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JAMES     S 1  I  R  L  I  N  G 


OXFORD    UNIVERSITY    PRESS 

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HUMPHREY  MILFORD 

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(^^iJaj  tk  cfn^^}^  2^  m^WJU^.   '>%  ^aU  htgyU^  <s^/e,- 
km^  /h^itc  Jatic  f'l*-  if  'V/ >^n^ ^^^^ ^  l^hm^^ ^  ^ W^ 


JAMES  STIRLING 


A  SKETCH  OF  HIS 


Htfc  auD  KHorfts 


ALONG    WITH   HIS 


SCIENTIFIC   CORRESPONDENCE 


BY 


CHARLES  TWEEDIE 

M.A.,  B.Sc,  F.K.S  E. 

CAKNEGIE   FELLOW,    1917-l'.t20;    LECTURER   IN    PURE    MATHEMATICS 
EDINBURGH    UNIVERSITY 


OXFORD 
AT     THE     CLARENDON     PRESS 

1922 


\  0  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  0 

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  0  on  it  such  that  '^OP  =  0. 

As  the  line  varies  in  position  0  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 
0  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  0 
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  0  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       0  +  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  =  0 

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 +  0 
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  0  <  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.  -  0 

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  =  0 

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  -  ?'  =  0 

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'' =  0  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  =  0  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  =  0 
'  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  0  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=  0 
x^-\0x*-\-Z0x^-iix^  +  32x-d  =  0 


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.  =  0  ; 

By  it  too  I  demonstrate  a  Theoreme  in  your  book  where 
a  quantity  is  expresst  by  a  Series  whose  coefficients  are  first, 
second,  third  fluxions  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=  0 
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  -  0  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,  0  &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=  0   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      0 

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+  |^,  =  0 


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  =  0  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  =  0   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  =  0 

(|ue  la  substitution  de  Ax"-  au  lieu  de  y,  change  en 

-^  +Ax"+'-2A^x^''+^  +  A^iiy'''  +  ' ^  =  0 

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-    ~^^  =  0 

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  =  0  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  =  0  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       ~  —  0 

till  .        .  .  ^ 

i^'  1  +  3      / -^  =  0  dont  la  multiplication  produit 

XX  —  2x  Vay  —  1 5  (<  ?/  =  0 
qui  est  la  fluente  de  la  fluxion  propos^e 

2xx  —  ix  ^<iy—  IG'iy  —  0 

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  =  0  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  =  0  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  ^  0  B-Ax  +  e-af—  0 

y  —  ^x  +  a  +  be-af-O  C—Bx  +  A  +  ae-bf=0 

8-yx+l3  +  ce-bf^  0  D-Cx  + B  +  be-cf  =^  0 

i-Sx  +  y  +  de-cf  =  0  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  0  maxime  distans  a 
recta  AE,  sitque  A  MO  Curva  quae 
transit  per  omnia  puncta  0 ;  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  =  0  laborem  calculi 
prolixiorem  reddunt,  quam  paret.  Sic  pro  Exemplo  Taylori 
in  Prop.  9.  Method.  Increm.  pag.  31  1 +sx- z^xx  —  x  =  0 
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  £  =  0  babetur  tuarum  Serierum  prima,  Si  ^  =  0  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  =  0  exsuri;-it  tua  tertia  Series,  et  si  fiat  x\  —  0 
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  =  0 
radix  y  est  = 

aA+(tiL       aaA—d"^       a^A  —  a^       a^A—a^         . 

quando  ^  =  0  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^  =  0 
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  =  0    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  =  0 

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  —  0    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)  =  0  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.  0  contains  only  extracts  from  letters  of  ]\iaelaurin, 
including  one  date<l  October  22,  1 728,  whicli  is  no  longer 
in  the  Stirling  collection. 

Ijetter  1.   7. 

In  the  spring  of  1921  I  had  the  good  fortune  to  obtain 
a  copy  of  Maclaurin's  reply  to  Campbell. 


NO'l^ES   T^PON   THE   CORRE>SPONDENCE        11)7 

Jt  is  entitled:  — 

'  A  Defence  ct"  the  Letter  i)ultlislic(I  in  the  Phihjsopliical 
Transactions  for  jMareh  and  April  1729,  concerning  the 
Impossible  Roots  of  Equations :  in  a  Letter  from  the 
Author  to  a  Friend  at  London. 

Qui  admonent  aniiee,  docemli  sunt  :  <|ui  inimice  infeetantur, 
repellendi. 

Cicero ' 

The  name  of  the  '  Friend '  is  not  oiven.  The  '  Defence ' 
consists  of  twenty  small  quarto  pao'es,  and  contains  numerous 
extracts  from  the  letters  to  Stirling;  and  towards  the  end 
Campbell's  statements  regarding  Maclaurin's  theorems  are 
refuted. 

Campbell  is  generally  referred  to  as  '  the  Autlior  of  the 
Remarks '  (on  Maclaurin's  Second  Letter  on  impossil)le 
roots) :  thougli  also  as  '  the  Remarker  '• 

Maclaurin  gives  the  extract  from  tlie  letter  of  October  1728 
(cf.  I.  6),  and  adds:  — 

'  See  the  2d  and  3d  Examples  of  the  Eighth  Fro'positlon  of  tlio 
Lineae  tertii  Ordiuis  Xewtonianae.' 

There  is  also  the  following  passage  containing  an  extract 
from  a  letter  by  Stirling,  not  otherwise  known : — 

'I  had  an  Answer  from  this  Gentleman  in  March,  from 
which,  with  his  Leave,  I  have  transcribed  the  following- 
Article  : 

"  I  shewed  your  Letter  (says  he)  to  Mr  Ma(hni,  and  we  were 
both  well  satisfied  that  you  had  carried  the  IMatter  to  the  greatest 
Height,  as  plainly  appears  b}^  what  3'ou  have  said  in  your 
Letter.  But  it  is  indeed  a  Misfortune,(,that  you  was  so  long 
in  giving  us  the  Second  Part,  after  you  had  delivered  some 
of  your  Principles  in  the  First: — Since  you  have  published 
Part  of  your  Paper  before  Mr  C — ^—11,  and  now  liavc  the  rest 
in  such  Readiness,  I  think  3'ou  have  it  in  j^our  Power  to  do 
j^ourself  Justice  more  than  any  Body  else  can.  I  mean  by 
a  speedy  Publication  of  the  remaining  Part :  For  I  am  sure, 
if  3'OU  do  that,  there  is  no  Mathematician,  but  who  must 
needs  see,  That  it  is  your  own  Invention,  after  the  Result 
of  a  great  Deal  of  Study  that  way." 

I  received  this  Letter  in  March,  and,  in  consequence  of  tliis 


198        NOTES   UPON    THE   CORRESPONDENCE 

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

Maclaurin  makes  it  clear  that  he  had  not  intended  his  First 
Letter  to  Folkes  to  he  published.  It  was  printed  without  his 
knowledge.  Had  lie  known  in  time,  he  would  liave  deferred 
its  publication  until  he  had  more  fully  investigated  additional 
theorems  Avhich  he  had  on  the  same  subject;  and  he  gives  an 
extract  from  a  letter  from  Folkes  in  corrolioration  of  his 
statement. 

Letter  I.  8. 

Letter  I.  8  is  reproduced  because  of  its  reference  to  an 
office  (in  the  Roj'al  Society)  for  which  Stirling  had  been 
thought  fit. 

Letter  I.  9. 

Letter  I.  9  announces  that  Maelaurin  has  started  to  write 
his  Treatise  of  Flnxi<ms.  His  conscientious  reference  to 
original  authorities  has  been  noted  b}"  Reitf  {GeschicJde  der 
UnendlicJien  Relheii).  The  earlier  proof-sheets  of  the  Treatise, 
at  least,  passed  through  Stirling's  hands. 

These  facts  bear  interesting  evidence  regarding  the  Ealer- 
Maclaurin  Summation  Formidn,  to  which  I  have  to  return  in 
connexion  with  the  correspondence  between  Stirling  and 
Euler  in  Letters  XL 

Simp.son,  referred  to  by  Maclaurin,  is  doubtless  his  old 
teacher,  Robert  Simson,  of  Glasgow  Univorsit}'. 

Letter  I.   10. 

Letter  I.  10,  which  is  a  mere  scrawl  written  on  the  outside 
of  the  copy  of  the  letter  from  Maupertuis  to  Bradley,  is  of 
interest  in  the  history  of  the  Royal  Society  of  Edinbui-gh, 
and  is  to  be  associated  with  the  two  letters  of  Maelaurin 
published  in  the  Scots  Magazine  for  June,  1804. 

The  date  of  the  letter  of  Maupertuis  shows  that  Maclaurin 
should  have  given  Feb.  4*',  173|  as  the  date  of  his  own. 

Maclaurin  was  more  successful  with  Stirling  than  with 
R.  Simson,  who  refused  to  become  a  member  after  IMaclaurin 
had  got  him  nominated.     (Scots  Mag.) 

Bradley's  translation  of  the  letter  of  Maupertuis  is  repro- 
duced  in   the   Works  and  Correspondence  of  Bradley,   1832 


NOTES   UPON   THK   CORRESPONDENCE        199 

(Ki^aud).     The  original   Fi-ench  Jetter  i.s  preserved  in  one  of 
the  hotter  hooks  of  the  Royal  Society  of  London. 

FuundatioR  of  the  Philosophical  Society  of  Edinhurgh. 

Letter  L  10  confirms  the  date  of  foundation  as  1737  (v. 
Forbes's  Hidory  of  the  Royal  Society  of  Edinburgh,  in  General 
Index  Trans.  R.S.E.  published  1890). 

But  at  the  date  of  this  letter  I.  10  the  Society  \vas  not 
complete  in  numbers,  for  Stirling  \Yas  not  yet  a  member. 

By  1739  the  Society  had  outrun  its  original  bounds,  having 
forty-seven  members  M'hose  names  are  given  (p.  26  of  Gen. 
Index  Trans.  R  S.E.). 

More  or  less  informal  meetings  were  held  in  1 737.  Maclaurin 
and  Dr.  Plummer,  Piofessor  of  Chemistry  in  the  University, 
were  the  Secretaries.  The  Rebellion  of  1745  seriously  affected 
the  activity  of  the  Society,  and  Maclaurin's  death  in  1746  was 
also  a  severe  blow. 

The  papers  read  before  the  Society  had  been  in  Maclaurin's 
hands,  but  only  some  of  these  were  found.  Three  volumes  of 
Easays  and  Observations,  Physical  and  Literary  (dated  1754, 
17nG,  1771),  were  published.  The  papers  in  Vol.  I  are  not  in 
chronological  order,  but  those  by  Plummer  are  fortunately 
dated,  the  first  bearing  the  date  January  3,  1738.  Dr.  Pringle, 
afterwards  President  of  the  Royal  Society  of  London,  followed 
in  Feljruary.  Then  it  was  Maclaurin's  turn  in  March,  when 
he  gave  two  papers,  one  being  on  the  Figure  of  the  Earth 
(Scots  Magazine). 

These  two  papers  are  not  printetl  in  the  Essays,  &c.  But 
among  the  Maclaurin  MSS.  preserved  in  Aberdeen  University 
there  is  one  entitled  '  An  Essay  on  the  Figure  of  the  Earth '. 

On  the  foundation  of  the  Royal  Society  of  Edinburgh  in 
1783  the  members  of  the  Philosophical  Society  were  assumed 
as  Fellows.  Maclaurin's  son  John  (Lord  Dreghorn)  is  one  of 
those  mentioned  in  the  original  charter  of  the  Royal  Society. 

Lttter  of  Mawpertuis. 

The  letter  of  Maupertuis  must  have  given  lively  satisfaction 
to  Maclaurin  and  Stirling.  Newton  had  assumed  as  a  postulate 
that  the  figure  of  the  Earth  is  approximatel}?"  that  of  an 
oblate  spheroid,  flatter  at  the  poles  than  at  the  Equator.     The 


m.)       NOTES   T^^OX   THE   COPIRESPOXDEXC'E 

Cassinis,  arguing  from  iiR'asui'eiiK'iit.s  oi'  tliu  arc  uL'  a  Mcriilian 
in  France,  maintained  that  the  figure  was  that  of  a  prolate 
splieroid.  There  were  thus  two  hostile  camps,  the  X'cwtonians 
and  the  Cassinians. 


Pol- 

Pole 

A 
Y 

titwroK 

CASbTNI 

The  French  expedition  to  Lapland  (173G-7)  with  Mauper- 
tuis  as  leader,  and  Clairaut  as  one  oF  the  party,  conclusively 
established  the  accuracy  of  Newton's  hypothesis.  In  the 
words  of  Voltaire,  Maupertuis  had  'aplati  les  Poles  et  les 
Cassinis '. 

Both  Stirling  and  Maclaurin  made  important  contributions 
to  the  subject,  and  the  rest  of  the  letters  preserved  as  passing 
betW'Cen  them  refer  mainly  to  their  researches  on  Attraction 
and  on  the  Figure  of  the  Eartli. 

Readers  who  are  interested  cannot  do  better  than  consult 
Todhunter's  History  of  the  Theory  of  Attraction  and  of  the 
Figure  of  the  Earth  for  full  details.  The  letters,  however, 
clear  up  some  difficulties  that  were  not  alwa3^s  correctly 
explained  by  Todhunter. 

Letter  I.  11. 

The  Dean,  near  Edinburgh,  jMaclaurin's  new  address,  now 
forms  a  residential  suburb  of  Edinburgh. 

De  Moivre's  book  is  doubtless  the  second  edition  of  tlie 
Doctrine  of  Chances  (1738). 

letter  T.  13. 

Tlie  remark  made  by  Stirling  towards  the  conclusion  that 
'  the  gravitation  of  the  particle  to  the  wdiole  spheroid  will  be 
found  to  depend  on  tlie  quadrature  of  the  circle '  seems  to 
have  given  Maclaurin  a  good  deal  of  trouble  (cf.  I.  14). 


NOTKS    T^'OX   THE    (X:)RRESPONDENCE       201 

]\Ia('l;uiiiirs  rct'crt'iKT  to  it  in  his  Fluxions,  §  647,  as  due  to 
Stirling-,  Mas  iiicxplicaltlc  to  'rodliunter,  as  Stii'lin<i^  never 
published  his  theorem.  But  Todliiniters  conjecture  {Hlttovij, 
vol.  i,  p.  139)  that  Maclaurin  ma}^  have  inadvertently  written 
Stirling  for  Simpson  is  of  course  quite  a  mistake. 

Letter  I.  15. 
Compare  the  correspondence  with  Machin  IX,  Clairaut  X, 
and  Euler  XL 

Letter  I.   16. 

This  letter,  dated  1740,  furnishes  ample  justitieation  of 
Todhunter's  contention  that  the  researches  of  Maclaurin,  '  the 
creator  of  the  theory  of  the  attraction  of  ellipsoids',  are  quite 
independent  of  those  given  by  T.  Simpson  in  his  Mathematical 
Diskertatioiis  (1 74.3).  Simpson  lays  claim  to  priority  in  certain 
theorems  of  the  Fluxions  on  the  ground  that  these  given  by 
himself  were  read  before  the  Royal  Society  in  1741. 

The  Treatise  of  Fluxions  so  near  completion  in  1740  was 
not  published  until  1742. 

II 

CUMING 

Sir  A.  Cuming  (1690?-17  75)  was  the  only  son  of  Sir  Alex- 
ander Cuming,  M.P,,  the  first  baronet  of  Culter,  Aberdeen. 
Cuming  went  to  the  Scotch  bar,  but  gave  up  his  profession 
on  receiving  a  pension.  In  1720  he  became  a  Fellow  of  the 
Royal  Society.  Though  no  mathematical  writings  of  his  are 
known,  he  seems  to  have  been  possessed  of  mathematical 
ability.  He  was  on  friendly  terms  with  De  J\Ioivre  and 
Stirling,  both  of  whom  acknowledge  their  indebtedness  to 
him  for  valuable  suggestions.  At  Aberdeen  there  is  preserved 
a  short  letter  (Nov.  3,  1744)  from  him  to  Maclaurin,  in 
which  he  shows  his  interest  in  the  controvers}^  regarding 
Fluxions. 

In  his  introduction  to  the  Methodns  Differential  is,  Stirling 
speaks  of  him  as  ■  Spectatissimus  Vir '.  Being  a  friend  of 
Campbell  he  had  a  share  in  the  dispute  between  Maclaurin 
and  Campbell. 

In  1729-30  he  was  in  the  American  Colonies,  visited  the 
Cherokees,  and  became  one  of  their  chiefs.     On  his  return  to 


202       NOTES   UPON   THE   CORRESPONDENT'E 

England    with    soiiii-   oi"   the  duel's    lie    was    iiistiuineiital    in 
sirrano-inor  n  treaty  for  liis  tribe. 

Later  he  fell  into  poverty,  and  was  confined  in  the  Fleet 
prison  from  1737  to  1765,  losing;  his  fellowship  in  the  Pvo^'al 
Society  for  neglecting-  to  pay  his  annual  fee.  In  1766  he 
obtained  admission  to  the  Charterhouse  and  died  tluM'e  in  1  7  7.'. 

Ill 

CRAMER   AND   STIRLING 

Gabriel  Cramer  was  born  in  1 704  in  Geneva,  where  his 
father  practised  medicine.  In  1724  he  was,  conjointly  with 
Calandrini,  entrusted  with  the  instruction  in  Mathematics  at 
the  University  of  Geneva.  In  172  7  he  started  on  a  two 
years'  tour,  visiting  Bale,  where  he  studied  under  John  Ber- 
noulli, and  England,  where  he  became  acquainted  with  Stirling 
and  De  Moivre,  and  returning  by  Paris.  He  became  F.R.S.  in 
1748,  He  died  in  1752.  He  is  best  known  through  his 
Introduction  d  I' Analyse  dcs  lignes  courhes  algebriques.  He 
also  edited  the  works  of  James  and  John  Bernoulli. 

Letter  IIL   1. 

It  is  unfortunate  for  us  that  Cramer  did  not  discover  before 
1732  that  he  wrote  'mi  Anglois  au&si  barbare '. 

Regarding  the  history  of  the  Probability  Problem  in  III.  1, 
see  Todhunter's  History  of  the  Theory  of  ProIxdnlUy  (p.  84). 
De  Moivre  gives  a  much  simpler  solution  in  the  Miscellanea 
Analytica  (1730). 

Letter  IIL  2. 

Compare  Lettei'  IV.  2  (Bernoulli). 

Letter  III.  3. 

In  this  letter  of  introduction  Cramer  in  the  address  describes 
Stirling  as  L.A.]\T.     I  do  not  know  what  these  letters  signify. 

Letter  III.  8. 

Letter  III.  8  contains  valuable  information  regarding  the 
manner  in  which  Stirling  wiote  his  Methodiis  Diferentialis. 
The  blank  made  for  the  formula  given  by  De  Moivre  was 
never  filled  u[) :  but  the  formula  in  (|uestion  is  of  course  easily 


NOTES   UPON    THE   CORRESPONDENCE       203 

obtained  i'roni  tlio  Suj^plement  to  the  M i scellanea  Auahjtica 
of  Do  Moivre.  We  have  also  tlie  important  information  that 
this  Supplement  appeared  after  tlie  pnlilieation  of  Stirling's 
own  Treatise. 

Letter  III.   10. 

One  will  note  Cramer's  difficulties  with  the  graph  of 
2/^"=  1 +.'-';  also  his  determination  of  (1 +a')^/-'''  as  x  tends 
to  zero. 

It  is  a  pity  there  is  no  indication  of  Stirling's  determination 
of  this  limit. 

Stirling' a  Series 
and  the  claims  to  priority  of  De  Moivre  and  Stirling. 
In  the  Bihlioteca  Matliematica  for  1904  (p.  207)  Enestrom 
makes  the  following  statement. 

'  Im  Anschluss  an  den  Bericht  liber  Stirling's  Formel  fiir 
die  Summe  einer  Anzahl  von  Logarithmen  ware  es  angezeigt 
mitzuteilen  dass  die  bekannte  Formel  dieser  Art  die  man  jetzt 
ziemlich  allgemein  gewohnt  ist  als  die  Stirlingsche  Formel  zu 
bezeichnen,  namlich 

log  (1  .  2  .  3  . . .  ./•)  =  -i  log  27r  +  {X  +  4)  log x 

.1        .1-2       , 

-x  +  A,,-  +  J.,  —^  +  eVc, 

"  X  X"" 

zuerst  von  Moivre  im  Anhange  an  der  Miec.  analytica  (17.30) 
angegeben  und  hergeleitet  wurde.  Moivre  berichtet  selbst 
dass  Stirling  ihm  brieflich  die  Formel 

log  (1  .  2  ...  a-)  =  i  log  273-  +  (a;  +  i)  log  (a;  +  I ) 


2.12(a;  +  i)       8.360(«  +  i)=^ 

mitgeteilt  liatte,  und  dass  er  selbst  dadurch  angeregt  wurde 
die  neue  Formel  auf  eineni  ganz  anderen  Wege  auf  zufinden.' 

Inasmuch  as  the  only  change  effected  by  De  Moivre  is  to  give 
the  expansion  of  log  {x  !j  in  descending  powers  of  x  instead  of 
descending  powers  of  x  +  ^,  which  has  no  special  advantage 
when  X  is  large,  the  priorit}^  of  De  Moivre  to  this  important 
formula  seems  to  me  to  rest  on  very  slender  foundations, 
unless   we   are   to   infer   from    Enestrom's   reference   to   the 


204        NOTES    UPON     THE   CORRESPONDENCE 

Sujiidemciit  Id  tlu'  Mli<rcflaiie(i  Aiialijllcc  thai  l)c  Moix  to 
pulilisliL'd  his  result  prioi-  to  Stirliui;-. 

Enestroins  stateiiR-nt  lias  had  considcraltle  influence  with 
subsequent  writers  (e.^-.  Czuher  and  Le  Roux,  Cuhul  des 
Probabllites;  Selivano\-  and  Andoj'er,  Calcul  des  Differences 
Finies,  in  the  well-known  Eacyc.  des  ^Sriences  Math. ;  Czuber, 
WaJir.  Recliuvivi,  1903,  s.  19),  wlio  refer  for  proof  to  tlie  Siipi>. 
Misc.  Anal,  of  De  Moivre. 

Aeainst  these  we  may  put  De  Moivre's  own  statement  in 
the  third  edition  of  the  Doctrine  of  Chances  (1756),  oiven  in 
tlie  Appendix,  p.  334,  where,  after  giving  a  table  of  ^'alue.s  for 
log  {x !)  for  numerical  \  alues  of  x  he  goes  on  to  add  : — 

'If  we  would  examine  these  numbers,  or  continue  the  Table 
farther  on,  we  have  that  excellent  Rule  communicated  to  the 
Author  by  jMr  James  IStirliiig,  published  in  his  Supplement  to 
the  Ulitcellanea  Analytica,  and  by  Mr  Stirling  himself  in  his 
Methodus  Different ialis,  Prop.  XXVIII. 

'Let  z  —  ^  be  the  last  term  of  any  Series  of  the  natural 
Numbers  1,  2,  3,  4,  5,  ...:-|;  (^  =  -43429448190325  the 
reciprocal  of  Neper's  Logarithm  of  10:  Then  three  or  four 
terms  of  this  Series 

a  7a  31 rt 

z     ogz-az-  2    12s  ^  8  .  3603^  ~  32  .  1260^5 

127ft 

H _-  —  (vc 

128.  1680-J 

added  to  0-39908993  il  79,  c^'C.  which  is  half  the  Logarithm  of 
a  Circumference  whose  Radius  is  Unity,  will  be  the  Sum  of 
the  Logarithms  of  the  given  Series ;  or  the  Logarithm  of  the 
Product 

1x2x3x4x5...  xz~\  &c.' 

There  is  thus  no  doubt  in  De  Moivre's  mind  that  the  dis- 
covery of  the  theorem  in  question  is  not  due  to  himself  but  to 
his  friend  Stirling. 

Date  of  ^u/rplement  to  the  Miscellanea  Aaahjtiai. 

At  first  sight  the  Supplement  appears  to  bear  the  date 
Jan.  7,  17|§.  Li  such  case  it  would  almost  certain]}'  be 
anterior  in  [)ublication  to  Stirling's  book. 


NOTES   UPON   THE   CORRESPONDENX'E       205 

Now  this  [supposition  is  quite  erroneous.  The  Miscellanea 
Aaalytica,  as  originally  published,  bears  the  above  date,  and 
contains  no  supplement.  (The  first  copy  I  consulted  has  no 
supplement.)  An  examination  of  a  copy  with  the  Supple'meiit 
shows  two  lists  of  Errata,  the  first  after  p.  250,  and  the 
f^econd  after  p.  22  of  the  Su^yjylemeiit,  the  latter  list  contain- 
ing Errata  observed  by  De  Moivre  and  his  friends  '  post  editum 
libruni  meum '. 

The  letter  III.  8  of  Stirling  puts  it  beyond  a  doubt  that  the 
Sa2}plemeiit  had  not  appeared  at  the  time  he  wrote  (September 
1730),  so  that  its  appearance  was  posterior  to  the  publication 
of  Stirling's  Methodas  Differe^dlalls. 

We  have  thus  the  following  events  in  chronological  order. 

De  Moivre  publishes  the  Mite.  Anal,  early  in  1730.  His 
friend  Stirling  points  out  to  him  the  poor  approximation 
he  gives  for  log  [x  !)  when  x  is  large  and  sends  him  a  formula 
of  much  greater  accuracy.  Stirling  publishes  his  Meth.  Diff. 
containing  the  famous  Stirling  Series.  In  the  meantime 
De  Moivre  busies  himself  with  Stirling's  formula,  and  obtains 
it  in  a  slightly  different  form  but  by  an  entirely  different 
process:  and  he  publi;  lies  his  result  as  a  Supijlement  to 
his  book  and  bound  with  it,  but  without  changing  the  date 
of  his  book.  He  explains  in  his  own  garrulous  way,  which 
makes  the  reading  of  his  works  so  attractive  nowadays,  how 
he  had  very  nearly  got  at  Stirling's  Theorem  before  he  had 
heard  from  Stirling. 

Will  any  scholar  be  bold  enough  to  assert  that  the  theorem 
is  due  to  De  Moivre  in  virtue  of  this  latter  statement,  pub- 
lished after  Stirling  had  given  the  theorem  in  all  its  generality 
in  the  Meth.  Dijf.  1  You  may  speak  of  De  Moivre's  form 
of  Stirling's  Theorem  if  you  please,  but  the  merit  of  discover- 
ing a  theorem  of  the  kind  eecms  to  rest  indisputably  with 
Stirling. 

IV 
N.  BERNOULLI  AND  STIRLING 

Nicholas  Bernoulli  was  born  in  1687  at  Bale  in  Switzerland, 
his  father  being  a  merchant  in  tliat  town.  His  two  uncles, 
James  Bernoulli  (1654-1705)  and  John  Bernoulli  (1607-1748), 
were  both  noted  mathematicians. 


20G       NOTES   UPON    THE   CORRESPONDENCE 

He  .studied  Hrst  under  the  t\)nuei'  at  f!ale  University,  and 
then  under  the  latter  at  Gri3nin»;"en,  returning-  with  his  uneh' 
John  to  Bale  in  1  705. 

He  devoted  hiniselt'  to  the  study  of  mathematics  and  law. 
He  became  F.R.S.  in  171."}.  (_)n  the  recommendation  ot" 
Leibniz,  he  was  in  1716  ap[)ointed  Professor  of  Mathematics 
at  Padua,  resignini^,-  in  171'.)  and  returning  to  Bale.  In  1722 
he  was  elected  to  the  chair  of  Logic,  and  in  1731  to  the  chair 
of  Law  in  Bale.     He  died  in  1759. 

His  cousins,  the  sons  of  Jolnij.  Nicholas  1G95-1726;  Daniel 
1700-82;  and  John^  1710-no  were  also  noted  mathematicians. 
Two  of  the  three  sons  of  Jolin^,  viz.  John,  and  James,  also 
showed  mathematical  ability,  so  that  we  have  here  a  remark- 
able instance  of  three  generations  of  distinguished  mathe- 
maticians in  one  family.  Venice  was  a  favourite  resort  of 
the  Beruoullis  about  the  time  that  Stirling  resided  there. 

Letter  IV.   1. 

Letter  IV.  1  is  the  earliest  of  the  letters  preserved  in  the 
mathematical  correspondence  of  Stirling.  When  the  actiuain- 
tance  betwc'ii  Bernoulli  and  Stirling  began  is  unknown,  but 
Bei'noulli  in  the  course  of  his  travels  spent  some  time  in 
Oxford  in  1712,  when  Stirling  was  still  an  undergraduate. 
One  is  strongly  tempted  to  suggest  that  it  was  at  Oxford 
that  they  first  met,  for  the  disparity  in  their  years  was  not 
very  great,  while  the  number  of  students  of  mathematical 
tastes  cannot  have  been  very  large.  The  fact  of  Bernoulli's 
presence  in  Oxford  I  have  discovered  in  the  Corvespoudaace 
Math,  et  Physique,  edited  by  N.  Fuss,  vol.  ii,  p.  183,  where,  in 
a  letter  to  Daniel  Bernoulli,  Goldbach  makes  the  remark : — 

'Cum  Oxonii  agerem  A.  1712,  atque  per  unum  alterumve 
diem  communi  diversorio  uterer  cum  consobrino  Tuo  CI. 
Nicolao  BernouUio,  donavit  me  dissertatione  (juadam  Jacobi 
Bernoulli  de  seriebus  intinitis  Arc' 

(Lettre  V  Goldbach  a  D.  BernouUi,  4  Nov.  1723) 

Licidentally  we  learn  an  interesting  fact  regarding  Goldbach 
that  has  escaped  the  notice  of  M.  Cantor,  who,  in  the  Vorwort 
to  the  second  edition  of  his  Gesdiivhte,  gives  1718  as  the 
earliest  <late  he  has  fonud  in  coiniexion  with  the  traxels 
of  Goldbach. 


NOTES   UPON   THE   CORRESPONDENCE       207 

Continuation  as  far  as  N.  Bernoulli  is  concerned  is  found 
on  p.  300  of  vol.  ii  of  Brewster's  Life  of  Newton.  He  (i.e. 
Bernoulli)  went  to  London  in  the  summer^  of  1712,  where  he 
met  with  the  kindest  reception  from  Newton  and  Halley, 
a  circumstance  which  he  speaks  of  with  much  gratitude  in  a 
letter  in  wdiich  he  thanks  Newton  for  a  copy  of  the  second 
edition  of  the  Principia.     (Letter  dated  Padua,  May  31,  1717.) 

Query  :   Did  Gold  bach  meet  Newton  1 

Taylor  ti  Problem. 

The  problem  &ent  by  Taylor  to  Montmort  was  a  cliallenge 
to  the  continental  mathematicians : — 

'  Problema  analyticum  omnibus  geonietris  non  Anglis  pro- 
positum :  Invenire  per  quadraturam  circuli  vel  hyperbolae 
Fluentem  liujus  quantitatis 

Taylor  had  obtained  it  in  the  posthumous  papers  of  Cotes, 
who  died  in  1716,  while  his  Harmoida  Mensarum,  in  which 
the  solution  is  given,  w^as  not  published  until  1722.  The 
limitation  on  A  was  given  by  Taylor  because  cnly  in  such 
a  case  had  Cotes  etiected  a  solution.  'I'he  challenge  was  really 
intended  for  John  Bernoulli. 

John  Bernoulli  published  a  solution  in  May  1719  (Leip.  Actis). 
Other  solutions  were  given  by  Hermann,  Professor  of  Mathe- 
matics at  Padua,^  and  by  Ganfredi.     (Montucla.) 

IV.   4. 

Letter  IV.  4  is  written  in  a  typical  Bernoullian  spirit  as 
a  reply  to  Stirling's  letter  IV.  3.  Bernoulli's  letter,  however, 
contains  a  number  of  valuable  criticisms  upon  the  tAvo  pub- 
lished works  of  Stirlini;  on  Cubic  Curves,  and  on  Serieti,  to 
which  Stirlinc;  would  have  had  to  a'ive  careful  attention  had 
second  editions  of  his  w^orks  ever  been  contemplated  by  him, 
and  to  which  I  may  have  to  advert  on  another  occasion. 

For  the  present  I  restrict  my  attention  to  the  discovery 
Bernoulli  makes  known  of  a  new  variety  of  cubic  omitted  by 

^  'Visit  to   England  duiing  the  months   of  September  and  October 
1712.'     (Edleston,  note,  p.  U2.) 
-  Formerly. 


208       NOTES   UPON   THE   CORRESPONDENCE 

l)otli  Newton  ;ui(l  Stirling  in  tliuir  uiiuiiiei-atiou  of  Cubic 
Curves.  (Newton's  error,  whicli  Bernoulli  points  out,  is  re- 
tained in  tlie  Horsley  edition.) 

In  the  enumeration  of  the  cubics  oiven  ])y  the  ecjuation 

only  four  of  the  six  possible  species  are  enunierateil  Ijy  Newton, 
and  by  Stirling;  following  Newton. 

Of  the  two  missing  species,  Nicole  in  1731  gave  one  (an 
oN'al  and  two  infinite  branches)  corresponding  to 

ay-=p-(x  +  (\-)  {x  +  l3') 

or  xy'  =  —  [r  {x  —  oi-)  {x  -  /3-) . 

N.  Bernoulli  here  announces  (in  173?)  the  discovery  of 
another,  consisting  of  an  acnode  and  two  infinite  branches  as 
given  by  the  equation 

XtJ-  =    +  y-  {X  +  Oi-f. 

Thus  Bernoulli  takes  precedence  of  Stone  173G,  Murdoch 
and  De  Gua  1740,  to  whom  reference  is  made  by  \V.  W.  R.  Ball, 
in  his  valuable  memoir  on  Xeivtons  Clasdjicatioii  of  Cuhic 
Carves  (Trans.  L.M.S.  1891). 

Murdoch  {Neivtoni  Genesis  Gurvariim  per  Umbras,  p.  87) 
has  the  remark  : — 

'  Speciem  hanc  No  Vlll  Analogam  apud  Newtonum  deside- 
rari  animadverterat  D.  Nic.  Bernoulli,  quod  me  dim  monuit 
I).  Cramer,  Phil,  et  Math,  apud  Genevenses  Celebris  Professor.' 

V 
CASTEL 

Louis  Bertrand  Castel  (1688-1757),  a  Jesuit  Father,  was 
the  autlior  of  Le  vrai  systcme  de  Newton.  He  became  F.R.S. 
in  1730. 

Stirling's  letter  \ .  2  contains  a  clear  exposition  of  what  he 
understands  by  geometrical  demonstration. 

VI 

CAMPAILLA 

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


NOTES   UPON   THE    CORRESPONDENCE       209 

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

VII 

BRADLEY 

J.  Bradley,  1692-1762,  was  a  distinguished  Astronomer. 
Like  Stirling,  he  studied  at  Balliol  College,  Oxford.  He 
became  F.R.S.  in  1718.  In  1721  he  was  appointed  to  the 
chair  of  Astronomy  in  Oxford,  in  succession  to  Keill.  He 
succeeded  Halley  as  Astronomer  Royal  in  1 742.  He  discovered 
the  aberration  of  the  tixed  stars  and  the  nutation  of  the 
earth's  axis. 

Both  the  letters  here  given  are  to  be  found  in  Rigaud's 
Bradley.  Stirling's  letter  is  taken  from  Rigaud  ;  and  Bradley's 
reply  is  among  the  letters  preserved  at  Garden. 

VIII 

KLINGENSTIERNA 

S.  Klingenstierna  was  Professor  of  Mathematics  at  Upsala. 
It  was  through  Cramer  that  he  was  introduced  to  Stirling 
(cf.  Letter  III.  3).  In  view  of  his  researches  in  Optics,  the 
letter  here  given  is  of  some  interest  He  became  F.R.S. 
in   17.30. 

IX 
JOHN  MACHIN 

John  Machin,  the  astronomer,  became  F.R.S.  in  1710  (the 
same  year  as  Poleni,  Professor  of  Astronomy  at  Padua,  men- 
tioned in  the  postscript  to  IV.  1),  and  acted  as  Sec.  R.S.  from 
1718  to  1747.  He  sat  on  the  committee  appointed  in  1712  to 
investigate  the  dispute  between  Newton  and  Leibniz.  In 
1713  he  became  Professor  of  Astronomy  at  Gresham  College. 
He  died  in  1751. 

Machin  used  the  formula 

tt/  4  =  4  arc  tan  -g-  —  arc  tan  gig 

to  calculate  tt  to  100  places  of  decimals.  His  result  is  given 
(1706)  in  Jones's  Syno2Jds  Palmariorum  Matheseos,  in  which 
the  .symbol  tt  is  first  used  for  the  number  3-14159  .  .  . 


:210       NOTES    LIPOX    THE   COKRESPUNDENCE 

His  'Laws  ot^  tlic  IVIoon's  Motion  aceordiiin^  to  Gravity'  is 
appended  to  Motte's  translation  of  the  Prineipia. 

A  greater  work  on  Lunar  Theory,  begun  in  1717,  was  never 
publislied :  and  relative  manuscripts  are  in  t\\c  possession  of 
the  Royal  Astronomical  Society-. 

Letter  IX.   1. 

Li  connexion  with  this  letter,  which  has  no  date,  see  the 
letters  from  Bernoulli  to  Stirling,  IV. 

Letter  IX.  2. 

Machin  was  keenly  interested  in  the  researches  of  Maclaurin 
and  Stirling  concerning  the  Figure  of  the  Earth,  though  his 
name  does  not  appear  to  find  a  place  in  Todhunter's  Hidory 
of  the  subject. 

The  book  by  Maupertuis  is  probably  one  on  the  Figure  of 
the  Earth  mentioned  by  Todhunter  (vol.  i,  p.  72  . 

Machin,  in  speaking  of  Stirling's  Proposition  concerning 
the  Figure  of  the  Earth,  cannot  refer  to  Stirling's  Memoir 
entitled  '  Of  the  Figure  of  the  Earth  and  the  Variation  of 
Gravity  on  the  Surface  ',  which  appeared  in  the  Phil.  Trans. 
for  1735-6. 

Compare  Stirling's  letter  to  Maclaurin  I.  15,  in  which  he 
refers  to  his  correspondence  with  Machin. 

I  do  not  quite  understand  Machin  in  his  reference  to  the 
invention  of  Eider's  Series,  though  Stirling's  letter,  if  it  could 
be  found,  would  explain. 

By  1738  Stirling  had  got  definitely  settled  as  Manager  of 
the  Lead  Hills  Mines  in  Scotland.  He  had  apparently  com- 
plained to  Machin  how  he  felt  the  isolation  from  his  scientific 
friends  and  their  researches  in  London.  Machin's  letter  to 
him  is  written  in  the  kindliest  spirit  of  warm  friendship. 

The  book  of  De  Moivre  mentioned  in  the  letter  is  doubtless 
the  second  edition  of  the  Doctrine  of  Chanceti  (1738j. 

X 

CLAIRAUT 

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


NOTES   UPON   THE   CORRESPONDENCE       211 

piiblii-lied  his  celebrated  '  Recherclics  siir  les  Courbcs  a  double 
Courbure '.  He  took  part  in  the  expedition  to  Lapland  under 
-Muupertuis  to  determine  the  length  of  the  arc  of  the  meridian. 
He  made  several  contributions  to  the  Theory  of  the  Figure 
of  the  Earth,  which  he  ultimately  embodied  in  the  classic 
work  entitled  Thcorie  de  la  Figure  de  la  Terre  (^1743).  His 
Theorle  de  la  Lune  appeared  in  1765,  shortly  before  his 
death.  He  was  also  the  author  of  Elihnents  de  la  Geometrie 
(1741),  and  of  an  Algebre  (1746i.  He  became  F.R.S.  in  1737. 
He  died  in  17G5. 

'  Clairaut  a  eu  pour  el  eve  et  pour  amle  la  celebre  Marquise 
de  Chatelet,  la  docte  et  belle  Emilie,  qu'il  a  aid^e  dans  sa 
traduction  du  Livre  des  principes'  (Marie,  Hist.  Math.),  a  state 
of  affairs  not  over-pleasing  to  Voltaire. 

In  the  letter  here  given  w^e  find  Clairaut  introducing  himself 
to  Stirling.  Cf.  I.  15.  Clairaut  had  frequent  correspondence 
with  Maclaurin,  and  several  of  the  letters  have  been  preserved. 

XI 
EULER 

Leonhard  Euler  (1707-83)  was  born  at  Bale  in  Switzerland. 
He  studied  Mathematics  under  John  Bernoulli,  having  as 
fellow -students  Nicholas  and  Daniel  Bernoulli,  the  two  sons 
of  John  Bernoulli.  The  two  brothers  were  called  to  Petrograd 
in  1725,  and  Euler  followed  in  1727.  In  1741,  on  the  invita- 
tion of  Frederick  the  Great,  he  went  to  Berlin,  returning 
again  in  1766  to  Petrograd,  where  he  died  in  1783.  For 
almost  the  whole  of  his  second  residence  in  Russia  he  was . 
totally  blind,  l)ut  this  misfortune  had  little  effect  on  his 
wonderful  production  of  mathematical  memoirs.  There  is 
hardly  a  department  of  pure  or  mixed  mathematics  wdiich 
his  genius  has  not  enriched  by  memoirs  of  far-reaching  impor- 
tance. A  complete  edition  of  his  works  has  been  undertaken 
by  a  Swiss  commission. 

We  are  here  only  concerned  with  his  relations  with  Stirling. 
Apparently  Euler  had  opened  the  correspondence  by  a  letter 
to  Stilling,  in  which  he  announces,  inter  alia,  the  theorem 
known  as  the  Euler-Maclaurin  Theorem  (Reiff',  Geschichte  der 
Unendlichen  Reiheii).     1'his  letter  is  not  preserved,  but  copies 

p2 


212       NOTES   UPON   THE   CORRESPONDENCE 

of  tlic  Icttei's  that  passed  between  Euler  and  Stirling  appear  to 
have  been  in  existence  at  Petrograd :  and  Professor  Enestrom 
in  his  Vorldujiges  VerzeicJniiK  der  Brlefe  von  und  an  L.  Eider, 
1726-41,  furnishes  the  following  dates  : 

(1)  Euler  to  Stirling,  9th  June,  17.3G, 

(2)  Stirling  to  Euler,  April,  1738, 

(3)  Euler  to  Stirling,  27th  July,  1738. 

The  letters  preserved  at  Garden  are  doubtless  (2)  and  (3). 

It  remains  to  l)e  seen  whether  the  letters  in  Petrograd  have 
survived  the  fury  of  the  Revolution  in  Russia. 

Stirling's  reply  was  much  belated,  for  his  time  was  now 
entirely  devoted  to  the  successful  development  of  the  Lead 
Hills  Mines,  of  which  he  had  been  appointed  manager  a  ye-AV 
or  two  before.  The  rounli  draft  of  it  is  all  that  Stirling 
preserved,  and  is  here  given  witli  all  his  corrections  and 
erasures.  Stirling  acknowledges  the  ini2:)ortance  of  Euler's 
Theorem,  and  remarks  that  his  own  theorem,  '  Theorema 
meum  ',  for  summing  Logarithms  is  only  a  particular  case. 
He  informs  Euler  that  Maclaurin  has  an  identical  theorem 
in  the  proof-sheets  of  a  Treatise  of  Fluxions  to  appear  shortly. 
At  the  same  time  he  offers  to  communicate  Euler's  results 
to  the  Royal  Society',  and  suggests  that  Euler  should  become 
a  Fellow. 

With  characteristic  modesty  and  absolute  freedom  from 
jealousy,  Euler  in  his  reply  waives  his  claim  to  priority  over 
Maclaurin,  and  proposes  that  the  Royal  Society  should  publish 
a  paper  on  the  Equation  of  liiccati,  which  he  had  sent  some 
3"ears  before  to  Sloane  the  President. 

There  can  be  little  doubt  that  luder  and  Maclaurin  dis- 
covered the  theorem  independently,  and  llie  suggestion  made 
by  Reiff  to  call  it  the  Euler-Maclaurin  Theorem  seems  fully 
justified. 

Maclaurin,  by  the  way,  does  not  refer  to  it  in  tlie  intro- 
duction to  liis  Fluxions,  but  on  p.  C91  of  his  Treatise.  Euler 
first  gave  his  theorem  without  pi-oof  in  his  Metliodus  generalis 
summandi  firoijressiones  (Jomm.  Pctrop.  ad  annos  1732, 1733: 
published  1738. 

The  proof  is  given  in  Invodio  ^ummae  cujusque  seriei  ex 
dato  termino  (jcnerali  (Jomm.  Peirop.,  173(1:  published  1741. 

Compare  Stirling's  letter  to  Maclaurin  I.  15. 


NOTES    UPON    THE    CORRESPONDENCE        213 

I  cannot  lierc  further  discuss  Euler's  letter,  wliicli  is  almost 
encyclopaedic  in  its  rany;e,  save  to  say  that  Stirling  had  sliown 
in    his   Me(h.    J^ijf.    lunv    to    approximate    with    any    desired 


00 


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


SSI  on 


n 

1 


as  77-/6. 

(See  letters  of  Dan  fJernoulli  to  Euler  in  Fuss,  Corr.  Math., 
t^'c.)     As  is  well  known,  Euler  became  F.R.S.  in  174G. 

XII 
M.  FOLKES,  P.R.S.,  TO  STIRLING 

This  is  the  letter  of  latest  date  in  the  correspondence.  It 
conveys  to  Stirling  the  news  that  he  had  been  made  a  member 
of  the  Royal  Academy  of  Science  at  Berlin,  an  honour  which 
has  not  hitherto  been  noted  in  any  of  the  biographies  of 
Stirling. 

May  the  Mr.  Mitchell  who  brings  the  letter  to  Stirling  not 
have  been  Maclaurin's  friend,  better  known  as  Sir  Andrew 
Mitchell,  who  afterwards  became  Ambassador  at  the  court  of 
Frederick  the  Great  ? 


PRINTED    IN    ENGLAND 
AT   THE   OXFORD   UNIVERSITY   PRESS 


Date 

Due 

MAY  1 

;  ?nnfi 

L.    B.    CAT.    NO.    1 

187 

M-'Jffl 


QA2').S6H3 


scni 


3  5002  00228  1975 

Stirling,  James  ,  ,     , ,        j 

James  Stirling;  a  sketch  of  his  life  and 


Science 

QA    29    . 

S69 

A3 

Stirling 

,     James 

!,     1692- 

■1770. 

James    St 

irling