserved at Garden and in Aberdeen L^niversity. The
o[)ening correspondence furnishes the best account we have
of the unfortunate dispute between Maclaurin and Campbell
regai'ding the priority of certain theorems in equations (vide
MalJi. Gazette, January 1919). Maclaurin placed great reliance
upon Stirling's judgment, and frequently consulted Stirling
while engaged in writing his Treatlte of Flnxlonis.
Their later letters are mainly concerned with their researches
upon the Figure of the Earth and upon the Theory of Attrac-
tion. In 1738, Stirling, at Maclaurin's special re<juest, joined
the Edinburgh Philosophical Society, in the foundation of
which Maclaurin had taken so prominent a part in 1737.
Maclaurin also begged for a contribution, but if Stirling gave
a paper to the Society it has not been preserved or printed.
In 1727 Gabriel Cramer, Profes&or of Mathematics at
Genev^a, received a welcome from the Royal Society on the
occasion of his visit to London. He formed a warm friendship
for Stirling, who was his senior by about twelve years, and
several of his letters to Stirling are preserved. A copy,
kept by Stirling, of a letter to Cramer furnishes interest-
ing information regarding his own views of his Methodus
Differeiitialis, and also regarding the date at which the
Supplement to De Moivre's Miacellanea Analytica was printed.
Stirling had sent two copies of his treatise to Cramer, one of
the copies being for Nich. Bernoulli, by this time Professor
of Law at Bale. Cramer liad requested to be the intermediary
of the correspondence between Bernoulli and Stirling in order
to have the advantage of their mathematical discussions.
A few letters from Bernoulli are preserved, the last bearing
the date 1733. In this letter Bernoulli pointed out several
errata in the works of Stirling, and observed the omission,
made by both Stirling and Newton, of a species in their
enumeration of Cubic Curves. Newton gave seventy-two
species, and Stirling in his little book of 1717 added four
16 LIFE OF JA1\IES STIRLING
more. But there were two additional species, one of which
was noted by Nicole in 1731. Murdocli in his Neiotoni
Genesis Garvarutn per Urnhraa (1740) mentions that Cramer
had told him of Bernoulli's discovery, hut without furnishinrr
a date. Bernoulli's letter not only conhrnis Cramei''s state-
ment, it also t;-ives undoubted precedence to Bernoulli over
Stone's discovery of it in 1736.
From 1730 onwards Stirling's life in London must have
been one of considerable comfort, as his 'aflairs' became
prosperous, while he was a familiar figure at the Royal
Society, where his opinions carried weight. According to
Ramsay he was one of the brilliant group of philosophers
that gathered round Polingbroke on his return from exile.
Of these Stirling most admired Berkeley. If he at all shared
the opinions of the disillusioned politician then he might still
be a Tory, but it was improbable that he retained any loyalty
to the Jacobite cause. When the Rebellion broke out in 1745
there is no trace of Stirling being implicated, though his uncle
of Cawdor was imprisoned by the government and thus kept
out of mischief His studies were now directed towards the
problem of the Figure of the Earth, the discussion of which
liad given rise to two rival theories, (i) that of Newton, who
maintained that the Earth was flatter at the Poles than at
the Equator, and (ii) that of the Cassirjis, who held exactly
the opposite view.
In 1735 Stirling contributed a short but important note on
the subject which appeared in the Philosophical Transactions
{vide Todhunter's History of the Theory of Attraction and the
Figure of lite Earth).
Return to Scotland
In 1735, a great change in his circumstances was occasioned
by his appointment to the Managers!] ij) of the Leadhills Mines
in Scotland.
A more complete change from the busy social life of
London to the monotonous and dreary moorland of Leadhills
can hardly be imagined. At first he did not break entirely
with London, but in a year or two he found it necessary to
reside permanently in Scotland, and a letter from Machin
to him in 1738, w^ould suggest that he felt the change keenly.
RETURN TO SCOTLAND 17
He was now well over forty years of age, l)ut, nothing daunted,
he .set liiniself to the discharge of his new duties with all the
energy and ability at his command.
The letters he exchanged with Maclaurin and Machin show
that his interest in scientific research remained unabated,
though the want of time due to tlie absorbing claims of his
new duties is frequently brought to our notice. He appears
to have discovered further important theorems regarding the
Figui'e of the Earth, which Machin urged him to print, but
he never proceeded to publication. His reputation abroad,
however, led the younger school of rising mathematicians
to cultivate his accjuaintance by correspondence, and to this
we owe a letter from Clairaut, and also a long and interest-
ing letter from Euler. Clairaut (1713-65), who had shown
a remarkable precocity for mathematics, was a member of
the French Commission under Maupertuis, sent out to Lapland
to investigate the length of an arc of a meridian in northern
latitudes, a result of which was to establish conclusively
Newton's supposition as against the Cassinians. As Voltaire
put it : Maupertuis ' avait aplati la Terre et les Cassinis.'
While still in Lapland Clairaut sent to the Royal Society
a paper, some of the conclusions in which had been already
connnunicated by Stirling. An apology for his ignorance of
Stirling's earlier publication furnished Clairaut with the
ground for seeking the acquaintance of Stirling in 1738, and
requesting his criticism of a second paper on the Figure of
the Earth.
The con-espondence with Euler in 1736-8, in connection
with the Euler-Maclaurin Theorem, has already been referred
to by me in the Math. Gazette. Euler (1707-83) is the
third member of the famous Swiss school of mathematicians
with whom Stirling had correspondence. From his letters
to Daniel Bernoulli (Fuss, Gorr. Math.) it is quite clear that
Euler was familiar with Stirling's earlier work.
Stirling was so much impressed by Euler's first letter that
he suggested that Euler should allow his name to be put up
for fellowship of the Royal Society. Euler's reply, which is
fortunately preserved, is remarkable for its wonderful range
of mathematical research ; so much so that Stirling wrote to
Maclaurin that he was 'not yet fully master of it.'
2447 C
18 LIFI^] OF JAMES STIRLING
Euler, who was at tlie tiiiiu iusLalled in Putrograd, ili<l not
then become a Fellow of the Iloyal Society. In 1741 he left
Ivussia for Berlin, where, in 1744, he was made Director of the
Mathematical Section of the Jjerlin Academy, and it is (juite
possible that he had a share in conferring upon Stirling the
honorary memVtership of tlie Academy in 1747. The informa-
tion is contained in a letter of that date from Folkes, P.R.S ,
conveying the message to Stirling with the compliments of
Maupertuis, the President, and the Secretary, De Formey.
The letter furnishes the last glimpse we have of Stirling's
connection with London. (He resigned his membership of the
Royal Society in 1754.)
Leadhills
Regarding Stirling's residence in Scotland we are fortunately
provided with much definite information. A detailed account
of his skilful management f the mines is given in the Gentle-
mans Magazine for 1853.^ He is also taken as one of the
best types of the Scotsmen of his day by Ramsay in liis
Scotland and Scotsmen,.
Ramsay, who always speaks of him as the Venetian, met
him frequently on his visits to Keir and Garden, and had
a profound regard for the courtly and genial society of the
Venetian, who by his long residence abroad and in London
had acquired to a marked degree la i/rande manicre, without
any trace of the pedantry one might have expected. Ramsay
also narrates several anecdotes regarding Stirling's keen sense
of humour.-
The association between V^enice and the l^eadhills in
Stirling's career is very remarkable. According to Ramsay,
before Stirling left Venice, he had, at the reijuest of certain
London merchants, acquired information regarding the manu-
facture of plate glass. Indeed, it is asserted by some that
owing to his discovery he had to flee from Venice, liis life
being in danger, though Ramsay makes no mention of this.
Be that as it may, his return to London paved the way for
further acipiaintance, with the result that about 1735 the
Scots IMining Company, which was controlled by a group
' 'Modern History of Leadliillh'. "^ I. c, vol ii.
LEADHILLS 19
of Loudon merchants, associated witli the Sun Fire Office,
selected him as manager of tlie Lcadhills mines. The company
had been formed some twenty years previously with the object
of developing the mining for metals, and had for managing
director Sir John Erskine of Alva, a man of good ideas, hut
lacking in business capacity to put them into practice. Leases
were taken in different parts of the country, but were all given
up, with the exception of that of the Leadhills mines, the
property of the Hopetoun family, which had already been
worked for over a century. When Stirling was appointed
the affairs of the Company were in a bad way.
For the first year or more Stirling only resided at the
mines for a few weeks, but about 1736 he took up definite
residence, devoting his energies entirely to the interests of
the Company. Gradually the debts that had accumulated
in ids predecessor's day were cleared off', and the mines became
a source of profit to the shareholders. But his scientific
pursuits had to be neglected. We find him, in his letters
to Maclaurin, with whom he still frequently corresponded,
complaining that he had no time to devote to their scientific
i-esearches, and when writing to Euler he tells him that
he is so much engrossed in business that he finds difficulty in
concentrating his thoughts on mathematical subjects in the
little time at his disposal.
The village in which he and the miners lived was a bleak
spot in bare moorland, nearly 1,300 feet above sea level. There
was no road to it, and hardly even a track. Provisions and
garden produce had to be sent from Edinburgh or Leith.
In spite of these disadvantages Stirling has left indelible
traces of his wise management, and many of his improvements
have a wonderful smack of modernity. The miners were
a rough, dissipated set of men, who had good wages but few
of the comforts of life. Stirling's first care was to add to
their comfort and to lead them by wise regulations to advance
their own physical and mental welfare.
Li the first place he carefully graded the men, and worked
them in shifts of six hours, so that with a six hours' day they
had ample time at their disposal. To turn their leisure to
profit they were encouraged to take up, free of charge, what we
should now call 'allotments', their size being restricted only
c 2
20 LIFE OF JAMES STIRLING
by the ability of the miners to cultivate. The j^ardens or
crofts produced fair crops, and in time assumed a value in
which the miner himself had a special claim, so that he could
sell his rioht to the ground to another miner without fear of
interference from the superior. In this way Stirling stimu-
la'ed their industry, while at the eame time furnishing them
with a healthy relaxation from their underground toil. The
mmers were subject to a system of rules, drawn up for their
guidance, by reference to which disputes could be amicably
settled. They had also to make contributions for the main-
tenance of their sick and aged. In 1740, doubtless with the
aid of Allan Ramsay, the poet, who was a native of the place,
a library was instituted, to the upkeep of which each miner
had to make a small subscription. Stirling is thus an early
precursor of Carnegie in the foundation of the free library.
When Ramsay of Ochtertyre visited Leadhills in 1790 the
library^ contained several hundred volumes in the different
departments of literature, and it still exists as a lasting
memorial to Stirling's provision for the mental improvement
of the miners.
On the other hand, Stirling's own re([uireuients were well
provided for by the Company, whose atfairs were so prosperous
under his control. They saw to it that he was well housed.
More than once they stocked his cellar with wines, while the
salary they paid him enabled him to amass a considerable
competency. When, with the increase of years, he became
(oo frail to move about with ease, they supplied him with
a carriaue.
The Glasgow Kettle
In the eighteenth century the rai)iilly ex])anding trade of
Glasgow and the enterprise of her merchants made it highly
desirable to have better water connnunication and to make
the city a Port, and in 175'2 the Town Council opened a
separate account to record the relative expenditure. The
' Of Stirling's private library two books have been preserved. One, on
Geometry, was presented to hitii by Bernoulli in 1719. The other (now
at Garden) is his co])y of Brook Taylor's Methodus Incremoitonim, which
he boujifht in 1725.
THE GLASGOW KETTLE 21
tii-st item in this account, which is headed ' Lock desi^n'd
upon the River of Clyde ', runs thus :
* Paid for a coniplinienfc made by the Town to James Stirling,
Mathematician for his service, pains, and trouble, in surveying
the River towards deepening by locks, vizt
For a Silver Tea Kettle and Lamp weighing 66^ oz
at 8/ per oz £26 10
For chasinu- & Enaravinu" the Towns arms 1 14 4
£^^8 4 4'
Stirling had evidently performed his task gratuitously but
with characteristic thoroughness; and to this day, when the
city holds festival, the Kettle is brought from Garden, where
it reposes, in grateful memory of the services that occasioned
the gift.
To this period there belongs only one paper by Stirling,
a very short article {Phil. Trans., 1745) entitled 'A Description
of a machine to blow Fire by the Fall of Water'. The
machine is known to engineers as Stirling's Engine, and
furnishes an ingenious mechanical contrivance to create a
current of air, due to falling water, sufficiently strong to blow
a forge or to supply fresh air in a mine. Its invention is
doubtless due to a practical difficulty in his experience as
a mining manager.
There is also preserved at Garden the manuscript of a treatise
by Stirling on Weights and Measures.
For thirty-five years Stirling held his managership. He
died in 1770, at the ripe age of seventy-eight, when on a visit
to Edinburgh to obtain medical treatment. Like Maclaurin
and Matthew Stewart, he was buried in Greyfriars Churchyard,
' twa' corps lengths west of Laing's Tomb V «'is the Register
Records grimly describe the locality.
By his marriage with Barbara Watson, daughter of
Mr. Watson of Thirtyacres, near Stirling, he had a daughter.
Christian, who married her cousin, Archibald Stirling of
Garden, his successor as manager of the mines ; and their
descendants retain possession of the estate of Garden.
' Laing'^ Tomb is a prominent mural tablet (1620) on the right wall
surrounding the churcb^-ard.
22 LIFE OF JAMES STIRLING
Thus closed a career filled with early romantic adventure
and brilliant academic distinction, followed in later years by
as marked success in the industrial field. As a mathematician
Stirling is still a livino- power, and in recent years there has
sprung up, more particularly in Scandinavian countries, quite
a Stirling cult. His is a record of successful achievement
of wliich any family might well be proud.
WORKS PUBLISHED BY J. STIRLING
(A)
ENUMERATION OF CUBICS
§ 1. His first publication, Lineae Tertii Ordinis Neutonianae
sive lilustratio Tradatas D. Neivtoni De Eimoneratlone
Linear am Tertii Ordinis. Cui suhjungitur, Solutio Trium
Prohlematum, was printed at the Sheldonian Theatre, at
Oxford, in 1717.
As the book^ is very scarce, I give a short account of its
leading contents.
By a transcendent effort of genius, Newton had, in the
publication of his Enumeration of Cubic Curves, in 1704, made
a great advance in the theory of higher plane curves, and
brought order into the classification of cubics.
He furnished no proofs of his statements in his tractate.
Stirling was the first of three mathematicians from Scotland
who earned for themselves a permanent reputation l;)y their
commentaries on Newton's work. Stirling proved all the
theorems of Newton up to, and including, the enumeration
of cubics. Maclaurin developed the organic description of
curves (the basis for which is given by Newton), in his
Geometria Organica (1720); and P. Murdoch ^ gave, in his
Genesis Gurvarum i^er Umbras (1740), a proof that all the
curves of the third order can be obtained by suitable pro-
jection from one of the five divergent parabolas given by the
equation ^ ■, ? » 7
^ 2/" = «i^ + bx^ + CX + d.
Stirling, in his explanatory book, follows precisely on the
lines suggested by Newton's statements, though I doubt
whether he had the assistance of Newton in so doing; for
" Edleston {CorresponfJene, &c., p. 2.3-5) refers to a letter from Taylor to
Keill, dated July 17, 1717, which gives a critique of Stirling's book.
^ Earlier proofs were given by Nicole and Clairaut in 1731 {Mem. de
I Acad, des Sciences).
24 WORKS PUBLISHED BY STIRLING
in that case why should he luivc stopped sliort \vith but hall'
of the theory ?
§ 2. Newton stated that the algebraic ecpiation to a cubic
can be reduced to one or other of the four forms (i) .ry'--\-cy,
or (ii) xy, or (iii) y"^, or (iv) y, = ax'-^ + hx'^ + rx + d ; and he
gave sufficient int'orniation as to the circumstances in -which
these happen.
The demonstration of this statement forms the chief diffi-
culty in the theory.
Stirling finds it necessary to devote two-thirds of his
little book of 128 pages to introductory matter. He bases
the analytical discussion on Newton's doctrine of Serie><,
and gives an adequate account of the use of the Parallelogram
of Newton for expanding y in ascending or descending powers
of X, X and y being connected by an algebraic equation. (He
also applies his method to fluxional or ditterential equations,
though here he is not always very clear.) With some pride
he gives on p. 32 the theorem ^
Let 2/ = ^ + ^^'' + ^^^'' + • • • '
then y may be expressed as
. xy x~y x^''y' „
y =^ A-\ *^ + ^, H , ., + &c.
^ 1 .rx 1.2 r-x^ 1.2.3. r'lr^
applicaljle when x is very large if r is negative, or when x is
very small if r is positive. As an example he establishes the
Binomial Theorem of Newton (p. 36).
Pages 41-58 are taken up with the general tlieor^^ of asymp-
totes. A rectilinear asymptote can cut the curve of degree ti
in, at most, n—2 finite points. If two branches of tlie curve
touch the same end of an asymptote, or opposite ends but on the
same side of the asymptote, then three points of intersection
go oti' to infinity.
A curve cainiot have more than u—l parallel asynq^totes,
and if it has n—\, then it cannot cut these in any finite point.
If the 2/-axis is parallel to an asymptote, the equation to the
curve can have no term in ?/". From this follows the inq)()rtant
corollary that the e(juation to a cubic curve ma}' alwa3s be
found in the form , , „ ,. , , , . ,
{x + a)y'' = yf^(x}+f.,{x).
^ Cf. Maclaurin's Theorem.
ENUMERATION OF CUBICS 25
For all lines of odd degree have real points at infinity.
Asymptotes may be found by the doctrine of sei-ies: but
not always.
Thus the quartie y ^ (ax^ + bx' + . . . + e)/ifx-' + (jx' + hx + k)
has the asymptote
ax hf— ag
" = 7^1^'
as found hy a series.
The rest of the asymptotes are given by x = a,
where a is any one of the roots of
fx^-V(jx^-Vhx + k = 0.
In the standard case of an ecjuation of degree n in x and y,
if we assume tlie series
y = Ax-vB+- + ^,-\-...
and substitute in the given equation we find, in general,
(1) an equation of degree n for A furnishing ii values of A,
(2) an equation involving A and B of the first degree in B,
(3) an equation in A, B, and C, of the first degree in C, &c.
So that in general- we may expect n linear asymptotes
y = Ax + B.
§ 3. Pages 58-69, with tlie diagrams, furnish quite a good
introduction to what we now call (jra'ph-tracing.
He thus graphs the rational function y = f(x) / (p{x) with
its asymptotes parallel to the y-i\\\^ found by ecjuating (p{x)
to zero.
The manner in which a curve approaches its asymptotes
is explained by means of series.
In the curves given by y — « + i^/ + . . . + A'*" there are only
two infinite branches which are on the same, or opposite, sides
of the ic-axis, according as n is even or odd. When x is large
the lower terms in x may be neglected as compared with kx^\
Then follows the graphical discussion of quadratic, cubic,
and quartie e(juations in x. The graph of 2/ = a;^ + "a? + h
shows that the roots of the corresponding quadratic equation
26
WORKS PUBLISHED BY STIRLING
in X arc real or imaginary according as the turning value
ot" y is negative or positive.
For the cuhic x''' + ax- + hx + c=Q he gives the excellent rule,
which has recently been resuscitated, that the three roots are
real and distinct only Avhen the graph of the corresponding
function has two real turning values opposite in sign. A similar
test is applied to discuss the reality of the roots of a quartic.
These results arc required later in the enumeration of cubic
curves.
On p. G9 he gives the important theorem that a curve of
degree n is determined by \n(n + 3) points on it.^
The demonstrations of Newton's general theorems in higher
plane curves are then given in detail.
An indication of some of these is interesting, and the
modern geometer \vill note the entire absence of trigonometry.
§ 4. Diameter Theorem.
Draw a line in a given direction to cut a curve in 1\,P^... P^^ ;
and find on it such that '^OP = 0.
As the line varies in position generates
a straight line.
Let the ecjuation to the curve be
y'' + {ax + h)f'-'^ + ... = 0. (1)
In the figure let AB = x, BC = y (so
that A is what we call the origin).
Take AF = —h/a\ and AE parallel
to BC, and equal to —h/n. Let ED = z,
DC =v; also let AB/ED = a.
Then x = occ,
y = DC—DB = V ■ 5
and substitution of these values in B leads to an ecjuation
V'' + v«- 2/2 («) &c. = 0,
in which the term in r"~' is awanting. Let D coincide with
and DC with OP. .: &c. Q. E. D.
Stirling adds the extensions, not given by Newton, to a
Diametral Conic, a Diametral Cubic, &c., corresponding to
when ^07] . OP, = 0, ^OP, . OP,. OP, = 0, S:c.
' Also stated by Hermann (Phoronotnia).
Fig. 1.
ENUMEKATION OF CUBICS 27
Neuions Rect((^i<lh Theorem for a Conic, and generalization.
The proof is made to dopciul on the theorem that il:
0(,, a.^, ... a, J are the roots of
(pix) = .(•" + ax"-'^ + . . . + A- = 0,
then (p(i)=(i-0(,)'i-o(.^-...ii-0(,).
In the case of the cubic
y^ + y- (ax + b) + y{cx^ + dx + e)+ fx^ + gx- + Itx + /• = 0.
XPi
Fm. 2.
Let F^OPr^, QiOQ., he drawn in fixed directions through
a point 0. Let i^i^ be the .r-axis, QiQ;. parallel to the y-axis,
and let be the point (^, 0).
Then 0Q,.0Q2.0(?3=/|'- + ryf + /.i + /.'
OP, . OP, .OF, = j (ft + ge + hi + k\
so that the quotient
OQ, . OQ, . OQJOP, . OP, . OP, = /(up to sign).
But a change to parallel axes does not change/. .•. Arc.
§ 5. After a brief enumeration of conies he proceeds to find
in Prop. XV (p. 83) the reduction of the equation of a cubic to
one or other of the four forms given by Newton.
The equation
{z + a) v^ = {hz^ + cz + d)v + e^ ^fz^ + gz + h ( 1 )
includes all lines of the third order, the r-axis being parallel
to an asymptote.
First Case. Let all the terms be present in (1).
Let A be the origin, AB any abscissa z, BO or BD the
corresponding ordinate v of the cubic. If F is the middle
point of CD
7> ET 1 / ^ ^^^ + cz+d
BF = \ (i\ + 1\,) = — J
28
WORKS PUBLISHED BY STIRLING
so that the locus of F is the conic
v= {h-J' + cz+(l)/2 z + a)
^vith real as3niiptotes GE •a\\^\ till.
Fig. 3.
Select these lines as new axes.
Call GE X, and EG or ED y.
Tlie cuhic o(jnation is of the same form as before, but EF
must =K/2x, where 7v is constant, by the nature of the
hyperbola. Therefore, the equation to the cubic is of the
^'oi'»i y^ - ey /x = ax^ + bx + c + d/x,
or xy'^ — ey = ax^ + hx'^ + cx + d. (I)
With a oood »"eal of inoenuity, the proof is indicated in the
other cases.
Prop. XVI (p 87).
When (I is positive in (I) all three asymptotes are real.
They are
(i) X = 0,
(ii) y = xVii +f>/ 2 -/(I,
(iii) y= —xVa — b/2va.
If I) = 0, the asymptotes are concuri'cnt.
ENUMERATION OF CUBICS 29
If I) zfz 0, they form a triangle, inside which any oval of the
cubic inuist lie, if there is an oval. The asymptotes (ii) and
(iii) cut on the .r-axis, which is also a median of the asymptotic
triangle. When c = 0, the point at infinity on the asjanptote
(i) is a point of inflexion, and conveisely : in that case the locus
of F reduces to a straight line, which is a ' diameter ' of the
curve. An inflexion at infinity and a diameter are always
thus associated. The condition that (ii) or (iii) cuts the curve
only at infinity is 6"- — 4((c' = ±^ae\/a.
Thus possible conditions for a diameter are
c = 0.
h" — \ac — iaeVa.
h'^ — 4a('= —^aeVa.
When any two of these are satisfied so is the third {a is
positive and not zero). Tiius a cubic may have no diameter, or
one diameter, or three diameters. It cannot have two.
§ G. The enumeration of cubics is then proceeded with in
the order given by Newton, to whose work the reader must
go for the figures, which are not given by Stirling. Newton
gave 72 species. To these Stirling added 4 species, viz. species
11, p. 99, species 15, p. 1 00, and on p. 102, species 24 and 25.
There still remained two species to be added (both arising
from the standard form xy'''' — ax^-'thx + c). One of them
was given by Nicole in 1731, and the other was communi-
cated by N. Bernoulli,^ in a letter to Stirling in 1733.
While sufficiently lucid, Stirling's reasoning is admirably
concise. He was never addicted to excess in the use of words,
and often drove home the truth of a proposition liy a well-
chosen example, especially in his later work.
The publication of his commentary on Newton's Cables gave
Stirling a place among mathematicians, and may have been
the ground on which he was invited by Tron to accept a chair
in Venetian l.erritor3^^
^ See note to Letter.
^ In connection with both Newton and Stirling see W. W. Rouse Ball
on 'Newton's (^Classification of Cubic Curves', London Math. Soc, 1891.
Another edition of Stirling's Lincae Terfii Ordinis was published in Paris
in 1787. (' Isaaci Newtoni Enunieratio Lincarum Tertii Ordinis. Sequitur
illustratio eiusdem tractatus lacobo Stirling.')
30 WORKS PUBLISHED BY STIRLING
(B)
JMETHODUS DIFFERENTIALIS. SIVE TRACTATUS
DE SUMMATIONE ET INTERPOLATIONE
SERIERUM INFINITARUM
■^ 7. The Mefhodus Differeidialis, as wc shall call it, is the
most important product of Stirling's genius, by which lie is
most generally known to mathematicians. The book is not,
as the title may suggest, a treatise on the Dilierential Calculus,
but is concerned with the Calculus of Finite Differences. It is
divided into: (1) the Introduction (pp. 1-13); (2) the Suvi-
mation of Series (pp. 14-84); (3) the Interpolation of Series
(pp. 85-153;.
In the Introduction he explains how the Series arc defined.
Denote the terms l)}' T, T' , T", &c., and write
s = T+r+r'+&c.
Suppose the terms arranged as ordinates to a line so that
consecutive terms are always at the distance unity.
Thus if T is at distance z from the origin, T' is at a distance
0+1, 2"' at distance z+2, &c. ; where z is not necessarily an
integer.
For example, in Brouncker's Series (p. 26)
1 1 ' 1
+ + — +...
1.23. 45. (3
any term is given by 1/45(5 +|) where 5 is, in succession,
2 ' ^ 2 ' "^2 ' ^^•
A series may sometimes be specified by a relation connecting
terms ;
e.g. if T'=-.^-^T,
then • y,.^5+;^+l^ ^^^^
s+1
Theorems of special interest arise when T can be ex-
pressed as
T^ A-^Bz + (Jz{z- \) + Bz{z-l}{z-2)+ c\:c.,
METHODUS DIFFEKENTIALIS 31
or as
T = A+ ^ + + Ac,
z 3(: + 1)
the latter bein*;- useful when z is a lar^-e number.
When T admits of either representation then after any
transformation it should be reduced again to the same form.
Thusif T ^ A+Bz + Gz{z~\)+...,
then Tz = {A+B)z^{B + 2C)z{z-\)
+ (C+3i))s(:-l)(^-2) + ...
To facilitate the reduction Stirling gives two formulae and
two numerical tables.
Let
x{x+\){x + 2) ...{x + n-\) ^ 6V*" + ^'„' *'*-'+...+ (^V' ^
and l/x{.c+\)...{x^,i-\)= 2 (-ir^nV ■«'''',
s=o
then
and ^=2 6V-"+Vs(.~+l)...(0 + r).
The first table (p. 8) furnishes the values of P/ for the lower
values of ii and s, and the second table (p. 11) the lower values
of CJ.
Owing to the importance of these results, and the applica-
tions which Stirling makes of them, it has been proposed by-
Professor Nielsen ' to call the numbers (7,/ the Stirling
Numhera of the First Species, and the numbers r„*' the Stirling
Numbers of the Second Species.
Nielsen has discussed their properties and indicated their
affinities with the Bernoullian numbers.
As an illustration Stirling deduces
1 1 1 - n
z'^ + nz z{z+l) z{z+l){z+2)
' Nielsen, Ann. di Mat., 1904 ; or Theorie der Gammafunktion (Teubner,
1906). An account in English is given by me in the Proc. Edin. Math. Soc,
1918-19. Lagrange used them in his proof of Fermat's Theorem.
32 WORKS PUBLISHED liY STIRLING
wliicli is L'tjuivalcnt to
1 _ 1 a <i((i + 1)
x — a~ X X {.c + 1 ) x{x + 1 ) (.c + 2) ■ " '
when it is usually spoken of as ^tlrllivjs Series; but it
had already been given before Stirling by Nicole and by
Montmort.
PARS I
SUMMATIO SeKIEHUM.
§ 8. Stirling explains that he is not so much concerned
with Scries the law of summation fur which is obvious or
well known, as with the transformation of slowly converging
series into scries tliat more rapidly converge, with their sum
to any desired degree of accuracy.
Let S = T ^ r+T"+...adoo,
S'= T'+T" + ...ad^,
S''= T"-\-...adrj,,&ii.
Any ditiercnce-equaticHi connecting *S', /S", ..., T, T', ...z,
may be transformed into another by writing for these, respec-
tively , ,,/ ,,„ rnff . , 1
But when the number of terms in the series is finite, he takes
T to be the last
{S= .., + T" + r + T),
so that >S"= S—T, and if S corresponds to s, >S" corresponds
to :-l.
On p. 16, he quotes a theorem of Newton,^ which furnishes
a key to several of the theorems that fijllow later in the
Mel hod as Dijfireidialls.
In modern garb it may be thus stated,
Z^>-\\-Z)'l-Kh =^'^^''-^^^^F{p + q, 1, p+\, Z).
where F((i, b, c, z) denotes the hypergeometric series
(6. h a{a+ 1) h{b ^ 1) ^
^■^ iTc^"^ i.2.c(cTTy"^ ^•••-
' See also p. 113 of Methodus Diffeicntkilis.
METH0DU8 DIFFERENTIALIS 33
When 5=1 we have, of cour.se, the Beta Function
Jo
Prop. I.
§9. If T = A + Bz + Cz{z-l) + ...
the sum of the first z terms is
A:+^Jz+l)z+^{z+l)z{z-l) + ...,
and Prop. II.
If r=^i^+ ^
z{z+l) z.z+1 .z + 2 ""
and s=T+T'+ . ad 00,
.1 ., A B C ,
then ^ = — H -^ ■ +, &c.,
z 2Z.Z+1 Sz.z+l .z + 2 '
were both given previously by Nicole and Montmort, but
Stirling carries their applications much further.
E.g. To sum
1 1 1
\- 1 f-
j2 ~ 2^ 3'-^ —
This Stirling effects in the following characteristic fashion
(pp 28, 29).
rp 1 1 1! 2! 3! „
52 z.z+l z.z+l .z + 2 z z + 3 &c.
Hence
o I 1! 2! „
^S = ~ + + + , &C.
z 2. Z.z+l 3. z.z+l. z + 2 '
Calculate >S' for 0=13.
■•• t|9 + iI6 + -- — -079,957,427.
Add thereto T + i + ... + m= 1-564,976,638.
The total is 1-644,934 065.
Stirling did not probably know that this is equivalent
to Itt^, until Euler sent him his well-known formulae for series
of the kind.
2447 X)
34 WORKS PUBLISHED BY STIRLING
Prop. III.
If r = a.-4^ + -A^ +...[,
( : z.z+1 )
then the sum (to infinity) is
^z^n\ '' , ^^-^^ , G-2BX l^
l{l-x)z {l-x)z.z+l {1-X)3.Z+1 .Z + 2) '\
where A, B, C, ... denote tlie coefficients of the terms preceding
those in which they occur. Thus
A ^(' 71 h — Ax
A = , B = 7- , &c.
His well-chosen example gives the summation of the Series
of Leibniz
^= 1-1+1-1 + .. .ad CO.
1,
Here T = (- 1)--^ ^ .^^ fo^j^j i,^ writing i, If, 2^, &c.
for z, so tliat 6=0, Sac. Calculate the sum for z = 12^ from
the formula. It is -020,797,471,9. Add thereto
l-^+.-.-is = -764,600,691^5,
so that the sum of the total series is -785,398,163,4, a result
which could never be attained by the simple addition of terms,
' id quod olim multum desiderabat Leihnitius '.
(Stirling sums the same series by another process on p. 66.)
This is an example of several numerical series, well known
in his day, the summation of which had hitherto proved
refractory, and which Stirling can sum to any desired degree
of accuracy.
Prop. IV is concerned with the problem of proceeding from
an equation in S and S\ say, to an equation in T and T'.
E. g. From (z-n)8= (s - 1 ) S\
he finds {z-n)T = zT'.
Prop. V is taken up with applications of IV.
§ 9. Prop. VI gives an interesting theorem (pp. 37-8).
If the equation connecting S and S' is
S{z' + az'-' ^- ...) = mS' {z' + kz'~' + ...),
METHODUS DIFFERENTIALIS 35
then the lust of the sums will be finite both ways only when
m = 1 and k = a.
In other words the infinite product
00
1 +
a
+ ...
11
'li
I
n
,!= 1
e +
+ ...
is finite both ways only when e = 1 and a — f.
This is one of the earliest general tests for the convergence
of an Infinite Product of which Wallis (' Wallisius noster '
as Stirling calls him in his earlier book) furnished an illustra-
tion, with rigorous proof, in the formula
TT _ 2 . 2 . 4 . 4 . 6 . 6 . . .
2 ~ 1.3.3.5.5.7... '
published in his Arltltvietira Iiijiiiitorum in 1655.
Prop. VII gives a remarkable transformation of a series, in
the discussion of which he has occasion to solve a Difference
Equation by the method so universally employed nowadays
of representing the solution by an Inverse Factorial Series.
As stated by Stirling it runs thus :
If the equation to a series is
{z-n)T+{in-l)zr{= 0),
, , „ m—l^ II A n+ 1 B n+2 G .
then S = Th + + + <S:c.
m z m + 1 m z+ 2 '>n
... 171 — 1 ,„ ^ . n A
(A IS T, B IB , &c.).
m z m
If we take T = 1 it becomes
f(.-«, 1, . ' ) = '-^^4«, 1, .,1).
^ 1 — m/ m V m/
[orF{a. l.y,^) = _Lf(y-a, 1, y,-^)]
As Professor Whittaker has pointed out to me, the theorem
in the latter form furnishes a remarkable anticipation of the
well-known transformations of the Hypergeometric Series
given by Kummer {Crelle, 15, 1836).
d2
36 WORKS PUBLISHED BY STIRLING
hi I'rups. Vlli to Xil ' Stilling rctunis again ami again to
the suuimation or transformation of the series delincd by
,^„^ z-m z-n ^,
z z—n+1
Professor Wliittaker suggests that the relative theorems
were doubtless invented to discuss the series
1 z — m 1 (z- m) (z — m+1) 1 „
-j- \. ^- '—^ : 1- etc.
z—n z z—n + l z.z+1 z — n+2
which (up to a factor) represents the remainder after s— 1
terms in the series
1 \ —Hi \ 1 . m, . 2 — m 1
+ -^—^ + ^^ o +•••.
1-rt 1 2 — n 1.2 3-/i
After the work of Euler this integral was calculated by
Gamma Functions.
§ 10. A number of theorems follow for summing a series
' accurate vel quam proxime ', all illustrated by well-chosen
examples. Then, to show that his methods apply to series
already well known, he takes up their application to the
summation of Recurring Series, the invention of bis friend
De Moivre, the Huguenot refugee, who lived and died in
London. He gives extensions to series when the terms at
infinity are approximately of the recurrent type.
Several examples are given of more complicated series such
as Hdj^oc^'' when
where A„, A,j j, ... are integral functions of n of degree r, and
for which he finds a difierential equation (jiujional he calls it)
af the rth order.
He would have been clearer had he adopted the repre-
sentation of integral functions as given by himself in the
I iifroductioii.
' Cf. Andoyer, Bn/L Soc. Math, de France, 1905.
METHODUS DIFFERENTIALIS 37
E.g. Suppose r = 2, and write the equation in the co-
efficients as
an((x + /3n + y n .??—!) + «„_j(a+ h .n—\ ^ c .n~\ .i\—2)
+ iV-c. = 0.
Let y = HGj^x'",
.'. y = ^'}irtjja;""\
y = Ii}i{n— l)rt„a:"~^
&c.
Hence (cny + fSxy + yx"^})),
+ X {ay + hxy + cx'^y)
+ &c.
= 0,
or differs from zero by a function of x depending on the
initial terms of the series, and easily calculated.
The differential equation being obtained, its solution has
next to be found when possible, and this he proceeds to do
(pp. 79-84) by means of power series. Unfortunately, in the
examples he takes he is not quite accurate in his conclusions.
In the last letter from N. Bernoulli referred to above (1733)
the latter remarks :
' Sic quoque observavi te non satis rem examinasse, quando
pag. 83 dicis, aequationem r'^ y^ — r- x^ — x- y^ nulla alia
radice explicabilem esse praeter duas exhibitas
y = x- x^ / 6r^ + x^ / 120 r^ + ...
y = Ax I -oj^-/2r^ + x^/24r'^+ ...
quarum prior dat sinum, et posterior cosinum ex dato arcu x,
et de qua posteriore dicis, quantitatem A quae aequalis est
radio r ex aequatione tiuxionali non determinari. Ego non
solum inveni seriem non posse habere banc foriiiam
A : Bx'' + Cx^+...
nisi fiat A = r, sed utramque a te exhibitam seriem compre-
hendi sub alia generaliori, quae haec est :
y = A-^Bx-^ Cx- + ...
rr 1.2. rr 2.3. r?
1.2. rr
B =
-, F=--
D
38 WORKS PUBLISHED BY STIRLING
in (ina coefficientes A, B, C, i^'c. hanc sequuntur relationem
j^^_ rr-AA ,. -A ,. -B
rr
E= ^--, F = — , &c.
3 . 4 . r?' 4.5. rr
Si fiat A = 0, habetur series pro sinu, sin autem A fiat = r
habetur series pro cosinu ; sin vero A alium habeat valorem
praeter hos duos, etiam alia series praeter duas exhibitas erit
radix aequationis. Similiter series illae quatuor. quae exhibes
pag. 84 pro radice aequationis
y + a-ij-xij-x^y =
sub aliis duabus generalioribus quae ex tuis particularibus
compositae sunt comprelienduntur.' Bernoulli adds his
solutions. (Vide Letter in question.)
PARS SECUNDA
DE Inteupolatione Serierum
§11. The second part contains the solution of a number of
problems in the treatment of which Stirling shows remarkable
analytical skill. Again and again he solves Difference Equations
by his method of Inverse Factorials. This is the method now
adopted by modern writers ^ when large values of the variables
are in question. In this short sketch I can only indicate very
briefly a selection of some of his conclusions.
A common principle applied is contained in the following :
Being given a series of equidistant primary terms, and
the law of their formation, intermediate terms follow the
same law.
Take for example the series
1 + 1 + 2 ! + 3 ! + 4 ! + &c.
in which the law is T\_^_^= zl\ (the law for the Gamma
Function). If a is the term intermediate between 1 and 1,
the corresponding intermediate terms are
2^^' 5 • 2"' ■2'2*'2^^' '^■C
or, as Stirling puts it,
h ■= |a, r = I A, (^'c. (Page 87)
' Cf. Wallenberg and Guldlierg, Thvorie dry linearen Differenzen-
GleichiDigen (Teubner, 1911).
METHODUS DIFFERENTIALIS 39
Prop. X\'IL Eveiy series admits of interpolation whose
terms consist of factors admitting of interpolation.
Thus, given the series
1, - A, , 7)', G, &c.
' p _/;+! ' 2J + 2 '
it will be sufficient to interpolate in
1 r r .r + \...,
1 'p p.p+l...,
and divide.
§ 12. Prop. XVIII is of fundamental importance in many
of the series discussed.
In the two series
r r + 1 ,
a, - a, 0, ....
' q q+1 ' '
if A and « are equal, then the term of the first series at the
distance q — r from the beginning is equal to the term of
the second series at the distance 2^ — '>' from the beginning.
The illustrations he gives can hardly furnish a proof, for
/J — ?' and q — r are not necessarily either integral or positive.
(The proof may be put in a couple of lines by the use of
Gamma Functions.)
Example. Consider the series
-I 2/1 4 D G P
which to meet the conditions must be written as
1^ 1+1 1+2
Suppose the term at distance ivi wanted.
Here p — r = — ^. Write q — r — m or q = m+ \, and
form the series
a 2b 3c
' m+l' m + 2' m + s'
Then the term wanted in the first series is that of this
second series which precedes 1 by the interval — |. Tliis
40 WOTIKS PUBLISHED T.Y STIRLING
artifice is often useful when m is a large number, provided
the second series can be easily interpolated.
He leaves these considerations to lay down the standard
formulae of interpolation already established by Newton,
viz. that known as Newton's Interpolation Formula
f{z)=f{0) + A,z + A,^-^^,&<^.,
and also the two formulae known as Stirling's Formulae,
though they are really due to Newton.
He also takes the opportunity to establish (p. 102) what is
called Maclaurin's Series. ' Et hoc primus deprehendit D.
Taylor in Mefhndo I ncre mentor um, et postea Hermanns in
Appendice ad PhoronomiaTn.'
§ 13. In Prop. XXI he teaches by examples how to inter-
polate near the beginning of a series. The second example
(pp. 110-12) furnishes by pure calculation a most remarkable
result, represented in modern rotation by the formula
r (i) = V^.
About the same time Euler had obtained the same result by
a different method (vide Fuss, Corresp. mathematique).
Stirling proposes to find the term midway between 1 and 1
in the series
1, 1, 2, 6, 24, 120, &c.
The law here is T,+, = z T^ and T^= \, T.,= \.
He interpolates between T^ and T-^^ to find jTnJ and then
he has to divide by lOf, 9|, ... 1| to obtain T^. Since the
numbers are rapidly increasing he uses their logarithms
instead and actually calculates log Tnj from which he finds
Tvl to be 11899423-08, so that Ti\ - -8862269251.
He adds Tj = 1-7724538502, and this number, he says,
is -/tt. (-/tt is actually 1-7724533509.)
Also the corresponding entry among the numbers 1, 1, 4,
36 576, &c. is it}
For inventive audacity Stirling's conclusion would be
difficult to match, and its skilful application led him to
' Is it not possible that ho thus d.'t.'ct d that Tj = ^/tt ?
METHODUS DIFFERENTIALS 41
results that aroused tlic admiration of his friend De Moivre.
(Vide Miscellanea Analytlca.)
In Prop. XXn, Ex. 1, it helps him in tlie interpolation of
the term at intinit}' in the series
1 2 A 4 R en
^j T-^^' "5^' ■5'^^' • • '
2 2.4 2.4.6 ,
or 1, -5 5 J (YC.
1 1.8 1.3.5
a problem which faces him again in Prop XXIII, in which
he gives a formula to find the ratio of the coefficient of the
middle term in (1 +xy to 2"^".
Binet in his Memoir^ (pp. 319-20) proved that of the four
solutions of the latter problem given by Stirling (1) and (3)
are correct, wdiile (2) and (4) are wrong. As a matter of fact
Stirling only proves (1) and (3) and leaves (2) and (4) to the
reader.
Binet, wanting h for the middle coefficient, gives the four
formulae
(1) (^y=7r^ii^(i I, n+\, 1).
Of these (1) and (3) ^ are also the first and third of Stirling's ;
while (2) and (4) replace the other tw^o given by Stirling, viz :
92 n 2
(^r (V) =
li'^n+r
V 12.32 „ -
1 h > &c.
2(2/1-3) 2.4(2/1-3) (2/1-5)
* Binet, Me»K fnir Jrs Tnfe(/)riJes definies EitJeriinnes.
"^ These are also the Kolutions ho gave in a letter to De Moivre to
publish in the Miscellanea Analytka. (See pp. 46-48.)
42 WORKS PUBLISHED BY STIRLING
nnl 2{2n-2) 2 . 4 (2 h- 2) (2u-4) J
Clearly (4)' must be wrong since the factors 2 9i — 2, 2 7i - 4, . . .
include zero in their number.
Binet remarks that the products of (1) and (4) and of (2)
and (3) furnish the first examples known of Gauss's law
Fia,l3,y, l)xF{~a,^,y-a, I) = 1.
§ 1 4. In Prop. XXIV the Beta Function is introduced (as an
Integral) for the interpolation of
r r(r+l)
and the conclusion drawn (in modern notation)
= ^^ (^J + 1 ) . . . (p + H - 1 ) / (yj + (/) (^j + (/ + 1 ) . . . ( /; + g + / 1 - 1 ) .
Again, on p. 139, he solves the associated difference equation
r +
obtaining T = AF{—n, —z, r, 1);
and Binet proves the interesting remark that had Stirling
T -\- Z 1 1
added the solution of u'= u, where tt'= 7h>j u = jr.t
r + z+ It r T
he would have obtained
A/T= F(n, -z, r + n, 1),
i. e. he would have established the Gaussian formula
F(a, h, c, l)xF(-a, h, c~a, 1)= 1.
STIRLING'S SERIES
§ 15. On pages 135-8 are given the formulae which have
rendered Stirling's name familiar whenever calculations in-
volvino- laroe numbers are concerned.
METHODUS DIFFERENTIALIS 43
STIRLING'S THEOREM
When ii is a lar<i,c niiinber the product
1 .2.3 ...n = ii''W2ii7r e ^-",
where < 6 < 1.
Stirling actually gives the formula
Log (1 . 2 . 3 ... .r) = i log (27r) + (x + i) log {x + ^)
J 1 7
-(■»+2)-2.i2.(^,,+ i) + 8. 360 (a; + !)•'" ""
with the law for the continuation of the series,
De Moivre (Sup2). MlbC. Anal.) later expressed this result
in the more convenient form
log (1 . 2 ... a;) = I log (27r) + (« + 1) log x
B. I 5o 1
T) 1
iM-l
(2/t-lj=^« ur'
Caueh}^ gave the remainder after the last term quoted as
^ ^ (-irO „^, 1
" (211 + 1) (2)1+ 2) a;-"+^
(5,, i?2, (^'C, denote the Bernoullian numbers.)
More particularly the series ^
A_i_ J^l
1.2a; 3 . 4 a;'^
has been called the Series of Stirling. It is one of the most
remarkable in the whole range of analysis to which quite
a library of mathematical literature has been devoted. The
series is divergent, and yet. in spite of this fact, when n is
very large and only a few of the initial terms are taken, the
approximation to log n ! found by it is quite suitable for
practical purposes.^ Its relative accuracy is due to the fact
^ See Godefroy, Tlieorie des Series, or Bromwicli's Treatise on Series.
44 WORKS PUBLISHED BY STIRLING
that the error coininitted at any sta(>;c, by neglecting i^„,
is always less in absolute value than the first of" the terms
neglected, which suggests that the series should be discontinued
when the minimum term is reached. Legcndre has shown
that if we write the !■ erics as Z (— l)""*"^ u„, then
'^n+i/Un <(2u-l)2>i/47r^r2,
and .-. < (n/3x)^.
The terms therefore decrease so long as n does not exceed 3.r.
When n = 3x the error is less in absolute value than
•393409... xa-^e-''-'^.
To later mathematicians, such as Gauss, who admitted onl}-
the use of convergent series, Stirlino-'s Series was an insoluble
riddle, but it now finds its place among the series defined as
Asymjjtotic Series}
To meet the objection to its divergence Binet (I.e., p. 22G)
gave the convergent representation.
log (a; — 1 ) ! = I log (2 tt) + (« — -1) log x — x
1
+ 2
>S' + — s., + — s,+ ...
2 5 A •^4.5 ■*
in which /S' denotes 1 1- ... ad oo.
" {xi-iy {x + 2f'
From this by the use of inverse factorials he deduces
(p. 231)
log(a;-l)!.= ilog(2 7r) + (^-|)log«-a^
1 1
+
12(a;+ 1) 12 (a; +!)(.« + 2)
59 1
360 (a;+l)(n'+2)(.f + 3)
227
480(a)+l)...(« + 4) "^ ^ ^'
§ IG. The conclusion of Stirling's book is taken up witli
various proltlems in intoi'polation, based partly on a papei- l)y
him in the ndlosophical Transactions for 1719, and partly
' Vide Toincaie, Acta Math., 1886.
STIRLING'S SERIES 45
on tlie researches of Newton and Cotes. It may be noted
that in Prop XXX he gives the expression of one of the roots
of a system of ii linear c(]uations in n variables, found ' per
Aly;ebram vnli^arem '.
A translation into English by Francis Holliday was published
in 1749 ' with the autlior's approbation '.
There was also a second edition of the original treatise
in 1764.
(C)
CONTRIBUTIONS TO THE
PHILOSOPHICAL TRANSACTIONS
§ 1 7, Though Ramsay (loc. cit.) refers to writings by Stirling
while in Italy, I am not acquainted with any such, save
the first of his three papers printed in the Philosophical
Transactions.
It is entitled Methodus Differentialis Newtoniana lllustrata
Aathore lacolo Stirling, e Coll. Balliol. Oxon., and furnishes
a useful commentary on Newton's Methodus Differentialis
published in 1711. Stirling restricts his attention entirely
to the case of equal increments and proves the three Inter-
polation Formulae already referred to above (p. 40). He
deduces a number of special formulae, several of which are
reproduced in his book of 1730. One of these may be noted
on account of the uncanny accuracy of its approximations in
certain cases.
Let a, ^, y, 8, ... be a series of quantities, and write down
the equations found by equating the differences to zero.
a~(3 = 0,
a-2/3 + y = 0,
a-3(3 + 3y-8 = 0, &c.
The assumption of any one of these will furnish a linear
equation in a, /3, &c., from which any one of these may be
determined when all the others are known :
e.g. to determine /dz / {\ -{■ z^),
consider {l+z-)-\ {l+z^f, {l+z'^)\ &c.
46 WORKS PUBLISHED BY STIRLING
The integrals of these oinittin<;- the first, are r, z-\rz'' j?,, &c.
Take the latter as /:J, y. kc , so that
a=/(/s/(l+s-^).
The above equations give in succession
tan-^s^:;; z-z''/^; s-sV^ + ^V^, &c.
Other examples are easily constructed.
Towards the end of his paper, while discussing a method of
approximating to a slowly converging series, he furnishes
what seems to be one of the earliest general tests for the
convergence of a series.
Consider the series of positive terms
If, in the long run,
1111
>
'^^n '^^n 1-1 '^n-V\ '^n + 2
the scries is convergent ; otherwise it is divergent.
There are also the two papers on the Figure of tJte Earth,
and on Stirling's Engine, to which reference has already been
made.
Li:tter fuom Stirling to De Moiyrk printed ix the
Miscellanea A nalytica}
(De Moivre was naturally much surprised by the intro-
duction of TT into the calculation of the ratio of the coefficient
of the middle term in (1 +.'•;" to the sum of all the coefficients.
Cf. p. 172.)
Quadrienium circiter abhinc, v'lr CI. cum significarem D.
Alex. Cuming Problemata de Interpolatione & Sunnnatione
Serierum aliaque cius generis (piae sub Analysi vulgo re-
cepta non cadunt, solvi posse per Methodum Differentialem
Newtoni ; respondit lUustrissimus vir se dubitare an Problema
a te aliquot ante annos solutum de invenienda Uncia media in
quavis dignitate Binomii solvi posset per Differentias. Ego
dein curiositate inductus, k confidcns me viro de ]\Iathcsi
bene merito gratum facturum, idem libentcr aggressus sum :
^ Miscellanen Anahjtica de Seriehus, pp. 170-2.
PHILOSOPHICAL CONTRIBUTIONS
47
& f.iteor ortas esse difficultates quae impediere (|Uoniinus ad
optatam conclusionem confestim pcrvenire potuerim, scd laboris
hand piget, siquidem tandem asseentus sura solntionem adeo
tibi probatam ut digneris eara propriis tnis scriptis inserere,
Ea vero sic se habet.
Si Index Dignitatis sit numerus par, appelletur ii ; vel si sit
impar, vocetur n—l; eritque ut Uneia media ad summam
onniinm eiusdem Dignitatis, ita unitas ad medium proportionale
inter semi-circumferentiam Circuli & Seriem scquentem
X +
A
+
9B
+
25(7
49X>
+
2X71 + 2 4xH + 4 6xn + 6 8x?i + 8
+
SlE
&c.
lOx-^+lO
Exempli gratia, si quaeratur ratio Unciae mediae, ad sum-
mam omnium in Dignitate centesima vel nonagesima nona, erit
n~ 100 (jui ductus in semiperipheriam Circuli
1 '5707963279 producit A primum terniinum
Seriei ; dein erit
B = ^^^A, C = ,f e 5, D = e¥^ G &c,
atque perficiendo computum ut in margine,
invenietursummaTerminorum 157-866984459,
cuius Radix quadrata 125645129018 est ad
unitatem ut summa omnium Unciarum ad
mediam in Dignitate centesima, vel ut summa
omnium ad alteram e mediis in Dignitate 157-866984459
nonagesima nona.
Problema etiam solvitur per reciprocam illius Seriei, etenim
suunna omnium Unciarum est ad Unciam mediam in sub-
duplicata ratione semiperipheriae Circuli ad Seriem
157-079032679
769998199
10658615
654820
37137
2734
246
26
3
1 A 9B 25C 49i)
— — , + =^ + == + --— + =^
n+l 2x/t + 3 4X)i+5 6xrt+7 8X91 + 9
+
81^
.&c.
10 X 71 +11
vel quod eodem redit, ponatur a — -6366197723676, quoto
scilicet qui prodit dividendo unitatem per semiperipheriam
Circuli ; & media proportionalis inter numerum a, & hanc
Seriem, erit ad unitatem, ut Uncia media ad summam omnium.
48 WORKS PUBLISHED BY STIRLING
•00€30316606304
3059789351
G5566915
2553229
143473
10470
934
98
12
1
Ut si sit li =100 ut Jintoa, eoiii})utuH
L'l'it ut ill niarj^iiic vidcs, nbi suuiuia tcriui-
iioruni prodit •00033144670787 cujus Rndix
quadrata -079589^:373872 est ad uiiitatcm
lit Uncia media ad suiiunam omnium in
Dignitato centesiina vel nonagesiina noua.
Sunt & aliae Series pro Solutione liujus
Problematis aequo simplices ac eae liacte-
nus allatae, sed paulo minus conver<4eiitcs.
ubi Index Biuomii est numerus exi^uus.
Caeterum in praxi non opus est recurrere
ad Series; nam suffieit suinere mediam pro- -00633444670787
portionalem inter semicircumferentiam Circuli & n + ^; liaec
enim semper approximabit propius quain duo priini Seriei
termini, quorum etiam primus solus pleruinque sutiieit.
Eadem vero Approximatio aliter i^ praxi accommodatior sic
euunciatur. Pone 2a = cr= 1-2732395447352 ; eritqueut summa
Unciarum ad mediam, ita unitas ad / quam proximo,
existente errore in excessu circiter / •
16nn V 2 71 + 1
c
St ii= 100, erit— — - = 006334525, ejusque radix quadrata
•07958973 accurata est in sexta decimali, quae si dividatur per
16 nn, id est per 160000 dabit correctionem -00000050, & haec
subducta de approximatione, relinquit numerum quaesitum
•07958923 ju&tum in ultima figura.
Similiter si sit n = 900, erit = -000706962545, cuius
Radix quadrata -026588767 superat verum biuario in nona
decimali, sin vero Correctio computetur ac subducatur de
approximatione, habebitur numerus desideratus accuratus in
decima tertia decimali.
En autem approximationem aeque facilem & magis accura-
tam, differentia inter logaritlimos numerorum n + 2 & u— 2
dividatur per 16, & quotus adjiciatur dimidio logaritlimi
Indicis n; liuic dein summa atljiciatur logaritlimus constans
•0980599385151 hoc est dimidium logarithmi seiniperipheriae
Circuli, & summa novissima est logaritlimus numeri (^ui est
ad unitatem ut summa omnium Unciarum ad mediam. St
n = 900 computus erit
f log 900 1-4771212547
16)Dif. log 902 t^' 898( -0001206376
Lost constans ...... -0980599385
Summa 1-5753018308
PHILOSOPHICAL CONTRIBUTIONS 49
Et haec siinuna veruni supcrat biiiario in ultima tigura ;
estque logaritliiiius numori 37-6098698 qui est ad unitatuiu
ut Suuiina Unciaruiu ad uicdiam in dignitatc 900 vol 899.
Et si vis illius nunieri reciprocuni, sumo complcmentum
logarithmi, scilicet -2-4246i)81692, & numerus eidem corre-
spondens inveuietur -0265887652.
Et hae sunt Solutiones quae prodierunt per Method uni
Differentialem Newtoni ; quarum demonstrationes jam non
attingo. cum in animo sit l)revi publico impertire Tractatum
quem de Interpolatione & Summatione serierum eonscripsi.
Tid Stu(liosissir}ii
10 Jim. 1729 Jac. Stirling
BIBLIOGRAPHY
(1) Sir D. Brewster: Life of Ne" ton.
(2) J. Brown: Epitaphs and Monumental In.'<criptions in Greyfriars
Churchyard, Edinhitrah. 1867.
(3) H. W. C. Davis: History of Balliol College, Oxford.
(4) Edleston : Newton's Corr. spondence tcith Cotes, &,c. 1850.
(5) W. Fraser: The Stirlin'js of Keir and their Private Papers. Privately
printed, 1858.
(6) Gentleman's Magazine for 1853: Modern Histnry of LeadhiUs.
(7) A. D. Godley's Oxford in the Eighteenth Century. 1908.
(8) T. Hearne : Hearne's Diary, edited by Bliss, 1869 ; also by the
Oxford Historical Society.
(9) Macclesfield: Correspondence of Scientific Men.
(10) G. 0. Mitchell: Old Glasgow Essays. 1905.
(11) J. Moir Porteous : God's Tnasure House in Scotland. 187t').
(12) J. Ramsay: Scotland and Scotsmen in the 18th Century. 1888.
(13) S. P. Rigaud : Miscellaneous Works and Correspondence of the Rec.
James Bradley, D.D. 1832.
(14) B.Taylor: Contemplatio Philo^ophica. 1793.
(15) W. W. R, Ball : Newton's Classification of Cubics, London Math. Soc.
1891.
(16) Historical works of Cantor, Chasles (Aper9u), Montucla ; articles on
Probability and Theory of Finite Differences in Encyclopedic des
Sciences math^matiques ; modern text-books on Finite Differences
by Markoff, Seliwanov, &c., and on Probability by Bertrand,
Czuber, &c.
(17) G. Cramer: Courhex algebriques.
(18) P. H. Fuss : Corr. math, et 2>hysique de quelques celebres geometres du
XVIir siecle. 1843.
(19) M. Godefroi : Th^orie des Series. 1903.
(20) C. Maclaurin : Treatise of Fluxions. 1742.
(21) De Moivre : Doctrine of Chances. 1756.
,, Miscellanea Analytica de Seriebus. 1730.
(22) R. Reiff: Geschichte der Unendlichen Eeihen. 1889.
(23) I. Todhunter: History of Probability and History of Attraction and
the Theory of the Figure of the Earth.
(24) Any student wishing to study Stirling's methods cannot do better
than read in the following order:
(i) J. Binet : Memoire sur les Integrales Euleriennes ; Jour. Ecole Poly.
1839.
(ii) N.Nielsen: Tlieorie der Gammafunktion. Teubner, 1906.
Also : Les Polynomes de Stirling. Copenhagen, 1820.
(lii; G. Wallenberg und A. Guldberg: Tlieorie der lincaren Dijferenzen-
gleichungen. Teubner, 1911.
STIRLING'S
SCIENTIFIC CORRESPONDENCE
E 2
INTRODUCTION
Much of the correfcpondence of James Stirling;- has been
preserved at the family seat of Garden. In tlie collection
are several letters from him to his iriends in Scotland, and
numerous extracts from them are to be found in the Family
History: — The Slirliags of Keir and their Private Papers,
by W. Fraser (Edinburgh, privately printed, 1858). Jn
addition to these are letters of a scientific character which
were with great courtesy placed at my disposal by Mrs. Stirling
in 1917. Of tl;e latter group of letters the earliest is one
from Nicholas Bernoulli in 1719, and the last is one from
M. Folkes, P.R.8. in 1747. Stirling enjoyed the acquaintance
of most of the British mathematicians of his day, while his
reputation and continental experience brought him into corre-
spondence with continental scholars like Clairaut, Cramer,
and Euler,
It is interesting to note that all of his correspondents save
Campailla were, or became Fellows of the Royal Society of
London. (It is clear from letter XI^ that Stirling suggested
to Euler that he should become a Fellow.) The dates when
they joined are indicated in the notes added to the letters.
One learns from the letters how much depended on corre-
spondence for the discussion of problems and the diH'usion
of new ideas, just as one would turn nowadays to the weekly
and monthly journals of science. Several of the letters in
the collection shed a good deal of light upon ob.'^cure points
in the history of Mathematics, as indicated in the notes.
]\laclaurin appears to have been Stirling's chief correspondent
and the letters between the two men are of particular interest
to students of Scottish Mathematics. They were warm friends,
though probably in opposite political camps, and Maclaurin
had the benefit of Stirling's judgment when engaged upon
his Treatise of Fluxions.
54 INTRODUCTION
There are not many letters of Stirlino-, and those are chiefly
copies made by Stirling himself.
I had the good fortune to find four original letters from
Stirling to Machiurin in the Maclaurin MSS. preserved in
Aberdeen, and they fit in admirably with the letters of the
Garden collection. But I am convinced that other letters by
Stirlino- are still to be found. Stirling is known to have had
frequent correspondence with R. Simson, G. Cramer, and
De Moivre, not to mention others, and the discovery of fresh
letters misht be the reward of careful search. Among letters
of Stirling already published may be mentioned his letter
to Newton in 1719 (Brewster's Newton), a letter to J. Bradley
reproduced in the Works and Correspondence of Bradley,
a letter to De Moivre in the Miscellanea Analytica de ^erichus,
and reference to a second letter in the Supplement to the
same work.
Ciamer's Letter III3 and the letter from Stirling to Castel
V^, are reproduced in the Stirling Family Histoiy.
CONTENTS
PAGE
I
CORRESPONDENCE WITH MACLAURIN, 1728-1740. 57
11 Letters from Maclaurin to Stirling.
4 „ ,, Stirling to Maclaui in.
1 Letter „ Gray to Maclaurin relative to Stirling.
The letters to Maclaurin have been obtained through
the courtesy of Aberdeen University.
Letter Ijo is a note attached to the translation of
a letter from Maupertuis to Bradley.
II
LETTER FROM SIR A. CUMING TO STIRLING, 1728 . 93
III
G. CRAMER AND STIRLING, 1728-1733 ... 95
10 Letters fiom Cramer to Stirling.
1 Letter ,, Stirling to Cramer.
IV
N. BERNOULLI AND STIRLING, 1719-1733 . . 181
3 Letters from Bernoulli.
1 Letter ,, Stirling.
L'ABBE CASTEL AND STIRLING, 1733 . . .151
1 Letter from Castel.
1 ,, ,, Stirling.
VI
CAMPAILLA AND STIRLING, 1738 .... 158
1 Letter from Campailla.
56 CONTENTS
PAGE
VI r
J. BKADLEY AND STIRLING, 1733 . . . .160
1 Letter from Stirling.
1 „ „ Bradley.
VIII
S. KLINGENSTIERNA AND STIRLING, 1738 . . 164
1 Letter from Klingenstierna, also solutions of cer-
tain ])roblems.
IX
MACHIN AND STIRLING, 1733 (?) and 1738 . . 172
2 Letters from Machin.
X
CLAIRAUT AND STIRLING; 1738 .... 176
1 Letter from Clairaut. *
XI
EULER AND STIRLING, 1736-1738 .... 178
1 Letter from Stirling, 1738.
1 „ ,, Euler, 1738.
(Euler's first letter to Stirling, prol)aI)]3" preserved at
Petrograd, was written in 1736.)
XII
M. FOLKES, P.R.S., AND STIRLING, 1747 . . 192
1 Letter from Folkes.
NOTES UPON THE CORRESPONDENCE . . .198
^:i/^^ i^ j-<^ ^f^K^y^^ r/ U£ ^ MZ'^ A^'ai^y 4^1^
I
COLIN MACLAUKIN AND STIRLING
(1)
MacJaiirin to Stifling, 1728
Mr James Stirling
at the Academy in
little Touer Street
London
Sir
Your last letter was very acceptable to me on several
Accounts. I intend to set about publishing the piece on the
Collision of Bodys very soon. I was obliged to delay it till
now having been very busy taking up my Classes in the
College. Your remarks on their experiments are certainly
just. I intend if I can get a good opportunity by any of our
members of parlia* to send you a copy of my remarks before
I publish them. I have seen Roberts's paper since I came
from Perthshire in August where I writ my remarks and find
he has made some of the same observations as I had made ;
nor could it well happen otherwise. I wish I had Mr Graham's
Experiment at full length with Liberty to insert it. I design
to write to him about this. I am much obliged to you for
your kind oifer and would accept of it if I was to publish this
piece at London.
I spoke to Col. Middleton and some others of influence
here and find they have better hopes of success to . . .
Mr Campbell in that Business than you have
I think some of his performances deserved to be taken
notice of. But as there is an imperfect piece of mine in the
transactions for 1726 on the same subject I wish you had
rather chose to publish some other of his pieces. I have been
at pains to soften some prejudices and Jealousies that may
possibly revive by it. It is true I have too long delayed
58 STIRLING'S SCIENTIFIC CORRESPONDENCE
piiblishinu- the rcnuiinder of my piece for wliieh I liave only
the excuse of much teacliino- and my desi<;n of (giving a
Treatise of Algebra where I was to treat that subject at
large.
I told you in my last I had the method of demonstrating
that rule by the Limits. In one of my Manusci-ipts is ye
following Article.
I et x" - 2)x'^-'^ + qx" -^ _ rx"-'' &c. -
be any equation proposed ; deduce from it an E(|uation for its
Limits
and from this last deduce an equation for its limits ; and by
proceeding in this manner you will arrive at the (piadratick
n X VI — 1 X x^ — 2 ( a - 1 ) ^xf + 2 7 =
whose roots will be imposf^ible if 'jf- be less than q
and therefor in that case at least two roots of ye proposed
Equation will be impossible. Afterwards I shew that if
1 n — 1
„ r X (^ be less than 'pv two roots must be impossible by
a quadratick equation deduced a little differently, and so of
the other terms. But this matter is so easy I do not think it
worth while to contend about it. I have some more concern
about a remark I make in my Algebra on the transformation
of Equations which has been of great use to me in demon-
strating easily many rules in Algebra which I am afraid may
be made use of in the paper you have printed because my
dictates go through everyltody's hands here.
The Observation is transform any Equation
a? — i)X^ + qx - ?' =
to another that shall have its roots less than the values of x
by any ditference e :
Let 2/ = x — e and
2/'' + 3e2/'- + 3C-2/ + C'' = where any Coefficient considered
— 'p[j^—2pe'\) — 'pe^ as an Ecjuation gives for its roots
+ qij+qe the limits of the following
— r Coefficient considered as an Equa-
CORRESPONDENCE WITH MACLAURIN 59
tion. This holds in Equations of all sorts and from this I
demonstrate many rules in a very easy manner.
By it too I demonstrate a Theorem in y[our] (?) book where
a Quantity is expressed by a series whose coefficients are first,
second, tliird fluxions, &e. I shall be vexed a little if he has
taken this from me. Pray let me know if there is any thing
of this in tlio paper you have printed.
I intended to have sent yoU one of my Theorems about the
Collision of many Bodj^s striking one another in different
directions in return for your admirable series. But I must
leave that to another occasion.
I expect to dispose of the six subscriptions I took for
Mr De Moivre's Book. Please to give my humble service to
Mr Machin and communicate what is above. I long foi' his
new Theory. I am with great Respect
Sir
Your most Obedient and Humble Servant
Colin Maclaukin
Edinburgh Dec' 7
1728.
(2)
Stirlwfj to MacJaurin, 1728
Sir
A few days ago I received 3'our letter of the 7'^ of this
Moneth and am very glad that your Book is in so great a
forwardness, but you have never yet told me in what language
it is, altho at the same time I question not but it is in Latine.
I should be very glad to see what you have done, and since
you mention sending a Copy, you may send it under Cover to
Mr Cuninghame of Balghane ; if I can do you any service as
to getting Mr Grahams Experiment I wish you would let me
know, I question not but that you may have liberty to print
it, because probably it will be in our Transactions very soon.
I am very glad that Coll. Midleton gives Mr Campbel
encouragement to come to London, no doubt but bread might
60 STIRLING'S SCIENTIFIC CORRESPONDENCE
be made by private teaching if a man had a rioht way of
niak[ing himself] known, but indeed I [quesjtion if Mr Campbel
will not want a prompter in that p . I am apt to
thi[nk that I ha]ve not given you a distinct account of his
paper about in [ ] ^ because you se[em to thi]nk that I
choose it out of a great many others to be printed [ ]
which indeed would not have been so very candid before you
had leasure to compleat your paper. But the Matter is quite
othervvays. For as soon as your paper was printed, Mr Campbel
sent up his directly to Mr Machine, who at that time being
very busy, delayed presenting it to the Society because the
Correcting of Press would divert him from prosecuting his
Theory of the Moon. Upon this delay Sir Alex. Cuming
complained grieveously to ]\Ir Machine that Mr Campbel was
ill used, this made Mr Machine present it to the Society,
upon which it was ordered to be printed, Mr Machine came to
me and desired I would take the trouble of correcting it
in the Press, which was all the Concern I had in it. And
now I hope you are convinced that I did no more than
yourself would have done had you been asked. Mr Campbels
Method is grounded on the following observation. Let there
be two equations x' + ^a;* + Bx' + Cx^ + Bx + E = and
Ez^ + Dz^ + Cz^ + Bz^ + ^0 + 1 = 0, where the reciprocals of
the Roots of the one are the Roots of the other, then it is
plain that the Roots in both are the same as to possibility and
impossibility. He deduces from erch of those a Quadratick
Equation for the limits the common way, and on that founds
his Demonstration. But he doth not use that property of
etjuations which you have been pleased to communicate, indeed
it is very simple and I can see at once what great use can be
made of it, I had observed that the last Term but one gave
the Fluxion of the e(|uation, but never any further before
you mentioned it. But Mr Campbell besides demonstrating
Sir Isaac Rule [ ] one of his own more general, he exempli-
fies it by an equation of 7 dinien[ ]ich his Rule discovers to
have 6 impossible Roots, wliereas S'' Isaac's disc()[ ]ly two
of the Six.
[I] shal now make a remai-k on some of those Gentlemen
who dispute for the new [njotion of Force to shew how
* Inipossible roots (V).
CORRESPONDENCE WITH MACLAURIN 61
mucli they depend one anotlicrs demonstrations which are to
convince their Adversarys.
Herman in his book page 113, I mean his Plioronomia, says
In hac virium a^stimatione, prseeuntem habcmus Illustrissimum
Leibnitium, qui eundeni non uno loco in Actis eruditorum
Lei})>.ia3 indicavit qiiidem non tamen dcmonstravit, etsi
apodictice demonstrari potest, ut forte alia id occasione
ostendemus — He denj^s then that his friend Leibnitz ever did
demonstrate it, but owns that it may be done and is in hope
one time or other to do it liimself .
Poleni in his Book de Castellis page 49 tells us that Leibnitz
demonstration was published ; and page 52 he mentions
Bernoulli demonstration [ ] as published in Woltius. And
page 53 [ ] that perhaps some and those not the most
scrupulous might doubt [ ] Leibnitz's and Bernoullis
demonstrations, and then page 61 he tells —is meaning in
plain words, Demonstrationem inventam fuisse reor non tamen
editam. So that it is very remarkable that a certain number
of men should run into an opinion ; and all of them deny one
another's proofs. For Herman denys Leibnitz demonstration,
and Poleni denys all that ever were given, and declares further
that he knows not possibly on what principles one should
proceed in such a Demonstration, but at the same time, he
resolves to be of the opinion : whether it be proved or not.
But no doubt you have observed many more of their
Absurdities as well as this. I have not seen Mr Machin since
I got your letter, but shal carry him your complements, I am
afraid it will be long before wee see his Theory, for Mr Hadly
and he do not agree about some part of it. We expect in the
first Transaction Mr Bradley's account of the new motion
observed in the fixt Stars. I wish you good success, and hope
to see your book soon, I am with all respect Sir
London Your most obedient
31 December humble servant
1728 James Stirling
62 STIRLING'S SCIENTIFIC CORRESPONUENCE
(3)
Madaurin to Stirling, 1729^
Mr James Stirling
at the Academy in
little Tower Street
London.
Sir
Last tuesday night I saw the philosophical Transactions
for the month of October for the first time. You may
remember I wrote to you some time ago wishing some of
Mr Campbell's papers might be taken notice of. I did not
indeed then know that Mr Machin had any paper of liis on
the impossible roots. But even when I heard of it from you
I was not much concerned because from a conversation with
the Author on the street I concluded his method was from the
equations for the Limits and never suspected that he had
followed the very track which I had mark'd out in my paper
in the transactions for May 1726 from the principle that the
squares of the differences of Quantities are always positive as
he has done in the latter part of this paper. As I never
suspected that he had followed that Method I had no suspicion
that he would prevent me in a Theorem that can be only
obtained that way but cannot be overlooked in following that
track. I cannot therefor but be a little concerned that after
I had given the principles of my method and carried it some
length and had it marked that my paper was to be continued
another pursuing the very same thought should be published
in the intervall ; at least I might have been acquainted that
I might have sent the continuation of mine before the other
was published.
You would easily see that the latter part of Mr Campbell's
paper after he has done with the limits is the very continuation
of my theorems if you had the demonstrations.
Let there be any Equation
+ Hx^-^ - /«;«-» + ira;"-iO- Ix^'^^ + Mx''-'^'^ &c =
' 1728 O.S. ; but 1729 N.S., cf. Letter !„.
CORRESPONDENCE WITH MACLAURIN 63
and ^^^^ X D^ will always exceed EG - FB + GA - 11
n — \ n — 2 11 — 3 „
if m = ii X —— X -— - X -— — cVc.
^ »5 t:
till you have as many factors as there are terms in the
Equation proceeding D.
I have had this Theorem by me of a long time : and it easily
arises from my Lemmata premised to my paper in the Trans-
actions for May 1726. An abridgment of my demonstration as
I have it in a book full of Calculs on these subjects is as follows.
The square of the coefficient of D consists of the squares of its
parts and of the double products of those parts multiplyed
into each other. Call the sum of the first of these P the sum
of the products Q and D"^ = P + 2Q. Now the number of
those parts is m and therfor by the 4*"^ Lemma of the paper
in the transactions for May 1726 (^i— 1)P must be greater
than 2 Q and D^ {= P + 2Q) must be greater than Q
or D^ greater than Q. Then I shew that
2m ^ ^
Q=EG-FB + GA-H
and thence conclude that -— — B^ always exceeds
2 m "^
EG-FB + GA-H
when the roots of the equation are all real.
I have a general Theoreme by which I am enabled to
compare any products of coefficients with any other products
of the same dimensions or with the Sums and Differences of
any such products which to shew you how much I have
considered this subject tho' I have been prevented when
I thought myself very secure I now give you. Let E and H
be any two coefficients and r/i the number of Terms from
E to H including both then shall
EH = F + ^i^^lQ+'''^.'^^^R + '21±l.'I!l+l.'I^^S
m+7 m+8 m + 9 m + 10 „ .
+ ^- -2- ^ — ^ *<=•
where P expresses the squares of the parts of E multiplyed
G4 STIRLING'S SCIENTIFIC CORRESPONDENCE
by the (liH.siunlai' parts of C. (a term as far distant from the
beginning of the E(|uation as H is from E) Q expresses the
squares of tlic parts of the coefficient iunnediately preceding E
viz. J) miiltiplycd by tlie dissimihir parts of the term next
followini;- (' but one viz. in this case E itself. K expresses the
s(|uarcs of tlie parts of the coefficient next precedin<^ E but one
that is G multiplyed by the dissimihir parts of the Term next
following C but three viz. G ; and so on. Where I mean by
the parts of a coefficient the terms that according' to the
connnon Genesis of Equations produce it; and by disi^inn'lar
parts those that involve not the same Quantitys.
Tills general Theorem opens to me a vast variety of Tlieorems
for comparing the products or S(juares of coefficients with one
another of which those Intherto published are only particular
Examples. Here I give you the theorem for comparing any
two products of the same dimensions as EI and CL. Let s
and m express the nundjer of terms that preceed C and / in
the Equation then let
71/ - 1 n — s — 1
P = r X 7r~ X
^ 6' + 1 s + 2
n
-s-2
&c.
6- + 3
X
■ii — m-
-2
, n — m n — m—l
and (7 = ; X — . .
•^ m + 1 m + 2 m + 3
continued in each till you have as many factors as there are
terms from to E including one of them only ; then shall
- X EI alway exceed (JL when the roots are all real.
Then I proceed to compare the ]3roducts of the Coefficients
with the sums or ditterences of other products & one of the
chief Theorems in that part is that mentioned above which
Mr Cami)bell also found by the same method as is very
apparent and could not miss in following the track I mark'd
out in the transactions.
I had observed that my rules gave often impossible roots in
the Ecjuations when Sir Isaac's did not in proof of which
I faithfully transcribe from my Manuscript the following
Article.
' In the Equation
x'-Ax^ + Bx-'-Cx'' + Dx-E=
x^-\0x*-\-Z0x^-iix^ + 32x-d =
CORRESPONDENCE WITH MACLAURIN 65
no impossible roots appear by Sir Isaac's rule. But i>'- x
2 III
here is less than AC—D i'or
n — 1 4 ,171—1 9
m — II X —~ — = 5 X - = 10 and == —
2 2 2m 20
now 2^0 X 30 X 30 is less than 44 x 10 — 32 the first being 405
the latter 408 so that there must be impossible roots by
our rule.'
After that I give other Examples
I believe you will easily allow I could not have invented
these Theorems since tuesday last especially when at present
by teaching six hours daily I have little relish left for such
investigations. I showed too my theorems to some persons,
who can witness for me. But I am afraid these things are
not worthy your attention. Only as these things once cost
me some pains I cannot but with some regret see myself
prevented. However I think I can do myself sufficient justice
by the length I have carried the subject beyond what it is in
the transactions.
I believe you will not find that Mr Campbell sent up his
paper or at least the latter part of it so soon after I sent up
mine which was in tlie beginning of 1726. One reason I have
is that Mr Machin never mentioned it to me tho' I spent
a whole day with him in September 1727 and talked to him
on this subject and saw some other papers of Mr Campbell's
in his hand at that time. So that I have ground to think
that the paper of May 1726 led the Author into the latter part
of his for October 1728.
When I was with Mr Machin in September 1727 I then
had not found a sufficient demonstration for the cases of
Sir Isac's rule when there may be six or seven impossible
roots arising by it. This part is entirely overlooked by this
Author: for all he demonstrates amounts only to some pro-
perties of Equations that have all their roots real ; from which
he says indeed all S"" Isac's rule immediately follows. But I
conclude from thence that he did not try to demonstrate
compleatly Sir Isac's rule. If he had tryed it new difficultys
would have arisen which he has not thought of.
The way he has taken to demonstrate Sir Isac's numbers
66 STIRLING'S SCIKNTJFJC CORKEsrO]N])ENCE
from tlie Limits is not so simple as tluit I luivc wliieli I may
semi you a;4aiii.
I now beg pardon lor this lung letter which I beg you
would communicate to Mr Machin not by way ol' complaint
against him fur whom 1 have more respect than for any
Mathematician whatsoever ; but to do me justice in the matter
of these impossible roots which I had thrown aside for ?ome
time and have now taken up with regret. I would have
justice done me without disputing or displeasing anybody.
At any [rate] in a few days I shall be very easy about the
whole Matter. I am with the greatest Respect
Sir
Your Most Obedient
Aflectionat Humble Servant
Edinburgh CoLiN Maci.AUUIN
febr. 11. 1728
Having room I send you here one of my Theorems about
the Collision of Bodys.
Let the Body G moving in the direction CD strike any
number of Bodys of any magnitude A, B, E, F, &c. and make
Fig. 4.
them move in the lines Ca, Gb, Ge, Cf &c. to determine ye
direction of G itself after the stroke.
CORRESPONDENCE WITH MACLAURIN 67
Suppose Da, Dh, De, Df &c. perpendicular to the directions
CA, GB, GE, CF, &c. Imagine the Bodys G, A, B, E, F &c.
to be placed in C, a, b, e, f &c. respectively ; find the centre of
Gravity of all those Bodys so placed and let it be P. Draw
DF and GG parallel to DF shall be ye direction of G after the
stroke if the Bodys are perfectly hard.
Adieu
(4)
Maclaur'ni to Stirlitig, 1729
J\Ir James Stirling
at the Academy in
little Tower Street
London
Sir
I delayed answering your last letter till I could tell you
that now I have sent Mr Folkes the remainder of my paper
concerning the impossible Roots of E(j[uations. I sent him
a part April 19 and the remainder last post. I thought to
have finished it in our Vacation in March but a Gentleman
compelled me to go to the Country with him all that time
where we had nothing but diversions of one sort or other, so
that I did not get looking into it once. However I am
satisfyed that any person who will read this paper and
compare it with Mr Campbell's will do me Justice. On
comparing them further myself I (find) lie has prevented me
in one proposition only ; which I have stated without naming
or citing him or his paper to be the least valuable. For I shew
that some other rules I have deduced from my Theorems
always discover impossible roots in an Equation when his rule
discovers any, and often when his discovers none. I wish you
could find time to read both the papers.
I am sorry to find you so uneasy about what has happened
in your last letter. It is over with me. When I found one
of my Propositions in his paper I was at first a little in pain ;
but when I found it was only one of a great many of mine
f3
68 STlHLlNCrS SCIENTIFIC CORRESPONDENCE
that he had liit upon; and reflected that the generality of my
TlieoreniR would satis- fy any judicious reader; I became less
concerned. All I now desyre is to have my paper or at least
the first part of it pnblisJied as soon as possible. I beg you
may put Mr ]\rachin in mind of this. I doubt not but you and
he will do what you can to liave this Justice done me. I could
not but send the second part to Mr Folkes having sent him
the first.
I have at the end of my paper given some observations on
Equations for the sake of those who may think the impossible
roots may not deserve all this trouble. Mr Folkes will shew
you the paper. I intend now to set about the Collisions
of Bodys.
The Proposition I sent you in my last letter is the foundation
of all my Theorems about the impossible Roots. I have
a little altered the form of it. It is the VI Proposition
as I have sent them to Mr Folkes the first five having been
given in 1726. I have made all my Theorems as I went over
them last and transcribed them more simple than they were
in my manuscripts ; and that occasioned this little delay : for
your advice about sending up my paper soon perfectly pleased
me. Abridgments and Additions that occurred as I transcribed
it took up my time but it was about the third or fourth of
April before I got beginning to it in earnest, and my teach-
ing in the Colledge continuing still as before with other
avocations ; you will allow I have not lost time.
I have a particular sense of the Justice and kindness you
have showed me in your last letter & will not forget it
if I ever have any opportunity of showing with how much
Esteem it afiection
I am Sir
Your Most Obedient
Humble Servant
Colin MacLauiun
Edinburgh May 1
1729
CORRESPONDENCE WITH MACLAURIN 69
(5)
Maclaiirin to Sfirliiuj, 1720
Mr James Stirling
at the Academy
in little Tower Street
London
Sir
Since I received your last I liave been mostly in the
country. On my return I was surprised with a printed piece
from ]\Ir Campbell ag-ainst me which the gentleman who
franked tlie letter told me he sent you a copy oft". Tlie
Gentleman indeed added he had not frank'd it it' he liad
known tlie nature of the paper; and was ashamed of it.
I wonder I had no message by a good hand from Mr Campbell
before he printed these silly reports he diverts himself with.
Good manners and prudence one M'ould think ought to have
led to another sort of conduct.
He has misrepresented my paper much and found things in
it I never asserted. I shall send you next post a fuller answer
to it. His friends here give out that 3'ou desyred him to
write against mo. I am convinced this is false.
Please to send me the letter I wrote to you in februar}^
if you have preserved it or a copy of it. I wish if it is not
too nuich trouble you would send me a copy of all I said
relating to IMr Campbell's taking the hint from my first paper
in my letters to you.
I wish you would allow me (if I print any defence) to
publish your letter to me of the date of febr. 2 7 whore you
have expressed yourself very cautiously. But I will not do
it without your permission.
I hope the paper Mr Campbell has sent you will have little
influence on you till 3'ou hear my repl3^ I have writ at large
to Mr Folkes by this post who will show you my letter if 3'ou
please. I assure you I am with great Esteem
Sir
Your ]\Iost Obedient
Edinburgh Most Humble Servant
nova's. 1729 CoLiN MacLauhin
70 STIRLING'S SCIENTIFIC CORRESPONDENCE
(6)
Stirling to Maclaurhi, 1720
To Mr Maclaiirin Professor of Mathematicks
in the Universitj^ of
Edenburgli
Out of 3^our Letter of October 22, 1728
I have other ways of demonstrating the Rule about
impossible roots & particularly one that was suggested to me
from reading your book in 1718 drawn from the limits of
Equations shorter than the one I have puldished. but according
to my taste not so elegant.
Out of Letter of December 7, 1728
Let ic"— pa;"'^ + 9^.i'"~^ — raj''"^ &c. = 0, be txny Equation
proposed, deduce from it an Equation for its Limits
9ia;""^ — 96- 1 X p.("-- + yt - 2 X g.c"-^ &c. = ;
By it too I demonstrate a Theoreme in your book where
a quantity is expresst by a Series whose coefficients are first,
second, third fluxions d'c.
A Cop3^ of 3' our Letter Fob 11, 172|.
S^
Last Tuesday night I saw the philosophical Transactions
for the month of October for the first time.
At an}' rate in a few days 1 shall lie verj- easy about the
whole matter. I am ^q..
S"" This is an exact cop}^ except the postscript which
containing a Theoreme about the collision of Podys 1 presume
is nothing to the present pui-pose. I am with all respect
Sr
Your most humble servant
Ja: Stirling
London 29 November 1 729
CORRESPONDENCE WITH MACLAURIN 71
(7)
Maclanrin to Stirling
Dear Sir
I send you witli this letter m}^ answer to Mr George
Campbell which I publish with regret being so far from
deliofhtino- in such a difference that I have the greatest dislike
at a publick dispute of this Nature. At the sfime time that
I own this Aversion I can assure you it flows not from any
Consciousness of any other wrong I have done this Author
than that I accepted of a settlement here that was proposed
to me when some persons at Aberdeen were persecuting me
and when a settlement here every way made me easy ; at the
same time that he had some hopes tho' uncertain in a course
of years of getting the same place.
I was sensible however of this and therefor made it my
great Concern to see him settled ever since I have been in
this place, nay after my business had proceeded so well that
it was indifferent to me whether he continued here or not in
respect of Interest.
However I have avoided everything that might seem writ
in his strain and have left out many things lest they might
look too strong, particularly in citing Mr Folkes's letter I left
out his words that Mr Campbell's paper was writ with the
greatest passion and partiality to himself, as you will see.
I sent the first sheet in Manuscript to have been communicated
to you above a fortnight ago by Mr Folkes that you might
let me know if you desyred to have anything changed and
have delayed the publication till I thought there was time for
an Answer to come to me. I have printed but a few Copys
intending only to take of as much (without hurting him) ^ the
Impression he endeavours to make as possible.
It was to avoid little skirmishing that I have not followed
him from page to page — but refuted the essentials of his
piece, overlooking his Imaginations and Strictures upon
them. I am at present in haste having several other letters
to write on this subject. I avoid things together towards the
^ Written above the line.
72 STIRLING'S SCIENTIFIC CORRESPONDENCE
end because it was like to have letjuired another hali'-sheet.
I am sure I have given more tlian the subject deserves.
I liave left out two or tliree paragraphs about his inconsistencys
his stor}^ of some that visited me and found me so and so
engaged Arc. This I answer in my manuscript letter sent to
you, Nov. 5. I am indeed tyred wnth this affixir.
I wished to have hoard from you what lie ol)jected to the
letter I wrote to you in the beginning of winter. I am truly
sorry Mr Campbell has acted the part he has pleased to act.
But my defence is in such terms after all his bad usage of me
as I believe to his own friends will shew I have no design to
do him wrong and have been forced into this ungrateful part.
It is true he speaks the same language ; with what ground
let the most partial of his friends judge from what I have
said in my defence.
You may remember that my desyre of doing him service
was what began our correspondence. I then could not have
imagined what has happened. Please to forgive all the trouble
I have given you on this Occasion and believe me to be Sir
Your Most Obedient
Humble Servant
Colin Mac Laurin
If you see Mr de Moivre soon, please to tell him I send him
by this post a bill for six guineas and a letter directed to
Slaughter's Coffee House. I did not know where else to
direct for him.
(8)
Gray to MacJaurhi, 17.1,2
London 2.3 Novom'" 1732
Dear Sir
I had the favour of yours yesterday (S: inclosed a part of
the abstract of your Supplement wilh a Letter to Mr Macliin,
which, as you desired, 1 copyed & gave to him. He is of
opinion that it will be iujproper to put any part of your
Abstract into our Abrigment, especially as matters stand.
He will take care to do you all the justice he can and desires
CORRESPONDENCE WITH MACLAURIN 73
his kiiul services to you. I am thinking that it will not be
impro})er to move the Society at their first meetino- that
Stirling be in Hodgson's room ; because he is nnich more
capable of judging than him ; but in this I will follow
Mr Machin's advice. I hope j^ou had m}' last, and am
persuaded you will do in that affiiir what is fit.
I have a great deal of business to do this evening. T will
therefore only assure you that I am most faithfully
Dear Sir
Your most obedient
i^' most humble Servant
Jno Gray
(9)
Madaurin to Stirlh/g, 1734
To
Mr James Stirling
at Mr Watt Academy
in little Tower Street
London
Sir
I was sorry on several accounts that I did not see you
again before you left this Country. Being in the Countrj^
your letter about the Variation did not come to my hand till
the time you said you had fix'd for your journey was so near
that I thought a letter could not find you at Calder.
I have observed it since I came to Town & found it betwixt
12 k 13 degrees westerlj^ ; the same had appeared in April
last. But I am to take some more pains upon it which if
necessary I shall communicate.
Upon more consideration I did not think it best to write
an answer to Dean Berkeley but to write a treatise of fluxions
which might answer the purpose and be useful to my scholars.
I intend that it shall be Liid before you as soon as I shall send
two or three sheets more of it to Mr Warrender that I may
have your judgment of it with all openness & liberty. This
74 STIRLING'S SCIENTIFIC CORRESPONDENCE
favour I am the rather ohli^eil to ask of you that I liad
no body to examine it here before I sent it up on whose
judgment I could perfectly depend. Robt. Simpson is lazy
you know and perhaps lias not considered that subject so
much as some others. But I can entirely depend on your
judgment. I am not at present inclined to put my name to it.
Amongst other reasons there is one that in my writings in
my younger years I have not perhaps come up to that
accuracy which I ma}^ seem to require here. When I was
verj^ young I was an admirer too of infinites ; and it was
Fontenelle's piece that gave me a disgust of them or at least
confirmed it toii'ethor with readino- some of the Antients more
carefully than I had done in my younger years. I have !-ome
thoughts in order to make this little treatise more compleat
to endeavour to make some of Mr De Moivre's theorems more
easy which I hope he will not take amiss as I intend to name
ever^diody without naming myself.
I have got some few promises as to Mr Machin's book and
one of my correspondents writes me that he has got two
subscriptions. I wonder at Dr Smith's obstinate delaj^ which
deprives me of the power of serving Mr Machin as yet so
much as I desyre to do. It is from a certain number of hands
that I get subscriptions of this kind. Peudjerton's book and
the Doctor's delay diminish my influence in that very much.
Looking over some letters I observed the other daj^^ that
you had once wrote to me you had got a copy from Mr Machin
of the little piece he had printed on the Moon for me. If you
can recollect to whom you sent it let me know ; for it never
came to my hand ; and I know not how to get it here. Nor
did the Copy of your treatise of Series come to my hand.
You need not be uneasy at this: Only let me know what you
can recollect about them. If Mr Machin's book happens to be
published soon you may always Ncnture to sett me down for
seven Copys. Jjut I hop(! to gett moi"e if 1 had once fairl}'
delivered Dr Smith's l>o(tk to the subscribeis. As to your
Treatise of Series i got a copy sent uk; IVom one Stewart
a Bookseller as a new book but about half a year after his
son sent me a note of my being due half a guinea for it which
I payed. But .as I said I only mention these things in case
you can recollect any thing further about them.
CORRESPONDENCE WITH MACLAURIN 75
I observe in our newspapers that Dr Halley has found the
longitude. I shall be glad to know if there is any more in
this than what was connnonly talk'd when I was in London
in 1732. Please to give my humble ser\ice to Mr Machin and
believe me to be ver}^ afFectionatly
Sir
Your Most Obedient
Edinburgh Most Humble Servant
Nov'-. IG. 1734. Colin MacLaurin
I have taken tlie libert}^ to desyre Mr Warrender to take
advice with you if any diffieultys arise about the publishing
the fluxions or the terms with a Bookseller. I would have
given you more trouble perhaps but he was on some terms
with me before you got to London.
(10)
Maclaurin to Stirling, 1738^
To
Mr James Stirling
at Lead hills
Dear Sir
This is a copy of Maupertuis's letter which I thought it
would be acceptable to } on to receive. I am told Mr Cassini
would willingly find some fault with tlie Oliservation to save
his father's doctrine, but is so much at a loss that he is obliged
to suppose the instrument was twice disordered. H" I can be
of any service to yon here in anything you may always
command
Dear Sir
Your Most Obedient
Humble Servant
Ed^. feb.'" 4. 1737. Colin Mac L.\urin
I forgot when yon was here to tell j^ou that last spring
1 1737 O.S. or 1738 N.S.
76 STIRLING'S SCIENTIFIC CORRESPONDENCE
some Gentlemen liad formed a design of a philosophical
society here wliicli tliey imagined might promote a spirit for
natural knowledge in this country, that you was one of tlie
meml)ers lirst thought of, and tliat Ld Hope & I were desyred
to speak to you of it. I liopc and intreat j^ou will accept.
The mnnhcr is limited to 45, of wliicli are L''^ Morton, Hope,
El[)lnnston, St Clair, Lauderdale, Stormont, L'' president &
Minto, S' Jolni Clark, D" Clark, Stevenson, St Clair, Pringle,
Johnston, Simpson, Martin, Mess. Munroe, Craw, Short,
Mr Will'" Carmichacl c^'c. I shall write you a fuller account
afterwards if you will allow me to tell them that you are
willing to be of the nundjer. If j^ou would send us an^-thing
it would lie most acceptable to them all iS: particularly to
yours iS:c
I had a letter from Mr De Moivre where he desyres to
give his humble service to 3'OU. His book was to be out
last week.
Mcmpertms to Bradley
A letter from Mons"" ]\Iaupertuis
To Professor Bradley
Dated at Paris Sepf 27^^ 1737 N.S.
[Translated fi-om the French]
Sir
The Rank You hold among the Learned (k the great
Discoveries with which you have enriched Astronomy, would
oblige' me to give j^ou an Account of the Success of an Under-
taking, which is of considerable consequence to Sciences (even
tho' 1 were not moved to do it by my desire of having the
honour to be known to }0u) by reason of the Share vou have
in the Work itself. Whereof a great part of the Exactitude
is owing to an Instrument made on the Modell of yours, and
towards the Construction of which I know you were pleased
to lend 3'our Assistance.
Wherefore I have the honour to Accpiaint You Sir, That we
are now returned from the Voycige we have made by Order of
His Majesty to tlie Poler Circle. We have been so hajipy as
CORRESPONDENCE WITH MACLAURIN 11
to be able, notvvithstaiKlino- the Severity ol* that Climate, to
measure from Tornea northward a Distance o£ 55023-47 Toises
on the Meridian. We had this distance by a Basis the longest
that ever has been made use of in this Sort of Work, &
measured on the most level surface, that is, on the Ice, taken
in tlie .middle of eight Triangles. And the small number
of these Triangles, together with the Situation of this great
Basis in the Midst of them, Seem to promise us a great Degree
of Exactness ; And leave us no room to apprehend any con-
siderable Accumulation of Mistakes ; As it is to be feared in
a Series of a greater Number of Triancrles.
We afterwards determined the Amplitude of this Arch by
the Starr (5 Draco als, Which we observed at each end with
the Sector you are Acquainted with. This Starr was first
observed over Kittis, one of the Ends, on the 4, 5, 6, 8, 10 of
October 1736.
And then we immediately carried our Sector by Water to
Tornea, with all the precaution requisite its being any way
put out of Order, And we observed the same Starr at Tornea
the 1. 2, 3, 4 & 5, of Nov'' 173G. By comparing these two
Setts of Observations we found, That the Amplitude of our
Arch (without making any other Correction than that which
The procession of the Equinox requires) would be 57'-25"07.
But upon making the necessary Correction according to your
fine Theoiy (Parallax of Light) of the Aberration caused by
the Motion of Light, This Amplitude by reason of the interval
of Time between the Mean of the Observations, was greater by
l'''-83 : & consequently our Amplitude was 57'.27''-9.
We were immediately Sensible that a Degree on the Meridian
under the Polar Circle was much greater than that which had
been formerly measured near Paris.
In Spring of the ensuing Year we Recommenced this whole
operation. At Tornea we observed Alpha Draconis on the 17,
18, & 19 of March 1737; and Afterwards set out for Kittis,
Our Sector was this time drawn in a Sledge on the Snow, and
went but a slow pace. We observed the Same Starr on the
4, 5 & 6 of Aprile 1737. By the Observations made at Tornea
& Kittis we had 57'.25''-19; to Which Adding 5''.35 for the
Aberration of this Starr during the time elapsed between
the Middle of the Observations, we found for the Amplitude
78 STIRLING'S SCIENTIFIC CORRESPONDENCE
of our Arch 57'-:50"-54 wliicli ditiers 3''i iVoiu tlio Amplitude
(k'torniined liy S (Delta).
Therefore takiu^j^ a Mean between these two amplitudes,
Our Arch will be 57'-28"-72 which beino- couipared with the
distance measured on the Earth, gives the Degree 57437'1
Toises; greater by 377-1 Toises than the J\li<ldle Degree of
France.
We looked upon the Verification which results from the
Agreement between our two Amplitudes deduced from two
ditterent (Setts of) Operations (Joined to the precautions we
had taken in the Carriage of the Sector) We looked (I say)
upon this Verification to be more certain than any other that
could be made ; and the more because our Instrument cannot
from its Construction serve to be turned Contrary Ways.
And that it was not recpiisite for our operation to knov/^
precisely the point of the Limb which answered to the Zenith.
We verified the Arch of our Instrument to be 15°^ by
a Radius of 380 Toises, and a Tangent both measured on
the Ice : and notwithstanding the great Opinion we had of
Mr Graham's Abilities we were astonished to see, that upon
taking the Mean of the Observations made by 5 Observers
which agreed very well together ; The Arch of the Limb
diff'ered but 1" from what it ought to be According to the
Construction. In fine, we Compard the degrees of the Limb
with one Another, and were surprized to find that between
tiie two Degrees wliicli we had made use of, there is a
Small Inequality, Which does not amount to l'\ & Which
draws the two Amplitudes, we had found, Still nearer one
Another.
Thus, Sir, You See the Earth is Oblate, according to the
Actual Measurements, as it has been already [found] by
the Laws of Staticks : and this flatness appears even more
considerable than Sir Isaac Newton thought it. I'm likewise
of Opinion, both from the experiments we Made in the frigid
Zone, & by those Which our Academicians sent us from their
Expedition to the E(|uator; that Gravity increaseth more
towards the Pole, and diminishes more towards the Line, than
Sir Isaac suppos'd it in his Table.
And this is all conformable to the Remarks you made on
Mr Campbell's Experiments at Jamaica. But 1 have one
CORRESPONDENC^l^ WITH MACLAURIN 71)
favour to Itei; of you, Sir, & hope you will not rcfus-c it lue;
Which is, to let lue know if you have any immediate Observa-
tions on the Aberration ot" our two Starrs 8 e^ a' DracoiitK;
and if we have made proper (\)rrecti()ns for this AbeiTation.
I shall have the honour, at Some Other time to eouumniicate
to you our Experiments on Gravity, & the Whole detail of dur
Operations, as soon as published.
I have the honour to be with Sentiments of the highest
Esteem
Sir
Yovu- Most humble iV' most Obedient Servant
Maupiktuis
I shall be much obliged to you if you will be pleased to
Communicate . . , the Royal Soe . . .
(11)
Maclaarin to IStirJiiig, 1738
Mr. James Stirling
at Lead Hills
Dear Sir
There is an ingenious young man here who I am very
sure will please you for what you write about. I have
promised him no more but that you will bear his charges in
going & returning & give him gome small thing besides
perhaps. I have not omitted to acquaint him that he will
have opportunity to improve himself with you. He is a quiet
modest industrious & accurat young man. I think I have
mentioned him to you as one who seems to have a natural
turn for making mathematical instruments, & deserves en-
couragement. But his father is a poor minister who has
ruined himself l)y lawsuits. If it will be time enough, it will
be more convenient for him to go about the middle or end of
May than just now.
I have a part of a letter I writ for you some weeks ago
in town, but some incidents hindered me from finishing it.
8;) SriRLTNG'S SCIENTIFIC CORRESPONDENCE
I shall write soon by the post. This j^oes by a student who is
to leave it for you at the lead hills.
I aiu Dear Sir
Your Most Obedient
Dean near Ed'" Humble Servant
April 1738. Colin MacLaurin
JMr Do Moivre's Ijook is come but I have not had time
to look much into it. I think you said you would send me
Mr Machin's piece. I say a little of the centripetal foi'ces l)ut
that part is now a printing off. Have you ever had occasion
to enquire into the tiuent of such a (juantity as this
X Va — XX Vh — XX Vc — X
The common methods do not extend to it.
My family is now come to this place, but I go every day to
town to the coUeire. The removini^; & some incidents occasioned
my delay in writing which I hope you will forgive.
(1:2)
Madimr'm to StirUng, 1738
To
Mr. James Stirling
at Leadhills
Dear Sir
This is to introduce Mr Maitland whom 1 have dispatched
sooner than I intended because of your urging it in a letter
I received on Monday last. I heartily thank you for
Mr Machin's piece, and that you may not be deprived of
the book bound in with it I shall send you my copy of it.
I am persuaded many things are wanting in the inverse
method of fluxions especially in what relates to fluents tliat
are not reduced & perhaps are not reducible to the logarithms
or circle. I give a chapter on these, distinguish them into
various orders, and shew easy constructions of lines by whose
CORRESPONDENCE WITH MACLAURIN
81
rectification they may be assigned, how to compare the more
complex with the more simple & other things of tliis nature.
But I suspect that some fluents (at least in some suppositions
of the variable quantity) may be reduced to the circle or
logaritlims that are not comprehended in the cases that have
been considered by Cotes <% De Moivre.
I could not hit upon a letter I had writ a great part of
to you in our vacation week when I sought for it today.
I shall mention sometliings of it as my memory serves.
I easily found as you observed that the
rigiit line AB attracts the particle P with
the same force as the ark GED but I could
make little use of this because when the figure
revolves on the axis PE, the attractions of the
circle generated by AE & of the spherical
surface generated by CE are not equal.
I found that what I had observed long ago
of the attraction of spherical surfaces holds
likewise of what is included betwixt two
similar concentric spheroidical surfaces in-
finitely near each other viz. That the attraction
of the part convex towards the particle is equal to the
attraction of the part concave towards it. This holds whether
the particle be in the axis of the spheroid or not.
Let EGKL be any solid, P the particle attracted, let PEK
Fig. 5.
Fig. 6.
meet the solid in E &, K and any surface GHL in H, let NH
be to EK in any invariable ratio, and the point N form
a surface GXL. Then the attraction towards the solid GNLH
shall be to the attraction of the solid EGKL in the same given
ratio of NH to EK.
Let ACE be a quadrant of a meridian, A tlie pole, E at the
2447 Q
8.2 STIRLING'S SCIENTIFIC CORRESPONDENCE
equator, it' T^M be the divectiou of the onivity at L then CM
sliall he to tlie ordinate LP in an invariable ratio. This ratio
I cannot preciseh^ recollect unless I had my
paper < which ai"e at the Dean. I remember
it is compounded of two ratios but how I can-
not suddenly recollect One of them I think
is the ratio of the o'ravity at A to the force
towards a sphere of the radius GA, the other
is the ratio of the gravity at E to the force
towards a sphere of the radius GE. I write
this in a haste at the college because Mr Mait-
land waits for it and I do not incline to
detain him.
On looking over the argument by which
I had thought to have proved that the earth is a spheroid,
I found that it supposed that in any right line GL from the
center the gravity at L is to the centrifugal force as the gravity
at / is to the centrifugal force. But this seems to need a proof.
I have some more propositions, if they be worth your while I
shall send them.
Having no time to go home for the book I was to send
I delay it till some carrier call to whom I shall give it. If
you will send me your receipt for De iNToivre I shall cause one
of the Booksellers get it down. In the mean time you may
command my copy if you please. I am
Dear Sir
Your Most Obedient
Ed-" May 12. Humble Servant
1738 Colin MacLaurin.
(13)
Stirling to JMucJanrbi, 1738
Leadhills 13 May 1738
Dear Sir
I am obliged to you for dispatching IMr Maitland, for
I am in a hast, ^ I hope he will do very well with smal
assistance.
CORRESPONDENCE WITH MACLAURIN
83
I shal be very gUul to see what you liave on tluents when
your book comes out, particularly it' you can reduce to the
area of a Conick Section, figured not comprehended in the
Theorems of S'" Isaac, Cotes, or Be Moivre, I readily agree
with you that great improvements may be in that piece of
knowledge ; but the way to it is so rugged that I am afraid
w^e arc not in the right path.
Fig. 8.
As to the attraction of an arch and its tangent being the
same, on a particle placed in the center, it was of no use
to me more than to you. What you say about the attraction
of the concave and convex part of a spheroidical surface,
being the same on a particle of matter, holds of any part
of a spheroid comprehended l)etwixt two similar, concen-
trical and similarly placed spheroidical surfaces, whether
the distance betwixt them l)e infinitely smal or finite ;
Suppose two such surfaces to l)e AEKB and GD^M, and a
particle P placed any where ; through P and F the center of
the spheroid, imagine a spherical surface to be described
similar and similarly placed with AEKB; and that surface
will cutt of the concave part from the convex part; and will
divide the W'hole spheroid into two parts, whose attraction on
P are equal ; which is true wdiether the particle P be w-ithout
or with the spheroid. The reason of it is because the ellipsis
passing through P and F, cutts all the lines AB and EK into
equal parts, if they converge to P. And from the same
principle follows what you say next in your letter, about the
attraction of solids being in a given proportion : because the
solides may be divided into cones wdiose vertex is the particle
attracted. And what you say about LP being in an invariable
ratio to CM is true ; but that ration cannot be assigned without
G 2
84 STIRLING'S SCIENTIFIC CORRESPONDENCE
the quadrature ol' the circle. And the Avhole problenic about
tlie variation of (gravity on the Surface
depends on it. When I firs solved that
prol)lem, I supposed the attracted particle
to be on the surface ; but now I am upon
solving it, when the particle is placed with-
out the spheroid on any distance, which
I have not had time yet to do, altho I know
I am master of it ; I have done it at the
equator, I mean when the particle is in
the plain of the equator produced ; Newton did it when it was
in the axis produced.
Suppose two ellipses similar described about the same center
whose axes are EK and ek, and GL and gl the diameters of
Fig. 9.
P-=
Fig. 10.
their equators whose difference I suppose infinitely little : Let
F the focus and C the center ; then if the elliptic ring revolve
ai)out the axis EK and generat a solid ; and P be a particle in
the axis produced, the gravitation of the particle P towards
the solid comprehended betwixt the spheroidical surfaces will
CORRESPONDENCE WITH MACLAURIN 85
be proportional to — jj-p^ — : that is in a ratio compounded of
the direct ratio of a rectangle under the axes, and in the
duplicate inverse ratio of the distance of the particle from
either of the foci : whence it follows that the gravitation of
the particle to the whole spheroid will be proportional to the
bigness of the spheroid and the diflerence betwixt the ai'ch
Or (described on the center C) and its tangent CF.
Again if M be a particle in the plain of the equator pro-
duced, it will gravitate to the part of the spheroid betwixt
the two spheroidical surfaces with a foi'ce propoitional to
— . And thence the gravitation of the particle
FGVPC'-CF^
to the whole sphseroid will be found to depend on the quadra-
ture of the circle, nay upon the forementioned difierence CF
and Cr. I have gone no further ; but could accomplish what
remains in a week or ?o if I had leisure. What I here send
you are conclusions hastily drawn, and therefore I would not
have them communicate because I have not yet examined
them to my own satisfaction, and I write in such hast that
I dont know if I have transcribed them right. I am in
great liast
DS. Your most obedient humble Servant
James Siirling.
(14)
Maclatirin to Stirling, 1738
To
Mr James Stirling
at Leadhills
Dear Sir
I believe you will find Mr Maitland utefull & exact and
am glad he has so good an opportunity of improving himself
under your eye. I wish you had time to finish what you are
doing relating to the figure of the earth. I am informed thst
something is soon to be published on that subject at London
by Celsius & others.
86 STIRLING'S SCIENTIFIC C0RRESP0NDENC1<:
The account 1 gave you ot" some propositions had occurred
to me on that subject was very imperfect. You may observe
from what follows it, that when I spoke of concentric surfaces
infinitely near I restricted it onl}'' that I mioht distinguish
the parts more properly into such as were convex and concave
towards the particle. I inquired into the ratio which I paid
was invariable & obtained it in a simple enough series which
I have not reduced to the quadrature of the circle, tho' I
conclude from your more perfect solution that it must be
reducible to it. I did not try the problem by the concentric
surfaces but in a different manner. And tho' I think 3'our
method must be better since an account of a different one may
be agreable to you I shall describe the principal steps I took.
Supposing PB the shorter axis, AC the transverse semi-
axis. I first computed the fiuxion of
the attraction of the solid generated by
PMB while the figure revolves about
the axis PB, and thence demonstrated
what Mr Cotes says of the attraction
of spheroids. By comparing what I
had found in this with 3'our account of
the attraction of P I drew immediately
on reading j^our letter this consequence
that seems worthy of notice. That if
PM meet a circle described from the
center P with the radius PC in N and
NR be perpendicular to PB in R, &: PE be taken equal to CR,
and EFG be a similar concentric semiellipse, then the attraction
of P towards the solid generated by EFGE revolving about
EG shall be equal to the attraction of P towards the solid
generated by the segment PAM revolving about PB. This
however I did not observe in the spheroid till I got your
letter, in the sphei-e it is obvious.
After I had made out Mr Cotes's theorems, I then proceeded
to consider the attraction at the equator, and still sought the
fluxion of the attraction of the solid which seemed then to me
to be more easily obtained than that of the concentric surfaces
in this case especially. I supposed therefore the solid to be
projected orthogi'aphically on the plane of the meridian
PA B D, the particle attracted I supposed to be directly over C,
CORRESPONDENCE AVITH MACLAURIN 87
and to bo in the pt)lc of the meridian FABD, 2s CM k nCm
to be any two infinitely near elHpses passing through the
particle ; and then I computed the attrac-
tion of the matter included betwixt these
two ellipses, or the fluxion of the attraction
of the solid represented by CPM. Thus
I found that if CP = a, GA = h, CF(F being
the focus of the generating ellipse) = c,
then the attraction of a particle at the
equator towards the spheroid is to the
attraction towards a sphere of the radius
36-2 9c*
CA as 7X1 +
I)
+
&c : is to unit.
106- ' SG/y-*
From this I computed the invariable ratio
I mentioned in my last, wherein the difi'er-
ence of the tangent OF & ark CZ entered
by Mr Cotes' s theorem already spoke of.
But by 3^our letter I perceive you have found the same
invariable ratio without a series, by the quadrature of the
circle only. From which I perceive that if the series I found
be legitimate, as I cannot doubt but it is, it must be assignable
by the circle. This perhaps would be easily found by
examining it, but since you have done this already in effect
I would willingly avoid the trouble. And only desyre you
will let me know if the proportion given by this series agrees
well enough with what you have found. I believe I might
have computed your proportion from what you sent me^ but
there are so many of my acquaintance in town this week
& I have had so little time that I have not got it done.
I have some suspicion from the fluxion that gave this series
that it is reducible to the circle, or to the square of it, by
a way I have sometimes made use of and I believe is not new,
of transforming a fluxion by the negative logarithms, but
I have not made the computation necessary to judge of this.
You may be assured that I will communicate nothing of
what you send me without your express alloiuance. I say
something on this subject in my book, and would willingly
add to it if you pleased, because since my book is grown to
such a bulk I would willingly have as much new in it on the
usefull problems as I can. I first proposed only to demon-
88 STIRLING'S SCIENTIFIC CORRESPONDENCE
strate Mr Cotes's theorems in a brief manner enough after
what Sir Isaac has on spheres, and so refer for the rest to
your piece in the transactions; but I wouki think it more
compleat to add this I have found since on the attraction
at the equator &: either suVyoin that 3'ou had a more compleat
solution which you would publish afterwards or mention,
if you inclined that solution itself. In this I sliall do just
as you pleaf-e.
I have not as yet tryed if the method I took for the attrac-
tion at the equator would succeed for computing the attraction
at any other part of the spheroid, and hardly think it worth
while to [ ] since you have a method that appears to be
much better. All I have mentioned I did before I received
your letter except the observation near [ j end of the
first page of this letter, else I had not taken so much p[ains]
about it. I was chiefly induced to try it, because I imagined
the method to be different from your's, and sometimes by
following a different method conclusions come out more
simj)le ; but it has not proved &o in this instance as far as
I can judge of your r[esult].
I told you there were some fluxions which I had ground to
suspect depended on the circle & hyperbola besides those
described already by authors but I did not say that I had
reduced these fluxions That I sent you is one of them, in
certain cases of the variable quantity. I resolve to try it,
but it is my misfortune to get only starts for minding those
things & to be often interrupted in the midst of a pursuit.
The enquiry, as 3'ou say, is rugged and laborious. This is
the first post as I am told to the lead hills since I got your
letter, and I shall be obliged to you if you will let me know
without delay whether the series I described agrees with your
solution by the circle which I imagine you will see at a look.
I am
Dear Sir
Your Most Obedient
Dean May 20. 1738 Humble Servant
Colin Maclaurin
I have not the transaction by me where your paper is, else
that perhaps would solve my question.
CORRESPONDENCE WITH MACLAURIN 89
(15)
Stirling to Maclanrin, 1738
To
Mr Maclaiirin Professor of Mathematicks
in
Edenburo-li
Leadhills 2G October 1738
D. S.
I was sorry that when I was last in Edenbnrj^li I could
not get time to wait on 3'ou. I got a letter this last summer
from IVIr Machin wholly relating to the figure of the Earth
and the new mensuration, he seems to think this a proper
time for me to publish my proposition on that Subject when
everybody is making a Noise about it : but I chuse rather
to stay till the French arrive from the South ; which I hear
will be veiy soon. And hitherto I have not been able to
reconcile the measurement made in the north to the Theory :
altho Dr Pound's and Mr Bradley's most accurate observations
on the Diameters of Jupiter agree to two thirds of a second
with m3^ computation. Mr Machin tells me you write to
him that you had hit on a demonstration to prove the figure
.of the earth to be a spheroid, on which I congratulate
you, for my part hitherto I can only prove it by a compu-
tation.
I have lately had a letter from Mr Euler at Petersburg!!,
who I am glad to find is under no uneasiness about your
having fallen on the same Theorem with him, because both
his and the demonstration were publickly read in the Academy
about four years ago ; which makes me perfectly at quiet
about it, for I was afraid of giving grounds of suspicion
because I had long neglected to answere his first letter : his
last one is full of a great many ingenious things, but it is
long and I am not quite master of all the particulars. I have
also heard lately from M. Clairaut, where he makes a great
many apologies for having taken no notice of my paper about
the figure of the earth when he sent his from Lapland to the
Royal Society ; and he tells me he has carried the matter
further since that time in a new paper which he has also sent
90 STIRLING'S SCIENTIFIC CORRESPONDENCE
to tliu Royal Society : now lie says he has heard that I have
been at some pains about that problemc and desires to have my
opinion on his two papers. Tlie first I barely saw l)efore it was
printed, and altho I had not time to read it thoroughly I soon
saw that it was not of a low rank, as for the second I never
saw it; and therefore I should be much ol)li<;ed to you if you
could favour me with a sight of both, that I might be able to
answere his letter. If you can, please send them to Mr Mait-
land who will give them to Mr Charles Sherrif at Leith with
w^hom I correspond weekly, and they shall be carefully and
speedily returnetl. I haxe yet had no time to medle with
that affair, and when I have, possibly I may not have inclina-
tion ; but I shal be very glad to hear what you are doing
& wdien we may expect to see your book
Sir
Your most obedient &
most humble servant
James Stirling.
(IC)
Maclaiirin to Sdrling, 1740
To
M"" James Stirling
At Leadhills
Dear Sir
I designed to have writ last Saturday, but having gone
to tlie country that forenoon, I did not get homo that day.
I am glad you are to send us a paper, and thank you for
allowing M' Maitland to come here for some days to help me
to forward the plates. I will acquaint him when I shall be
ready for him, that I may make that my only business
(besides my Colleges) while he is here. We have some daj^s
of vacatioii about Christmas, if that time be not inconvenient
for you I can find most leisure to apply to the figures then.
I have so much drudgery in teaching, that I am commonly
so fatigu'd at night I can do little business.
M'' Short writes that an unlucky accident has happened
to the frencli Mathematicians in Peru. It seems they were
CORRESPONDENCE WITH MACLAURIN 91
shewing some iTeiieli guUantiy to the natives wives, wlio have
murdered their sei'vants destroyed their Instruments & burn't
their pajK'rs, the Gentlemen escaping narrowly themselves.
What an ugly Article will this make; in a journal
M"" Sliort saw the satellite of Venus Oct''. 23 for an hour in
the morning, the phas is similar to that of Venus, but writes
that he has never been able to sec it since. His account
agrees with Ca&sini"s. It is a very shy planet it seems.
M'' Graham has found that Brass has some influence on the
magnetic needle, but I have not got a particular account of
the experiments.
I wish I had an opportunity to shew j^ou all that I have
printed in my book relating to the attraction of spheroids and
the figure of the Earth. In the mean time I shall give you
some of the chief articles. 1. I begin with what I sent
you two years ago, but the demonstration is somewhat
difierent. 2. I give a general proposition concerning the
attraction of a slice of a solid the figure of the section and
position of the particle being given. 3. I apply this to
spheres in a few words, and then to a spheroid.
The attraction at the pole is measured by an area easily
reduced to the circle. The attraction at the equator by the
complement of this area to a certain rectangle.
Here I take notice that you was the first that measured the
attraction at the equator by a circle. 4. I easily reduce
the al traction in the axis or equator produced to the attraction
at the Pole and circumference of the equator, without any
computation or new quadrature. 5. I apply this doctrine to
the late observations & mensurations. G. The result of this
leads me to shew that a density increasing towards the center
accounts for a greater increase of gravitation from the ecpiator
to the poles but not for a greater variation from the spherical
figure ; and that it is the contrary, when the density decreases
towards the center. I then compute both in several hypotheses
of a varial)le density, and then propose it as a query whether
D"" Halley's hypothesis may not best account for the increase
of gravitation & of the degrees at the same time. I afterwards
treat of Jupiter, and find that supposing his density to increa,se
with the depth uniformly so as to be 4 times greater at the
center than at the surface, the mean of D'' Pound's ratios will
92 STIRLlNcrs SCIENTIFIC CORUESPONDEXCF
result. I find tlie variation i'rom Kepler's law in the periods
of his satellites arising from the splieroidieal tigure of the
primary cannot be sensible. I shall send you the proposition
you mention and would have sent it today, but I have been
somewhat out of order. It would be better to send j'ou the
2 or 3 sheets that relate to this subject if I could find a proper
oi)portunit3\ 1 know not any particular reason for }tV Machin's
printing that piece of late. M'" Short who engaged to send me
the transactions has not as yet sent me M"" Clairaut's 2*^ paper.
I have printed all my book, excepting the 3 last sheets. The
printers are very slow in the algebraic part, and I have little
time at this season of the year. This with the figures will
retard the publication I believe to the spring. I am
Dear Sir
Your Most Obedient
Humble Servant
Colin MacL.vuiux.
Edinburo-h: Dec'. 6. 1740
II
SIR A. CUMING AND STIRLING
Cuming to Stirling, 1728
Kensington July 4*'' 1728
These were transmitted me from Scotland this day by
M"^ George Campbell. I am
Dear M'" Stirling
Your most obedient humble
Servant
Alex"". Cuming
Let water run out of y^ circular hole NBRD whose radius
BC = r. Let AC y® constant height of y°
water above G y® center of y" hole be = a,
and let Q = y"^ quantity of Water which
wou'd be evacuated thro y^ same hole in any
given time t ; providing y° water was to run
out at all parts of y" hole with y" celerity at
y° center C. Then y° quantity of water
which will be evacuated in y° same time
will be =
B
N( cHr
D
Fig. 13.
1 -
3-
^^^-2^i ^7^+4^6 ^
13 5-7 Or'',
+ tX-x-- X — X X— T + IVC
4 G 8 10 12 a''
Let A DP be y^ elliptick Orbit which any of y" planets
describes about y® Sun placed in one of y^ foci S, let i^ be
94- STIRLING'S SCIENTIFIC CORRESPONDENCE
y" other I'ocus, C its center, ^1 y aphelion, P the perihehon,
SMy^' mean distance of y" planet from y" sun,
and let 7) be any place of y° planet. Let SM
or CA he = r, ye lesser semi Axe CM = c,
r — c = (l, the excentricity tiC = a, k let m
represent y" de<^rees in an arch of a circle
equal to y' radius or m = 57-29578. Let u
be y*^ sine of y" angle AFl), and x the sine
of its double y" radius being — r. Then y''
difference between y" angle AFD (which is
y*' mean acquate anomaly) and y" mean anomaly
Fig. 14.
belonging to it, will be
_ 2 md-^u^ 4 ma^u^
- 3 c3^:5 5 ~
+
•nid
2r'
c'r"
X 1
+
6 ma'' lb'
iVc
7 c:"!'^
9c"-^f^ + 8c(:Z- + 2(P
+
100c3cZ2+145c2cZ3+72c# + 12c?-^ li}
13c^ r*
-c^c.
From whence is deduced an easie method of determining y®
true anomaly from y° mean anomaly being given.
Let the angle Y be found which beaieth y° same proportion
to an angle of 57-29578 degrees which half y® difference
between y" semi axes bears to y'^ greater semi axe. Let also
y® angle Z be found bearing y° same proportion to y® angle
of § of 57-29578 degrees or .38-1971 degrees which y^ cube of
y" eccentricity bears to y^ cube of ludf y" greater semi axe.
Take an angle T proportional to y'^ time in which the Arch
^D is described or equal to y® mean anomaly. Then let y''
angle V be to y" angle Y as y'' sine of twice y'^ angle T is to y®
radius, let also y ' angle X be to y*^ angle Z as y® cube of
y® sine of T is to y° cube of y" radius, then y° mean acquat
anomaly or AFD will be very near T+X+ F when T is less
than 90, but T^X-V when T is more than 90° and less
than 180=.
Let z represent y^ ratio of y^ centripetal force at y^ acquator
of any planet to y" power of gravity there, tlius in y" case of
y° Earth z— gig- Then ye aequatorial diameter will be to
the Polar, as 1 is to 1 — l-s 4 ■^%z- — -Mn%z'' kc.
Ill
G. CRAMER AND STIRLING
(1)
Cramer to Stirling^ 17 28
To
Mr James Stirling F.R.S. in y° Academy
in little Tower Street
London
Sir,
Tis time to break ofFy^ silence vvich I kept so long, the'
unwillingly. The wandering life of a traveller, and a long and
tedious distemper, have been the only reason, why I did differ
so long from giving you thanks for all the kindnesses and
tokens of friendship you bestow'd upon me during my sojourn
in London, and from making use of the permission you gave .
me of writing to ye, and inquiring into the litteral news of
your countr\', but chiefly into the news of your health wich is
very dear to me.
The very day of my departure I received a Letter from
M"" Nicolas Bernoulli desiring me to present you his duties.
In the same he demonstrates in an easy way, a General
Principle whence it is not difficult to derive all y'' Propositions
of M"" de Moivre about his Serus recur rentes. The principle
is such. Let m + n + p + q, be the Index of y^ Series, and
inquire into y'' Roots of y'^ Equation z'^ — mz^ — nz^—2)z — q=
Let them be z, y, x, v: And make four Geometrical Series
the Indices of whom be z, y, x, v. The Sum of y" respective
Terms of these Geometrical Series is the respective Term of y^
Series recurrent four terms of wich may be given, because y®
four first Terms of y'' Geometrical Serieses are taken ad
1)6 STIRLING'S SCIENTIFIC CORRESPONDENCE
libitum he demonstrates also liis method for findin<^ the Com-
ponent quantities of a Binomimu like 1 + :" hy y° Division of
ye Circle
I would fain know your opinion of this demonstration
I found of M"" de Moivre's first Lemma in his Doctrine of
Chances. The Lemma is such
The number of chances for casting 7; + 1 points, with n
Dices of / faces each is
^:) .p— 1 .^) — 2 ...p — 91 + 2 n q .q—\ ...q — n + 2
1.2. 3. ..91^1 1 1 . 2...n-l
n{n—\)r.T—\ ...r — n-\-2
1-2 1.2 ...n-l
{<l=P~f
n.n—ln—2 s . s— 1 ... s — ?i + 2 „ .
X = (VC V' = q — t
1.2.3 1.2. ..71-1 ^ .
[s = r-f
&c.
The Series is abrupted when one Terra comes to be nought
or negative.
My demonstration is grounded upon that principle that the
number of chances for casting ^? + 1 points with n Dices is
equal to the number of chances for casting p and 2^—1 and
p — 2 &c. to 2^~f+ 1 = 5'+ 1 points with n— 1 Dices. For it
follows that y® number of chances for casting p points with
one Dice is p^ — q^, wich is equal to nought if q is positive
that is if p is bigger than/ and equal to one if |? = vel < /.
Now the number of chances for casting p + l points with
two Dices is equal to y® number of chances for casting 2^
with one Dice = 2^^ — <l^+ to y" number of chances for casting
2? — 1 with one Dice = p — 1 —q—1
&c &c &c &Q
to y° number of chances for casting j>—/+ 1 with one Dice
that is q + 1 — r+1
The Sum of y^ P* Col. p-q
of the 2-^ Col. -q + r
Total sum 2) — 2 (7 + r
CORRESPONDENCE WITH CRAMER
97
I couVl proceed in the .same manner to the case of three
Dices, then to four, and so forth, and if 1 will, demonstrate in
general that if the Lennna holds for the case of n—1 Dices it
holds too for n Dices.
M'' 'S. Gravesandc, who is wholly employ 'd about y®
Doctrine of forces, did comnuinicate me the following con-
struction for the laws of percussion.
Let A and B be two bodies Elastic or not Elastic. AL, BL
their respective velocities before the shock. Let D be their
Fig. 15.
center of gravity, and DC be drawn perpendicular to AB
of an indeterminate length. Draw AC, BG to be prolong'd if
it needs.
Now if the bodies are not Elastic, QC will be the common
velocity after y® percussion. If they are Elastic, take Cs — GB
and GT= CA and PT shall be the velocity of y« Body A,
and PS the velocity of the Body B after y" Concussion.
If they are imperfectly Elastic, take C'y to GS and Gt to GT
as y® elasticity to the perfect elasticity and Gt, Gs shall be the
velocitys of the Bodies A and B. In his opinion about the
forces of the Bodies, this construction is very commodious, for
before the percussion ALM represents the force of y® Body A,
and BLN the force of y® Body B. But after y^ percussion
CTM and GX are the forces of the bodies A and B, if they are
elastic, and CQM GQN are these forces if they are not elastic,
and AGB is the force lost in y® percussion
M'' 'S Gravesande demonstrates it, by this proposition, That
y" instantaneous mutations of forces in the two bodies, are
proportional to their respective velocities. But I found that
2447 H
98 STIRLING'S SCIENTIFIC CORRESPONDENCE
it cou'd be proved, witliout the new notion of forces, by this
proposition. That y'' contemporaneous mutations of velocities
of the two bodies are reciprocal to their masses wicli can be
evinc'd in several manners, and very easily, if granted that
the connnon center of gravity does not alter its velocity by
the percussion.
I am just ariived at Paris, and so have no news from france
to impart with ye. You'll oblige me very much, if you vouch-
safe to answer to this, and inform me about your occupation
and those of your Royal Society and its learned members.
Did M"" Machin publish his Treatise about y*" Theory of y*^
Moonl Is M'" de Moivre's Book ready to be published? Is
there nothing under the press of S"" Isaac's remains? What
are you about? Can we flatter ourselves of the hopes of
seeing very soon your learned work about y'^ Series? All
these and other news of that kind, if there are some, will be
very acceptable to me ; and 111 neglect nothing for being able
of returning you the like, as much as the sterility of the
country I live in, and my own incapacity will allow. In the
meanwhile, I desire you to be fully persuaded, I am, with all
esteem and consideration
Sir
Your most humble
Most obedient Servant
Paris, this i| X'^'° 1728 G. Cramer
You can direct y° Answer
A Messieurs Rilliet & Delavine, rue Grenier S* Lazare pour
rendre ii M"^ Ci'amer a Paris.
(2)
Cramer to Stirling, 1729
To
M-- James Stirling F R.S. at the
Academy in little Tower Street
London
Here is, Dear Sir, a Letter from M"" Nich. Bernoulli in answer
to yours, wich I received but t'other day. I send with it,
CORRESPONDENCE WITH CRAMER 99
according to Ins Orders a Copy oi' his method of resolving y°
quantity ^ ~ 27^ ii^ i^s component fractions the former
part of wich he sent me to Paris, by M'" Klingenstiern the
supplement I had but in the same time with your Letter.
I hope you have lately received from me an answer to your
kind Letter brought l)y M"" Sinclair. I am with a great
esteem
Your most humble
and obedient Servant
Geneva the 6^^ January, 1729. N.S. G, Cramer.
Methodus resolvendi quantitates l+gs" + s"^" in factores
duarum Dimensionum, Auctore U*". Nicolao Bernoulli.
Prob. I Resolvere quantitatem 1+^2-" + 5'-'* in factores
duavum Dimensionum.
Solut. Sit unus ex factoribus 1 —xz + zz
& productum reliquorum
l+az + bz- + cz\.. + )'z''-^ + s^"-- + tz''-'^ + S5« + rz''+\. .
Ex comparatione terminorum homogeneorum product!
horuni factorum cum terminis propo&itae quantitatis invenitur
a = X, b = ax—l, c= bx — a & ita porrho usque ad t = sx — r,
item ±q = 2s — tx, adeo ut quantitates 1, a, b, c, ... r, s, t con-
stituant Seriem recurrentem in qua quilibet terminus per x
multiplicatus est aequalis Summae praecedentis & eequentis.
Jam vero si Chorda complement! BD
alicujus arcus AD vocetur x & ladius
AC = 1 Chordae arcuum multiplorum
ejusdem arcus AD exprimentur respec-
tive per eosdem terminos inventae
Seriei recurrentis 1, a, b, c, &c. multi-
plicatos per Chordam AD. Hinc .'i
arcus AE i^\i ad arcum AD ut 11, ad 1,
erit Chorda AE s.& Chordam AD ut / ad 1, id est AE — t x AD,
& Chorda DfJ— s x AD. Ex natura vero quadrilateri ADEB
h2
]()() STIRLING'S SCIENTIFIC CORRESPONDENCE
circiilo iiisciipti est AL' . l)B — AB . DE + AJ) . BK id est
tx.AD ^ 2s. AD + AD. BE
sive t.c = 2i> + BE = (quia + ry = 2s - tx) tx±'j+ HE,
hinc BE =+q.
Ex (|U0 sequitur (piod si arcus liabens pro Chortla complementi
+ () dividatur in u partes aeqnales quarum una sit arcus AD,
hujus complementi Chorda futura sit x : vel si rem per Sinus
conficere malinuis, dividendus est arcus habeus pro Cosinu
+ ^g in n partes aequales, qui si vocetur A, erit cosinus arcus
— = -X Inv'ento valore ipsius x cognoscitur 1 —xz + zz unus
ex factoribus (|uantitatis propositae l+(/c"+,:^". Sed & re-
liqui factores liinc cognoscuntur. Si enim tota circumferentia
vocetur C, habebunt onnies sequentes arcus A, C—A, C + A,
2 C- A, 2C+ A, 3 C— J., 3 6' + ^, &c pro Cosinu +^q, quorum
singuli in partes aequales divisi determinabunt totidem diversos
valores ipsius x.
Coroll. 1. Per methodum serierum recurrentium invenitur
X — radici hujus aequationis
Coroll 2. Si capiatur arcus AH aetjualis alicui sequentium
A G-A C + A 2C-A 20 + A
il'C & fuerit
arcuum
n n 10 n a
GG = z erit GH — radici quadratae factoris 1 —.xz + zz. Quia
enim CF = \x erit GF = \x — z,
^ . FH= ^n-lx' & proinde GH
— y 1 —xz+ zz.
Coroll 3. Si g = 0, erit A = ^C,
& reli(pii arcus dividendi ^C, |C, ^C,
1 6* &c. Hinc si dividatur tota cir-
cumferentia in 4 II partes aequales
AH, HI, IK, tV'C & ad singulos im-
pares divisionis terminos H, K, j\f,
iv'C. ex puncto G ducantur rectae GH,
GK, &c erit horum onmium productum 1 +5^".
CORRESPONDENCE WITH CRAMER 101
Probl. II
Resolvere quantitateiu 1 +c2?t+i jj^ factorcs cluarimi Diuicn-
sionum.
Solut. Sit uniis ex factoribus l—xc + sz, & productum
reliquorum
1 + (13 + bz^ + cz^... rz''-^ + s:"-- + f ;"-^ + tz"" + sc^+i + rs"+2_ _ ^
& invenitiir ut antea a = x, h = ax—l, c = hx — a, & ita
porrlio usque ad t ~sx — r. Sed loco aequationis ±q= 2s — tx
invenietur haec t = tx + s - id est, si ponatur arcus AD ad
A V DP
arcuin AE, ut 1 ad n, erit (quia t = . ^ & s = -r— , il' x = BD)
' AD AD
AE- AE . BD + DE = 0. Sive DE =AE.BD- AE
& aequatione in analogiam versa
DE:AE= BD-IA = (facta DF = DC = AC = 1) BF: CB.
Hinc tiianoula ADE, CFB, ob angulos ad ^ & i? aequales,
erunt similia & angulus BCF = DAE.
Ergo ang. BCF +ang CBF = ang DFG
= ang DCF = ang DAE+ ang CBF Sed
& ano- CDF = ang CBF. Hinc omnes
tres anguli Trianguli CDF sunt aequales
2 ang. DAE +3 ang Ci?i^ ipsorum que
mensura, id est, semieircumferentia
= ^G= arc DE + ^ arc. AD
= (quia arc DE = n-1 arc A D) — - — arc AD.
C
ideoque arcus AD := . Si imtur circumferentia Circuli
^ 2/t+l ^
dividatur in 2?^+ 1 partes aequales, (j[uarum una sit arcus AD,
erit Chorda BD = x, vel si semieircumferentia in totidem
partes aequales dividatur, erit cosinus unius partis -|a; unde
cognoscetur factor 1 —xz + zz. Quia vero tot factores duarum
dimensionum inveniendi sunt quot unitates i^unt in numero lu
habebit totidem diversos valores qui erunt dupli cosinus 1, 3,
5, 7 &c partium semicircumferentiae in 2/1+1 partes aequales
divisae : invenitur enim arcus ^D = singulis sequentibus
C 36' 5(7 76' , .
arcubus ? ? j 5 (fee, quia arcus Ah
2/1+1 2)i+l 2/t + l 2/t + l ' ^
102 STIRLING'S SCIENTIFIC CORRESPONDENCE
qui est ad arcuiii AB, ut n ad 1. potest iiitellioi auctus Integra
Circuinferentia vel ejus multiplo, hoc niodo igitur resolvetur
(|Uantitas proposita 1 + c^"''"^ in n
factorcs duaruni diniensionum &^
unum factorcm 1 + - unius dinien-
sionis.
Coroll. Si fuerit
CG = z, AC= CB= 1
& Circunit'erentia circuli dividatur
in 4n + 2 partes aequales AH, HI,
IK, i*(c ad si noulosini pares divisionis
terniinos H, K, M, Ac ducantur rectae
Gil, GK, GM, &c, erit horum omnium productum aequale
Probl. Ill Re?olvere quantitntem l — -2»+i \y\ factores
duarum Dimensionum.
Solut. Sit unus ex i'actoribus \—xz + z: & productum
reliquorum
& invenietur s = t + 1x : roliqua vero se habcnt ut prius.
Positis igitiir ut in Prob II arcu AJJBE = n arc A D, x = BD,
AF I)F
t = -^, s = ,~^^, erit DE= AE+AE. BD.
AD AD
Hinc DE:AE= BD+1 : 1 = (facta DF =^ DC = \)BF:CB
Proinde triangula ADE, CFB habentia angulos ad E (l- B
aequales erunt similia, & ano;BCF = ang DAE: quamobrem
ano-: F = ang: DCF = ang 5Ci^- ano-: BCD = ang: DAE-
ang: BCD. Hinc omnes trcs anguli triano-uli BGF sunt =
ang: B + 2ang: D^i^"— ang: BCD: ipsorum que mensura
I C = 1 arc :AD-\ arc : DBF- arc : BD = |arc :AD + arc : BE
= n + ^ arc. AD-^ C.
C 2G
Hinc C = n + h- arc AD, & ai-c AD =
2
It T o
(J ^
cuius dimidii, nempe , cosinus erit ^x. Si arcus ADBE
^ 2 71 + 1
CORRESPONDENCE WITH CRAMER 103
intelligatur auctiis inteij,n\ circumferenticl vel ejus multiplo
invenieiitiu- reliijui valores ipsius -|.i' aequales cosiuibus arcuuin
2C 3C 46'
, , , &c. Et
2*1+1 2 >i + 1 2ii + l
sic resolvetur quantitas proposita
1— s-""*^ in n factores duarum
Diuiensionum, & uniiin factorem
I —: uiiius dimensionis.
Coroll. Si in tio. Coroll. praeced.
ad singulos pares terniinos /, L,
N, &c. ducantur rectae GI, GL,
GN, GO, &c. erit harum omnium productum — 1 — s-"+'.
Probl IV Resolvere quantitatem 1 — c-" in factores duarum
Dimensionum.
Solut. Sit unus ex factoribus 1—xz + zz & productum
reliquorum
1 + az -I- &5^ . . + rz''-^ + sz''-^ ± tz'"''^ - i-z'' - rs"+^ . . - bz^^''^
Hie quia terminus tz^^~'^ debet affici signo tam affirmative
quam negativo, opportet esse t — 0, adeoque si ponatur arcus
AE
AD ad arcum AE ut 1 ad n, & per consequens t = ~xy\'
erit AE — 0, k arcus huic Chordae respondens = vel C, vel
2C, vel 3C &c. Proinde arcus ^£' = alicui sequentium
C 2C 30 , , , . ., C 2C 3(7
arcuum -5 — , — > ivc. iS: ia; = cosinibus arcuum -—} -— > —- >
n U' a 2n 2 n 2 it
&c qua ratione resolvitur quantitas 1— ,s-" in n—l factores
duarum Dimensionum similes huic 1 —xz + zz,& alium factorem
duarum dimensionum, nempe 1—zz.
Coroll. Si in fig. Cor. 2 & 3, Probl I ad singulos pares
terminos divisionis I. L, B, 0, Q, A, Ducantur reciae GI, GL,
GB lI'c, erit harum omnium productum = 1 — j^".
Coroll. geneiale. Si Circumferentia Circuli dividatiir in 2m
partes aequales AH, HI, IK, &c, & ducantur rectae GH, GI,
GK &c sive m sit numerus par, sive impar semper erit
GHx GKxGM &c = l +z'", & GA x GIxGL &c = 1 -s'».
Quod est Theorema Cotesii memoratum
Act. Erud. Lips. 1723, pag. 170 et 171.
104 STIRLING'S SCIENTIFIC CORRESPONDENCE
Supplemcntniu Eodein Auctore
Probl. V Dividere fractionem :; — in fmctiones plures,
1 + qz^ + c-'*
quarum denominatores ascendant tantum ad duas Dimen-
sioncs.
e — fz
Solut. Sit una quaesitarum fractionum , = & sumnia
^ 1—ccz + sz
oc + ^z + yzz + 8z'^ + ez'^ + &c
^^^^^^'^''^ l+az + bz- + cz^ + dz^ + &c
Valor ipsius x determinatur in Problemate primo, & quan-
titates 1, a, h, c, d, &c designant ut ibidem terminos Seriei
recuiTcntis 1, x, xx—l, x^ — 2x, x* — 3xx+l, S:c. Valores
autem ipsarum e & f post eliminationem ipsarum a, /?, y, 8 &c
inveniuntur ut sequitur : neinpe si /? = 2, id est si
1 e—fz oc + ^z
l+qz'^ + z'^ 1—xz + zz 1+xz + zz
invenitur e = i & / = - x - Si 7i = 3, id est si
1 _ e-p Oi + ^z + yzz + Sz^
l+qz^' + z'' 1-xz + zz l+xz + xx-lzz + xz'^ + s^
1 X
invenitur e = 4, & f = -: si n = 4, id est si
•^ * 3 XX — 1
1 e-fz
l±qz + c'' 1 -XZ + ZZ
a + Pz + yzz + Sz^ + es'^ + ^z"^
+
1+xz + xz-l zz + x-^-2xz-^ + xx—l z'^ + xz-' +
1 XX — 1
invenitur e -— ^ & /= - —, — -— : similiter si n = 5
invenitur e — | k f = -^ .' ~ ' , , ^ .neneraliter ob i-atio-
1 1 .s-
neni prosfressionis jam satis nianifestam ent r = - ct/ = - »
ubi s &, t significant duos postrenios tei-niinos Seriei rccurrentis
1, a, b, c, d, iVc. Hinc si in fi<;. Probl 1 .sit Chorda BJ'J = +q
CORRESPONDENCE WITH CRAMER 105
& arciis AD = , erit s:t = DE.AE per ibi demonstrata,
n
& per consequens / = j, , ipsaque quaesita fractio
1 _ DE
6— /^ n~ n. AE'^
1-xz + zz i-BJJz + zz
Si porrho intelli(;atur arcus AE auctus Integra circumferentia
vel ejus multiplo, ita ut inutentur valores ipsarum BD & DE,
e — fz
mutabitiir quoque valor fractionis -^ — — invenienturque
successive omnes fractiones in quas proposita fractio
l+n-n^^2n resolvi potest Q.E.F.
Coroll. Si q = 0, DE= DB = x, AE = AB ^ 2, fractio
I X ^
resolvitur in fractiones banc formam /«, 2n habentes.
1+s
\—XZ + Z'^
Scbol I Solutio inventa congruit cum ea quam Pemberton
ex calculo valde operoso deduxit in Epist. ad amicum pag. 48
& 49 & ejus appendice pag. 11, 12. Est quoque simplicior
quani Moivraei qui invenit fractiones banc formam habentes
I a — le
II it — uii^ ubi a — \x — sinui f arcus BD, I — +\q = 9,mM\
1—xz + zz
^ arcus BE, e = cosinui ^ arcus DE, potuisset enim adbibere
1 ez
banc simpliciorcm expressionem n nV 1—u intelligendo per
l — 2az + zz
e non cosinum sed ipsum sinum i arcus DE
Scbol II Non absimili methodo resolvi possunt fractiones
vel
I I -•^rt— 1 I ^^n
Schol III Methodus praeced. supponit q minorem binario,
quando autem a > 2, fractio :. ^j- resolvi potest ut
106 STIRLING'S SCIENTIFIC CORRESPONDENCE
oc 8
ostendit Moivraeus, in duas lias + poiiendo
1 + X" 1+2/"
£c" = s" X |(/ + V^qq- 1 & 2/" = -" X i 7 - ^iW - ^ >
+ — ^ ^ and /3 = ^
Schol. IV Sint a\ a^ a;", .r'", a;'^' &c valores omncs ipsius x
sen radices hnjus aeqnationis,
\/r/r/ — 4 =: |iz;+ V^xx—i'^ — \x— V\xx—i'
vel potins hnjns
±q - ^X+ V'iu:;^;— l" + i«- V^XX-l^
in Coroll. Prob I inventae, et significent e, /, s, ^, idem qnod
supra, per ea quae Peniberton non sine magno labore in\ enit
in Epist. pag. 49. est
t «
x — x^ . « — a;" . a;— a;"i &c a; - *• . a^ — ^c" . a; — a;"' &c
Denominator harum fractionum invenitur per Regulam
Moivraei dividendo differentialem quantitatis
^x+ V^Xf:-l\ +^x— V^xx—1
per dx & liabebimus
w *A/ • tA/ ~~" tA.' • t^ "^~ tX' tVO
11
ia' + V^xx —ir — n^x— y/ixx — 1
2yAa;a;-l
= (per methodum Serierum recurrentium) nt.
Hinc e = =: _, ^^- /■ — ut supra.
/li yt lit
Schol. V Ut Regula Moivraei quae i'acillimc deducitur ex
art. 163 de I'Analyse des Infinim petits possit applicari, oportet
aequationem esse dehite praeparatam, id est, ita comparatam
ut nulla mutationc, multiplicatione vel divisione opus sit ad
inveniendum terminum pure cognitum, qui prodit quando
CORRESPONDENCK WITH CRAMER 107
Radix ah omiii vinculo liberatiir i^' tcrinimis altissiniac dioni-
tatis iiuUo coefHcientc afficitur ut coutiugit in ista aequatione
non aiitem in altera
^x+ \/i.r.r— 1 1 —^x— ^/^xx—1 I = Vqq — 4.
(3)
Cramer to Siirluuj, 1729
Viro Clarissimo, Doctissinio
Jacobo Stirling
L.A.M. & R.S. Socio
Gabriel Cramer
S.P.D.
Dominum Klingnestierna Matheseos Professorem Vpsalien-
sem aniicum meum intimuni eo digniorem e^^se familiaritate
tua intelliges, quo tibi intimius innotescet.
Is cum apud Germanos baud vulgaris Mathematici famam
reportas&et & a Job. Bernoullio mibi magnopere commendatus
mecum Parisiis degcret ; in Angliam profecturus est ut Matbe-
maticorum tuique in primis consuetudine uteretur. Ubi tuum
in me amorem intellexit, confidit his meis literis se apud te
gratiosum fore quae ne spes cum fallat vehementer rogo te :
Sed ut ad eam voluntatem quam tua sponte erga ipsum
habiturus esses, tantus cumulus accedat commendatione mea,
quanti me a te tieri intelligo. Hoc mibi gratius facere nihil
potes. Vale.
Dabam Genevae ad diem 20 Junii 1729.
Mr James Stirling F.R.S. at y'' Academy
in little Tower Street
London.
108 STIRLING'S SCIENTIFIC CORRESPONDENCE
(4)
Cramer to Stirling, 1729
Mr James Stirling at the Academy
in little Tower Street
London
Sir
I received some days ago your dear letter, wicli in such
a Town, and such a Time of Carnaval, I could not find any
proper moment to answer sooner. I wrote this morning to
Mr Nich. Bernoulli and presented him your compliments. I
gave him advise too of your Mind of writing to him. As for
his direction, if you will be so kind as to permit me to be the
Mediator of that correspondence I'll be infinitely obliged to ye :
and you ought l)ut to send me the Letter, wich shall arrive
safe to him.
I don't know whether he has thought upon that difficulty
wich you made me advert to ; of finding any term whatsoever
of a Series recurrens, when y" Divisor by wich it is produced
being put equal to nought, has impossible roots : but I found
an easy way of determining it by y'' help of Tables of Sines
already calculated. For it is known that cich equation wich
has impossible roots, has an even number of them and con-
sequently may be reduced to as many (juadratick equations as
many couples of impossible roots it has : therefore y'' fraction
by y'' division of wich y" Series is produced may be reduced
to as many fractions whose denominator shall be (jujidratick ;
besides, perhaps, some others whose denominator is simple.
Let the fraction whose denominator is quadratick be repre-
sented by that ii:eneral expression where, in v"
-^ ^ i 1 + mx + iixx ''
case of two impossible roots n, is positive and mm less than 4 ;(,.
Now in order to find any term whatsoever of the Series
produced by that fraction for inst, y" term /*'' in order.
Let V II be y'' Radius of a CircU^ and — l>e y'" Cosine of an
Arch z of that Circle: take the Sine of y'' Arch c, multiply
/-I
it by H ^ , and divide it by y'' Sine of y'^ Arch z. 'Vhc quotient
CORRESPONDENCE WITH CRAMER 109
will be y" Term reciiiired. The Deiuonstratioii follows easily
from that Observation, that 1 being the first term ; and the
sine of an Arch z y° second term of a Series recurreus, whose
index is 2c — rr (<• being y*" cosine of y" arch z, and r y^ radius)
each term I is etjual to ye Sine of y ' Arch Iz multiplied by y"
l—\ power of Y radius. Where 'tis to be observed, that
if m, be positive, you needs but to render all y*^ even Terms
negatives.
I am glad that M"" de Moivre's Lemma is by me demon-
strated in a manner that pleases ye ; and since you have seen
M'' De Moivre's own demonstration, I am anxious to know
how far it agrees or differs from mine.
I'll !-ee with a great pleasure M. Maclaurin's Book about
vivid forces, but I fear it shall pass a long time before it
comes into my hands, because English books come abroad
very late : unless you wou'd be so good as to procure one
to M'" Caille where I did lodge in Alderniary Churchyard,
he shou'd pay for it, and find some way of sending it to me
here in Paris. I'll be very obliged to ye for that trouble, and
will be very glad to render ye any Services, when you'll judge
fit to command.
Shall M"" Bradley's account of y" newly observed motion of
y fixt Stars appear in y*" Philosophical transactions, or by
itself % If so, I desire you to take the same trouble about it;
as about M^" Maclaurin's book.
I long after seeing your book about Series, and intreat you
not to put off y*^ printing of it, being sure that whatever set
forth from yowx: hands is excellent, and will be very welcome
in Publick.
I desire you to be so kind as to give me advice, when
M'' de Moivre's book shall be published, because M"" Caille has
got a Subscription for me, and I'll be glad to peruse y" book as
soon as it shall be publish'd.
A learned friend of mine, M'' de Mairan, I should much
oblige, if I cou'd by your help, give him an account of a Letter
wich D"^ Halley wrote about twenty years ago, to M'' Maraldy,
in answer to a Discourse, wich this printed in y® French
Academy's Memoirs A° 1707. against y^ commonly received
opinion of y" Successive propagation of Light : wherein he
endeavours to argfue ao-ainst M'^' Roemer's and S'' Isaac Newton's
110 STIRLING'S SCIENTIFIC CORRESPONDENCE
demonstration drawn iVoni y"" Observations of y'' Satellites
of Jupiter'.s Emersions and Inniiersions. M'' de Mairan wishes
to know, in what time exactly y" Letter was written, and its
contents. If you cou'd help me to a copy of it, or, at least, to
a short abstract of what is most material in it, I shou'd think
myself infinitely oblidged t'ye.
I am asliamed to trouble ye with so much business, but
I hope your friendship will excuse me, and that in like cases,
you will be not sparing of my trouble, wich I shall very
willingly take, being with a great esteem and a sincere
affection.
Your most humble and
obedient Servant
G. Cramer
Paris y W March 17 29. N.S.
(5)
Cramer to Stirling, 1729
To
M' James Stirling, F.R.S. at the Academy
in little Tower Street
London
Geneva, y' if May 1729.
Sir
The place whence I date this Letter, wall be, I hope,
a sufficient excuse for having been so long in your Debt.
I return you my humble thanks for all the trouble you took
on my occasion, and shou'd think myself happy to find some
opportunity of doing you any Service. I received, since y"
last time, I wrote ye, a Letter from M' Nicolas Bernoulli who
seems to be very glad of your correspondence and expects
your Letters impatiently. My direction is now, A Moaslcur
Cramer, Professeur en Mathematique a Geneve. You may
spare y" trouble of freeing them, from London to Paris, if
you'll wrap them in a sheet of Paper directed, A ]\[onsieur
le Fevre Coinmis de la Poste, a Paris.
I grant ye, my way of assigning a Term of a Recurring
Series, wdien y^ Denominator of y ' Fraction hath impossible
CORRESPONDENCE WITH CRAMER 111
Roots is not general enougli : for I thought not of y'' Case you
make mention of: but I doubt veiy much of y*^ Possibility of
a general Solution, for it seems to include a CJeneral Solution
of any Equation.
I have seen lately a Dissertation that M"" Daniel Bernoulh,
IM"" John Bernoulli's son, did read in y'' Petersburg's Academy
concerning the recurring Serieses. What seem'd to me most
material and, I believe, new is that he deduces from this
Serieses, an easy and elegant way of founiling by approxi-
mation two Roots of any Equation, viz: the greatest and y^
smallest. The Method is such.
Let the Equation be disposed after this form —
1 = ax + hx^ + ex"' + Sec,
and make a recurring Series beginning by as much arbitrary
Terms as dimensions The Equation has, and y" index of y®
Series be « 4 6 + c + &c : and any Term divided by y'' subse-
quent shall be equal or very near to y^ Smallest root. The
greatest root is found in y'' same manner if this is y*' form
of y® Equation
and any Term of y'^ Series whose index is a + b + c be divided
by y*^ precedent. The further you continue y^ Series y*^ better
is y" Approximation.
I think myself very oblidg'd t'ye for y'^ account you gave
me of M"" Bradley's discovery, wich is indeed very noble, and
pleased very much y" French Mathematicians, wich I com-
municat'd it to. It seems wondrous now that those who
made some attempts to determine y^ Parallax of y" fixt Stars,
took no notice of y® successive propagation of y^ Light. This
is very surprising too what he observed of the different
variation of declination, of y'' Stars, greater for those wich
are near y*^ Equinoxes, less for y^ Stars near y'' Solstices. It
is plain, that the precession or change of Longitude being y°
?ame for two Stars, the one in or near y- Solstitial Colure,
the t'other in or near y® Equinoxial Colure, the mutation of
Declination of this shall be greater than y® mutation of
Declination of y*" first. But, I suppose, M"" Bradley took into
consideration this Difference, wich arises only from their
112 STIRLING'S SCIENTIFIC CORRESPONDENCE
situation and found the true mutation of Declination more
difi'erent than it shoud he if no extraordinary cause did
inthie in it.
I render you thanks too for y" account of D'' Halley'.s Letter
to M"" Maraldy. INP de Mairan is very satisfied and ohligcd to
ye. He hid me to offer ye liis Thanks and humhle respects.
I long for receivin<;' news of your hook heing under y**
press. My thirst of seeing it is rather increased, than quenched,
hy the noble Theorem, you vouchsaf'd to comnuinicate me.
I found indeeil a Demonstration of it, but as by chance, and,
I think, not very general, and so your Method will give me
a great pleasure.
Here is my demonstration.
It is known and easy to demonstrate that
XX'
X I — x'i is equal —
1-x'i
'/+i i
1
m
m + q + l
x"" —
m
.m-l
vi + q+1 .m + q
m . m—1
m + q + 1 .m + q m+q—1
m.7)l—l .771—2
= x"
&c.
m + q+ 1 . VI + q . m + q — 1 . vi + q — 2
wich Series may be terminated to any Term, viz., to
on .ni—l &c usque ad m — z + 2
^m-z+ 1
m + q + 1 . m + q (kc uscjue ad m + q — z + 2
if you add this quantity
m . m—1 i^c us(|ue ad in — z + I
X 1-x'i
7)i + q + l . m + q &c us(jue ad m + q — z + 2
In the case of \—x= all the terms become ecjual to
nought, but this last quantity, and it is
icx"* X \ —x'l
Til .m—\ &c usque ad m — z+\
m + (/ + 1 . m + q . . . m + q — z + 2
XX'
•X l-x'J
CORRESPONDENCE WITH CRAMER
113
Let m bo equal o + 7"— 1, and iit+q + l be r-r?.)-!, or
q =j) — r—l, 3'ou'll havu
xx'' + '~^
xl-xP-
z + r—\ .z-\- r—2 ...z + r-
z+'p-l .5 + /J-2...-; + |)-
xx'''^x(l-'X)i'-''-^
Then
XX-''''' ' X l—x>^'
xx'''^ X (1 —a;)
p r-l
z + r—\ . z + r—2 ... z+r-
^^^^r.r+l.,.r + z-l^^^
z+2^-\ .z + p-2 ...z + 'p-z' ' ' p.'p+\...'p + z-\
I am with a oreat esteem and affection
Sir
Your most humble, most
Obedient Servant
G. Cramer
(6)
Cramer to StirUufj^ 1729
M"" James Stirling at the Academy in
little Tower Street
London
Sir
I received indeed in due time your last letter, with the
inclosed for M"" Nichob Bernoulli which I sent him imme-
diately; but several indispensable affairs, together with
receiving no news from him, were the cause of my long
delay in answering your most agreable Letter. I began to
reproach myself my Laziness, when your worthy friend came
with your dear Letter to awake me. I'll be very glad to find
some opportunity to show him, by any Service I am able to
do him, how much I am sensible of your kindnesses to me.
I told you already I had no news from M'" Nicli. Bernoulli,
since I sent him yonv learned Letter. I believe he is medi-
tating you an answer : however I write to him to warn him
it is high time to do it. I received in the meanwhile several
letters from his Uncle : D"" John Bernoulli, who is always
114 STmLlNG'S SCIENTIFIC CORRESPONDENCE
contriving again and again new Arguinunts lor liis Opinion
about vivid forces. I don't know you liavc read what
AU 'S Gravesande publisli'd in the Journal Lllteraire about
that matter. 'Pis all metaphysical reasoning, in answer chiefly
to the late D"" Clarke and M/' MacLaurin.
I read with a great pleasure your Elegant Series for finding
the Middle Uncia of any Power of a Binomial, and for sum-
ming a slow converging Series, but cannot imagine what
pi'inciples have 1)rought ye to these Series. Tis nothing like
3'our Theorem for interpoling any Term in that Series
A, A. B, 6' (^^c. :
I sent all that to Mr Bernoulli.
I render 3'e thanks for the account you gave M' Bernoulli
of M"" Machin's Theorems. They peem indeed ver}^ well
contrived for clearing S"" Isaac Newton's Theory of the Motion
of the Moon and easily computing that Motion. I was
mightily pleased with that Elegant improvement of Kepler's
Proposition, of Areas described in Proportional Times, and
the more pleased I was, that the Demonstration is so easy
that I wondered no body, before M'' Machin, had thought
of that Theorem.
I wrote 3'ou in so few words oT M"" Dan. Bernoulli's Waj^
of approximating to y° greatest and smallest root of any
given Equation by the help of a recurrent Series, that I
was almost unintelligible. Now here are his own words.
' Methodus inveniendae minimae radicis aequationis cujus-
cumque tam numericae tarn algebraicae. Concilietur aequa-
tioni propositae haec forma 1 = ax + h.v^ + ca"^ + ex* + (^'c.
Dein formetur Series incipiendo a tot terminis arbitrariis
quot dimensiones habet Equntio, hac lege, ut si A, B, (J, D, E
denotent terminos se invicem directo online consequentes,
sit ubique "^ — aD-\-hG + cB-\-eA-\-&c sintcjue in hac Serie
satis continuata duo termini proximi M & N, erit terminus
antecedens M divisus per consequentcm N proximo aequalis
Radici minimae quaesitae.' And after some cautions to be
observed in several cases he goes on. ' Ut inveniatur Radix
aequationis maxima, Proposita sit aequatio Catholica sic
disposita .1;'" = ua;'" ^ +^.^"'~- + ca;'"""-t-&c Formetur Series
CORRESPONDENCE WITH CRAMER 115
incipiendo a tot tcrniiuis arbitrariis quot dimensionum est
aeqiiatio, eaque talis, ut si A, B, G, D, E denotent tenninos
directo ordine e Serie excerptos & contigiios, sit ul)i<iuc
^ = iiD + hC+cB + cA +&:c, sintque in liac Saric satis con-
tinuata duo termini proximi M & N, erit terminus N divisus
per praccedentem M proxime aecjualis radici maximae.'
Tlie demonstration oi wich I conceive to bo tlius. Let the
Roots of the Equation 1 = <i.v -\-hx" ■{■ ex" -\- &c. be -, -, - , Sec
^ X y
and of tlie Equation x'^' = ((x"' ^ + hx^"~'^ + cx^'^~^ + (S:c be x, y, z,
kc : and if the term M is in order I of the recurrent Series
whose index is a -\- h -^ c + ^c this term M will be, for the
values rt, h, c, c^'c of the lirst Equation -, + -i + -7 +&:g. and.
;>,.• y' c'
for the values a, b, c, S:c in the second Ecpiation pJ + (jy^ + rz^ ;
and the next term in order l+\, and called N shall be, for the
first Equation -^^ + -^— + ^j—^ + (^^c and for the second
X y z
Ecjuation ^>a;' + ^ +q]/'^^ + 7-:^"^^ + &c. Now if x be the o-reatest
and the smallest root the "•reater is /, or the further is that
X
term M from the beoinnino- of the Series, the oreater is - in
coniparii-on with the other terms -^ + -j ^c, and -j^ in com-
parison with -^— + -^^-f +^c. So that if I be infinite the
terms -j + -^ &:c and -j^^ + t^j + (^'c are not to be considered
but — , and -/xr make up the Terms M and N, the foi-mer of
a;' «'■•"' ^
wich being divided by the latter gives you x. In the
other Equation 'px^ and j9a;^+^ being infinitely greater than
qyl + rz^ + Szc and qy^^^ +rz^'^^ +&:c make up the Terms M
N _ px^^'^ + ii^c
M ^»* + &c
I am with a great respect
Sir
Your most humble and most
Obedient Servant
Geneva y^ 2G Decemb 1729 N.S. G. Cramer.
I 2
and i\', and -^ = ' j — ^ — = « the greatest root.
116 STIRLING'S SCIENTIFIC CORRESPONDENCE
As poon aa yours and IM"" de Moivre's books are printed,
you'll oblige me very much to give notice of it to M'' Caille,
that he may get them and send them to me. I believe he has
changed his lodgings, but he uses to go to Bridge's Coft'ee
house over against y'' Royal Excliange.
Cramer to Slirliiu/, 1730
M' James Stirling F.R.S. at the
Academy in little Tower Street
London
Sir
As there is no less than a j'ear, since I have no Letter
from 3'e, I don't know, whether I must not fear the Loss of
a Letter wich I sent ye about that time, containing a Letter
from M'' Nich. Bernoulli in answer to yours, together with
a Copy of his Method for finding y*" component quantities
of a Binomium like this 1 +3" by the Division of the Circle.
Extraordinar}^ businesses have, from that time hindred me
always, from having the Pleasure of writing ye, and intjuiring
after tlie Philosophical and Mathematical news of wich there
is abundance in England in any time. I don't know whether
your learned book about Serieses is published, but I wish and
I hope it is. and y*^ Publick is not prived of your fine Inven-
tions. I heard M'' de Moivre's book is out, but I have not
seen it yet.
You know without any doul)t, that M'" 'S Gravesande had
made fome little improvement to your metliod, given in your
book Enumeratio linearum 3" Ordinis (S:c for finding the
difference of exponents Aritinnetically proportional in an
infinite Series formed from a given equation : wich impro\e-
ment he publishe<l at the end of his Mathescos universalis
Elementa : but I found his Method wants yet a little correction,
for it can induce into luTor, if the given equation, besides
X and y contains their fiuxions. Let, for instance, the
E(juation be
Z|L +x'yy''-2x^yy \-u'y+ |^, =
CORKliSPONDENCE WITH CRAMER 117
ami by S'' Is-tuic's Method ot! Parallolo^raiu, you'll tinu in
the Series resulting {>/ = Ax^^ + Bx"^'' + &:c) n= 1, and sub-
stitutin<;- ./; instead of y, and x instead of // : the indices shall
be 9.4. 1 .4. 14. Whence, by D' Taylor's ]\lethod, r Ixjino-
the common divisor is 1. By your methotl, the first term
shall be AAx-2Ax+lx = or ^1^1 — 2^1 + 1=0, where
^1 has two equal valors, and therefore, by your method
?• = — = -• Mr 'S Gravcsande's Method "ives for r's value
p 2 ^
2^. But really r may be taken = 5, and the form of the
Series is y = Ax + Bx^' + Cx^^ + &:c. This valor of r = 5, is
deduced from this Rule, wich may be substituted to othei's.
haviny; found, by the Parallelogram, the greatest terms of tho
E([uation, and thereby the valor of ii ; see whether these
terms give for y, or y, or y &c many eijual valors, and let 'p
design the number of these equal valors of y, or y cV^c. Then
substitute for y and y, y &c, x", a,"~', a;" - &c and write down
the indexes of all the terms. Subtract them all from y"
greatest, or subtract the smallest from all the others; accord-
ing as the Parallelogram gave you the greatest or the least
index. Divide the least of these differences by p, & of this so
tjivided, and of all others, find the greatest common divisor.
This shall be the valor of r.
So in the Example cited, the Parallelogram gives for the
greatest terms of y^ Equation x'y dy'^—2x''ydy + x^dy — 0,
wieh divided by xUly, gives y(iy—2y + x = 0, where y has
not many equal values, Theref. pj — I. The indexes are
9.4.4.4.14, The difference 5.10, The common Divisor 5.
Whence r = 5.
I wou'd gladly know from ye, how one can find the nundjer
of Roots of an exponential E([uation, like this y-'' = I +x for
the method you give in the 6 Coroll. of y" 2"^' Prop, of your
book Enuineratio &g p. 18 does not succeed in this case.
It is a thing pretty curious, that in the Curve represented
by that Equation y'' = 1 +a;, or y — I +x' , the abscissa being
= 0, the ordinate y is not 1, but of a very different value, tho'
it seems at the first siirht, it must l)e 1, beino: 1 +o".
I have happily conserved a Copy of M'' Bernoulli's Letter,
118 STIRLING'S SClb:NTIFIC COIUIESPONDENCE
so iliat I can suiul it yc, il' you have not received y ' (jri^inal,
wicli I pi'a}^ 1 may know I'roin ye, as soon as you can witliout
any trouble at all.
I am, with a jj;reat esteem and respect
Sir
Your most humble
Geneva, the 22 X''"" 1730 N.S. most obedient Servant
G. Cramer.
(8)
Slirliiifi io (■rauier, 1730
Copy of a Letter .scut to M'' Cramer at
Geneva September 1730
Sir
I Ijeg a thouf-and pardons for delaying so long to return
you an ans^wer. I was designing it every day but unluckily
hindi'ed by unexpected accidents. So that now I am quite
ashamed to begin, and must intirely depend on your goodness.
I send two Copies of my Book, one for yourself and y° other
for M"^ Bernoulli which I hope you will transmit to him along
with the letter directed to him. I have left it open for your
perusal, and you will find a letter which M"" Machin pent me
being an answer to what M'' Bernoulli write about his Small
Book.
The first part of my Book you see is about y" Suming
of Series where I have made it my chief business to change
them that conversfe slow into others that converge fast : but
that I might not seem quite to neglect the suming of those
which are exactly sumable, I have shown how to lind a
tluxionary Equation which shall have any proposed Series
for its root, by the Construction of which Equation the series
will be sumed in the simplest manner possible, I mean either
exactly or reduced to a Qua(irature perhaps, by wiiich means
I take this matter to be carryed farther than it was before :
this you will !-ec is the 15 Proposition and its Scholien
I have taken an opportunity of clearing up a difiiculty about
the extracting the Root of a fluxionary Equation, wliich is
the only one that Sir Lsaac left to be done. This first part
CORRESPONDENCE WITH CLIA]\1ER 119
lias l>i'en written 8 or 9 years ago, so that il" I were to write
it again I should Scarce cliange anytliing in it; Ijut indeed
that is nujre than I can say ior the Second part, because
tliere was not above one halt' ot* it finished when the begining
of it was sent to the Printer. And altho' I am not conscious
of any Errors in it but Typographical ones, yet I am sensible
that it miglit have been better done.
The 20 Prop: about y° Suming of Logarithms has been
Considered by M"" Dc Moivre since y" publication of my Book,
and he lias found a Series more simple than mine which
is as follows. Let there be as many naturall numbers as
you please 1, 2, 3, 4 ... c; whereof the last is z. ]\Iake
/, : = Tabular log. of z, I, c=log. of 6-28318 which is the
Circumference of a Circle whose Radius is unity, a — '43429 ...
which is y- reciprocal of y'^ Hyperbolick Log of 10. and y"
sum of y" Logarithms of the proposed numbers will be ^
whereas you will see that in my Series y'' Numerators are
y^ alternate powers of 2, diminished by unity: the degree
of convergency is y'' same in both, and indeed there is seldome
occasion for above three Terms, reckoning — za the first :
M"" De Moivre is to publish this with his manner of finding it
out, which is (|uite different from mine, whicli is done by an
old and well known principle, namely the taking of the differ-
ence of the succes!-ive values of quantitys as you will see in y'^
Book, about which I shall be glad to have your opinion : and
I hope you will write to me soon after this comes to hand,
else 1 shall take it for granted that you have not forgiven me.
I shall be always glad to hear of your wellfare, and to know
your news of any kind whatsoever. I am with the greatest
respect
D. Sir
Your most Obedient &
most humble Servant
London September 1730 James Stirling.
^ The gap occurs in Stirling's copy of the letter.
120 STIRLING'S SCIEX'J'IFIC COKRESPONDENCE
(9)
Cramer to S/irUi/fj, 1731
31' Jiuues Stirling R.S.S. at the Academy
in little Tower Street
London
Sir
I guess Ity the date of your Letter you must be very
angry with me, thinking, as you may well, my negligence
in returning you an Answer quite unpardonable. But I
beseech you to believe, I cou'd not be so ungrateful! as not
to rendring you due thank for your fine present, wich I re-
ceived but from five days. The chief reason of tliat accident
is the forgetfullness of a Merchant to whom M'" Caille gave
the two Exemplarys of your Book for sending them to me,
then his sickness, then the violence of the winter, than I know
not what, so that, to my great misfortune, they came here
but the 12**^ of June. As soon as I received them, 1 sent
M"" Bernoulli his Exemplary together with the Letter for
him and the inclosed Letter of Mt Machin. And I resolved
to write }ou even before the perusing of your book that
I coud justify myself of a so long and unexcusable delay.
As far as I can see, by a superficial Lecture of the Titles
of your Propositions, this Treatise is exceedingly curious, and
carries far beyond what has been done heretofore a Docti'ine
of the utmost importance in the Analysis. I rejoice before-
hand, for the advantages I shall reap from an attentive
Lecture of it. and I Hatter myself you shall be so kind as
to permit me to improve this benefit by the correspondence
you vouchsafe to keep with me.
You shall know ]\I' N. Bernoulli has been this month elected
Piofessor of the Civil Law, in his own University, wich I fear
will perhaps interrupt his Mathematical Studies. I have
perused, as you permitted, your Letter to him, and, in my
opinion you are in the right as to your objections against his
,. . , ,. ,, ,, . T .r^-h.r+2h ... r + zh-h
maimer ol intcnjonng tlie Series ; ^ -. j
2).p + o. /> + 2U ... p + zb — b
1 ii- -4 1 . r.r + b ...p-b
by putiing it ('(lual to ; r-^- ,
or =
CORRESPONDENCE WITH CRAMER 1*21
p + zb . . . zb + r — 1> , . , , 1 1 i- •
i — 5 which cannot succeed l)Ut m some
p . p + b ...O' — b
few cases, wich have no difficulties. His Theorem sent to
M'' Montmort seems to be usefull in many cases. I have
found a demonstration of it very simple, and made it more
general, in that manner. The Series
1
a.a + b.a+2b ... a+2J—lb
n
a + c. a -\- (■ -\- b . a -^ c + 2b . . . a + c +2> —'^b
n n—1
+
a + 2c.a + 2c + b...a + 2c+i:>-lb
n 11 — 1 n — 2
1 >< -2- ^ ^^
a
+ 3c.a + 3c + b ...a + 3c+2:>—ib
n n — 1 n — 2 n—3
T '^ ^ >< 3- "" "1-
+ kc.
« + 4c.a + 4t-|-& ... a + ^c + 'p—lb
(by putting ^-,5=^/1, 6' = ^^' B, D = -~— G, &c
and
Az + Bz' + a. ■■'> + Dz^ + etc = Hz" + is" + ^ + it s« + ^ + Xo" + '■' + etc)
will be reduced into this
p .f>+\ .p ^2 ...'[) ^n—\ jj_ p .p+l .p +2 . . .p + Uj
a .a + b ... a +/> + n— lb a . a+b ... a +p + nb
^ P-P+^ ...y + n + l j^_ p.p+l .../) + n + 2 ^ ^^
a . a -^ b . . . ii + J) + n + 1 b a . a + b ... a +p + ii + 2 b
or, (if you like rather to have but the sign + and not
alternately + & —) into This
p.p+l .p + 2 ... p + n-1 „
a + nc -{ p—\b . a + nc+p—2b ... a+ nc — n b
'p .p+1 .p + 2 ... p + n J
+
a + nc +p) —lb . . . a + nc — 11 + I b
p.p+J\^p + 2...j, + n+l j^ ^ ^,^^
a + nc + p — lb ... a + nc — n+2h
1.2:2 STIRLING'S SCIENTIFIC CORRESPONDENCE
where if c = h, A l.e'iMM- = //', ;ui<l B = C = X» = t^'c = all
the Series is reduced to the first term
(t . a + h .a + 2h ... (i + 'p+ ii — \b
and, moreover, if you put again p = 1, you'll ha\ e M' Ber-
noulli's Theorem I have also read over M"" Machin's Letter,
but I cannot judge of their difference having not seen his
Book. M"" Caille cou'd not find it. I am glad for what you
say to M' Bernoulli, he is preparing for the press a compleate
Treatise about it. I conjure you to make me know as soon as
it shall come forth, where it is })rinted, for I shall read it with
a great pleasure.
I had willingly dehiyed this letter till 1 had some news for
ye, but I chusc rather to send this empty answer, than to put
off any longer to tell ye I am with the greatest esteem and
respect
Sir
Your moi^t humble, mo.st obedient
Geneva 18*^^ June 1731. and most faithfull Servant
G. CllAMEH.
(10)
Cramer to StirUug, 1732
W James Stirling. K.S.S.
at the Academy in little Tower Street
London.
Geneve, ce 22° Fevrier, 1732.
Ne Soyes pas surpris, mon cher Mon.sieur, de recevoir si
fcard la Keponse ji Voire chere Lcttre du Mois de May 1731,
puisqu'il n'y a (jue tres pen de jours que Monsieur Bernoulli
me I'a fait remettre. J'espere aus.si tpie vous me permeterds
de vous t^crirc dans ma Langue nuiLernelle, puisque je sais que
vous I'entendds fort bien. Et je crois vous eniuiyer moins
en vous parlant une Langue qui vous est un peu dtrangcrc
qu'en vous obligeant <\ lire un Anglois aussi barbare que celui
que je pourrois vous ecrire. Je continue li vous rgndre mille
graces pour le present (jue \'ous aves daigne me faire de votre
CORRESPONDENCE WITH CRAMER 1:23
excellent Ouvrage, doiit jc vous ai accuse la reception dans
unc Lettre ({ue vous dcvds avoir rt^u depuis I'envoy de la
Votre. On ne peut ricn trouver dans le livre que d'excjuis
pour ceux qui se plaisent aux Spc^culations dont vous a\'es
enriclii les Mathematiques. Je n'en dirai davantage de peur
(le paroitre vous flatter, quoiqu'assurement ce que j'en pourrois
dire seroit fort au dessous de ce que j'en pense, et de ce que
j'en devrois dire.
La Regie de D"" Taylor pour trouver la forme d'une Serie
iloit etre proposee, commc vous le remarqut^s sous une forme
ditferente de celle qu'il a donnee, en ce que r doit etre, non le
plus grand connnun diviseur des indices, mais bien celui des
Differences des Indices. Mais pour qu'elle puisse s etendre a
tous les cas possibles, M"" Gravesande dit qu'ayant substitue
dans TEquatiou, yl,t" au lieu do y dkc il faut chercher la
Valeur de A & s'il se trouve qu'il ait plusieurs valeurs
egales, il faut prendre pour r le plus grand commun diviseur
des Differences, mais tel qu'il mesure la plus petite par le
nombre des valeurs egales de A ou par un nmltiple de ce
nombre II en donne I'exemple suivant.
|ni + c^y - 2 .^'^2/^ + xy^ - ^5 =
(|ue la substitution de Ax"- au lieu de y, change en
-^ +Ax"+'-2A^x^''+^ + A^iiy''' + ' ^ =
Done les indices sont 14, n+3, 2ii+2, 3)1+1, 9 ii. Par le
Parallelogramme de M'' Newton on trouve pour la forme de
la suite d'autant plus convergente que x est muindre, 11= 1,
ce qui change les indices en 14, 4, 4, 4, 9. Otant le plus petit
des autres, les differences sont 5, 10. Le plus grand counnun
diviseur est 5 ; Ain?i selon la Regie de M'' Taylor corrigee,
la forme de la suite doit etre Ax + Bx''' + Cx^'^ + &c. Mais selon
M"" 'S Gravesande si Ton veut determiner la valeur de A par
le moyen des plus grands termes de I'equation (}ui sont
Ax"+''^-2A\r^''+^ + A-^x-'" + \ ou Ax*-2A-x'^ + A'''x* dgales a
zero et divises par a;* on trouve (ju'il a 2 valeurs egales.
Done /' doit diviser les 2 differences 5 & 10, et entr'autres
la plus petite par 2 ou 4, ou 6, &c.
1:24 STlllLING'8 SCIENTIFIC CORRESrONDENCE
.
\
«
Ml
H
•
Fig. 21.
Ainsi r doit ctrc 2|, ct la forme dc la Seric sera
Ax + Bar^ + Cx'' + I)x^^- + <S:c.
Mais ccttc Re<;le de M' 'S. Gravesande iie paroit pas encore
assess generale, car il peut aisement arriver dans les Ecpuitions
tluxionelles que A ait plusieurs Valeurs
e<^ales, sans (ju'ii y faille faire ancune
attention. Ainsi quoi(iiie sa Regie donne
toujours una Suite propre a determiner
la Valeur de y, cependant elle ne donne
pas toujours la plus simple. II faloit
done (^tablir la Regie ainsi. Si les plus
grands termes de I'l quation determines
])ar le Parallelogramme de M"" Newton,
etant egalt^s a zero, font une Equation
dans laquelle y ou quelcune de ses
Fluxions ait plusieurs Valeurs ^gales,
Divis^s la plus petite difference des Indices par le nombre de
ces Valeurs ^gales, Et le plus grand commuu divifeur du
Quotient et des autres Differences sera le nombre r clierche.
Par exeniple, si I'Ecjuation cy-dessus avoit 6te
-^ +x'y-2x'yy + x-y-y- ~^^ =
on auroit trouve la nieme valeur de \i-= 1, les memes indices 14,
4, 4, 4, 9, les memes differences 5, 10, que cy-devant, k A auroit
aussi deux Valeurs. Done selon la Regie de M'' 'S Gravesande
ou auroit la meme forme de Serie, ^^ + i?a;'** + 6V + ("('c, Au
lieu (|ue suivant la Regie (pie je viens de poser, les plus grands
termes de TEcpiation x'^y—2x''yy-\-xSfy, (^gales a zero et
divis^s par x' y donnent x—2y + yy=iQ (pii ne donne pas
deux v^aleurs cgales de y ou y Ainsi il faudra simplement
prendre pour r le plus grand conaiuni diviseur 5 des diff'erences
5, 10, Et la forme de la Serie est Ax + Bx^' + Cx^'^ +(%c. Ainsi
si I'on calcule selon la forme de M"" 'S Gravesande, on trouve
tons les Coefticiens des Termes pairs dgaux a zero.
C'est h\ la Regie Generale. Mais il se rencontre (lueLpiefois
des cas, ou il n'est pas si facile de I'appliquer. Les Termes
places sur le Parallelogramme de M"" Newton peuvent se trouver
sur une nieme ligne Verticale. Alors on ne peut en les
CORRESPONDENCE WITH CRAMER 125
coinparant (letenniner la Valeiir <le Toxposant ii. Mais en
supposant (pie le tenne le ])lu.s "^rand est celui (Hii a le plus
yi'and oil le phis petit exposaut selon ([ii'oii vent (jue la Suite
eonver<;'e, d'autant plus (pie x est plus petite ou plus oraiide :
On determine par cette supposition la Valeur de // i^' la forme
de rEt[uation. Mais la valeur du premier r et sonvcnt de (juel-
([ues autres coelficiens reste indctermiiK^e.
Done si tons les termes places sur le Parallelof^ramme de
^P Newton se trouvent dans une meme Hone ol»li(jue, ou ce
(pii revient au nic^'me, lorscpi'ayant substitue dans rp](]uation
Ax" au lieu de y, & nAx'"~'^ au lieu de y, S:c les indices dcs
termes resultans so peuvent tons rencontrer entre les Termes
d'une Progression Arithmeti(|ue : alors re(|uation est a une ou
plusieurs Paraboles, ou bien h une ou plusieiirs hyperboles,
(|u'il est facile de determiner.
Soit par exemple I'eriuation 2xx — 4xVay—15(iy = iH^
apres la substitution de Ax^^ au lieu de y, les indices seront 1,
^ a, n — \, qui sont en Progression Arithmetifpie. les supposant
egaux on trouve n = 2. Soit done y — Ax^ et apres la Sub-
stitution I'equation devient 2xx — 4xxVaA — ZQuAxx = ou,
divisant par xx, 2—4 VaA — 30«^ = 0. Done les Racines sont
1—5 V<iA — Q, k 1+3 \/(7Z = 0. Dans ces Racines mettant
ail lieu de A sa valeur "^ , elles se chanoent en 1 — 5 ~ —
till . . . ^
i^' 1 + 3 / -^ = dont la multiplication produit
XX — 2x Vay — 1 5 (< ?/ =
qui est la fluente de la fluxion propos^e
2xx — ix ^<iy— IG'iy —
Or cette equation designe deux demi Paraboles decrites sur
le meme axe & du meme Sommet, les branches tirant d'un
meme C(")te, dont la superieure a pour Paramelre 25a, ^
I'inferieure 9(/ ; L'abscisse commune est y, ^ I'ordonn^e de la
premiere est x, celle de la seconde —x.
Quant a I'Equation de la Courbe y-'^ = \ + x, voici la difficult^
qui m'avoit portt^ a vous demander si elle n'a qu'une ou deux
brcUiches. C'est que quand x est un nombre pair, il semble
que y doive avoir 2 Valeurs egales, I'une positive I'autre
negative, puisque toute puissance paire a deux Racines. Par
126 STIRLING'S SCIENTIFIC CORRESPONDENCE
Exeniplo qiiaud a; = 2, I'equjitioii dcvicnt y' = S, Done y = + V 3
& — v' 3. Mais quand x est impair, je ne trouve plus qu'une
Valeur pour y. Car, par exemple, quand x = 3, I'cq nation
2/^=4 n'a qu'une racinc reelle, sea voir y = v^4 les deux autres
Raeims y=-^V^6+ 7-1^10, & 7/ - -i ^10- v/-^^16
rtant iniaoinaires, II somble done qn'outre le Rameau on
la l>ranche (pii est du Cote ou Ton prend les y positives,
I'Ecjuation dcsio-no quelcpies points par-ci par-la du eute
negatif, kK' non pas une Iti'anche entiere et continue ce (|ui est
absurde. La difficulte est la nienic quand 1+x est ncoatif.
Car a en juger par I'Ecjuation il .'•enibie ([uc y aura alterna-
tivement des Valeui's rdelles et imaginaires, selon que x sera
impair ou pair. La meme difticultc^ se preeente dans toutes
les Courbes exponentielles sans en excepter la Logarithnn(iue.
Je ne vois pas (jue personne ait donnd 1;\ dessus quelque
(^elaircissement. Je soubaiterois que vous vous donnass-ies la
peine de m'expli(|uer un peu pUis au lonij^ sur (|uel t'ondenient
il vous paroit que y a deux valours egalcs mais avec des Signes
contraires.
En rcduisant en suite I'equation y^ = 1 + x je erois qu'on ne
trouve qu'une seule suite, ce qui n'indiqueroit qu'une valeur.
Mais le ( 'alcul est si lon<;', (pie je n'ai ni le courage ni le terns
de I'entreprendre pour mioux m'assurer de ce soup^.on.
Votre determination de la Valour do y (piand x est zero, est
conforme h cello quo j'ai aussi trouvc'e par la memo nianiere et
encore par ({uelques autres. Par Exemple. On pout ainsi
construire la Courbe
Sur I'Asymptoto CD soit decrite la
Logaritbrnicjue, dont la Foutano-entc
soit I'unite. Soit I'Oi-donnde AB eo-ale
a la Soutano-ente ou j\ I'unite. Soit
prise unc abscisse ([uelconque AAf—x.
Pour trouver I'Ordonncc correspon-
dante j\fP = y, jo trace la perpen-
diculairo PMF rencontrant la
Logaiitbrnique au point F. Par les
points F k A je tire la Clioi'de ou
secantc FAI, (pii I'oncontro I'Asymptotc en I. Je i)rens
BK = BI ct elevant la perpendiculaiie K L je fais il/P = KL.
Le point P est ii la Courbe l^Q cbercbeo. Car puisque
CI B K D
Fig. 22.
CORRESPONDENCE WITH CRAMER 127
j\M=x, BM = x+l, Sc MF =Lx+l. Soit MP = LK = y
tl' BK = BI = Lj/. Lcs Triangles Semblables AMF, ABI
donncnt FM {L1,^\) : MA (x) : : BI (Ly) : BA (1 ) Done xLy
= L.r + 1, ou y^' = x+\ Or (luand x = la Secante FAI
devient la Tano-entc AC, iK' pieiiant BD = BG = \ (la i^ou-
tano-ente) la Perpendiculairo J)E (i[\u est le nombre dont le
Lo-aritlime est I'unite = 2-71 8281828450 i^x) sera r-ale a
rOrdoiinee AQ.
N.B. que cette Construction ne donne qu'une l)ranclie ] torn-
la Courbe so. PQ.
Mais ce qui i'ornie une nonvolle ditticulte, c'est (ju'en eher-
chant la Soutano-ente an point Q il sernble (ju'il y ait deux ou
3 rameaux (|ui se coupent en ce point la. Car I'expression
orenerale de la soutansjente est '- . Or cette
x-1+x.ll+x
expression devdent (en substituant au lieu de x la valeur= 0) •
Done suivant I'art. 1G3 de TAnalyse des infininient petits,
prennnt la Difterentielle ou iluxion du Numerateur et du
3 XX + 2 X
Denoniinateur on trouve la soutan!>ente an point Q = — 7— :
'^ ^ ^ —l{l+x}
qui est encore ^. Done differentiant de nouveau, on trouve
cette soutnngente = —Qxx — Sx-2= —2 (puisque x = 0). Or
les Autheurs posent qu'on n'est oblige k ces differentiations
que lorsque 2 ou plusieurs Rameaux de Courbe se coupent
dans le point ou Ton cherche la soutangente Voyds Memoires
de I'Academie de Paris. Annde 171G p. 75 & Ann(^e 1723
pag. 321. Edit, de Coll. Voyes aussi Fontenelle Elements de
la Geometric de I'infini, p. 418 & 99.
Votre Probleme du jet des Bombes est de la derniere im-
portance par raport a cette branche de la Mechanique. Je serai
infiniment curieux d'aprendre le re.sultat de vos Experiences
& de Vos Calculs. J 'ai lu cet article de votre Lettre a plusieurs
de mes Amis Oiliciers d'Artillerie, ches qui il a excite une
merveilleuse curiosite. Ce que vous dites de la facilit(^ de
votre solution ne pique pas moins la mienne, puisque la Solu-
tion de M"- Jean Bernoulli (Acta Erud. 1719. p. 222, & 1721.
p. 228) est si compliquee et inapliquable h la pratique. Je
vous suplie, si vous aves compose quelque chose la dessus
de daigner me la commiiniquer.
128 STIRLING'S SCIENTIFIC CORRESPONDENCE
Jo voudrois bien en ecliaii(;u do votrc belle Lettre vous
iiuli(iner aussi quel(juecliose digne de votre attention Mais
il n'est pas donne a tout le nionde do Voler si luiuL. Je nxo
rabaisse a do plus petits Sujets. Voici uu Problenie qui m'a
oceupe ces jours passes, ct qui sera peut-dtre du oout de Mr de
Moivre. Vous ne savi'^s peut-ttre pas ce que nous ap])ellons en
Francois le jeu du Franc Carreau. Dans une chanibre pav^e
de Carreaux, on jotte en I'air un Ecu. S'il retonibe sur un
seul carreau, on dit (|u'il tombe franc, et celui qui I'a jettd
gai^ne. S'il tonibe sur deux ou plusieurs Carreaux, c'est ti dire,
s'il tombe sur la Raye qui separe deux Carreaux, celui qui I'a
jette perd. C'est un Prob1i''nio ix resoudre d' (jui n'a point de
difficult^. Trouver la Probabilite de gagncr ou de perdre, Les
Carreaux & I'Ecu ^tant donn(^es, Mais si au lieu de jeter en
I'air un Ecu qui est rond, on jettoit une Piece Quarrde, Le
Probldnie m'a paru asses difficile, soit qu'il le soit naturellement,
soit (jue la voye par laquelle je I'ai resolu ne soit pas la
meilleure. Au reste j'ai re9u le Livre que Mr de Moivre m'a
envo} 4 en present. J'ai pris la Libert*^ de lui en faire mes
remercimens dans une Lettre dont j'ai charged un jeune liomme
d'ici, qui est parti il y a quelques mois pour I'Angletei-re. Je
ne scais s'il la lui aura remise n'en ayant eu dcpuis aucune
nouvelle Je vous prie, quand vous le verrds de vouloir bien
I'assurer de mes hund^les re-^pects, (S: de ma reconoissance.
Temoign^s lui combien je suis sensible aux Marques publi(|ues
qu'il m'a donn^es de son amitid. II ne sera pas trompt- dans
sa Conjecture, quand il a cru que la 2^ Methode de M'' Nicolas
Bernoulli est la meme que celle de Mr Stevens. II y a plus
d'un an tiue je n'ai aucune nouvelle de ce dernier. Sa nouvelle
Profession 1 oceupe entidrement. II a poui-tant rec;,n votre
Livre avec vos Lettres, et vous aura sans doute repondu.
Je suis avec une estime et une consideration toute parLiculiere
Monsieur
Votre tres lunnblo, iV' tres obdissant Serviteur
G. Cram EH.
CORRESPONDENCE WITH CRAMER 129
(11)
Cramer to Stirliii;/, 1733
M' Jcuues Stirling. F.R.S.
at the Academy in little Tower Street
London
Monsieur
Voici uue Lettre que je viens de re^evoir pour vous de la
part de M"" Nicol. Bernoulli. Elle est venue enfin aprds s'etre
fait longtems attendre. Un nombre considerable d'occupations
ni'empi^che d'avoir I'lionneur de vous ecrire plus au long.
Voici seulement un Extrait de ce qu'il me marque touchant sa
nouvelle Maniere de calculer les Numerateurs des fractions
simples auxquelles se rciduit la fraction ^7- 1—~ . Soit
suppose
1 e-fz
+
z-" + 2l-J'+l 1-JCZ+::
oc + ^z+yz- + 8z^'' +... + jUs"-- + J/o"-^ + . . . + CV^"-5 + Bz""-^ + Az^''-^^
1 + az + 6s- + c;^ . . + rz"-'-" + sz'^-'' + tz'"-'^ + az" ... + az^''-"" + z^''-'^
et reduisant ces deux fractions au commun denominateur, en
multipliant en Croix, & faisant oc+e — 1, ^^ les autres coeffi-
ciens = on aura les Equations de la Tabl. I lesquelles apres
avoir substitu^ pour x, ax, hx, ex, &:c respectivement a, 1+6,
« + <?, b + d cV:c selon la nature de la suite r(^currente, I, a, h,c,d,
&c se cliangeront en celles de la Tabl. II
Tabl. I
a+e:= 1 A-f^O
13-oiX + ae-f ^ B-Ax + e-af—
y — ^x + a + be-af-O C—Bx + A + ae-bf=0
8-yx+l3 + ce-bf^ D-Cx + B + be-cf =^
i-Sx + y + de-cf = E—Dx + C + ce — df=0
ike &c
2447 K
130 STIRLING'S SCIENTIFIC CORRESPONDENCE
Tabl. II
0^=1-6 A=f
(3 = a-2<ie+f B = 2af-c
y = b-3bc + 2af-e C= 3hf-2ae+f
S = c-4ce + 3bf-2ae+f D = 4cf-3he + 2(if-e
e = d - 5de + icf- 3bc + 2af- e E - 5df-ice + 3hf-2ae+f
fi = s—n—lse + n — 2rf M = n—lsf—n — 2re
— n—3qe + &c. +n—3qf-S:c.
M—t — nte + n—lt<f fi = ntf— n —Ise
— 70 — 2 re + (See. +n—2 rf— &c.
Ces deux differentes valeurs de M ^galees ensemble donnent
t — nte = 0, ou e = - & les deux valeurs de u donnent s — vlf
n r- J
OU f— -- , coninie i'ai trouve par induction dans la Solution
^ at *' '■
de mon Probl. 5.
Je vous soupplie, Monsieur, de vouloir bien me faire la
grace de me donner de Vos nouvelles, & de m'informer de ce
qui s'est publi(^ nouvellement en Angleterre en fait de Philoso-
phie & de Mathematique. Soy^s persuade que je suis avec une
extreme consideration & un Veritable attachement,
Votre tres humble & tres ob^issant
Serviteur
G. Cramer.
Geneve ce 10*^ Avril, 1733
IV
N. BERNOULLI AND STIRLING
(1)
N. BeiiioulU to Stirling, 1719
D"° niihi plurimuni colende
Peugratum milii fuit iiudius tertius accipere epistolam
tuam, qua me ad mutuuiii epistolariim coiiiercium invitare
voluisti, gaudeoque quod ea, de quibus ante hac Venetiis
egimus, consideratione tua digna esse judices, quia igitur ea
tibi in memoriam revocari cupis petitioni tuae libenter morem
geram, quod attinet primo ad difficultateni illam, quam de
resistentia pendulorum movebam, ea hue redit. Posita gravi-
tatis vi unifornii et resistentia proportionali veloeitati, non
potest corpus grave oscillari in Oycloide ; hoc quidem inveni
per calculum, sed quomodo ista impossibilitas a i^riori ex
rationibus physicis demonstrari possit, adhucdum ignoro.
Rogo igitur ut banc rem sedulo examines et quaeras construc-
tionem Curvae, in qua abscissis denotantibus spatia oscillatione
descripta (i.e. arcus Cycloidis interceptos inter punctum quietis
et punctum quodvis ad quod mobile oscillando pertingit)
apphcatae denotent resistentiam vel velocitatem mobihs in
fine illorum spatiorum. D"^^ Newtonus pag. 282. dicit hanc
Curvam 'proxl'nie esse Ellipsi Problema quod a D"" Taylor
Geometris propositum mecum coiliunicavit D. Monmort, est
sequens. Invenire per quadraturam circuli vel hyperbolae
fluentem hujus quantitatis — -r- ^, ubi S significat
numerum quemlibet integrum aftirmativum vel negativum, et
X numerum aliquem Imjus progressionis 2, 4, 8, 16, 32 &c,
petitur autem, ut hoc fiat sine ulla limitatione per radices
k2
132 STIRLING'S SCIENTIFIC CORRESPONDENCE
imaf:^inarias. Doniquc quod attinet ad Thcorcina Patrui mei
pro conjiciendis Ciirvaruni areis in Scries convergciites, tuaiii
que contra ejus generalitateni factam oppositionem, in ea re
adliucduni tecum dissentio, et in mea opinione finnatus sum,
post([uam nuper exemplum a te o))latum, et alia calculo
subduxi ; deprehendi enim seriem, licet in infinitum abeat,
tamen esse suulabilem, si area invenienda sit quadrabilis.
De rebus aliis novis Matlicmaticis aut Philosophicis nihil,
coiiiunicadum habeo, nisi quod Patruus meus miserit Lipsiam
solutionem Proljlematis D' Taylori ((|Uod et ego jamdudum
solvi) cum subjuncta appendice infra scripta. Quod superest
Vale et fave.
Dabam Patavii d. 29 Apr. 1719
Iipuus Polenus me enixe D"'^ Tuae
rogavit ut suis verbis tibi Servo humillimo
plurimam Salutem dicerem Nicolao Bernoulli
Appendix Patrui
Adjicere lubet quaedam milii inventa Theoremata, quae in
reductionibus utilitatem suam liabent non exiguam. Demon-
strationes eorum brevitatis gratia jam supprimo: Erunt inter
Geometras qui facile invenient, quocirca illis eas relinquo.
Definltio. Per q et I intelligo numeros qualescun(|ue in-
tegros, fractos, attirmativos, negativos, rationales, irrationales.
Per p intelligo tantum numerum integrum et atHrmativum,
vel etiam cyphram. Sed per n et k intellectos volo numeros
quoslibet integros affirmativos cxclusa cyphra.
Theorema I
+ !•■
(IX : {e +fx'i)'' est algebraice cpiadrabilis.
r . - +'!■+■■■
Theor. II Generalius, \x>"'(lx:{e+Jx'i)'' est algebraice
quadi'aV»ilis.
Theor. Ill
-'U::
x!"i '^dx: (e +fx'i) '^ est algebraice quadra-
bilis : Adeoque existente p = 0, erit etiam
Ix^^-'i-'dxiie+fx'if''^"
algebraice (piadrabilis.
CORRESPONDENCE WITH BERNOULLI 133
Theor. V
Theor. VI
Theor. IV x^''dx:{e+fx'jy^ dependet a quadratnra luijiis
dx : (e +fx'i).
'«(/.<• : (c+/c'/)" dependet a quadratura ejusdem
dx:{e+fx'i).
I'P^+^dx ; (<> +fx'iy^ dependet a quadratura hujus
j xhfx:(e+fx'i).
Theor. VII Siimtis 8 et A in Casu Taylori erit
'/-I
dz:{e + fz'if
quadrabilis per circuluni vel h^^perbolam.
Corolloria (piae ex liisce Theorematibus deduci possent
pulchra et miranda non minus quam utilia nunc omitto, sicut
et plura alia ad quadraturaruni reductioneni spectantia, quae
olim inveni ac passim cum Amicis coinunieavi. Ex. gr. Ex
collatione Theorr. V et VI sequitur inveniri posse duos coeffi-
cientes a et /3, ita ut
algebraice quadrabilis.
(a.r-p? + ^xf"i+'')dx : (e +fx'i)'"- sit
(2)
Bernoulli to Stirling^ 1729
Viro Clarissimo Jacobo Stirling
S.P.D. Nic. Bernoulli.
Pergrata fuit epistola, quam per coiliunem amieurn D. Cra-
merum mihi baud pridem transmisisti et ad (|uam citius
respondissem, si per varia impedimenta licuisset Gaudeo te
valere et rem Mathematieam per impressionem libri de &uma-
tione et interpolatione Serierum novis inventis locupletare.
Gratias tibi ago pro illis quae prolixe narrasti de nova theoria
Lunae a D. Machin inventa, cujus hac de re libellum nuperrime
mihi donavit D. de Maupertuis, (pii nunc apud nos versatur.
Pauca quidem in eo intelligo, quia nullam adhuc operam
134 STIRLING'S SCIENTIFIC CORRESPONDENCE
collocavi in lectione tertii libri Princi})iuiuiii J). Newtoni ;
videris tamen mihi haud recte in cpistola tiia explicuisse (juid
ipse vocat an E(juaiit. Verba sua sunt liaec :
' lie constructs a figure whose
Sector CDF is proportional to
the an^^'le ASB, and finds the
point C which will make the
fioure CD nearest to a Circle '.
Existinio dicenduni fuisse ' ho
constructs a fiourc, whose Sector
CDF is equal to the area ASB,
and finds the point 7^ which will make the fio-ure 67) nearest
to a Circle.' Ceterum etiam si inveniatur punctum ali(jUod F
ex quo xelocitas Planetae in utraque apside constituti eadem
appareat ex hoc non sequitur ae(|uantem CD maxime accedere
ad circulum, vel punctum i^ esse illud, ex (|U0 motus Planetae
maxime uniformis appareat, ut D. Machin asserit pag. 41.
Nam locus ex (pio Planeta in ^4 et P (fig. seq.) constitutus
aeque velox apparet non est unicum punctum F sed Integra
linea tertii ordinis FAffPf cu]us aequatio est
Fig. 23.
a — x . yy = a + l> — x .b — x.x
positis AS — a, SF = h, Ag — x, flf = y- In hac igitui- linea
et quidem in ejus ramo Pf datur fortassis punctum /. ex quo
Fio. 24.
Planeta apparet aeque velox in tribus punctis A, P, et D:
adeoque ejus motus magis regularis vel uniformis ((uam ex
puncto F. In ead. pag. 41. lin. IG omissa est vox reciprocally ;
praeter hunc errorem in cadem pag. notavi, (juod Auctor
videatur coihittere paralogismum, dum areas descriptas a
corpore moto per arcum AR circa puncta S et F, item areas
descriptas a lineis Fp et Fli dicit esse in duplicata ratione
CORRESPONDENCE WITH BERNOULLI 135
perpendiculaiiuia in taugeiitoni (;ul punctuui R) deinissaium
ex S et F; haec enim ratio obtinet tantum in barum arearuni
fluxionibus, a qiiaruni proportioiialitate ad proportionalitatem
ipsaruni areanun ar^iiinentari iion licet, ut scis ine olim (juoque
ex alia oceasione monuisse ; nihilominus consecjuentia, quod
area a linea Fp descripta aeqiialis sit areae a linea SR descriptae
vera manet. Theorema illud, quod corpus ad duo fixa puncta
attractum describat solida aequalia circa rectain conjungentcm
ilia duo centra virium teniporilais aequalibus, verum esse
deprehendo. Reli(]ua examinare non vacat.
r r + I
Theorema tuum pro interpolatione Seriei A, -A, B,
/> p+ 1
C, D, &c per quadraturas Curvaruni deduci potest
p+2 ' p+3 ' ^ ^
ex isto altero theoremate quod ante 19. annos cum D. de
Monmort coiiiunicavi,
1 n n.n—l n .n—1 .11—2
a ~ a+b "^ 1 T2 .a + 2b ~ 1 . 2 . 3 . « + 3 6
a. ii-l.n-2 .n-3 , 1 . 2 . 3 . 4 . 5 ... /<6"
+ tvc =
1 . 2 . 3 . 4 . a + 4 6 a.a + b .a + 2h ...a + nb
T
Sed et sine quadraturis interpolatur facillime Series A, - A,
r + 6 „ r + 2b r + 3b ^ . -, ^^
B, ; C, ; D, Arc ponendo
p + b p + 2h /) + 3b ^
r .r + b .r + 2b ...r + zb — b r .r -\-b . r + 2b ... p — b
2) .p + b . p + 2b ... 2:> + zb — b r-\-zb.r + zh + b...zb+p — b
, ,. p + zb .p + zb + b ... zb}-r — b
vel etiam = ^ --. ; — ,
'p .p + b .p) + ^'^ ... r — h
prout p) major vel minor est quam r.
Ex. gw Si s = 2^ erit terminus inter tertium
7 B et quartum yC medius =
p-\-b ^ p+2b
r . r + b.r+2b ...p-b , p + 2\b .p+3\b ... l^/> + r
vel
r+2|6.r+3i6 ... l^b+p p.p + b.p + 2b ... r-b
Aliud vero est interpolare ejusmodi Series quando valor
ipsius z non est numerus integer, aliud invenire per approxi-
mationem aliquam earundem Serierum terminos non tantum
136 STIRLING'S SCIENTIFIC CORRESPONDENCE
quail Jo z t'st imiiienis fractus, sed et quando differentia inter
2) et r est numerus magnus, quod ultimum, ut et valorem
Seriei alicujus lente convergentis, ope Serierum quarundam
infinitaruni pronitc convcrgcntium a te inveniri, ex littoris
D"* Cramer intellexi, quaruin Serierum ddiionstrationein
libenter videbo.
Optarem spei tuae satisfacere tibi vieissim impertiendo nova
fjuaedam inventa, sed dudum est quod Mathesis parum a me
excolitur, ncc nisi in gratiam amicorum me subindo ad solu-
tionem quorundam Problematuin accinxi, (juorum solutiones
in Scbedis meis dispersae latent, et quoad maximam partem
vix tanti sunt ut tecum coinunicari mcrcantur. D"™ Cramer
rogavi, ut tibi transmittere velit Specimen method! meae
(Pembertiana multo facilioris et cujus ipsnm participem feci)
resolvendi fractioncm in i'ractiones luiius formae
, ; 1 + qz"" + Z-"
a + bz • I •
\ +CZ + ZZ
Dfis de Maupertuis Patruo meo nuper proposuit sequens
Problema: A et B sunt duoignes quorum intensitates sunt ut
p ad q, quaeritur per quam Curvain CD homo in dato loco G
Fig. 25.
constitutus recedere debeat, ut scntiat miniiinim ealorein, posito
rationem cujusque ignis in objectum aH(juod esse in ratione
reciproca duplicata distantiaium.
Hujus Prol)lematis se(|urntem constructionem inveni.
Centris ^ et i^ descriltantnr circuli acg, lulh aequalium
CORRESPONDENCE WITH BERNOULLI 137
radioriiin Aa, Bh, juny.-mtiir AC, BC, secaiites circumferentiam
horum circiiloruni in c, d et in eas demittantur perpendiculares
ae, If, ex pnnctis c et (/ abscindantur arciis cff, dh, ea lege ut
demissis perpendieularibiis a I, hi in radios Ag, Bh et perpen-
dieularibus Ik, Im, in radios Aa, Bh et ductis en.fo, parallelis
ad Aa, Bh sit eg— in ad dh — lo ut q ad p, erit, prodnctis radiis
Ag, Bh, intersectio D punctiim Cnrvae quaesitae CD.
En aliud Problcnia a Patruelo moo qui Potrol)urgi agit, mihi
propositum.
Circa punctuin A rectae positiono
datae AE rotatur Curva ANOD, et
ill quolibot Curvae ANO situ intelli-
gatur punctuni maxime distans a
recta AE, sitque A MO Curva quae
transit per omnia puncta ; oportet
invenire quaenam sint hae Curvae
ita ut segmentum A MO A sit semper ad segmentum ANO A in
ratione data in ad n. Ego iiiveni utramque Curvam OFse
algebraicam.
Idem proposuit sequens Problema cujus solutionem quoque
inveni : Ex tribus altitudinibus stellae et duobus intervallis
temporum invenire declinationem illius et elevationem poli.
In Actis Lips, praeteriti anni pag. 523 : extat Problema,
cujus solutionem talem dare possum, ut pro qualibet data
Curva CBA possim invenire aliam Ahc, ita ut grave descendens
ex quolibet Curvae CBA puncto B
et descensu suo describens arcum
BA, posteaque cum velocitate ac-
quisita ascendens per Curvam Ahc
integro suo ascensu describat arcum
Ah aequalem arcui descensus BA ;
motus autem fiat in medio resistente in ratione duplieata
velocitatis. Sed invenire Curvam CBA talem, ut altera Ahc,
in qua fit ascensus sit ips-a Curva descensus CBA ad alteram
partem continuata, sive ut CBA et Ahc sint duo rami ejusdem
Curvae videtur esse res altioris indaginis. Vale.
D. Basileae d. 22. Xbris 1729.
138 STIRLING'S SCIENTIFIC CORRESPONDENCE
(3)
Sthihif) to BernotiUi, T/'PtO
Cop3'' of a Letter sent to M"* Nicholas
Bcrnouilli September 1730
Sir
I was vcvy glad to hear of your welfare hy ^'our most
obhgino- Lettei' ami luivc delayed answering it hitherto for no
other reason but that I might he able at length to answer you
in ever}^ particular : for seeing you desired the Demonstrations
of the two Series which M' Ci'amer sent you, and these
Demonstrations are such as could not be conveniently brought
within the bounds of a Letter, I thought it was best to stay
till ni}' book was read}' to be sent j'ou ; for you will find in it
the principles explained by which I found these and such
Series. Indeed I might have sent you my Book somewhat
sooner, but unluckil}' I was taken up with an affair which
obliged me far against my inclination to defer my answer till
this time.
As to M"" Machin's Treatise it was written in great hurry
and designed only to shew wdiat mny be expected from his
larger Treatise on that Subject & therefore it is no great
wonder if you met with some difficulties in it, especially
considering that not only his propositions but aho the prin-
ciples from which most of them are deduced are new. I have
prevailed on him to write an answer to that part of your
Letter which relates to himself, which 1 now send you and
hope it will satipfie you intirely till you shall see the Book he
is now preparing for the press, which I am Confident will
please j'ou extremely, as it clears up the Obscure parts of
Newton's third Book of principles, and carrj^s the Theory
of Gravity further than even Sir Laac himself did. And
it is somewhat strange that altho the principles have been
]»ublished above 40 years, that no body has read further than
the two first Books, altho they be barely Speculative and
were written foi' no other reason but that the third might
be understood.
The Theoi-eme which M'' Cramer sent you for Interpoling
by Quadratures may as you observe be deduced from one
CORRESPONDENCE WITH BERNOULLI 139
which you sent to M"" Moninort 15 years ago, and so may
it as easily be deduced from a more simple one which
D' Wallis pulilished 75 years ago namely that -a;" is the
Area of a Curve, whose Ordinate is a;""^ and I value it so
much the more because the Demonstration of it is so very
easy. But neither your Theoreme nor that of D"" Wallis is
sufficient except in that case when the Series is so simple as
to admit of Interpolation by a Binomial Curve, for if a Trino-
mial or more Compound Curve be required we must liave
recourse to the Comparing of Curves according to the 7 & 8
Propositions of Newton's Quadraturus, that being the generall
principle for this kind of Interpolation.
r r + h r + 2h
I agree with you that the Series A, ~ A, , B, r C,
*= -^ p p + b p + 2h
&c. may be Interpoled without Quadratures, as you will see
by many Examples in the 21, 22, 2G, &: 28 Propositions of my
Book : but I am still at a loss to find out that it is to be done
after the manner you propose by putting
Indeed it is true that the Terms may be expressed by a Frac-
tion, but to what purpose I know not ; for if the Term required
be an Intermediate one, both the Numerator and Denominator
of the Fraction will consist of an Infinite number of Factors,
and therefore that is no Solution, for it is as Difficult, nay it
is the very same Probleme, to find the Value of such a Fraction
as to find the Value of the Term proposed. The fraction no
more gives the value of a Term whose place is assigned, than
the place of a Term being assigned gives the Fraction. Besides,
that Method would not even give a primary Term which
stands at a great distance from the begining of the Series :
for the Number of Factors, tho not infinite, yet would be so
great as to render the work altogether impracticable.
But here I except the case where the difference betwixt
2^ tV r is not much greater than h, and at the same time is
a multiple of it ; this is the only case when your Method will
do, as far as I understand it ; but when this happens, the
Series is interpoleable by the bare inspection of the Factors,
even without the help of common algebra : and therefore
140 STIRLING'S SCIENTIFIC CORRESPONDENCE
I hope 3'ou did not imagine tliat I designed to trouble a
Gentleman of M"" Cramer's abilities with such a simple Ques-
tion, or that I pretended to reduce it to Quadratures, altho
perhaps I might take it for an Example of the general
solution.
I cannot but think that one of us has misunderstood the
other, and therefore I should be glad to have your Method
explained to me: for instance in the Series 1, ^A, ^B, |(7, |i),
(^'c. which is the Simplest of all those which do not admit
of an exact interpolation: how do 3'Ou find out that the Term
which stands in the midle betwixt the first & second is equal
to the ni)ml)er 1-570796 &c'l You know I find it to be such
from the method of Quadi-aturcs, which demonstrates it to be
double the area of a (.'ircle whose Diameter is Unity. And
how doth your method give a Term remote from the l)egin-
ing ; for instance the pi-oduct of a million of these Fractions
I X f X I X f X -^ X ... f iggi^l which I can find in the quarter
of an hour to be the number 1772-454 0724, as you may try by
the Series which was sent you for finding the proportion
which the midle Uncia in the Binomial has to the Sum of all
the Unciae of the same Power.
Altho you are pleased to say that you have not spent much
time on Mathematicks of late, it would rather seem to be
otherwise from the ingenious Problems which you mention;
for my part, as their Solution depends not on new principles,
and since I know not for what design they were proposed,
I have not thought about them especialy since you say you
have solved them ab-eady. M'' Klingcnstierna shewd me
a Construction of the Probleme about two fires different from
yours and Extremely Simple. He has also constructed the
Probleme about a Curve revolving about a point, and whereas
you have said without any limitation that you found both the
Curves to be algcbi-aical, he observes that it is so only when
the Areas mentioned in the Probleme are to one another as one
number is to anotlici-. He has also solv'd the Piolileme about
a Body falling down in a Curve, and afteiwards rising either
in another or in the same continued; of which last 3'ou say
vkletur esse res <iltioris indagiiiis:
And as to the Probleme about finding the Latitude of the
place (^- declination of a Star from having three altitudes of it,
CORRESPONDENCE WITH BERNOULLI 141
and the times betwixt them, it is evident at first sight how
it may be brought to an equation.
M"" Klingenstierna had shewed me that part of your Demon-
stration of Cotes's Theoreme which you liad ready when he
left you ; and M"" Cramer sent me the same with the remaining
part which j^ou sent to him about the begining of this Year :
indeed I take it to be an elegant Demonstration and far
Superior to that of the person you mentioned. But I suppose
you know that M' De Moivre found out his Demonstration
of the same Theoreme very soon after M"" Cotes's Book was
published, which is now many years ago, and I am of opinion
that it will please you, as it requires no Computation.
And now I come to beg pardon for this long Letter and to
assure you that I am with the greatest respect
Sir
Your most obedient
most humble Servant
James Stirling.
(4)
Bernoulli to Stirling, 1733
Viro Clarissimo Jacobo Stirling Nicolaus Bernoulli
S.P.D.
Epi.stolam tuam die 30 Scptembris 1730, seriptam una cum
inclusa D'" Machin et cum eximio tuo (pro quo debitas ago
gratias) Tractatu de SuilTatione et Interpolatione Serierum
Infinitarum post annum fere accepi eo tempore, quo novae
Stationi in nostra Academia Professioni nempe Juris admotus
variisque occupationibus implicitus fui, quae me ex illo tem-
pore a rerum Mathematicarum studio abduxerunt, et ab
attenta et seria lectione Libri tui avocarunt. Est et alia
dilatae responsionis causa. Perdideram epistolam tuam inter
Schedas meas latentem, eamque multoties frustra quaesitam
non nisi ante paucos dies inveni. Ignosce quaeso tam diuturnae
morae. Alacrior quoque ad respondendum fuissem, si quae-
dam a me dicta, quae tamen nunc sub silentio praetereo, paulo
aequiori animo a te et a D"° Machin excepta fuissent.
142 STIRLING'S SCIENTIFIC CORRESPONDENCE
Quae D""^ Machin rcgcssit contra objectioiiem mcam circa
definitioneiii loci, ex quo Planetae uiotus maxime uniformis
apparet verissima sunt. Fateor mo non attendissc ad motum
medium aut ad motum retrogi'adum Planetae, sed studio id
feci. Ego nunquam credidi Planetae motum apparere magis
rcgularem aut magis uniformem eo ex loco, ex quo motus in
tribus orbitae punctis aequalis apparet quam eo ex loco,
ex quo motus in duobus tantum orbitae punctis aequalis
apparet, id est, motum ex primo loco apparentem minus
ditierre a motu medio, quam motus ex secundo loco apparens.
Objectio mea erat tantum argumentum, ut vocant, ad hominem.
Credebam Dnum Machin esti masse regularitatem vel uni-
formitatem motus ex eo quod Planeta in utraque apside ex
centro aequantis visus aeque velox appareat ; et ad hoc
credendum me induxerunt haec verba pag. 42. ' The said
center F will be the place about which the body will appear
to have the most uniform motion. For in this case the point F
will be in the middle of the figure LpD (which is the e(|uant
for the motion about that point). So that the body will appear
to move about the center F, as sivift ivhen it is iti its sloiuest
motion in the remoter a2)sis A, as it does ivhen it is in its
siuiftest motion in the nearest apsis P' quae verba sane alium
sensum fundere videntur, quam sequentia quae habet in sua
responsione : ' I did not conclude this to be the place of most
uniform motion, because it is a place that reduces the velocity
in two or three or more points to an equality, but because the
motion throughout the revolution differs the least possible
from the mean motion.'
Obscuritatem verborum pag. 41. ubi Dnus Machin demon-
strat acqualitatem arearum FjjL et SRA quae ansam praeluiit
suspicandi paralogismum, non puto natain esse ex praeli crrato,
sed ex festinatione, quam ipse Auctor se adhibuisse dicit ;
si quidem non solum particula and cum in locum, quern dicit
Auctor, transponenda est, sed delenda etiam particula sequens
therefore, ego in meo exemplari locum sic correxi Pag. 40. lin.
pen. pro areas scripsi fluxions of the areas LFp and AFR.
Pag. 41. lin. 4. pro the areas scripsi and the fluxions of the
areas ASR and AFR Pag. 41. lin. 8 delevi And therefore
Ead. lin. post area adjunxi LFj). Ead. pag. lin. 10. pro that
scripsi the area ASR.
CORRESPONDENCE WITH BERNOULLI 143
Vehementer cnpio videre, qiioniodo theoreiiui tuuin pro
interpolatione Seriei A, -A, B, G (Sec aeoue txcile
((]uod to per jocuin dixissc puto) dedvicatur ex theoremate
Wallisii ante 75 annos publicato, quod neinpe - x^ sit Area
Curvae en jus ordinata est cc""^ ac ex isto ineo theoremate
quod me ante 15 annos ]\Ionmortio misisse scripseram, nimi-
rum quod
1 . 2 . 3 . 4 . . . ii X 6"
a .a-^b .a + 2b ... a-\- nb
1 n )i .n—\
a a+b 1 . 2 . ct + 2 6
n . n—l . n—2 .
1.2.3.«+36+*'^
Sane cum haec Series sit aequalis areae curvae cujvis
ordinata est x"~'^ x 1 —xh^'' in casu x = I, sola substitutionc
terminorum a te adliibitorum res immediate conHcitur Nam si
lino-amus duasCurvas,unamcujus ordinata est a;'"' ^ x 1 —x^^~^~ >
alteram cujus ordinata est x~'^''~^ x l—x''~^~ , faciendo b ~ 1,
a =r at = z + r, n = p — r— 1 erunt istarum Curvarum Areae
per theorema meuni
1 . 2 . 3 .4 ... 7>-?'-l , 1 . 2 . 3 . 4 . ... «-r- 1
ct
r .r+\ .r+2 ... r+p — r — \ r-\- z .r + z+\ ... r-\-z +p - r — 1
adeoque prima ad secundam ut 1 ad
r .r+\ ... r+ n~r—\ . .r . r + 1 ... ?' + s— 1
sive ad
r-\-z.r + z+\...r+p — r—\ p) .p)-\-\ ... p + z — l
id est, ut primus Seiiei interpolandae terminus ad alium
cujus distantia a pi-imo = z, Dcmonstratio haec ubique
supponit idipsum alterum theorema quod allegasti, nempe
quod — a;'* = areae Curvae cujus ordinata est a^"^ (theorema
melius notum ex methodo fluxionum quam ex Arithmetica
Infinitorum Wallisii) quomodo enim potuissem dicere Seriem
r + t^'C, esse aream curvae cuius ordinata et-t
a a + b "^
x^'^xl - c^''" in casu .r = 1, nisi scivissem modum eruendi
areas ex datis ordinatis? Sed hoc ipsum alterum theorema
solum neutiquam sufficiens est etiam in istis Seriebus quae
144 STIRLING'S SCIENTIFIC CORRESPONDENCE
adiuiitunt iiitci'[)()l;itioiicin per cnrvas binomialcs. Siinili
iiiodo potiiisscs (licorc diiticillinia tlieoreiuata Nevvtoni et
alioruiii do tiuadraturis ex dicto Wallisii facili dcduci posse.
Quae dixisti de interpolationibus quae requirunt Curvas trino-
iniales aut magis coinpositas, (]Uod nempe recurrenduui sit
ad 7 et 8 Prop. Newt, de Quadraturis, ea non magis tangunt
meuiu quain tuuni theoreina ; inihi animus non fuit tractatum
scribere de interpolationibus, aut nieum theorema pro generali
interpolationum remedio venditare, sed tantum tuuni a DiTo
Cramero mihi missum theorema demonstrare.
Quod attinet ad alterum uiodum interpolandi Seriem
. r . r + h r + 2h . • ,-, • i
A, -A, i B, , 6, ivc. (lui consistit m ponemlo
r .r + b .r+2h ... r + sh — h__ r .r + h .r + 2h ...p — b
p.^ + b .'p + 2h ...'p-\-zb — h r + zb.r + zb + b ...zb+2) — b
, p + zb.p + zb + b ... zh + r — b .
vel = — ; } i — ' lateor ilium non succedere
J) . 2) + b . 'p + '^b ...r — o
nisi iis in casibus, ubi differentia inter ^j et r est divisibilis
per b, et sinuil nunierus non admodum magnus, quod ultimum
in praecedentibus meis literis ipse jam agnovi. Fateor prae-
terea sensum tlieorematis tui non recte intellexisse, credebam
/j> T + 1 ■?■ + 2
enim in hac Serie A, ~A, ~ B, -G, &c. (luam Unus
p p+1 p+2
Cramer tancjuam formulam generalem, non tanquam exem-
plum alius generalioris mihi miserat, ^) et r significare numeros
integros ; unde non capiebam cur hacc Series, utpote (juae
accurate posset interpolari, ad quadraturas reduceretur. Sed
his majora te praestitisse vidi cum voluptate in tuo libro, cujus
Propositio 18 continet, ni i'allor, idipsum quod ego per modo
dictum alterum interpolandi iiio(hnn monere volebam. In
exemplo 1. Prop. 25. ubi tradis interpolationein unciarum
binomii ad dignitatem indeteriuinatam elevati, inveni theorema
non nmltum absimile praedicto meo theoremati. Si iractionis
1 .2.3.4... 71 X 6" ^ ,• -1 i 171 i. • „i;
;- -. r numerator dividatur per b^\ et sniguii
a . a + b .a + 2b ...a + nb
factores denoniinatoris cxcepto })rin)o per b, et ipsa fractio
nmltiplicetur per prinunii I'actorem a, proveniet reciprocus
it
terminus unciae ordine n+ 1 in binomio ad dignitatem r + '"'
CORRESPONDENCE WITH BERNOULLI 145
elevato ; liinc per theorenui incum, ut Area ordinatao
x^'' 'x 1—x" ad -, ita imitas ad dictam unciaui. Ex. g-r. si
a ^
ponatur a = 5, b = 1, it = 4 erit area ordinatae x'^xl—x , id
est, i — l + f — f+l sive glo '^^1 | ut 1 ad 126 unciam termini
quinti in dignitate noiia. Si a = 1, h = 2, it, = ^, erit area
i
ordinatae *" x l—xx , id est, quadrans circuli ciijus radius = 1,
sive area circuli cujus diameter = 1, ad 1 sive ad quadratum
circumscriptum, ut unitas ad terininum Wallisii Q inter-
ponendem inter primum et secundum terminum Seriei 1, 2, 6,
20, 70, &:c quae continet uncias medias dionitatum parium,
sive ad terminum qui consistit in medio inter duas uncias
1 et 1 in potestate simplici binomii ; sicut tu quoque invenisti
in exemp. 2. dictae Prop. 25.
Laboriosa quidem sed elegans est methodus per quam in-
venisti ope Logarithmorum interpolationem Seriei 1, 1, 2, 6,
24, 120, &c in Ex 2. Prop. 21. Ceterum frustra quaesivi
modum, quem dixisti in sequentibus monstrari, interpolandi
hujiismodi Series absque Logarithmis, quod autem a te prae-
stare posse nullus dubito. Termiimm qui consistit in medio
inter duos primos 1 et 1 ope Theorematis mei sic eruo. Sit in
dicto theor. a = n+1, b = 1 , eritque area ordinatae
„ -- n 1.2.3.4 .11 1 .2.3.4...nx 1 .2.3...'M
^" X 1 X = =
n + 1 . it + 2...2ii+l 1 .2.3 ... 2/H-l
Fiat n=^ eritque area ordinatae Vx — xx i.e. area semicirculi,
cujus diameter = 1, aequalis dimidio quadrato quaesiti termini.
Hinc quo(|ue deducitur interpolatio terminorum intermediorum
in hac Serie 1, 1, 3, 15, 105, 945, &c. Nam si liat a=l,b = 2,
II 12 3 n X 2'*
erit area ordinatae x*'xl—xx = ^ — '- — ~ ; sed in
1.3. 5 ... 1+2 II
casu II = i praedicta area sit ae(|ualis areae circuli cujus
diameter = 1, et numerator fractionis sit aequalis radici quad-
ratae duplae istius arcae, per niodo ostensa, denominator autem
fractionis sit aequalis termino qui consistit in medio inter
secundum et tertium Seriei 1, 1, 3, 15, 105, 945, cVc proinde ut
radix quadrata dimidiae areae circuli ad 1, ita unitas ad
ternnnum ilium intermedium, qui per binarium divisus dabit
medium inter duos primos 1 et 1 dictae Seriei.
2H7 L
146 STIRLING'S SCIENTIFIC CORRESPONDENCE
Do iiKxlo invciiit^'iKli radiccin acquationis thixionalis per
Seriem infinitarn, do ({110 a^is in Scliolio Piop. iilt. I 'art. I.
ctiam e^o aliquoties cogitavi, at liac dc re scri])tuiii aliquoil
comimicavi cuin Diio de Maupertuis ciun apiid 110s a<^eret, in
quo sequentia observavi. Posse inveniri Series ^eneraliores
quani quae inveniuntnr per parallelograiniini Newtoni ; non
necesse esse ut indices dignitatiim in terniinis Seriei qiuiesitae
aut aequationis transformatae cadant in eandem progressioneni
aritlinietieain ; posse aliqnos indices esse irrationales ; et prop-
terea tani Taylori reyulani in Prop 9 (juani tuani in Enunierat.
Linear, tertii ordinis datam, pro deterniinanda forma Seriei
fallere ; posse per terminos solitarios in ae(piatione trans-
formata noiiunquani aliquid determinari, aljsque ut oinnes
coefficientes fiant aequales nihilo ; non necesse esse, ut Serierum
in aequatione transformata provenientium ad minimum duorum
terminorum primorum indices inter se aequentur, ut deter-
minetur coefficiens primus A, quia hie nonunquam potest ad
arbitrium assumi ; posse evitari terminos superfluos, quorum
coefficientes in methodo Taylori evadentes = laborem calculi
prolixiorem reddunt, quam paret. Sic pro Exemplo Taylori
in Prop. 9. Method. Increm. pag. 31 1 +sx- z^xx — x =
sequentes 4 Series inv^eni ; quarum tres priores sunt genera-
liores illis quas Taylorus invenit.
40-
7 . 14 1 ^ -5+^165 , „ 288 , .
2^- «^=^2^"^-20^'+^' 4 +^^ 5¥87^'*'
3«. x= 2z^ + B-\z-'-rlBz-i-'-i^BBz-'' + -i^B'--i^z-i<kQ
4^ « = -z~'^~z-i — ^-iz-^-^%^z--^'i\c.
Sic quoquc ol»servavi te non satis accurate rem examinasse,
quando pag. 83 dicis, ae(|uationem r'^y- = 'rx^ — x-y'^ nulla alia
radice explicabilem esse praeter duas exhibitas
y = •^- 6-^ + 120,. - 5040^> + ^^'" "^
2/ = ^lxl--.+ ^.-7-^^o+&c
CORRESPONDENCE WITH BERNOULLI 14-7
quaiuiu })rior dat .siiiuui, et posterior cosiiiuiii ex dato arcu x ;
et de (lua posteriore dicis, quantitatein A cjiiae aequalis est
radio )• ex aequatione llnxioiiali nou deterininari. Ego
11011 solum iiiveni, Sericm iioii posse habere banc forniain
A + Bx^ + Cx* + Dic" iS:c nisi fiat A = r, sed utranique a te
exbibitam Seriem couiprebeiidi sub alia general iori, quae haec
est : y = A-\- Bx + Cxx + Dx^' + Ex^ + &c in qua eoefficientes
A, B, G, D i^'c banc sequuntur relationem
BB="-^'\ C=-^—, D=-^-^,
rr 1.2. rr 2.3. ?-/•
c =
A
D —
1.2.r/
E= -
C
3 . 4 . 7'r 4 . 5 . r?'
Si fiat ^ = 0, babetur Series pro Sinu ; sin autem A fiat= r,
babetur Series pro cosinu; sin vero A alium habeat valorem
praeter bos duos, etiam alia Series praeter duas exbibitas erit
radix aequationis fluxionalis propositae. Similiter Series
illae quatuor, quas exbibes pag. 84. pro radice aequationis
y + a^y — xy — x'^y = 0, sub aliis duabus generalioribus quae
ex tuis particularibus compositae sunt, comprebenduntur.
Duae nempe priores sub bac
y = A+Bx + Cxx + Da;"' + Ex"^ + &c.
in qua eoefficientes A et B babent valores arbitrarios, reliqui
autem C, D, E, (kc sequentem ad priores babent relationem
C'.^H^^, i)=L^^5, E^tz^c, F = ^-z:^D&c.
1.2' 2.3' 3.4' 4.5
Si £ = babetur tuarum Serierum prima, Si ^ = babetur
secunda. Duae posteriores comprebenduntur sub bac generali
forma y = Ax^ + Bx''' + C^"-^ + J)x-"-'^ + ^a;"-* + Fx-^-* + &c.
ubi iterum A et B babent valores arbitrarios,
g=-'''''-'a, e=-"-^-''-^c,
4 . « - 1 8 . (( - 2
12.«-3 ' ' 4.a+l '
^^ a+2.a + 3 ^^a + 4.a4-5
8.a+2 ' 12.a + 3 '
l2
148 STIRLING'S SCIENTIFIC CORRESPONDENCE
Si tiat B = exsuri;-it tua tertia Series, et si fiat x\ —
exsur^it ([Uarta, in ([ua teniiini |)cr si^nuni + non per sinnuni
— connccti debeiit. Jiicoinodum (juoiiiie est in tuis Si'riolnis,
(|Uod Literae A, B, C, D iVrc mox pro coefficientilais terniinoruni,
mox pro ipsis terniinis usurpentur.
Hac (lata occasione describani hie ea quae ad (jiias lam tiias
Series in Libro tuo de Enunieratione Linearuni tertii ordinis
contentas notaverani eo tempore, (jiio hiinc Librum a l)no do
Maupertuis comodatiim luibebam. Eum (|uidem nunc non
habeo, sed in quadam mea Scheda liaec notata reperio. In
Escemjylo 2 ikkj. 22. aequationis x'y + ayxJc -\- a-xJo — 2a^ Jo =
radix y est =
aA+(tiL aaA—d"^ a^A — a^ a^A—a^ .
quando ^ = provenit Stirlingii solutio ; sed quando A -— a
2aa
exsurgit y = a-\ •
In Ex. 4. pag. 26. y-x-—3x'^xi/ + 2x-x' — a.i'>'r + (rx^ =
radix y est = x + BxP— BBu.^ + - +B'''x''+ -aB—B*.x'^
36 4 10
+ iL aaB + ,— aB' + W . x"^ &c
324 180
2aa , 6rt'^ , 88a* ., „
item 1/ = 2a' + (:/— x ' + x~- x " &.C.
'' 7 35 637
Pag. 28. aequationis y'-^ — (nj^ + o'-ij — a"' + x-y = radix y
^..2 ^.4 2 ^.G -,.8
non est := tt 4- r --^, (vc sed a H ^ H _ <xc.
2a 2tt' 2a \^a^ 32 a^
Pag 31. y = x+ —. „ ., + 7^—7—, — ^^^
a^ 2a" 7(('-' . . , ,„„
Pag. 34 Ex. 1. vSatisfacit etiam
x' ic* 4*" Ax'^ „
7/ = .1' + — ^- + — g- tVc.
aa w^ a^ a"
CORRESPONDENCE WITH BERNOULLI 149
In eadem Scheda iiotataiii reperio Speciein aliciuaui linearuni
tertii ordinis a te et a Newtono oiiiiseani. Neinpe in Libri tui
pag. 112, Sp. 58 ubi pro aeqaales et ejutdcin si(jnl legi debet
aeqaales ajjirmativae: nam si radices sint aequales negativae,
ligura non evadit cruciformis, sed habet crura ut in fig. 57. et
praeterca piinctiiin conjiigatum in dianietio AB, ijuod reperitur
faciendo abscif-sam = r .
2l>
V.t igitur haec nova Species est di versa k Specie 53
Newtoni, apud quem in mentione Speciei 54 pro Imposibiles
etiam legi debet aequales ajffirmaiivae.
Probleniata de qui bus in fine epistolae meae nicntionem
injeci, eum in finem subjunxi, ut petitioni tuae ali(|U0 modo
obedirera inipertiendo nova quaedam Mathematica. Mos iste
Probleniala proponendi et alios ad eoruni solutioncm amice
iiivitandi, non est omnino culpandus, si is nempe scopus
propo&itionis sit, ut coiiiunicatis invicem methodis solutionum
Ars Analj'tica incrementum capiat. Dictorum Pi ol>lematum
solutiones Patruus mens et ego cum Dnu Klingenstierna tum
apud nos degente coraunicavimus ; hinc credo cons.tructionem
quam hie tibi ostendit, Problematis de Curva recessus intra
duos ignes, et quam tamquam valde simplicem laudas, non
aliam esse quam Patrui mei, qui hoc Problema ope Trajec-
loriarum Orthogonalium ingeniose quidem solvit, sed ipsius
trajectoriae orthogonalis sive curvae (piaesitae constructionem
non dedit, De problemate circa curvam circa datum punctum
revolventem recte monitum est utramque curvam esse alge-
braicam ; si areae de (piibus in Problemate sermo est, sint
ut numerus ad numerum ; haec limitatio tanquam facile
animavertenda a me studio omissa fuit. Vix est ut credam
Problema in Act. Lips. 1728. pag. 523 propositum a d. Klin-
genstierna solutum fuisse eo etiam in cas^u, de quo dixi, videri
I'em esse altioris indaginis.
Moivraei demonstratio Theorematis Cotesiani sive rosolutio
fractionis ~ -, in fractiones simpliciores habentes
z~>'-2lz" + l '■
denominatores duarum dimensionum milii perplacet, quamvis
ob concisum sermonem explicatione (juadam opus habeat,
Posteriorem partem demonstrationis meae, quam ex coinuni-
catione Dm Crameri vidisti ab inductionis vitio liberavi
150 STIRLING'S SCIENTIFIC CORRESPONDENCE
substitiita liquida et ricrida demonstratione, quam ad eundem
D Crainerum Amicmn nostrum mitto in epistola cui banc
ad te perferendam inehido. Vale.
Dab. Basileae d. 1. Aprilis. 1733.
P.S, Nescio quo fato acciderit ut nonien nieum in Cataloj^'o
Socioruni R. S. omissum sit. Conjicio id factum esse bae
ratione ; primum nomen meum mutatum fuisse in nomen
Adgnati mei Nic. Bern. Professoris turn Bernensis, postea
Petroburgensis ; dein ex catalogo expunctum post hujus
obitum. Spero bunc errorem emendatum iri.
V
CASTEL AND STIRLING
(1)
Castel to Stirling^ 1733
Doctissime Vir
Ltbenteh vidi quae de me in epistola ad clarissimum
amicum D. de Ramsay seripsisti, et gratias pro benevolentia
tua in me habeo quam plurimas. Jamdudum professus sum
quanti sit apud me. Vidisti baud dubie quae in commentariis
Trivoltiensibus seripsi circa opusculum tuum ultimum de
seriebus infinitis tum summandis tum interpolandis.
Quod nunc attinet ad aequabilitatem arearum Newtonianam,
noUem mibi tribuisses errorem adeo crassum quasi lineam
eandem duobus aliis non parallelis parallelam afficerem. Vel
ipsa verba mea reclamant, licet verbis figura non satis re-
spondet. Supposui enim statim cum Newtono lineolam Cc
parallelam SB] et deinde distincte supposui lineolam aliam
CR parallelam BT. Relegere potes liaec ipsissima verba mea
pag. 539 (et tirant OR parallele a BT) quae si advertittas
aliter profecto rem accepisses, neque demonstrationis meae
errorem sed demonstrationis Newtonianae vitium deprehen-
disses ; vitium dico non quidem geometricum sed pliysicum,
quod plerisque summi illius geometrae demonstrationibus
accidit, quae quidem geometrice verae sunt, a veritate physica
autem omnino aberrant. Sensus itaque demonstrationis meae
iste est.
Suppono constructionem et demonstrationem Newtonianam
circa puuctum >S'.
En meum circa punctum T. Duco CR parallelam TB,
et dico ATB = BTC, atque BTC = BTR. Ergo quod erat
demonstrandum. Tam vera est haec demonstratio quam
152 STIRLING'S SCIENTIFIC CORRESPONDENCE
Demonstratio Ncwtonii et si ([uithjuid circa illain dixi paginis
totis 531, 32. 33. 34. 35. 3G. 37. 38 39. 40. 41, dignatus esses
legere, sensisses noii in toto trium liiicai'uin iion parallclaniin
parallelisiiio rem stare, sed in ipsa pi'aecipue curvariim
geometricariim natura, qiiaruni latera infinitesimalia sunt
omnino indeterminata ut hoc vel illo inodo physico resolvantur
in determinationes laterales numero infinitas,
Conclusio autem tua, non est mea, quam mihi affingis. Non
sequitur ex mea demonstratione sectores
D^ AED, DEB quos satis scio esse inaequales,
esse aequales. A finito ad infinitum, ab
infinitesimali ad finitum non valet eon-
^ JL — ^ — J sequentia. Diversa elemcnta, diversae
y pq fluxiones dant fluentes omnino diver>as. In
priori figura CB non est = RB, ncc lortasse
TAB = SAB. Diversae sunt etiam vires centripetae Cc, GR.
Vera est autem observatio Kepleri vera est Demonstratio
Newtoni : sed non vere ista demonstratio Imic observationi
applicatur: vel potius vera est utraque inclusiva non autem
exclusiva, Punctum S centrum esse bine ita demonstratur ;
ut centrum sit et T eodem modo, et quodvis punctum aliud,
nullum enim est ad quod non dirigatur vis centripeta ut ipse
adstruit Newtonus, varias versus vaiia puncta ciirvac dcfinicns
vires centripetas.
Excidit mihi superius plerasque Newtoni assertiones geo-
metrice veras, physice falsas esse. Parce vir doctifcsime huic
ingenuitati meae. admiror Newtonum nullum novi geometram
illi anteferendum. Pbysicae vitiuni est : nimis geometrice
tractari renuit, (juamvis tota sit geometrica, natura, ut ajunt,
geometrizat senqier: sed geometria sesc infinitis acconnnodat
CORRESPONDENCE WITH CASTEL
153
hypothesibus ; nee qnidiiuid o-eoiiietriciim est, coiitinuo pliy-
sicum esse convincitur. geonietria circa absti'acta v^crsatur,
circa possibilia, possibilia autein sunt nnmero infinita: unicuiu
est in quolibet plienonieno naturae systema: nee a possibili ad
actum valet consequentia.
A quindecini circiter annis opusculum composui (juo pb}^-
sicum Newtoni convellere totuni niihi videbar. Praelo paratuni
erat opus ; suuinia mea Newtoni reverentia coliibuit ne publice
illud juris facereni : nee faciam credo equidem tanta in animo
meo insidet summi illius viri existimatio. Vale vir claris.>ime,
meque tui observantissimuni, ^ervumque bumillimum habe.
LuDOVic Casti L.
Parisiis die 25 Martiis 1733
P.S.
Status quaestionis est. vult Newtonus aequabilitatem
arearum acquabili tempore descriptarum signum esse eertis-
FiG. 30.
simum, propriuni, unicum centri respectu cujus ea regnat
aequabilitas. contend© ego signum illud esse omnino aequi-
vocum. nee unam hac de re demonstrationem assero unicam
impugnas clarissime vir. omnes sunt impugnandae si asser-
tionem Newtonianam salvam velis. nam vel ea quae circa hie
appositam figuram versatur totum systema Newtoni convellit:
demonstrat enim 1", sine ulla vi centripeta, et sine ullo centre
154 STIRLING'S SCIENTIFIC CORRESPONDENCE
aequabiles esse tanien areas circa pnnctuin E. 2 '. iiifiiiita esse
puncta circa quae haec vigeat aequabilitas. at(|uc eiiim curvis
eidi'iii olitinet iiuletenninato. ruit ergo propositio hace i'unda-
mentalis Newtoniani systeinatis.
(^^)
Stirling io CasfcJ
Reverendo Patri D° Ludovico Castel.
Doctissiine Celeberrimeque Vir
Gratias ago maximas propter epistolam quam nuper ad me
scribere dignatus es, cui certe responsum antcliac dedissein,
si per varia negotia licuisset. Commentaria trivoltiensia ad
manus meas nondum pervenere, fateor tamen me pluribus
noininibus tibi devinctum propter ea quae in aliis tuisoperibus
de me scripta videram. Cur ego ad amicum commnnem
D. Ramsay ea scripsi quae tibi paulo liberius videntur, in
causa fuit tua erga me publico attestata benevolentia, quam
certe credebam me satis renmnerari non posse, agnoscendo
lil)rum tuuin de gravitate esse multiplici cruditione refertum
si non libere etiam tecum communicarem objcctioncs quasdam
mea opinione liaud male f undatas ; hoc enim ni fallor non
minus quam illud munus est amici.
Quantum ad aecjualitatem arearum circa centrum virium,
ego in pagina 539 tui lil)ri credebam CR fuisse errorem praeli,
Fig. 31.
si quidem nulla istius modi linea extat in schemate ; et pro
eadem legebam CV. Et procul dubio opportet CR et Cc esse
unam atque eandem tam magnitudine (juam positione nisi
fingas duas e.sse vires centripetas ut in tua epistola. Ibi
suj^ponis demonstrationem Newtoni pro accjualitate arearum
circa punctum ^, dein proi'ers propriam pro areis circa punctum
CORRESPONDENCE WITH CASTEL 155
T, qiiam ;iis tain verain esse qnam earn Newtoni ; quod ego
libeiiter concedo. Nam si existente >S' centro viriuin areae
circa idem aecpiales sint per demonstratioiiem Newtoni ; annon
per eandem demonstratioiiem areae erunt aequales circa aliud
quodvis punctum T modo idem supponatur esse centrum
virium ? Sed quid lioc ad nostram controversiam ago sane
iioiidum percipio. Tuum est demonstrare areas esse aequales
circa punctum quod non est centrum virium, alias inconcussa
manebit Veritas propositionis Newtonianae.
Inquis me si perlegerem paginas 531, 532 &c ' sensurum non
in solo triura linearum parallelarum parallelismo rem stare,
sed in ipsa praecipue eurvai-um geometricarum natura, quarum
latera infinitesimalia sunt omiiino indeterminata ut hoc vel
illo modo physico resolvantur in determinationes laterales
numero infinitas '. Sed post lectas sedulo paginas mihi recom-
mendatas, minime sentio rem stare in natura curvarum,
etiainsi resolvi possint in latera infinitesimalia ad libitum.
Et si CR et Gc supponantur non coincidere erunt duae vires
centripetae, quo in casu nihil probari potest contra Newtonum.
Ut autem coincidant est impossibile, quoniam aS'J. et TA non
sunt parallelae.
Revolvatur jam corpus in semicirculo ADB cujus centrum G,
et E punctum quodvis in diametro AB,
cui normalis sit GD. Dico impossibile esse
areas circa puncta G 8c E descriptas esse
temporibus proportionales. Sit enim si
fieri potest. Itaqu« ex hypothesi erit ut
tempus quo arcus AD describitur ad tem-
pus quo arcus DB describitur ita quadrans
AGD ad quadrantem DGB; et eadem de causa ut tempus quo
describitur arcus ^D ad tempus quo describitur arcus DB ita
area AED ad aream DEB; unde ex aequo ut quadrans ad
quadrantem ita sector AED ad sectorem DEB, unde ob
quadrantes ejusdem circuli sibi invicem aequales, erit area
AED aequalis DEB. Quod est absurdum. nam prior excedit
quadrantem, posterior vero ob eadem deficit triangulo GDE.
Haec autem deducitur consequentia non arguendo a finite ad
infinitum aut ab infinitesimali ad finitum, sed argumentando
per aequalitatem rationis.
Et in quacunque curva deferatur corpus, geometrice semper
156 STIRLING'S SCIENTIFIC CORRESPONDENCE
deinonstrari potest, impossibile esse ut aieae eirca duo puncta
descriptae sint teniporibus proportionales.
A is verain esse observationcin Kepleri et veraiii esse deiiion-
strationein Newtoni sed iion vore applicatam huic observationi
quod ultimuiu veliin ostendes. Deinde ais ' punctum ^centrum
esse ita deiiioustratnr ut centrum sit et T eodem iiiodo, et
quodvis punetuni aliud, nullum enim est ad (juod non dirigitur
vis centripeta, ut ipse adstruit Newtonus, varias versus varia
puncta curvae definiens vires centripetas '.
Newtonus ut demonstret vim qvia planetae retinentur in
orbibus tendere ad centrum Solis, ostendit per prop. 2. lib. I
corpus onnie quod movetur in ciirva, et radio ad punctum
innnobile ducto describit areas Temporibus proportionales,
urgeri a vi centripeta tendentc ad idem punctum ; quumque
Keplerus observasset planetas describere areas circa solem
temporibus proportionales, concludit vires quibus planetae
retinentur in orbibus tendere ad centrum Solis. Et liaec est
legitima argumentatio quoniam unicum tantum est punctum
circa quod areae descriptae sunt temporibus proportionales.
Unde constat nee punctum T nee aliud quodlibet probari posse
centrum virium nisi prius ol)servetur areas circa idem descriptas
esse temporibus proportionales.
Newtonus definivit legem vis centripetae tendentis ad punc-
tum (juodvis in genere, et exinde non sequitur eum adstruere
vim centripetam tendere ad omnia puncta, e contra tota vis
demonstrationis propositionis 1"'^° Lib I de aecjuabilitate
arearum pendet ex hoc (juod vis centripeta dirigatur ad unicum
punctum id(|ue immobile. Nam si dirigerentur ad punctum
mobile, vel ad duo aut plura puncta propositio esset falsa. Et
si vis centripeta tenderet ad duo puncta immobilia, turn
triangulum confectum lineis jungentibus puncta ilia duo et
centrum corporis moventis describeret solida proportionalia
tempoi'ibus, ut paucis abliinc annis invenit D. IVIacliin. Lex
autem pro pluribus centris (juam duobus nondum est I'eperta:
aequalitas arearum ad unicum centrum pertinet.
LKjuis pleras(|ue Newtoni assertiones esse geometrice veras,
& physice i'alsas; banc distinctionem i'ateor me non capcrc.
Nam secundum me assertio geometrice vera est propositio
demonstrata; haec erit semper et ubi(|ue vera, nee falsa
physice aut metaphysice, aut alio quovis modo. Fieri quidem
CORRESPONDENCE WITH CASTEL 157
})ote.st propo^iitionem geometricam in rerimi natura locum noii
habere propter ali(|uam suppositionein quae in natura non est,
sed inde non seqnitur propositioneni esse falsani. Exempli
<jjratia si nulla existat linea absolute recta in rerum natura,
(um nullum cxstabit triangulum cujus tres anguli aequantur
duobus rectis ; attamen est propositio vera non solum geo-
metrice sed et in omnibus scientiis, quod tres anguli trianguli
aequantur duobus angulis rectis modo latera ejus sint lineae
rectae. Si tantum velis, non sequi conclusiones geometrice
inventas existere nisi per cxperimenta vel observationes con-
stiterit hypotheses quibus innituntur haec conclusiones existere,
inficias non ibo.
Si habes opusculum apud te quo physica Newtoni tota
convelletur, oro te meo et omnium nostratum nomine ut
eundem illico mandes praelo, neve patiare Newtoni reverentiam
te cohibere a propaganda veritate; cujus amor apud nos ante-
cellit reverentiam qua colimus mortalium quemvis.
In conclusione dicis P"° sine ulla vi centripeta et sine ullo
centre aequabiles esse tamen areas circa punctum E. In cujus
contrarium aio demonstrationem Newtoni in eo fundari, quod
sit vis centripeta continue agens, et quod vis ilia senq^er tendat
ad unicum immobile centrum. Secundo dicis intinita esse
puncta circa quae haec vigeat inaequalitas ; liujus autem
impossibilitas geometrice demonstrari potest, de quo itaquo
non est mihi disputandum. Adeoque post omnia quae ad me
scripsisti, non percipio propositionem fundamentalem New-
tonianae systematis mere ; ignoscas interim oro si tibi assentire
nequeo, et obsecro ut tu legas hanc epistolam eodem animo
quo ego eandem tcripseram. Quod supcrest valeas illustrissime
Vir, meque tibi devinctissinmm et obhcquentissimum credas
Jacob. Stuiling
Londini Julii 1733 S.V.
VI
CAMPAILLA AND STIRLING
(1)
CamiKdlla to Stirling, 1738
Clarissiine, k Doctissime Domine
QuAM priinum ad me successive pervenerunt quaedam
Opera Insignis Scientiaruin Antistitis, & in Mathesi loiige
praestantissimi Aequitis Angli Isaaci Newton votis annuente
candido Amico nullani pati moram tanti Viri apud Vos illustre
nonien, quiii oeius ea perlustrarem fecit, lit ut cximia nie
tenuerit jucunditas, dum perlegerem mathematica Phjdoso-
phiae Principia, nee minus dein Opticae libros, in nonnuUas
incidi du])itationes, quas calamo inermi in binos includere
Dialogos, lubuit. Praelo evulgare formidavi, neve mihi petu-
lantis notam inureret, quam longe patet, Sapientum Respublica
& indignationem apud Vestrates incurrerem ; (juod auderem
censoria virga phylosophicam tangere hypothesim, (juam litera-
rius Orbis eximio prosequitur lionore, raagnaq: reverentia
colit. Tandem timorem ex animo prorsus excussit admodum
Reverendus e Societate Jesu Pater Melchior Spedaleri, qui per
Epistolam significavit, te mira, qua ornaris ingenuitate, ac
candore ad Patreni Castel, hisce, quae subdo verbis scripsisse,
([uibus petisses, ut difficultates, quas adversus Newton haberet,
typis statim mandaret, sicq: talia fando, eum adhortatus
fuisti : ' Oro to, meo & omnium Nostrum nomine, ut illud
praelo statim mandes: neve patiare reverentiam Newtoni
plus apud te valere, (|uam amor Veritatis : nam certo apud
Nos plus valet amor veritatis, quam reverentia, qua columus
Mortalium quemvis'. Revocato igitur animo ab tui consilii
heroica sinceritate, qui inter caeteros, quibus decoratur Societas
Regia Londinensis Mathematicos & Phylosoplios, emicas cele-
LETTER FROM CAMPAILLA 159
benimus, constitui nedum publice juris faceie, vcruin inodo
Opuseuluin liot-ce qualecuuui: nieuin ad to traiisinittcrc. Unuin
ab in<;eiiita Humanitatc tna euixo depraecor, Vir Clarissinie
ne deiboncris ISapicntiae tuac dubia me cdocere; ab to uno
enim solidain accipcre !-ententiani potcro ccrte ; eruiitq: inihi
& dogmata, & oracula. Oalleo prorsus, ut rem porgratam,
diu: praestes exoptatam hand valere famulatus mei offieia ;
at reeorderis, oportet, (pios siiblimlori Sapientia ditavit natura,
quaecumqiic agenda suscipinnt, virtute propria peragere, quae
sibimet sola praemia dat. Vale interim, faelieissime vive, &
dum to docentem habere obsecro, tuo noniini in omne aevum
suscipe
Motueae die sexta Mensis Maii 1738
Addictissimum & Obsequentissimum
TlIOMAM CaMI'AILLA
VII
BRADLEY AND STIRLING
(1)
Stirling to Bradley, 1733 '
Tower-street, London, Nov. 24, 1733.
Dear Sir,
I was very sorry that I did not see you when last in
town, because I wanted very much to have conversed about
the experiment made in Jamaica, which I hear you have
considered, as indeed I have also done. If the pendulum
went slower there than here by 2' 16" in a sidereal day, and
only 9" or 10" are to be allowed for the lengthening of it by
heat, as Mr. Graham tells me, thence it would follow that the
earth's diameters are as 189 to 190, or thereabouts, in which
case the force of gravity at the equinoctial would be to the
centrifugal force as 237^ is to unity; which is impossible,
unless the diameter of the earth were above 9000 miles, and
that differs so much from the measures of Norwood, Picart,
and Cassini, that it cannot be admitted, nor consequently the
experiment from whence it is deduced : and besides, I can
prove from undoubted observations in astronomy, that Cas-
sini's measure is very near the truth, for the diameter of the
earth can be found surer by them than by any actual
mensuration. If 29" could be allowed for the len;i-thenin<;:
of tlie pendulum l)y heat, tliis experiment made at Jamaica
would agree with other things, but Mr. Graham says that he
cannot allow that by any means. I am very far from think-
ing that the experiment was not exactly made, and indeed
a greater absurdity would follow from llicher's experiment
made in the island of Cayenna, which is the only one that can
be depended on, which is mentioned in sir Isaac's Principia.
' Pp. 398-400 oi'Miscell. Worh>> S; Corrofp. of James Bradley.
CORRESPONDENCE WITH BRADLEY 161
Althouy,h 1 have treated oi" the problem of the fi<;'ure of the
earth in a manner which is new, yet I am still obliged to
suppose the figuire of it to be an exact spheroid, and althoui^h
I be sensible that this supposition is not sufficient to determine
the number of vibrations to 8'' or 9" in a day, yet I know that
the error cannot be so great as the Jamaica experiment makes
it. If Mr. Graham be certain that not above lO'' can be
allowed for the heat, it is as certain either that the mountains
have a sensible effect on the pendulum, or some other thing,
which will render the experiment entirely precarious.
I find that sir Isaac in his 3d edit. Princip. mentions three
observations of Dr. Pound, which )nake Jupiter's diameter
about S?''; I want to know if that be the greatest diameter
of Jupiter ; because if it be, then the lesser would be about 34'^
which would make too great an odds in the thing for which
I want it. And I should be glad to know if you can help me
to any observation which ascertains the moon's middle distance
from the earth, which I could depend more on than the common
ones ; if you could inform me of these things, I should be able
quickly to make an end of what I shall say about the figure
of the earth, which I would the more willingly do, because not
only Mairan, but also Hugens, Herman, and Maupertuy, have
all of them entirely mistaken the matter. I heartily wish
j'ou all happiness, and the sooner I hear, the more you will
oblige.
Sir, your most humble servant,
J. SlIRLINti.
(2)
Bradley to Stirling, 1733
To
Mr James Stirling
at the Academy in Tower Street
London
Dear Sir
When I was last in London an unexpected accident obliged
me to return hither sooner than I intended ; and hindred me
from waiting on you, as I proposed to have done ; having been
informed that you were then examining into the Dispute
2447 M
162 STIRLING'S SCIENTIFIC CORRESPONDENCE
conccrniug the Figure of the Eartli. Not that I had much
more to tell you, than what is contain'd in the Account of the
Jamaica Experiment, which I left with M' Graham ; wherein
I have stated the Facts as well as I could, and made such
allowance for the lengthening of the Pendulum by Heat as
former Observations and Experiments would warrant.
The Result of all seem'd to be that tlie Clock went l'-58"
p Diem slower in Jamaica than at London. I allowed only
8 2" on account of the different degrees of Heat, having no
Authority from former experience to mal<e any greater Abate-
ment ; so that I apprehend this Retardation of the Clock
(so much greater than what is derived by a Computation
founded on the Principles of Gravity and an uniform Density
in ye several parts of the Earth) must be rather ascribed to
an inequality in the Density of the parts of ye Earth near
which the Clock is fix'd, than to the greater Heat. For the
greatest part of the force of Gravity upon any particular
Body arising from the parts of the Earth that are near it
(the Action of ye remote parts being but small) does it not
thence seem likely that a Body placed near a great Quantity
of rarer Matter as Water &c: will not be attracted with so
much Force as if it were in the midst of a large quantity
of Denser Matter, as in a great Tract of Land (S:c ? and may it
not thence follow that Clocks (tho' in the same Latitude) may
yet not go alike, when placed on y*" Continent and on Islands
or on larger and smaller Islands ? or may not the Mountains
(as you observe) according as they contain Matter more or less
Dense, contribute something towards such Inequalities. These
considerations do at least suggest the necessity of a great
variety of exact experiments made in difiercJit Places, situated
in the same, as well as in different Latitudes, and I have (for
this reason) proposed in the fore-mentioned Account, to have
the Experiment repeated in several Places, in order to discover
whether the Density of Different Regions be uniform or not ;
for till that Point is settled, we may be at a loss for the true
cause of this Difference between the Theory S: Experiment.
As to the Diameters of Jupiter, 1 find from the Mean of
several Observations which I made with the R. Society's Glass
of 123 feet focus, that the greater Diameter is to the Lesser
(when both were measured with a Micrometer) as 27 to 25.
CORRESPONDENCE WITH BRADLEY 163
the greatest Diaiiietei- (at ll." mean Distance Ironi y'' ICarth
or Sun) being just 39". This is the Case when ye Diameter
was actually measured with the Micrometer; but by other
observations of the Time of the Passage of some of the Satellites
over ll^ Disk, compared with their greatest apparent Elonga-
tions taken with a Micrometer, the Diameter of 1/ comes out
only 37'' or 38", the ditterencc arising (as I conceive) from y°
Dilatation of bight i^c.
Having never made any Observations myself particularly
with a view to determine the Moons mean Distance I can
give you no information relating to that Point, but believe
M"" Machin has examined that matter and lix'd it with all the
accuracy that the best Observations we have, would enable
him to do it.
You would have had my Answer sooner, had I not been
engaged in a Course iVc upon y'' conclusion of which I have
taken the first opportunity of assuring you that I am with
great Respect
S' Your )nost obedient
Oxford \ humble Serv*
Dec. 2^^^ Ja: Bradley,
1733
M 2
VIII
KLINGENSTIERNA AND STIRLING
(1)
Klingenstierna to StirUiifj, 1738
Viro Clarissimo, Doctissi
moque Domino
Jacobo Stirlingio
Londinium
at y- Academy in little
Tower Street.
Clariss, Viro
Jacobo Stirlingio
Sam. Klingenstierna
S. p. d.
Daplici nomine indulgentia Tua maximopere me egere
sentio uno, quod multis singularis cujusdam benevolentiae
documentis a te aftectus per tantum temporis spatium silucrim :
altero quod nunc tandem silentium rumpens non dubitaverim
negotiorum nonnullorum demandatione tibi esse molestus.
Sed quemadmodum Te persuasissimum esse velim, me ofHcia
it studia in me Tua, quae dum Londini agerem, multis modis
expertus sum, prolixiori animi afi'ectu (|uam verborum apparatu
agnoscere, seniper(j[ue agniturum esse: Ita spero ctiam te
non aegre laturum, (juod Tibi amicorum optimo harumque
rerum intelligentissimo ejusmodi negotia demandem, (juac ad
comunium studiorum ([ualecunc^ue incrementum aliquid forte
conferre poterunt. Constitui nimirum apparatum Instru-
mentoruin Physicae Experimentali inservientium quam potcro
perfectissimum niihi comparare. Eumque in fineni instrumenta
(juae apud nos per })eritiam artificum fabricari possunt, confici
LETTER FROM KLINGENSTIERNA 165
cuiavi. Ceteruni (jumii iiistniineiita optica millibi terrarum
meliora quaiii Londini conficiantur, te etiaiu atque etiaiu oro,
lit optima eoruni, quae se(|uens designatio exliibet, pro lue
eliuas, A' Domino Claesson (cui curam numoruin pro iis solven-
dorum, cV- transmittendorum Holmiam instrumentorum comisi)
tradi facias. Certissimiis ero me bona habiturum instrumenta,
si tu, harum rerum intellioentissimus Judex ea elet^eris &
approbaveris. Si aliqua fuerint, quae apud artifices statim
haberi non poterunt, ea mihi primum transmittas quae haberi
nou poterunt, ea mihi primum transmittas quae haberi possunt,
reliqua etiam missurus, quam primum parata fuerint. Optarim
enim, ut ante hyemem, quam potero pluriina habeam. Si
aliqua ratione heic locorum utilis tibi esse potero senties
gratam animi vohmtatem mihi nunquam defuturam.
Designatio Instrum.
Vitra ad Tubum Astron. 16 pedd. circiter.
Vitra ad Tub. Astron. 8 ped.
Prismata et Lentes ad Newt. Theoriam Colorum demon -
strandam.
Laterna Magica cum figuris necessariis.
Lens pro Camera ob&cura 4 ped.
Specula Conica iVr Cylindrica cum picturis deformibus.
Plana vitrea inter quae aqua ascendit in figura hyperbolica.
Oculus artificialis.
Tubus vitreus amplus pro electricitate vitri monstrar.da.
Microscopium duplex cum apparatu necessario.
Instrumenta pro Legibus Refractionis t*^ Reflexionis dete-
gendis.
Duo vitra concava pro Myopibus foe. unius pedis,
Diaboli Cartesiani.
Praeterea etiam libros nonnullos novos apud vos noviter
editos libenter desideraverim, ut D°' Smith Systeme of Opticks:
D"' MacLaurin Systema Algebrae, & si qui alii recens editi
fuerint in Mathematicis, novi quid continentes, quales credo
in Anglia, ingeniorum feracissima non deesse. ante alios
aveo scire, utrum D°' Machin Theoria Gravitationis lucem
viderit, vel quando videbit & quomodo valeant insignes viri
IGG STIRLING'S SCIENTIFIC CORRESPONDENCE
fautoresque inei lionoratissiivii D"' HallLyus, Moivreus, Machin,
(jiiibus meis verbis salutem plurinuun iinpertias. Vale iiiteriia
ife fave
Tui Studiosissinio
Holmiae d 19 S. Klingenstierna
Septembris 1738.
Problems of Klinoenstierna (1733?)
Prohlema Sint in A & a duo ignes, quorum vires cale-
faeiendi in distantiis aequalibus sint in data ratione AF ^i\iif,
& creseentibus distantiis decrescant in ratione quadratorum
distantiarum. Quaeritur per quam viam ab i<;nibus illis
reeedere debeat viator in loco aliquo dato & eonstitutus, ut
minimum sentiat calorem.
Solutio Sit BD particula quam minima viae, qua viator
a puncto quocunque B reeedere debet, ut ab ignibus A et a
minimum calorem sentiat. Centro B intervallo BD describatur
eircumferentia circuli DK, (S: erit intensitas caloris in D minor
intensitate, ejus in quovis alio circumferentiae DK puncto.
Quare si in eircumferentia ilia sumatur punctum (/ puncto D
proximum, calor in d per naturam minimi aequalis eenseri
potest calori in D. Sed calor in /) per h^^poth. est -rjr^ H j^^
AJJ Q/U
. , . J AF of ^ AF at AF af
& calor in d, -j— + —-, Ergo . y^ + -y- = -—■, + -^,
Ad" ad^ ^ AD" aD- Ad- ad-
„ , , AF AF af of
& transponendo ^, - ^^, = ^^ - ^^, •
Centris A k a intervallis AD k ad describantur arcus
Dj) & dP, rectis Ad & aD occurrentes in ^) & 7\ k per
principia methodi infinitesimalis erit -rr.., :r-r. = ..-,..> &
^ AD- Ad^ AD^
1 1 2DP . . AF AF af af
,., — ,v, = ~i^ > a( ooque ae(|uatio-7-;— — -r-r = ' ^ — — —
ad' uD' uD- ^ ^ AD- Ad- («P aW
^ ^ . , 2dp.AF 2DP.af „ ,. ., ,
mut'itur HI haiic, — ~ri\^r— = rrr^ ■> k dividendo per 2, ac
AD' aD' ^
, ,, ,, ., 1 .„ n dp.AF DP.af
pro AJ) aD, scribendo AB aB, ' „ = — . -•' .
^ AB' aB-"
LETTER FROM KLINGENSTIERNA
1G7
Centris A & a intervallis AB, uB describantur arcus BE
& Be rectis AD iSc aD occuiTentes in IiJ & e, & erit trian^uluin
BBE simile triano-. Bdj), triangulum DBa simile triany;.
dDP. Quare DB : Dd = BE: dp, & DB : Dd = Be : DP, adeo-
que ex aequo BE -.dp = Be: DP Si itaque in aequatione
' p, — — ^~ pro d/p & DP substituantur earum propor-
.. , DX.X o 1 1 . BE.AF Be. of
tionales BE & Be, habetur — -r-r, — = — -^ •
AB' aB-^
Centris A & a intervallis AF & af describantur eircum-
ferentiae FQ & fq, rectis AB, AD, atque aB, aD occurrentes
in /, L, & i, I, eritque ob similitudinem triangulorum ABE,
n F' AW
AIL, AB:BE=AI (id est AF): IL, unde *
AB
= IL.
168 STIRLING'S SCIENTIFIC CORRESPONDENCE
Similiter ob similitudinein trianguloruin uBe, ail, erit
(iB : Be = ai (id est af) : il, unde -rr = il.
a B
^ .. . BE.AF Be.af BE.AF , Be.af
Ergo SI in aequatione ^^,^ = -^^ pro ^^ ^v -^
IL il
substituantur IL ^ il, transit ilia in liane : -t-tto = ^w *
AB^ (iB-
Ad reetara Aa demittantur normales LH, IG, BO, ig, Ih,
ipsique Aa parallelae IN, in, rectis LH, Ih occurrentes in iY, n.
Propter similitudinem triangulorum ABG, AIG, LIN, est
AB:BC = AI (id est AF) : IG,
& AB:BG= . . LI.IN;
quare terminis ordinatim in se ductis
AB^:BC^ = AFx LI: IGx IN, unde ^,= i^ti^, '
AB- AF.BC^
Similiter propter similitudinem triangulorum aBG, aig, tin,
est
aB : BG = ai (id est af) : ig
aB : BG = . . li : in ;
quare terminis ordinatim in se ductis
aB^ : BG^ = . . af. li:ig. in ; unde -^, = ^^.'l^\.^
ah- af.BG^
^ , . , , IL il IG.IN ig.in
Sed mventum erat ^^ = ^,, ergo ^^^^ = -j-^^.
& multiplicando per BG'-, ^. — = ^ • Est vero IG . IN
elementum circuli IGHL, i\: ig.in elementum circuli igJil
IGHL ighl , AF IGHL
quare — ^^" = -^ ' adeoque — r; = . , , •
^ AF af ^ af ^gkl
Sit >S' locus datus unde prodit viator. Jungantur AS, aS
circumfercntiis FQ,fq occurrentes in II, r i^ demittantur RT, rt
perpendiculares ad Aa. Et cum per jam dumonstrata, elementa
IGHL, ighl ubique sint in data ratione AF ad af, erit etiam
componendo, Summa IGHL, id est spatium RTHL, ad suiliam
omnium ighl, id est spatium rthl, in eadem data ratione AF ad
Af, unde scqucns prodit Gonstrudio.
LETTER FROM KLINGENSTIERNA
169
Centris A S: a descriptis circulis FQ,fq, quorum radii AF, af
sint proportionalos virilms calefacicndi io-nium A iS: a, jung-antur
AS iV aS, circulis illis occurrentes in R iS: r, & demittantur
Fig. 34.
Rt, rt, normales ad Aa Rectis LH, Ih, itidem normalibus ad
Aa, abscindantur Sj)atia TRLH, trih, quae sint in ratione AF
ad af. Jungantur i^' producantur AL & al, donee conveniant
in D, A' erit punctum D in curva quaesita 8D.
Prohlema. In venire curvas AGBG i^ AHBl, quarum talis
est ad se invicem relatio, ut curva prior AGBC rotata circa
polum fixum A semper secetur ab altera AHBl in punctis
summis 5, h, iV ut segmenta AGS A, AHBA semper sint in
data ratione m ad n.
Solutio. Rotetur curva AGBG circa punctum fixum A,
donee perveniat in situm proxiuium AFDG, in quo situ secetur
a curva AHBl in b. Centro A intervallo AB dcscribatur
arcus BD curvae occurrens in D, ^ jungantur AD, Ah, quarum
haec occurrat arcui BD in E. Et quia AGBA : AHBA - m : n,
& AFhA : AHhA = rii : n, erit etiam dividendo
A FbA - A GBA : AHbA - AHBA = m : n,
id est, Triangulum ADb : triang. ABb = m : n. unde ob basin
communem Bb, erit DE.BE = m : n.
„ o^DE
Dicatur AD, x\ Eb, dx; DE, dy; d' erit EB =^
iV BD = DE 4- = dy.
m m
m
170 STIRLING'S SCIENTIFIC CORRESPONDENCE
Et quoniani per hyp. tautens ciirvae JUli(J in Ji paiallela
est tan<;enti ejusdem in b, erit angulus rotationis BAD aequalis
angulo (luem duae rectae ad curvaiu nonnales in punctis B
& /; constituunt in centre circuli osculatoris. Ergo AD.DB
= radius curvedinis in J) : ad elementuni curvae Dh, id est
.,. , ,,, , , 97i + 9«, , xda'^dx -,
(dicto Db — ds)x : (hi = -. — , — ^ ^ — ^-r '• ds,
m dxdyds — xdydds
dy . X ds'^ dx ds dx
adcoque xds = ^ — j — -. vel 1 =
dxdyds — xdydds dsdx — xdds'
11 nde ds dx — xdds =■ ds dx,
m
, , n -, -, . . n dx dds
sen —xdds = — dsdx, nine = 9— >
m ni x ds
sumtisque lorarithmis — I- = I -r i iV' perficiendo quod restat
^ ^ m a ds ^ ^
reductionis :
11
=. = dy
2 n 2 n
Centro A, intervallo a describatur circulus, eujus elemontum
rectis AD, Ab comprehensum dicatur dz, eritque x -.dy = a : dt,
xdz
unde dy = ' — » & hoe valore substituto in aequatione modo
inventa
x'^'dx
= dy,
1 2n in
^jd^-X^
transformatur
ilia
in banc
m
X " dx
xdz
/ 2w 2m
\a"' -X'"
a
seu
ax"' dx
1 211 2 n
= dz.
LETTER FROM KLINGENSTIERNA
171
Ponatur x — a
~V
m
\- aequatio transibit in lianc;
adv
n y
^ck
ua — vv
quaosequentem snppeditat Pi'oblematis Constructioncm. Cen-
tro A intervallo quovis AB describatur circuint'erentia circuli, in
qua hinc inde a puncto quovis dato B sumantur arcus BC, BD
Fig. 35.
in ratione n ad m. Jungantur AG, AD & a puncto C demittatur
CE normalis ad radium AB. In AC i^ AD sumantur AF
CE "
& AG aequales AB . -j^ 6^ erit punctum F in curva fixa
AHBT, d' punctum G in curva rotatili AGBC.
Coroll. Si fuerit m ad n ut numerus ad numeruin, utraque
curvarum est A]oebraica,sive minus, earum constructio dependet
a multisectione anguli tV rationis, seu quod idem est quadratura
circuli & hj'perbolae.
JX
MACHIN AND STIRLING
(1)
MacMn to Stirling {1733?}
To
M"" Stirlino^ at the Academy
in little Tower Street
Dear Sir
I intend to give you some short notes upon M'" Bernoulli's
Letter, w*'' if you approve of it shall be addrest in a Letter
to yourself. It shall be ready against the beginning of next
week, unless anything material happen to hinder it. I have
reason to believe that if he be a man of any candour, I shall
be able to give him entire satisfaction as to every objection
that he makes, iS: do intend withal to oblige him w*^ the
solution of a Problem w"'' I now percieve he had proposed to
himself but quitted rather than be at the pains to go through
w*** it. And that is whether there be a point in his locus
from whence the Planet will appear to move equally swift in
the Apsides i^' one of the middle distances. And where it is
that y*^ point \yes. As I apprehend he may have communi-
cated some of his remarks to others as well as yourself or may
have hinted that he has made some ; I should be glad to
a word or line
know by the bearer, whether you will give me leave to shew
this Letter to the Society upon the foot of there being some
new Problems in it, w*^'' may furnish me w^'' the oi:)portunit3'
of saying that his Objections are to be answered. I do not
mean to have the Letter read, but only to have the Contents
of it mentioned iV especially the Problems since lie seems to
have sent those on purpose to be proposed to others. I shall
CORRESPONDENCE WITH MACHIN 173
herein behave according to the directions you are pleased
to give.
E"". Your most faithful
Friend & very humble Serv*
Thursday morning J. Macuin.
(2)
Macliin to Stirling^ 1738
„ ^. Gresham College June 22. 1738
Dear Sn- ^
The date of your obliging Letter when I cast my eye
upon it gives me great concern. I was ashamed when I
received a Letter from you to think you had prevented me
in paying ni}'' respects to you first, but am now confounded
in the reflection of having slipt so long a time without return-
ing an answer to it. Sure I am in the case of Endymion !
But every day has brought its business and its impertinence
to engage me and to interrupt me. Were there time I could
plead perhaps more things in my excuse than you may be apt
to imagine. This long vacation which begins today, appears,
if it deceive me not in my expectation, as one of y'^ greatest
blessings I have long since enjoyed. If I am tardy after this,
then believe (what would grieve me if you should believe)
that you are one that are not in my thoughts. Think not
that you are singular in your retirement from y° world. There
may I can assure you be as great a solitude from acquain-
tance k conversation in a Town as in a Desert. But of this
sufficient.
Mons'^ Maupertuis has sent you a present of his book
which I have deliverd to M'' Watts for you. It contains
a complete account of the measurement in the North.
M"" Celsius likewise published two or 3 sheets on y'' same
subject chiefly to shew that Cassini's measurement was far
inferior to this in point of exactness, and which I suppose you
will need no argument to prove when you have read over
M. Maupertuis's book.
We have also had from time to time scraps of accounts
communicated to us, still in expectation of something more
perfect, w"^"^ I intended to have sent to you, but this book has
rendered it unnecessary.
174 STIRLING'S SCIENTIFIC CORRESPONDENCE
Tliurc have been great wrangles and disputes in France about
this measurement. Cassini has endeavoured to bring the
exactness of it into Question. Because the Gentlemen did
not verify the truth of their astronomical observations, by
double observation with y" face of their Instrument turned
contraryways. So that M'' Maupertuis was pdt to the necessity
of procuring from England a certificate concerning the con-
struction of M' Graham's Instrument, to show that it did not
need that sort of verification.
You will see that this measurement in y" North, if it l)e
compared with y* in France, will serve to prove that y*^ figure
is much more oblate than according to y*^ rule. But perhaps
it will be safer to wait for the account from Peru before any
conclusion be drawn. These Gentlemen have also compleated
their work and are returning home where they are expected
in a short time.
Mons' De Lisle has published a Memoir read in tlie Academy
at Petersburg w*^^ contains y'' scheme of a Grand Project
of the Czarina for making a compleat Mapp of her whole
Empire, and in w*^'' there is a design of making such a measure-
ment not only from North to South but from East to West
also as will far surpass any thing that was ever yet thought
of ; it being to contain above 20 degrees of y® meridian and
many times more in the parallels.
Your Proposition concerning y® figure (wherein all my
friends can witness how much I envy you) could never find
a time to appear in the world with a better grace than at
present, Now when y® great Princes of y'' Earth seem to have
their minds so fix't upon it.
But for other reasons I should be glad if your Proposition
could be published in some manner or other as soon as possil)le,
but not without some investigation at least : unless you have
hit upon a Demonstration w*'^ would be better, because I find
several people are concerning themselves upon that subject.
I have kept your paper safe in my own custody, nor has any
one had the perusal of it.
Nor shall I believe that any one will find it out till I
see it. But M"" Macklaurin in a Letter to me dated in febry
last, (and w*^^ was not deliver'd to me but about a month
ago, the Gentleman being ill to whose care it was entrusted)
taking occasion to speak of y'" figure of y ' Earth, and that
CORRESPONDENCE WITH MACHIN 175
S' Is. had supposed but not demonstrated it to be a Splieioid,
proceeds on in the following words, ' M' Stirling; if I reniendjer
right told nie in April that none of those who have considered
this subject have ghewed that it is accurately of that figuie.
I hit upon a demonstration of this since he spoke to me w'"'
seems to be pretty simple.'
I have given you his own words for fear of a mistake,
because I am surprised you did not take that opportunity
to inform him, that you had found it to be of that figure.
For that nobody has yet shewn it to be so is what I thonght
everybody had known. But I shall take this opportunity to
advise him to connnunicate his demonstration to you.
And if he has found out a simple demonstration for it,
I think it ought to l)e highly valued, for it does not seem easy
to come at it. I own I have not had time to pursue a thought
I had upon it, and which I apprehended and do still apprehend
might lead to the demonstration and shall be very glad if he
or any one else by doing it before shall save me that trouble.
As to y'^ Invention of M'" Euler's Series were I in your case
I would not troul)le myself about it, but let it take its own
course, if anything should arise your Letter to me w'"^ I shall
keep will be a sufficient acquittal of yourself.
M'' Moivre's Book is now published but I have not got it yet
nor have I been able to see him but once since I reced your
Letter and as to this conveyance I was but just now apprized
of it and have but just time to get this ready before M'' Watts
goes out of Town,
As to y'^ moon's Distance I have now materials to fix y*^
moon's Parallax, and chiefly by means of an Observation of
the last Solar Eclipse at Edinburgh by M'" Macklaurin, and
will take care as soon as I can make y° calculation to send
it to you.
There are some other matters whereto I should speak which
I must now defer to another opportunity, and only say now
that I am with affectionate regard
Your most faithful friend
& very humble Servant
John Machin.
X
CLAIRAUT AND STIKLING
(1)
Clairmit to StirUng, 1738
Monsieur
En cus qu'un Menioire sur la Figure de la Terre que
j'envoyai de la Laponie a la Societe Royale, soit parv^enu
juscjua vous et que vous I'ayes daigne lire, vous y aures
reconnu plusieurs Theoremes dont vous avit^s donne auparavant
les enoncds, parmi les belles decouvertes dont est rempli un
morceau que vous av(^s insere dans les transact. Philosoph.
de I'ann^e 1735 ou 1736. Vous aur^s ^t^ peut-etre etonne
que traitant la meme matiere que vous je ne vous aye point
eit^. Mais je vous supplie d'etre persuade que cela vient de ce
que je ne connoissois point alors votre Memoire, et que si je
I'eusse lu je me serois fait autant d'honneur de le eiter que j'ai
ressenti de plaisir lorsque j'ai appris que je m't^tois rencontre^
avec vous.
Depuis le terns o\x j'ai donne cette Piece j'ai pouss<^ mes
recherches plus loin sur la nieme matiere, et j'envoye actuelle-
ment mes nouvelles decouvertes a la Society Royale. Apr^s
vous avoir fait ce recit Monsieur et vous avoir prid d'cxcuscr
la liberti^ que j'ai pris de vous ecrire sans avoir I'lionneur
d'etre connu de vous, oserois je vous demander une grace, c'est
de vouloir bien jetter les ycux sur nion second Memoire que
M"" Mortimer vous remcttra si vous le dai^'nes lire.
Ce n'est pas seulement I'envie d'etre connu de vous qui
m'engage a vous prier de me faire cette grace, Mais c'est que
j'ai appris par un ami qui a vu a Paris un Gcometre anglois
appelld M. Robbens que vous avies depuis pen travaille sur la
memo matiere.
LETTER FROM CLAIRAUT 177
Jc souluiitcrois done cxtroincment de scavoir si j'ui ete asses
heureux encore pour m'etro rencontre avec voiis. Si au con-
traire jc m'etois tronipe je vous serois infinimeiit obligd de me
le dire i'rancliement afin ({ue je men corrigiasse. Qnoi (ju'il en
soit si vous daign(^s me donner quelques momens, vous aurds
bientut vii de quoy il est question et si mon memoire m'attire
une reponse de vous je serai cliarmd de I'avoir fait parce qu'il
y a deja longtems que je souhaite d'etre en liaison avec vous.
Qufkpi'envie que j'en aye ne croyes pourtant pas Monsieur
que je soye asses indiscret pour vous importuner sou vent
par des lettres inutiles pleines de simples complimens.
M'' Mortimer pourra vous dire ({uelle est ma conduite a son
egard, J'en oserai de memo avec vous si vous me le permettes.
En attendant j 'ai I'lionneur d'etre avec estime et respect
Monsieur
Votre tres humble et tres
a Paris le 2 Octobre 1738 obeissant Serviteur
Claikaut.
P,S. En cas que vous veuillt^s me faire reponse il faudra
avoir la bont(^ de remettre votre lettre a M. Mortimer. Si
vous n'aimes a ecrire en francois, je decliifFre asses d'anglois
pour entendre une lettre et quand ma science en cette langue
ne suffiroit pas, j'aurois facilement du secours.
XI
EULEK AND STIRLING
(1)
Stirling to Eider, 1738^
Celeberrimo Doctissimocjue Viro
Lconhardo Eiiler
S.P.D
Jacobus Stirling
mihi
Tantuin teinporis elapsuin est ex quo dignatus es (ad me)
scribere, ut jam reseribere vix ausiin nisi tua humanitate
fretiis. Per hosce duos annos plurimis negotiis implieitus
sum, quae occasionem mihi dederunt frequenter eundi in
Scotiam et dein Londinum redeundi. Et haec in causa fuerunt
turn quod epistola tua sero ad manus meas pervenit, turn quod
in liunc usque diem vix suppeterat tempus eundem perlegendi
ea qua meretur attentione. Nam postquam speculationes sunt
diu interruptae, ne dicam obsoletae, patientia opus est ante-
quam induci possit animus iterum de iisdem cogitare. Hanc
igitur primam corripio occasionem testandi meam in te Obser-
vantiam et sinnil (gratias) agendi gratias dudum debitas
propter literas eximiis inventis refertas.
Gratissimum mihi fuit Theorema tuum pro suinmandis
Seriebus per aream Curvae et differcntias sive Huxiones
Statim
Terminorum quippe generale et praxi expcditum. (lllius)
percepi item extendi ad phirima serierum genera, et (|Uod
celerrime
praecipuum et A plerunujue (celeriter) approximat. Forte
non observasti theorema meum pro summandis Logarithmis
Tui
nihil aliu<l esse quam casum particularem tui Tlieorematis a
* This is only StirUng's rough draft with all his corrections. Erasures
are indicated by bracket!?.
CORRESPONDENCE WITH EULER 179
eo
generalis ; (quod ingenue fateor). Sed ct A gratius mild fuit
quod
(tuum) hunc invcntuni, (quoniam) de eodem (ego) quoque ego
olim cogitaveram ; sed ultra prinium terminum non proeessi,
approximav pro libitu
et per euni solum (perveni satis expedit(i) ad valores Serierum
satis expedite
A scilicet per repetionem calculi, ut in resolutione aequa-
tionum affectaruni ; cujusque specimen dedi (plurimis abliinc
annis) in philosophicis nostris transactionibus :
Quae liabes de inveniendis Logarithmis per Seriem Harmo-
saltem
nicam (non percipio, propter novi) obscura mihi a videntur,
notationem
quoniam a non recte intelligo (notationem.)
Imprimis autem mihi placuit methodus tua summandi
quasdam Series per potestates periferiae circuli, (quarura
indices sunt numeri pares). Hoc fateor (omnino novum et)
et omnino novum habeat
admodum ingeniosum a nee video quod A quicquid
commune methodis receptis
(affin habeat) cum (iis quae hactenus publicantur,) adeo ut
ciedara hausisse
facile (concedam) te idem (hausisse) ex novo fonte A ((et
nuUus dubito to hactenus observasse, aut certe ex fundamento
tuo facile percipies, alias series tuis tamen affines summari
posse per potestates periferiae quarum indices sunt numeri
impares. Verbi gratia, denotante ]) periferia,
l^j = 1 — -3- + -I — f + 1 — &c ut vulgo notum
13,111 1 n
32^ 3- 5' 7^ 9-*
5 , 1 1 1 1 P
1536 3^ 5^ 7'' 95
c^-C.))
continentur in
Series tuae (comprehenduntur sul^) forma generali
1 1 1 1 1 P
ubi n est numerus par) eadem (tamen ad formulam scquen-
n2
180 STIRLING'S SCIENTIFIC CORRESPONDENCE
tein) nullo iiegotio reducitur, (scilicet) re<lucitur ad ronmdain
sequcntem,
1111 1 o
1+ ■Sn+ 5"+ r'+ 9^+ 11" + ^^''
(iibi teniiiiii altcnii desiint, ct oniiies sub luic iorina compre-
ot Ikuic sunimare
liciisas suimiiarc) A doces A per potcstatem periferiae
cujus index est n modo sit Ceterum si
((piando n est) imuicius par, (Si jam iiiutentur) signa
tenninoruni alternoruin inutcntur ut cvadat Scries
1111 1 ,
3" 5" 7" 9" 11"
Haec inquain seii)})er suimiiari potest (at(|Ui! liaec Series,
qiiaudo li est luunerus iiiipar suiliari potest) per dignitatem
modo sit Humerus impar
periferi (circiili) ciijiis index est h. (verbi gratia) uticpie si sit
n = 1, (erit) ^p = 1 — -3 + i — t + 9 — tt + ^^ ut vulgo notum
' 32 ^ 3^ 5-^ 7-^ 9-^ ll'^
5 . 11111.,
' 153G ' 3 5^ 7^ 9' 11^^
&c.
Et nuUus dul)ito te liactenus idem ubserxasse, aiit saltem
facile observatiir ex fundamcnto tuo (jnod lil)enter videbo,
quando (animus erit tibi idem impertire) ita tibi vi^nm fuerit.
monendus es Mathcsoos
Hie autem (ae(}uum est ut te moneam) D. Maclaurin a pro-
fessorem (Matliescos) Edinburgi, post alicjuot tenq)us (brcvi)
jam
editurum lil)rum de fluxionibus cujus paginas ali(|uot (liactenus)
imprcssas (niecum) mecum connnunicavit in (juibus duo liabet
Theorcmata pro summandis seriebus per differentias termi-
norum, (juorum alterum ipsissimum est (juod tu dudum
mild
(ad me) misisti, (et cujus ego cum illico certiorem feci). Et
etiam si illc libenter promiserat se idem testaturum in sua
praefationc, judicio tamcn tuo submitto annon velles (edcre)
edere
tuam epistolam A in nostris philosophicis trausactionibus.
CORRESPONDENCE WITH EULER 181
Et si vis (luaedam illustrare vul demonstrare, (ant plura
ut lucem videat
adjicere, egoaiit) et cito inihi rescribere, ciirabo (tuam epistolaiii
viseram lucem diu) Mntecjiiaiii ejus liber prodierit. Quod si
animus erit hac (data) oecasione eligi unus ex Sociis nostrae
Societatis
(Academiae) Regiae, idem reli(iuis gratum (esse non) procul
quando viderint praeclara tua inventa
dubio gratum erit (postquam inventa tua viderint Et) mihi
vero semper gratissinuim ut amicitiam (mihi licet immerenti)
continuare difjneris
Edinburgi IG Aprilis 1738
(2)
Elder to Stirlinfj, 1738
lllustrissimo atque Celeberrimo Viro
Jacobo Stirling
S. P. D.
Leonhard Euler
Quo majore desiderio litteras a To Vir Celeb, expectavi, eo
majorc gaudio me responsio Tua liumanissima afiecit, qua,
eo magis sum delectatus, quod non solum litteras meas Tibi
non ingratas fuisse video, sed Temet etiam ad commercium
hoc inceptum continuandum invitare. Gratias igitur Tibi
habeo maximas, quod tenues meas mcditationes tam benevole
accipere Tuumque do iis judicium mecum communicare
volueris. Epistolam autem meam a Te dignam censeri, quae
Transactionil)Us Vestris inseratur, id summae Tuae tribuo
humanitati, atque in hunc tinem nonnullas amplificationes et
dilucidationes superaddere visum est, quas pro arbitrio vel
adjungere vel omittere poteris. Hac autem in re quicquam
laudis Celeb. D. Maclauriu derogari minime vellem, cum is
forte ante me in idem Theorema seriebus summandis oserviens
incident, et idcirco primus ejus Inventor nominari mereatur.
Ego enim circiter ante quadriennium istud Theorema inveni,
quo tempore etiam ejus demonstrationem et usum coram
Academia nostra fusius exposui, quae dissertatio mea pariter
ac ilia, quani de Suumiatione Serierum per potestates peri-
\S2 STIRLING'S SCIENTIFIC CORRESPONDENCE
phcriac circiili coinposui in nostris Coininentariis, (jui (jiiotiuinis
prodeunt, brevi lucein publicam aspicict. In Connnontariis
autom nostris jam editis ali(|Uot extant aliae nietliodi nicae
Series sununandi quaruui (juaedani niultuni liabent Siniilitu-
dinis cum Tuis in ej^regio Tuo opere traditis, sed (|uia tum
temporis Tuum methodum diffcrentialem nondum videram,
ejus quo(jue mcntionem facere non potui, uti debuissem. Misi
etiam jam ante eomplures annos ad Illustris, Praesidem
Vestrum D. Sloane schediasma quodpiam, in (pio <^eneralem
constructioncm liujus aequationis
y = yyx -\- ax^" X
dcdi, quae acquatio ante multum erat agitata, at paucissimis
tantum casibus cxponentis m constructa. Haee io^itur Diss-er-
tatio, si etiamnum praesto esset, simul tanquam specimen
produci posset, coram Societate vestra, quando me pro mendn-o
recipere esset dignatura, quem quidem honorem Tibi Uni Vir
Celeber, deberem. Sed vereor ut Incl3^tae Societati expediat
me Socium eligere, qui ad Academiam nostram tam arete sum
aliigatus, ut meditationes meas qualescunque hie primum pro-
ducerc tenear,
Ut autem ad Theorema, quo summa cujusque Seriei ex ejus
termino dicto generali inveniri potest, rev'ertar, perspicuum
est formulam datam eo majorem esse allaturam utilitatem,
quo ejus plures termini habeantur, summa autem difficile esse
videtui-, eam quousque lubuerit, continuare. Equidem ad
plures quam duodecim terminos non pertetigi, quorum ultimos
non ita pridem demum inveni ; hacc autem expressio se habet
ut sequitur.
Si Seriei cujuscunque terminus primus fuerit A, secundus B,
tertius C, etc. isque cujus index est x sit = X : erit sunnna
hujus progressionis, puta
^ + i)'f C'+etc... +X ^
„ , X dX
Xclx+ - -^ +
1.2 1 .2.3.2dx
d^X df'X
- +
1.2.3.4.5. Gt/u,'^ 1.2.3.4.5.6.7. iSdx'
Zd'X bd'X
+
1 .2.3 ...9. lOc/x'' 1.2.3 ... 11 .Grfic'-'
CORRESPONDENCE WITH EULER ISJJ
1.2.3... 13.210(/a;i^ "*" 1 . 2.3 ... \5 .2dx^ '
3617tZ'5X 43867(^^^Y
1 .2.3 ... 17.30(/.t;'^ 1 . 2 . 3 .. . 19 . 42(/a;'^
1222277cZ^^X
1.2.3 ... 21.110(/a;i-*
etc.
ubi fluxio dx constans est posita.
Haec aiitein expressio parumper luutata etiam ad summam
seriei a tenuino A" in infinituin u&(|ue invcnienJam acconimodari
potest. Hujus vero forinae praeter insigneni facilitatcm, quam
siippoditat ad summas proxime inveniendas, oxiinius est usiis
in veris sunnnis serieruni algebraicarum investigandis, quarum
quidem sumniae absolute exhiberi possunt, ut si quaeratur
summa hujus progressionis potestatum
erit X = x'\ [Xdx = ^x''\ ^ = 12a;^\
J 13 dx
iP X d^'^X
—r- =10.11.12.0;", et ita porro, donee , ,.,
dx^ ' dx^'
una cum sequentibus Terminis =
Hinc igitur resultabit summa quaesita =
x^" .1-12 „ lla-^ 22.Z' 33.r^ 5x^ 691 «
f- 4-.l''^ 1 + J
13 2 07 10 3 2730
quam summam nescio, an ea per ullam aliam methodum tarn
expeditam inveniri queat. Potest autem hac ratione aeque
commode definiri summa hujus progressionis
l+22i + 3-i + 4'-^i + ...+0'2',
quod per alias vias labor insuperabilis videtur.
Sin autem seriei propositae termini alternativi signis + et —
fuerint aft'ecti, tum theorema istud minus commode adhiberi
posset, quia ante binos terminos in unum eolligi oporteret.
Pro hoc igitur serierum genere aliud investigavi Theorema
priori quidem fere simile, quod ita se habet.
Si quaeratur summa hujus seriei
A-B + C-D+...+X,
184 STIRLING'S SCIENTIFIC CORRESPONDENCE
iibi A' t>it tcniiiniis ciijiis exjJOiiL'iis sen index est ./', luiLetque
sigimin vel + vel — proiii ,'■ nuinerus erit vel inipar vel par.
Dico auteiu liujus progressionis siiuimam esse
^ , /X dX cPX
= Const. ± (p +
+
2 1.2.2dx 1 .2.3.4 .2da"''
3d'X \7,rX
1.2,3... G.2dx^ 1 .2.. 3 ... 8.2(^.^7
155d'>X 2073d^^X
+
1 .2.3 ... 10.2(/.^'-' 1 .2.3 ... 12.2(/rt;i'
33227cZ^3X
^ 1.2.3 ... 14. 2(/./,'^
Constantem autem ex uno casu, quo summa est eognita,
determinari oportet.
At si series sunimanda eonnexa sit cum Geomctriea pro-
gressione hoc modo
An + Bii^ + Cn" + . . . + Xn""
turn minus congruo utrumquc praecedentium tlieorematum
adhibeietur. Summa enim coiinnodius invcniotur ex liac
expressionc
^./nX (xdX ^d'^X
Const. + n- (^_^_ J - J ^,^_ 1^,^/ ,. + 1 :2{n^rfdx''
y(F'X Sd'X
1.2.3. (vi - l)*(/,t^ '*' 1 .2.3.4(/t-l)
^-— i — etc. )
t-l)'(/.r'* /
valores autem coefficientium a, /3, y, 8, etc sunt sc(|uentes
oc = n
^ = n^ + n
y — n' -^-^n- -\-n
S ~ Qi* + 1 1 ,v- + 1 1 n~ + n
€ = lV'-\-2Qii' + GG/t'' + 2GH^ + li
etc.
cujus progressionis legem facile inspicies. En igitur tivia liujus
generis TheorematM, (juac singula cortis easibus exiiiiiaiii liabe-
bunt utilitatem ad sunmias serierum indacfandas.
CORRESPONDENCE WITH EULER 185
Quod Jeiiule attinet ad suiiiiuatiout'S liujusmodi serierum,
(juae contineutur in liac
1111
1 + ^. + ^. + ^7. + ^. + ^'tc.
cxistente n nnmero pari eas duplici operatione sum consecutus,
(juarum alteram uti recte conjectus Yir Celeb, dcduxi ex
scrie 1+- h 1 + etc. altera vcro immediate mihi
3" 5" 7"
illius summam praebuit. Priore modo utique summas etiam
hujusmodi serierum I-^j+t ^i+^~ ^^^- existentc n
numero impare detexi, invenique eas se habere, prorsus ac
Tu indicas. Sunt autem summae tam pro paribus quam im-
paribus exponentibus n sequentes
;? ^ 1 1 1 1
-^=1 1 \ etc.
4 3 5 7 \)
p^ 1111,
— = H H . + —. -\ — ^+ etc.
?/ 1111,
— = 1 — — + -T, ; + —. — etc.
32 3' 5' 7- 9"
,/ 1111^
5^/' 1111,
— ^— = 1 r + —r ^+77^ — etc.
1536 3'^ 5^ 7' 9'
«« 1111,
9^=^+3^+5«+r^+9^ + ^^^-
61// 1111,
— = 1 ^H — ^ ?-l — ^ — etc.
194320 3^ 5' 7' 9^
17p8 1 1 1 1 „
'-— r= 1 + — + ^- + — . H ; + tVC
161280 3^ 5-^ 7^ 9«
etc.
quae series omnes continentur in una hac generali :
i+(-ir+(+ir+(-7r+(+if+etc.
existente n numero integro. Si enim n est numerus par, turn
186 STIRLING'S SCIENTIFIC CORRESPONDENCE
omnes terinini luibijltunt siyimm + ; sin auteni n sit iinpar,
turn signa scsc altcrnatiiii insequcntur.
Oinnes auteiii has suininas dcrivavi ex liae ae(|iiatione
infinita ;
^ , s s^ s^ ,
0=1 + + etc.
I .a \ .2.3.(1 } .2. 3. 4. 5. a
qua relatio inter arcum ,s ejusque siniim <i expriniitur in
circulo cujus radius est 1. Quoniani igitur cideni sinui (i
iimumerabiles areus s respondent, necesse est. Si s consideretur
tan(juam radix istius aequationis, earn habituram esse infinites
valores, eos(]ue oninos ex circuli indole cognitos. Sint ergo
A, B, G, 1), etc. omnes illi arcns, (|uoruni idem est sinus a erit
ex natura aequationum
1 + + etc.
1 . a 1 . 2 . 3 . a 1 . 2 . 3 . 4 . 5 . a
= (>'1)('-b)C-5)*-
Posita nunc ista fractionum serie ex omnibus illis arcubus
formata —3 -7-, » -, » -^r etc. perspicuum est suillam banc f'rac-
A B (J D ^ ^
tionum aequari coefficienti ipsius — s qui est = -; seu fore
- = + T, + 7^, + 7; + etc. Simili modo summa factorii ex
a A B U V
binis fractionibus aequatur coefficienti ipsius a^ qui est = 0,
unde erit
1/1 1 1 . \^ 1 / 1 1 1 . \
' = 2 (Z + 5 + C + 'W - 2(^-^ + i^^ + t- + '^V'
1 ' 1 1,1,^
Porro summa factoru ex ternis fractionibus aecpialis esse
1
3
6(t
debet coefficienti ipsius — s^, qui est = — r-' undo deducitur
summa cultoru illarum fractionu,
1111,11
2. + 5^+(7^ + 2}3 + ^tc = ^--;
CORRESPONDENCE WITH EULER 187
atque ita procedendo Miinniac reperientur oiuniuin serierum
m hac <^enerali T7i + /jy, + 7*7^ + Jul + ^^^- compreliensaruin
dummodo pro n siimatur numerus integer affirmativus. Si
nunc pro sinu indelinito it ponatur sinus totus 1, illae ipsae
oriuntur series quas Tecum cunununicavi. In istis autem
summis notari meretur insignis afKnitas inter coefticientes
numericos haruni suIITaru, atque terminos superioris progres-
sionis, quani priniuui ad series quascun(|ue sumniantlas dedi,
nempe liujus
V 7 -Y (IX
A ax + H etc.
1.2 1 .2.3.2(/a;
Quo autem haec affinitas clarius pcrspiciatur, summas ipsas
congruo modo expressas repraesentare visum est.
2M ., 1111^
I^:^2^^"=^+2^^+3^^+4^^+5^^+^^^-
1111.
^ ='+ 2^^+31 + ^^ + ^^ + ^t^-
r , 1111.
s , 1 1 1 1
?^^-l+.7s+3-. + ^4^+p + etc.
1.2.3.'.:il.6 ^'" = ' + ^'" + ^" + ^ + o^> + ^^^•
2^691 ^, _ 1 1 J_ 1_
1.2.3... 13.210^' " - 1 + ^. + gl^ + ^2 + 5!^ + etc.
21^35 ,, 1111^
-,l^* = 1 + ^4 + :7u + 7r4 + M4 + etc.
2-'
.1
1 .
.2.3.
2'\ 1
4.
5.6
1
.2.3
.4.5
2".
.6.
3
7.G
1 .
2.3.
2^
.. 9
, 5
. 10
1 . 2 . 3 ... 1 5 . 2
2^^3017 ,, 1111,
1.2.3...17.30 ^^ =^+^'+i^+4l^' + ?^^ + ^^^-
2^". 43867 ,_ , 1 1 1 1 ^
/P''=^+^s+ ^+ in+ 77^ + etc.
1.2.3.... 19.42^
2''\ 3222277 „„ , 1 1 1 1 ^
r7Yy^2>"= 1+^0+ 3.0 + ^.+ ^ + etc.
1.2.3. ... 2
etc
188 STIRLING'S SCIENTIFIC CORRESPONDENCE
Hac scilicet convenientia aiiiiuadvcrsa mihi iiltcrius progredi
licuit, quaiii si niethodo genuiiia inveniciidi coefficientes
potestatu ipsiiis p, usus fuissein quippe qua labor niniis
cvaderet operosus. Quamobrem non dul)ito, quin nexu hoc
mirabili penitius cognito (mihi euim adluic sola constat obser-
vatione) praeclara adjiimenta ad Analyseos proinotioneni sose
sint proditura. Tu forte Vir Celeb, non difticultcr ncxum
hunc ex ipsa rei natura derivabis.
Dum haec scribo, accipio a Cel. Nicolao Bernoulli Prof.
Juris Basiliensi et Membro Societatis Vestrae singularom
deuionstrationem suniniae huius seriei \ ^ — o H — r, H — :, + etc.
J 3- 5- 7^
(juam deducit ex suuiiua hujus notae 1— ^ + i — y+ etc. illam
considerans tanquam hujus quadratum niinutum duplis factis
binoru terminorum. Haec autem dupla facta seorsim con-
templans multiferiam transforniat, tandenique ad seriem
quandam regularem perducit, quam analytice ostendit pariter
a Circuli quadratura pendere. Sed hac niethodo certe Viro
Acutissimo non licuisset ad sunnuas altiorum potestatum
pertingere.
Eodem incommodo quoque laborat alia quaedam methodus
mea, qua directe per solam analysin hujus seriei sunniiani
111,.. -. ,, XM-
1+ -r, A — r, + -S + etc. inveni, ex riua pariter nullam utili-
tatem ad sequentes series suinmandas sum consecutus. Haec
autem methodus ita se habet : Fluentem hujus fluxionis
—z 5 qua arcus circuli cxprimitur cuius sinus est = ic
V{l-a:x) ^ ^ •'
existente sinu toto = 1, inultiplico per ipsam fluxionem
^-5 quo prodeat facti Hucns = ^ss, posito .s- pro arcu
V{l—xx)
illo cujus sinus est = x Si ergo post summationem peractam
ponatur x = I, fiet s = -■> denotante p 'i'^ 1 ratifmem peri-
pheriac ad diametrum ; ita ut hoc casu hal>eatur -^ • Fluens
. . X . .
autem ipsius — — , per seriem est
' V{\—xx)^
1 , 1.3 . 1.3.5 „
= X + .» ■' H X' H X' + etc.
2.3 2.4.5 2.4.6.7
CORRESPONDENCE WITH EULER 189
Ducautur nunc sinmili termini in tluxionem -—- — — - efc
" v(i — XX)
suniantur tiuentes ita nt tiant — posito x — 0, turn vero
= 1_ J{\-xx) = 1,
ponatur x—\. Ita rcpcrietiir
posito X — \.
Siniili niodo erit - — ~
'Jil-XX)
1
s'{\-xx) 3.3
1.3 r xJ'x 1
''*'*1"'' 2. 4. 5 J ^{\~xx)~ 5.5
et ita porro, adoo ut tandem obtineatur
ir 111,
— = 1 + -", + - 7, + - ., + etc.
8 ^ 3^ 5- 7-
Sed huic argumento jam ninnum sum innnoratus, ijuocirca
Te ro<^o Vir Celeb, ut quae Ipse hac de re es meditatus, mecum
benevole communicare veb's.
Incidi alicpiando in banc expressionem notatu .satis (Hgnam :
3.5. 7. 11 . 13. 17. 19.23.29.31 . 37.41
4.4.8. 12. 12. 16.20.24.28.32. 36 . 40
cujus numeratores sunt omnes nunieri primi naturaU ordine
sese insequentes, denominatores vero sunt nuuieri pariter
pares unitate distantes a numeratoribus. Hujus vero ex-
pressionis valorem esse aream circuli cujus diameter est = 1,
demonstrare possum. Quamobrem baec expressio aequalis
erit huic Wallisianae
2.4.4.6.6.8.8. 10. 10 etc
3.3.5.5.7.7.9 . 9 .11 etc.
Ut autem novi quiddam Tibi Vir (Jeleb. perscribam Tuoque
acutissimo subjiciam judicio, communicabo quaedam proble-
mata, quae inter Viros Celeberrimos Bernoullios et me al)
alic{U0 tempore sunt versata. Proponebatur autem mihi inter
alia problemata hoc, ut inter omnes curvas iisdem terminis con-
tentas investigarem eam, in qua r"^s haberet valorem
minimum, denotante s curvae arcum, et r radium curvaturae,
quod problema ope consuetaru methodorum, quales Bernoulii,
100 STIRLING'S SCIENTIFIC CORRESPONDENCE
Heniuiniius ct Taylorus Vcstcr dcdcrc, resolvi iioii potest,
(jiiia in r fluxiones secundae ingrediuntur. Invcni autem
jam ante incthodum universaleiii omnia huinsuiodi prohlomata
solvendi, quae etiam ad fiuxioncs cujusqiie ordinis uxtunditur,
cujus ope pro cnrva quaesita sequcntcm dedi acquationem
c6"*./j + 6'"2/ = ("* + 1) >'"'« i" <i^^'^ •'■ *'t y coordinatas ortho-
gonales hujus curvae denotant. Hinc autem sequitur casu,
quo ';n = 1, cycloidem quacstioni satisfacere.
Deinde etiam (^uaerebatur inter omnes tantum curvas
cjusdem longitudinis, quae per duo data puncta duci possunt
ea, in qua 7'"*s esset minimum ; hancque curvam deprehendi
ista aequatione indicari a'"a; + />'"?/ + c'"s = (m+ 1) r'"b'.
Praeterea quaerebantur etiam oscillationes sevi vibrationes
laminae elasticae parieti firmo altero termino infixae, cui
quaestioni ita satisfeci, ut primo curvam, quam lamina inter
vibrandum induit, determinarem, atque secundo longitudinem
penduli simplicis isochroni definirem, ([uod aequalibus tem-
poribus oscillationes suas absolvat ; hinc enim intelligitur (]Uot
vibrationes data lamina dato tempore sit absolutui'a.
Ego vero contra inter alia problema istud proposui, ut
inveniantur super dato axe duae curvae algebraicae non
rectificabiles, sed (juarum rectificatio a datae curvae quadratura
pendeat, (|uac tamen arcuum eidem abscissae respondentium
summam habeant ubique rectificabilem ; cujus problematis
difKcillimi visi, neque a Bernoullio soluti, sequentem adeptus
sum solutionem. Posita abscissa utrique curvae communi = x ;
sit alterius curvae applicata = y ; alterius vero = z. Assumatur
nova variabilis u ex qua et constantibus variabiles x, y Gt z
■
definiri debent, atque exprimat Vii illam quadraturam, a qua
rectificatio utriusque curvae pcndere debet; sintque p et q
quantitates (juaecunque algebraicae ex u et constantibus
compositae. Quibus pro lubitu sumtis fiat
V{\ +2^p) + V{l+qq) = r: V{1 ^-X)p) - V{\+ qq) = s,
tum quaerantur scquentes valores
P P P
CORRESPONDENCE WITH EULER 191
item i> = -; E=^ cb F=^'
A A b
Ex liis quantitatibus porro forinentur istac
V ^'''- n ^' f p ^
i = ^ U = - ■ et U = -r '
F D A
Ex his deniqiic valoribus, (|ui omncs erunt algebraici sumta
coininiini abscissa = — 3
j)
fiat y=2^-R atque z = ^lAzi^ +Q;
V V
liacque ratione, cum p et q sint quantitatcs arbitrariae pro-
blemati infinitis modis satisfieri poterit. Erunt enim ambae
curvae algeljraicae, atque utriusque rectificatio pendcbit a
fluente hujus fluxioiiis Fa. Summa vero amboru arcuum
algebraice exprimi poterit. Est enim summa arcuum
differentia vero eorum est
= sx - OR -^EQ-FP+ f Fa .
Detexi autem pro resolutione hujusmodi problematum pecu-
liarem methodum, (juam Analysin infinitoru indeterminatam
appellavi, atque jam maximam partem in singulari tractatu
exposui. At tam longam epistolam scribendo vereor ne
patientiam Tviam nimis fatigem : quamobrem rogo, ut pro-
lixitati meae veniam des, eamque tribuas summae Tui existi-
mationi, quam jamdudum concepi. Vale Vir Celebcrrime,
meque uti coepisti amicitia Tua dignari perge.
dabam Petropoli
ad d. 27 Julii 1738.
XII
FOLKES AND STIRLING
(1)
Folkes to Stirling, 1747
Dear Sir
After so many years absence I am proud of an oppor-
tunity of assuring you of my most sincere respect and good
wishes for your prosperity and happiness of all t-orts.
I received the day before yesterday of a Gentleman just
arrived from Berlin, the enclosed Diploma which I am desired
to convey to you with the best respects of the Royal Academy
of Sciences of Prussia, and more particularly of M'" de
Maupertuis the President and IVP' de Formey the Secretary.
M"" Mitchell going your way I put it into his hands for you
and congratulate you Sir upon this mark of the esteem of
that Royal Academy upon their new establishment under
their present President. Our old ffriend M"" IMontagu is well
and we often talk of you together, and our old Master
de Moivro whom we dined with the other day on the occasion
of his compleating his eightieth year. I remain with the
truest esteem and affection
Dear Sir
Your most obedient humble servant
London June 10. 1747 M. ]<"i'OLKks. Pr. R.S.
member of the Royal Academies
of Sciences of Paris and Berlin,
and of the Society of
M"^ Stirling
Edinburgh
NOTES
UPON THE CORRESPONDENCE
MACLAURIN (1698-1746), F.R.8. 1719
Colin Maclaiiiin was born at Kihnodan in Argyleshire,
and attended Glasgow University. He became Professor ot"
^Mathematics at Aberdeen in 1717, and in 1725 was appointed
to the chair of Mathematics in Edinburgh University. He
died in 1746.
His published works are Geometrla Organica, 1720;
Treatise of Fluxions, 1742; Treatise of Algebra, 1748, and an
Accoant of Xewtoit's Fidlosophical Discoveries, 1748.
His Treatise of Fluxions, which made a suitable reply to
the attack by Berkeley, also gives an account of his own
important researches in the Theory of Attraction.
T/ie Dispute betiveen 3Iaclaurin and Camphell.
Letters I. 1 to I. 7 are mainly concerned with a dispute
between Colin Maclaurin and George Campbell, a pretty
full account of which is given in Cantor's GesvJiicJde der
Matliematik.
But the correspondence before us gives a good deal of fresh
information, as well as practically the only details known
regarding George Campbell, about whom the Histories of the
Campbell Clan are silent, in spite of the fact that he was
a Fellow of the Royal Society, being elected in 1730. From
Letter I. 1, it would appear that when Maclaurin, glad to leave
Aberdeen University owing to the friction arising from his
absence in France, and conse(|uent neglect of his professorial
duties, accepted the succession to Professor Gregory in the
Chair at Edinburgh, he had in a sense stood in the way of
Campbell for promotion to the same office. Feeling this, he
had done his best to advance Campbell's interests otherwise
and had corresponded to this intent with Stirling, who
194 NOTES UPON THE CORRESPONDENCE
suggested that Campbell iniglit gain a livelihood in Loudon
by teaching. Some of Campbell's papers were sent to
London. One, at least, was read before the Royal Society,
and, through the intluence of that erratic genius, Sir A. Cuming,
ordered to be printed in the Transactions. Stirling himself
read the paper in proof for the Society. When the paper
appeared Maclaurin was much perturbed to find that it
contained some theorems he had himself under discussion
as a continuation of his own on the Impossible Roots of an
Equation.
He wrote letters to Folkes explaining his position, and
o'ivinoj fresh additional theorems. But the matter did not
end here. For Campbell in a jealous mood wrote and published
an attack upon Maclaurin, who found himself compelled to
make a similar public defence. An attempt was also made
to embroil Stirling with Maclaurin, fortunately without
success. Practically nothing further is known regarding
George Campbell (who is not to be confused with Colin
Campbell, F.R.S., of the Jamaica Experiment, mentioned in
Letter I. 10). The names of G. Campbell and Sii* A. Cuming
are given in the list of subscribers to the MUcellauea Analyttca
de Ser'iehus of De Moivre (1730).
Xewtou's Theorem regarding the nature of the Root3
of an Algebraic Equation.
Neither Campbell nor Maclaurin attained the object aimed
at, — to furnish a demonstration of Newton's Theorem^ stated
without proof in the Avitliinetica U nlveraaH^.
Other as eminent mathematicians were to try and fail, and
it was not until the middle of the nineteenth century that
a solution was furnished by Sylvester, who also gave a generali-
zation. {Phil. Tran,s. 1864: Phil. Mag. 18GG.)
Newton's Theoi'em may be stated thus (vide Todhunter's
T/ieorg of E(inailont>).
Consider the equation
/{x) = a,x^' + ^^C\a,x^^-' + ... + ,^C,.a,.x"-'- + ... I a„ = 0.
Form the two rows (^f (pianiities
A^ A, A.,...A,^
NOTES UPON THE CORRESPONDENCE 195
where
Call «,. «^.+i
an associated couple of siiccet-sions. In such a couple tlic
signs of «,. and (t,.+i may be alike and represent a Permanence,
P ; or unlike, and represent a Variation V.
Similarly for A^. and A^.^^.
An associated couple may thus give rise to
( 1 ) a double Permanence,
(2) a Permanence- Variation,
(3) a Variation-Permanence,
(4) a double Variation.
Then we have New^ton's Rule : —
The number of double Permanences in the series of couples
is a superior limit of the number of negative roots ; and
the number of ^'ariation-Permanences is an upper limit of
the positive roots ; to that the number of Permanences in the
Series
A, A^...A^,
is an upper limit to the number of the real roots of /(.i) = 0.
Sylvester (v. Cdleded Works) was the first to fui'ni&h
a demonstration of Newton's Theorem, and he gave the
following generalization.
Write /(a; + A) in the form
and form the table
f'o "i "n
Jy Ay A|^
(where ^4^, ... A^^ are as before).
Denote the numljer of double Permanences arit-ing therefrom
by PP (A).
Similarly denote by PP(n) the number corresponding to
/(.r + /z).
Then if /i > X, PPl/z)— PP(A) is either equal to the
o2
196 NOTES UPON THE COREESPONDENCE
number of ix'al roots ot" f(x) = butwet'ii jx ;ui(l A, or exceeds
it by an even number.
Letter I. 1.
On p. 19 of his Defence (against Campbell) Maclaurin
makes the statement : —
'In a Treall^e of Algebra, wliicli T composed in the Year
172G, and which, &ince that Time, lias been very publick in
this Place, after giving the same Demonstration of the Doctrine
of the LtinLtt<, as is now published in my second Letter, I add
in Article 50 these Words, iVc'
Maclaurin appears to be referring here to a course of
lectures to his students.
Maclaurin's Algehra did not appear until 174 8, after his
death. It was in English, but contained an important appendix
in Latin on the Properties of Curves. De Moivre's book
referred to is his Miscellauea Aiudgtica, 1730. In 1738
appeared the second edition of his Doctrine of Chances, also
referred to in the letters.
Letter 1 3.
This letter, dated by Maclaurin as Februar}' II"', 1728,
should have been dated as February ll*'"', 172f. i.e. 1728 Old
Style, or 1729 New Style.
Stirling makes this correction in I. G, which consists of
extracts from letters by Maclaurin. Lentil this had been
noted, the first three letters seemed hopelessly confused.
Maclaurin shows the same slovenliness in the important note
of his, I. 10, attached to the letter from Maupertuis to
Bradley.
Letter I. C.
Letter I. contains only extracts from letters of ]\iaelaurin,
including one date<l October 22, 1 728, whicli is no longer
in the Stirling collection.
Ijetter 1. 7.
In the spring of 1921 I had the good fortune to obtain
a copy of Maclaurin's reply to Campbell.
NO'l^ES T^PON THE CORRE>SPONDENCE 11)7
Jt is entitled: —
' A Defence ct" the Letter i)ultlislic(I in the Phihjsopliical
Transactions for jMareh and April 1729, concerning the
Impossible Roots of Equations : in a Letter from the
Author to a Friend at London.
Qui admonent aniiee, docemli sunt : <|ui inimice infeetantur,
repellendi.
Cicero '
The name of the ' Friend ' is not oiven. The ' Defence '
consists of twenty small quarto pao'es, and contains numerous
extracts from the letters to Stirling; and towards the end
Campbell's statements regarding Maclaurin's theorems are
refuted.
Campbell is generally referred to as ' the Autlior of the
Remarks ' (on Maclaurin's Second Letter on impossil)le
roots) : thougli also as ' the Remarker '•
Maclaurin gives the extract from tlie letter of October 1728
(cf. I. 6), and adds: —
' See the 2d and 3d Examples of the Eighth Fro'positlon of tlio
Lineae tertii Ordiuis Xewtonianae.'
There is also the following passage containing an extract
from a letter by Stirling, not otherwise known : —
'I had an Answer from this Gentleman in March, from
which, with his Leave, I have transcribed the following-
Article :
" I shewed your Letter (says he) to Mr Ma(hni, and we were
both well satisfied that you had carried the IMatter to the greatest
Height, as plainly appears b}^ what 3'ou have said in your
Letter. But it is indeed a Misfortune,(,that you was so long
in giving us the Second Part, after you had delivered some
of your Principles in the First: — Since you have published
Part of your Paper before Mr C — ^—11, and now liavc the rest
in such Readiness, I think 3'ou have it in j^our Power to do
j^ourself Justice more than any Body else can. I mean by
a speedy Publication of the remaining Part : For I am sure,
if 3'OU do that, there is no Mathematician, but who must
needs see, That it is your own Invention, after the Result
of a great Deal of Study that way."
I received this Letter in March, and, in consequence of tliis
198 NOTES UPON THE CORRESPONDENCE
kind A(lvic(\ resolved to send n]i my Second Paper as i-oon as
possible."
Maclaurin makes it clear that he had not intended his First
Letter to Folkes to he published. It was printed without his
knowledge. Had lie known in time, he would liave deferred
its publication until he had more fully investigated additional
theorems Avhich he had on the same subject; and he gives an
extract from a letter from Folkes in corrolioration of his
statement.
Letter I. 8.
Letter I. 8 is reproduced because of its reference to an
office (in the Roj'al Society) for which Stirling had been
thought fit.
Letter I. 9.
Letter I. 9 announces that Maelaurin has started to write
his Treatise of Flnxi<ms. His conscientious reference to
original authorities has been noted b}" Reitf {GeschicJde der
UnendlicJien Relheii). The earlier proof-sheets of the Treatise,
at least, passed through Stirling's hands.
These facts bear interesting evidence regarding the Ealer-
Maclaurin Summation Formidn, to which I have to return in
connexion with the correspondence between Stirling and
Euler in Letters XL
Simp.son, referred to by Maclaurin, is doubtless his old
teacher, Robert Simson, of Glasgow Univorsit}'.
Letter I. 10.
Letter I. 10, which is a mere scrawl written on the outside
of the copy of the letter from Maupertuis to Bradley, is of
interest in the history of the Royal Society of Edinbui-gh,
and is to be associated with the two letters of Maelaurin
published in the Scots Magazine for June, 1804.
The date of the letter of Maupertuis shows that Maclaurin
should have given Feb. 4*', 173| as the date of his own.
Maclaurin was more successful with Stirling than with
R. Simson, who refused to become a member after IMaclaurin
had got him nominated. (Scots Mag.)
Bradley's translation of the letter of Maupertuis is repro-
duced in the Works and Correspondence of Bradley, 1832
NOTES UPON THK CORRESPONDENCE 199
(Ki^aud). The original Fi-ench Jetter i.s preserved in one of
the hotter hooks of the Royal Society of London.
FuundatioR of the Philosophical Society of Edinhurgh.
Letter L 10 confirms the date of foundation as 1737 (v.
Forbes's Hidory of the Royal Society of Edinburgh, in General
Index Trans. R.S.E. published 1890).
But at the date of this letter I. 10 the Society \vas not
complete in numbers, for Stirling \Yas not yet a member.
By 1739 the Society had outrun its original bounds, having
forty-seven members M'hose names are given (p. 26 of Gen.
Index Trans. R S.E.).
More or less informal meetings were held in 1 737. Maclaurin
and Dr. Plummer, Piofessor of Chemistry in the University,
were the Secretaries. The Rebellion of 1745 seriously affected
the activity of the Society, and Maclaurin's death in 1746 was
also a severe blow.
The papers read before the Society had been in Maclaurin's
hands, but only some of these were found. Three volumes of
Easays and Observations, Physical and Literary (dated 1754,
17nG, 1771), were published. The papers in Vol. I are not in
chronological order, but those by Plummer are fortunately
dated, the first bearing the date January 3, 1738. Dr. Pringle,
afterwards President of the Royal Society of London, followed
in Feljruary. Then it was Maclaurin's turn in March, when
he gave two papers, one being on the Figure of the Earth
(Scots Magazine).
These two papers are not printetl in the Essays, &c. But
among the Maclaurin MSS. preserved in Aberdeen University
there is one entitled ' An Essay on the Figure of the Earth '.
On the foundation of the Royal Society of Edinburgh in
1783 the members of the Philosophical Society were assumed
as Fellows. Maclaurin's son John (Lord Dreghorn) is one of
those mentioned in the original charter of the Royal Society.
Lttter of Mawpertuis.
The letter of Maupertuis must have given lively satisfaction
to Maclaurin and Stirling. Newton had assumed as a postulate
that the figure of the Earth is approximatel}?" that of an
oblate spheroid, flatter at the poles than at the Equator. The
m.) NOTES T^^OX THE COPIRESPOXDEXC'E
Cassinis, arguing from iiR'asui'eiiK'iit.s oi' tliu arc uL' a Mcriilian
in France, maintained that the figure was that of a prolate
splieroid. There were thus two hostile camps, the X'cwtonians
and the Cassinians.
Pol-
Pole
A
Y
titwroK
CASbTNI
The French expedition to Lapland (173G-7) with Mauper-
tuis as leader, and Clairaut as one oF the party, conclusively
established the accuracy of Newton's hypothesis. In the
words of Voltaire, Maupertuis had 'aplati les Poles et les
Cassinis '.
Both Stirling and Maclaurin made important contributions
to the subject, and the rest of the letters preserved as passing
betW'Cen them refer mainly to their researches on Attraction
and on the Figure of the Eartli.
Readers who are interested cannot do better than consult
Todhunter's History of the Theory of Attraction and of the
Figure of the Earth for full details. The letters, however,
clear up some difficulties that were not alwa3^s correctly
explained by Todhunter.
Letter I. 11.
The Dean, near Edinburgh, jMaclaurin's new address, now
forms a residential suburb of Edinburgh.
De Moivre's book is doubtless the second edition of tlie
Doctrine of Chances (1738).
letter T. 13.
Tlie remark made by Stirling towards the conclusion that
' the gravitation of the particle to the wdiole spheroid will be
found to depend on tlie quadrature of the circle ' seems to
have given Maclaurin a good deal of trouble (cf. I. 14).
NOTKS T^'OX THE (X:)RRESPONDENCE 201
]\Ia('l;uiiiirs rct'crt'iKT to it in his Fluxions, § 647, as due to
Stirling-, Mas iiicxplicaltlc to 'rodliunter, as Stii'lin<i^ never
published his theorem. But Todliiniters conjecture {Hlttovij,
vol. i, p. 139) that Maclaurin ma}^ have inadvertently written
Stirling for Simpson is of course quite a mistake.
Letter I. 15.
Compare the correspondence with Machin IX, Clairaut X,
and Euler XL
Letter I. 16.
This letter, dated 1740, furnishes ample justitieation of
Todhunter's contention that the researches of Maclaurin, ' the
creator of the theory of the attraction of ellipsoids', are quite
independent of those given by T. Simpson in his Mathematical
Diskertatioiis (1 74.3). Simpson lays claim to priority in certain
theorems of the Fluxions on the ground that these given by
himself were read before the Royal Society in 1741.
The Treatise of Fluxions so near completion in 1740 was
not published until 1742.
II
CUMING
Sir A. Cuming (1690?-17 75) was the only son of Sir Alex-
ander Cuming, M.P,, the first baronet of Culter, Aberdeen.
Cuming went to the Scotch bar, but gave up his profession
on receiving a pension. In 1720 he became a Fellow of the
Royal Society. Though no mathematical writings of his are
known, he seems to have been possessed of mathematical
ability. He was on friendly terms with De J\Ioivre and
Stirling, both of whom acknowledge their indebtedness to
him for valuable suggestions. At Aberdeen there is preserved
a short letter (Nov. 3, 1744) from him to Maclaurin, in
which he shows his interest in the controvers}^ regarding
Fluxions.
In his introduction to the Methodns Differential is, Stirling
speaks of him as ■ Spectatissimus Vir '. Being a friend of
Campbell he had a share in the dispute between Maclaurin
and Campbell.
In 1729-30 he was in the American Colonies, visited the
Cherokees, and became one of their chiefs. On his return to
202 NOTES UPON THE CORRESPONDENT'E
England with soiiii- oi" the duel's lie was iiistiuineiital in
sirrano-inor n treaty for liis tribe.
Later he fell into poverty, and was confined in the Fleet
prison from 1737 to 1765, losing; his fellowship in the Pvo^'al
Society for neglecting- to pay his annual fee. In 1766 he
obtained admission to the Charterhouse and died tluM'e in 1 7 7.'.
Ill
CRAMER AND STIRLING
Gabriel Cramer was born in 1 704 in Geneva, where his
father practised medicine. In 1724 he was, conjointly with
Calandrini, entrusted with the instruction in Mathematics at
the University of Geneva. In 172 7 he started on a two
years' tour, visiting Bale, where he studied under John Ber-
noulli, and England, where he became acquainted with Stirling
and De Moivre, and returning by Paris. He became F.R.S. in
1748, He died in 1752. He is best known through his
Introduction d I' Analyse dcs lignes courhes algebriques. He
also edited the works of James and John Bernoulli.
Letter IIL 1.
It is unfortunate for us that Cramer did not discover before
1732 that he wrote 'mi Anglois au&si barbare '.
Regarding the history of the Probability Problem in III. 1,
see Todhunter's History of the Theory of ProIxdnlUy (p. 84).
De Moivre gives a much simpler solution in the Miscellanea
Analytica (1730).
Letter IIL 2.
Compare Lettei' IV. 2 (Bernoulli).
Letter III. 3.
In this letter of introduction Cramer in the address describes
Stirling as L.A.]\T. I do not know what these letters signify.
Letter III. 8.
Letter III. 8 contains valuable information regarding the
manner in which Stirling wiote his Methodiis Diferentialis.
The blank made for the formula given by De Moivre was
never filled u[) : but the formula in (|uestion is of course easily
NOTES UPON THE CORRESPONDENCE 203
obtained i'roni tlio Suj^plement to the M i scellanea Auahjtica
of Do Moivre. We have also tlie important information that
this Supplement appeared after tlie pnlilieation of Stirling's
own Treatise.
Letter III. 10.
One will note Cramer's difficulties with the graph of
2/^"= 1 +.'-'; also his determination of (1 +a')^/-''' as x tends
to zero.
It is a pity there is no indication of Stirling's determination
of this limit.
Stirling' a Series
and the claims to priority of De Moivre and Stirling.
In the Bihlioteca Matliematica for 1904 (p. 207) Enestrom
makes the following statement.
' Im Anschluss an den Bericht liber Stirling's Formel fiir
die Summe einer Anzahl von Logarithmen ware es angezeigt
mitzuteilen dass die bekannte Formel dieser Art die man jetzt
ziemlich allgemein gewohnt ist als die Stirlingsche Formel zu
bezeichnen, namlich
log (1 . 2 . 3 . . . ./•) = -i log 27r + {X + 4) log x
.1 .1-2 ,
-x + A,,- + J., —^ + eVc,
" X X""
zuerst von Moivre im Anhange an der Miec. analytica (17.30)
angegeben und hergeleitet wurde. Moivre berichtet selbst
dass Stirling ihm brieflich die Formel
log (1 . 2 ... a-) = i log 273- + (a; + i) log (a; + I )
2.12(a; + i) 8.360(« + i)=^
mitgeteilt liatte, und dass er selbst dadurch angeregt wurde
die neue Formel auf eineni ganz anderen Wege auf zufinden.'
Inasmuch as the only change effected by De Moivre is to give
the expansion of log {x !j in descending powers of x instead of
descending powers of x + ^, which has no special advantage
when X is large, the priorit}^ of De Moivre to this important
formula seems to me to rest on very slender foundations,
unless we are to infer from Enestrom's reference to the
204 NOTES UPON THE CORRESPONDENCE
Sujiidemciit Id tlu' Mli<rcflaiie(i Aiialijllcc thai l)c Moix to
pulilisliL'd his result prioi- to Stirliui;-.
Enestroins stateiiR-nt lias had considcraltle influence with
subsequent writers (e.^-. Czuher and Le Roux, Cuhul des
Probabllites; Selivano\- and Andoj'er, Calcul des Differences
Finies, in the well-known Eacyc. des ^Sriences Math. ; Czuber,
WaJir. Recliuvivi, 1903, s. 19), wlio refer for proof to tlie Siipi>.
Misc. Anal, of De Moivre.
Aeainst these we may put De Moivre's own statement in
the third edition of the Doctrine of Chances (1756), oiven in
tlie Appendix, p. 334, where, after giving a table of ^'alue.s for
log {x !) for numerical \ alues of x he goes on to add : —
'If we would examine these numbers, or continue the Table
farther on, we have that excellent Rule communicated to the
Author by jMr James IStirliiig, published in his Supplement to
the Ulitcellanea Analytica, and by Mr Stirling himself in his
Methodus Different ialis, Prop. XXVIII.
'Let z — ^ be the last term of any Series of the natural
Numbers 1, 2, 3, 4, 5, ...:-|; (^ = -43429448190325 the
reciprocal of Neper's Logarithm of 10: Then three or four
terms of this Series
a 7a 31 rt
z ogz-az- 2 12s ^ 8 . 3603^ ~ 32 . 1260^5
127ft
H _- — (vc
128. 1680-J
added to 0-39908993 il 79, c^'C. which is half the Logarithm of
a Circumference whose Radius is Unity, will be the Sum of
the Logarithms of the given Series ; or the Logarithm of the
Product
1x2x3x4x5... xz~\ &c.'
There is thus no doubt in De Moivre's mind that the dis-
covery of the theorem in question is not due to himself but to
his friend Stirling.
Date of ^u/rplement to the Miscellanea Aaahjtiai.
At first sight the Supplement appears to bear the date
Jan. 7, 17|§. Li such case it would almost certain]}' be
anterior in [)ublication to Stirling's book.
NOTES UPON THE CORRESPONDENX'E 205
Now this [supposition is quite erroneous. The Miscellanea
Aaalytica, as originally published, bears the above date, and
contains no supplement. (The first copy I consulted has no
supplement.) An examination of a copy with the Supple'meiit
shows two lists of Errata, the first after p. 250, and the
f^econd after p. 22 of the Su^yjylemeiit, the latter list contain-
ing Errata observed by De Moivre and his friends ' post editum
libruni meum '.
The letter III. 8 of Stirling puts it beyond a doubt that the
Sa2}plemeiit had not appeared at the time he wrote (September
1730), so that its appearance was posterior to the publication
of Stirling's Methodas Differe^dlalls.
We have thus the following events in chronological order.
De Moivre publishes the Mite. Anal, early in 1730. His
friend Stirling points out to him the poor approximation
he gives for log [x !) when x is large and sends him a formula
of much greater accuracy. Stirling publishes his Meth. Diff.
containing the famous Stirling Series. In the meantime
De Moivre busies himself with Stirling's formula, and obtains
it in a slightly different form but by an entirely different
process: and he publi; lies his result as a Supijlement to
his book and bound with it, but without changing the date
of his book. He explains in his own garrulous way, which
makes the reading of his works so attractive nowadays, how
he had very nearly got at Stirling's Theorem before he had
heard from Stirling.
Will any scholar be bold enough to assert that the theorem
is due to De Moivre in virtue of this latter statement, pub-
lished after Stirling had given the theorem in all its generality
in the Meth. Dijf. 1 You may speak of De Moivre's form
of Stirling's Theorem if you please, but the merit of discover-
ing a theorem of the kind eecms to rest indisputably with
Stirling.
IV
N. BERNOULLI AND STIRLING
Nicholas Bernoulli was born in 1687 at Bale in Switzerland,
his father being a merchant in tliat town. His two uncles,
James Bernoulli (1654-1705) and John Bernoulli (1607-1748),
were both noted mathematicians.
20G NOTES UPON THE CORRESPONDENCE
He .studied Hrst under the t\)nuei' at f!ale University, and
then under the latter at Gri3nin»;"en, returning- with his uneh'
John to Bale in 1 705.
He devoted hiniselt' to the study of mathematics and law.
He became F.R.S. in 171."}. (_)n the recommendation ot"
Leibniz, he was in 1716 ap[)ointed Professor of Mathematics
at Padua, resignini^,- in 171'.) and returning to Bale. In 1722
he was elected to the chair of Logic, and in 1731 to the chair
of Law in Bale. He died in 1759.
His cousins, the sons of Jolnij. Nicholas 1G95-1726; Daniel
1700-82; and John^ 1710-no were also noted mathematicians.
Two of the three sons of Jolin^, viz. John, and James, also
showed mathematical ability, so that we have here a remark-
able instance of three generations of distinguished mathe-
maticians in one family. Venice was a favourite resort of
the Beruoullis about the time that Stirling resided there.
Letter IV. 1.
Letter IV. 1 is the earliest of the letters preserved in the
mathematical correspondence of Stirling. When the actiuain-
tance betwc'ii Bernoulli and Stirling began is unknown, but
Bei'noulli in the course of his travels spent some time in
Oxford in 1712, when Stirling was still an undergraduate.
One is strongly tempted to suggest that it was at Oxford
that they first met, for the disparity in their years was not
very great, while the number of students of mathematical
tastes cannot have been very large. The fact of Bernoulli's
presence in Oxford I have discovered in the Corvespoudaace
Math, et Physique, edited by N. Fuss, vol. ii, p. 183, where, in
a letter to Daniel Bernoulli, Goldbach makes the remark : —
'Cum Oxonii agerem A. 1712, atque per unum alterumve
diem communi diversorio uterer cum consobrino Tuo CI.
Nicolao BernouUio, donavit me dissertatione (juadam Jacobi
Bernoulli de seriebus intinitis Arc'
(Lettre V Goldbach a D. BernouUi, 4 Nov. 1723)
Licidentally we learn an interesting fact regarding Goldbach
that has escaped the notice of M. Cantor, who, in the Vorwort
to the second edition of his Gesdiivhte, gives 1718 as the
earliest <late he has fonud in coiniexion with the traxels
of Goldbach.
NOTES UPON THE CORRESPONDENCE 207
Continuation as far as N. Bernoulli is concerned is found
on p. 300 of vol. ii of Brewster's Life of Newton. He (i.e.
Bernoulli) went to London in the summer^ of 1712, where he
met with the kindest reception from Newton and Halley,
a circumstance which he speaks of with much gratitude in a
letter in wdiich he thanks Newton for a copy of the second
edition of the Principia. (Letter dated Padua, May 31, 1717.)
Query : Did Gold bach meet Newton 1
Taylor ti Problem.
The problem &ent by Taylor to Montmort was a cliallenge
to the continental mathematicians : —
' Problema analyticum omnibus geonietris non Anglis pro-
positum : Invenire per quadraturam circuli vel hyperbolae
Fluentem liujus quantitatis
Taylor had obtained it in the posthumous papers of Cotes,
who died in 1716, while his Harmoida Mensarum, in which
the solution is given, w^as not published until 1722. The
limitation on A was given by Taylor because cnly in such
a case had Cotes etiected a solution. 'I'he challenge was really
intended for John Bernoulli.
John Bernoulli published a solution in May 1719 (Leip. Actis).
Other solutions were given by Hermann, Professor of Mathe-
matics at Padua,^ and by Ganfredi. (Montucla.)
IV. 4.
Letter IV. 4 is written in a typical Bernoullian spirit as
a reply to Stirling's letter IV. 3. Bernoulli's letter, however,
contains a number of valuable criticisms upon the tAvo pub-
lished works of Stirlini; on Cubic Curves, and on Serieti, to
which Stirlinc; would have had to a'ive careful attention had
second editions of his w^orks ever been contemplated by him,
and to which I may have to advert on another occasion.
For the present I restrict my attention to the discovery
Bernoulli makes known of a new variety of cubic omitted by
^ 'Visit to England duiing the months of September and October
1712.' (Edleston, note, p. U2.)
- Formerly.
208 NOTES UPON THE CORRESPONDENCE
l)otli Newton ;ui(l Stirling in tliuir uiiuiiiei-atiou of Cubic
Curves. (Newton's error, whicli Bernoulli points out, is re-
tained in tlie Horsley edition.)
In the enumeration of the cubics oiven ])y the ecjuation
only four of the six possible species are enunierateil Ijy Newton,
and by Stirling; following Newton.
Of the two missing species, Nicole in 1731 gave one (an
oN'al and two infinite branches) corresponding to
ay-=p-(x + (\-) {x + l3')
or xy' = — [r {x — oi-) {x - /3-) .
N. Bernoulli here announces (in 173?) the discovery of
another, consisting of an acnode and two infinite branches as
given by the equation
XtJ- = + y- {X + Oi-f.
Thus Bernoulli takes precedence of Stone 173G, Murdoch
and De Gua 1740, to whom reference is made by \V. W. R. Ball,
in his valuable memoir on Xeivtons Clasdjicatioii of Cuhic
Carves (Trans. L.M.S. 1891).
Murdoch {Neivtoni Genesis Gurvariim per Umbras, p. 87)
has the remark : —
' Speciem hanc No Vlll Analogam apud Newtonum deside-
rari animadverterat D. Nic. Bernoulli, quod me dim monuit
I). Cramer, Phil, et Math, apud Genevenses Celebris Professor.'
V
CASTEL
Louis Bertrand Castel (1688-1757), a Jesuit Father, was
the autlior of Le vrai systcme de Newton. He became F.R.S.
in 1730.
Stirling's letter \ . 2 contains a clear exposition of what he
understands by geometrical demonstration.
VI
CAMPAILLA
Thomas Campailla was born at IModica in Sicily in 16G8.
and died in 17-10. He studied in succession law, astrology'.
NOTES UPON THE CORRESPONDENCE 209
and philosophy, and tinally devoted himself entirely to the
Natural Sciences and Medicine. He was not a Fellow of
the Royal Society.
VII
BRADLEY
J. Bradley, 1692-1762, was a distinguished Astronomer.
Like Stirling, he studied at Balliol College, Oxford. He
became F.R.S. in 1718. In 1721 he was appointed to the
chair of Astronomy in Oxford, in succession to Keill. He
succeeded Halley as Astronomer Royal in 1 742. He discovered
the aberration of the tixed stars and the nutation of the
earth's axis.
Both the letters here given are to be found in Rigaud's
Bradley. Stirling's letter is taken from Rigaud ; and Bradley's
reply is among the letters preserved at Garden.
VIII
KLINGENSTIERNA
S. Klingenstierna was Professor of Mathematics at Upsala.
It was through Cramer that he was introduced to Stirling
(cf. Letter III. 3). In view of his researches in Optics, the
letter here given is of some interest He became F.R.S.
in 17.30.
IX
JOHN MACHIN
John Machin, the astronomer, became F.R.S. in 1710 (the
same year as Poleni, Professor of Astronomy at Padua, men-
tioned in the postscript to IV. 1), and acted as Sec. R.S. from
1718 to 1747. He sat on the committee appointed in 1712 to
investigate the dispute between Newton and Leibniz. In
1713 he became Professor of Astronomy at Gresham College.
He died in 1751.
Machin used the formula
tt/ 4 = 4 arc tan -g- — arc tan gig
to calculate tt to 100 places of decimals. His result is given
(1706) in Jones's Syno2Jds Palmariorum Matheseos, in which
the .symbol tt is first used for the number 3-14159 . . .
:210 NOTES LIPOX THE COKRESPUNDENCE
His 'Laws ot^ tlic IVIoon's Motion aceordiiin^ to Gravity' is
appended to Motte's translation of the Prineipia.
A greater work on Lunar Theory, begun in 1717, was never
publislied : and relative manuscripts are in t\\c possession of
the Royal Astronomical Society-.
Letter IX. 1.
Li connexion with this letter, which has no date, see the
letters from Bernoulli to Stirling, IV.
Letter IX. 2.
Machin was keenly interested in the researches of Maclaurin
and Stirling concerning the Figure of the Earth, though his
name does not appear to find a place in Todhunter's Hidory
of the subject.
The book by Maupertuis is probably one on the Figure of
the Earth mentioned by Todhunter (vol. i, p. 72 .
Machin, in speaking of Stirling's Proposition concerning
the Figure of the Earth, cannot refer to Stirling's Memoir
entitled ' Of the Figure of the Earth and the Variation of
Gravity on the Surface ', which appeared in the Phil. Trans.
for 1735-6.
Compare Stirling's letter to Maclaurin I. 15, in which he
refers to his correspondence with Machin.
I do not quite understand Machin in his reference to the
invention of Eider's Series, though Stirling's letter, if it could
be found, would explain.
By 1738 Stirling had got definitely settled as Manager of
the Lead Hills Mines in Scotland. He had apparently com-
plained to Machin how he felt the isolation from his scientific
friends and their researches in London. Machin's letter to
him is written in the kindliest spirit of warm friendship.
The book of De Moivre mentioned in the letter is doubtless
the second edition of the Doctrine of Chanceti (1738j.
X
CLAIRAUT
Born at Paris in 1713, Clairaut showed a wonderful pre-
cocity for mathematics, and at eighteen years of age he
NOTES UPON THE CORRESPONDENCE 211
piiblii-lied his celebrated ' Recherclics siir les Courbcs a double
Courbure '. He took part in the expedition to Lapland under
-Muupertuis to determine the length of the arc of the meridian.
He made several contributions to the Theory of the Figure
of the Earth, which he ultimately embodied in the classic
work entitled Thcorie de la Figure de la Terre (^1743). His
Theorle de la Lune appeared in 1765, shortly before his
death. He was also the author of Elihnents de la Geometrie
(1741), and of an Algebre (1746i. He became F.R.S. in 1737.
He died in 17G5.
' Clairaut a eu pour el eve et pour amle la celebre Marquise
de Chatelet, la docte et belle Emilie, qu'il a aid^e dans sa
traduction du Livre des principes' (Marie, Hist. Math.), a state
of affairs not over-pleasing to Voltaire.
In the letter here given w^e find Clairaut introducing himself
to Stirling. Cf. I. 15. Clairaut had frequent correspondence
with Maclaurin, and several of the letters have been preserved.
XI
EULER
Leonhard Euler (1707-83) was born at Bale in Switzerland.
He studied Mathematics under John Bernoulli, having as
fellow -students Nicholas and Daniel Bernoulli, the two sons
of John Bernoulli. The two brothers were called to Petrograd
in 1725, and Euler followed in 1727. In 1741, on the invita-
tion of Frederick the Great, he went to Berlin, returning
again in 1766 to Petrograd, where he died in 1783. For
almost the whole of his second residence in Russia he was .
totally blind, l)ut this misfortune had little effect on his
wonderful production of mathematical memoirs. There is
hardly a department of pure or mixed mathematics wdiich
his genius has not enriched by memoirs of far-reaching impor-
tance. A complete edition of his works has been undertaken
by a Swiss commission.
We are here only concerned with his relations with Stirling.
Apparently Euler had opened the correspondence by a letter
to Stilling, in which he announces, inter alia, the theorem
known as the Euler-Maclaurin Theorem (Reiff', Geschichte der
Unendlichen Reiheii). 1'his letter is not preserved, but copies
p2
212 NOTES UPON THE CORRESPONDENCE
of tlic Icttei's that passed between Euler and Stirling appear to
have been in existence at Petrograd : and Professor Enestrom
in his Vorldujiges VerzeicJniiK der Brlefe von und an L. Eider,
1726-41, furnishes the following dates :
(1) Euler to Stirling, 9th June, 17.3G,
(2) Stirling to Euler, April, 1738,
(3) Euler to Stirling, 27th July, 1738.
The letters preserved at Garden are doubtless (2) and (3).
It remains to l)e seen whether the letters in Petrograd have
survived the fury of the Revolution in Russia.
Stirling's reply was much belated, for his time was now
entirely devoted to the successful development of the Lead
Hills Mines, of which he had been appointed manager a ye-AV
or two before. The rounli draft of it is all that Stirling
preserved, and is here given witli all his corrections and
erasures. Stirling acknowledges the ini2:)ortance of Euler's
Theorem, and remarks that his own theorem, ' Theorema
meum ', for summing Logarithms is only a particular case.
He informs Euler that Maclaurin has an identical theorem
in the proof-sheets of a Treatise of Fluxions to appear shortly.
At the same time he offers to communicate Euler's results
to the Royal Society', and suggests that Euler should become
a Fellow.
With characteristic modesty and absolute freedom from
jealousy, Euler in his reply waives his claim to priority over
Maclaurin, and proposes that the Royal Society should publish
a paper on the Equation of liiccati, which he had sent some
3"ears before to Sloane the President.
There can be little doubt that luder and Maclaurin dis-
covered the theorem independently, and llie suggestion made
by Reiff to call it the Euler-Maclaurin Theorem seems fully
justified.
Maclaurin, by the way, does not refer to it in tlie intro-
duction to liis Fluxions, but on p. C91 of his Treatise. Euler
first gave his theorem without pi-oof in his Metliodus generalis
summandi firoijressiones (Jomm. Pctrop. ad annos 1732, 1733:
published 1738.
The proof is given in Invodio ^ummae cujusque seriei ex
dato termino (jcnerali (Jomm. Peirop., 173(1: published 1741.
Compare Stirling's letter to Maclaurin I. 15.
NOTES UPON THE CORRESPONDENCE 213
I cannot lierc further discuss Euler's letter, wliicli is almost
encyclopaedic in its rany;e, save to say that Stirling had sliown
in his Me(h. J^ijf. lunv to approximate with any desired
00
accuracy to ^ -^, > without hein^- aware ol" its expre
SSI on
n
1
as 77-/6.
(See letters of Dan fJernoulli to Euler in Fuss, Corr. Math.,
t^'c.) As is well known, Euler became F.R.S. in 174G.
XII
M. FOLKES, P.R.S., TO STIRLING
This is the letter of latest date in the correspondence. It
conveys to Stirling the news that he had been made a member
of the Royal Academy of Science at Berlin, an honour which
has not hitherto been noted in any of the biographies of
Stirling.
May the Mr. Mitchell who brings the letter to Stirling not
have been Maclaurin's friend, better known as Sir Andrew
Mitchell, who afterwards became Ambassador at the court of
Frederick the Great ?
PRINTED IN ENGLAND
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Date
Due
MAY 1
; ?nnfi
L. B. CAT. NO. 1
187
M-'Jffl
QA2').S6H3
scni
3 5002 00228 1975
Stirling, James , , , , j
James Stirling; a sketch of his life and
Science
QA 29 .
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A3
Stirling
, James
!, 1692-
■1770.
James St
irling