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LIBRARY 

UNiVERSlTY  OF  CALIFORNIA 

PAVl§ 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/correspondenceofOOnewtrich 


This  "0-P  Book"  Is  an  Authorized  Reprint  of  the 
Original  Edition,  Produced  by  Microfilm-Xerography  by 
University  Microfilms,  Inc.,  Ann  Arbor,  Michigan,  1965 


COERESPONDENCE    ^ 


OF 


SIR    ISAAC    NEWTON 

AND 

PROFESSOR  COTES, 


INCI.UD1NO 


LETTERS  OF  OTHER  EMINENT  MEN, 

NOW  FIRST  PUDLISIIED  FROM  THE  ORIGINALS  IN  THE  LIBRARY 
OF  TRINITY  COLLEGE,  CAMBRIDGE; 


TOOETHER  WITH 

AN    APPENDIX, 

OONTAININO 

OTHER  UNPUBLISHED  LETTERS  AND  PAPERS 
BY    NEWTON; 

^\1TH  NOTES,  SYNOPTICAL  VIEW  OF  THE  PHEOSOPHEH'S  LIFE, 

AND 

A  VARIETY  OF  DETAILS  ILLUSTRATIVE  OP  HIS  HISTORY, 


BY 

J.  EDLESTON,  M.A., 

FELLOW   OP  THIKTTY  OOLLEOE,  CAMBRIDGE. 


LONDON: 

JOHN   W.  PARKER,  WEST  STRAND. 

CAMBRIDGE :  JOHN  DEIGHTON. 


M.DCCO.L. 


LIBRARY 

TTNTVERSITY  OF  CALIFORNIA 


./ 


X 


19r(nUti  St  ttt  dntOtrsits  Vrcss. 


CONTENTS. 


Words,  or  parts  of  words,  enclosed  wUhIn  {  }  hare,  with  on©  exception  in  p.  218, 
been  added  by  the  Editor  for  the  purpose  of  illiutmtion  or  to  supply  omissions  in 
the  original  MS. 


Preface         

Synoptical  View  op  Newton's  Lipr        ..... 

N'oTEs xli 


PAOB 

ix 
xxi 


Newton's  Dividends  and  Weeks  of  Rteidenco 

Exits  and  Redits    .        .        .        . 

Buttery  Bills       .... 

Lectures  on  Optics 

Algebra    . 

Motion 

System  of  the  World 


LETTEn 

I. 

Bentley  to  Cotes,  May  21, 1700 

11. 

Cotes  to  Newton,  Aug.  18,  ..... 

III. 

Newton  to  Cotes,  Oct.  11,  

IV. 

Cctes  to  Newton,  April  15, 1710 

V. 

30,  

VI. 

Newton  to  Cotes,  May    1, 

VII. 

Cotes  to  Newton, 7,  

VIII. 

Newton  to  Cotes, 13,  

iX. 

Cotes  to  Newton, 17,  

X. 

20,  

XI. 

Newton  to  Cotes, 80, 

XII. 

Cotes  to  Newton,  June    1,  ...... 

XIII. 

Newton  to  Cotes, 8,  

XIV. 

Cotes  to  Ne\vton, 11,  

XV. 

Newton  to  Cotes, 15,  

XVI. 

Cotes  to  Newton, 30, 

Ixxxii 
Ixxxv 

Ixxxvi 
xci 
xcii 
xcv 

ycviil 


1 

3 

4 

8 

12 

14 

10 

10 

20 

24 
25 
27 
20 
30 
31 


415479 


IV 


CONTENTS. 


LfTTEn 

XVII.  Nowton  to  Cotos,  Juno  31,1710. 

XVIII.  Cotes  to  Newton,  Sept.   4, 

XIX.  Newton  to  Cotes, 13,  

XX.  Cotes  to  Newton, 21, 

XXI.  Newton  to  Cotes, 30,  

XXII Oct.  27, 

XXIII Mar.  24, 1711. . 

XXIV Juno   7, 

XXV.        Cotes  to  Newton, 9, . 

XXVI.  Newton  to  Cotes, 10, 

XXVII.  Cotes  to  Newton, 23, 

XXVIII July  19,  

XXIX.  Newton  to  Cotea?^,. . .  S8,  . ...... 

XXX.  Cotes  to  Newton, 30,  

XXXI.        Sept.    4, 

XXXII.  Newton  to  Cotes,  Feb.    2, 1712 

XXXIII.  Cotes  to  Newton, 7. 

XXXI V.  Newton  to  Cotes, 12, 

XXXV.  Cotes  to  Newton, IG, 

XXXVI.  Newton  to  Cotes, 19, 

XXXVII.  Cotes  to  Newton, 23, 

XXX VI 1 1.     Nowton  to  Cotes, 2G, 

XXXIX.      Cotes  to  Newton. 20, 

XL.  Cotes  to  Newton,  Mar.  13, 

XLI.         Newton  to  Cotes, 18, 

XLII April   3, 

XLIII 0, 

XLIV.        Cotes  to  Newton, 14,  

XLV 16, 

XLVI.       Newton  to  Cotes, 22, 

XLVI  I.       Cotes  to  Newton, 24, 

XLVIII 20, 

XLIX.        Newton  to  Cotes, 24,  (?  29) 

L.  Cotes  to  Newton,  May  day, 

LI May  3,  

LII.         Newton  to  Cotes, 10,  

Lin.        Cotes  to  Newton, 13,  

LIV.         25,  

LV.         Newton  to  Cotes, 27,  


CONTENTS. 


I.K1TI!R 

FAOI 

LVI. 

Cotes  to  Newton,  July  20, 1712 

118 

LVII. 

Aug.  10 

121 

LVIII. 

Newton  to  Cotes, 12,  

122 

LIX. 

l^.aH) 

12G 

LX. 

Cotes  to  Newton, 17, 

127 

LXI. 

Newton  to  Cotes, 2G, 

120 

LXII. 

Cotes  to  Newton, 20 

132 

LXIII. 

Newton  to  Cotes,  Sept.   2, 

134 

LXIV. 

Cotes  to  Newton, 7, 

13(5 

LXV. 

Newton  to  Cotes,  Sept.  13,  

139 

LXVI. 

Cotes  to  Newton, 15,  ...... 

140 

LXVII. 

Newton  to  Cotes, 23,  

141 

LXVIII. 

, Oct.  14,  

LXIX. 

; 21 

143 

LXX. 

Cotes  to  Newton,  ..;  ..  23,  

... 

LXXI. 

Nov.   1 

144 

LXXII. 

23,  ...... 

145 

LXXIII. 

Newton  to  Cotes,  Jan.    G,  1713 

... 

LXXIV. 

Cotes  to  Newton,  13,  

146 

LXXV. 

Newton  to  Cotes,  Mar.  2,  

147 

LXXVI. 

Newton  and  Bentlcy  to  Cotes,  Mar.  5, 1713 

148 

LXXVII. 

Cotes  to  Newton,  Mar.  8, 1713 

... 

LXXVIII. 

Cotes  to  Bentley,  10, 

•        •        •        • 

149 

LXXIX. 

Bentley  to  Cotes,  12,  

150 

LXXX. 

Cotes  to  Newton,  Feb.  (?  March)  18, 1713 

151 

LXXXI. 

Newton  to  Cotes,  Mar.  28, 1713 

154 

LXXXII. 

31 

. 

156 

LXXXIII. 

Cotes  to  D'  Clarke,  June  25, 

158 

LXXXIV. 

Newton's  Paper  of  Corrigenda  and  Addenda,  Dec  1713 

160 

LXXXV. 

Cotes  to  Newton,  Dec.  22, 1713 

166 

LXXXVI. 

Newton  to  Keill,  (respecting  a 

proposed  answer  to 

a  Leibnizlan  paper)  Apr.  2, 1714       .... 

169 

LXXXVII. 

Newton  to  Keill,  April  20, 1714 

•        1        •        • 

•170 

LXXXVIII. 

May  11, 

174 

LXXXIX. 

15, 

•        .        •        • 

176 

•  Since  the  sheet  containing  note  "  p.  171  was  printed  off,  I  have  wen  a  paper  by 
Brinkley  on  the  origrin  of  the  error  in  Newton's  lat  solution  of  the  refllstance  problem 
{Royal  Irish  Trantaetions,  1810,  p.  45)  in  which  the  mistake  is  traced  to  its  true  source. 


VI 


CONTENTS. 


tKrrER 

Xa  Cotes  to  Newtdi  (after  Apr.  25, 1716) 

XCL  May  13, 1716   . 

XCII.  Halley  to  KeiU,  Oct.  3, 

XCIII.  Newton  to  Keill,  May  2, 1718    . 

XCIV.  Newton  to  Arland,  Oct.  22, 1722    . 

XCV.  Cotes  to  his  Uncle  Smith,  Doc.  31,  1698 

XCVI.  Smith  to  Cotes,  Aug.  30, 1701 

XCVII.  Cotes  to  Smith,  Sept.  9, 

XCVIII Feb.  10, 1708 

XCIX Nov.  30, 1710    . 

C.  Cotes  to  Ewer,  Dec.  26  or  27,  1710 

CI.  Cotes  to  Ilallcy  (exact  dato  uncertain) 

CII.  Cotes  to  Jones,  Feb.  16, 1711 

cm.  Jones  to  Cotes,  Sept.  17, 

CIV.  Cotes  to  Jones, 30, 

CV.  Jones  to  Cotes,  Oct.  26, 

CVI.  Cotes  to  Jones,  Nov.  11, 

CVII.  Jones  to  Cotes, 16, 

CVIII.  Cotes  to  Jones, 25, 

CIX.  Jones  to  Cotes,  Jan.  1, 1712 

ClX.(bis)  Newton  to  J.  Smith,  May  8, 1676 

ex.  Cotes  to  Jones,  Jan.  1712 

CXI.  Jones  to  Cotes,  Feb.  6, 1713   . 

CXII.  Cotes  to  Jones, 13, 

CXIII.  Jones  to  Cotes,  Apr.  29, 

CXIV.  Cotes  to  Jones,  May  3, 

CXV.  Jones  to  Cotes,  July  11, 

CXVI.  Cotes  to  Whiston,  March,  1716 

CXVII.  Cotes  to  Lord  Trevor,  Jan.  10, 1716 

CXVIII.  Cotes  to  Dannye,  March  15, 1716 

CXIX.  Brook  Taylor  to  Prof.  Smith,  Nov.  27, 1718 

CXX Dec.  11,  .. 

Notice  of  three  Letters  from  Taylor  to  Keill 

CXXI.  Voltaire  to  Prof.  Smith,  Oct.  10  (N.  S.),  1739 

CXXII.  Duke  of  Cumberland  to  Prof.  Smith,  July  3, 1740 


CONTENTS. 


Til 


APPENDIX. 


I.        Oldenburg  to  Newton,  Jan.  2, 1072     . 
II.       Newton  to  Oldenburg,  Mar.  10,  ..... 

Ill 19. 

IV 20. 

V ......30 

VI.       Apr.  13, 

VII Junell, 

VIII July30, 

VIII.(bis)01denburg  to  Newton, 10, . 

IX.  Newton  to  Oldenburg,  June  23, 1073     .... 

X.  Examples  of  Algebraical  Reduction  given  by  Newton  to 

Flamatced  at  Lecture,  Midsummer,  1074 

XI.       Newton  to  Oldenburg,  Nov.  13, 1076      . 

XII 30 

XIII May  11, 1070      . 

XIV Aug.22, 

XV.      Oct.  20, 

XVI Nov.14, 

XVII.    Newton  to  D'  Maddock.  Feb.  7, 1670      . 
XVIII.  Newton  to  Hooke,  Dec.  3, 1080  .       .        . 

XIX.    Newton  to  D'  Briggs,  June  20,  1682,  on  Vision 

XX Sept.  12, 

XXI. Apr.  26, 1085 

XXI  I.    Directions  given  by  Newton  to  Bentley  respecting  his  mathe- 

matical  reading,  1091    . 
XXII  I.   Newton  to  Locke,  July  7, 1092      . 
XXIV.    Newton  to  Leibniz,  Oct.  10, 1093 

XXV.     Newton  to  Hawes,  May  25, 1094,  on  Course  of  Mathematical 
Studies  at  Clmst's  Hospital      .        .        .        .        . 

XXVI 20, 

XXVII June  14,  1096    . 

XXVIII.Wallis  to  Newton,  AprU  10 

XXIX.  Newton  to  Harington  (a  young  Oxford  student)  May  30, 1098 

XXX.   Paper  by  Newton  on  Time  of  Vernal  Equinox,  Apr.  1700 

a  2 


Till  CONTENTS. 

PAIIE 

XXXI.  Nowton  to  hio  coutiin.  Sir  John  Newton,  Apr.  1707    .        .        307 

XXXII.  Critique  by  Newton  on  Leibniz's  three  Papers,  1712? 

XXXIII.  Newton's  Abstract  of  Paper  on  the  different 'forms  of  the 

year  in  use  among  the  nations  of  antiquity,  Nov.  1713    .        314 

XXXIV.  Newton  to  Lord  Townshend  on  a  criminal  under  sentence 

ofdeath,  Aug.  24, 1724 316 


PREFACE. 


Newton's  PhilosopMce  Naturalis  Principia  Mathema^ 
tica,  the  most  remarkable  production  of  the  human  intel- 
lect that  has  yet  been  seen  on  the  earth,  whose  mysterious 
path  through  space  was  first  explained  in  its  pages,  was 
published  about  the  middle  of  the  year  1C87,  a  few 
weeks  after  his  appearance  before  James's  Ecclesiastical 
Commission,  as  the  upholder  of  the  rights  of  his  Univer- 
sity and  the  laws  of  the  realm,  against  the  aggressions  of 
arbitrary  power.  We  are  not  informed  how  many  copies 
of  the  work  were  printed,  but  the  number  probably  was 
not  large.  If  the  extent  of  the  impression  had  been 
rigorously  limited  to  the  number  of  persons  likely  to 
comprehend  its  contents,  the  volume  would  now  have 
been  one  of  excessive  rarity.  The  work,  however,  seems 
to  have  found  a  readier  sale  than  the  abstruse  nature  of 
the  subject  and  the  engrossing  interest  of  politics  at  that 
crisis  of  our  history  might  have  prepared  us  to  expect ; 
and  the  sensation  which  it  produced  was  long  remem- 
bered, even  by  those  who  saw  but  darkly  that  tlie  veil 
was  now  raised  from  the  face  of  nature,  which  succes- 
sive generations  of  philosophers,  from  the  first  dawn  of 
science,  had  vainly  endeavoured  to  draw  aside.  It  is 
true  that,  in  a  legal  argument  by  Lord  Mansfield,  when 
Solicitor-General,  the  names  of  Locke  and  Newton  are 
coupled  with  that  of  the  author  of  Paradise  Lost,  as 
affording  instances  of  the  neglect   shewn  to  works  of 


X  PREFACE. 

genius  for  a  considerable  time  after  their  being  given  to 
the  world.  Dugald  Stewart  has  assigned  good  reasons 
for  doubting  the  correctness  of  the  statement  with  re- 
spect to  the  Essay  on  the  Human  Understanding,  and 
I  believe  the  assertion  to  be  equally  unfounded  as  predi- 
cated of  the  Princi'pia,  except  so  far  as  the  slow  recep- 
tion of  the  Newtonian  doctrines,  in  some  parts  of  the 
continent,  may  be  considered  as  supplying  ground  for 
affirming  the  fact.  Doubtless  there  were  others  besides 
Locke  who  tried  to  master  the  first  principles,  read  the 
enunciations  of  the  propositions,  and  accepted  them 
either  on  the  faith  of  the  author's  own  word,  or  in  re- 
liance upon  the  judgment  of  some  known  mathematician; 
nor  was  Bentley,  we  may  rest  assured,  the  only  person 
in  that  inquisitive  age  who  was  struck  with  the  wonder- 
ful truths  developed  by  the  new  philosophy,  and  strove 
to  attain  to  an  intellectual  appreciation  of  them.  Locke's 
more  popular  book  appeared  in  1G90,  and  a  second  edi- 
tion was  published  in  1G04.  The  Principia  seems  to 
have  been  sold  off  with  almost  equal  rapidity.  In  1691 
we  hear  of  an  improved  edition  of  it  as  being  in  contem- 
plation. In  1C94  Newton  renewed  his  attack  on  the 
lunar  and  planetary  theories  with  a  view  to  a  new  edi- 
tion of  his  book.  And  if  Flamsteed,  the  Astronomer- 
Royal,  had  cordially  co-operated  with  him  in  the  humble 
capacity  of  an  observer  in  the  way  that  Newton  pointed 
out  and  requested  of  him,  (and  for  his  almost  unpardon- 
able omission  to  do  so  I  know  of  no  better  apology  that 
can  be  offered  than  that  he  did  not  understand  the  real 
nature  and,  consequently,  the  importance  of  the  re- 
searches in  which  Newton  was  engaged,  his  purely  empi- 
rical and  tabular  views  never  having  been  replaced  in  his 


PREFACE.  xi 

mind  by  a  clear  conception  of  the  Principle  of  Universal 
Gravitation,)  the  lunar  theory  would,  if  its  creator  did 
not  overrate  his  own  powers,  have  been  completely  in- 
vestigated, so  far  as  he  could  do  it,  in  the  first  few 
months  of  1695,  and  a  second  edition  of  the  Principia 
would  probably  have  followed  the  execution  of  the  task 
at  no  long  interval.  But  science  and  the  world  were  not 
destined  to  such  good  fortune.  Flamsteed's  infirmities 
of  temper  and  bodily  health  conspired  to  thwart  Newton's 
plans  for  the  first  half  of  the  year  just  mentioned ;  and 
the  imperfect  manner  in  which  the  Astronomer-Royal 
then  met  his  wishes,  leaves  it  uncertain  whether  we  are 
to  attribute  the  entire  blame  of  the  non-completion  of 
the  lunar  theory  in  the  latter  half  of  the  year  to  the 
circumstance  of  steps  being  at  last  taken  by  Newton's 
friends  to  provide  for  his  material  interests.  His  ap- 
pointment to  the  Wardenship  of  the  Mint  in  March, 
1696,  was  a  bar  to  the  further  prosecution  of  his  re- 
searches in  physical  astronomy.  Henceforward  his  ofii- 
cial  duties  made  it  impossible  for  him  to  work  continuously 
at  his  former  pursuits ;  his  studies  in  mathematics  and 
natural  philosophy  were  by  snatches  and  in  the  intervals 
of  business.  We  shall  accordingly  find,  when  at  length 
his  consent  to  a  new  edition  o^i\\Q  Principia  was  wrung 
from  him,  that  his  necessary  avocations  seriously  inter- 
fered with  the  progress  of  the  work  through  the  press. 
But  his  removal  to  a  new  sphere  of  labour  did  not  abate 
his  zeal  for  the  promotion  of  science :  the  starving 
mathematician  found  in  him  a  kind  and  liberal  patron, 
and  he  was  always  ready  with  his  purse  and  counsel  to 
encourage  any  rational  attempt  to  extract  from  nature 
more  of  her  secrets. 


xii  PREFACE. 

Probably  as  good  an  idea  raay  be  formed  of  the 
actual  feeling  which  prevailed  with  reference  to  the 
demand  for  a  republication  of  the  Principia,  until  the 
time  when  a  new  edition  was  finally  determined  upon, 
as  would  be  conveyed  by  any  description  that  I  could 
give,  if  I  cite  a  few  notices  referring  to  the  subject, 
extracted  from  various  contemporary  letters  and  journals. 

1691  Dec.  1 8.     Fatio  writing  to  Huygens  from  London 

says:  "  Mr.  II  est  assez  inutile  de  prier  M""  Newton 
de  faire  une  nouvelle  edition  de  son  livre.  Je 
I'ai  importune  plusieurs  fois  sur  ce  sujet,  sans 
I'avoir  jamais  pu  flechir,  Mais  il  n'est  pas  im- 
possible que  j'entreprenne  cette  edition ;  a  qiioi 
je  me  sens  d'autant  plus  portd,  que  je  ne  crois 
pas  qu'il  y  ait  personne  qui  entende  a  fond  une 
si  grande  partie  de  ce  livre  que  moi,  graces  aux 
peines  que  j'ai  prises  et  au  temps  que  j'ay 
employe  pour  en  surmonter  I'obscurite.  D'ail- 
leurs  je  pourrois  facilement  aller  faire  un  tour 
k  Cambridge,  et  recevoir  de  Mr  Newton  meme 

Texplication  de  ce  que  je  n'ai  point  entendu " 

Again,  on  Feb.  5,  1692  he  writes :  "  Je  n'ai  encore  ni  aban- 
donne,  ni  embrasse  absolutement  la  pensee  de 
faire  une  seconde  edition  du  livre  de  Mr  New- 
ton." 

1692  "  Mr  Newton  is  preparing  a  new  System  of  Philo- 

sophy, which  will  be  much  larger  and  plainer 
than  his  Principia  Philosophiaj  Naturalis  Phy- 
sico-Mathcmatica."  (De  la  Croze's  Works  of 
the  Learned  for  Jan.  169^.  p.  269,  under  the  head 
of  "  Cambridge.") 
"According  to  the  best  of  our  advices  nothing 
considerable  is  doing  new  at  Cambridge,  but 


PREFACE.  xiii 

Mr  Newton's  new  System  of  PhUoPophy,  and 
Mr  Barnes's  edition  of  Euripides."  (Id.  for 
March  and  April  1692,  p.  398.) 

1694  May  29.  Huygcns,  in  a  letter  to  Leibniz,  speaks  of 
**la  nouvelle  edition*'  of  the  .Frincipia,  "que  doit 
procurer  D.  Grcgorius." 
Nov.  1.  "I  desire  only  such  observations  as  tend 
to  perfecting  the  theory  of  the  planets,  in  order 
to  a  second  edition  of  my  book."  Newton  to 
Flamsteed  (Baily,  p.  138.) 

1697  Dublin,  Nov.  4.  "I  hear  Mr  Newton's  Phil.  Nat, 
Prin,  Math,  is  out  of  press,  and  that  he  designs 
a  2nd  Edition.  Pray  advise  him  to  make  it  a 
little  more  plain  to  Readers  not  so  well  versed 
in  Abstruse  Mathematicks,  a  few  Marginal  Notes 
and  references  and  Quotations  would  doe  the 
business."  (P.  S.  to  a  letter  from  W.  Molyneux 
to  Sloane.      Orifj.  Lett.  Dk.  Roy.  Soc.  M.  i.  99.) 

1699  July  15.  J.  Monroe,  writing  from  Paris,  says  that 
Malebranche  **  mightily  commends  ISIr  Newton, 
adding  at  the  same  time  that  there  were  many 
things  in  his  book  that  passed  the  bounds  of  his 
penetration,  and  that  he  would  be  very  glad  to 
see  Dr  Gregory's  critick  upon  it."  Orig.  Lett. 
Bk.  iloy.  Soc.  M.  ii.  10.  (Comp.  Addison's  ac- 
count of  his  visit  to  Malebranche  at  Paris,  in 
the  latter  half  of  the  year  1700.  "His  book  is 
now  reprinted  with  many  additions,  among  which 
he  shewed  me  a  very  pretty  hypothesis  of  colours, 
which  is  diflcrent  from  that  of  Cartesius  or  ^Mr 
Newton,  tho  they  may  all  three  be  true.  He 
very  much  praised  Mr  Newton's  mathematics, 
shook  his  head  at  the  name  of  Hobbcs  and  told 
me  he  thought  him  a  pauvre  esprit.'*  Letter  to 
Bp.  Hough  from  Lyons,  Aikiii's  Life,  i.  91.) 


XIV  PREFACE. 

1700  Febr.  ,|.      "  tTai  appris   aussi  (je  ne  s^ai  ou)  qu'il 

donnera  encore  quelque  chose  sur  le  mouvement 
de  la  Lune;  et  on  m'a  dit  aussi  qu'il  y  aura 
une  nouvelle  edition  de  scs  principes  de  la 
nature."  (Leibniz  to  T.  Burnet,  Opp,  Tom.  vi. 
pars  I.  p.  2GG.) 
July  4.  "  The  Royal  Society  have  laboured  to  get 
his  Theory  of  the  Moon,  Book  of  Colours  &c. 
printed,  but  his  excessive  modesty  has  hitherto 
hindered  him,  but  the  Society  will  do  what 
further  they  can  ivith  him."  (Sloane  to  Leibniz, 
Orig,  Lett.  Bh,  Roy.  Soc.  S.  ii.  14.) 

1701  In  some  MS.  memoranda  by  David  Gregory,  dated 

Oxon.  21  May,  of  a  variety  of  points  upon  which 
he  wished  to  consult  Newton  we  find  the  follow- 
ing :  "  To  see  if  he  has  any  design  of  reprinting 
his  Principia  Mathematica  or  any  other  thing." 
(Rigaud,  Appendix  to  Essay,  p.  80.) 

{1702,  Monday}  Nov.  30.  "He  owns  there  are  a  great 
many  faults  in  his  book,  and  has  crossed  it, 
and  interleaved  it,  and  writ  in  the  margin  of 
it,  in  a  great  many  places.  It  is  talked  he 
designs  tcr=Teprint  itr  though  he  would  not 
own  it.  I  asked  him  about  his  proof  of  a 
vacuum,  and  said  that  if  there  is  such  a  matter 
as    escapes    through    the   pores   of  all   sensible 

bodies,  this  could  not  be  weighed I  find  he 

designs  to  alter  that  part,  for  he  has  writ  in 
the  margin.  Materia  sensibilis ;  perceiving  his 
reasons  do  not  conclude  in  all  matter  what- 
soever." Bd.  Greves  to  Lord  Aston  (Tixall 
Letters,  n.  152),  giving  an  account  of  a  visit 
which  he  had  paid  to  Newton  the  preceding 
Thursday    in   company   with   Sir    E.   Southcotc 


PREFACfc.  XY 

at  the  request  of  Lord  Aston,  "a  great  lover 
of  the  mathematics,  who  would  gladly  be  satis- 
fied in  a  difficulty  or  two  of  that  science." 
1704  Nov.  15.  "  The  book  {Newton's  Optics}  makes 
no  noise  in  town,  as  the  Principia  did,  which 
I  hear  he  is  preparing  again  for  the  press  with 
necessary  corrections."  (Flamsteed  to  Pound, 
Greenwich  MSS.  xxxiii.  81.) 

The  book  had  now  become  extremely  scarce,  and 
proportionately  dear.  Sir  William  Browne,  who  took 
his  B.A.  degree  in  1711,  states  that  when  he  was  at 
Cambridge,  he  gave  two  guineas  for  a  copy,  "  which  was 
then  esteemed  a  very  cheap  purchase."  (Speech  at 
Royal  Society,  Nov.  19,  1772,  when  he  was  eighty  years 
of  age,  in  Nichols's  Literary  Anecdotes,  iii.  322.)  Its 
original  price  seems  to  have  been  10^.  At  last,  in  the 
beginning  of  1709,  Bentley's  importunity  prevailed  over 
the  scruples  of  the  author,  and  induced  him  to  entrust 
the  superintendence  of  a  new  edition  to  the  care  of  a 
promising  young  mathematician,  Roger  Cotes,  Fellow 
of  Trinity  College,  and  recently  appointed  Professor  of 
Astronomy  and  Experimental  Philosophy.  "  Itaque  cum 
Exemplaria  prions  Editionis  rarissima  admodum  et  im- 
mani  pretio  coemcnda  suporcssent;  suasit  llle  crebris 
efflagitationibus  ct  tantum  non  objurgando  perpulit  dcni- 
que  Virum  Praistantissimum,  ncc  modestia  minus  quara 
eruditione  summa  Insignem,  ut  novam  hanc  Operis  Edi- 
tioncm,  per  omnia  elimatam  dcnuo  et  ogrcgiis  insupcr 
acccssionibus  ditatam,  suis  sumptibus  et  ausplciis  prodire 
pateretur:  Mihi  vero,  pro  jure  suo,  pensum  non  ingra- 
tum  demandavit,  ut  quam  posset  emendate  id  fieri  cura- 
rcm."     (Cotes,  Pref.  to  2nd  ed.)     In  a  letter  to  Professor 


XVI  PREFACE. 

Sike,  dated  March  31,  1706  (the  true  date  of  which,  I 
apprehend,  from  internal  and  external  evidence,  which 
it  is  not  necessary  to  adduce  here,  to  be  1709),  Bentley 
says:  **Pray  tell  Professor  Cotes,  that  the  book  in  your 
parcel,  directed  to  him,  is  presented  by  Sir  Isaac  New- 
ton ;  let  him  read  it  over  with  care,  and  I  will  tell  him 
further  of  it  in  a  particular  letter.  The  bundle  of  wood 
cuts  were  found  by  Sir  Isaac  in  his  study,  some  of  which 
he  thinks  may  belong  to  the  future  sheets  of  his  book. 
In  the  printed  book  are  folded  the  MS.  sheets  that  Sir 
Isaac  has  now  finished."  (Bentley's  Correspondencey 
p.  231.  Lond.  1842.)  The  book  here  alluded  to  was  pro- 
bably a  copy  of  the  Pinncipia,  containing  Newton's  MS. 
corrections  and  additions.  This  does  not  seem  to  have 
been  the  copy  from  which  the  second  edition  was  printed, 
unless  it  was  sent  back  to  Newton  for  further  modifica- 
tion. In  May  following,  Cotes  received  intimation  from 
Bentley  that  Newton  would  be  glad  to  see  him  in  town, 
and  to  put  into  his  hands  part  of  his  revised  copy  of  the 
Principia,  The  reader  is  now  at  the  point  where  the 
Correspondence  now  offered  to  the  public  commences. 
This  Correspondence,  consisting  of  the  letters  which 
passed  between  Newton  and  Cotes  relative  to  questions 
that  arose  connected  with  the  new  edition  of  the  Prin- 
cipia, in  the  course  of  its  passage  through  the  press,  is 
preserved,  with  some  of  the  MS.  sheets  of  Newton's  in- 
terleaved copy  of  the  first  edition,  and  various  mathe- 
matical papers  in  Cotes's  handwriting,  in  the  library  of 
Trinity  College.  It  was  "collected  from  amongst  the 
loose  papers  bequeathed "  by  Dr  Robert  Smith  to  the 
Rev.  Edward  Howkins,  Fellow  of  Trinity  College,  who 
in  1779  demised  the  Collection,  with  a  profile  of  New- 


PREFACE.  XY 

at  the  request  of  Lord  Aston,  "a  great  lover 
of  the  mathematics,  who  would  gladly  be  satis- 
fied in  a  difficulty  or  two  of  that  science." 
1704  Nov.  15.  "The  book  {Newton's  Optics}  makes 
no  noise  in  town,  as  the  Principia  did,  which 
I  hear  he  is  preparing  again  for  the  press  with 
necessary  corrections."  (Flamstced  to  Pound, 
Greenwich  MSS.  xxxiii.  81.) 

The  book  had  now  become  extremely  scarce,  and 
proportionately  dear.  Sir  AVilliam  Browne,  who  took 
his  B.A.  degree  in  1711,  states  that  when  he  was  at 
Cambridge,  he  gave  two  guineas  for  a  copy,  "  which  was 
then  esteemed  a  very  cheap  purchase."  (Speech  at 
Royal  Society,  Nov.  19,  1772,  when  he  was  eighty  years 
of  age,  in  Nichols's  Literary  Anecdotes^  iii.  322.)  Its 
original  price  seems  to  have  been  10^.  At  last,  in  the 
beginning  of  1709,  Bentley's  importunity  prevailed  over 
the  scruples  of  the  author,  and  induced  hira  to  entrust 
the  superintendence  of  a  new  edition  to  the  care  of  a 
promising  young  mathematician,  Roger  Cotes,  Fellow 
of  Trinity  College,  and  recently  appointed  Professor  of 
Astronomy  and  Experimental  Philosophy.  "  Itaque  cum 
Exemplaria  prioris  Editionis  rarissima  admodum  et  im- 
niani  prctio  cocmcnda  supcrcssent;  suasit  111c  crebris 
eflflagitationibus  ct  tantum  non  objurgando  perpulit  deni- 
que  Virum  Pra}stantissimum,  nee  modestia  minus  quam 
eruditione  sunima  Insiguem,  ut  novam  banc  Operis  Edi- 
tioncm,  per  omnia  elimatam  dcnuo  ct  cgrcgiis  insuper 
acccssionibus  ditatam,  suis  sumptibus  et  ausplciis  prodire 
pateretur:  Jlihi  vero,  pro  jure  suo,  pensum  non  ingra- 
tura  demandavit,  ut  quam  posset  emendate  id  fieri  cura- 
rcm."     (Cotes,  Pref.  to  2nd  ed.)     In  a  letter  to  Professor 


XVI  PREFACE. 

Sike,  dated  March  31,  1706  (the  true  date  of  which,  I 
apprehend,  from  internal  and  external  evidence,  which 
it  is  not  necessary  to  adduce  here,  to  be  1709),  Bentlcy 
says:  **Pray  tell  Professor  Cotes,  that  the  book  in  your 
parcel,  directed  to  him,  is  presented  by  Sir  Isaac  New- 
ton ;  let  him  read  it  over  with  care,  and  I  will  tell  him 
further  of  it  in  a  particular  letter.  The  bundle  of  wood 
cuts  were  found  by  Sir  Isaac  in  his  study,  some  of  which 
he  thinks  may  belong  to  the  future  sheets  of  his  book. 
In  the  printed  book  are  folded  the  MS.  sheets  that  Sir 
Isaac  has  now  finished."  (Bentley's  Correspondence^ 
p.  231.  Lond.  1842.)  The  book  here  alluded  to  was  pro- 
bably a  copy  of  the  Principia,  containing  Newton  s  MS. 
corrections  and  additions.  This  does  not  seem  to  have 
been  the  copy  from  which  the  second  edition  was  printed, 
unless  it  was  sent  back  to  Newton  for  further  modifica- 
tion. In  May  following,  Cotes  received  intimation  from 
Bentley  that  Newton  would  be  glad  to  see  him  in  town, 
and  to  put  into  his  hands  part  of  his  revised  copy  of  the 
Princ'Jpia.  The  reader  is  now  at  the  point  where  the 
Correspondence  now  offered  to  the  public  commences. 
This  Correspondence,  consisting  of  the  letters  which 
passed  between  Newton  and  Cotes  relative  to  questions 
that  arose  connected  with  the  new  edition  of  the  Prin- 
cipia,  in  the  course  of  its  passage  through  the  press,  is 
preserved,  with  some  of  the  MS.  sheets  of  Newton's  in- 
terleaved copy  of  the  first  edition,  and  various  mathe- 
matical papers  in  Cotes's  handwriting,  in  the  library  of 
Trinity  College.  It  was  "collected  from  amongst  the 
loose  papers  bequeathed"  by  Dr  Robert  Smith  to  the 
Rev.  Edward  Ilowkins,  Fellow  of  Trinity  College,  who 
in  1779  demised  the  Collection,  with  a  profile  of  New- 


PREFACE.  xvii 

ton,  a  lock  of  his  hair,  and  other  objects  of  interest,  to 
the  Society.  The  papers  had  come  into  Smith's  posses- 
sion on  the  death  of  Cotes,  who  was  his  cousin.  In  their 
original  state  they  contained  among  other  things,  which 
were  afterwards  lost,  about  twenty  or  thirty  letters, 
written  by  Newton  to  Cotes  "  during  the  printing  of  the 
2nd  edition  of  the  Principia"  which  were  borrowed 
from  Smith  by  Conduitt,  who  was  collecting  materials 
for  a  Life  of  Newton,  and  were  never  returned.  They 
will,  I  suppose,  be  found  among  the  papers  which  have 
descended  with  other  property  of  Newton's  from  his 
niece,  Catharine  Barton  (who  married  Conduitt),  to  the 
Earl  of  Portsmouth.  Smith,  in  1757,  endeavoured,  with 
the  assistance  of  a  friend,  to  obtain  a  clue  to  these  letters 
which  had  belonged  to  him,  and  instituted  inquiries, 
which  were  equally  unsuccessful,  respecting  a  common- 
place book  of  Newton's,  "bound  in  green  parchment," 
which  he  had  formerly  seen  in  the  hands  of  William 
Jones,  the  father  of  the  celebrated  orientalist.  Some 
correspondence  which  took  place  with  reference  to  this 
subject  is  bound  up  with  the  Newtonian  Letters  and 
Papers. 

The  late  Mr  Kidd,  in  1796,  saw  in  the  possession  of 
the  Rev.  Thomas  Jones,  Fellow  of  Trinity  College,  a  copy 
of  the  Principia,  "  with  an  astonishing  quantity  of  addi- 
tions and  corrections"  in  Newton's  hand.  "Numerous 
loose  papers  of  4to  form  covered  with  diagrams  and 
writing  were  placed  between  the  leaves  in  different  parts 
of  the  volume,"  which  contained  also  "a  loose  copy  of 
Ilalley's  laudatory  verses  on  the  Principia,  corrected 
throughout  by  the  hand  of  D*"  Bentley."  Jones  stated 
that  this  interesting  volume  was  given  to  him  by  Mr 


xviii  PREFACE. 

Davies,  Senior  Fellow  of  Trinity  College,  who  received 
it  from  Smith,  and  he  from  Newton.  All  attempts  that 
have  been  recently  made  to  discover  its  existence  have 
hitherto  failed.  I  am  inclined  to  think  that  it  may  be 
the  identical  volume  alluded  to  in  Bentley's  letter  to 
Sike,  quoted  above,  in  which  case  a  link  must  be  in- 
serted in  the  chain  of  its  transmission  between  Newton 
and  Smith. 

Of  the  other  letters  in  the  Trinity  College  Newtonian 
Collection  which  have  been  admitted  into  this  publica- 
tion, those  which  were  not  written  by  or  to  Newton 
will  bo  foui\d,  with  few  exceptions,  to  refer  to  him  in 
some  way  or  other,  and  to  throw  light  upon  the  scien- 
tific history  of  the  time. 

The  Appendix  contains  various  letters  and  papers, 
of  more  or  less  interest,  from  Newton's  pen,  collected 
principally  from  original  sources.  For  details  of  these, 
and  of  other  matter  which  is  placed  before  the  Corre- 
spondence, the  reader  is  referred  to  the  Table  of  Con- 
tents. 

The  Portrait  which  accompanies  this  Work  is  taken, 
by  the  obliging  permission  of  the  Master  and  Fellows  of 
Magdalen  College,  from  an  original  drawing  in  Indian 
ink,  which  is  preserved  in  the  Pepysian  Collection.  It 
is  uncertain  when  Pepys  first  became  acquainted  with 
Newton,  but  there  is  reason  to  think  that  their  acquaint- 
ance began  a  short  time  previous  to  the  Revolution,  and 
they  are  known  to  have  been  on  intimate  terras  in  1691 
and  1693.  The  absence  of  Newton's  name  from  the 
long  list  of  persons  who  received  at  Pepys's  funeral,  in 
1703,  some  token  in  memory  of  the  deceased,  may  create 


PREFACE.  xix 

a  suspicion  that  their  intimacy  did  not  ripen  into  a 
friendship  that  continued  unbroken  to  the  last ;  a  cir- 
cumstance which  need  not  excite  much  surprise  when 
we  reflect  that  neither  the  politics  nor  the  morality  of 
the  Secretary  of  the  Admiralty,  under  the  two  last 
Stuart  kings,  were  at  all  congenial  to  Newton  s  taste. 
In  assigning,  therefore,  the  date  of  the  portrait  to  the 
period  of  a  few  years  on  either  side  of  1G91,  we  shall 
not  perhaps  be  very  wide  of  the  truth.  If  this  supposi- 
tion be  well-founded,  this  portrait  may  be  considered  as 
the  most  interesting  of  all  the  known  portraits  of  our 
philosopher,  as  representing  him  at  a  time  of  his  life  the 
least  remote  from  those  memorable  eighteen  months 
which  it  cost  him  to  produce  the  great  work  that  has 
immortalized  his  name. 

The  public  is  indebted  to  the  liberality  of  the  Master 
and  Seniors  of  Trinity  College  for  the  appearance  of  the 
present  volume. 

Trinity  College,  Cambridge, 
October  IS50. 


SYNOPTICAL    VIEW 

OF 

NEWTON'S   LIFE. 


Quo  fit  ut  omnis 
Votiv4  pateat  veluti  de&cripta  tabellA 


1642    Dec.  25.     Isaac  Newton  born  at  TVoolsthorpe,  near  Grantham, 

Lincoln8liire(^). 
1655    Sent  to  Grantham  School. 

}656    Taken  away  from  school  and  put  to  a;;ricultural  employment. 
Reads  matliematics  while  watching  the  sheep,  and  in  consequence 

1660  Sent  back  to  school  with  the  view  of  his  going  to  College. 

1661  Jun.  5.     Admitted  Subsizar  at  Trin.  Coll. 
July  8.     Matriculated  Sizar  f')  (Quadrantarius). 

1664  Feb.  19.     Observations  on  two  halos  about  the  Moon(*). 
Thursday,  Apr.  28.     Elected  Scholar  (41<  vacancies). 

1665  Jan.     Takes  B.A.  degree  with  25  other  Trinity  men(*). 

May  20.     Paper  on  fluxions  ('),  in  which  the  notation  of  potntf 

is  used. 
Nov.  13,     "Discourse"  on  fluxions  and  their  applications  to 

tangents  and  curvature  of  curves  ('). 

1666  In  the  beginning  of  this  year  (the  year  beginning  March  25) 

"applies  himself  to  the  grinding  of  Optic  glasses  of  other 
figures  than  spherical,"  and  "procures  a  triangular  glass  prism 
to  try  therewith   the  celebrated  Phrenomena   of  Colours :" 

DISCOVERS    THE    UNEQUAL     REFRANGIBILITY    OP    LIGHT  (^),    and 

abandoning  in  consequence  the  idea  of  improving  the  refract- 
ing telescope, leaves  oft" his  "glassworks,"  and  turns  his  atten- 
tion to  "  Reflections,"  but  while  engaged  thereon  is  "  forced 
from  Cambridge  in  {June}  by  the  intervening  plague (*),  and 
it  was  more  than  two  years  before  he  proceeded  further." 

May  16.     Another  paper  on  fluxions. 

First  idea  of  gravity  occurs  to  him  from  observing  the  fall  of  an 
apple  (®)  in  the  garden  at  Woolsthorpe ;  proves  (from  Kepler's 
3d  law)  that  it  must  vary  inversely  as  the  square  of  the 
distance. 
b 


XXII  SYNOPTICAL  VIEW  OF 

1(}G6  Octob.  Small  tract  on  fluxions  and  fluents  with  their  applica- 
tions to  a  variety  of  problems  on  tangents,  curvature,  areas, 
lengths,  and  centres  of  gravity  of  curves ("*). 
Nov.  Small  tract  similar  to  the  preceding,  but  apparently  more 
comprehensive  (").  (Notation  by  pointa  in  first  and  second 
fluxions.     Basis  of  his  larger  tract  of  IO7I). 

1667  Oct.  1.     Elected  minor  fellow  (").     Spiritual  Chamber  ("). 

1668  March  16.     Admitted  major  fellow. 
July  7.     Created  M.A.(^*) 

Makes  a  reflecting  telescopc(")  (probably  towards  the  end  of  the 
year) :  is  interrupted  until  the  autumn  of  1671. 

1669  Feb.  23.    Describes  his  Reflecting  Telescope  in  a  letter  to  a  friend. 
May  18.     Letter  of  advice  to  his  friend  Francis  Aston. 

July  31.     His  I)e  Anali/si  sent  by  Barrow  to  Collins. 
Oct.  29.     Ap})oiiitcd  Lucasian  Professor  ("). 
Dec.     "Writes  notes  upon  Kinkhuysen's  Algebra  sent  by  Collins 
through  Barrow. 

1670  Jan.  19.     Letter  to  Collins (").     (Summation  of  harmonic  series. 

Solution  of  equations  by  tables.     Is  writing   notes   at  his 

leisure  upon  Kinkhuysen's  Algebra). 
Feb.  6.     Letter  to  Collins.     (Solution  of  annuity  problem,  given 

all  tho  other  quantities,  find  tho  rate  per  cent.    Kinkhuysen's 

Algebra  not  worth  the  pains  of  a  formal  comment). 
18.     Letter  tO  Collins  (").     (Could  give  exacter  solutions 

of  tho  annuity  problem,  but  has  no  leisure  for  computations. 

Sees  also  a  way  of  summing  a  harmonic  series  by  logarithms). 
July  11.     Letter  to  Collins  (with  his  notes  upon  Kinkhuysen's 

Algebra)("'). 
—  16.     Letter  to  Collins  (proposing  to  make  further  additions 

to  Kinkhuysen's  Algebra,  which  is  accordingly  sent  back  for 

the  purpose). 
Sept.  27.     Letter  to  Collins  (two  mean  proportionals  cannot  be 

found  by  trisecting  an  arc.    General  methods  best  adapted  for 

instruction.     Kinkhuysen's  Algebra  not  so  imperfect  as  ho 

had  thought). 

1671  July  20.    Letter  to  Collins.  (Prevented  by  a  sudden  fit  of  sick- 

ness from  visiting  him  at  the  Duke  of  Buckingham's  installa- 
tion as  Chancellor.  Will  not,  he  fears,  have  time  to  return 
to  discourse  of  infinite  series  before  winter.  Approximate 
sum  of  harmonic  series). 
Autumn.  ^Makes  his  2nd  Reflecting  Telescope  (in  its  essential 
parts  like  tho  former) :  it  is  sent  up  in  December  "  for  his 
Majesty's  perusal  ('^)." 


NEWTON'S  LIFE.  xxiii 

1671  Dec.  21.     Proposed  candidate  at  the  Royal  Society  by  Dr  Scth 

"NVard,  Bishop  of  Salisbury. 
(Towards  the  end  of  the  year)  occupied  in  enlarging  his  method 
of  infinite  soric8("),  and  preparing  20  Opticall^ctures  for  the 
press. 

1672  Jan.  6.     Letter  to  OIdenburg("),  "altering  and  enlarging  the 

{Latin}  descrij)tion  (■')  of  his  instrument  which  had  been  sent 
him  for  his  review  before  it  should  go  abroad"  to  lluygens  at 
Paris. 

11.     Elected  Fellow  of  the  Royal  Society.     His  telescope 

the  subject  of  conversation  at  the  meeting:  the  revised  de- 
scription of  it  read  ("). 

10.     Letter  to  Oldenburg  on  "  a  fit  metalline  matter"  for 

the  specula ;  (announces  his  intention  of  sending  to  the  Royal 
Society  "  an  account  of  a  philosophical  discovery,"  "  being  the 
oddest,  if  not  the  most  considerable  detection,  which  hath 
hitherto  been  made  in  the  operations  of  nature,"  viz.  the  con)- 
position  of  light). 

—  29.  Letter  to  Oldenburg  on  the  proportions  of  arsenic  and 
bell-metal  for  specula. 

Feb.  6.  Letter  to  Oldenburg  communicating  his  discovery  of 
the  unequal  refrangibility  of  the  rays  of  light  (read  to  the  Soc. 
Feb.  8:  printed  in  tlie  Trans,  for  Feb.  19). 

—  10.  Letter  to  Oldenburg,  in  acknowledgment  of  fho  flat- 
tering reception  of  his  letter  of  Feb.  6,  and  acceding  to  the 
wish  of  the  Society  that  it  should  bo  printed. 

—  20.  Letter  to  Oldenburg,  "  promising  an  answer  to  Mr 
Ilooke's  observations  upon  his  new  theory  of  light  and  colour," 
and  acknowledging  "the  handsome  and  ingenious  remarks'* 
in  Huygens's  letter  on  his  telescope  (read  to  the  Soc.  Feb. 
22). 

March  16.     Letter  to  01denburg("). 

19.  Letter  to  Oldenburg,  "containing  several  particu- 
lars relating  to  his  new  telescope  ("),"  (read  to  the  Soc  ^Farch 
21 :  printed  in  Trans,  for  March  25). 

« 26.  Letter  to  Oldenburg,  "  containing  some  more  par- 
ticulars relating  to  his  new  telescope  ("),"  (read  to  the  Soc. 
March  28  :  printed  in  Trans,  for  Apr.  22). 

— —  30.  Letter  to  Oldenburg,  "  containing  his  answer  to  the 
difficulties  objected  by  M.  Auzout  against  his  reflecting  tele- 
scope ;  as  also  the  queries  of  M.  Denys  concerning  it;  together 
with  his  proposal  of  a  way  of  using,  instead  of  the  little  oval 

62 


XXIV  SYNOPTICAL  VIEW  OP 

metal,  a  crystal  figured  like  a  A'  prism  (")."    (Read  to  Soc. 

Apr.  4 :  extract  printed  in  Trans,  for  Apr.  22). 
1672    Apr.  13.     Latin  letter  to  Oldenburg,  in  answer  to  the  objections 

of  Pardies  (professor  in  the  college  of  Clermont,  in  Paris) 

against  his  theory  of  light  and  colours  (read  to  the  Soc.  Apr. 

18:  printed  in  Trans,  for  June  17). 
Same  date.     Letter  to  Oldenburg,  "  answering  some  experiments 

proposed  by  Sir  Robert  Moray,  for  the  clearing  of  his  theory 

of  light  and  colours  C-")"  (read  to  the  Soc.  Apr.  18:  extract 

printed  in  Trans,  for  May  20). 
May  4.     Letter  to  Oldenburg,  "  containing  his  judgment  of  M, 

Cassegraine's  telescope  "  (read  to  the  Soc.  May  8 :  printed  in 

Trans,  for  May  20). 

—  25.  Letter  to  Collins  (does  not  intend  to  publish  his 
lectures)  ('"'.). 

Juno  11.  Letter  to  Oldenburg  ("),  accompanying  (l)  his  2nd 
answer  to  Pardies,  who  is  satisfied  by  it,  (printed  in  Trans. 
July  ir>),  and  (2)  his  answer  toHooke's  "considerations  upon 
his  discourse  on  light  and  colours  "  (part  of  it  read  to  the  Soc. 
June  12:  printed  in  the  Trans.  Nov.  18). 

—  19.     Letter  to  Oldenburg  from  Woolsthorpe. 

July  0.  Letter  to  Oldenburg  from  Stoke,  in  Northamptonshire, 
in  answer  to  an  inquiry  concerning  refraction,  and  containing 
8  queries  to  test  his  theory  of  light  and  colours  (partly  printed 
in  English  and  Latin  in  the  Trans.  July  15). 

—  8.  Letter  to  Oldenburg  from  Stoke  (containing  remarks 
upon  Huygens's  letter  of  July  1,  N.  S.)("). 

—  13.     Letter  to  Collins  from  Stoke. 

..         —       . Oldenburg in  which  he  repeats  his 

inquiry  about  the  4  feet  telescope,  and  desires  to  know  the 
terms  on  which  Cox  will  make  one. 

■  30.  Letter  to  Collins  with  a  copy  of  his  edition  of  Vare- 
nius's  Geography  ("). 

Letter  to  Oldenburg  ("). 

Sept.  21.  Letter  to  Oldenburg,  in  answer  to  one  from  Olden- 
burg of  the  17th,  inquiring  whether  the  duplicate  of  July  16 
had  come  to  hand  :  (had  drawn  up  some  experiments  adapted 
for  determining  the  queries  in  his  letter  of  July  6,  and  had 
intended  from  them  to  prove  various  propositions  relating  to 
colours  by  means  of  definitions  and  axioms,  but  prevented  by 
other  business  from  carrying  out  his  design.  But  if  the  answer 
to  Ilooko  will  conduce  to  the  determination  of  any  of  the 
queries,  it  may  be  published).         ^ 


NEWTON'S  LIFE.  XXV 

IG72  Doc  10.  Letter  to  Collins,  containing;  (1 )  Rn  account,  requested 
by  Collins  in  a  letter  received  two  days  before,  of  his  method 
of  tangents ('*),  and  (2)  "a  long  scribble"  on  James  Gregorj-'s 
observations  upon  his  paper  on  Cassegrain's  telescope,  (Very 
glad  to  have  Barrow  again,  especially  as  Master). 

1673  March  5.     Joins  in  a  protest  against  the  claim  of  tho  Heads  of 

Houses  to  nominate  for  tho  Public  Oratorship.     Votes  for 
Isaac  Craven  of  Trin.  Coll.  (not  nominated)  (**). 
■         8.     Letter  to  Oldenburg  (desires  to  withdraw  from  the 

Royal  Society)  C'). 

Apr.  .3.  Letter  to  Oldenburg,  in  answer  to  Iluygens's  letter  of 
Jan.  14  (read  to  the  Soc.  Apr.  9:  printed  in  Trans.  Oct.  6). 

9.     Letter  to  Collins  (containing  remarks  upon  Gregory's 

"candid  reply"). 

May  20.     Letter  to  Collins  (»"). 

June  23.  Letter  to  Oldenburg,  thanking  Ifuygens  for  tho  pro- 
sent  of  his  Horologium  Oscillatoriuniy  and  replying  to  his 
remarks  (in  his  letter  of  Jun.  10)  upon  Newton's  letter  of 
Apr.  3  (partly  printed  in  Trans.  July  21  )(="). 

Sept.  17.  Letter  to  Collins:  (postpones  further  discussion  of 
telescope  until  Gregory  pays  his  expected  visit  to  Cambridge), 

1674  June  20,     Letter  to  Collins :  (horizontal  velocity  of  a  bullet  not 

uniform.     Value  of  y  in  y'  +  a*y  -  i'  =  0). 

Nov.  17.  Letter  to  Collins:  (mentions  rules  for  solving  incom- 
plete equations  by  logarithms). 

Dec.  ■^..  Letter  to  Oldenburg :  declines  to  take  any  notice  of 
Linus's  "  conjecture :"  however  Oldenburg  may  direct  him  to 
the  figure  in  the  2nd  answer  to  Pardies,  and  signify  "  but  not 
fiom  me,"  that  tlip  experiment  with  tho  prism  was  made  on 
clear  days,  with  the  prism  close  to  the  hole  and  tho  coloured 
image,  not  parallel  but  trans"'^r8e  to  the  axis  of  the  prism, 
(A  letter  was  written  by  Oldenburg  accordingly,  and  printed 
without  Newton's  knowledge  in  the  Trans.  Jan.  25,  1G75), 

1675  Chemical  pursuits ("). 

Jan.  22.     Letter  to  IMichael  Dary  (length  of  an  elliptic  arc). 
—  28.     Excused  the  weekly  payments  to  tho  Royal  Society("), 
Feb.  18.     Admitted  F.R.S.(^'). 
Apr.  27.     Obtains  from  the  Crown  a  patent  allowing  tho  Luca- 

,sian  Professor  to  hold  a  fellowship  without  being  obliged  to 

go  into  orders  (*'). 
May  8.    Letter  to  John  Smith  (construction  of  tables  of  square, 

cube,  SiC.  roots)  (**). 


XXVI  SYNOPTICAL  VIEW  OF 

1675  July  24.  ) 

A       27  I  ^®*^®™  *^  ^^*®  ^"*®  ^"  extraction  of  roots. 

Nov,  13.  Letter  to  Oldenburg,  with  minute  directions  for  Linus 
how  to  make  the  spectrum  experiment :  (communicated  to 
the  Soc.  Nov.  18:  principal  part  of  it  printed  in  Trans.  Jan. 
24,  167G)  (**).     Offers  to  send  a  paper  on  colours. 

— -  30.  Letter  to  Oldenburg  (is  adding  "  an  hypothesis  "  con- 
cerning light  to  his  paper  on  colours.  Description  of  ear- 
trumpet)  ("). 

Dec.  1.  Gives  a  copy  of  Irenteus  (Paris,  1G75)  to  College 
Library. 

Decemb.  Sends  to  the  Roy.  Soc.  his  papers,  containing  (1)  his 
Hypothesis  explaining  the  properties  of  light,  (2)  his  explana- 
tion of  the  colours  of  thin  plates,  and  of  natural  bodies ("). 

Dec.  14.  Letter  to  Oldenburg  (suggesting  that  the  glass  in  the 
electrical  experiment  should  be  nearer  the  table  than  he  had 
stated  in  his  paper). 

—  21.  Letter  to  Oldenburg,  with  (1)  further  directions 
respecting  the  electrical  experiment  (read  to  the  Soc.  Dec.  30(**), 
and  the  experiment  ordered  to  be  made  at  next  meeting),  and 
(2)  remarks  on  Hooke's  "  insinuation." 

Communicates  to  Mercator  his  explanation  of  the  Moon's  li- 
hration  (**). 

1676  Jan.  10.     Letter  to  Oldenburg,  containing  (l)  suggestions  re- 

specting the  electrical  experiment,  (2)  remarks  upon  Hooke's 
"  insinuations,"  (3)  further  directions  for  Gascoines  how  to 
make  the  spectrum  experiment.  Oldenburg  (Jan.  18)  sends 
them  to  Gascoines,  who  requests  Lucas  (Linus's  successor  in 
the  mathematical  chair  at  Liege)  to  make  the  experiment. 
(Last  part  of  the  letter  printed  in  Trans.  Jan.  24,  1070). 
'  13.  At  the  meeting  of  the  Royal  Society,  the  electrical 
experiment  being  made  according  to  Newton's  "more  par- 
ticular directions  succeeded  very  well."  "  It  was  ordered  that 
Mr  Newton  should  have  the  thanks  of  the  Society  for  giving 
himself  the  trouble  of  imparting  to  them  such  full  instructions 
for  making  the  experiment." 

20.  On  the  reading  of  the  first  15  "observations"  of 
Newton's  discourse,  the  Society  were  "  so  well  pleased  "  with 
them,  that  Oldenburg  was  ordered  to  desire  him  "  to  permit 
them  to  be  published  together  with  the  rest." 
A  passage  was  also  read  from  his  letter  of  Dec.  21,  "stating  the 
difference  between  his  hypothesis  and  that  of  Mr  Hooke,"  in 


NEWTON'S  LIFE.  xxvii 

allusion  to  what  had  fallen  from  Hooke  at  the  meeting  of 
Doc.  16. 
167c  Jan.  25.  Letter  to  01clenbur<»,  in  acknowledgement  of  the 
favourable  reception  of  his  papers  (•*),  with  alterations  to  be 
made  in  them.  (Read  to  the  Soo.  Jan.  27). 
Feb.  3.  On  the  reading  of  Newton's  observations  on  colours, 
a  discussion  arose  as  to  whether  the  difference  of  colour  in  the 
rays  of  light  was  not  to  bo  attributed  to  the  different  velocities 
of  the  pulses  rather  than,  as  lie  tliought,  to  a  connate  difference 
of  refrangibility  in  the  rays  themselves.  Ilooke  expressed 
himself  in  favour  of  the  former  explanation.  See  Newton's 
Letter  of  Fob.  15. 

—  15.  Letter  to  Oldenburg,  answering  the  objection  that 
had  been  raised  at  the  meeting  of  Feb.  3. 

—  29.  Letter  to  Oldenburg,  occasioned  by  his  having  read 
in  the  Trans,  for  Jan.  24,  Linus's  letter  of  Feb.  25,  1075:  it 
contains  a  particular  answer  to  that  letter,  followed  by  expla- 
natory remarks  for  the  behoof  of  Linus's  friends.  (Printed  in 
Trans.  March  25). 

Apr.  20.  letter  to  Oldenburg,  thanking  him  for  "  motioning 
to  get "  the  spectrum  experiment  tried  before  the  Royal  So- 
ciety (").  Remarks  upon  Boyle's  paper  on  the  incalescence  of 
gold  and  mercury. 

May  11.  Letter  to  Oldenburg,  thanking  him  for  getting  the 
experiment  tried :  during  the  summer  may  possibly  work  at 
his  long-projected  discourse  about  the  prismatic  colours("). 

Juno  8.  At  a  meeting  of  the  Soc.  a  letter  from  Lucas  to  Olden- 
burg (Liege,  May  27)  was  read,  containing  partly  an  account 
of  the  success  of  the  spectrum  experiment,  partly  some  new 
objections  against  Newton's  theory  of  light  and  colours.  A 
copy  of  the  letter  ordered  to  be  sent  to  Newton  immediately : 
printed  in  Trans,  for  Sept.  25. 

— —  13.  Letter  to  Oldenburg,  containing  a  general  answer  to 
Lucas  with  a  promise  of  a  particular  one,  and  also  "some 
communications  of  an  algebraical  nature  for  M.  Leibniz,  >vho 
by  an  express  letter  to  Mr.  Oldenburg  had  desiicd  thorn." 
(read  to  the  Soc.  Juno  15:  the  part  for  Leibniz(")  was  sent 
to  him  at  Paris,  July  20). 

Aug.  22.  Letter  to  Oldenburg  (")  (accompanied  by  another 
dated  Aug.  18,  the  latter  being  an  answer  to  Lucas,  printed 
in  Trans,  for  Sept.  25). 

Sept.  5.     Letter  to  Collins.     (Infinite  Scrie3  of  no  great  uso  in 


xxviu  SYNOPTICAL  VIEW  OP 

the  nutnorical  solution  of  equations.  Tho  University  press 
cannot  print  Kinkhuysen's  Algebra :  the  book  is  in  the  hands 
of  a  Cambrid/^e  bookseller  with  a  view  to  its  being  printed : 
shall  add  nothing  to  it.  Will  alter  an  expression  or  two  in 
his  paper  about  infinite  series,  if  Collins  thinks  it  should  be 
printed). 

1676  Oct.  24.     Latin  letter  to  Oldenburg  (")  for  Leibniz,  who  desired 

explanation  with  reference  to  some  points  in  tho  letter  of 
June  13. 

—  26.  Letter  to  Oldenburg,  with  corrections  for  his  letter  of 
Oct.  24,  &c.(") 

Nov.  8.  Letter  to  Collins,  thanking  him  for  copies  of  the  letters 
of  Leibniz  and  Tschirnhaus,  with  remarks  shewing  that  Leib- 
niz's method  is  net  more  general  or  easy  than  his  own("). 

14.     Letter  to  Oldenburg  (cider-fruit-trees :  Lucas's  2nd 

letter:  further  alterations  of  his  letter  of  Oct.  24)("). 

—  18.  Letter  to  Oldenburg  (answer  to  Lucas  will  not  be 
ready  so  soon  as  he  intended.  Will  never  publish  anything 
more  on  philosophy,  after  he  has  got  clear  of  this  dispute. 
Letter  to  Boy^C"). 

28.     Rejoinder  to  Lucasf"*). 

Subscribes  £40  towards  New  Library. 

1677  March  5.     Letter  of  Collins  to  him  (")• 
Sept.     Death  of  Oldenburg. 

1679  Feb.  7.    Letter  to  Dr  Maddock("). 

—  H.  Sir  Thomas  Exton,  Master  of  Trin.  Hall,  and  James 
Vernon,  of  Trinity,  (the  Duke  of  Monmouth's  Secretary,) 
elected  M.P.  for  the  University.  Newton  plumps  for  the 
former("). 

'         28.     Letter  to  Boyle  (physical  qualities  of  bodies)  (**). 
Nov.  8.     Charles  Montagu  entered  a  feUow-commoner  at  Trinity 

College  ("). 
December.     Determines  (in  consequence  of  a  letter  from  Hooke) 

the  curve  described  by  a  body  under  tho  action  of  a  central 

force,  and  applies  his  theorem  to  the  case  of  an  ellipse ("). 
Gives    copy  of  Huet's   Demonstratio  Evangelica  to    College 

Library. 

1680  Jan.  21.     Collins  offers  to  print  Newton's  Algebra  (along  with 

Wallia's  and  Baker's),  if  the  Society  would  take  60  copies, 
which  the  Council  two  years  and  a  half  afterwards  agreed  to 
do  (July  12,  1682),  bv^the  design  was  carried  out  only  with 
^    respect  to  Baker  and  Wallis. 


NEWTON'S  LIFE.  xxix 

Lends  the  Collefije  £lOO.  for  the  New  Library  (sometime  bo- 
twe;n  Dec.  1079  and  Michaelmas  IGOO)^. 

1680  Dec.  3.    Letter  tolloolcen. 

Gives  copy  of  Grcw's  Muswum  Jlegalii  Socictatlt  to  College 
Library. 

1681  Jan.     Promises  to  assist  Adams  (probably  by  advice  and  calcu- 

lations) in  a  survey  of  Enf!;land("). 

Feb.  28.    Letter  of  Flamsteetl  (through  Crompton,  Follow  of 
Jes.  Coll.)  about  the  Cometh'). 

Apr.  16.     Letter  to  Flamsteed  about  tho  Comet ("). 
1082    Apr.  3.     Testimonial  to  Edw.  Paget,  Fellow  of  Trin.  Coll., 
candidate  for  tho  IMathematical  Mastership  of  Christ's  Hos- 
pital (^*)*. — Letter  to  Flamsteed  (introducing  Paget). 

^    1   lo  r  Letters  to  Briggs  on  Vision ("). 

1683  Nov.  7.     Votes  for  James  Ilalman  of  Caius  College,  tho  success- 

ful candidate  for  tho  Registraryship. 

10.     Death  of  Collins. 

Dec.  2%     Letter  to  Aubrey,  who  had  offered  some  books  for 

sale  to  Trinity  College  or  the  University  ("). 

1684  Jan.  19.     Votes  for  James  Manfoild  of  Trinity,  the  successful 

candidate  for  tho  Librarianship. 
August.  Halley  on  a  visit  to  him,  "  learns  the  good  news  that 
he  had  brought  the  demonstration "  of  "  tho  laws  of  the  ce- 
lestial motions  to  perfection."  Newton  cannot  lay  his  hands 
upon  his  papers,  but  works  them  over  again,  and  sends  them 
in  November  by  Paget  to  Halley  in  tho  form  of  4  theorems 
and  7  problems (^*).  Halley  "thereupon  takes  another  jour- 
ney to  Cambridge,  on  purpose  to  confer  with  him  about 
them(")." 

1685  Feb.  23.     Letter  to  Aston  (unsuccessful  attempt  to  establish 

a  philosophical  society  at  Cambridge (").     Thanks  for  regis- 
tering at  the  Royal  Society  his  "  notions  about  motion"). 
Apr.  25.     Letter  to  Briggs  ("). 

DETERMINES   THE   ATTRACTIONS    OP   MASSES  AND    THUS    COMPLETES 
THE  DEMONSTRATION   OP   THE  LAW  OP   UNIVFJISAL  GRAVITATION. 

Summer.     The  2nd  book  of  the  Principia  finished. 

Sept.  10.     Certificate  of  approval  of  Mabbot's  Tables  for  renewal 

of  leases  ("). 
■         19.     Letter  to  Flamsteed  (is  about  to  calculate  the  orbit 

of  the  comet  of  1080  from  3  observations.     Tides  at  solstices 

and  equinoxes)  ("). 


XXX  SYNOPTICAL  VIEW  OF 

1G85     Oct.  14.     Letter  to  Flamstecd  (acknowledging  the  receipt  of 
Flamsteed's  two  letters  in  answer  to  the  preceding). 

Dec.  30.  Letter  to  Flamsteod  (with  thanks  for  information 
about  comet  of  1680  and  Jupiter's  satellites.  Kepler  makes 
Saturn's  orbit  too  small.  Requests  the  greatest  elongations 
of  any  of  Jupiter's  satellites,  and  of  Saturn's  satellite)  (*"). 
J  686  Jan.  13.  Letter  to  Flamstecd  (wishes  to  know  the  major  axes 
of  the  orbits  of  Jupiter,  Saturn  and  his  satellite)  ("). 

22.     Votes  for  John  Laughton,  of  Trinity,  the  successful 

candidate  for  the  Lihrarianship. 

Apr.  28.  First  Book  op  the  Principia  exhibited  At  the 
KoYAL  Society  ("*). 

May  19.  At  a  meeting  of  the  Society  it  was  ordered  "  that  M' 
Newton's  PhilosopMw  Naturalis  Principia  Mathematlca  be 
printed  forthwith  in  4to.  in  a  fair  letter ;  and  that  a  letter  be 
written  to  him  to  signify  the  Society's  resolution,  and  to  do- 
sire  his  opinion  as  to  the  print,  volume,  cuts,  &c."("). 

June  2.  Ilalley  undertakes  the  publication  of  the  Principia  at 
his  own  expense  ("). 

—  20.  Letter  to  Ilalley  (demolishing  the  claim  set  up  by 
Uooke  of  having  communicated  to  him  the  law  of  decrease  of 
gravity  according  to  the  inverse  square). 

— —  30.  At  a  meeting  of  the  Council  of  the  Royal  Society, 
the  President  was  desired  to  license  the  Philosophias  Natnralis 
Principia  Mathematica. 

July  14.  Letter  to  Ilalley  (approves  of  the  suggestion  of  having 
wood-cut-^.     Conciliatory  remarks  respecting  Ilooke). 

■■■        27-     Letter  to  Ilalley  (further  remarks  on  ITooke's  claim). 

Ar  .  20.  Letter  to  Halley  (with  Cor.  2  and  3  of  Prop.  xci. 
Lib.  I.  of  Princip.  on  the  attraction  of  a  spheroid  on  a  point 
in  its  axis  produced,  and  on  au  internal  point)  (**). 

Sept.  3.  Letter  to  Flamstecd  (  sinian  satellites.  Cassini's 
observation  of  Jupiter's  oblate:   '   ). 

Autunm.     Second  Book  of  Princiwut  made  ready  for  the  pres8("). 

Oct.  18.     Letter  to  Ilalley  (corrections  of  Scholium  to  Prop.  31.  • 
Lib.  I. :  transformation  of  a  trapezium  into  a  parallelogram). 
1687    Feb.  18.     Letter  to  Ilalley  (may  have  the  second  book  of  Prin- 
cipia when  he  pleases :  has  the  sheets  up  to  M  :  thanks  him 
for  putting  forward  the  press  again)  ("). 

March  1.  Tuesday.  Letter  to  Ilalley,  advising  him  that  the 
2nd  book  will  arrive  on  Thursday  night  or  Friday,  by 
coach  :  obliged  to  him  for  pushing  on  the  edition  because  of 


NEWTON'S  LIFE.  xxxi 

people's  expectation^  tliougli  otherwise  he  could  bo  as  well 
satisfied  to  let  it  rest  a  year  or  two  lonjifor  (read  to  tlie  Soc. 
3Iarch  2). 
1C87  Marcli  11.  Deputed  with  Billers,  tlie  Public  Orator,  to  carry  to 
the  Vico-Chancellur  tlie  opinions  of  the  Non-Regent  House 
respecting  King  James's  second  mandate,  requiring  the  Uni- 
versity to  confer  upon  Alban  Francis,  a  Benedictine  monk, 
tho  degree  of  M.A.  without  the  usual  oaths(*''). 

Apr.  0.  The  3rd  book  of  the  Principia  "produced  and  presented" 
to  the  Royal  Society  (*"). 

11.     Appointed  one  of  eight  delegates  to  represent  tho 

Senate,  in  conjunction  with  the  Vice-Chancellor,  before  tho 
Ecclesiastical  Commission ("'). 

Publication  op  the  Principia  (about  Midsummer) ("). 

1688  Spring.     Cliarles  IMontagu  vacates  his  fellowship  ("), 

Dec.  15.     Votes  for  Archbishop  Sancroft  (for  Chancellor  of  tho 
"    Ufiiversity)  who  declines  the  office. 

1689  Jan.  15.     Elected  one  of  the  representatives  of  the  University  in 

tho  Convention  Parliament  (""). 
First  acquaintance  with  Locke.     Furnishes  him  (March)  with 

an  easy  proof  of  elliptic  motion  about  a  centre  of  force  in  ono 

ofthefociC*). 
June  12.     Iluygens  and  Newton  at  tho  Royal  Society  ("). 
Aug.  20.     Parliament  prorogued. 
■         —      Contemplated  appointment  to  tho    Provostship   of 

King's  College  C^'^). 
Oct.  19.     Meeting  of  Parliament  ("). 

1690  Jan.  27.     Parliament  prorogued. 
Feb.  G.     Parliament  dissolved. 

— -  21.  Sir  Robert  Sawyer,  who  had  been  expelled  the  IIouso 
of  Commons,  Jan.  20,  for  having  been,  as  Attorney-General, 
one  of  the  ])rosccutors  of  Sir  Thomas  Armstrong  in  1084,  re- 
elected M.P.  for  the  University.     Newton  votes  for  him. 

Oct.  28.  Letter  to  Locke :  (will  send,  as  desired,  his  "  llisto* 
rical  Account  of  two  notable  corruj)tion3  of  scripture.**  Ac- 
knowledgments to  Lord  and  Lady  Monmouth  for  their 
endeavours  to  procure  him  preferment)  ("). 

Nov.  14.     Letter  to  Locke,  with  the  "  Historical  Account." 

1691  Feb.  7«     Letter  to  Locke  (Daniel  and  Apocalypse). 

Juno  30.  Letter  to  Locke.  (Locke's  good  offices  in  trying  to 
get  him  tho  place  of  comptroller  of  the  Mint  Effijcts  of  look- 
ing at  tho  Sun's  image  in  a  mirror). 


xxxii  SYNOPTICAL  VIEW  OP 

1691  July  (London).    Testimonial  to  David  Gregory,  recommending 

him  for  tlie  vacant  chair  of  Astronomy  at  Oxford  (•'). 

Directions  to  Bcntley  about  reading  the  Principia,  (p.  273). 

Aug.  10.  (London).  Letter  to  Flamstecd  (introducing  David 
Gregory.  Hopes  Flamsteed  will  publish  his  catalogue  of  tho 
fixed  stars  before  long.  "Would  willingly  have  his  observa- 
tions of  Jupiter  and  Saturn  for  next  4  or  5  years  at  least,  or 
rather  for  tho  next  12  or  15,  before  thinking  further  of  their 
theory.  Does  the  light  of  Jupiter's  satellites,  immediately 
before  eclipse,  incline  either  to  red  or  blue,  or  become  ruddier 
or  paler  than  before  ?) 

Dec.  13.  Letter  to  Locke.  (Declines  "making  a  bustle"  for 
the  ^lastership  of  the  Charter-House)  (""). 

1692  Jan.  26.     Letter  to  Locke.    (Charles  Montagu  a  false  friend. 

Desires  to  have  his  "  Historical  Account"  returned.) 

Feb.  16.  Letter  to  Locke.  (Desires  the  translation  and  impres- 
sion of  tho  "  Historical  Account "  to  be  stopped.     Miracles). 

May  3.     Letter  to  Locke  (glad  of  his  intended  visit.  Miracles). 

June.     Observations  on  three  halos  about  the  Sun(^*^). 

July  7»  Letter  to  Locke  (Boyle's  recipe  for  producing  gold  by 
means  of  red  earth  and  mercury)  (^°'). 

Aug.  2.  Letter  to  Locke  (Boyle's  recipe.  Discourages  Locke 
from  trying  it). 

^      \L'  >  Letters  to  Wallis,  with  illustrations  of  tho  calculus 
Sept.  17.  i 

of  fluxions  and  fluents,  sent  at  Wallis's  request  (***'). 
Nov.  21.     Election  of  a  Member  for  the  University  in  the  place 

of  Sir  Robt.  Sawyer,  deceased.     Votes  for  the  unsuccessful 

candidate,  Dr  Brookbank,  of  Trin.  Hall("*). 
Dec.  10.     First  letter  to  Bentley. 
Paper  on  Acids  (exact  date  uncertain)  communicated  to  a  friend 

this  year(»**0- 

1693  Jan.  I7.     Second  letter  to  Bentley (^'»«). 

Feb.  11.     Third  letter  to  Bentley.  ,^^ 

25.    Fourth  letter  to  Bentley(*")- 

March  14.  Letter  to  Fatio  (proposing  to  make  him  such  an 
allowance  as  might  make  his  subsistence  at  Cambridge  easy 

tohim)C"0- 
September.     Bad  state  of  health. 
13.     Letter  to  Samuel  Pepj's  (desiring  to  "  withdraw  from 

his  acquaintance")  C**0« 
■         16.     Letter  to  Locke  (beg^png  his  pardon  for  having  had 
«  hard  thoughts"  of  him)  ("0- 


NEWTON'S  LIFE.  xxxiii 

1693  Oct.  15.     Letter  to  Locke  (explaining  tlio  circumstoncca  under 

which  the  letter  of  Sept.  16  was  written)  ("')• 

16.    Letter  to  Leibniz,  (p.  276). 

Nov.  23. ) 

D     '  16    I  ^^^^^^"^  ^^  Pcpys  on  a  problem  in  chances  (""). 

1694  May  7.     Haunted  house  ('"). 

1 1 .     Charles  Montagu,  Chancellor  of  the  Exchequer. 

—  25.  Letter  toITawes  (explaining  his  views  relative  to  the 
old  and  new  schedules  of  niathcniatical  studies  at  Clirist'a 
Hospital)("*). 

—  26.     Letter  to  Hawes  (supplementary  to  preceding). 
May.     David  Gregory  at  Cambridge ("*). 

July.  Requested  by  the  Royal  Society  to  publish  his  optical 
and  other  treatises  ("*). 

Sept.  1.  Visits  Flamstced  at  Greenwich,  who  shews  him  up- 
wards of  150  lunar  observations,  and  a  comparison  of  them 
with  the  places  as  calculated  from  tables  ("^).  Consequent 
correspondence  between  them,  extending  from  Oct.  until  Sept. 
of  the  following  year("*). 

Oct.  7.  Letter  to  Flamsteed  (describing  what  further  observa- 
tions h(-  will  want,  with  which  he  believes  he  can  "  set  right 
the  moon's  theory  this  winter"). 

—  24.  Letter  to  Flamsteed  (thanking  him  for  his  letter  of 
Oct.  11,  and  particularly  for  the  table  of  the  difference  of 
refractions  of  Sun  and  Venus.     Parallactic  Equation)  ("*). 

Nov.  1.  Letter  to  Flamsteed  (errors  in  some  of  his  observations. 
Lunar  inequalities.     Sun's  menstrual  parallax)  ('-**). 

■  17.     Letter  to  Flamsteed  (will  send  back  the  two  synopses 

of  the  Moon's  places  the  next  day,  together  with  a  table  of 
refractions.  His  method  of  proceeding  in  determining  the 
Moon's  motions.  Requests  to  have  the  JMoon's  right  ascen- 
sions and  meridian  altitudes  just  as  they  are  observed  without 
>,  any  correction :    if  Flamsteed  will  do  him  this  favour,    ho 

desires  them  as  Flamsteed  had  observed  them  for  tho  last 
six  months). 

Dec.  18.  Letter  to  Flamsteed  (Table  of  refractions  not  so  accu- 
rate as  it  may  be  made :  intends  to  correct  it  and  send  a 
new  copy  of  it.  Thanks  Flamsteed  for  complying  with  his 
request  of  sending  the  Moon's  right  ascensions  and  meridian 
altitudes  unreduced  :  begs  her  places  on  certain  days  which  ho 
names :  observations  in  this  and  next  month  or  two  of  great 
importance). 


xxxiv  SYNOPTICAL  VIEW  OF 


Deo.  20.  Letter  to  Flamsteed  (theoroni  upon  which  his  table 
of  refraction  is  founded.  Equations  of  the  mean  motions  of 
Jupiter's  satellites.  "  What  you  say  about  my  having  a  mean 
opinion  of  you  is  a  great  mistake  "). 
1695  Jan.  15.  Letter  to  Flamsteed  (thinks  he  has  discovered  a  new 
theorem  in  refractions,  but  intends  to  consider  it  a  littlo 
further.  Thanks  Flamsteed  for  two  lunar  observations  sent 
him,  and  as  Flamsteed  has  calculated  the  ^loon's  places  in 
these  and  the  other  three  observations  of  last  month,  will  be 
glad  to  have  a  synopsis  of  the  calculations.  But  for  the 
rest  of  the  observations,  he  merely  wants  the  observed  places ; 
at  the  same  time  is  obliged  to  Flamsteed  for  offering  to  be  at 
the  pains  of  calculating  them.  Suggestions  respecting  the 
kind  of  time  to  be  employed  in  taking  the  observations). 

—  20.  Letter  to  Flamsteed  (answer  to  Flamstced's  childish 
question  respecting  a  book  which  Flamsteed,  two  or  three 
years  before,  had  intended  as  a  present  to  him.  IMoon's  hori- 
zontal parallax.  Mas  at  last  found  out  a  new  theorem  in 
Refractions:  is  at  present  a  little  indisposed  but  hopes  in  a 
few  days  to  be  well  enough  again  to  finish  the  subject.  The 
two  observations  mentioned  in  the  last  letter  ("').  Promises 
to  send  a  table  of  a  small  equation  of  Moon's  parallax.  If 
Flamsteed  would  rather  have  the  observations  perfectly  his  own 
in  all  respects,  by  calculating  them  himself,  will  stay  his  time). 

Feb.  16.  Letter  to  Flamsteed  (with  thanks  for  the  observations 
of  Dec.  and  Ja-n.  Has  been  engaged  since  ho  wrote  last  upon 
making  a  new  table  of  refractions,  and  has  not  yet  finished  it. 
Manly  answer  to  Flamsteed's  ungenerous  suspicions  of  his 
observations  having  been  communicated  to  Halley). 

March  15.  Letter  to  Flamsteed  (Candidates  for  mathematical 
mastership  at  Christ's  Hospital.  Encloses  a  copy  of  table  of 
refractions  now  finished  ("^).  Will  send  the  other  tables  ho 
promised  in  a  few  days). 

Apr.  23.  Letter  to  Flamsteed  (with  the  promised  tables  of 
^loon's  horizontal  parallax,  equations  of  apogee  and  eccentri- 
cities). 

25(*").    Letter  to  Flamsteed  (in  reply  to  some  remarks 

on  the  tables  sent  with  his  last  letter). 

Jun.  14.  Letter  to  Ilawcs  (with  new  scheme  of  mathematical 
reading  for  Christ's  ITo8pital)(^"). 

^—-  29.  Letter  to  Flamsteed  (with  thanks  for  solar  tables.  As 
Flamsteed's  health  and  other  business  will  not  permit  him  to 


NEWTON'S  LIFE.  xxxy 

calculnto  the  Moon's  places  from  olxjervations,  he  proposes 
once  more  that  Flamstced  slioiild  ecnd  tlio  bare  observntions 
and  first  of  all  thoso  of  1G1)2.  If  not,  let  him  propose  some 
other  way  of  supplying;  the  desired  observationsi,  or  say  plainly 
that  he  will  not  send  any.  Kecommends  equestrian  exercise). 
1GD5  July  9.  Letter  to  Flamstced  (thankfully  accepts  the  offer  of  the 
observations  prior  to  1690.  Parallactic  equation.  Points 
out  the  kind  of  observations  that  he  wants). 

—  20.  Letter  to  Flamstced  (has  written  to  contradict  tho 
report  about  llamsteed's  not  communicating  his  observations. 
Thanks  for  tho  lunar  observations.  Has  not  yet  compassed 
tho  small  equations,  and  bec^s  him  not  to  bo  impatient  for 
them.  Forbears  to  take  notice  of  some  querulous  expressions 
of  Flamstecd's.     "  Pray  take  care  of  your  health  "). 

— —  27.  Letter  to  Flamstced  (is  glad  that  all  misunderstind- 
ings  are  composed.  Describes  the  observations  that  he  wants. 
Remuneration  to  Flamstecd's  servant)  ('"). 

Sept.  14.  Letter  to  Flamstced  (Ilalley's  calculated  orbit  of  tho 
comet  of  1683  agrees  with  his  own  and  Flamstecd's  observa- 
tions to  a  minute.  Is  going  on  a  journey  and  will  not  therefore 
have  time  to  consider  the  lunar  theory  for  a  month  or  above. 
Hopes  he  gets  ground  of  his  distemper). 

Oct.  25.  In  the  contest  for  the  University  plumps  for  tho  Hon. 
II.  Boyle. 

Nov.     Rumour  of  his  appointment  to  Mastership  of  Mint("*). 

1696  Feb.  19.     Votes  for  W.  Ayloffo  of  Trin.  successful  candidate 

for  the  Public  Oratorship. 
March  14.     Letter  to  Ilalley  (is  not  engaged  upon  the  longi- 
tude.    Not  a  candidate  for  any  place  in  the  Mint,  nor  would 
accept  the  Comptroller's  place,  if  offered)  ('"). 

—  19.  Letter  from  Charles  Montagu  announcing  his  appoint- 
ment to  Wardenship  of  ]\Iint. 

1697  Jan.  30.     Solution  of  John  Bernoulli's  two  problems  ("*):  (read 

to  the  Soc.  Feb.  24 :  printed,  without  his  name,  in  Trans,  for 

Jan.). 
Feb.  11 .    Letter  to  Ilalley :  (has  proposed  Ilalley  as  a  fit  person 

to  teach  the  mathematical  principles  of  engineering)^"). 
End  of  June,  or  beginning  of  July.     Examines  boys  at  Christ's 

Hospital  ("»). 

1698  May  30.     Letter  to  Ilarington,  p.  302. 

July  25.     Votes  for  Hon.  II.  Boyle  (re-elected) ('»')• 
Dec.  4.     Visit  to  Flamstced,  in  order  to  obtain  12  computed 
places  of  the  Moon  ('"). 


xxxvi  SYN6HE1CAL  VIEW  OF 


Jan.  6.  Letter  to  Flamsteed  (explaining  why  he  did  not  wish 
his  name  to  bo  meiitionod  in  the  letter  to  Wallis,  and  stating, 
that  there  may  be  cases  in  which  "  friends  should  not  be  pub- 
lished without  their  leave  ")("^). 

Feb.  11.     Made  Associe-Etranger  ol"  the  French  Academy  ("*). 

Aug.  16.    Exhibits  at  the  Royal  Society  an  improved  form  of 
his  sextant  (commonly  called  Iladlcy's) (*''*). 
r  Nov.  30.     Chosen  member  of  Council  of  Royal  Society  (*"). 

This  year  the  great  re-coinage  of  silver  was  completed,  having 
occupied  the  greater  part  of  this  and  of  the  three  preceding 
years  (*"). 

Contributes  towards  the  expenses  of  Lhuyd's  Lithophylacii 
Britannici  IcImographia("*). 

1700  Apr.     Paper  on  time  of  vernal  equinox  (p.  304). 

July  24.  His  opinion  of  the  method  proposed  by  an  Italian 
mathematician  for  trisecting  an  angle,  doubling  the  cube,  and 
squaring  the  circle  by  means  of  a  spiral  line(""). 

1701  Jan.  27.     "NV^histon  begins  his  Astronomical  Lectures,  as  New- 

ton's deputy,  receiving  "  the  full  profits  of  the  place." 
May  28.     His  scale  of  heat  read  to  the  Society  (""),  (printed  in 

the  Trans,  for  ]\rarch- April). 
Nov.  26.     Elected  M.P.  for  the  University("')- 
Dec.  10.     Resigns  his  Professorship,  and  his  Fellowship  shortly 

aftcrC"). 

1702  May  2.5.     Parliament  prorogued. 

(About  June)  his   *'Lunie   Theoria"   published  in   Gregory's 

Astronomy  (*"). 
July  2.     Parliament  dissolved. 
Autumn.     On  a  visit  to  Locke  at  Gates  ("*). 

1703  J\Iay  15.     Letter  to  Locke  (giving  his  opinion  of  Locke's  MS. 

papers  on  the  Epistles  to  the  Corinthians,  and  criticising  his 
paraphrase  on  the  1st  Ep.  vii.  14). 
Nov.  30.    Elected  President  of  the  Royal  Society  f"). 

1704  Jan.  20.     JVIentions  to  the  Royal  Society  his  burning-glass  ("'). 
Feb.     Publication  of  Optics  ("0- 

Dec.  5.  Note  to  Sloane  (desiring  him  to  be  in  readiness  on  the 
7th,  the  day  fixed  for  tlieir  introduction  to  Prince  George,  for 
the  purpose  of  having  the  honour  of  his  signature  in  the 
Statute  book  of  the  Society,  of  which  he  was  elected  a  mem- 
ber, Nov.  30). 

■  7'  Waits  on  the  Prince,  and  takes  the  opportunity  of 
giving  him  a  copy  of  Flamsteed's  estimate  of  his  Obser- 
vations. 


NEWTON'S   LIFE.  xxxvii 

1704  Dec.  18.    Letter  to  Flanisteed  (inviting  him  to  dinner  to  meet 

the  gentlemen  appointed  by  Prince  George  to  inspect  hift 
papers,  and  requesting  liini  to  Lring  his  papers,  or  specimens 
of  tluMn  for  the  referees  to  examine). 

20.    Letter  to  Flamsteed  (begging  him  to  bring  his  papers 

for  the  referees  to  examine). 

1705  Jan.  1.     (^N.  S.)    Equivocal  expressions  in  tho  review  of  his 

tract,  Do  Quadratura  Curvarum,  in  tho  Leipsic  Act8("*). 
(Origin  of  dispute  on  the  priority  of  discovery  of  the  new 
analysis). 

—  23.  Report  to  Prince  George  recoinn»ending  tho  publica- 
tion of  Flamstecd's  Observations  ("'). 

March  2.  Letter  to  Flamsteed  (earnestly  desiring  him  to  attend 
a  meeting  of  the  referees,  in  order  to  agree  about  an  amanu- 
ensis, calculators^  and  what  else  he  has  to  propose  for  dispatch- 
ing the  work). 

7.     Presents  Royal  Society  with  tho  1st  Vol.  of  Rymer's 

Foedera,  lately  published  (''"). 

Visit  to  Cambridge^'') 

Subscribes  £60.  towards  the  repairs  of  Trin.  Coll.  Chapel  ('"). 

April     Returns  to  London  (about  the  5th). 

10.     Knighted  by  Queen  Anne  at  Trinity  College. 

24  or  25.     Goes  to  Cambridge  to  contest  the  University. 

May  17.     Defeated  in  tho  contest  for  the  University("'). 

June  8.  Note  to  Flamstend  (m  .itlng  him  to  meet  the  referees 
at  dinner,  'Hhat  wo  may  sot  tho  press  a  going  as  soon  as 
possible  "). 

Sept.  14.  Noto  to  Sloane  (begging  him  to  get  Ilauksbee  to 
bring  his  air-pump  some  evening  to  his  house.  "  I  can  then 
get  some  philosophical  friends  to  see  his  experiments,  who  will 
otherwise  be  difficultly  got  together  ")("*). 

—  17.  Letter  to  Flamsteed  (urging  him  to  put  his  papers  to 
press.  "  If  you  stick  at  anything,  pray  give  Sir  Chr.  Wren 
and  mo  a  meeting  as  soon  as  you  can  conveniently,  that  what 
you  stick  at  may  bo  removed"). 

—  Noto  to  Sloane  (desiring  llauksbeo's  experiments  to  bo 
put  off  for  a  while,  as  Lord  Halifax,  Archbishop  of  Dublin, 
and  Robartes  are  out  of  town). 

Nov.  14.  Noto  to  Flamsteed  (inviting  him  to  meet  tho  referees 
at  dinner,  to  finish  the  agreement  and  sign  the  articles  about 
printing  his  book). 

20.     Signature  to  pedigree  ('"). 


XXXViii  SYNOPTICAL  VIEW  OP 

1 700    J^tin  edition  of  Optics  ("•). 

Sept.  13.  Note  to  Sloano  (tliinks  Bishop  Wilkiiia's  Legacy  of 
£400  in  1072  should  bo  defended  at  any  cost)C"). 
1707  Jan.  14.  Date  of  statutes  of  recently  founded  Plumian  Pro- 
fessorship, drawn  up  partly  under  his  eye("*). 
Apr.  9.  Note  to  Flamsteed  (requesting  him  to  meet  the  referees, 
tliat  all  things  may  he  now  settled  and  adjusted,  and  to  bring 
liis  bill  of  disbursements). 

—  Letter  to  Sir  John  Newton  (recommending  a  poor  kins- 
man as  undertaker  to  conduct  tho  funeral  of  his  cousin 
Coko)("''). 

1709  Jan.  12.     Gives  tho  Royal  Society  £20  ('""). 

Oct.  11.  Commencement  of  his  correspondence  with  Cotes 
relative  to  the  2nd  ed.  of  tho  Principia,  extending  from  this 
date  to  March  31,  1713(''"). 

1710  Sept.  13.     Note  to  Sloane  (glad  that  Sir  Christopher  and  Mr 

Wren  like  tho  house  in  Crane  Court,  proposed  to  bo  pur- 
chased for  the  Royal  Society,  and  hopes  they  will  like  the 
price  also). 
Dec.  14.     Promises  to  give  £100  towards  the  easing  of  the  debt 
for  tho  house,  besides  the  £20  mentioned  Jan.  12,  1709. 

1713  Midsummer.     Second  edition  of  Principia("-). 

Nov.  Paper  on  tho  different  kinds  of  years  in  use  among  tho 
nations  of  antiquity  ('**). 

1714  Apr.  2.     Letter  to  Keill  (respecting  an  answer  to  be  made  to 

Leibniz's  "charta  volans"  as  reprinted  with  remarks  in  tho 
Journal  Literaire)  Q^*). 

— —  20.     Letter  to  Keill  (on  same  subject). 

May  11.     Letter  to  Keill  (on  same  subject). 

Letter  to  Chamberlayne  in  reply  to  one  from  Leibniz  of 

Apr.  28,  (if  it  can  be  pointed  out  where  he  has  wronged 
Leibniz,  he  will  endeavour  to  make  satisfaction,  but  he  cannot 
retract  what  he  knows  to  be  true,  and  believes  the  Committee 
of  tho  Royal  Society  has  not  wronged  Leibniz)(*"). 

—  15.     Letter  to  Keill  (in  continuation  of  his  letter  of  tho 

11th). 
^lay — June.     One   of  Bishop  Moore's  Assessors  at  Bentley's 

trial  ('«"). 
End  of  ^May  or  be-  )  Evidence  before  a  Committee  of  the  House 

ginning  of  June.  /  of  Commons,  on  the  different  methods  of 

finding  the  longitude  at  sea(*"). 
Woodward's  Classification  of  Fossils  dedicated  to  him("*). 


NEWTON'S  LIFE.  xxxix 

lyjfi    Feb.  26.     Letter  to  Conti  in  answer  to  one  from  Ix>ibniz("*). 

May  18.     Observations  upon  Leibniz's  rej)!}' ("''). 

Juno  .'5.     Death  of  Cotcs('^')- 
J7I7     May  1().     Presents  his  portrait  to  the  Royal  Society^'*). 

Sept.  2L     Report  on  the  state  of  the  Coin  ('"). 

Nov.  2.3.     Another  Report  on  tlio  CoinC'^. 

1 718  Second  edition  of  Optics('"). 

Jan.  2L     At  the  IIouso  of  Lords  with  accounts  relating  to  the 

coinC^"). 
^lay  2.     Letter  to  Keill  (will  John  Benioulli's  denial,  in  a  jm- 

vate  letter,  of  tlio  autliorship  of  the  Epistola  jtro  eminent^ 

MathematkOy  satisfy  him  ('")  ?) 
Oct.  22.     Observations  on  the  state  of  the  Coin("'). 
Gift  of  £70  to  the  Royal  Society  ('^O- 

1719  July  1.3.     Present  to  Pound  the  Astronomer  ("")• 
Letter  to  Monmort,  enclosing  one  to  Bernoulli C^")*. 

1721  Third  edition  of  Optics  ("'). 

1722  Attack  of  stone. 

Oct.  22.  Letter  to  Arland  the  artist  (thanking  him  for  his  pro- 
fessional  services  in  tlic  matter  of  a  .j)luto  in  the  Frencli 
translation  of  the  Optics)  (*''■). 

1723  Jan.  17.     Appoints  (at  a  meeting  of  the  Council  of  the  Royal 

Society)  IMartin  Folkes  liis  Deputy  or  Vice-  President. 

1724  Apr.  27.     Report  on  Wood's  Halfpence  and  Farthings ("'). 
Jun.  25.     Imprimatur  for  new  edition  of  Ray's  Sf/nopsig  Plan- 

taritm  Britannicarum, 

g^"^^*  I  Delisle  in  England C^^). 

Aug.  25.  Letter  to  Lord  Townshend  (respecting  a  criminal 
under  sentence  of  death  for  coining:  thinks  the  law  should 
take  its  course)  ("*). 

Dec.  3.  Letter  to  Halley  (reque  ting  l)im  to  examine  two  of 
the  calculated  places  in  the  elliptic  orbit  of  the  Coniet  of 
1080,  and  to  calculate  anotlier  place,  supposing  the  orbit  a 
parabola)  C^"). 

1725  Jan.     Violent  cough  and  inflammation  of  tho  lungs.     Prevailed 

upon  to  take  a  house  at  Kensington. 

Feb.  Fit  of  tho  gout  in  both  his  feet  (liad  had  a  slight  attack 
a  few  years  before).     Improved  health  after  it. 

Letter  to  Mason,  Rector  of  Colsterworth,  notifying  his  subscrip- 
tion of  £12.  towards  erecting  a  gallery  in  Colsterworth 
church  C"). 

C2 


xl  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

1725  .  March  7.    Conversation  with  Conduitt  on  tho  formation  of  the 

planetary  bodies  ("•). 

—  25.  Grant  of  rents  (£25)  for  four  years  of  tho  ancestral 
part  of  his  estate  at  "Woolsthorpo  to  his  god-son  Isaac  Warner. 

May  12.  Letter  to  Mason  (very  glad  to  under8tanc\  that  tho 
gallery  in  Colsterworth  church  is  finished.  The  stirj/las  in 
I^Iason's  hands  belonging  to  him  to  bo  applied  "  to  the  use  of 
the  young  people  of  tho  parish  that  are  learning  to  sing 
Psalms,"  according  to  Mason's  desire). 

May  12.  Letter  to  his  tenant  Percival  of  Woolsthorpe,  agree- 
ing to  a  proposed  distribution  of  tho  commons  there  and  at 
Colsterworth  ("»). 

27.     Refuses  his  sanction  to  Freret's  Translation  of  his 

Chronological  Summary  ('•*•). 

July  1.     Visit  of  Abbe  AlariC"). 

Date  not  given.  Letter  to  Maclaurin  (glad  that  he  has  a  pros- 
pect of  being  joined  to  James  Gregory  in  the  Professorship  of 
Mathematics  at  Edinburgh,  and  heartily  wishes  him  good 
success)  ("'^). 

Date  not  given.  Letter  to  Lord  Provost  of  Edinburgh  (is  ready 
to  contribute  £20  per  ann.  towards  a  provision  for  Maclaurin, 
if  ho  will  act  as  assistant  to  Gregory). 

Towards  the  end  of  the  year.  Remarks  upon  Freret's  observa- 
tions in  his  unauthorised  translation  of  Newton's  Chronologi- 
cal Summary  ("=•). 

1726  Third  Edition  of  the  Principia  ("*)• 

May  10.  Letter  to  Mason  (with  note  for  £3  for  repair  of  the 
floor  of  Colsterworth  church). 

1727  Feb.  4.    Letter  to  Mason  (has  procured  assays  to  be  made  of 

the  pieces  of  ore  left  with  him  by  a  Woolsthorpo  friend  of 
Mason's,  but  they  contain  no  metal). 

Feb.  16.     Writes  Imprimatur  for  Hales's  Vegetable  Statics. 

March  2.  Present  for  the  last  time  at  a  meeting  of  the  Royal 
Society,  at  which  he  calls  attention  to  the  fact  of  the  Astrono- 
mer-Royal (Halley)  having  omitted  to  send  to  the  Society  a 
copy  of  his  annual  observations,  as  required  by  the  late  Queen's 
letterf"). 

— —  20.     Monday,  between  1  and  2  a.m.  Dies(""). 


NOTES. 


(')  *' Natus  est  Isaacus  Neutonu8...1ioril  primA  veJ  secunilli  post  mediam  noctemt 
idque  tempore  ipso  Plenilunii.  Capillis  eflloruit  sensim  in  gummam  canitiem  veraw, 
Annum  ajtatis  inter  trigcsimum  &  quadrngesimum.*'  (Nicolas  Fatio,  in  a  printed  copy 
of  Latin  Hexameters,  entitled  Neutonus  EclogOt  inserted  in  his  copy  of  the  3rd  ed.  of  the 
Principia  which  is  preserved  in  the  Bodleian  Library.) 

For  a  description  of  liis  person  and  liabits  sec  hi;  nephew  Conduitt*s  account  in 
Tumor's  Grantham  (pp.  103, 165),  or  Brewster's  Newton^  pp.  340—342. 

According  to  Hamsteed  he  was  short-sighted.  "  I  happened  once  {during  the  year 
1707}  to  visit  the  press  while  he  was  there,  and  took  the  opportunity  to  shew  him  how 
ill  the  compositor  hud  placed  the  types  of  the  figures  {in  Flamstecd's  Observations}... 
He  put  his  head  a  little  nearer  to  the  paper,  hut  not  near  enough  to  see  the  fault,  (for 
he  is  very  near  sighted,)  and  making-  a  slighting  motion  with  his  hand,  said,  •  Mcthinks 
they  are  well  enough.' "    (Baily,  p.  83.) 

(')  This  class  of  students  were  required  to  perform  various  menial  services,  which 
now  seem  to  be  considered  degrading  to  a  young  man  who  is  endeavouring  by  the  force 
of  his  intellect  to  raise  himself  to  his  proper  position  in  society.  The  following  extract 
from  the  Conclusion  Book  of  Trinity  College,  while  it  uftbrds  an  example  of  one  of  their 
duties,  will  also  serve  to  illustrate  tlie  rampant  buoyancy  of  the  Academic  youth  at  the 
period  of  the  Restoration.  "Jan.  16.  1660-1.  Ordered  also  that  no  bachelor  of  what 
condition  soever,  nor  any  undergraduate,  come  into  the  upper  butteries,  save  only  a 
Sizar  that  is  sent  to  see  liis  Tutor's  quantum,  and  then  to  stay  no  longer  than  is  requisite 
for  that  purpose,  under  penalty  of  6cU  for  every  time ;  but  if  any  shall  leap  over  the 
hatch  or  strike  a  butler  or  his  servant,  upon  this  account  of  being  hindered  to  come 
into  the  butteries,  he  shall  undergo  the  censure  of  the  Master  and  Seniors." 

(3)   Optics,  Bk.  II.  Part  iv.  Obs.  13. 

(*)  The  persons  appointed  (in  conjunction  with  the  Proctors,  John  Slader  of  Cath. 
Hall  and  Benj.  Pulleyu  of  Trin.  Newton's  tutor)  to  examine  the  Questionlsts,  were 
John  Eachard^(the  satirical  author  of  The  Grounds. ..of  the  contempt  of  the  Clerf^y...) 
of  Cath.  Hall  and  Tho.  Gipps  of  Trinity.  I  am  sorry  that  I  cannot  gratify  the  curiosity 
of  those  who  may  expect  to  find  here  a  notice  of  the  Academical  estimate  formed  of  the 
acquirements  of  tlie  most  illustrious  candidate  that  ever  offered  himself  for  a  degree,  as 
the  "  Ordo  Senioritalis"  of  the  Bachelors  of  Arts  for  this  year  is  provokingly  omitted  in 
the  Grace  Jiook. 

(*)    Shewing  how  to  take  the  fluxion  of  (or  to  differentiate)  an  equation  connecting 
any  number  of  variables.    It  is  referred  to  in  a  paper  which  peems  to  be  part  of  a 
draught  of  his  observations  on  Leibniz's  letter  of  Apr.  9,  1716.    (Kigaud's  AppendiSf 
p.  23,  compared  with  llaphson's  llittunj  of  Fluxions,  p.  116). 
(")    lligaud  and  Raphson,  u.  s. 

C)  The  recipe  described  in  the  subjoined  extiact  is  at  least  as  worthy  of  being 
recorded  as  Tasso's  malmsey,  or  Blackstone's  port.  "  I  have  been  credibly  infonned 
that  Sir  Isaac  Newton,  when  he  applied  himself  to  what  is  esteemed  the  greatest  stretch 


xlii  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

of  human  invention  and  penetration  (viz.  the  study,  investigation  and  analysis  of  the 
theory  of  light  and  colours)  to  quicken  his  faculties  and  fix  his  attention,  confined  him- 
self to  a  small  quantity  of  bread,  during  all  the  time,  with  a  little  sack  and  water,  of 
which,  without  any  regulation,  he  took  as  he  found  a  craving  or  failure  of  spirit^" 
Cheyne*8  Natural  Metfiod  of  curing  diseases  of  the  body  and  disorders  of  mind,  8^c. 
Lond.  1742,  p.  81. 

(■)  The  College  was  **  dismissed"  June  22  on  the  reappearance  of  the  plague.  The 
Fellows  and  Scholars  were  allowed  their  commons  during  their  absence.  Nev/ton 
received  en  this  account  3s.  4d.  weekly,  for  13  weeks  in  the  quarter  ending  Mich*  W66. 

12 Dec.  21 

5 Lady  Day,  1667. 

The  College  had  been  also  dismissed  the  previous  year,  Aug.  8,  on  the  breaking  out  of 
the  plague,  but  Newton  must  have  left  Cambridge  before  that,  as  his  name  does  not 
appear  in  the  list  of  those  who  received  extra  coes  for  6.^  weeks  on  the  occasion.  "  Aug. 
7,  1665.  A  month's  commons  (beginning  Aug.  8)  allowed  to  all  Fellows  and  scholars 
which  now  go  into  the  country  upon  occasion  of  the  pestilence."  (Conclusion  Book). 
On  the  continuance  of  the  scourge  we  find  him,  with  others,  receiving  the  allowance  for 
commons  for  12  weeks  in  the  quarter  ending  Dec.  21,  1665,  and  for  13  weeks  in  that 
ending  Lady-Day,  1666. 

(*)  'I'o  the  authorities  for  this  anecdote  (Biot,  Journal  des  Savans,  1832,  p.  265) 
may  be  added  Green  {Philosophy  of  Erpansive  and  Contractive  Forces,  p.  972),  whose 
information  on  the  point  was  derived  from  a  very  good  source :  **  quaj  sententia...origi- 
nem  ducit,  uti  omnis,  ut  fc-rtur,  Cognitio  nostra,  a  Porno,  idquud  accepi  ab...amicissimo 
Martino  Folkes."  For  the  sentiment,  compare  the  following  from  the  meditations  of  a 
modern  speculatist:  "  plcbiautcm  vis  gravitatis  cognituplacuit... quia... corpora  coelestia 
in  orbes  revolvi  praisertim  per  tritissimam  illam  pomi  coram  Newtone  delapsi  historiam 
edocta  securitatem  adversus  cocluni  hausit,  oblita  scilicet,  universaj  generis  humani, 
deinde  Troja;  mlseriaj  prliicipiis  poinum  adfuisse,  malum  etiam  scientiis  philosophicis 
omen."  Hegel's  Dissertatio  Fhibsophica  de  Orbitis  Planetarum — an  exercise  written  at 
the  age  of  31,  jjro  licentia  docendi.    Werke,  Band  16,  p.  18.  Berlin,  1834. 

(''')  In  this  tract  his  previous  method  of  taking  fluxions  is  extended  to  surds.  The 
area  of  a  curve,  whoso  ordinate  is  y,  i»  denoted  by  n  !/•  (Higaud's  Append,  p.  23.) 
(")  Raphson,  p.  116.  Wilson's  Appendix  to  Robins' Tracts  (II.  351— 356). 
(")  There  were  nine  fellowships  vacant;  among  them  those  of  Duport,  Thorn- 
dike,  and  Cowley  (the  last  by  <lcath  in  .luly,  1667).  Two  of  the  other  vacancies  were 
caused  by  the  parties  falling  down  staircases,  one  of  which  was  that  in  which  Newton 
subsequently  "kept."  All  the  nine  successful  candidates  were  in  their  last  year.  One 
of  the  middle  bachelors  had  procured  a  King's  letter  for  his  election,  but  an  order  was 
passed  by  the  Seniority  putting  him  oflT  until  the  following  year.  Besides  Pearson,  the 
Master,  Babington  and  Lynnet  were  probably  two  of  the  examiners  at  this  election.  It 
is  very  improbable  that  Barrow  examined :  he  was  thirteenth  on  the  list  of  fellows,  and 
by  the  absence  of  one  of  the  Seniors,  and  the  exclusion  of  another  (Barton)  on  the 
ground  of  mental  aberration,  he  became  temporarily  the  eleventh,  but  it  is  not  likely 
that  he  would  come  within  the  first  eight  on  so  important  an  occasion,  though  in  the 
preceding  June  he  had  sat  upon  the  Seniority  which  ejected  Barton  from  College. 

In  a  MS.  calendar,  drawn  up  by  Lynnet,  of  the  routine  events  of  an  academical  life, 
we  find  the  following  memorandum  relative  to  the  fellowship-examination;  it  was 
written  five-and-twenty  years  or  more  posterior  to  the  period  under  consideiation,  but 
the  practice  had  probably  undergone  little  change  in  the  interval.  "  The  fellowes  on  the 
3''  day  of  lluir  sitting  must  have  a  theme  given  them  by  the  ]Master,  w''  the  chappel- 
clerk  fetcheth  for  them:  they  sit  3  dayes  being  excused  the  4th  for  their  theme. 

"Thoy  sit  from  7  till  10,  &c  from  one  to  4,  each  writing  his  name  his  age  &  his 
country;  as  doe  the  scholars,  &  also  y"  Masters  of  Arts,  w*"''  papers  are  carried  to  y* 


NOTES.  xliii 

Master  &  Vice-M',  the  first  morning  so  soon  as  all  have  written... Oetoh.  l...by  y 
tolling  of  ye  little  bell  at  8  in  y^  morning  y«  scniours  are  called  ic  the  day  after  at  one 
o'clock  to  swear  them  y*  are  chosen " 

Tliere  was  no  election  of  fellows  in  the  years  1665  and  1666,  probably  on  account  of 
the  plague.  At  the  election  in  1G64,  there  were  seventeen  fellows  chosen,  seven  out  of 
the  middle  year,  and  five  out  of  each  of  the  other  years. 

(")  It  WHS  usual,  in  Trinity  College,  as  rooms  fell  vacant  to  distribute  them  amonjf 
the  fellows  in  the  order  of  their  seniority,  and  the  chamber  so  assif^ned  to  a  person  was 
called  his  "  seniority ''  or  "  fellowship  chamber."  A  few  of  the  papers  containing:  a 
schedule  of  the  succession  to  the  various  rooms  at  these  periodical  distributions  are  ttill 
preserved  in  the  archives  of  the  College,  and  among  them  is  the  one  which  was  arrnnged 
on  Sept.  30, 1667,  with  Pearson's  signature,  confirming  the  arrangement:  **Oct.  5,  K367. 
I  confirme  this  Succession  of  Chambers.  Jo.  Pearson  Master."  The  last  line  on  this 
paper  runs  thus:  "to  S""  Newton  —  Spirituall  chamber,"  n  locality  with  respect  to 
which  the  only  conjecture  that  I  have  to  offer  (and  it  i,j  not  altogether  free  from  objec- 
tion) is  that  the  apartment  so  designated  may  have  been  the  ground-room  next  the 
Ch;ipel,  in  the  north-east  corner  of  the  great  court.  There  is  some  reason  for  suppos- 
ing that  this  room  was,  previously  to  1640,  the  vestry,  and  that  it  is  the  same  as  that 
which  is  denominated  the  "vestry,"  or  "vestry  chamber,"  in  the  Junior  Bursar's 
Books  of  1648  and  1649.  Though  "  spiritual  chamber "  is  put  down  in  the  schedule  as 
the  habitation  assigned  to  Newton,  it  docs  not  follow  that  he  actually  dwelt  there  ;  if  he 
did  not  occupy  the  room  himself,  he  would  receive  the  rent  of  it  from  the  person  who 
was  his  tenant. 

ITie  rooms  that  he  occupied  before  he  was  elected  fellow — the  scene  of  the  experi- 
ments by  which  ho  analysed  light — are  not  known.  There  is  no  mention  of  them  in  the 
Junior  Bursar's  books  during  that  period.  Neither  b  it  known  in  what  part  of  the 
College  he  lived  from  the  epoch  just  mentioned  to  1683.  He  himself  states,  that  in  June, 
1673,  John  Wickins  (a  fellow,  two  years  junior  to  him)  was  his  chaml>er-fellow  (Jitter 
to  IldUey,  July  27,  1686).  But  in  the  Junior  Bursar's  Book  for  the  year  ending  at 
Michaelmas,  1673,  we  find  the  two  entries  "  for  selling  M'  Newton's  chamber,"  *'  for 
mending  the  slating. ..over  M'  Wickins,"  from  which  perhaps  we  may  infer  that  one  of 
them  had  changed  his  rooms  in  the  interval  between  June  and  September*.  In  1678  ho 
had  a  sizar  living  with  him :  "  for  mending  over  M""  Newton's  sizar's  chamber."|(  JiiMim- 
Bursar's  Hook.)  The  first  notice  of  Newton's  rooms  which  fixes  their  position,  occurs 
in  the  Junior  Bursar's  Book  for  the  year  ending  ut  INlichaelmas  1603,  and  wo  then  find 
him  inliabiting  the  rooms  which  well-informed  tradition  still  points  out  to  the  stranger 
(the  rooms  on  the  first  floor  to  the  north  of  the  Great  Gateway) :  "For  mending  the 
wall  betwixt  Mr  Newton's  garden  and  St  John's"  (probably  a))Out  the  end  of 
1682).  I  am  unable  to  determine  satisfactorily  the  date  of  his  taking  these  rooms, 
but  the  most  probable  supposition  is  that  he  went  into  tliein  in  the  summer  of  1679 1. 
Herbert  Thorndikc  preceded  him  in  the  occupation  of  them  (with  one  or  two  removes): 
when  Newton  left  Cambridge  in  1696,  they  seem  to  have  come  into  the  possession  of 


•  If  It  wai  Newton  that  change*!,  we  may  find  in  th.it  fact  a  foundation  for  the  atntcmpnt 
made  by  a  grandson  of  Wickins,  who,  in  making  mention  of  a  woo<>en  pint  flagon  given  to  his 
grandfather  by  Newton,  says:  "  This  with  the  whole  furniture  of  the  chambers  devolved  u|>on  my 
ancestor  upon  Sir  Isaac's  leaving  the  college,  and  hath  with  some  other  articles  remained  in  the  family 
ever  since."  (Gent.  Mag.  Apr.  H)<>2.)  Wickins  vacated  his  fellowship  In  ICa^  (eleven  years  before 
Newton  left  College),  and  had  i-cascd  to  reside  for  several  years.  Yet,  curiously  enough,  In  Walker's 
account-book,  quoted  p.  xnv,  in  the  statement  of  the  "  income"  of  his  rooms,  there  \%  the  following 
hem  (date  I/IG):  "  I'aid  !)•  Wickins  a  bill  for  rcpiiring  what  Mr  Hanbury's  brother  UtoV.  away, 
£\.  85."  «'D»  Wickins"  was  a  son  of  Newton's  fiiend,  and  had  Just  taken  his  bachelor's  degree. 
Perhaps  he  had  occupied  part  of  the  rooms  jointly  with  llanbury, 

♦  A  view  of  Newton's  rooms  from  the  eai>t,  with  the  garden  attached,  may  be  seen  In  Loggan's 
plate  of  the  College.     The  following  chronological  notices,  in  conjunction  with  Loggan'i  plates. 


xliv  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE 

Daniel  Hopkins,  whom  Bentley  describes  as  "  a  Fellow  of  Trinity  College  and  a  very 
useful  person  in  it,  having  the  greatest  number  of  pupils  of  any  one  amongst  us  "  (CVr< 
retpondencey  p.  185);  Nat.  Ilanbury  (see  p.  192)  took  them  in  1704,  and  was  succeeded 
in  1715  by  *'  Our  hat"  Walker,  who  continued  in  them  until  his  death  in  1764.  Cum- 
berland, who  came  up  a  freshman  in  1747,  speaking  of  the  kindness  shewn  to  him  by 
Walker,  who  was  Vice-Master,  says:  *'  He  frequently  invited  me  to  his  rooms,  which 
1  had  80  often  visited  as  a  child,  and  which  had  the  further  merit  with  me  as  having  been 
the  residence  of  Sir  Isaac  Newton,  every  relic  of  whose  studies  and  experiments  were 
respectfully  preserved  to  the  minutest  particular,  and  pointed  out  to  me  by  the  good  old 
Vice-Master  with  the  most  circumstantial  precision.  lie  had  many  little  anecdotes  of 
my  grandfather  {Dentley},  which  to  me  at  least  were  interesting-,  and  an  old  servant 
Deborah,  whom  he  made  a  kind  of  companion,  and  who  was  much  in  request  for  the 
many  entertaininf;  circumstances  she  could  narrate  of  Sir  Isaac  Newton,  when  she 
waited  upon  him  as  his  bedmaker,  and  also  of  D""  Bentley,  with  whom  she  lived  for 
several  years  after  Sir  Isaac  loft  college,  and  at  the  death  of  my  grandfatlier  was  passed 
over  to  D'  Walker,  in  whose  service  she  died."  {Memoirs,  p.  73.)  What  the  '*  relics  " 
alluded  to  were  I  cannot  exactly  say.  It  happens  that  Walker's  private  account-book 
has  been  preserved.  It  contains  a  statement  of  what  is  called  the  "income"  of  his 
rooms,  and  an  inventory  of  the  furniture  and  movables  in  them  and  in  the  garden.  In 
the  list  there  appears  a  "  thermometer,"  "  a  bureau  bought  of  r)'  Smith  {the  Mastjr}," 
a  "  violoncello  (sold),"  *'  a  picture  of  Vandyke,"  *'  a  barometer,"  and  10  pounds'  worth 
of  books,  but  there  is  nothing  to  indicate  that  any  of  these  or  the  other  articles  ever 
belonged  to  Newton.  In  1730  Walker  made  considerable  alterations  in  the  rooms. 
The  same  book  contains  his  accounts  with  his  bedmaker,  Betty  Baxter,  and  on  her 
death,  in  Feb.  1744,  with  her  sister  "  Deb."  They  seem  to  have  been  both  women  of 
thrift,  and  improved  their  capital  by  loans  to  their  master.  Deborah  did  not  profit  by 
her  attendance  upon  Newton  to  learn  the  art  of  writing:  in  Walker's  book,  instead  of 
her  signature,  slie  appends,  like  our  early  kings,  her  mark. 

('*)  lie  wjw  23rd  on  the  list  of  148  signed  by  the  Sen.  Proctor  (Thomas  Burnet, 
author  of  Theoria  Telluris  Sacra). 

(")  It  was  6  inches  long,  aperture  something  more  than  an  inch,  depth  of  plano- 
convex eye-pflass,  one-sixth  or  one-seventh  of  au  inch,  magnifying  power  about  40. 
(Letter  of  Feb.  23,  1669  in  INIacc.  Corr.  ii.  289.    Comp.  Brewster's  Newton,  p.  27.) 

(")  The  Lucasian  statutes,  dated  Dec.  19,  1663  (they  are  printed  in  the  Appendix 
to  Whiston's  Account  of  his  Prosecution,  ed.  1718-9)  require  the  Professor  to  lecture  at 
least  once  a  week  during  term-time,  on  some  portion  "Geometriaj,  Arithmeticje,  Astro- 
nomic, Geographia;,  Optica;,  Staticaj  aut  alterius  alicujus  Mathematicae  Disciplinae" 
...••  per  unius  circiter  horje  spatium,"  and  also  two  days  in  the  week  during-  term-time 
(and  during  vacation  one  day,  if  the  Professor  is  in  residence)  "  per  duas  horas...omni- 


will  enable  the  academical  reader  to  plctiue  to  himself  the  College  as  it  was  when  Newton  walked 

to  and  fro  within  Its  courts  : 

1670-1    Oerrard's  Hostle  rebuilt  at   the  expense  of  Bishop  Hacket  and  thence  called   Bishop's 

Hostle. 
167(1    Peb.    Foundation  of  new  Library  dug. 
1678    Rooms  over  eight  arches  next  the  Library  in  north  cloister  finished,  those  next  the  library 

being  built  out  of  the  subscriptions  for  the  Library,  those  next  to  them  to  the  east  at  the 

expense  of  Sir  Thomas  Sclatcr. 
1681-S    Rooms  over  right  arches  next  the  Library  in  south  cloister  built,  those  adjoining  the  library 

out  of  the  library  subscriptions,  the  others  at  the  expense  of  D*  Humfrey  Babington. 
,  1G81    May  J.    Four  statues  on  the  top  of  the  library  by  Cibber  for  which  he  received  £80. 
1685    Feb.    New  Library  ceiled. 
UJ86    Library  floor  laid  down. 
16S7-8    Library  paved. 

1694    Ruinous  part  of  King's  Hostle  pulle<l  down. 
169.'5    Books  removed  from  the  old  library  to  the  new. 


NOTES.  xlv 

bus  ilium  consulturU  vacare,  libcrum  adeuntibus  aperto  cubiculo  acceHsum  praibere, 

circa  propositas  ipsi  qua?stioncs  &  difKcultates  baud  f^rravato  respoudert'." Tbi»  la»t- 

mentioned  part  of  tbe  Professor's  prescribed  duties  explains  a  p&Mage  in  the  Life  of 
Henry  Wharton  (15.A.  in  1G84),  wlio,  we  are  told,  "  attained.. .no  mean  skill  in  mathe- 
matics. Which  last  wa<<  much  increased  by  the  kindness  of  M'  Isaac  Newton,  Fellow 
of  Trinity  College,  the  incomparable  Lucas- ProfLssor  of  Mathematics  in  the  University, 
who  V  as  pleased  to  pive  him  further  instructions  in  that  noble  science,  amongst  a  select 
company  in  his  own  private  chamber."  Life  of  Wharton,  prefixed  to  his  Sermons, 
2nd  ed.  1700. 

The  Letter  of  Charles  IL  (confirming  the  Lucasian  statutes),  dated  Jan.  18,  1664, 
further  ordered  that  all  Undergraduates  after  their  2nd  year,  and  all  Uachulors  of  Arts 
"  usque  ad  annum  tertium,"  should  attend  the  l*rofessor'8  lectures:  it  also  allowed  the 
Professor  to  hold  a  Fellowship  along  with  his  Professorship,  but  forbad  him  "  DecaiM, 
Thesaurarii,  Seneschalli,  aut  Lectoris  cujusvis  in  suo  CoUegio  munus  capessat,  aut... 
inibi  Tutorem  se  gerat  (nisi  Nobilium  forte  vel  (Jenerosorum  Sociis  Commcnsalium), 
vel...  Procurator  is,  Taxatoris,  aut  altcrius  cujuslibet  I^etoris  publicum  in  Academii 
Olficium  8Ustineat...Ab  omnibus  et  singulis  Muneribus  istis  prwdiclis  liberatum  volu- 
mu3  et  exemptum."  (Baker  AISS.  xxix.  403.)  This  prohibition  will  account  for  our 
not  finding  Newton's  name  at  any  time  among  the  College  or  University  OflTicers.  He 
availed  himself  of  the  privilege  of  taking  Fellow-Commoners  as  pupils  in  two  instance:! 
only :  viz.  Mr  George  Markham  (son  of  Sir  llobt.  INIarkham,  of  Sedgebroke,  NotU), 
afterwards  Pnronet  and  F.R.S.,  entered  Jun.26, 1680,  and  Mr  Robt.  Sacheverell,  whose 
mother  was  daughter  of  the  2nd  Sir  John  Newton,  and  sister  of  the  3rd  IJuronet  of  the 
name  (to  whom  Letter  No.  XXXL  Ajjpendix,  is  addressed)  entered  Sept.  16,  1687. 
Wo  also  find  Mr  St  Leger  Scroope  (possibly  connected  afterwards  by  marriage  with 
Sir  John  Newton's  family)  entered  Fellow-Commoner  under  him  Apr.  2,  1669,  before 
he  was  appointed  Lucasian  Professor. 

In  1675  Newton  obtained  a  Royal  Patent  allowing  the  Professor  to  remain  Fellow  of 
a  College  without  being  obliged  to  go  into  orders,  as  the  statutes  of  some  Colleges  re- 
quire.    Sec  below,  under  that  year. 

In  packet  No.  E.  of  the  Lucasian  MSS.  there  is  a  copy  (with  a  few  clerical  errors) 
of  the  Statutes  and  the  King's  Confirmation  of  them  in  Newton's  handwriting  on  a  folio 
sheet  doubled  twice.    On  the  last  page  he  has  written  the  following,  as  a  help  to  his 
memory,  the  almanacs  not  having  yet  begun  regularly  to  register  the  information: 
Termini  durant  1.  a  10°  Octob.  ad  16'""  Decemb. 

2.  a  13"  Jan.  ad  10  ante  Paschn 

3.  ab  IP  post  Pascha  ad  diem  veneris  Comitia  scquentem. 

(")  This,  like  most  of  Newton's  letters,  is  in  answer  to  questions  proposed  to  him. 
(")  In  this  letter  he  says:  **That  solution  of  the  annuity  problem  {in  letter  of 
Feb.  6}  ...you  have  my  leave  to  insert  it  into  the  Philos.  Trans,  so  it  be  without  my 
name  to  it.  For  I  see  not  what  there  is  desirable  in  public  esteem,  were  I  able  to 
acquire  and  maintain  it.  It  would  perhaps  increase  my  acquaintance,  the  thing  which 
1  chiefly  study  to  decline."    Mace.  Corr.  ii.  296. 

(")  Newton  wishes  his  name  to  be  suppressed  in  connexion  with  the  improve- 
ments made  in  the  book,  and  suggests  that  in  the  title-page,  after  the  words  '*  Nunc  e 
Belgico  Laline  versa,"  some  such  words  as  **  et  ab  alio  authore  locupletata"  should  be 
added. 

(20)  Collins,  writing  to  Vernon  at  Paris,  Dec.  26,  says :  "  As  to  Mr  Newton's 
Telescope,  I  suppose  Mr  Bernard  {  of  Oxford  }  writ  the  same  to  you  as  he  did  to  me 
upon  the  authority  of  one  IMr  Gale  of  Cambridge  {  Fellow  of  Trin.  Coll.  afterwards 
Dean  of  York}  :  since  it  hath  been  brought  up  for  his  Majesty's  perusal,  i!c  I  have 
seen  an  object  in  it,"  &c.  He  then  proceeds  to  give  a  description  of  the  instrument. 
(Royal  Soc.  MSS.  i.xxxi.)  Compare  Collins  to  Vernon,  Dec.  14,  in  Mace.  Corr.  i. 
176.    This  instrument  is  in  the  possession  of  the  Hoyal  Society.    The  instrument  in 


xlvi  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

Trinity  Colleg«  Library,  which  is  usually  shewn  to  visitors  as  Newton's  own  telescope, 
1  believe  to  have  belonged  to  Robert  Smith,  and  to  be  that  which  is  described  in  his 
OpticSf  p.  304,  note.  The  inscription  upon  it,  "  Sir  Isaac  Newton's  Telescope,"  merely 
means  **  a  Newton'an  Telescope.** 

(«»)  It  was  uever  finished.  It  was  published  by  Ilorsley,  i.  391—518,  under  the 
title  of  Geometria  Analytica.  It  first  appeared  in  1736,  in  Colson's  translation,  with  the 
title,  "  The  Method  of  Fluxions  and  Infinite  Series,  with  its  Application  to  tlie  Geome- 
try of  Curve-Lines.  By  the  Inventor,  Sir  Isaac  Newton. ..translated  from  the  Author's 
Latin  Original  not  yet  made  public...**  Peraberton,  in  speaking  of  the  treatise,  tells  us 
that  he  had  prevailed  upon  Newton  *♦  to  let  it  go  abroad."  "1  had  examined  all  the 
calculations  and  prepared  part  of  the  figures ;  but  as  the  latter  part  of  the  treatise  had 
never  been  finished,  he  was  about  lettinor  me  have  otlier  pnpers  in  order  to  supply  what 
was  wanting.  But  his  death  put  a  stop  to  that  design."  (Preface  to  View  of  Newton's 
Philosoifnj,  Lond.  1728.) 

(23)  In  answer  to  Oldenburg's  letter  of  Jan.  2,  printed  in  the  Appendix,  No.  I. 
The  opening  and  concluding  paragragJia_are  transcribed  here,  principally  on  account  of 
the  touching  modesty  of  the  closing  words  of  the  latter. 

"At  the  reading  of  your  letter  I  was  surprised  to.  see  so  much  care  taken  about 
securing  an  invention  to  me,  of  which  1  have  hitherto  liad  so  little  value.  And  tliere- 
fore  since  the  Royal  Society  is  plcaseii  to  think  it  worth  the  patronising,  1  must  acknow- 
ledge it  deserves  much  more  of  them  for  that,  than  of  me,  who,  hud  not  the  communica- 
tion of  it  been  desired,  might  have  let  it  still  remain  in  private  as  it  hath  already  done 
some  years. 

*•  I  am  very  sensible  of  the  honour  done  rae  by  t?ie  Bishop  of  Sarum  in  proposing 
me  candidate,  and  which  I  hope  will  be  furtlier  conferred  upon  me  by  my  election  into 
the  Society.  And  if  so,  I  shall  endeavour  to  testify  my  gratitude  by  communicating 
what  my  poor  and  so"itary  endeavours  can  effect  towards  the  promoting  your  philoso- 
phical designs."    Mace.  Coit,  ii.  311,  313. 

(■")  A  copy  of  this  description,  with  Newton's  alterations  added  by  Oldenburg,  is 
preserved  at  the  Royal  Society.  Orig.  Lett.  Bk.  N.  i.  37.  It  is  printed  in  Ilorsley 's 
NeicUm,  IV.  270. 

Voltaire  informs  us  that  he  had  seen  a  little  work  by  a  German  Jesuit,  published 
about  this  time,  "dans  lequel,  en  purlant  du  telescope  de  Newton,  on  le  prend  pour  un 
liinetier:  Artifex  quidam  Anglus  nomine  Newton.  La  posterit6  I'a  bien  veng^."  (Diet. 
Philos,  and  some  editions  of  tlie  Lettres  Fhibs.) 

{"*)  *'  It  was  ordered  that  a  letter  should  be  written  by  the  secretary  to  Mr  Newton 
to  acquaint  him  of  his  election  into  the  Society,  and  to  thank  him  for  the  communica- 
tion of  his  telescope,  and  to  assure  him  that  the  Society  would  take  care  that  all  right 
should  be  done  him  with  respect  to  this  invention."  Birch,  iii.l.  Picart's  recent 
measure  of  the  earth  was  also  communicated  at  the  same  meeting  in  a  letter  from  Vernon 
to  Oldenburg,  dated  Paris,  Jan,  P,  but  Oldenburg  does  not  seem  to  have  made  any 
allusion  to  it  in  the  letter  which  he  was  directed  to  write  to  Newton. 

(25)    Appendix,  No.  II. 

(2«)    Appendix,  No.  III. 

(27)    Appendix,  No.  IV. 

(2«)    Appendix,  No.  V. 

(•*)    Appendix,  No.  VI. 

(30)  i"  Finding  already,  by  that  little  use  I  have  made  of  the  press  that  I  shall  not 
enjoy  my  former  serene  liberty  till  I  have  done  with  it,  which  I  hope  will  be  so  soon  as 
I  have  made  good  what  is  already  extant  on  my  account."  He  adds  that  he  may  possi- 
bly complete  his  method  of  infinite  series,  "the  better  half  of  which  was  written  last 
Christmas."    Mace.  Corr.  i\.  322. 

Under  this  date  may  be  given  the  anecdote  related  in  Nichols's  History  of  Hinckley 
(p.  61,  note),  if,  as  is  probable,  it  refers  to  the  action  between  the  English  and  Dutch 


NOTES.  ^  xlvii 

fleets  in  South  wold  bay  on  the  28th  of  May.  **  There  U  a  traditional  story  at  Cambridge 
...  { that }  Sir  Isaac  Newton  came  into  the  hall  of  Trinity  College  and  told  the  other 
fellows  that  there  had  been  an  action  just  then  between  the  Dutch  and  Enjflish,  and 
that  the  latter  had  the  worst  of  it.  Ikinf^  asted  how  ho  came  by  his  knowledge,  ho 
said  that  being  in  the  observatory,  he  heard  the  report  of  a  great  firing  of  cannon,  such 
as  could  only  be  between  two  great  ileets,  and  that  as  the  noise  grew  louder  and  louder 
he  concluded  that  they  drew  nearer  to  our  coasts  and  consequently  that  we  had  the 
worst  of  it,  which  the  event  verified."  Jones,  in  his  Physiological  Disquisitiims,  p.  299 
(quoted  ib.),  says  that  he  had  been  infonned  "  that  the  great  engagement  l>etwecn  the 
Dutch  and  English  at  sea  in  1672  was  heard  by  the  people  who  were  out  at  work  in  the 
fields  to  the  very  centre  of  England  :  Mr  Dcrham  says  it  was  heard  200  miles."  The 
"observatory"  in  the  passage  quoted  alwve  is  a  prolepsLs  for  the  "great  gateway," 
which  was  not  converted  into  an  observatory  until  several  years  after  Newton  had  left 
Cambridge. 

(8>)    Appendix,  No.  VII. 

("5)  He  also  says, "  I  should  be  glad  to  hear  whether  Mr  Cox  hath  finished  the 
4  feet  telescope  and  what  its  efTects  are...  1  Jut  I  know  not  whether  I  shall  make  any 
further  trials  myself,  being  desirous  to  prosecute  some  other  studien."  Mace.  Corr.  ii. 
329. 

(«')  For  a  character  of  this  work  see  Humboldt's  Kmwm,  Vol.  i.  The  edition  of 
1681  seems  to  be  almost  a  reprint  of  the  preceding  one,  in  spite  of  the  ' '  auctior  et  emen- 
datior  "  of  the  title-page. 

(»*)    Appendix,  No.  VIII. 

(8'^)  This  part  of  the  letter  is  cited  in  the  3rd  edition  of  the  Principiot  p.  246,  instead 
of  the  letters  to  Leibniz  referred  to  in  the  two  first  editions.  Its  contents  were  sent  to 
Leibniz  July  26,  1676,  alon<ir  with  Newton's  letter  of  June  13  of  that  year.  There  is  a 
copy  of  it  at  the  lloyal  Society  (Miscell.  MSS.  lxxxi.)  written  in  a  tremulous  hand, 
a  consequence  probably  of  the  endeavour  of  the  copyist  to  imitate  Newton's  writing.  It 
has  an  address  in  Newton's  hand,  "These  to  his  ever  Honoured  flViend  M'  John  Col- 
lins...," and  bears  the  post-mark  of  May  27  (probably  1676).  This  transcript  may  be 
conjectured  to  have  been  made  at  CoUins's  request  for  the  purpose  of  accompanying  the 
other  papers  which  he  was  preparing  to  send  through  Oldenburg  to  Leibniz.  See  Com- 
mere.  Epjs?.  p.  47.  (128,  2nd  ed.)  Doubts  have  been  expressed  whether  these  papers 
were  actually  sent  to  Leibniz.  We  have  however  Collins's  own  testimony  that  they 
were  sent  as  had  been  desired  (Comm.  Epist.  p.  48,  or  129,  2nd  ed.),  besides  I^ibniz's 
and  Tschirnhaus's  acknowledgments  of  the  receipt  of  them,  {lb.  pp.  58, 66,  or  129, 142.) 
It  may  also  be  observed  that  the  papers  actually  sent  (in  a  letter  dated  July  26,  1676) 
to  Leibnij;  by  Oldenburg  have  been  recently  printed  from  the  originals  in  the  Royal 
Library  at  Hanover  (Leibn.  Math,  Schrift.  lierlin,  1849),  and  that  in  them,  as  in  Col- 
lins's draught,  which  is  preserved  at  the  lloyal  Society  (*'  To  Leibnitz  the  14th  of  June 
1676  About  Mr  Gregories  remains  "  MSS.  lxxxi.),  we  find  the  contents  of  Newton's 
letter  of  Dec.  10, 1672,  except  that  instead  of  the  example  of  drawing  a  tangent  to  a 
curve,  there  b  merely  allusion  made  to  the  method.  Collins's  larger  paper  (called 
"Collectio"  and  •'  Historiola"  in  the  Commercium  Epistolicum),  of  .which  the  paper 
just  quoted  "About  Mr  Gregories  remains"  is  an  abridgment,  and  which  contains 
Newton's  letter  of  Dec.  10  without  curtailment,  is  stated  in  the  second  edition  of  the 
Commercium  to  have  been  sent  to  Leibniz,  but  whether  that  was  the  cjise  may  l)e  fairly 
questioned.  This  paper  was  intended  by  Collins  to  be  deposited  in  the  archives  of  the 
lloyal  Society,  where  it  is  still  preserved,  with  the  title  "tlxtiacts  from  Mr  Gregories 
Letter"  (MSS.  lxxxi.),  consisting  of  thirteen  sheets.  A  copy  of  Newton's  letter  was 
sent  to  Tsehirnhaus  in  May,  1675,  in  Collins's  paper  "About  Descartes"  (14  folio 
leaves,  Roy.  Soc.  MSS.  lxxxi.) 

(3")    On  the  Public  Oratorship  becoming  vacant  by  the  resignation  of  Ralph  WiJ- 
driiigton,  the  mode  of  electing  his  successor  became  a  subject  of  dispute  between  the 


xlviii  SYNOPTICAL  VIEW  OP  NEWTON'S  LIFE. 

Masters  of  Colleges  and  the  Senate.  The  Statuteii  of  Elizabeth  contain  no  express  pro- 
vision for  the  election  of  Orator,  but  the  Heads  (under  the  40th  Statute,  which  enacts 
that  •*  Nominationcs  et  electiones  lectorum,  bedellorum,  stationariorum,  gageatorum, 
vinopolarum  et  aliorum  ministrorum  seu  olHciariorura  academiffi  quorumcunque  de  qui- 
bus  aliter  a  nobis  non  est  provisum  sequentur  moduni  et  formam  in  electione  procancel- 
larii  praescriptara  fientquo  intra  xiv  dies  post  vacationem  nisi  aliter  staiutis  nostris  aut 
fundatione  cautum  sit ")  claimed,  as  had  been  usual,  the  right  of  nominating  two  persons, 
one  of  whom  was  to  be  elected  by  the  Senate.  The  Senate,  however,  maintained  that 
the  proper  mode  of  procedure  was  by  an  open  election,  as  directed  by  the  Statuta  Anti- 
gua, which  they  contended  were  still  in  force,  except  upon  points  where  they  were  con- 
trary to  the  Elizabethan  code.  The  Chancellor  ("  great  Villiers")  endeavoured  to 
effect  an  arrangement  between  the  contending  parties.  "Being- informed,"  he  writes, 
"that  there  may  be  a  contest  between  the  Heads  of  the  Colleges  and  the  body  of  the 
University  about  the  manner  of  electing  an  Orator,. ..he  thinks  it  becomes  his  duty  and 
affection  to  the  University  to  communicate  his  thoughts  :...he  thinks  that  the  election  of 
Orator  should  be  regulated  by  the  statute  of  Henry  VIII.  made  only  for  that  purpose 
rather  than  by  that  of  Queen  Elizabeth."  He  suggests  an  expedient,  which  he  says 
"  I  hope  may  for  the  present  satisfy  both  sides.  I  propose  tliut  the  Heads  may  for  this 
time  nominate  and  the  Body  comply,  yet  interposing  (if  they  think  fit)  a  Protestation 
concerning  their  plea  that  this  election  may  not  hereafter  pass  for  a  decisive  Precedent 
in  prejudice  to  their  claim."  And  "whereas  I  understand  that  the  whole  University 
has  chiefly  a  consideration  for  D'  Paman  of  St  John's  and  Mr  Craven  of  Trinity  College 
I  do  recommend  them  both  to  be  nominated.  For  it  is  very  reasonable  that  in  this  nomi- 
nation, before  the  difference  be  determined  between  you,  the  Heads  should  have  regard 
to  the  inclination  of  the  Body,  especially  seeing  you  all  agree  in  two  men  that  are  very 
worthy  and  very  fit  for  the  place."  (Letter  read  to  the  Senate,  March  3.  Mandates  in 
Registr.  Office,  Vol.  ii.  p.  251*.)  These  conciliatory  suggestions  were  not  attended  to. 
A  majority  of  the  Heads  nominated  Paman  and  a  Mr  lialph  Sanderson,  likewise  of  St 
John's,  on  the  day  after  the  letter  was  read,  and  on  the  next  day  121  Members  of  the 
Senate  recorded  their  votes  in  favour  of  Craven  and  98  for  Paman.  On  the  morning  of 
the  election,  before  the  polling  commenced,  the  following  protest  was  read  and  entered 
in  the  Regent  House :  "  Nos  Antonius  Marshall,  Georgius  Chamberlaine,  Ilumfredus 
Babiogton,  Gulielraus  Lynnet,...Ioannes  Hawkins,  Isaacus  Newton. ..aliique  quorum 
nomina  sunt  infra  scriptu,  coram  Matlhajo  Whinn,  Notiirio  Publico,  Protestamur  de 
invaliditate  et  nullitate  Nomiuationis  et  Notationis  per  puncta  Prfcfectorura  CoUegio- 
rum  ad  Officium  Oratoris  hujus  Academiai.  Etiara  et  de  nullitate  oinnis  actus  exin 
facti  aut  faciendi."  The  Vice-Chancellor  admitted  Paman  the  same  morning;  Craven, 
as  "legitime  electus...per  majorem  partem  suffragantium  secundum  statutum  de  elec- 
tione Oratoris,"  gave  in  a  protest  against  the  validity  of  his  competitor's  election  and 
admission,  and  there,  so  far  as  our  information  goes,  the  matter  seems  to  have  ended. 

The  reader  who  wishes  to  see  what  may  be  said  on  both  sides  of  the  question  may 
consult  an  anonymous  pamphlet,  entitled  An  Argument  to  prove  that  the  39th  section  of 
the  C^lh  chapter  of  the  statute,  given  by  Queen  Elizabeth. ..includes  the  Old  Statutes  [by 
Mr  Burford,  fellow  of  King's]... «;j7/t  an  Answer  to  the  Argument  [by  Bentley]  and... 
[Burford's]  Reply.  London,  1727.    Comp.  Monk's  Bentley,  pp.  524—6. 

(37)  '*  Since  I  see  I  shall  neither  profit  them,  nor  (by  reason  of  this  distance)  can 
pau-take  of  the  advantage  of  their  assemblies."    Mace  Corr.  ii.  348. 

(s«)  It  begins,  "1  received  your  two  last  letters  with  Heuret's  Optics,  which  (not 
being  so  ready  in  the  French  tongue  myself,  as  to  read  it  without  the  continual  use  of 
a  dictionary)  I  committed  to  the  perusal  of  another..." 

Here  may  be  mentioned  the  myth  respecting  his  not  being  elected  into  the  law- 
fellowship,  which  became  vacant  Feb.  14,  in  this  year,  by  the  death  of  Dr  Robert 
Crane.  The  story  as  told  by  a  great-grandson  of  the  person  who  was  selected  to  fill  the 
vacancy  is,  that. Newton  and  Robert  Uvedale  (who  was  two  years  senior  to  Newton,  and 


NOTES.  xlix 

would,  in  the  usual  course  of  thino^,  vacate  his  fellowship  in  a  few  months)  were  candi- 
dates for  the  fellowship  in  question ;  and  that  "  Mr  Rarrow  {  who  had  been  admitted 
Master  on  Feb.  27  )  decided  it  in  favour  of  Mr  U.  saying  that  Mr  U.  and  Mr  N.  being 
Cat  that  time)  equal  in  literary  attainmeuts,  he  must  give  the  fellowtihip  to  Mr  U.  as 
senior."  {Gentleman's  Mag.  Supplement  for  1799,  p.  1186.)  I  apprehend  the  tenure  of 
the  law-fellowship  of  Trinity  College  was  considered  to  be  scarcely  compatible  with  the 
efficient  dischdrpe  of  the  duties  of  the  Mathematical  Professor,  and  I  believe  that  it 
would  arpTue  much  misconception  of  the  characters  of  the  two  preat  men  concerned  to 
suppose  them  capable  of  beinp  parties  to  a  lax  interpretation  of  the  statutes  which  they 
had  Bworn  to  obey.  The  person  who  holds  this  fellowsliip  is  required  "  operam  dare 
juri  civili,"  and  accordln^rly  we  find  UvodaIe,on  receiving  the  appointment,  excused  by 
the  University  from  appearing,  according-  to  an  announcement  made  in  April  previous 
to  his  election,  as  Respondent  in  the  Theological  Schools  on  June  26  (the  fellow  next 
below  him  beinp  called  upon  to  perform  the  exercise),  the  pround  assif^ned  for  the 
exemption  being  that  "jam  interea  temporis  .Juris  Civilis  studio  scse  addixerit  et  ad 
ejusdem  facultatis  professionem  virtutesodnlitiisui  praidictocollegio  teneatur..."  (Grace 
Book,  June  11,  1673.)  The  turn  given  in  the  above  story  to  the  real  facts  of  the  case 
(viz.  that  Uvedale  was  appointed  to  a  lay-fellowship,  and  that  Newton  would  have 
been  glad  to  have  one)  is  a  very  natural  family  embellishment. 
(=">)  Appendix,  No.  IX. 
(")    We  hear  of  those  incidcntjdly  from  a  letter  of  Collint  to  James  Gregory,  dated 

Oct.  19, 1675.    •'  Mr  Newton I  have  not  writ  to  or  seen  these  11  or  12  months,  not 

troubling  him  as  beinff  intent  upon  chemical  studies  and  practices,  and  both  he  and  Dr 
Barrow  beginning  to  think  mathematical  speculations  to  grow  at  least  dry,  if  not 
somewhat  barren."    ISIacc.  Cojt.  ii.  280. 

(")  Jan.  28.  At  a  meeting  of  the  council  '•  INIr  Oldenburg  having  mentioned, 
that  Mr  Newton  had  intimated  his  being  now  in  such  circumstances,  that  he  desired 
to  be  excused  from  the  weekly  payments  {h.},  it  was  agreed  to  by  the  council,  that  he 
should  be  dispensed  with,  as  several  others  were."  It  seems  probable  that  the  "  inti- 
mation" respecting  Newton's  altered  "  circumstances"  is  to  be  referred  to  the  expected 
vacating  of  his  fellowship,  which  in  the  usual  course  of  things  would  expire  in  the 
following  autumn. 

(*3)  On  March  11,  partly  in  consequence  of  Linus's  second  letter  (Feb.  25.  N.S.) 
*' containing  assertions  directly  opposite  to  those  of  Mr  Newton,"  Ilooke  was  ordered 
by  the  Royal  Society  to  have  the  apparatus  ready  for  the  next  meeting  in  order  to  make 
the  spectrum  experiment,  but  the  day  proved  unfavourable.  Newton  was  present  at 
both  meetings.  While  Newton  was  in  I^ndon,  Oldenburg  shewed  him  Linus's  letter, 
but  ui»on  reading  it,  he  did  not  think  it  worth  noticing.  However,  on  the  old  man's 
writing  again  on  the  subject  (Sept  11),  Newton  was  induced  to  send  him  in  a  letter 
to  Oldenburg  (Nov.  13)  further  directions  for  performing  the  controverted  experiment. 
Linus's  3rd  letter  is  preserved  in  the  Royal  Society  Collection  ( L.  5.  89).  The 
writer  feeling  the  disadvantageous  position  in  which  the  publication  of  his  first  letter 
with  Oldenburg's  rider  left  him,  requests  that  his  2nd  letter  may  be  printed.  It  accord- 
ingly appeared  in  the  2'rans.  for  Jan.  24,  167G  in  company  with  Newton's  letter  of 
Nov.  13. 

(*')  A  draught  of  the  patent  (probably  Newton*s  own  composition)  from  a  paper 
in  his  handwriting  among  the  Lucasian  ]\1SS.  (No.  E.)  is  here  subjoined. 

"Carolus  secundus  Dei  gratia  Angliaj  Scotiae  ffranciaj  et  Hibernia  Rex,  fidei 
Defensor,  &c. :  Omnibus  et  sioRulis  has  literas  visuris  salutem. — Cum  muni'  Profes- 
soris  Mathematifi  in  Academia  nra  Cantabrigicnsi  a  Consult©  Viro  Henrico  Lucas  non 
ita  pridem  institutum  authoritate  nostra  veg'n  et  Literis  Patentibus  stabiliverimus,  et 
Ordinationes  ad  idem  munus  ftpcctanU."!  rd*i6ce  erimus,  et  ad  petitionem  executorum 
cum  consilio  Procancellari'  et  Prnfectorum  privilegia  insuper  nonnuUa  eidem  Profes- 
sori  Mathematico  in  perpetuum  concesserimus :  Inmr  quee  statuimus  ut  dictus  Professor 


1  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

eligi  posfiit  in  Socium  cujusvis  CoUegii  non  vetante  Professione  sua,  et  ne  Is  sodalitio 
suo,  si  quod  ante  susccptum  hoc  munus  obtinuit  aut  postea  obtinebit,  vel  uUis  so{da}litii 
sui  emolumcntis  aut  privileglis  eo  tantum  nomiue  seu  causil  privetur  quovis  cujuscunque 
collcgii  statuto  non  obstante.  Quod  privilcfirlum  ea  intensione  {sic}  illi  indulsiinus  ut 
cidem  Professori  liceret  quod  vis  sodalitium  capeseere  et  retinere.  Quod  ut  debitum 
Bortiatur  eflcctuni  nee  restrictioni  alicui  in  damnum  aut  praejudicium  ejusdem  Pro- 
fessoris  pateat  indulgentia  nostra;  Insuper  volumus  &  statuimus  ut  verba  nostra 
praedicta  in  favorem  dicti  Professoris  semper  accipiantur,  ut  non  eo  tantum  sed  nee 
alio  quovis  nomiae  aut  causa  sodalitio  suo  aut  ejus  emolumeuto  privetur  nisi  quod 
quemlibet  ejusdem  CoUegii  Socium  cujuscunque  professionis  &  ordinis  merito  privare 
debeat.  Et  speciatim  volumus  et  ordinamus  ut  onlines  sacros  non  nisi  ipse  voluerit, 
Buscipiat,  nee  ob  defectum  sacrorum  ordinum  sodalitio  cedere  ipse  tencatur  aut  ab 
aliis  quibuscunque  cogntur,  sed  ca  immunitate  quamdiu  suo  munere  fungitur  gaudeat 
et  fruatur  quo  quililnjt  socius  Medicinas  aut  Juri  Civili  vel  Canonico  dicatus  frui  solct 
quovis  cujuscunque  Colleg-ii  Statuto  aut  consuetudine  vel  interpretatione  quacunque 
non  obstante.  In  cujus  rei  Testimonium  has  Literas  nostras  fieri  fecimus  patentes. 
Teste  meii)so  apud  Westmonasterium  vicesimo  septimo  die  Aprilis,'  Anno  llegni  nostri 
viccsimo  septimo. 

Per  Breve  de  Privato  Sigillo 

Pigrott." 
After  the  above  comes  the  following,  lUso  in  Newton's  hand  : 

"  Whitehall,  March  2,  1674  {0.  S.}. 
His  Ma'y  being  willing  to  give  all  just  encouragement  to  learned  men  who  are  & 
shall  be  elected  into  y«  said  Profetisorship,  is  graciously  pleased  to  refer  this  draught  of 
a  Patent  unto  M'  Atturney  Generall  to  consider  y"  same,  &  to  report  his  opinion 
what  his  INIa^y  may  lawfully  do  in  favour  of  y^  said  Professors  as  to  y«  indulgt^nce  & 
dispensation  proposed  &  desired.    And  then  his  Ma'X  will  declare  his  further  pleasure. 

A.  COVENTRY.'* 
The  above  draught  was  adopted:   the  actual  instrument,  (coinciding  with  the 
draught  except  in  two  unimportant  particulars),  with  the  broad  seal  attached,  b  in  the 
Registrary's  office  (Box  21.  G.  1.  2»)  : 

**  A  grant  to  the  Mathematical  Professor  in  Cambridge. 

Pigott." 
A  transcript  of  it  will  be  found  in  a  large  folio  copy  of  the  Elizabethan  statutes  of 
Trinity  College,   preserved  in  the  College  Archives,  with  the  heading  '*  Indulgentia 
Kegia  Profesuori  Mathematico  concessa,  dignissimo  Viro  M""*  Isuaco  Newton,  hujus 
CoUegii  Socio,  istud  munus  tunc  temporis  obeunte." 

Newton's  visit  to  London  in  February  may  have  been  connected  with  his  application 
to  the  Crown. 

Towards  the  end  of  the  preceding  year,  Francis  Aston  endeavoured  to  obtain  a 
similar  dispensation  on  his  own  individual  account,  and  was  backed  by  the  interest  of 
Sir  Joseph  WiUiamson,  Principal  Secretary  of  State.  There  is  extant  in  the  State 
Paper  Office,  (Domestic,  No.  102),  a  characteristic  letter  from  Barrow  to  WiUiamson 
on  the  subject  (Dec.  4,  1674),  in  which  he  gives  his  reasons  for  resisting  the  applica- 
tion. One  short  extract  from  it  may  be  given  here :  "  Indeed  a  Fellowship  with  us  is 
now  so  poor,  that  I  cannot  think  it  worth  holding  by  an  ingenuous  person  upon  terms 
liable  to  so  much  scruple." 

(")    Letter  CIX  (bis)  in  this  work. 
(**)    Appendix,  No.  XI. 
(")    Appendix,  No.  XII. 

(47)  ♦<Dec.  9.  There  was  produced  a  MS.  of  Mr  Newton,  touching  his  theory  of 
light  &  colours,  containing  partly  an  hypothesis  to  explain  the  properties  of  light 
discoursed  of  by  him  in  iiis  former  papers,  partly  the  principal  phenomena  of  the 
various  colours  exhibited  by  thin  plates  or  bubbles,  esteemed  by  him  to  be  of  a  more 


NOTES.  li 

difficult  consideration ;  yut  to  depend  also  on  the  said  properties  of  lifr^t."  Se«  Birch 
III.  247,  seqq.  One  experiment  mentioned  in  the  "  hypotliesia"  relative  to  the  effecta  of 
glass  electrised  by  friction  particularly  struck  some  of  the  members,  and  it  waa  ortlercd 
to  bo  tried  at  the  next  meeting.  The  paper  was  read  by  instalments,  the  "  hypothcais" 
on  Dec.  9  and  16,  the  "  observations"  respectinjf  colour:!  on  Jan.  20,  Feb.  3,  and  10. 
The  "  observations"  afterwards  forme«l  part  of  the  2nd  Uook  of  his  Optics.  Tlio  "  hy- 
pothesis" has  been  lately  reprinted  in  the  P/u7.  Mag,  for  Sept.  184(5,  pp.  187—213. 

**  Dec.  16.     Mr  Newton's  exiKirinient  of  gliu?s  rubbed  to  cause  various  motions  in 

bits  of  paper  underneath  was  tried,  but  did  not  succeed This  trial  was  made  upon 

the  reading  of  a  letter  of  his  to  Mr  Oldenburg  (Dec.  14)  in  which  he  gives  some  more 
particular  directions  about  that  exi)eriment."  Oldenburg  was  ordered  to  write  to  him 
again  upon  the  subject  "  &  desire  him  to  send  his  own  apparatus,  as  also  to  en- 
quire wjicther  he  had  secured  the  papers  from  being  moved  by  the  air,  that  nught 
somewhere  steal  in." 

On  the  second  part  of  Newton's  hypothesis  being  read,  Hooke,  according  to  his 
wont,  said  that  the  main  of  it  was  contained  in  his  Microf^aphia, 

(**)  At  the  meeting  on  Dec.  30,  there  was  also  rea<l  n  letter  from  John  Gascoines 
(Liege,  15  Dec.  1675)  to  Oldenburg,  acquainting  him  with  the  death  of  Linus  from 
the  prevailing  epidemic,  "and  with  the  resolution  of  Mr  Linus's  disciples  to  try  Mr 

Newton's  experiment  concerning  light  and  colours,  more  clearly  and  carefully" 

according  to  his  directions  of  Nov.  13 :  "  intimating  withal  that  if  the  said  experiment 
be  made  before  the  Royal  Society,  and  be  attested  by  them  to  succeed,  as  Mr  Newton 
affirmed,  they  would  rest  satisfied.  It  was  ordered  that  when  the  sun  should  lierve, 
the  experiment  should  be  made  before  the  Society." 

(*»)  IIarum....librationura  causas  Ilypothesi  elegantissimii  explicavit  nobis  vir  CI. 
Isaac  Newton,  cujus  Humanitati  hoc  et  aliis  nominibus  plurimum  debere  me  lubens 
profiteer.  Mercator's  Jnstitutiones  Astronomiac  (p.  286)  published  in  the  beginning  of 
1676.  See  Princip.  (3d  ed.)  Lib.  3.  Prop.  17.  Mtcan.  C6le$t.  Tom.  v.  p.  279.  Newton 
seems  to  have  been  in  possession  of  his  explanation  in  1673.  See  his  letter  to  Olden- 
burg, June  23  of  that  year,  Ilorsley  iv.  343.    Rigaud,  Append.  42. 

(*<•)  He  returns  his  hearty  thanks  for  **  the  favour  of  the  Society  in  their  kind 
acceptance  of  his  late  papers  ;"  "  that  he  knew  not  how  to  deny  any  thing  which  they 
desired  should  be  done,  but  he  requested  that  the  printing  of  his  observations  about 
colours  might  be  suspended  for  a  time,  because  he  had  some  thoughts  of  writing  such 
another  set  of  observations. ..which  ought  to  precede  those  now  in  the  Society's  pos- 
session."   Mace.  Corr.  ii.  388. 

(*•)  We  find  the  following  notices  in  the  Journal  Book  upon  this  subject.  On 
March  2,  Oldenburg  reminded  the  Society  that  the  sky  was  favourable  for  making  the 
experiment.  Ilooke  said  that  he  had  an  apparatus  ready  whenever  it  should  be  called 
for.  March  16.  The  experiment  ordered  to  be  made  at  next  meeting  if  the  weather 
should  prove  favourable.  Apr.  6.  A  committee  appointed  to  try  the  experiment  and 
repeat  it  before  the  Society,  Apr.  27.  The  experiment  tried  with  success,  of  which 
Oldenburg  sends  an  account  to  Gascoines  (May  4). 

(")    Appendix,  No.  XIII. 

(")  It  was  afterwards  printed  in  Wallis's  0pp.  iii.  622—629.  (Oxf.  1699),  and, 
from  that  work,  in  the  Commercium  Epistolicum,  where  the  typographical  error  of  26 
Junii  for  Julii,  which  is  corrected  in  Wullis's  errata^  is  also  copied  in  the  heading  of 
the  letter. 

(")    Appendix,  No.  XIV. 

(**)  The  original  letter  extending  over  14  folio  pages  is  in  the  British  Museum 
(MSS.  Birch  4294).  It  was  accompanied  by  a  note  to  Oldenburg  (Mace.  Carr.  ii. 
400)  in  a  postscript  to  which  he  observes:  "I  hope  that  this  will  so  far  satisfy  M. 
I/jibnitz  that  it  will  not  bo  necessary  for  me  to  write  any  more  about  this  subject;  for 


lii  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE.     , 

having  other  things  in  my  head,  it  proves  an  unwelcome  interruption  to  me  to  be  at 
this  time  put  upon  considering  these  things."  Newton  sent  some  corrections  by  the 
next  post  (Appendix,  p.  257).  A  copy  of  the  letter  so  corrected  was  not  despatched  to 
Leibniz  until  May  2  of  the  following  year,  the  delay  arising  from  Oldenburg's  anxiety 
to  send  this  ** Thesaurus  Newtonianus"  by  a  safe  hand.  Leibn.  Mathem.  Schrift.  i.  1. 
151  (Berlin,  1849). 

On  Nov.  14  he  desired  Oldenburg  to  make  some  further  corrections,  (Appendix, 
No.  XVII.)  which,  however,  were  not  introduced  into  the  copy  sent  to  Leibniz,  which 
was  made  ten  days  before. 

This  letter,  like  its  predecessor  of  June  13,  was  printed  in  the  3rd  Volume  of  Wallis's 
Opera,  from  which  it  wa^  copied  into  the  Commercium  EpistoUcum.  Wallis  says  that 
he  obtained  his  copies  of  the  two  letters  from  Oldenburg. 

Leibniz  wrote  two  letters  in  answer  (June  21,  July  12,  1677)  in  the  former  of  which 
he  gives  examples  in  differentiation.  Oldenburg  acknowledged  the  receipt  of  these 
Aug.  9,  observing,  "  Non  est  quod  dicti  Newtoni  vel  etiam  CoUinii  nostri  responsum 
tarn  cito  ad  eas  expectes,  cum  et  urbe  absint,  et  variis  allis  negotii?  distincantur." . 
(Leibn.  Math.  Schrift.  u  i.  107,  Berlin,  1849).  Oldenburg  died  the  following  month, 
but  there  is  no  reason  to  think  that,  if  that  event  had  riot  taken  place,  Newton  would 
have  departed  from  his  intention  of  not  continuing  the  correspondence.  Leibniz's 
answers  will  be  found  in  Wallis's  3rd  volume,  the  Commercium  Epistolicum  and  his 
Works. 

(")    Appendix,  No.  XVI. 

{")    Mace.  Corr.  ii.  403. 

(")    Appendix,  No.  XVII. 

(")    Mace.  Ct>rr.  ii.  405.    See  next  note. 

(«o)  Lucas  replied  to  Newton's  letter  of  Aug.  18  in  a  letter  of  four  pages  closely 
written,  dated  Oct.  23,  "containing  further  objections  and  experiments  against  Mr 
Newton's  theory  of  light  and  colours  with  an  examination  of  his  experimentum  crucis :" 
among  other  things  he  professes  to  prove  that  the  red  rays  sulfer  the  same  refraction  as 
the  blue  ones.  Newton  sent  an  answer  to  this  (Nov.  28),  but  with  a  determination 
that  It  should  close  the  controversy.  In  a  letter  to  Oldenburg  (Nov.  18),  he  writes : 
*♦  1  see  I  have  made  myself  a  slave  to  philosophy,  but  if  1  get  free  of  Mr  Linus's 
business,  I  will  resolutely  bid  adieu  to  it  eternally ;  excepting  what  I  do  for  my  private 
satisfaction,  or  leave  to  come  out  after  me ;  for  I  see  a  man  must  either  resolve  to  put 
out  nothing  new,  or  to  become  a  slave  to  defend  it."    Mace.  Cotr.  ix.  405. 

His  opponent,  however,  was  not  satisfied  with  the  answer,  and  indited  another  letter 
(Feb.  2,  1677  N.  S.),  the  sole  value  of  which  to  us  consists  in  its  preserving  for  us  a  few 
words  out  of  Newton's  letter  of  Nov.  28.  "  In  his  last  of  Nov.  28,"  writes  the  Liege 
professor,  "  I  still  meet  with  new  demurs.... He  is  pleased  to  quarrel  with  my  examining 
his  Experimentum  Crucis,  representing  it  '  a  jostling  out  of  the  point  in  dispute  by  a 
new  attempted  digression,'  or  as  he  is  pleased  to  term  it  *  a  running  from  one  thing  to 
another.*  He  tells  us  '  that  he  intends  to  take  into  consideration  one  or  two  of  my 
experiments,  w  Inch  I  shall  recommend  for  the  best :  and  when  there  appears  to  be  no 
weight  in  them,  let  others  judge  what  there  may  be  in  the  number  of  the  rest'."  Lucas 
closes  his  epistle  with  a  desire  that  the  whole  of  his  previous  letter  of  Oct.  23  may  be 
printed,  but  the  request  was  not  attended  to.  The  matter  does  not  seem  to  have  alto- 
gether dropt  here,  for  in  Oldenburg's  letter  to  Leibniz  of  May  2,  accompanying 
Newton's  letter  of  Octob.  24  preceding,  we  read,  "  Ad  alia  nunc  distrahitur  Newtonus 
ab  iis,  qui  Leodii,  Francisco  Lino  succenturiati,  novam  ipsius  de  Lumine  et  Coloribus 
Theoriam  vehementer  insectantur:  qua  de  re  brevi  plura  accipies,  ni  raliones  meas 
male  subduxi,"  but  our  information  extends  no  further. 

Goethe,  in  his  "Geschichte  der  Farbenlehre"  (Werke,  Baud  55.  Stuttg.  1833)  gives 
an  account  of  the  reception  of  Newton's  discovery  of  the  composition  of  light,  which 
does  not  indicate  a  very  intimate  acquaintance  with  the  circumstances  of  the  history. 


NOTES.  liii 

For  example,  he  does  not  know  that  the  three  penions  whose  suggestions  or  objections 
accompanied  by  Newton's  answers  are  printed  without  their  names  in  the  Phibti^hieal 
Transactions  were  Moray,  Hooke  and  Huygens.  One  of  tliem,  indeed,  he  conjectures 
riffhtly  enough  to  be  Hooke,  the  loss  of  whose  pai>er  of  "considerations,**  he  says,  is 
preatly  to  be  regretted.  It  will,  however,  be  found  in  a  book  which  he  himself  quotes 
not  many  pnpres  before,  viz.  Birch's  History  of  the  Royal  Society,  in.  10—15.  In  p.  56 
he  confounds  John  Gnscoines,  Linus's  pupil,  with  William  Ga«coiffne,  the  inventor  of 
telescopic  sights,  who  fell  at  the  age  of  23  at  Marston  Moor  fighting  on  the  Royalist 
side.  Again,  Newton,  in  his  answer  to  Lucas  (Phil.  Trans.  Sept.  1076,  p.  703)  says 
that  the  principal  experiments  which  Lucas  had  sent  him  were  detailed  in  a  **  tractate" 
which  he  had  written  upon  light.  Goethe,  in  quoting  the  passage,  for  the  word 
*'  tractate"  writes  "  Optical  Lectures,'*  and  adds  that  the  statement  "  keineswegs  der 
Wahrheit  gemiiss  ist"  (p.  64).  It  is  true  tliat  the  treatise  in  question  conrtisted  in  the 
main  of  the  Optical  Lectures,  but  it  would  not  have  been  amiss  to  have  ascertained  the 
perfect  identity  of  the  two  works  before  using  language  like  that  which  has  just  been 
quoted.  For  Goethe's  speculations  on  colours,  see  Whewell's  Hist.  hid.  Sci.  ii. 
Wilde's  Geschichte  der  Optik,  Theil.  u.  p.  153  sqq.  (Berlin,  1843),  and  the  works 
referred  to  by  him. 

(*•)  Printed  in  Wallis's  Works,  in. 646  (extracts  from  it  in  the  Commercium  Epist.). 
At  tlie  end  of  the  letter  Collins  says:  "  Narrat  mihi  D.  Loggan  (Chalcographus)  quod 
Effigiem  tuam  delineavit  ille,  in  ordine  ad  Sculpturam ;  Quae  praifigenda  sit  libro  tuo 
de  Lumine,  Coloribus,  DioptriciSf  &c.  quem  edecidum  intendis.  Qua  de  re  desideramus 
esse  certiores."    Nothing  further  is  known  of  the  *'  effigies"  here  spoken  of. 

We  may  mention  here  Loggan's  Dedication  of  his  Plate  of  St  Mary's  Church. 
Its  date  is  uncertain,  as,  though  Loggan's  Cantabrigia  lllustrata  was  published  in 
1690,  the  dates  of  the  separate  plates  range  over  a  period  of  several  years.  "  Clariss", 
Viro  D".  Isaaco  Newton  Matheseos  apud  Cuntubrigienses  Profeisori  l.ucasiano  A'6'*. 
Trinitatis  CoW^.  ibidem,  et  Regice  Societ'.  Socio,  Mathematieo,  Phihsopho,  Chymico 
consummatisso.  Nee  minus  suavitate  Morum  et  Candore  Animi,  Cum  rerum  Huma- 
narum  Divinarumq:  Periti^  spectabili,  Hanc  Tabulam  Observantiaa  ergo  D.  D.  C.  Q. 
Dav.  Loggan."    Loggan  had  the  use  of  a  room  in  Trinity  College  for  his  press. 

(")   Appendix,  No.  XVII. 

(")  In  this  and  other  instances  where  Newton  is  mentioned  as  voting  at  Univer- 
sity elections  of  Members  of  Parliament  or  Officers,  our  information  is  derived  from  the 
actual  slips  of  paper  on  which  each  voter  recorded  his  suffrage,  and  which  are  still 
preserved  in  the  Registrary's  office.  A  copy  of  Newton's  voting  paper  on  this  occasion 
is  given  as  a  specimen.  *♦  Isaacus  Newton  eligit  Thomara  Exton  Militem  in  Burgensem 
hujus  AcademisB  in  Regni  Comitiis." 

(««)  Boyle*s  Life  (by  Birch)  prefixed  to  his  Works,  p.  70.  Mace.  Corr.  ii.  407. 
and  elsewhere. 

(«>)  A  very  pretty  story  is  told  of  him  by  his  biogrrapher— how  that  in  1682  when 
his  schoolfellow  George  Stepney  was  elected  scholar  from  Westminster  to  Trinity  Col- 
lege, Montagu,  unable  to  l)ear  the  tlioughts  of  l)eii)g  separated  from  his  **dearist 
friend,"  went  to  College  a  year  before  the  proper  time — but,  like  many  other  pretty 
stories,  it  will  not  stand  the  test  of  dates.  Montagu  was  matriculated  Dec.  18, 1679, 
the  "chamber"  in  which  he  "kept"  in  1680  and  following  years  is  known,  being 
the  same,  in  fact,  in  which  these  lines  are  written,  and  on  Oct.  6, 1681,  he  was  made 
M.A.  by  Royal  Mandate. 

(««)  Newton  seems  to  have  been  requested  to  give  his  opinion  on  a  wild  hypothesis 
of  the  heavens,  which  a  Frenchman  of  the  name  of  Mallemont  had  sent  to  the  Royal 
Society.  His  judgment  was  given  briefly,  and  with  some  reluctance,  in  a  letter  to 
Hooke,  one  of  the  Secretaries,  (Nov.  28,  read  to  the  Soc.  Dec.  4),  in  which,  to  make 
amends  for  the  curtness  of  his  answer,  he  suggested  **  an  experiment  whereby  to  try 
whether  the  earth  moves  with  a  diurnal  motion  or  not,  viz.  by  the  falling  of  a  body  from 

d 


liv  SYNOPTICAL  VIEW  OP  NEWTON'S  LIFE. 

t  considerable  height,  which,  he  alleged,  must  fall  to  the  eastward  of  the  perpendicular, 
if  the  earth  moved.  This  proposal  was  highly  approved  of  by  the  Society,  &  it  wa» 
desired  that  it  might  be  tried  as  soon  as  could  be  with  convenience." 

At  the  meeting  of  the  Soc.  Dec.  1 1,  Hooke  read  his  answer  to  Newton's  letter,  in 
which  he  shewed  that  the  path  of  the  falling  body  would  not  be  a  spiral  "as  Mr.  New- 
ton seemed  to  suppose,"  and  that  it  would  fall  "  not  directly  east,  but  to  the  south-east 
&  more  to  the  south  than  the  east.  It  was  desired  that  what  was  tryable  in  this  expe- 
riment might  be  done  with  the  first  opportunity." 

At  the  meeting  on  Dec.  18,  Hooke  read  his  answer  again,  and  also  a  reply  to  it  from 
Newton,  "containing  his  farther  thoughts  and  examinations  of  what  had  been  pro- 
pounded by  Mr  Hooke."  He  also  gave  an  account  of  three  trials  that  he  had  made 
of  the  experiment. 

At  the  meeting  on  Jan.  8,  1680,  Hooke  read  another  letter  of  his  to  Newton,  giving 
*♦  a  further  account  of  his  theory  of  circular  motion  6c  attraction,  as  also  several  obser- 
yations  6c  deductions  from  it."  Newton  declined  answering  this  letter.  At  the  same 
meeting  Hooke  "  was  desired  to  make  his  trials  "  of  Newton's  experiment  as  soon  an 
possible. 

(«)  Library  Account  Bk.  for  year  from  Dec.  22,  1679,  to  Dec.  22,  1680.  The 
charge  for  the  bond  appears  in  the  Sen.  Bursar's  Bk.  for  year  ending  Mich.  1G80.  The 
money  seems  to  have  been  repaid  Nov.  12,  1688.    Conclus.  Bk.  Feb.  5,  1C89. 

(«9)    Appendix,  No.  XVIII. 

(«»)  Birch,  IT,  65.  A  letter  of  his  to  his  kinsman  Sir  John  Newton,  introducing 
Adams,  is  printed  in  Tumor's  Grantham,  p.  85,  note. 

C")  Gen,  Diet,  vii.  788.  The  originals  of  th'is  and  the  other  letters  to  Flamsleed 
down  to  1698,  are  preserved  in  the  Library  of  Corpus  Christi  College,  Oxford. 

(]})    Gen.  Diet,  yii.lBX, 

(»)•  "  Cambridge  April  y«  3«»  1682 

"These  are  to  signify  y*  M'  Ellis  advising  w**>  me  ab'  a  person  fit  to  be  intrusted  w"» 
y«  Charge  of  teaching  Navigation  toy*  Boys  of  y*  King's  late  foundation,  I  propounded 
M'  Edw*  Paget  Master  of  Arts  &  ftellow  of  Trinity  College  in  this  University,  as  y« 
most  promising  person  for  this  end  I  could  think  of;  and  that  upon  these  considerations. 
He  is  of  a  temper  very  sober  &  industrious,  as  I  am  confident  all  that  know  him  are 
ready  to  testify.  He  understands  y«  several  parts  of  Mathematics,  Arithmetic,  Geometry, 
Algebra,  Trigonometry,  Geography,  Astronomy,  Navigation,  &  w**  is  y«  surest 
character  of  a  true  Mathematical  Genius,  learned  these  of  his  own  inclination  &  by  his 
own  Industry  without  a  Teacher :  And  to  make  him  y*  readier  in  practicall  Matters,  hLs 
hand  is  very  steady  &c  accurate,  as  well  as  his  fancy  &  apprehension,  good  ;  as  may  be 
•een  by  his  writing  &  drawing  w*'»  his  Pencil  very  well :  Perfections  w^h  I  conceive 
considerable  for  making  y»  Boys  accurate  &  curious  in  their  Draughts  of  Charts,  Mapps 
&  Prospects  from  Sea,  wc>»  joyn'd  w*'»  his  knowledge  in  perspective  and  projections  of  ye 
Sphere  will  enable  him  to  contrive  &  draw  schemes  after  y*  best  manner  for  y«  Boys 
apprehension,  &  perswades  me  y'  he  will  not  only  be  dexterous  &  nice  in  y«  use  of 
Instrum**  but  improve  them :  His  long  acquaintance  also  wi'»  variety  of  Learning 
here,  will  help  him  to  be  methodical  &  clear  in  his  teaching  j  w^*"  much  conduces  to  y«^ 
Boys  ready  &  distinct  apprehension  of  what  they  are  taught.  So  y»  tho  it  may  be  easy 
to  find  persons  valuable  for  some  of  these  Qualifications,  yet  considering  him  in  all 
rcsixicts  as  I  could  not  think  of  ony  other  person  in  this  University  so  fit  in  my 
Opinion  to  be  intrusted  w'''  a  place  of  so  great  concerne  as  that  of  preparing  Boys  to 
make  more  skilful  Navigators  than  formerly,  so  I  believe  it  will  be  difficult  to  meet  w'*> 
fitter  persons  abroad  for  that  purpose.  These  things  made  me  forward  to  propound 
him  to  y«  Electors ;  but  to  compare  him  w***  other  Competitors  6c  chuse  y«  best  I  leave 
wholly  to  their  judgment. 

Is.  Newton,  Profess.  IMath.  Luc.** 
( Pepysian  MSS.  2612.  p.  536). 


NOTES.  Iv 

Newton  also  wrote  to  his  friend  Collins  requesting  him  to  use  hi«  interest  in  behalf  of 
Paget.  There  is  in  the  same  MS.  volume  from  which  the  above  is  taken^  a  copy  of  the 
letter  which  Collins  wrote  in  consequence,  enumerating  from  Newton's  letter  to  him 
Paget's  qualifications,  and  dwelling  upon  the  weight  which  the  recommendation  of  the 
greatest  mathematician  of  the  age  ought  to  have  with  the  electors. 

(")    Appendix,  Nos.  XIX.  XX. 

(")    **The  charge  of  building"  the  College  Library,   "disables  us  from  buyiog 

books  at  present We  know  not  yet  whether  the  University  will  purchase  them,  their 

chest  being  at  present  very  low."     Gentlcman''t  Magazine^  LXl.  504. 

C*)  The  propositions  here  mentioned  as  sent  to  Ilalley,  have  been  printed  by 
Rigaud  from  the  copy  in  the  Register  Uook  of  the  Royal  Society,  vi.  218.  (Appendix 
to  Essay  on  Publication  of  Principiu,  No.  I.)  It  is  to  be  observed,  however,  that  the 
title  which  Rigaud  g'ives  to  the  Paper  ( Newtoni  Propositionet  de  Motu)  is  not  to  bo  found 
in  the  MS. 

('*)  At  the  Meeting  of  the  Royal  Society,  Dec.  10,  "  Mr  Halley  gave  an  account 
that  he  had  lately  seen  M""  Newton  at  Cambridge,  who  had  shcwcil  him  a  curious  treatise, 
De  Motu  {  drawn  up  since  August }  ;  which,  upon  Mr  Ilalley's  desire,  was,  ho  said, 
promised  to  be  sent  to  the  Society,  to  bo  entered  upon  their  Register.  Mr  Ilalley  was 
desired  to  put  Mr  Newton  in  mind  of  his  promise  for  the  securing  his  invention  to 
himself  till  such  time  as  he  could  be  at  leisure  to  publish  it.  Mr  Paget  was  desired 
to  join  with  Mr  Ilalley."    Birch,  iv.  347. 

The  treatise  De  MotUf  mentioned  here,  was  probably  the  same  as  that  of  which  a 
copy  is  preserved  in  the  University  Library  (Dd.  IX.  46,)  beginning  "De  motu  cor- 
porum  Liber  primus,  Definitiones,"  &c.  consisting  of  the  Lectures  which  he  delivered 
as  Lucasian  Professor,  (the  first  of  them  is  dated  Octob.  1684),  and  forming,  to  a  certain 
extent,  the  first  draught  of  the  Principia.  (See  Letter  CIV.).  The  paper  which  New- 
ton sent  up  to  Haiiey,  in  Nov.  1684,  was  the  germ  of  this  treatise.  It  is  probable  that 
Halley  produced  the  paper  at  the  meeting  on  Dec.  10,  though  the  fact  is  not  recorded 
in  the  Journal  Book.  The  treatise  was  never  registered,  but  the  paper  was,  apparently  in 
February  1685,  with  the  date  Dec.  10, 1684. 

Rigaud*s  idea  that  the  paper  which  he  has  printed  from  the  Register  of  the  Royal 
Society  (consisting  of  4  theorems  and  7  problems)  is  different  from  the  paper  which 
Newton  sent  to  Halley,  and  that  it  was  sent  to  the  Society  in  Feb.  1685,  is  founded  upon 
what  I  conceive  to  be  a  misapprehension  of  a  passage  in  Newton's  letter  to  Aston, 
(Feb.  23, 1685).  The  words  are  as  follow  :  '•  I  thank  you  for  entering  in  your  Register 
my  notions  about  motion.  I  designed  them  for  you  before  now,  but  the  examining 
several  things  has  taken  a  greater  part  of  my  time  than  I  expected,  and  a  great  deal  of 
it  to  no  purpose.  And  now  I  am  to  go  into  Lincolnshire  for  a  montli  or  six  weeks.  After- 
wards I  intend  to  finish  it  as  soon  as  I  can  conveniently,"  &c.  We  possess  only  a  part 
of  the  letter,  and  that  in  a  copy.  We  cannot  therefore  be  sure  that  the  grammar  is 
Newton's.  It  seems  clear  to  me  that  what  he  "designed"  for  the  Society  "  before 
now,"  was  not  yet  finished  and  sent  to  the  Society  :  that  he  was  in  fact  working  at  his 
Treatise  De  Motu  with  a  view  to  fulfil  the  promise  whicli  he  had  made  to  Halley,  that 
he  would  *'  send  it  to  the  Society  to  be  entered  upon  their  register." 

That  the  paper  sent  to  Halley  is  identical  with  that  which  we  find  in  the  Register  of 
the  Royal  Society,  is  evident  from  the  whole  tenor  of  our  information  on  the  subject :  it 
is  sufficient  to  refer  to  Halley'sown  statement  (Rigaud,  Appendix  to  Kssay,  p.  37), and 
a  letter  of  his  to  Wallis,  dated  Dec.  11,  1686,  in  which  he  says:  "  JSIr  Is.  Newton  about 
two  years  since  gave  me  the  inclosed  propositions,  touching  the  opposition  of  the  medium 
to  a  direct  impressed  motion  and  to  falling  bodies,  upon  supposition  that  the  opposition 
is  as  the  velocity  ;  which  tis  possible  is  not  true  ;  however,  I  thought  any  thing  of  his 
might  not  be  unacceptable  to  you,  and  I  beg  your  opinion  thereupon,  if  it  might  not  be 
(especially  the  7th  problem)  somewhat  better  illustrated."  (The  original  of  this  letter 
is  in  the  collection  of  Dawson  Turner,  Ksq.     Compare  Birch,  iv.  514.    Rigaud,  77.) 

d2 


hi  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

The  probability  is  that  Halley  saw  no  immediate  prospect  of  obtainiDg  the  treatise  De 
Motu,  and  determined  to  secure  the  author's  rights  by  at  once  reeristering  the  11  propo- 
sitions which  he  bad  received  in  November. 

(»«)  Birch,  IV.  370.  Rigaud  (Appendix  to  Essay,  p.  24).  Newton  observes  that 
•*  that  which  chiefly  dashed  the  businesa  was  tlie  want  of  persons  willing  to  try  experi- 
ments, he  whom  we  chiefly  relied  on  refusing  to  concern  himself  in  thatkind I  should 

be  very  ready  to  concur  with  any  persons  for  promoting  such  a  design,  so  far  as  I  can  do 
it  without  engaging  the  loss  of  my  own  time  in  those  things." 

(»)    Appendix,  No.  XXI. 

(»»)  Tables  for  renewing  and  purchasing  of  the  leaset  of  Cathedral  Churches  and 
Colleges,  H^e.y  Cambridge,  1686.  Newton's  certificate  prefixed  to  this  work,  the  author  of 
which  was  manciple  {mancipium}  or  caterer  of  King's  College,  runs  as  follows:  "  Metho- 
dus  hujus  Libri  recte  se  habet,  nuroerique  ut  ex  quibusdam  ad  calculum  revocatis 
judico,  satis  exacte  computantur.  Is.  Newton,  Math.  Prof.  Luc."  The  later  editions 
on  the  strength  of  this  testimonial  were  published  under  the  title  of"  Newton's  Tables." 

In  the  treasury  of  Trinity  College  in  a  book  labelled  "  Notitia  E,"  which  belonged 
to  Humfrey  Babington,  as  Bursar  (1674 — 1678),  containing  "a  true  particular  of  the 
rents  and  leases  belonging  to  Trin.  Coll.  1674-5,"  there  is  a  table  and  an  explanation  of 
it  in  Newton's  handwriting,  of  the  fines  to  be  paid  for  renewing  any  number  of  years 
lapsed  in  a  lease  for  20  years.  It  is  entitled  Tabula  redemptionalis  ad  reditus  Collegii 
SS.  Trinitatis  accommodata.  It  is  constructed  on  the  hypotliesis  that  a  lease  for  20  years  b 
worth  7  years'  purchase,  and  that  for  the  renewal  of  7  years  lapsed,  one  year's  purchase 
must  be  paid,  (This  is  equivalent  to  allowing  the  lessee  between  12  and  13  per  cent, 
for  his  money).  This  table  vvrhich  was  apparently  drawn  up  by  Newton  for  Babington's 
oflicial  use,  continued  to  be  employed  by  the  College  until  1700,  when  Bentley,  on  his 
appointment  to  the  Mastership,  introduced  the  10  per  cent,  tables.  The  innovation 
however,  according  to  Vice-lNIaster  Walker,  was  unpalatable  to  the  Seniors  and  Ofl^cers, 
whose  "greediness  for  present  sealing  money"  superadded  to  "quarrels  in  the  Col- 
lege," compelled  a  return  to  the  old  system,  and  occasionally  the  granting  of  terms 
still  more  favourable  lo  the  tenant.  On  Dr  Robert  Smith's  succeeding  to  the  IMas- 
tership  in  1742,  the  10  per  cent,  tables  were  introduced,  and  these  were  replaced  in 
1750,  by  9  per  cent,  tables. 

(")  Gen.  Diet.  vii.  793,  where  also  the  next  four  letters  lo  Flamsteed  will  be 
found.  -=:-t-_:l,- 

(8")  "  You  seem  to  insinuate  as  if  Saturn  had  not  yet  any  more  satellites  than 
one  discovered  by  Hugeuius.  I  should  be  glad  to  know  if  it  be  so."  If  Flamsteed 
returned  an  answer  to  this  question,  it  seems  to  have  been  still  in  the  negative. 
Writing  to  him  on  Sept.  3,  of  the  fallowing  year,  Newton  says :  "  He  [Mr  Philips]  tells 
me  he  apprehended  by  some  of  your  discourses,  tliat  you  had  seen  two  of  Cassini's 
Hew  planets  about  Saturn.  Ilugenius  with  a  sixty  foot  glass  could  see  none  of  them. 
Mr  Halley  (who  was  lately  here)  I  find  still  suspicious  of  them,  notwithstanding  what 
Cassini  has  lately  published  of  two  more.  I  was  glad  to  hear  two  of  them  confirmed  by 
youT  observation."  IMr  Philips'  information  does  not  appear  to  have  been  correct,  for 
in  a  paper  in  Cotes's  handwriting  (Trin.  Coll.  Newtonian  INISS.  No.  382)  which  is 
apparently  a  memorandum  of  a  conversation  which  he  had  had  with  Flamsteed  some 
time  between  1706  and  1716)  it  is  stated  "that  he  (Flamsteed)  thought  there  were 
but  3  satellites  of  Saturn,  himself  had  never  seen  above  one." 

The  first  discovered  satellite  of  Saturn  (now  the  6th,  reckoning  outwards)  was  ob- 
served by  Huygens  March  25,  1655.  In  1671,  2,  3  Cassini  discovered  what  is  now  the 
8th,  in  1672,  3  (while  in  pursuit  of  the  last-mentioned  one)  the  5th,  (see  Phil.  Trans, 
Iklarch25, 1673),  and  in  1684  the  3d  and  4th  :  (an  account  of  this  last  discovery,  given  in 
the  Journal  des  Savons  for  April  1686,  was  mentioned  at  the  Royal  Society  April  28, 
communicated  at  their  next  meeting,  and  printed  in  (he  Transactions  for  May  25 :  a 
letter  from  Cassini  to  Halley,  dated  Oct.  10,  giving  more  correct  elements  of  the  then 


NOTES.  Ivii 

known  5  satellites  was  read  to  the  Society  Nov.  3,  and  published  in  the  Traniactiona  for 
Apr.— June  of  the  following  year.) 

In  the  first  edition  of  the  Principia  Newton  mentioned  only  the  Hug«nian  tatellite, 
but  in  the  second  he  introduced  the  others,  availing  himself  of  Cassini's  paper  in  the 
Memoiret  of  the  Academy  for  1705,  published  in  170G  (comp.  p.  49  of  this  work).  Pound 
(in  1718)  was  the  first  English  astronomer  who  succeeded  in  observinj?  the  Cassinian 
satellites :  this  he  did  by  means  of  corrected  elements  supplied  by  the  younger  Cassini, 
in  the  Mimoires  for  1714  (published  in  1717),  and  a  telescope  with  an  object-glass 
of  123  feet  focal  length,  which  Huygeus  hod  presented  to  tlic  Royal  Society  in  1G91. 
(See  Phil,  Trans.  Jan.— Apr.  1718.  Delisle's  "  Second©  Lcttre  sur  Ics  Tables  Astro- 
noraiques  de  M.  Halley..."  Journal  des  Savans,  June,  1750).  Flamsteed,  however, 
was  not  convinced.    (See  his  letter  to  A.  Sharp,  Sept.  13, 1718,  Baily,  p.  331). 

(**)    The  date  is  taken  from  the  post-mark,  which  is  Jan.  14, 

(82)  f  Dr  Vincent,  {  Fellow  of  Clare  Hall }  presented  to  the  Society  a  manuscript 
treatise  intitled,  Phibsophiie  Naturalis  principia  mathematicaf  and  dedicated  to  the 
Society  by  ]Mr  Isaac  Newton,  wliercin  he  gives  a  mathematical  demonstration  of  the 
Copernican  hypothesis  as  proposed  by  Kepler,  and  makes  out  all  the  phenomena  of 
the  celestial  motions  by  the  only  supposition  of  a  gravitation  towards  the  center  of 
sun  decreasing  as  the  squares  of  the  distances  therefrom  reciprocally. 

It  was  ordered  that  a  letter  of  thanks  be  written  to  Mr  Newton ;  and  that  the 
priming  of  his  book  be  referred  to  the  consideration  of  the  Council :  and  that  in  tlie 
mean  time  the  book  be  put  into  the  hands  of  JNIr  llalley,  to  make  a  reiwrt  thereof 
to  the  council."    Birch,  iv.  479. 

For  some  account  of  Dr  Vincent,  see  Whiston's  MemoirSf  who  was  his  sizar.  It 
may  perhaps  prevent  further  currency  being  given  to  the  supposition  of  his  being 
the  husband  of  the  lady  to  whom  in  early  life  Newton  is  said  to  have  been  attached, 
if  I  state  that  he  was  a  Senior  Fellow  of  Clare  Hull  at  the  time  of  his  death  (March 
1722). 

(»=>)    See  Birch,  iv.  484. 

(")  At  a  meeting  of  the  Council  of  the  Royal  Society  "it  was  ordered  that  Mr 
Newton's  book  be  printed,  &  that  Mr  llalley  undertake  the  business  of  looking  after 
it,  &  printing  it  at  his  own  charge,  which  he  engaged  to  do."    Birch,  iv.  406. 

(86)  J\ly  knowledge  of  this  letter  is  derived  from  a  memorandum  by  Halley,  on  the 
back  of  Newton's  letter  of  July  14,  mentioning  among  Newton's  letters  one  of  this 
date.  The  contents  as  stated  above  are  purely  conjectural,  and  founded  upon  a  sen- 
tence  in  Newton's  letter  of  Febr.  18,  1686-7,  ('*  I  hope  you  received  a  letter  with  two 
corollaries  1  sent  you  in  autumn,")  coupled  with  the  fact  that  the  two  corollaries  above- 
mentioned  are  not  found  in  Newton*8  MS. 

(*«)  It  had  been  finished  in  the  summer  of  the  preceding  year.  Writing  to  Halley 
June  20, 1686,  he  says  that  it  "  only  wants  transcribing  and  drawing  the  cuts  fairly." 

(87)  "I  think  I  have  the  solution  of  your  problem  about  the  sun's  parallax,  but 
through  other  occasions  shall  scarce  have  time  to  think  further  on  these  tilings:  and 
besides,  I  want  something  of  observation."  The  "occasions"  may  refer  to  the  anti- 
cipated effects  of  James's  mandate,  which  had  been  received  in  Cambridge  nine  days 
before.    See  under  March  11. 

(88)  The  first  mandate  was  dated  Febr.  7,  received  by  the  Vice-Chancellor  on 
the  9th,  and  read  to  the  Senate  on  the  21st,  the  second  was  dated  Febr.  24,  and  read 
March  11. 

(89)  "  It  contained  the  whole  system  of  celestial  motions,  as  well  of  the  secondary 
as  i>riraary  planets,  with  tlie  theory  of  comets,  which  is  illustrated  by  the  example  of 

the  great  comet  of  1680-1,  proving  that  which  appeared  in  the  morning  in  Nov to 

have  been  the  same  that  was  observed  in  Dec.  and  Jan.  in  the  evening."  Birch,  iv.530. 

The  MS.  sheets  of  the  Principia  (without  the  preface)  have  been  bound  up  into  a 
Volume  which  is  preserved  at  the  Royal  Society.    It  is  from  no  wish  to  detract  from  the 


Iviii  SYNOPTICAL  VIEW  OP  NEWTON'S  LIFE. 

value  of  this  treasure  that  I  etate  that  I  do  not  think  the  MS.  to  be  in  Newton*8  auto- 
lirraph.  I  believe  it  to  be  written  by  the  same  hand  as  the  first  draught  of  the  Principia 
in  the  University  Library.  The  author's  own  hand  is  easily  recogrnised  in  both  MSS, 
in  additions  and  alterations. 

The  Preface  in  the  first  edition  has  no  date.  The  date  "  Dabam  Cantabrigise  e 
Collegio  S.  Trinitatis,  Maii  8,  1686,"  first  appeared  in  the  second  edition  in  1713.  See 
ndte  to  Febr.  1704. 

(•0)  The  following  are  the  dates  of  the  proceedings  connected  with  this  affair.  Apr. 
21.  Vice-Chancellor  and  delegates  appear  before  the  Commissioners.  Apr.  27.  Give 
in  their  plea.  May  7.  Plea  discussed.  Vice-Chancellor  sentenced  to  be  deprived  of 
his  office,  and  suspended  from  his  Mastership.  May  12.  The  delegates  reprimanded. 
Jeffreys  wound  up  his  address  to  them  with  the  words :  "Therefore  I  shall  say  to  you  what 
the  scripture  says,  and  rather  because  most  of  you  are  divines ;  '  Go  your  way  and  sin  no 
more,  lest  a  worse  thing  come  unto  you."*  See  State  Trials,  or  Cooper's  Annals  of 
Cambridge,  Newton  does  not  appear  at  all  as  a  speaker  during  the  proceedings.  The 
Chancellor  alludes  twice  to  his  having  himself  formerly  been  a  member  of  the  Univer- 
sity. Until  some  other  College  can  establish  a  claim  to  him,  Trinity  College  is  liable 
to  the  suspicion  of  having  had  him  for  an  alumnus,  A  "  Georgius  Jeffrys  "  was 
admitted  pensioner  there  March  15, 1661-2,  under  Mr  Hill,  and  he  would  therefore  be  a 
year  junior  to  Newton. 

Under  this  date  may  be  given  the  following  entry  in  the  College  Account  Book  of 
the  building  of  the  New  Library,  which  probably  refers  to  our  philosopher.  "  May  28, 
1687.  P^'-.-for  erecting  a  scaffold  for  ]Mr  Newton  to  measure  the  fret  work  of  the  stair- 
case :  4s.  6d," 

We  may  also  notice  under  this  year  an  elegant  method  given  by  him  of  finding 
(by  infinitesimals)  the  volume  of  a  segment  of  a  parabolic  conoid  cutoff  by  a  plane  per- 
pendicular to  the  axis.  "Construction  and  Demonstration  as  I  received  it  from  M. 
Isaac  Newton,  Prof,  of  the  Mathematics,  in  Cambridge."  Guager*s  Magazine,  by  Wm. 
Hunt,  Loud.  1687. 

(0^)  Rigaud,  81,  82.  Tlie  copy  which  he  gave  to  the  College  Library  does  not 
contain  his  autograph.  In  a  copy  in  Emmanuel  College  Library  is  written,  "  Ex 
dono  Authoris  suirie  docti  lulii  13"°.  1687."  The  copy  in  Keill's  catalogue  of  his  books 
is  priced  at  10s.,  as  also  is  a  copy  in  Clare  Ilall  Library,  given  by  Cornelius  Crownfield 
to  Cotes's  friend  Morgan,  of  which  however  the  price  at  the  time  of  the  gift  is  put  5s. 
There  is  in  the  same  Library  a  copy  of  the  Theses  printed  at  Edinburgh,  in  the  first  half 
of  1690,  by  James  Gregory,  of  St  Andrew's,  containing  a  compend  of  the  Principia^ 
alluded  to  in  the  Museum  Criticum,  ii.  518,  note,  and  Brewster's  Newton,  p.  174,  note. 

The  following  anecdote  of  Demoivre's  first  introduction  to  the  Principia  may  not 
be  altogether  out  of  place  here.  The  scene  is  probably  to  be  laid  in  the  year  after 
its  publication,  when  Newton  is  known  to  have  been  out  of  College.  (See  Table  of 
Exit?  and  Redits).  Demoivre,  then  about  21,  was  earning  a  livelihood  in  London  by 
teaching  mathematics,  in  which  he  thought  himself  a  perfect  master.  "  II  en  fut  bientOt 
et  bien  singulierement  desabus6.  Le  hasard  le  conduisit  chez  Mylord  Devonshire  dans 
le  moment  oil  M.  Newton  venoit  de  laisser  chez  ce  Seigneur  un  exemplaire  de  ses 
Principes.  Le  jeune  Math<imaticien  ouvrit  le  livre,  et,  s6duit  par  la  simplicity  appa- 
rente  de  Touvrage,  se  persuada  qu'il  alloit  I'entendre  sans  difficult^  j  mais  il  fut  bien 
surpris  de  le  trouver  hors  de  la  port^e  de  ses  connoissances,  et  de  se  voir  oblig6  de 
convenir  que  ce  qu'il  avoit  pris  pour  le  faite  des  INIathematiques  n'C'toit  que  Tentr^e 
d'une  longue  et  ptnible  carriere  qui  lui  restoit  d  parcourir.  II  se  procura  cependant 
le  livre,  et  comme  les  lemons  qu'il  6toit  oblig6  de  donner  I'engageoient;  a  des  courses  pres- 
que  coDtioueUes,  il  en  d^chlra  les  feuillets  pour  les  porter  dans  sa  poche  et  les  6tudier 
dans  les  intervalles  de  ses  tnivaux."    Eloge,  Hist,  de  I'Acadimie,  1754. 

(*2)    He  did  not  give  up  his  rooms  until  Midsummer.     On  Sept.  14,  a  donation  y 
of  £50  towards  the  New  Library  was  received  from  him. 


NOTES.  lix 

(M)  la  many  of  the  votingr  papers  his  name  u«  preceded  by  the  wordi  *'prwcla- 
rura  vin.  j,"  in  some  tlxe  adjective  is  "doctiRsimum,"  "  integerrimura,"  "  venera'jilem/* 
"  reverendum.**    Pulley  n,  his  old  tutor,  calls  him  "suramum  virum." 

Thirteen  letters  from  Newton  to  the  Vicc-Chancellor,  written  between  Febru  iry  and 
May  1689,  on  matters  connected  with  the  University  as  nfiectcd  by  llie  new  order  of 
thingrs,  have  been  recently  printed  by  Dawson  Turner,  Esq.  from  the  originals  in  his 
possession. 

Laplace,  in  speaking  of  the  publication  of  the  Prineipia,  observes :  "  Les  principes 
du  systeme  social  furent  poses  dans  Tann^-e  suivant,  et  Newton  concourut  a  leur  £ta- 
blissement."    Syst.  du  Mond.  p.  372,  Paris,  1824. 

(»*)    See  Lord  King's  Life  of  Locke,  i.  389  (2nd.  ed.) 

(»»)  "  Mr  Iluygens  of  Zulichem  being  present  gave  an  account  that  he  himself  was 
now  about  publishing  a  Treatise  concerning  the  cause  of  gravity,  and  another  about 
Refractions  giving  amongst  other  things  the  reasons  of  the  double  refracting  Island 
Crystal. 

Mr  Newton  considering  a  piece  of  the  Island  Crystal  did  observe  that  of  the  two 
species  wherewitli  things  do  appear  through  that  body,  the  one  suffered  no  refraction 
when  the  visual  ray  came  parallel  to  the  oblique  sides  of  the  parol lelepii^ed ;  the  other, 
as  is  usual  in  all  other  transparent  bodies,  suffered  none,  when  the  beam  came  perpen- 
dicular to  the  planes  through  which  the  object  appeared."   Journ.  lik. 

The  first  mentioned  observation  of  Newton  is  due  to  Erasmus  Bartholinus,  but  was 
found  by  Iluygens  not  to  be  rigorously  true,  {Traite  de  la  Lutnure^  1690,  p.  57). 

I  take  this  opportunity  of  offering  my  grateful  acknowledgments  to  the  President 
and  Council  of  the  Royal  Society  for  their  liberality  in  granting  me  access  to  their 
Archives.  Perhaps  I  may  be  permitted  in  this  place  to  express  my  opinion  of  the  obli- 
gation which  that  illustrious  body  would  confer  upon  the  world  by  tlie  continuation  of 
Birch's  History  of  the  Society,  at  least  down  to  the  close  of  Newton's  Presidentship. 
Independently  of  the  value,  great  or  small,  of  such  a  work  to  the  historian  of  science,  it 
would  give  us  an  opportunity  of  meeting  our  philosopher  once  or  twice  a  week  for  the 
twenty  three  last  years  of  his  life.  The  following  extracts  from  tlie  Journal  Books  of  the 
period  are  given  not  as  specimens  of  their  contents,  but  are  selected  solely  for  the  local 
allusions.  "  JNlarch  31,  1720.  The  President  ...mentioned  a  reiaarko.ble  exi)crimr.nt  he 
made  formerly  in  Trinity  College  kitchin  at  Cambridge,  upon  the  heart  of  au  eel  which 
he  cut  into  three  pieces,  and  observed  every  one  of  them  beat  at  the  same  instant  and 
interval :  putting  spittle  upon  any  of  the  sections  had  no  effect,  but  a  drop  of  vinegar 
utterly  extinguished  its  motion."  (He  had  mentioned  the  same  experiment  more  briefly 
at  the  meeting  on  Nov.  13,  1712).  "  Febr.  20,  1723-4.  The  President  upon  reading 
this  I  a  letter  containing  an  account  of  the  effects  of  a  violent  thunderstorm  }  made 
mention  of  an  accident  much  like  it  which  he  once  saw  at  Trinity  College  in  Cambridge. 
He  was  suddenly  surprized  with  a  violent  strong  flash  of  lightning  which  was  so  exceed- 
ing bright  that  he  was  forced  immediately  to  guard  his  eyes  with  his  hands.  And  at 
the  same  instant  a  violent  clap  of  thunder  broke  down  the  window  in  the  next  room,  and 
forced  some  splinters  out  of  tjje  floor  which  darted  against  tlie  cieling,  and  there  being 
another  window  opposite  to  that  which  was  broke  down  they  observed  it  to  be  bowed 
outwards  by  the  violence  of  the  shock." 

(»o)  «« Aug.  29,  1689.  Before  the  King  &  Council  was  heard  the  matter  of  King's 
College  about  Mr  Isaac  Newton,  why  he  or  any  other  not  of  that  foundation  should  be 
Provost,  &  after  the  reasons  shewed  &  argued  M'  Newton  was  laid  ai<ide."  (Alderman 
Newton's  Diary  among  Bowtell  ]\ISS,  at  Downing  College.)  The  Statutes  of  King's 
College  require  the  Provost  to  be  in  Priest's  Orders  and  to  be  chosen  from  the  existing 
or  former  fellows  of  the  Society,  Newton  therefore  was  disqualified  for  the  post. 

(")  The  following  entry  among  the  gratuities  given  by  the  College  in  the  course 
of  the  year  ending  at  INlichnelmas  1690,  is  probably  to  be  referred  to  the  end  of  1689,  or 
beginning  of  1690,  when  Newton  was  in  London  in  attendance  on  his  parliamentary 


Ix  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

duties.  **  To  Mr  John  Lainb»  commended  by  Mr  Newton,  lately  an  operator  to  the 
Royal  Society.    \0a." 

(•»)  This  and  the  other  letters  to  Locke,  except  that  of  July  7,  1692,  will  be  found 
in  Lord  Kingr's  Life  of  Locke.  This  letter  is  dated  "  Sept."  by  mistake,  the  London 
post  mark  being  *•  Oc.  29." 

(»•)    Nichols's  Illustrat,  Lit,  Hist.  xiii.  49. 

(>««)  "  Besides  a  coach  which  I  consider  not,  it  is  but  200^  per  annum,  with  a 
confinement  to  the  London  air,  &  to  such  a  way  of  living  as  I  am  not  in  love  with." 

(w»)    Optics,  Bk.  2.   Part  iv.  Obs.  la 

(loj)    Appendix,  No.  XXIII. 

(»o»)    Wallis,  0/ip.  II.  391.  seqq. 

(>o*)  Brookbank  was  originally  of  Trinity  College.  The  successful  candidate  was 
the  Hon.  H.  Boyle,  "  a  near  relation"  of  the  Chancellor,  (Duke  of  Somerset)  who  wrote 
a  letter  (Sept.  6)  recommending  him  to  the  University.    (Baker  MSS.  xxx.  355). 

(iM)  It  was  read  at  a  meeting  of  the  Royal  Society,  Febr.  15,  1710,  and  ordered  to 
be  printed  in  the  Transactions.  It  was  printed  in  the  Introduction  to  Vol.  ii.  of 
Harris's  Lex.  Techn.  1710. 

(io«)  It  may  be  doubted  whether  this  letter  is  in  Newton's  handwriting.  The 
conclusion  "  S'  I  am"  &c.,  and  the  address,  are  evidently  in  his  hand. 

(107)  The  four  letters  to  Bentley  were  given  to  tlie  College  by  Cumberland.  They 
were  printed  in  1756,  and  reviewed  by  Johnson  in  the  Literary  Magazine.  See  Monk's 
Bentley,  p.  33  j  Brewster's  Newton,  p.  286.  They  first  appeared  in  their  correct  order 
in  Ikntley's  Correspondence  (Lond.  1842),  the  third  and  fourth  having  previously 
changed  places. 

(108)  '*i  have  now  received  the  box  of  rulers,  with  your  receipt  of  £14.  I  sent 
you  that  money  because  I  thought  it  was  just  j  &  therefore  you  compliment  me  if  you 
reckon  it  an  obligation.  The  chamber  next  me  is  disposed  of ;  but  that  which  I  was 
contriving  was  ...  to  make  you  such  an   allowance,  6cc."     Gentleman's  Magazine, 

LXXXIV.  3. 

(io»)  Brewster's  Life  of  Newtont  p.  232.  In  this  letter  he  says :  '•  I  have  neither 
ate  nor  slept  well  this  twelvemonth,  nor  have  my  former  consistency  of  mind."  A  fort- 
night ttfteryvards  he  apologized  through  a  common  friend  for  having  written  such 
••  a  very  odd  letter,"  saying,  "  that  it  was  in  a  distemper  that  much  seized  hb  head, 
&  that  kept  him  awake  for  above  five  nights  together.**  lb.  p.  234. 
I  ("0)    Dated  "At  the  Bull,  in  Shoreditch."    When  he  wrote  this  letter,  he  **had 

i  not  slept  an  hour  a  night  for  a  fortnight  together,  &  for  five  nights  together  not  a  wink." 
I  See  his  letter  of  Oct.  15,  in  which  he  explains  the  cause  of  this  state  of  his  health.  *•  The 
\  last  winter,  by  sleeping  too  often  by  my  fire,  I  got  an  ill  habit  of  sleeping ;  &  a  dis- 
j  temper,  which  this  summer  has  been  epidemical,  put  me  farther  out  of  order."  Lord 
j  King's  Life  of  Locke,  i.  420,  Brewster's  Life  of  Newton,  p.  240,  where  the  date  is 
printed  by  mistake,  Oct.  5. 

Intelligence  of  liis  being  out  of  health  was  conveyed  in  a  very  exaggerated  form  to 
Huygens  in  INIay  of  the  following  year  by  a  Scotchman,  of  whom  we  know  nothing 
whjitever  except  that  his  name  was  Colm,  (M.  Biot's  Colin):  this  person's  information 
as  recorded  in  a  sort  of  journal  by  Huygens,  who  was  himself  troubled  at  the  time  with 
symptoms  which  in  little  more  than  a  year  afterwards  terminated  fatally,  and  would 
drink  in  with  a  morbid  sympathy  the  tale  of  the  affliction  of  a  kindred  spirit,  is  in  the 
following  terras:  "29  Muj.  1694.  Narravit  mihi  D.  Colm  Scotus  virum  celeberriraum 
ac  summum  geometram  Is.  Neutonum  in  phrenesin  incidisse,  abhinc  anno  et  6  mcnsi- 
bus.  An  ex  niraia  studii  assiduitate  an  dolore  infortunii,  quod  incendio  laboratorium 
chymicum  et  scripta  quxdam  amiserat?  Cum  ad  Archiepiscopum  Cantabrigiensera 
venissct,  ea  locutum,  qua)  alienationem  mentis  indicarent.  Deinde  ab  amicis  curam 
ejus  susccptam,  domoque  clauso  rcincdia  volenti  nolenti  adhibita,  quibus  jam  sanitalem 
■•*    recuperavit,  ut  jam  rursus  librum  suum  Principiorum  Philosophiaj  Mathcraaticorum 


NOTES.  Ixi 

intelHgere  incipiut."  (Hugcnii  Exereitationei..„Vy\eahioek,  Fa«cic.  ii.  p,  171.  Hag. 
Com.  1833).  This  extract  was  first  published  by  M.  liiot  in  the  Dufgruphie  Vnivtr$elte 
(art  Newton,  p.  168).  Sir  David  Brewster  hiis  pointed  out  the  improbability  of  the 
story  and  shewn  the  impossibility  of  reconciling  it  with  known  facts,  {Life  of  JVeirfon, 
p.  230  foil.)  but  not  to  M.  lliot*s  sutisfaction.  We  will  first  quote  at  length  an  anecdote 
which  has  been  brought  to  bear  upon  the  question,  which,  however,  I  think  an  atten- 
tive perusal  will  prove  to  refer  to  a  period  some  years  antecedent  to  the  ej.>och  under 
consideration.  It  b  found  in  a  MS.  diary  written  by  a  member  of  Su  John's  College, 
who,  at  the  date  of  the  entry  about  to  be  quoted,  was  in  his  second  year  of  residence  at 
Cambridge.  He  seems  to  have  heard  the  anecdote  in  company,  and  immediately 
chronicled  it  in  his  journal.  He  does  not  tell  us  who  waa  his  informant,  and  therefore 
we  do  not  know  the  precise  correction  to  be  applied  in  this  instance  to  an  under- 
graduate's story.  We  shall  not,  however,  probably  err  much  in  believin(;  in  the 
substantial  truth  of  the  narrative.    It  runs  as  follows : — 

"  1692.  i'eb.  3d.  What  1  heard  to-day  I  must  relate.  There  is  one  Mr  Newton 
(whom  I  have  very  oft  seen)  Fellow  of  Trinity  College,  that  is  mighty  famous  for 
his  learning,  being  a  most  excellent  Mathematician,  Philosopher,  Divine,  &c.  He 
has  been  fellow  of  the  Royal  Society  these  many  years,  ic  amongst  other  very  learned 
Books  &c  Tracts  he's  written  6ne  upon  y«  mathematical  principles  of  Philosophy, 
which  has  got  him  a  mighty  name,  he  having  received  especially  from  Scotland 
abundance  of  congratulatory  letters  for  the  same :  but  of  all  the  Books  that  he  ever 
wrote  there  was  one  of  colours  &c  light  established  upon  thousands  of  Experiments 
which  he  had  been  20  years  of  making,  &t  which  had  cost  him  many  hunda-d  of 
pounds.  This  Book  which  he  valued  so  much,  &c  which  was  so  much  talked  of,  had 
the  ill  luck  to  perish,  6c  be  utterly  lost  just  when  the  learned  Author  was  almost  at 
putting  a  conclusion  at  the  same,  after  this  manner : — 

In  a  winter's  morning  leaving  it  amongst  his  other  Papers,  on  his  Study  table 
whilst  he  went  to  Chapel,  the  Candle  which  he  had  unfortunately  left  burning  there 
too,  catched  hold  by  some  means  of  other  papers,  &  they  fired  the  aforesaid  Book, 
&c  utterly  consumed  it,  &  several  other  valuable  writings,  &  which  is  most  wonderful 
did  no  further  mischieC 

But  when  Mr  Newton  came  from  Chapel  and  had  seen  what  was  done,  every  one 
thought  ho  would  have  run  mud,  he  was  so  troubled  thereat  that  he  was  not  himself  for 
a  Month  after.  A  long  account  of  this  his  system  of  light  £c  colours  you  may  find  in 
the  Transactions  of  the  Royal  Society  which  he  had  sent  up  to  them  long  before  this 
sad  mischance  happened  unto  him."  (Abraham  do  la  Prymc's  Dtary,in  the  possession 
of  Prof.  Pryme). 

The  foregoing  narrative  is  shewn  by  Sir  David  Brewster  to  be  irreconcileable  with 
Huygens's  memorandum,  on  the  supposition  that  they  both  refer  to  the  same  circum« 
stance.  But,  as  I  have  stated,  I  believe  De  la  Pryme's  anecdote  to  refer  to  an  earlier 
period  not  exactly  known  but  admitting  of  being  fixed  within  certain  limits,  as  I  will 
hereafter  endeavour  to  point  out.  The  discrepancy  between  the  two  statements  is 
adverted  to  here  solely  for  the  purpose  of  noticing  the  singular  hold  which  a  traveller's 
gossip  has  acquired  over  M.  Biot.  '*  Nous  trouvons  au  contraire,"  observes  that 
distinguished  philosopher,  "  entre  ces  dates  un  parfait  accord,"  and  twits  Sir  David 
Brewster  willi  having  overlooked  the  difference  of  calendar  (^Journal  de»  Savant  1B32, 
p.  325).  M.  Biot  tells  us  that  in  English  documents,  previous  to  the  change  of  style  in 
the  middle  of  last  century,  we  arc  to  add  I  to  the  year  of  our  Lord  for  dates  Iwitween 
January  1  and  March  25,  in  order  to  find  the  year  according  to  the  present  reckoning, 
and  that  therefore  1692  in  the  above  extrac:  is  what  would  now  be  written  1693.  It 
does  not  require  a  very  extensive  acquaintance  with  the  literature  of  our  diaries  and 
correspondence  to  know  that  this  rule  is  by  no  means  a  safe  one  to  follow.  In  the 
case  before  us  it  is  a  matter  of  fact  that  the  author  of  the  diary  commences  the  year  in 


IxU  SYNOPTICAL  VIEW  OP  NEWTON'S  LIFE. 

January*:  (ex,gr.  the  death  of  Charles  II.  ia  placed  in  Febr.  1685;  under  Jan.  1692, 
which  follows  1691,  the  writer  laments  the  loss  of  Robert  Boyle  who  died  Dec.  31) 
1691 ;  Dec,  1692  is  followed  by  **  1693  Jan.  1.  This  year  begins  very  ill,  &c."  ;  Dec. 
1693  is  followeil  by  "  1694  Jan.    This  month  we  sat  for  our  degrees,  &cc.") 

Sir  David  Brewster  points  to  the  fact  that  Newton  wrote  his  four  celebrated  letters 
to  Bentley  during  the  time  when  Colm*8  gossiping  statement  represents  him  as  having 
fallen  into  "phrenesis."  Upon  which  M.  Biot  says,  "nous  admettrons  volontiers 
roaintenant  qu'il  {the  fire  which  consumed  Newton's  papers}  est  posterieur  a  la  premiere 
lettre"  (Journ.  de  SavanSt  p.  332),  and  proposes  to  place  the  catastrophe  between  the 
10th  and  30th  of  Dec.  1692.  "  C*e8t  d  cela  sans  doute,"  he  remarks,  "  que  se  rnpporte 
le  passage  Buivant  des  oeuvres  de  Wallis  imprimOes  en  1693....*  Quara  (metliodum) 
speraverim  Neutonum  ipsum  aliquaudo  fusius  traditurum;  ct  quidem  audio  ilium 
hujusmodi  aliquid  prelo  parutum  habuisse  anno  1671,  sed  quod  (infortunio  quodam) 
flammis  periit.'  Wallis,  Tom.  ii.  p.  390.  Le  temps  present  du  verbo  audio,  (-crit  en 
1693,  ne  peut  s*appliquer  qu'a  un  accident  recent,  tel  que  celui  que  les  autres  docu- 
mens  nous  attestent."  Now  the  extract  here  quoted  from  Wallis  is  merely  a  trans- 
lation of  what  had  originally  appeared  in  English  in  his  Algebra  some  years  before. 
(Wallis's  Algebra  bears  the  booksellers'  date  of  1685.  The  bulk  of  the  work  was 
sent  to  London  to  be  printed  in  1676  or  7,  but  the  printing  was  not  proceeded  with 
until  about  the  beginning  of  Aug.  1683,  some  additions  having  been  made  to  it  in  the 
mean  time.  The  Preface  is  dated  Nov.  20,  1684.)  The  passage  alluded  to  is  as 
follows:  "But  I  here  only  give  some  specimen  of  what  we  hope  Mr  Newton  will 
himself  publish  in  due  time.  And  it  was,  I  hear,  near  ready  for  the  press  in  1671. 
But  most  of  those  papers  have  since  (by  a  mischance)  been  unhappily  burned"  (p.  347). 
It  is  the  more  remarkable  that  M.  Biot  should  have  fallen  into  such  an  error,  as  nine 
lines  below  in  tlie  same  page  from  which  he  has  taken  the  above  extract,  Wallis  goes  on 
to  say,  "  Atque  ha;c  sunt  quaj,  ex  memoratis  Newtoni  Uteris  excerpta,  inserueram  in 
editione  Anglicana  1685." 

M.  Biot  makes  another  application  of  his  chronological  rule  to  Newton's  fourth 
letter  to  Bentley,  dated  Febr.  11,  1693,  which  he  affirms  to  mean  our  1694,  and  that 
**  les  propres  expressions  de  celleci  et  sa  relation  avec  les  autres"  shew  that  it  was 
written  a  long  time  after  the  third,  dated  Febr.  25,  169^.  Now  the  letters  here  called 
the  third  and  fourth,  though  printed  in  that  order  until  the  appearance  of  Bentley's 
Correspondence  in  1842,  are  wrongly  placed.  The  four  letters  are  endorsed  by  Bentley 
in  the  order  in  which  they  were  received :  on  the  back  of  the  letter  of  Febr.  11,  1693  he 
has  written  "A  3«i  Letter  from  M'  Newton,"  and  on  that  of  Febr.  25,  169^  he  has 
written  "  A  4''»  Letter  from  Mr  Newton."  Besides,  it  can  be  sliewn,  I  think  satisfac- 
torily, that  Bentley's  two  last  sermons  were  printed  in  1693,  and  as  Newton  must  have 
known  that,  his  words  in  his  letter  of  Febr.  11,  **if  this  come  not  too  late  for  your  use" 
would  have  no  meaning  if  they  were  written  in  1694. 

By  way  of  supplement  to  Sir  David  Brewster's  refutation  of  the  statement  in 
Huygens*8  journal,  it  may  bo  observed  that  the  words  "  Archiepiscopum  Cantabri- 
giensem'*  (probably  a  mistake  for  Cantuariensem)  imply  that  the  crisis  of  Newton's 
"phrenesis"  took  place  in  London.  A  glance  at  the  Table  in  p.  lxxxix.  will  shew  that 
ho  was  not  absent  from  College  for  more  than  a  fortnight  at  a  time  in  1692  and  1693,  and 
therefore  if  the  calamity  which  IVI.  Biot  first  made  known  to  the  world  really  occurred, 
Newton  must  have  been  brought  down  to  Cambridge  very  soon.  Now  if  this  hud  been 
the  case,  we  should,  almost  to  a  certainty,  have  found  Newton's  name  among  the 
invalids  in  the  Steward's  Books,  where  a  record  is  kept  of  the  *' commons"  allowed  to 
sick  fellows  in  their  own  rooms.     I'or  example,  in  the  year  in  question,  ending  at 


•  I  am  enabled,  by  the  kindness  of  the  family  in  whose  possession  the  diary  now  it,  to  state  tliii 
distinctly. 


NOTES.  Ixiii 

Michaelmas  1693,  we  find  one  valetudinarian  fellow  allowed  his  commons  in  his  rooms 
("ex.  CO,'*)  for'  8  weeks,  another  for  1 :  in  1(394  one  for  6  weeks,  another  for  2;  in 
1692  one  for  19  weeks,  a  second  for  15  and  a  third  for  20A  ;  in  1691  one  for  9  weeks, 
another  for  1^,  three  others  for  half  a  week  eich,  and  another  for  3. 

But  probably  the  most  elaborate  and  complete  refutation  will  have  less  weight  with 
the  mj^jority  of  persons  than  the  testimony  of  a  trustworthy  contfmi)orary  witness.  I 
will  therefore  lay  before  the  reader  an  extract  from  a  letter  of  Dr  Wallis  to  Waller,  the 
Secretary  of  the  Royal  Society,  dated  I^Iay  31,  1695,  from  which  by  the  way  it  will  be 
observed  what  "strength"  Colm's  story  had  "  acquired'*  in  the  course  of  its  circulatioa 
to  this  country.  Wullis  had  sent  a  copy  of  the  second  Volume  of  his  Works  as  a 
present  to  Sturm  a  Professor  at  Altorf.  Sturm  wrote  to  thank  Wallis  for  tlie  present, 
and  it  is  tins  letter  of  thanks  which  Wallis  alludes  to  in  the  beginning  of  the  following 
extract :  "  I  have,  since,  one  from  Sturmius,  which  signifies  that  ho  had,  some  weeks 
before,  received  the  Book  I  sent  him.  lie  sends  me  word  of  a  Rumor  amongst  them 
concerning  Mr  Newton  as  if  his  House  &  Books  &  all  his  Goods  were  Burnt,  fie 
himself  so  disturbed  in  mind  thereupon,  as  to  be  reduced  to  very  ill  circumstances. 
Which  being  all  false,  I  thought  fit  presently  to  rectify  that  groundless  mistake"  { in  a 
letter  which  he  desires  Waller  to  forward  } .   (Lett.  Bk.  Roy.  Soc.  W.  2.  50.) 

I  may  observe  that  I  should  not  have  devoted  so  large  a  space  to  so  transparent  a 
piece  of  exaggeration  but  for  the  remarkable  fact  of  its  adoption  by  M.  l^iot,  whose 
veneration  for  the  creator  of  Natural  Philosophy  will  not,  I  hope,  suffer  diminution  by 
this  exposure  of  an  idle  traveller's  tale.  ("  Et  si  le  sort  eftt  vdulu  lo  frapper  aussi 
cruellement,  quel  sentiment  dcvrait  fuire  naitro  en  nous  son  infortune,  sinon  do 
plaiudre  et  de  v6uerer  davantage  cet  autre  Tiresias,  dont  I'intelligence  se  serait  aiasi 
aveugld'C  pour  avoir  vu  de  trop  pres  les  secrets  des  dieux  ?  Toute  autre  pens^'e  serait 
un  sacrilege."  Biot  in  Jour,  des  Sav,  Apr.  1836,  p.  216). 

A  word  may  be  added  on  the  probable  date  of  the  fire  in  Newton's  rooms.  The 
notice  which  we  have  given  above  respecting  the  publication  of  Wallis's  Algebra  shews 
that  the  accident  happened  before  Aug.  1683.  The  superior  limit  is  the  winter  of 
1677,  16T8  as  Wallis  believed  copies  of  Leibniz's  letters,  the  last  of  which  was  dated 
June  21,  1677,  to  have  perished  in  the  flames.  (Letter  to  Lcibn.  Dec.  1,  1696).  One 
of  the  winters  therefore  from  1677  to  1682  (excluding  perhaps  that  of  1680, 1681  during 
which  we  know  a  little  more  of  Newton's  movements  than  in  the  others)  may  bo  fixed 
upon  as  the  probable  date  of  the  occurrence. 

The  version  of  the  story  in  which  "  Diamond"  is  made  to  play  a  prominent  part, 
and  according  to  which  the  scene  is  laid  in  Newton's  latter  years,  and  consequently  in 
London,  may  perhaps  deserve  a  place  here.  "His  temper  was  so  mild  and  eqval, 
that  scarce  any  accidents  disturbed  it.  One  instance  in  particular,  which  is  authenti- 
cated by  a  person  now  living,  [1780,]  brings  this  assertion  to  a  proof.  Sir  Isaac  being 
called  out  of  his  study  to  a  contiguous  room,  a  little  dog,  called  Diamond,  the  constant 
but  incurious  attendant  of  his  master's  researches,  happened  to  be  left  among  the  papers, 
and  by  a  fatality  not  to  be  retrieved,  as  it  was  in  the  latter  part  of  Sir  Isaac's  days, 
threw  down  a  lighted  candle,  which  consumed  the  almost  finished  labours  of  some 
years.  Sir  Isaac  returning  too  late,  but  to  behold  tlie  tlreadful  wreck,  rebuke<l  the 
author  of  it  with  an  exclamation  (ad  sidera  pahmts)  *  Oh  Diamond  !  Diamond !  thou 
little  knowest  the  mischief  done  !' — without  adding  a  single  stripe."  (Notes  to  AIaude*» 
IVensleydale,  p.  102.  4th  ed.  1816.) 
(»>»)    See  under  Sept.  16. 

("'')  A  Mr  Smith  "  took  a  journey"  to  Cambridge  for  the  purpose  of  consulting 
Newton  on  a  problem  in  chances  which  had  its  origin  in  a  lottery  recently  drawn,  and 
brought  with  him  a  letter  of  introduction  from  Pepys.  The  Ist  of  Newton's  letters  is 
principally  occupied  with  settling  the  meaning  of  the  question  (What  are  the  chances  of 
throwing  1  six  with  6  dice,  2  sixes  with  .12  dice,  and  3  sixes  with  18  dice  1),  The  2nd 
contains  his  "easy  computation."     See  Pepys's  Coirespondence, 


Ixiv  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

(u«)  4*  On  {the)  Monday  (night)  likewise  there  being  a  great  number  of  people  at 
the  door  {of  the  haunted  house,— it  was  a  house  opposite  St.  John's  College  in  the 
occupation  of  Valentine  Austin)  there  chanced  to  come  by  Mr  Newton,  fellow  of 
Trinity  College,  a  very  learned  man,  and  perceiving  our  fellows  to  have  gone  in  {three 
fellows  of  St  John's  with  a  fellow -commoner  of  that  college  had  rushed  in  armed  with 
pistols) ,  and  seeing  several  scholars  about  the  door,  Oh  ye  fools!  says  he,  will  you 
never  have  any  wit  1  Know  you  not  that  all  such  things  are  mere  cheats  and  impos- 
tures ?  Fie !  fie  !  go  home  for  shame.  And  so  he  left  them,  scorning  to  go  in."  ( De 
la  Pryme*(*  MS.  Diary,  where  there  is  a  full  account  of  the  proceedings  of  the  "spirit'* 
which  the  writer  of  the  diary  had  received  in  a  letter  from  Cambridge.) 

("♦)    Appendix,  No.  XXIV. 

(116)  "  Quoniam  varii  errores  in  Prop.  37  &  38  (Lib.  ii.)  irrepsere,  illos  omnes 
restitutos  hie  apponam,  prout  in  autoris  exemplari  inveni,  ineunte  Maio  1694,  dum  Can- 
tabrigiae  hsrerem,  consulendi  divini  autoris  gratik."  MS.  of  Dav.  Gregory  (Rigaud. 
p.  100). 

("«)  «« July  4.  Ordered  that  a  letter  be  written  to  Mr  Isaac  Newton  praying  that 
he  will  please  to  communicate  to  the  Society  in  order  to  be  published  his  Treatise  of 
light  &  colours  &  what  other  Mathematical  or  Physical  Tnatises  he  has  ready  by  him." 
Joum,  Bk. 

(U7)  ««Mr  Newton  coming  to  see  me  Sept.  1, 1694,  and  discoursing  of  the  theory 
of  the  moon,  to  let  him  see  wha:  1  had  done  in  order  to  restore  her  motion,  I  produced 
and  shewed  him  these  3  sheets  {  or  synopses  )  of  her  observed  and  calculated  places 
compared.'*  Flamsteed  ap.  Baily,  p.  191.  Shortly  afterwards  Flamsteed  lent  him  copies 
of  two  of  the  synopses,  of  which  Newton  made  transcripts  at  Cambridge.  A  copy  of 
the  3d  was  sent  Oct.  29. 

("«)  The  whole  of  the  known  correspondence  is  printed  in  Baily's  Flamsteed, 
pp.  133 — 160.  Newton's  letters  are  preserved  in  the  library  of  Corpus  Christi  College, 
Oxford,  to  which  Society  they  were  givein  in  1764  by  S.  Adee,  M.D.,  formerly  Scholar 
of  the  College. 

Mr  Baily  has  attempted  from  this  correspondence  to  shew,  in  opposition  to  a  prevail- 
ing opinion,  that  Flamsteed  manifested  no  unwillingness  to  furnish  Newton  with  the 
observations  necessary  to  enable  him  to  complete  the  lunar  theory,  but,  on  the  contrary, 
freely  communicated  every  observation  that  Newton  required.  (Supplement  to  Flam- 
steed's  History,  pp.  708 — 720.)  I  regret  that  I  cannot  concur  in  Mr  Baily's  conclusion. 
Assuming,  what  is  far  from  clear,  that  up  to  December,  1694,  Flamsteed  sent  Newton 
all  the  observations  that  he  asked  for,  I  think  that  in  the  following  month,  and  after- 
wards, we  discover  traces  of  a  feeling  which  is  scarcely  compatible  with  Mr  Baily's 
hypothesis.  The  following  particulars  are  gleaned  from  Newton's  letters,  and  Flam- 
steed's  rough  draughts  or  notes  j  additional  light  will  be  thrown  upon  the  subject  when 
the  correspondence  between  them  is  made  complete  by  the  discovery  of  Flamsteed's 
actual  letters,  which  it  is  hoped  may  be  found  among  the  Portsmouth  papers : 

1694  Dec.  6.    Flamsteed  promises  to  send  Newton  the  observations  that  he  wants  after 

the  Christfnas  holidays, 

1695  Jan.  15.    Newton  acknowledges  the  receipt  of  two  observations  uncalculated, 

and  as  Flamsteed  had  calculated  these  and  the  other  three  of  last  month,  he 
desires  a  synopsis  of  the  calculations,  merely  to  save  himself  the  trouble  of  doing 
what  was  already  done.  But  as  regards  the  rest  of  Flamsteed's  observations, 
he  repeats  what  he  had  said  in  his  letter  of  Nov.  17,  that  he  desires  only  the 
naked  observations. 

— —  19.  Flamsteed  wrote  back,  "  but  no  observations  imparted... I  have  not 
time  to  send  the  synopsis  now ;  may  do  it  hereafter :  but  would  gladly  see  what 
places  you  have  derived  from  the  given  Right  Ascensions  first.  Shall  give 
more  hereafter." 

— —  26,    Newton  replies :  "  Since  I  perceive  you  have  a  mind  to  see  whether  we 


NOTES.  Ixv 

can  compute  correctly,  if  you  please  to  «enil  me  the  latitude  of  Greenwich,  I'll 
send  you  what  you  desire."..."  I  told  you  in  autumn  that  it  would  be  neces- 
sary  to  have  about  half  of  the  ol)servations  in  your  synopses  set  right  by  the 
correct  places  of  the  fixt  stars.  If  you  please  to  do  it  at  your  leisure,  I'll  send 
you  a  catalogrue  of  the  observations."  This  request  is  agrain  alluded  to  by 
Newton  in  his  letters  of  Apr.  23  and  July  9,  but  was  never  complied  with. 
*•  One  thing,"  he  continues,  and  we  now  come  to  an  important  part  of  the  corre- 
spondence as  affecting  the  question  under  discussion,  "  I  did  not  consider. 
The  olwervations  being  yours,  perhaps  you  had  rather  have  them  perfectly  your 
own  in  all  respects,  by  dcterminingr  the  moon's  longitude  and  latitude  from 
them  all  yourself.  If  so  (for  that's  what  you  have  a  very  just  right  unto) 
I  will  stay  your  time.  And  when  I  have  ^i  a  little  further  in  the  theory. ..I'll 
make  a  new  table  of  the  moon's  eccentricities  and  equations  of  her  apogee  for 

finding  her  mean  anomaly,  and  send  you  a  copy  of  it Chuse  you  therefore 

whether  you  will  compute  the  moon's  places  from  the  observbtions  or  leave 
that  work  to  me." 
Tliis  was  answered  in  haste  on  the  day  on  which  it  was  received,  but  we  do  not 
know  in  what  terms.    Flamsteed  sent  a  fuller  answer,  Feb.  7,  with  some  lunar 
observations  calculated  and  reduced,  (among  them  the  three  mentioned  by 
Newton  Jan.  15,  but  not  the  two  others.)    In  his  draught  of  this  answer  he 
says:  "  I  shall  mind  my  business  of  the  fixt  stars  and  give  him  an  account  of 
my  progress,  whilst  he  is  employed  on  the  moon :  and  shall  be  very  well 
pleased  with  an  account  of  his  success."    Flamsteed  accepted  Newton's  pro- 
posal with  respect  to  the  observations,  hinting,  at  the  same  time,  that  he  should 
devote  himself  to  his  catalogue  of  the  fixt  stars.    At  this  point  therefore  New- 
ton's labours  upon  the  lunar  theory  are  suspended  while  he  is  "  staying  the 
time  "  of  the  Astronomer  Royal. 
March  2.    Flamsteed,  in  a  draught  of  an  answer  to  Newton's  letter  of  Febr,  16, 
has  these  words :  •'  Vindication  of  myself  for  not  imparting  my  observations, 
and  an  account  of  my  northern  correspondence." 
Apr.  23.     Newton  writes :    **  When  I  have  your  materials,  1  reckon  it    { the 
moon's  theory  }  will  prove  a  work  of  about  three  or  four  montlis :  ond  when 
I  have  done  it  once  I  would  have  done  with  it  for  ever." 
June  29.    Newton,  who  is  still  staying  the  Astronomer's  time,  thanks  him  for 
sending  his  solar  tables  (which  Newton  does  not  seem  to  have  wanted):  "  IJut 
these,  and  almost  nil  your  communications  will  be  useless  to  me,  unless  you 
can  propose  some  practicable  way  or  other  of  supplying  me  with  observations. 
For  as  your  health  and  other  business  will  not  permit  you  to  calculate  the 
moon's  places  from  your  observations, so  it  was  never  my  inclination  to  put  you 
upon  such  a  task,  knowing  that  the  tediousness  of  such  a  design  will  make  me 
as  weary  with  expectation  as  you  with  drudgery...!  will  therefore  once  more 
propose  it  to  you   {  as  he  had  done  Nov.  17  and  Jan.  15  }    to  send  me  your 
naked  observations  of  the  moon's  right  ascensions  and  meridional  altitudes ;  and 
leave  it  to  me  to  get  her  places  calculated  from  them.    If  you  like  thin  propo- 
sal, then  pray  send  me  first  your  observations  for  the  year  1692,  and  1  will  get 
them  c:dculated,  and  send  you  a  copy  of  the  calculated  places.    IJut  if  you  like 
it  not,  then  I  desire  you  would  propose  some  otlier  practicable  method  of  sup- 
plying me  with  observations ;  or  else  let  me  know  plainly  that  I  must  be  con- 
tent to  lose  all  the  time  and  pains  I  have  hitherto  taken  about  the  moon's 
theory  and  about  the  table  of  refractions." 
July  2.     Hamsteed,  stung  to  the  quick,  offers  not  the  mural  arc  observations  of 
1692,  but  the  sextant  observations  from  1677  to  1690.    It  would  also  seem, 
from  a  statement  written  by  Flamsteed  on  the  back  of  Newton's  Ittter,  as  if  he 
had  sent  at  the  same  tinjc  the  30  obscrvaiions  which  he  had  made  from  Febr.  8 


kvi  SYNOPTICAL  VIEW  OF  NEWTOJTS  LIFE. 

to  June  25  in  the  current  year.  But  as  Newton  makea  no  mention  of.  having 
received  them, merely  saying,  •♦  when  you  have  computed  your  30  observations, 
you  will  know  no  more  of  it  { the  parallactic  equation  }  than  at  present,"  I 
suspect  that  there  is  some  mistake  in  Flamsteed's  memorandum. 
1695  July  9.  Newton  writes:  "  After  I  had  helped  you  where  you  stuck...  {  he  par- 
ticularly mentions  the  table  of  refractions,  which  he  says  )  cost  me  above  two 
months'  hard  labour  which  I  should  never  have  undertaken  but  upon  your 
account,  and  which  I  told  you  I  undertook  that  I  might  have  something  to 
return  you  for  the  observations  you  then  gave  me  hopes  of,  and  yet,  when  1  had 
done,  saw  no  prospect  of  obtaining  them*  or  of  getting  your  synopses  rectified, 
I  despaired  of  compassing  the  moon's  theory,  and  had  thoughts  of  giving  it 
over  as  impracticable,  and  occasionally  told  a  friend  so  who  then  made  me 
a  visit.  But  now  you  offer  me  those  observations  which  you  made  before  the 
year  1690, 1  thankfully  accept  of  your  oflTer,  and  will  get  as  many  of  them  com- 
puted as  are  sufficient  for  my  puipose." 
>— —  13.  Flamsteed  sends  his  observations  from  Jan.  to  July  1677. 
— —  20.  Newton  says,  "The  report  you  mention  {  which  was  current  in  Lon- 
don about  Flamsteed's  not  furnishing  Newton  with  observations  }  was  mucli 
against  my  mind,  and  I  have  written  to  put  a  stop  to  it.  I  thank  you  for.. .your 
lunar  observations." 
—  27.  Newton  says,  •'  The  other  day  I  had  an  excuse  sent  me  for  what  was 
said  at  London  about  your  not  communicating,  and  that  the  report  should  pro- 
ceed no  furtlier.  I  am  glad  all  misunderstandings  are  composed."  He  then 
specifies  the  further  observations  (out  of  the  sextant  stock  )  that  he  wants. 
Sept.  14.  Newton  returned  to  Cambridge  on  Sept.  10,  and  went  away  again  on 
the  14th :  before  leaving,  he  writes,  "  I  have  not  yet  got  any  time  to  think  of 
the  theory  of  the  moon  nor  shall  have  leisure  for  it  this  month  or  above :  which 
I  thought  fit  to  give  you  notice  of,  that  you  may  not  wonder  at  my  silence." 
He  however  returned  in  a  fortnight,  but  had  sublunary  matters  to  attend  to, 
was  named  by  rumour  shortly  afterwards  as  Master  of  tlie  Mint,  and  in  the 
March  of  next  year  was  actually  appointed  Warden. 
■»-  17.  In  Flamsteed's  draught,  written  on  Newton's  letter,  we  read,  *'  My 
exercise  will  devour  no  small  part  of  my  time,  and  therefore  I  shall  desire  my 
friends  to  excuse  me  if  I  answer  not  their  letters  so  fully  nor  readily  as  for- 
merly;  however,  when  you  want  more  of  my  lunar  observations  {  i.e.  those 
made  before  1690  with  the  sextant,  not  those  which  he  had  made  or  was  making 
with  the  mural  arc  ]   I  shall  cause  them  to  be  transcribed  and  it  will  be  no 

trouble."    Mr  Bwly  has  printed  the  words  "however no  trouble"  in 

italics  ;  the  preceding  part  of  the  sentence  is  not  however  destitute  of  signi- 
ficance. 
Here  the  correspondence  terminates.  There  are  several  allusions  to  it  in  Flamsteed's 
extant  memoranda,  two  of  which  are  produced  here  as  evidence  in  the  question  we  are 
examining :  "  {  Mr  Newton  }  ceased  not  to  importune  me  (though  he  was  informed  of 
my  illness)  for  more  observations,  and  with  that  earnestness  that  looked  as  if  he  thought 
he  had  a  right  to  command  them,  and  had  about  50  more  imparted  to  him.  But  I  did 
not  think  myself  obliged  to  employ  my  pains  to  serve  a  person  that  was  so  inconsiderate 
as  to  presume  he  had  a  right  to  that  which  was  only  u  courtesy.  And  I  therefore  went 
on  with  my  business  of  the  fixed  stars ;  leaving  Mr  Newton  to  examine  the  lunar  obser- 
vations over  again :  which  had  he  done,  he  had  found  that  he  needed  not  be  so  importu- 
nate for  new, — the  old  would  have  been  sufficient  for  the  purpose  and  design  for  which 


•  Flamsteed  htu  written  on  tlie  letter  "  My  sickness  has  hindercil."    But  wc  shaU  see  by  and  by 
fh>m  his  own  statement  that  that  was  not  the  sole  cause. 


NOTES.  Lxvii 

I  had  imparted  them  to  him.  I  was  therefore  forced  to  leafo  off  my  corre«pondence 
with  him  at  that  time."  (Baily,  p.  63.)  Ajrain:  **  I  continued  since  fumi«hin(r  him 
with  lunar  observations,  as  I  gained  them,  until  Midsummer  1G95,  when  bcin^  troubled 

with  a  distemper I  was  forced  to  intermit  my  correspondence  with  him."    (lb. 

p.  191.) 

Upon  the  whole,  I  think,  we  may  conclude  that  the  combined  action  of  Flamsteed's 
bad  temper  and  bad  health,  for  which  great  allowance  muet  be  made,  coupled  with  his 
professional  jealousy  of  Halley*  and  his  exaggerated  opinion  of  the  value  of  hit.  own 
astronomical  labours,  has  robbed  us  of  the  lunar  theory  in  the  form  tliat  its  creator  would 
have  given  it,  and  tiiat  the  following  words  contain  more  truth  than  is  sometimes  to  be 
met  with  in  epistolary  statements :  '*  Flamstcedius  suas  de  Luna  obscrvalioncs  Newtono 
negaverat.  Inde  factum  aiunt  quod  hie  qua;dam  in  motu  Lunari  adhuc  indeterminata 
reliquit."  (Leibniz  to  Roemer,  Oct.  4, 1706.  0pp.  Tom.  iv.  Pan  ii.  p.  126.) 

.     .  »,.  .....  .      ,      sin  0*«  parallax 

("»)  This  inequality  m  the  Moon's  longitude  is  proportional  to  -7-^- , 

sin  ])'»  parallax 

its  argument  being  ])'s  mean  anpulia.r  distance  from  ©.  •'  On  la  consld6rer...avec 
raison  comme  unc  des  applications  les  plus  delicates  de  I'analysc  moderne.**  (Hiot, 
Journ.  des  Sav.  Apr.  1836,  p.  218.)  In  his  letter  of  July  9,  1695,  Newton  says  that  its 
maximum  value  scarce  exceeds  2  or  3,  or  at  most  4  minutes,  Biirg  ( Mtcan.  Cel.  Tom. 
III.  p.  282)  gives  it  2',  2",  38.  Compare  Pont^coulant,  iv.  605,  who  (ib.  xiv.  note)  does 
not  seem  to  be  aware  that  this  equation  was  known  to  Newton.  M.  Biot  says  that  this 
equation  is  omitted  in  thesccond  edition  of  the  Priricij^ta,  and  suggests  reasons  to  account 
for  the  omission.  But  see  p.  120  of  this  work,  where  the  "  Variatio  secunda"  is  de- 
scribed. 

("0)  This  is  now  called  the  lunar  equation  of  the  Sun,  "et  Ton  avail  tout  lieu  de  la 
considerer  comme  une  des  corrections  les  plus  delicates  des  tables  modernes."  (Biot, 

Journ.  des  Sav.  Apr.  1836,  p.  220.)    It  =  J^J!^  .  dist.  of^from  Q^  ^  ^.^  difference 

ti^'s  mass    dist.  of  Q)  from  tP 
of  longitudes  of])  and  ©•    The  coefficient  is  given  8",83  in  the  M/lcan.  Cel.  Tom.  iir. 
p.  108.    Newton  in  the  above  letter  says  that  he  had  not  yet  ascertained  its  magnitude, 
but  that  it  may  be  assumed  16"  or  20"  until  it  be  determined  more  exactly.    Comp. 
Pontecoulan'.,  iv.  653. 

(1")  Flamsteed's  coquetry  about  his  two  observations  draws  from  Newton  a  little 
playful  irony— an  indulgence  extremely  rare  with  him :  "  The  places  of  the  moon  from 
your  two  observations  I  have  not  yet  computed :  for  I  thought  it  superfluous  to  do  what 
you  had  done  to  my  hands ;  and  desired  a  copy  of  your  computations  only  to  save  myself 
that  labour.  But  since  I  perceive  you  have  a  mind  to  see  whether  we  can  compute 
exactly,  if  you  please  to  send  me  the  latitude  of  Greenwich  I'll  send  you  what  you 
desire."   (Baiiy,  p.  149.) 

(122)  This  is  the  table  afterwards  published  by  Ilalloy  in  the  Phil.  Trans.  May— Aug. 
1721,  ••  such  as  I  long  since  received  it  from  its  Great  Author."  See  Biot's  third  article 
on  Baily*s  Flamsteed  in  the  Journal  des  Suvans  for  Nov.  1836,  which  he  commences  by 
observing  that  he  is  in  arrcar  with  the  article,  "et  pourtant,  depuis  environ  ncuf  mois 
que  mon  second  article  a  paru,  je  n*ai  pas  6t6  cccup6  d'autre  chose  que  de  sa  con- 
tinuation. Ma'is,  pendant  tout  co  temps,  je  puis  dire  en  v6rit6,  comme  Jacob,  que 
j*ai  lutt6  avec  l'esprit."  For  the  results  of  the  struggle  see  that  article,  and  his  paper 
**  Analyse  des  Tables  de  refraction  construites  par  Newton,  avec  I'indication  des  pro- 
cadi's  num6riques  par  lesquels  il  a  pu  les  calculer."    ( lb.  pp.  73S— 754.) 


*  The  torrents  of  vituperation  poured  by  Flamsteed  upon  thia  illutlrioui  man  are,  I  tielieve,  to  be 
explained  on  the  principle  alluded  to:  (Krpa/bi(v<  Kepafitt..)  At  the  meeting;  of  the  Itnyal  (kK'icty, 
June  1,  161)2,  Hallcy  read  a  paper  vindicating  his  St  Helena  Observations  "  from  some  groundluu  ex. 
ccptlons"  of  Flamsteed's. 


Ixviii  SYNOPTICAL  VIEW  OP  NEWTON'S  LIFE. 

(»«•)  Some  delay  occurred  in  sending  this  letter.  Flamsteed  did  not  receive  it  until 
May  6. 

(>««)    Appendix,  No.  xxvi. 

("»)  •'  I  shall  order  Will  Martin...to  pay  him  two  guineas,  if  you  please  to  let 
him  call  for  them,  or  to  pay  it  to  his  or  your  order  in  London  if  you  please  to  let 
Bie  know  where."  The  words  in  this  extract  which  follow  '•  pay  him  "  are  crossed 
out  in  the  MS.  and  tiie  word  "  guineas"  altered  into  "shillings  "  apparently  by  Flam- 
steed.  The  words  after  "for  them,"  to  the  end  of  the  passag-e,  are  conjectural,  tlie 
original  writing  bein&  most  skilfully  blotted  out.  I  believe  however  that  it  might  be 
made  out  on  a  bright  day,  if  it  were  thought  worth  the  trouble.  What  motive  Flam- 
•teed  could  have  had  for  disguising  any  part  of  the  above  sentence  I  do  not  pretend  to 
divine.  It  is  curious  that  Mr  Rigaud,  who,  at  Mr  Daily's  request,  examined  the  MS. 
with  reference  to  this  very  point,  should  have  overlooked  the  original  "  guineas." 
(Baily,  p.  16.9,  note.) 

(»3«)    Wallis,  writing  to  Ilalley  from  Oxford.  Nov.  26,  says:  *•  We  are  told  here 
that  he  is  made  Master  of  the  Mint,  which  if  so,  I  doe  congratulate  to  him  and  am 
his  &  your  &c."    Orig.  Lett.  Bk.  Roy.  Soc.  W.  2.  66.    See  Appendix,  p.  302. 
(»7)    Mace.  Co)T.  II.  419. 

("8)  The  original  MS.  with  the  address,  "  For  the  Right  Honourable  Charles 
Montagu  Esq.  Chancel  lour  of  the  Exchequer,"  is  preserved  at  the  Royal  Society, 
Orig,  Lett,  Bk,  N.  l.eib.  The  problems  are  (1)  To  determine  the  brachistochron 
between  two  giv^n  points  not  in  the  same  vertical  line :  (2)  APF'  is  a  straight  line 
passing  through  a  fixed  point  A,  and  meeting  a  curve  in  P,P':  to  find  the  curve  such 
that  ilP"  +  i4P'"  =  constant.  One  of  the  two  identical  papers  (a  printed  folio  half- 
sheet)  which  were  sent  to  Newton  by  Bernoulli,  containing  the  problems,  still  exists 
in  the  Archives  of  the  Royal  Society,  (Volume  lettered  "Arithmetic,  Algebra,"  &c, 
13).  At  the  bottom,  in  Newton's  hand,  are  the  words  "  Chartam  hanc  ex  Gallia 
missam  accepi  Jan.  29,  169^." 
(»2»)  Mace.  Carr.  n.  420. 
(•M)    See  Appendix,  p.  299. 

("1)  "  Isaac  Newton  chuseth  the  Hon**'*  Henry  Boyle  Esq",  Burghess  of  this 
University."  The  votes  were  given  in  English  on  account  of  the  election  occurring 
during  the  vacation. 

("2)  James  Hodgson  had  calculated  these  12  places  for  Newton  by  Flamsteed's 
orders,  during  the  absence  of  the  latter  in  Derbyshire,  and  sent  them  to  him  Sept, 
8.  Flamsteed  on  examining  them  Nov.  11,  "  found  them  all  false,"  and  computed 
them  afresh.  The  results  of  these  last  calculations  were  communicated  to  Newton 
on  his  visit  to  Greenwich,  Dec.  4. 

On  December  29  Flamsteed  sent  him  a  correction  of  the  time  of  one  of  the  observa- 
tions, and  after.vatds  found  that  his  results  required  further  modification.  "  I  acquainted 
him,"  he  says,  "there  was  a  further  fault  in  them,  when  I  was  last  with  him.  He  is 
reserved  to  me,  contrary  to  his  promise,  I  lie  under  no  obligation  to  be  open  to 
him."  (Baily,  p.  166).  Flamsteed  was  in  London  on  Dec.  30  and  31,  (Friday  and 
Saturday),  and  the  words  '*  when  I  was  last  with  him,"  probably  refer  to  one  of  those 
days.  Newton  was  then  aware  of  the  liberty  which  Flamstce<l  had  taken,  in  men- 
tioning his  name  in  connexion  with  the  Lunar  Theory,  in  the  Letter  to  Dr  Wallis. 
Hence  we  may  explain  the  "  reserve  "  of  which  Flamsteed  complains,  and  to  which  Mr 
Baily  has  attached  a  different  meaning,  (p.  710,  note). 

(13S)  In  a  letter  to  Dr  Wallis  on  annual  parallax,  which  was  to  appear  in  the  3d 
volume  of  the  Doctor's  Works,  Flamsteed  alluded  to  his  having  supplied  Newton  with 
lunar  observations.  On  being  informed  by  David  Gregory  of  the  fact,  Newton  desired 
him  to  request  Dr  Wallis  not  to  print  the  paragraph  containing  the  allusion  in  question. 
When  Flamsteed,  who  does  not  seem  to  have  anticipated  that  there  could  be  any  objec- 
tion to  his  making  public  use  of  Newton's  name  without  previously  obtaining  permission 


NOTES.  Ixix 

to  do  80,  received  intimation  of  this  from  Wallia,  he  wrote  to  Newton  on  the  wbject, 
( Monday,  Jan.  2,)  and  nyain  on  the  Gtli.  Newton  in  his  answer,  dated  Jan,  6,  states 
his  reasons  for  having  requested  the  suppression  of  tl»e  paragraph.  "I  was  concerned," 
he  says,  "to  be  publicly  brought  upon  the  stftfro  about  what,  perhaps,  will  never  l«  fitted 
for  the  public,  and  tliereby  the  world  put  into  an  expectation  of  what,  perhaps,  they  are 
never  like  to  have.  I  do  not  love  to  be  printed  on  every  occasion,  much  less  to  be 
dunned  and  teased  by  foreigners  about  mulliematical  things,  or  to  be  thought  by  our 
own  people  to  be  trifling  away  my  time  about  tl»em,  when  I  should  bo  about  the  King's 
business."    (The  great  re-coinage  of  silver  was  not  yet  complete<l). 

("*)  The  eight  foreign  Associates  created  on  the  re-modelling  of  the  Academy  in 
1699,  were 

1.  Leibniz,  ^ 

2.  Gugliclmini,  I    p^^^^  ^^ 

3.  Ilartsoeker,  i 

4.  Tschirnhausen,      i 

5.  James  Bernoulli,   )    p  u     i^ 

6.  John  Bernoulli,     * 

7.  Newton,  I   F  b    21 

8.  Roemer,  J 

The  first  four  seem  to  have  been  nominated  by  the  King,  the  rest  by  the  Academy. 

(MS)  it  Mr  Newton  shewed  a  new  instrument  contrived  by  him  for  observing  the 
moon,  stars  and  {  so  finding  the  }  longitude  at  sea,  being  the  old  instrument  mended 
of  some  faults,  with  which  notwithstanding  Mr  Halley  had  found  the  longitude  better 
than  the  seamen  by  other  methods."  Jmimal  Book.  (Plooke,  as  usual,  at  the  next 
meeting  of  the  Society,  Oct.  25,  laid  claim  to  the  discovery).  A  paper,  in  Newton's 
hand,  describing  the  instrument,  headed  "  An  instrument  for  observing  the  distance  of 
the  moon  from  the  fixt  stars  at  sea,"  is  preserved  in  No.  LXXXI.  MSS.  Roy.  Soc.  It 
was  found  among  Halley 's  papers  after  his  death,  and  was  published  in  the  Transactions 
for  Oct.~Nov.  1742. 

The  following  extract  from  a  letter  of  Charles  Montagu  to  Sloane,  dated  Aug.  7, 
1699,  refers  to  the  "mending"  of  the  "faults"  of  the  "  old  instrument."  After 
stating  that  he  was  to  have  waited  on  the  Lord  Chancellor  (Somcrs)  at  Gresham 
College,  next  Wednesday,  he  says :  '•  But  I  understand  that  Mr  Newton's  experiment 

will  not  be  ready  by  that  time I  hear  the  engine  will  not  be  made  within  10  days, 

and  then  I  believe  my  Lord  will  wait  upon  you.'*    (Sloane  MSS.  Briu  Mus.  4063). 

("«)  With  Aston  and  Flamsteed.  Lord  Chancellor  Somers  was  re-elected 
President. 

("' )  Ruding's  Annals  of  the  Coinage,  ii.  427. 

(188)  120  copies  of  the  work  were  printed  "impensis  illustrissimorum... Somers... 
Dorset.. .Car.  Montagu... Newton..."  and  five  others,  including  Sloane  and  Aston. 

(139)  The  method  was  sent  by  a  M.  du  Verger,  in  a  letter  from  Rome,  with  a 
description  of  an  instrument  for  solving  the  three  problems.  (Regist,  Bk.  ix.  12.)  At  a 
meeting  of  the  Royal  Society,  Apr.  8,  Sloane  was  **  ordered  to  give  the  letter  and 
demonstrations  to  Mr  Newton,  to  have  his  opinion  and  answer,"  At  the  next  meet- 
ing, Apr.  15,  Sloane  "  promised  to  take  care  to  deliver  "  them.  On  July  24,  was  read 
a  letter  from  Sloane  to  du  Verger,  containing  Newton's  report  concerning  his  papers. 
The  following  is  an  extract  from  it :  *'  Ipsissimo  quo  chartas  acccpil  momento  exumi- 
nandas  comraLsit  uni  e  Sociis  in  hisce  rebus  versatissiroo,  qui  nuper  opinionem  suam 
Societati  retulit  modum  nimirum  describendi  volutam  accuratum  satis  videri  et  in  rebus 
mechanicis  usui  futurum,  nee  tamen  geometrice  dcmonstratum  esse  existimat ;  et  jiroinde 
anguli  trisectionemt  dupUcationem  eubi  et  quadraturam  circuli  non  esse  mathematice 
investigata."  Letter  Bk.  xii.  328. 

(1*0)  •«  Tabula  quantitatura  et  graduum  Caloris."  Orig,  Lett.  Bk.  Roy,  Soc.  N. 
1.  62.   Comp.  Brewster's  Newton,  297. 


Ixx  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

(i«i)  The  poll  stood  as  follows :  Right  Hod.  II.  Boyle,  (Trin.)     180 
Mr  Newton,  (Trm.)  161 

Mr  Hammond,  (Joh.)  64 

Dr  Bainbriggr*  Vice-Master  of  Trinity,  in  voting  for  Newton,  calls  him  "virnm 
optimum,"  Dr  Stubbc,  one  of  the  Seniors,  and  afterwards  Vice-Master,  terms  him 
*•  clarlssiraum  virum :"  in  some  of  the  voting  papers  the  epithet  is  "dignissimus"  or 
•*  doctissimua."  Bishop  JNIonk,  (Life  of  nentleif,  p.  122,)  says  that  Bentley  "had  the 
iatisfaction  of  assisting  in  the  return  of  his  illustrious  friend  Sir  Isaac  Newton."  lient- 
ley'  voting-paper  however  is  not  found  among  those  of  any  of  the  three  candidates. 
Newton  himself  voted  for  Boyle. 

(1")  His  resignation  of  the  Professorship  in  his  own  handwriting,  is  preserved  in  the 
Ilegistrary*8  ofBce.  With  respect  tu  tlic  resignation  of  his  fellowship,  see  p.  Lxxxii. 
note  §. 

(It*)  It  appeared  in  English,  separately,  the  following  August,  also  in  Harris's 
Lexicon  Techniciim,  1704,  (a  work  to  which  Newton  was  a  subscriber),  and,  with  a  few 
corrections  by  Newton  in  the  table  of  Errata,  in  the  Miscellanea  Cmiosa,  1705,  (tliis 
is  the  date  of  the  1st  ed.,  not  1708  as  stated  by  M:  Baily  in  his  Supplement  to  Flam- 
$teed*t  History,  p.  688,)  with  the  title  of  "The  Ffimous  Mr  Isaac  Newton's  Theory  of 
the  Moon."  With  respect  to  ]Mr  Baily's  renewed  assertion  (ib.  p.  735)  that  "  in  the 
Theoria  Luntc  there  is  not  a  single  allusion  made  to  Flamsteed,**  it  may  be  observed 
that  in  the  three  above  mentioned  English  reprints  the  mention  of  Flamsteed's  name 
comes  after  the  title  of  the  tract,  not  before  it  as  in  Gregory's  Astronomy.  Not  that  this 
is  a  point  of  any  great  consequence,  for  the  acknowledgment  of  Flamsteed's  services  in 
supplying  Observations  is  much  the  same  in  eilher  case.  It  is  extremely  improbable 
that  the  essay  was  communicated  to  Gregory  in  the  naked  form  in  which  it  stands  within 
inverted  commas  in  his  Astronomy :  it  must  have  been  accompanied  by  some  notice  of 
Flamsteed's  Observations  and  their  near  agreement  with  the  results  derived  from  the 
Theory,  the  substance  of  which  Gregory  chose  to  embody  in  an  introductory  paragraph, 
then  prefixing  the  title  "  Lunre  Theoria  Newtoniana,"  and  finally  giving  us  the  actual 
Theory  in  its  author's  own  word&— a  bare  numerical  statement  of  facts  and  rules,  in 
which  complimentary  phrases  would  scarcely  find  an  appropriate  place. 

("*)  During  this  visit  Locke  shewed  him  his  Essay  xtpoii  the  Corintliians,  with  which 
*•  he  seemed  very  well  pleased,  but  had  not  time  to  look  it  all  over."  Locke  sent  it  to 
him  before  Christmas  for  his  more  careful  perusal,  and  not  hearing  anything  from  him, 
towards  the  end  of  March,  1703,  sent  him  a  further  communication.  Receiving  no  an- 
swer, Locke,  who  was  now  old  and  infirm,  became  impatient  to  learn  something  of  the 
fate  of  his  papers,  and  in  a  letter  dated  Apr.  30,  commissioned  his  nephew  Peter  King 
(afterwards  Lord  Chancellor)  to  wait  upon  the  Master  of  the  Mint,  with  a  letter 
which  he  had  written  for  the  purpose.  "  He  lives  in  German  St.  You  must  not  go 
on  a  Wednesday,  for  that  is  his  day  for  being  at  the  Tower.  The  reason  why  I  desire 
you  to  deliver  it  to  him  yourself  is  that  I  would  fain  discover  the  reason  of  his  so  long 
silence.  I  have  several  reasons  to  think  him  truly  my  friend,  but  he  is  a  nice  man 
to  deal  with,  and  a  little  too  apt  to  raise  in  himself  suspicions  where  there  is  no  ground  ; 
therefore  when  you  talk  to  him  of  my  papers,  and  of  his  opinion  of  them,  pray  do 
it  with  all  the  tenderness  in  the  world,  and  discover,  if  you  can,  why  he  kept  them 
80  long  and  was  so  silent.  But  this  you  must  do  without  asking  why  he  did  so,  or 
discovering  in  the  least  that  you  are  desirous  to  know.. ..Acquaint  him  that  you  in- 
tend to  see  me  at  Whitsuntide,  and  shall  be  glad  to  bring  a  letter  to  me  from  him, 
or  any  thing  else  he  will  please  to  send.. ..Mr  Newton  is  i-eally  a  very  valuable  man, 
not  only  for  his  wonderful  skill  in  mathematics,  but  in  divinily  too,  and  his  great 
knowledge  in  the  Scriptures,  wherein  I  know  few  his  equals.  And  therefore  pray 
manage  the  whole  matter,  so  as  not  only  to  preserve  me  in  his  good  opinion,  but  to 
increase  me  in  it;  and  be  sure  to  press  him  to  nothing,  but  what  he  is  fonvard  in 
himself  to  do."    Lord  King's  Life  of  Locke,  ir.  38. 


NOTES.  Ixxi 

Newton  accordingly  sent  an  answer,  apparently  in  the  m»nner  luggMted,  (it  in 
dated  May  15,  the  day  before  Whitsunday),  the  first  clause  of  which  shews  that  the 
grouodlcsu  suspicions  were  on  the  part  of  I^cke.  "  Upon  my  first  receiving  your 
papers,  I  read  over  those  conccniing-  the  first  Epistle  of  the  Corinthians,  but  by  so 
many  intermissions,  that  I  resolved  to  go  over  them  ng-ain,  so  soon  n»  I  could  got 
leisure  to  do  it  with  more  attention.  I  have  now  read  it  over  a  second  time,  and 
gone  over  also  your  papers  on  the  second  Kpistle."    Jb.  i.  420. 

0*^)  He  succeeded  Lord  Somers,  who  had  held  the  office  five  years,  lie  wa«  re- 
elected annually  during  the  remainder  of  his  life, 

(**«)    "The  President  said  he  had  thought  of  a  contrivance  for  buming.glasscs,  by 
uniting  several,  {  probably  apropos  of  a  i)aper  by  Lowthorp  on  the  subject }  ....The 
President  was  desired  to  give  directions  to  make  such  glasses  as  he  shall  think  proper. 
May  17.    The  President  shewed  a  piece  of  silver  money  and  iron  wire,  part  of  which 
were  melted  in  the  focus  of  a  metallic  speculum,  ficc.  &c. 

24.    The  President  said  that  he  had  tried  the  addition  of  a  reflecting  speculum, 

and  ho  thought  the  focus  of  the  burning-glass  too  near  to  produce  the  desired  efTect. 

31.    The  President  shewed  a  piece  of  red  tile  {  vitrified  by  the  burning-glass  ) ,  &c. 

June  21 .    The  President  tried  some  nc  w  experiments  with  his  speculum. 
July  12.    The  President  gave  the  speculum  lately  contrived  by  him  to  the  Society. 
Nov.  15.    Mr  Halley  was  desired  to  draw  up  an  account  of  Mr  Newton's  buming-spe» 
culum.**     (Jmirnal  Bk.) 

Tiie  burning-glass  given  by  Newton  to  the  Society  is  described  by  Harris  (Lexicon 
Technicuwy  Vol.  ii.),  as  consisting  of  7  concave  glasses  (each  about  11 J  inches  in  diam.)^ 
with  their  foci  coincident,  6  of  them  being  placed  round  the  7th  and  in  contact  with  it, 
and  forming  a  sort  of  segment  of  a  sphere,  whose  subtense  is  about  34  ^  inches.  The 
central  glass  lies  about  an  inch  lower  or  farther  in  than  the  rest.  The  common  focus  is 
about  22  J  inches  distant,  and  about  A  inch  in  diam.  It  vitrifies  brick  or  tile  in  a  mo- 
ment, and  melts  gold  in  about  ^  a  minute.    Comp.  Hutton's  Math.  Diet. 

Under  the  date  of  Febr.  2,  maybe  mentioned  the  examination  of  the  pseudo-For- 
mosan,  George  I'salmanazar,  at  the  Royal  Society.  In  the  British  JNIuseum  there  is  a 
letter  from  John  Chamberlayne  to  Newton,  dated  Febr.  2,  1703-4,  reminding  him  of 
**  the  famous  conference  appointed  to  take  place  this  afternoon  at  Gresham  College,  be- 
tween Mr  George,  the  Formosan,  the  bearer  hereof,  and  Le  Pere  Fontenay,  a  Jesuit, 
lately  come  from  China.  I  have  engaged  Mr  George,  and  am  to  can^  him  thither  this 
ailernoon  in  my  coach,  but  without  telling  him  the  reason.  I  beg  therefore  the  same 
caution  and  security  on  your  side."  (MSS.  Birch,  4292).  Newton  does  not  seem  to 
have  attended  the  meeting.  Psalmanazar  gives  an  account  of  the  conference  in  the 
Preface  to  his  Description  of  Formosa.  (Lond.  1704.  p.  vii.).  The  impostor  quailed 
under  the  searching  scepticism  of  Halley,  Mead  and  Woodward.  (Memohifp.  196. 
Lond.  1764).  For  a  brief  account  of  this  singular  person,  who  at  32  repented  of  his 
ways,  and  in  after  life  became  a  large  contributor  to  the  Universal  History,  and  won  the 
respect  of  Johnson,  see  Chalmers's  Biogr.  Diet. 

(147)  «  Febr.  16,  the  President  presented  his  book  of  Optics  to  the  Society ;  Mr 
Halley  was  desired  to  peruse  it,  and  to  give  an  abstract  of  it ;  and  the  Society  gave  the 
President  thanks  for  the  book  and  for  being  pleased  to  publish  it"    {Jonrn.  Bk.) 

The  Preface  in  the  first  edition  bears  no  date.  In  the  second  edition  ( 1718)  the  date 
••April  1,  1704,"  was  added.  There  is  a  similar  peculiarity  about  the  Preface  to  the 
Prineipia.  (See  p.  tviii.)  The  dispute  with  Leibnit  had  probably  taught  our  philosopher 
the  importance  of  dates. 

(148)  The  words  are:  "Pro  differentiis  igitur  Leibnitianis  D.  Newtonus  adhibet 
semperque  adbibuit  yfurionM... iisque  tum  in  suis  Principiis  Naturae  Mathematicis,  turn 
in  aliis  postea  editis  eleganter  est  usus,  quemadmodum  et  Honoratus  Fabrius  in  sua  Sy- 
nopsi  Geometrica  motuum  progressus  Cavallerianaj  Methodo  substituit."  (p.  35).  Ludo- 
vici   {Ilistorie  der   Leihnizisehen  Philosophies    quoted   by   Guhrauer),  and   Guhrauer 

e2 


Ixxii  SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

{Biographie  of  Leibniz, u  311,  Breslau,  1846,)  inform  us  that  no  other  person  than  Leibniz 
himself  was  the  writer  of  the  review  in  question,  for  that  in  the  Pauline  Library  at 
Leipsic  there  is  a  copy  of  the  Acts   in  which  Leibniz's  name  is  addod  in  writing  to 
several  of  his  anonymous  articles,  and  to  this   among  others.    Keill,  in  a  paper  on 
central  forces,  {Phibt.  Tram.  Sept.  Oct.  1708,  p.  185,)  took  occasion  to  retort  in  the' 
following  terms.    "  Hsec  omnia  sequuntur  ex  celebratissimii  nunc  dierum  Fluxionum 
Arithmetic^,  quam  sine  omni  dubio  Primus  Invenit  Domitius  Newtonus,  utcuilibet 
ejus  Epistolas  u  Wallisio  editas  legenti,  facile  conetabit,  eadem  tamen  Arithmetica  postea 
mutatis  nomine  et  notatiouis  modo  a  Domino  Leibnitio  in  Actis  Eruditorura  edita  est." 
On  receiving  from  Sloaue,  Secretary  of  the  Royal  Society,  the  Volume  containing  Keill's 
article  (the  Volume  for  1708  and  1709,  published  in  1710),  Leibniz,  who  was  at  Berlin, 
wrote  to  Sloane  (March  4,  1711,  N.  S.)  complaining  of  the  imputation  cast  upon  him, 
and  begging  the  Society  to  interfere.    **  Nempe  aiquum  esse  vos  Ipsi  credo  judicabitis, 
ut  D.  Keillius  testetur  publice,  non  fuisse  sibi  aninium  imputuudi  mihi  quod  verba  in- 
sinuare  videutur,  quasi  ab  alio  hoc  quicquid  est  Inventi  didicerim  et  mihi  attribuerim." 
A  synopsis  of  the  proceedings  of  the  Society  in  relation  to  this  uffuir  is  subjoined.     1711, 
March  22.  President  in  the  chair.    Part  of  Leibniz's  letter  was  read,  and  Sloane  ordered 
to  write  an  answer  to  him.     Newton,  before  the  article  in  the  Acts  was  shewn  to  him, 
was  annoyed  at  what  Keill  had  said,  but  at  the  meeting  on  Apr.  5,  Keill  drew  attention 
to  the  *'  unfair  account"  of  Newton's  tract.     "  Upon  which  the  President  gave  a  short 
account  of  that  matter,  with  the  particular  time  of  his  first  mentioning  or  discovering  his 
invention,  referring  to  some  letters  published  by  Dr  Wallis;  upon  which  Mr  Keill  was 
desired  to  draw  up  an  account  of  the  matter  in  dispute  and  set  it  in  a  just  light."    Apr. 
12,  "The  fonner  minutes  being  read  gave  occasion  to  further  discourse  of  the  matter 
mentioned  in  the  Leipsic  Acts.    The  President  was  pleased  to  mention  his  letters  many 
years  ago  to  Mr  Collins  about  his  method  of  treating  Curves,  &c.,  and  Mr  Keill  being 
present  was  again  desired  to  draw  up  a  paper  to  assert  the  President's  right  in  this 
matter."    May  24.   Keill's  reply  read,  and  a  copy  of  it  ordered  to  be  sent  to  Leibniz,  and 
to  be  printed  in  the  Transactions  on  the  receipt  of  Leibniz's  answer  to  it.    At  the  next 
meetintf,  May  31,  at  which  Newton  was  not  present,  Sloane  read  his  letter  to  Leibniz, 
which  was  approved  of.     1712  Jan,  31.    Leibniz's  answer  (Dec.  29,  1711)  read  and 
delivered  to  Newton.    (See  p.  55).    Febr.  7.    *'  The  President  not  coming  there  was  no 
account  given  of  M.  Leibniz's  letter  to  Dr  Sloane."     March  6.    In  consequence  of 
Leibniz's  letter  a  committee  was  appointed  consisting  of  Arbuthnot,  Hill,  Halley,  Jones, 
Alachin  and  Burnet,  to  inspect  the  letters  and  papers  relating  to  the  dispute,  and  make 
a  report  to  the  Society.    On  INIarch  20,  Francis  Robartes,  March  27,  Bonet  the  Prussian 
ISIinister,  and  on  Apr.  17,  Demoivre,  Aston  and  Brook  Taylor  were  added  to  the  Com- 
mittee.    Apr.  24.  Tlie  Report  of  the  Committee  read.    (See  Commerc.  Epistol.  p.  120, 
p.  241, 2d  ed.    Tumor's  Grantham,  p.  185.    Brewster's  Newton,  p,  207.     Weld's  Royal 
Soc.  I.  410.)    The  Committee  conclude  their  Report  as  follows :  "For  which  reasons  we 
reckon  Mr  Newton  the  first  Inventor,  and  are  of  opinion  that  Mr  Keill,  in  asserting  the 
same,  has  been  noways  injurious  to  Mr  Leibniz.    And  we  submit  to  the  judgment  of  the 
Society,  whether  the  extracts  of  Letters  and  Papers  now  presented,  together  with  what 
is  extant  to  the  same  purpose  in  Dr  Wallis's  3rd  Volume,  may  not  deserve  to  be  made 
public.**    The  Report  was  unanimously  adopted,  and  it  was  "ordered  that  tlie  whole  of 
the  matter  from  the  beginning,  with  the  extracts  of  all  the  letters  relating  thereto,  and 
]Mr  Keill's  and  Mr  Leibniz's  letters,  be  published  with  all  convenient  speed  that  may  be, 
together  with  the  Report  of  the  said  Committee,"    (Joum.  Bk,  Roy.  Soc.)    The  collec- 
tion accordingly  appeared  early  in  1713,  under  the  title  of  "  Commercium  Epistolicum  D. 
Johannis  Collins  et  aliorum  de  Analysi  promota:  jussu  Socletatis  Regia)  in  lucem 
cditum,"    The  printing  of  the  work  was  entrusted  to  Halley,  Jones  and  Machin.    "  1713 
Jan.  8.  Some  copies  of  a  book  entitled  Commercium  Epistolicum,  &c..,. being  brought, 
the  President  orderefl  one  to  be  delivered  to  each  person  of  the  Committee,  appointed  for 
that  purpose,  to  examine  it  before  its  publication."    (Joum.  Bk.)    It  appears  from  the 


NOTES.  Ixxiii 

Minutes  of  the  Council,  that  on  Jan.  29,  it  was  "  ordered  by  ballotingr  that  the  Treasurer 
pay  the  charpes  of  printing  the  Commercium  EpistoUeum'  and  that  on  June  II,  the  sum 
of  £22.  2j.  6</.,  was  ordered  to  be  disbursed  to  Halley,  "  being  money  he  had  paid  for 
printinj?**  it.  Only  a  few  copies  of  tlie  book  were  printed,  and  they  were  principally 
distributed  as  presents  to  Universities  or  distin^ished  scientific  men,  (see  p.  221)  but 
not  entirely  so,  as  b  shewn  by  the  followin^^  extract  from  the  Journal  Book.  **  1714  June 
17.  The  President  in  the  chair.  Dr  Keill  acquainteil  the  Society  that  Mr  Johnson,  Book* 
seller  at  the  Ilague^  desired  a  parcel  of  the  Commercium  EpistoUcum  at  a  certain  price, 
and  that  he  would  return  the  money  upon  the  receipt  of  the  books.  Ordered  that  25 
complete  books  be  delivered  by  Mr  Thomas  to  Dr  Keill  to  be  transmitted  to  Mr  Johnson 
accordinprly,  at  3*.  per  book." 

At  the  meetingr  of  the  Society  on  Apr.  24,  Keill  "  said  he  would  draw  up  an  answer 
to  Mr  Leibniz's  h\st  letter,  it  rdiiting  chiefly  to  himself,  which  he  was  also  desired  to  do, 
and  that  it  should  be  read  at  a  meeting  of  the  Royal  Society."  We  hear  no  more  of  this 
contemplated  answer  of  KeilKs. 

("»)  Sif^ned  by  Robartes,  Wren,  Newton,  &c.  On  this  recommendation  Prince 
Geor^  most  liberally  offered  to  defray  the  expenses  of  the  work.  Flamsteed  instead  of 
feelingr  grateful  for  Newton's  intervention  in  his  In'half,  was  annoye<l  at  the  thought  of 
any  other  opinion  than  his  own  being  taken  on  the  propriety  of  publishing  his  OlMserva- 
tions,  and  when  the  rt-forees  proceeded  in  the  discharge  of  their  trust,  to  take  stepn 
witli  reference  to  the  publication,  he  naturally  enough  wished  to  have  his  own  wny 
in  the  management  of  it,  and  by  his  jKfrverseuens  in  this  resjiect,  gave  them  (to  use 
their  own  language)  **  a  great  deal  of  trouble." 

It  is-not  necessary  to  enter  further  into  this  question  here  :  the  reader  will  find  in  Mr 
Raily's  Account  of  Flamsteed  a  multiplicity  of  details  upon  the  subject,  through  which 
the  clue  just  given  will  guide  him  with  tolerable  safety.  But  I  may  remark  tliat  among 
the  documents  that  are  still  wanting  to  comploteour  knowledge  of  the  circumsUmces  thiit 
attended  the  passage  of  Flamsteed's  work  through  the  press,  there  is  one  which  it  is 
hoi)ed  will  yet  be  discovered — the  paper  of  Articles  actually  agreed  upon  preliminary  to 
the  printing  of  the  book.  And  yet  Mr  Buily  (p.  xlii.  note)  has  ventured  to  assert  in 
contradiction  to  1 1  alley,  that  it  was  not  agreed  that  the  Catalogue  should  be  prefixed  to 
the  first  volume  or  book.  It  Is  true  that  we  have  a  private  memorandum  of  Flamsteed's 
(Baily,  p.  253)  stating  that  he  "signed  the  Articles,  but  covenanted  that  the  Catalogue 
of  the  fixed  stars  mentioned  to  make  a  part  of  the  first  volume  should  not  be  printed,  but 
with  the  last ;"  but  thij  implies  that  the  point  **  covenanted  "  about  did  not  form  one  of 
the  Articles,  and  we  have  no  proof  that  the  "  covenant"  was  accepted  by  the  referees. 
Flamsteed  uses  the  same  phrase  on  a  similar  occasion.     (lb,  p.  86). 

(150)  On  the  11th  of  July  following  Lord  Halifax  gave  to  the  Society  the  2nd  Vol.  of 
the  work. 

(151)  Probably  on  business  connected  with  the  approaching  election.  Parliament 
would  expire  under  the  triennial  Act  the  following  August,  but  that  event  was  antici- 
pated (after  a  prorogation  on  March  14)  by  dissolution  on  April  5.  Flanisteed  in  a 
letter  written  on  the  last-mentioned  day,  which  I  think  there  can  be  no  doubt  was  in- 
tended for  Newton,  though  Mr  Baily  (p.  238)  descril)e8  it  as  "  probably  addressed  to  Mr 
Hodgson,"  says :  "  Good  success  in  your  affairs :  health  and  a  happy  return  is  heartily 
wished  you  by,  Sir,  your  obliged  and  humble  servant."... 

('")  In  the  Senior  Bursar's  Book  for  the  year  1707  in  a  "particular  account  of 
several  Benefactions  received  for  the  use  of  the  Chapel  and  Library,  by  tlic  R**  Mr 
Nicholas  Spencer....  {who  was  Bursar  from  December  1701,  to  June  1705,}  never 
yet  accounted  for  to  the  College  from  the  Sen'  Bursar's  Office  "  we  find,  "  R**  the 
Gift  of  Mr  Isaac  Newton  £60."  I  have  ventured  to  assume  that  this  donation  was 
intended  for  the  Chapel,  as  he  had  already  in  1676  subscribed  liberally  to  the  fund 
for  building  the  Library.  The  date  of  the  subscription  may  probably  be  assigned  to 
his  electioneering  visit  to  Cambridge. 


Ixxiv  SYNOPTICAL  VIEW  OP  NEWTON'S  LIFE. 

(»»•)  The  numbers  were  "  Hon.  A.  Annesley,  (Magd»)  182 
Hon.  D.  Windsor,  (Trin.)  170 
Hon.  Fra.  Godolphin,  (King^s)  162 
Sir  L  Newton.  {Trin,)        117." 

Bentley  voted  for  him. 

In  a  letter  to  A.  Sharp,  Apr.  24,  Flamsteed  writes :  **  Mr  Newton  is  knighted : 
stands  for  parliament  man  at  Cambridge ;  and  is  going*  down  thither,  thb  day  or  to- 
morrow, in  order  to  his  election.  'Tis  something  doubtful  whether  he  will  succeed  or 
no,  by  reason  he  put  in  too  late."  The  Tory  election  cry  was  "  the  Church  in  danger." 
In  the  debate  in  the  House  of  Lorda  on  the  subject  of  this  alleged  danger  the  follow- 
ing December,  Patrick,  Biuhop  of  Ely,  \s  reported  as  moving  that  the  Judges  *•  might 
be  consulted  what  power  the  Queen  had  in  visiting  the  Universities,  cc;  .plaining  of 
the  heat  and  passion  of  the  gentlemen  there,  which  they  inculcated  into  their  pupils ;... 
that  at  the  election  at  Cambridge,  it  was  shameful  to  see  a  hundred  or  more  young  stu- 
dents, encouraged  in  hollowing  like  schoolboys  and  porters,  and  crying,  No  Fanatic, 
No  occasional  Conformity,  against  two  worthy  gentlemen  -that  stood  candidates." 
Cobbetfs  Pari.  Hiit.  vi.  496. 

(^'^)    The  originals  of  this  and  five  other  notes  to  Sloane  are  in  the  British  Museum. 

("»)    Tumor's  Grantham,  p.  169. 

(*'•)    With  alterations  and  additions  (among  others,  seven  new  queries). 

The  translation  was  made  by  Samuel  Clarke,  who  was  rewarded  by  the  author  with  a 
present  of  il500.    A  second  edition  of  it  appeared  in  1719. 

Demoivre  is  stated  to  have  "  revQ  et  conduit  la  traduction  latine  de  I'optique  de 
Newton,  pour  laqucllo  il  n*^'pargua  ni  soins  ni  peines;  aussi  ce  grand  homme  lui 
avoit>il  accordC*  toutc  sa  confidence.  II  alloit  tous  les  soirs  I'attendre  dans  un  cafe  (pro- 
bably Slaughter's  Coffee  House  in  St  Martin's  Lane  }  ou  M.  Moivro  se  rendoit  des  qu'il 
avoit  fini  ses  lemons,  et  d'ou  il  I'emmenoit  chez  lui  pour  y  passer  la  soiree  dans  des  tete-a- 
t^te  philosophique."    ( Eloge,  1754). 

(167)  •*  I  thank  you  for  giving  me  timely  notice  of  the  caveat,  and  think  we  should 
stick  at  no  charge  for  defending  the  legacy.  What  money  shall  be  wanting  for  this 
purpose  I'll  advance  till  the  Council  shall  be  called.  If  you  see  Dr  Harwood  before 
me,  pray  desire  him  to  have  an  eye  upon  this  matter.  I  do  not  know  the  method 
of  proceeding  in  these  cases ;  but  he  can  tell  us.  I  will  take  the  first  opportunity 
to  inform  myself  of  what  is  to  be  done."  (Sloane  MSS.  Brit.  Mus.  4054 ;  printed 
without  the  date  in  Nichols's  Illustrations  of  Lit.  Hist.  xiii.  59).  The  note  in  the 
same  volume,  dated  Thursday  night,  ("  Lady  Betty  Gayer  being  engaged  for  to- 
morrow, and  at  liberty  on  Monday  or  Tuesday,  I  beg  the  favour  we  may  wait  on 
you  on  either  of  those  days  at  tliree  o'clock,  and  that  you  will  let  us  know  which  of 
those  two  days  you  are  most  at  leisure,")  is  recommended  to  the  attention  of  those 
who  are  versed  in  the  "fashionable  arrangements"  of  Anne's  reign. 

(*")  The  trustees  appointed  under  Plume's  will  (Covel,  Bentley,  Whiston,  Fra. 
Thompson  of  Caius)  were  directed  to  frame  statutes  for  the  regulation  of  tlie  Professor- 
ship, "with  the  advice  of  Sir  John  Ellis,  (Master  of  Caius),  Sir  Isaae  Newton  and 
Dr  { sic  \  Flamsteed."  Cotes,  the  first  professor,  was  elected  Oct.  16,  1707.  Flam- 
steed  wrote  to  Whiston  Febr.  13,  1705-6,  (compare  Baily,  p.  258,)  recommending  his 
assistant  Mr  John  Witty  for  the  Professorship.  (Flamsteed's  MSS.  at  Greenwich, 
XXXIII.  65).  In  Vol.  lxix.  of  the  same  Collection,  there  is  a  long  letter,  dated  Dec. 
31,  1706,)  from  Ellis  to  Thomi)son,  on  the  subject  of  the  Professorship,  in  which  Cotes  is 
spoken  of  in  very  high  terms,  and  in  Vol.  xxxin.  p.  74,  there  is  an  answer  to  it,  in 
which  Flamsteed  is  reported  as  saying  that  "Trinity  Gatehouse  is  not  fit  for"  an  ob- 
servatory, (see  p.  200)  "  and  that  that  of  St  John's  is  preferable,  and  that  the  Virtu  tis 
Gotcway  at  Caius  is  better  than  either."  Flamsteed  wished  a  separate  building  to  bo 
devoted  to  tlie  purjjose. 

The  substance  of  a  note  written  by  Prof.  Smith  on  the  fly-leaf  of  his  copy  of 


NOTES.  Ixxv 

I!uyBCM*8  Cosmvtheoros  (Hag,  Com,  1698)  and  dated  1764,  i*  worth  preserving.  ••  I 
have  been  well  informed  that  Dr  Plume,  Archdeacon  of  llochest«r»  was  so  pleased 
with  this  book,  which  the  celebrated  Mr  Flamsteed  had  recommeudcd  to  him,  as  to 
leave  by  his  will  £1800  to  found  the  Pluraian  Professorship  of  Astronomy  and  £zperi- 
mental  Philosophy,  which  1  held  many  years  after  Mr  Coles's  decease." 

(»')    Appendix,  No.  XXXI. 

(»«o)  •'  Instead  of  the  like  sum  he  intended  after  his  death.  It  was  ordered  to  b« 
put  up  by  itself  and  to  be  subject  to  such  end  or  benefaction  as  tite  President  shall 
direct."  This  no  doubt  is  the  foundation  for  Thomas  Ileamc's  scandal, "  ho  promUcd 
to  become  a  benefactor  to  the  Royal  Society,  but  foiled."  See  under  Doc.  14,  of  the 
following  year. 

(ii^)    It  (ills  pp.  4—157  of  the  present  volume. 

('")  Mr  Do  Morgan,  in  his  sketch  of  the  life  of  Newton,  says  that  in  the  2nd  etlitton 
Flamsteed's  name  was  "erased  in  all  the  passages  in  which  it  appeared  (we  have 
verified,  for  this  occasion,  eight  or  nine  places  ourselves)."  The  name  however  will  still 
be  found  in  pages  441,  443,  455,  458,  4G5,  478  and  479 :  the  lost  two  references  occur 
in  some  additional  matter  on  comets,  which  was  put  into  Cotes's  hands  in  October  1712. 
(See  p.  141  of  this  work.)  I  question  very  much  whether  the  suppression  of  Flomsteetrs 
name  in  several  places  where  it  had  appeared  in  tlie  1st  edition  was  not  such  -as  was 
necessary  in  the  process  of  improving'  the  work.  Newton's  own  experiments  on  the  old 
echo  in  Trinity  College  cloister  give  way  in  the  2nd  edition  to  more  accurate  researches, 

("»)  The  original  of  this  paper  is  in  the  Britisli  Museum,  Add.  MSS,  6489.  fol.  67. 
('♦  ex  dono  D°*  Sharp"),  It  is  printed  in  the  Gentleman*t  Mag.  for  Jan.  1755,  pp.3 — 5, 
(Compare  his  Chronology,  p.  71,  sqq.)  In  the  same  MS.  volume  (fol.  69)  is  an  abstract 
of  the  paper  in  Newton's  hand,  (printed  in  the  Appendix  to  this  Work,  No.  XXXIII.), 
which  was  embodied  in  a  letter  to  Bishop  Lloyd  by  an  unknown  writer,  dated  Nov.  7, 
1713,  of  which  the  draught  is  preserved  in  the  volume  referred  to  (fol.  G5, 66),  beginniniif 
"  I  had  the  honor  to  receive  and  the  pleasure  to  read  the  papers  your  Lordship  directed 
to  the  Dean  of  Norwicli  {  Prideaux  }  :  and  before  I  sent  them  forward  I  communicatefl 
them  to  Sir  Isaac  Newton,  according  to  your  Lordship's  order  by  Mr  Archdeacon  : 
when  Sir  Isaac  brought  them  back,  he  told  me  that  he  found  many  excellent  observa- 
tions in  them  about  the  ancient  year,  and  at  the  same  time  acquainted  me  that  ho  had 
formerly  discoursed  with  your  Lord.ship  about  that  year  of  360  days,  and  represented" 
&c.  (See  Appendix,  p.  314).  Trininell,  IVishop  of  Norwich,  may  possibly  have  been 
the  writer  of  this  letter,  as,  three  years  before,  he  was  the  organ  of  communication 
between  Lloyd  and  Prideaux,  conveying  to  the  latter  Lloyd's  scheme  of  Daniel's  70 
weeks.  {PrUeuux's  Life,  p.  237).  It  would  appear  that  Newton's  abstract,  and  not  the 
paper  itself,  was  sent  to  Lloyd,  but  it  docs  not  seem  very  clear  why  the  abstract  was 
drawn  up  at  all. 

("*)    This  and  four  other  letters  to  Keill  are  printed  in  this  volume,  p.  169,  foil. 

('^^)  John  Chamberluyne  was  endeavouring  to  reconcile  the  two  philosophers.  He 
sent  Newton's  letter  to  Leibniz,  who  replied  in  a  letter  dated  Vienna,  Aug.  25,  (Leibn. 
0pp.  III.  491)  part  of  which  was  read  by  Chamberlayne  ut  the  meeting  of  the  Uoyal 
Society  on  Nov.  11.  In  it  Leibniz  "  desires  that  some  letters  and  papers  of  Mr  Oldenburg 
and  Mr  Collins  which  he  supposes  to  be  in  the  custody  of  the  Royal  Society  may  bo 
communicated  to  him  in  order  to  his  publishing  a  Commercium  Kpistolicum  in  defence  of 
himself  ut  his  return  from  Vienna  to  Hanover.  The  Society  was  of  opinion  that  Mr 
Leibniz  ought  either  to  make  good  his  charge  against  Dr  Keill  or  to  ask  pardon  of  the 
Society  for  suspecting  their  judgment  and  integrity  in  the  Commerciura  Kpistolicum 
alreiidy  published  by  their  order  and  approbation.  But  Mr  Chamberlayne  saying  that 
Mr  Leibniz  designed  in  a  sliort  time  to  be  in  England,  the  farther  consideration  of  this 
affair  was  referred  to  some  other  opportunity."  Journ.  Bk.  There  w  in  the  BritisJi 
Museum  (MSS.  Birch,  4284)  a  copy  in  Newton's  hand  of  Leibniz's  letter  of  Aug.  25. 

("«)    The  other  assessors  were  Sir  James  Montagu,  Dr  Cannon,  Prebendary  of  Kly, 


Ixxvi  SYNOPTICAL  VIEW  OP  NEWTON'S  LIFE. 

Dr  Samuel  Clarke,  Dr  Henry  Newton,  Chancellor  of  the  Diocese  of  London,  and  Dr 
Johnson,  Chancellor  of  the  Diocese  of  Ely.  (Colbatch**  MSS.)  The  trial  after  con- 
tinuing about  B\x  weeks,  the  Court  holding  its  sittings  two  evenings  in  the  week,  ended 
on  June  14.    See  Monk's  Bentley,  pp.  281—286. 

(»«»)  Commons*  Journals,  xvii.  677,  716.  I  do  not  consider  M.  Biot's  abstract  of 
the  proceedings  on  this  occasion  (Biog,  Univ,  art.  Newton,  pp.  192, 193)  as  a  model  of 
accurate  condensation  t  I  will  therefore  exhibit  Whiaton^s  statement  as. nearly  as  may  be 
in  his  own  words.  In  1714  Whiston  and  Ditton  commuLicated  to  Newton  their  method 
of  discovering  tlie  longitude  at  sea  by  signals,  aiid  at  his  desire  to  HoIIey,  as  also  to  Sam. 
Clarke  and  Cotes,  and  soon  had  their  approbation- ^o  tar  as  to  encourage  them  to  apply  to 
the  House  of  Commons  for  a  reward  to  such  as  should  discover  the  Longitude.  A  Com- 
mittee was  appointed  to  examine  into  the  matter,  and  the  four  persons  just  mentioned 
were  summoned  to  attend.  "  As  soon  as  the  Committee  was  set,  which  was  a  very  large 
one,  Newton,  Halley,  Clarke  and  Cotes  appeared.  A  chair  was  placed  for  Sir  I.  Newton 
near  the  Chairman  j  ]Mr  Clayton,  M.P.  for  Liverpool } ,  and  I  stood  at  the  back  of  it. 
What  the  rest  had  to  say  they  delivered  by  word  of  mouth,  but  Sir  I.  Newton  delivered 
what  he  had  to  say  in  a  paper  {  referred  to  above  }  .  Upon  the  reading  of  this  paper, 
the  Committee  were  at  a  loss,  as  not  well  understanding  its  contents:  Sir  I.  Newton 
sitting  still  and  saying  nothing  by  way  of  explication.  This  gave  the  chairman  an  op- 
portunity which  it  was  perceived  he  wanted  of  trying  to  drop  the  bill ;  which  he  did  by 
declaring  his  own  opinion  to  be  that  '  Unless  Sir  I.  Newton  would  say  that  the  method 
now  proposed  was  likely  to  be  useful  for  the  discovery  of  the  Longitude,  he  was  against 
making  a  bill  in  general  for  a  reward  for  such  a  discovery* ;  as  Dr  Clarke  had  particularly 
propose<l  to  the  Committee.  Ujwn  this  opinion  of  his,  not  contradicted  by  any  other  of 
the  Committee ;  and  upon  Sir  I.  Newton's  silence  all  the  while,  I  saw  the  whole  design 
was  in  the  utmost  danger  of  miscarrying.  I  thought  it  therefore  absolutely  necessary  to 
speak  myself:  which  I  did  nearly  in  these  words,  *Mr  Chairman,  the  occasion  of  the 
puzzle  you  are  now  in  is  nothing  but  Sir  L  Newton's  caution.  He  knows  the  usefulness 
of  the  present  method  near  the  shores'  [which  are  the  places  of  greatest  donger]. 
Whereupon  Sir  Isaac  stood  up  and  said  that  '  He  thought  this  bill  ought  to  pass, 
because  of  the  present  method's  usefulneKS  near  the  shores.'  Which  decUu-atiou  of 
his  was  much  the  same  with  what  he  had  said  in  his  own  paper,  but  which  was  not 
understood  by  the  Committee,  and  determined  them  unanimoasly  to  ogree  to  such  a 
bill."  Historical  Preface,  date  probably  1742,  inserted  in  some  copies  of  his  "  Longitude 
disco vered,..Lond.  1738,"  p.  v. 

I  will  now  leave  it  to  the  reader,  who  will  of  course  make  the  requisite  allowance  for 
the  forwardness  and  vanity  of  the  reporter,  to  judge  whether  M.  Biot's  term  **  pres<rie 
pu6rile  **  be  a  proper  epithet  to  apply  to  the  part  that  Newton  took  on  the  occasion. 

("«)  "  Redit  nunc  demum  Tibi,  Vir  illustris!  quod  sane,  si  non  omnino  Tuum 
sit,  Ortum  saltern  suum  Tibi  debet ;  nempe  Opusculum  de  Methodo  Fossiliura,  te  as- 
sidue  hortautti,  inceptum,  provectum,  absolutum,"  etc.  Nuturalis  Historia  Tellurisy  &c 
Lond.  1714.  The  letter  is  given  in  English  in  the  same  author*8  "  Fossils  of  all  kimls," 
&c.   Lond.  1728. 

(i«»)  Raplison's  Hist,  of  Fluxions,  pp.  100—103.  Des  Maizeaux*8  Recue'U... Tom.  ii, 
Amsterd.  1720.    Leibn.  0pp.  iii.  451—455. 

("o)  Raphson's //ist.  o/F/iaJoHS,  pp.  HI— 123.  Des  Maizeaux's  Becuei/.  L«ibn. 
0pp.  III.  474—488.  The  French  Translation  of  Newton's  letters  of  Febr.  26  and  May 
18,  as  given  by  Des  Maizeaux,  had  tlie  benefit' of  Newton's  supervision.  His  correc 
tions  of  the  press  (in  his  own  hand)  are  preserved  in  the  British  Museum,  MSS.  Birch, 
4284.  fol.  235. 

(iTi)  '« Mr  Roger  Cotes  Astronomy  Professor  &  Fell,  dyed  upon  a  Relapse  into  a 
Fever  attended  with  a  violent  Diarrhoea  and  constant  Delirium.  He  was  bury'd  on  y« 
gth.  yre  y,n  20  rings  of  20i.  each  &  30  at  IO5.  each."  (Rud's  Diary.)  Cotes  **tout- 
u-la-fois  gdonietre,  astronome  et  physicien"  (see  Dclambre's  Hist.  Astron,  18  siecle. 


NOTES.  Ixxvii 

p.  449,  Mathieu*8  note)  was  born  at  Burbage  in  Leioesterahire,  July  1%  1682.  He  was 
entered  pensioner  at  Trin.  Coll.  Apr.  6,  ItiUQ,  from  St  Paul's  School.  His  name  stands 
in  the  admission  book  immediately  after  that  of  Conyem  Middleton.  They  were  elected 
Bcholare  togrether  in  May,  1701,  took  their  B.A.  degree  in  1703,  and  were  sworn  in  minor 
fellows  of  the  College  on  Octob.  3, 1705.  An  accurate  life  of  Cotes  is  given  in  the  Gent^ 
ral  Dictionaryy  partly  from  materials  supplied  by  his  cousin  Robert  Smith.  See  also 
Knight's  Life  of  CoUty  (Lond.  1724)  who  says  (p.  430)  *'I  could  run  out  many 
pages  in  the  just  character  of  this  extraordinary  man,  being  very  intimate  with  him,  and 
having  the  opportunity  of  knowing  him  perfectly,  by  being  his  chamber-fellow  many 
years  in  Trinity  College  in  Cambridge,  but  am  obliged  not  to  exceed  the  boumla  of  a 
short  account  "...and  IMonk's  Bentley  (p.  314  and  elsewhere).  Bentley's  inscription  on 
his  monument  has  been  frequently  printed.  The  authority  for  the  well>known  saying 
attributed  to  Newton  on  the  premature  death  of  this  promising  mathematician  is  Robert 
Smith,  who  in  his  copy  of  the  Uannonia  Mensurarum,  under  Cotcs's  epitaph,  has  written 
the  words  *'  S'  Isaac  Newton,  speaking  of  Mr  Cotes,  said,  *  If  He  had  lived  we  might  have 
known  something  '.'*  In  his  Optics,  (Vol.  ii.  art.  465,  remarks^  p.  16)  he  gives  the  saying 
in  exactly  the  same  words,  where  in  allusion  to  a  theorem  on  the  image  of  an  object  seen 
through  a  number  of  lenses  he  says:  "That  noble  and  beautiful  theorem...  was  tlie  last 
invention  of  that  grreat  Mathematician  Mr  Cotes,  just  before  his  death  at  the  ago  of  32: 
upon  which  occasion  I  am  told  Sir  Isaac  Newton  said  "...The  autlior  of  Cotes's  Life  in 
the  Biographie  Universelte,  who  has  been  followetl  by  Delambre  (p.  457),  seems  to  have 
misunderstood  this  passage,  taking  Newton's  remark  to  apply  to  the  discovery  of  the 
optical  tlieorem.  Parne,  who  was  six  years  junior  to  Smith,  in  his  Collections  fcr  Hist, 
of  Trill,  Coll.  p.  351,  gives  the  saying  with  the  single  variation  of  "had"  for  "might 
have  :"  "  On  the  death  of  Mr^otes  Sir  Isaac  Newton  b  said  to  have  expressed  himself 
in  these  honourable  and  remarkable  words....** 

(179J  "The  President  in  the  chair.  The  President  grave  the  Society  his  picture 
drawn  by  Mr  Jervase  for  which  he  had  their  thanks."   Joum.  Bk, 

("«)  In  pursuance  of  an  Address  to  the  king  it  was  laid  before  the  House  of  Lords 
on  Jan.  21,  1718. 

In  consequence  of  this  Re])ort  a  Proclamation  was  issued  in  December  1717,  reducing 
guineas  from  2l5.  6d.  to  2I5. 

("*)  This  Report  was  accompanied  with  an  Account  of  the  Gold  and  Silver  coine<l 
from  Jan.  1,  1702,  to  Nov.  20, 1717,  and  with  the  Report  of  Sept.  21,  was  laid  Mom 
the  House  of  Commons  on  Dec.  21,  in  pursuance  of  an  address  to  the  King.  Both  Re- 
ports will  be  found  in  the  Commons'  Journals,  xviii.  664 — 6.  That  of  Sept.  21,  was 
printed  in  the  Daily  Courant^  Dec.  30, 1717,  and  may  also  be  seen  in  The  Political  Statt 
of  Great  Britain,  Tindal'3  Continuation  of  Rapin,  and  Mace.  Corr.  11.  424. 

(176J  With  additions  (among  others,  eight  new  queries,  from  the  17th  to  the  24th.) 
The  Advertisement  is  dated  July  16,  1717. 

(178)  *•  The  House  being  informed  *  That  Sir  Isaac  Newton  attended  at  the  Door,' 
he  was  called  in ;  and  delivered  at  the  Bar  pursuant  to  the  Address  of  this  House  to  his 
Majesty  of  Thursday  last :  *  An  Account  of  the  Silver  Monies  coined  in  the  four  years 
ending  at  Christmas  1699,  by  weight.*  Also  '  An  Account  of  the  Gold  and  Silver 
Monies  coined  yearly  from* Christmas  1699  to  Christmas  1716,  by  wjjight."  And  then 
he  withdrew."  (Lords'  Journals.)  The  last  "  Account*  is  printed  in  Mace,  Corr,  11.  434. 

(1")    p.  185. 

("8)    Mace,  Corr,  11.  430. 

0*")  Nov.  6, 1718.  "  The  Treasurer  acquainted  the  Council  that  Sir  Isaac  Nevr. 
ton  { who  was  present }  had  lately  paid  him  as  a  gift  to  the  Society  £70."  ( Council 
Minutes,) 

(180)    «•  1719  July  13,  to  a  free  gift  rec«*.  from  Sir  I.  Newton  £52  lOi. 

1720  Apr.  28,  to  a  gift  rec''.  of  Sir  I.  Newton  £52  lOi.**  Pound's  Account  Book, 
quoted  by  Rigaud  (Bradley,  p.  iii.)  These  instances  of  Newton*ft  liberality  were  pro- 
bably  in  acknowlcdgincut  of  i^tronomicul  observations  supplied  by  Pound  (ex,  gr. 


Ixxviii         SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

th«  magnitude  of  Jupiter's  diamet^»  Princip»  ed.  3.  p.  416).  Found  was  Dradley't 
uncle, 

("0)*    It  is  written  in  a  most  peace-loving  spirit    See  p.  187,  note  t. 

(">)  A  fourth  edition  was  published  in  1730  from  a  copy,  it  is  suid,  of  the  third  cor- 
rected by  the  autlior's  own  hand,  and  left  before  his  death  with  the  bookseller. 

(>")    p.  188. 

("')  Made  jointly  with  E.  Southwell  and  J.  Scroope,  Esqrs.  Printed  in  Hibernian 
Patriot,  **  being  a  Collection  of  the  Drapier's  Letters,"  ficc.  1730,  p.  244.  Comp.  Scott's 
Swift,  VI.  392,  ed.  2. 

(iM)  Xewton  was  then  occupied  with  the  3rd  edition  of  the  Principia.  Delisle  tells 
us  that  Newton  assured  him  that  "si  M.  H alley  avoit  eu  6gard  { in  constructing  his 
Lunar  Tables }  aux  moindres  Equations  dont  il  a  fait  mention  dans  sa  Theorie,  et  qu*il 
eut  ajout6  une  minute  etdemie  a  la  longitude  de  la  Lune  pour  son  ucc616ration  physique 
dans  notre  teraps,  il  n*auroit  trouv6  aucune  difference  sensible  entre  sea  observations  et 
le  calcul."    Journal  det  Savans,  June  1750,  p.  428. 

(  "»)    Appendix,  No.  XXXIV. 

(}**)  Mace  Corr,  n.  435.  Newton  wanted  the  calculations  for  the  3rd  ed.  of  the 
Principia.  If  Ualley  re-examined  the  two  calculations,  the  examiuation  led  to  no  new 
result,  and  if  he  performed  the  calculation  for  the  place  iu  the  parabolic  orbit,  no  use  was 
made  of  it  in  the  3rd  ed.  as  had  been  intended. 

(18T)  Gentleinan'i  Mag.  lix.  775,  (with  three  other  letters  to  the  same  person).  It 
begins  "  A  bad  state  of  health  makes  me  averse  from  minding  business." 

(iM)  Tumor's  Grantham,  p.  172.  Brewster's  Newton,  p.  363.  "Just  after  he  was 
come  out  of  a  fit  of  the  gout... ;  he  was  better  after  it  and  his  head  clearer  and  his  me- 
mory stronger  than  I  had  known  them  for  some  time." 

(189)     Tumor's  Grantham,  p.  158. 

(»M)  J»/ii/.  TraM*.  for  1725,  p.  315.  Brewster's  Netutou,  p.  262.  The  summary  is 
entitled  **  A  Short  Chronicle  from  tlie  first  memory  of  things  in  Europe  to  the  conquest 
of  Persia  by  Alexander  the  Great,"  and  was  afterwards  published  in  his  Chronology.  It 
was  drawn  up  in  a  few  days  at  the  request  of  the  Princess  of  Wales.  Conti,  at  her  desire* 
was  allowed  to  have  a  copji  of  it,  from  which  when  he  went  to  France  other  transcripts 
were  made. 

Newton's  CAronoIo^i/,  (Lond.  1728)  appeared  towards  the  end  of  1727.  Conduitt's 
Advertisement  stales  that  it  "  was  writ  by  the  author  many  years  since ;  yet  he  lately 
revised  it,  and  was  actually  preparing  it  for  the  press  at  the  time  of  hisdeath."  Martin 
Folkes  writing  to  Morgan,  Masterof  Clare  Hall,  Jan.  6,  1727-8,  says:  '*  I  am  glad  you 
have  been  so  well  entertained  with  Sir  Isaac's  book,  and  at  the  same  time  to  find  my  own 
opinion  of  it  so  entirely  confirmed.. .but  indeed  I  have  had  that  satisfaction  from  several 
hands,  and  I  even  hear  your  Neighbour  of  the  great  College  {  Bentley  }  who  spoke 
very  slightingly  of  the  performance  before  it  appeared  begins  not  to  talk  so  magisterially 
as  he  did  before,  but  \V.  W.  {  VVhiston  }  continues  in  the  same  way,  and  declares  he 
shall  overturn  it  so  easily  that  he  shall  not  be  able  to  extend  the  whole  confutation  to  a 
sheet  of  paper." 

(»»i)  "  Pendant  les  deux  mois  que  I'abb^  Alari  passa  a  Londres  {  1725  }  ,  il  visita 
I'universit^  de  Cambridge,  et  le  grand  Newton,  qui  joulssait  alors  dans  la  capitale  de 
I'Angleterre,  de  I'estirae  g^n^rale  de  I'Europe  et  de  cinquante  mille  livres  de  rente,  en 
qualit6  d'intendant  des  monnaies.  L'abb6  6tant  all6  chez  lui  a  neuf  heures  du  matin, 
I'Anglais  debuta  par  lui  apprendre  qu'il  avait  quatre-vingt-trois  ana.  On  voyait  daiis  sa 
chambre  le  portrait  du  Lord  Halifax,  son  protecteur  et  celui  de  l'abb6  Varignon  dont  il 
estimait  les  ouvrages  de  g6om6trie.  Varignon  et  le  ptre  S^bastien  carme,  sont,  dit-il, 
eeux  qui  ont  le  tnieux  enlendu  mon  tyittme  sur  les  couleurs.  La  conversation  tomba 
ensuite  sur  I'histoire  ancienne,  dont  Newton  s'occupait  alors.  L'abb6,  qui  6tait  plein 
de  la  lecture  des  auteurs  grecs  et  latins,  I'ayant  satisfait,  il  le  pria  a  diner.  Le  repas 
fut  detestable ;  Newton  6tait  avare,  et  il  ne  fit  boire  ii  son  convive  que  des  vinsde  Palme 
ou  de  Madere,  qu'il  rccevait  en  pr^sens.    Apres  le  diner,  il  mena  I'abb^  a  la  Societe 


NOTES.  Ixxix 

royale  de  Londres,  dont  il  ^tait  president,  et  le  fit  aMeotr  4  la  droits.  La  t^aiiM  eora* 
mcnya  et  Newton  s'endonnit.  A  la  fm  dc  la  seance,  tout  le  nionde  •iffna  le  reRiatre,  et 
I'nbbe  comino  lus  aulrea.  Newton  le  ramena  ensuile  chcz  lui,  oil  il  le  garda  jusqu'^  neuf 
hcurcB  du  soir."  (  Eisai  H'utorique  tur  liolingbroktt^  compiled  by  General  Grinioard)  in 
LeUre$Historique$...de...Boliugbroke,  i.  155, 1'aria.  1808). 

Alari  was  born  in  1089  ;  he  was  a  friond,  at  leant  for  soroe  time,  of  Dolinf^broke'i) 
and  instructor  of  Louis  XV.  The  intelligent  roadur  will  make  allowance  for  the  wpico 
added  to  give  pungtjncy  to  the  story.  The  following  is  the  simple  record  in  the  Journal 
Book  of  Alari 's  visit,  '  Mr  Mildmay  had  leave  to  be  prcsent|  as  also  Mr  Petre  Joseph 
Alary,  a  French  Gentleman." 

C*^)  Life  of  Maclaurin,  prefixed  to  bis  Account  of  Sir  Isaae  Neu'(on*i  PhUosophieal 
Discoveries. 

( »")  Printed  in  the  Phil.  Trans,  for  1725,  pp.  315—321.  Comp.  Brewster's  New. 
ton,  pp.  261— 2G5.  The  MS.  written  in  a  fine  copper-plato  hand  is  preserved  in  the 
Archives  of  the  Royal  Society,  and  is  endorsed  "  read  about  the  latter  end  of  1725,** 
In  this  paper  he  incidentally  informs  us  that  when  he  lived  at  Cambridge  he  used  some- 
times to  refresh  himself  with  History  and  Chronology  for  a  while,  when  he  was  weary 
with  other  studies. 

(»*)  The  Preface  is  dated  Jan.  12,  1725-6.  Twelve  copies  are  stated  to  have  been 
printed  on  large  paper,  (Rigaud's  Bradley^  p.  xi.),  of  which  there  is  one  in  Trinity 
College  Library,  another  in  that  of  Queens'  College,  (a  presentation  copy  from  the 
author  to  his  friend  J.  F.  Fauquier,)  and  a  third  in  the  Library  of  the  Royal  Society,  of 
which  we  find  the  following  naive  notice  in  the  Journal  Book.  "  March  31,  1726.  Mr 
Folkes  in  the  name  uf  the  President  gave  the  Society  a  Book  richly  bound  in  morocco 
leather  aa  a  present  for  the  Library,  entitled  Philosophiie  Naturalis  Principia  Matht' 
miiticof  printed  at  London  1726.  The  Society  ordered  thanks  to  be  rendered  to  the 
President  for  this  invaluable  present."  It  is  to  be  hoped  that  the  correspondence  which 
passed  between  Newton  and  his  editor  (Henry  Pemberton,  M.D.)  during  the  progress 
of  the  work  through  the  press  will  yet  be  discovered.  See  Rigaud's  Essay,  p.  107. 
PMlos.  Mag.  May  1836,  p.  441. 

We  may  give  here  the  anecdote  quoted  by  Mr  De  Morgan  firom  Maty*s  Memoirs  of 
Demoivre  {Phil.  Trans,  1846,  p.  109.)  *'  Comme  tout  ce  qui  regarde  les  grands  hommes 
peut  £tre  interessant,  on  sera  peut-etre  bien  aise  de  savoir  que  Newton  a  souvent  dit  h 
Mr.  de  Moivre  que  s'il  avoit  ^te  moins  vieux  il  auroit  ^t6  tent6  de  revoir  sur  les  dernierea 
observations  sa  th^orie  de  la  Lune,  ou  comme  il  s*exprimoit  de  Vattaquer  de  tiouveau  {to 
have  another  pull  at  the  moon).    Je  tiens  ceci  de  Mr.  de  Moivre  lui-rafime." 

("*)    Baily,  Memoirs  of  Astron.  Soe,  viii.  188. 

(iM)  u  JMarch  23.  The  chair  being  vacant  by  the  death  of  Sir  Isaac  Newton  there 
was  no  meeting  this  day."  {Journal  Bk.)  For  the  reflections  which  his  death  suggested 
tosome  minds,  see  Boyer's  Po/itica/  State  of  Great  J3ritain  (or  Mzitch  1727,  (Vol.  xxxiii« 
pp.  327 — 330),  In  Mist's  Weekly  Journal  for  IMarch  25,  the  obituary  opens  with  "  Sir 
Isaac  Newton,  the  greatest  Mathematician  that  the  World  ever  knew."  Thomson's 
"  Poem  siicred  to  the  IMemory  of  Sir  Isaac  Newton,"  (dedicated  to  Walpole)  seems  to 
have  had  a  large  circulation.     I  have  a  copy  before  me  of  the  5th  edition,  dated  1727. 

I  wish  that  I  had  been  able  to  contribute  more  local  information  respecting  Sir  Isaac 
Newton  than  it  has  been  my  fortune  to  meet  with.  But  the  age  of  "conversations 
with"  and  "reminiscences  of  had  not  yet  arrived,  and  we  do  not  know  that  any 
fellow  of  his  College  kept  a  diary.  Thomas  Parne,  who  took  his  B.A.  degree  in  1718, 
collected  materials  for  the  history  of  Trinity  College,  and  had  opportunities  of  conversing 
with  men  who  had  been  contemporaries  of  Newton  (for  example,  George  Modd  who 
was  two  years  junior  to  Newton,  and  lived  in  College  until  his  death  in  1722).  He  has 
given  us  many  particulars  of  more  or  less  interest  relating  to  Ray,  Thorndike,  Pearson, 
Barrow,  Duport  and  other  members  of  the  College,  but  the  only  allusions  to  its  chief 
pride  and  boast  that  I  have  found  in  his  MSS.  are  the  following':  under  the  head  of 


Ixxx  SYNOPTICAL  VIEW  OP  NEWTON'S  LIFE. 

**  Writers**  the  name  *'  Newton"  stands  first  in  the  list;  the  dates  of  his  return  as 
M.P.  for  the  University  and  of  his  unsuccessful  contest,  (in  the  latter  of  which  the 
majority  agrainst  him  is  erroneously  stated)  are  given,  and  an  anecdote  is  preserved 
of  his  absence  of  mind  in  these  terms :  "  Newton  hath  come  into  the  Hall  without  his 
Dand,  and  went  towards  St.  Maries  in  his  surplice ;"  for  which  Tame  quotes  as  his 
authority  a  "  Mr  Burwell,"  (perhaps  Alexander  Burrell,  eleven  years  senior  to  Parne, 
who  may  have  been  a  connexion  of  the  Alexander  Burrell  who  took  his  B.  A.  degree  in 
1670,  and  was  chaplain  of  the  College  from  Oct.  1673  to  June  1681.)  I  do  not  know 
that  I  can  find  a  more  appropriate  place  for  a  similar  anecdote  which  has  already 

o 

appeared  in  English.  It  was  told  to  the  Swedish  Professor  Bjornstahl  at  Basle  by  John 
Bernoulli,  son  of  the  famous  John,  on  Nov.  5,  1773 :  "  sagte  uns,  Newton  sey  eben- 
falls  sehr  zerstreut  gewesen,  und  babe  einmahl  den  Fing-er  eineti  Frauenzimmers  genom- 
men,  urn  seine  Tabakspfeife  nachzustopfen."  {Brief e  auf...lUk.en.  Leip«.  und  Ilostock. 

1777—1783.  v.  46).  On  Dec.  8,  in  the  following  year  Bjornitahl  paid  a  visit  at  Am- 
sterdam to  the  "gelehrten  llerrn  Fontein,"  an  Anabaptist  preacher  and  scholar  of 
HemsterhuLs  and  Albert  Schultens.  "  Ira  Jahr  1738  hat  er  eine  lleise  nach  Kiigluiid 
gemacht  und  mit  dem  grosseu  Bentley  Bekauntschaft  unterhalteu.  Zu  Cambridge  hat 
er  verschiedne  Auekdolen  von  Newton,  welcher  bervihmte  Mann  ncun  oder  zehn  Jalir 
vorher  gestorben  war,  gehtirt,  unter  andern :  Newton  habe  geglaubt,  dass  Mahomed  von 
Gott  gesandt  worden  sey,  um  die  Arabcr  von  der  Finsterniss  zuriick,  und  zum  Glauben 
an  einen  Gott  zu  fiihren  u.  s.  w.  (Dies  haben  ihm  wenigstens  die  Professoren  oder 
Fellows  zu  Cambridge  als  eine  besondre  Merkwlirdigkeit  aus  Newtons  Geschichte 
erziihlt ;)  die  im  Koran  und  Mahomeds  Leben  vorkommendcn  Fabelu  und  Wunder 
jcdochhabe  dieser  aufgeklarte  Mann  nicht  geglaubt.  £r  sagte  mir,  Newton  habe  eine 
Abhandlung  herausgegeben,  um  zu  beweisen,  die  Stelle  1  Joliann.  v.  7.  sey  nicht 
iicht,  und  der  Text  habe  ohne  diesea  Vers  einen  weit  bessern  Zusammenhang.**  ( lb, 
462). 

The  Professor  was  in  England  from  April  1775  to  March  1776.  Writing  from  Oxford 
Oct.  24, 1775,  after  saying  that  he  passes  over  many  remarkable  objects,  such  as  the 
Marmora  Oxoniensia,  Cromwell's  scull,  Guy  Faux's  lantern,  Blenheim,  Stowe,  6cc,  he 
proceeds :  "  Dagegen  aber  will  ich  einen  Umstand  mclden,  der,  wie  ich  mit  Ueberzeugung 
weiss,  bisher  in  keinem  Buche  vorkommt :  diesen,  dass  wir  unter  andern  in  der  hiesigen 
Nachbarschaft  ausdrlicklich  zudcm  Ende  eine  Reise  gethan  haben,  um  die  eigne  Blicher- 
samralung  des  grossen  und  unsterblichen  Ritters  Newton  zu  sehen.  Jetzt  besitz  sie  Herr 
Doctor  Musgrave...  Rector  zu  Chinnor,  achtzehn...Meilen  von  Oxford.  Sie  hat  ihm  un- 
gefehr  vier  hundert  Pfund  sterling  gekostet.  Hier  findet  man alle  Ausgaben  von  Newton's 
Werken,  und,  welches  das  merkwUrdigste  ist,  am  Rande  mit  seinen  eigenhiindigen 
Anmerkungen  angefUllt,  und  bisweilen  mehrere  Blatter  am  Schlusse  der  BUcber  von 
ilim  gauz  vollgeschrieben.  Ich  zweifle  nicht,  dass  ein  Newtonianer  hier  nicht  viel 
Vergniigen  und  manche  ErlUuterung  antreffen  wUrde.  Hier  sah  ich  auch  di\s  seltne 
Buch  von  Herr  Jones  Vater,  woven  ich  oben  angemerkt  habe,  das  der  Sohn  selbst  es 
nichteinmahl  besitze.  Der  Titel  ist:  Epitome  of  the  Art  of  Practical  Navigation... Lon- 
don, 1706.  Noch  ein  sehr  seltnes  Buch  von  eben  diesem  Jones:  (dies  ist  ganz  ausser- 
ordentlich  rar:)  Synopsis  Palmariorum  Matheseos... London  1706...Uebngen8  sieht 
man,  dass  Newton  eine  vortrefllche  Bibliothek  gehabt  hat.  Alle  griechirichen  und  latcin- 
ischen  classischen  SchriftstoUer  finden  sich  daselbst.  Sonst  habe  ich  verschiedne 
eigenhiindige  Briefe  von  Newton  an  Flamsteed  gesehen,  die  in  der  Corpus-Christi-Bibli- 
othek  zu  Oxford  aufbewahrt  werdon.  Zu  Cambridge  werden  noch  mehr  Ilandschrifteu 
von  ihm  angetroflTen."    (in.  288.) 

I  have  no  means  of  confirming  or  impugning  the  accuracy  of  the  account  given  by 
the  simple-hearted  Swede  of  the  disposal  of  Newton's  Library.  A  statement  of  its 
magnitude  will  be  found  at  the  end  of  the  subjoined  extract  from  Maude's  Wemley- 
dale  (p.  106.) 


NOTES.  Ixxxi 

Newton'g  nephew,  Benjamin  Smith,  ••  left  a  small  ivory  bust  \  of  hw  uncle  ]  of  ad- 
mirable workmanship  by  that  celebrated  artist,  Marchand,  which  from  its  elef^nce, 
similitude  and  placid  expression  is  truly  valuable.  It  is  said  to  have  cost  Sir  Isaac 
100  guineas  and  is  specified  in  an  authentic  inventory  of  his  effects,  taken  by  virtuo  of 
a  commission  of  appraisement  in  April  1727,  now  in  my  possession.  It  appears  that 
his  personal  estate  amounted  to  jC31,82I  16s.  \0d.  which  was  distributed  among  eight 
relations,  Sir  Isaac  dying-  intestate:...  as  a  proof  of  his  benevolence. ..at  his  death  there 
was  owing  him  by  one  tenant  £60  for  3  years  rent,  and  by  another  for  2  )  years  a  smaller 
sum. ...  {  Ilis  }  wardrobe  and  cellar. ..in  the  valuation  stand  thns.  Item,  wearing 
apparel,  woollen  and  linen,  one  silver  hiltcd  sword,  and  two  canes,  £8.  3j.  Item,  in 
the  wine  vault,  a  parcel  of  wine  and  cider  in  bottles,  JC14. 16>.  6d.  The  furniture  and 
luxuries  of  his  house  bearing  nearly  the  like  proportion,  his  library  excepted,  which 
consisted  of  2000  volumes  and  100  weight  of  pamphlets." 

It  does  not  fall  within  the  scope  of  our  Chronological  Synopsis  of  Newton's  life 
to  notice  the  great  political  events  of  his  time,  and  I  am  therefore  compelled  to  place 
here  an  extract  from  an  in^renious  French  writer  which  lui^^ht  otherwise  have  been  given 
under  a  more  convenient  head.  1  leave  it  to  future  inquirers  to  ascertain  the  precise 
embarrassment  alluded  to  in  it,  and  to  determine  the  probable  extent  to  which  we  aro 
indebted  for  the  story  to  tlie  play  of  a  lively  imagination. 

'*  Pour  faire  voir  que  runiversaliti*  des  talents  est  une  chimSrc,  je  ne  teux  pas 
chercher  mcs  autorit^-s  dans  la  classe  commune  des  esprits ;  roontons  jusqu'i^  la  sphere 
de  ces  g^nies  rares  qui,  en  faisant  honneur  h  I'humanit^,  humilient  les  hommes  par 
la  comparaison.  Newton,  qui  a  devin6  le  systeme  de  Tunivers,  du  moins  pour  quelque 
temps,  n'etoit  pas  regarde  comme  capable  de  tout  par  ceuz  mdmes  qui  s'honoroient 
de  I'avoir  pour  compatriote. 

Guillaume  III,  qui  se  connoissoit  en  hommes,  £toit  embarrass^  sur  une  affaire 
politique;  on  lui  conscilla  de  consulter  Newton;  Newton,  dit-il,  n'est  qu'un  grand 
philosophe.  Ce  titre  (;toit  sans  doute  un  ^loge  rare ;  mais  enfin,  dans  cctte  occasion- 
Ihf  Newton  n'etoit  pas  ce  qu'il  falloit,  il  en  {'toit  incapable,  et  n'6toit  qu'un  grand 
philosophe.  II  est  vraisemblable,  mais  non  pas  d^'montr6,  que,  s'il  eflt  appliqu6  h  la 
science  du  gouvernement  Ics  travaux  qu'il  avoit  consacr^s  k  la connaissance  de  I'univers, 
le  roi  Guillaume  n'edt  pas  d6daign6  ses  conseils. 

Dans  combien  do  circonstances,  sur  combien  de  questions  le  philosophe  u'eOt-i'l 
pas  rd'pondu  h  ceux  qui  lui  auroient  conseill6  de  consulter  le  monarque :  Guillaume 
n'est  qu'un  politique,  qu'un  grand  roil" 

(Duclos's  Considerations  sur  Us  Moeurs,  (Euvres,  1. 160.  Paris,  1820.) 


Ixxxii 


SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 


DIVIDENDS  RECEIVED   BY   NEWTON,   AND    NUMBER   OP 

WEEKS  HE  RESIDED  EACU  YEAR  WHEN  FELLOW 

OF  TRINITY  COLLEGK 


Tear  emllng 

Weeks  rcaiJed 

Year  ending 

Weeks  rcBldod 

Mlchaelimw. 

Hli  DlWdend. 

by  him. 

Blicliaclnuu. 

nia  Dividend. 

by  him. 

1668* 

£15 

AUin  long 
^IvacuUon.) 

1685 

£25 

51 

9 

25 

52 

6 

12.10 

52 

70 

20 

m 

7 

12.  10 

45 

1 

16.  IS*.  4i. 

48 

8 

Nil. 

45 

2 

16.  IS*.  4d. 

48  J 

Of 

Nil. 

19 

3 

Nil. 

49i 

90 

Nil. 

294 

4 

25 

51 

1 

12.  10 

441 

5 

25 

46 

2 

12.  10 

49 

6 

25 

50X 

3 

25 

49i 

7 

12.  10*. 

43| 

4 

Nil. 

49 

8 

25 

49 

5t 

34 

50 

9 

25 

38 

6 

34 

274 

80 

25 

36^ 

7 

Nil. 

0 

1 

25 

49 

8 

34 

4 

2 

12.  10 

46J 

9 

34 

0 

S 

12.  10 

46 

1700 

37 

0 

4 

25 

52 

1§ 
2 

40 

0 
2 

The  dividend  was  voted  at  the  annual  audit  in  December,  and  paid 
by  the  Bursar  "  as  money  came  to  his  hands,"  generally  at  the  end  of  6 
or  12  months,  but  sometimes  the  payment  was  still  further  delayed. 

Newton's  own  receipts  for  his  dividend,  livery  and  stipend  for  the 
four  years  of  Humfrey  Babington's  Bursarship  are  to  be  found  in 
Babington's  Day-Book,  which  is  one  of  two  or  three  that  are  still  pre- 


•  Steward's  bill  unpaid  19j.  7  Jd. 

As  an  illustration  of  the  scrupulous  exactness  and  regrularity  which  characterised 
Newton  in  all  matters  of  business,  it  may  be  mentioned,  that  in  two  instances  only  was  he 
in  arrear  with  h's  Steward's  bill,  viz.  the  one  before  us  when  he  had  just  become  Master 
of  Arts,  and  pnibably  did  not  know  the  proper  mode  of  paying  the  bill  until  after  the 
accounts  for  the  year  were  made  up,  the  other  when  he  was  absent  in  London  as  a 
member  of  the  ('onvention  Parliament. 

t  Steward's  bill  unpaid  i;&12«. 

X  The  augrmented  dividend  of  this  and  subsequent  years  is  in  consequence  of  New- 
ton's increased  standing  in  the  College. 

$  lie  must  liave  resigned  hb  fellowship  before  Dec.  21, 1701,  otherwise  the  Bursar's 
Book  would  have  contained  a  record  of  his  receiving  dividend  for  the  quarter  ending 
then.  At  the  time  of  his  resignation  he  stood  10th  on  the  list:  had  he  remained  fellow 
until  August  of  the  year  next  but  one  following,  he  would  have  been  elected  a  senior. 


DITIDENDS  RECEIVED  BY  NEWTON.  Ixxxiii 

eerved  in  the  Muniment  Room.     Tho  first  of  these  rocoipta  wo  give  at 
full  length :  the  others  are  added  for  tho  sake  of  tlie  dates : — 

Oct  11,  1675.     Rec''  then  my  wages  as  fellow  for  tho  \  £a  -to  a 

whole  year  ending  Mich,  last ; 

My  livery  for  the  same  year 1   IS  4 

Pandoxator  voted  1G73. 5    0  0 

i  dividend  voted  last  audit  1674 12  10  0 


In  all £21   16    8 


By  me,  Isaac  Neavton. 

Nov.  20,  1675    R"  tho  later  moiety  of  ^Ir  Newton's  div....  £l2  10  0 

By  me,  John  Battely. 

July  8,  1676      Wages  for  f  year  ending  Midsummer £2     0  0 

Livery  for  1676 1    13  4 

Pandoxator's  div.  granted  at  audit  1674....  5  0  0 
idiv last  audit  1675....     12  10  0 

21     3  4 


Nov.  16,  1676    Rec**  the  later  i  of  div.  granted  1675 £l2  10  0 

"Wages  for  quarter  ending  IMich*.  last 0  13  4 

Dec.  13,  1677     Rec**  wages  for  year  ending  Mich' £2  IS  4 

Livery 1   13  4 

Pandox.  Div.  Audit  1675 5    0  0 

Do 1676 5    0  0 

Firstjof  div.  Audit  1676 12  10  0 

26  16  8 

Nov.  22,  1678    Wages  as  fellow  for  year  1678 £2  13  4 

Livery 1   IS  4 

*•       -         Later  J  of  div.  granted  Audit  1676. 12  10  0 

Pandox.  div 1677* 5    00 

Later  ^  of  Mr  Wickins's  div.   granted 


jr  ^  01  j»ir   wicKinss  aiv.   grantea  \ 

Audit  1676 /    ^^  ^^ 


34     6  8 

Dec.30,  1678     Rec^div.granted  Audit  1677 ^12  10  0 

Also  Mr  Wickins's 12  10  0 

Besides  the  dividend  Newton  was  in  receipt  of  the  following  emoluments 
from  the  College : 

1.  Pandoxator's  dividend  (from  the  profits  of  tho  bakehouse  and 
brewhouse)  £2  10*.  for  year  ending  Mich'.  1668,  and  £5 
annually  afterwards  except  when  ho  did  not  reside  tho  major 
part  of  the  year  as  in  1689,  1697,  &c. 


Ixxxiv        SYNOPTICAL  VIEW  OF  NEWTON'S  LIFE. 

2.    3*.  4d.  weekly  during  residence  "  pro  pane  et  potu."   (This  sum 
represents  10  penny  loaves,  10  quarts  of  sraall  beer  at  Irf.  a 
quart  and  10  quarts  of  ale  at  2d,  a  quart.) 
a    13*.  4d.  for  livery  for  year  ending  Mich'.  1668  and  £1  13*.  id, 
annually  afterwards  until  Mich*.  1701.    £2  3s,  4d.  for  stipend 
or  wages  for  year  ending  Mich*.  1668,  and  I3^f.  id,  a  quarter 
until  the  quarter  ending  Dec.  21,  I7OI. 
If  to  these  sources  of  income  it  be  added  that,  as  fellow,  he  had  no* 
thing  to  pay  for  his  dinners  or  room-rent,  that  his  hereditary  estate 
brought  him  in  £80  and  his  professorship  £100  a  year,  every  reader  can 
form  his  own  opinion  on  the  condition  of  Newton's  worldly  circum- 
stances while  he  was  a  resident  fellow  of  Trinity  College. 


NEWTON'S  EXITS  AND  REDIT8, 


Ixxxv 


NEWTON'S  EXITS  AND  REDIT8. 
[From  tlio  book  in  which  the  Fellows  entered  their  names  on  going 
out  of,  or  returning  to,  College.  Tlio  entries  are  generally  in  Newton's  own 
hand,  but  sometiraea  in  that  of  North,  tho  Master,  or  of  Lynnot  when 
Vice-Master,  and  occasionally  they  seem  to  have  been  written  by  a 
servant.  There  is  a  4to.  book  in  tho  Muniment  Room  containing  tho 
Exits  and  Redits  of  the  Bachelor  Fellows  and  Scholars,  commencing 
with  Ootob.  1667.  The  first  six  names  in  it  are  those  of  tho  six  follows 
of  Newton's  year  senior  to  him :  tho  second  leaf  of  tho  book,  at  the  top 
of  which  Newton's  nam©  stood,  with  the  dates  of  his  Exits  and  Redits 
from  Octob.  1667  until  Midsummer  of  tho  following  year,  has  been  cut 
out,  tho  lower  portion  of  tho  D  belonging  to  the  "  Ds"  prefixed  to  his 
name  being  the  only  part  of  tho  entry  relating  to  him  that  is  left.] 


Year. 

Exit 

Rcdlt. 

Tear. 

Exit 

RtMltt. 

1668 

Sept.  29 

1682 

Feb.  21 

Feb.  28 

1669 

Nov.  26 

Dec.  8* 

Apr.  8 

Apr.  29  J 

1671 

Apr.  17 

May  11 

May  10 

1672 

Jun.  18 

Jul.  19 

1683 

]\Iarch  27 

Mays 

1673 

March  10 

Apr.  1 

May  21 

1674 

Aug.  28 

Sept.  5 

1685 

]\Iarch  27 

Apr.  11 

1675 

Feb.  9 

March  I9 

Jun.  11 

Jun.  20 

Oct.  14 

Oct.  23 

1687 

March  25 

1676 

]\Iay  27 

Jun.  1 

1688 

March  SO 

Apr.  25 

1677 

Feb.  20 

March  3 

Jun.  22 

Jul.  17 

March  26 

1690 

Feb.  4 

Apr.  26 

May  22  + 

March  10 

Apr.  12 

Jun.  8 

Jun.  22 

Jul.  2 

1678 

May  6 

May  27 

1691 

Sept.  12 

Sept.  19 

1679 

May  15 

May  24 

Dec.  31 II 

Jul.  19 

1692 

Jan.  21 

Jul.  28 

Nov.  27 1 

169311 

May  SO 

Jun.  8. 

168O 

March  11 

1695 

Sept.  10 

Apr.  28 

May  29 

Sept.  14 

Sept.  28 

1681 

March  15 

March  26 

1696 

March  23 

May  23 

Apr.  20 

*  Newton  was  makings  this  entry  under  tlie  Exits  and  had  written  more  than  half  the 
first  letter  of  his  name  when  he  found  out  his  mistake. 

+  Newton  has  also  entered  DrLynnet's  Redit,  who  returned  to  College  the  same  day. 

$  Two  entries,  one  in  Newton's  hand,  the  otlier  by  North.  Newton  had  been 
down  in  Lincolnshire,  and  a  friend  of  hw  availed  himself  of  his  return  to  Cambridge  to 
employ  him  on  a  small  commission,  which  it  will  be  seen  he  lost  no  time  in  executing. 
•*  Nov.  28,  1679.  Retf*  £11.  15s.  7d.  by  the  hand  of  Mr  Isaac  Newton  from  Mr  W. 
Walker,  Rector  of  Grantham  School."  Dr  Babington's  Day-liook.  Walker  received 
the  money  from  Mr  Edw.  Pawlet  and  he  from  James  Thompson  who  owed  it  to  Babington. 

§  Newton  had  made  nearly  the  whole  of  this  entry  under  the  year  1677,  where  on 
turning  over  the  leaves  of  the  book  ample  room  offered  itself,  but  when  about  to  write  the 
"  8  "  he  discovered  the  mistake. 

II  He  was  in  London  in  Jan.  1692.  On  the  9th  of  that  mouth  we  find  Pepys  inviting 
Evelyn  to  his  usual  Saturday  evening  party  to  meet  Dr  Gale  and  Mr  Newton,  (3/«mojri, 
V.  181.2nded.) 

^  Newton  had  entered  an  Exit  for  Apr.  15  ofthis  year,  but  it  was  afterwards  crossed  out. 


Ixxxvi  NEWTON'S  WEEKLY  BUTTERY  BILLS. 


NEWTON'S  WEEKLY  BUTTERY  BILLS, 

From  October  1686,  to  February  1694,  and  from  June  1698  to 
March  1702,/ar  Breads  Beert  i^c. 

These  relics  of  Newton's  household  expenditure  are  extracted  from 
two  mutilated  Buttery  Books  in  Trinity  College  Muniment  Room. 
The  Fellows'  Buttery  Books  for  the  remainder  of  the  period  of  his 
residence  and  all  those  of  the  Scholars  during  the  time  when  he  was 
undergraduate  and  bachelor,  have,  I  fear,  been  destroyed  by  some  per- 
son or  persons,  who,  it  is  to  be  hoped,  could  not  be  supposed  to  know 
that  books  apparently  so  useless  were  indispensable  for  a  correct  history 
of  the  discovery  of  the  new  calculus  and  of  the  true  theory  of  the 
world. 

Explanation  op  Abbreviations. 

no  CO  means  not  in  commonst  i.  e.  not  in  residence,  or  out  of  college. 

di  or  dimi  means  half  the  week, 

0.  P.  or  Com.  P.  means  the  fine  for  not  delivering  a  Common  Pktce 
in  chapel  after  morning  prayers.  This  Fine  continued  to  be  levied  until 
1830,  when  the  system  of  compounding  was  introduced. 

M  is  supposed  to  stand  for  manf  i.  e.  servant.* 

Ton.  stands  for  tonsor  (the  College  barber),  ch.  probably  for  chapel, 
Lett,  for  letters. 


*  Newton  mentions  his  servant  twice  in  his  correspondence  with  Flamsteed.  (Baily, 
pp.  139,  157).  "  As  for  the  places  calculated  from  the  tables,  I  will  give  you  no  trouble 
about  them :  my  servant  has  lately  learnt  arithmetic,  and,  if  I  go  on  with  this  business  of 
the  moon,  he  shall  learn  astronomical  calculations  and  examine  them,  and  I  will  send 
you  his  corrections."  (Letter  of  Nov.  17,  1694).  *'I  want  not  your  calculations,  but 
your  observations  only.  For  besides  myself  and  my  servant,  S'  Collins  { of  Catharine 
Hall }  (whom  1  can  employ  for  a  little  money,  which  I  value  not)  tells  me  that  he  can 
calculate  an  eclipse,  and  work  truly."  (Letter  of  June  29, 1695).  This  may  have  been 
the  John  Perkins  "  Astrologus  Cantabrigiensis,"  to  whom  Vincent  Bourne  addressed 
a  copy  of  elegiacs,  beginning 

Lusit,  amabiliter  lusit  Fortuna  jocosa, 

Et  tunc,  siquando,  tunc  oculata  fuit; 
Cum  tibi,  Joannes,  Newtoni  sternere  lectum; 

Cum  tibi  museum  verrere  diva  dedit. 
And  ending  Nee  melior  lex  est,  ncc  convenientior  aequo, 

Quam  siet  astronomo  scrvus  ut  astrologus. 


NEWTON'S  WEEKLY  BUTTERY  BILLS. 


Ixxxvii 


•luUug 

9. 

rf. 

1686 

Oct.  15 

2 

5i 

22 

7 

2i 

29 

2 

9i 

Nov 

.  5 

3 

H 

12 

3 

oi 

19 

3 

loi 

26 

2 

9 

Dec 

.  3 

5 

0 

10 

10 

Oi 

17 

3 

2 

24 

2 

Hi 

31 

2 

9i 

1687 

Jan. 

7 

2 

11 

14 

7 

0 

21 

10 

6 

28 

3 

11 

Feb 

4 

3 

5i 

11 

2 

5i 

18 

3 

H 

25 

3 

Oi 

March 

4 

4 

9 

11 

14 

5J 

18 

5 

8i 

25 

3 

loi 

no  CO. 

Apr. 

1 

0 

8 

no  CO. 

8 

... 

... 

no  CO. 

15 

... 

... 

di.  no  CO. 

22 

2 

*i 

no  CO. 

29 

• . . 

... 

no  CO. 

May 

6 

... 

... 

no  CO. 

13 

... 

... 

di.  no  CO. 

20 

1 

10 

27 

2 

7 

Juno 

3 

5 

2i 

10 

10 

0 

17 

3 

7i 

24 

3 

lOi 

July 

1 

4 

5i 

8 

2 

H 

15 

2 

7i 

22 

2 

11 

euillug  *•      **• 

1687  July   29  2     9i 
Aup^.    5  3     Oi 

12     3     2i 

19  3     1 

26  4     24 
Sept.    2  3     3i 

9  16  lOj 

16  13     6J 
23     3     5i 

30  4     4I 

Oct.    7  2  lOi 

14  4     3 

21  3     5J 

28  3     4i 

Nov.    4  5     7i 

11  6     14 

18  3     0 

25  2     5i 

Dec.    2  3  lOj 

9  3  lOf 

16  2  11 

23  3     Oj 
30  3     1 

1688  Jan.    6  3     5J 

13  3     5J 

20  16     1 

27  3     Oi 
Feb.    3  3  11 

10  3     5J 

17  3     Oi 

24  15     7i 
March    2  8     Oi 

9  10    7 

16  3     1 

23  3     4i 

6.8.  C.  P.     30  19     2 

no  CO.  Apr.    6  1     2i 

no  CO.  13  2     0 

20  

di.  no  CO.      27  0    9i 

May    4  3     1 


1688 


no  CO. 
no  CO. 
no  CO. 
di.  no 


1689 


no  CO. 
no  CO. 
no  CO. 
no  CO. 


May    11  3     5J 

18  4     2i 

25  8     2 
June    14    6 

8  9     7i 

15  12    7 

22  2  lOi 
8*.m.  29  0  11 

July   6     

13     

CO.      20  1     1 

27  3     2 
Aug.    3  4     3 

10  2  10} 

17  3     8 
24  10     5} 
31  3     2 

Sept.    7  18     8 

14  10  7i 
21  5     5J 

28  4     0 
Oct.    5  6     Ij 

12  4  Hi 

19  2     8 

26  3     2 
Nov.    2  3     1 

9  3     1 

16  2  lOi 

23  4  lOi 
SO  4j^.6 

Dec    7  2     8i 

14  2  lOi 
21  3  3i 
28  6     8 

Jan.    479 

11  4     li 

18  8     5 
25     

Feb.    1     

8     

15     

/2 


Ixxxviii         NEWTON'S  WEEKLY  BUTTERY  BILLS. 


WMk 

•odlug 

1689 

no  CO.  Feb.  22 

no  CO.  March  1 

no  CO.  Ton,  .  ^ 
8 

15 


10  CO.  Ton.  \ 
7#.  Ch.  3*.j 
no.  CO. 
no.  CO.  22 

no.  CO.  29 

no  CO.  Apr.    5 
no  CO  12 

no  CO.  19 

no  CO.  26 

no  CO.  May    3 
no  CO.  10 

no  CO.  17 

no  CO,  24 

no  CO.  31 

"^^^-  I  June  7 
Ton.j 

no  CO.  14 

no  CO.  21 

no  CO.  28 

no  CO.  Jnly    5 

no  CO.  12 

no  CO.  19 

no  CO.  26 

no  CO.  Aug.    2 

no  CO.  9 

no  CO.  l6 

no.  CO.  23 

no.  CO.  30 

no  CO.  Sept.    6 

13 

20 

27 

Oct.    4 

11 

di.no  CO.      18 

no  CO.  25 

no  CO.  Nov.    1 

no  CO.  8 


I    6 
10    0 


1     6 
5     0 


Week 
MuUng 

1689 

no  CO.  Nov.  15 
no  CO.  22 

no  CO.  29 

no  CO.  Dec.    6 
no  CO.  13 

no  CO.  20 

no  CO.  27 

1690 
no  cou^  Jan.    3 


*.  rf. 


1     6 


10 
17 
24 
31 


Feb. 


6  8 

1  6 

2  1 

2  lOi 

3  6i 

3  1 

2  9h 

5  Sl 


no  CO. 
no.  CO. 
no  CO. 
no  CO. 
dimi.  1 
no  CO. ! 

14 

21 

28 

March    7 

dimi.  no  co.  14 

no  CO.  21 

no  CO.  28 

no  CO,  Apr.    4 

no  CO.  1 1 

dimi.  18 

25 

May    2 

9 

16 

dimi  no  co.  23 

30 

June    6 

13 

20 

27 

dimi. 

no  CO  . 

11 
18 
25 


1     9h 

3  114 
6  2J 
5     sl 

4  1 
4     7i 


Wttk 

•udlng 

1690  Aug.    1 
8 

15 

22 

29 

Sept.    5 

Ton.  10*.      12 

no  CO.  19 

no  CO.  26 

Oct.    3 

10 

17 

24 

31 


68.  8d.     \ 

Com.  P.  f 

Nov 


7 
14 
21 
28 
Dec.  5 
12 

19 

26 


3  5i 

3  2 

2  7J 

3  2i 
3  5i 
2  llj 
6  8 

IS  3 

10  lOj 

8  6i 

2  si 


\july   4    2     3i 


19    6| 
15    6^ 

7     5i 


1691 

dimi.  Jan.      2 

dimi.  no  co.  9 
16 
23 
30 
Feb.  6 
13 
20 
27 

Ton.  March  6 
13 
20 
27 
Apr.  3 
10 
17 
24 


*.  d. 
6    5i 
11  11 
3  3i 
3  3J 

3  lOi 
5  0 

14  2i 

2  114 

4  li 
4  lOi 
4  34 
4  2I 

9  2i 

4  2i 

3  04 

4  114 
3  lol 

16  2I 

3  11 

3  8 

4  5i 

1  104 
4  04 

4  4 

5  24 
4  6 

4  10 

5  0 

3  9 
8  14 

13  3 

4  2 
3  44 


7 

74 

74 
54 

14 


NEWTON'S  WEEKLY  BUTTERY  BILLS. 


Ixxxix 


WMk 

•odlug 

s.   d. 

Wr«k 

•adiug 

s.  d. 

WMk 

•ndlnc 

*.  d. 

1691  May  1 

5     0 

1692  Feb.  5 

7  Oi 

1692  Nov.  18 

4  0 

8 

5     0 

12 

4  0 

25 

5  Si 

15 

4  6i 

19 

5  0 

Dec  2 

6  2 

22 

5  li 

no  CO.     26 

1  11 

9 

S  7i 

29 

6  Oi 

March  4 

7  6 

16 

7  9 

To"-ljune  5 

108.   / 

13  6i 

11 

3  8 

23 

5  Hi 

18 

5  4 

SO 

7  2i 

12 

3  lli 

25 

4  6i 

1693 

Jan.  6 

4  10 

19 

S  9i 

Apr.  1 

4  3i 

13 

8  4i 

26 

4  Si 

8 

10  7 

20 

6  1 

July  3 

3  7i 

15 

4  10 

27 

13  6i 

10 

7  0 

22 

4  0 

Feb.  3 

6  0 

17 

7  1 

29 

5  9 

10 

6  1 

no  CO.     24 

0  6i 

May  6 

4  8i 

17 

11  6i 

no  CO.     31 

13 

8  Oi 

24 

7  0 

no  CO.  Aug.  7 

20 

3  10 

March  3 

8  7i 

dimi.no  co.  14 

r"6i 

27 

3  6 

10 

5  6i 

21 

4  0 

Juno  3 

6  2 

17 

7  5i 

28 

3  0 

10 

5  li 

24 

6  Oi 

Sept.  4 

4  2i 

17 

5  1 

31 

4  11 

11 

5  7i 

24 

3  7 

Apr.  7 

5  6i 

no  CO.     18 

1  4 

July  1 

4  6 

14 

4  6 

25 

3  5i 

8 

6  6 

21 

8  0 

Oct.  2 

4  5 

15 

4  8i 

28 

7  li 

9 

3  3i 

22 

5  li 

May  5 

8  3 

16 

4  8i 

29 

4  6 

12 

4  8i 

23 

^    9k 

Aug.  5 

5  0 

19 

5  0 

30 

4  4i 

12 

5  Oi 

26 

7  2i 

Nov.  6 

S  9i 

19 

4  4i 

no  CO 

I  June  2 

7  0 

13 

3  Oi 

26 

5  0 

dimi. 

20 

2  74 

Sept.  2 

3  9i 

no  CO 

.  dimi.  9 

2  4 

27 

3  101 

9 

5  8i 

16 

5  9i 

Dec.  4 

6  7 

16 

4  0 

23 

5  5i 

11 

7  2i 

23 

4  8i 

no  CO 

.  dimi.  30 

3  Si 

18 
25 

5  0 
3  9 

30 
Oct.  7 

5  6 
3  10 

no  CO 
dimi. 

•  1  July  7 

3  7i 

1692  Jan.  1 

3  9 

14 

6  2i 

14 

5  3 

no  CO.      8 

1  Oi 

21 

7  0 

21 

7  0 

no  CO.     15 

28 

5  8i 

28 

5  Oi 

22 

Nov.  4 

5  6 

Aug.  4 

11  2 

29 

4  0 

11 

5  0 

11 

4  5i 

xo 


NEWTON'S  WEEKLY  BUTTERY  BILLS. 


WMk 

Mdlug 

s. 

rf. 

Wwk         -        J 
ending       '•      ". 

JX     *.    d. 

1693  Aug.  18 

5 

3i 

1694   Jan.  26    5    7i 

1699 

25 

4 

10 

Feb.    266 

no  CO.  Sept.  8     16 

Sept    1 

6 

0 

962 

no  CO.  Dec.     1     1     6 

8 

6 

3 

1700 

noco.dim{.  15 

4 

0 

1698  [In  this  and  fol- 

no CO.   Mar.  116 

no  CO.  M.     22 

0 

6i 

lowing  years  wo  have 

no  CO.  May  31     16 

noco.  dimi.  29 

2 

Hi 

copied     only     those 

no  CO.  Sept.  6)1  6 
CP.                 J6    8 

Oct.    6 

5 

0 

dates  where  a  charge 

13 

5 

4 

is  put    opposite    his 

noco.  Nov. 29     1     6 

20 

5 

3i 

name.  The  sums  con- 

1701 

27 

4 

5i 

sist     principally     of 

noco.  Mar.    1     1     6 

Nov.    3 

4 

3i 

quarterly  payments.] 

no  CO.  May  SO     1     6 

10 

5 

111 

no  CO.  June    3     1     6 

noco.  Sept.  12     1     6 

17 

8 

H 

no  CO.  July  22     

Nov.  21     6    4 

24, 

6 

0 

no  CO.  dimi.  29     3     6 

Nov.  28     £1  0  10 

Dec.    1 

7 

H 

no  CO.  Aug.    5     

no  CO.  Dec.  5)1  6 
Lett.                 J  0     5i 

8 

3 

7 

no  CO.  Sept.    9     1     6 

15 

5 

7i 

no  CO.  Dec.     2     1     6 

noco.C.  P.  26    6    8 

22 

() 

2 

1^99 

1702 

29 

9 

0 

no  CO.  Feb.  24     1     6 

no  CO.  Mar.  6     1     6 

1694    Jan.    5 

5 

0 

no  CO.  June   3     16 

May  8,  name  disappears 

12 

5 

5] 

--}june23  6     8 

from  list  of  fellows. 

19 

5 

6 

TABLE  OF  NEWTON'S    LECTURES 

AS    LUCASIAN   PROFESSOR. 


NEWTON'S  LECTURES  ON  OPTICS 
(MS,  Univ.  Libr.  Dd.  9.  67.) 

[The  numbers  on  the  right  designate  the  pages  in  tho  MS.,  those  on  the 
left  tho  pages  in  tho  work  as  printed  Lend.  1729.] 


Jan.  I6J. 

1—  IS 

Optica?  pars  1* — varia. 

Lect.    1(1—6) 

13—  25 

Ex  codem— exi^ipt. 

Lect    2  (  6— 12) 

2f>—  34 

Jam  liquet — determinentur. 

Lect.    3(12—17) 

35—  41 

Sectio  2***— reflexes. 

Lect.    4(17—21) 

42—  52 

Cum  eandem — attolluntur. 

Lect.    5(21—28) 

5^—  62 

Probleniatis— de  aliis. 

Lect.    6(28—33) 

62—  73 

Ad  eundcm — videar. 

Lect.    7(33—39) 

74—  85 

Sectio  3**" — proxime. 

Lect.  8(39—44) 
Octob.  1670. 

85—  95 

Prop.  12 — ajqualos. 

Lect.    9  (45—49) 

95—105 

TiCmma  5—~ixxv.  q  .  e.  d. 

Lect.  10  (49—54) 

105—116 

Prop.  17 — sufficiant. 

Lect.  11  (54—60) 

116—125 

Do  radiorum— G^A"//. 

Lect.  12(60—64) 

126—136 

Sectio  4*" — possunt. 

Lect.  13  (64—69) 

137—146 

Prop.  32— dcfinitur. 

Lect.  14(69—75) 

146—152 

Prop.  SQ — censeam. 

Lect.  15(74—77) 

[145]-153 

Opticas  pars  2*^ — disceptaturus. 

Lect.    1(1—5) 

153—164 

Prop.  1 — nequeant. 

Lect.    2(  5— 11) 

164—171 

Prop.  2 — censeam. 

Lect.  3(11—17) 
Octob.  1671. 

171—181 

Prop.  3 — commisceantur  sibi. 

Lect    4(17—23) 

182—189 

Adhaec — judicaveris 

Lect.    5  (23—29) 

189—197 

Verum — manifestum  est. 

Lect    6  (29—34) 

197—207 

Quinetiam — Prisma. 

Lect    7(34—41) 

207—215 

Ad  hiec — cogantur. 

Lect    8  (41—46) 

215—226 

Prop.  5 — subjicient. 

Lect    9(46—54) 

227—239 

Sect.  2^*— emergcntis. 

Lect  10  (54—63) 

239—247 

Antequam — Hceat. 

Lect  11  (63—68) 

248—260 

2  De  Plh-enomenis — posaint. 

Lect  12  (69—77) 

261—269 

3  De  Phajnomonis^licere. 

Lect  13  (78—84) 

XCll 


TABLE  OF  NEWTON'S  LECTURES. 


Octob.  1672. 
269 — 277    4  De  Phronomenis— patebunt.  Lect  14  (84 — 90) 

278 — ^285     Notiasimum — inforioris.  Lect  15  (90 — 96) 

285—291     Superest— decrevi.  Lect.  I6  (96— 101) 

The  MS.  does  not  seem  to  bo  in  Newton's  hand,  except  some  cor- 
rections hero  and  there,  almost  all  the  marginal  notes,  the  diagrams  and 
between  2  and  3  pages  at  the  end.  It  was  put  into  the  hands  of  the 
Vice-Chancellor  and  delivered  by  him  to  Robert  Peachy  to  be  placed 
in  the  University  Library,  Octob.  21,  l674. 


1 — ^9    Computatio  vel  fit — in  eadem  ratione. 


11—15    Do  Additione — '-QOa^  Jaa-xx. 

3 
5* 


1 5— '1 7    De  Subductione ^^3  4- 


LECTURES  ON  ARITHMETIC  AND  ALGEBRA. 

(MS.  Univ.  Libr.  Dd.  9.  68.) 

[The  numbers  on  the  left  refer  to  the  pages  in  the  edition  published 
by  Whiston,  Cantab.  1707.] 

Octob.  1673. 

Lect.  1(1—5) 

Lect.  2(5—8) 

Lect.  3  (  8—10) 

Lect.  4  (10—12) 

Lect.  5  (13—15) 
Lect.  6  (15—18) 
Lect.  7  (18—21) 

Octob.  1674. 
Lect.  1  (21—22) 
Lect.  2  (22—24) 
Lect.  3  (25—27) 

Lect.  4  (27—28) 

Lect.  5  (28—29) 
Loot.  6  (30—32) 
Lect.  7  (32—33) 
Lect.  8  {33^35) 
Lect.  9  (35—37) 
Lect.lO  (37—38) 


18—21  De  Multiplicatione— . 

22 — 25  De  Divisione — homogeneas. 

25 — 30  Quod  si  quantitas — sufiicit. 

31 — 34  De  extractione  Radicum — 279- 

34—37  Extractionem  radicis— observandum  est. 

37—40  E  simplicibus — radicibus. 

41,42,51,52*     De  Reductione ^hc. 

fit 

53 — 55  Quod  si  divisor —  — - . 

55 — 57  De  reductione  Radicalium— et  sic  in  aliis. 

62 — 66  De  forma  iEquationis —  =  x*. 

66 — 68  Reg.  4— docero. 

69 — 72  De  duabus — linquo. 

72 — 74  Exterminatio — x  df=  0. 

74—76  Reg.  3 — asymmetria. 


•  The  part  De  Inventione  Divisorum— totam  quantitatem,  pp.  42—51,  is  taken  from 
the  end  of  the  MS. 


TABLE  OP  NEWTON'S  LECTURES. 


xciu 


76 —  79  Quomodo  Qurestio—detorminandte. 

79—81  Ciutcrum  ut  liujusinodi— <r«ll. 

81—  85  Prob.  5--sivo  24. 

85^  89  Prob.  7—2  solidis. 

89'-'  91  Prob.  10—36  bobua. 

91—94  Prob.  12— dantur. 

94—  96  Prob.  16 — solutioncs. 

97 — 104  Quomodo  Quwstiones— conetur. 

104 — 109  Sed  ut  hujusmodi —  +  2a6(7. 

109 — 1 1 3  Caeterum  ut  patcat — scquentibus. 


1 13 — 1 16    Quod  ad  Geomotriam — ^tradere. 
116—119    Prob. 


1. «x. 

a 


1 1 9—122  Prob.  4— &  BA  C, 

122—124  Prob.  7—JiT+Tv, 

124 — 126  Prob.  9— secabit. 

127 — 130  Prob.  11 — conditiones. 

130 — 133  Prob.  13 — elucescet 

133 — 136  Prob.  14 — quinquisoctionem,  &o. 

136—139  Prob.  16— rarius. 

139 — 142  Prob.  18 — parallclogrammum. 

142 — 145  Prob.  20— erit  Ellipsis. 

145 — 151  Prob.  22— quantitatcm  y. 

151—152  Prob.  24— ad  BE. 

152—156  Prob.  26— qucTsitum  />. 

156 — 159  Prob.  29 — proportionales. 

159—161  Prob.  31— satisfaciet. 

162—164  Prob.  33— =  0. 

165—168  Prob.  35— aequales. 

16*8 — 171  Idem  brevius — altcrius  ad  A, 

1 7 1  —1 72  Prob.  37— rectam  FE, 

172—174  Prob.  38— longitudinem  DC. 

174—176  Prob.  39— Q.  E.  F. 

176 — 179  Prob.  40 — manifestura  est. 

1 79—1 82  Prob.  41 — oportuit. 

1 82—1 86  Prob.  42—  =  VZ. 

1 86 — 1 89  Analyseos — invcnicnda. 

189— 192  J^rob.  43— j-j-^. 


Octob.  1675. 
Lect.  I  (39—41) 
Loct  2  (41—42) 
Lect.  3  (42—44) 
Loct.  4  (44 — 47) 
Lect.  5  (47 — 48) 
Lect.  6  (48—50) 
Loct.  7  (51—52) 
Lect.  8  (52—58) 
Lect.  9  (58—62) 
Loct.  10  (62—65) 

Octob.  1676. 
Lect.  1  (65—67) 

Lect.  2  (67—69) 

Loct.  3  (69—70) 
Lect.  4(70—71) 
Lect.  5  (71—72) 
Lect.  6  (72—74) 
Lect.  7  (74—76) 
Lect.  8  (76—78) 
Lect.  9  (78—79) 
Lcct.lO(80— 81) 

Octob.  1677. 
Lect.  1  (81—83) 
Lect.  2  (83—85) 
Lect.  3  (85—86) 
Lect.  4  (86—89) 
Lect.  5  (89—91) 
Lect.  6  (91—93) 
Lect.  7  (93—94) 
Lect  8  (94—96) 
Lect.  9  (96^99) 
Lect.  10  (99—100) 

Octob.  1678. 
Lect.  1  (100—101) 
Lect  2  (101—102) 
Lect.  3  (102—104) 
Lect.  4  (105—106) 
Lect.  5  (106—109) 
Lect.  6(109— 111) 

Octob.  1679. 

Lect.  1  (111—113) 


XCIV 


TABLE  OP  NEWTON'S  LECTURES. 


192—195 
195—197 
197—199 
199—201 
201—202 

203—204 
205—207 
207—209 
209—211 
211—213 
213— Sl5 
215—218 
218—221 


221—224 

224—227 
227—230 
231—234 
234—240 
240—243 
243—247 
247—250 

250—253 
253—257 


257— 

45— 

47—50 

50— 
259—262 
263—266 
266—272 
272—277 


Prob.  44— cognosci  potest. 
Prob.  45 — super  basem. 
Prob.  47 — ^latera  trianguli. 
Prob.  48 — ^triangiilum  quaesitum. 
Prob.  49 aabb^O. 

Prob.  51 — ilia  quresita. 
Prob.  52 — trajecit. 
Prob.  53—^.  E.  I. 
Prob.  54 — quaesitae. 
Prob.  55 — iequalis  est 
Prob.  56— et  EO. 
Hoc  modo— Conica  Sectio. 
Quod  si  quatuor — potest. 


Lect.  2  (114—115) 
Loot.  3  (116—117) 
Lect.  4(117— 118) 
Lect.  5(118— 119) 
Lect.  6(119— 120) 

Octob.  168O. 
Lect.  1  (120—121) 
Lect.  2  (121—124) 
Lect.  3  (124—125) 
Lect.  4  (125—127) 
Lect.  5  (127—128) 
Lect.  6  (128—130) 
Lect.  7  (130—132) 
Lect.  8  (132—134) 

Octob.  1681. 

Lect.  1  (134—137) 

Lect.  2  (137—140) 
Lect.  3  (140— 143) 
Lect.  4  (143—147) 
Lect.  5  (147—153) 
Lect.  6(153—157) 
Lect.  7  (157—161) 
■limitibus  cequationum. 

Lect  8  (161—164) 
Ex  -^uationum— remotissimam  -  5.  Lect.  9  (l64 — 167) 
Si  inter  summam—coDsistunt.  Lect.lO(l67 — 172) 


Prob.  58 21^^'^  ^^' 

Prob.  59 — invenire. 
Prob.  60— ^ar. 
Prob.  61— oportebit. 
Quomodo  iEquationes — sit  par. 
Sunt  tamen  radices — impossibiles  dua?, 
Ubi  termini  duo — detegi  possunt. 
Eadem  operation 


ITorum  vero — investigandoa  sunt. 

Si  jiuUus  occurrit a:  +  7  =  0. 

Si  nuUus  inveniri 66^  =0. 

Si  quantitatis — kx-\-l  (p.  258). 
Exempli  gratia—  J  a  a  +  cc. 
Hactenus — proponebatur. 
Si  requatio— quadratico. 
Adjungere — potuorunt. 


♦Octob.  1682. 
Lect  1  (172—176) 
Lect  2(176—180) 
Lect  3  (180—185) 
Lect  4(185—188) 
Lect  5  (188— 192) 
Lect  6  (192—196) 
Lect  7(197—203) 
Lect.  8  (203—207) 


•  The  matter  in  Lectures  1 — 4  is  given  in  a  modified  form  at  the  end  of  the  MS.,  with 
a  direction  that  it  should  be  inserted  at  an  earlier  part  of  the  vohime.  For  tluse  four  lec- 
tures, therefore,  it  is  impossible  to  give  exact  references  to  the  pages  of  the  printed  book. 


TABLE  OP  NEWTON'S  LECTURES. 


xcv 


277,58--6l 
279—284 


284—289 
289—293 
293—297 
297—299 
299—302 
303—307 
307—310 
311—317 
317—321 
321—326 


Ilactcnus  8?quationum — 1 1  +  ^125.    Loct.  9*  (207 — 210) 
Ilactenus  cequationum  proprictatca— adhibctur. 

liCct.  10  (211— 215) 
Octob.  I683t 
Lcct.  1  (216—219) 
Lect.  2  (219—222) 
Lcct.  3  (223—226) 
Lect.  4(226—228) 
Lect.  5(228—231) 
Lcct.  6(231—235) 
Lect.  7(235—237) 
Lect.  8  (238—243) 
Lect    9  (244—247) 


Si  cui — immoror. 
Proponatur  jam— conStruenda. 
Solvuntur  etiam — dcmonstrantur. 
Construonda  jam — demonstrandi, 
Proponatur  jam — compinguntur. 
Ilto  8unt — Prop.  XXXII. 
Quod  si — nimius  sum. 
Ilactenus — habeatur  —  r. 
Demonstratur— et  BC. 
Scholium — proportionalium  «.  . 


Lect.  10  (247—251) 


At  the  end  of  the  Volume  are  corrections  and  additions  by  Newton, 
and  " De Inventione  divisorura — nihil  relinquit"  (pp.42 — 51  of  printed 
book.) 


LECTURES  DE  MOTU  CORPORUM 
{MS.  Univ,  Libr.  Dd.  9.  46.) 

[The  numbers  on  the  left  denote  the  pages  in  the  1st  ed.  of  tbo 
Principia :  those  on  the  right  the  leaves  in  the  MS.] 

The  title  is  "  De  motu  corporum  Liber  primua."     It  forms  the 
draught  of  tlie  Ist  book  of  the  Principia,  see  p.  20^,  note. 

Octob.  1684. 
1 — 11   Dcfinitiones...Tractatuw  pcquentem  composui. 

Lect.    1(1—9) 
12 — 20  Axiomata  sive  Leges  Motus...et  motus  oorum  inter  so. 

Lcct.    2  (10—16) 
20 — 29  Schol.  Hactenus  principia  tradidi... 

...in  ultima  rationc  partis  ad  partem.     Lect.    3  (16—20) 


•  In  the  corrections  at  the  end  of  tlie  MS.  part  of  this  Lecture  is  ordftred  to  be  trans- 
ferred  to  an  earlier  place  in  the  Volume,  and  accordingly  it  appears  in  pp.  68 — 61  of  the 
printed  book. 

t  The  MS.  in  Lambeth  Library,  No.  592,  (quoted  by  Rigaud,  Etsay,  p.  97,  note) 
entitled  "Trigonometriae  Fundameuta  a  Viro  01.  Iswico  Newtou,  Mathesoos  Profes- 
sore,  anao  1683  data/'  contains  merely  rules  for  the  solution  of  plane  and  sphoricjU 
triangles  given  to  flenry  Wharton  probably  at  one  of  those  private  lensons  mentioned 
in  p.  xlv.  It  consists  of  two  folio  leaves  (i.e.  of  two  pages  and  seven  lines  on  the  last 
page,  the  second  being  blank),  forming  part  of  a  volume  entitled  '^Scripta  Acaderaica 
&CC.  annos  inter  1682  et  1686,  a  me  facta  "  &c.  in  Wharton's  handwriting. 


xcvi  TABLE  OP  NEWTON'S  LECTURES. 

S9"~  S6    Lemma  v.  Similium  figuraram... diminuendos  sine  limito. 

Lect.    4(20—25) 
37—  49    Artio.  11.  De  Inventione  Virium  Centripetanim... 

...in  ccntrifugam  versa.     Lect.    5  (25 — 31) 
50—  56    Artie,  m.  Do  motu  corporum  in  Conicis  Soctionibus  cxecn- 
trici8...majoribu8  axibus  Ellipsodn.      Lect.    6(31—36) 
56—  63     Prop.  XVI.  Tlicorcina  viii.  lisdem  positis... 

...nimis  obvias  non  adjungo*.    Loot.    7  (36'^39) 
67—  Lemma  xvt.  A  datia  tribu8...pergo  jam  oxponero. 

Lect.    8(39-42) 
115—  Prop.  XXI.  Prob.  XIII.  Posi to  quod  vis.... Lect.    9(42—    ) 

breaks  off  at  fol.  44,  in  Prop.  xxiv.  with  the  words  "gyrando,  arcum 
Kk."  Frinctp.}).  118. 

Octob.  1685. 
79—  88     Prop.  xxii.  Prob.  xiv....Problema  impossibile  est. 

Lect.    1  (58—63) 
89 —  98     Prop.  XXVI.  Prob.  xviii....8irailem  et  a?qualem.  q.  e.  p. 

Lect.    2(63—68) 
98 — 107    Lemma  xxvn....irrationalem  ut  sequitur. 

Lect.    3(68—73) 
107 — 114    Prop.  XXXI.  Prob.  xxin....pergo  jam  exponere. 

Lect.    4(73—76) 
115 — 125    Artie,  vii.  Do  corporum  a8censu...ad  area,m  DLme. 

Lect.    5(76—83) 
(whole  of  7th  section.) 
125 — 131     Artie,  vin.  continens  Inventionem  orbium  in  quibus... 

...adjiciamus  pauca.     Lect.    6(83 — 86) 
(whole  of  8  th  section.) 
132 — IS**     Artie,  vni.  {error  for  ix.}... oblique  ascenderc. 

Lect.    7(86—89) 


*  **  Demonstrationes  hujus  et  pnecedentis  ut  nimis  obvias  non  adjungo."  In  the 
Prtficipia  the  demonstrations  of  these  two  propositions  (17th  and  18th  MS.,  18th  and 
I9th  Frxnc'x-p.)  are  given  complete. 

t  This  is  Lemma  xvi.  of  the  ^rxncxif,  p.  67.  The  Prop,  which  follows  it  in  the  MS. 
B  Prop.  XIX.  Prob.  xi.,  being  Prop.  xxi.  Prob.  xiii.  of  the  Vr'xncip.  pp.  68,  69.  The 
reference  to  Lahire  is  not  in  the  MS.  having  probably  been  suggested  by  Halley. 
(Newton  to  Halley,  Octob.  18,  1686.  Rigaud's  Appendix,  p.  47). 

Prop.  XX.  Prob.  xii.  in  the  MS.  is  Prop,  xxx,  Prob.  xxii.  in  Princip.  p.  104.  Tlie 
difference  in  the  numbers  of  the  propositions  arises  from  the  circumstance  of  the  5th 
section  which  contains  eight  propositions  having  been  afterwards  inserted. 

After  Prop.  xx.  comes  a  scholium  containing  the  approximate  solution  of  the  same 
problem  for  the  ellipse  and  hyperbola.  Then  follows  the  clause  •'  Hactenus... exponere  " 
as  in  Princip,  p.  114. 


TABLE  OP  NEWTON'S  LECTURES.  xcvii 

137 — 144    Prop.  xlv.  Prob.  xxxi....9ubindo  dotonninamus. 

Lcct    8(90—95) 
145 — 152     Artie. x.  De  Motibus  Corporum... semper  peragcnt.  q.  e.  d. 

Lect.    9(95—99) 
153 —  Prop.  Lii.  Prob.  xxxiv....  Lect  10  {9<) —    ; 

breaks  off  in  Prop.  Liv.  with  tho  words  "  qua  vis  altitudino  CT  per.** 
Princip,  p.  159. 

Tho  MS.,  it  will  be  seen,  is  imperfect,  ending  abmptly  at  tho  second 
page  of  fol.  102.  Foil.  37—^4  aro  rcpeate<l,  ono  set  being  tho  first 
draught,  tho  other  as  printed  in  the  Principia,  pp.  57 — 73.  The  na- 
ture of  tlio  former  will  be  understood  from  the  following  outline.  After 
Corel.  6*.  In  Parabola,  &c.,  and  tho  other  corollaries  comes 

Prop.   XVI.    Prob.  VIII.  being  Prob.   xvu.    Prob.  DCofPriViop. 

then,    XVII ix xviii x 

without  demonstration. 

then   Prop,    xvm x.    xix xi 

without  demonstration. 

then   Lem.      xv Lem.    xvi.  otPrincip, 

Prop,  xix Prop,    xxi 

XX XXX.    ...(seenotefp.xcvi). 

XXI xxxii 

xxn , xxxin. 

XXIII XXXIV 

xxiv.a3{ara8"arcumKk"...  xxxv 

The  latter  sot  and  foil.  55—58  as  far  as  "  absurdum  est.  q.  r.  d.** 
{Princip.  p.  79)  ar©  not  divided  into  Lectures.  Fol.  45  is  numbered  55 
apparently  by  a  clerical  error,  which  is  propagated  through  the  remainder 
of  the  MS. 

In  binding  the  volume  the  sheets  seem  to  have  been  taken  at  ran- 
dom. "When  the  disjecta  membra  are  brought  together  they  form  a 
whole,  as  follows : 

1—57     De  motu  corporum... ad  tangentem.       (  1—36) 

57 —  73     Corel.  6.... in  rectam  qua  quajvis  (37—44) 

(The  other  37 — 44  in  the  MS.  is  the  rough  draught  of  this.) 

73 —  88     quajvis  ex  punctis...duoe  evadent  {55 — 62) 

88 — 118     parallel£e...arcum  Kk  {63 — 78) 

118 — 133     dcscribere...inoveri  possunt  (79 — 86) 

133 — 144    est  in  triplicata...usurpamu3  plana        (87 — 94) 

1 44.— 1 59    his  parallela. . .altitudino  CT  per  (95— 102) 


xcvm  TABLE  OP  NEWTON'S  LECTURES. 

LECTURES  ON  THE  SYSTEM  OP  THE  WORLD. 
From  a  Copy  in  Cotet^a  hand  in  Trin,  Coll.  Library,  (R.  16.  39). 

Db  Motu  Corporum  Liber*. 

1—  8t  Fixas  in  supreniis — Aatronomi.  Preelect.  1.     Sept.  29.  l687. 

8—1 6  Martem — duplicatcm.  Lcct.  2. 

16 — ^22  Stabilita — fuligino.  Lect.  3. 

22 — 27  Analogioe — modum.  Lect.  4. 

27 — 33  Designet — intelligetur,  Lect.  5. 

Here  Cotes's  copy  ends.  The  remainder  of  the  treatise,  however, 
(not  divided  into  Lectures)  is  bound  up  in  the  same  volume,  and  was 
probably  obtained  by  Professor  Smith  from  Charles  Morgan  of  Clare 
Hall,  for  in  the  Library  of  that  College  there  is  a  MS.  copy  of  the 
treatise  which  belonged  to  Morgan,  who  states  in  a  note  that  the  first  5 
Lectures  were  communicated  to  him  by  Smith,  and  the  remainder  by 
Martin  Folkcs. 


•  This  is  the  title  in  the  MS.,  not  "  De  Mundi  Systemate  "  as  in  the  printed  book 
(Lond.  1731).  This  tract,  drawn  up  "method©  populari  ut  a  pluribus  legeretur"  was 
intended  to  form  the  3d  book  of  the  Principia,  but  readers  who  have  not  mastered  the 
principles,  says  the  author,  *'  vim  consequentiarum  minime  percipient,  neque  praBJu- 
dicia  deponent  quibus  k  multis  retro  annis  insueverunt,"  and  therefore  "  ne  res  in  dis- 
putationes  trahatur,  suramam  libri  ilUua_Jranstuli  in  Propositiones,  more  Mathematico, 
ut  ab  iis  solis  legantur  qui  principia  prius  evolverint."  (Introduction  to  3d  book  of 
Princip.) 

t  The  numbers  refer  to  the  pages  in  the  printed  book. 


CORRESPONDENCE 

{ 

I  OP 

SIR    ISAAC    NEWTON 


AND 


PROFESSOR    COTES, 


CORRESPONDENCE  &c. 


LETTER  I. 
BENTLEY  TO  COTES. 


Dear  Sir, 
I  waited  to  day  on  S^  Isaac  Newton,  who  will  be  glad 
to  see  you  in  town  here,  and  then  put  into  your  hands  one 
part  of  his  Book  corrected  for  y*  press.  I  shall  get  of  him 
a  Character  of  M*"  Hussey;  but  we  both  apprehend  y* 
Interest  rather  than  Merit  will  prevail  in  y*  Election,  &  y* 
one  Coleson  has  y*  best  friends.  D*"  Ayloff  I  suppose  has 
given  you  a  Bill*  of  100*''  payable  here  in  London  at  14'^y» 
sight ;  I  must  desire  you  to  transfer  y'  Bill  to  W  Smallwell 
in  part  of  payment ;  for  y"  former  bill  I  gave  him  upon 
y*  Marquiss  of  Dorchester's  Steward  will  not  be  p**  yet. 
So  y*  if  you  send  the  Bill  by  Mascal  y*  Carrier  to  have  it 
accepted,  &  from  thence  to  bring  it  to  me,  I  will  take 
Smallwells  receipt  for  so  much  money.  Pray  let  me  know, 
when  you  think  of  coming  up  hither. 

I  am, 
Your  affectionate  friend  &  Serv* 

Cotton  House.  {May  21. 1709.}  Bi :  Bentley. 

For  Mr  Cotes  Fellow  of  Trinity 
College  in  CamhHdy. 

Tho  post  mark  of  this  letter,  though  at  first  sight  scarcely  legible, 
may  I  think  be  pronounced  to  be  May  21,  and  the  year  is  pretty 
clearly  1709.  About  tho  middle  of  July  Cotes  is  in  London  (in  his 
letter  of  Feb.  15,  1711>  to  Jones,  lie  mentions  his  having  been  last 


*  I  can  discover  no  traces  of  this  bill  in  any  of  the  College  Account  Bookn. 
It  may  possibly  have  come  into  the  Chapel  Account,  for  which  Cotes,  as  supcrin- 
lendcnt  of  tho  repairs  of  the  Chnpel,  kept  a  separate  book  of  rerripts  and  disburse- 
ments.    Whether  ths  book  is  still  in  existence  1  am  unable  to  say. 


2  CORRESPONDENCE  OF 

in  town  "about  a  year  and  a  half  ago")  drawn  thither,  no  doubt, 
by  this  noto  of  Bontlcy's,  and  expecting;  to  take  with  him  down  to 
Cambridge  tho  first  instalment  of  Newton's  corrected  copy  of  the 
Principia.  Newton  however  is  still  reluctant  to  part  with  it,  having 
probably  some  further  improvements  to  make,  but  promises  to  send 
the  copy  down  in  about  a  fortnight.  So  Cotes  returns  to  Cambridge 
without  tho  "one  part  of  tho  book  corrected  for  the  press,"  which 
Bentley's  letter  had  informed  him  was  ready  to  be  put  into  his  hands 
some  eight  weeks  before.  Tlio  copy  docs  not  arrive  in  that  fort- 
night, nor  in  tho  next.  The  long  vacation  being  nearly  half  over 
and  no  signs  of  tho  promised  copy  appearing,  tho  young  editor  be- 
comes impatient.  Ilenco  his  letter  of  Aug.  18,  which  however  pro- 
duced no  apparent  effect,  until  his  next-door  neighbour  Winston,  one 
evening  probably  in  September,  newly  arrived  from  London,  (ho  is 
known  to  have  been  in  Cambridge  on  tho  29th  of  that  montli)  put 
into  his  hands  "  the  greatest  part  of  tho  copy  of  tho  Principia,"  end- 
ing at  Prop.  XXXI II  Cor.  2  Lib.  ii  p.  .320.  That  is  followed  some 
time  afterwards  by  Newton's  letter  of  Oct.  11,  which  apparently  did 
not  come  through  the  post,  being  brought  perhaps  by  some  member 
of  the  University  coming  up  on  the  beginning  of  term.  Whiston, 
whose  autobiography  records  so  many  other  things  certainly  of  not 
greater  importance,  makes  no  mention  of  his  being  employed  as  a 
messenger  on  this  occasion:  so  absorbed  was  he  in  his  Arian  heresy 
and  Apostolical  Constitutions,  with  regard  to  which  ho  tells  us  "his 
best  friends  began  to  be  greatly  affrighted  this  summer  at  what  they 
had  heard  he  was  going  about."  It  is  not  likely  that  he  found  his 
old  patron  wanting  in  tho  duties  of  friendship  at  this  critical  period 
of  his  life,  and  it  is  not  impossible  that  Sir  Isaac,  in  delivering  to  him 
a  portion  of  a  work  containing  so  much  close  and  profound  reasoning, 
may  have  dropt  a  word  of  caution  into  his  ear. 

The  "election"  referred  to  in  this  letter  is  probably  that  of  a 
Head  Master  of  Sir  Joseph  Williamson's  Free  ]\Iathematical  School 
at  Rochester,  tho  electors  to  which  post  are  some  17  in  number, 
consisting  of  tho  Mayor,  Recorder,  eldest  Resident  Prebendary,  &c. 
The  Rev,  John  Colson  was  the  first  Head  Master  of  this  school,  and 
was  appointed  June  1,  1709.  He  resigned  the  place  March  1,  I7I5, 
on  being  elected  •  Lucasian  Professor.  He  was  entered  at  Emmanuel 
April  23,  1728,  and  was  one  of  the  71  persons  in  the  King's  list 
(William  Warburton  was  another  of  the  number)  on  whom  the  de- 
gree of  M.A.  was  conferred  at  George  II's  visit  to  the  University, 
April  25,  1728.  On  coming  to  reside  as  Lucasian  Professor  at  Cam- 
bridge, ho  was  appointed  Taylor  Lecturer  at  Sidney  College,  where 
he  was  admitted  "in  convictum  sociorum"  11  March  I755,  «tat.  60. 


NEWTON  AND  COTES.  S 

It  was  for  the  purpose  of  boardin;^  with  this  same  CoUon,  and  bcinf^ 
instructed  by  him  "in  ^lathematics  and  Philosophy  and  humane 
Icamin^^,"  tliat  Garrick  set.  out  from  Lichfield  on  the  morning  of 
^larch  2,  173}  for  London,  accompanied  by  **one  Mr  Johnson,"  who 
was  going  "to  try  his  fate  with  a  tragedy,  and  to  see  to  got  him- 
self employed  in  some  translation,  either  from  the  Latin  or  French," 

Christopher  Ilussey  was  a  senior  Bachelor  of  Arts  of  Trinity  Col- 
lege, and  was  elected  Fellow  the  following  October.  On  Winston's 
expulsion  from  the  University  (Oct.  30,  1710),  he  was  appointed  by 
him  as  his  deputy  in  the  Lucasian  Chair,  and  "  was  ready  to  perform 
his  duty,  had  not  the  heat  of  that  time  prevented  him."  Winston's 
Memoirs  I.  312.  lie  was  afterwards  an  unsuccessful  candidate  for 
the  Professorship  against  "Blind"  Saunderson  (Nov.  20,  1711).  Soo 
letter  cviit,  note. 

A  slightly  different  date  is  assigned  to  this  letter  in  the  Bontley 
Correspondence  (p.  370),  and  a  widely  different  one  is  mentioned  as 
being  suggested  by  Bishop  Monk  (ib.  p.  7^7). 


LETTER  II. 
COTES  TO  NEWTON. 

S'.  Cambridge  August  18*^  1709. 

The  earnest  desire  I  have  to  see  a  new  Edition  of  y 
Princip.  makes  me  somewhat  impatient  'till  we  receive  Y' 
Copy  of  it  which  You  was  pleased  to  promise  me,  about 
the  middle  of  the  last  Month,  You  would  send  down  in 
about  a  Fourtnights  time.  I  hope  You  will  pardon  me  for 
this  uneasiness  from  which  I  cannot  free  my  self  &  for 
giveing  You  this  Trouble  to  let  You  know  it.  I  have  been 
so  much  obliged  to  You  by  Y"^  self  &  by  Y"^  Book  y*  (I 
desire  You  to  beleive  me)  I  think  my  self  bound  in  grati- 
tude to  take  all  the  Care  I  possibly  can  that  it  shall  be 
correct.     Some  days  ago  I  was  examining  the  2^  *  Cor :  of 


•  In  this  Corollary  is  determined  the  Attraction  of  a  Spheroid  on  a  point  in  its 
axis  produced,  the  attractive  force  of  each  particle  varying  inversely  as  the  square  of 
the  distance.  A  paper  by  Cotes  containing  the  investipfation  is  still  preserved  in  the 
volume  from  which  these  letters  are  taken  Nos.  24  and  25, 

1—2 


A  CORRESPONDENCE  OP 

Prop  91  Lib  i  and  found  it  to  be  true  by  y*  Quadratures 
of  y«  1'*  &  2**  Curves  of  y«  8*^  Form  of  y*  second  Table  in 
Y'  Treatise  De  Quadrat,  At  the  same  time  I  went  over 
y*  whole  Seventh  &  Eighth  Forms  which  agreed  with  my 
Computation  excepting  y®  First  of  y*  Seventh  &  Fourth  of 
y*  Eighth  which  were  as  follows 

4de  -=-  -  2rf/ SdeecT  +  4>dfg8 

Form  :  7.  1.     ^ — «  t 

^*jeg  -  vff 

Form:  8.4.  +S6de/^   ^Sdegg        -28dc/>      -iGdeeg 
„   8  a)  XV  XV  V 

'\5dp      "^dffg         +}Odf'         +I0deff    _^ 

24>rjeg^  -  Grjffgg 

I  take  this  Oportunity  to  return  You  my  most  hearty 
thanks  for  Y'  many  Favours  &  Civilitys  to  me  who  am 

Y'  most  Obliged  humble  Servant 

Far  S'  Isaac  Newton  at  his  House  RoGfiR  CoTES. 

in  Jermin  Street  near  St  Jameses 
Church  Westminster. 


LETTER  III. 

NEWTON  TO  COTBS. 
S-" 

I  sent  you  by  M*"  Whiston  the  greatest  part  of  y*  copy 
of  my  Principia  in  order  to  a  new  edition.  I  then  forgot 
to  correct  an  error  in  the  first  sheet  pag  3  lin  20>  21,  &  to 
write  plusquam  duplo  for  quasi  quadruplo  &  plusqtiam  de- 
cuplo  for  quasi  centuplo*. 


■  These  two  corrections  are  not  adopted  literally  in  the  2d  edition,  the  "quasi"  for 
which  Newton  here  substitutes  "plusquam"  being  still  retained  in  it.  Perhaps  Cotes 
had  already  altered  the  "quadruplo"  and  "centuplo"  before  receiving  this  letter,  as 
so  obvious  an  error  could  scarcely  have  escaped  his  attention.    In  the  passage  referred 


NEWTON  AND  COTES.  5 

I  forgot  also  to  add  the  following  Note  to  the  end  of 
Corol.  1  pag.  55  lin  6.  Nam  datis  umbilico  et  puncto 
contactus  &  positione  tangentis,  describi  potest  Sectio 
conica  qua)  curvaturam  datam  ad  punctum  illud  habcbit. 
Datur  autem  curvatura  ex  data  vi  centripeta:  et  Orbes  duo 
se  mutuo  tangentes  eadcm  vi  describi  non  possunt. 

I  thank  you  for  your  Letter  &  the  corrections  of  y* 
two  Theorems  in  y*  treatise  de  Quadratura.  I  would  not 
have  you  be  at  the  trouble  of  examining  all  the  Demon- 
strations in  the  Principia.  Its  impossible  to  print  the  book 
w^'^out  some  faults  &  if  you  print  by  the  copy  sent  you, 
correcting  only  such  faults  as  occurr  in  reading  over  the 
sheets  to  correct  them  as  they  are  printed  off,  you  will 
have  labour  more  then  it's  fit  to  give  you. 

M'.  Livebody  is  a  composer  (I  mean  M'^  Livebody  who 

made  the  wooden  cutts)  &  he  thinks  that  he  can  sett  the 

eutts  better  for  printing  off  then  other  composers  can,  and 

offers  to  come  down  to  Cambridge  &  assist  in  composing 

if  it  be  thought  fit.     When  you  have  printed  off  one  or 

two  sheets,  if  you  please  to  send  me  a  copy  of  them  I  will 

send  you  a  further  supply  of  wooden  cutts. 

I  am 

Yo'  most  humble  &  faithful  servant 

London.  Octob.  11.  1709.  Is.  Newton. 

For  M'  Cotes  Professor  of  Astronomy 
in  the  University  of  Cambridt/e  at 
his  Chamber  in  Trinity  College. 

Shortly  after  the  date  of  tho  above  letter,  Newton  changed  liis 
residence  from  Jermyn  Street  to  Chelsea.     Flamsteed,  writing  to  Ab. 


to,  Newton,  speakiDgr  of  a  ball  shot  horizontally  with  a  eriven  velocity  from  the  top  of  a 
mountain  to  a  distance  of  two  miles  before  it  reaches  the  ground,  says,  (as  the  words 
stand  in  the  2nd  and  3rd  editions)  '*  dupla  cum  velocitate  quadi  dupio  longius  per- 
geret,  et  decupla  cum  velocitate  quasi  decuplo  longius."  When  he  wrote  "quadruplo  " 
and  *' centuplo,"  he  was  probably  thinkingr  of  oblique  projection.  The  passage  in 
question  occurs  in  8omo  additional  remarks  in  illustration  of  J)cf.  v.,  which  were  not 
given  in  tlie  Ist  edition.  The  MS.  of  them,  unfortunately,  does  not  appear  in  the 
Newtonian  \'olunie. 


6  CORRESPONDENCE  OF 

Sharp  Oct.  25,  (Baily,  p.  272)  says :  "  He  |  Sir  Isaac  |  is  now  re- 
moving to  Chelsea,  and  has  been  lately  much  talked  of ;  but  not  much 
to  his  advantage.  Our  Society  \  the  Royal  Society  j  is  ruined  by  his 
close,  politic,  and  cunning  forecast;  I  fear  past  retrieving,  for  our 
Doctor's  |Sloano|  Transactions  have  been  twice  burlesqued  publicly; 
and  now  we  have  had  none  published  I  think  this  four  months."  This 
burst  of  spleen  would  seem  to  bo  in  anticipation  of  the  resolution  passed 
by  the  Council  of  the  Society  Nov.  9,  ordering  Flamsteed's  name  "  to 
be  left  out  of  the  list  of  the  Society  for  next  year  for  not  having  com- 
plied with  the  order  of  Council  made  12  Jan.  }*JOl"  relative  to  the 
payment  of  arrears. 

Here  there  is  a  break  of  6  months  in  the  correspondence  until  we 
come  to  Cotes's  letter  of  April  15,  17^0,  by  which  time  nearly  half 
of  the  whole  work  was  printed  off,  the  part  then  finished  ending  at 
p.  224,  (2nd  ed.)  in  the  middle  of  the  Lemma  (II  Lib.  2)  in  which 
the  principles  of  fluxions  are  explained. 

A  note  by  Mr  Howkins  states  that  there  is  wanting  a  letter  of 
Cotes  to  Newton,  dated  Apr.  9,  1710,  *"*  de  Cor.  1  and  6  Prop.  ix. 
Lib.  2."  No.  33  contains  a  draught  in  Cotes's  writing  of  these  two 
Corollaries,  and  two  additional  steps  in  the  proof  of  the  Proposition, 
but  not  (with  the  exception  of  the  latter  of  the  two  steps)  as  they 
stand  in  the  second  edition.  On  the  same  paper  Cotes  has  also  written 
"  dele  Cor.  4  and  5,  Prop,  viii,"  which  are  accordingly  omitted  in  the 
2nd  ed.  It  is  probable,  therefore,  that  if  this  missing  letter  of  Apr.  9 
referred  to  the  Corollaries  mentioned  by  Mr.  Howkins,  the  proposed 
omission  of  Cor.  4  and  5,  Prop.  viii.  and  the  introduction  of  the  two 
steps  into  the  reasoning  of  Prop.  ix.  in  order  to  avoid  a  reference  to  the 
latter  of  the  cancelled  corollaries  would  also  form  a  part  of  its  contents. 

But  besides  this  letter  of  Apr.  9  and  Newton's  answer  to  it,  there 
is  good  reason  for  supposing  that  at  least  one  other  pair  of  letters 
passed  between  them  during  the  interval  from  October  to  April.  For 
(1)  it  seems  probable  that  Cotes  would  return  some  answer  to  Newton's 
letter  of  Oct.  11,  in  explanation  of  his  not  adopting  the  precise  language 
of  the  emendations  contained  in  it ;  and  at  all  events  he  would  attend 
to  Newton's  request  to  have  one  or  two  sheets  sent  to  him,  to  say 
nothing  of  the  presumption  that  ho  would  feel  himself  called  upon  to 
take  some  notice  of  Mr  Livebody's  offer  of  his  services.  (2)  The  2nd 
method  of  finding  the  force  to  the  centre  of  an  ellipse  given  in  p,  40 
2nd  Ed.  is  so  much  altered  (in  the  opening  part  of  it)  from  the  form 
in  which  it  stands  in  Newton's  MS.  (No.  9),  that  Cotes  would  scarcely 
have  changed  it  without  some  communication  from  Newton  on  tho 
subject.     At  the  head  of  this  2nd  method  Cotes  has  written  **  vid.  fol. 


NEWTON  AND  COTES.  7 

scqu,"  but  the  leaf  referred  to  is  not  to  bo  found.  (3)  From  a  mark  in 
No.  11  it  appears  that  the  first  word  in  p.  40  in  tho  proof  sheet  (H) 
was  "corporis"  which  is  now  in  the  eightli  and  ninth  lines  lower  down; 
so  that  some  additional  matter  must  have  been  introduced  in  a  pro- 
cetling  page  after  the  proof  of  II  was  printed*,  and  tins  almost  neces- 
sarily implies  tho  receipt  of  instructions  from  Newton  to  that  cft'ect, 
(perhaps  at  tho  end  of  November  or  beginning  of  December,  if  we  may 
judge  from  tho  rate  at  which  tho  press  was  working). 

The  loss  of  any  letters  in  this  interval  is  tho  more  to  be  regretted, 
because  if  ever  the  celebrated  Scholium  to  Lemma  II.  Book  2  was 
touched  upon  in  the  correspondence  between  Newton  and  his  Editor, 
the  place  for  doing  so  would  lie  within  this  period.  Tho  missing  letter 
of  April  9,  as  has  been  said,  may  have  contained  remarks  connected 
with  Prop.  VIII.  which  immediately  follows  that  Scholium.  The  only 
alteration  in  the  Scholium  t  made  in  the  2nd  Edition,  consists  in  tho 
addition  of  tho  words  "  ct  Idea  generationis  quantitatum"  after  **  nota- 
rum  formulis."  Tho  "annis  abhinc  decem"  referring  to  his  second 
letter  through  Oldenburg  to  Leibniz,  in  Oct.  1076,  is  still  retained, 
though  20  years  intervened  between  tho  publication  of  tho  Ist  and  2nd 
editions. 

In  this  interval,  it  may  bo  remarked,  the  quarrel  between  Bentley 
and  the  Seniors  broke  out,  and  we  read  of  Cotes  being  present  at  two 
conferences  at  the  lodge  between  tho  conflicting  parties,  as  a  friend  of 
the  Master's.  (See  Monk's  Bentley,  p.  187.)  On  Jan.  18,  1710, 
15entley  cut  INIiller's  name  out  of  tho  boards.  On  Feb.  10,  Miller 
presented  tho  petition,  signed  by  thirty  of  tho  fellows,  to  the  Bishop  of 
Ely.  Great,  however,  as  was  tho  delay  which  retarded  the  second 
*;dition  of  the  Principia  in  its  passage  through  the  press,  Cotes  had 


*  A  comparison  of  Newton's  MS.  with  sheet  G  of  the  2nd  Ed.  shewH  that  the  addi- 
tion must  have  been  made  in  some  sheet  preceding  that,  but  it  is  impossible  to  fix  the 
exact  place,  as  the  part  of  the  MS.  which  is  preserved  only  begins  with  Prop.  vi. 
Theor.  v. 

t  This  Scholium  was  completely  remodelled  in  the  3rd  Ed.  and  Leibniz's  name  sup- 
pressed. The  reader  of  these  pages  will  smile  at  the  followingr  piece  of  information 
with  which  Montucla  favours  us  (xii.  108):  "On  se  demandera  jicut  etrc  pourquoi 
cette  suppression  ne  fut  pas  faite  loi-s  de  I'edition  des  jtrincipes  de  1713,  puisquc  alors  la 
querelle  £'toit  encore  dans  toute  sa  chaleur ;  en  voici  la  raison,  qui  est  une  anecdote 
asst'Z  i>eu  connue  et  que  je  tiens  de  la  mOme  main  que  ce  que  j*  ai  dit  ci-dessus  {the 
*  bonne  main '  that  had  informed  him  that  the  notes  on  the  Cominerciuvi  EpistnUcum 
were  written  by  Newton}.  C'est  que  cette  Edition  fut  faite  u  Cambridge,  loin  de 
Neuton  et  presque  en  cachette,  par  les  soins  do  Cotes  et  de  lientley,  et  que  Ncuton 
en  fut  tres-m^content  C  est,  en  cffet,  un  proc6<l6  assez  6trange  de  la  part  de  ce« 
deux  hommes,  d'  ailleurs  c^-lebres,  que  d'  imprimer  un  ouvrage  du  vivantdesod  auteur 
sans  prendre,  pour  ainsi  dire,  son  attache  sur  les  changcmens  ou  additions  ti  y  fairc." 


8  CORRESPONDENCE  OF 

brought  his  labours  upon  it  to  a  conclusion  nearly  a  year  before  Bent- 
ley's  trial  came  on. 

It  may  assist  us  still  further  to  fill  up  this  gap  of  six  months,  and 
to  imagine  the  direction  which  the  thoughts  and  conversation  even  of 
men  engaged  on  a  new  edition  of  the  Principia  would  occasionally  take, 
if  wo  remember  that  it  was  during  this  same  period  that  the  kingdom 
was  plunged  into  the  Sacheverell  excitement,  (the  2nd  of  the  two 
obnoxious  sermons  was  preached  on  Nov.  5,  1709,  tlio  trial  began 
Febr.  27,  17^0,  and  on  March  21  the  Doctor  was  suspended  from 
preaching  for  three  years);  and  that  Marlborough,  yielding  f/O  the 
solicitations  of  Oodolphin,  whose  ministerial  difficulties  called  for  the 
support  and  authority  of  the  Great  Captain's  presence,  arrived  from  the 
Hague  on  Nov.  8,  and,  after  experiencing  in  several  mortifying 
instances  the  eifocts  of  Masham  influence,  against  whicli  even  Mai- 
plaquct's  recent  laurels  were  powerless,  was  sent  back  to  Holland 
towards  the  end  of  February,  and  that,  on  the  failure  of  the  negotia- 
tions with  which  Louis  had  been  amusing  the  allies  at  Gertruydcnberg, 
he  and  Eugene  (duofulmina  belli)  opened  their  magnificently  planned 
campaign  of  1710,  by  passing  the  French  lines  on  tlie  morning  of 
Monday  April  10,  and  proceeding  to  the  investment  of  Douay. 


LETTER    IV. 
COTES   TO  NEWTON. 


S^  j  Saturday  I  Apr.  16.  1710. 

We  have  printed  so  much  of  y®  Copy  You  sent  us  y*  I 
must  now  beg  of  You  to  think  of  finishing  the  remaining 
part  assoon  as  You  can  with  convenience.  The  last  sheet 
y*  is  printed  off  ends  at  y®  251**  page  of  y*  old  Edition  & 
y«  224*^  page  of  y"  new  Edition.  The  whole  y*  is  finished 
shall  be  sent  You  by  the  first  oportunity.  I  have  ventured 
to  make  some  little  alterations  my  self  whilst  I  was  cor- 
recting the  Press  such  as  I  thought  either  Elegancy  or 
Perspicuity  or  Truth  sometimes  required.  I  hope  I  shall 
have  Y"^  pardon  if  I  be  found  to  have  trusted  perhaps  too 
much  to  my  own  Judgment,  it  not  being  possible  for  me 
witliout  great  inconvenience  to  y*  work  &  uneasiness  to 


NEWTON  AND  COTES.  9 

Y'  self  to  have  Y'  approbation  in  every  particular.  The 
Pages  which  are  next  to  be  printed  being  somewhat  more 
y"  usually  intricate  I  have  been  looking  over  them  before 

hand.     Page  270*  Reg.  1  I  think  should  begin  thus Si 

servetur  turn  Medii  densitas  in  A  turn  vclocitas  quacum 

corpus  projicitur  &  mutetur .1  must  confess  I  cannot 

be  certain  y*  I  understand  the  design  of  Reg.  4  &  y*  last 
part  of  Rq^.  7  and  therefore  dare  not  venture  to  make  any 
alteration  without  acquainting  You  with  it.  I  take  it 
thus,  y*  in  y*  4***  Rule  You  are  shewing  how  to  find  a  mean 
among  all  y*  Densitys  through  which  y*  Projectile  passes, 
not  an  Arithmeticall  mean  between  y*  two  extream  Den- 
sitys y*  greatest  and  least,  but  a  mean  of  all  y*  Densitys 
considered  together,  which  will  be  somewhat  greater  than 
y*  Arithmeticall  mean,  y*  number  of  Densitys  which  are 
greater  y"  it  being  greater  y"  y*  Number  of  Densitys 
which  are  lesser  y"  y*  same.  If  this  be  Y"^  design  I  would 
alter  the  4**^  Rule  thus,  with  Y*"  consent.  Quoniam  Den- 
sitas Medii  prope  verticem  Hyperbola)  major  est  quam  in 
loco  Ay  ut  habeatur  Densitas  mediocris  debet  ratio  minima) 
tangentium  GT  ad  tangentem  AH  inveniri,  &  Densitas  in 
A  augeri  in  ratione  paulo  majore  quam  semisumma)  harum 
tangentium  ad  minimam  tangentium  GT,  The  latter  part 
of  y«  7*^  Rule  I  understand  thus.  Simili  mcthodo  ex 
assumptis  pluribus  longitudinibus  AH  invenienda  sunt 
plura  puncta  N  &  per  omnia  agenda  Curva  linea  regularis 
NNXN  secans  rectam  SMMM  in  X*  Assumatur  dcmum 
AH  lequalis  abscisssB  SX  Si  inde  denuo  inveniatur  longitudo 
AK\  &  longitudines  quae  sint  ad  assumptam  longitudinem 
AI  &  banc  ultimam  AH  ut  longitudo  ^A'per  experimentum 
cognita  ad  ultimo  inventam  longitudinem  AK  erunt  vera) 
ilia)  longitudines  At  ^  AH  quas  invenire  oportuit.  Hiscc 
Vero  datis,  dabitur  &  resistentia  Medii  in  loco  A  quippe 
qua)  sit  ad  vim  gravitatis  ut  AH  ad  2  AI^   augenda   est 


•  pp.  270—274,  (Schol.  to  Prop.  x.  Lib.  2.)  contain  Rulw  for  the  approximate  «lc. 
tetinination  of  the  motion  of  a  projectile  in  the  iiir. 


10  CORRESPONDENCE  OF 

autem  densitas  Medii  per  Reg.  quartam  &  resistentia 
modo  inventa,  in  eadem  ratione  aucta  fiet  accuratior. 
About  y*  end  of  the  8*^  Rule  are  these  words — quorum 
minor  eligendus  est — which  I  would  either  leave  out  or 
print   thus — quorum   minor  potius   eligendus    est.     Page 

274. 1:  2  should  be — - — ;  there  are  some  others  like 

nn  ^  fi  X  GV 

this  which  I  will  not  trouble  You  with.    Prop,  xiv  Prob  iv 

should  be  Prop  xiv  Theor  xi.    Two  lines  lower  are  these 

words — est  ut  summa  vel  differentia  area)  per  quam — I 

would  leave   {out}    summa  vel.   Corol.  page  281  I  would 

print   thus.      Igitur   si    longitudo    aliqua    V  sumatur   in 

ea  ratione  ad  duplum  longitudinis  J/,  qua)  oritur  appli- 

cando  aream  DET  ad   BD,  quam    habet   linea   DA    ad 

lineam   DE ;  spatium  quod  corpus  ascensu  vel  descensu 

toto    in    Medio    resistente    describit,    erit    ad    spatium 

quod  in  Medio  non  resistente  eodem  tempore  describere 

BD  X  VV 
posset,  lit  arearum  illarum  differentia  ad  — —     —  ,  ideoq: 

At  AB 

ex  dato  tempore  datur.     Nam  spatium  in  Medio  non  re- 
sistente est  in  duplicata  ratione  temporis,  sive  ut  FF,  & 

BD  X  VV 

ob  datas   BD   &  AB^  ut   -— — .     Momentum    hujus 

4f  AB 

.     ^  .  ,.     DAq  ^  BDxM'       ,     , 

areas,  sive  huic  aQqualis  — ^r-^r -— —  ,  est  ad  momentum 

DEq X  AB 

,.«,  .  -^^rr,    o         ^,    mrr^        .      DAO'KBD'KQM^fn 

differentia)  arearum  DET  8z  AbNK  ut ^- — 

DEq X  AB 

^    AP^BD^m      ,            ^      ^    DAq  ^  BD  x  M      ,    , 
ad    ^^ ,     hoc   est,  ut   ^^^ ad   1 

BD  X  AP  sive  ut  -^  in  DET  ad  DAP,  adeoq :  ubi  area) 
DEq 

DET  &  DAP  quam  minima)  sunt  in  ratione  ajqualitatis. 

Jiqualis  igitur —     Page  286. 1:  6*  must  be  thus  corrected 


•  Prop.  XV.  Lib.  2,    On  the  motion  of  a  body  in  a  logarithmic  spiral  in  a  resist* 
ing  medium,  (force  cc     .     .^ ,  resist.oc(vel.)*). 


NEWTON  AND  COTES.  11 

Rr  &  TQ,  seu  ut  '^ — — &  ~,p     ^"^^  simul  generant, 

hoc  est,  ut  VQ  &  PQ  seu  OS  &  OP.  This  Corollary  being 
thus  corrected,  the  following  must  begin  thus.  Corel.  4, 
Corpus  itaq :  gyrari  nequit  in  hac  Spirali,  nisi  ubi  vis  resis- 
tentia)  minor  est  quam  vis  centripeta.  Fiat  resistentia 
sequalis  vi  centripeta)  &  Spiralis  conveniet  cum  linea  recta 
PSt  inq :  hac  recta — &c.  Tis  evident  by  y*  !■*  Corollary 
that  y®  descent  along  y®  line  PS  cannot  be  made  w*^  an 
uniform  velocity.  Tis  as  evident  I  think  y*  it  must  be  with 
an  uniform  velocity  because  y*  resistance  &  force  of  gravity 
being  equall,  mutually  destroy  each  other's  effect  and  con- 
sequently no  acceleration  or  retardation  of  motion  can  be 
produced.  I  cannot  at  present  see  how  to  account  for  this 
difficulty  &  I  choose  rather  to  o>vn  my  ignorance  to  You 
y"^  to  run  y®  hazard  of  leaving  a  blemish  in  a  book  I  so 

OP 

much  esteem*.  Cor.  6.  lin.  ult.  I  would  print  thus — ut  ~-  , 

id  est,  ut  secans  anguli  ejusdem,  vel  etiam  reciproce  ut 
Medii  densitas.  If  I  mistake  not  y®  design  of  y®  8***  Co- 
rollary, I  would  alter  it  thus — Centro  S  intcrvallis  continue 
proportionalibus  6M,  SB,  SC,  &c.  describe  circulos  quot- 
cunq:  &  statue  tempus  revolutionum  omnium  inter  peri- 
metros  duorum  quorumvis  ex  his  circulis,  in  Medio  de 
quo  egimus,  esse  ad  tempus  revolutionum  omnium  inter 
eosdem  in  Medio  proposito,  ut  Medii  propositi  densitas 
mediocris  inter  hos  circulos  ad  Medii  de  quo  egimus 
densitatem  mediocrem  inter  eosdem  quam  proximo ;  sed 
&  in  eadem  quoq :  ratione  esse  secantem  anguli  quo  Spiralis 
pra?finita  in  Medio  de  quo  egimus  secat  radium  ^15*  ad 
secantem  anguli  quo  Spiralis  nova  secat  radium  eundcm  in 
Medio  proposito:  Atq;  etiam  ut  sunt  corundem  angulo- 
rum  tangentes  ita  esse  numeruni  revolutionum  inter  circu- 


•  See  the  next  and  three  following'  Letters. 


12  CORRESPONDENCE  OP 

los  eosdem  duos  quam  proximo.  Si  ha3C  passim —  Prop. 
16  must  be  altered  for  by  my  reckoning  if  y*  centripetall 

force  be  as  op<.+i  ^^^®  llesistance  will  be  as  ^^pnli* 

the  Velocity  as  -—-r  ,  &  therefore  y*  Density  as  -^n'~^h  • 

With  y  consent  I  would  add  this  Corollary.     Si  vis  cen- 

tripeta  sit  ut  ■      ^^ ,  erit  1  -  ^  n  «  0,  adeoq :  Resistentia  & 

Densitas  Medii  nulla  erit  ut  in  Prop  ix  Lib  1.  Another 
Corollary  might  be  added  to  shew  in  what  cases  y®  Resist- 
ance is  affirmative  and  in  what  cases  negative.  I  beg  of 
You  to  pardon  the  freedom  of  this  Letter. 

Y"^  &c. 


LETTER  V. 
COTES  TO  NEWTON. 
S'  Apr.  30*'»  1710 

I  suppose  M'  Crownfield  our  Printer  has  delivered  to 
You  all  y®  Sheets  that  are  already  printed  off.  I  desired 
him  to  wait  upon  You  before  he  returned  to  Cambridge 
y*  I  might  have  Y"^  answer  to  my  former  Letter  or  at  least 
to  y*  first  part  of  it.  The  difficulty  which  I  proposed  to 
You  concerning  y*  4***  Corollary  of  Prop,  xv  I  have  since 
removed.  Upon  examination  of  y*  Proposition  I  think  I 
have  observed  another  mistake  in  Cor.  3.  which  ballances 
y^  which  I  before  mentioned  *  to  You  in  y*  Corollary.  For 
if  I  be  not  deceived  y*'  force  of  Resistance  is  to  y®  Centri- 
petall force  AS  ^  Rr  to  TQ  not  as  i?r  to  TQ.  You  will 
see  my  reasons  in  y®  following  alterations  which  I  propose 
to  You.     Page  284.  1:  6  Ponantur  qu89  in  superiore  Lem- 


viz.  TQ  being  erroneously  put  =  -— —  in  the  Ist  ed.  instead  of  .5  J*^  . 


NEWTON  AND  COTES.  13 

mate,  &  producatur  »VQ  ad  V  ut  sit  SV  ajqualis  SP»  Tem- 
pore quovis,  in  Medio  resistente,  describat  corpus  arcum 
quam  minimum  FQ,  &  tempore  duplo  arcum  quam  minimum 
PR ;  &  decrementa  horum  arcuum  ex  resistentia  oriunda, 
sive  defectus  ab  arcubus  qui  in  Medio  non  resistente  iisdem 
temporibus  deseriberentur,  erunt  ad  invicem  ut  quadrata 
temporum  in  quibus  generantur:  est  itaq:  decrementum 
arcus  PQ  pars  quarta  decrementi  arcus  Pli.  Postquam 
vero  descriptus  est  arcus  PQ  in  Medio  resistente,  si  area) 
PSQ  aaqualis  capiatur  area  QSr,  erit  Qr  arcus  quem  tem- 
pore reliquo  corpus  describet  absq :  ulteriore  resistentia, 
arcuumq:  QR,  Qr  differentia  Rr  dupla  erit  decrementi  arcus 
PQ ;  adeoq :  vis  resistentia)  &  vis  centripeta  sunt  ad  invicem 
ut  lineolce  ^Rr  &  TQ  quas  simul  generant.  Quoniam  vis 
centripeta,  qua  corpus  urgetur  in  P  est  — .  Pag.  285.  1 :  5 
—  ^  FQ  fit  aiqualitatis.  Quoniam  decrementum  arcus  PQ, 
ex  resistentia  oriundum,  sive  hujus  duplum  Rr  est  ut  re- 
sistentia &  quadratum  temporis  conjunctim ;  erit  Eesisten- 

Rr 

tia  ut  -— ; .     Erat  autem  PQ  ad — Page  286  1 :  4. 

PQq  X  SP 

i  FQ  X  PQ 

Nam  vires  illze  sunt  ut  \Rr  &  TQ  sive  ut  '^  -  —  & 

SQ 


hPQq 


,  hoc  est,  ut  i  FQ  &  PQ  seu  \0S  Si  OP.— I  satisfied 


SP 

my  self  more  fully  y*  I  am  not  mistaken  in  my  reasoning 
after  y"  manner.     If  (as  in  Prop  xvi)  y*  Centripetall  force 

be  as  ■ ,  the  force  of  Resistance  will  be  to  y*  Centri- 

petall force  as ^Rr  to  TQ  ie  as  1  -  Jw,  OS  to  OP.  Put 
y®  Centripetall  force  as  — ,  &  You  will  have  «  =  O,  &  con- 
sequently 1  -iw,  OS  to  OP  as  OS  to  OP,  Therefore 
when  y®  Spiral  coincides  with  y*  line  PS  y®  Resistance  will 
be  equall  to  y*  Centripetall  force  &  y*  Body  will  descend 
"with  an  uniform  Velocity  as  it  ought  to  do,  by  Cor.  1  Prop 


14  CORRESPONDENCE  OP 

X  V,  &  Cor  5.  Prop  iv  Lib  r.  compared  together,  and  also 
upon  y*  consideration  y*  y*  velocity  in  y®  Spiral  of  Prop 

XVI  is  as         .   ,  i  e,  as  -— —  .     I  have  some  things  further 

to  propose  to  You  about  y*  remaining  -part  of  Y""  copy, 
which  I  will  not  trouble  You  with  till  I  have  Y'  answer  to 
my  former  Letter 

Y"^  &c. 


LETTER  VI. 
NEWTON  TO  COTES. 
S'  Chelsea  near  London  May  1'*  1710. 

I  thank  you  for  your  letter  with  your  remarks  upon  the 
papers  now  in  the  Press  under  your  care.  As  soon  as  I 
could  get  some  time  to  think  on  things  of  this  kind,  from 
V^*  I  have  of  late  years  disused  myself,  I  examined  them*, 
&  all  your  corrections  may  stand  till  you  come  at  page 
287.  In  page  28G  lin  4  for  ^  OS  read  OS.  In  the  same 
page  let  Corol.  4  stand  thus.  Corpus  itaq :  gyrari  nequit 
in  hac  spirali  nisi  ubi  vis  resistentia)  minor  est  quam  vis 
centripeta.  Fiat  resistentia  oBqualis  vi  centripetaD,  et  spi- 
ralis conveniet  cum  linea  recta  PS,  et  motus  corporis  ces- 
sabit.  In  page  287  &  288  the  8*^  Corollary  may  remain 
as  in  the  Copy  I  sent  you.     In  page  289  let  the  16"*  Pro- 

Rr 
position  end  thus  et  resistentia  in  P  ut  — — — —  ,  sive  ut 

PQ.SP^.SQ'  "^'"^^  "'  OptsP^^*  ^^"^  ""''  ("^   ^"- 

turn   ^~ ]  reciproce  ut  SP^^K    Et  propterea  densi- 

tas  in  P  est  reciproce  ut  SP^If, 

*  Newton  does  not  seem  to  have  worked  the  problem  out  himself,  but  to  have  taken 
Cotes's  results  (in  Letter  IV.)  for  granted. 

t  The  *'  5P-  "  is  no  doubt  copied  inadvertently  from  the  Ist  Ed.    It  should  be  SP. 


NEWTON  AND  COTES.  15 

Corol.  1.  Si  vis  centripcta  sit  reciproce  ut  SP^\  crit 
1  -  J  n  «  0,  adeoq :  resistentia  et  densitos  Medii  nulla  erit 
ut  in  Propositione  nona  Libri  prirai. 

Corol.  2.  Si  vis  centripcta  sit  reciproce  ut  radii  SP 
dignitas  aliqua  cujus  index  est  major  numcro  3,  resistentia 
affirmativa  in  negativam  mutabitur. 

When  you  sent  me  the  sheets  last  printed  off,  I  hap- 
pened to  be  from  home,  but  a{t}  night  found  them  left  at 
my  house,  and  thank  you  for  them.  I  am  going  to  finish 
the  next  part  of  the  copy  I  am  to  send  you,  &  I  hope  to 
have  it  ready  in  due  time  if  some  experiments*  succeed. 
I  thank  you  once  more  for  your  corrections  &  for  your 
care  of  the  edition. 

I  am 

S'  Your  most  humble  &  most  obedient  servant 

Is.  Newton. 

After  the  ^vriting  of  this  Letter  I  received  your  second 
Letter  dated  Apr.  29.  In  the  alterations  you  propose  to 
be  made  in  Prop  xv  you  say.  Postquam  vero  descriptus 
est  arcus  PQ  in  Medio  resistente,  si  areso  PSQ  ajqualis 
capiatur  area  QSr,  erit  Qr  arcus  quem  tempore  reliquo 
corpus  describet  absq:  ulteriore  resistentia.  And  this 
would  be  true  if  the  velocity  of  the  body  at  Q  were  the 
same  as  when  the  arch  PQ  is  described  in  the  same  time 
in  Medio  non  resistente.  But  the  velocity  at  Q  being 
less  in  Medio  resistente  then  in  non  resistente,  the  arch 
Qr  will  be  less  in  the  same  proportion  &  thereby  reduce 
Rr  to  half  the  bigness,  &  make  the  resistance  to  the  cen- 
tripetal force  as  Rr  to  TQ.  I  hope  therefore  that  what  I 
have  written  on  the  other  page  of  this  Letter  is  right  & 


•  Probably  experiments  with  glass  balls  dropt  from  the  dome  of  St  Paul's  with  a 
view  to  test  his  theory  of  the  resistance  of  fluids.    See  Letter  XXV.  fin,  and  note. 


16  CORRESPONDENCE  OF 

that  yo'  difficulty  will  be  removed  by  the  words  &  motus 
corporis  cessabit. 

I  am  Yo" 

♦May  2^  I.  N. 

For  the  R"*'t  M'  Roger  Cotes  Professor 
of  MatJiematieh  and  Fellow  of  Triniti/ 
College  in  Cambridge. 


LETTER  VII. 
COTES  TO  NEWTON. 
S',  May  7.  1710 

I  received  Y'  Letter  by  y®  last  Post.  I  am  not  satis- 
fied that  Y"^  words  [et  motus  corporis  cessabit]  will  remove 
y*  difficulty  proposed.  They  cannot  in  my  opinion  be 
reconciled  with  Cor.  1.  I  acknowledge  Y'  objection  to  be 
just  against  those  words  of  mine  [erit  Qr  arcus  quem  tem- 
pore reliquo  corpus  describet  absq :  ulteriore  resistentia]  I 
remember  y*  I  inserted  them  into  my  Letter  as  I  was 
hastily  transcribing  y*  passage  from  another  paper  &  was 
myself  sensible  of  y®  mistake  soon  after  my  Letter  was 
gone  from  me.  The  alteration  which  I  proposed,  as  it 
stood  in  y^  Paper,  was  thus  J.  [Ponantur  qua3  in  superiore 
Lemmate  et  producatur  SQ  ad  V  ut  sit  SV  sequalis  SP, 
Tempore  quovis  in  Medio  resistente,  describat  corpus  ar- 
cum  quam  minimum  PQ,  &  tempore  duplo  arcum  quam 
minimum  PR ;  &  decrementa  horum  arcuum  ex  resistentia 
oriunda,  sive  defectus  ab  arcubus  qui  in  Medio  non  resis- 
tente iisdem  temporibus  describerentur,  erunt  ad  invicem 


•  The  post  mark  is  May  4. 

t  Though  addressed  under  thw  title  by  Newton  here,  and  in  the  remainder  of  the 
correspondence,  Cotes  was  not  ordained  until  three  years  afterwards,  (deacon,  May  29, 
1713,  priest  the  following  day). 

t  As  may  still  be  seen  in  tlie  MS.  of  Letter  V.(N6.  41),  the  words  "  Unde  etiam 

erit  decreraentum  arcus  PQ  xquale  dimidio  lineolaa  Kr,"  being  crossed  out  and 

replaced  by  those  which  we  have  printed  in  p.  13,  line  8,  &c.  "  Postquam  vero,  &c." 


NEWTOX  AND  COTES.  17 

ut  quadrata  temporum  in  quibus  gencrantur.  Est  itaq : 
(lecrcmcntum  arcus  PQ  pars  quarta  dccrcmenti  arcus  PR. 
Unde  ctiam  si  area)  PSQ  roqualis  capiatur  area  QSr,  erit 
decrementum  arcus  PQ  roquale  dimidio  lineola)  Rr\  adcoq: 
vis  resistentife  &  vis  centripcta  sunt  ad  inviccm  ut  lincoldo 
^Rr  Si  TQ  quas  simul  gencrant.]  I  am  yet  of  opinion  y* 
this  alteration  is  just  &  that  the  resistance  is  to  y*  centri- 
petall  force  as  }^Rr  to  TQ\  Your  own  objection  does  I 
think  if  You  carefully  consider  it  prove  it  to  be  so.  To 
avoid  further  misunderstanding  I  m  ill  put  down  my  demon- 
stration more  at  large  thus 

P  Q    K  U        r        L 


Tempore  quovis  in  Medio  resistente  describat  corpus  arcum 
quam  minimum  PQ  &  tempore  duplo  f  rcum  quam  minimum 
PR ;  &  decrementa  horum  arcuum  ex  resistentia  oriunda 
sive  defectus  [QK,  RL]  ab  arcubus  [PK,  PL]  qui  in  ]\Iedio 
non  resistente  iisdem  temporibus  describercntur  erunt  ad 
invicem  ut  quadrata  temporum  in  quibus  generantur ;  Est 
itaq:  decrementum  IQK]  arcus  PQ  pars  quarta  dccrcmenti 
RL  arcus  PR.  Unde  etiam  si  area)  PSQ  a)qualis  capiatur 
area  QSr  erit  decrementum  IQK]  arcus  PQ  tequale  dimidio 
lineola)  Rr.  [Nam  ut  SQ  ad  SP  ita  PK  ad  KL  ita  PQ  ad 
Qr  ita  dividendo  QK  ad  KL  -  Qr;  ergo  componendo  PK 
ad  PL  ut  QK  ad  {QK  +  KL  -  Qr  sive) rL,  unde  rL^ZQKi 
sed  erat  RL  =  ^QK,  itaq:  Rr  ^2  QK]  adcoq:  vis  resistentia) 
&  vis  centripeta  sunt  ad  invicem  ut  lineolao  QK  vel  ^7?r  & 
TQ  quas  simul  gencrant.  This  I  take  for  a  direct  demon- 
stration of  the  truth  of  what  I  proposed,  &  if  You  will  be 
pleased  to  consider  what  I  offered  at  y*  end  of  my  second 
Letter,  You  will  {find  J  that  also  to  amount  to  a  demon- 
stratio  per  absurdum.  I  did  there  assume  y*  proportion  of 
y®  Resistance  to  y®  Centripetall  force  to  be  as  ^iJr  to  TQ 
&  from  y*  assumption  I  deduced  a  consequence  whose  truth 
is  very  evident  upon  other  considerations.  But  if  You 
2 


18  CORRESPONDENCE  OF 

take  the  proportion  to  be  as  Rr  to  TQ  or  any  other  way 
diflferent  from  y*  of  ^Rr  to  TQ,  the  consequence  will  be 
as  evidently  false;  Therefore  the  proportion  can  be  no 
other  than  y*  of  ^72r  to  TQ.  You  say  in  Y'  Letter  y*  the 
8***  Corollary  may  remain  as  in  Y'  copy,  but  in  Y"^  copy 
there  are  no  alterations  of  y®  first  Edition.  That  You  may 
see  the  reason  I  had  for  the  alteration  I  proposed,  I  will 
put  JV  for  the  number  of  Revolutions,  T  for  y**  Time  of 
those  Revolutions,  D  for  y**  Density  of  the  Medium,  t  for 
y®  tangent  of  y®  Angle,  a  for  y®  secant  of  y®  same.     Now 

in  Cor.  6  You  put  N  as  t,   3r  as  -  or  «,  but  in  Cor  8  You 

put  iV  as  --  or  ^,   T  as  a.     The  alteration  which  I  proposed 

was  to  make  y®  8^''  Corollary  agree  w*^  y®  6^\  for  I  am 
satisfied  of  y*  truth  of  y®  6^\  In  my  first  Letter  I  took 
notice  of  two  mistakes  in  Prop  xvi,  You  have  consented  y* 
one  of  'em  may  be  amended  by  putting  l  -  ^w  for  ^n. 
The  other  You  seem  not  to  have  observed  which  was  y*  y® 
Density  is  not  reciprocally  as  SP'  but  reciprocally  as  SP : 

For  the  Resistance  in   P  being    as  ^^p«+i    ^"^  ^^ 

Velocity  in  P  as  -— ^^ ,  it  follows  y*  y''  Density  in  P  is  as 


\^in,  OS       ^         1  -Iw,  OS    ^,      ^      ,,     .    .  g 

op\ SP  "*'*  "^  OP.  SP' '  '^^  °'"''*y  ''"'"«  "'  y 

Resistance  directly  &  y®  square  of  y*  Velocity  inversly.  If 
You  consent  to  this  correction  as  I  do  not  doubt  You  will, 
I  desire  You  to  send  me  the  words  of  y®  Proposition  as 
You  would  have  them  altered.  It  seems  to  me  not  im- 
proper to  add  somewhere  in  this  xvi  Prop,  or  in  a  Corollary 
to  it  That  y®  force  of  resistance  is  to  y*  eentripetall  force 
as  1  -|w,  OS  to  OP 

Y'  &c. 


NEWTOX  AND  COTES.  19 

LETTER   VIII. 
NEWTON  TO  COTES. 

This  letter  is  either  misdated  or  was  an  unusually  long  time  in 
arriving  at  its  destination.  It  had  not  reached  Cotes's  hands  when  ho 
penned  his  short  note  of  May  17.  It  lias  no  address,  and  was  probably 
sent  by  a  private  hand,  perliaps  by  Bentloy. 

M'  Professor  Chelsea.  13  May.  1710. 

I  have  reconsidered  the  15***  Proposition  with  its  Corol- 
laries &  they  may  stand  as  you  have  put  them  in  yo*"  Let- 
ters. But  in  pag.  285  lin.  13  after  the  word  coincident  add 
the  words,  et  angulus  PS  V*  Jit  rectus. 

Let  the  16*^  Proposition  stand  thus 
Prop.  XVI.     Theor.  xii. 

Si  Medii  densitas  in  locis  singulis  sit  reeiproce  ut  dis- 
tantia  locorum  a  centro  immobili,  sitq:  vis  centripeta  reei- 
proce ut  dignitas  quajlibet  ejusdem  distantia):  dico  quod 
corpus  gyrari  potest  in  spirali  quro  radios  omnes  a  centro 
illo  ductos  intersecat  in  angulo  dato. 

Demonstratur  eadem  methodo  cum  Propositione  supe- 
riore.  Nam  si  vis  centripeta  in  P  sit  reeiproce  ut  distan- 
tia9  SP  dignitas  qua3libet  SP'"^^  cujus  index  est  n  +  1  ;  col- 
ligetur  ut  supra,  quod  tempus  quo  corpus  describit  arcum 
quemvis  PQ,   erit   ut   PQ  x  SP"],  et  resistentia  in  P  ut 

hoc  est,  ob  datum  ^-^ ,  reeiproce  ut  SP'^K     Et 

propterea  cum  velocitas  sit  reeiproce  ut  SP^*,  densitas  in 
P  erit  reeiproce  ut  SP. 

Corol.  1.     Resistentia  est  ad  vim  centripetam  ut 
1  -  ^  n  X  0*9  ad  OP, 


•  Cot€S  has  written  PVQ,  in  the  margin, 
t  Cotes  haa  written  SP^'  in  the  margin. 

2—2 


20  CORRESPONDENCE  OF 

Corel.  2.  Si  vis  ccntripeta  sit  reciproce  ut  SP^"^^,  erit 
1  -  ^  n  -  0,  adeoq :  rcsistcntia  et  densitas  Mcdii  nulla  erit, 
ut  in  Propositione  nona  Libri  primi. 

Corol.  3.  Si  vis  centripeta  sit  reciproce  ut  dignitas 
aliqua  radii  SP  cujus  index  est  major  numero  3,  rcsistcntia 
affirmativa  in  negativam  mutabitur. 

Pag.  289,  lin.  14.  ffor  data  lege,  read  data  velocitatis 
lege. 

Your  most  humble  servant 

Is.  Newton 


LETTER  IX. 
COTES  TO  NEWTON. 
S'.  Cambridge  May  l?***  1710. 

After  I  had  received  Y'  Letter  I  wrote  to  You  again 
about  a  week  ago,  about  some  difficultys  which  still  remain 
with  me.  The  Compositor  is  now  at  a  stand,  &  I  dare  not 
let  him  go  on  till  You  shall  be  pleased  to  send  me  Y"^  an- 
swer. 

Y'^  most  Obedient  and  Faithfull  Serv*. 

EooER  Cotes. 


LETTER  X. 
COTES  TO  NEWTON. 
S^  May  20.  1710 

I  thank  You  for  Y*"  Letter  which  came  very  season- 
ably. I  now  beg  leave  to  propose  to  You  some  few 
alterations  in  the  remaining  part  of  Y'  Copy.  Page  293. 
1:1  —  secunda  BFK  (per  Prop  xix)  pro  mensura  sua 
cequaliter  premuntur.  1:  4  Hac  pressione,  pro  mensura  sua, 


NEWTON  AND  COTES.  21 

&  insuper — Page  303.1:  G  —  nisi  forte  per  particulos  in- 
termedias  virtute  ilia  auctas  —  I  think  these  words  were 
better  left  out ;  for  as  I  apprehend  it,  they  alter  y*  case  of 
y®  Proposition.  1:  11  Ut  si  particula  unaqusoq:— quadrato- 
cubi  Densitatis.  I  think  also  y*  this  whole  Period  ought 
to  be  omitted,  the  two  propositions  containd  in  it  seeming 
to  me  to  be  erroneous,  fdess  I  mistake  the  sense  of  Y"^ 
words.  Page  304.  Coroll:  5  &  G  for  [quadratum  temporis 
directe]  You  have  substituted  in  Y'  copy  [quadrato-quad- 
ratum  temporis  directe]  I  find  >vritten  in  y*  margin  of  Y' 
book  by  a  different  hand  [quadr.  quadratum  temporis 
(credo)]  This  marginal  note,  not  Y'  own  judgment,  was  I 
beleive  y®  occasion  of  Y*^  making  the  alteration.  Page  308 
1 :  10  I  would  omit  y®  words  [si  verbi  gratia  arcus  alter 
sit  altero  duplo  major].  With  Y*^  leave  I  would  begin  the 
311  page  thus*.  [Est  itaq:  incrementum  vclocitatis  ut 
V-RSi  particula  ilia  temporis  in  qua  factum  est  conjunctim: 
Sed  &  velocitas  ipsa  est  ut  incrementum  contemporaneum 
spatii  descripti  directe  &  particula  cadem  temporis  inverse. 
Unde  cum  resistentia  (per  Hypothesin)  sit  ut  quadratum 
velocitatis,  incrementum  resistentia)  erit  (per  Lem :  ii)  ut 
velocitas  &  incrementum  velocitatis  conjunctim,  id  est,  ut 
momentum  spatii  &  F— 72  conjunctim;  atq:  adeo  si  mo- 
mentum —  In  my  Opinion  this  alteration  is  necessary  to 
make  the  Demonstration  accurate.  When  I  first  looked 
over  this  passage  upon  account  of  it  I  thought  the  whole 
construction  erroneous.  I  therefore  set  my  self,  after  the 
following  manner,  to  examine  how  it  ought  to  be,  which  I 
here  put  down  for  a  further  use  I  have  of  it.  Taking  .r, 
Xy  V  for  quantitys  analogous  to  the  Force  arising  from  y' 
gravity  of  y®  Pendulous  body,  the  force  of  resistance,  &  y* 


•  In  Prop.  XXIX.  Lib.  2.    "  Posito  quod  corpori  in  cycloide  oscillanti  resistitur  in 
duplicata  rationo  vclocitatis :  invenire  rcsistentiam  in  locis  singulis."    This  Proposition 

contains  the  geometrical  construction  of  the  expression  ^  (2k$+  I  -  2ka  + 1  «-»*"-•), 
n  being  the  first  arc  of  descent. 


22  CORRESPONDENCE  OP 

velocity  in  />,  tis  evident  y*  the  arch  CD  will  also  be  as  op, 
&  the  Fluxion  of  y*  space  BD  already  described  will  be  as 
-  0?.  If  therefore  t  be  put  for  y®  moment  of  time  in  which 
the  fluxion  of  y®  space  -  w,  the  fluxion  of  y*  velocity  v,  the 

fluxion  of  y*  resistance  x  are  generated ;  You  will  have 

• 

V  II  —^t    t>  II  d?  -  »  X  #     But  »  II  «v  &  therefore  x  ||  vv  || 


—  d7  X  a?  -  «  II  «^  -  aw.  Assuming  therefore  the  determi- 
nate quantity  [a]  of  a  just  magnitude  You  will  have  this 
Equation  ax  ^  xoo  ^  ooa.  To  construct  this  a3quation  I 
introduced  another  indeterminate  quantity  \\j\  putting 
jr  =  j9  -f  ^.r  +  ry  &  X  ^  qoo  ^-  ry  \  which  values  o^  x  h  x 
being  substituted  in  y®  former  roquation  I  obtained  this 
other  aqoo  +  ary  =  px  +  qxx  +  ryx  —  cox.     Then   puttmg 

ay 
9=1,   p  "  «,  I  had  the  two  following  a3quations  —  « ar, 

jif««  +  a?  +  ry  &  y®  construction  of  these  two  [equations 
agreed  intirely  with  Y'  own  Solution  of  y®  Problem*.  Being 
satisfied  by  this  Analysis  of  y®  truth  of  Y^  conclusion  I 
easily  saw  y*  my  former  difficulty  lay  in  y®  ambiguity  of  y® 
word  [data]  in  line  1  &  5,  &  y®  word  [detur]  in  line  6. 
which  I  think  may  be  remedied  by  the  alteration  which  I 
propose.  Page  312.  1:  21  I  would  leave  out  y®  word 
[quamproxime].  Page  313.  1 :  29  f  I  would  conclude  the 
Demonstration  thus  —  et  ex  aequo  perturbate  Fh  seu  MN 


*  The  analysis  and  construction  of  the  problem  will  be  found  in  Cotes*8  LogorMtria^ 

(Philos.  Trans.  Jan.— March,  1714,  pp.  40 — 42.    Harmonia  Mensurarum,  pp.  36 — 38.) 

t  In  Prop.  XXX.  Lib.  2.    This  Proposition  contains  the  geometrical  construction  of 

jUie  equation  ~  (a' -6')  =  k  i  v'ds  (h  being  the  first  arc  of  ascent),  which  is  ob- 
tained by  one  integration  from  the  equation  of  motion  —  — j—  =  -j .  a  —  fcu",   Cotes*8  sugr- 

gestion  leads  to  further  correspondence  (see  the  next  five  letters).  This  and  the  preced- 
ing proposition  may  give  us  an  idea  of  the  trouble  that  Newton  would  take  to  exhibit 
hi«  results  in  a  synthetical  form. 


NEWTON  AND  COTES.  23 

ad  Dd  ut  DK  ad  CF  seu  CM;  Ideoq:  summa  omnium 

MNx  CM,  id  est,  ^CAq--  ^Caq  seu  Aa  x^aB  icqualis 

erit   summj©  omnium  Dd  x  DK,   id  est,   arero  BKk  VTa, 

quam  rectangula  omnia  Dd  x  DK  seu  DKkd  componunt. 

Q.  E.  D   I  was  further   satisfied  y*  there   is  no  mistake   in 

the    Proposition    or  in    this   way   of  concluding  it  thus. 

Taking  a:  for  CD  &  %  for  DK  by  y*  abovementioned  ooqua- 

tion  ass  wa  «ai  ^  xof  it  appeares  y*  ax  +  ^ir^r  is  equall  to  the 

Fhient  of  za,    AVhence  I  conclude,  if  CL  be  taken  on  y* 

other  side  of  y*  point  C  equall  to  Ca  &  y*  ordinate  LQho 

erected,  y*  the  indeterminate  area  DKVTa  is  equall  to 

DK 

■j-prx  LQTa^-\CDq-\Caq  &  y*  whole  Area  BKVTa 

is  equall  to  ^CBq  ^\Caq  or  Aax  ^aB,  Page  315.  1:7  I 
would  read  thus — &  Ellipsis  aBRVS,  centro  O,  semiaxibus 
OB,  OV — 1:  22  Thus.  Nam  cum  Ellipsis  vel  Parabola 
aBRVS  congruat  —  1:  24  thus  alterutram  BRV  vel  VSa 
exccdit  figuram  —  lin.  penult,  I  would  leave  out  [quam- 
proxime].  pag.  319. 1 :  13  You  say  [cum  distantijo  particu- 
larum  Systematis  unius  sint  ad  distantias  correspondentes 
particularum  alterius,  ut  diameter  particula)  vel  partis  in 
Systemate,  priore  ad  diametrum  particula)  vel  partis  cor* 
respondentis  in  altero.]  The  same  thing  is  implied  in  the 
Demonstration  of  Prop.  32.  I  think  it  ought  also  to  be 
expressed  in  y®  words  of  y®  32  Proposition. 

Y'  &c. 


24  CORRESPONDENCE  OF 

LETTER   XI. 
NEWTON  TO   COTES. 
S'  CholBca.  May  30.  1710. 

The  corrections  w***  you  have  sent  me  in  your  Letter 
of  May  20  are  right.  But  I  fear  least  that  w*^^  relates  to 
Prop.  XXX  may  render  the  Demonstration  thereof  too 
obscure.  And  therefore  I  think  that  the  Proposition  with 
its  Demonstration  may  stand,  &  in  the  end  of  it,  after  the 
words  et  sic  eidem  a3quabitur  quam  proximo,  may  be  added 
these  two  sentences.  Quinimo  eidem  ajquabitur  accurate, 
ideoq:  conclusiones  pra3dicta9  sunt  accuratoe.  Nam  si  ad 
alteras  partes  puncti  C  capiatur  CL  jequalis  ipsi  Ca,  et 
erigatur  normaliter  Z,Q  ad  Curvam  aTVKB  terminata,  et 
pro  Curvaa  hujus  area  inde terminata  a  TVQL  ad  ordina- 
tam  LQ  applicata  scribatur  litera  M ;  area  inde  terminata 
a  T  VKD  roqualis  invenietur  quantitati  M,  DK  +  \  CD'i  -  ^ 
Ca^t  et  area  tota  aTVKB  quantitati  ^  C5' -  ^  Ca^,  seu 
A  a  X  \aB, 

The  Scholium  Generate  w*^**  in  the  former  edition  was 
printed  in  the  end  of  the  seventh  Section,  I  would  havr 
printed  in  the  end  of  the  sixt  section  next  after  Prop. 
XXXI.     But  it  wants  the  following  corrections 

Pag.  330.  lin  21,  22,  23  &c  read 

Scholium  generale 

Ex  his  Propositionibus  per  oscillationes  Pendulorum 
in  Mediis  quibuscunq:,  invenire  licet  resistentiam  Medio- 
rum.  Aeris  vero  resistentiam  investigavi  per  Expcrimenta 
sequentia.  Globum  ligneum  pondere  unciarum  Roma- 
narum  57-^^,  diamctro  digitorum  Londinensium  6^  fa- 
b{r}icatum,  filo  tenui  &c. 

Pag.  340.  lin  24,  25,  blot  out,  omnino  ut  in  Corollariis 
Propositionis  xxxii  demonstratum  est. 

Pag  341  lin.  18  for  resistentia  read  resistentioe. 

Pag  342  lin  21  blot  out,  Undo  cum  corpus  tempore, 
&  what  follows  to  the  end  of  the  words,  longitudincm 
duplam  30,556  digitorum. 


NEWTON  AND  COTES.  25 

Pag.  343  lin  6  for  pedum  read  digitorum.  lb  lin  8 
read  vi8  rcsistentico  eodem  tempore  uniformiter  continuata. 
lb  lin  12  read  posset. 

Pag  344  lin  13,  14  for  prima,  secunda,  tertia  read  ter- 

tia  quinta  septima  &  for  —  read  -^  , 
^  ^  193  193 

Pag.  345  lin  7,  25  for  dimidiata  read  subduplicata. 
lb.  lin.  8  read  Nam  ratio  7^  -  ^  ad  1  -  ^  seu  10  J  ad  1,  non 
longe  * 

Pag.  349  blot  out  the  lines  17,  18,  19,  20,  21,  22,  23. 
24,  25,  2G,  27 

Pag.  350  lin.  32  blot  out  Quare  eum  globus  aqueus  in 
aere  movendo  &  what  follows  to  the  end  of  the  words, 
probe  tamen  cum  prajcedcntibus  congruebat. 

Pag  354  blot  out  the  lines  11,  12,  13,  14,  16. 

In  the  beginning  of  Sect  vii  pag.  317  lin.  5  after  the 
words  similes  sint,  insert  the  words  &  proportionales. 

I  am 
Your  most  humble  servant. 

For  the  11°^  M'  CoTE»,  Professor  Is.   NewtoN 

of  Astronomy  t  ^  Fellow  of  Trinity 
College  in  Cambridge. 


LETTER  XII. 

COTES  TO  NEWTON. 
S'  Juno.  !•»  1710 

I  received  Your  Letter  last  night,  by  which  You  give 
Y"^  consent  to  the  other  alterations  which  I  proposed,  but 
seem  to  fear  least  y*  which  relates  to  Prop  xxx  may  render 
the  Demonstration  thereof  too  obscure  &  therefore  at  the 
end  of  y®  Corollary  after  the  words  [et  sic  eidem  aequa- 
bitur  quamproxime]  You  add  [Quinimo  eidem  lequabitur 


•  The  words,  "  lb.  lin.  8 non  longe",  are  crossed  out,  apparently  by  Cotes,  in 

pursuance  of  Newton's  orders  in  letters  X  III .  and  XV. 


26  CORRESPONDENCE  OP 

accurate,  &c]  I  beleive  You  designed  those  two  sentences 
to  be  inserted  pag.  314  Hn  18  after  the  words  [erit  etiam 
ajquale  areiB  BKTa  quamproxime,  &  y*  by  some  inadver- 
tency in  y  Letter  You  ordered  them  to  be  placed  in  page 
315  1 :  25  after  y*  words  [eidem  ajquabitur  quamproxime.] 
For  though  the  Proposition  it  self  &  the  first  part  of  the 
Corollary  ending  w*^  the  words  [omnino  ut  in  Propositione 
XXVIII  demonstratum  est]  be  accurate,  yet  as  I  understand 
it  the  remaining  part  of  the  Corollary  is  still  but  an  Ap- 
proximation, the  Ellipsis  &  Parabola  mentioned  in  the 
latter  part  of  y®  Corollary  not  agreeing  perfectly  with  the 
Figure  BKVTa ;  but  by  placing  those  two  sentences  as  in 
Y*"  Letter,  even  this  latter  part  of  the  Corollary  is  declared 
to  be  accurate.  I  beg  leave  to  express  my  self  freely  to 
You,  I  fear  it  will  be  look'd  upon  as  a  blemish  in  Y"^  book 
first  to  Demonstrate  y*  the  Proposition  is  true  &  afterwards 
to  assert  it  to  be  true  accurate,  I  am  of  opinion  y^  the 
alteration  which  I  proposed  pag.  313.  1 :  29  does  make  the 
Demonstration  compleat  to  an  intelligent  Header.  If  You 
think  good  it  may  be  put  down  more  at  large  some  such 
way  as  this  which  follows  —  et  ex  soquo  perturbate  {Fh  seu) 
MN  ad  Z)d  ut  DK  ad  {CF  seu)  CM\  ideoq:  summa  om- 
nium MN  X  CM  aaqualis  erit  summio  omnium  Dd  y.  DK, 
Ad  punctum  mobile  M  erigi  semper  intelligatur  Ordinata 
rectangula  asqualis  indeterminata)  CM^  qua?  motu  continuo 
ducatur  in  totam  longitudinem  Aa\  &  trapezium  ex  illo 
motu  descriptum  sive  huic  a^quale  rectangulum  Aa  x  ^aB 
rsquabitur  summge  omnium  MN  x  CM  adeoq:  summflD  om- 
nium 7)d  X  DK,  id  est,  arcfe 
BKkVTa.  q.e.d.  Or  if 
You  think  the  Demonstra- 
tion will  even  this  way  be 
too  obscure,  a  new  Scheme 
may  be  cut  with  y*'  addition 
of  y "  lines  here  drawn  &  the 


A  MN 


NEWTON  AND  COTES.  27 

Demonstration  may  end  thus  —  &  ex  roquo  perturbate 
(FA  sen)  MN  ad  Dd  ut  DK  ad  {CF  seu)  CMx  idcoq: 
MN  X  C3/  iequabitm*  Z)d  x  DK,  Erigantur  normales  AX, 
aZ  sequales  ipsis  AC^  aC  &  jungatur  XZ  occurrens  nor- 
malibus  il/F,  iW  in  F  &  /;  &  erit  MY  lequalis  ipsi  CM 
atq:  adeo  MN  x  YM  ajquale  />d  x  DK,  &  summa  omnium 
MN  X  YM,  id  est,  trapezium  A  a  ZX  sive  huic  rcquale  rect- 
angulum  Aa  x  ^aB  aequabitur  summoe  omnium  Dd  x  DK, 
hoc  est,  areoe  BKkVTa  q  e.d.  I  think  the  first  of  these 
two  way  3  sufficiently  clear ;  but  will  wait  for  Y'  resolution 

Y'  &c. 


LETTER  XIII. 
NEWTON  TO  COTES. 

I  have  reconsidered  your  emendation   of  the   xxx** 

Proposition  w*^  the  Demonstration  &  approve  it  after  the 

manner  you  propose  in  the  first  of  the  two  ways  set  do>vn 

in  your  Letter  of  June  1**.     In  my  last  letter,  as  I  was 

sending  it  away,  I  crossed  out  four  lines  &  should  have 

struck  out  also  these  words  relating  to  them  [lb.  lin.  8, 

read,  Nam  ratio  7^  -  ^  ad  1  -  ^  seu  10|  ad  1,  non  longe] 

I  am 

Yo'  most  humble  Servant 

Chelsea  Jun  8.  *  Is.  Newton. 

1710. 

I  thank  you  for  mending  the  Proposition 

Fw  the  Rot'"*  M'  Cotes  Professor  of 
Astronomy  and  fellow  of  Trinity 
College  in  Cambridge, 

This  letter  and  the  next  must  have  crossed  on  tlio  road. 


The  poHt  mark  is  Jun.  10* 


28  CORRESPONDENCE  OF 

LETTER  XIV. 
COTES  TO  NEWTON. 
S'  {Sunday}  June  U  1710. 

I  received  Y'  Letter  of  May  30**».  In  that  which  1 
wrote  to  You  by  y®  next  Post  instead  of  y®  alteration  in 
page  310.  1:  29  which  You  thought  too  obscure,  I  proposed 
the  following — et  ex  asquo  perturbate  Fh  seu  MN  ad  Del 
ut  Dk  ad  CF  seu  CM\  ideoq :  summa  omnium  MN  x  CM 
jequalis  erit  summa)  omnium  Dd  x.  DK.  Ad  punctum 
mobile  M  erigi  semper  intelligatur  Ordinata  rectangula 
ajqualis  indeterminatao  CM^  qua)  motu  continuo  ducatur  in 
totam  longitudinem  Aa\  k  trapezium  ex  illo  motu  descrip- 
tum  sive  huic  a)quale  rectangulum  Aay.^aB  cequabitur  sum- 
ma9  omnium  MN  x  CM  adeoq :  summae  omnium  Dd  x  DK^ 
id  est,  area9  BKkVTa.  q.e.d.  We  are  now  at  a  stand 
expecting  Y**  resolution.  You  gave  me  orders  in  Y'  Let- 
ter to  print  the  Scholium  Generale  after  y®  sixth  section  & 
sent  me  Y"^  corrections  of  it.  I  have  not  had  leasure 
since  I  received  Y"*  Letter  to  examine  all  the  Calculations 
of  y*  Scholium,  being  at  this  time  engaged  in  a  Course 
of  Experiments  &  having  some  other  Buisness  upon  my 
Hands,  but  I  have  read  it  over  &  considered  the  reason- 
ing of  it.  Page  345.  1:  26  You  say — {Si  longitudo  pen- 
duli....augeretur....arcuum  descensu  &  subsequente  as- 
censu  descriptorum}  differentia  0,4475  diminueretur  in 
ratione  velocitatis,  adeoq:  evaderet  0,4412.  I  do'nt  see  any 
reason  for  this  diminution,  but  think  it  ought*  to  remain 
0,4475  notwithstanding  y'  y®  length  of  y®  Pendulum  is  in- 
creased in  the  ratio  of  126  to  1221,  &  thereby  the  time 
increased  &  y*  velocity  diminished  in  y®  subduplicate  ratio 


•  This  is  also  clear  from  the  fact  that  the  equation  which  connects  the  arcs  of 
ilcsceut  and  ascent  (the  resistance  varying  as  the  square  of  the  velocity)  does  not 
involve  the  length  of  the  pendulum. 


NEWTON  AND  COTES.  29 

of  126  to  122 J.  You  will  see  my  reasons  in  what  follows. 
QutB  tradita  sunt  in  Prop  xxxi  &  ejus  Corollariis  obtinent 
ubi  Oscillationes  sunt  Isochronae.  At  si  oscillationum  tern- 
pus  quoq :  mutctur,  differentia  inter  areum  desccnsu  de- 
scriptum  &  areum  subsequente  ascensu  deseriptum  crit  ut 
resistentia  &  quadratum  temporis  conjunetim :  Nam  totius 
retardationis  particulne  singula?  ex  quibus  differentia  ilia 
componitur  sunt  in  hac  ratione  per  Lem :  x  Libr.  1, 

Unde  si  detur  longitudo  arcus  descripti  &  resistentia 
sit  ut  quadratum  veloeitatis ;  manebit  differentia,  utcunq : 
mutetur  tempus  atq:  adeo  veloeitas  eorporis  oscillantis. 
Nam  ob  datam  longitudinem  arcus  descripti,  tempus  erit 
ut  veloeitas  inverse ;  adeoq :  differentia  ilia,  cum  sit  ut 
resistentia  &  quadratum  temporis,  erit  ut  quadratum  velo- 
eitatis directe  &  quadratum  veloeitatis  inverse,  ac  proinde 
magnitudinem  datam  habebit. 

Idem  aliter.  (vide  Fig  Prop  xxx)  Manente  longitudinc 
areus  descripti  aD  augeatur  longitudo  Penduli.  Si  mutata 
longitudine  Penduli  mancret  Resistentia,  maneret  quoq : 
ratio  resistentise  ad  vim  gravitatis  atq :  huic  ajqualis  ratio 
Ordinata)  DK  ad  longitudinem  Penduli ;  adeoq :  augenda 
esset  Ordinata  DK  in  ratione  longitudinis  Penduli.  Verum 
ob  auctam  Penduli  longitudinem  augetur  quoq :  tempus  in 
ratione  ejus  subduplicata  adeoq:  diminuitur  veloeitas  in 
eadem  ratione  subduplicata,  &  resistentia  atq :  huie  propor- 
tionalis  Ordinata  DK  diminuitur  in  ratione  integra.  Itaq : 
Ordinata  DK  diminuitur  in  eadem  ratione  qua  prius  auge- 
batur  ac  proinde  manet  ejusdem  longitudinis,  manetq : 
adeo  magnitudo  area)  BKVTa  atq :  huic  roquale  rectangu- 
lum  Aa  X 1  ajB  &  differentia  ilia  Aa.  If  You  admit  of  this 
reasoning,  it  will  not  only  affect  this  place  in  page  345  but 
also  pag.  348  1:  1  and  Pag.  353.  1:  27  and  page  341.  1:  10. 
In  Page  346:  1:  23  You  cite  the  Corollarys  of  Prop  xl 
which  are  now  to  come  after  the  Scholium ;  there  being  no 
alteration  of  this  place  among  the  corrections  You  sent 
me,   I   do  not  know  whether  You   took  notice   of  it  & 


30  CORRESPONDENCE  OF 

have  therefore  mentioned  it  to  You.  Page  348  1:  7  &c. 
You  seem  to  confound  the  Differentia  arcuum  with  y®  Be- 
sistentia  Glohi;  the  former  is  represented  hy  AV  •{-  CV^  & 
y*  latter  ought  I  think  to  be  represented  by  ^-^K+fCF*. 
I  desire  Y'  answer  to  tfiitXetter,  when  I  receive  it  I  will 
examine  Sc  alter  y®  Calculation,  if  there  be  occasion,  ac- 
cording to  Y'  direction  _ 

Y'-&c. 


LETTER  XV. 
NEWTON  TO   COTES. 

I  sent  you  a  letter  the  last  week  in  w*^^  I  approved  your 
correction  of  Prop  xxx  w*^  its  demonstration  according  to 
the  first  of  the  two  ways  w'^^  you  sent  me  in  your  Letter  of 
June  1**  &  have  now  repeated  in  yours  of  June  11*^  w*^^  I 
received  last  tuesday  morning  {the  13th.}  I  thank  you 
for  that  correction.  In  my  last  letter  but  one  I  crossed 
out  four  corrections  w*^^  I  had  'wrote  down  in  it,  &  should 
have  crossed  out  a  fift  w*^^  related  to  those  four  &  was  in 
these  words.  Pag.  345  lin.  8  lege,  Nam  ratio  7^  -  -J-  ad 
1  -  1  seu  lOf  ad  1. 

The  correction  in  the  Scholium  p.  345  lin  26,  sent  me 
in  your  last,  is  right,  &  I  beg  the  favour  that  you  would 
alter  the  calculations  accordingly. 

In  pag.  346  lin  23  strike  out  the  words  et  propterea 
(per  corollaria  Prop  xl  Libri  hujus)  resistentia  quam  Globi 
majores  &  velociores  in  acre  movendo  sentiunt  &  so  on  to 
the  end  of  the  sentence 

In  pag.  348  lin  7,  14,  15,  16  for  yl  &  C  put  other 
letters*  suppose  F  &  G,  writing,  Designet  jam  FV -{- GV^ 
resistentiam  Globi  &c  because  AV  +  CV*  was  used  before 
for  the  differentia  arcuum. 


"  Not  adopted.    Cotes  altered  this  part  of  the  Scholium  in  conformity  with  his 
remarks  at  the  close  of  the  preceding  Letter. 


NEWTON  AND  COTES.  81 

You  need  not  give  your  self  the  trouble  of  examining 
all  the  calculations  of  the  Scholium.     Such  errors  as  do 
not  depend  upon  >vrong  reasoning  can  be   of  no  great 
consequence  &  may  be  corrected  by  the  Reader. 
I  am  w*^  many  thanks 

S"^  Your  most  humble  servant 
Chelsea  June  15***  1710  Is.  Newton. 

For  tlie  il"**  M'  Cotes  Professor  of  Astronomy 
^  Fellovj  of  Trinity  College  in  Cambridge 
Cambridgeshire. 


LETTER  XVI. 
COTES  TO  NEWTON. 
S"^  Juno  30  1710 

We  have  now  finished  all  Y'  Copy  &  y*  Scholium  Ge- 
nerale.  I  received  Y*"  Letter  of  June  15***  in  which  You 
consent  to  y®  alterations  y*  I  proposed  in  y*  Scholium.  I 
have  examined  the  whole  Calculation  &  done  it  anew  where 
I  thought  it  necessary.  The  discourse  it  self  is  also  a  little 
altered  in  those  places  which  I  mentioned  in  my  last,  as 
You  will  perceive  by  y®  2  inclosed  sheets  {Oo  &  Pp}. 
They  are  not  yet  printed  ofl*,  but  will  stay  for  Your  cor- 
rections if  You  shall  think  fit  to  make  any,  I  could  wish 
You  would  be  pleased  to  look  *em  over,  for  I  fear  I  may 
possibly  have  injured  You.  The  Press  being  now  at  a 
stand  I  will  take  this  oportunity  to  visit  my  Relations  in 
Lincolnshire  &  Leicestershire.  I  hope  I  shall  come  back 
again  to  College  in  5  or  6  weeks.  \VTien  I  return  I  will 
Avrite  to  You  to  desire  y*  remaining  part  of  Y'  Copy. 

Y'&c. 


32  CORRESPONDENCE  OP 

LETTER  XVII. 

NEWTON  TO  COTES. 

S'  Chelsea  Juno  31*.  1710. 

I  received  yours  of  June  30  this  noon  with  the  two 
inclosed  proof  sheets,  &  have  perused  them  without  ob- 
serving any  faults  except  in  the  last  page  of  the  second 
sheet  lin  28  where  vires  autem  motrices  should  be  vires 
autem  acccleratrices.  And  in  the  preceding  page  (pag. 
295)  upon  reconsidering  the  words  of  Prop,  xxxiii,  I  think 
the  words  will  be  better  understood  if  they  run  as  in  the 
former  edition,  viz*  lisdem  positis,  dico  quod  Systematum 
partes  majores  rcsistuntur  in  ratione  composita  &c.  The 
remaining  part  of  the  copy  will  be  ready  against  your 
return  from  the  visit  you  are  going  to  make  to  your 
friends.  I  am  w*^  my  humble  service  to  yo""  Master  & 
many  thanks  to  yo*^  self  for  your  trouble  in  correcting 
this  edition,  S*" 

Yo'^  most  humble  servant 

For  the  R"*  M'  Cotes  Professor  of  Is.   Newton. 

Astronomi/  ^  Fellow  of  Trinity 
College  in  Cambridge. 

"Wanting— a  Letter  from  M'.  Cotes  to  S'.  Is.  Newton— dated  ll'\ 
July  1710..."  Note  by  Mr  Howkins:  who  liero  and  elsewhere  informs 
tis  of  the  absence  of  letters,  the  dates  of  which  wo  should  otherwise 
(from  any  thing  that  can  be  gathered  from  the  correspondence  itself) 
have  been  in  ignorance  of.  Smith  had  probably  made  a  list  of  all  the 
letters,  and  Ilowkins  on  collecting  and  arranging  them  when  they  came 
into  his  possession,  noted  such  as  were  missing. 


"  This  means  July  1.  Newton  was  not  always  exact  in  dating  his  letters.  It  may 
serve  to  make  the  lapius  in  this  case  less  incredible,  though  most  persons  will  be  able 
to  supply  instances  for  themselves,  if  I  mention  that  the  letters  which  were  delivered 
by  the  morning  post  at  Cambridge,  on  July  1,  1847,  were  stamped  June  31,  and 
that  one  of  them,  written  the  previous  day  by  a  distinguished  prelate,  was  dated 
April  30. 


NEWTON  AND  COTES.  S3 

LETTER  XVIII. 
COTES  TO  NEWTON. 
S^  Monday  Sopt  4***  1710. 

I  hope  to  be  at  Cambridge  again  oa  Wednesday  next. 
I  have  been  somewhat  longer  in  y*  Country  y"  I  at  first  in- 
tended, I  hope  You  will  exeuse  me :  For  the  future  I  shall, 
I  hope,  be  ready  without  any  further  intermission  to  attend 
upon  y®  Edition  of  Y'  Principia.  I  desire  You  to  send 
the  remaining  part  of  Y""  Copy  assoon  as  You  ean. 
Y'  most  Humble  Servant 

For  S'  Isaac  Newton  at  his  House  Roger   Cotes. 

near  the  College  in  Chelsea  near 

London 
On  his  return  to  college  Cotes  would  find  that  a  slight  change  had 
just  been  introduced  into  the  daily  habits  of  the  place,  which,  for  the 
sake  of  those  for  whom  the  fact  may  possess  an  interest,  may  bo 
recorded  here.  "  Sept.  4,  at  night  Dr  Smith  the  Senior  Dean  began 
the  custom  of  standing  at  grace,  chiefly  upon  my  sollicitation,  and  all 
the  HaJl  readily  complied  with  the  alteration."     Rud's  Diary. 


LETTER  XIX. 

NEWTON  TO  COTES. 
S' 
This  Letter  accompanies  the  next  part*  of  the  Prin- 
cipia.    I  am  not  certain  that  you  have  all  y®  cutts  in  wood, 
but  if  any  be  wanting  pray  send  me  a  draught  in  paper  of 
what  is  wanting  &  I'le  get  them  cut  {in}  wood. 

I  am  S' 
Yo'^  most  humble  Servant 
Chelsea.  Sept  13  1710.  Is.  Newton. 

For  the  R"**  M'  R.  Cotes  Professor  of 
Mathematlcks  ^  Fellow  of  Trinity  CoU 
lege  in  y*  University  of  Cambridge. 


•  Beginning  at  p.  321,  with  part  of  Cor.  2,  Prop,  xxxiii.  Lib.  2,  and  ending  at 
|».  432,  with  Prop.  xxiv.  Lib.  3. 


34  CORRESPONDENCE  OP 

LETTER  XX. 
COTES  TO  NEWTON. 
S'.  Sept.  21"*  1710 

I  have  received  y*  second  part  of  Y'  Copy,  there  arc 
wanting  only  two  wooden  cutts  which  I  can  get  done  at 
Cambridge.  I  have  read  over  what  relates  to  y®  resistance 
of  Fluids,  I  thank  You  for  the  satisfaction  I  have  received 
in  seeing  y*  Theory  so  perfectly  compleated.  I  confess  I 
was  not  a  little  surprized  upon  y®  first  reading  of  Prop. 
36* ;  but  I  now  begin  to  be  better  reconciled  to  it.  One 
of  my  greatest  difficulties  was  an  Experiment  of  Mons^ 
Marriotte  which  he  says  (page  245  Traite  du  Mouvment 
des  Eauxf)  he  often  repeated  with  great  care.  By  his  Ex- 
periment I  concluded  y*  y*  Velocity  of  y®  effluent  water 
was  equall  to  y*  gotten  by  an  heavy  body  falling  but  from 
half  y*  Height  of  y*'  Vessel.  He  tells  us  y*  14  Paris  Pints 
of  water  were  evacuated  in  a  IMinute  of  time  through  a 
circular  aperture  of  J  Inch  diameter,  the  altitude  of  y* 
Vessel  being  13  feet.  He  describes  the  Paris  pint  to  be  y® 
35*^  part  of  y*  Cube  of  y*  Paris  foot. 

Therefore    the    water    evacuated   in    a    second   was 

or Cubick  inches.   The  Area  of  y®  aperture 

35  ^  60  ^5  •^      ^ 

was ~  inches.     Hence  y*  length  of  a  Cylinder  equall 

in  magnitude  to  y®  evacuated  water  &  having  y®  above 

,     .  n       ..      T>     .      •       14  X  l6  X  2  X  144. 

mentioned   Aperture   for   its    Basis   is    — 

^  11  X  25 

Inches,  and  this  length  is  y*  space  described  in  a  second 
of  time  with  y*  uniform  velocity  of  y®  water  as  it  passes 

•  Making  the  velocity  of  efflux  of  a  fluid  through  an  orifice  in  the  base  of  a  cylindri- 
cal vessel  to  be  that  due  to  the  height  of  the  surface  of  the  fluid  above  the  orifice,  a 
result  first  stated  by  Torricelli,  and  adopted  by  him  as  a  principle,  (  De  motn  Projecto- 
rum,  Florcnt.  1644.  p.  191.)  In  the  Ist  Ed.  (Prop,  xxxvii.)  the  velocity  had  been  made 
that  due  to  half  the  height.  The  MS.  of  tl»e  Prop,  which  Cotes  had  before  him  when  he 
wrote  this  Letter  is  wanting. 

t  New  Edit.  Paris  1700.    The  Ist  Ed.  is  dated  1686. 


NEWTON  AND  COTES.  35 

through  the  aperture.  The  space  described  in  a  second 
of  time  with  y*  uniform  Velocity  acquired  by  any  falling 
body  in  y*  same  time  is  (according  to 

Tlio  remainder  of  this  letter  is  wanting:  at  the  point  where  it  breaks 
off  Cotes  is  saying  that,  according  to  Iluygens's  penduhim  experiments, 
the  velocity  generated   by   gravity   in  l"07)  =  30j    Paris   feet;    and 

.'.  the  height  duo  to  the  velocity  of  efflux  ■=  \  — Tr—^i — )  •  t-  in  feet, 

which  lies  between  GJ  and  6J. 

Some  of  the  contents  of  the  deficient  part  of  the  letter  are  men- 
tioned in  Newton's  letter  of  ^larch  24,  171?.  The  letter  which  was 
actually  sent  will  probably  bo  found,  with  others  that  are  wanting  to 
complete  this  correspondence,  in  the  Portsmouth  Collection. 

The  above-mentioned  re:?ult  of  IMariotte's  experiment  had  been 
brought  before  the  notice  of  the  Royal  Society  by  Ilalloy  at  their  meet- 
ings on  March  18  and  25,  1C91.  On  the  latter  day  an  experiment 
(inaccurately  described  in  the  Journal  Book)  was  made,  in  which  the 
jet  was  found  to  rise  "  far  above  the  middle  of  the  height  of  the  liquor, 
whence  it  is  to  bo  noted  that  there  is  a  mistake  in  the  .37th  Prop,  of 
Mr  Newton's  2nd  Book,  whereof  it  was  ordered  that  Mr  Newton 
should  be  certified."  (It  was  probably  in  consequence  of  this  sugges- 
tion that  Newton  revised  the  Proposition,  and  put  it  into  the  shape 
which  surprised  Cotes.)  On  Hallcy's  recommendation,  further  experi- 
ments were  made  with  a  view  to  ascertain  the  cause  of  the  discrepancy 
between  the  results  derived  from  the  observed  height  of  the  jet  and 
the  quantity  of  fluid  discharged,  but  they  only  served  to  establish  the 
fact,  which  remained  unaccounted  for  until  Newton  (towards  the  end 
of  1710  and  beginning  of  1711),  compelled  by  the  statements  of  Cotes's 
letter  of  Oct.  5,  I7IO,  to  investigate  the  subject  afresh,  found  the  true 
explanation  in  the  difference  between  the  velocities  at  the  orifice,  and 
at  that  part  of  the  vein  of  issuing  fluid  where  it  ceases  to  contract. 
See  Newton's  letter  of  March  24,  171t.  For  an  account  of  what  has 
been  done  in  this  branch  of  Hydrodynamics,  since  Newton's  time,  see 
Rennie's  Report  to  the  British  Association  (meeting  1833)  with  the 
works  there  referred  to,  to  which  add  Navier's  Resume  des  Le9on8... 
sur  r Application  de  la  Mdcanique.-.Part.  2,  1838;  and  D'Aubuisson's 
Traite  d'Hydraulique,  2nd  Ed.  1840. 


3—2 


36  CORRESPONDENCE  OF 

LETTER  XXI. 
NEWTON  TO  COTES. 
S'  London.  Sept.  30.  1710 

Since  the  receipt  of  your  Letter  I  have  been  removing 
from  Chelsea  to  London,  w*^**  has  retarded  my  returning  an 
answer  to  yo"^  last.  I  have  not  seen  Mariots  book  con- 
cerning the  motion  of  running  water,  but  certainly  there  is 
something  amiss  in  his  experiment  w*^^  you  give  me  an  ac- 
count of.  ffor  I  have  seen  this  experiment  tried  &  it  has  been 
tried  also  before  the  Royal  Society  *,  that  a  vessel  a  foot  & 
an  half  or  two  foot  high  &  six  or  eight  inches  wide  with  a 
hollow  place  in  the  side  next  the  bottom  &  a  small  hole  in 
the  upper  side  of  the  hollow,  being  filled  with  water ;  the 
water  w*^^  spouted  out  of  the  small  hole,  rose  right  up  in  a 
small  streame  as  high  as  the  top  of  the  water  w*^^  stagnated 
in  the  vessel,  abating  only  about  half  an  inch  by  reason  of 
the  resistance  of  the  air.  The  small  hole  was  made  in  a 
thin  plate  of  sheet  tin  and  well  polished,  that  the  water 
might  pass  th{r}ough  it  with  as  little  friction  as  possible. 
It  was  about  the  bigness  of  a  hole  made  with  an  ordinary 
pin. 

The  corrections  you  have  made  are  very  well  &  I  thank 
you  for  them,  &  am  glad  that  the  Theory  of  the  resistance 
of  fluids  does  not  displease  you  provided  the  xxxvi*^  Propo- 
sition be  true,  as  I  think  it  is. 

Direct  your  next  Letters  to  me  in  S*  Martins  street 
neare  Leicester  fields. 

I  am  Yo*"  most  humble  Servant 

For  the  R"**  M'  Cotes  Professor  of  Astro-  Is.   New  TON 

nomt/f  ^  Fellow  of  Trinity   College  in 
Cambridge  in  Cambridgeshire. 


•  An  experiment  of  this  kind  attended  with  the  same  result  was  tried  by  Ilooke  at  a 
meeting  of  the  Royal  Society,  April  1,  1691.  The  velocity  of  efflux  was  also  the  sub- 
ject of  experiment  or  discussion  at  several  other  meetin(r<*  in  that  year.  See  the 
Journal  Book,  March  18,  25.    April  8,  22. 


NEWTON  AND  COTES.  37 

*•  Wanting,  two  letters  from  Mr  Cotes  to  Sir  Isaac  Newton,  dated 
5th  and  2()th  Oct.  1710,  concerning  Prop,  xxxvi.  Lib.  ii. 

I  remember  to  have  Been  the  wliolo  of  this  Prop,  as  it  ia  now 
printed  in  the  2nd  Edition,  fairly  written  in  Mr  Cotcs's  own  hand ; 
but  I  fear  it  is  lost,  or  inadvertently  destroyed;  as  I  cannot  find 
it  now. 

E.  HOWKINS,  1770.** 


LETTER  XXn. 
NEWTON  TO  COTES. 


I  received  both  your  Letters  &  am  sensible  that  I  must 
try  three  or  four  experiments  before  I  can  answer  your 
former*.  My  time  has  been  taken  up  partly  with  remov- 
ing to  this  house,  partly  with  journeys  about  purchasing  a 
house  t  for  the  Royal  society  &  partly  w*^  settling  some 
matters  in  the  Mint  in  order  to  go  on  w*^  y®  coynagej  that 
I  have  had  no  time  to  take  these  matters  into  considera- 
tion but  hope  w*4n  a  fortnight  to  try  the  experiments  & 


*  Of  Oct.  5,  containing  probably,  among  other  thini^,  experiments  Cotes  had  been 
tiiaking  on  fluids  issuing  from  an  orifice  in  a  vessel,  and  which  went  to  confirm 
Mariotte's.    See  letter  of  Newton  in  Macclesfield  Corrtip.  ii.  437. 

+  In  Crane  Court.  The  Society  met  there  for  the  first  time  on  Nov.  8,  having 
previously  held  tlieir  meetings  at  Grcsham  Colleg'e.  The  change,  as  is  usual,  was 
opposed  by  ?ome  of  the  members.  In  1782,  Government  assigned  the  Society  apart- 
ments in  Somerset  House.  See  Weld's  Hist,  uf  Royal  Soc.  i.  389,  seqq. ;  Kllis*s  Let- 
ters of  Eminent  Literary  Men,  346,  (where  C.  Wren's  letter  should  evidently  be  dated, 
1711.) 

I  The  following  table  of  gold  and  sUver  coined  yearly  from  Christmas,  1708,  to 
Christmas,  1713,  will  shew  approximately  the  times  at  which  Newton's  duties  at  the 
Mint  would  experience  a  pressure  during  the  years  over  which  this  correspondence  on 
the  Principia  extends. 


1709 

OOt.D. 

lbs. 
2468 

8II.VF.U. 

lbs. 
25423 

(in  preceding  year,  3751) 

1710 

3716 

817 

1711 

9324 

24768 

1712 

2855 

1784 

1713 

13137 

2333 

Macclesfield  Corre$p.  u.  434. 
In  the  beginning  of  March  1711  the  Royal  Society  changed  their  day  of  meeting  to 
Thursday  at  4,  the  President  "  being  obliged  to  attend  the  Mint  on  Wednesdays.** 


88  CORRESPONDENCE  OP 

settle  the  matters  in  doubt  &  beg  the  favour  that  you  will 
let  the  press  stay  till  you  hear  from  me  again. 

1  am  Yo'  most  faithfuU  friend 

Si  humble  Servant 

London.  Octob  27*.  1710.  Is.  Newton 

For  the  R°*  M'  Cotes  Professor  of  Astro- 
nomi/f  at  kU  chamber  in  Trinity  College 
in  Cambridge. 


LETTER    XXIII. 
NEWTON   TO    COTES. 
S*  Martins  street  by  Leicester  ffiolds.  Mar.  24*^  171?. 

I  send  you  at  length  the  Paper  for  w^^  I  have  made 
you  stay  this  half  year.  I  beg  your  pardon  for  so  long 
a  delay.  I  hope  you  will  find  the  difficulty  cleared,  but  I 
know  nott  whether  I  have  been  able  to  express  my  self 
clearly  enough  upon  this  difficult  subject,  &  leave  it 
to  you  to  mend  any  thing  either  in  the  expression  or 
in  the  sense  of  what  I  send  you.  And  if  you  meet  w*^ 
any  thing  w^^  appears  to  you  either  erroneus  or  dubious, 
if  you  please  to  give  me  notice  of  it  I  will  reconsider 
it.  The  emendations  of  Corol.  2  Prop  38  &  Prop  40  are 
your  own.  You  sent  them  to  me  in  yours  of  Sept.  21, 
1710,  &  I  thank  you  for  them.  That  you  may  have  the 
clearer  Idea  of  the  experiments  in  the  beginning  of  the 
inclosed  paper,  let  ABCD  represent  a  vessel  full  of 
water  perforated  in  the  side  with  a  small  hole  EF  made 


•  Post  Mark  28. 

t  It  18  doubtful  whether  the  "  not"  has  not  been  added  by  another  hand.  If  it  be 
in  Newton's  handwriting,  it  is  about  the  nearest  approach  to  an  instance  of  his  crossing 
a  *  t',  that  I  remember  to  have  seen. 


NEWTON  AND  COTES.  39 

in  a  very  thin  plate  of 
sheet  tin.  And  con- 
ceive that  the  water 
converges  towards  the 
hole  from  all  parts  of 
the  vessel  &  passes 
it"  through  the  hole  with 
a  converging  motion  &  thereby  grows  into  a  smaller  stream 
after  it  is  past  the  hole  then  it  was  in  the  hole.  In  my  trial 
the  hole  EF  was  1^^^'  of  an*  inch  in  diameter  &  about  half  an 
inch  from' the  hole  the  diameter  of  the  stream  US*  was  but 
^  of  an  inch.  And  therefore  the  streame  had  the  same 
velocity  as  if  it  had  flowed  directly  out  of  a  hole  but  JJ  of 
an  inch  wide.  And  so  in  Marriotts  experim*  the  stream  had 
the  same  velocity  as  if  it  had  flowed  directly  out  of  a  hole 
but  -^^  of  an  inch  wide.  In  computing  the  velocity  of  the 
water  w*^^  flows  out  we  are  not  to  take  the  diameter  of  the 
hole  for  the  diameter  of  the  streame,  but  to  measure  the 
diameter  of  the  streame  after  it  is  come  out  of  the  hole  & 
has  formed  itself  into  an  caven  &  uniform  stream.  And 
the  velocity  thus  found  will  be  what  a  body  would  get  in 
falling  from  y*  top  of  the  water  :  as  is  manifest  also  by  the 
distance  CG  to  which  the  stream  will  shoot  it  self,  &  also 
by  the  stream's  ascending  as  high  as  the  top  of  y*  water 
stagnating  in  the  vessel,  if  the  motion  be  turned  upwards. 

I  am 
Your  most  humble  &  most  obliged  Servant 

For  the  R"*^  M'  Roger  Cotes  Professor  of  Is.   Newton 

Astronomy  at  his  Cliamber  in  Trinity 
College  in  tJie  University  of  Cambridge, 


•  JRS  b  the  diameter  of  the  "sectio  venae  contractae,"  (a  terra  first  used  by  Jurin, 
Philosoph.  Transact.  Sept.— Oct.  1722,  p.  185;  and  afterwards  by  Dan.  Bernoulli, 
Hydrodynam.  p.  65.  Jurin  also  uses  **  vena  contracta"  to  denote  the  same  thing,  and 
the  expression  b  still  retained  in  works  on  Hydrostatics,  though  differently  defined  by 
different  writers,  most  of  tliem  describing  it  as  that  part  of  the  issuing  fluid  between 
the  orifice  and  the  section  whose  diameter  b  RS.) 


40  CORRESPONDENCE  OF 

The  "  Paper"  mentioned  in  the  above  letter  seems  to  have  consisted 
of  four  folio  sheets,  and  to  have  included  from  Prop,  xxxvi.  to  Prop. 
XXXIX.  with  part  of  Prop,  xl.,  and  a  page  of  corrections  (No.  111.)  to 
be  made  in  the  conclusions  of  "the  Experiments  set  down  in  the 
Scholium  to  the  40th  Proposition  sent  you  formerly."  The  first  three 
leaves  are  wanting,  the  portion  which  is  preserved  beginning  with  the 
latter  part  of  the  37th  Prop.  (No.  72). 

Tliero  were  several  things  in  this  **  Paper"  which  did  not  satisfy 
Cotes.  (1)  His  "  difficulty"  about  the  SGth  Prop,  was  not  yet  com- 
pletely removed.  This  probably  led  to  his  (missing)  letter  of  March 
31,  which,  if  no  other  letter  passed  between  them  in  the  interval, 
brought  him  a  satisfactory  answer  from  Newton.  This  answer,  sent 
apparently  in  a  parcel  from  Bcntley,  is  also  wanting.  (2)  Besides 
making  other  alterations  of  a  minor  character.  Cotes  has  crossed  out 
what  is  left  of  Prop,  xxxvii.,  and  written  the  Proposition  out  on 
another  piece  of  paper  (Nos;=?0^71)  as  it  now  stands  in  the  2nd  Ed. 
with  this  note  at  the  top  :  "  Print  this  instead  of  wliat  is  blotted  out 
in  Prop,  xxxvii."  He  has  also  modified  part  of  the  Scholium  of  this 
Prop,  though  not  to  the  extent  that  Ilorsloy  {Ncwtoni  Op.  ii.  412) 
attributes  to  him.  He  has  drawn  his  pen  through  almost  the  wliole  of 
Prop,  xxxviii.  and  part  of  its  2nd  Cor.  and  re- written  the  parts  struck 
out  as  they  now  stand  in  the  2nd  Ed.  These  were  the  materials  of  his 
letter  of  June  9.  See  introductory  remarks  to  the  fragment  of  that 
letter. 


"Wanting,  a  letter  from  Mr  Cotes  to  Sir  Isaac  Newton,  dated  31  st 
March  1711.   Another  dated  4th  June  1 711."   Note  by  Mr  Howkins. 


LETTER    XXIV. 
NEWTON  TO   COTES. 

Yo"  of  June  4***  I  received  the  next  day  &  thank 
you  for  it.  I  am  glad  you  received  what  D*"  Bentley  sent 
you  &  that  you  think  the  difficulty  removed,  except  what 
you  mention  about  the  manner  of  delivering  y®  37*^  Pro- 
position, ffor  clearing  the  sense  of  the  first  &  second 
Paragraphs,  these  words  may  be  added  to  the  end  of  the 
second  Paragraph  after  the  word  locatum.     Circellus  av^em 


NEWTON  AND  COTES.  41 

sustinendo  vim  aquce  defluentis  mimiet  ejua  velocitatem,  idq : 
in  ratione  qua  minuit  spatium  per  quod  aqua  Jam  transit. 
Nam  {per  Cas,  5.  Prop  xxjcn^  ^  efus  CoroL  6)  aqua  jam 
transihit  per  spatium  annulare  inter  circellum  fy  latera  canalis 
eadem  velocitate  qua  prius  transibat  per  canalis  cavitatem 
totam^. 

And  a  little  after  where  I  have  these  words  [augcatur 
velocitas  circelli  in  eadem  ratione  et  resistentia  ejus  auge- 
bitur  in  ratione  duplicata]  may  be  written  these  [augeatur 
veloeitas  circelli  in  eadem  ratione  &  resistentia  ejus  auge- 
bitur  in  eadem  ratione  bis,  nempe  semel  ob  auctam  quan- 
titatcm  aquas  in  quam  circellus  dato  tempore  agit  &  semel 
ob  auctum  motum  quem  circellus  in  singulas  aqua3  partes 
imprimit.  Nam  partes  fluidi  similibus  motibus  agitabuntur 
atq:  prius  sed  velocioribus  et  minore  tempore*.] 

But  since  you  are  considering  how  to  set  this  xxxvii^** 

Proposition  in  a  cleare  light  I  will  suspend  saying  any 

thing  more  about  it  till  I  see  your  thoughts.     I  am 

Yo'  humble  servant 

London  7*^  June  1711  Is.  Newton, 

For  the  R"**  M'  RooEit  Cotes  Pro/mor  of 
Astronomy  at  hia  Chamber  in  Trinity 
College  in  Cambridge 


LETTER  XXV. 
COTES  TO  NEWTON. 


This  is  only  the  concluding  part  of  a  letter,  which  a  note  by  Mr 
Howkins  states  to  have  been  dated  June  9th,  I7II.  In  the  words  with 
which  this  fragment  opens,  Cotes  is  giving  his  emendation  of  Prop. 
XXX vni.  as  it  stood  in  the  "  Paper"  wliich  Newton  sent  liim  in  liis 
letter  of  Marcli  24.  The  former  part  of  the  letter  must  have  contained 
Prop,  xxxvii.  in  the  form  in  which  Cotes  had  at  last  put  it,  and  also 


Not  adopted. 


42  CORRESPONDENCE  OP 

hia  modification  of  the  construction  in  the  Scholium,  whore  Newton 
had  made  the  latug  rectum  of  the  smaller  parabolas  S  AB  and  tliat  of 
the  others  32  CD. 

Ilorsley  saw  some  of  Cotes's  actual  letters  in  the  Portsmouth  Col- 
lection, and  this  of  the  9th  of  June  among  others.  In  a  note  on  the 
37th  Prop,  he  says  (ii.  404) :  Iliec  demonstratio  a  Cotesio  tota  est. 

Ilorsley  also  says  tliat  the  Oth  Cor.  of  Prop,  xxxvii.  is  due  to 
Cotes,  and  that  in  the  letter  in  which  it  was  sent  (what  the  date  of  the 
letter  was  docs  not  appear  certain)  after  explaining  this  Corollary  he 
adds:  "Hoc  Corollarium  lucem  aliquam  tuo  in  Corollario  dccimo 
guantum  sentio  offundere  possit."  This  6th  Corollary  in  Cotes's  hand 
written  on  a  slip  of  paper  is  still  preserved  (No.  07). 

)June  9.  1711.} 

•  •*«**## 

{et  propterea}  Vis  ilia  qu89  tollere  possit  motum 
omnem  Cylindri  interea  dum  Cylindrus  describat  longi- 
tudinem  quatuor  diametrorum,  Globi  motum  omnem  toilet 
interea  dum  Globus  describat  duas  tertias  partes  hujus 
longitudinis,  id  est,  octo  tertias  partes  diametri  proprijo. 
Resistentia  autem  Cylindri  est  ad  banc  vim  quamproxime 
ut  densitas  Fluidi  ad  densitatem  Cylindri  vel  Globi,  per 
Prop  XXXVII,  &  resistentia  Globi  sequalis  est  resistentioB 
Cylindri  per  Lemm :  v.  vi.  vii.  I  will  remember  to  alter  the 
2**  Corollary  of  this  Proposition  which  You  had  forgotten 
to  do  in  Your  last  Copy.  I  have  computed  y®  Table  pre- 
ceeding  y®  Scholium  of  Prop,  xl  &  find  some  of  the  numbers 
to  be  amiss  which  I  will  take  care  to  rectify ;  as  over 
against  0,9  G  the  space  should  be  0,7 1 96609  F;  over  against 
3G  the  space  should  be  4^,6186570  F ;  over  against  4>G  should 
be  6,6U3765F,  I  computed  also  all  the  Experiments  & 
found  my  Calculations  to  agree  nearly  enough  with  Yours 
except  in  the  1**  Experiment  which  I  will  alter  throughout. 
Of  the  rest  the  greatest  difference  was  in  the  11  *^  in  which 
y®  result  was  46 -g^  oscillations  not  46  as  You  make  it  in 
Your  corrections*,  I  took  care  to  make  a  right  allowance 


•  Sent  March  24.    See  p.  40. 


NEWTON  AND  COTES.  4S 

for  y*  narrowness  of  the  Vessel.  I  desire  You  to  send  mo 
the  Altitude  from  which  the  Globes  fell  in  the  9***  Experi- 
ment. You  had  forgotten  to  mention  it  in  Your  Copy. 
The  Six  Experiments  in  y®  Air  *  agree  also  very  well  with 
my  computation,  in  the  5*^  the  space  should  be  225^.  5**. 

Your  most  humble  Servant 

B.  C. 


LETTER  XXVI. 
NEWTON  TO  COTES. 

I  have  read  over  &  considered  your  alterations,  & 
like  them  very  well  &  return  you  my  thanks.  In  y*  end  of 
Exper.  9,  add,  desanbentes  altitudinem  digitorum  182.  I 
thank  you  also  for  correcting  the  numbers.  I  hope 
there  will  be  no  more  occasion  of  stopping  the  press. 
After  you  have  read  the  objection  of  Muysf  taken  from 


•  These  experiments  were  made  by  Hauksbee,  June  9, 1710,  with  glass  balU  let  fall, 
from  the  top  of  the  Cupola  of  St  Paul's,  (nearly  220  feet).  See  Phitosoph.  Trans.  Oct. 
— Dec.  1710,  p.  198.  An  account  of  them  was  read  at  a  meeting  of  the  lloyal  Society, 
June  14,  at  which  Newton  presided.  At  the  previous  meetinpr,  June  7,  (the  President 
then  also  in  the  chair)  Hauksbee  read  a  paper  on  some  experiments  of  the  same  kind, 
which  are  described  in  the  article  in  the  Phil.  Trans,  just  referred  to.  Newton  assisted 
at  similar  experiments,  made  by  Uesaguliers,  April  27  and  July  27,  1719,  from  tl»e 
upper  gallery  in  the  luntern  on  the  top  of  the  Cupola,  a  height  of  272  feet.  He  with 
some  other  persons  was  below,  and  noted  the  difference  in"  the  time  of  fall  of  the  leaden 
and  of  the  lighter  bulls.  See  Phil,  Trans.  Sept._Oct.  1719,  pp.  1071—1078.  The  ex- 
periments made  on  the  latter  day  are  introduced  into  the  3rd  Ed.  of  the  Principia, 
p.3o3. 

t  In  EUmenta  Physices  methodo  Mathematlca  Demonstrata,  &c.  Amstelod.  1711: 
a  heavy  quarto,  reviewed  in  the  Leipsic  Acts  for  Sept.  171 1,  and  severely  criticised  by 
Leibniz  and  John  Bernoulli  in  their  Correspondence. 

In  the  ist  Ed.  of  the  Principia  (p.  337),  there  is  a  Lemma  which  states  that  if  a 
spherical  or  other  vessel,  filled  with  fluid f  move  rectilineurly  with  an  accelerated  velocity, 
the  molecules  of  the  contained  Jluid  participating  equally  in  the  motion  of  the  ve^elwill 
remain  at  rest  among  themselves.  Muys  (p.  355),  in  opposition  to  this  quotes  a  pitisago 
from  the  4th  Dialogue  of  Galileo's  System.  Cosmic,  (p.  315,  Lyons,  1641,)  where  Sal- 
viati,  in  attempting  to  explain  the  tides,  takes  the  case  of  a  vessel^  which  contains  water. 


44  COllRESPONDENCE  OF 

Galileo's  experiment  of  the  motion  of  a  bucket  full  of 

water  you  will  scarce  expect  very  much  from  that  author. 

I  am  S' 

Yo*"  very  humble  servant 

SV  Martins  street  London. 

June  18*''  1711.  Is.  Newton 

For  the  R"**  Af  Cotes  Profmor  of  Astro, 
notntff  at  his  chamber  in  Trinity  College 
in  the  University  of  Cambridge. 


LETTER  XXVII. 


COTES  TO  NEWTON. 
S'  Juno  23^  1711 

I  received  Your  Letter  &  have  delivered  Y'  Papers  to 
the  Printer.  I  hope  we  shall  now  go  on  without  any  fur- 
ther intermission.  As  for  Muys,  I  have  look'd  over  what 
relates  to  the  resistance  of  Fluids.  He  acknowledges  that 
what  he  offers  upon  y*  subject  at  present  is  but  crude  & 
indigested  &  I  am  very  willing  to  agree  with  him.  His 
Objections  as  far  as  I  can  understand  *em  do  not  in  any 
wise  affect  Your  Book,  much  less  the  new  Edition  of  it. 
One  M""  Green  of  Clare-Hall  has  now  in  the  Press  a  book* 


moving  horizontallif^  and  says  that,  If  a  farce  be  applied  to  retard  the  vessel^  the  moleculet 
of  the  fluid  will  still  retain  their  velocity,  and  the  water  mil  rise  at  the  anterior  yart  of 
the  vessel.  Ify  on  the  contrary,  the  velocity  of  the  vessel  be  increased^  the  water  will  lag 
behind,  and  so  will  be  higher  at  the  hinder  than  at  the  fore  part  of  the  vessel.  This  fact 
the  speaker  proceeds  still  further  to  illustrate  by  referring^  to  the  boats  used  to  convey 
fiesh  water  from  Lizza  Fusina  to  Venice. 

•  The  Principles  of  Natural  Philosophy,  in  which  is  shewn  the  Insufficiency  of  the 
Present  Systems^  Sec.  &c.  Camb.  1712.  With  a  Latin  Tract  at  the  end,  entitled, 
Geometria  Solidorum^  &c.  This  eccentric  writer  also  published  A  Demonstration  of  the 
Truth  and  Divinity  of  the  Christian  Religion,  &c.  Camb.  1711,  and  a  large  thick  folio, 
(pp.  981)  with  the  title  of  The  Principles  of  the  Philosophy  of  the  Vxpansive  and  Con- 
tractive Forces,  &c.  Camb.  1727.  In  the  Preface  to  this  last  work  he  says:  *'  Our  Phi- 
losophy, as  it  is  now  received  and  embraced,  is  the  product  of  Popish   countries, 


NEWTON  AND  COTES.  45 

of  the  like  nature  with  Muys  wherein  I  am  informed  he 
undertakes  to  overthrow  the  Principles  of  Your  Philosophy. 
I  do  not  expect  very  much  from  him,  &  I  beleive  You  will 
not  Your  self  when  I  have  told  You  he  is  a  Person  who 
pretends  to  have  solv'd  y*  grand  Problem  of  y*  Quadrature 
of  the  Circle.  That  the  Press  may  not  stop,  I  am  now 
looking  over  Your  Copy  beforehand.  I  find  nothing  amiss 
till  I  come  to  Prop :  48.  I  will  choose  to  make  my  Objec- 
tion against  the  Corollary,  wherein  You  have  these  words 
[Nam  lineola  Physica  67,  quamprimum  ad  locum  suum 
primum  EG  redierit,  quiescet;]  This  assertion  cannot  I 
think  be  reconciled  with  what  You  assert  &  prove  in  the 
Proposition  [&  propterea  vis  acceleratrix  lincoloj  Physicto 
67  est  ut  ipsius  distantia  a  medio  vibrationis  loco  Q]  I 
propose  to  alter  the  whole  Proposition  thus  if  You  approve 
of  it.  [Propagentur  pulsus  in  plagam  BC  a  B  versus  C 
&  designet  BC  intervallum  corundem  ab  invicem.  Sint  E, 
Fy  G  puncta  tria  Physica  Medii  quiescentis  in  recta  BC 
ad  aaquales  distantias  sita  \  ee,  //*,  gg  spatia  sc-qualia  pcr- 
brevia  per  qua9  puncta  ilia  motu  rcciproco  singulis  vibra- 
tionibus  eunt  &  rcdeunt;  6,  0,  7  loca  qua)vis  intermedia 
eorundem  punctorum ;  &  EF,  FG  lineola?  Physica)  seu 
Medii  partes  lineares  punctis  illis  intcrjectio  &  successive 


imported  to  us  from  Italy  and  France All  therefore  which  I  design  and  intend,  is  to 

propose  a  Philosophy,  which  is  truly  EngUth,  a  Cuntnhriguni,  and  a  Claren$ian  one,  as 

it  was  born,  and  educated,  and  studied  in  those  places; And  as  my  Name  is  not 

much  worse  in  the  Letters  which  belong  to  it,  than  those  of  Galileus  or  Des-Cartes, 

I  shall  venture  to  call  the  GIIKENIAN."  Mr  Green  was  not  altoorcther  a  stranger  to 
Newton  when  Cotes  introduced  a  notice  of  him  in  this  letter.  On  makin^f  the  discovery 
that  the  area  of  a  circle  is  equal  to  four-fiftlis  of  the  square  of  its  diameter,  shortly  after 
taking  his  B.  A.  degree  (1700),  •' Dominum  Newtonum  accessi  ut  consulercm,"  says 
he,  "orantem  qui  chartulas  perlepreret,  ipsis  intactis,  ne  inspectis  certe,  rejecit,  ag- 
gressus  sum  dein  epistola,  recusavit,  (in  the  Preface  to  his  Ceometria  SoUdornm,  his 
phrase  is  '  rescripsit  nihil,')  quid  posthrcc  arbitremini  me  putassel  Saltern  vel  con- 
temptum  me  vel  Problema."  (lb.  p.  940.  1st  Lecture  "ad  Clarensem  juvcntutcm.*'^ 
On  the  publication  of  Green's  "  Natural  Philosophy"  in  1712,  where  his  quadrature 
of  the  circle  was  asserted,  he  tells  us  that  Cotes  was  "so  kind  and  obliging  as  to  com- 
municate to  me  with  great  candour  and  friendship  a  demonstration  aprainst  it,"  which 
will  be  found  lb.  pp.  924-5.    Cf.  Letter  CVL 


46 


CORRESPONDENCE  OF 


translate  in  loca  «0,    (py  &  ef,  fg.     Recta?     ......^ 

ee  aoqualis  ducatur  recta  PS,  bisecetur  eadem 
in  O,  centro  O  &  intervallo  OP  describatur  cir- 
culus  SIP  if  &   agatur  diameter  QR  ad  diame- 
triim  PS  perpendicularis.     Per   circuli   hujus 
circumferentiam  totam  cum  partibus  suis  expo- 
natur   tempus    totum  vibrationis  unius  cum  ip- 
sius  partibus  proportionalibus ;  sic  ut  complete 
tempore  quovis  QII  vel  QESh, 
si  demittatur   ad   PS  perpendi- 
culum   HL  vel   hi,   &    capiatur 
Ee  lequalis    OL  vel    01,    punc- 
tum  Physicum  JE  reperiatur  in 
€.     Ilac  lege  punctum   quodvis 
E  eundo  ab  E  per  6  ad  e  atq :  inde  redeundo, 
iisdem  accelerationis  ae  retardationis   gradib® 
vibrationes  singulas  peraget  cum  oscillante  Pen- 
dulo.  Probandum  est  quod  singula  Medii  puncta 
Physica  tali  motu  agitari  debeant.     Fingamus 
igitur  Medium  tali  motu  a  causa  quacunq :  cieri, 
Sc  videamus  quid  inde  sequatur. 

In  circumferentia  PQSB  capiantur  cequales 
arcus  ///,  IK  vel  hi,  ik  cam  habentes  rationem 
ad  circumferentiam  totam  quam  habent  ajquales 
rectaB  EF,  FG  ad  pulsuum  intervallum  totum 
BC*  Et  demissis  perpendiculis  IM,  KN  vel  im, 
kn  ;  quoniam  puncta  jE,  F,  G  motibus  similibus 
successive  agitantur  &  vibrationes  suas  integras  ex  itu  & 
reditu  compositas  interea  peragant  dum  pulsus  transfertur 
a  i?  ad  C,  si  QH  vel  QllSh  sit  tempus  ab  initio  motus 
puncti  E,  erit  QI  vel  QISi  tempus  ab  initio  motus  puncti 
F,  &  QK  vel  QKSk  tempus  ab  initio  motus  puncti  G ;  & 
propterea  Ee,  F(p,  Gy  erunt  ipsis  OL,  OM,  ON  in  itu 
punctorum,  vel  ipsis  01,  Om,  On  in  punctorum.  reditu 
lequales  respective.     Unde  €7  seu  EG  -f  Gy  ^  Ee  in  itu 


NEWTON  AND  COTES.  47 

punctorum  roqualis  erit  EG  -  LN,  in  reditu  autcm  aequalis 
EG  +  In,  Sed  ey  latitude  est  seu  expansio  partis  Medii 
EG  in  loco  ey,  Sc  propterea  expansio  partis  illius  in  itu,  est 
ad  ejus  expansionem  mcdiocrem  ut  EG  -  LN  ad  EG ;  in 
reditu  autcm  ut  EG  +  In  seu  EG  +  LN  ad  EG,  Quare 
cum  sit  LN  ad  KIT  ut  JM  ad  radium  O/,  &  KH  ad  JEG  ut 
eircumferentia  PQSBP  ad  i?C,  id  est,  (si  ponatur  V  pro 
radio  circuli  peripheriam  habentis  cequalem  intervallo  pul- 
suum  BC)  ut  0/ad  F,  et  ex  a?quo  LN  ad  EG  ut  Llf  ad  V: 
erit  expansio  partis  EG  punctive  Physici  F  in  loco  ey  ad 
eo'pansionem  mediocrem  qvam  habet  in  loco  8uo  primo  EG  ut 
V-  IM  ad  V  in  itu,  utq  :  F  +  im  ad  V  m  reditu.  Unde 
vis  Elastica  puncti  F  in  loco  €7  est  ad  vim  ejus  Elasticam 

mediocrem  in  loco  EG  ut  ,^ — ---  ad  —  in  itu,  in  reditu 

K  -  IM         V 

vero  ut  — — ; —  ad  — .     Et  eodem  arffumento  vires  Elas- 
r+  im         V 

tic£B  punctorum  Physicorum  G  &  J^  in  itu  sunt  ad  vires 

mediocres  ut  --- — — -  &  r= — rrrr  ad  •- ,  &  virium  differen- 
V  -  KN       V  -  HL        V 

tia .  sive  excessus  vis  Elastica3  puncti  y  supra  vim  Elasti- 
cam puncti  e  est  ad  Medii  vim  Elasticam  mediocrem  ut 

KN-IIL  1    ^       u  .      . 

ad  —  ,    hoc  est,  ut 


VV-Vy.KN-Vy.HL'^KN  y,  HL  F' 

vv ^^   v  ^^^^  "**  ^^ "  ^^  ^^   ^*  ®^  modo   (ob 

angustos  limites  vibrationum)  supponamus  KN  &  ILL  in- 
definite minores  esse  quantitate  V,  Quare  cum  quantitas 
V  detur,  excessus  ille  est  ut  KN  -  HL,  hoc  est  (ob  pro- 
portionals KN  -  HL  ad  HK  &  031  ad  07,  datasq :  HK 
&  OF)  ut  OM,  id  est,  ut  F(p,  Et  eodem  argumento  ex- 
cessus vis  Elastic©  puncti  7  supra  vim  Elasticam  puncti  e 
in  reditu  lineolae  Physicso  6  7  est  ut  F(p.  Sed  excessus  ille 
est  vis  qua  hsBC  lineola  accelcratur ;  &  propterea  vis  acce- 


48  CORRESPONDENCE  OF 

leratrix  lineolf©  PhysicaD  €y  est  ut  ipsius  distantia  a  medio 
vibrationis  loco  F,  Proinde  tempus  (per  Prop  xxxviii 
Libr.  1)  recte  exponitur  per  arcum  QI;  &  Medii  pars 
linearis  ey  perget  lege  praoscripta  moveri,  id  est,  lege 
oscillantis  Penduli :  &  par  egt  ratio  partium  omnium  linea- 
rium  ex  quibus  Medium  totu  componitur.  q.e.d.]  I  was 
going  to  propose  an  alteration  of  the  Corollary  but  I  choose 
rather  to  leave  it  to  Your  self.  It  must  be  made  to  cor- 
respond mth  what  You  have  at  the  end  of  Page  372  where 
You  cite  it.  I  propose  to  alter  Prop.  49  as  follows, 
[p.  368.  1 :  28  —  ad  lineola)  illius  pondus  ut  IIK  ^  A  ad 
V  X  EG  sive  ut  PO  y^  A  ad  FF,  nam  HK  erat  ad  EG  ut 
PO  ad  F.]  [1 :  32  —  urgente  vi  ponderis  in  subduplicata 
ratione  W  ad  PO  x  A  atq :  adeo  — ']  [1 :  ult  —  in  sub- 
duplicata ratione  W  ad  PO  ^  A  &  subduplicata  ratione 
PO  ad  A  conjunctim,  id  est,  in  ratione  integra  F  ad  A, 
Sed  tempore  vibrationis  unius — .]  [Ergo  tempus —  & 
reditu  composita)  ut  V  ad  A,  id  est,  ut  BC  ad  circumferen- 
tiam  circuli  &c.]  I  propose  to  add  the  2  following  Corol- 
laries to  Prop  49. 

Cor.  1.  Velocitas  pulsuum  ea  est  quam  acquirunt 
Gravia  a)qualiter  accelerato  motu  cadendo  ct  casu  suo 
describendo  dimidium  altitudinis  A,  Nam  tempore  casus 
hiyus,  cum  velocitate  cadendo  acquisita,  pulsus  percurret 
spatium  quod  erit  sequale  toti  altitudini  A,  adeoq :  tempore 
oscillationis  unius  ex  itu  &  reditu  compositoB  percurret 
spatium  a3quale  circumferentia)  circuli  radio  A  descripti ; 
est  enim  tempus  casus  ad  tempus  oscillationis  ut  radius 
circuli  ad  ejusdcm  circumforentiam. 

Cor.  2.  Unde  cum  altitudo  ilia  A  sit  ut  Fluidi  vis 
Elastica  directe  &  dcnsitas  ejusdcm  inverse ;  velocitas  pul- 
suum erit  in  ratione  composita  ex  subduplicata  ratione 
densitatis  inverse  &  subduplicata  ratione  vis  Elasticro  di- 
recte.    I  think  the  47*^  Proposition  is  out  of  its  place  :  for 


NEWTON  AND  COTES.  49 

the  Demonstration  of  it  proceeds  upon  the  supposition  of 
the  truth  of  the  48*^,  &  therefore  it  ought  to  follow  the 
48*^  &  besides  the  48*''  serves  to  form  some  Ideas  which 
are  necessary  to  the  understanding  of  the  47*^*.  If  You 
agree  that  these  Propositions  should  change  places  I  would 
add  the  following  words  at  y®  end  of  the  47*^  which  will 
then  be  the  48*^  [Hjcc  Propositi©  ultcrius  patebit  ex  con- 
structione  scquentis],  I  see  nothing  further  in  the  2**  Book 
which  I  could  wish  might  be  altered.  In  the  3^  Book 
under  Phoonom :  1,  The  Periodical  times  should  be 

I**.  18''.  27'.  34>"      3**.13^13'.  42".      7**.  s\^2\  36'\      1(3^16''. 32'. 9" 

and  the  Distantif©  ex  temporibus  periodicis  may  be 

5,667  9,017  14,384  25,299 

I  perceive  You  have  made  use  of  Cassini's  Tables  of  Ju- 
piter's Satellits  printed  in  1693  in  the  Recudl  d*  ObservO' 
tions  faites  en  plusieurs  Voyages  ^c.  But  Your  numbers 
give  the  times  of  the  Revolutions  to  Jupiters  shadow,  not 
to  y®  same  point  of  y®  Ecliptick.  The  Revolutions  to  the 
same  point  of  the  Ecliptick  are  (by  those  Tables)  as  I  have 
set  'em  down.  Y""  time  of  the  Revolution  of  Saturns  outer- 
most Satellit  differs  from  the  time  assigned  by  Hugenius 
in  his  Cosmotheoros  &  by  Cassini  in  the  Philosophical 
Transactions  but  I  find  it  is  y®  time  which  was  afterwards 
determin'd  by  Cassini  in  y*  Memoires  de  VAcadem.  1705. 
You  have  made  an  addition  to  the  3**  Proposition  in  which 
are  these  words  [Hicc  ratio  obtinet  in  Orbe  Luna)  nostra). 


•  The  object  of  Prop,  xlviii.  is  to  shew  that  when  pulses  or  undulations  are  pro*)a> 
gated  in  a  fluid,  the  particles  vibrate  accordingr  to  the  law  of  an  oscillating  pendulum. 
Prop.  xLvii.  shews  how  the  velocity  of  propagation  varies,  and  Prop.  xlix.  determines 
its  quantity,  the  expression  for  which  (\/^ .  heighTof'homog.  atmosph.)  Laplace  wai 
the  first  to  prove,  must  (in  the  case  of  sound)  be  multiplied  by 

A /spec,  heat  of  air  under  a  constant  pressure 

volume* 

Mtcan.  Ctlest.  <  121,  129.     Poisson,  Mtcan.  ii.  716.     VVhewell's  lint.  Ind.  Sri  ii. 

4 


50  CORRESPONDENCE  OF 

In  minore  Orbe  motus  Aphelii  minor  esset  in  triplicata 
ratione  minoris  distantisD  Lunze  a  Terra,  &  Fractio  ^|^ 
diminui  deberet  in  eadem  ratione.  Et  propter  banc  dimi- 
nutionem  vis  qua  Luna  retinetur  in  Orbe  suo  est  ad  vim 
eandem  in  superficie  Terra)  ut  1  ad  D-sa  quamproxime,  uti 
computum  ineunti  patebit]  1  sbould  be  glad  to  understand 
tbis  place  if  it  ^vill  not  be  too  great  a  trouble  to  make  it 
out  to  me.  I  do  not  at  present  so  mucb  as  understand 
what  it  is  that  You  assert. 

I  am  S"  y  &c. 


LETTER  XXVIII. 
COTES  TO  NEWTON. 
S'.  July  lO***  1711 

I  wrote  to  You  about  a  Month  ago  concerning  the  48**^ 
Proposition  of  Y*^  second  Book,  &  the  last  week  I  ordered 
the  Printer  to  send  You  all  the  sheets  which  were  printed 
off.  If  You  have  received  these  sheets  You  will  perceive 
by  *em  that  the  Press  is  now  at  a  stand.  But  having  no 
Letter  from  You  I  fear  the  sheets  have  miscarried.  The 
Compositor  dunn's  me  every  day,  &  I  am  forc'd  to  write 
to  You  again  to  beg  ^^^nswer  to  my  former  Letter.  I 
have  received  the  last  part*  of  Your  Copy  by  D'  Bently. 
I  have  now  read  over  and  examined  all  the  calculations  of 
the  former  part  which  ends  in  y®  432^  page.  I  will  write 
to  You  concerning  it  assoon  as  1  receive  Your  answer  to 
my  last  Letter. 

I  am  S^  Y-"  Sec, 


*  Beginning  at  p.  433,  with  part  of  Prop,  xxiv..  Lib.  3,  and  terminating^  at  p.  510 
with  Prop.  xLii.  (end  of  1st  Ed.)  Bentley  returned  to  College  on  the  7th,  (llud'fl 
Diary.) 


NE^VTON  AND  COTES.  51 

LETTER  XXIX. 
NEWTON  TO  COTES. 
S*  Martins  Street  ia  Leicester  ffiolds  London  July  28*^  1711. 

I  received  your  Letters  &  the  papers  sent  mc  by  the 
Printer  But  ever  since  I  received  yours  of  June  23  I  have 
been  so  taken  up  with  other  aflairs  that  I  have  had  no  time 
to  think  of  Mathematicks.  But  now  being  obliged  to  keep 
my  chamber  upon  some  indisposition  w*^^  I  hope  will  be 
over  in  a  day  or  two*  I  have  taken  your  letter  into  con- 
sideration. You  think  that  in  the  Corollary  to  the  48*** 
Proposition  these  words  [Nam  lineola  Physica  ey  quampri- 
mum  ad  locum  suum  primum  redierit,  quiescet]  consist 
not  w*^  what  I  assert  &  prove  in  the  Proposition,  viz*  [& 
propterea  vis  acceleratrix  lineolso  Physic89  ey  est  ipsius 
distantia  a  medio  vibrationis  loco  Q]  But  I  suspect  that 
you  take  the  words  [ad  locum  suum  primum]  in  another 
sence  then  I  might  intend  them,  ffor  when  all  the  lineola; 
physical  ey  are  returned  to  their  first  places  or  places  in 
w*^^  they  were  before  the  vibrations  began,  the  medium  will 
be  uniform  as  before  &  the  vis  acceleratrix  of  the  lineola 
physica  ey  will  cease,  whether  that  lineola  arrived  to  its 
first  place  in  the  beginning  middle  or  end  of  the  vibrations. 
For  making  the  Corollary  more  intelligible,  these  words 
may  be  added  to  the  end  of  it.  Partes  fluidi  non  quies- 
cent nisi  in  locis  suis  primis.  Quamprimum  in  loca  ilia 
motu  retardato  redierint,  component  Medium  uniforme 
quietum  quale  erat  ante  vibrationes  excitatas. 

In  altering  the  48*'*  Proposition  you  have  shortned 
the  Demonstration.  If  you  had  proposed  your  alteration 
of  the  Corollary  I  should  have  been  better  able  to  compare 
the  whole  w*^  mine. 


•  He  was  sufficiently  recovered  by  the  followinfir  'Jhursday,  (Aug.  2,)  to  preside  at 
a  meeting  of  the  Council  of  the  Royal  Society  on  that  day. 

4—2 


52  CORRESPONDENCE  OP 

Your  emendations  of  Prop  49  are  very  well  &  the  two 
Corollarys  you  propose  may  be  added  to  it.  And  the  47*^ 
&  48*^  Propositions  may  change  places,  &  at  the  end  of  the 
47*^  these  words  may  be  added  [Hsdc  Propositio  ulterius 
patebit  ex  constructione  sequentis. 

I  will  write  to  you  about  {the}  third  book  in  my  next. 

I  am  S'  Your  very  humble  servant 

For  the  Roror"'*  M'  Roger  Cotes  Professor  Is.   Newton. 

oj'  Astronomy,  at  his  Chamber  in  Trinity 
College  in  Cambridge. 


LETTER   XXX. 
COTES  TO  NEWTON. 
S'.  July  30*''  1711. 

I  have  read  Y"*  Letter  &  find  my  self  obliged  to 
trouble  You  once  more.  I  must  beg  leave  to  tell  You  I 
am  not  as  yet  satisfied  as  to  the  Inconsistency  which  I 
mention'd  in  my  former  Letter.  You  seem  to  say  that 
when  the  Lineola  Physica  €7  is  return'd  to  its  first  place, 
which  You  take  to  be  the  beginning  of  the  Vibration,  the 
Medium  will  be  uniform  as  at  first  &  consequently  its  Vis 
acceleratrix  will  cease.  If  upon  the  return  of  the  Lineola 
to  its  first  place  it  be  granted  that  the  Medium  ^vill  be  uni- 
form 1  confess  it  must  also  be  granted  that  the  Vis  Accele- 
ratrix will  cease :  but  then  if  the  Vis  acceleratrix  does 
cease  in  this  place  it  must  likewise  be  granted  that  its 
quantity  is  less  than  in  places  nearer  the  middle  of  y® 
Vibration  where  it  does  not  cease,  &  of  consequence  its 
quantity  will  not  be  proportionable  to  the  distance  of  the 
Lineola  from  ttie  middle  of  the  Vibration,  for  to  be  pro- 
portionable it  ought  not  to  cease  in  the  beginning  of  the 
Vibration,  but  on  the  contrary  it  should  be  greater  there 
than  in  any  other  place,  &  if  it  be  greater  there  than  in 
any  other  place  the   Medium  will  not  then  be  uniform. 


NEWTON  AND  COTES.  53 

This  consideration  was  to  me  the  occasion  of  altering  the 
Proposition.  By  making  the  middle  of  the  Vibration  the 
loctts  primus  I  saw  this  inconsistency  might  be  avoided. 
But  besides  this,  it  appeares  altogether  reasonable  upon 
other  accounts  that  the  loctis  primus  should  be  the  middle 
of  the  Vibration.  Suppose  a  Musical  Chord  to  be  put  into 
motion ;  tis  certain  its  locus  primus  is  the  middle  of  its 
Vibration  &  consequently  also  y®  locus  primus  of  any  lineola 
Pliysicd  of  Air  which  is  contiguous  to  the  Chord  is  in  the 
middle  of  its  own  space  of  Vibration ;  for  the  motion  of 
this  Lineola  Physica  follows  &  depends  upon  the  motion 
of  the  contiguous  Chord.  And  for  the  same  reason,  a 
second  Lineola  Physica  not  contiguous  to  the  Chord  but  to 
the  first  Lineola  will  have  its  locus  primus  in  the  middle  of 
its  own  Vibration,  since  its  motion  depends  upon  the  first 
as  the  first  did  upon  the  Chord  it  self;  &  the  same  may  be 
said  of  other  Lineolce  which  are  yet  more  remote  from  the 
Chord.  Now  assoon  as  the  motion  of  the  Chord  ceases  in 
its  locus  primus  ie,  in  the  middle  of  its  Vibration,  though  it 
should  perhaps  be  said  y*  the  motion  of  the  first  Lineola 
would  not  cease  of  it  self  at  the  same  time  with  it,  yet  tis 
evident  it  will  be  made  to  cease  by  the  resistance  of  the 
Chord,  for  being  contiguous  to  the  Chord  when  it  is 
arriv'd  at  its  locus  primus  or  the  middle  of  its  Vibration  it 
can  proceed  no  further  towards  the  Chord  whilst  y*  Chord 
maintains  its  rest,  &  it  cannot  return  back  again  from  the 
Chord  as  having  no  Vis  Acceleratrix  or  acquired  Impetus 
that  way.  And  as  this  first  Lineola  ceases  by  y®  resistance 
of  y®  Chord,  so  y®  second  ceases  by  y®  resistance  of  y®  first, 
&  so  on.  By  this  You  ^vill  understand  how  I  would  alter 
the  Corollary;  but  I  chose  rather  to  refer  it  to  Your  self, 
as  fearing  I  could  not  express  my  thoughts  with  sufficient 
clearness  &  brevity  &  exactness  at  y®  same  time.  What  I 
have  represented  above  is  not  so  exact  as  it  should  be,  for 
y®  motions  of  the  Lineolce  must  be  supposed  gradually  to 


64  CORRESPONDENCE  OP 

cease  with  the  motion  of  the  Chord ;  but  I  chose  to  ex- 
press my  self  as  I  have  done  that  You  might  the  more 
clearly  understand  me.  In  altering  the  Proposition  I 
altered  the  4*^  line  of  Page  366  by  putting  P/,  Pw,  Pn 
instead  of  Pn,  Pm,  PI;  &  in  the  2^  line  of  Page  367  in- 
stead of  [ob  brevitatem  pulsuum]  I  have  put  it  [ob  angus- 
tos  limites  vibrationum]  for  it  would  be  truer  &  more  to 
the  purpose  to  say  ob  magnam  pulsuum  di8tantia\m}  than 
to  say  oh  brevitatem  pulsuum.  In  Your  Example  taken  from 
M'  Sauveur  the  latitude  of  the  Pulse  is  about  10  foot,  when 
perhaps  y®  space  of  Vibration  is  not  above  y*  10^^  of  an 
Inch  at  y®  utmost.  If  You  consent  to  my  Alteration  of 
the  Proposition  the  Figure  must  be  altered.  I  propose  to 
have  it  cut  like  y®  Figure  I  sent  You,  which  docs  better 
express  the  disproportion  of  y®  breadths  of  y®  Pulses  & 
Vibrations  than  the  former  Figure. 

I  am  S^  Y*-  &c. 


LETTER  XXXI. 
COTES  TO  NEWTON. 
S""  Sept.  4*»»  1711 

I  received  a  Letter  from  you  about  a  Month  ago,  & 
sent  You  an  Answer  to  it  the  next  day  by  y®  Carrier,  in 
which  I  gave  You  my  reasons  why  I  was  not  yet  satisfied 
as  to  y'  Inconsistency  in  the  48*^  Proposition  &  its  Corol- 
lary which  I  formerly  mention'd  to  You.  I  have  not  heard 
from  You  since  y*  time,  &  therefore  I  fear  that  either  my 
Letter  or  Your  Answer  to  it  has  miscarried.  I  shall  be 
glad  to  know  Your  resolutions  concerning  this  48^^  Propo- 
sition assoon  as  You  have  leasure  that  the  Press  may  go  on. 
There  were  some  things  relating  to  the  3"^  Book  in  my 
former  Letter,  I  hope  You  will  not  forget  to  let  me  know 
Your  mind  concerning  them  also. 

I  am  S--  Y^  &c. 


NEWTON  XND  COTES.  55 

Newton's  occupations  at  tho  Mint  (soo  note  J  p.  37)  coupled 
with  liis  duties  as  President  of  the  Royal  Society  will  probably  bo 
sufficient  to  account  for  his  not  having  had  leisure  to  attend  to  tho 
two  preceding  letters  until  after  the  lapse  of  5  or  6  months  (Feb.  2, 
1712,  tho  date  of  the  next  letter).  The  following  dates  will  give  ua  a 
glimpse  or  two  of  him  during  some  of  these  months. 

October  16,  1711:  "The  President  |of  tho  Royal  Society |  ap- 
pointed  a  Council  {a  meeting  of  the  Committee  of  Visitors  of  Greenwich 
Observatory}  to  be  called  on  Friday  come  sevennight  (the  26th)  when 
Mr  Hunt  is  ordered  to  desire  Mr  Flamsteed  to  meet  the  Council  on  that 
day  at  11  o'clock,  at  their  house  in  Crane  Court  in  Fleet  Street;  to 
know  of  him  if  his  instruments  be  in  order,  and  fit  to  carry  on  tho 
necessary  celestial  observations."  (Baily's  Flamsteed,  p.  96,  97  note). 
Three  accounts  of  this  meeting  from  Flamsteed's  pen  are  extant,  (Daily, 
p.  96,  228,  294),  which  bear  painful  marks  of  his  unhappy  temper 
soured  by  the  mortification  he  felt  at  having  a  board  of  Vieitors  "  set 
over  him." 

Jan.  31,  171  j}.  Leibniz's  2nd  letter  to  Dr  Sloano  (dated  29  Dec. 
1711),  complaining  of  "  Keill's  unfair  dealing  with  him  in  his  last 
letter,  relating  to  the  dispute  between  him  and  Sir  Isaac  Newton,  was 
read :  the  letter  was  delivered  to  the  President  to  consider  tho  contents 
thereof."  (Journal  Book  of  Royal  See.)  Tliis  letter,  in  which  Leib- 
niz, speaking  of  tho  obnoxious  passage  in  the  Leipsic  Acts  for  Jan. 
1705,  in  tho  review  of  Newton's  tract  De  Quadratura,  says  "  in  illis 
circa  banc  rem  quicquam  cuiquam  dctractum  non  rcperio,  sed  potius 
passim  suum  cuique  tributum,"  led  to  the  appointment  of  a  Committee 
(March  6,  171 2)*  to  inspect  the  letters  and  papers  relating  to  the  sub* 
ject,  who  delivered  in  their  Report,  Apr.  24. — 

A  great  part  of  Cotes's  correspondence  with  Jones  falls  within  this 
interval  (letters  cm — ex)  and  may  bo  conveniently  read  hero  as  con- 
tributing towards  filling  up  tho  blank. — 

With  the  next  letter  the  correspondence  begins  to  be  carried  on  with 
briskness.  In  a  letter  of  Saunderson  to  Jones,  March  16,  17^2,  (Mac- 
clesfield Corrcsp.  i.  264,  where  it  is  printed  out  of  its  chronological 
place,)  a  postscript  adds  that  "  Sir  Is.  Newton  is  much  more  intent 
upon  his  Principia  than  formerly,  and  writes  almost  every  post  about 
it,  80  that  we  are  in  great  hopes  to  have  it  out  in  a  very  little  time." 


56  CORRESPONDENCE  OP 

LETTER  XXXII. 
NEWTON  TO  COTES. 
S'  London  2^  Feb.  ITli 

I  have  at  length  got  some  leasure  to  remove  the  diffi- 
culties w*^^  have  stopt  the  press  for  some  time,  &  I  hope 
it  will  stop  no  more,  ffor  I  think  I  shall  now  have  time  to 
remove  the  rest  of  your  doubts  concerning  the  third  book 
if  you  please  to  send  them. 

In  reveiwing  yo'  letters  I  do  not  see  but  that  y®  xlviii*^ 
Proposition  of  the  second  Book  with  its  Corollary  may 
stand,  ffor  the  particles  of  air  go  from  their  loca  prima 
with  a  motion  accelerated  till  they  come  to  the  middle  of 
the  pulses  where  the  motion  is  swiftest.  Then  the  motion 
retards  till  the  particles  come  to  the  further  end  of  the 
pulses.  And  therefore  the  loca  prima  are  in  the  beginning 
of  the  pulses.  There  the  force  is  greatest  for  putting  y® 
particle  into  motion  if  any  new  pulses  follow.  But  if  no 
new  pulse  follows  the  force  ceases  &  the  particle  continues 
in  rest.  In  this  Proposition  pag.  366.  lin.  12,  this  emenda- 
tion may  be  made.  Quare  cum  sit  LN  ad  KII  ut  IM  ad 
radium  OP,  et  KII  ad  EG  ut  circumferentia  PHShP  ad 

OP  X  BC 

BC;  id  est  (si  circumferentia  dicatur  %  et  dica- 

OP  X  BC 

tur  F*,)  ut  OP  ad seu  OP  ad  V.     Et  ex  a?quo 

LN  ad  EG  ut  IM  ad  V:  erit  expansio  partis  EG,  punc- 
tive  physici  F,  in  loco  €7,  ad  expansionem  mediocrem 
quam  pars  ilia  habet  in  loco  suo  prime  EG  ut  V  -  IM  ad 
Finitu,  utq:  V  +  im  ad  Fin  reditu.  Vnde  vis  elastica 
puncti  F  in  loco  ey  est  ad  vim  ejus  elasticam  mediocrem 


*  Cotes  did  not  adopt  the  part  where  t  is  brought  in,  but  printed  it  as  he  proposed 
in  his  Letter  of  June  23,  "(si  ponatur  V  pro  radio  circuli  circumferentiam  habcntis 
xquulem  intervallo  pulsuum  BC),  &c."  His  suggestion  of  "ob  angustos  limites  vi- 
brationum,"  (Letters  June  23,  July  30,)  of  which  Newton  takes  no  notice,  is  also 
introduced  into  the  2nd  Ed. 


NEWTON  AND  COTES.  57 


in  loeo  EG  ut  -- — j^^  ad  —  in  itu,  in  reditu  vero    ut 
F—  IM         V 

1  1 

ad  — .      Et  eodem  argumento  vires  elastica)  &c 


V  +  im         V 
See  lin  27. 

You  stuck  at  a  difficulty  in  the  third  Proposition  of  the 
third  Book.  I  have  revised  it  &  the  next  Proposition  & 
sent  you  them  inclosed*  as  I  think  they  may  stand.  What 
further  Observations  you  have  made  upon  the  third  Book 
or  so  many  of  them  as  you  think  fit  if  you  please  to  send 
in  yo*"  next  Letters,  I  will  dispatch  them  out  of  hand.  I 
shall  be  glad  to  have  them  all  because  I  would  have  {the} 
third  Book  correct. 

I  am  Yo'  most  humble  Servant 

For  the  R''^M.^CoTE3,  Professor  of  Astro^  Is.   Newton 

nomi/t  at  his  chamber  in  Trinity  College 
in  Cambridge. 


LETTER    XXXin. 
COTES   TO  NEWTON. 


I  have  received  Your  Letter  &  as  to  the  buisness  of 
sounds,  I  do  intirely  agree  with  You  upon  considering 
that  matter  over  again.  By  Your  alteration  of  y*  3**  Prop : 
of  y®  3^  Book,  it  is  now  very  intelligible.  What  I  have 
observed  concerning  the  remaining  part  of  Your  Copy 
I  will  send  You  in  the  most  convenient  order  I  can.  I 
begin  with  the  37*^*  Proposition,  in  the  3**  section  of  which 
You  have  these  words  [Eo  autem  tempore  Luna  distat 
a  Sole  \5^\qt,  circitcr.      Et  Sol  in  hac  distantia  minus 


•  A  folio  sheet,  Nos.  127—129.  To  the  4th  Proposition,  a  Scholium  beginningf 
"  Pkartut  mensurando  arcum,  &c.*'  is  subjoined,  which  is  a  modification  of  what  he 
had  previously  sent  down  in  the  second  instalment  of  his  copy  of  the  Principia^  Sept. 
13,  1710.  He  afterwards,  (Letter  XL!.),  determined  on  omitting  this  Scholium,  and 
placing  it  after  Prop,  xxxvii.  Eventually,  however,  part  of  it  was  transferre<l  to 
Prop.  XIX.,  and  a  smaller  part  lo  Cor.  7  of  Prop,  xxxvii. 

t  It  should  be  15|,  as  it  stands  in  Newton's  MS.  No.  193.    Sec  p.  78. 


58  CORRESPONDENCE  OP 

auget  ac  minuit  motum  maris  a  vi  Luni©  oriundum  quam 
in  ipsis  Syzygiis  &  quadraturis,  in  ratione  Radii  ad  co- 
sinum  distantitc  hiyus  duplicata)  seu  anguli  30igr.  hoc 
est,  in  ratione  7  ad  6  eirciter  ideoq :  in  superiore  Analogia 
pro  S  scribi  debet  -!^-  S,  1  suppose  You  intended  to  have 
said  .[in  ratione  duplicata  Radii  ad  cosinum  distantio) 
hiyus]  or  [in  ratione  diametri  ad  sinum  versum  duplicati 
complementi  hujus  distantit©].  After  the  same  manner 
in  y®  foregoing  proposition,  at  y®  bottom  of  y®  4G3  page, 
You  have  added  these  words*.  [In  aliis  solis  positionibus 
vis  ad  mare  attollendum  est  ut  cosinus  dupla)  altitudinis 
Solis  supra  horizontem  loei  directe  &  cubus  distantio9  Solis 
a  Terra  inverse]  I  suppose  You  intended  to  have  said 
[ut  sinus  versus  dupla)  altitudinis].  This  alteration  being 
made  in  Prop  37,  You  will  have  \-^  S  instead  of  ^  ^S*, 
whence  *S'  will  be  to  Z  as  l  to  5  •^,  &  in  y®  4^'*  Corollary 
You  will  have  a  different  proportion  from  y*  of  1  to  38. 
In  y®  3'^  Corollary  You  make  use  of  3i' .  2?"  &  32' .  12" 
for  y®  apparent  diameters  of  y®  Sun  &  Moon :  I  query 
whether  it  would  not  be  more  adviseable  to  use  y®  numbers 
of  Your  new  Theory f  32'.  15"  for  y®  Sun,  3l' .  l6"^  for  y® 
Moon.  Making  use  of  these  numbers,  &  of  5?' .  5"  for 
y®  Moons  Horizontal  Parallax,  &  taking  y®  density  of  y® 
Sun  to  be  to  y®  density  of  y®  earth  as  100  to  398  -jL.  as  my 
computation  gives  it;  the  quantity  of  matter  in  y®  Moon 
will    be   to  y®   quantity    of  matter  in    y®    Earth  as   1    to 

176  f    X  — ,  or  asl    to   S4|-.      This  alteration  will  very 

much  disturb  Your  Scholium  of  y®  4*^  Proposition  as  it 
now  stands;  neither  will  it  well  agree  with  Proposition 
39*^,  in  which  I  further  observe  that  You  take  y®  pro- 
portion of  y®  semidiameters  of  y®  earth  to  be  as  689  to 
692;    But  if  their  difference   be  32  JVIilcs,  there  will  be 


•  No.  191. 

+  "  Lunaj  Thcoria  Newtoniana,"  printc*!  in  David  Gregory*H  Aitronomia:  Elemenln, 
(Oxford,  1702),  p.  332. 


NEWTON  AND  COTES.  59 

another  proportion,  &  I  query  whether  here  ought  not  to 
be  some  allowance  made  upon  that  score. 

I  have  not  examin'd  all  the  calculations  of  y*  Scholium 
to  y*  iv^^  Proposition  but  I  formerly  observ'd  a  small 
difference  from  Your  Numbers  as  to  y*  descent  of  heavy 
bodies.  If  y®  length  of  a  Pendulum  which  vibrates  seconds 
be  3  feet  &  8^  lines,  the  descent  in  that  time  will  be 
15  feet  1  inch  2 -jJ^  lines:  You  have  it  2^  lines.  And 
when  I  examin'd  y®  xix*^  Proposition  I  found  the  vis 
centrifuga  to  be  in  proportion  to  the  vis  gravitatis  as  1  to 
288  ^,  You  have  it  as  i  to  290  -*-.  In  this  computation 
I  took  y'  measure  of  a  degree  to  be  57200  Toises  as  You 
had  formerly  stated  it,  the  descent  of  heavy  bodies  in 
a  second  to  be  15,0976  feet,  the  time  of  y*  earths  revolution 
to  be  23^. 56'. 4".  If  this  Vis  centrifuga  be  increased 
in  y*  proportion  of  57230  to  57200,  it  will  be  to  y®  vis  gravi- 
tatis as  I  to  288 1^.  I  will  send  You  some  things  further  as 
I  can  recollect  them  from  my  loose  papers  of  y*  computa- 
tions which  I  made  about  1  an  Year  ago;  In  Your  next 
You  may  be  pleasd  to  send  me  Your  Answer  to  what 
I  formerly  proposed  concerning  y®  periodical  times  of  y* 
Satellits,  for  I  do  not  yet  know  Your  resolution  as  to  that 
part  of  my  Letter. 


LETTER    XXXIV. 
NEWTON   TO   COTES. 
S""  London  Fob.  12.  ITJl- 

In  the  third  Book  under  Phaenom.  I,  the  periodical 
times  may  be 

1** .  18^  27'  34".  3**  13^  13'  42".  7**  3^  42'  36^\  l6^  l6^  32'  9" 
&  the  distances,  ex  temporibus  periodieis  5,6C7  9,017 
14,384  25,299  as  you  have  put  them  in  yo"  of  June  23  last. 
But  the  numbers  in   the  Corollaries   of  Prop,  viii   must 


60  CORRESPONDENCE  OP 

be  altered  accordingly.  And  so  must  one  or  two  of  y« 
numbers  in  Prop,  xii  &  xiii. 

In  y'  3**  section  of  y®  xxxvii*^  Proposition,  I  think  my 
proportion  is  right,  ffor  the  force  of  the  Sun  increases  the 
force  of  the  Moon  in  the  Syzygies,  diminishes  it  in  the 
Quadiatures  &  neither  increases  nor  decreases  it  in  the 
Octants :  &  therefore  the  distance  of  the  Moon  from  the 
Sun  must  be  doubled  that  the  cosine  thereof  may  vanish 
in  the  Octants. 

In  the  3**  Corollary  of  that  Proposition  lin  5,  6,  the 
words  should  run  thus  [et  cubus  diametri  Lunse  ad  cubum 
diametri  Solis  inverse,  id  est,  (cum  diametri  mediocres 
apparentes  Luna?  et  Solis  sint  31 '  2?"  &  32'  12")  ut  &c.] 
But  instead  of  the  Moons  mean  diameter  31 '  27"  may  be 
written  Si'  16 J,  &  the  Suns  mean  diameter  32'  12"  may 
be  every  where  retained,  even  in  the  Moons  Theory.  For 
32'.  15"  is  too  bigg. 

In  the  Scholium  to  the  iv*^  Proposition,  if  the  length 
of  a  Pendulum  w*^''  vibrates  seconds  in  vacuo  be  put  3  feet 
&  8  J,  the  descent  in  that  time  will  be  15  feet  1  inch  & 
2j  lines. 

And  in  the  xix**^  Proposition  the  vis  centrifuga  may  be 

put  in  proportion  to  the  vis  gravitatis  as  I  to  289,  &  then 

these  corrections  must  be  made.     Neare  the  end  of  the 

Scholium  of  Prop  iv.   for  the  numbers  290  -J-,  669  &  ^J-^ 

write  289,   G65,  &  ^J^.     Also   pag  422  lin  9  write,  ut  1 

ad  289.     lin.  13,  ut  289  ad  288.     lin   15,  289.     lin   16,  288. 

Pag  423  lin  27,  ut  1  ad  288.     lin  28,  pars  -^^.     lin  31,  vis 

centrifuga  ^^.    lin  ult.  pars  tantum  ^^.    Pag.  424  lin  1, 

ut  229  ad  228.    lin  3, 19^24224,  seu  millia Jr  J  ium  3035,    lin  5, 

29  X  1  X  5 

pedum  86101  seu  miUiarium  17.    lin  16,  ut ad  1, 

*  5  X  228 

seu  1  ad  8.     lin  29,  ut  "229  ad  228. 

The  xxxix*^  Proposition  must  be  corrected  by  putting 

the  scmidiamcters  of  the  earth  as  228  to  229  instead  of  689 


NEWTON  AND  COTES.  61 

to  692,  or  perhaps  as  S919  to  3951  the  difference  being 
32  miles.  I  think  [228  to  229]  should  be  put  for  [689  to  692] 
&  the  difference  of  32  miles  may  be  allowed  for  in  the 
latter  part  of  the  Proposition.  But  I  have  lost  my  copy 
of  the  emendation  I  made  to  that  Proposition  &  the 
Lemmas  preceding,  &  so  know  not  how  to  make  this  cor- 
rection. If  you  can  mend  the  numbers  so  as  to  make 
y®  precession  of  the  Equinox  about  50"  or  5l'\  it  is  suf- 
ficient.    I  am 

Yo'  most  humble  Servant 

For  the  R"**  M'  Cotes  Professor  of  Is.   Newton 

Astronomy t  at   his  chambei'  in 
Trinity  College  in  Cambridge, 


LETTER  XXXV. 
COTES  TO  NEWTON. 
S'  Cambridge  Fob  16*''  17ii 

I  received  Your  last  of  y®  12*^  of  this  Month.  Tis  very 
evident  that  y®  3**  section  of  Proposition  xxxvii*^  ought  not 
to  be  altered.  I  had  observed,  that  in  an  addition  which 
You  have  made  at  y®  bottom  of  page  463,  cosinus  ought  to 
be  chang'd  into  sinus  versus;  &  thereupon,  (without  any 
further  consideration),  I  had  applied  the  same  change  to 
y®  3^  section  of  y®  following  Proposition.  I  ^vill  observe 
Your  directions  as  to  y®  Diameters  of  y*  Sun  &  Moon  in 
Corel.  3 ;  retaining  in  all  other  places  32'.  12"  for  y*  Sun. 
In  y®  Scholium  of  iv*^  Proposition  I  think  the  length  of  y* 
Pendulum  should  not  bo  put  3  feet  &  8|^  lines;  for  the 
descent  would  then  be  15  feet  1  inch  1^  line.  I  have 
considered  how  to  make  y*  Scholium  appear  to  the  best 
advantage  as  to  y®  numbers,  &  I  propose  to  alter  it  thus. 
To  take  57220  Toises  for  y*'  measure  of  a  degree,  instead 
of  57230 ;  for  57220  is  y®  nearest  round  number  to  a  mean 


62  CORRESPONDENCE  OF 

amongst  57060,  57292,  57303.  To  take  3  feet  8 1^  lines  for 
y*  length  of  y*  Pendulum ;  for  y*  French  sometimes  make 
it  8^  sometimes  8^,  &  8-j-g^  is  a  mean  betwixt  these  mea- 
sures. To  take  48^.  50'  for  y"  Latitude  of  Paris  instead  of 
48''.  45'  as  You  had  put  it.  From  these  principles  the  fol- 
lowing alterations  may  be  made,  [semidiamctcr  Tcrraj 
19670787  ped]  [distantia  mediocris  Luna)  a  Terra  1190082614 
pedj  [distantia  D*  a  communi  ccntro  gravitatis  1159567675 
ped]  [Sinus  Versus  ped.  14,  dig.  9,  Hn.  5j^]  —  id  est,  in 
ratione  1  ad  3680,84502  ;  ideoq:  corpus  ad  superficiem  Terr© 
vi  ilia  cadendo  describet  pedes  Parisienses  15,  dig.  1, 
lin.  5J-. 

Observatum  est  longitudinem  Penduli  ad  minuta  se- 
cunda  oscillantis  in  vacuo,  esse  pedum  trium  Parisiensium 
&  linearum  8^  seu  linearum  8^.  Sumatur  longitudo  me- 
diocris pedum  trium  &  linearum  8J-g^:  &  altitudo  quam 
grave  in  vacuo  cadendo  tempore  minuti  unius  secundi  de- 
scribit,  (cum  sit  ad  dimidiam  longitudinem  Penduli  hujus 
in  duplicata  ratione  circumferentia3  ad  diametrum..circuli, 
ut  indicavit  Hugenius,)  erit  pedum  Parisiensium  15,  dig.  1, 
lin.  1-5^.  Hie  est  descensus  gravium  in  Latitudine  Lu- 
tetiso  Parisioru  seu  48^.  50'. 

Ad  iEquatorcm  vis  ccntrifuga  corporum  a  diurna  rota- 
tione  Terra)  oriunda  est  ad  vim  gravitatis  ut  1  ad  289 
circitcr ;  &  in  Latitudine  Lutetij©  minor  est,  idq :  in  du- 
plicata ratione  sinus  complementi  Latitudinis  48^  50'  ad 
Radium  adeoq :  est  ad  vim  gravitatis  ut  1  ad  667.  Et  hac 
vi  descensus  gravium  in  latitudine  Lutetiaa  diminuitur. 
Descensus  igitur  pedum  15,  dig.  1,  lin.  1-^^  augcatur  parte 
^-g  seu  lineis  3^,  &  habebitur  totus  gravium  descensus 
pedum  15,  dig.  1,  lin.  5^  quem  gravitas  sola,  tempore  mi- 
nuti unius  secundi  in  Latitudine  48*^^  50'  efficere  posset,  si 
modo  Terra  quiesceret. 

I  have  gone  over  the  computation  of  y°  viii*^  Propo- 
sition again  taking  32'.  12"  for  y*'  Suns  diameter,  for  I  had 


NEWTON  AND  COTES.  63 

formerly  made  use  of  32'.  15".  I  propose  these  alterations. 
[Satellitis  extimi  Jovialis  tcmpus  periodicum  dierum  16  & 
horarum  16-j^]    Pondera  ad  roquales  distantias  a  centris 

Solis,   Jovis,    Saturni   ac   Terra)  l.    .    .    . 

1033        2411       227512 

Semidiametri  Solis,  Jovis,  Saturni  ae  Terroo  loooo.  1077. 
889.  104..  Pondera  ad  superficies  Solis,  Jovis,  Saturni  ac 
Terra)  10000.  835.  525.  410.  Dcnsitatcs  Solis,  Jovis,  Saturni 
ae  TerrsD  100.  78.  59.  396*, 

The  xii*^  Proposition  may  be  altered  thus  [Nam  cum, 
per  Corol.  2.  Prop  viii.  materia  in  Sole  sit  ad  materiam  in 
Jove  ut  1033  ad  1,  &  distantia  Jovis  a  Sole  sit  ad  semi- 
diametrum  Solis  in  ratione  paulo  majore ;  incidet  commune 
centrum  gravitatis  Jovis  &  Solis  in  punctum  paulo  supra 
superficiem  Solis.  Eodem  argumento  cum  materia  in  Sole 
sit  ad  materiam  in  Saturno  ut  2411  ad  1,  &  distantia  Sa- 
turni a  Sole  sit  ad  scmidiamctrum  Solis  in  ratione  paulo 
minore :  incidet  &c.]  The  xiii*^  Proposition  may  be  altered 
thus,  pag.  419.  1 :  18  [ut  1  ad  1033].  lin:  21.  [ut  81  ad  iGx  1033 

seu  1  ad  204  circiter]    lin:  antepenult.  [&  — 


•  All  the  figures  which  Cotes  proposes  in  this  paragraph,  duly  appear  in  their  pluccft 
in  the  first  three  Corollaries  of  Prop,  viii.,  in  the  2nd  Edit;,  though  Newton  in  hw 
answer  to  this  Letter  takes  no  notice  of  his  suggestions  with  respect  to  them. 

Cotes  has  made  about  half  a  dozen  other  alterations  (adopted  in  the  2nd  Ed.)  in 
the  MS.  of  the  four  Corollaries  of  this  8th  Prop.,  which  are  not  noticed  in  this  rough 
draught,  though  some  of  them  would  probably  be  mentioned  in  the  letter  actually  sent. 
The  most  important  of  them  are  the  following,  (Nos.  133,  134)  : 

In  Cor.  1.  The  last  sentence  is,  "Pondera  corporQ  in  superficie  Lunne  fere  duplo 
minora  esse  quam  pondera  corporum  in  superficie  Terras  dicemus  in  sequentibu8,"  as  it 
stands  in  the  1st  Ed.  Cotes  has  altered  it  to  "  Quanta  sint  pondera  corporum  in  super- 
ficie Lunro  dic&mus  in  sequentibus.V 

In  Cor.  3,  the  words  "  Dcnsitas  Terra;  hie  posita  non  pendet  a  parallaxi  Solis,  &c.** 
are  altered  to  "  Densilns  Terra;  qutr  prodit  ex  hoc  computo  non  pendet,  &c.** 

In  Cor.  4,  Newton  had  written  '*  Scd  et  densiores  sunt  Planetsp,  cjeteris  paribus, 
qui  sunt  Soli  propiores;  ut  Jupiter  Saturno,  et  Terra  Jove.  Oritur  utiq:  densitas  ma- 
teria; ex  calore  solis  earn  decoquentis.  Et  collocandi  erant  Planeta;  in  diversis  a  Sole 
distantiis  ut  quilibet  pro  gradu  dcnMtatis  calore  solis  majore  vel  minore  frueretur.** 

Cotes  has  drawn  his  pen  through  the  words  "  Oritur Planeta;,"  and  has  altered  the 

last  clause  to  "  In  diversis  utiq  :  a  Sole  distantiis  collocandi  erant  Planeta;,  ut  quilibet, 
Sec."  In  the  Ist  Ed.  the  last  clause  runs  thus  :  "  Collocavit  igitur  Deus  Planetas  in 
divcraia  distantiis  a  Sole vel  minore  fruatur." 


64  CORRESPONDENCE  OF 

seu  124986]  lin :  ult.  [ut  65  ad  124986  seu  1  ad  1923].  I 
observe  that  You  have  added*  to  the  xiv*^  Proposition  a 
Scholium  concerning  y®  motion  of  y*  Aphclia  of  y®  Planets, 
in  which  by  supposing  y'  of  Mars  to  go  forward  35'  in  100 
Yeares  You  deduce  the  motion  of  y®  Earths  Aphelium  to 
be  18',  3(^'.  I  should  be  glad  to  know  whether  You  have 
found  these  motions  to  be  nearly  so  by  Observations  or 
whether  these  numbers  are  proposed  barely  as  an  Example ; 
for  in  Your  new  Theory  published  by  D''  Gregory  You 
make  y®  motion  of  y*'  Earths  Aphelium  to  be  2l'.  40"  in  an 
100  Yeares  The  Rule  delivered  in  this  Scholium  puts  me 
in  mind  of  a  mistake  in  the  New  Edition  of  Your  book 
which  I  did  not  observe  till  it  was  too  late.  In  y®  16*^ 
Corollary  of  y®  lxvi*^  Prop  :  of  Lib :  1,  or  in  page  166,  line 
9th  ^£  ye  ^Qyf  Edition  You  will  find  ut  quadratum  temporis 
periodici  corporis  P  directe  &c.  So  You  had  altered  it  in 
Your  Copy,  but  I  think  it  should  be  as  in  y®  former  Edition 
ut  tempus  periodicu.  Over  against  Your  alteration  there  is 
written  in  y*  margin  with  a  black  lead  pencil  by  another 
hand  quadr,  temporis  period,  which  I  suppose  You  depended 
upon  without  considering  the  thing  Your  self.  I  will  write 
to  You  concerning  the  xix*^  &  xx*^  propositions  in  my 
next.  I  come  now  to  y®  xxxix*^  Proposition,  it  stands  thus 
in  Your  Copy,  pag :  470.  lin :  10  dele  reciproce,  lin :  26. 
ut  474721  ad  4143  seu  114584  ad  1000  pag:  471.  lin:  20 
[evaderet  minor  quam  prius  in  ratione  2  ad  5.  Ideoq: 
annuus  aequinoctioru  regressus  jam  esset  ad  20^.  ll'.  46".  ut 
1  ad  7330,  ac  proinde  fieret  9",  55'",  ^'\  Casterum  hie  motus, 
ob  inclinationem  plani  iEquatoris  ad  planum  Ecliptica), 
minuendus  est,  idq :  in  ratione  &c.]  You  have  left  out  all 
from  pag:  471.  1:  22  to  pag.  473.  lin:  13.  Then  in  pag. 
473.  lin :  27  You  have  [diminuendus  est  motus  9".  55'",  8'" 
in  ratione  sinus  91706  (qui  sinus  est  complementi  graduum 


No.  137. 


NEWTON  AND  COTES.  65 

23^)  ad  radium  100000.     Qua  rationc  motus  iste  jam  fiet 

g'\5"\4!6^\     Ha?c  est  annua  Prrocessio  -^Equinoctiorum  a  vi 

Solis  oriunda. 

Vis  autem  Lunre  ad  marc  movendum  &c.* 

I  should  be  glad  to  have  this  Proposition  settled  before 

we  print  any  thing  which  may  in  any  wise  relate  to  it 

y  humble  Servant 

EooER  Cotes. 

Before  I  conclude  this  Letter,  I  will  take  notice  of  an 
objection  which  may  seem  to  be  against  y*  3**  Corel :  of 
Prop  :  VI.  Lib  :  in  Itaq  :  Vacuum  necessario  datur  &c.  Let 
us  suppose  two  Globes  A  &  S  o£  equall  magnitudes  to  be 
perfectly  fill'd  with  matter  without  any  interstices  of  void 
Space ;  I  would  ask  the  question  whether  it  be  impossible 
that  God  should  give  different  vires  inertiue  to  these  Globes. 
I  think  it  cannot  be  said  that  they  must  necessarily  have 
y®  same  or  an  equal  Vis  inertiae.  Now  You  do  all  along  in 
Your  Philosophy  &  I  think  very  rightly  estimate  the  quan- 
tity of  Matter  by  the  vis  inertiae  &  particularly  in  this  vi*^ 
Proposition,  in  which  no  more  is  strictly  proved  than  y' 
y®  Gravitys  of  all  Bodies  are  proportionable  to  their  Vires 
inertia?.  Tis  possible  then  that  the  equal  spaces  possessed 
by  the  Globes  A  &  B  may  be  both  perfectly  filld  with 
Matter  so  as  no  void  interstices  may  remain  &  yet  that 


*  This  being  merely  the  draught  of  a  letter,  Cotes  has  not  taken  the  trouble  of 
transcribing  the  whole  of  the  passage,  though  of  course  in  the  letter  which  was  actually 
sent,  he  would  copy  it  entire.     It  stands  as  follows  in  Newton's  IMS.  No.  204. 

Vis  autem  Luna:  ad  marc  movendum  erat  nd  vim  Solis  ut  4*  ad  1  circiter.  Et  in 
eadem  proportione  est  vis  Lunas  ad  vim  Solis  ad  /Equinoxia  movenda.  Indcq:  prodit 
annua  /Equinoctiorum  Prajcessio  a  vi  Luna;  oriunda  42" .  52'" .  54'*',  ac  tota  Prajcessio 
annua  a  vi  utraq :  oriunda  51" .  58"' .  40'". 

Si  vis  Luna;  ad  mare  movendum  esset  ad  vim  Solis  ut  4*  ad  1  (nam  proportionem 
harum  virium  nondum  satis  accurate  ex  phainomenis  definire  licuit)  prodiret  annua 
iEquinoctiorum  prajcessio  50" .  40"' .  43'".  Quod  cum  phrenomenis  consrruit.  Nam 
praecessio  ilia  ex  observationibus  Astronomicis  est  50"  vel  51"  circiter. 

Descripsimus  jam  systema  Solis  Terrs  &  Planetarum ;  superest  ut  de  Cometis 
nonnulla  adjiciantur. 


(yO  CORRESPONDENCE  OF 

the  quantity  of  matter  in  each  space  shall  not  be  the  same 
Therefore  when  You  define  or  assume  y*  quantity  of  matter 
to  be  proportionable  to  its  vis  inertias  You  must  not  at 
the  same  time  define  or  assume  it  to  be  proportionable  to 
y*  space  which  it  may  perfectly  fill  without  any  void  in- 
terstices unless  You  hold  it  impossible  for  the  two  Globes 
A  &  B  to  have  different  Vires  Inertiie.  Now  in  y®  3** 
Corollary  I  think  You  do  in  effect  assume  both  these  things 
at  once 


LETTER  XXXVI. 
NEWTON  TO  COTES. 

In  the  scholium  to  y*  iv***  Proposition  I  should  have  put 
the  length  of  y®  Pendulum  in  vacuo  3  feet  &  8f  lines.  It 
was  by  an  accidental  error  that  I  wrote  8|^  lines.  The 
Pendulum  must  be  something  longer  in  Vacuo  then  in 
Acre  to  vibrate  seconds.  You  may  put  it  either  8f  or  8|^ 
as  you  shall  think  fit,  the  difference  being  inconsiderable. 
If  you  chuse  S]^,  the  numbers  computed  from  thence  may 
stand. 

In  the  new  Scholium  to  the  xiv*^  Proposition,  I  took 
the  motion  of  the  Aphelium  of  Mars  to  be  what  D""  Halley 
had  computed  it  &  thence  deduced  the  motion  of  the 
Earth's  Aphelium  to  be  18'.  56"  in  an  100  years.  D'  Halley 
had  formerly  given  me  the  motion  of  y®  Aphelium  of  ^  40' 
in  100  years  &  thence  I  computed  the  motion  of  the 
Earths  Aphelium  2l'.  40":  but  I  account  the  latter  reccon- 
ing  to  be  more  confided  in,  &  therefore  in  the  Theory  of 
y®  Moon  you  may  put  the  motion  of  y*  earths  Aphelium 
18'.  3(j'  in  100  years. 

In  y^  16*^  Corollary  of  Prop,  lxvi  Lib.  1  (or  in  pag  166 
lin  9  of  y®  new  Edition)  it  should  be   [ut  tempus  periodi- 


NEWTON  AND  COTES.  67 

cum  corporis  P  directe  &c]  as  you  well  observe,  &  not  [ut 
quadratum  temporis  periodici]  as  it  is  now  printed. 

In  the  xxxix*^  Proposition  these  emendations  may  be 
made.  Pag.  470  lin  26  [ad  diametrum  mnjorem  AC  ut  228 
ad  229)  ut  51984  ad  457  seu  11375  ad  100.]  Pag  471  lin  1 
[ut  100  ad  11375  et  lOOOOOO  ad  925275  conjunctim,  hoc  est, 
ut  1000  ad  105042,  ideoq:  motus  annuli  esset  ad  summam 
motuum  annuli  et  globi  ut  1000  ad  106042.]  lb.  lin  7  [ut 
1000  ad  106042;]  lb.  lin  10  [ut  1436  ad  39343  et  1000  ad 
106042  conjunctim,  id  est,  ut  1  ad  29 19.  lb.  lin.  20  [eva- 
deret  minor  quam  prius  in  ratione  2  ad  5.  Ideoq:  annuus 
sequinoxiorum  regressus  jam  esset  ad  20**  ll'  46"  ut  l  ad 
7298,  ac  proinde  fieret  9"  57'"  42  .]  Pag  473  lin  27  [Cum 
igitur  inclinatio  ilia  sit  23^  graduum,  diminuendus  est 
motus  9"  5/"  42'^  in  ratione  sinus  91706  (qui  sinus  est  com- 
plementi  graduum  23^)  ad  radium  100000.  Qua  ratione 
motus  ille  jam  fiet  9"  8'"  8"".  And  a  little  after.  Pra5ces- 
sio  a  vi  Luna9  oriunda  43".  4"'.  4'^^,  ac  tota  Pnecessio 
annua  a  vi  utraq:  oriunda  52"  12'".  13'\ 

Si  vis  Lungs  ad  Mare  movendum  esset  ad  vim  Solis  ut 
4^  ad  1  (nam  proportio  harum  virium  nondum  satis  accu- 
rate ex  phsenomenis  definire  licuit)  prodiret  annua  aequi- 
noxiorum  prsecessio  50'*  14'".  45'^.  Quae  cum  phsonomenis 
congniit.  Nam  praecessio  ilia  ex  observationibus  Astrono- 
micis  est  vel  50"  vel  5l"  circiter. 

Si  altitude  Terras  ad  -^quatorem  superet  altitudinem 
ejus  ad  polos  milliaribus  plusquam  17»  materia  ejus  rarior 
erit  ad  circumferentiam  quam  ad  centrum,  et  prcecessio 
ajquinoxiorum  ob  altitudinem  illam  augebitur  &  vicissim  ob 
raritatem  diminuetur. 

Descripsimus  jam  sy sterna  Solis  Terra?  et  Planetarum : 
superest  ut  de  Cometis  nonnulla  adjiciantur. 

ffor  obviating  the  objection  you  make  against  the  3* 
Corollary  of  Prop,  vi  Lib.  in,  you  may  add  to  the  end  of  that 
Corollary  these  words.     Hoc  ita  se  habebit  si  modo  ma- 

5—2 


68  CORRESPONDENCE  OF 

teria  sit  gravitati  suae  proportionalis  &  insuper  impenetra- 
bilis  adeoq:  ejusdem  semper  densitatis  in  spatiis  plenis. 

I  am  Y'  most  humble  Servant 
London  Feb.  19  17-J-J.  Is.  Newton. 

For  tJie  R*"*  M""  Cotes  Professor  of  Astronomy, 
at  his  Chamber  in  Trinitt/  College  in  Cam- 
bridffe. 


LETTER  XXXVII. 
COTES  TO  NEWTON. 
S'.  Febr.  23'*  Vr$  Cambridge 

I  have  received  Your  last.  As  I  reviewd  the  xx*'* 
Proposition  I  perceiv'd  it  was  by  a  slip  of  the  Pen  that 
You  had  put  8|-  instead  of  8|  lines  in  Your  former  Letter. 
I  choose  this  number  rather  than  8]-§-  for  the  reason  which 
You  gave  &  because  the  fraction  is  more  simple  &  already 
in  use  amongst  the  French.  I  am  satisfied  that  these  exact- 
nesses, as  well  here  as  in  other  places,  are  inconsiderable 
to  those  who  can  judge  rightly  of  Your  book :  but  y®  gene- 
rality of  Your  Readers  must  be  gratified  w*^  such  trifles, 
upon  which  they  commonly  lay  y®  greatest  stress.  I  thank 
You  for  the  information  You  have  given  me  concerning 
the  new  Scholium  to  the  xiv*^  Proposition.  You  have  very 
easily  dispatched  the  32  Miles  in  Prop.  xxxIx*^  I  think  You 
have  put  that  matter  in  the  best  method  which  the  nature 
of  the  thing  will  bear. 

Your  addition  to  y^  3^  Corollary  of  Prop,  vi*''  does  not 
seem  to  come  fully  up  to  y®  Objection.  Your  words  are 
[Hoc  ita  se  habebit  si  modo  materia  sit  gravitati  sua?  pro- 
portionalis &  insuper  impenetrabilis  adeoq:  ejusdem  semper 
densitatis  in  spatiis  plenis].  Now  by  materia  You  mean 
the  quantity  of  Matter  &  this  You  had  always  estimated 


NEWTON  AND  COTES.  69 

by  its  Vis  inertia?,  &  therefore  it  will  bo  supposed  that  You 
do  in  this  place  so  estimate  it :  but  if  materia  be  here  taken 
in  this  sense  the  Objection  will  not  be  obviated.  Perhaps 
wfth  some  alteration  of  my  words,  which  You  may  be 
pleased  to  make,  the  addition  may  stand  thus  [Hoc  ita  se 
habebit  si  modo  magnitudo  vel  extensio  materia?  in  spatiis 
plenis,  sit  semper  proportionalis  materioe  quantitati  &  vi 
Inertia?  atq:  adeo  vi  gravitatis:  nam  per  banc  Proposi- 
tionem  constitit  quod  vis  inertia?  &  quantitas  materitE  sit 
ut  ejusdem  gravitas] 

In  the  xix*^  Proposition  pag.  422.  lin  9  I  will  put  [l  ad 
289]  &  in  line  13"*  [ut  289  ad  288]  in  line  15*^  [289],  in  line 
16*'*  [288]  according  to  Your  former  directions*.  In  the 
25*^  &  28*^  lines  I  would  omit  y*  fractions  -^  &  write  [ut 
126  ad  125]  &  [ut  125  ad  126] :  for  my  computation  makes 
the  former  proportion  to  be  126,44024  ad  125,44024  &  the 
latter  to  be  124,80397  ad  125,80397.  In  Page  423  line  11*^  I 
would  put  [ha?  tres  rationes  126  ad  125,  126  ad  125^,  &  JOG 
ad  101].  lb.  lin  27*'*  [ut  l  ad  289].  lin  28*'*  [est  tantura 
pars^^]  line  31**  [vis  centrifuga  ^l^]  in  y*  last  line  [pars 
tantum  g^^].  Page  224.  line  1**  I  would  put  [per  polos 
230  ad  229]  &  y®  rest  accordingly  taking  the  measure  of  a 
mean  degree  to  be  57230  Toises. 

In  the  xx*'*  Proposition,  page  425,  line  8***,  You  have 
altered  thus  [Undo  tale  confit  Theorema — vel,  quod  pe- 
rinde  est,  ut  quadratum  sinus  recti  Latitudinis.  Et  in 
eadem  circiter  ratione  triplicataf  augentur  arcus  graduum 
Latitudinis  in  ^leridiano.  Ideoq:  cum  Latitudo  Lutetioe 
&c.]  I  think  the  word  tripUcata  ought  to  be  omitted :  it 
should  be  [Et  in  eadem  circiter  ratione  augentur  arcus 
graduum  &c].  I  suppose  by  some  inadvertency  the  mis- 
take arose  from  this.  That  the  degree  under  y*  /Equator  is 


•  In  letter  of  Feb.  12. 

t  In  Newton's  MS.  the  word  is  triplic*  (No.  138.) 


70  CORRESPONDENCE  OF 

to  y*  degree  under  the  Pole  as  CP  cub  to  CA  cub  (fig : 
page  422).  This  proportion  is  no  where  mentioned  in 
Your  additional  papers,  but  I  guess  You  designed  to  have 
added  it  or  something  to  y*  same  effect  to  make  Your 
Rule  compleat  for  finding  the  measure  of  a  degree  under 
any  Latitude. 

When  I  was  formerly  upon  this  place  I  made  the  fol- 
lowing alteration  in  order  to  examine  the  numbers  of  Your 
Table,  [Undo  tale  confit  Theorema,  quod  incrementum 
ponderis  ut  et  mensuraa  gradus  unius  in  Meridiano  per- 
gendo  ab  iEquatore  ad  Polos  sit  quam  proximo  ut  sinus 
versus  latitudinis  duplicatoe,  vel,  quod  perinde  est,  ut  quad- 
ratum  sinus  recti  latitudinis.  Nam  si  M  ponatur  pro 
AB  X  PQ  cub  -  PQqq  ABqq  -  Pdqg 
AB^q '    ^  P™  ABqq         '  ^    ^  P"^** 

^j- (vid:  fig:  p.  422)  erit  gravitas  sub  JEquatore 

ad  excessum  gravitatis  in  alio  quovis  loco  cujus  sinus  rec- 

M 
tus  latitudinis  est  S  existente  R  radio,  ut  1  ad  -5--  SS 

RR 

+  — — —  o^  +  — =-—  o®  +  &c.     Mensura  vero  gradus  unius  m 
R*  Rr 

Meridiano  ad  iEquatorem,  erit  ad  excessum  ejus  in  alio 

loco  ut  1  ad  -—-  SS  +  - — -— f  S^  +  - — - — ^-z^S"  +  &c. 
2RR  2x4/?^  2x4x  6R^ 

Itaq:  cum  sit  AB  ad  PQ  ut  230  ad  229,  &  Lutetioe  Parisi- 

orum  in  latitudine  48^^  50'  longitudo  penduli  singulis  minu- 

tis  secundis  oscillantis  sit  pedum  trium   Parisiensium  & 

linearum    8^ ;    longitudines   vero    pendulorum    a?qualibus 

temporibus  in  locis  diversis  oscillantium  sint  ut  gravitates : 

longitudo    penduli    sub   iEquatore    erit    pedum    trium   & 

linearum  7,4-8,  sub  Polo  erit  pedum  trium  &  linearum  9,S9 : 

mensura  vero  gradus  unius  ad  iEquatorem  erit  Hcxapeda- 

rum  56733,  ad  Polum  erit  Hexapedarum  57530,  si  modo 

inter  gradus  latitudinis  48  &  49  ponatur  esse  Hexapedarum 

57200.     Et  simili  computo  confit  Tabula  sequens.] 


NEWTON  AND  COTES.  71 

In  making  these  rules  I  take  the  measure  of  a  degree 
at  any  point  of  the  Meridian  to  be  proportionable  to  y* 
Radius  of  the  curvature  of  y*  Ellipsis  at  that  point,  or 
which  is  y*  same  thing  to  be  proportionable  to  y*  Cube  of 
y*  part  of  the  Radius  of  y*  curvature  which  is  intercepted 
between  y®  point  proposed  in  y*  Ellipsis  &  the  point  where 
the  Radius  intersects  y*  greater  Axis ;  and  y*  angle  made 
by  that  intersection  I  take  for  the  measure  of  the  Latitude, 
Thus  I  had  then  altered  y®  place,  but  I  think  this  exact- 
ness is  not  necessary;  for  y®  following  terms  of  these 
series  are  inconsiderable  in  respect  of  the  first,  &  the  figure 
of  the  Earth  is  not  exactly  Elliptical  &  the  solution  of  the 
Problem  will  be  more  simple  without  it,  by  taking  y* 
length  of  y®  Pendulum  under  the  ^Equator  to  y*  length 
under  the  Poles  in  the  proportion  of  229  to  230,  &  the 
Measure  of  a  degree  at  the  ^Equator  to  y*  measure  at  y* 
Poles  in  the  triplicate  proportion  of  229  to  230  or  as  228  to 
231  or  76  to  77,  &  in  both  cases  by  making  the  increment 
from  the  -Equator  to  be  as  the  square  of  y*  sine  of  y*  Lati- 
tude or  as  the  versed  sine  of  the  doubled  Latitude. 

As  to  the  Table  of  the  lengths  of  Pendulums  &  the 
measures  of  Degrees  I  beleive  Your  Readers  would  rather 
desire  it  were  computed  to  y®  difference  of  32  Miles  than 
to  that  of  17  JMiles,  &  I  do  not  see  any  use  of  it  as  it  now 
stands  for  which  the  Table  made  to  the  difference  of  32 
Mies  may  not  serve.  If  You  agree  to  this  Proposal,  I  will 
compute  it  as  you  shall  direct  either  by  the  Series  or  the 
other  way.  It  must  be  placed  after  Your  account  of  the 
Observations  &  thereby  some  small  changes  will  be  made 
in  the  context  which  You  may  be  pleased  to  send  me. 

IVhat  I  have  further  observed  as  to  this  Proposition  is 
as  follows.  You  have  put  down  Goreje  Latitudo  14^.  15'.  by 
y®  observations  of  Dcs  Hayes  tis  14®.  40',  In  Your  account 
of  Picard's  experiment  of  an  heated  wire  You  say  [in  igne 
posita]  De  la  Hire  says  only  [car  M :  Picard  ayant  expose 


72  CORRESPONDENCE  OF 

les  corps  a  gelee,  les  mettoit  ensuite  aupres  du  feu]  or  near 
the  fire.  By  my  eomputation  the  observation  at  Guada- 
loupe  redueed  to  the  JSquator  gives  the  differenee  of  2,29 
lines,  that  at  Martinique  2,31  lines,  exceeding  Your  limit  of 
2^  lines ;  the  rest  fall  within  Your  limits.  After  [auctus 
in  ratione  differentiarum  fiet  milliarium  32]  I  would  add 
[&  diameter  secundum  ajquatorem  erit  ad  diametrum  per 
polos  ut  123  ad  122]  for  as  1,C/  to  2  so  is  -^^-^  to  Yj-jf. 
Speaking  of  the  Shadow  of  the  Earth  in  Lunar  Eclipses 
You  say  [diameter  ejus  ab  Oriente  in  Occidentem  ducta, 
major  erit  quam  diameter  ejus  ab  Austro  in  Boream  ducta 
excessu  56''  fere]  I  think  it  should  be  4l" ;  for  the  mean 
Horizontal  Parallax  of  y®  Moon  in  Syzygiis  being  57'.  30", 
the  Parallax  of  y®  Sun  10'',  &  the  Suns  mean  diameter 
32'.  12";  the  diameter  of  y^  Shade  will  be  4988",  add  70" 
upon  account  of  the  Atmosphajre  &  the  diameter  will  be 
5058",  which  divided  by  123  gives  4l".  At  the  end  of  this 
Paragraph  You  have  [Et  distantia  mediocris  centrorum 
Terra)  &  Luna3  erit  60^  semidiametrorum  Terrro]  which  I 
do'nt  well  understand.  In  y^  last  Paragraph  You  have  [et 
Pendula  isochrona  longiora  forent  in  Observatorio  Regio 
Parisiensi  quam  ad  iEquatorem  excessu  semissis  digiti  cir- 
citer]  I  suppose  it  should  be  [longiora  forent  ad  iEquatorem 
quam  in  Observatorio]  And  a  little  lower  You  have  [Sed 
&  diameter  umbra3  Terras  — major  foret — excessu  2'.  45" 
seu  parte  duodecima  diametri  Lunso]  I  think  it  should  be 
[excessu  2',  seu  parte  decima  sexta  diametri  Lunao] 

In  the  Memoires  of  the  Royale  Academic  for  the  Year 
1708  there  are  one  or  two  observations  of  the  lengths  of 
Pendulums,  besides  those  which  You  have  related  in  Your 
History  from  other  Memoires  &  from  the  Observations 
faitcs  en  plusieurs  voyages. 

Taking  y*  semidiameters  of  the  Earth  to  be  as  229  & 
230  instead  of  228  &  229,  I  have  made  a  small  alteration  in 
Proposition  xxxix**'  which  I  will  not  trouble  You  with  since 


NEWTON  AND  COTES.  73 

I  think  I  do  understand  Your  thoughts  as  to  that  Proposi- 
tion. The  conclusion  of  it  puts  me  in  mind  of  an  allowance 
which  ought  to  be  made  in  Prop,  xxxvii***  on  account  of 
the  Moons  coming  nearer  to  y*  Earth  in  Syzygiis  &  going 
further  from  it  in  Quadraturis  than  in  her  mean  distance 
at  the  Octants.  But  this  allowance  would  increase  the 
number  45-  so  much  as  to  give  some  disturbance  to  the 
xxxix*^  Proposition  &  the  Scholium  of  the  iv*^  as  they  now 
stand,  unless  You  think  fit  to  ballance  it  some  other  way, 
for  there  is  a  latitude  in  that  xxxvii***  Proposition. 
I  am,  S*",  Your  most  Humble  Servant 

For  S'  Isaac  Newton  at  his  House  Roger  CotES. 

in  S*  Martin's  Street  in  Leicester 
.    Fields  London 


LETTER  XXXVIII. 
NEWTON  TO  COTES. 


S' 
I  have  reconsidered  the  third  Corollary  of  the  vi*^  Pro- 
position. And  for  preventing  the  cavils  of  those  who  are 
ready  to  put  two  or  more  sorts  of  matter  you  may  add 
these  wordjs}  to  the  end  of  the  Corollary.  Vim  inertice 
proportionalem  esse  gravitati  corporis  constitit  per  experi- 
menta  pendulorum.  Vis  inertise  oritur  a  quantitate  materia) 
in  corpore  ideoq:  est  ut  ejus  massa.  Corpus  condensatur 
per  contractionem  pororum,  &  peris  destitutum  (ob  impeni- 
trabilitatem  materia?)  non  amplius  condensari  potest ;  ideoq: 
in  spatiis  plenis  est  ut  magnitudo  spatii.  Et  concessis 
hisce  tribus  Principiis  CoroUarium  valet. 

Your   emendations  of  the  xix*'^  Proposition  may  all 
of  them  stand. 

In  the  emendation  of  the  xx***  Proposition  pag  425 
lin.  8  the  word  triplicata  should  be    struck  out  as  you 


74  CORRESPONDENCE  OP 

obBcrve.  The  rest  may  stand  unto  the  words  [Et  simili 
computo  fit  Tabula  sequens]  correcting  only  the  numbers 
as  you  propose  &  putting  the  numbers  229  &  230  instead  of 
689  &  692.  The  Table  is  computed  to  y®  excess  of  17  miles 
rather  then  to  that  of  32  miles,  because  that  of  17  is  the 
least  that  can  be  &  is  certain  upon  a  supposition  that  the 
earth  is  uniform,  that  of  32  is  not  yet  sufficiently  ascer- 
tained, &  I  suspect  that  it  is  too  big. 

After  the  last  observations  of  Des  Hayes  ending  w*** 
these  words  [et  quod  in  insula  S.  Dominici  eadem  esset 
ped.  3,  lin.  7]  add  this  Paragraph. 

Deniq:  anno  1704,  P.  Fuelleus  invenit  in  Porto-belo  in 
America  longitudincm  Penduli  ad  minuta  secunda  oscil- 
lantis  esse  pedum  trium  Parisiensium  et  lin  ear  um  5-j3^,  id 
est  tribus  circiter  lineis  breviorem  quam  in  Latitudine 
Lutetiae  Parisiorum;  &  subinde  ad  insulam  Martinicam 
navigans  invenit  longitudincm  Penduli  isochroni  esse  pe- 
dum trium  Parisiensium  et  linearum  5^, 

Latitude  autem  Paraibae  est  6^  38'  in  austrum  et  ea 
Porto-beli  9^^  33'  in  boream,  et  Latitudines  insularum  &e. 
You  may  here  put  the  Latitude  of  Goree  14^  40'.  I  have 
not  books  by  me  to  examin  it. 

Let  the  next  Paragraph  run  thus.     Observavit  utiq: 

ad  ignem  calefacta  evasit  pedis  unius  cum  quarta 

parte  lineae In  priore  casu  calor  major  fuit  quam  in 

posteriore,  in  hoc  vero  major  fuit  quam  calor  externarum 
partium  corporis  humani.     Nam  metalla  ad  solem  restivum 

valde  incalcscunt sed  excessu  quartam  partem  linear 

unius  vix  superante differentia  prodiit  non  minor 

quam  l|^-  lineae  non  multo  major  quam  linearum  2^.  Et 
inter  hos  limites  quantitas  mediocris  est  2^.  Propter 
calores  locorum  in  Zona  torrida  negligamus  tres  decimas 
partes  lineae  et  manebit  differentia  duarum  linearum  cir- 
citer   jam  autus  in  ratione  differcntiarum  fiet  milli 

arium  plus  minus  32.     Est  igitur  excessus  ille  non  minor 


NEWTON  AND  COTES.  75 

quam    milliarium     17,    non    multo    miyor     quam    milH- 
arium   82. 

I  think  the  words  [excessu  56"  fere]  are  right,  ffor  the 
Moons  parallax  57'  SO"  must  have  an  increase  in  the 
proportion  of  32  miles  to  the  earths  semidiameter,  that  is 
an  increase  of  28",  w^^  doubled  give  56^'  to  be  added  to  y* 
diameter  of  the  earths  shadow,  ffor  the  Suns  diameter  & 
parallax  remain  without  sensible  alteration.  And  for  y* 
same  reason  I  take  [excessu  2'  45"]  to  be  right. 

In  the  calculation  of  the  Moons  force  (Prop,  xxxvii) 
your  scruple  may  be  eased  (I  think)  by  relying  more  upon 
the  observation  of  the  tyde  at  Chepstow  then  on  that 
at  Plymouth,  but  I  have  mislaid  my  copy  of  the  calcu- 
lation. If  the  nearer  access  of  the  Moon  to  the  earth 
in  the  Syzygies  then  in  the  Quadratures  create  any  diffi- 
culty be  pleased  to  send  me  a  copy  of  the  calculation  &  I 
will  reconsider  it.   The  Latitude  of  Paris  should  be  48^  50', 

I  am  S"- 
Yo"^  most  humble  Servant 
London  Feb.  2G  17  f^.  Is.  Newton 

For  the  R"**  M'  Roger  Cotes  Professor 
of  Astronomy  at  his  chamber  in 
Trinity  College  in  Cambridge. 


LETTER    XXXIX. 
COTES   TO   NEWTON. 
S'  Febr.  28^  17ll 

I  have  look*d  over  Your  new  addition  to  y*  3**  Co- 
rollary of  y®  vi***  Proposition,  but  I  am  not  yet  satisfied 
as  to  the  difficulty,  unless  You  will  be  pleased  to  add, 
that  it  is  true  upon  this  concession  that  the  Primigenial 
particles  out  of  which  the  world  may  be  supposed  to  have 
been  fram'd  (concerning  which  You  discourse  at  large  in 


76  CORRESPONDENCE   OF 

y*  additions  to  Your  Opticks  pag.  343  &  seqq.)  were  all  of 
them  created  equally  dense,  that  is,  (as  I  would  rather 
speak,)  have  all  the  same  vis  Inertia?  in  respect  of  their 
real  magnitude  or  extension  in  spatio  pleno.  I  call  this  a 
concession,  because  I  cannot  see  how  it  may  be  certainly 
proved  either  a  priori  by  bare  reasoning  from  the  nature 
of  the  thing,  or  be  inferrd  from  Experiments.  I  am 
not  certain  whether  You  do  not  Your  self  allow  the 
contrary  to  be  possible.  Your  words  seem  to  mean  so  in 
pag :  347.  lin :  5  Optic :  [forte  etiam  &  diversis  densita- 
tibus  diversisq:  viribus] 

I  do  not  clearly  understand  how  You  would  have  y® 
alteration  settled  in  Prop :  xx*'',  I  mean  that  which  begins 
with  [Unde  tale  confit  Theorema]  &  ends  with  [et  simili 
computo  confit  Tabula  sequens].  You  may  be  pleased  to 
send  me  a  transcript  of  y®  Context  leaving  void  spaces  for 
the  Numbers.  You  may  let  me  know  at  y®  same  time 
time  whether  You  choose  57200  or  57230  Toises  for  the 
Measure  of  a  degree  between  the  Latitudes  48**.  49°.  I 
suppose  You  retain  8^  lines  for  y®  length  of  y®  Pendulum. 

I  am  satisfied  that  56"  is  the  right  increase  of  y®  shadow 
of  y®  Earth,  'twas  my  oversight  in  making  the  figure  of  y® 
shadow  to  be  similar  to  that  of  y®  Earth. 

As  to  the  xxxvii***  Proposition,  I  take  it  that  the 
Moons  force  must  be  augmented  in  her  Syzygies  & 
diminished  in  her  Quadratures  in  the  proportion  of  47  to 
46  nearly.  Whence  by  my  computation,  if  nothing  else  be 
altered  in  the  Proposition,  S  will  be  to  L  nearly  as  1  to 
5|.  To  make  5*  to  L  as  1  to  4^^  or  4|-;  instead  of 
putting  L  +f  S  to  '^-L-fSas  7  to  4,   it  may  be  put 

^L  +  ^-Sto     ^*     Zr  -  ?-  /S'  as  1 1  to  6.    But  this  proportion 

47,  7 

of  1 1  to  6  falls  without  y®  Limits  at  Bristol  &  Plymouth.  I 
shall  therefore  leave  it  to  Your  self  to  settle  y®  whole 
Proposition  as  You  shall  judge  it  may  best  be  done.     In 


NEWTON  AND  COTES.  77 

y*  xxviu*^  Proposition  I  shall  hereafter  take  notice,  that  I 
find  the  proportion  to  be  as  ^9^  to  70^  instead  of  CS-J-J 
to  ^9^.  I  think  69  to  70  may  every  where  be  used. 
Your  Copy  of  y*  xxxvii*^  Proposition  is  as  follows*.  {Vis 
Luna9  ad  mare  movendum  colligenda  est  ex  ejus  pro- 
portione  ad  vim  Solis,  et  hroc  proportio  colligenda  est 
ex  proportione  motuum  maris  qui  ab  his  viribus  priuntur. 
Ante  ostium  fluvii  Avonce  ad  lapidem  tertium  infra  Bris- 
tolianif  tempore  verno  et  autumnali  totus  aquar  ascensus 
in  conjunctione  et  oppositione  Luminal ium,  obscrvante 
Samuele  Sturmio,  est  pedum  plus  minus  45,  in  Quadraturis 
autem  est  pedum  tantum  25.  Altitudo  prior  ex  summa 
virium  posterior  ex  eorundemf  differentia  oritur.  Solis 
igitur  et  Lunai  in  iEquatore  versantium  et  mediocriter 
a  Terra  distantium  sunto  vires  S  et  L,  et  erit  L  +  S  ad 
L  -  iS^nt  4^  ad  25  sen  9  ad  5. 

In  portu  Plymuthi  aestus  maris  (ex  observatione  5a- 
muelis^  Colepressi)  ad  pedes  plus  minus  sexdecim  altitudine 
mediocri  attollitur,  ac  tempore  verno  et  autumnali  altitudo 
jestus  in  syzygiis  superare  potest  altitudinem  ejus  in  quad- 
raturis pedibus  plus  septem  vel  octo.  Si  maxima  hanim 
altitudinum  differentia  sit  pedum  octo,  erit  Z.  +  5  ad  i  -  5^ 
ut  20  ad  12  sen  5  ad  3.  Donee  aliquid  certius  ex  phae- 
nomenis  constiterit,  assumamus  L  -k-  S  esse  ad  L  —  S  (pro- 
portione mediocri)  ut  7  ad  4. 

Caeterum  ob  aquarum  reciprocos  motus  eestus  maximi 
non  incidunt  in  ipsas  Luminarium  syzygias  sed  sunt  tertii  a 
syzygiis  ut  dictum  fuit,  et  incidunt  in  horam  Lunarcm  plus 
minus  tricesimam  sextam  a  syzygiis,  id  est,  in  horam 
solarem  tricesimam  septimam  circiter.  Oritur  hie  rcstus 
ab  actione  Lunae  in  ejus  proecedente  appulsu  ad  meridianum 


"  I  have  transcribed  the  Proposition  from  Newton's  MS.  Nos.  193,  194,  Cotes  not 
having  copied  it  into  this  drauprht  of  his  letter.  The  headini;  is  "  Invenire  vim  Lunie 
ad  Mare  movendum." 

1  sic. 


78  CORRESPONDENCE  OF 

loci  et  hie  appulsus  praecedit  cesium  in  portu  Bristolios 
lioris  plus  minus  septem,  ideoq :  incidit  in  horam  solarem 
post  syzygias  et  quadraturas  tricesimam  eirciter.  Eo 
autem  tempore  Luna  distat  a  Sole  15^  gr.  eirciter.  Et 
Sol  in  hac  distantia  minus  auget  ac  minuit  motum  maris  a 
vi  Lunse  oriundum  quam  in  ipsis  syzygiis  et  quadraturis, 
in  ratione  Radii  ad  cosinum  distantise  hujus  duplicatae  seu 
anguli  30  ^  gr.  hoc  est,  in  ratione  7  ad  6  eirciter ;  ideoq :  in 
superiore  analogia  pro  S  scribi  debet  |-  S 

Sed  et  vis  L  in  Quadraturis  ob  declinationem  Lunse 
diminui  debet.  Nam  Luna  in  Quadraturis  tempore  verno 
et  autumnali  extra  sequatorem  in  declinatione  graduum 
plus  minus  23  ^  versatur,  et  Luminaris  ab  iEquatore  decli- 
nantis  vis  ad  mare  movendum  diminuitur  in  duplicata 
ratione  sinus  complementi  declinationis  quamproxime,  & 
propterea  vis  Lunae  in  his  Quadraturis  est  tantum  |-  L.  Est 
igitur  L  +  fS  ad  J  L  -  ^~  S  ut  7  ad  4.  Et  inde  fit  S  ad  L 
ut  7  ad  33  vel  1  ad  4 1 . 

Est  igitur  vis  Solis  ad  vim  Lunae  ut  l  ad  4|-  quam 
proximo.  Et  hanc  proportionem  donee  aliquid  eertius  ex 
observationibus  accuratius  institutis  constiterit,  usurpare 
licebit.  Undo  cum  vis  Solis  sit  ad  vim  gravitatis  in 
superficie  Terrae  ut  1  ad  12868162,  vis  Lunae  erit  ad  vim 
gravitatis  ut  1  ad  272961 0  eirciter. 

Corel.  1.  Cum  aqua  maris  vi  Solis  agitata  ascendat  ad 
altitudinem  pedis  unius  &  undecim  digitorum  cum  quad- 
rante,  eadem  vi  Luna9  ascendet  ad  altitudinem  pedum 
novem,  &  vi  utraq :  ad  altitudinem  pedum  undecim  eirciter, 
et  ubi  Luminaria  sunt  in  perig83is,  ad  altitudinem  pedum 
duodecim  &  ultra,  prajsertim  ubi  eestus  ventis  spirantibus 
adjuvantur.  Tanta  autem  vis  ad  omnes  maris  motus 
excitandos  abunde  sufficit,  et  quantitati  motuum  probe 
respondet.     Nam  in  maribus . . . .} 


NEWTON  AND  COTES.  79 


LETTER   XL. 


COTES   TO  NEWTON. 
S'  March  13*^  17-}i 

I  received  Your  last  of  the  2G***  of  February  in  due 
time  &  by  the  next  post  I  sent  You  w*^  one  or  two  other 
things  a  Transcript  of  the  xxxvii*^  Proposition  as  it  now 
stands  in  Your  Copy.  Having  received  no  Letter  from 
You  since  that  time  I  fear  there  has  been  some  miscarriage. 
About  two  sheets  of  the  in**  Book  are  composed,  but 
expecting  Your  answer  I  have  not  yet  given  leave  to  print 
them  off.  Your  most  humble. 


LETTER  XLI. 
NEWTON  TO  COTES. 


I  have  not  yet  been  able  fully  to  settle  the  Theory  of 
the  xix*^,  xx^^  xxxvi^^  xxxvii*^  &  xxxix*^  Propositions  & 
that  of  the  Scholium  to  the  iv***.  But  I  think  to  let  the 
Scholium  of  iv*^  Proposition  be  set  at  the  end  of  the 
xxxvii*^  because  it  depends  on  a  Corollary  of  that  Propo- 
sition. And  therefore  you  may  let  the  Press  go  on  at 
present  without  it  &  set  it  aside  till  you  come  to  the 
xxxvii*^  Proposition.  But  let  the  new  Corollary*  to  y*  in** 
Proposition  be  printed  at  the  end  of  that  Proposition. 
And  in  the  third  Corollary  to  y*  v*^  Proposition  strike  out 
the  word  [novissimam,]  &  let  the  words  in  the  latter  part 
of  y®  Corollary  run  thus  [Et  hinc  Jupiter  &  Saturnus  prope 
conjunctionem  se  invicem  attrahendo  sensibiliter  perturbant 
motus  mutuos,  Sol  perturbat  &c].     In  my  copy  it  is  prope 


•  Sent  Feb.  2.    See  p.  .57,  note  •. 


80  CORRESPONDENCE  OF 

conjunctionem  novissimam.  If  it  be  so  in  yours,  the  word 
novissimam  is  better  omitted. 

I  thank  you  for  explaining  yo**  objection  against  y® 
third  Corollary  of  the  sixt  Proposition.  That  Corollary  & 
the  next  may  be  put  in  this  manner.  Corel.  3.  Spatia 
omnia  non  sunt  ODqualiter  plena.  Nam  si  spatia  omnia 
icqualiter  plena  essent,  gravitas  speeifica  fluidi  quo  regio 
aeris  impleretur,  ob  summam  densitatem  materis9,  nil  ce- 
deret  gravitati  specificro  argenti  vivi  vel  auri  vel  corporis 
cujuscunq :  densissimi,  et  propterea  nee  aurum  neq :  aliud 
quodcunq :  corpus  in  acre  descendere  posset.  Nam  corpora 
in  fluidis,  nisi  specifice  graviora  sint,  minime  descendunt. 
Quod  si  quantitas  materia)  in  spatio  dato  per  rarefactionem 
quamcunq :  diminui  possit,  quidni  diminui  possit  in  infini- 
tum ?  Corel.  4.  Si  omnes  omnium  corporum  particuloe 
solidflo  sint  ejusdem  densitatis  neq:  absq:  poris  rarefieri 
possint,  Vacuum  datur.  Ejusdem  densitatis  esse  dico 
quarum  vires  inertiie  sunt  ut  magnitudines.  Corel.  5.  Vis 
gravitatis  diversi  est  generis  a  vi  magnetica.  Nam  attractio 
magnetica  non  est  ut  materia  attracta.  Corpora  aliqua 
magis  trahuntur,  alia  minus,  plurima  non  trahuntur;  Et 
vis  magnetica  in  uno  et  eodem  corpore  intendi  potest  & 
remitti,  estq :  nonnunquam  longe  major  pro  quantitate 
materia)  quam  vis  gravitatis,  et  in  recessu  a  magneto  de- 
crescit  in  ratione  distantia3  non  duplicata  sed  fere  triplicata 
quantum  ex  crassis  quibusdam  observationibus  animadver- 
tere  potui*. 

In  the  tenth  Proposition  pag.  417  lin  11  for  [viginti  et 
unius]  read  [triginta.]  &  lin.  12  for  [320]  read  [459]  &  lin 
17  for  [800]  read  [850]. 


•  At  the  meeting  of  the  Royal  Society  two  days  afterwards,  Newton  proposed  that 
Halley  and  Hauksbee  should  make  experiments  with  '*  the  great  loadstone,"  in  order 
to  find  the  tiue  law  of  the  decrease,  "which  he  believed  would  bo  nearer  the  cubes 
than  the  squares."    See  also  Journal  Book,  March  27,  Apr.  3,  May  15,  Jun.  12,  26. 

Phil.  Trans.  Jul Sept.  1712.    June— Aug.  1715.    Coulomb's  experiments  with  the 

Torsion  Balance  first  established  the  law  to  be  as  the  squares. 


NEWTON  AND  COTES.  81 

I  hope  to  send  you  the  xix  &  xx*^  Propositions  emend- 
ed within  a  Post  or  two.     I  am  S' 

Yo*^  most  humble  Servant 
♦Mar.  IS^**  17 f;**^  Is.  Newton. 

For  the  R"<*  M'  Cotes  Professor  of 

Astronomy  in  the  University  of 

Cambridge 
To  be  left  at  Trinity  College, 


LETTER  XLII. 
NEWTON  TO  COTES. 
S'  London  Apr.  3  1712. 

I  have  been  diverted  a  few  days  w*^  some  other  inter- 
vening business,  but  now  send  you  the  emendations  of  y* 
xix*^  xx*^  &  xxv^^f  Propositions,  as  follows. 

Prop.  XIX.  Prob.  ii. 

Invenire  proportionem  axis  Planetce  ad  diametros  eidem 

perpendicular  es, 

Picartus  mensurando  arcum   gradus  unius  et   22'.  55" 

inter  Amhianum  &  Malvoisinam,  invenit  arcum  gradus  unius 

esse    hexapedarum    Parisiensium    5706o.      Unde    ambitus 

TerrsQ  est  pedum  Parisiensium  123249G00,  ut  supra.      Sed 

cum  error  quadringentesimse  partis  digiti  tam  in  fabriea 

instrumentorum  quam  in  applicatione  eorum  ad  obscrva- 

tiones  capiendas  sit  insensibilis,  et  in  Sectore  decempcdali 

quo    Gain   observarunt   Latitudines    locorum    respondeat 

minutis  quatuor  secundis,  et  in  singulis  observationibus  in- 

cidere  possit  tam  ad  centrum  Sectoris  quam  ad  ejus  cir- 

cumferentiam,  et  errores  in  minoribus  arcubus  sint  mfyoris 


"  The  date  is  in  Cotes's  hand. 

t  This  is  an  oversight,  as  this  letter  does  not  contain  any  emendations  of  Prop.  xxv. 
and  in  his  next  letter  he  speaks  of  his  having  sent  his  corrections  of  the  19th  and  20th 
Propositions,  making  no  mention  of  the  25th, 


82  COnKilSPONDENCE  OP- 

momenti:  ♦idco  Cgwm'/im*  jussu  Regio  •  vyj  jr  t  '  A 
mensuram  Terras  per  iiiajora  locorum  demioB  liegim  Scientiaro. 
intervalla  aggressus  est,  et  subinde  "^^w^  1700. 
per  distantiam  inter  Obscrvatorium  Regium  Parisiense  et 
villain  Colioure  in  Roussillon  &  latitudinum  differentia 
&^.  18',  supponendo  quod  figura  Terra)  sit  sphierica,  invenit 
gradum  unum  esse  hexapedarum  57292,  prope  ut  Noi^woochis 
noster  antea  invenerat.  Hie  enim  eirea  annum  1635  men- 
surando  distantiam  pedum  Londinensium  905751  inter 
Londinum  et  Ehoracnm  &  observando  differentiam  Lati- 
tudinum 2^"".  28'  collegit  mensuram  gradus  unius  esse  pedum 
Londinensium  367196,  id  est,  hexapedarum  Parisiensium 
57300.  Ob  magnitudinem  intervalli  a  Cassino  mensurati, 
pro  mensura  gradus  unius  in  medio  intervalli  illius  id  est 
inter  Latitudines  45^"^  &  46*^'"  usurpabo  hexapedas  57292. 
Unde,  si  Terra  sit  splia3rica,  semidiameter  ejus  erit  pedum 
Parisiensium  1 9695539. 

Penduli  in  Latitudine  Lutetice  Parisiorum  ad  minuta 
seeunda  oscillantis  longitudo  est  pedum  trium  Parisiensium  & 
linearum  W^,  Et  longitudo  quod  \sic]  grave  tempore  minuti 
unius  secundi  cadendo  describit  est  ad  dimidiam  longitu- 
dinem  peuduli  hujus  in  duplicata  ratione  circumferentias 
circuli  ad  diametrum  ejus  (ut  indicavit  Hugenius)  ideoq :  est 
pedum  Parisiensiu  15,  dig.  1,  lin.  2^t»  seu  linearum  2l74jj. 

Corpus  in  circulo,  ad  distantiam  pedum  \9^055S0  a 
centro,  singulis  diebus  sidereis  horarum  23 .  bG).  4"  unifor- 
miter  revolvens,  tempore  minuti  unius  secundi  describit 
arcum  pedum  143,6223 :(:,  cujus  sinus  versus  est  pedum 
0,05236558,  scu  linearum  7,54064.  Ideoq :  vis  qua  gravia 
descendunt  in  Latitudine  Lutetice  est  ad  vim  centripetam 
corporum  in  JEquatore  a  Terrf©  motu  diurno  oriundam  ut 
21J4jt  ad  7,54064. 


t  i  is  altered  by  Cotes  in  the  MS.  to  ,^. 
X  Altered  by  Coles  to  1436,223. 


NEWTON  AND  COTES.  83 

Vis  centrifuga  corporum  in  iEquatore  est  ad  vim 
centrifugam  qua  corpora  directe  tcndunt  a  Terra  in 
Latitudine  Lutetice  in  duplicata  ratione  Radii  ad  sinum 
complcmenti  Latitudinis  illius,  id  est,  ut  7,54064  ad  S,27*. 
Addatur  ha9e  vis  ad  vim  qua  gravia  descendunt  in  Lati- 
tudine LutetioR,  et  corpus  in  Latitudine  Ltttetioi  vi  tota 
gravitatis  cadendo,  tempore  minuti  unius  secundi  dcscribet 
lineas  21 77,52  f  seu  pedes  Parisienses  15,  dig.  1,  &  lin  5,52  f. 
Et  vis  tota  gravitatis  in  Latitudine  ilia  erit  ad  vim  centri- 
petam  corporum  in  iEquatore  Terrao  ut  21 77,52  f  ad  7,54064, 
seu  289  ad  l. 

Unde  si  APDQ  figuram  Terra)  designet  jam  non  am- 
plius  sphajrica  sed  revolutione  Ellipseos  circum  axcm 
minorem  PQ  genitam,  sitq :  ACQqca  canalis  aqujD  plena, 
a  polo  Qq  ad  centrum  Cc,  &  inde  ad  -^quatorem  A  a  per- 
gens :  debebit  pondus  aquaa  in  canalis  crure  ACca  esse  ad 
pondus  aqua)  in  crure  altero  QCcq  ut  289  ad  288,  e6  quod 
vis  centrifuga  ex  circulari  motu  orta  partem  unam  e  pen- 
dens partibus  289  sustinebit  ac  detrahet,  et  pondus  288  in 
altero  crure  sustinebit  reliquas.  [In  the  rest  of  the  xix*** 
Proposition  proceed  according  to  the  former  corrections 
untill  you  come  at  page  484  f,  where  read]  ad  ipsius 
diametrum  per  polos  ut  230  ad  229.  Ideoq :  cum  Terra) 
semidiameter  mediocris  juxta  mensuram  Cassini  sit  pedum 
Parisiensium  \9Q955SC),  seu  milliarium  3939  (posito  quod 
milliare  sit  mensura  pedum  5000)  Terra  altior  erit  ad 
iEquatorem  quam  ad  Polos  excessu  pedum  85820,  seu  mil- 
liajri}um  17-J-. 

Si  Planeta  major  sit  vel  minor  quam  Terra  manente 
ejus  densitate  ac  tempore  periodico  revolutionis  diurna), 
manebit  proportio  vis  ccntrifugj©  ad  gravitatem,  &  prop- 
terea  manebit    etiam  proportio   diametri    inter  polos  ad 


•  Altered  by  Cotes  to  3,267.  f  Altered  by  Cotes  to  32. 

*  This  should  be  424. 

6—2 


84  CORRESPONDENCE  OP 

(liametrum  secundum  a?quatorem.  At  si  motus  diurnus  in 
ratione  quacunq :  acceleretur  vel  retardetur,  augebitur 
vcl  niinuctur  vis  centrifuga  in  duplicata  ilia  ratione,  et 
propterca  differentia  diametrorum  augebitur  vel  minuetur 
in  eadem  duplieata  ratione  quamproxime.  Et  si  densitas 
PlanetsB  augeatur  vel  minuatur  in  ratione  quavis,  gravitas 
ctiam  in  ipsum  tendons  augebitur  vel  minuetur  in  eadem 
ratione,  et  differentia  diametrorum  vicissim  minuetur  in 
ratione  gravitatis  aucta)  vel  augebitur  in  ratione  gravitatis 
diminuta).  Undo  cum  Terra  respectu  fixarum  revolvatur 
horis  23 .  50^  Jupiter  autem  horis  9 .  56\  sintq :  temporum 
quadrata  ut  29  ad  5  et  densitates  ut  5  ad  1  :  differentia 
diametrorum  Jovis  erit  ad  ipsius  diametrum  minorem  ut 

—  X  -  X  —  ad  1,   seu  1  ad  8  quamproxime.     Est  igitur 

diameter  Jovis  ab  oriente  in  occidentem  ducta  ad  ejus 
diametrum  inter  polos  ut  9  ad  8  quamproxime,  et  propterea 
diameter  inter  polos  est  ^5"^.  Hase  ita  se  habent  ex  hy- 
pothesi  quod  uniformis  sit  Planetaru  materia.  Nam  si 
materia  densior  sit  ad  centrum  quam  ad  circumferentiam, 
diameter  quae  ab  oriente  in  occidentem  ducitur  erit  adhuc 
major. 

Jovis  vero  diametrum  quae  polis  ejus  interjacet  minorem 
esse  diametro  altera  Cassinus  dudum  observavit,  et  Terroe 
diametrum  inter  polos  minorem  esse  diametro  altera  pate- 
bit  per  ea  qua?  dicentur  in  Propositione  sequente. 

In  the  XX*''  Proposition  page  425  lin.  8,  read.  Unde 
tale  confit  Theorema,  quod  incrementum  ponderis  pergendo 
ab  iEquatore  ad  Polos,  sit  quam  proximo  ut  sinus  versus 
Latitudinis  duplicatoe,  vel,  quod  perinde  est,  ut  quadratum 
sinus  recti  Latitudinis.  Et  in  eadem  circiter  ratione  au- 
gentur  arcus  graduum  Latitudinis  in  Meridiano.  Ideoq : 
cum  Latitudo  Lutetice  Parisiorum  sit  48^'.  50\  ea  locorum 
sub   iEquatore   00^.  oo',    et   ea  locorum  ad  Polos  90^''   & 


NEWTON  AND  COTES. 


85 


duplorum  sinus  vcrsi  sint  11334,  00000  ct  20000,  existcntc 

Radio  10000,  et  gravitas  ad  Polum  sit  ad  gravitatem  ejus 

sub  iEquatore  ut  229  ad  228,  &  excessus  gravitatis  ad  polum 

ad  gravitatem  sub  /Equatorc  ut  1   ad  228 :  crit  excessus 

gravitatis  in  Latitudine  Lutetice  ad  gravitatem  sub  iEqua- 

11334 

tore,  ut  I  X ad  228  seu  5667  ad  2280000.  Et  propterea 

20000 

gravitates  tota?  in  his  locis  erunt  ad  invicem  ut  2285667  ad 

2280000.     Quare   cum  longitudines  pendulorum  ocqualibus 

temporibus  oscillantium  sint  ut  gravitates,  et  in  Latitudine 

Lutetice  Parisiorum  longitudo  penduli  singulis  minutis  se- 

cundis  oscillantis   sit  pedum  trium  Parisiensium  &  8^   li- 

nearum,  longitudo  penduli   sub  /Equatore    superabitur   a 

longitudine   synchroni   penduli  Parisiensis,  excessu   linea; 

unius  et  92  partium  millesimaru  lineai.     Et  simili  computo 

confit  Tabula  sequens. 


Latitudo 
Loci. 

longitudo 
Penduli. 

Mennura 
gradus  unius 
in  Meridiano 

Grad. 

Ped. 

hin. 

llexaped. 

0 

3  . 

7,463 

56907 

5 

3  . 

7,478 

56913 

10 

3  . 

7,521 

56930 

15 

3  . 

7,592 

56957 

20 

3  . 

7,689 

56995 

25 

3  . 

7,808 

57041 

30 

3  . 

7,945 

57095 

35 

3  . 

8,0f)8 

57154 

40 

3  . 

8,260 

572 18 

45 

3  . 

8,427 

57283 

46 

3  , 

8,461 

57296 

47 

3  . 

8,494 

57309 

48 

3  . 

8,528 

57322 

49 

3 

8,561 

57335 

50 

3  . 

8,594 

57348 

55 

3 

8,756 

.57412 

60 

3 

8,909 

57471 

65 

3  . 

9,046 

57525 

70 

3  . 

9,16*5 

57571 

75 

3 

9,262 

57602 

80 

3 

9,333 

57626 

85 

3 

9,376 

57653 

90 

3 

9,391 

57659 

86  CORRESPONDENCE  OP 

Constat  autem  per  banc  Tabulam  &c 
Haec  ita  se  habent  ex  bypotbcsi  quod  Terra  &c 
Jam  vero  Astronomi  aliqui  in  longinquas  rcgiones  &c. 
Deinde  anno  1682  D.  Varini  &c. 
Postbac  D.  Couplet  filius  anno  1697 
Annis  proximis  (1699  &  1700)  D.  Des  Hayes  &c 
Annoq:   1704  P.  Feuellcus  invenit  in  Po{r}tobelo  in 
America  Longitudinem  Penduli  ad  minuta  secunda  oscil- 
lantis  esse  pedum  trium  Parisiensium  et  linearum  tantura 
5^r^f   id  est  tribus  fere   lineis  breviorem   quam  Lutetia) 
Parisiorum,  sed  errante   Observatione.     Nam   deinde   ad 
insulam  Martinicam  navigans  invenit  longitudinem  Penduli 
isocbroni  esse  pedum  tantum  trium  Parisiensium  et  linea- 
rum 5  }^. 

Latitudo  autem  Paraibce  est  C^'*"  38'  ad  austrura  et  ea 
Portobeli  ff^  33'  ad  boream,  et  Latitudines  insularum 
Cayennoj,  Goreuj,  Guadaloupoe,  Martanica3,  Granadse,  S*^ 
Christopbori  &  S"  Dominici  sunt  respective  4'^''  55\  14^  40', 
U^"*  oo',  14>«^  44',  12«^''  C,   l?^'  19'  &   19^'  48'  ad  boream.     Et 

excessus  longitudinis  Penduli auxerint. 

Observavit  utiq :  D.  Picartus  quod  virga  ferrea,  qure 
tempore  hyberno  ubi  gelabant  frigora  erat  pedis  unius 
longitudine,  ad   ignem   calefacta   evasit  pedis   unius  cum 

quarta  parte  lineaj.     Deinde  D.  de  la  Hire cum  dua- 

bus  tertiis  partibus  linesa.  In  priore  casu  calor  major  fuit 
quam  in  posteriore,  in  boc  vero  major  fuit  quam  calor 
externarum   partium  corporis  humani.     Nam  metalla  ad 

solem  sestivum  valde  incalescunt.     At  virga  penduli 

quam  hyberno,  sed  excessu  quartam  partem  linese  unius 

vix  superante.     Proinde differentia  ilia  prodiit  baud 

minor  quam  1^^-  linea;,  baud  major  quam  2^  linearum.  Et 
inter  bos  limites  quantitas  mediocris  est  2-^  linearum. 
Propter  calores  locorum  in  Zona  torrida  negligamus  -^^ 
partes  linea?  et  manebit  differentia  duarum  linearum. 

Quare  cum  differentia  ilia  per  Tabulam  prtecedentem 


NEWTON  AND  COTES.  87 

cx  hypothesi  quod  Terra  ex  materia  uniformiter  clensa 
constat,  sit  tantum  l  yS^ij*  lincaa:  cxecssus  altitudinisTerrje 
ad  ajquatorem  supra  altitudincm  ejus  ad  polos,  qui  erat 
milliarium  17  J,  jam  auotus  in  rationo  difFerentiarum,  fiet 
milliarium  31  ^f*  ^am  tarditas  Penduli  sub  iEquatore 
defectum  gravitatis  arguit;  et  quo  levior  est  materia  eo 
major  esse  debet  altitudo  ejus  ut  pondere  suo  materiam 
sub  Polis  in  ajquilibrio  sustineat. 

Hinc  figura  umbne  Terne  per  eclipses  Lunas  determi- 
nanda,  non  erit  omnino  circularis  sed  diameter  ejus  ab 
oriente  in  occidcntem  ducta,  major  erit  quam  diameter 
ejus  ab  austro  in  boream  ducta,  excessu  55"  eirciter.  Et 
parallaxis  maxima  Luna?  in  Longitudinem  paulo  major  erit 
quam  ejus  parallaxis  maxima  in  Latitudinem.  Ac  Terra» 
semidiameter  maxima  erit  pedum  Parisiensium  197(54030, 
minima  pedum  1J)()09860  &  mediocris  pedum  19686945  quam 
proximo. 

Cum  gradus  unus  mensurante  Picarto  sit  hexapedarum 
57060,  mensurante  vero    Cassino    sit  hexapedarum  57^^92 : 

Buspicantur  aliqui scu  parte  duodecima  diametri  Lunto. 

Quibus  omnibus  experientia  contrariatur.  Certe  Cassinus, 
definiendo  gradum  unum  esse  hexapedarum  57292,  medium 
inter  mensuras  suas  omnes,  ex  hypothesi  de  «;qualitate 
graduum  assumpsit.  Et  quamvis  Picartus  in  Gallioi  limite 
boreali  invenit  gradum  paulo  minorem  esse,  tamen  Nor- 
woodus  noster  in  regionibus  magis  mensurando  majus 
intervallum,  invenit  gradum  paido  majorem  esse  quam 
Cassinus  invenerat.  Et  Cassinus  ipse  mensuram  Picarti 
ob  parvitatem  intervalli  mensurati  non  satis  certam  & 
exactam  esse  judicavit  ubi  mensuram  gradus  unius  per 
intervallum  longe  majus  definire  aggressus.  Dificrcntiro 
vero  inter  mensuras  Cassini,  Picarti  &  Norwoodi  sunt  propo 


•  Newton  liad  written  92,  but  Cotes  has  altered  it  to  87.    See  Cotet'i  next  letter. 
t  Cotes  has  druwn  a  line  round  the  \  and  written  ,^  by  the  bide  of  it. 


88  CORRESPONDENCE  OP 

insensibiles   &   ab  insensibilibus   observationum  errorlbus 
facile  oriri  potuere,  ut  nutationem  axis  Terraj  pra)teream. 

Pag.  424  lin  penult,  read  229  ad  228. 

The  rest  of  the  Propositions  to  Prop,  xxxvi,  may 
continue  as  they  are,  w^^  y®  corrections  already  sent  you. 
I  will  speedily  send  you  the  corrections  of  y®  xxxvi,  xxxvii, 
&  XXXIX  Propositions. 

I  am 

Yo'  very  humble  Servant 

Isaac  Newton. 
The  following  is  in  Cotea's  hand. 

"Maxima  19767630  19688725 

Minima  196O982O  197U886 

Mediocris  19688725  304036lT 

Sem-.  Sph :  iEqu  :  I97U886  197018O5  Media  Mcdianim." 


LETTER    XLIII. 
NEWTON  TO  COTES. 
S'  London  Apr  S***  1712. 

I  sent  you  by  D"^  Bently  my  emendations  of  the  19*^  & 
20*^  Propositions,  &  now  send  you  those  of  the  36**^  &  37*^ 
When  you  have  perused  them  I  should  be  glad  to  have 
your  thoughts  upon  them,  &  if  any  thing  else  want  to  be 
corrected  before  you  come  at  y®  39*''  Proposition.  In  my 
next  I  intend  to  send  you  my  emendations  of  that  Propo 
sition. 

I  am 

Yo*"  most  humble  Servant 

For  the  R°**  M'  Cotes  Professor  of  Astro-  Is.   Ne  WTON. 

nomyy  at  his  cTiamber  in  Trinity  College 
in  Cambridge. 

All  that  is  preserved  of  the  emendations  of  Prop,  xxxvi.  is  contained 
in  a  small  slip  of  paper  (No.  192);  it  relates  to  the  Corollary  and  is 
as  follows : 


NEWTON  AND  COTES.  89 

"  In  Prop.  XXXVI.  pag.  4G4  lin.  3,  read  85820 ;  &  lin.  9  road,  et 
digitorum  undecim  cum  trieuto,  Est  enim  hwc  mcnsura  ad  mensuram 
pedum  85820  ut  1  ad  44038." 

Cotes,  however,  afterwards  (letter  of  Apr.  20)  altered  the  numbers 
in  tlio  Corollary  otherwise,  and  the  changes  together  with  his  other 
suggestions  were  approved  of  by  Ne>vton  (letter  of  May  10). 

The  emended  fonn  of  Prop,  xxxvii.  coincides  with  Newton's  pre- 
vious copy  (a  transcript  of  which  Cotes  sent  him  Feb.  28),  as  far  as  the 
middle  of  the  2nd  paragraph  except  that  "  carundem*  appears  in  the 
right  gender.  It  is  not  necessary  therefore  to  print  that  common  part 
again,  but  it  will  be  sufficient  to  begin  our  transcript  at  the  point  where 
the  first  correction  shews  itself.  (Nos.  195-198)  ...  "Si  maxima  harum 
altitudinum  differentia  sit  pedum  novem,  erit  L+Sad  L-Sut  20?  ad 
llf  seu  41  ad  23.  Qua)  proportio  satis  congruit  cum  priore.  Ob 
magnitudincm  astus  in  Portu  BUtoliw*^  observationibus  Stunnii 
magis  fidcndum  esse  videtur,  ideoq :  donee  aliquid  certius  constiterit, 
proportionem  9  ad  5  usurpabimus. 

Cieterum  ob  aquarum  reciprocos  motus,  a?8tus  maximi  non  incidunt 
in  ipsas  Luminarium  syzygia?,  scd  sunt  tertii  a  syzygiis  ut  dictum  fuit, 
seu  proximo  tertium  Lunie  post  syzygias  appulsum  ad  meridianum 
loci,  vel  potius  tertium  post  tertiam  circiter  vel  quartam  a  syzygiis 
horam  appulsum  ad  meridianum  loci.  ^Estas  ct  hyems  maximo  vigent, 
non  in  ipsis  solstitiis,  scd  ubi  sol  distat  a  novissimis  solstitiis  decima 
circiter  vel  undecima  parte  totius  circuitus,  seu  gradibus  plus  minus 
35.  Et  similiter  maximus  spstus  maris  oritur  ab  appulsu  Luna?  ad 
meridianum  loci  ubi  Luna  distat  a  Sole  decima  vel  undecima  parte 
motus  totius  ab  a^stu  ad  lestum,  seu  gradibus  plus  minus  septendecim 
cum  dimidio.  Et  Sol  in  hac  distantia  minus  augct  vel  diminuit  motum 
maris  a  vi  Lunte  oriundum  quam  in  ipsis  syzygiis  et  quadraturis  in 
rationo  Radii  ad  sinum  complemcuti  distantia*  hujus  duplicatn?  seu 
anguli  graduum  3.5,  hoc  est,  in  ratione  1000000  ad  819152;  ideoq: 
in  analogia  superioro  pro  ^  scribi  debet  0,8191 52  >S'. 

Sed  ct  vis  Luna3  in  Quadraturis,  ob  Declinationem  Luna?  ab  -^qua- 
tore,  diminui  debet.  Nam  Lima  in  Quadraturis  vel  potius  in  gradu 
17j  post  Quadraturas,  tempore  /Equinoctiorum,  in  Doclinatione  gra- 
duum plus  minus  22  &  21'  versatur.  Et  Luminaris  ab  Tlilquatoro 
Declinantis  vis  ad  mare  movcndum  diminuitur  in  duplicata  ratione 
sinus  complcmenti  Declinationis  quamproxime.  Et  propterea  vis 
Luna)  in  his  Quadraturis  est  tantum  0,85539968  X.  Est  igitur 
Z/  +  0,81952*/S'adO,855399C8/;-0,81952*AS'ut9ad  5. 

Pra?terea  diamctri  Orbis  in  quo  Luna  abs^j :  excentricitate  movori 


90  CORRESPONDENCE  OF 

deberct  sunt  ad  iiivicom  ut  6y  ad  70  (per  Prop,  xxviii,)  ideoq :  distantia 
Lun«  a  Terra  in  Syzygiis  est  ad  distantiam  ejus  in  Quadraturis  ut  69 
ad  70  ca»teri8  paribus.  Et  distantia  ejus  in  gradu  17  j  a  8yzyp;iis  ubi 
a?8tu8  maximuB  genoratur  est  ad  distantium  ejus  in  gradu  175  a  Quad- 
raturis  ubi  cestus  minimus  generatur  ut  83,8317  ad  84',8317,  id  est,  ut 
1  ad  1,0119286  vel  0,9882125  ad  1.  Undefit  l,0119286X  +  0,819152/S' 
ad  0,9882125  x  0,85539908/.-  0,8 19152 >S'  ut  9  ad  5.  Et  aS'  ad  Z  ut  1 
ad  45-. 

Corol.  1.  Cum  igitur  aqua  vi  Solis  agitata  ascendat  ad  altitudinem 
pedis  unius  et  digitorum  undecim  cum  triento,  eadem  vi  Lunae  a«cendct 
ad  altitudinem  pedum  octo  et  digitorum  novcm.     Tanta  autcm  vis  &c. 

Corol.  2.     Cum  vis  Lunte  ad  mare  movcndum  &c. 

Corol.  3.  Quoniam  vis  Lunaj  ad  mare  movendum  est  ad  Solis  vim 
consimilcm  ut  4j  ad  1,  et  vires  ilia*  (per  Corol.  14  Prop.  LxviLibr.  i) 
sunt  ut  densitates  corporum  Lunto  &  Solis  &  cubi  diametrorum  appa- 
rentum  conjunctim :  erit  densitas  Lunrc  ad  densitatem  Solis  ut  4j  ad  1 
directo  et  cubus  diametri  Lunte  ad  cubum  diametri  Solis  inverse,  id 
est,  (cum  diametri  mediocres  apparentes  Lunte  et  Solis  sint  31'.  1 6" 
et  32'  12")  ut  49112*  ad  10000.  Densitas  autem  Solis  erat  ad 
densitatem  Terrse  ut  100  ad  396  et  propterea  densitas  Lunae  est  ad 
densitatem  Terrae  ut  491 12*  ad  396OO  sen  31  ad  25.  Est  igitur  corpus 
Lunaj  donsius  et  magis  terrestre  quam  Terra  nostra. 

Corol.  4.  Et  cum  vera  diameter  Lunte  (ex  observationibus  Astro- 
nomicis)  sit  ad  voram  diametrum  Terras  ut  100  ad  365,  erit  raassa 
Lunio  ad  massam  Tome  ut  1  ad  39i  • 

Corol.  5.  Et  gravitas  accelcratrix  in  superficio  Luna3  orit  triplo 
minor  quam  gravitas  accelcratrix  in  superficio  Terrae. 

Corol.  6.  Et  distantia  centri  Luna?  a  centre  Terne  erit  ad  distan- 
tiam centri  Lunae  a  communi  gravitatis  centro  Lunie  ac  Terrse  ut  40  s 
ad  39\\ 

Corol.  7«  Et  distantia  mediocris  centrorum  Lunje  ac  Terrae  aequalis 
erit  maximis  Terra^  semidiametris  6*0j  quam  proximo.  Nam  Terrae 
semidiametor  maxima  fuit  pedum  Parisiensium  19764030.  Et  hujus- 
modi  semidiametri  60  J-  lequantur  pedibus  1190782815.  Et  si  htec  sit 
distantia  centrorum  Solis  et  Luna?,  eadem  (per  Corollarinm  novissimu) 
erit  ad  distantiam  centri  Luna?  a  communi  gravitatis  centro  Luna?  ac 
Terra;  ut  40^  ad  395-,  quae  proindo  est  pedum  Il6ll6l352.  Et  cum 
Luna  revolvatur  respectu  fixanim  diebus  27  horis  7  &  minutis  primis 
435-,  sinus  versus  anguli  quem  Luna  tempore  minuti  unius  primi  motu 
8U0  medio  circa  commune  gravitatis  centrum  Lunae  ac  Terrae  describit 


•  The  last  two  figures  are  altered  by  Cotes  to  51.    The  "  n"  in  "  sint"  (lin.  18), 
the  "2"  ill  lin.  37  &  the  •*  5"  in  lin.  6  (p.  91)  seem  also  clue  to  him. 


NEWTON  AND  COTES.  91 

est  1,275235,  existonto  Radio  100,000000,000000.  Et  nt  iUdius  est 
ad  hunc  einiim  vcrsum  ita  sunt  pedes  II6II61 35  ad  pedes  14,807536. 
Luna  igitur  vi  ilia  qua  rctinetur  in  orbe,  tempore  niinuti  unius  priini 
cadendo  describeret  pedes  1 4,807536.  Et  lure  vis  (per  Corol.  Prop,  iii 
est  ad  vim  gravitatis  nostra;  in  orbe  Lunie  ut  177S  ad  178^;  proindeq : 
corpus  grave  in  orbe  Lunoe  ad  distantiam  pedum  1190782815  a  centre 
Terra%  vi  gravitatis  nostra)  in  Torram  cadendo,  tempore  minuti  unius 
primi  describeret  pedes  14,8J)08,  &  ad  sexagctumam  partem  distantiw 
illius,  id  est  ad  distantiam  pedum  1984638  a  ceutro  Terrtie,  vi  gravitatis 
in  Tcrrara  cadendo  tempore  minuti  unius  secundi  describeret  etiam 
pedes  14,8908,  ot  ad  distantiam  pedum  1 9694278  a  ccntro  Terne 
cadendo  eodem  tempore  minuti  unius  secundi  describeret  pedes 
15,1217  seu  pedes  15,  dig.  1,  et  lin.  55 .  Et  bac  vi  gravia  cadunt  in 
supcrficio  Terra*  in  Latitudine  urbis  Lutetia)  Parisiorum,  ut  supra 
ostensum  est.  Et  distantia  pedum  19694278  paulo  major  est  quam 
Torrte  semidiameter  mediocris,  et  paulo  minor  quam  scmidiameter  globi 
cui  Terra  requalis  est,  suntq:  diffcrentine  insonsibiles ;  ac  proindo  vis 
qua  Luna  rctinetur  in  Orbe  suo  ad  distantiam  prtedictam  semidiame- 
trorum  60j-,  si  desccndatur  in  Terram,  congruit  cum  vi  gravitatis  quam 
experimur  in  superficie  Terrw. 

Corol.  8.  Distantia  mediocris  centrorum  Lunae  ac  Terras  oequalis 
est  mediocribus  Terrce  semidiametris  60j  quamproxime.  Nam  tot 
semidiametri  mediocres  sunt  pedum  II9IO6OI72. 

Siquando  mcnsuru)  graduum  in  meridiano,  longitudes*  pendulorum 
isocbronorum  in  diverais  parallelis  Terruj,  leges  fluxus  8c  refluxus  marisi 
diametri  apparentes  SoHs  et  Luna',  &  Luniu  parallaxis  borizontalis  ex 
pluenomenis  accuratius  dctcrminatuj  fuerint :  liccbit  calculum  huno 
omnem  accuratius  repetere." 


LETTER  XLIV. 
COTES  TO  NEWTON. 
S^  Cambridge  Aprill  y*  H*"*  1712 

I  have  received  Your  Letter  by  D*"  Bcntley  &  the  other 
which  You  ^\Tote  since.  I  have  sent  You  two  Proof 
Sheets  t  for  Your  re  visa!,  having  made  some  alterations  in 
them  different  from  Your  Copy. 

In  Page  379  line  6  I  have  put  [lin.  2  ■^']  instead  of 


t  Ccc,Ddd,  pp.  377-392. 


92  CORRESPONDENCE  OF 

[lin.  2  J].     In  line  10***  1436,223  instead   of  143,6223.     In 
line  21■^  2177,32  instead  of  2177,52 

In  Page  382  I  have  put  the  proportion  of  230  to  229 
instead  of  229  to  288*  and  altered  the  latter  part  of  y*' 
Page  accordingly  &  computed  the  Table  anew  in  the  next 
Page.  The  Latter  Column  supposes  the  measure  of  a  degree 
at  y*  Latitude  of  45^.  41 '  to  be  57292  Toises  as  I  think  You 
put  it  in  Your  Table.  The  two  extreme  numbers  are  as 
the  Cubes  of  230  &  229,  In  y®  rest  the  increment  from  y® 
^'Equator  is  as  the  Versed  Sine  of  y®  doubled  Latitude. 

In  Page  38C  lin:  penult.  1  j^^^  for  l  -j-J-g^.  Page  387 
lin,  1  31  j3^  for  31  ^.  Line  11"'  I  have  put  other  numbers 
for  y®  semidiamcters  of  the  Earth,  wliich  I  desire  You 
would  examine,  since  there  are  different  ways  of  coming 
at  those  numbers  &  I  may  not  possibly  have  taken  that 
which  You  like  best.  Line  21"*  I  put  95  Miles  for  94. 
Line  27*^  2'.  46"  for  2'.  45,  Line  32  Norivoodus  noster  in 
regionibus  magis  borealibus,  the  word  borealibus  or  some- 
thing to  y*  effect  was  omitted  in  Your  copy 

In  Page  389 :  line  20*'^  I  have  put  8**.  24  for  9**.  34.  In 
the  last  Period  of  y®  same  xxiii**  Proposition  I  have  made 
an  alteration  which  You  will  see. 

I  think  You  have  much  improved  the  Method  of  the 
whole,  but  there  seemes  to  be  a  mistake  in  y*  Section  of 
Prop  xxxvii  which  begins  with  Prceterea  diametri  Orbis  in 
quo  Luna  ^c.  The  Moons  force  in  her  Syzygies  &  Quad- 
ratures should  be  increased  &  diminished  in  the  tripli- 
cate proportion  of  those  distances  to  her  mean  distance 
reciprocally,  Your  correction  is  nearly  according  to  y® 
duplicate  proportion.  I  am  streightned  in  time  at  present, 
&  will  explain  myself  more  fully  in  my  Next 

Your  most  humble  Serv* 

R  C 


•  Slip  of  the  pen  for  228. 


NEWTON  AND  COTES.  93 

LETTER    XLV. 
COTES  TO  NEWTON. 
S**.  Cambridge  April  IB***  1712." 

I  hope  You  have  received  the  sheets  sent  You  hy  the 
Carrier  for  Your  examination,  with  my  Letter.  I  come 
now  to  the  xxv*^  Proposition  wliich  I  think  were  better  to 

end  thus ad  dies  3G5.  G^  9'.  id  est,  ut  1000  ad  178725 

seu  1  ad  178  |{J.  Unde  ex  proportione  linearum  TM,  ML, 
datur  etiam  vis  TM:  &  ha)  sunt  vires  Solis  quibus  Lunso 
motus  perturbantur.  q.e.i.  The  two  Periods  which  are  left 
out  may  be  removed  to  Prop:  xxxvi  for  I  think  they  are 
of  no  use  till  we  come  to  that  Proposition.  If  You  remove 
them  I  suppose  You  will  at  the  same  time  alter  them,  by 
putting  in  line  14*^  instead  of  y®  proportion  of  6ol*  to  60 
the  proportion  of  40  J-  to  39-J-,  if  this  be  the  Proportion 
which  may  at  last  stand  in  Corol.  G^^  of  Prop.  xxxviA 
Now  because  the  Proportion  of  40  -J-  to  39  }  is  made  out  in 
y®  xxxvii*^  Proposition,  the  xxxvi  &  xxxvii*^  Propositions 
ought  to  change  places,  but  this  they  cannot  do  because 
the  xxxvii*^  does  in  other  respects  depend  upon  y®  xxxvi^*^. 
Whence  it  appeares  that  there  ought  to  be  a  further 
alteration  in  y®  Form  of  these  Propositions,  that  the  former 
may  not  depend  upon  the  latter.  This  may  easily  be  done 
&  I  think  the  whole  would  be  clearer  &  more  !RIethodical 
if  in  y*"  former  Proposition  the  Problem  were  to  find 
neither  y®  force  of  y®  sun  nor  the  force  of  y®  moon,  but 
only  their  proportion  to  each  other,  &  in  y®  latter  the 
Problem  were  to  find  the  proportion  of  both  forces  to  y* 
force  of  Gravity.  And  thus  y^  3^  4}\  5'\  G'\  r\  &  8*^ 
Corollarys  of  y®  xxxvu*^  will  belong  to  y®  former,  &  the 
Corollary  of  y®  xxxvi*^  together  with  the  1"*  &  2^  corollarys 
of  y*  xxxvii*^  will  belong  to  y®  latter.     There  will  be  this 


•  It  should  be  ^. 


94  CORRESPONDENCE  OF 

further  advantage  in  the  change,  That  in  y**  7^**  Corollary 
of  y"  XXXVII*''  which  will  then  be  annex'd  to  y®  former 
Proposition  a  good  foundation  may  be  laid  for  making  out 
y*  latter.  In  my  Letter  which  I  yesterday  wrote  to  You 
I  was  somewhat  in  haste,  I  just 
mention'd  a  difficulty  in  Prop : 
XXXVII.  Let  iST  be  the  Moons 
distance  from  y®  Earth  when 
she  is  17°^  from  her  Syzygies 
&  QThe  her  distance  at  1?°^ 
from  her  Quadratures  St  MT 
her  mean  distance  in  y^  Octants. 

I  think  the  force  of  y®  Moon  must  be  increased  at  S  in  the 
proportion  of  MT  cub  to  *S'7'  cub,  &  diminished  at  Q  in  the 
proportion  of  MT  cub  to  QT  cub.  Your  last  corrections 
increase  it  at  S  in  y®  proportion  of  QT'  to  ST,  which  is 
nearly  in  the  proportion  of  MTq  to  ST  quad,  &  diminish 
it  at  Q  in  y®  same  proportion.  I  could  wish  when  the 
whole  is  settled  that  the  proportion  of  4  J-  to  1  may  be 
retain'd  for  the  sake  of  Proposition  xxxix.*  I  think 
there  is  no  Proposition  in  Your  Book  which  does  more 
deserve  Your  care. 


LETTER  XLVI. 

NEWTON  TO   COTES. 

S"^  London  Apr.  22.  1712. 

I  have  run  my  eye  over  the  two  proof  sheets  &  approve 

yo'  corrections.   The  sheets  may  be  printed  off.   The  xxv*^ 

Proposition   may  end  thus. — ad  dies  365,  6^'  9',  id  est  ut 

1000  ad  178725  seu  1  ad  178  |-g-.     Invenimus  autem  in  Pro- 

positione  quarta  quod,  si  Terra  et  Luna  circa  commune 

gravitatis  centrum  revolvantur,  earum  distantia  mediocris 


•  "Invenire  Prnccessionem  -i'quinoctiorum." 


NEWTON  AND  COTES.  95 

ab  invicem  erit  Co^  semidiametrorum  mediocrium  Tern© 
quamproxime.  Et  vis  qua  Luna  in  Orbe  circa  Tcrram 
quiescentem  ad  distantiam  semidiametrorum  60  revolvi 
posset  {est  ad  vim  qua  eodem  tempore  ad  distantiam 
semidiametrorum  60  revolvi  posset*,}  ut  60^  ad  60  &  hsoc 
vis  ad  vim  gravitatis  apud  nos  ut  1  ad  60  x  60.  Ideoq:  vis 
mediocris  ML  est  ad  vim  gravitatis  in  superfieie  TerroD  ut 
1  X  6iU^  ad  60  X  60  X  60  X  178 1 g,  seu  1  ad  638092,6.  Undo 
ex  proportione  linearum  TM,  ML,  datur  etiam  vis  TM,  Et 
ha3  sunt  vires  Solis  quibus  motus  Lunno  perturbantur.  q,  e.  i. 
I  here  referr  the  summ  of  y®  forces  uponf  the  Sun  upon 
the  earth  &  ^loon  to  the  ]Moon  alone  &  therefore  consider 
the  earth  as  resting  &  referr  its  motion  to  the  Moon. 

I  am  satisfied  that  the  force  of  the  ^loon  upon  the  Sea 
is  in  a  triplicate  ratio  of  her  distance  reciprocally  &  have 
altered  the  calculations  accordingly,  w^^  I  send  you  in  the 
inclosed  paper  together  with  the  emendation  of  the  SQ*** 
Proposition. 

T  am 
Yo"^  most  humble  Servant 

For  tlie  IC'^W  Cotes,  Professor  of  Is.   Newton. 

Astronomy,  at  his  Chamber  in 
Trinity  College  in  Cambridge. 

The  "  inclosed  paper"  mentioned  at  the  end  of  this  letter  is  a  folio 
sheet  (Nos.  202,203,208),  and  contains  Newton's  further  corrections  of 
Prop.  XXXVII.  called  for  by  the  two  preceding  letters,  and  also  those  of 
the  39th  Prop,  which  he  had  promised  in  his  letters  of  Apr.  3d  and 
8th,  (compare  letters  of  March  18,  Feb.  19,  Feb.  I6,  and  Feb.  12). 
It  is  not  necessary  to  copy  the  whole  of  what  relates  to  Prop,  xxxvn. 
Every  useful  end  will  be  answered  by  giving  only  those  parts  of  it 
where  it  differs  from  the  copy  which  Newton  had  recently  sent  (Apr.  8), 
leaving  blanks  to  represent  what  is  common  to  the  two.  The  paper 
begins  as  follows :  **  In  Prop  xxxvii  read  Ca'terum  ob  aquarum  reci- 


•  I  have  added  the  words  between  braces  from  the  1st  Ed.  The  identity  of  temii< 
nation  of  the  two  clauses  with  "  revolvi  posset,"  combined  with  a  little  hurry  in  tran- 
scribing,  will  readily  account  for  their  omission. 

t  This  should  be  "  of." 


96  CORRESPONDENCE  OF 

procos  motu8...8eu  proximo  sequuntur  tertium  Luna;...vol  potius  (ut  a 
Sturmio  notatur)  sunt  tertii  post  diem  noviliinii  vel  plonilunii,  scu 
post  lioram  a  novilunio  vel  plenllunio  plus  minus  duodecimam,  adcoq  : 
incidunt  in  boram  a  novilunio  vel  plenilunio  plus  minus  quadragcaimam 
tertiam.  Incidunt  voro  in  hoc  portu  in  lioram  septimam  circitcr  ab 
appulau  Lunne  ad  meridianum  loci,  ideoq :  proximo  scquuntur  appul- 
8um  Lunne  ad  meridianum  ubi  Luna  distat  a  Sole  vol  oppositione  Solis 
gradibus  plus  minus  octodecim  vol  novcndccim  in  consequentia.  iTlstas 
. . .  Sol  distat  a  solstitiis  dccima  circitcr  parte  totius  circuitu)?  seu 
gradibus  plus  minus  30  vel  37  ...  a  Solo  dccima  circitcr  parte  motus 
totius  ab  fpstu  ad  a}stum.  Sit  distantia  ilia  graduum  plus  minus  18z}. 
Et  vis  Solis  in  liac  distantia  Luute  a  syzygiis  &  quadraturis,  minor 
erit  ad  augcndum  et  ad  minuondum  motum  . . .  scu  anguli  graduum  37, 
hoc  est,  in  rationc  10000000  ad  798()3.'55.  Idcoq  :  . .  .debet  0,7980.355/9. 
...in  gradu   18^  post  Quadraturas,  in  Declinatione  graduum   plus 

minus   22.13'   vcrsatur est  tantum   0,8570328//.     Est   igitur 

L  +  0,7980355  S  ad  0,8570328  L  -  0,7980355 ^S*  ut  9  ad  5 ut  Q9 

ad  70;  ideoq:  distantia....  ca^teris  paribus.  Et  distantia)  ejus  in 
gradu  18|  a  syzygiis  ...  maximus  generatur,  &  in  gradu  18^  a  qua- 
draturis  ubi  testus  minimus  generatur,  sunt  ad  mediocrcm  ejus  distan- 
tiam  ut  09,100082  &  09,899318  ad  6c)^^.  Vires  autem  Luna^  ad  mare 
movendum  sunt  in  triplicata  ratione  distantiarum  inverse,  ideoq  :  vires 
in  maxima  et  minima  barum  distantiarum  sunt  ad  vim  in  medi{o}cri 
distantia  ut  O,98280l6  et  1,017342  ad  1.  Undo  fit  1,017342  Z 
+  0,7980355  aS'  ad  O,98280l0  x  0,8570328 Z  -  0,7980355 /S'  ut  9  ad  5. 
Et  X  =  4,4824  S. 

Corel.  1  &  2,  as  before. 

Corol.3.  ...  ut  4,4821.  ad  l...ut  4,4824  ad  1  ...  sint  31'l0"i... 
ut  4892  ad  1000  ...  ad  densitatem  Terrw  ut  4892  ad  39OO  seu  21 
ad  17.     Est  igitur  ... 

Corel.  4.  ...  ad  massam  Terra?  ut  1  ad  S^^SQS, 

Corel.  5.  . . .  erit  quasi  triple  minor. . . 

Corel.  0.  . . .  ut  40,303  ad  30,^63, 

Corel.  7'  Et  mediocris  distantia  centrj  Lunae  a  centro  Terrae  erit 
semidiametrorum  maximarum  Terrae  60J  quam  proximo.  Nam  semi- 
diameter  maxima  Terra)  fuit  pedum  Parisiensium  I97O7O3O,  et  medi- 
ocris distantia  centrorum  Terra)  et  Lunae  ex  hujusmodi  semidiametris 
Go  J  constans,  a?qualis  est  pcdlbus  1190999707.  Et  ha?c  distantia  (per 
Corollarium  superius)  est  ad  distantiam  . . .  centro  Terrao  et  Lunas  ut 
40,303  ad  3%3G3,  qua?  proinde  est  pedum  llOl 492740.  Et  cum 
Luna  . . .  centrum  Terras  et  Luna?  describit  est  1275235,  existente  . . . 
pedes  llOl 492740  ad  pedes  14,811702.  Luna  ...  in  Orbe,  cadendo 
in  Terram,  tempore  minuti  unius  primi  describet  1    Jes  14,811702. 


NEWTON  AND  COTES.  97 

Et  81  ha?c  vis  augeatur  in  rationo  1775  ft^  178S  liabebitur  vi«  tota 
gravitatis  in  Orbe  Luna?  per  Corol.  Prop.  iii.  Et  hac  vi  Lima  cadendo, 
tempore  minuti*  iiniiia  primi  dcscribero  debcrot  pedes  1 4,89.') IS.  Et  ad 
sexagesimam  partem  hiijus  distantiw,  id  est,  ad  distant iam  pedum 
19849995  a  eentro  Terra?  corpus  grave  cadendo,  tempore  minuti  unius 
secundi  describere  deberct  etiam  pedes  14,89513.  Diminuatur  ha»c 
distantia  in  subduplicata  ratione  pedum  14,89513  ad  pedes  15,12028, 
et  habebitur  distantia  pedum  19701651  a  qua  grave  cadendo,  codem 
tempore  minuti  unius  secundi  describet  pedes  15,12028,  id  est  \Hn\ca  15, 
dig  1,  lin  5,32.  Et  hac  vi  ...  urbis  Lutethn  Parisiorum^  lit  supra 
ostensum  est.  Est  autem  distantia  pedum  19701651  paulo  minor 
quam  semidiametor  globi  huic  Terne  wqualis  et  paulo  major  quam 
Terrae  hujus  semidiamctcr  mediocris  ut  oportet.  Sed  differentia  sunt 
insensibiles.  Et  propterea  vis  qua  Luna  ...  ad  distantiam  maximanim 
Terras  semidiametrorum  60  J,  ea  est  quam  vis  gravitatis  in  superficio 
Terras  requirit. 

Corol.  8 centrorum   Terrae  et   Luna?  est  mediocrium  Terr® 

semidiametrorum  60^  quam  proximo.  Nam  semidiameter  mediocris 
qucB  erat  pedum  19688725  est  ad  semidiametrum  maximam  pedum 
19767630,  ut  60^  ad  60^  quamproxime. 

In  liis  computationibus  attractionem  magneticam  Terras  non  con- 
sideravimus,  cujus  utiq :  quantitas  perj)arva  est  et  ignoratur.  Siquando 
veroha?c  attractio  invcstigari  potent,  et  mcnsura  graduum  in  merldiano, 
ac  longitudines  . . .  parallelis,  legesq :  motuum  maris,  &  parallaxis  Lunro 
cum  diametris  apparentibus  Solis  et  Luna?  ex  pha?nomenis  ..." 

The   following  are  the   corrections   of  the  39tli  Prop.     "  In  the 

xxxix'*"  Proposition  pag  470  lin   23   write r-  id  est   (cum  Terra) 

diameter  minor  PC  vol  aCsit  ad  diametrum  majorem  AC  wi  229  ad 
230,)  ut  52441  ad  459;  si  annulus  iste  Terram  secundum  /Equa- 
torcm  cingeret  &  utcrq:  simul  circa  diametrum  annuli  revolveretur, 
motus  annuli  esset  ad  motum  globi  interioris  (per  hujus  Lemma  iri) 
ut  459  ad  52441  et  1000000  ad  925275  conjunctim,  hoc  est,  ut  4590 
ad  485223,  ideoq:  motus  annuli  esset  ad  summam  motuum  annuli 
ac  globi  ut  4590  ad  489813.  Vndo  si  annulus  globo  adhaereat,  & 
motum  Buum  quo  ipsius  Nodi  seu  puncta  a^quinoctialia  regrediuntur, 
cum  globo  communicet:  motus  qui  restabit  in  annulo  erit  ad  ipsius 
motum  priorem  ut  4590  ad  489813;  et  propterea  motus  punctorum 
aequinoctialium  diminuctur  in  eadem  ratione.  Erit  igitiir  motus  an- 
nuus  punctorum  ajquinoctialium  corporis  ex  annulo  et  globo  compositi 


•  "  minuti "  here  6c  "  quam  "  p.  98.  lia.  20  have  been  added  by  Coles,  who  has  made 
a  number  of  other  alterations  in  the  MS.,  the  principal  of  wliich  are  mentioned  in 
Letter  XLVllI. 

7 


98  CORRESPONDENCE  OF 

ad  motum  20«*  11'  46",  ut  1436  ad  39343  ot  4590  ad  489813  con- 
jiinctim,  id  est,  ut  100  ad  292368.     Vires  autem  quibus  &c. 

Pag.  471  lin  19  write atq:  adeo  ad  movenda  puncta  osquinoc- 

tialia  ovaderet  minor  quam  prius  in  rationo  2  ad  5.  Ideoq :  anniuis 
iEquinoctiorura  rcgressus  jam  esset  ad  20«'  11'  46"  ut  10  ad  73092, 
ac  proinde  fieret  9"  56"'  50' ^ 

Cwterura  hie  motus  ob  inclinationom  plani  iEquatoris  ad  planum 
Ecliptica}  mimicndu8  est,  idq:  in  ratione  sinus  91706  (qui  sinus  est 
complementi  graduum  23 J)  ad  Radium  100000.  Qua  rationo  motus 
iste  jam  fiet  9".  7"'.  20'*'.  Ilajc  est  annua  Prajcessio  i^quinoctiorum 
a  vi  Solis  oriunda. 

Vis  autem  Luna)  ad  mare  movendum  erat  ad  vim  Solis  ut  4,4824 
ad  1  circiter.  Et  vis  Lunss  ad  iEquinoctia  movenda  est  ad  vim  Solis  in 
eadem  proportione.  Indeq:  prodit  annua  ^quinoctiorum  Pracessio  a 
vi  Lunoe  oriunda  40"  53"'  22",  ac  tota  Proecessio  annua  a  vi  utraq: 
oriunda  50" .  00"' .  42' '.  Et  hie  motus  cum  phaenomenis  congruit. 
Nam  Praecessio  requinoctiorum  ex  Observationibus  Astronomicis  est 
minutorum  secundorum  plus  minus  quinquaginta 

Si  altitudo  Terrte  ad  iEquatorem  superet  altitudinem  ejus  ad  Polos 
milliaribus  plus  quam  I7J,  materia  ejus  rarior  erit  ad  circumferen- 
tiam  quam  ad  centrum :  et  Prajcessio  jEquinoctiorum  ob  altitudinem 
illam  augeri,  ob  raritatem  diminui  debet. 

Descripsim  usjam  Systcma  Solis,  Terra?,  Lunro,  et  Planetarum: 
superest  ut  do  Cometis  nonnulla  adjiciantur." 


LETTER  XL VII. 
COTES  TO  NEWTON. 

I  have  received  Your  last,  but  have  not  yet  had  time 
to  try  the  Calculations  of  the  inclosed  sheet.  I  am  satisfied 
as  to  the  xxv*^  Proposition,  upon  reconsidering  it. 

In  Page   441,  lin:    25,  the   first  &  last  numbers    are 
368682  &  362046:  they  should  be  368676  &  362047.   The  iEqua- 
tion*  which  results  from  hence  will  be 
88487,19  -  12307251,44.r+75578,14.T?^-5082017,44cP''+ 42456,19**  =  0, 


•  The  following  is  on  a  separate  piece  of  paper,  (No.  209)  : 
iEquatio  fit  88487,19  -  12307251,44  r  +  75578,14ii  -  5082017,44x«  +  42456,19 jc*  =  0. 

Inde  X  =  0.00719,    CT  =  1,00719,    AT  =  0,99281  adeoq :  CT  ad  AT  ut  70,041  ad 
69,041,  sive  ut  70.i  ad  69,-,  vol  70.-3  a^  ^^h* 

Vera  Radix  ilerato  examine  est,  0071900057  ter  exam  : 


NEWTON  AND  COTES.  99 

of  which  I  find  the  Root  to  be  0,0071900057.  If  You 
approve  of  it  I  would  alter  the  bottom  of  the  Page  thus 
[obtinetur  w  roqualis  0,007 19»  &  inde  semidiameter  CT  fit 
1,00719  &  semidiameter  AT  0,99281,  qui  numeri  sunt  ut 
70^  &  69^  quam  proximo.  Est  igitur  distantia  Luna)  a 
Terra  in  Syzygiis  ad  ipsius  distantiam  in  Quadraturis 
(seposita  scilicet  Eccentricitatis  consideratione)  ut  C9^  ad 
70^  vel  numeris  rotundis  ut  69  ad  70]  This  will  cause 
an  alteration  in  the  xxix*^  Proposition  &  in  the  xxxi"*, 
page  450. 

I  have  not  computed  the  alterations  for  the  xxix**"*, 
not  knowing  whether  You  will  chuse  the  whole  numbers 
69  and  70  or  the  fractions  69^^  &  10-^. 

As  for  the  other  place  in  page  450*^  I  took  the  numbers 
69  &  70  that  I  might  find  what  alteration  would  arise  in  the 
conclusion  of  y®  xxxii**  Proposition.  The  result  of  my 
computation  is  as  follows.  Pag:  450.  lin:  18  [69  ad  70] 
Lin:  20,  [si  capiatur  angulus  16".  2l'".  s'\  30*]  Page  452**. 
lin:  5,  [erat  32".  42'".  7'']  Lin:  8,  [illud  est  17'".  43'".  11"] 
Lin:  10,  [relinquit  16".  16"'.  37' ^  42*']  Page  453,  Lin:  22, 
[fit  39^  38'.  7".  50'"]  Lin:  23,  [l9^  49'.  3".  55""]  Lin:  ult: 
[seu  S9fiS55'\  Page  454,  Lin:  3,  [id  est,  ut  9,0827646 ^ T^] 
Page  455,  Lin:  4  [prodibit  0,ll88502]t  Lin:  6,  [est  1°.  29'. 
58".  3'"]  Lin:  7  [subductis  relinquit  18°.  19'.  5".  52'"]  Lin:  9 
[relinquit  341°.  40'.  54".  8'"]  Lin:  12  [qui  propterea  erit 
19°.  18'.  1".  22'"] 

In  finding  the  Number  0,1188502, 1  supposed  y*  ordinate 
e  Z  to  bisect  y**  base  NT  by  which  meanes  the  series  for  y® 
Area  TZeF  converged  quicker  than  the  other  for  the 
Area  Ne  Z,  so  y*  on  account  of  this  Latter  I  would  not 
depend  upon  the  last  figure  2,  I  think  the  other  are  right. 

•  Tliese  alterations  of  Prop.  xxix.  form  the  subject  of  Letter  L. 

X  This  correction,  though  approved  by  Newton,  was  subsequently  modified  (as  also 
the  four  following  corrections  which  depend  upon  it),  1  he  result  wliich  w  substituted 
for  it  in  the  2nd  Ed.  leads  to  the  value  .1188496  for  the  area  of  the  curve  f^eVn  :  in 
the  Ist  Ed.  it  is  .1188478. 

7—2 


100  CORRESPONDENCE  OF 

In  Line  14*^  You  have  19°.  20'.  Si",  i'"  from  Flamsteeds 
Tables.  By  Your  Theory  in  D'  Gregory  tis  19^  2l'.  22".  3'"*. 
So  in  the  following  Proposition,  page  45G.  Lin  13  You 
have  9^  lO'.  40" ;  by  Your  Theory  tis  9°.  n'.  3". 

There  will  need  some  other  alterations  in  Prop,  xxxiu** 
&  its  Corollary  upon  account  of  those  in  the  preceding 
Proposition.  You  seem  to  depend  too  much  upon  Your 
Readers  quickness  when  you  say  [ut  rem  perpendenti  con- 
stabit]  I  hope  when  You  review  the  whole  You  will  make 
it  easier  to  apprehend  the  agreement  of  the  two  Con- 
structions. 

I  do  not  rightly  understand  line  12*^  of  page  458 
[Inclinationis  autcm  Variatio  tantum  augebitur  per  decre- 
mentum  sinus  IT,  quantum  diminuitur  per  decrementum 
motus  Nodorum] 

I  think  I  had  observed  nothing  further  before  we  come 
to  y®  XXXV i**^  Proposition. 

I  am,  S', 
Your  most  Humble  Servant 
Trinity  College  Apr.  24**^  1712  Eoger  Cotes. 

For  S'  Isaac  Newton  at  his  House 
in  St  Martin's  Street  in  Leicester 
Fields  London. 


LETTER  XLVIII. 


COTES  TO  NEWTON. 
S'.  April  26*'*  1712. 

I  have  examin*d  your  last  Emendations  f  of  the  xxxvii*** 
Proposition.    I  am  very  glad  to  see  the  whole  so  perfectly 


•  Newton,  in  his  next  letter,  adopts  this  correction  and  the  followingf  one.  After- 
wards, however,  (Letter  LIT.)  apparently  forgetting  that  he  had  already  given  direc- 
tions about  them,  he  orders  19*. 21'.  20". 45'"  to  be  written  in  p.  455,  and  9°.  11'. 3"  in 
p.  456.  Cotes,  in  his  reply,  (Letter  LIII.)  proposes  to  write  190.21'. 21". 50'"  in 
p. 455,  which  Newton  approves,  (Letter  LV.) 

Flamsteed's  Tables  here  referred  to,  are  printed  at  the  end  of  his  Doctrine  of  the 
Sphere^  London,  1680. 

t  Sent  in  the  Letter  of  Apr.  12. 


NEWTON  AND  COTES,  101 

well  settled  &  fairly  stated,  for  without  regard  to  the 
conclusion  I  think  y*  distance  of  18^  degrees  ought  to  be 
taken  &  is  much  better  than  17^  or  15\  &  the  same  may 
be  said  of  y*  other  changes  in  y*  principles  from  which  the 
conclusion  is  inferr'd. 

In  examining  Your  Numbers  I  found  it  necessary  to 
alter  most  of  them,  I  here  send  you  others  {instead  of 
them  \  for  your  approbation. 

Proeterea   diametri    Orbis    in    quo    Luna...... sunt   ad 

mediocrem  ejus  distantiam  ut  69,098747  &  69,897345  ad 
69^,  Vires  autcm  Lunnc.ad  vim  in  mediocri  distan- 
tia  ut  0,9830427  ct  1,017522  ad  1.  Undo  fit  1,017522  L 
+ '0,7986355  S  ad  0,9830427  x  0,85703271*  -0,7986355*9  ut  9  ad 
5.  Et  S  ad  L  ut  1  ad  4,4815.  Itaq:  cum  vis  Solis  sit  ad 
vim  gravitatis  ut  1  ad  12868200  vis  Lunoe  erit  ad  vim 
gravitatis  ut  1   ad  2871400. 

CoroL  1.  Cum  igitur*  aqua  vi  Solis  agitata  asccndat 
ad  altitudinem  pedis  unius  &  undecim  digitorum  cum 
octava  parte  digiti,  eadem  vi  Lunio  ascendet  ad  altitudi^ 
ncm  octo  pedum  &  digitorum  octo.     Tanta  autem  vis — 

Corol.  2.  Cum  vis  Luna3  ad  mare  movendum  sit  ad 
vim  gravitatis  ut  1  ad  2871400 — 

Corol:  3.     Quoniam  vis  Lunai  ad  mare  movendum  est 

ad  Solis  vim  consimilem  ut  4,4815  ad  1 et  32'.  12")  ut 

4891   ad   1000.     Densitas   autem   Solis ad  densitatem 

Terra)  ut  4891  ad  3960  seu  21  ad  17.     Est  igitur 

Corol:  4 ad  massam  Terro9  ut  1  ad  39,  371. 

Corol:  C ut  40,371  ad  39,371. 


•  The  word  "  igitur**  is  omitted  in  the  2nd  Ed,,  neither  does  it  appear  in  Newton's 
first  copy  of  the  Prop,  whicli  k  given  at  the  end  of  Letter  XXXIX. 

After  the  words  "  digitorum  octo,"  the  sentence  is  continued  as  follown  in  the  2nd 
Ed.,  "&  vi  utraque  ad  altitudinem  pedum  decem  cum  semisse,  &  ubi  Luna  est  in 
Perigajo  ad  altitudinem  pedum  duodecim  cum  semisse  &  ultra,  praisertim  ubi  ii-^stus 

ventis  spirantibus  adjuvatur.    Tanta  autem  vis "  corresponding:  to  Newton's  copy 

just  referred  to.  Cotes's  omission  of  these  words  in  this  draught  of  his  letter  probably 
arose  from  the  fact  of  Newton*s  having  omitted  the  passage  in  the  emendations  sent  in 
his  Letter  of  Apr.  8. 


102  CORRESPONDENCE  OF 

Corol:  7 ut    40,371    ad    39,371,    quce   proinde    est 

pedum  1161498340 ita  sunt  pedes  u6l4i.8340  ad  pedes 

14,811833 Et  hac  vi  Luua  eadendo,  tempore  minuti 

unius  primi  describere  deberet  pedes  14,89517 et  ha- 

bebitur  distantia  pedum  19701 678  a  qua  grave  eadendo, 
eodem  tempore  miimti  unius  secundi  describet  pedes 
15,12028... 

In  the  XXXIX*''  Proposition.     Vis  autem  Lunae  ad  mare 

movendum  erat  ad  vim  Solis  ut  4,4815  ad  1  eirciter 

Prsccessio  a  vi  Lunae  oriunda  40".  52'".  52'^  ac  tota  Prse- 
cessio  annua  a  vi  utraq:  oriunda  50".  oo'".  12'*^.  Et  hie 
motus 

The  xxxvi*''  Proposition  depends  upon  the. latter  part 
of  the  xxv*^,  &  must  therefore  stand  as  in  the  former 
Edition.     I  have  altered  the  Corollary  of  it  thus 

Corol.     Cum  vis ."SS^vim  gravitatis  lit  1  ad  289.,... 

mensura  pedum  Parisiensium  85820,  vis  Solaris  de  qua 
egimus,  cum  sit  ad  vim  gravitatis  ut  1  ad  12868200  atq: 
adeo  ad  vim  illam  eentrifugam  ut  289  ad  12868200  seu  l  ad 

44527,  efficiet  ut mensura  tantum  pedis  unius  Parisien- 

sis  &  digitorum  undecini  cum  octava  parte  digiti.  Est 
enim  ha3C  mensura  ad  mensuram  pedum  85820  ut  1  ad 
44527. 

I  have  altered  the  xxxviii*^  Proposition  thus.  Pag:  467. 
lin:  10  [id  est,  ut  39,371  ad  1  &  100  ad  365  conjunctim,  seu 
1079  ad  100.  Unde  cum  mare  nostrum  vi  Lunso  attollatur 
ad  pedes  8f ,  fluidum  Lunare  vi  Terras  attolli  deberet  ad 

pedes  93 J excessu  pedum  187 

Your  very  Humble  Servt. 

R  Cotes. 


NEWTON  AND  COTES.  103 

LETTER  XLIX. 
NEWTON  TO  COTES. 

The  corrections  made  in  yo'  last  of  Apr.  24***  may  all 
stand.  In  y*  xxix*^  you  may  use  either  y*  whole  numbers 
69  &  70  or  the  fractions  69^^  &  70^.  In  pag  455  lin  14  & 
pag  45G  I  have  put  the  motion  of  the  Nodes  of  Moon  from 
y®  Equinox  &  should  have  put  it  from  y®  fixt  starrs.  In  y* 
first  place  therefore  for  I9^'20'3l"  l'"  ^vrite  19^21^22".3"' 
In  y®  second  for  9°.  lO'  40"  write  9^  u'.  s". 

In  pag.  458  lin  11.  write.  [Et  in  eadem  ratione  minue- 
tur  etiam  Inclinationis  Variatio.]  And  strike  out  the  rest 
to  the  end  of  the  Paragraph. 

In  y®  xxxiii**  Proposition,  pag  456,  instead  of  y*  words 
[ut  rem  perpendenti  constabit]  may  be  written  [ut  rem 
pcrpendenti  &  computationes  instituenti  constabit.]  And 
the  numbers  in  this  Proposition  are  to  be  suited  to  y* 
alterations  made  in  y®  preceding  Proposition  as  you  men- 
tion. 

I  am 

London  Apr.  24****  Yo"^  most  humble  Servant 

1712  Is.  Newton 

For  the  R"**  M'  Roger  Cotes  Professor 
of  Aatronomrff  at  his  Chamber  in 
Trinity  College  in  Cambridge. 


LETTER  L. 
COTES  TO  NEWTON. 


S' 
I  have  received  Your  last,  &  taking  the  whole  numbers 
69  &  70,  the  alteration  in  Pag:  442.t  lin.  penult,  will  be 


•  The  post  mark  is  Ap.  29. 

t  Prop.  XXIX.    Invenire  Variationem  Luni 


104  COIIRESPONDENCE  OF 

[68,6877  o-d  numerum  69.  Quo  pacto  tangens  anguli  CTP 
jam  erit  ad  tangentem  motu8  medii  ut  68,6877  ad  70,  & 
angulus  CTP  in  Octantibus,  ubi  motus  medius  est  45^'. 
invenietur  44^^  27'.  28":  qui  subductus  de  angulo  motus 
medii  45°  relinquit  Variationem  maximam  32'.  32".     Hcec 

ita  se  habercnt  si  &  Variatio  maxima  qua;  sccus 

esset  32  .  32".*  jam  aucta  in  eadem  ratione  fit  35\  l0".t] 
You  go  on  thus  J.  Hwc  est  ejus  magnitudo  in  mediocri 
distantia  Solis  a  Terra,  neglectis  differentiis  quso  a  curva- 
tura  Orbis  magni  majoriq:  Solis  actione  in  Lunara  falcatam 
et  novam  quam  in  gibbosfi  &  plenam,  oriri  possint.  In 
aliis  distantiis  Solis  a  Terra,  Variatio  maxima  est  in  ratione 
qua?  componitur  ex  duplicata  ratione  revolutionis  Syno- 
dicae  Lunaris  (dato  anni  tempore)  directe,  et  ratione  anguli 
CTa  directe,  &  triplicatii  ratione  distantia*  Solis  a  Terra 
inverse ;  id  est,  ex  triplicata  ratione  revolutionis  synodicae 
Lunaris  directe  et  triplicata  ratione  distantiae  Solis  a  Terra 
inverse.  Ideoq:  in  Apogwo  Solis  Variatio  maxima  est 
S3'.  11"  &  in  ejus  Perigaeo  37'.  24",  si  modo  eccentricitas 
Solis  sit  ad  Orbis  magni  semidiametrum  transversam  ut 
16J-5-  ad  1000. 

Ilactenus  Variationem  investigavimus  in  Orbe  non 
eccentrico  in  quo  utiq:  Luna  in  Octantibus  suis  semper  est 
in  mediocri  sua  distantia  a  Terra.  Si  Luna  propter  eccen- 
tricitatem  suam,  magis  vel  minus  distat  a  Terra  quam 
si  locaretur  in  hoc  Orbe,  Variatio  paulo  major  esse  potest 
vel  paulo  minor  quam  pro  Regula  hie  allata :  sed  excessum 
vel  defectum  ab  Astronomis  per  Phaenomena  determinan- 
dum  relinquo. 

I  was  going   to  diminish  §  Your   numbers   33'.  1 1",  & 


•  32'.  34"  in  Newton'fl  MS. 

t  35'.  12"  in  Newton's  IMS. 

t  N08.  149,  150. 

S  In  the  margin  of  Newton's  MS.  (No.  149,)  Cotes  has  actually  made  this  diminu- 
tion, as  he  has  done  above,  in  the  case  of  the  numbers  32'.  34"  and  35'.  12"  at  the  end  of 
the  extract  inclosed  within  biackets. 


NEWTON  AND  COTES.  105 

37'.  24"  by  2"  which  is  nearly  the  diminution  if  those 
numbers  are  right,  which  I  am  forc'd  to  take  upon  trust 
not  knowing  how  to  state  the  proportion  of  the  Moon's 
Periodical  Revohitions  nor  consequently  of  her  Synodical 
in  the  Apogee  &  Perigee  of  y*  Sun.  But  I  cannot  fully 
satisfy  my  self  about  Your  Rule.  As  I  take  it,  the  dupli- 
cate ratio  of  y®  Synodical  revolution  of  y*  Moon  &  y* 
simple  ratio  of  y''  angle  CTa  compose  not  the  triplicate 
ratio  of  y*  Synodical  revolution  alone  but  this  triplicate 
ratio  directly  &  y®  simple  ratio  of  y®  periodical  revolution 
inversly :  the  angle  CTa  being  as  y®  Synodical  revolution 
directly  &  y®  Periodical  revolution  inversly.  I  have  besides 
some  scruple  about  introducing  y*  ratio  of  y*  angle  CTa, 
I  have  not  throughly  considered  the  thing,  but  I  quajry 
whether  it  will  not  be  sufficient  to  make  the  compounded 
ratio  consist  only  of  y®  duplicate  ratio  of  y®  Synodical 
revolution  directly  &  y®  triplicate  ratio  of  y®  Sun's  distance 
inversly  according  to  y®  11*^  Corol:  of  Prop:  lxvi*^  Lib.  1. 
I  have  transcribed  y®  whole  y^  You  may  review  it  and 
order  it  as  You  think  it  should  stand. 

Your  &e. 
May  day  1712.  R  C. 

In  liis  answer  to  this,  (May  10,)  Newton  adheres  to  the  statement 
that  the  Variation  is  proportional  to 

(Moon's  synodical  period)^  dato  anni  tempore  x  /  CTa 
(distance  between  Sun  &  Moon)' 
Cotes  then  (May  13)  further  explains  his  reasons  for  thinking  that  tlie 
/  CTa  should  bo  cancelled.  Not  receiving  an  answer,  he  writes  again 
(May  25)  to  draw  his  attention  to  the  point,  and  has  the  gratification 
of  finding  (see  letter  of  May  27)  that  Sir  Isaac  has  been  convinced  by 
liis  arguments. 


106  CORRESPONDENCE  OF 

LETTER  LI. 
COTES  TO  NEWTON. 

I  fear  I  give  You  too  much  trouble  with  my  Letters, 
but  I  think  this  will  be  my  last  till  we  come  to  the  Theory 
of  Comets.  In  the  Corollary  of  the  xxxiii'*  Proposition  I 
put  16".  19'".  27".  instead  of  16".  18"'.  41'" i.  I  am  not  cer- 
tain  how  You  would  compute  that  motion,  &  therefore  I 
mention  it  to  You,  I  found  it  by  this  Proportion:  As 
19'.  18'.  01".  23'"  to  19".  21'.  22".  3'"  so  16".  16'".  37".  42"  to 
16".  19'".  26'".  56'. 

In  Your  last  letter  You  order  page  458.  lin  11.  thus. 
[Et  in  eadem  ratione  minueter  etiam  Inclinationis  Variatio] 
This  will  cause  some  alteration  in  the  following  Corollarys 
&  in  the  xxxv*''  Proposition  unless  You  design  to  consider 
the  Moons  Inclination  only  as  moving  in  Orbe  circulari. 

At  the  bottom  of  Page  461  You  make  use  of  5^  17'.  46" 
&  5°  for  the  extream  Inclinations ;  In  D""  Gregorys  Astro- 
nomy You  have  5\  17'.  20"  &  4^  59'.  35".  AVhich  I  suppose 
You  find  to  be  more  agreeable  to  observations. 

In  the  first  Paragraph  of  y®  New  Scholium*  to  Prop; 
XXXV*''  You  have  [ad  11'.  5o"  circiter  ascendit,  &  additur 
medio  motui  Lunaa  ubi  Terra  pergit  a  Perihelio  suo  ad 
Aphelium  &  in  opposita  orbis  parte  suhducitur']  As  I  take 
it,  the  words  additur  &  suhducitur  should  change  places. 
You  have  not  mentioned  how  to  find  this  -Equation  in 
every  place. 

In  the  second  Paragraph  concerning  the  Annual  Equa- 
tions of  the  Moon's  Apogee  &  Node  You  have  forgotten 
to  mention  when  they  must  be  added  &  when  substracted. 

In  the  third  Paragraph  You  say  [Per  Theoriam  gravi- 
tatis  constitit  etiam  quod  actio  Solis  in  Lunam  paulo  major 


•  See  the  remarks  which  follow  the  Letter. 


NEWTON  AND  COTES.  107 

sit  ubi  transversa  diameter  Orbis  Limaris  transit  per  Solem 
quam  ubi  eadem  ad  rectos  est  angulos  cum  linea  Terram 
&  Solem  jungente  &  propterea  Orbis  Lunaris  paulo  minor 
est  in  priore  casu  quam  in  posteriore]  I  think  it  should  be 
[paulo  major  est  in  priore] 

In  the  fourth  Paragraph  concerning  y*  ^Equation  of  y* 
Moon  arising  from  y®  position  of  her  Nodes  which  You  call 
Semestris  secunday  You  have  [additur  vero  medio  motiu 
Lunje  dum  Nodi  transeunt  a  Solis  Syzygiis  ad  proximas 
Quadraturas  &  stthducitur  in  eorum  transitu  a  Quadraturis 
ad  Syzygias]  As  I  apprehend  it  y®  words  additur  &  subduci- 
tur  should  change  places. 

The  sixth  Paragraph  I  do  not  understand.  The  yEqua- 
tion  which  You  there  describe  seems  to  be  established  not 
so  much  from  Observations  as  from  the  Theory  of  gravity, 
but  I  cannot  perceive  how  it  answers  Your  design  ex- 
press'd  in  these  words.  In  Perihelio  Terra?  propter  majo- 
rem  vim  Solis  Apogajum  Lunoe  velocius  movetur  in  epicy- 
clo  circum  centrum  D  (I  suppose  it  should  be  centrum  C) 
quam  in  Aphelio,  idq:  in  triplicata  ratione  distantia)  Terra) 
a  Sole  inverse.  Ob  ajquationem  centri  Solis  in  argumcnto 
annuo  comprehensam  Apogajum  Lunaj  velocius  movebitur 
in  epicyclo  in  duplicata  ratione  distantise  Terrce  a  Solo 
inverse.  Ut  idem  adhuc  celerius  moveatm*  in  ratione  sim- 
plici  distantia)  inverse,  sit  &c.*     Now  the  ^Equation  which 


•  We  will  add  the  remainder  of  the  paragraph  from  Newton's  MS.  ( No.  170) : 
'  eit  TD  excentricitas  primo  sequata,  et  producatur  TD  ad  E  ut  sit  D£  ad  TD  ut 


daplutn  excentricitatis  Solis  ad  radium  Orbis  magni  seu  33|  ad  1000.  Capiatur  angulus 
EDF  squalis  argumento  annuo,  vel  quod  perinde  est,  agantur  parallelae  TS  ac  DF 
solem  versus,  et  sit  DF  ipsi  DK  aqualis,  et  erit  DTF  xquatio  annua  apogxi  Lunas 
fi  FTS  distantia  Solis  ab  apogxo  Luniu  ter  xquata,  &  TF  excentricitas  Lunx  bis 
.Tquata  in  apogxum  Luna;  ter  aquatuin  tendens." 


108 


CORRESPONDENCE  OF 


You  describe  in  what  follows  does  not  in  the  least,  as  I  see, 
depend  upon  the  Sun's  Anomaly  but  intirely  upon  y* 
Annual  Argument  of  the  Apogee.  You  will  perhaps  more 
easily  perceive  my  difficulty  if  I  tell  You  how  I  think  the 
^Equation  should  be  stated  to  answer  what  was  propos'd. 
Let  CTD  be  y®  JEquatio  Semestris  describ'd  in  y®  preced- 
ing Paragraph ;  produce  CD  to  E,  so  y*  DE  may  be  to 
CD  as  33^  to  1000;  make  the  angle  EDF  equal  to  the 
Sun*s  Anomaly,  &  the  line  DF  equall  to  DE,  &  joyn  TF: 
then  will  DTF  be  the  second  annual  Equation  of  y® 
Apogee  &  TF  be  the  Eccentricitas  Lunse  bis  aquata  in 
Apogaium  Lunas  ter  roquatum  tendens. 


The  following  Paragraph  concludes  thus*.     Ducantur 


•  The  former  part  of  this  parayraph  is  as  follows,  (No.  170) :  "Per  eandem  gravi- 
tatis  Theoriam  Sol  fortius  agit  in  Lunam  annuatim  ubi  apogaeum  Lunae  et  perigaeum 
Solis  conjunguntur  quam  ubi  oppoauntur.  Etinde  oriuntur  aequationes  duae  periodicae, 
una  medii  motua  Lunae,  altera  apogxiejus:  quae  quidetn  acquationes  nullae  sunt  ubi 
apogacura  Lunae  vel  conjungitur  cum  perigso  Solis  vel  eidem  opponitur,  et  maximae  in 
apogaeorum  quadraturls.  In  aliis  apogaeorum  positionibus  datam  habent  proportionem 
ad  iuvicem,  suntq  :  ut  sinus  distantiae  apogaeorum  ab  invicem.  /Equatio  prior  subduci- 
tur  et  posterior  additur  tibi  apogaeum  Lunae  minus  distat  a  perigaeo  Solis  in  consequen- 
tia  quam  gradibus  180;  prior  voru  additur  &  posterior  subducitur  ubi  distantia  ilia  fit 
major,  llarum  a>quationum  quantum  sentio,  /Equatio  maxima  apoga;i  asccndit  ad 
15'  vel  20'  circiter,  sed  a>quatio  maxima  motus  medii  Lunae  vix  asceudit  ad  30",  et  ob 
piirvitatem  negligi  potest  donee  quantitas  ejus  ex  observationibus  determinetur,  Pro- 
ducatur  excentricitas  Lunaj  bis  acquata  TF  ad  G  ut  sit  FG  sinus  aiquationis  maximae 
periodica)  apogaii  Lunae  15'  vel  20'  ad  radium  TF.    Ducantur,"  &c. 


NEWTON  AND  COTES.  109 

rcctre  dusD  parallelce  TP,  FII  in  Perigroum  Solis  tendcntcs, 
vel  quod  perinde  est,  capiatur  angulus  GFJI  sequalis  distan- 
tiai  Perigaji  Solis  ab  Apogrco  Liinro,  &  sit  FII  ipsi  FG  ajqua- 
lis ;  et  angulus  FT II  crit  ajquatio  Periodica  Apoga'i  Luna;, 
&  angulus  PT// distantia  Apogrci  Luno)  quarto  aequati  a 
Perigajo  Solis  et  Til  cccentricitas  tertio  wquata  in  apo- 
gseum  quarto  a*quatum  tendens.  Instead  of  which  I  pro- 
pose the  following  alteration,  leaving  out  y®  line  TP  in  the 
Figure.  Capiatur  angulus  GFII  aiqualis  distantio3  Apogaei 
Luna)  a  Perigico  Solis  in  consequentia  et  sit  FH  ipsi  FG 
aqualis,  &  angulus  FTII  erit  yEquatio  periodica  Apogaii 
Luna}  &  Til  cccentricitas  tertio  aquata  in  Apoga;um 
quarto  a3quatum  tendens.  This  Alteration  will  agree  with 
what  You  lay  down  a  little  before  in  the  same  Paragraph, 
where  speaking  of  this  Periodical  ^'Equation  of  y*  Apogee 
You  say  additur  ubi  Apogyeum  Lunje  minus  distat  a  Pe- 
rigaio  Solis  in  consequentia  quam  gradibus  180  &  subduei- 
tur  ubi  distantia  ilia  fit  major,  Which  Rule  I  think  is  right 
but  not  agreable  to  the  conclusion  of  the  Paragraph  which 
I  therefore  propose  to  alter. 

In  the  last  Paragraph  but  one  You  say  [pono  medio- 
crem  distantiam  centri  Luna;  a  centro  Terra  in  Octantibus 
jpqualem  esse  6o|^  semidiametris  maximis  Terrae]  I  desire 
to  know  whether  You  will  here  retain  6o^  or  put  instead 
of  it  6o\  as  in  Corol  7*^  of  Prop  xxxvii**> 

Your  &c. 
May.  3^  1712  R  C. 

The  "  New  Scholium  to  Prop,  xxxv.**  whicli  forms  so  largo  a  part 
of  the  subject  of  the  preceding  letter  is  a  Scliolium  on  the  Lunar 
Theory,  containing  a  statement  of  the  origin  and  quantity  of  various 
Limar  Inequalities,  and  occupying  the  place  of  a  short  Scholium  in  th« 
Ist  Ed.  relative  to  the  motion  of  the  Moon's  Apogee.  It  is  written  on 
three  sides  of  a  sheet  of  foolscap  (Nos.  109 — 171)  which  seems  to  liave 
been  doubled  up  and  placed  loosely  between  the  pages  of  Newton's 
interleaved  copy  of  the  1st  Ed.  It  was  probably  sent  to  Cotes  with 
the  third  and  last  division  of  Newton's  copy^tho  first  week  in  July 


110  CORRESPONDENCE  OF 

1711.  (Letter  xxviii  and  note).  The  reason  why  the  Scholium  appears 
on  folio  paper  is,  no  doubt,  that  there  was  not  room  for  it  on  tho 
quarto  leaf  in  the  interleaved  copy :  that  quarto  leaf  is  still  preserved, 
and  its  first  page  (No.  190)  is  headed  "  Scholium"  and  is  devoted  to 
the  opening  words  of  it  followed  by  an  "  &c."  thus :  **  llisce  motuum 
Lunarium  computationibus  ostendero  volui  quod  motus  Lanares  per 
Tlieoriam  gravitatis  &c."  indicating  that  tho  Scholium  was  to  bo  found 
written  out  on  another  paper.  The  second  page  of  tho  leaf  contains 
some  supplementary  matter  to  be  added  to  Prop,  xxxvi.  These 
minutim  are  mentioned  for  tho  purpose  of  limiting  the  date  of  tho  com- 
position of  tho  Scholium,  as  the  circumstance  of  its  being  written  on  a 
folio  sheet  might  have  led  one  to  suppose  that  it  was  sent  down  to 
Cambridge  not  as  part  of  the  copy,  but  as  an  emendation  of  copy  pre- 
viously sent.  Tho  quarto  leaves  of  Newton's  handwriting  in  tho 
Newtonian  Volume  all  formed  part  of  his  interleaved  copy  of  tho 
Principia :  those  in  folio  were  sent  down  in  letters  as  corrections.  The 
only  exception  to  this  remark  that  I  have  noticed  is  the  sheet  now 
referred  to,  which  contains  the  Scholium  on  the  Lunar  Theory. 

A  distinct  idea  of  tho  contents  of  this  Scholium  (or  "  first  draught 
of  tho  Moon*s  theory,"  as  it  is  afterwards  called),  as  it  stood  before 
undergoing  the  alterations  which  Newton  made  in  it  in  consequence  of 
the  above  letter  from  Cotes,  may  be  obtained  from  tho  following  out- 
lino  of  it.  It  consists. of  twelve  paragraphs,  which,  for  convenience  of 
reference,  I  will  number  in  the  order  in  which  they  present  themselves. 

1.  "  Hisce  motuum  Lunarium  computationibus . . .  jequatio  maxima 
erit  1 1' .  52"."    (Annual  Equation). 

2.  "Inveni  etiam . . .  tequatio  maxima  medii  motus  nodorum  9'. 27''." 
(Annual  Equations  of  mean  motion  of  apogee  and  nodes.) 

3.  "  Per  tlieoriam  gravitatis. .  .quadratura  ad  radium."  (iEquatio 
semestris,  the  argument  of  which  is  =  twice  the  distance  of  apogee 
from  Sun,  i.  e.  twice  tho  annual  argument). 

4.  "Per  eaudcm  gravitatis  tlieoriam ...  ad  49"  circiter  ascendit." 
(iEquatio  semestris  sccunda,  the  argument  of  which  is  =  twice  the 
distance  of  node  from  Sun). 

Tho  four  preceding  paragraphs  stand  as  they  are  printed  in  tho 
2nd  Ed.  with  the  exception  of  the  modifications  introduced  in  confor- 
mity with  Cotcs's  suggestions  in  the  above  letter.  (See  Letter  lviii). 
In  the  2nd  the  word  "  inverse"  is  also  omitted  after  "  si  motus  SoHs 
essct  in  triplicata  ratione  distantia\" 

5.  "Per  eandcm  gravitatis  Theoriam  apoga?um  Luna?... in  apo- 
gajum  secundo  rcquatum  tendens".  (The  Equation  of  the  centre  and 
Evection  combined,  giving  tho  a?quatio  semestris  of  the  apogee  and 
first  correction  of  the  eccentricity). 


NEWTON  AND  COTES.  Ill 


Same  as  in  2nd  Ed.  with  two  exceptions :  (1)  the  upper  foeui  of 
the  moon's  orbit  (and  not  its  centre  according  to  Newton's  subsequent 
correction  in  his  paper  of  alterations,  see  letter  Lvii.)  is  represented  as 
describing  the  epicycle  BDA  ;  by  a  similar  inadvertence  in  paragraph 
6  the  apogee  is  made  to  move  in  that  epicycle.  (2)  In  the  2nd  Ed. 
there  is  a  clause  "  llabitis  autem. .  .per  methodos  notissimas"  added  at 
the  end  of  the  paragraph,  which  is  in  part  transferred  from  paragrapli  9. 

Cotes  has  drawn  two  other  lines  in  the  figure  (no  doubt  on  receiving 
Newton's  paper  of  alterations)  viz.  DE  to  the  right,  parallel  to  A  /i, 
and  DF  making  an  acute  angle  with  it  (not  an  obtuse  angle  as  in  the 
figure  in  the  2nd  Ed.) 

6.  "In  porihelio  terraa. .  .in  apoga?um  Lunn?  ter  a?quatum  tendens." 
(Third  correction  of  the  place  of  the  apogee  and  second  of  the  excen- 
tricity  by  an  "  annual  equation"  whoso  argument  «  annual  argument). 

This  paragraph  is  given  in  the  preceding  lett(?r  and  note.  It  was 
completely  remodelled  in  Newton's  paper  of  alterations.  Two  para- 
graphs were  substituted  for  it  explanatory  of  what  he  says  may  bo 
called  **  (equatio  centri  secunda"  depending  on  the  argument  "  dis- 
tance of  moon  from  sun  +  dist.  of  moon's  apogee  from  sun's  apogee." 
The  latter  of  them  merely  contains  an  approximation  to  its  value. 
Newton's  mode  of  determining  the  position  of  the  centre  of  the  moon's 
orbit  in  a  secondary  epicycle  with  centre  D  became  the  subject  of  an 
active  correspondence  between  him  and  Cotes  (letters  lvii-lxvi). 

7.  "  Per  eandem  gravitatis  Theoriam  Sol  fortius  agit ...  in  apo- 
ga?um  quarto  a3quatum  tendens.  (Fourth  correction  of  the  place  of 
the  apogee  and  third  of  the  excentricity  by  a  "  periodical  equation" 
whose  argument  =  distance  of  apogee  from  Sun's  perigee.  ^lention  is 
also  made  of  a  "periodical  equation"  of  the  Aloon's  mean  motion 
depending  on  the  same  argument  having  barely  30/'  for  its  maximum  : 
Damoiscau  gives  it  2",  Plana  O'' .  466,  Pontecoulantl". 496-1".  108^ 
0''.388,  Burckhardt  0".l.  See  Pontdcoulant,  torn.  iv.  pp.  451-465, 
580,  604,  626  ;  the  two  terms  of  which  it  is  composed  are  of  the  fourth 
and  fifth  orders.) 

This  paragraph  will  also  be  found  in  the  preceding  letter  and  proper 
note. 

a  "  Si  tres  anguli  CTD,  DTP  &  FTH  ad  singulos  gradus 
angulorum  BCD,EDFci  Gi«V/ computentur  &  in  Tabulas  rcfcrantur. 


112  CORRESPONDENCE  OP 

et  81  logarithmi  quoq:  trium  distantianim  TD^  TF  &  Tffad  radios 
TC  TD  et  TF  in  partes  100000  divisos  sinml  compiitentur  &  in 
Tabulas  reforantur :  aggrcgatum  trium  angulorum  sub  signis  suis  +  & 
-  erit  a?quatio  tota  apogwi,  et  aggregatum  trium  Logarithmorum  erit 
Logarithmus  excentricitatis  verw." 

This  and  the  preceding  paragraph  were  not  given  in  Newton's 
paper  of  alterations,  whore  another  paragraph  ("  Si  computatio  accu- 
ratior...non  multum  errabitur")  appeared  relating  to  the  "variatio 
secunda,'*  which  was  omitted  in  the  3rd  Ed. 

9.  "  Ilabitis  autem  Lunce  motu  medio  &  apogaeo  et  excentricitate 
ultimum  a^quatis,  ut  et  Orbis  diametro  transversa  partium  200000 ;  ex 
his  eruetur  verus  Lunae  locus  in  orbe,  et  distantia  ejus  a  Terra,  idq :  per 
methodos  notissimas.  Deinde  per  Variationem  et  Reductionem  ad 
EcHpticam  dabitur  ejus  loiigitudo  et  latitude  vera." 

10.  "  Diximus  orbem  Lume  a  viribus  Solis  per  vices  dilatari  et 
contrahi  &  a^quationcs  quasdam  motuum  Lunarium  indo  oriri.  Indo 
etiam  oritur  variatio  aliqua  parallaxeos  Lunae,  sed  quam  inscnsibilem 
esse  judico ;  idcoq :  in  computationibus  motuum  Luna?,  pro  mcdiocri 
ejus  distantia  a  centro  Ternu  semper  usurj)o  numcrum  100000,  &  pro 
Orbis  diametro  transversa  numerum  200000,  et  ad  parallaxim  inves- 
tigandam  pono  mediocrem  distantiam  centri  Luna?  a  centro  Terroe  in 
Octantibus  iequalem  esse  60j  semidiametris  maxirais  Terra?.  Semi- 
diametrum  ejus  maximam  voco  qua)  a  centjr|o  ad  cequatorem  ducitur, 
minimam  qua*  a  centro  ad  polos.  Et  hinc  fit  Lunoo  parallaxis  horizon- 
talis  mcdiocris  apparens  in  Octantibus  57'  5'\  in  Syzygiis  57'  30"  in 
quadraturis  56'  40".  Luna?  voro  diameter  mcdiocris  apparens  in 
Syzygiis  31 .30  in  Quadraturis  31  .3  usurpari  potest  &  Solis  diameter 
mcdiocris  32  . 1 2."  -=^^ 

11.  "Et  cum  atmosphajra  Terra?  ad  usq  :  altitudinem  milliarium 
35  vel  40  refringat  Lucem  Solis  et  refringendo  spargat  eandem  in 
umbram  Terra?,  &  spargcndo  lucem  in  confinio  umbrte  dilatet  umbram: 
ad  diametrum  umbra)  qua)  per  parallaxim  prodit,  addo  minutum  unum 
primum  in  eclipsibus  Lunae,  vel  minutum  unum  cum  triente." 

12.  "  Theoria  vero  Luna?  primo  in  Syzygiis,  deinde  etiam  in  qua- 
draturis" &c.  as  in  the  2nd  Ed.  except  (1)  as  regards  the  changes  in 
some  of  the  figures  mentioned  in  Letter  lxvi.,  and  (2)  the  addition  of 
the  clause  "  &  diffcrcntiam  meridiauorum  Observatorii  hujus  &  Obsor- 
vatorii  Rcgii  Parisiensis  0''°'-  Q*"'"-  20**"-."  at  the  close  of  the  paragraph, 
which  does  not  appear  here. 


NEWTOX  AND  COTES.  113 

LETTER  LII. 

NEWTON  TO   COTES. 

*"Prop.  De  Variatione  Lunro  p.  402."  }2'*  Ed.}. 

I  have  received  three  letters  from  you  since  my  last. 
And  the  corrections  w*^''  you  send  me  in  the  two  first 
of  them  may  all  stand.  In  the  second  of  them  dated 
May  1**,  you  cite  my  words.  In  aliis  distantiis  Solis  a 
Terra  Variatio  maxima  est  in  ratione  qua)  componitur  ex 
duplicata  ratione  [temporis]  rcvolutionis  sy{n}odicfl3Lunaris 
(dato  anni  tempore  directe,  et  ratione  anguli  CTa  directe, 
et  triplicata  ratione  distantiaQ  Solis  a  Terra  inverse.  Ideoq: 
in  Apogoco  Solis  Variatio  maxima  est  S3\  \l"  et  in  ejus 
Periga30  37'  24"  si  modo  excentricitas  Solis  sit  ad  Orbis 
magni  semidiametrum  transversam  ut  16-}-^-  ad  lOOO.  Here 
S3  11  &  37  24  may  be  diminished  by  2"  &  the  word  tem- 
poris may  be  inserted  where  you  see  it  w*^in  the  brackets. 
The  Variatio  maxima  is  composed  of  the  ratios  of  the 
time,  the  angle  CTa,  &  the  sun's  force,  as  above ;  because 
if  any  one  of  the  three  ratios  be  enlarged  while  the  rest 
remain  given,  the  variation  will  be  enlarged.  If  the  time 
alone  be  enlarged  the  Variation  will  be  enlarged  in  a 
duplicate  proportion,  as  may  be  gathered  from  the  descent 
of  falling  bodies  in  a  greater  or  less  time.  If  the  angle  be 
enlarged  the  Variation  w^*'  is  a  proportional  part  of  y* 
Angle  will  be  inlarged  in  the  same  simple  proportion,  & 
the  force  also  w*^^  is  reciprocally  as  the  cube  of  y*  Suns 
distance  enlarges  the  Variation  in  proportion  to  it  self. 

In  pag  445  write.  Idem  per  Tabulas  Astronomicas  est 
19.  21.  20.  45  f.  Differentia  minor  est  parte  fere  quadrin- 
gentesima  motus  totius,  et  ab  Orbis  &c. 


•  Tn  Cotes's  hand. 

t  This  is  the  mean  motion  of  the  Moon*8  nodes  in  a  Julian  year.  Dut  it  is  the 
mean  notion  in  a  sidereal  year  that  is  required  in  the  place  referred  to.  See  Cotcs*i 
annwer. 

8 


lU  CORHESPONDEXOli:  OF 

Pag  450  liii  13  write  9^'.  ll'.  3".  &  lin  28  in  Qiiadra- 
iuris  aiitcm  rcgrcdiuntur  motu  horario  l6"  19'"  5\'\  I 
compute  it  thus.  As  JB  to  JD  +  Ali  so  is  the  mean 
horary  motion  of  the  Node  to  K)'.  if/".  51'^ 

1  am  S*" 
Yo*"  most  humble  Servant 
London  lO'*^  May  1712  Is.  Newton. 

At  the  bottom  of  pag  461  you  may  put  the  numbers 
5^'.  17'.  20"  &  4^"^  59'  35" 

Pag  45G  lin  1  instead  of  38  ^^  write  SS^^. 
The  Lunar  systeme  must  be  altered " 

To  M'  Cotes  Professor  of  Astronmny 
at  hia  Chamber  in  Trinity  College 
in  Cambridge 


LETTER  LIII. 


COTES  TO  NEWTON. 
S'.  May  13***  1712 

I  have  received  Your  last,  but  I  am  not  yet  clear  that 
the  ratio  of  y®  angle  CTa  ought  to  be  introduced  in  y® 
xxix*^  Proposition,  though  I  do  fully  understand  the  reasons 
You  give  for  it.  As  I  apprehend  it  the  duplicate  ratio  of 
y®  Synodical  time  does  itself  account  for  the  dilatation  of 
the  Angle,  &  therefore  it  ought  not  to  be  again  ac- 
counted for.  According  to  the  reasoning  of  the  16*^ 
Corollary  of  Prop :  lxvi.  Lib.  i,  the  Variatio  maxima  which 
is  the  angular  Error  of  y®  moon  whilst  she  describes  the 
half  of  y®  Arch  Cpa,  is  as  the  Square  of  y*'  time  imploy'd 


•  This  is  all  the  notice  that  Newton  at  present  takes  of  Cotes's  remarks  upon  the 
Scholium  on  the  Lunar  Theory.  The  necessity  of  an  "alteration"  in  "the  Lunar 
Systeme"  points  to  the  6th  and  7th  paragraphs  of  the  Scholium,  especially  the  former. 
About  the  end  of  June,  we  are  to!d,  he  intended  to  send  down  his  corrections  "  very 
soon,"  but  even  with  tlie  stimulus  of  a  letter  from  Cotes  (July  20),  it  is  only  a  little 
before  A\ig.  10  tlmt  they  are  despatched  to  Cambridge,  (Letters  LVL,  LVIL) 


NEWTON  AND  COTES.  115 

in  describing  that  half  Arch  directly  &  y*  Cube  of  y" 
distance  from  y*  Sun  inversly :  Or  as  the  Square  of  y* 
Synodical  time  directly  &  y*  Cube  of  y*  distance  invcrsly. 
Now  I  think  the  dilatation  is  accounted  for  by  taking  the 
angular  Error  which  arises  in  the  time  of  describing  half 
y®  arch  Cpa,  instead  of  y*  Error  which  would  arise  in  y* 
time  of  describing  half  y®  arch  CPA,  The  thing  may  be 
considered  another  way  which  perhaps  will  give  more  light 
to  y®  understanding  of  my  difficulty  The  true  Variatio 
maxima  35\  lo"  arises  from  y®  arch  Cjoa,  but  the  Variatio 
maxima  32'.  32"  arises  from  the  arch  CPA,  Now  this  latter 
by  y®  16*^  Corollary  of  Prop  lxvi  Lib  1  must  be  altered 
with  y®  Square  of  y®  Periodical  time  directly  &  the  Cube 
of  y®  distance  invcrsly,  &  so  it  will  be  more  correct ;  after 
it  is  thus  corrected,  the  corrected  true  Variatio  maxima 
will  be  dcduc'd  from  it,  by  enlarging  or  dilating  it  in  y* 
proportion  of  y®  Angle  CTa  to  y®  Angle  CTJ  Oi  in  the 
proportion  of  y®  Synodical  to  y®  Periodical  time.  There- 
fore the  corrected  true  Variatio  maxji}ma  will  be  as  the 
Square  of  y®  Periodical  time  directly,  the  Cube  of  the 
distance  invcrsly,  the  Synodical  time  directly  &  the  Peri';- 
dical  time  inversly :  that  is,  as  the  Periodical  &  Synodical 
times  directly  &  the  cube  of  y«  distance  inversly.  In  this 
latter  way  I  scruple  not  to  account  for  the  dilatation,  but 
in  the  former  I  think  it  is  already  accounted  for  by  taking 
the  Square  of  the  Synodical  time  instead  of  the  Square  of 
y®  Periodical.  If  You  find  the  Objection  to  be  of  any 
moment,  I  desire  you  to  send  me  otlier  numbers  instead 
of  33' .  il".  &  37'.  21.".  If  You  choose  to  let  the  place 
stand,  yet  still  there  must  be  a  further  alteration  of  those 
numbers  besides  y®  diminution  by  2",  for  the  Square  of  y' 
Synodical  time  compounded  with  y*  ratio  of  y®  angle  CTa, 
makes  not  the  triplicate  ratio  of  y®  Synodical  time  (upon 
which  those   numbers  were   computed  but  that  triplicate 

8—2 


lit)  CORRESPONDENCE   OF 

ratio  directly  &  y*  ratio  of  y*  Periodical  time  inversly  as  I 
observed  in  my  former  Letter. 

In  Page  455  You  direct  me  to  write.  Idem  per  Tabulas 
Astronomicas  est  19°.  21'.  20".  45'".  Differentia  minor  est 
parte  fere  quadringentesima  motus  totiiis  &c.  I  would 
choose  to  put  it  thus.  Idem  per  Tabulas  Astronomicas 
est  l9^  21'.  21".  50"\  Differentia  minor  est  parte  trecen- 
tesima  &c.  For  according  to  Flamsteed's  Tables  the 
motion  of  y®  Nodes  from  y®  Fix't  stars  in  20  Yeares  or 
7305  Days  is  l*"*^.  0*'».  27°.  6'.  53",  and  therefore  in  365^  6\  9™ 
it  is  19^  21'.  21".  50'". 

The  mean  horary  motion  of  y®  Nodes  by  the  same 
Tables  is  7".  56"\  56""  and  as  AB  to  AD  +  AB  or  as  373  to 
766  so  is  7".  56'".  56"'  to  16".  19'".  26".  Therefore  in  Pag: 
456,  lin:  28,  I  would  write  16-".  19'".  26".  Unless  You 
find  other  reason  for  writing  16".  19'".  51'*^  as  You  put  it 
in  Your  Letters. 


LETTER    LIV. 
COTES   TO  NEWTON. 
S^  Trin:  College  May  26**»  1712 

I  have  not  yet  received  an  answer  to  my  last  of  May 
13***  concerning  the  xxix*^  Proposition;  I  am  therefore 
afraid  it  has  miscarried. 

I  sent  You  by  D*^  Bentley  a  small  Treatise*  of  my  own 


•  This  was  afterwards  published  in  the  Philosophical  TransactionSi  (Jan — March, 
1714),  and  subsequently  formed  the  first  part  of  Cotes's  Harmonia  Mensurarurrif  Cantab. 
1722,  edited  by  his  cousin  Rob.  Smith.  There  is  prefixed  to  it  a  short  address  to  Ilalley 
as  Secretary  of  the  Royal  Society,  the  first  sentence  of  which  is :  "  Mitto  tibi,  hortatu 
IlIustriBsimi  Prnsidis  Newtoni,  quo;  aliquot  abhinc  annis  conscripseram  de  Rationibus 
dimetiendis."  Cotes  had  succeeded  in  integratingr  some  general  expressions,  the  inte- 
prrals  of  which  involve  logarithms.  His  Logometria  contains  the  application  of  the 
results  to  the  solution  of  a  variety  of  problems.    Compare  Letter  CX.  fin. 


NEWTON  AND  COTES.  J 17 

concerning  Logaritlims,  of  which  the  Title  is,  Elementa 
Logometrice  together  with  the  Figures  belonging  to  it,  I 
desire  the  favour  of  You  to  deliver  'em  to  M*"  Livebody  to 
be  cut  in  Wood  &  to  give  him  Your  directions  if  he  meets 
with  any  difficulty.  1  fear  You  are  at  this  time  taken  up 
with  other  buisness,  otherwise  I  would  beg  of  You  to  pe- 
ruse the  Treatise.  You  will  find  I  am  there  proposing  a 
new  sort  of  Constructions  in  Geometry  which  appear  to 
me  very  easy,  simple  &  general.  But  I  am  fearfull  of  re- 
lying upon  my  own  Judgment  alone,  which  possibly  in  this 
matter  may  be  too  much  byass'd.  What  I  think  to  be 
right,  may  to  others  appear  whimsical  &  of  no  use  &  I 
would  not  willingly  give  them  the  satisfaction  of  laughing 
at  my  Dreams.  If  You  think  I  may  venture  to  publish  it, 
I  shall  be  glad  to  know  what  may  want  to  be  corrected  or 
altered  either  in  the  Matter  or  Expression.  I  have  been 
forc'd  to  use  some  new  Terms,  as  Modulus,  Ratio  modu- 
laris,  &c.  If  others  more  proper  occur  to  You  upon 
reading  the  Papers,  I  shall  be  very  willing  to  make  any 
alteration.  I  hope  You  will  pardon  this  Trouble  I  give 
You.     I  am  Sir 

Your  most  Obliged  &  Humble  Servant 

For  S'  Isaac  Newton  at  his  House  RooER   CoTES, 

m  8^  Martin*s  Street  in  Leicester- 
Feilds  London, 


LETTER  LV. 
NEWTON  TO  COTES. 


S' 
I  have  reconsidered  what  you  write  about  the  Varia- 
tion &  agree  to  it.     You  may  leave  out  the  words  [et 
ratione  anguli   CTa  dirccte]   &  instead  of  the  numbers 


118  CORRESPONDENCE  OF 

33'  iT'  &  37'  24"  diminished  by  2",  write  33'  14"  &  S?'  ll". 
fTor  so  I  found  them  upon  computing  them  anew. 

Also  in  pag  455  lin  14  you  may  write.  Idem  per  Ta- 
bulas  Astronomicas  est  Iff ^  2l'.  2l".  50"'.  DifFerentia  minor 
est  parte  treeentcsima  &c  And  pag  456  lin.  28  you  may 
write  16".  19'".  2()". 

I  received  yo*"  papers  by  D*"  Bently  &  have  run  my  eye 
over  them.  I  intend  to  read  them  over  again  &  get  the 
cuts  done  for  you  as  soon  as  I  can  find  out  M*^  Livebody. 

I  am  Yo**  most  humble  Servant 

London  May  27  1712  Is.  Newton. 

For  the  U"*^  }ii*  RooKU  Cotes  Professor  of 
Astronomy  at  ?iis  Chamber  in  Trinity 
College  in  Camhridge 

Brouglit  probably  by  Bcntley. 


LETTER  LVL 

COTES  TO   NEWTON. 

S'.  Cambridge  July  20***  1712 

It  is  now  about  three  Weeks  since  D*"  Bentley  return'd 
from  London.  He  told  me,  You  then  intended  to  send 
down  Your  Emendations  of  the  Lunar  Theory  very  soon. 
I  have  not  received  any  thing  from  You  since  that  time, 
&  am  therefore  apprehensive  of  some  miscarriage.  He  in- 
form'd  me,  You  had  thoughts  of  adding  something  further 
upon  the  Subject  of  Comets*,  &  besides  a  small  Trea- 
tise concerning  the  Methods  of  Infinite  Series  &  Fluxions. 
I  hope  You  will  go  on  with  Your  design :  it  were  better 
that  the  publication  of  Your  Book  should  be  dcferr'd  a 
little,  than  to  have  it  depriv'd  of  those  additions.    I  thank 


•   This  was  (lone  (sec  Letter  LXVIIT.),  but  the  project  with  respect  to  scries  and 
fluxions  w«s  ahandoned. 


NEWTON   AND  COTES.  119 

You  for  tlTe  Picture  which  I  have  received  of  him :  'tis 

rauch  better   done  than   tlie  former;    but  I  could  have 

wish*d-it  htid  been  taken  from  the  first  of  M""  Thornhill's. 

I  am  Sir  Your  most  Humble  Servant 

For  S'  Isaac  Newton  at  his  //biw  RonEU   CoTES. 

in  S*  Mayt}n*s  S(rM  in  Leicester 
Feilds  London 

On  the  back  of  Cotes's  draught  of  Apr.  20,  there  is  the  drauj;hi  of  a 
letter  from  him  to  Newton,  wliich,  from  the  alhision  to  the  intended 
treatise  on  scries  and  fluxions,  seems  to  liavo  been  written  about  the 
same  time  as  the  letter  wo  have  just  been  reading.  lie  probably 
never  sent  it,  but  replaced  it  by  the  above,  suppressing  the  suggestions 
and  remarks  which,  upon  second  thoughts,  he  may  have  considered 
as  out  of  place.  We  need  not,  liowever,  withhold  it  here.  It  is  as 
follows : 

"I  am  glad  to  understand  by  D'  Bcntley  that  You  have  some 
tliouglits  of  adding  to  this  Book  a  small  Treatise  of  Infinite  Series  & 
the  Method  of  Fluxions.  I  like  the  design  very  well,  but  I  beg  leave 
to  make  another  proposal  to  You.  "When  this  Book  shall  be  finished 
I  intended  to  have  importun'd  You  to  review  Your  Algebra  for  a  better 
Edition  of  it  &  to  have  added  to  it  those  things  which  are  published 
by  M'  Jones  &  what  others  You  liave  by  You  of  the  like  nature. 
These  together  will  make  a  Volume  nearly  of  y*  same  size  >vith  Your 
Principia  &  may  be  printed  in  the  same  Character.  Your  Trcatiso 
of  y"  Cubick  Curves  should  be  reprinted,  for  I  think  the  Enumeration 
is  imperfect,  there  being  five  cases  of  ^Equations  viz:  xytf-¥ey-\ 
yy  +  gxxy  -\xxy-^cy-\xy=\y-\  I  should  have  acquainted  You 
with  this  before  M'  Jones's  book  was  published,  if  I  had  known  any 
thing  of  the  Printing  of  it,  for  I  had  observed  it  two  or  three  yeares  ago. 
I  think  there  are  some  other  things  of  less  moment  amiss  in  the  same 
Treatise. 

I  am  S'  Your  most  Humble  Sorv' 

R.  Cotes" 


120  CORRESPONDENCE  OF 

Here  we  miss  two  conniunicationB  from  NewtoD,  one  of  which 
accompanied  the  MS.  of  the  "  Elementa  Logometriae"  on  its  return 
to  its  author,  conveying  his  opinion  of  the  tract  in  terms,  the  gist  of 
which  may  be  perceived,  though  more  dimly  than  one  could  wish, 
through  Cotes's  litotes  of  "  I  am  glad  you  are  not  displeased  with  it." 
(Next  letter).  The  other  coutained  his  corrections  of  the  Scholium  on 
the  Lunar  Theory,  (see  note  on  the  postscript  of  Letter  LI  I).  The 
nature  of  this  lost  paper  may  he  easily  collected  from  the  correspond- 
ence that  passed  relative  to  parts  of  its  contents. 

Newton  overlooked  Cotes's  suggestions  on  the  first  four  paragraphs 
of  the  Scholium,  and  commenced  his  paper  of  alterations  with  paragraph 
5,  probably  with  the  words  ^^  Horroxhis  noster. .  .Halleim  suporiorem 
ElHpscos  umbilicum,"  &c.  The  three  last  words  are  inadvertently 
copied  from  his  first  draught ;  they 
ought  to  be  "centrum  Ellipseos," 
as  Cotes  points  out  in  the  next 
letter.  The  diagram  belonging  to 
this  and  two  following  paragraph)?, 
(the  "new  figure"  mentioned  in  the 
next  letter)  seems  to  have  been  as 
represented  in  the  annexed. 

Next  came  paragraph  6,  ("  In  perihelio  Terras"  &c.)  as  it  stands  in 
the  2nd  Ed.  with  the  exception  of  the  clerical  errors  rectified  in  the 
letter  of  Aug.  12,  and  the  further  correction  (Aug.  20),  in  the  mode  of 
determining  the  z  EDF. 

After  that  there  was  a  new  paragraph  beginning  "Computatio 
hujus  motus  dilBcilis  est"  &c.  containing  an  approximation  of  the  pre- 
ceding paragraph.  (The  "a}quatio  centri  secunda,"  whose  argument  is 
dist.  of  ]\Ioon  from  Sun  +  dist.  of  IMoon's  apogee  from  Sim's  apogee). 
This  paragraph,  in  consequence  of  the  difficulties  which  Cotes  found 
in  it,  was  alterwards  rendered  more  perspicuous  in  the  paper  of 
Aug.  2G. 

Tlien  followed  another  new  paragraph  describing  the  "  Variatio 
pccunda,"  as  it  is  printed  in  the  2nd  Ed.  except  that  "  Aphelii"  was 
twice  written  by  mistake  for  "Apogtei."  The  Variatio  secunda 
=  -  (2'1  - cosi'!^'+  1')  sin /),  if  PE=  dist.  of  ])'8  apogee  from  ©'s peri- 
gee and  D  =  dist.  of ))  from  0. 

And  lastly,  (omitting  paragraphs  9,  10,  11,  the  first  of  which  was 
j)artly  removed  to  the  end  of  paragraph  5)  came  the  concluding  para- 
graph "  Thcoria  vero  Lunoa"  &c.  as  printed  in  2nd  Ed. 

Compare  the  account  of  the  first  draught  of  tlie  scholium  which  we 
have  given  after  Cotes's  letter  of  May  3.  pp.  1 10 — 112. 


NEWTON  AND  COTES.  121 

LETTER  LVII. 
COTES  TO  NEWTON. 
S'  Cambridge  August  10'''  1712 

I  thank  You  for  Your  care  of  the  Wooden  Cutts  which 
I  received  of  the  Carrier  together  with  the  Manuscript*. 
I  am  glad  You  are  not  displeased  with  it,  &  I  wish  You 
had  signified  what  Emendations  might  be  made  in  it. 

In  my  Letter  of  May  the  3**,  I  mentioned  some  alter- 
ations in  the  former  part  of  Your  Lunar  Theory.  You 
have  left  me  uncertain  as  to  Your  resolution  about  them, 
by  taking  no  notice  of  them  in  Your  Last  in  which  Your 
correction  of  the  latter  part  of  the  Theory  is  set  down. 

I  observe  in  the  beginning  of  it,  You  have  chang*d  [et 
circulus  BDA  centro  C  intervallo  CD  descriptus  erit  Epi- 
cyclus  ille  in  quo  superior  Ellipseos  umbilicus  locatur]  for 
[Epicyclus  ille  in  quo  centrum  Orbis  Lunaris  locatur].  I 
quaBry  whether  [Halleius  superiorem  Ellipseos  umbilicum 
in  Epicycle  locavit]  shoidd  not  be  also  chang'd  into  [Hal- 
leius centrum  Ellipseos]  I  have  not  D"^  Halley's  little 
Treatise  by  me  concerning  the  Lunar  Theory. 

I  do  not  yet  understand  the  Paragraph  beginning  with 
[In  Perihelio  Terra),  propter  majorem  vim  Solis  &c.]  As 
I  apprehend  it,  the  angle  EDF  in  Your  new  Figure,  should 
be  equall  to  the  excess  of  y®  doubled  annual  argument  of  the 
Apogee  above  the  Sun's  mean  Anomaly  as  I  had  suppos'd 
it  in  my  Letter  of  May  y®  3**.  Your  Rule  concerning  that 
angle  is  this ;  [Et  capiatur  angulus  EDF  cequalis  exccssui 
argumenti  annui  supra  distantiam  Aphelii  Lunro  ab  Aplielio 
Solis.]  I  am  uncertain  how  You  understand  the  words 
[argumenti  annui] ;  they  may  signify  either  the  Annual 
argument  of  y®  moons  apogee  or  the  annual  argument  of 
the  Sun,  i.  e,  the  Sun's  mean  Anomaly.     I  am  also  uncer- 


•  Of  the  KUmenta  Logometria: , 


122  CORRESPONDENCE  OF 

tain  about  y®  words  [Aphelii  Lima3  ab  Aphclio  Solis]  I 
suppose  it  should  be  wrote  [Apogcoi  I^una)  ab  Apogsoo 
Solis].     About  the  end  of  this  Paragraph  You  say  [Et 

concipe  centrum  orbis  Lunro interca  revolvi  dum  punc- 

tum  D  revolvitur  circum  centrum  C]  I  do  not  perceive 
why  it  should  be  thus. 

The  following  Paragraph*  is  rather  more  obscure  to 
me.  I  find  I  cannot  form  any  conceptions  of  it,  unless 
You  will  be  pleased  to  give  some  further  light  to  it.  The 
^Equation  which  You  here  call  JEquatio  centri  sccunda  is  I 
perceive  the  same  with  that  which  in  D"^  Gregories  Astro- 
nomy You  call  zEquatio  loci  Lunce  sexta  I  shall  be  very 
glad  to  learn  from  You  more  distinctly  the  reasoning  by 
which  it  is  established. 

I  am  S'  Your  oblcged  Frcind 

&  most  Humble  Servant 


LETTER   LVIII. 
NEWTON  TO  COTES. 


Upon  the  receipt  of  yo"  of  Aug.  10*^  I  have  looked 
back  upon  yo*'  of  May  3'^  w^^'  I  had  forgotten.  In  the  first 
paragraph  of  y®  new  Scholium  to  Prop  xxxv,  where  I  have 
[ad  iT  50"  circitcr  asccndit  &  additur  medio  motui  Luna) 
ubi  Terra  pergit  a  Pcrihclio  suo  ad  Aphelium  et  in  oppo- 
sita  Orbis  parte  suhducitur]  the  words  additur  &  subducitur 
should  change  places,  &  after  the  word  ascendit  let  these 
words  be  added  [in  aliis  locis  a)quationi  centri  solis  propor- 
tionalis  est,] 

In  the  end  of  the  second  Paragraph  add  these  words. 
Additur   vero  ajquatio  }«'ior  &   subducitur   posterior  ubi 


•  Ikglnninej  "  Computatio  hujus  motup,"  &cc. 


NEWTON  AND  COTES.  123 

Terra  pergit  a  Perihclio  suo  ad  Aphclium,  &  contrarium  fit 
in  opposita  Orbis  parte. 

In  the  third  Paragraph  the  words  [paulo  minor  est  in 
priore  casu]  are  in  my  copy  [paulo  major  est  in  priorc 
casu]  &  should  be  so  in  yours. 

In  the  fourth  Paragraph  the  words  additur  &  subduci* 
tur  should  ehange  places. 

In  the  beginning  of  the  correction  of  the  latter  part  of 
the  Moons  Theory  you  may  write  [Halleius  centrum  Ellip- 
seos  in  Epicyclo  locavit.] 

In  the  next  Paragraph  beginning  w**'  the  words  [In 
Aphelio*  Terroe  &c]  after  the  first  sentence  of  the  Para- 
graph the  word  Aphclium  is  written  five  times  erroneously 
for  the  word  Apoga)um.    Write  therefore  [recta  DE  versus 

Apoga3um    Luna) cxcessui  Argumenti   annui  Apogroi 

Lunoo  supra  distantiam  Apoga)i  Luna)  ab  Apogfco  Solis,  vel 
forte  oequalis  cxcessui  Argumenti  annui  &  360^  supra  dis- 
tantiam Apog^ei  Lunso  ab  Apogroo  Solis Solis  ab  Apo- 

gioo  Luna) Solis  ab  Apogajo  proprio  conjunctim.     The 

Equation  described  in  this  Paragraph  I  had  first  from 
observations  of  Lunar  Eclipses,  &  afterwards  found  that 
it  answered  the  Theory  of  gravity  in  the  manner  here 
described.  Its  quantity  when  greatest  came  to  about 
2'  lo"t  by  Eclipses.  By  y®  Theory  tis  2'  25".  I  suppose  you 
understand  that  the  force  of  y*^  Sun  for  disturbing  the  Moons 
motions  is  reciprocally  as  the  cube  of  the  distance  of  the 
earth  from  y®  Sun.  The  motion  of  the  center  of  the  Moons 
Orb  in  y*'  cycle  BDAB  arises  from  the  force  of  the  Sun,  & 
as  this  force  varies,  the  motion  of  the  center  of  y^  jNIoons 


•  Apparently  a  slip  of  the  pen  for  "  Periljelio." 

t  This  is  the  value  given  in  the  Luna:  Theorut  Newtoniana,  in  Gregory's  Attronomif, 

In  Mayer  (modified  by  Lulande)  it  is  2'  9" ;  Clairaut  ijives  it  only  -2()",8 ;  Damoiscau 

-28",67;  Plana  -28",811 ;  Pont6coulant  -28" ,511 ;  Burckhardt  -27", 6.    The  terms 

which  compose  it  arc  of  the  3rd  and  higher  orders  ( Ponti'coulant  iv.  pp.  577,  602),thc 

15 
first  term  Itcing  -  -r  '"  .  <"''  =  -  53"  174.    Sec  Letter  LX  V. 


121  CORRESPONDENCE  OF 

Orb  should  vary  in  this  cycle  both  as  to  the  length  of  the 
radius  DC  &  as  to  y®  velocity  of  the  rotation  of  this 
radius  about  the  center  C,  supposing  the  suns  annual 
motion  to  be  always  equal  &  uniform,  &  that  his  distance 
from  the  earth  only  changed.  But  because  the  suns  annual 
motion  accelerates  &  retards  in  a  duplicate  proportion  of 
the  Suns  distance  reciprocally,  &  this  acceleration  & 
retardation  is  allowed  for  in  the  angle  BCD  so  as  to  make 
the  point  D  accelerate  &  retard  in  the  same  proportion 
in  y*  cycle  J3DAB,  here  is  a  variation  of  the  motion  of  the 
center  of  the  Moons  Orb  in  the  cycle  DDAB  in  a  duplicate 
proportion  of  the  suns  distance  reciprocally  &  this  without 
altering  the  length  of  the  radius  CD.  Had  this  variation 
been  in  a  triplicate  proportion  there  would  have  been  no 
need  of  any  further  ajquation,  but  because  it  is  only  in  a 
duplicate  proportion,  there  wants  a  further  allowance  in  a 
single  proportion.  And  this  allowance  must  be  made  w*^ 
respect  to  the  Sun's  motion  &  true  place.  If  the  suns  true 
motion  could  be  accelerated  &  retarded  in  this  proportion, 
I  would  accelerate  &  retard  the  motion  of  the  point  D  in 
y^  Epicy  {c}le  BDAB  in  the  same  proportion.  But  because 
this  cannot  be  done,  I  make  the  allowance  by  the  rotation 
of  the  line  DF  about  y®  center  />,  so  that  the  center  of  the 
Moons  orb  may  revolve  about  the  center  D  in  an  Epicycle 
described  by  the  point  F,  &  about  y®  center  C  in  a  curvilinear 
Orb  with  a  velocity  reciprocally  proportional  to  the  cube  of 
the  distance  of  the  earth  from  the  Sun,  or  directly  as  the 
force  of  y®  Sun  w*^''  causeth  this  velocity ;  or  that  the  velo- 
city of  the  point  F  in  the  said  curvilinear  Orb  be  to  the 
velocity  of  the  point  D  in  the  Orb  BDAB  reciprocally  as 
the  distance  of  the  earth  from  the  Sun.  And  this  "will 
come  to  pass  quam  pro{x}ime  by  determining  y®  length 
DF  &  the  angle  EDF  as  in  the  Theory. 

The  next  Paragraph  beginning  with  the  words  [Com- 
putatio  motus  hujus  diflicilis  est]  contcins  only  an  approxi- 


NEWTON  AND  COTES.  125 

mation  of  the  former  paragraph,  by  computing  the  angle 
at  y"  earth  w*^**  the  line  DF  subtends  at  the  Moon  in  her 
mean  distance  from  the  earth.  For  the  translation  of  the 
center  of  the  Moons  Orb  from  D  to  F,  creates  the  same 
translation  of  the  whole  orb  of  the  INIoon  &  of  the  Moon  in 
its  Orb  from  the  place  in  w^**  they  would  otherwise  be,  & 
so  makes  an  equation  or  angle  at  the  Earth  w*^**  the  line 
DF  subtends  at  the  Moon. 

If  the  Sun  did  not  act  upon  the  Moon  tlie  center  of 
the  Moons  orb  would  be  in  the  point  C,  By  the  action  of 
Sun  it  is  transferred  from  the  center  to  the  circumference 
of  the  Epicycle  BDAD,  If  the  earth  moved  uniformly  in 
a  concentric  circle  about  the  Sun  so  that  y*  action  of  the 
Sun  upon  the  Moons  Orb  might  be  uniform,  the  center  of 
her  Orb  would  move  uniformly  in  y*  Epicy  { c }  le  BDAB,  15y 
the  inequality  of  the  Suns  action  the  center  of  the  Moons 
orb  is  transferred  from  the  center  to  the  circumference  of 
a  secondary  epicycle  described  with  y®  radius  DC*  about 
the  point  D.  If  the  inequality  of  the  Suns  force  or  action 
on  y®  Moons  orb  arose  only  from  the  variation  of  the  dis- 
tance of  the  earth  from  y®  Sun  &  the  angular  motion  of 
the  earth  about  the  Sun  was  uniform,  the  point  D  would 
move  uniformly  in  the  epicycle  BDAB,  the  angle  BCD 
w*^^  is  double  to  the  argumentum  annuum  increasing  uni- 
formly &  the  center  of  the  Moons  orb  would  move  uniform- 
ly about  the  point  D  in  an  Epicycle  whose  radius  is  3DF, 
But  the  angular  motion  of  the  earth  about  the  Sun  not 
being  uniform,  the  angular  motion  of  the  radius  CD  about 
the  Center  C  is  not  uniform.  If  the  angular  motion  of  the 
earth  about  the  Sun  was  as  the  cube  of  the  distance  of  the 
earth  from  the  Sun  reciprocally,  that  is  as  the  force  of  the 
Sun  upon  the  Moons  Orb,  the  angular  velocity  of  the 
Radius  CD  about  the  center   C  would  be  in  the  same  pro- 


■  A  slip  of  the  pen  for  DF, 


126  CORUESPONDENCK  OF 

portion,  &  the  center  of  the  Moons  orb  being  placed  in  the 
point  D  would  have  a  velocity  in  the  Orb  BDAB  propor- 
tional to  the  force  of  the  Sun  w^''  causeth  it,  &  there  would 
be  no  need  of  a  secondary  Epicycle  about  the  center  />. 
But  because  the  angular  motion  of  the  earth  about  the 
Sun  is  but  in  a  duplicate  proportion  of  the  distance  of  the 
Sun  reciprocally,  the  motion  of  the  point  D  in  the  epicycle 
BDA  will  {be}  but  in  a  duplicate  propoi*tion  &  for  making 
up  this  proportion  a  triplicate  one,  the  center  of  the 
Moons  Orb  must  be  placed  not  in  the  point  I)  but  in  an 
Epicycle  about  the  point  Z),  &  the  radius  of  the  Epicycle 
must  be  but  a  third  part  of  such  a  Radius  as  would  make 
the  epicycle  alone  answer  to  a  triple  proportion,  so  that 
the  motion  of  the  center  of  the  Moons  orb  in  this  Epi- 
cycle &  of  the  point  D  about  the  center  C  may  together 
compound  a  motion  in  a  triplicate  proportion  of  the  distance 
of  the  earth  from  the  Sun  reciprocally. 

In  yo*"  papers*  1  met  w*^'  nothing  w^^'  appeared  to  me  to 
need  correction. 

I  am 

Yo'  most  humble  Serv* 

London  Aug.  12.  1712.  Is.  Newton. 

For  the  11"''  M'  Roger  Cotes  Pro/mov 
of  Astroiiomy  at  his  Cliamhcr  in 
Trinity  College  in  Canihrid(je, 


LETTER  LIX. 
NEWTON  TO  COTES. 
S'  London.  16 1  Aug.  1712. 

In  the  Letter  I  wrote  to  you  two  days  ago,  the  words 
[Apoga)i  Luna)]  were  interlined  after  the  words  [excessui 


"  The  Elemejita  Logometria; 
t  The  post  mark  is  Aug.  14. 


NEWTON  AND  COTES.  127 

Argumcnti  annui.]*  Its  better  to  strike  out  the  interlined 
words,  &  at  the  end  of  the  Paragraph  to  add  this  sen- 
tence. [Per  Argumentum  annuum  intelligo  excessum  qui 
relinquitur  subduccndo  medium  locum  Apogaji  Lunro  semel 
ajquatum  a  vero  loco  Solis,  vel  a  summa  veri  illius  loci  et 
360^. 

Yo""  humble  Servant 

For  the  R"'^  M'  Cotks  Professor  Is.  Nkwton 

of  Astronomy  at  his  chamber  in 
Trinity  College  in  Camhrllgc, 

The  directions  given  in  this  billet  were  supcrsedetl  l»y  the  coniinu- 
nication  of  Aug.  20. 


LETTER  LX. 
COTES  TO  NEWTON. 
S"  Cambridge  August.  17***  1712 

I  have  received  two  Letters  from  You  by  the  last  Post 
&  the  foregoing.  I  thank  You  for  the  trouble  You  have 
given  Your  self  to  make  the  thing  clearer  to  me,  but  am 
sorry  to  find  You  had  mistaken  my  difficulty.  I  was  very 
well  satisfied  as  to  the  design  of  introducing  a  secondary 
Epicycle  about  y®  point  D:  the  motion  which  You  had 
given  the  point  F  in  that  Epicycle  was  what  I  stuck  at,  & 
consequently  Your  manner  also  of  determining  the  angle 
EDF,  By  making  the  angle  BCD  equal  to  the  doubled 
annual  argument  of  y®  Moons  Apogee  the  motion  of  the 
point  D  in  the  primary  Epicycle  DDAD  was  not  yet 
enough  accelerated  in  the  Earths  Perihelium  nor  enough 
retarded  in  the  Earths  Aphelium :  the  secondary  Epicycle 
was  therefore  added  that  the  velocity  might  be  in  a  trip- 
licate instead  of  a  duplicate  proportion,  &  an  increase  of 
velocity  be  made  in  y®  Earths  Perihelium  &  a  decrease  be 


•  All  these  five  wouls  are  interlined. 


128  CORRESPONDENCE  OF 

made  in  its  Aphelium.  Hence  it  seem'd  evident  to  nie, 
that  the  motion  of  y*  point  F  in  the  secondary  Epicycle 
ought  to  be  such  that  it  might  arrive  at  y®  place  of  its 
nearest  distance  from  y®  point  C  in  y®  enrths  Perihclium  & 
there  by  its  motion  conspiring  with  y®  motion  of  the  point 
D  might  render  the  compound  of  both  the  swiftest  & 
again  that  it  might  arrive  at  y*  place  of  its  furthest  dis- 
tance from  the  point  C  in  y®  earths  Aphelium  &  there  by 
its  motion  contrary  to  y®  motion  of  y®  point  D  might 
render  the  compound  of  both  the  slowest.  Wherefore* 
if  CD  be  produced  to  G  so  that  DG  ha  equal  to  DF  & 
on  the  other  side  between  D  &  C,  DH  be  also  taken  equal 
to  DFx  tis  evident  that  in  the  Earths  Aphelium  DF  will 
coincide  with  DG  &  in  y®  Earths  Perihclium  DF  will 
coincide  with  DII  so  revolving  about  y®  centre  D  y*  the 
angle  GDF  may  always  be  equal  to  the  suns  mean 
Anomaly.  Hence  the  angle  EDF  or  EDG  -  GDF  or 
BCD  -  GDF  will  be  equal  to  the  excess  of  y®  doubled 
Annual  argument  above  y®  suns  mean  Anomaly  as  I  ob- 
serv'd  in  my  last.  This  is  the  only  way  according  to 
which  I  can  apprehend  the  motion  of  y®  point  F  in  the 
secondary  Epicycle  to  be  regulated ;  but  I  cannot  perceive 
how  it  may  be  reconcil'd  with  Your  way  of  determining 
the  angle  EDF]  or  with  the  time  You  Assign  for  its 


•  Cotes  does  not  give  any  figure :  the  annexed  is  added  for  the  convenience  of  tlie 
reader. 


t  t.  e,  by  making  it  =  annual  argument  -  dist.  of  Moon's  apogee  from  Sun's  apogee, 
=  twice  annual  argument  -  Sun*s  anomaly. 
Cotes  himself  afterward!)  (letter  of  Sept.  7)  contends  for  this  mode  of  determining  the 
l  EDFf  taking  the  l  GDF  =  Sun's  true  anomidy,  not  its  mean,  as  he  makes  it  in  this 
and  former  letters. 


NEWTON  AND  COTES.  129 

revolution  by  making  it  equal  to  y*  time  in  which  y*  point 
Z>  revolves  about  y*  centre  C, 

What  I  have  here  said  will  also  affect  the  following 
Paragraph  beginning  with  [Computatio  motus  hiyus  dif- 
ficilis  est  &c.]  But  besides  this  there  were  two  other 
difficultys  containd  in  this  Period  [Et  hncc  recta  [DF^ 
Rubtendit  angulum  ad  Terram  quern  translatio  ccntri  Orbis 
Lunoo  aloeoi^  ad  locum  Fgcncrat,  &  cujus  duplum  proj)- 
terea  dici  potest  ^l^quatio  ccntri  secunda.]  The  angle  at 
the  Earth  which  DF  subtends  is  y'  angle  DTF  compre- 
hended by  y^  lines  TDy  TF.  I  understood  You  thus,  but 
I  perceive  by  Your  Letter  that  You  do  not  mean  the 
angle  DTF,  but  an  angle  at  y°  Earth  which  is  subtended 
by  a  line  at  the  Moon  equal  &  parallel  to  DF\  so  y*  I  can 
now  understand  what  follows  [Et  ha?c  ajquatio  est  ut  sinus 
anguli  quem  recta  ilia  DF  cum  recta  a  puncto  F  ad 
Lunam  ducta  continet  quam  proximo]  which  I  could  not 
before.  However  I  am  still  at  a  loss  to  understand  why 
You  take  the  double  of  that  angle  for  the  JEquatio  centri 
secunda. 

The  following  Paragraph  describes  the  Variatio  se- 
cunda. I  suppose  it  was  derived  from  Observations.  In  it 
the  word  Aphelium  is  twice  used  instead  of  Apogaoum. 

I  am  S*"  Your  &c. 


LETTER    LXI. 
NEWTON    TO    COTES. 
S'  London  Aug  20.  1712. 

For  removing  the   difficulties  in  the   Theory  of  the 
Moon  mentioned  in  yo"  of  Aug.  17  I  have  sent  you  the 
inclosed  paper  conteining  some  alterations  in  the  descrip- 
tion of  the  latter  part  of  that  Theory.    I  had  by  mistake 
9 


130  CORRESPONDENCE  OF 

writ  [Aphelio  Solia]  &  changed  it  to  [Apogoeo  Solis*]  & 
should  have  changed  it  to  [Perigajo  Solis,]  as  I  have  done 
in  this  paper  inclosed.  By  considering  that  the  angle 
CDF  is  the  complement  of  y**  Suns  Anomaly  to  a  circle  (as  I 
have  exprest  it  in  the  paper  inclosed)  you  may  perceive  that 
whenever  the  Sun  is  in  his  Apoge  the  point  F  will  fall  be- 
tween the  points  D  &  C  8c  so  will  be  in  its  slowest  motion 
in  the  Curve  line  w*^^  it  describes  about  the  center  C  If 
the  line  DF  kept  parallel  to  it  self  the  points  F  8c  D  would 
have  equal  motions :  but  by  the  revolving  of  the  point  F 
about  the  point  D  according  to  the  order  of  the  signes  this 
motion  of  the  point  F  is  subducted  from  the  motion  of  the 
point  Z>,  &  the  difference  is  the  motion  of  the  point  F  in  the 
said  curve  line,  w^^  motion  is  therefore  the  slowest  that  it 
can  be.  And  on  the  contrary,  in  the  Sun's  Perige  the  line 
DF  will  lye  in  directum  witli  the  line  DC,  &  the  motion  of 
the  point  Ff  in  the  said  curve  line  will  be  at  the  swiftest 
being  thef  summ  of  the  two  motions.  By  the  inclosed 
paper  you  will  understand  also  why  I  took  the  double  of 
the  angle  subtended  by  a  line  at  the  Moon  equal  &  parallel 
to  DFy  £or  the  Equatio  centri  8ecu\n]da.  The  line  must  be 
doubled  at  the  superior  focus  of  the  Moon's  Orb  &  carried 
thence  to  the  Moon. 

I  am  Yo*"  most  humble  Servant 

For  the  R"**  M'  Cotes  Professor  of  Is.   Newton. 

Astronomy  at  his   Cliamher  in 
Trinity  College  in  Cambridge 

Paper  inclosed  in  the  above. 

Capiatur   angulus     BCD    a3qualis   duplo    argumento 

annuo,  seu  dupla)  distantia)  veri  loci  Solis  ab  Apogteo  Luna3 
semel    sequato,   et    erit    CTD  eequatio  secundaj   Apogiei 


In  his  letter  of  Aug.  12,  adoptingr  the  conjecture  thrown  out  by  Cotes  in  his 
letter  of  Aug.  10. 

t  The  "  F  "  and  part  of  "  the  "  are  covered  by  the  wax. 

\  lu  the  fair  copy  of  the  Scholium  which  Cotes  made  for  the  printer  (No.  173),  he 


NEWTON  AND  COTES.  131 

Luna9  et  TD  excentricitas  Orbis  ejus.  Habitis  autcm 
Lunoo  motu  medio  et  Apogroo  et  cxcentricitatc,  ut  et  Orbis 
axe  majore  partium  200000 ;  ex  his  eruetur  verus  Luna} 
locus  in  Orbe  et  distantia  ejus  a  Terra  idq:  per  methodos 
notissimas. 

In  perihelio  Terrio,  propter  majorem  vim  Solis  centrum 
Orbis  Lunjo  velocius  movetur  in  epicyclo  BDA  circum 
centrum  C  quam  in  Aphclio,  idq :  in  triplicata  rationc  dis- 
tantia? Terro9  a  Sole  inverse.  Ob  acquationem  centri  Solis 
in  argumento  annuo  compreliensam,  centrum  Orbis  Luna) 
velocius  movetur  in  Epicyclo  illo  in  duplicata  rationc  dis- 
tantiai  Tcrrrc  a  Sole  inverse.  Vt  idem  adhuc  velocius 
movcatur  in  ratione  simplici  distantiaj  inverse;  ab  Orbis 
centro  D  agatur  recta  DE  versus  Apogaeum  Lunne  sou 
recta?  TC  parallela,  et  capiatur  angulus  EDF  aqualis  ex- 
cessui  Argument!  annul  pra?dicti  supra  distantiam  Apogaei 
Luna*  a  PerigaiO  Solis  in  conscquentia ;  vel  quod  perinde 
est,  capiatur  angulus  CDF  ajqualis  complcmento  Anomaliae 
verae  Solis  ad  gradus  s6o,  Et  sit  DF  ad  DC  ut  dupla  ex- 
centricitas Orbis  magni  ad  distantiam  mediocrem  Solis  a 
Terra  et  motus  medius  diurnus  Solis  ab  Aphelio*  Lunae 
ad  motum  medium  diurnum  Solis  ab  Apogaeo  proprio  con- 
junctim,  id  est,  ut  S3|  ad  1000  et  52'.  27".  16'"  ad  58'.  8".  10'" 
conjunctim,  sive  ut  3  ad  100.  Et  concipe  centrum  Orbis 
Lunae  locari  in  puncto  is  et  in  Epicyclo  cujus  centrum  est 
D  et  radius  DF  interea  revolvi  dum  punctum  D  progredi- 
tur  in  circumferentia  circuli  DABD,  Hac  enim  ratione 
velocitas  qua  centrum  orbis  Lunse  circum  centrum  C  in 
linea  quadam  curva  movebitur,  erit  reciproce  ut  cubus  dis- 
tantiae  Solis  a  Terra  quamproxime,  ut  oportet. 

Computatio  motus  hujus  difficilis  est,  sed  facilior  red- 


has  altered  "secunda"  into  "semestris",  and  added  the  words  "  in  Apogaium  secundo 
icquatum  tendens"  after  "Orbis  ejus",  in  both  instances  returning  to  the  phraseology 
of  the  first  draught  from  wliich  Newton  had,  probably  without  intending  it,  departed. 
•  Altered  by  Cotes  to  Apoga'o. 

9—2 


132  CORRESPONDENCE  OF 

detur  per  approximationem  sequentem.  Si  distantia  mc- 
diocris  Lunoe  a  Terra  sit  partium  100000,  et  excentricitas 
TC  sit  partium  5505  ut  supra :  recta  CB  vel  CD  invenietur 
partium  ll72j,  et  recta  DF  partium  S5\,  Et  haec  recta  ad 
distantiam  TC  subtendit  angulum  ad  Terram  quem  trans- 
latio  centri  Orbis  a  loco  D  ad  locum  F  generat  in  motu 
centri  hujus ;  et  eadem  recta  duplicata  in  situ  parallelo  ad 
distantiam  superioris  umbilici  Orbis  Luna)  a  Terra,  sub- 
tendit eundem  angulum,  quem  utiq:  translatio  ilia  generat 
in  motu  umbilici,  et  ad  distantiam  Lunse  a  Terra  subtendit 
angulum  quem  eadem  translatio  generat  in  motu  Lunse, 
quiq:  propterea  roquatio  centri  secunda  dici  potest.  Et 
ha»c  aequatio  in  mediocri  Lunaj  distantia  a  Terra  est  ut 
sinus  anguli  quem  recta  ilia  DF  cum  recta  a  puncto  F  ad 
Lunam  ducta  continet  quamproxime,  et  ubi  maxima  est 
evadit  2'  25".  Angulus  autem  quem  recta  DF  et  recta  a 
puncto  F  ad  Lunam  ducta  comprehendunt,  invenitur  &c. 

In  the  next  Paragraph  but  one*  write  Ajyogoii  twice 
for  Aplielii, 


LETTER  LXII. 

COTES  TO  NEWTON. 

S*"  CamlMdgo  Aug:  28*'*  1712 

I  received  Yours  with  the  inclosed  paper,  but  cannot 

yet  agree  with  You.    In  my  former  Letters  I  had  suppos'd 

the  point  F  to  come  the  nearest  to  C  in  y*'  Suns  Perigee  & 

to'be  the  furthest  from  C  in  the  Suns  Apogee :  You  on  the 

contrary  suppose  it  to  be  y®  the  nearest  in  y®  Suns  Apogee 

&  the  furthest  in  the  Suns  Perigee.     According  to  your 

supposition  the  motion  of  y®  point  F  in  its  curvilinear  Orb 


"  The  words  "but  one"  are  added  by  mistake.    They  led  Cotes  to  suspect  that 
Newton's  coi»y  contuincd  an  ndditionul  paragraph  which  was  not  in  his. 


NEWTON  AND  COTES.  133 

will  then  be  the  swiftest  when  that  point  is  at  its  greatest 
distance  from  y*  Centre  C,  &  slowest  at  its  least  distance 
from  the  same,  for  we  agree  that  tis  the  smftest  in  the 
Suns  Perigee  &  slowest  in  his  Apogee :  whereas  according 
to  my  supposition  the  smfltest  motion  accompanys  the  least 
distance  &  y®  slowest  the  greatest,  as  I  think  it  ought  to 
do. 

By  considering  that  the  angle  CBF  is  the  complement 
of  y®  Suns  Anomaly  to  a  circle,  You  say,  I  may  perceive 
that  whenever  the  Sun  is  in  his  Apogee,  the  point  F  will 
fall  between  the  points  7)  &  C,  &  so  will  be  in  its  slowest 
motion  in  the  Curve  line  which  it  describes  about  the  cen- 
tre C.  I  do  indeed  perceive  that  y®  point  F  will  fall  be- 
tween y®  points  D  k  C,  but  I  think  it  will  then  be  in  its 
swiftest  motion  not  its  slowest.  For  since  y*  angle  CDF 
is,  by  supposition,  the  complement  of  the  suns  Anomaly  to 
a  circle;  it  follows,  that  as  that  Anomaly  is  continually 
increasing  its  complement  must  be  continually  decreasing. 
Therefore  the  line  DF  does  so  revolve  to  the  line  DC  as 
by  its  motion  to  diminish  continually  the  angle  CDFi 
Whence  it  appeares  that  in  respect  of  y*  line  DC  the  line 
DF  does  revolve  with  a  motion  contrary  to  y*  order  of  y* 
signcs  1  say  in  respect  of  y*'  moveable  line  /)C,  not  in 
respect  of  y®  Fixt  Stars  &  it  is  in  respect  of  y®  line  DC 
that  its  motion  must  be  estimated  in  order  to  compound  it 
with  the  motion  of  y®  point  D  in  the  circle  ADD,  The 
motion  then  of  y®  point  F  in  its  passage  over  y**  line  DC 
or,  by  supposition,  in  the  Suns  Apogee  does  conspire  with 
y*^  motion  of  y®  point  D  &  therefore  the  sum  of  y*  two 
motions  renders  the  motion  of  y*'  point  F  in  its  Curvilinear 
Orb  the  swiftest  in  the  Suns  Apogee,  which  ought  not 
to  be. 

I  think  I  apprehend  Your  meaning  very  well  where 
You  say,  The  line  DF  must  be  doubled  at  y*'  superior 
Focus  of  the  Moons  Orb,  &  carried  thence  to  the  Moon : 


134  CORRESPONDENCE  OF 

but  I  cannot  see  any  reason  why  y®  doubled  line  at  y*  supe- 
rior Focus  rather  than  the  single  line  at  y*  centre,  should 
be  carried  to  the  Moon,  excepting  that  Observations  may 
require  it. 

Your  &c.  It.  C. 

By  Your  Letter  I  suspect  that  in  Your  copy  there  is  a 
Paragraph  between  that  beginning  with  Computatio  mottis 
htij'us  dijficilis  &c.  &  that  beginning  with  Si  computatio  accu- 
ratior  desideretur ;  they  immediately  follow  one  the  other 
in  my  Copy. 


LETTER  LXIII. 
NEWTON  TO  COTES. 

The  reason  why  the  doubled  line  at  the  superior  focus 
rather  then  the  single  one  at  the  center  should  be  carried 
to  the  Moon  is  this.  The  angles  about  the  superior  focus 
are  (quamproxime)  proportional  to  the  times,  those  about 
y*  Center  are  not.  And  therefore  if  the  superior  focus  be 
translated,  the  line  drawn  from  it  to  y®  Moon  will  keep  its 
parallelism,  &  by  doing  so  will  make  the  same  translation 
in  the  Moon. 

As  for  your  other  difficulty,  if  the  line  DF  kept  parallel 
to  it  self,  so  as  being  produced  to  cut  the  line  TB  in  a 
given  angle  the  motion  of  the  points  D  &  F  would  be  al- 
ways equal  to  one  another.     I  do  not  speak  of  the  angular 


motion  of  the  lines  CD  and  GF  about  the  center  C  but  of 


NEWTON  AND  COTES.  1S5 

the  local  motion  of  the  points  D8cF  in  their  curvilinear  Orbs 
w***  in  this  case  '\>'ill  be  two  equal  circles.  Let  the  circle 
FMN*  be  described  w*'*  the  center  Cf  &  radius  DF  &  be  cut 
by  the  line  CD  in  the  point  H  Sc  hy  the  line  CD  produced 
in  the  point  J\L  And  if  the  line  DF  keep  parallel  to  it  self, 
the  increase  of  the  angle  MDF  >vill  be  equal  to  the  increase 
of  the  angle  BCD,  I  mcane  that  y*  two  angles  will  increase 
w*'*  equal  swiftness  or  have  equal  augmentations  in  equal 
times.  And  in  this  case  the  motions  of  the  points  D  &  F 
will  be  equal.  But  if  the  angle  MDF  increase  but  half  so 
fast  (w*^^  is  the  case  of  the  Theory),  the  motion  of  the 
point  F  will  be  accelerated  neare  M  &  retarded  nearc  JV*. 
When  the  line  DF  keeps  parallel  to  it  self  &  has  no  angu- 
lar motion,  its  motion  in  it{s}  orb  will  be  equal  to  that  of 
the  point  Z>.  But  if  it  has  an  angular  motion  according  to 
the  order  of  the  letters  FMIIF  (as  in  the  Theory)  that 
angular  motion  will  accelerate  the  point  F  neare  M  & 
retard  it  neare  N*,  You  seem  to  consider  the  angular 
revolution  of  the  line  DF  or  CF  in  respect  of  the  line  DC, 
I  consider  not  the  relative  angular  motion  of  the  line  DF 
or  CF  but  the  absolute  linear  motion  of  the  point  F  in  its 
linear  orb  described  about  the  point  C  in  the  unmoved 
plane  of  the  Moons  orb  w^'^out  any  relation  to  the  angular 
motion  of  the  line  CD. 

There  is  no  Paragraph  between  that  w*^^  begins  w^**  Conu' 
putatio  motus  hujus  difficilis  &c  &  that  w*^*^  begins  w*^  Si 
computatio  accuratior  desideretur  &c  If  the  words  of  the 
paper  inclosed  in  my  last  are  not  right,  pray  correct  them. 
After  these  two  Paragraphs  there  is  or  should  be  a  Para- 
graph concerning  the  refraction  of  the  Atmosphere  whereby 
the  Diameter  of  the  earths  shadow  is  enlarged  in  Lunar 


"  The  **  N  "  should  be  "  //  "  if  wc  follow  the  figure,  as  it  w  aUo  in  Coles'*  figure, 
(letter  LX.)    It  would  miiurally  drop  from  the  pen  after  " M** 
•f-  A  slip  of  the  pen  for  D. 


13G  CORRESPONDENCE  OF 

Eclipses.       That    Paragraph    was    (I    think)    in    the   first 

draught  I  sent  you  of  the  Moons  Theory*. 

I  am  Yo"^  most  humble  Servant 

London  Sept  2*^  1712.  Is.  Newton. 

For  the  R""*  M'  Roqeu  Cotes  Pro/mor  of 
Astrimottii/  at  his  Chamber  in  Trinity 
College  in  Camhridije 


LETTER  LXIV. 
COTES  TO  NEWTON. 


S' 

I  received  Your  last,  by  which  I  do  at  length  perceive, 
that  You  consider  the  absolute  linear  motion  of  the  point 
F  in  its  linear  Orb  described  about  the  centre  C,  &  not 
the  angular  revolution  of  the  line  CF  about  the  same 
centre,  which  I  had  before  supposed  You  to  do. 

I  am  satisfied  that  this  linear  motion  of  the  point  jf^will  be 
accelerated  near  M  &  retarded  near  N  &  therefore  if  it  be 
the  linear  motion  which  ought  to  be  considered  in  Your 
Theory  &  not  the  angular  You  do  rightly  in  making  the 
angle  CDF  equal  to  the  complement  of  the  Suns  Anomaly 
to  a  Circle,  or  which  is  the  same  thing,  in  making  the 
angle  EDF  equal  to  the  excess  of  the  Annual  Argument 
above  the  distance  of  the  Moons  Apogee  from  the  Suns 
Perigee. 

But  I  am  of  opinion  that  You  ought  rather  to  consider 
the  angular  motion  of  the  point  F  than  the  linear.  And  if 
so,  because  the  angular  revolution  of  y^  line  CF  about  the 
centre  C  in  the  unmoved  plane  of  the  Moons  Orb,  is 
accelerated  near  N  &  retarded  near  M\  the  angle  MDF 
nmst  be  taken  equal  to  the  suns  Anomaly,  or  which  is  the 
same  thing,  the  Angle  EDF  must  {be}  taken  equal  to  the 


*  It  is  parngrnph  11.    Sec  p.  112. 


NEWTON  AND  COTES.  137 

excess  of  the  Annual  Argument  above  the  distance  of  the 
Moons  Apogee  from  the  Sun's  Apogee* 

I  Avill  not  set  down  other  reasons  for  considering  the 
Angular  motion  rather  than  the  linear,  >yhieh  may  admit  of 
dispute.     What   I   ofTer  is  as   follows.     I  suppose  these 
words  at  y®  end  of  the  Paragraph  answer  to  observations 
[ — subducendam  si  summa  ilia  sit  minor  semicirculo,  ad- 
dendam  si  major.     Sic  habebitur — ^]     But  these  words  are 
not  true  by  the  Theory  if  the  angle  EDF  be  taken  equal 
to  the  excess  of  the  annual  Argument  above  the  distance 
of  the  Moons  Apogee  from  the  Suns  Perigee,  as  it  must  be 
taken  if  the  linear  motion  be  considered.     And  they  arc 
true  by  the  Theory  if  the  angle  EDF  be  taken  equal  to 
the  excess  of  y^  Annual  Argument  above  y*  distance  of  the 
Moons  Apogee  from  the  suns  Apogee,  as  it  must  be  taken 
if  the  angular  motion  be  considered.     Therefore  tlie  angu- 
lar motion  ought  to  be  considered  rather  than  the  linear, 
that  the  Theory  may  answer  to  the  Observations. 

F 


Let  DL  be  a  line  drawn  from  the  point  D  to  the 
Moon,  then  will  the  JEquatio  ceiitri  secimda  be  as  the  sine 
of  the  angle  FDL.  I  suppose  You  agree  with  me  that  the 
^Equation  must  be  substracted  whenever  the  angular  dis- 
tance of  y*  line  DL  from  the  line  DF  taken  according  to 
the  order  of  the  signs  is  less  than  a  semicircle  &  be  added 
whenever  y*  distance  is  bigger,  or  in  other  words,  that  it 


•  This  13  precisely  the  value  which  Newlon  gave  to  the  Z  KDF  by  mistuke  in  Ww 
Letter  of  Aug.  12,  (see  his  Letter  of  Aug.  26),  and  agalmt  which  Cotes  argues  iu  \m 
Letter  of  Au^.  17,  where  he  takes  MDF  =  Sun's  mean  anomaly,  not  its  true,  as  hire. 


138  CORRESPONDENCE  OF 

must  be  substracted  whenever  the  excess  of  the  Moons 
Anomaly  above  the  angle  EDF  is  less  y"  a  semicircle  &  be 
added  whenever  that  excess  is  bigger. 

If  then  the  angle  EDF  be  taken  equal  to  y®  excess  of 
the  Annual  argument  above  the  distance  of  the  Moons 
Apogee  from  the  Suns  Perigee :  the  excess  of  the  Moons 
Anomaly  above  the  angle  EDF  will  be  equal  to  the  sum  of 
distances  of  the  Moon  from  the  Sun  &  of  the  Moons  Apo- 
gee from  the  Suns  Perigee,  &  therefore  the  ^Equation 
must  be  substracted  when  this  sum  is  less  y"  a  Semicircle 
&  added  when  it  is  greater.  Now  this  sum  is  less  than  a 
Semicircle  when  the  sum  of  the  distances  of  the  Moon 
from  the  Sun  &  of  the  Moons  Apogee  from  the  Suns 
Apogee  is  greater  than  a  Semicircle,  and  on  the  contrary 
the  first  sum  is  greater  than  a  Semicircle  when  the  second 
is  less.  Therefore  the  ^Equation  must  be  substracted 
when  the  second  sum  is  greater  than  a  semicircle  &  added 
when  it  is  less.  But  this  Rule  derived  from  the  Theory  is 
contrary  to  Your  Eulc  at  the  end  of  the  Paragraph  derived 
from  Observation.  From  which  contrariety  I  think  it  is  evi- 
dent that  the  angle  EDF  ought  not  to  be  taken  equal  to  y*' 
excess  of  the  Annual  Argument  above  the  distance  of  the 
Moons  Apogee  from  the  Sun's  Perigee  &  consequently  the 
linear  motion  of  the  point  F  ought  not  to  be  considered 

but  its  angular  motion. 

I  am  Your  &c. 

Cambridge  Sep*.  T^  1712  11  C 


NEWTON  AND  COTES.  139 

LETTER  LXV. 

NEWTON  TO  COTES. 
S'  London  Sopt.  IS***  1712. 

If  it  could  be  supposed  that  the  force  of  the  sun  upon 
the  Moon  for  disturbing  her  motions  could  be  increased 
w%ut  altering  the  periodical  times  of  the  sun  &  Moon,  & 
that  the  Orb  of  the  earth  was  concentric  to  the  Sun :  the 
line  DF  would  vanish  &  the  radius  DC  would  be  increased 
in  proportion  to  the  Sun's  force  without  altering  its  angular 
motion  about  the  center  C.  By  the  increase  of  the  Suns 
force,  the  linear  motion  of  the  point  D  would  bo  increased 
by  its  moving  in  a  larger  orb,  but  its  angular  motion  about 
the  center  C  would  remain  the  same  as  before.  But  the 
earths  orb  being  excentric  &  the  excentricity  causing  a 
variation  of  the  Suns  force  upon  the  INIoon  greater  then  in 
proportion  to  the  variation  of  the  Suns  velocity,  I  compen- 
sate the  excess  or  defect  of  the  force  by  a  secondary  epi- 
cycle described  w*^'  the  radius  DF  about  the  center  D,  so 
that  the  distance  CF  may  increase  or  decrease  accordingly 
as  there  is  an  excess  or  defect  of  the  suns  force  &  by  in- 
creasing or  decreasing  cause  the  linear  motion  of  the  point 
F  in  the  plane  of  the  Moons  Orb  to  be  greater  or  less  then 
the  linear  motion  of  the  point  D  in  the  circle  DDA  in  pro- 
portion to  the  said  excess  or  defect  of  the  suns  force. 

I  thank  you  for  putting  me  upon  examining  the  words 
[ — suhducendam  si  summa  ilia  ait  minor  semicirculo,  addenda 
si  major.  Sic  hahehitur  ^-c]  I  have  compared  them  with 
my  calculations  of  the  Moons  place  in  Eclipses  &  fmd  that 
they  must  be  corrected  &  put  [ — addendam  si  summa  ilia 
sit  minor  semicirculoy  suhducendam  si  major.  Sic  hahehitur 
Sfcl  The  Equation*  I  gathered  from  Observations  many 
years  ago  &  put  it  when  greatest,  to  be  2'  lo".     The  last 


Compare  p.  123. 


140  CORRESPONDENCE  OF 

year  I  gathered  its  quan  {ti}  ty  from  observations  to  be  2'  25" 
wli.en  greatest,  but  in  describing  it,  committed  the  mistake 
w^**  I  have  now  corrected  by  reviewing  my  old  calculations. 

I  am  S' 

Yo""  most  humble  Servant 

F(yr  tlie  R"**  M'  Roger   Cotes  Professor  Is.   Newton. 

o/  Astronomy  J  at  his  Chamber  in  Trinity 
College  in  Cambridge 


LETTER  LXVI. 
COTES  TO  NEWTON. 


I  have  received  Your  last  Letter.  &  am  now  sufficiently 
satisfied  as  to  the  JEquatio  ceiitri  secunda,  I  hope  the  de- 
scription of  the  Variatio  secunda  is  accurate.  The  Para- 
graph concerning  the  refraction  of  the  Atmosphere  in 
Eclipses  was  in  Your  first  draught,  but  was  left  out  in 
Your  Alteration*  of  it.  There  was  also  another  Para- 
graph before  it  describing  the  dimensions  of  the  Sun's  & 
Moon's  Diameters  &  Parallaxes  which  was  also  omitted  in 
Your  Paper  of  Alterations.  I  am  uncertain  whether  You 
would  have  both  of  them  inserted  or  that  only  concerning 
the  Eifect  of  y®  Atmosphere.     They  stood  thusf. 

Diximus  Orbem  Luna?  a  viribus  Solis  &c. 

Et  cum  Atmosphtera  Terrse  &c. 

I  suppose  You  would  omit  the  first  of  these  Paragraphs 
since  the  substance  of  it  is  in  other  parts  of  Your  Book, 
excepting  that  You  have  6oJ  semidiameters  in  Corol.  7. 
Prop,  xxxvii.  Lib.  iii  instead  of  Go|-.  Be  pleased  to  send 
what  You  would  have  inserted. 


•  Soe  the  introduction  to  Letter  LVII.  p.  120. 

t  These  form  paragraphs  10  and  II  in  the  first  draught  of  the  Lunar  Theory,  and 
will  \>e  found  in  tlie  account  which  we  have  given  of  it,  (p.  112.)  This  being  only  the 
draujfht  of  his  letter,  Cotes  has  not  copied  them  at  full  length. 


NEWTON  AND  COTES.  141 

In  the  last  Paragraph  I  suppose  You  have  designedly 
altered  Your  first  draught  by  putting  yf  20®.  43'.  40"  for 
V^*  20".  43'.  50",  and  ;sy  150.  20'.  00"  for  :c:  15^l9^50'^  and 
X  8".  20'.  00"  for   K  8®.  18'.  20''. 

Sept.  15. 1712  Your  &c.  II  C. 


LETl'ER  LXVir. 
NEWTON  TO  COTES. 


S' 
I  beleive  it  will  be  suflleient  to  insert  only  the  last  of 
the  two  Paragraphs  w*^^  you  have  copied  in  your  last,  viz* 
that  w*^^'  concerns  the  refraction  of  the  Atmosphere.  The 
alterations  made  in  the  last  Paragraph  of  the  Scholium 
were  advisedly.  The  description  of  the  Variatio  secunda 
is  derived  only  from  phfcnomena  &  wants  to  be  made 
more  accurate  by  them  that  have  Icasure  &  plenty  of 
exact  observations.  The  public  must  take  it  as  it  is.  It 
brings  the  Moon  nearer  to  the  Sun  in  both  the  Quadra- 
tures. 

I  am  Yo'  most  humble  Servant 

London.  Sept.  23  1712.  Is.  Newton. 

For  tJte  R"^  M'  Cotes  Professor  0/ Astro- 
nomy in  the  University  of  Cambridge  At 
his  chamber  in  Trinity  College. 


LETTER  LXVIII. 
NEWTON  TO  COTES. 


S' 
I   send   you  the   conclusion*   of  the  Theory  of  the 
Comets  to  be  added  at  y®  end  of  the  book  after  the  words 


•  No3.  252 — ^255,  beg-inning  "  Cieterum  Cometarum  revolventium,  &c."  and  ending 
"primus  omnium  quod  sciam  deprehendit,"  (pp.  476 — 461  of  2nd  Ed.) 


142  CORRESPONDENCE  OF 

[Dato  autem  Latere  transverso  datur  etiam  tempus  periocli- 
cum  Cometa)  Q.  J^.  /.] 

There  is  an  error*  in  the  tenth  Proposition  of  the 
second  Book,  Prob  III,  w*^^  will  require  the  reprinting  of 
about  a  sheet  &  an  half.  I  was  told  of  it  since  I  wrote  to 
you,  &  am  correcting  it.  I  will  pay  the  charge  of  reprint- 
i  ig  it,  &  send  it  to  you  as  soon  as  I  can  make  it  ready. 
With  my  service  to  D*"  Bentley 

I  remain  Yo'  most  humble  Servant 
London  14  Octob.  1712.  Is.  Newton. 

For  the  R"**  M'  Roger  Cotes  Professor  of 
Astronomy  at  his  Chamber  in  Trinity 
College  in  Cambridge 


*  Tliis  error  in  finding  the  value  of  the  resistance  to  the  motion  of  a  projectile  in  the  air 
(see  Letter  LXXIV.)  was  pointed  out  to  Newton  hy  Nicolas  Bernoulli  (John's  nephew), 
who  was  on  a  visit  to  England  dtiring  the  months  of  September  und  October,  1712. 
*'Monente  tandem  D.  Nic.  Bernoulli  quod  error  aliquis  admissus  fuissct  in  I'rop.  x. 
Lib.  It.  constructioncm  propositionis  correxi  ct  correctam  ei  ostendi,  et  imprimi  curavi 
non  subdole  sed  eo  cognoscente."     Letter  of  Newton  in  Macclesfield  Corr.  ii.  437, 
Newton's  result,  when  the  curve  described  is  a  circle,  had  been  previously  shewn  to  be 
erroneous  by  John  Bernoulli,  in  a  Letter  to  Leibniz,  in  August,  1710,  (see  their  Cor- 
respondence, II.  231),  and  in  a  communication  made  to  the  French  Academy,  in  Jan. 
1711,  (see  Memoires  for  1711,  pp.50 — 56,  not  published  until  1714,)  in  an  appendix  to 
which  his  nephew  corrects  two  others  of  Newton's  examples,  und  professes  to  explain 
the  origin  of  the  mistake  (en  examinant  avec  soin  sa  solution  gcaerale,  j'  en  ay  trouv6 
Torigine).    John  afterwards  resumed  the  inviting  subject  in  the  Leipsic  Acts  for  Feb.  and 
»     March,  1713,  (see  Letters  LXXXIL,  LXXXVIL)    It  is  remarkable  that  both  of  these 
I     mathematicians  mistook  the  source  of  the  error.     They  imagined  that  Newton  had 
/     taken  the  coefficients  of  the  successive  powers  of  h  in  the  expansion  of  (j  +  /i)"  for  the 
successive  fluxions  of  a".  This  was  one  of  the  points  upon  which  Keill  was  subsequently 
engaged  in  controversy  with  John  Bernoulli  or  his  partisans,  who  worked  their  crotchet 
with  wearisome  pertinacity  in  the  Leipsic  Acts.     Keill  informs  us  that  Newton  told 
Nicolas  that  the  mistake  did  not  arise  from  the  use  of  series.  Newton,  through  Nicolas, 
thanked  the  sturdy  professor  of  Buide  for  the  timely  notification  of  the  error,  sent  him  a 
;  copy  of  his  Analysis,  &c.,  pubiishcd  hy  Jones  in  1711,  and  nine  days  after  the  date  of 
I  this  letter,  proposed  him  as  a  member  of  the  Royal  Society,  into  wliich  he  was  accord- 
ingly elected  on  the  1st  of  December  following. 


NEWTON  AND  COTES.  U3 

LETTER  LXIX. 
NEWTON  TO  COTES. 

I  sent  you  last  tucsday  a  sheet  inclosed  in  a  Letter. 
It  concerned  the*  The  Theory  of  Comets  to  be  added  to 
y®  end  of  the  book.  I  should  be  glad  to  hear  that  it  came 
to  your  hands.  I  mentioned  also  an  error  that  I  was 
lately  told  of  &  yv^^  wants  to  be  set  right.  I  have  heard 
nothing  from  you  this  month  or  above  &  should  be  glad  of 
a  line  to  know  in  what  forwardness  the  Press  is. 

I  am  Yo"^  most  humble  Servant 
London.  Octob.  21.  1712.  Is.  Newton 

For  the  R*"*  M'  Roger  Cotes  Professor  of 
Astronomy  at  his  Chamber  in  Trinity 
College  in  Cambridge 


LETTER    LXX. 


COTES   TO  NEWTON. 
S'  October.  23.  1712. 

I  received  both  Your  last  Letters,  together  with  the 
Sheet  to  be  added  at  the  end  of  the  Book,  which  was 
inclosed  in  the  former.  You  mention'd  an  Error  in  the 
x*^  Proposition  of  the  ii^  Book,  which  will  require  the 
reprinting  of  about  a  Sheet  &  an  half.  I  have  not  re- 
vised that  Proposition  to  see  if  I  might  find  it  out,  but 
shall  stay  for  Your  corrections.  The  sheet  which  is  now 
under  the  Press,  ends  in  Page  492  of  y®  old  Edition,  and 
Page  456  of  the  new  Edition.  I  have  not  observed  any- 
thing of  moment  which  may  be  altered  in  the  Theory  of 
Comets.    In  the  new  fourth  Corollary  f  of  Prop,  xl  I  have 


t  No.  245. 


144  CORRESPONDENCE  OF 

inserted  after  the  first  line  [&  quadratum  radii  illius  ponatur 
esse  partium  lOOOOOOOO].  Pag.  490,  lin.  5,  I  have  put  [in 
subduplicata  ratione  SQ  ad  St]  instead  of  [in  subduplicata 
ratione  St  ad  SQ]  In  the  last  Page  of  the  Book,  lines  8  & 
9,  I  design  to  i^ut  2G  -  2C  &  2T -  2S  for  G -  C  &  T^S, 
unless  You  forbid  it.  I  suppose  the  Astronomical  compu- 
tations relating  to  the  Comets  are  exact,  having  been  exa- 
mined both  by  Your  self  &  by  D""  Halley. 

I  should  have  given  You  notice  sooner,  that  I  had-  re- 
ceived Your  additional  Sheet  at  the  end  of  the  Book, 
but  that  I  expected  D"*  Bentley  would  have  seen  You  before 
this  time,  for  he  once  intended  to  have  been  at  London  a 
week  sooner.  I  am  S^ 


LETTER  LXXI. 
COTES   TO   NEWTON. 


I  here  send  You  the  Sheets  as  far  as  they  are  Printed 
off,  that  Your  self  or  some  freind  may  revise  them,  in  order 
to  see  what  Errata  may  be  put  in  a  Table.  I  know  not 
whether  You  have  got  the  Copper-plate  of  the  Comet  yet 
done.  The  Printer  tells  me  there  will  be  750  requisite* 
The  next  week  I  shall  be  in  the  Countrey,  when  I  return 
I  suppose  You  will  have  the  corrections  ready  which  You 
mention'd  for  the  Sheet  to  be  reprinted 

I  am  Sir 
Your  most  Humble  Serv* 
Nov.  1"*.  1712  Roger  Cotes 

For  S'  Isaac  Newton  at  His 
House  in  S*  Martin' s-street 
Leicester  feilds  London 


NEWTON  AND  COTES.  145 

LETTER    LXXII. 
COTES  TO  NEWTON. 

I  hope  You  have  received  the  Sheets  which  I  sent  last, 
ending  in  Page  456  of  the  New  Edition,  We  have  since 
printed  off  3  Sheets  more,  which  take  in  the  whole  Book 
with  the  Additional  Sheet,  excepting  about  20  lines.  To 
fill  up  the  following  Sheet  may  be  added  a  Table  of  the 
Contents  of  each  Section,  if  You  think  fit.  D""  Bentley  was 
proposing  to  have  subjoyned  an  Index  to  the  whole,  but 
particularly  to  the  Third  Book.  If  You  approve  of  it, 
such  an  Index  may  soon  be  made.  If  Your  alterations  in 
the  Second  Book  arc  finished  I  desire  You  will  be  pleased 
to  send  'em. 

I  am  Sir,  Your  most 

Humble  Servant 
Cambridge  Nov^^  23**.  1712  Roger  Cotes 

For  S'  Isaac  Newton  at  his 
House  in  S*  Martin's  Street 
Leicester- Feilds  London 


LETTER  LXXIIL 

NEWTON   TO   COTES. 

S' 

I  send  you  enclosed*  the  tenth  Proposition  of  the 
Second  book  corrected.  It  will  require  the  reprinting  of 
a  sheet  &  a  quarter  from  pag  230  to  pag.  240.  There  is 
wooden  cut  belonging  to  it  w*^**  I  intend  to  send  you  by  the 
next  Carrier.  I  think  this  Proposition  as  it  is  now  done 
will  take  up  much  the  same  space  as  before.     If  not,  the 


•  No«.  262-265. 

10 


146  CORRESPONDENCE  OF 

space  about  the  cuts  may  be  made  a  little  wider  or  a 
little  narrower,  or  the  number  of  lines  in  a  page  may  be 
increased  or  diminished  by  a  line.  When  this  sheet  &  a 
quarter  is  printed  oft'  I  liope  your  trouble  of  correcting 
will  be  at  an  end.  As  for  making  a  Table  to  the  book 
I  leave  it  to  you  to  do  what  you  think.  I  beleive  a  short 
one  will  be  sufficient.  I  shall  send  you  in  a  few  days  a 
SchoIiu{m}*  of  about  a  quarter  of  a  Sheet  to  be  added  to 
the  {endj"*^  of  the  book:  &  some  are  perswading  me  to 
add  an  Appendix  concerning  the  attraction  of  the  small 
particles  of  bodies.  It  will  take  up  about  three  quarters 
of  a  Sheet,  but  I  am  not  yet  resolved  about  it.     I  am 

Yo"*  humble  &  obedient 

Servant 

London.  Jan.  6.  171  §.  Is.  Newton 

For  the  R"**  M'  Cotes  Professor  of 
Astronomy  at  his  Chamber  in 
Trinity  College  in  Cambridge. 


LETTER  LXXIV. 

COTES  TO  NEWTON. 

S'.  Cambridge  Jan.  IS***  1713. 

I  have  considered  Your  alteration  of  Prop,  x,  Lib.  ii. 

and  am  well  satisfied  with  it.     I  observe  that  You  have 

increased    the    Resistance   in   the   proportion   of  3  to   2, 

which  is  the  only  change  in  Your  Conclusions,  arising  from 

hence  (as  I  apprehend  it)  that  in  the  new  Figure  LH  is 

to  NI  as  Roo  to  lloo  ^  sSo^,  whereas  in  y®  former  Figure 

kl  was  to  FG  as  Roo  to  Roo  +  2So^,     Some  things   in 

Your  Paper   I  have  altered,   they  are   not    worth  Your 

•  These  four  letters  within  {  }  have  disappeared  with  the  wax. 


NEWTON  AND  COTES.  147 

notice,  being  only  faults  in  transcribing*.    I  have  this  day 
received  the  "Wooden  Cut.   I  shall  expect  the  Scholium  at 
y®  end  of  the  Book  &  the  Appendix  at  Your  leasure. 
I  am  Sir 

Your  Obliged  Freind 
&  Humble  Servant 

Roger  Cotes. 
For  S'.  Isaac  Newton  at  his 
House  in  S*  Martin*s  Street 
Leicester.  Feilds  London 


LETTER   LXXV. 
NEWTON  TO  COTES. 

The  inclosed!  is  the  Scholium  w^^  I  promised  to  send 
you,  to  be  added  to  the  end  of  the  book.  I  intended  to 
have  said  much  more  about  the  attraction  of  the  small 
particles  of  bodies,  but  upon  second  thoughts  I  have  chose 
rather  to  add  but  one  short  Paragraph  about  that  part  of 
Philosophy.  This  Scholium  finishes  the  book.  The  cut 
for  the  Comet  of  1G80  is  going  to  be  rolled  off.  I  am 
Yo'^  most  humble  &  obedient  Servant 

London  2**  March  t  171f.  Isaac  Newton. 

For  the  Roy"**  M'  Roger  Cotes  Professor  of 
Astronomy,  at  his  Chamber  in  Trinity  College 
in  Cambridge. 


•  Cotes,  however,  besides  making  the  alterations  alluded  to  here,  has  (perhaps  from 
want  of  room)  omitted  a  paragraph  at  the  beginning  of  the  Scholium  of  the  Prop, 
(p.  269,  Ed.  1,  p.  240,  Ed.  2.)  in  which  Newton  points  out  another  mode  of  viewing 
the  problem  which  is  the  subject  of  the  Proposition.    The  parai^raph  runs  as  follows  : 

"  Fingere  liceret  projectilia  pergere  in  arcuum  GH,  HI,  IK  chordis  &  in  Holis  punctis 
Gf  H,  J,  K  per  vim  gravitatis  £c  vim  resistcntite  ngitari,  perinde  ut  in  Propositione 
prima  Libri  primi  corpus  per  vim  ccntripetam  intermittentem  agitabatur,  deinde  chordas 
in  infinitum  diminui  ut  vires  reddantur  continue.  Et  solutio  Problematis  hac  ratione 
facillima  cvadcrct." 

t  Nos.  269, 270,  272. 

%  The  Post  mark  is  March  3,  (1  uesday.) 

10—2 


148  CORRESPONDENCE  OP 

LETTER  LXXVI. 

NEWTON  AND   BENTLEY   TO  COTES. 
S' 

I  sent  you  by  last  tuesdays  Post  the  last  sheet  of  y* 

Prineipia,  &  told  you  that  the  cut  for  y®  Comet  of  1G80 

was  going  to  be  rolled  off.     But  we  want  the  page  where 

it  is  to  be  inserted  in  the  book.     I  think  y®  page  is  462  or 

463.     Pray  send  me  w*^^  it  is,  that  it  may  be  graved  upon 

the  Plate  for  directing  the  Bookbinder  where  to  insert  it. 

I  am  Yo'^  most  humble  Servant 

London  5  March  171|.  Is.  Newton 

I  have  S""  Isaac's  Leave  to  remind  you  of  what  You 
and  I  were  talking  of,  An  alphabetical  Index,  &  a  Preface 
in  your  own  Name ;  If  you  please  to  draw  them  up  ready 
for  y®  press,  to  be  printed  after  my  Return  to  Cambridg, 

You  will  oblige 

Yours 

For  the  R"^  M'  Roger  Cotes  Pro/mor  of  R  Bentley. 

Astronomy^  at  his  Chamber  in  Trinity 
College  in  Cambridge 


LETTER  LXXVIL 
COTES  TO  NEWTON. 


S'. 

I  received  both  Your  Letters  with  the  last  sheet  of  the 
Book  inclosed  in  the  former  of  them.  The  Paragraph 
beginning  with  Cceterum  Trajectoriam  quam  Cometa  descrip- 
sit  &c.,  which  is  in  the  497"'  page  of  the  former  Edition, 
falls  in  the  465*^  page  of  the  new  Edition.  This  is  the  place 
to  which  I  suppose  You  would  refer  the  Cut  for  the  Comet. 

I  intend  in  a  day  or  two  to  set  about  the  Alphabetical 
Index.     I  will  Avrite  to  D""  Bentley  concerning  the  Preface 
by  y*  next  Post. 
March.  8.  17^  I  am  S^  Your  &c. 


NEWTON  AND  COTES.  149 

LETTER  LXXVIII. 
COTES  TO  BENTLEY. 

To  D""  Bentley  March.  lO***.  171? 

S^ 
I  received  what  You  wrote  to  me  in  S'  Isaac's  Letter. 
I  will  set  about  the  Index  in  a  day  or  two.  As  to  the 
Preface  I  should  be  glad  to  know  from  S'  Isaac  with  what 
view  he  thinks  proper  to  have  it  written.  You  know  the 
book  has  been  received  abroad  with  some  disadvantage,  & 
the  cause  of  it  may  easily  be  gues3*d  at.  The  Commer- 
cium  Epistolicum  lately  published  by  order  of  the  R.  So- 
ciety gives  such  indubitable  proof  of  IVIr  Leibnitz's  want  of 
candour  that  I  shall  not  scruple  in  the  least  to  speak  out 
the  full  truth  of  the  matter  if  it  be  thought  convenient 
There  are  some  peices  of  his  looking  this  way  which 
deserve  a  censure,  as  his  Tentamen  de  Motuum  Ccelestium 
causis*.  If  S**  Isaac  is  willing  that  something  of  this  nature 
may- be  done,  I  should  be  glad  if,  whilst  I  am  making  the 
Index,  he  would  be  pleas'd  to  consider  of  it  &  put  down  a 
few  notes  of  what  he  thinks  most  material  to  be  insisted 
onT^  This  I  say  upon  supposition  that  I  write  the  Preface 
my  self.  But  I  think  it  will  be  much  more  adviseable  that 
You  or  He  or  both  of  You  should  Avrite  it  whilst  You 
are  in  To>vn.  You  may  depend  upon  it  that  I  will  own  it  & 
defend  it  as  well  as  I  can  if  hereafter  there  be  occasion. 

I  am  S'  &c. 


*  Newtoo  had  himself  drawn  up  some  strictures  upon  this  piece,  which  were  made 
use  of  by  the  editors  of  the  Commercium  Epistolicum  (p.  97).  See  the  paper  entitled 
"  Ex  Epistola  cujusdam  ad  Amicum,*'  printed  in  the  Appendix  to  tliis  work. 


J  50  CORRESPONDENCE  OF 


LETTER    LXXIX. 

BENTLEY  TO  COTES. 

Dear  Sir,  At  S'  Isaac  Newton's  March  12. 

I  communicated  your  Letter  to  S'.  Isaac,  who  happend  to 
make  me  a  visit  this  morning,  &  we  appointed  to  meet  this 
Evening  at  his  House,  &  there  to  write  you  an  Answer. 
For  y®  Close  of  your  Letter,  w*^^  proposes  a  Preface  to  be 
drawn  up  here,  and  to  be  fatherd  by  you,  we  will  impute 
it  to  your  Modesty ;  but  You  must  not  press  it  further, 
but  go  about  it  your  self.  For  y®  subject  of  y®  Preface, 
you  know  it  must  be  to  give  an  account,  first  of  y®  work 
it  self,  2^*'^  of  y®  improvements  of  y®  New  Edition ;  &  then 
you  have  S^  Isaac's  consent  to  add  what  you  think  proper 
about  y°  controversy  of  y®  first  Invention.  You  your  self 
are  full  Master  of  it,  &  want  no  hints  to  be  given  you : 
However  when  it  is  drawn  up,  You  shall  have  His  &  my 
Judgment,  to  suggest  any  thing  y*.  may  improve  it.  Tis 
both  our  opinions,  to  spare  y®  Name  of  M.  Leibnitz,  and 
abstain  from  all  words  or  Epithets  of  reproch ;  for  else,  y* 
will  be  y®  reply,  (not  that  its  untrue)  but  y*  its  rude  & 
uncivil.    S^  Isaac  presents  his  service  to  you. 

I  am  Yours 

ForM*.  Roger  Cotes  Professor  of  It.  Bentley* 

Astrcyncymy  at  Trinity  College  in 
Cambridg. 


•  The  original  of  this  Letter,  which  has  been  already  printed  in  the  Bentley  Corre- 
spondence (p.  460),  is  in  the  possession  of  Dawson  Turner,  Esq.,  who  has  kindly  fur- 
nished me  with  a  new  transcript  of  it. 


NEWTON  AND  COTES.  151 

LETTER  LXXX. 

COTES  TO  NEWTON. 
S' 

I  have  received  D"^  Bentlys  Letter  in  ansiyer  to  that 
which  I  wrote  to  him  concerning  the  Preface.  I  am  very 
well  satisfied  with  the  directions  there  given,  &  have 
accordingly  been  considering  of  the  Matter.  I  think  it 
will  be  proper  besides  the  account  of  the  Book  &  its  im- 
provements, to  add  something  more  particularly  concerning 
the  manner  of  Philosophizing  made  use  of  &  wherein  it 
differs  from  that  of  Descartes  and  Others,  I  mean  in  first 
demonstrating  the  Principle  it  employs.  This  I  would  not 
only  assert  but  make  evident  by  a  short  deduction  of  the 
Principle  of  Gravity  from  the  Phaenomena  of  Nature  in  a 
popular  way  that  it  may  be  understood  by  ordinary  readers 
&  may  serve  at  y®  same  time  as  a  specimen  to  them  of 
the  Method  of  y*  whole  Book.  That  You  {may}  y*  better 
understand  what  I  aim  at  I  think  to  proceed  in  some 
such  manner.  [Tis  one  of  y*  primary  Laws  of  Nature, 
that  all  bodys  persevere  in  their  state  &c.  Hence  it  follows 
that  Bodys  which  are  moved  in  curve-lines  &  continually 
hindred  from  going  on  along  the  tangents  to  those  curve- 
lines  must  incessantly  be  acted  upon  by  some  force  suffi- 
cient for  that  purpose.  The  Planets  (tis  matter  of  fact) 
revolve  in  Curve-lines,  therefore.  &c.  [Again,  tis  Mathe- 
matically demonstrated  that  Corpus  omne,  quod  movetur  ^c. 
Prop,  2  Lib  1,  &  corpus  omne,  quod  radio  ^c,  prop,  3  Lib  1, 
Now  tis  confess'd  by  all  Astronomers  that  the  Primary 
Planets  about  y®  Sun  &  the  Secondary  about  their  re- 
spective primary  doe  describe  areas  proportional  to  the 
times.  Therefore  y*^  force  by  which  they  are  continually 
diverted  from  the  tangents  of  their  Orbits  is  directed  & 
tends  towards  their  central  Bodies;  which  force  (from  what 
cause  soever  it  proceeds)  may  therefore  not  improperly  be 


152  CORRESPONDENCE  OP 

call{ed}  Centripetal  in  respect  of  the  revolving  Bodies  & 
Attractive  in  respect  of  y*  central  ones.  [Furthermore  tis 
Mathematically  demonstrated  that.  Cor.  6,  Prop.  4.  Lib.  1  & 
Cor.  1,  Prop.  45,  Lib.  1.  But  tis  agreed  upon  by  Astro- 
nomers that  &c.  or  &c.  Therefore  the  centripetal  forces 
of  the  Primary  Planets  revolving  about  the  Sun  &  of  the 
Secondary  Planets  revolving  about  their  Primary  ones,  are 
in  a  duplicate  proportion  &c.  In  this  manner  I  would  pro- 
ceed to  the  4*^  Prop  of  Lib.  iii  &  then  to  the  5*^.  But 
in  the  first  corollary  of  this  5*^  Proposition  I  meet  with 
a  difficulty*,  it  lyes  in  these  words  [Et  cum  attractio 
omnis  mutua  sit]  I  am  persuaded  they  are  then  true  when 
the  Attraction  may  properly  be  so  called,  otherwise  they 
may  be  false.  You  will  understand  my  meaning  by  an 
Example.  Suppose  two  Globes  A  &  B  placed  at  a  distance 
from  each  other  upon  a  Table,  &  that  whilst  y®  Globe  A 
remaines  at  rest  the  Globe  B  is  moved  towards  it  by  an  in- 
visible Hand;  a  by-stander  who  observes  this  motion  but 
not  the  cause  of  it,  will  say  that  y®  Globe  B  does  certainly 
tend  to  the  centre  of  y®  Globe  A^  &  thereupon  he  may  call 
the  force  of  the  invisible  hand  the  centripetal  force  of 
B  &  the  Attraction  of  A  since  the  effect  appeares  the  same 
as  if  it  did  truly  proceed  from  a  proper  &  real  Attraction 
of  A,  But  then  I  think  he  cannot  by  virtue  of  this  Axiom 
£Attractio  omnis  mutua  est]  conclude  contrary  to  his  sense 
&  Observation  that  the  Globe  A  does  also  move  towards 
the  Globe  B  &  will  meet  it  at  the  common  centre  of  Gravity 
of  both  bodies.     This  is  what  stops  me  in  the  train  of 


*  Tlie  difRculty  raised  by  Cotes  here  affords  an  instance  of  the  temporary  haze 
which  may  occasionally  obscure  tlie  brightest  intellects.  Compare  the  story  told  of 
Lagrange  by  Biot  (Journal  des  Savantty  1837,  p.  84):  "Lagrange  tira  un  jour  de  sa 
poche  un  papier  qu'il  lut  a  I'Acad^mie,  et  qui  contenait  une  d6monstration  du  fameux 
Pflstulatum  d*Buclide,  relatif  a  la  th^orie  des  parallcles.  Cette  demonstration  reposait 
8ur  un  paralogisme  Evident,  qui  parut  tel  u  tout  le  monde ;  et  probablement  Lagrange 
aussi  le  reconnut  pour  tel  pendant  sa  lecture.  Car,  lorsqu^il  eut  fini,  il  remit  son  pa- 
pier dans  sa  poche,  ct  n'en  parla  plus.  Un  instant  de  silence  univereel  suivit,  et  Ton 
passa  aus8it6t  a  d'autrcs  objcts." 


NEWTON  AND  COTES.  153 

reasoning  by  which  I  would  make  out  as  I  said  in  a  popular 
way  Your  7***  Proposition  of  y*  in**  Book.  I  tihall  be  glad 
to  have  Your  resolution  of  the  difficulty,  for  such  I  take  it 
to  be.  If  it  appeares  so  to  You  also,  I  think  it  should  be 
obviated  in  the  last  Sheet  of  Your  Book  which  is  not  yet 
printed  off  or  by  an  Addendum  to  be  printed  with  y*  Errata 
Table.  For  till  this  objection  be  cleared  I  would  not  un- 
dertake to  answer  any  one  who  should  assert  that  You  do 
Hypothesim  fingerCt  I  think  You  seem  tacitly  to  make  this 
supposition  that  y®  Attractive  force  resides  in  the  Central 
Body 

After  this  Specimen  I  think  it  will  be  proper  {to}  add 
somethings  by  which  your  Book  may  be  cleared  from  some 
prejudices  which  have  been  industriously  laid  against  it. 
As  that  it  deserts  Mechanical  causes,  is  built  upon  IMiracles, 
&  recurrs  to  Occult  qualitys.  That  You  may  not  think  it 
unnecessary  to  answer  such  Objections  You  may  be  pleased 
to  consult  a  Weekly  Paper  called  Memoires  of  Literature 
&  sold  by  Ann  Baldwin  in  Warwick -Lane.  In  the  18*** 
Number  of  y®  second  Volume  of  those  Papers  which  was 
published  May  5*^  1712*  You  will  find  a  very  extraor- 
dinary Letter  of  Mr  Leibnitz  to  Mr  Hartsoekcr  which  will 
confirm  what  I  have  said.  I  do  not  propose  to  mention 
Mr  Leibnitz's  name,  twere  better  to  neglect  him,  but  the 
Objections  I  think  may  very  well  be  answered  &  even 
retorted  upon  the  maintainers  of  Vortices.  After  I  have 
spoke  of  Your  Book  it  will  come  in  my  way  to  mention 
the  Improvements  of  Geometry  upon  which  Your  Book  is 
built,  &  there  I  must  mention  the  time  when  those  im- 
provements were  first  made  &  by  whom  they  were  made. 
I  intend  to  say  nothing  of  ]Mr  Leibnitz,  but  desire  You 
will  give  me  leave  to  appeal  to  the  Commercium  Epis- 


•  p.  137.    Leibniz.  0pp.  Torn.  ii.  Pare  ii.  p.  60.     The  letter  is  dated,  Ilaoover, 
Feb.  10, 171 1.    I^eibniz  does  not  mention  Newton's  name. 


154  CORRESPONDENCE  OP 

tolicutn  to  vouch  what  I.shall  say  of  Your  self  &  to  insert 
into  my  Preface  the  very  words  of  the  Judgment  of  the 
Society  (page  120^^  Com.  Ep)  that  foreigners  may  more 
generally  be  acquainted^ with  the  true  state  of  the  Case. 
Feb.  ''^  18.  171| 

The  plan  of  the  Preface  sketched  in  the  above  letter  was  afterwards 
modified.  The  Indices  compiled  by  Cotes  supplied  the  place  of  "an  account 
of  the  book",  and  the  short  preface  wliicli  Newton  nent  him  in  liis  letter 
of  March  31  made  it  unnecessary  to  enter  into  a  detail  of  "  its  improve- 
ments." The  intended  notice  of  the  method  of  fluxions  and  of  the 
dispute  relative  to  its  discovery  was  abandoned,  whether  in  consequence 
of  Newton's  declaration  at  the  close  of  the  letter  just  quoted  that  he 
"  must  not  see  it/'  or  from  a  feeling  that  it  was  better  to  leave  the 
evidence  in  the  Commercium  Epistolicum  to  work  its  own  way,  wo 
have  no  precise  information.  Coten's  Preface  therefore  is  confined  to  an 
ex])osition  of  **  the  manner  of  philosoj)liizing  made  use  of"  in  the  work, 
and  to  an  examination  of  the  objections  of  Leibniz  (without  mentioning 
his  name)  and  of  the  system  of  Vortices. 

Leibniz  in  a  letter  (Apr.  9,  I^Hj.  N.S.)  written  under  excitement, 
(it  is  his  reply  to  Newton's  raking  fire  of  Feb.  26.)  calls  this  Preface 
*'plcine  d'aigreur,"  an  expression  which  may  be  taken  as  a  measure 
of  that  extraordinary  man's  sensitiveness  at  the  time. 


LETTER  LXXXI. 
NEWTON  TO   COTES. 


I  had  yo*"*  of  Feb  W\  &  the  Difficulty  you  mention  w^^ 
lies  in  these  words  [Et  cum  Attractio  omnis  mutua  sit]  is 
removed  by  considering  that  as  in  Geometry  the  word 
Hypothesis  is  not  taken  in  so  large  a  sense  as  to  include 
the  Axiomes  &  Postulates,  so  in  Experimental  Philosophy 
it  is  not  to  be  taken  in  so  large  a  sense  as  to  include  the 


*  It  is  clear  that  this  is  a  mistake  for  Marchf  though  Newton  himself  in  his  answer 
to  this  letter  speaks  of  it  as  "  yo""*  of  Feb.  18." 


NEWTON  AND  COTES.  155 

first  Principles  or  Axiomes  w^^  I  call  the  laws  of  motion. 
These  Principles  are  deduced  from  Pha)nomena  &  made 
general  by  Induction :  w*^^  is  the  highest  evidence  that  a 
Proposition  can  have  in  this  philosophy.  And  the  word 
Hypothesis  is  here  used  by  me  to  signify  only  such  a  Pro- 
position as  is  not  a  Phenomenon  nor  deduced  from  any 
Phajnomena  but  assumed  or  supposed  w%ut  any  experi- 
mental proof.  Now  the  mutual  &  mutually  equal  attrac- 
tion of  bodies  is  a  branch  of  the  third  Law  of  motion  & 
how  this  branch  is  deduced  from  Phenomena  you  may  see 
in  the  end  of  the  Corollaries  of  y®  Laws  of  ^lotion,  pag.  22. 
If  a  body  attracts  another  body  contiguous  to  it  &  is  not 
mutually  attracted  by  the  other:  the  attracted  body  will 
drive  the  other  before  it  &  both  will  go  away  together  w*** 
an  accelerated  motion  in  infinitum,  as  it  were  by  a  self 
moving  principle,  cotrary  to  y®  first  law  of  motion,  whereas 
there  is  no  such  phjcnomenon  in  all  nature. 

At  the  end  of  the  last  Paragraph  but  two  now  ready  to 
be  printed  off  I  desire  you  to  add  after  the  words  [nihil 
aliud  est  quam  ffatum  et  Natura.]  these  words :  [Et  hoic 
de  Deo:  de  quo  utiq:  ex  pha)nomenis  disserere,  ad  Philo- 
sophiam  experimentalem  pertinct.] 

And  for  preventing  exceptions  against  the  use  of  the 
word  Hypothesis  I  desire  you  to  conclude  the  next  Para- 
graph in  this  manner  [Quicquid  enim  ex  phrenomenis  non 
deducitur  Hypothesis  vocanda  est,  et  ejusmodi  Hypotheses 
seu  INIetaphysicae  seu  Physicaj  seu  Qualitatum  occultarum 
seu  Mechanic®  in  Philosophia  experimentali  locum  non 
habent.  In  hac  Philosophia  Propositioncs  dcducuntur  ex 
phaenomenis  &  redduntur  generales  per  Inductionem.  Sic 
impenetrabilitas  mobilitas  &  impetus  corporum  &  leges 
motuum  &  gravitatis  innotuere.  Et  satis  est  quod  Gravi- 
ias  corporu  revera  existat  &  agat  secundum  leges  a  nobis 
cxpositas  &  ad  corporum  coelcstium  et  maris  nostri  motus 
omnes  sufficiat. 


156  CORRESPONDENCE  OP 

I  have  not  time  to  finish  this  Letter  but  intend  to 
write  to  you  again  on  Tuesday. 

I  am 
Yo'  most  humble  Servant 
London.  28  March  {Saturday}  1713.  Is.  Newton 

For  the  RoTorcnd  M'  Roger  Cotks  Professor 
of  Astronomy/,  at  hia  Cfiamber  in  Trinitij 
College  in  Cambridge. 


LETTER  LXXXII. 
NEWTON  TO  COTES. 

S'  London.  31  Mar.  1713. 

On  Saturday  last  I  wrote  to  you,  representing  that 
Experimental  philosophy  proceeds  only  upon  Phenomena 
&  deduces  general  Propositions  from  them  only  by  Induc- 
tion. And  such  is  the  proof  of  mutual  attraction.  And 
the  arguments  for  y*  impenetrability,  mobility  &  force  of 
all  bodies  &  for  the  laws  of  motion  are  no  better.  And 
he  that  in  experimental  Philosophy  would  except  against 
any  of  these  must  draw  his  objection  from  some  experi- 
ment or  phaenomenon  &  not  from  a  mere  Hypothesis,  if 
the  Induction  be  of  any  force. 

In  the  same  Letter,  I  sent  you  also  an  addition  to  the 
last  Paragraph  but  two  &  an  emendation  to  the  last  Para- 
graph but  one  in  the  paper  now  to  be  printed  off  in  the 
end  of  the  Book. 

I  heare  that  M*"  Bernoulli  has  sent  a  Paper*  of  40 


•  Part  of  it  appeared  in  the  Number  for  Feb.  1713,  pp.  77—95,  the  remainder  in  the 
March  number,  pp.  115—132.  See  Comm.  Epistol.  Leibn.  and  Bernoull.  ii.  299. 
Bernoulli  afterwards  (Letter  to  Leibniz,  Feb.  i?,  1714),  in  consequence  of  his  not  re- 
ceiving a  copy  of  the  Commercium  Epistolicum,  and  of  the  2nd  Ed.  of  the  Principia, 
which  Demoivre,  in  Newton's  name,  had  promised  more  than  a  year  before  to  send  him, 
fancied  that  Newton  was  offended  at  his  animadversions,  and  seems  to  have  stated  his 
suspicions  to  Demoivre ;  but  the  tone  of  the  article  did  not  prevent  the  author  of  the 
Principia  from  expresbing  his  sense  of  the  merits  of  Bernoulli's  solution  of  his  problem. 


NEWTON  AND  COTES.  157 

pages  to  be  published  in  the  Acta  Leipsica  relating  to 
what  I  have  >vritten  upon  the  curve  Lines  described  by 
Projectiles  in  resisting  Mediums.  And  therein  he  partly 
makes  Observations  upon  what  I  have  written  &  partly 
improves  it.  To  prevent  being  blamed  by  him  or  others 
for  any  disingenuity  in  not  acknowledging  my  oversights 
or  slips  in  the  first  edition  I  believe  it  will  not  be  amiss  to 
print  next  after  the  old  Proefatio  ad  Lectorem,  the  follow- 
ing Account  of  this  new  Edition. 

In  hac  secunda  Principiorum  Editione,  multa  sparsim 
emendantur  &  nonnulla  adjiciuntur.  In  Libri  primi  Sect,  ii, 
Inventio  virium  quibus  corpora  in  Orbibus  datis  revolvi 
possint,  facilior  redditur  et  amplior.  In  Libri  sccundi 
Sect.  VII  Thcoria  resistentiro  fluidorum  accuratius  invcsti- 
gatur  &  novis  expcrimentis  confirraatur.  In  Libro  tertio 
Thcoria  Luna)  &  Pra^cessio  JEquinoctiorum  ex  Principiis 
suis  plenius  deducuntur,  et  Thcoria  Cometarum  pluribus 
et  accuratius  computatis  Orbium  exemplis  confirmatur. 
28  Mar.  1713.  I.  N. 

If  you  write  any  further  Preface  ft  I  must  not  see  it  J. 
for  I  find  that  I  shall  be  examined  about  it.  The  cuts  for 
y«  Comet  of  1680  &  1681  are  printed  off  &  will  be  sent  to 
D'  Bently  this  week  by  the  Carrier. 

I  am 
Yo'  most  humble  Servant 

For  tU  R"**  M'  Cotes  Professor  of  Astro-  IsAAC  Newton 

nomy  in  the  University  of  Cambridge,     At 
his  Cluimber  in  Trinity  College  in  Cambridge 


**  J'ai  vu  Mr.  Neuuton,  qui  m*a  dit,  qu*il  avoit  lu  avec  beaucoup  de  plaiHir  vdtre 
methodede  resoudre  le  probleme  de  la  resistance,  il  vous  rend  justice  en  Homme,  qui 
n'  est  nullement  offens^,  il  dit  qu*  elle  est  admirablement  belle,  6c  roeme  qu*  elle  est 
commode  pour  des  expressions  finies."  Extract  from  a  Letter  of  Demoivre  to  Bernoulli 
in  Leipsic  Acts  for  July  1716,  p.  309. 

t  Newton  seems  to  have  particularly  in  his  eye  Cotefl*fl  proposed  allusion  to  the 
dispute  about  the  invention  of  fluxions. 

:J:  Compare  Commerc.  Epistol.  2nd  Ed.  ad  Lectorem  pag.  penult     "Quae  nov» 


158  CORRESPONDENCE  OF 

This  is  tho  last  letter  in  the  Trin.  Coll.  collection  that  passed  be- 
tween Newton  and  his  editor  while  the  work  was  in  the  press.  The 
proof-sheet  however  of  the  Scholium  Generale  must  have  been  sent 
up  to  Newton,  as  there  is  a  paper  (No.  271)  in  his  handwriting  con- 
taining some  alterations  of  the  Scholium,  in  which  the  pages  and  lines 
are  referred  to  as  wo  find  them  in  the  printed  book. 

The  Index  was  finished  in  April  (letter  cxiii),  and  tho  Preface  is 
dated  May  12.  In  his  letter  of  May  3  to  Jones  (letter  cxiv),  Cotes 
**  hopes  the  whole  book  may  bo  finished  in  a  fortnight  or  3  weeks :" 
**  it  might  have  been  done  by  this  time"  but  for  indisposition.  It  was 
not  however  until  about  June  18  that  tho  impression  was  finished. 
(See  next  letter). 

It  was  probably  about  this  time  that  the  Cambridge  Aristarchus 
made  his  emendations  of  Ilalley's  verses  prefixed  to  tho  Principia. 
See  Rigaud's  Essay,  pp.  86,  87. 


LETTER    LXXXIII. 
COTES  TO  D'  SAM.   CLARKE. 
S'  Cambridge  June  25*'*  1713. 

I  received  Your  very  kind  Letter.  I  return  You  my 
thanks  for  Your  corrections  of  the  Preface,  &  particularly 
for  Your  advice  in  relation  to  that  place  where  I  seem'd  to 
assert  Gravity  to  be  Essential  to  Bodies.  I  am  fully  of 
Your  mind  that  it  would  have  furnish'd  matter  for  Cavil- 
ling, &  therefore  I  struck  it  out  immediately  upon  D' 
Cannon's  mentioning  Your  Objection  to  me,  &  so  it  never 
was  printed.  The  impression  of  the  whole  Book  was 
finished  about  a  week  ago. 

INIy  design  in  that  passage  was  not  to  assert  Gravity  to 
be  essential  to  Matter,  but  rather  to  assert  that  we  are 
ignorant  of  the  Essential  propertys  of  Matter  &  that  in  re- 


Principionim  editiuni  praemissa  sunt,  Newtonus  non  vidit  aDtequam  Liber  in  lucem 
prodiit."  Dalemhert's  misstatement  on  this  point  ("preface  faite  sous  les  yeux  de 
Tauteur,"  Encycloped.  i.  854)  is  noticed  by  Wilson  (Robins's  Tracts,  Appendix, 
II.  334). 


NEWTON  AND  COTES.  159 

spect  of  our  Knowledge  Gravity  might  possibly  lay  as  fair 
a  claim  to  that  Title  as  the  other  Propertys  which  I  men- 
tion'd.  For  I  understand  by  Essential  propertys  such  pro- 
pertys without  which  no  others  belonging  to  the  same 
substance  can  exist :  and  I  would  not  undertake  to  prove 
that  it  were  impossible  for  any  of  the  other  Properties  of 
Bodies  to  exist  without  even  Extension. 

Be  pleased  to  present  my  humble  Service  to  S'  Isaac 
when  You  see  him  next,  &  let  him  know  that  the  Book  is 
finished* 

I  am  S^ 

Your  much  Obliged  Freind 

&  Humble  Servant 
To  D*^  Clark  H  C 

It  appears  from  the  above  letter  that  a  meaning  has  been  given  to 
expressions  in  Cotes's  Preface  wliich  he  did  not  intend  them  to  convey. 
He  has  been  understood  to  assert  that  gravity  is  an  essential  property 
of  bodies:  his  words  are  *^  Inter  primarias  qualitates  corporum  univer- 
sonim  vel  Gravitas  habobit  locum ;  vel  Extensio,  Mobilitas  &  Impeno- 
trabilltas  non  habebunt."  His  supposed  views  are  controverted  by  D' 
Whewell  (Philosophy  of  the  Inductive  Sciences,  i.  249,  or  258  2nd  Ed.), 
and  are  quoted  with  approbation  in  a  recent  work  (Le  Cartesianisme  ou 
la  veritable  renovation  des  sciences,  par  Bordas-Deraoulin,  Paris  1843,— 
a  work  less  remarkable  for  accuracy  than  for  liveliness  of  declamation). 
Though  Newton,  says  this  last  writer,  had  not  the  true  idea  of  attrac- 
tion, "citte  notion  perce  ct  triomphe  deju  chez  quelques-uns  deses  dis- 
ciples immediats,  tcls  que  Roger  C6tes."  (i.  304).  He  also  refers  to 
Maupertuis  and  Lalande  as  holding  the  same  opinion.  "  Pour  moi,  dit 
Lalande,  je  pense  avec  M.  Maupertuis  et  la  plupart  des  m^taphysicicns 
anglais,  que  I'attraction  depend  d'une  propriete  intrinseque  do  la  ma- 
tiere."     Astron.  ed.  2.  art.  3384." 


•  On  Monday  July  27  Newton  waited  on  the  Queen  with  a  copy  of  the  new  edition 
of  his  book.  (Daily's  Flamsteed,  p.  98.)  Jones's  letter  of  thanks  for  a  presentation  copy 
(letter  cxv)  is  dated  July  11.  Compare  Bentley*a  Correspondence,  p.  465.  Flumstceti 
gave  \Ss.  for  a  copy  (Baily,  p.  305).  In  Clare  Hall  Library  are  two  copies  of  the  book, 
one  of  which  belonged  to  Cotes's  friend  Charles  INIorgan  "  Kx  dono  Clariss'.  Editoris 
Pr.  P.  1*.  C."  and  the  other  to  Rob.  Green  "  Pret.  15j."  In  a  catalogue  of  Keill's 
library  in  his  own  hand-writing  among  the  Lucasian  papers  the  price  of  a  copy  is  put 
down  at  £1. 


160  CORRESPONDENCE  OP  ^ 

Newton  was  obliged  on  several  occasions  to  protest  against  the  doc- 
trine of  innate  gravity  being  ascribed  to  him.  See  letters  to  Bcntley, 
Jan.  17.  Feb.  25.  1G9§.  Advertisement  to  2<*  Ed.  (in  English)  of  his 
Optics,  July  16,  1717 :  "  And  to  shew  that  I  do  not  take  Gravity  for 
an  essential  Property  of  Bodies,  I  have  added  one  question  {the  21"} 
concerning  its  Cause,  chusing  to  propose  it  by  way  of  a  Question,  be- 
cause I  am  not  yet  satisfied  about  it  for  want  of  Experiments*."  & 
his  letter  in  Macclesfield  Corrcsp.  11.  437. 


LETTER    LXXXIV. 


This  is  not,  properly  spealcing,  a  letter,  but  a  paper  of  Corrections  and 
Additions  sent  by  Newton  to  Cotes  through  Cornelius  Crownfteld, 
the  University  Printer,  six  months  after  the  publication  of  the  book. 
See  next  letter. 

Corrigenda  et  Addenda  in  Lib.  i. 

Pag  7.  lin.  8,  post  veriore  tempore  adde  mensurent. 
P.  10,  1.  6,  post  sed  adde  sunt  P.  10.  1.  17,  lege  difficillimum 
est. 

P.  15  1  16  lege  in  plana,  ut  /)iVad  pll.  lb.  1.  20  veri- 
tatem  ejus. 

P.  17  1.  20  pro  communis  lege  corporis.  P  31,  1.  38 
AD  et  DB,  P.  36,  1.  6  kge  Cor.  5.  P.  38  1  24  lege  Corol. 
2.  1.  26  lege  Corol.  4. 

P.  41  1  5  lege  P  et  Q.  P.  42,  1.  8  post  vol  adde  circu- 
lum  concentrice  tangit,  id  est. 

P  44  1  23  lege  QR  x  RN  +  QN.  P.  45, 1.  14  post  hoc 
est  adde  (ob  datam  specie  figuram  illam)  lb.  1.  21  post 
Spiralem  adde  concentrice.  P.  46  1.  30  post  intelligatur 
adde  recta.  lb.  1.  ult.  post  PvxuV  lege,  Adde  rectangulum 
uPv  utrinq:  et  prodibit  quadratum  chorda?  arcus  PQ 
ajqnale  rectangulo  VPv.     P.  47  1  4  post  conica  in  P,  lege 


•  This  declaration  was  probably  drawn  from  him  by  the  recent  controversy  between 
Leibniz  and  Clarke. 


NEWTON  AND  COTES.  161 

adeoq:    ex    natura    Sectionum    Conicnrum,    circuli    hi\jiis 

chorda  PT  tequalis  erit  —^^  .     P.  52,  1.  16  dele  per. 

P.  54,  1.  4,  jyoat  area  QT  ^  SP  adds  quso  dato  tempore 
describitur. 

P.  57,  1.  25  post  si  ea  adde  sit.  P.  59,  1.  7  7>o^«  oxi 
principali  ^qwtvq,  adde  id  est  axi  in  quo  umbilici  jacent. 

P.  61,  1. 12  lege  ita  ut  sit  GA  ad  AS  et  Ga  ad  a^S*  ut  est 
KB  ad  jBiS*,  et  axe  A  a, 

lb.  1.  15,  16  %e,  et  cum  sit  GA  ad  ^5*  ut  Ga  ad  aS, 
erit  divisim  Ga  -  GAy  seu  ^a  ad  aS-AS  seu  iS*//  in 
eadem  ratione.     P.  86,  1.  7,  ^o^f  biseca  adde  in  JT/  et  N, 

P.  87  1.  7  lege  per  Prob.  xiv. 

P.  89  &  90  in  Figura  jungatur  FD, 

P.  92,  in  Figura  jungantur  FG  et  HL  P.  101,  1.  6,  7, 
8,  %e.  Nam  centro  O  intervallo  OA  dcscribatur  semicircu- 
lus  AQB  rectae  LP  si  opus  est  products,  occurrens  in  Q, 
junganturq:  SQ^  OQ,  quarura  OQ  producta  occurrat  arcui 
EFG  in  F,  et  in  eandem  OQ  demittatur  perpendiculum 
SE»  lb.  1.  36  post  qujo  adde  per  punctum  P  transit  et. 
P.  109,  1.  1  post  Hyperbola  adde  rectangula.  lb.  in  Sche' 
mate  pro  liter  a  O  scrihatur  liter  a  H,  P.  117.  1  15  lege  prio- 
ns in  /.  Et  stantibus.  P.  121  in  Schemate  e  regione  liter w 
p  acHhatur  litera  K  in  Orhe  VPK,  P.  127, 1.  7, 9  graduum. 
P.  131  1.  17  lege^  m  roqualis  1  et  n, 

P.  136,  1.  2  pro  Bp  scribe  BP.  P.  137,  1.  16  post  sinus 
versus  adde  est.  P.  139,  1.  10  post  adeoq:  ad  adde  globi 
exterioris.  1. 12  post  habet  ad  adde  globi  interioris.  P.  148, 
1.  4  post  distantisB  adde  corporum.  lb.  1.  7  pro  tcrminos 
suos  communi  scribe  terminum  suum  communem.  P.  151, 
1.  8,  21  scribe  ad  primum  duorum.  P.  151, 1.  18  scribe  ut 
primum  duorum.  P.  156  1 31  scribe  maximo.  Nam.  P  158 
1  32  Post  atq:  adde  ut,  et  post  proportionalitate  deh  ut.  lb. 
1.  36  post  non  sit,  adde  reciproce. 

P.  166,  1.  9  dele  quadratum  temporis  periodici  et  scribe 
11 


162  CORRESPONDENCE  OP 

tempus  periodicum.  P.  169,  1.  26,  33,  34,  &  P.  170  1 3  pro 
C  scribe  O,  et  in  schemate  inter  P  ac  T  scribe  literam  O, 
P.  184  1  21  post  area  adde  ABNA,  P.  187  1  4  pro  duplo 
eju8  scribe  ejus  duplo.  P.  190,  1.  15  pro  simUia  scribe  con- 
tinue proportionales  SI,  SB,  SP,  similia  sunt.  lb.  1.  19, 
post  PE*  adde,  (ob  proportionales  IE  ad  PE  ut  IS  ad  SA) 

P.  191,  1.  7,   %e  corpus  P  erit  ut      ^„    . — - — . 

P  196  1.  25  post  qua  annidi  acZtZe  centro  A  intervallo  AE 
in  piano  praidicto  descripti.  P.  197  1  24  pro  diametro  lege 
semidiametro. 

Corrigenda  et  addenda  in  Lib.  ii. 
Pag.  213,  lin.  10,  12  Pro  EC  et  BD  scribe  BACH  et 
BADE,  lb.  lin  14  post  partes  adde  recta)  AB,  lb.  1.  24 
jyro  BC  scribe  BACH.  lb.  1.  26  pro  AH  scribe  BACH 
P.  214,  1.  33  post  gravitatis  qua  adde  corpus  illud.  P.  223 
1.  20,  22  pro  sesquialtera  scribe  sesquiplicata.  P.  229, 1.  7, 
8  lege  omne  ascendendi  ad  locum  summum  ut  Sector  Cir- 
culi,  et  tempus  omne  descendendi  a  loco  summo  ut  Sector 
Hyperbolae.  lb.  1.  13,  14,  15  post  Circularis  AtDui  tem- 
pus lege  omne  ascendendi  ad  locum  summum,  &  Sector 
Hyperbolicus  ATD  ut  tempus  omne  descendendi  a  loco 
summo ;  si  modo  Sectorum.  lb.  1.  21,  post  ut  lege 
qDp  y^  ^2)quad. 

- —        ^^j ,  id  est,  ob  datam  tD,  ut.     lb.  1.  26  post 

pjj 

mento  adde  velocitatis.  1.  30,  post  est  ut  adde  tempus  totum 
ascendendi  ad  locum  summum.  q.e.d. 

P.  233,  lin.  ult.  pro  ZQRo  lege  2  QRo\    P.  240  1.  27  pro 
MX  lege  NX.     P.  241,  1.  13  Parabola  prjedicta?. 

P.  244,  1.  22  lege  FG.    Pag.  248,  1.  2  lege  sit. 

n  —  2 

lb.  1. 10,  pro  omnis  futuri  lege  totius    lb.  1.  23,  ^2  pro  futuri 

lege  totius.    P.  249,  1.  20  post  tempus  adde  totum     P.  251, 

1.  32  post  et  AB  ut  adde  area.     P.  255  1.  8  pro  sit  lege  est. 

P.  285  1.  17  ^^05^  arcubus  adde  vel.     P.  290  1.  31  pro  aero 


NEWTON  AND  COTES.  163 

scribe  aqua.  lb.  1.  34  pro  aqua  scribe  acre.  P.  300  1.  11 
pro  CD  scribe  AB,  P.  301, 1.  7  post  axis  sui  ndde  unifomii- 
ter  progrediendo.  lb.  1  9  post  diaraetri  sua)  adde  uuiformi- 
ter  progrediendo.  lb.  1.  12  pro  totum  globi  motum  lege 
motum  globi.  lb.  1.  lb ^  post  diametri  sua)  adde  uniforiniter 
progredicndo 

p.  317  I.  penult,  pro  maximam  G  lege  maximam  //. 
Corrigenda  ct  Addenda  in  Lib.  in. 

Pag.  358,  1.  3,  4  lege  affirraatur.  Corpora  plura  dura 
esse  experimur ;  oritur  autem. 

P.  367,  1.  14  lege  foret.  P.  378  1.  28  pro  circa  annum 
lege  anno. 

P.  379  1.  13,  23  pro  centripetam  lege  centrifugam. 
P.  387  1.  22  lege  quam.  P.  396,  1.  17  pro  erit  Kk  ad  lege 
erit  FK  aqualis  TK  &  Kk  erit  ad.  lb  1  19  post  FKkf 
adde  erit.  P.  399  1.  6  post  Solem  adde  vel  ab  ea  supcratur. 

P.  415  1.  12,  15  pro  annua  et  annu«3  scribe  scmcstris  et 
semestri.  P  422  1.  34  post  hajc  a)quatio  adde  maxima. 
P.  425  1.  23  dilatet.  P  444  1.  33  dele  formata  est,  et  post 
inter  se  adde  formata  sunt.  P  450  1.  16  lege  ad  ejus  velo- 
citatem.  P.  453  1  17  lege  quorum  AM,  P  457  1.  penult. 
&  ult.  post  manentem  dele  parum  diligenter  definivit.  Nam 
Cometa,  ^  scribe  ex  observationibus  definire  neglexit. 
Cometa  autem.  P.  459.  1.  3  lege  partium  100000.  P  459 
proxime  post  Tabulam  lege  Apparuit  etiam  hie  Cometa 
mense  Noverabri  praicedende*  in  signis  Virginis  &  Libra) 
ut  Stella  secundae  vel  tertiae  magnitudinis,  &  Florentia?  qui- 
dem  ad  horam  octavam  Italicam  ea  nocte  quae  mensis 
hujus  diem  vigcsimum  &  vigesimum  primum  intercessit,  st. 
novo,  id  est,  decimum  &  undccimum  st.  vet.  visus  fuit  in 
signo  Virginis  sub  stellis  in  sinistro  pede  [vel  fcmure] 
Leonis  cum  Ascentione*  recta  graduum  165,  referente  Cas- 
sino.     Erat  igitur  Cometa  in  mj   13^  circiter.     Nam    et 


11—2 


164  CORRESPONDENCE  OP 

Hillus  quidam  hora  quinta  matutina  die  12  vel  potius  10 
Novembris,  Cantuaria)  in  Anglia  distantiam  ccepit*  hi\ju3 
Cometaj  a  Corde  Lconis  graduum  scptcndccim  in  Orien- 
tem  et  a  Cauda  Lconis  paulo  plusquam  graduum  undecim 
in  austrum.  Unde  Cometa  tunc  crat  in  -njj  12^  24'  cum 
latitudine  borcali  2^  circitcr.  Crassissiraoe  fuerunt  ha; 
obsen'ationes ;  meliores'^glmt  quae  scquuntur.  Pag.  459  lin 
35  post  Galletius  etiam  scribe  Avenioni.  lb.  1.  39  Cellius 
in  =a.  13.  so'  lb.  1.  40  dele  Roma;.  P.  460  1.  33  post  Au- 
strali  l'^  16'  adde  Cellius  in  ^  28.  lb.  1. 37  post,  id  est  2^  2' 
vice  linearum  quinq:  sequentium  adde.  Eodem  die  ad  horam 
quintam  matutinam  Ballasora;  in  India  Oricntali,  capta 
est  distantia  Cometa;  a  Spica  i^jt  7^  35'  in  Orientem.  In 
linea  erat  recta  inter  Spicam  et  Lancem  australem,  ideoq : 
versabatur  in  ===  26^'.  58',  cum  Latitudine  australi  l'^'"  ll' 
circiter ;  et  propterea  post  horas  5  &  40',  ad  horam  scilicet 
quintam  matutinam  Londini  erat  in  ^  28^^  ll'  cum  Lati- 
tudine australi  1'^''  iG'  circiter.  Pag.  462  lin  30  post  factoe 
videntur  adde  Die  22  ubi  Cometa  ex  observatione  Monte- 
nari  crat  in  nt  S*^*"  36'  Venetiis,  &  propterea  in  ii\  2^"  48' 
eadem  hora  matutina  Londini :  Hookius  nostcr  eundcm 
locavit  in  ir|^  3 .  3o'  ut  supra.  Montenarus  in  dcfectu 
Hookius  in  cxccssu  errasse  videntur.  Nam  et  Ballasoraj 
eodem  die  ante  ortum  Solis,  Cometa  obscrvabatur  in  ni  ^^^ 
50',  ideoq:  eadem  hora  matutina  Londini  erat  in  nj,  3^"^  5'. 
Die  24  ad  horam  quintam  matutinam  Ballasora;  Cometa 
obscrvabatur  in  ii|  ll^**  45',  ideoq:  ad  horam  quintam  Lon- 
dini erat  in  m.  13^  circiter.  Pag.  463  m  Tabula  priore  pro 
^  27  .  52',  m  2  56,  rri  12  .  58,  letje  ^^  28  .  0.  m  3  .  5.  rr^  13  . 0. 
lb.  initio  secundoi  Tabuloe  addantur  Novem.  9.  17  |  101551  | 
m,  12 .  25  .  50 1 0 .  43 .  30  Bor.  Pag  472  lin  27  lege  cadent. 
Pag  474  lin  23,  inter  Et  et  similis  lege  in  Chronico  Saxo- 
nico.     lb.  dele  llOl  vel.     lb.  lin.  26  post  habet  adde  etiam. 


NEWTON  AND  COTES.  165 

P.  478,  1.  25  pro  prima  lege  sccunda.  P.  482  1.  2,  post  spa- 
tiis  adde  ob  defectum  neris.  lb.  lin  18  lege  ut  se  mutuo 
quam  minime  trahant.  lb.  1.  29  lege  non  in  corpus  pro- 
prium  (uti  scntiunt  quibus  Dcus  est  anima  mundi,)  scd  in 
servos.  P.  483  1  36,  post  Fatum  et  Natura.  adde,  A  neces- 
sitate Metaphysica,  qua?  utiq:  eadem  est  semper  et  ubiqr, 
nulla  oritur  rerum  variatio.  Omnis  ilia  quae  in  mundo 
conspicitur  pro  locis  ac  temporibus  diversitas  a  voluntate 
sola  Entis  necessario  existcntis  oriri  potuit,  Dicitur  au- 
tem  Deus  per  Allegoriam  videre,  audire,  loqui,  ridere, 
amare,  odio  habere,  cupcre,  dare,  accipere,  gaudere,  irasci, 
pugnare,   fabricare,    condere,    construere,   &  intclligentes 

(vitam  infundendo)  *<2:enerare.     Nam  sermo 

.                             '      c                 ^                .     .  .  •  Job  38  7 

omnis  de  Deo  a  rebus  humanis  per  similitu-  ,     *  „  *   * 

*  ^  Luc.  3.  38. 

dinem    aliquam   desumi    solet.     Et   hajc   de 

Deo;  de  quo  utiq:  ex  pba?nomenis  disserere  ad  Philoso- 

phiam  cxperimentalem  pertinet. 

The  following  notes  are  in  Cotes's  hand  :  tlioy  are  the  elements  of 
the  next  letter. 

p.  3.  1:  14 

p.  41.  1:  3 

p.  47  1:  penult. 

p.  47.  1:  4  non  emend. 

p.  109.  in  schem.  non  H  pro  O 

p.  148.  1.  7  n. 

p.  151.  1.  8.  18,  21  n 

p.  191. 1.  7  w. 

itjj  .12«.25'.50"  non  lit 

p.  230,  1.  penult,  post  incremento  adde  vclocitatis 

p.  460.  p.  462  n  intell. 


166  CORRESPONDENCE  OF 

LETTER   LXXXV. 

COTES  TO  NEWTON. 

*    S' 

I  lately  received  from  You  by  M'  Crownfeild  a  Paper 
of  Errata,  Corrigenda  &  Addenda  to  be  printed*  &  bound 
up  with  Your  Principia.  I  take  leave  to  send  You  some 
observations  upon  them. 

By  comparing  Your  Catalogue  with  my  Table  of 
Corrigenda,  I  find  you  have  omitted  that  of  pag:  3. 
lin :  14.  I  think  it  convenient  to  make  some  such  alteration, 
that  You  may  not  seem  to  assert  what  is  false.  You  have 
also  omitted  that  of  pag,  47.  lin.  penult,  which  I  think  is 
requisite  to  determine  Your  meaning.  "Whilst  that  Sheet 
was  printing  I  remember  I  did  not  understand  what  it  was 
that  You  there  asserted,  &  not  having  then  time  to  ex- 
amine the  thing  to  the  bottom,  I  was  forc'd  to  let  it  go. 
Soon  after  I  considered  it,  &  found  in  what  sense  You{r} 
words  could  be  true  &  accordingly  made  the  Alteration. 
Since  Your  book  has  been  published  I  have  been  ask'd  the 
meaning  of  that  place  by  one  who  told  me  he  knew  not 
what  sense  to  put  upon  Y'  words:  I  referr'd  him  to  the 
Table  of  Corrigenda  &  then  I  perceiv'd  he  understood 
You. 

Your  addition  of  pag.  47  lin.  4  should  I  think  be 
omitted.  For  if  that  addition  be  made  the  8  preceding 
lines  are  to  no  purpose  &  ought  to  be  omitted.     Tis  very 

evident  that  PF  is  equal  to        _,     by   pag.  46   lin.   ante- 

-»  o 

penult. 

In  pag  109  You  direct  to  put  H  in  the  Figure  instead 
of  O.  You  mean  instead  of  the  lower  O  which  bisects 
the  transverse    diameter    of  the  Hyperbola.     If  this  be 


•  I  am  not  aware  that  this  table  of  Errata  was  ever  printed.    Cotes  does  not  seem 
to  Imvc  been  altogether  pleased  at  the  receipt  of  so  formidable  a  list. 


NEWTON  AND  COTES.  167 

done,  then  the  Figure  will  not  agree  with  the  second  line 
of  this  page,  nor  indeed  with  the  whole  Demonstration  as 
it  relates  to  the  Hyperbola. 

In  pag.  148 :  lin,  7.  I  think  the  alteration  should  not 
be  made.  There  arc  three  different  distantice,  &  three 
different  termini  &  one  common  angular  motion. 

Pag.  151.  You  change  prima  the  Fajminine  into  pri- 
mum  y®  Neutre.  Tis  my  Opinion  that  this  alteration  is 
not  necessary.  I  understand  the  printed  text  thus :  prima 
duantm  medie  proportionalium  quantitatum.  If  it  were  ad- 
viseable  to  make  an  alteration,  I  woiUd  rather  choose  the 
Masculine  &  put  it ;  primus  duorum  medie  proportionalium 
terminorum  inter  ^c, 

Pag.  191.  lin.  7  I  think  wants  no  correction.  I  cannot 
understand  by  what  reasoning  You  make  one;  You  "will 
be  pleas'd  to  reconsider  it.  If  Your  correction  be  true,  it 
will  be  very  necessary  to  explain  it  more  fully. 

Page  463  in  the  beginning  of  the  second  Table  I  sup- 
pose You  intended  to  put  itK  12^  25'.  50"  not  vri  12,  25.  50 
as  it  is  in  Your  written  copy 

You  order  the  3  last  lines  of  page  460,  &  the  2  first  of 
page  461  to  be  struck  out ;  &  in  their  room  You  place 
what  follows.  [Eodem  die  ad  horam  quintam  matutinam 
Ballasora9  in  India  Orientali,  capta  est  distantia  Cometa; 

a  Spica  trR  7^.  35' Londini,  erat  in  .^  28^.  ll'  cum 

Latitudine  australi  1^.  l6  circiter.]  I  suppose  You  intended 
to  make  this  addition  at  the  end  of  the  Paragraph  which 
begins  with  Nov,  21.  PonthoBtis  4-c.  &  would  not  have  the 
5  first  lines  of  the  following  Paragraph  struck  out. 

I  observe  You  have  put  down  about  20  Errata  besides 
those  in  my  Table.  I  am  glad  to  find  they  are  not  of  any 
moment,  such  I  mean  as  can  give  the  reader  any  trouble. 
I  had  my  self  observed  several  of  them,  but  I  confess  to 
You  I  was  asham'd  to  put  'em  in  the  Table,  lest  I  should 
appear  to  be  too  diligent  in   trifles.       Such  Errata  the 


168         CORRSSrONDENCE  OF  NEWTON  AND  COTES. 

Reader  expects  to  meet  with,  and  they  cannot  well  be 
avoided.  After  You  have  now  Your  self  examined  the 
Book  &  found  these  20,  I  beleive  You  will  not  be  surpriz'd 
if  I  tell  You  I  can  send  You  20  more  as  considerable, 
which  I  have  casually  observ'd,  &  which  seem  to  have 
escap'd  You :  &  I  am  far  from  thinking  these  forty  are  all 
that  may  be  found  out,  notwithstanding  that  I  think  the 
Edition  to  be  very  correct.  I  am  sure  it  is  much  more  so 
than  the  former,  which  was  carefully  enough  printed ;  for 
besides  Your  own  corrections  &  those  I  acquainted  You 
with  whilst  the  Book  was  printing,  I  may  venture  to  say 
I  made  some  Hundreds,  with  which  I  never  acquainted 
You 

I  am  S*- 

Your  very 

Humble  Serv* 
Dec.  22^  1713.  II.  Cotes 


END  OF  CORRESPONDENCE   ON  THE  PRINCIPIA. 


LETTERS  OF  NEWTON  TO  KEILL. 


LETTER  LXXXVI*. 
NEWTON  TO  KEILIi. 

Yo'  Letter  of  Feb.  S***  I  delayed  to  answer  till  tlie 
Journal  Literaire  for  November  and  December  should 
come  out.  It  is  just  come  from  Holland  &  I  desired  M' 
Darby  to  send  you  a  copy  w*^'*  I  doubt  he  has  not  done 
because  he  sent  one  to  me  this  morning  w*^^  I  reccon  to  be 
for  you  &  I  designe  to  send  it  to  you  the  first  opportunity 
by  the  Carrier.  JM""  Leibnitz  in  August  last,  by  one  of  his 
correspondents  published  a  paperf  in  Germany  conteining 
the  judgment  of  a  nameless  Mathematician  J  in  opj)osition 
to  the  judgment  of  the  Committee  of  tlie  Hoyal  Society, 
with  many  reflexions  annexed.  This  paper  hath  been  sent 
to  ]\r  Johnson  with  remarks  prefixed  to  it.  And  the 
whole  is  printed  in  the  journal  Literaire  pag.  445.     And 


*  Letters  LXXWI,,  XCII.,  XClll.  were  formerly  among  the  papers  belonging  to 
the  Luodsian  ]*rofessor. 

t  A  "cliarta  volans,"  ilated  29  Jul.  1713,  without  name  of  place,  printer,  or 
author. 

4:  i.e.  John  Bernoulli  nx  the  letter  of  June  7,  1713,  to  Leibniz.  There  are  two 
circumstances  connected  with  this  letter — one  of  them  affecting  the  writer  of  it,  the 
other  his  correspondent— which  are  not  calculated  to  add  lustre  to  either  of  tliese  great 
names.  To  mention  the  latter  first:  l^einoulli  accompanied  the  letter  with  the  request 
that  in  any  use  that  might  be  made  of  it,  his  name  mig-ht  not  be  mixed  up  with  the  con- 
troversy.  Leibniz  observed  his  friend's  injunction  of  secrecy  at  the  time,  but  between 
two  and  three  years  afterwards,  without  Bernoulli's  permission  or  knowledge,  he  quoted 
the  letter  with  Bernoulli's  name,  in  letters  to  Count  Bothmar  and — {qux  legal  ipia 
Ijycoris) — Madame  lu  Comtesse  de  Kilmansegg,  He  had  shortly  before  intimated  the 
fact  in  the  plainest  terms  in  his  letter  of  April  0,  1716,  to  Conti  for  Newton.  The 
other  point  alluded  to  wears  a  more  serious  aspect.  Though  Bernoulli  was  confessedly 
the  writer  of  tl>e  letter,  (which  accordingly  appears  in  his  Correspondence,  published 
during  his  lifetime),  he  afterwards  (1719),  in  a  letter  which  he  sent  to  Newton,  dis- 
avowed the  authorship  of  it.  The  following  references  will  be  sufficient  to  enable 
any  reader  to  form  his  own  judgment  upon  these  two  points.  Leibn.  ami  Bernoull. 
Cimmefi.  ii.  311,  323,  330,  334,  37}3.  Leibniz.  0pp.  iii.  459,  4()2.  Macclesfield  Cor- 
respomtenre,  ii.  4'M).     T)es  Mai/.eaux  to  Conti,  MSS.  Birch,  42«1.  fol.  '222,  Brit.  Mur. 


170  LETTERS  OP 

now  it  is  made  so  publick  I  think  it  requires  an  Answer. 
It  is  very  reflecting  upon  the  Committee  of  the  Eoyal 
Society,  &  endeavours  to  derogate  from  the  credit  of  some 
of  the  Letters  published  in  the  Commercium  Epistolicum 
as  if  they  were  spurious.  If  you  please  when  you  have  it, 
to  consider  of  what  Answer  you  think  proper,  I  will  within 
a  Post  or  two  send  you  my  thoughts  upon  the  Subject, 
that  you  may  compare  them  w*^  your  own  sentiments  & 
then  draw  up  such  an  Answer  as  you  think  proper.  You 
need  not  set  your  name  to  it.  You  may  write  either  in 
English  or  in  Latine  &  leave  it  to  M*^  Johnson  to  get  it 
translated  into  F{r}ench.  M""  Darby  will  convey  yo'  An- 
swer to  the  Hague. 

I  am 

Yo**  most  humble  Servant 

London.  2  Apr.  1714.  Is.  Newton 

For  D'  JoiiN  Keh^l,  Professor  of 
Astronomy,  at  his  house  in  Ox^ 
ford. 


LETTER  LXXXVII*. 

NEWTON  TO  KEILIi. 
S'^ 
I  am  glad  you  have  read  both  the  pieces  concerning 
the  Commercium  inserted  in  the  Journal  Literaire  &  are  of 
opinion  that  they  must  be  immediately  answered  &  are 
thinking  of  an  Answer.  As  to  what  you  want  to  know 
concerning  things  in  the  Principia  contrary  to  the  doctrine 
of  fluxions  or  differences  I  take  it  to  be  this.  In  the 
Scholium  of  y*  10*^  Proposition  of  the  second  book  of  the 
Principia  I  have  made  use  of  y®  method  of  Infinite  Series 
for  determining  the  Curves  in  w*^''  Projectiles  will  move  in 


•  This  and  the  two  following  Letters  were  "the  gift  of  Mr  Watson,  fellow  of  the 
College,  1771,"  (afterwards  Bishop  of  LlandafT).  They  were  formerly  placed  in  a 
folio  volume,  which  is  now  marked  R.  4.G9. 


NEWTON  TO  KEILL.  171 

a  resisting  Medium  such  as  is  air.  John  Bernoulli  has 
published  in  the  Acta  Eruditorum  for  Febr.  &  March  was 
a  twelve  month,  a  I*aper  upon  that  Scholium,  in  w***  ho 
represents  that  the  Method  there  used  is  the  Method  of 
fluxions,  &  that  it  appears  thereby  that  I  did  not  under- 
stand y*  2**  ffluxions  when  I  ^vrotc  that  Scholium  because 
(as  he  thinks)  I  take  the  second  terms  of  the  series  for  the 
first  fluxions,  the  third  terms  for  the  second  fluxions  &  so 
on*.  But  he  is  mightily  mistaken  when  he  thinks  that  I 
there  make  use  of  the  method  of  fluxions.  Tis  only  a 
branch  of  y®  method  of  converging  series  that  I  there 
make  uses  of.  The  Acta  Eruditorum  for  the  last  year  are 
but  just  come  to  London,  &  I  find  thereby  that  John  Ber- 
noulli is  the  great  Mathematician  f  who  accuses  me  on  this 
account.  But  I  beleive  it's  better  not  to  reflect  upon  hira 
for  it  nor  so  much  as  to  name  him  any  otherwise  then  by 
the  general  name  of  the  great  Mathematician.  They  are 
seeking  to  pick  a  quarrell  with  me  &  its  better  to  lett 
them  begin  it  still  more  openly  without  a  provocation. 

There  is  another  great  Mathematician  J  to  whom  Leib- 
nitz referred  the  examination  of  the  Commercium  Episto- 
licum.     He  makes  use  of  two  arguments  against  me.    One 


•  See  p.  142  note.  An  abortive  attempt  haa  been  made  to  revive  this  delusion  by 
M.  Jean  Trembley  (Berlin  M6moires,  1798)  in  a  paper  which  professes  to  overthrow 
Lagrange's  explanation  of  the  real  source  of  the  error  in  the  expression  for  the  resist- 
ance given  in  the  1st  edition  of  the  Principia.  Lagrange  has  shewn  (Th^orie  dea 
Fonctions,  Paris,  1813.  pp.  339—349  :  see  also  p.  6)  that  if  powers  of  0  (the  time  of 
describing  a  small  arc)  above  the  square  be  neglected,  we  get  Newton's  first  result,  but 
that  if  we  include  terms  involving  0^  we  obtain  the  correct  value.  He  has  not,  liowever, 
pointed  out  in  what  respect  Newton*s  ^omctrical  expression  is  erroneous,  or  at  what 
step  of  the  demonstration  the  fallacy  is  introduced.    The  error  consists  in  substitulinjc 

FG  (which  =  /lo»+5o»=  Jg0»-J^^  03)  for //r  (which  -  Wu'+tJS.)""  a  if  0"+i^  *^  e''). 

where  r=  resist,  and  u  =  vel.  I  am  fully  sensible  of  the  danger  of  dissenting  from  that 
great  geometer  on  a  point  of  mathematics,  but  1  think  that  a  remarit  to  the  effect  just 
stated  would  have  been  less  open  to  objection  than  his  mode  of  arriving  at  the  correct 
expression  by  substitution  in  an  erroneous  formula  (p.  347.  lines  15,  16,  17.) 

Lncroix  (Calc.  Uiff.  et  Int.  torn.  3.  p.  644.  Parw.  1819)  does  not  seem  to  have 
read  the  part  of  the  Principia  in  question  with  mixh  attention. 

t  i.e.  the  "emincns  quidam  Mathematicus,"  quoted  in  the  Charta  Vohns.  Sec 
next  page,  line  3. 

:J:  John  HernouUi.    Sec  preceding  Letter  and  note. 


172  LETTERS  OP 

that  I  made  no  use  of  the  prickt  letters  till  of  late,  the 
other  that  when  I  wrote  the  Principia  I  understood  not 
the  second  fluxions  as  a  certain  great  Mathematician  (Ber- 
noulli) has  observed*.  The  Answer  is  that  I  use  any  nota- 
tion for  fluents  &  any  other  notation  for  fluxions,  &  an 
unit  for  the  fluxion  of  time  or  its  exponent  &  the  letter 
o  for  the  moment  of  time  or  of  its  exponent,  &  the  rect- 
angles of  the  fluxions  &  the  moment  o  for  the  moments  of 
other  fluent  quantities.  That  in  the  Analysis  per  cequatio- 
nes  nuraero  terminorum  infinitas  I  represent  fluents  by  the 
areas  of  figures,  time  by  the  Abscissa  flowing  uniformly, 
the  fluxions  of  fluents  by  the  Ordinates  of  curves,  the 
moments  of  fluents  by  the  rectangles  under  the  Ordinates 
&  0  the  moment  of  the  Abscissa :  but  do  not  confine  my 
self  to  any  certain  symbols  for  the  Ordinates  or  fluxions. 
That  I  do  the  same  in  the  book  of  Quadratures  &  even  to 
this  day.  That  where  I  use  prickt  letters  they  signify  not 
moments  or  differences  w^^  are  infinite  little  quantities  but 
fluxions  or  the  Ordinates  of  curves  as  the  exponents  of 
fluxions  w^**  are  finite  quantities,  unless  they  be  multi- 
plied by  the  symbol  o  (either  exprest  or  understood)  to 
make  them  infinitely  little :  but  it  is  not  necessary  that  the 
Ordinates  of  curves  should  be  represented  by  prickt  letters 
Such  letters  may  be  a  convenient  sort  of  notation  but  not 
necessary  to  the  method.  That  prick  letters  are  older 
symbols  for  fluxions  then  any  used  by  M'  Leibnitz :  for  he 
has  no  symbols  for  fluxions  to  this  day.  That  the  rect- 
angles under  the  Ordinates  of  curves  &  the  moment  O 
are  older  symbols  for  moments  or  differences  then  any 
used  by  ISV  Leibnits  they  being  used  by  me  in  my  Analy- 
sis abovementioncd  communicated  by  D'  Barrow  to  M*" 
Collins  in  the  year  16G0  &  the  symbols  dx  Si  dy  being  not 
used  by  M"*  Leibnitz  before  the  year  1C77.  And  whereas 
IVl'  Leibnits  prnefixes   the  letter  /  to  the   Ordinate  of  a 


•  "yueiiiadiuodum  nb  cininento  quo<lain  MHtlteinatico  duduni  notutum  est."    These 
words  were  inserted  in  lU'rnoulli's  letter  in  the  Chartu  Volans  by  (.eibniz. 


NEWTON  TO  KEILL.  173 

curve  to  denote  the  Summ  of  the  Ordinates  or  area  of  the 
Curve,  I  did  some  years  before  represent  the  same  thing 
by  inscribing  the  Ordinate  in  a  square  as  may  be  seen  in 
the  Analysis.  My  Symbols  therefore  (so  far  as  I  have  used 
any  particular  symbols)  are  the  oldest  in  the  kind. 

The  other  argument  used  by  the  great  Mathematician, 
is  that  when  I  wrote  my  Princijjia  I  understood  not  the 
second  difFcrences,  as  ^a  certain  great  Mathematician  (viz' 
Bernoulli)  has  noted,  meaning  in  the  Scholium  to  y*  lO*** 
Proposition  of  y''  second  Book.  But  this  great  Mathema- 
tician is  grosly  mistaken  in  taking  the  method  there  made 
use  of,  w*^^  is  a  branch  of  the  method  of  converging  scries 
to  be  the  method  of  fluxions.  The  Elements  of  the 
method  of  fluxions  are  set  down  in  y*  2**  Lemma  of  the 
second  Book  &  are  very  diflerent  from  y*  method  made 
use  of  in  this  Scholium. 

The  author  of  the  Remarks*  cites  D*"  Wallis  as  favour- 
ing M""  Leibnitz  &  yet  D""  Wallis  in  the  Preface  to  the  first 
Volume  of  his  works  a.d.  1G95  writes  that  in  my  two 
letters  of  June  13  &  Octob.  24,  1676  I  expounded  my 
method  of  ffluxions  to  M"^  Leibnitz  found  by  me  ten  years 
before. 

In  my  Letter  of  10  Deccm.  1672  sent  to  M'^  Collins,  in 
writing  of  a  method  whereof  the  method  of  Tangents  of  « 
Slusius  was  but  a  Corollary,  &  which  stuck  not  at  surds,  & 
w^^  was  therefore  the  method  of  fluxions,  I  represented 
that  this  method  was  very  general  &  amon{g}st  other 
things  extended  to  the  determining  the  curvature  of 
Curves.  Whence  its  manifest  that  I  then  understood  the 
second  fluxions  or  diflcrcnccs  of  diflcrcnccs. 

I  received  yo"^  Letter  this  afternoon  at  three  of  the 
clock  &  have  time  to  add  no  more  but  that  I  am 
Yo*"  most  humble  Servant 
London  20  April  {Tuesday}  1714.  Is.  Newton 

*  In  the  Journal  Litcraire.    See  a»(ru,  p.  169. 


174 


LETTERS  OF 


In  the  book  of  Quadratures  where  1  use  prickt  letters 
for  fluxions  I  solve  some  Problems  in  the  Introduction  to 
y*  book  without  making  use  of  such  Letters  &  therefore 
did  not  then  confine  the  method  of  fluxions  to  such  Let- 
ters. 

For  the  R*"*  D'  John  Keill  Professor 
of  Astronomy  in  the  University  of 
Oxford. 


LETTER   LXXXVIII. 
NEWTON  TO   KEILL. 


S' 


I  have  read  over  your  Letter  &  find  it  right.  The 
Marquess  de  L'  Hospital  in  his  Treatise  de  Infinitement 
Petits  teaches  that  if  the  Ordinates  AB,  CD,  EF  be  at 
Equal  distances,  &  the  chord  BD  be  produced  till  it  cuts 
the  Ordinate  EF  produced  in  N,  the  line  jPTNT  shall  be  the 


second  difference  of  the  three  Ordinates.  And  the  points 
B,  D  being  infinitely  neare,  perhaps  Burnoulli  may  take 
BD  for  a  tangent  of  the  Curve  at  Z)  &  so  reccon  that  the 
distance  between  the  Curve  &  y®  Tangent  is  the  second 
difference :  whereas  BDN  is  not  a  tangent  but  cuts  the 
Curve  at  Z),  &  the  tangent  at  D  is  parallel  to  the  chord 
BF  &  bisects  the  second  difference  FN,  suppose  in  G. 
So  that  the  line  FG  w*^^'  lies  between  the  Curve  &  the 
tangent,  &  is  equal  to  the  third  term  of  the  series,  is  but 
half  the  second  Difference,  as  I  have  put  it.    IM""  Burnoulli 


NEWTON  TO  KEILL.  175 

therefore  is  mistaken  in  affirming  that  I  put  the  third 
terme  of  the  series  equal  to  the  second  difference,  &  I  am 
in  the  right  in  putting  it  equal  to  y*  line  between  the 
Curve  &  the  Tangent  &  by  consequence  to  half  the  second 
difference  as  you  observe.  And  I  think  yo'  Demonstration 
is  good. 

I  have  corrected  a  paragraph  in  y*  11***  page  of  y"  papers 
you  sent  me  &  put  it  thus.  ,  [3^'^^  We  do  not  dispute  about  the 
,  antiquity  of  the  symbols  of  flluents  Fluxions  &  Moments, 
,  Summs  &  Differences  used  by  31'  Newton  &  M'  Leibnitz, 
,they  being  not  necessary  to  the  method,  but  liable  to 

,  change.     And  yet  the  symbol 


6ix 


used  by  M'  Newton 


,in  his  Analysis  for  fluents  or  summs  is  much  older  then 

,the  symbol  /  - —  used  by  M"^  Leibnitz  in  the  same  sense. 

,And  some  of  the  symbols  of  fluxions  used  by  M""  Novton 
,are  as  old  as  his  said  Analysis,  whilst  M'  Leibnitz  has  no 
,  symbols  of  fluxions  to  this  day.  And  the  rectangles  under 
,the  fluxions  &  the  letter  o  used  by  M*"  Newton  for  mo- 
,ments  are  much  older  then  the  symbols  dx  &  dij  used 
,by  M"^  Leibnitz  for  the  same  quantities.  But  these  are  only 
,ways  of  Notation  &  signify  nothing  to  y®  method  it  self 
^^ych  j^oy  ]^Q  without  them].  I  have  made  this  alteration 
to  avoyd  quoting  my  Manuscripts  w*^**  are  not  upon  record. 
And  for  the  same  reason  the  last  leaf  of  the  papers  you 
sent  me  must  be  altered.  But  I  have  time  to  add  no 
more  at  present  but  that  I  am 

Yo'  most  humble  Sen'ant 

London  May  11*^*  1714.  Is.  Newton 

For  the  IV^  D'  JoiLV  Keill  Professor 
of  Astronomy  in  the  University  of 
Oxford. 

*  The  post  mark  is  13  Ma. 


176  LETTERS  OP 

LETTER   LXXXIX. 
NEWTON  TO   KEILL. 
S'  London  May  16  1714. 

I  wrote  to  you  on  Tuesday  that  the  last  leafe  of  the 
papers  you  sent  me  should  be  altered  because  it  refers 
to  a  Manuscript  in  my  private  custody  &  not  yet  upon 
Record.  For  setting  right  this  leafe  it  is  to  be  considered 
that  altho  I  use  prickt  Letters  in  the  first  Proposition  of 
the  book  of  Quadratures,  yet  I  do  not  there  make  them 
necessary  to  the  method.  For  in  the  Introduction  to  that 
book  I  describe  the  method  at  large  &  illustrate  it  \v^*' 
various  examples  without  making  any  use  of  such  letters. 
And  it  cannot  be  said  that  when  I  wrote  that  Preface 
I  did  not  understand  the  method  of  fluxions  because  I 
did  not  there  make  use  of  prickt  letters  in  solving  of 
Problems.  The  book  of  Quadratures  is  ancient,  many 
things  being  cited  out  of  it  by  me  in  my  Letter  of 
24  Octob  1G76.  A  copy  of  the  first  Proposition  where 
letters  with  pricks  are-  used,  was  at  the  request  of  D"" 
Wallis  sent  to  him  in  the  year  1G92  &  the  next  year  pub- 
lished in  the  second  Volume  of  his  works.  And  in  the 
Principia  Pholosophia?  {sic}  pag  254  the  Notarum  formula? 
used  in  those  days  in  explaining  this  Proposition  are  re- 
ferred unto. 

ffluxions  &  moments  are  quantities  of  a  different  kind, 
ffluxions  are  finite  motions,  moments  are  infinitely  little 
parts.  I  put  letters  with  pricks  for  fluxions,  &  multiply 
fluxions  by  the  letter  o  to  make  them  become  infinitely 
little  &  the  rectangles  I  put  for  moments.  And  wherever 
prickt  letters  represent  moments  &  are  without  the  letter 
o  this  letter  is  always  understood.  Wherever  x,  J/,  y,  y  &e 
are  put  for  moments  they  are  put  for  a;o,  yoy  yoo,yo^.  In 
demonstrating  Propositions  I  always  write  down  the  letter 
o  &  proceed  by  the   Geometry  of  Euclide  &  Apollonius 


NEWTON  TO  KEILL.  177 

without  any  approximation.  In  resolving  Questions  or  in- 
vestigating truths  I  use  all  sorts  of  approximations  w***  I 
think  will  create  no  error  in  the  conclusion  &  neglect  to 
write  down  the  letter  o,  &  this  do  for  making  dispatch. 
But  where  x,  y,  y,  y  are  put  for  fluxions  without  the  letter 
o  understood  to  make  them  infinitely  little  quantities,  they 
never  signify  differences.  The  great  Mathematician  there- 
fore acts  unskilfully  in  comparing  prickt  letters  with  the 
marks  dx  &  rfy,  those  being  quantities  of  a  different  kind. 
MT  Leibnitz  has  no  mark  for  fluxions  &  therefore  prickt 
letters  are  older  marks  for  fluxions  then  any  used  by  him 
&  so  are  others  \$ic]  marks  used  by  mo  for  fluxions.  The 
rectangles  under  fluxions  &  the  moment  o  being  my  marks 
for  moments  are  to  be  compared  with  the  marks  dx  &  dy 
of  M"^  Leibnitz  &  are  much  the  older  being  used  by  me  in 
the  Analysis  communicated  by  D"*  Barrow  to  M""  Collins  in 
the  year  1669. 

The  Author  of  the  Remarks  represents  that  D"^  Wallis 
was  for  M^  Leibnitz  &  yet  the  D*"  in  the  Preface  to  the 
first  Volume  of  his  works  represents  that  I  in  my  Letters 
of  June  13  &  Octob  24, 1676  explained  to  M'^  Leibnitz  this 
method  found  out  by  me  ten  years  before  or  above,  that 
is  in  the  year  1666  or  1665. 

I  am 

Yo"^  most  humble  Servant 

For  the  R"**  D'  John  Keh^l  Professor  Is.  Newton 

of  Astronomy  in  tlve   University  of 
Oxford, 

Keill's  "Answer". to  the  Leibnizian  cartel,  drawn  up,  as  wo  see  by 
the  four  preceding  Letters,  with  Newton's  assistance,  appeared  in  llio 
Journal  Literaire,  for  July  and  August,  1714,  (Tom.  rv.  p.  319),  and 
produced  an  anonymous  reply  in  tlio  Lcipsic  Acta  for  July,  1716,  under 
the  title  of  Epistola  pro  eminente  MathematicOy  Dn.  Johanne  Bar- 
12 


178  LETTERS  OP  NEWTON  TO  KEILL. 

noulliOj  contra  quendam  ex  Anglia  antagcmutam  scrlpta*.  Among 
tlio  Lucasian  papers  (packet  No.  5)  are  found  the  draught  and  fair 
copy  of  an  answer  t  to  this  "Epistola,"  by  Keill,  in  French,  probably 
intended  for  insertion  in  the  Journal  Literaire,  but,  as  far  as  I  am 
aware,  never  published.  Newton's  Lctner  of  May  2, 1718,  (q.  v,  p.  185.) 
may  have  led  to  its  suppression. 


*  Tht8  was  in  reality  Bernoulli's  own  production,  though  in  a  disgtiised  form.  In 
its  original  shape  it  formed  almost  the  entire  contents  of  a  letter  to  Christian  Wolf  (dated 
Apr.  8,  1716),  who,  jointly  with  Leibniz,  interpolated,  abridged  and  otherwise  altered  it 
{e.g.  by  changing  the  first  person  into  the  third,  and  writing  antagonista,  Anglus  iste  or 
antagonitta  andax  for  Keiliui)  previous  to  its  insertion  in  the  Acts.  See  two  papers  by 
a  grandson  of  IJernoulli  in  the  Berlin  Memoirs  for  1799—1800  and  1802,  in  the  latter  of 
which  a  comparison  is  exhibited,  in  parallel  columns,  of  the  Epistola  and  the  MS.  copy 
of  Bernoulli's  letter  to  Wolf.  Bernoulli  was  extremely  anxious  to  preserve  a  strict 
incognito,  **  ingratum  enim,"  he  observes,  "  mihi  valde  foret  a  Keilio  bile  sua  perfricari 
et  contumeliose  traduci,  ut  solcnt  ejus  antagonists,  postquam  ille  me  hactenus  satis 
humaniter  tractavit."  Hermann  suspected  that  he  was  the  author,  **  quod  tamen,"  says 
Wolf,  in  announcing  the  fact,  "hactenus  constanter  negavi."  All  the  precautions, 
however,  that  had  been  taken  to  elude  detection  were  defeated  by  the  unlucky  "  meam" 
which  had  been  overlooked  in  the  process  of  transforming  the  letter  (See  p.  185  note  and 
p.  186).  It  was  more  than  a  year  before  Bernoulli's  attention  was  directed  to  the  over- 
sight, when  he  desired  Wolf  (Sept.  18, 1717)  to  insert  in  the  Errata  "pro  meam  legen- 
dam  esse  earn,"  adding  "sed  hoc  tamen  non  satis  quadrat;  vellem  itaque  ut  invenires 
modum  commodiorem,  quo  culpa  in  typothetam  plausibiliter  rejici  posset."  But  Wolf 
was  in  no  great  hurry  to  meet  his  wishes,  and  ten  months  later  we  find  Bernoulli  em- 
ploying his  son  Nicolas  as  his  mouthpiece  in  an  explanatory  statement  upon  the  subject, 
in  which  he  attempts  to  effect  his  escape  under  cover  of  the  change  which  his  letter 
had  undergone  in  the  editorial  hands  of  the  friend  to  whom  it  was  addressed.  See 
p.  185  note. 

i*  The  title  of  it  is  Lettre  de  Mr.  Jean  Keill...iL  Jean  Bernoulli,  This  may,  possibly, 
be  the  piece  alluded  to  by  J.  Bernoulli  in  his  article  on  Keill's  problem,  about  the 
path  of  a  projectile  in  the  air,  (Leipsic  Acts,  May,  1719,  p.  218.  0pp.  u.  395): 
"  Taceo  alia,  ut  rumor  fert,  dictu  horrenda,  ex  quibus  nuper  conflavit  libellura,  (editum 
an  ineditum  nescio)  qucm  tum  manuscriptum  circumferebat  prailo  destinatum.  Fue- 
runt,  ut  mihi  scribitur,  inter  ipsos  adversa;  partis  sequaces,  qui  perlegendo  cohor- 
ruerunt." 


LETTERS  OF  COTES  TO  NEWTON. 


LETTER  XC. 
COTES  TO  {NEWTON.     After  Apr.  26.  1718}. 
S'. 
I  think  it  my  duty  to  send  You  what  Observations 
I  could  make  of  the  late  Eclipse 

I  beg  Your  pardon  for  troubling  You  with  so  large  an 
account  of  my  Method  for  correcting  the  Pendulum.  I 
must  confess  to  You,  I  have  a  design  in  it  for  the  advan- 
tage of  our  yet  imperfect  Observatory.  The  Clock  which 
I  used  was  borrowed  of  a  Clock-maker  in  this  Town  who 
took  it  for  a  very  good  one.  Not  expecting  so  great 
inaequality  in  its  motion  I  was  very  much  surpriz*d  to  find 
it  by  the  Observations,  &  since  I  have  found  it  I  cannot 
think  of  making  use  of  such  ordinary  workmanship  again, 
unless  in  case  of  necessity.  To  speak  plainly,  I  beg  of 
You  to  let  that  excellent  Clock*  be  now  sent  do^vn  to  us 
which  You  order'd  to  be  made  for  the  use  of  our  Obser- 
vatory. I  cannot  think  of  a  more  accurate  Instrument 
for  the  setting  of  it,  than  such  an  one  as  I  have  been 
describing :  f  having  it  therefore  by  me  I  think  I  am 
prepar'd  to  receive  Your  Noble  gift.  I  have  written  to 
M'  Street  to  wait  upon  You  for  Your  resolution 
I  am  Sir  Your 

Obliged  Humble  Serv* 

Roger  Cotes. 
I  will  send  You  an  account  of  what  was  observed  at 
Cambridge  during  the  total  Obscuration  in  another  Letter, 


•  See  Letter  XCVIII. 

t  The  description  of  his  mode  of  adjusting  a  telescope  for  the  purpose  of  finding  the 
time  by  the  method  of  corresponding  altitudes  is  wanting  in  the  MS.,  which  is  only  a 
rough  draught  of  a  letter:  it  will  however  be  found  in  Smith's  Optics,  Vol.  ii.  p.  328. 

12—2 


3.     xxm.    6\47'.29"1^   Upper  limbl.p.^ 
6  .  50  ,  58  J  Lower  limbj 


180  LETTERS  OP 

On  tlie  opposite  side  of  the  leaf  is  the  following : 
Day 
1.      XXI.      4*^.01'. 21"  pm.  Sun's  upper  limb  observ'd 

at  y®  3*^  Pin 
2       XXII.     6^. 48'. 41     am.  Upper  limb    2**  Pin 
.    6  .  52  .  09    am.  Lower  limb    2"*  Pin 

] 

4      XXIII.    7^.5l'.l0"   am.  Upper  limb.  3*  Pin 
6       XXV.     6^.  44'.  53")  Upper  limb]    .  _. 

6  .  48  .  22  J  Lower  limbj 

6       XXV.     5^.  08'.  18"!  Lower  limb  I    J 

5  .11  .47  J  ^"^    Upper  limbj 
Allowing  for  the  variation  of  Declination  I  find 
By  y«  2**  &  3^  the  length  of  y«  Solar  day  measured  by 

the  Clock  was  24**.  oo'.  is". 

By  y*  3**  &  5*^  the  length  of  2  Solar  days  measured  by 

y*  Clock  was  48^.00'.  18"    Which   2  deductions  shew  the 

Clock  inequal.  of  motion 

By  y«  !■*  &  4*^  the  Meridian  of  y*  xxii**  day  was  at  11**.  5?'.  32" 

By  y«  5*^  &  6***,  the  Meridian  of  y®  xxv  day  was  at  11  .58'.  02" 
And  therefore  the  Meridian  of  y*  xxii  at       1 1  .  57  .  26 
I  put  the  correct  Meridian  of  y®  xxii  day  at  11  .  57  .  29 

The  "  Eclipse"  of  this  and  the  following  Letter  is  the  total  eclipse 
of  the  Sun  which  occurred  Apr.  22,  1715.     See  letter  cxvi. 

In  an  account  of  this  eclipse  by  Halley  (Phil.  Trans.  March— May 
1715 :  see  also  Number  for  Sept.  and  Oct.)  he  states  that  Cotes  "had 
the  misfortune  to  bo  opprcst  by  too  much  company,  so  that,  though  tho 
Heavens  wore  very  favourable,  yet  he  miss'd  both  tho  time  of  tho 
Beginning  of  the  Eclipse  and  that  of  total  Darkness.  But  ho  observed 
tlie  Occultations  of  tho  three  spots. .  .the  End  of  total  Darkness. .  .and 
tho  exact  End  of  the  eclipse  at  10''.  21'.  57' V  Some  of  its  popular 
effects  are  described  by  Mead  in  his  "Do  Imperio  Solis  ae  Lunoj  in 
Corpora  Humana"  Lend.  1746.  pp.  65^  66. 

llud  in  his  diary  under  tho  date  Apr.  11,  after  noticing  the  time  of 
the  middle  of  the  eclipse  as  calculated  by  Whiston  and  Halley,  adds 
♦♦M'  Robt.  Smith  T.C.C.fe.  says  at  7  min:  past  9.  but  I  suppose  Ho 


COTES  TO  NEWTON.  181 

calculates  for  Cambridge ;  whereas  they  calculate  for  London.    Obscrvo 
who  is  nearest  the  truth." 

In  the  Memoirs  of  the  French  Academy  for  1715  there  are  no  fewer 
than  seven  papers  on  the  subject  of  this  eclipse,  not  to  mention  several 
others  relating  to  the  luminous  ring  round  the  Mofm's  disk  during  tho 
time  of  total  obscuration,  which  tho  writers  endeavour  to  account  for 
without  having  recourse  to  tho  hypothesis  of  a  lunar  atmosphere,  to 
which  Louvillo  and  Ilallcy  attributed  the  phenomenon.  One  of  these 
papers  by  Maraldi  commences  with  the  remark  that  this  eclipse  "  est 
mdmorablo  par  sa  grandeur,  par  la  rencontre  d*  uno  Tacho  qui  s'est 
trouvee  dans  lo  Soleil,  &  par  les  Personages  Augustes  qui  I'ont  obser- 
vee" — tlio  King,  the  Duke  of  Orleans  and  a  brilliant  Court.  It  was  tho 
last  eclipse  that  had  tho  honour  of  being  observed  by  the  Grand  Mon- 
arque.     Louis  died  on  the  21st  of  August  following. 


LETTER  XCI. 
COTES  TO  {NEWTON}. 


D"^  Bentley  has  told  me,  You  have  been  pleas'd  to  give 
orders,  that  the  Clock  may  be  sent  to  Cambridge.  I  take 
this  oportunity  of  returning  You  my  hearty  thanks  for  it, 
&  of  giving  You  an  account  of  what  was  observ'd  by  Us 
during  the  time  of  the  sun's  total  obscuration  in  the  late 
Eclipse,  so  far  as  I  judge  it  to  be  of  any  moment.  The 
sky  was  perfectly  clear  all  the  Morning  till  about  two  or 
three  minutes  afler  the  recovery  of  the  suns  light.  It 
surpriz'^d  us  to  find  so  great  a  quantity  {of}  Light  re- 
maining in  the  middle  of  the  Eclipse :  I  think  it  did  very 
much  exceed  the  brightness  of  the  clearest  Moon-light 
nights.  A  Freind  assur'd  me  He  could  very  easily  &  dis- 
tinctly read  the  smallest  letters  engrav'd  about  ]VIr  Whistons 
Scheme  of  the  Heavens,  which  he  had  in  his  hands  at  that 
time.  We  saw  the  Planets  Jupiter,  Mercury,  &  Venus, 
with  some  fixed  stars,  but  they  appear'd  with  far  less 
splendour  &  fewer  in  number  than  we  expected,  or  than 
they  might  have  done  by  Moon-light.  I  took  the  greatest 
part  of  this  remaining  light   to  proceed  from  the  King 


182  LETTERS  OF 

which  incompa98*d  the  Moon  at  that  time.  As  nearly  as 
I  could  guess,  the  breadth  of  this  lling  was  about  an  eighth 
or  rather  a  sixth  part  of  tlie  Moons  Diameter,  the  light  of 
it  was  very  dense  where  it  was  contiguous  to  the  Moon 
but  grew  rarer  continually  as  it  was  further  distant,  till  it 
became  insensible :  its  colour  was  a  bright  clear  white. 
I  saw  this  King  begin  to  appear  about  five  seconds  before 
the  total  immersion  of  the  suns  body,  &  it  remain'd  visible 
to  me  as  long  after  His  emersion.  I  did  not  apply  my  self 
to  observe  whether  it  was  of  the  same  breadth  in  all  its 
parts  during  the  total  Obscuration.  Mr  Walker*  a  Fellow 
of  our  College  whom  I  can  very  well  depend  upon  assur'd 
me  He  was  very  certain  it  was  not.  He  says  He  took  notice 
with  a  great  deal  of  attention  that  at  first  the  Eastern  part 
was  very  sensibly  broader  &  brighter  than  the  Western, 
afterwards  they  became  equal,  &  some  time  before  the 
emersion  the  Western  side  was  manifestly  broader  & 
brighter  than  the  Eastern.  His  design  in  attending  so 
diligently  to  such  an  Observation  was  this;  He  thought, 
as  he  afterwards  told  me,  that  I  might  desire  to  note 
the  Time  of  the  middle  of  the  Obscuration ;  &  being  in 
the  same  Room  with  me.  He  was  willing  to  assist  me  in 
judging  of  that  Time,  &  beleiv'd  the  method  which  He 
took  to  be  the  propcrcst  for  it ;  accordingly  I  do  remember 
that  I  heard  him  call  out  to  INIc,  Noiv's  the  3Iiddle,  though 
I  knew  not  at  that  time  what  he  meant.  I  think  this 
Observation  of  M*"  Walkers  is  of  moment,  I  have  therefore 
been  very  particular  in  giving  You  the  circumstances  of  it 
that  You  may  Your  self  judge  how  far  it  may  be  depended 
upon,  for  my  part  I  cannot  see  any  reason  to  doubt  of  it. 
Besides  this  Ring  there  appeared  also  Rays  of  a  much 
fainter  Light  in  the  form  of  a  rectangular  Cross :  I  have 
drawn  You  a  Figure  which  represents  it  pretty  exactly, 


•  Richard  Walker,  afterwards  (in  1734)  Vice-Master,  Bentley's  devoted  adiierent. 
Though  four  years  junior  to  Coles,  in  acodcmical  etandingr,  he  was  six  ycai>j  oKlcr, 
having  been  entered  ut  the  mature  age  of  27. 


COTES  TO  NEWTON.  183 

IIS  it    appeard  to  Me.      The    longer  &  brighter    braneh 


of  this  Cross  lay  very  nearly  along  the  Eeliptick,  the 
light  of  the  shorter  was  so  weak  that  I  did  not  con- 
stantly see  it.  The  colour  of  the  Light  of  both  was  the 
same :  I  thought  it  was  not  so  white  as  that  of  the  Ring 
even  in  it's  fainter  parts,  but  verg'd  a  little  towards  the 
colour  of  very  pale  copper.  You  may  observe,  that  in  my 
Figure  the  branches  of  the  Cross  are  represented  as 
bounded  by  parallel  lines,  for  so  it  was  they  appear'd  to 
me.  But  there  are  others  here,  who  saw  a  very  differ- 
ent form.      I   have  therefore  sent  You    another  Figure 


fy 


m^MMMkmi\4\mmA¥*iim'^ , 


184  LETTER  OF 

the  mo8t  remote  of  any  I  have  met  with  from  my  own, 
This  was  dra^vn  by  a  very  ingenious  Gentleman  represent- 
ing the  appearance  as  seen  by  himself.  He  differs  also 
from  me  in  this  particular,  viz'  that  he  takes  the  Cross 
light  to  be  only  a  continuation  of  the  Ring  whereas  I  make 
'em  to  be  intirely  distinct  from  each  other.       I  am  Sir. 

May  13.  1715. 


Edmund  Halley  bom  in  London  1656,  died  1742. 

.      LETTER  XCII. 
HALLEY  TO  KEILL. 
Dear  S"^  London  Octob  3°  1716 

We  have  printed  a  French  translation  of  y®  account  of 
the  Commercium  given  in  the  Transactions*,  in  order  to 
send  it  abroad :  S""  Isaac  is  desirous  it  should  be  publisht 
in  the  Journal  Literaire,  and  M"^  Gravesant  has  promised 
to  gett  it  done,  but  cares  not  to  do  it  as  of  his  own  head ; 
and  therefore  proposes~EIiat  you  would  signifie  to  M*"  John- 
son at  the  Hague,  by  a  letter  enclosed  either  to  S'^  Isaac 
or  me,  that  you  are  desirous  that  the  said  French  paper  be 
inserted  in  his  Journal,  as  containing  the  whole  state  of  y® 
controversy  between  you  and  M"^  Leibnitz.  S'  Isaac  is 
unwilling  to  appear  in  it  himself,  for  reasons  I  need  not 
tell  you,  and  therfore  has  ordered  me  to  write  to  you 
about  it,  who  have  been  his  avowed  Champion  in  this 
quarrell ;  and  he  hopes  you  will  gratifie  him  in  this  matter 
by  the  first  opportunity  f 

I  have  rec*^  Cloaks  Lady  days  rent,  but  hear  not  one 


•  For  Jan.  and  Feb.,  1715,  pp.  173—224.  "An  Account  of  the  Book  entituled 
Commercium  E}HStoUcum " 

t  Keill,  gladly  enough,  no  doubt,  complied  witli  the  request.  The  French  transla- 
tion of  the  "Account"  or  Abstract,  alluded  to,  was  inserted  in  the  7th  Vol.  of  the 
Journal  Literaire^  pp.  114—158,  and  344—365.  A  Latin  translation  of  the  "Account" 
was  prefixed  to  the  2nd  Kd.  of  the  Commercium  Epiftolicum,  (1722). 


IIALLEY  TO  KEILL.  185 

word  of  Spctty;  Pmy  let  mo  know  what  I  shall  say  to 

him  about  the  Lease,  and  I  will  endeavour  to  make  him 

pay  the  Years  rent  due  at  Lady  day,  or  at  least  the  best 

part  of  it,  before  I  come  down  to  you,  which  will  not  be 

long. 

I  am 

Dear  S'  your  most  faithfull  Serv* 

Edm:  Halle y. 


LETTER  XCIII. 

NEWTON  TO  KEILL. 
D'  Keill 
I  received  about  a  month  ago  the  inclosed  Letter  from 
M'  Monmort*.  It  conteins  some  extracts  of  Letters  to 
him  from  IVr  Bernoulli  &  his  son.  The  chief  point  is  that 
W  Bernoulli  denies  f  that  he  is  the  author  of  y*  Memoir 
entituled  Epistola  pro  eminente  &c  that  is  inserted  in  the 


•  Born  1678,  died  Oct.  7,  (N.S.)  1719.  Ho  acted  as  a  sort  of  messenger  between 
tlio  Codes  of  the  Leibnizian  bridge,  as  Fontenelle  calls  Demoulii,  and  some  of  the 
English  mathematicians.  See  his  Eloge  by  Fontenelle.  We  see  him  here-,  and  on 
another  occasion  (p.  187),  in  the  amiable  character  of  a  peacemaker.  The  extracts 
from  his  letters,  which  were  emulously  published  against  each  other  after  liis  death,  by 
the  belligerent  parties,  shew  that  he  could  go  considerable  lengths  in  adapting  his  lan- 
guage to  suit  the  different  tastes  of  his  correspondents.  His  pen  has  left  us  an  im- 
passioned tribute  to  the  beauty  and  accomplishments  of  Newton's  niece,  Miss  Catha- 
rine Barton.    Letter  to  Taylor,  Apr.  1716,  in  Coutemp.  Philos.  p.  93. 

t  In  the  Leipsic  Acts  for  the  following  June,  by  way  of  Appendix  to  a  paper  on 
trajectories,  Bernoulli's  eldest  son,  Nicolas,  then  23  years  of  age,  took  occasion  to  refer 
to  the  subject  of  the  "  Epistola  pro  eminente  Mathematico,"  and  to  express  his  father's 
annoyance  at  the  rumour  which  attributed  it  to  him.  He  admits,  says  Nicolas,  that  at- 
the  request  of  a  friend,  he  put  down  in  writing,  *'  sine  ulla  animi  commotione,"  the 
main  of  the  facts  contained  in  the  Letter,  but  his  responsibility  did  not  extend  to  the 
"modus  scribendi"  and  form  in  which  the  Letter  appeared.  In  confirmation  of  this, 
Nicolas,  whose  Latin,  at  this  stage  of  his  explanation,  becomes  somewhat  obscure, 
points  to  the  ludicrous  oversight  into  which  the  toi-dhant  writer  falls  towards  the  close 
of  his  diatribe,  where  the  mask  drops  and  Bernoulli  is  found  speaking  in  his  own 
person.  "  Examinentetiam  considerentque,  quam  brevi  via  quamque  diversa  a  New- 
toniana  incesserit  Bernoullius,  {in  the  solution  of  the  inverse  problem  of  central  forces}, 
dicantque  postea,  an  alius  quispiam  praiter  antagonUtam  sibi  persuadere  possit,  meam 
formulam  ex  Newtoniana  esse  dcsumtam."    Leipsic  Acts  for  July,  1716,  p.  314. 


186  LETTER  OF 

Acts  of  Leipsic  1716.  The  Memoir  it  self  lays  it  upon 
M*^  Bernoulli  by  the  words  meam  solutianem,  &  if  M*"  Ber- 
noulli is  injured  thereby  it  is  not  you  but  the  author  of 
the  Memoir  who  has  injured  him.  The  injury  is  public 
&  in  justice  requires  a  public  satisfaction,  not  from  you 
but  from  him  that  has  done  the  injury.  The  question  is 
therefore  whether  you  will  take  notice  of  M*^  Bernoulli's 
excusing  himself  in  private  or  leave  him  to  do  it  in  publick. 
I  have  not  yet  returned  any  Answer  to  M'  Monmort,  be- 
cause I  thought  it  best  to  stay  till  I  had  your  sense  upon 
this  matter.  I  think  to  discourse  also  your  friends  D^ 
English*  &  D"  Bower  about  it.     I  am 

Your  faithful  friend  & 

humble  Servant 

London.  2  May.  1718.  {Friday}.  Isaac  Newton 

I  pray  return  M*"  Monmorts  Letter  by  D""  Halley  be- 
cause I  am  to  answer  it. 

For  D'  John  Keill,  Professor  of 
Astronomy  at  Oxford. 

This  letter,  as  has  already  been  observed,  p.  1 78.  may  have  been  the 
means  of  inducing  Keill  to  suppress  the  answer  whicli  he  had  prepared 
to  the  '*  Epistola  pro  cniinente  Mathcmatico."  Fragments  of  it,  how- 
ever, may  bo  discerned  in  a  Latin  dross  in  the  first  few  pages  of  a  sub- 
sequent publication,  the  origin  of  which  may  claim  a  notice  here. 


•  Keiira  cousin,  John  Inglb,  M.D.  Among  the  Lucasian  MSS.,  (packet  No.  3,) 
there  are  two  short  letters  from  him  to  Keill.  In  the  first  of  them,  (Dec.  19,  1717), 
after  congratulating  him  on  his  marriage,  the  writer  proceeds  as  follows :  "  Your  papers 
have  been  in  Sir  Isaac's  hands  ever  since  they  came  into  mine,  and  as  yet  I  have  heard 
nothing  about  them  ;  but  as  soon  as  1  receive  them,  I  shall  endeavour  to  forward  them 
to  Holland  by  the  first  sure  hand."  These  "  papers  *'  were  probably  Keill 's  answer  to 
the  Epistola  pro  eminente  Malhematico.  See  antea,  p.  178.  The  second  letter,  (Jan. 
14,  171^),  also  relates  to  the  aforesaid  "papers."  "  1  acquainted  Sir  Is.  Newton  that 
you  was  fully  satisfyd  with  his  corrections,  and  referr'd  the  whole  to  his  judgement ; 
which  he  received  very  kindly,  though  he  had  been  impatient  to  hear  from  you.  But 
you  have  forgott  to  send  me  back  his  paper,  as  we  had  done  to  take  a  copy  of  it,  and 

therefore  you  must  send  it  me,  to  free  Sir  Is.  of  the  trouble  of  going  over  it  again 

Doctor  Bower  is  yours."    Bower  was  M.D.  and  Professor  of  Mathematics  at  Aber- 
deen.   Ho  and  Inglis  were  Fellows  of  the  Royal  Society. 


NEWTON  TO  KEILL.  187 

In  the  Journal  Literaire  for  1716  Keill  had  published  an  article*  in 
defence  of  Newton  against  some  remarks  of  John  Bernoulli  and  his 
nephew  relative  to  the  inverse  problem  of  central  forces  and  the  error  in 
the  10th  Prop.  Book  2,  of  the  1st  ed.  of  the  Principia.  An  answer  to 
this,  framed  under  Bernoulli's  eye  by  a  pupil  of  the  name  of  Crusius, 
appeared  in  the  Leipsic  Acts  for  October  1718,  which  had  the  effect  of 
rousinn;  Keill  once  more,  lie  drew  up  a  reply  to  it  in  the  shape  of  a 
Latin  letter  to  Bernoulli,  but  while  the  brochure  was  passing  through 
the  press,  Newton  shewed  him  a  letter  which  he  had  received  (July 
1719)  from  Bernoulli  through  Monmort,  disavowing  the  authorship  of 
the  famous  letter  of  June  7»  1713.  Upon  talking  the  matter  over, 
Keill  seems  to  have  consented  to  proceed  no  further  with  the  publica- 
tion of  his  pamphlet t.  His  pacific  intentions,  however,  were  scattered  to 
the  winds  by  the  arrival  of  the  May  number  of  the  Leipsic  Acts  (1719) 
coutaining  a  paper  by  Bernoulli^  in  which  that  mathematician  ushers 


*  There  is  a  MS.  copy  of  this  among  the  Lucasian  papers,  (packet  No.  5)  :  it  is  en- 
titled **  Apjlogle  pour  le  Chevalier  Newton,  dans  laquelle  on  repond  aux  remarquet 
de  Messieurs  Jean  et  Nicolas  Bernoully  inserees  dans  les  M^moires  de  I'Academie 
Hoyale  des  Sciences  pour  les  ann^es,  1710  &c  1711,  par  J.  Keill...**  It  appears  tliat  on 
Jan.  19, 1716,  Halley  wrote  to  Fontenelle  with  a  view  to  this  moreeau  of  Keill's  being 
inserted  in  ihe  Mimoires  de  l*Academie,  where  the  papers  against  which  it  was  directed 
had  appeared.  Monmort  spoke  in 'favour  of  the  application,  but  the  feeling  of  the 
majority  of  the  members  was  adverse  to  it.  (See  Coutemplatio  Phib$i*phicaf  p.  85.) 
Fontenelle  in  his  answer,  (dated  March  8,)  a  copy  of  which,  ia  Keill's  hand,  is  extant 
in  a  folio  book  in  the  custody  of  the  Lucasian  Professor,  says,  *<  Nous  ne  ccdons  point 
ici  aux  Anglois  meme  en  estime  et  en  veneration  pour  M'  Newton.  Kt  PAcademie 
voudroit  fort  qu'  il  fust  possible  "  to  insert  Keill's  paper  in  their  Memoirs,  but  that  it 
was  their  invariable  rule  to  admit  only  articles  written  by  members  of  their  body. 

t  Quantum  sentio,  a  litibus  in  posterum  abstinebit,  (draught  of  a  letter  of  Newton  in 
Macclesfield  Corres.  u.  437. )  1  assume  that  the  letter,  of  which  the  draught  is  printed 
in  the  work  referred  to,  without  date  or  address,  was  addressed  to  Monmoit,  (about  the 
end  of  July,  1719,)  though  the  editor  (Preface,  p.  x)  states  that  "  it  was  found  impossi* 
ble  clearly  to  make  out  the  date."  '1  he  point  may  be  set  at  rest,  if  the  letter  to  which 
thU  is  an  answer,  should  turn  up  among  tlic;  Portsmouth  Papers. 

X  Joatinis  Bernoulli  Hespamio  ad  Non  neminii  Provocatumem,  ejutque  solutio  quit$ti' 
onis  ipai  ab  eodem  propositu:  de  invenienda  Linea  curva  quain  describit  projectile  in  medio 
resistente.  Lt'ipsic  ActSj  May,  1719,  p.  216.  BernouU.  0/>/>.  ii.  393.  The  tone  and 
language  of  this  piece  are  such,  that  even  Ikrnoulli's  friends,  tlic  conductors  of  the 
Acti,  thought  it  necessary  to  apologize  for  inserting  it  without  modification. 

Injustice  to  Keill,  it  ought  to  be  observed,  that  the  problem  which  led  to  this  ex- 
plosion does  not  appear  to  have  been  sent  as  a  challenge  to  Bernoulli,  and  still  less  to 
foreign  mathematicians,  as  has  been  represented.  It  was  mentioned  incidentally  in  a 
pri^ate  letter  of  his  to  Taylor,  in  which  he  expressed  a  wish  that  Bernoulli  would 
apply  his  skill  to  questions  of  real  utility  (as,  for  instance,  the  one  referred  to,  which 
Leibniz  had  attempted  in  vain),  instead  of  wasting  it  upon  such  problems  as  that  of 
Trajectories.  An  extract  from  this  letter  was  (contrary  to  Keill's  intention,  and  with- 
out his  knowledge)  sent  by  Taylor  to  Monmort,  who  forwarded  it  to  Bernoulli.  Keill 
seems  to  have  intimated  to  Monmort,  his  dissatisfaction  at  the  extract  being  communi- 


188  LETTER  OP 

in  a  construction  which  he  gives  of  a  generalization  of  Keill's  projectile 
problem  by  a  most  violent  attack  upon  its  proposer.  Forbearance  was 
out  of  the  question :  Keill  let  loose  his  "  Epistola  ad  Jo . . .  Bernoulli," 
(London  1720)  and  gave  further  vent  to  his  feelings  in  an  **  Additamen- 
tum"  appended  to  it,  which  he  closed  with  some  stinging  extracts  from 
Monmort's  letters  to  Taylor  who  kindly  supplied  them  for  the  pur- 
pose,— a  species  of  weapon  which  enabled  Bernoulli  afterwards  to  take 
ample  revenge  by  turning  it  upon  Taylor  (Leips.  Act.  May  1721,  p.  207 
seqq.  BemouU.  0pp.  ii.  493.  seqq.). 

There  are  rough  draughts  of  Keill's  letter  in  English  and  Latin  among 
the  Lucasian  papers,  and  part  of  it  was  read  by  Ilalloy  (no  doubt  in  the 
original  English)  at  a  meeting  of  the  Royal  Society  May  28,  1719  at 
winch  Newton  presided.  Before  publishing  it,  Keill  laid  a  complaint 
before  the  Royal  Society  agiiinst  his  adversary  "for  affronting  him 
with  scurrilous  language,"  and  called  upon  the  Society  to  take  steps  "  to 
shew  their  dislike  of  such  foul  proceedings."  "The  President  ordered 
that  the  consideration  of  this  complaint  be  deferred  till  Dr  Halley 
(Secretary)  comes  to  town,  &  that  enquiry  be  made  into  precedents  for 
the  better  information  &  direction  of  the  Society."  Journal  Book,  May 
26,  1720.  The  Society  does  not  seem  to  have  moved  any  further  in  the 
matter. 


J.  A.  Arlaud  or  Arland,  an  eminent  painter,  bom  at  Geneva  1668, 
died  1764.  "Newton  fut  son  ami,  et  lui  fit  present  de  la  version 
fran9aiso  de  son  Optique ;  il  etait  en  correspondance  avec  lui."  Biogr. 
Univ.     At  the  age  of  20  he  went  to  live  at  Paris. 

LETTER  XCIV. 

NEWTON  TO  ARLAND. 

Vir  celeberrime, 

Gratias  tibi  debeo  quam  maxinias  quod  Schema  expe- 

rimenti  quo  lux  in  colores  primitivos  &  immutabiles  sepa- 

ratur,  emendasti,  et  longe  elegantius  reddidisti  quam  prius. 

Sed  et  me  plurimum  obligasti  dum  Schema  illud  in  lamina 


cated  to  Bernoulli,  for  among  the  Lucasian  papers,  (packet  No.  2)  we  find  a  very  civil 
letter  from  Monmort  to  Keill,  (it  is  not  dated,  but  bears  the  London  post  mark, 
**Nov.  5,"  probably  in  1718),  in  answer  to  one  from  Keill  to  him,  (dated  Sept.  3)  in 
which  he  states  that  he  thought  that  the  extract  was  intended  to  be  sent  on  to  Bernoulli, 
and  protests  that  if  he  had  had  any  idea  of  the  offence  that  he  should  give,  he  would 
never  have  sent  it. 


NEWTON  TO  ARLAND.  189 

senea  incisum  &  inter  imprimendum  obtritum,  refici  curasti, 
ut  impressio  libri*  elegantior  redderetur.  Gratias  itaque 
reddo  tibi  quas  possum  amplissimas.  Quod  invcnta  mca 
de  natura  lucis  &  colorum  viris  summis,  D"°  Cardinali 
Polignacf  &  D^^^  Abbati  Bignon  non  displiceant,  valde  gau- 
deo.  Utinam  hajc  vestratibus  non  minus  placercnt  quam 
elegantissimso  vestrie  &  perfectissime  delineatce  picturro 
nostratibus  placuerunt.  Ut  Deus  te  liberet  a  doloribus 
capitis  &  salvum  conservet,  ardentissimo  precatur 

Servus  tuus  humillimus 

&  obsequentissimus 
Dabam  Londini  22  Oct.  1722.  Isaacus  Newton  J. 

Celeberrimo  Viro  D°°  Arland 


•  Peter  Coste's  French  translation  of  Newton's  Optics^  Paris,  1722. 

t  Born  1661,  died  1741.  Author  of  Anti- Lucretius  (a  posthumous  Latin  poem). 
It  is  said  that  he  took  great  pains  to  have  Newton's  fundamental  experiments  on  light 
properly  performed  in  France,  and  had  the  honour  of  receiving  a  letter  of  thanks  from 
our  philosopher  in  consequence. 

X  The  original  is  in  the  Library  at  Geneva,  to  which  institution  Arlaud  bequeathed 
several  medals,  paintings,  &c. 


COTES'S  CORRESPONDENCE  WITH  HIS  UNCLE. 


LETTER  XCV. 
COTES  TO  JOHN  SMITH. 
Written  while  CotoB  was  at  St  Paul's  School. 
S'.  {London  Dec.  31*.  1698} 

I  am  now  very  well  recovered ;  and  am  I  thank  God 
in  as  good  health  as  ever.  As  for  y*  works  of  Kepler,  and 
GalilsDO  as  far  as  I  can  learn  they  are  dispersed  in  divers 
Volumes,  put  forth  at  different  times.  I  have  from  severall 
choice  Catalogues,  as  Draudius's  Bibliotheca  classica,  A 
Catalogue  of  y*  Mathematical!  books  in  y®  Savilian  Library 
at  Oxford,  and  y*t  immense  one  of  D"^  Francis  Bernard's 
Library  which  is  now  under  y®  Auctioners  Mallet  at  Lon- 
don and  is  Like  to  continue  so  for  many  Months,  and 
severall  others  collected  what  I  could  find  of  those  t{w}o 
Learned  Authors.  I  send  'em  you  here  in  y®  latest  Edi- 
tions y*  I  could  find  there  set  down.  You  may  from  hence 
pitch  upon  those  you  most  like  of,  &  I  shall  be  very  glad 
to  use  my  utmost  endeavours  to  procure  'em  for  You— 

[Here  follows  in  the  MS  a  long  list  of  Kepler's  and  Gali- 
leo's works,  which  it  has  not  been  considered  necessary  to 
print}. 

I  suppose  there  might  be  added  to  each  Catalogue 
especially  to  y'  of  Galilaeus.  Perhaps  this  is  more  than 
You  expected  of  theire  Works.  The  first  Tome  of  Gali- 
liBus's  Works  translated  into  English  came  out  some 
Yeares  ago  Jin  1601}  ;  but  y®  Second  is  as  yet  unpublished 


•  The  day  of  the  month  is  taken  from  the  post  mark. 

t  "  A  Catalogue  of  the  Library  of  the  late  learned  D'  Francis  Bernard,  Fellow  of 
the  CollcKe  of  Physicians,  and  Physician  to  S.  Bartholomew's  Hospital....which  will  be 
sold  by  Auction  at  the  doctor's  late  Dwelling  House  in  Little  Britain :  the  Sale  to  l)egin 
on  Tuesday,  Octob.  4.  1698." 


COTES'S  CORRESPONDENCE  WITH  HIS  UNCLE.      191 

and  perhaps  will  never  see  light*.  I  have  my  self  Galileo's 
Nuncius  Sidereus  put  out  at  London  in  8^°  together  with 
Kepler's  Dioptricks  and  Gassendus's  Astronomy  ;  if  you 
please  I  ^nll  send  you  'em.  You  wrote  of  y*  Quadrature 
of  Curve's,  as  yet  I  cannot  enquire  of  any  Mathematician 
about  'em.  S""  Edw :  Sherbourn  in  his  Appendix  to  his 
Translation  of  INIanilius's  Astronom:  {Lond.  1G75}  tcU's  us 
y*  from  M*"  Isaac  Newton  is  expected  a  New  general  Ana- 
lytical method  by  infinite  Series  for  y*  Quadrature  of  Cur- 
vilinear figures.  I  have  D*"  Wallis's  Algebra  {London 
1685  j  I  think  I  bought  it  very  cheape  I  am  very  well 
pleased  w*^  y®  Book.  The  D*"*'.  Buisness  therein  is  to  shew 
y®  Original,  Progress  &  Advancement  of  Algebra  from 
time  to  time,  and  by  what  steps  it  hath  attained  to  y' 
height  at  which  it  now  is  he  give{s}  us  a  full  Account  of 
y®  Methods  used  by  Vieta  Harriot  Oughtred  De-Chartes 
and  Pell  &  others  and  of  y®  several  methods  of  exhaustions, 
Indivisibles,  Infinites,  Approximations  &c.  amongst  other 
things  he  speak's  of  squaring  Curves  and  after  other  ways 
of  approximations  shewed  he  show's  you  this  of  M"^  New- 
tonf  he  determin's  it  impossible  to  do  y®  buisness  exactly. 
In  my  mind  there  are  many  pretty  things  in  y*  book  worth 
looking  into.  If  you  have  a  mind  to  see  it,  or  have  not 
seen  it  already  I  will  send  it  w*^.  Galila)o'9  Nuncius  I 
thank  you  for  your  Directions  about  Instruments  in  your 
last  letter  dated  December  21  You  your  self  put  me  ofof 
y®  Instrumentary  way  while  I  was  with  you  but  I  meant  In 
my  Letter  such  Instrument's  y*  were  not  superseded  by 
calculation  or  some  more  exact  way ;  as  a  Quadrant  is 
♦  ♦  ♦  bigg  as  y®  *         themselves 


•  It  was  published  in  1665,  but  nearly  the  whole  impression  was  destmyed  by  the 
fire  of  T>ondon.    See  Macclesfield  Coiresp,  i.  120. 

t  From  the  famous  Letters  of  June  13  and  Oct.  24,  1676,  to  Oldenburg,  to  be  for- 
warded to  Leifiniz. 


192  OOTES'S  CORRESPONDENCE 

sometimes  be  at  a  loss  for  But  I  will  not  be  so  bold  as  to 
ask  my  Grandfather  for  y*  larger  size.  I  wi{ll}  *  • 
•         ♦  *         little  one  in  a  concave  case  with  y*  Cir- 

cles only  which  will  serve  y^  end  as  well  as  y®  largest  size 
it  will  als{o}  *  *  »  ♦      pocket  and  ready 

upon  all  occasions. 

I  am 
your  very  Obedient  Servant  and  Nephew 

These  For  y*  Reverend  W  Smith  of  Lea  R.  CoTES. 

nere  Oainshorough  IN  Lincolnshire  ^ 
Newark  Bagg 

The  lower  part  of  the  second  leaf  of  the  letter  lias  been  torn  off. 


LETTER  XCVI. 
JOHN    SMITH  TO  COTES. 

Dear  Cos:  Roger  Aug:  30, 1701. 

I  was  very  glad  to  hear  of  your  welfare  by  your  Father 
who  befriended  us  w*^  his  company  about  a  fortnight  ago ; 
he  showed  us  your  letter  in  w^^  you  expressed  a  feeble 
inclination  to  come  and  see  us  in  y®  Country,  we  thank  you 
for  y*,  and  count  it  a  favour  y*  you  can  spare  us  any  share 
of  your  affection  from  your  dear  M"  Mathesis ;  I  am  glad 
to  hear  y*  she  so  easily  yields  to  your  courtship,  and  has 
procured  you  such  signal  marks  of  favor  from  great  men  as 
D'  Bently  M'  Hanbury*;  I  am  sorry  y*  gentleman  is  so 


•  Nathaniel  Hanbury,  elected  from  Westminster  School  to  Trinity  College,  in  1677, 
admitted  Minor  Fellow,  Sept.  17,  1683,  (Charles  Montagu  was  admitted  Major  Fellow 
on  the  following  day).  He  published  Horologia  Scioterica  Pralibata. ..L-ond.  1683  ; 
and  Supplementum  Analyticum  ad  JEquationes  CartesianaSy  Cantab.  1691.  A  paper  by 
him  on  a  mode  of  approximating  to  the  value  of  7r  by  the  continual  subdivision  of  an 
arc  of  60",  was  produced  at  a  meeting  of  the  Royal  Society,  August  17,  1698.  He 
filled  various  College  offices,  and  we  are  told  by  Middleton,  that  Bentley  "  took  oc- 
casion to  convict  him,  in  a  solemn  manner,  by  the  testimony  of  all  the  College,  of 


WITH  HIS  UNCLE.  193 

overlookt  as  not  to  be  Vice-pro  {fe  |  ssor  instead  of  M^  Whis- 
ton;  for  I  believe  he  has  far  greater  Mathemat:  accom- 
plishments ;  I  hear  he  has  a  great  respect  for  you ;  con- 
sidering therefore  y*  favorable  fair-promising  circumstances 
you  are  under  I  cannot  forbear  presaging  in  your  behalf, 
w*  Ovid  did  to  his  friend,  Scena  manet  dotes  grandis  Amice 
tuas.  Divines  you  know  are  stiled  prophets,  as  well  as  y* 
poets  are,  &  I  fimcy  I  shall  be  a  true  one  in  this;  pro- 
vided you  so  moderate  your  studies  as  not  to  impair  your 
health ;  a  journey  into  y*  countrey  once  a  year  would  do 
well  for  y*  purpose ;  what  ?  I  warrant  you,  you  have  forgot- 
ten your  old  Ne  quid  nimis,  &  Interpone  tuis  &c.  ♦  but  I  am 
resolved  to  remember  you  of  em  now  &  then ;  I  had  writ 
to  you  before  but  expected  ever  &  anon  to  have  seen  you 
here  r  there  is  in  y®  monthly  accounts  of  y*  works  of  y* 
learned,  for  y®  year  1700,  month  December,  a  method  for 
finding  two  middle  proportional  lines,  w*^^  to  me  is  false, 
there  being  a  great  error  in  y®  demonstrat:  pray  look  upon 
it  a  little ;  I  should  be  glad  to  hear  of  you,  &  of  any  new 
discovery ;  I  never  saw  yet  what  discoveries  M*"  Plally  has 
made  in  his  voyage,  pray  comunicate  to  me  if  there  be  any 
thing  worth  while ;  &  you  will  much  oblige 

Your  most  affectionate  friend  &  uncle 

J :  Smith. 
My  wife  &  son  &  daughter  remember  their  kind  love 
to  you. 

For  W  Roger  Cotes  at  Trinity 
Colkdg  in  Cambridg 
Deliver  this  in  at  Caxton  to  go  to  Camhridg 


being  a  common  ncearer  8f  habitual  drunkard,  aT;d  without  inflicting  the  least  cen- 
sure upon  him  for  all  this,  made  him  not  long  after  |in  1712  &  1713}  the  Senii)r  Dean.'* 
Miscellaneous  Works,  iii.  356.  He  was  curate  of  St  Michael**  for  many  years.  He 
died  in  Nov.  1715,  and  Colbatch  was  elected  Senior  in  his  place. 

•  From  that  once  popular  school-book  Dionysii  Catonis  Disticha  dt  Moribut  ad 
Filium. 

Interpone  tuis  interdum  gaudia  curis, 

Ut  possis  animo  quemvis  sufferrc  laborem. 

13 


194  COTES'S  CORRESPONDENCE 

ITalloy  was  appointed  (Aug.  19,  1698)  to  the  command  of  the 
Paramore  Pinky  with  orders  to  make  a  series  of  observations  with  a 
■view  to  ascertain  the  law  of  the  variation  of  the  compass,  "  to  call  at 
his  majesty's  settlements  in  America  &  make  such  observations  as  are 
necessary  for  the  better  laying  down  the  longitude  &  latitude  of  those 
places,  &  to  attempt  the  discovery  of  what  land  lies  to  the  south  of 
the  western  ocean."    lie  set  out  on  his  expedition  Nov.  29,  and  was 
carrying  on  his  observations  some  degrees  south  of  the  line  when  the 
insubordination  of  his  officers  compelled  him  to  return  :    he  reached 
England  at  the  end  of  June  1C99.     In  the  following  September  he  em- 
barked again  in  the  Pink.   In  this  second  voyage  after  penetrating  beyond 
the  52d  degree  of  south  latitude  where  ho  was  stopt  by  icebergs,  he 
turned  his  course  northwards,  visiting  among  other  places  St  Helena  (a 
spot  familiar  to  him  by  his  sojourn  there  more  than  20  years  before), 
Pernambuco,   Barbadoes,   Bermudas  and   Newfoundland.      After  an 
absence  of  12  months  he  arrived  in  the  Thames  in  Sept.  1700.     His 
observations  were  embodied  in  a  General  Chart  which  he  published  in 
1701  "shewing  at  one  view  the  variation  of  the  Compass  in  all  those 
seas  where  the  English  Navigators  were  acquainted."     The  Journals  of 
his  two  voyages  were  published  by  A.  Dalrymple  (London  177»'>«  4to). 
These  are  the  voyages  to  which  Smith  alludes,  and  about  which  the 
young  Cambridge  student  could  give  him  no  information.     But  while 
Smith  was  writing  this  letter,  the  Captain  was  again  afloat  and  en- 
gaged upon,  if  ho  had  not  finished,  another  undertaking  in  which  his 
activity  and  spirit  of  enterprise  sought  employment.     At  a  meeting  of 
the  Royal  Society,  Juno  18,  I7OI  "the  Vice-President  (Sir  John  IIos- 
kyns)  informed  the  Society  that  Mr  Halley  was  gone  on  a  new  voyage, 
as  he  heard,  having  designed  to  make  nice  observations  on  the  Tides  & 
Currents  in  the   Channel,   for  the  Improvement  of  Navigation,  that 
thereby  by  their  different  times,  the  going  out  of  the  Channel  might  bo 
more  easy  against  contrary  winds."    And  on  July  30,  a  letter  was  read 
from  Halley  (Guernsey,  18  July)  "giving  an  account  that  the  weather 
having  been  fair  for  a  Month  past,  he  had  made  a  great  progress  in  the 
designs  he  had  in  making  this  voyage."    The  fruit  of  this  voyage  was  a 
largo  map  of  the  British  Channel  published  in  1702. 


WITH  HIS  UNCLE.  J  95 

LETTER  XCVIL 
COTES  TO  JOHN  SMIIH. 
Hon:  S"  !  Cambr:  Sopt  9.  1701. 

I  heartily  thank  You  for  Your  kind  Lett',  &  as  heartily 
beg  Your  Pardon  for  suffering  my  self,  by  so  long  delay, 
to  be  as  it  were  forc'd  to  returne  You  an  Answ'.  You  are 
pleased  to  express  a  greate  deale  of  Kindness  to  Me  in 
Your  Prophecies,  as  You  call  'em,  or,  as  I  would  rath'  have 
it,  Y'  Wishes ;  und'  which  Name,  not  y*  other,  I  again 
thank  You  for  it.  I  am  sorry  You  should  suspect  me  of 
forgetting  my  Ne  quid  nimia.  I  have  learn't  y*'  lesson  too 
p'fectly  &  'twould  be  more  adviseable  (for  y®  Accomplish- 
ment of  Y'  Prophecies)  to  rememb'  me  of  my  Old  Midta 
tulitfecitq:  &c.  The  Mesolabe  of  y'  wretched  pretend',  y' 
Quack  Geomet'  S*  Julicn  Potier,  one  cannot  but  admire 
for  it*s  grosness;  &  much  more  y*  laborious  confutation 
of  it  published  in  y*  same  paper  some  JMonths  after.  M' 
Halley's  late  discoveries  I  am  wholly  ignorant  of.  Surely 
You  mistake  Cambridg.  Wee  are  situated  in  as  dark 
a  Corner  of  y®  Land  (in  these  Matters)  as  can  well  be  de- 
sired. You  have  often  mentioned  to  me  y*  Quadratures 
of  Curves ;  &  particularly  (which  I  now  call  to  mind)  You 
have  wished  to  be  satisfied  in  pag.  374  of  Ne\vton.  I  per- 
suaded my  self  therefore  y*  something  concerning  this 
Matter  might  perhaps  make  amends  for  M'  Halley's  Story 
And  y*  I  may  be  as  short  as  is  possible,  I  desire  You  to 
Consid'  2  Lem.  2  Lib.  in  which  &  it's  Converse  y®  grounds 
of  his  Method  of  Fluxions  are  contained.  To  come  to  an 
Instance. 

Let  AMD  be  any  Curve;  AMy  AP,  PM,  any  Chord, 
Abscist,  Ordinate  of  this  Cu{r}ve  which  w***  y®  Arch  AM 
are  all  unstable.  Flowing,  increasing  or  decreasing  Quanti- 
tys ;  and  y^  too  after  a  certain  Law,  in  a  certain  proportion 
among  themselves,  according  as  y*  Nature  of  y*  Curve 

13—2 


196 


OOTES'S  CORRESPONDENCE 


requires.    Let  then  for  once  AP  stret*ch  it  self,  &  it's  very 
first  increase,  it's  primum  nascem  incrementumy  it's  Fluxion, 


it's  moment  be  an  infinitely  little  Pp ;  w*^  it  y*  Ordinate, 
Chord,  &  Arch  shall  also  change  themselves  into  jt?m,  Amy 
A  Mm  &  their  Moments  or  Fluxions  will  be  Rm,  Sm,  Mm. 
The  Area  AMP  will  also  have  it's  differentiola  or  Fluxion 
MPpm  differing  from  y«  D  MPpR  by  y«  A^®  MRm  infi- 
nitely little  in  respect  of  MP pR  which  is  it  self  infinitely 
little  in  respect  of  AMP :  Now  quantities  whose  difference 
is  infinitely  little  ought  to  be  look'd  upon  as  equall  by 
1  Lem.  1  Lib.  Newt.  For  y®  same  reason  y®  Sectour  AmM 
(which  is  y*'  Fluxion  of  y®  Bilinear  Figure  AM)  may  be 
account'd  equall  to  y®  A'®  AMS.  Now  naming  y®  Abs.  Ord. 
Ch.  &  Arch  X,  F,  Uy  Z.  Pp,  Rm,  Smy  Mm,  will  be  Xy  y, 
w,  Zy  according  to  y®  second  Lemma :  or  rath*"  let  us  name 
y®  Magnitudes  themselves  Xy  y,  w,  z.  &  their  Fluxions 
x-y.Uy  z.  'Tis  evident  y*  y®  Fluxion  of  y^  Area  will  be 
«  UMPpR^yx  To  particularize;  let  this  Curve  be  y® 
Parabola,  whose  Area  we  know  very  well  otherways. 
ax^yyy  Sx^  ^yy  orx^so  ^yx  ^  Fluxion  of  y®  Area  But 
y®  Fluent  of  ahx^x  (by  Lem.  2  Lib  2  convers.)  =  |-a^a;5 
«  l^jy  «=  Area).  In  Newton's  Hyperboloeid  c?  ^  x^y  or 
(j?x~^ «  y  now  in  our  case  y®  Fluxion  of  y®  Abscist  runs 
backward  &  is  therefore  =  -  a?  and  Fluxion   of  y®  Area 

=  -  y^  «  -  c^x~^x  whose    Fluent  y®  Area  «=  d^x'^  or  ~ 

X 

or  xy  is  reciprocally  as  x. 


WITH  HIS  UNCLE.  197 

This  may  p'haps  serve  as  a  Specimen  of  y*  Method  of 
Fluxions  applied  to  y*  buisness  of  Quadratures  tho  it's  uses 
seem  to  be  as  inexhaustible  as  they  are  Natural!  &  Easy 
for  by  it  y*  great  Geometers  of  our  Age  are  enabled  To 
draw  Tangents,  To  rectifie,  To  find  y"  Evolutes,  The  Caus- 
ticks  by  reflection  &  refraction  of  all  sorts  of  Curves,  To 
measure  y''  Surfaces  generated  by  their  rotation.  The  solids 
they  comprehend,  The  Centers  of  Gravity,  Oscillation  & 
Percusn.  of  all  these  To  resolve  all  sorts  of  Questions  de 
Max  &  Min.  To  find  y*  Points  of  Inflection  &  Rebrous- 
sement  (as  y®  French  term  it)  in  all  Curves  &  y*  Converse 
of  all  these  &  many  more  But  what  wonders  docs  it  not 
do  when  applied  to  Nature !  where  it  Triumphs  alone  & 
admitts  of  no  Partner  But  I  transgress  y*  Bounds  of 

a  Letf 

Pray  S""  pay  my  humble  respects  Y'  very  &c. 

to  my  Aunt ;  and  my  Love  to  Cozz".  R  Cotes 

These  to  the  Reverend  W  Smith  Rector 
of  Gate- Burton  tiear  Gainsborough 
by  Newark  '^  Caxton. 

This  letter  in  which  Cotes  gives  his  old  master  an  insight  into  the 
powers  of  the  new  Calculus  was  written  in  the  long  vacation  between 
his  2(1  and  3d  years.  It  is  a  very  creditable  performance  for  a  junior 
soph. 


LETTER  XCVIII. 
COTES  TO  JOHN  SMITH. 

Cotes  is  now  Plumian  Professor.    His  appointment  took  place 
Oct  16.  1707. 

Honoured  Unckle 
I  have  lately  been  at  London ;  I  found  Y'  Letter  at 
Cambridge  upon  my  return.    The  occasion  of  my  going  up 


198 


OOTES'S  -CORRESPONDENCE 


thither  was  partly  to  view  a  large  Brass  Sextant*  of  5  foot 
Radius  (y*  had  been  makeing  for  us  &  is  now  finished) 
before  it  should  be  sent  down.  Whilst  I  was  in  Town  S*" 
Isaac  Newton  gave  orders  for  y®  making  of  a  Pendulum 
Clock  which  he  designs  as  a  present  to  our  new  Observa- 
tory. The  Sextant  will  cost  y*  CoUedge  150^*^  &  I  bcleive 
S'  Isaac's  clock  can  cost  him  no  less  y"  SO'**.  We  have 
another  Instrument  in  hand  for  takeing  y®  Transits  of  Stars 
or  y®  Sun  &  Moon  over  y®  Meridian  &  then  we  shall  be 
pretty  v^ell  furnished  for  makeing  Observations.  All  Alti- 
tudes You  know  may  as  well  be  taken  by  a  Sextant  as  a 
Quadrant.  We  want  another  200'^^  if  we  can  procure  it  in 
y®  University  to  raise  up  another  Story  over  y®  gate  for 
Astronomical  uses.  I  have  lately  hit  upon  a  contrivance 
which  I  bcleive  will  be  of  very  good  use  for  observing 
Eclipses.  You  will  easily  understand  it  by  this  rude 
draught     The  Telescope  a  &  is  to  be  so  directed  as  to  look 


at  y*  Pole  of  y®  World  &  thereby  its  axis  will  be  parallel 


•  December  10,  1707  :  "The  President  in  the  chair.  A  draught  of  a  Sextant  made 
for  the  use  of  the  Astronomical  Professor  in  Cambridge  was  produced.  Dr  Harris  & 
M'  Halley  reported  that  it  was  very  exactly  done  by  Mr  Rowley."  Jounutl  Book  of 
the  Royal  Society.  A  plate  with  a  description  of"  this  noble  instrument'*  is  given  in 
Harris's  Lexicon  Technicum,  Vol.  ii.  Lend.  1710. 


WITH  HIS  UNCLE.  199 

to  y*  Axis  of  y*  Earth  in  which  position  it  must  be  fixd  by 
y*  rings  ef  gh  hl,cdis  o.  looking  glass  reflecting  y*  Object 
into  the  Telescope  Tlien  if  y*  Telescope  revolve  about 
its  Axis  within  the  rings  with  a  motion  correspondent  to 
that  of  y'  Earth  about  its  Axis  the  Object  will  constantly 
be  in  y*  Telescope  for  a  whole  day  togather  as  You  will 
easily  understand  by  considering  y*  y*  looking  Glass  parti- 
cipates of  y*  same  uniform  motion  by  being  fixt  to  y*  Tube. 
I  have  not  described  y*  method  of  altering  y*  Inclination 
of  y*  glass  according  to  y*  different  Declination  of  y* 
Object  from  y®  ^Equator  You  will  easily  find  out  how  y* 
may  be  done  as  also  how  a  piece  of  Clock  work  if  it  be 
thought  needfull  may  communicate  to  y*  Telescope  its  due 
motion  about  its  Axis*.  I  thank  You  for  y*  kind  Judg- 
ment You  made  concerning  my  Paper  about  Projectiles. 
I  have  by  me  another  such  a  Paper  concerning  y*  motion 
of  Pendulums  which  I  drew  up  about  y*  same  time  with  y*t. 
This  or  any  thing  else  You  know  You  have  a  right  to 
command  from  mc  haveing  taught  me  all  y'  little  which  I 
understand  in  these  matters.  I  am  glad  Coz  Rob'  has 
made  so  good  progress  in  Mathematicks  &  y*  he  has  a 
genius  suited  to  those  Study  s  as  I  all  ways  thought  he  had 
but  I  fear  You  are  too  diffident  of  his  Abilitys.  It  will 
undoubtedly  be  more  for  his  satisfaction  &  advantage  to 
be  admitted  Pensioner  y"  Sizer,  the  other  way  if  I  can  gett 
him  a  Poor  Schollars  place  will  be  about  lO''^  cheap"^  I  wish 
You  would  resolve  Y"^  self.  I  should  be  very  sorry  to  have 
advised  You  amiss  &  I  cannot  now  be  certain  of  futuritys 


*  It  will  be  seen  from  this  that  Cotes  anticipated  'a  Gravc^ande  in  the  principle  of 
the  lleliostAt,  by  upwards  of  thirty  yenrs.  lioth  liowovcr  had  been  forestallwl  by 
Ilooke  and  Halley.  llegist.  Bk.  Roy.  Soc.  ix.  23.  For  a  description  of  that  instrument 
see  's  Uravesande's  Physices  V.lementa  Mathematica,  3rd  Ed.  1742,  p.  715;  Biot's  TraitS 
de  Phyaique,  iti.  188.  Compare  Novi  Coinmentarii  Petropol.  i.  291  ;  Co<Idington'8 
Opticsy  (1st  or  2nd  Ed.)  Letter  of  Volluire  to  's  Gravesande,  June  1,  1741,  Cin  some 
editions,  1738),  a  paper  by  Hachette  in  thf  Journal  de  I' Kcole  Polytechn.  Torn,  ix, 
p.  263.  and  Liouville's  Journal,  1844. 

♦  These  papers  are  printed  among  Cotes'*  Ojiiiicuia  J\/a(AenM(ica  at  theeod  of  the 
Ilannonia  Mensuraruin,  pp.  80 — 91. 


200  COTES'S  CORRESPONDENCE 

However  I  will  consult  with  some  freinds,  y*  I  may  better 
know  how  to  direct  You*.  Pray  give  my  hearty  respects 
to  my  Aunt  &  my  Love  &  Service  to  Coz  Rob* 

I  am  Y'  &c. 
Cambridge  Febr.  10  1708  IIogeii  Cotes. 

I  lately  heard  y*  Coz  Tho  Summerfield  is  dead  at 
Ghent 

By  "our  now  Observatory"  are  meant  the  leads  of  tho  King's  Gate 
which,  by  a  College  order  dated  Febr.  5.  170?,  were  granted  to  the 
Plumian  Professor.  The  additional  "story"  mentioned  a  few  lines 
further  on  was  tho  work  of  several  years,  and  Cotes  did  not  live  to  see 
it  finished. 

Bentley  in  his  Letter  to  the  Bishop  of  Ely  (Febr.  1710)  boasts  of 
"  the  College  Gate  House  rais'd  up  &  improv'd  to  a  stately  Astrono- 
mical Observatory,  well  stor'd  with  the  best  instruments  in  Europe," 
and  in  another  letter  (Christmas  1712)  he  calls  this  erection  "the  com- 
modiousest  building  for  that  use  in  Christendom,  &  without  charging 
the  College,  paid  for  by  mo  &  my  friends."  In  ono  of  tho  articles 
aga'nst  him  laid  before  tiie  Bishop  of  Ely  in  July  1710  ho  is  accused  of  ^ 
"  applying  money,  which  ought  to  be  applied  only  for  tho  use  of  tho 
Library  towards  buying  instruments  for  an  Observatory,  which  ho 
caused  to  be  built  by  his  own  authority" — a  charge  which  is  true  with 
resi)ect  to  the  sextant. 

From  Blomer's  "Full  View  of  Bentley's  Letter"  (July  1710)  it 
appears  that  the  "  Finishing"  of  the  observatory  was  then  "going  on 
very  slowly  for  want  of  money  to  pay  the  workmen"  and  that  Bentley's 
estimate  of  the  expense  was  less  than  one  third  "  of  what  it's  like  to 
come  to."  (p.  120). 

On  Juno  8,  1717  an  order  was  made  by  the  Master  and  Seniors 
that  the  payments  of  the  Doctors  of  the  College  for  their  degrees  (£20 
per  man)  should  bo  "for  tho  present  applied  to  finish  the  Observatory" 
under  tho  superintendence  of  Prof.  Smith,  Cotcs's  successor  and  the 
"  Coz  Rob'"  of  the  above  letter. 

On  May  30,  1792  the  Vice-Chancellor  (Postlcthwaite,  Master  of 
Trinity)  and  the  other  Plumian  trustees,  having  before  them  the  fact 
that  the  Plumian  Professor  had  **  neither  occupied  the  said  rooms  & 
leads  nor  fulfilled  tho  conditions  for  at  least  50  years"  and  that  "  the 


•  "  Coz.  Robt."  was  admitted  a  Pensioner  under  Mr  L'dw.  Rud,  May  28  following, 
annos  natus  18.. .e  scholu  Leiccstriensi." 


WITH  HIS  UNCLE.  201 

observatory  &  the  instruments  belonging  to  it  were  through  disuse, 
neglect  and  want  of  repairs  so  mucli  dilapidated  as  to  be  entirely  unfit 
for  tlio  purposes  intended,"  agreed  to  give  up  all  claim  to  the  rooms  and 
leads  at  the  King's  Gate  and  to  allow  tlio  I^Iaster  and  Fellows  to  take 
tho  Observatory  down  or  convert  it  to  any  use  they  thought  proper. 

This  memorial  of  Bentlcy's  zeal  for  tho  promotion  of  science  was 
pulled  down  in  1797.  . 


LETTER   XCIX. 
COTES  TO  JOHN  SMITH. 
S^  Cambridge  Novomb*.  30  1710 

I  thank  God  we  go  on  very  well.  I  hope  You  arc  all 
in  good  health  notwithstanding  this  very  sickly  season.  I 
suppose  my  Cozen  told  You  in  his  Letter,  which  he  wrote 
on  Tuesday  last,  that  he  has  received  the  10'**  which  You 
sent  him.  I  talk'd  with  M""  Winston  to  day  &  gave  him 
Your  advice  of  making  a  recantation,  for  which  he  thanks 
You,  but  will  not  accept  it.^^  I  have  been  long  ago  well 
satisfied  y*  no  advice  from  any  private  person  can  possibly 
have  any  effect  upon  him  :  I  asked  him  therefore  whether 
y®  Judgment  of  y®  Convocation  might  not  be  a  sufficient 
ground  for  him  to  alter  his  Opinions  &  whether  he  should 
not  think  himself  obliged  to  desist  if  he  should  chance  to 
be  censured  by  them :  He  answered  me  in  the  Negative, 
unless  they  would  prove  to  him  that  his  Opinions  were 
wrong.  I  afterwards  told  him  y^  the  Church  must  in  3  or 
4  Yeares  recover  it's  Primitive  purity,  according  to  his  own 
Exposition  of  the  Kevelations ;  and  y*  therefore  it  would 
be  perhaps  adviseable  for  him  to  stay  till  y*  time  &  expect 
the  Issue  with  patience.  Upon  this  he  could  not  help 
discovering  himself  (as  1  imagined  he  would  do)  &  told  me 


•  VVhiston  had  been  expelled  a  month  before.  "Oct.  30,  1710.  This  time  M' 
W'histou  wjis  expelled  aa  an  obstinate  htretick  by  the  Heads,  after  he  hud  thrice  coii- 
vented  before  them."     Ilud'iJ  Diary. 


202       COTES'S  CORRESPONDENCE  WITH  HIS  UNCLE. 

y*  the  completion  of  y*  Prophecy  might  he  beleiv'd  depend 
in  good  measure  upon  y*  reviving  of  those  antient  Doc- 
trines in  which  he  was  at  present  engaged;  He  bid  me 
consider  what  answer  S*  Paul  would  have  given  to  one  y* 
should  have  dissuaded  him  from  preaching  the  Gospell, 
upon  this  reason  ;  y*  it  was  certainly  foretold  y*  the  Gospell 
should  be  preached  to  all  Nations.  You  may  easily  un- 
derstand, by  these  Answers,  upon  what  grounds  he  is  so 
very  resolute,  I  am  persuaded  'tis  in  vain  to  endeavour  to 
reclaim  him  till  y®  term  of  that  Prophecy  be  expired. 

I  am  Your  very  dutifuU  Nephew 

R  Cotes 

Pray  present  my  humble  respects  to  my  Aunt  &  my 
hearty  Love  to  my  Cozen. 

On  the  back  of  this  letter  besides  some  arithmetical  computations 
such  as  Mr  Smith  has '  written  on  Cotes's  first  two  letters  to  him  there 
are  also  notes  for  a  sermon  in  his  hand. 


LETTER  C. 

COTES  TO  {ISAAC  EWER.} 

S\  {Dec.  26  or  27.  1710} 

I  have  this  day  paid  to  M"  Medley  Ten  pounds  & 

inclosed  M'  Herring's  Bill  for  Fifty  two  pounds  which  is 

in  fiUl  of  Y'  dues  from  the  Jun'  Bursar's  OflSce.    I  cannot 

at  present  pay  y®  Interest  of  y®  Thousand  pounds  not 

having  Money  in  my  hands.     I  hope  in  a  very  short  time 

I  may  do  it  for  tis  reported  y^  the   Seniors  design  at  a 

meeting  this  day  to  order  the  Principal  to  be  paid  You  & 

to  vote  two  dividends  &  an  half  &  to  leave  (after  this  is 

done)  a  Thousand  pounds  in  Stock.   Tis  said  y*  M*"  Bathurst 

will  be  chosen  Senior  Bursar.     D*"  Ayloffe  &  M**  Barwell 


LETTER  OP  COTES  TO  E^VER.  208 

were  talked  of  for  Jun'  Bursar  &  Steward.  I  do'nt  hear 
who  is  to  be  y*  Pandoxator  unless  M'  Eden  be  y*  person 
intended. 

I  am  S' 

Y'  faithfull  freind 

&  humble  Servant 

Roo:  Cotes. 

This  letter  was  written  on  one  of  the  above  stated  days  as  will  bo 
seen  from  the  following  extract  from  Rud's  Diary,  "  I7IO.  Dec :  26 
was  appointed  the  day  for  voteing  Div.  but  when  they  were  mett  Mr 
Hanbury  objected  that  whatever  they  should  do  before  the  Seniority 
were  filled  up,  {a  Senior  fellowship  was  vacant  by  the  death  of  M' 
Mayer  on  Nov.  2}  would  be  unlawful  &  void ;  and  Ho  prevail'd,  so 
that  they  adjoum'd  to  the  Chappie  next  morning;  when  Mr  Cooper 
was  sworn  (ho  was  chosen  upon  ]Mr  Hawkyns's  death  in  Apr.  before) 
and  Mr  Hanbury  was  chosen  to  succeed  Mr  Mayer.  After  noon  they 
proceeded  to  vote  zj  a  Div.  for  1708,  &  2  whole  ones  for  the  2  next 
years.  The  first  Moyety  was  paid  in  the  Beginning  of  January." 
(Tliis  will  servo  to  correct  two  or  three  slips  in  Monk's  Bentley  pp. 
221,  222  note.) 

Bathurst  was  chosen  Sen.  Bursar,  Bar  well  Jun.  Bursar,  Whitfield 
Steward  and  Modd  Pandoxator.  The  statutable  day  for  swearing  in 
these  officers  is  the  day  following  the  dies  computiy  so  that  this  year 
they  ought  to  have  been  sworn  in  on  Dec.  28,  whereas  in  the  Admission 
BooK'the  date  is  Dec.  31  (Sunday).  If  this  date  bo  correct,  the  cause 
of  the  delay  is  probably  to  be  sought  for  in  the  dissensions  with  which 
the  college  was  distracted.  Modd  l^ad  filled  the  office  of  Sen.  Bursar 
8inc¥June  23,  1705  and  Cotes  that  of  Jun.  Bursar  since  Dccemb.  19, 
1707.  It  is  not  unlikely  that  Coles's  resignation  of  that  office  was 
connected  with  what  had  occurred  af-  tho  election  of  officers  and  lecturers 
in  October,  when  Bentley  was  overruled  by  the  Seniors  in  all  his  nomi- 
nations. "  They  had  taken  a  piquo  against  Mr  Whitfield  for  being  so 
desirous  of  that  office  {the  Latin  Lectureship}  &  therefore  pass'd  him 
by,  on  pretence  that  he  liad  one  place  already  {he  was  Steward} ;  l^Ir 
Cotes  was  also  past  by  on  tho  same  account,  &  they  choso  Mr  Pilgrim 
Lect.  Math,  in  his  room."     Rud's  Diary. 

The  sum  of  £1000  was  borrowed  by  the  College  in  I7OO  at  5  per 
cent,  to  be  appropriated  to  tho  repairs  of  the  Chapel.  It  was  advanced 
by  Bentley  out  of  his  wife's  fortune  on  a  bond  to  his  trustee  Mr  Isaac 


204  LETTER  OP 

Ewor  of  Lincoln's  Inn  (to  whom  this  letter  was  probably  addressed) 
and  was  repaid  by  instalments  in  ]711,  12,  13  and  14.  Conclusion 
Book  Sept.  6,  1706.  Sen.  Bursars  Books.  Lease  Book  p.  82.  Bent- 
ley's  Letter  to  Bp.  of  Ely  p.  19.  Blomer's  Full  View  p.  137.  Monk's 
Bentley  p.  103.    Articles  laid  before  Bp.  Moore  xxviii,  lb.  Appendix 

p.  XVIII. 


LETTER  CI. 
COTES  TO  {HALLEY}. 


This  letter  is  not  dated,  but  the  circumstance  of  its  being  written 
upon  the  same  sheet  of  paper  as  Letters  c,  cii  shews  that  it  is  sepa- 
rated by  no  long  interval  from  them.  It  is  clear  from  its  contents  that 
Ilalley  was  the  person  to  whom  it  was  addressed, 

S'. 
Tis  now  about  two  Yeares  since  I  wrote  to  You,  in 
behalf  of  M'  Jurin  a  Fellow  of  our  College,  to  desire  y* 
he  might  have  Your  leave  to  annex  some  of  Y*^  Treatises 
to  his  Edition  of  Varenius's  Geography.  You  was  pleased 
to  consent  to  it  &  to  promise  some  additional  improve- 
ments &  besides  a  new  Treatise  concerning  Coelestial 
Refractions.  I  hope  You  have  lately  received  a  Letter 
from  him  to  remind  You  of  Y'  promise,  &  to  desire  y*  a 
freind  of  his  may  wait  upon  You  for  Y'  Papers  assoon  as 
You  shall  have  leisure  to  finish  *em.  He  further  desires  if 
any  new  Figures  must  be  inserted  or  any  alterations  made 
in  y*  old  ones  y*  You  will  be  pleased  to  send  them  first  & 
y*  You  will  be  so  kind  as  to  send  him  word  what  he  had 
best  do  with  y®  iMap  of  y^  Trade  Winds  &  Variations  { of 
the  Compass}  &  whether  He  may  take  that  in  the  Miscel- 
lanea Curiosa  with  the  English  names  as  they  stand  there. 
The  greater  part  of  Varenius  is  already  printed  off,  we 
do  therefore  beg  of  You  to  finish  Y""  Papers  assoon  as 


COTES  TO  HALLEY.  205 

You  have  convenient  leasure.     I  beg  Your  pardon  for  the 
trouble  I  give  You. 

lamS' 

Y'  much  Obliged  &  Humble  Scrv' 

HooEit  Cotes. 

Jurin's  edition  of  Varenius  dedicated  to  Bentley  who  had  oncouragod 
him  to  undertake  the  work  bears  date  1712,  though  a  notice  of  it  ap- 
pears in  the  "  3Ioinoira  of  Literature"  for  Sept.  I7II.  The  copy  of  it  in 
Trin.  Coll.  Library  has  Cotes's  autograph  "  Donum  Amicissiini  Edi- 
toris." 


CORRESPONDENCE  OF  COTES  AND  JONES. 


William  Jones  born  1680,  died  1749.  See  life  of  Sir  William  Jones 
(his  son)  by  Lord  Teignmouth,  whore  six  of  these  seven  letters  of  Jones 
and  one  of  Cotes  are  printed,  but  very  inaccurately. 

LETTER  CII. 

COTES  TO  JONES. 

S'  Febr.  16.  1711 

I  yesterday  received  Your  most  valuable  &  accepta- 
ble gift*  togather  with  Y*"  very  kind  Letter  I  return  You 
my  most  hearty  thanks  for  'em  both.  You  have  highly 
obliged  the  Mathematical  part  of  y®  World  by  collecting 
into  one  Volume  those  curious  &  useful!  Treatises  which 
were  before  too  much  dispersed  but  more  especially  by  y® 
publication  of  y®  Analysis  per  iEquationes  infinitas  &  the 
Methodus  DifFerentialis.  I  could  heartily  wish  y*  nothing 
of  S^  Isaac's  might  be  lost,  I  hope  You  will  endeavour  as 
You  find  an  Oportunity  to  persuade  him  to  publish  some 
other  Papers  for  I  believe  he  has  yet  many  excellent  things 
in  reserve.  About  a  Year  &  an  half  ago  (when  I  was  last 
in  Town)  I  acquainted  Mr  Ralphson  y*  You  had  some  Pa- 
pers of  S'  Isaacs  in  Y""  hands  which  were  communicated 
long  ago  to  Mr  Collins.  I  thought  they  might  have  been 
pertinent  to  his  design  of  writing  y®  History  of  y°  Method 
of  Fluxions.  I  afterwards  understood  y*  You  gave  him  a 
sight  of  those  Papers,  &  y*  he  thought  'em  not  to  be  for 
his  purpose,  which  I  do  now  very  much  wonder  at,  if  his 
intention  was  to  do  justice  to  S"^  Isaac.  If  y*  was  not  his 
Intention  I  think  Your  Preface  has  already  sufficiently  de- 


•  A  quarto  volume,  edited  by  Jones,  containing  some  opu$cida  of  Newton's,  It  is 
entitled  Analysli  per  Quanlitatum  Series^  Fluxiones  ac  Differentias  cum  Enumtrat'ume 
Linearum  Tertii  Ordinis.     Lond,  1711. 


CORRESPONDENCE  OP  COTES  AND  JONES.    207 

feated  all  his  attempts.  AVe  are  now  at  a  stand  as  to  S' 
Isaac's  Principia,  he  designs  to  make  some  few  Experiments 
before  we  proceed  any  further.  The  first  Book  &  y*  six 
first  Sections  of  y'  Second  are  already  printed  off.  The 
inclosed  Paper  ^''  is  what  I  wrote  about  3  Yeares  ago  &  read 
to  my  Auditors  in  our  Schools  in  1709.  I  have  sent  it  to 
You  as  it  relates  to  y*  Methodus  Diffcrentialis  but  more 
particularly  as  a  small  acknowledgment  of  my  gratitude 
for  having  received  y*  and  the  other  excellent  Treatises 
from  Your  hands  &  as  a  token  of  my  hearty  freinilship  & 
sincere  good  will  to  You 

I  am  S'  Y*^  most  obliged  freind 

&  humble  Servant 

II  Cotes. 

Not  having  heard  any  thing  of  y*  book  till  I  saw  it  I 
received  it  with  y®  additional  pleasure  of  a  Surprize. 

Printed  in  tlie  Gen.  Diet,  iv.  443.  Macclesfield  Corr,  i.  257. 


LETTER  cm. 
JONES  TO  COTES. 
[Extract] 
S'.  London  Soptemb'.  17.  1711 

The  paper  concerning  S^  Is,  Newton's  method  of  In- 
terpolation, which  you  have  bin  pleas'd  to  send  me,  being 
done  so  very  neat,  that  it  wou'd  be  an  injury  to  the  Curious, 
in  these  Things,  to  be  kept  any  longer  without  it ;  there- 
fore must  desire  you'd  grant  me  leave  to  publish  it  in  the 
Phil.  Trans,  you  may  be  assur'd,  that  I  don't  move  this  to 
you,  without  S^  Isaac's  approbation,  who  I  find  is  no  less 


•  Prlntcil  among  hi»  Opera  Miscellanea  at  tl»e  cod  of  the  Ilanuouia  Menturantm, 
pp.  23—33. 


208  CORRESPONDENCE  OF 

willing  to  have  it  done.  The  new  Edition  of  the  Prineipia 
is  what  we  wait  for  with  a  great  deal  of  impatienee ;  tlio', 
at  the  same  time,  I  believe  the  Book  will  be  far  more  valu- 
able than  if  it  had  bin  done  in  a  hurry,  Since  I  find  the 
interruptions  are  necessary,  and  Such  as  will  render  it 
Compleat.  We  have  nothing  considerable  in  hand  here  at 
present,  only  W  De  Moivre's  Treatise  of  Chance  ^^  which 
makes  a  whole  Transaction,  he  is  very  fond  of  it,  &  we 
may  expect  it  well  done  :  M^  Raphson  has  printed  off  four 
or  five  Sheets  of  his  History  of  Fluxions,  but  being  shew'd 
S'.  Is.  Newton,  (who,  it  seems,  wou'd  rather  have  them 
write  against  him,  than  have  a  piece  done  in  that  manner 
in  his  favour,)  he  got  a  Stop  put  to  it,  for  some  time  at 
least.  D^  Halley  has  almost  finish'd  the  printing  of  the 
Greenwich  Observations  f,  which  will  be  a  work  of  good 
use ;  especially  as  it  is  now,  free'd  from  the  trifls  it  was 
loaded  with.  S"^.  I  have  one  thing,  which  I  wou'd  trouble 
you  with  further,  &  that  is,  to  let  me  know,  what  Lectures, 
or  other  Papers  of  S*".  Is.  Newton's,  remain,  in  your  Uni- 
versity, unpublish'd,  this  may  be  done  at  your  leasure  : 


*  "  De  INIensura  Sortis."  Phil.  Trans.  Jan March,  1711.     Comp.  Letters  CVII., 

CVIII.  Demoivre  was  born  at  Vitri  in  Champagne,  in  1667.  On  the  revocation  of  the 
edict  of  K antes,  he  settled  in  f2n{?hind.    He  died  Nov.  27, 1754. 

t  The  Observations  here  referred  to  (made  with  a  mural  arc)  form  the  2nd  Rook  of 
Flamsteed's  Historia  Calestis,  published  in  1712.  The  Observations  contained  in  the 
Ist  Book  (made  with  a  sextant)  were  printed  under  Flamsteed's  superintendence,  at 
Prince  George's  expense,  and  with  a  trifling  exception,  were  wrought  off  before  Christ- 
mas, 1707  ;  but  in  consequence  of  his  misunderstanding  with  the  Prince's  referees, 
which  seems  to  have  arisen  principally  from  his  objection  to  print  his  catalogue  of  the 
Fixed  Stars  before  the  2nd  I3ook  of  Observations,  the  task  of  editing  those  parts  of  the 
work  was  confided  to  Halley.  In  Flamsteed's  MS.  of  the  2nd  Book,  the  Observations 
stood  recorded  as  they  were  made,  but  Halley  arranged  them  under  the  heads  of  the 
Moon  and  planets  to  which  they  related,  not  giving  the  whole  of  the  Observations,  but 
retaining  only  those  of  such  Stars,  as  in  passing  the  meridian,  had  nearly  the  same 
right  ascension  and  declination  as  a  planet.  (See  Halley's  Pref.  to  Hist.  Cal,  and 
Baily's  Pref.  to  Account  of  Flumsteed,  p.  xli.) 

Some  years  elapsed  before  Flamsteed  had  any  other  means  of  revenging  himself 
upon  his  editor,  than  by  unsparing  abuse.  At  length,  in  April  1716,  having  got  pos- 
session of  300  copies  of  his  work,  he  separated  the  "  very  sorry  abstracts  "  of  his  Ob- 
servations, and  the  "corrupted  Catalogue"  from  the  part  which  he  had  himself 
superintended,  and  committed  nearly  the  whole  of  them  to  the  flames,  "  as  a  sacrifice 
to  Heavenly  Truth."     Baily,  pp.  101,  321,  322. 


COTES  AND  JONES.  209 


LETTER  CIV. 
COTES  TO  JONES. 

[Extract.] 
S".  {Cambridge,  Sept.  30,  1711.} 

I  return  You  my  thanks  for  Your  Letter  &  the  Infor- 
mation You  gave  me  concerning  the  State  of  Mathematicks 
at  present  in  London.  I  shall  be  glad  to  see  M'  De  Moi- 
vre's  Treatise  of  Chance  when  it  comes  out;  his  things 
are  always  very  neat  and  curious.  We  have  nothing  of  S' 
Isaac's  that  I  know  of  in  Manuscript  at  Cambridge,  besides 
the  first  draught  of  his  Principia  as  he  read  it  in  his  Lec- 
tures*, his  Algebra  Lectures  which  are  printed  &  his  Op- 
tick  Lectures  the  substance  of  which  is  for  y*  most  part 
contained  in  his  printed  Book  but  with  further  Improve- 
ments. I  thank  You  for  Your  kind  offer  of  recommend- 
ing my  Paper  to  the  Publick ;  but  I  am  of  opinion  that  it 
is  not  of  so  great  use  as  to  deserve  to  be  printed  after  S*" 
Isaac's  Methodus  DifTerentialis. 

*#**♦♦♦ 

I  am  very  desirous  to  have  the  Edition  of  S*"  Isaac's 
Principia  finish'd,  but  I  never  think  the  time  lost  when  we 
stay  for  his  further  corrections  &  improvements  of  so  very 
valuable  a  book,  especially  when  this  seems  to  be  the  last 
time  he  will   concern  himself  with   it.     I  am  sensible  his 


•  The  folio  volume  marked  Dd.9.46  in  the  University  Library,  corresponds  to  this 
description,  but  it  has  the  book-plate,  whicli  indicates  it  to  have  been  one  of  lUshop 
Moore's  books,  eriven  to  the  University  by  George  I.,  in  1715.  If,  then,  this  be  the  volume 
which  Cotes  means,  either  tlie  book-plate  has  been  pasted  in  by  mistake,  or  the  book 
must  have  found  its  wuy  somehow  into  the  Bishop's  library.  See  more  of  this  JNIS.  in  the 
notes  to  the  Synoptical  View  of  Newton's  Life,  under  August  1684,  and  Table  of  his 
Lectures  for  that  year. 

Newton's  presentation  copies  of  his  Optical  and  Algebra  Leeture$,  the  latter  in  his 
own  hand-writing,  are  still  in  the  University  Library,  marked  Dd.9.67  and  68. 

14 


210  CORRESPONDENCE  OF 

other  Business  allows  him  but  little  time  for  these  things 
&  therefore  I  ought  not  to  hasten  him  so  much  as  I  might 
otherwise  do,  I  am  very  well  satisfied  to  wait  till  he  has 
leasure. 

Printed  in  Gen»  Diet,  iv.  444.     Macclesfield  Corr.  i.  iJ58. 


LETTER  CV. 
JONES  TO    COTES. 
Dear  S'  London  Octob'.  26*'*.  1711 

The  favour  of  your  account  of  S^  Isaac's  papers  left  at 
Cambridge,  I  return  you  my  hearty  thanks  for;  And  as 
you  have  some  further  Considerations  about  the  Doctrine  of 
Differences,  I  am  assured,  they  cannot  but  be  valuable ;  and 
if  a  few  Instances  of  the  application  were  given,  perhaps  it 
wou'd  n't  be  amiss :  Having  tarried  some  time  for  a  con- 
venient opportunity,  I  was  at  last  oblig'd  to  send  you 
Mouton's  Book  by  the  Carrier ;  tho  it  will  only  satisfy  you 
that  D*".  Gregory  had  but  a  very  Slender  notion  of  the 
design,  extent,  &  use  of  Lem.  5.  Lib.  3  of  the  Principia;  I 
hope  it  will  not  be  long  before  you  find  leasure  to  send  us 
what  you  have  further  done  in  this  curious  subject ;  no  ex- 
cuse must  be  made  against  the  publishing  of  them  ;  Since, 
with  respect  to  Reputation,  I  dare  say,  'twill  be  no  way  to 
your  disadvantage. 

I  have  nothing  of  news  to  send  you ;  only  the  Germans 
and  French  have  in  a  violent  manner  attack'd  the  Philo- 
sophy of  S""  Is:  Newton*,  and  seem  resolv'd  to  stand  by 
Cartes ;  M*"  Keil*,  as  a  person  concern'd,  has  undertaken  to 
answere  &  defend  some  things,  as  D^  Friend*,  &  D'Meadf, 


•  See  Letter  CVII. 

t  Mead  was  concerned  as  the  author  of  a  work  De  imperio  Soils  ac  Lunet  in  corpora 
htimanaf  Lond.  1704. 


COTES  AND  JONES.  211 

does  (in  their  way)  the  rest :  I  wou'd  have  sent  you  y*  wliole 
Controversy,  was  not  I  sure  that  you  know,  those  only  are 
most  capable  of  objecting* against  his  Writings,  that  least 
understand  them  ;  however,  in  a  little  time,  you'l  see  some 
of  these  in  y*  Philos.  Transact. 


LETTER  CVI. 
COTES  TO  JONES. 

[Extract.] 

The  controversy  concerning  S*"  Isaac's  Philosophy  is  a 
piece  of  News  that  I  had  not  heard  of  unless  Muys's  late 
Book  be  meant.  I  think  that  Philosophy  needs  no  defence, 
especially  when  tis  attack't  by  Cartesians.  One  M*"  Green* 
a  Fellow  of  Clare  Hall  in  our  University  seems  to  have 
nearly  the  same  design  with  those  German  &  French  ob- 
jectors whom  You  mention.  His  book  is  now  in  our  press 
&  is  almost  finished.  I  am  told  he  will  add  an  Appendix 
in  which  he  undertakes  also  to  square  the  circle.  .  I  need 
not  recommend  his  performance  any  further  to  You. 
Nov.  11*^  1711 

Printed  in  Gen.  Diet.  iv.  444.     Macclesfield  Corr.  i.  261. 


LETTER  CVIL 

JONES  TO  COTES. 
D^  S^  Not.  16»\  1711 

I  receiv*d  yours  of  the  11*^.  instant,  and  am  glad  to  find 
you've  finish'd  your  second  Paper,  and  do  hope  it  will  not 


•  Sm  Utter  XXVII.  note. 

14—2 


212  CORRESPONDENCE  OP 

be  long  before  I  receive  it :  I  have  taken  this  opportunity 
of  p*senting  you  with  one  of  M*"  De  Moivre's  late  Tracts, 
tho  the  Author  himself,  perhaps,  may  send  you  another; 
how  well  he  has  handled  this  subject,  is  what  I  shall  not 
have  time  soon  to  consider.  The  Objections  of  y®.  writers 
of  the  Leipsic  Transactions,  against  the  Philosophy  intro- 
duced in  D\  Friend's  Chimical  Lectures*,  together  with 
his  answere,  as  also  those  of  Wolfius,  and  of  M^  Saurin 
of  the  Fr.  Academy,  against  y*.  same  Philosophy,  with  an 
answere  by  ^P.  Keilf,  are  now  in  the  Press  here,  and  nearly 
finish'd,  I  shall  not  be  wanting  to  send  them  you.  I  am 
concern'd  to  find,  by  S*".  Isaac,  that  his  Book  does  not  go 
forward,  *tis  a  great  grieveans  to  be  so  long  depriv'd  of  it, 
I  am,  S*",  very  much 

Your  friend  and  Servant 

W":  Jones 

S'.  you  need  not,  if  you  please,  make  known  to  the 
Person  that  brings  this,  that  I've  sent  you  Moivre's  Book. 

To  W  Roger  Cotes 
Thia 

Christian  "Wolf  (an  eminent  philosopher  and  mathematician,  bom 
I679,  died  1754,  at  tho  date  of  this  letter  a  Professor  at  Halle ;  see  Ten- 
nemann's  Hist,  of  Phil,  and  life  by  Degerando  in  the  Biographie  Uni- 


•  Freind's  PraUctiones  Chymica:,  LonJ.  1709,  dedicated  in  most  complimentary 
terms  to  Newton. 

"  Nov.  15,  1711.  The  President  in  the  Chair...The  editors  of  the  Acta  Ernditoi-um 
having  published  {September  1710,}  a  reflecting"  paper  upon  Dr  Freind's  Chymistry,a. 
Discourse  was  now  read  of  Dr  Freind's  in  vindication  of  his  book,  and  the  principles 
therein  maintained.  This  Discourse  was  ordered  to  be  published  in  the  Transactions, 
and  the  thanks  of  the  Society  returned  to  the  Dr."    Journal  Book  of  Royal  Soc. 

Freind's  defence  appeared  in  the  Phil.  Trans,  for  July — Sept.  1711,  pp.  330—342, 
under  the  title  of  "  Prajlectionum  Chymicarum  Vindiciae  in  quibus  objectiones  in  Actis 

Lips contra  Vim  materia)  Attractricem  allatie  diluuntur."   lie  shews  the  impropriety 

of  calling  attraction  "  an  occult  quality,"  and  takes  occasion  to  criticise  some  of  Leib- 
niz's opinions,  but  uses  only  the  initial  letter  of  his  name.  (Freind  afterwards,  in  1726, 
reprinted  the  article  of  the  Leipsic  Reviewers,  accompanied  by  his  Answer,  as  an 
Appendix  to  the  2nd  Ed.  of  his  Lectures.)  A  reply  was  published  in  the  Acts  for  June, 
1713,  pp.  307—314. 

t  See  the  remarks  at  the  end  of  the  Letter. 


COTES  AND  JONES.  213 

vorsclle)  in  his  "  Aoromotrice  Elomenta. . .  1709**  attacked  nn  argument 
wliicli  Kcill  in  liia  Lectioncs  Physicw  COxf.  1702)  had  advanced  in 
proof  of  a  vacuum,  founded  on  tho  fact  that,  abstracting  from  the  re- 
sistance of  the  air,  all  bodies  fall  from  equal  heights  in  tho  same  time, 
Keill  answered  his  objections  in  a  letter,  part  of  wliich  was  printed  in 
the  T^ipsic  Acts  for  Jan.  1710  (pp.  11 — 15),  to  which  his  antagonist 
replied  in  tho  following  Number  (pp.  7^^ — CO),  A  rejoinder  was 
prepared  by  Keill,  tho  first  portion  of  which  exists  in  MS.  among  the 
Lucasian  papers  (a  folio  sheet  in  packet  11).  This  seems  to  be  the 
piece  to  which  Jones  refers  in  the  above  I-K;tter,  though  I  do  not  remem- 
ber jto  have  ever  seen  it  in  print.  In  the  4th  page  of  this  last-men- 
tioned paper  Keill  proceeds  to  notice  some  of  the  views  propounded  by 
Saurin  in  a  Memoir  read  before  tho  Academy  of  Sciences  in  1701)  ("  Kx- 
amen  d'une  difficulte  cousiderablo  proposee  par  M.  Iluyghcns  contro  lo 
Systeme  Cartesien  sur  la  cause  do  lii  Pesanteur."  i^Iemoirs  for  that 
year,  p.  131,  published  in  1711.  The  ditliculty  alluded  to  is  that  if 
Descartes's  celestial  matter  circulates  with  the  enormous  velocity  that  it 
ought  to  have  in  order  to  produce  the  observed  effects  of  gravity,  it 
ought  to  hurl  away  all  the  bodies  on  tho  earth's  surface — (jnippe  fcrat 
rapidd  sccum  vcrratque  per  auras.  He  returned  to  the  subject  in  a 
supplementary  Memoir  in  171 C,  in  which  he  notices  the  allusion  which 
Malcbranche  in  the  last  ed.  of  his  "  Do  la  Recherche  do  la  Verite  *  had 
made  to  the  former  Memoir.)  Joseph  Saurin,  bom  1059,  died  1737»  was 
a  fervent  believer  in  the  system  of  Vortices,  the  impossibilities  of  which 
seem  to  have  had  a  piquancy  for  him  that  stimulated  his  faith.  lie 
frankly  admits  the  difficulties  that  surround  the  hypothesis,  and  the 
course  of  his  investigations  leading  him  to  an  absurd  consequence,  ho 
says,  "  il  semble  qu'il  n'y  auroit  pas  d'autre  parti  a  prendre,  que  do  la 
digerer  cette  absurdite,  comme  on  est  oblige  d'en  digerer  tant  d'autri'S. . . 
dans  presque  tons  les  objets  de  nos  connaissances."  A  remark  toward? 
the  end  of  his  Memoir  does  not  impress  us  with  a  favourable  opinion  of 
the  extent  of  his  acquaintance  with  the  Newtonian  philosophy :  "  II 
(Newton)  aime  mieux  considerer  la  Pesanteur  comme  uno  qualite  inhe- 
rente  dans  les  corps,  &  ramener  les  idees  tant  decrides  de  qualite  oceulte, 
&  d'attraction."  If  we  abandon  mechanical  princij)les,  ho  continues, 
**  nous  voil'i  replongez  de  nouveau  dans  les  anciennes  tdndbres  du  Peri- 
patetisme,  dont  le  Ciel  nous  veiiille  preserver."  Ho  started  in  life  by 
following  his  father's  profession  of  a  Calvinist  minister,  was  then  carried 
off  by  the  invited  pounce  of  the  "  eagle  of  Meaux,"  and  about  fourteen 
months  before  Jones  mentioned  him  in  this  letter  the  malice  of  a  poet 
threw  him  into  a  dungeon.  For  the  events  of  his  strange  life  see  his 
Elogo  by  Fontenelle,  and  the  Biographic  Universelle.  Comp.  Vie  de 
J.  B.  Rousseau  (Beuchot's  Voltaire,  xxxvii.  505). 


214  CORRESPONDENCE  OF 

LETTER  CVIII. 

COTES  TO  JONES. 

[Extract.] 

S' 
I  thank  you  for  M"'  De  Moivre's  Treatise  concerning 
Chance :  I  have  not  yet  had  leasure  to  go  over  it.  M"" 
Sanderson*  by  whom  You  sent  it,  was  on  Tuesday  last 
elected  our  [Mathematical  Professor  in  the  room  of  M' 
Whiston.  I  am  not  perfectly  acquainted  with  him,  he  seems 
as  far  as  I  can  judge  of  him  to  have  an  extraordinary  good 
Genius.  The  want  of  his  sight  is  certainly  an  insuperable 
disadvantage  to  him  in  several  respects  but  I  believe  in 
some  others  he  has  an  advantage  from  it. 

Nov.  25*^  1711 

Printed  in  Macclesfield  Corr,  i.  261. 


LETTER    CIX. 
JONES  TO   COTES. 
D""  S^  London  Jan.  1'*:  17j^ 

I  have  sent  you  here  inclos'd,  the  Coppy  of  a  Letter, 
that  I  found  among  M"^  Collins's  papers,  from  S*".  Is.  New- 
ton to  one  ]VP.  Smith ;  the  contents  thereof  seems  to  have, 
in  some  measure,  relation  to  what  you  are  about,  as  being 
the  application  of  the  Doctrine  of  Differences  to  the  mak- 
ing of  Tables ;  and  for  that  reason  I  thought  it  might  be 
of  use  to  you,  so  far  as  to  see  what  has  bin  done  already : 


•  "  Nov.  19.  A  Mandate  from  the  Queen  to  make  Mr  Nicolas  Saunderson  (a  blind 
man  from  his  infancy,  but  who  liad  taught  ]Mathematics  in  Christ's  College  about  four 
years),  Master  of  Arts.  It  did  not  command,  but  only  recommended  him ;  and  yet  he 
was  immediately  admitted  and  created,  without  reading  any  Grace  for  it."  Kud's 
Diary.  He  was  chosen  Professor  on  the  20th,  having  six  votes  against  his  competitor's 
(MrHussey  of  Trinity)  four,  and  made  his  inauguration  speech  on  Jan.  21  follow- 
ing,   lb. 


COTES  AND  JONES.  215 

I  shew'd  this  to  S'.  Isaac,  he  remembers  y*.  he  apply'd  it  to 
all  sorts  of  Tables,  but  has  nothing  by  him,  more  than 
what  is  printed :  I  have  more  papers  of  ^r.  Mercator's  and 
others,  upon  this  subject,  tho,  I  ';hink,  none  so  material,  to 
your  purpose,  as  this.  I  shou'a  be  very  glad  to  see  what 
you  have  done  of  this  kind  all  publisli'd ;  And  I  must  con- 
fess, that,  unless  you  design  a  considerable  large  Volume, 
'twere  much  better  to  put  them  into  the  Transactions ;  for 
that  wou'd  sufficiently  preserve  them  from  being  lost, 
which  is  y*.  common  fate  of  small  single  Tracts ;  and  at  y*. 
same  time  save  the  trouble  and  expense  of  printing  them, 
since  the  subject  is  too  curious  to  expect  any  profit  by  it : 
and  besides,  now,  as  the  R.  Society  having  done  them- 
selves the  honour  of  choosing  you  a  Member*,  something 
from  you  cannot  but  be  acceptable  to  them :  S*"  Isaac  him- 
self expects  those  things  of  yours  that  I  formerly  men- 
tion'd  to  him  as  your  promise. 

I  am,  S"".  your  much  oblig'd 

friend,  &  humble  ServS 

AV.  Jones. 


LETTER  CIX.  (bis) 
NEWTON  TO  J.  SMITH. 
[Copy]. 
Enclosed  in  Letter  CIX. 
S'.  Trin.  Coll.  Cambridge,  May  8*^  1675. 

I  have  consider'd  y®  buisiness  of  computing  Tables  of 
Square,  Cube,  &  Sq.  Sq^  Roots;  and  y*.  best  way  of 
pTorming  it,  y'.  I  can  think  of  is  y^  which  follows : 

If  y\  wo'd  compute  a  Table  to  8  decimal  places,  let  y*. 


*  JoDes  had  himself  been  chosen  on  the  same  day,  (Nov.  30).    Cotes  was  not  ad- 
mitted until  May  20, 1714.    Newton  presided  on  both  occasions. 


216 


CORRESPONDENCE  OP 


roots  of  every  hundredth  number  be  extracted  to  ten 
decimal  places,  and  then  compute  every  tenth  numb',  and 
afterwards  every  number  by  the  following  methods. 


Tab.  1. 

Tab.  2. 

n-50 

A 

*0 

a 

n-Q 

4£ 

0 

m 

o 

5e 

>i-40 

B 

P 

op 
m 

/J 

IT 

n-5 

5EF5 

n-30 

C 

pq 

7 

«-4 

i^4 

n-20 

h 

m 
qr 

1 

n-3 

II 

r 

m 

P 

C2 

n-10    ' 

E 

rs 

€ 

n-2 

i^2 

s 

m 

or 

m 

£1 

n 

F 

St 

C 

To 

n-l 

>I 

n+10 

t 
G 

m 
tv 

T 

n 

S 

^/ 

V 

m 

V 

ir 

100 

n  +  20 

II 

vx 

e 

M  +  1 

li^ 

X 

m 

4> 

2C 

n  +  30 

I 

xy 

1 

n  +  2 

2F 

I 

in 

^ 

3C 

n  +  40 

y^ 

< 

n  +  3 

3F 

z 

in 

w 

S 

n  +  50 

L 

5* 

\ 

n  +  4 

5r 

n  +  5 

5FG5 

t]4> 

n  +  6 

G4 

»;3 

n+7 

G3 

»/2 

«  +  8 

G2 

^1 

n  +  g 

Gl 

n  +  10 

n 
G 
If] 

tv 
100 

M+ll 

IG 

2»/ 

n  +  12 

2G 

In  the  First  Table, 

Let  n  signify  every  100*^  numb^  &  F  its  root,  wheth'. 
Square,  Cube,  or  Sq.  Square ;  &  ri  -  50,  n  -  40,  w  -  30,  &c. 


COTES  AND  JONES.  217 

every  10***.  numb"^;  and  J,  J?,  C,  D,  &c,  their  roots;  and 
o,  p,  7,  r,  &c,  the  differences  of  these  roots ;  &  op,  pq,  qr, 
&e.  their  second  differences,  (that  is  op,  the  diff.  of  o  &  p, 
pq  the  diff.  of  p  &  q,  &c.)  and  7?*  their  third  difference,  that 
is,  y®.  common  difference  of  >|<o,  &  op,  op  &  pq,  pq  8c 
qr,  &c. 

Further,  let  a,  /3,  7, 5,  &c.  signify  y*.  differences  of  these 
Roots  from  those  next  less,  namely  a  the  difference  of  A 
y^  root  of  n  -  50  &  y*«  like  root  of  fi  -  51,  /3,  the  diff.  of 
y®.  roots  w-40  &  n-4i,  ^  the  diff.  of  y*.  roots  of  w  & 
n  -  1,  rj  the  diff.  of  y^  roots  of  n  +  10  &  n  +  9.  &c.  And 
let  o,  TT,  x>  P*  ^^  signify  the  diff.  of  a,  /3,  7,  S,  &c.     And 

—  the  common  diff.  of  o,  tt,  v,  «,  &c. 
10  ^  ^ 

In  the  Second  Table, 

Let  w  -  6,  n  -  5,  «  -  4,  n  -  3,  &c  signify  y'.  single 
numbers, 

4>E,  5E  or  F5,  F4>,  Fs,  &c.  their  Roots, 

5  €y  ^4,  ^3,  ^2,  &c  the  diff.  of  those  roots ; 

8t 

—  the  common  diff.  of  those  differences  for  y*.  ten 

100  ^ 

numbers  between  w  -  5  &  n  +  5, 

And  so  for  y®.  ten  numbers  between  n  +  5  &  n  +  15 ; 
let  G5,  G4,  Gs,  &c.  signify  y®.  roots;  174,  >;3,  i;2,  &c,  their 

first  differences,  and    -    their  second  differences;  and  the 
100 

like  for  every  denarie  between  n  -  50  &  w  +  50. 

This  explication  of  the  Tables  being  p'mis'd,  you  may 
compute  them  thus ; 


218 


CORRESPONDENCE  OF 


Out  of  «,  (Square] 
extract    I  Cube    jRoot,  make  ; 
F  y\      ISq.  Sq.) 


r  loF 

2n 
lOF 


10ft) 
20  ft) 


«/, 


S0«^ 


in 


am. 


50s^ 
3n  3n  3n 


lOF 


SOo)  lOst 

),      ='8t, W. 


4» 


471 


1  1  ^        i*^ 

^  «  10       100  6000 


+ *a  r,    and  —  +   e  (T. 

^  10       1000 


And  these  quantities  Fy  st,  m,  s,  ^,  &  <7,  being  thus  found, 
y\  rest  are  given  by  Addit".  &  Subduct. 

For  8t+m^r8y  rS'\-mcsqr,  &c.  st'-m='tv,  tv-m^vwt  &c. 

Again  fi  +  r««r,   r  +  qr^q,  &c.   s-st'^ty  t-tv^Vy  &c. 

And  F^a^Ey  jE-r-A  &c.  F+t^Gy   G  +  v^Hy&c. 

Further 

m  mm  *»  « 

10     '^     ^      10      '^  10  10 

Lastly  ^  +  o-  =  e,  e  •fj'^^y  &c,  ^  -  t  «  jy,   »;  -  v  «=  ^,  &C. 

These  quantities  being  thus  computed,  in  y®.  first  Table, 
to  every  lO***.  number,  tlie  roots  may  be  computed  in  y®. 
2**  Table  to  every  numb^  by  Addition  and  Subduction 
only ; 

For  r+-ii  =  n,  ti  +  — =  r2,&c. 

5»       100      ^        ^  100      ^ 

y^fL^ir    i^_*^  «2t,  &c. 

^        inn  2>'         ^       inn  V 


100 


100 


Again  F  -  ^  =  /'I,    /"I  -  ^1  «  Fg,  &c. 

/'+ 1^=17^,     l/'+2^«2/',  &c. 


•  I  have  added  the  *qq^'^  I  have  also  corrected  some  other  errors  of  transcription. 


COTES  AND  JONES.  219 

Thus  you  must  proceed  to  five  Figures  on  either  hand, 
and  then  do  the  like  in  the  next  ten  Figures,  saying 

tv  ^^  o 

n  + ■»  Jjl,    tjl  + ■•  »72,   &C. 

'       100       '         '         100       '  ' 

And  the  like  for  every  Denarie  between  n  -  50  &  n  +  50, 

In  these  Computations,  Note,  1**.  That  they  must  be 
done  every  where  to  10  or  11  decimal  places,  if  you  will 
have  a  Table  of  Roots  exact  to  8  of  these  places. 

gdiy  If  5p  Sc  G5,  the  roots  of  n  +  5  found  two  ways 
agree  to  8  decimal  places,  it  argues  the  whole  works  from 
which  they  were  derived,  to  be  true.  And  so  of  y*.  roots 
of  w  +  15,  w  +  25,  n  -  5,  &c.  And  also  of  y*.  Terms  J,  *  o, 
&  a  ;  Z,  ;» >K,  &  X,  where  two  works  meet.  Let  this  there- 
fore be  y®.  Proof  of  y®.  work. 

This  S'.  is  w*.  has  occurr'd  to  me  about  your  design, 
which  I  hope  will  do  your  business,  the  whole  work  being 
p'form'd  by  Addit.  &  Subduct :  excepting  y*.  in  y*.  com- 
putation of  every  100**^.  number,  there  is  required  y*. 
Extraction  of  one  root,  &  three  divisions,  to  find  F,  a>, 
stf  &  m, 

S^  I  am 

Your  humble  Scrv' 

Is.  Newton. 

The  person  to  whom  this  letter  is  written  may  be  conjectured  to  be 
"John  Smith,  Philo-Accoinptant"  author  of  Stereometrie^  Lond.  1673. 
(He  must  not  be  confounded  with  Cotes's  uncle).  In  the  Macclesfield 
Correspondence,  ii.  370 — 374,  tliere  are  two  other  letters  on  the  ex- 
traction of  roots  from  Newton  to  this  same  person  (not  to  Collins,  as 
tliere  printed)  dated  July  24  and  Aug.  27, 1075,  in  the  fonner  of  which 
he  refers  to  the  metliod  given  in  tlie  foregoing  letter.  Mr  J.  Smith 
seems  to  have  had  a  design  of  constructing  Tables  of  Square,  Cube  and 
Biquadr.  Roots,  and  consulted  Newton  as  to  the  best  mode  of  com- 
puting them.  The  Tables,  if  ever  made,  do  not  appear  to  have  been 
published.  The  earliest  I'ables  of  Roots  are  Briggs's  IMS.  Tables  of  the 
Square  Roots  of  Numbers  up  to  1000  mentioned  in  Mayne's  Merchant' $ 
Companion  (London,  1674),  p.  80. 


220  CORRESPONDENCE  OF 

LETTER    ex. 

COTES   TO  JONES. 

Answer  to  Letter  CIX,    No  date. 

I  have  received  Your  Letter  with  the  inclosed  Paper 
of  S'  Isaac  Newton  for  which  I  return  You  my  hearty 
thanks.  His  method  seems  to  be  excellently  well  suited 
to  those  particular  purposes  for  which  he  designed  it,  &  I 
do  not  doubt  I  shall  find  it  very  curious  when  I  have  lea- 
sure  to  examine  it  to  y*^  bottom.  What  I  intend  to  print 
will  make  but  a  small  Volume,  I  cannot  say  it  will  be  big- 
ger than  that  of  S"^  Isaacs  which  You  lately  j)ublished.  It 
^vill  contain  the  Lectures  I  have  hitherto  read  in  Publick, 
together  with  those  which  I  shall  read  this  Year,  all  of 
which  amount  to  no  more  than  Ten,  for  by  the  Statutes  of 
my  place  I  am  obliged  annually  to  make  but  two.  I  can- 
not indeed  expect  any  profit  from  the  Publication,  twill  be 
sufficient  if  y®  expense  of  it  can  be  defrayd.  I  have 
already  put  y®  University  to  the  charge  of  Types  for  some 
new  characters  which  I  have  occasion  to  make  use  of  & 
therefore  for  that  reason  as  well  as  some  others  I  cannot 
now  draw  back.  What  You  mention  that  y®  R:  Society 
have  chose  me  one  of  their  Members  is  altogether  a  peice 
of  news  to  me.  If  it  be  so,  I  shall  be  very  sensible  of  the 
Honour  they  have  done  me.  That  Title  may  recommend 
my  papers  to  y®  Publick  though  they  be  printed  at  Cam- 
bridge. If  You  insist  upon  my  Promise  of  sending  those 
things  to  You  before  they  are  printed  I  shall  be  ready  to 
make  it  good.  What  I  have  further  concerning  y®  subject 
of  differences  consists  of  Ten  Propositions  whereof  the  Six 
first  are  particular  &  fitted  for  use  &  are  sufficient  for  all 
cases  that  comonly  hapjicn,  the  other  four  are  general. 
You  will  be  able  to  judge  of  my  Method  by  y^  first  Propo- 


COTES  AND  JONES.  221 

sition  which  I  here*  send  You.  You  may  shew  it  to  S' 
Isaac  if  You  think  it  proper  but  I  desire  You  would  not 
shew  it  to  others. 

I  cannot  so  easily  give  You  an  Idea  of  my  other  peice 
concerning  Logarithms  but  I  find  room  enough  in  this 
Page  to  send  Youf  one  thing  out  of  it  as  a  curiosity  which 
may  be  understood  independently  of  the  rest. 

Rectificatio  Logarithmica) 

— Oblata  sit  igitur  Logarithmica  &c. 


LETTER  CXI. 
JONES    TO   COTES. 


Accompanying  4  copies  of  tho  Commorcium  Epistolicum, 

[Extract.] 
S'  London  Feb  Q'\  17}§ 

The  R.  Society  having  order'd  one  of  their  Books  for 
you,  &  another  for  M"".  Sanderson,  also  one  for  Trinity 
College  Library,  &  one  for  the  University  Library ;  I  wou'd 
not  miss  the  opportunity  of  paying  you  my  respects  by 
sending  them :  I  need  not  tell  the  occasion  &  design  of 
that  Collection  :  you'l  see  readily  that  it  affords  such  light 
concerning  what  it  relates  to,  as  cou*d  not  easily  have  bin 
discovered  any  other  way :  and  also  shews  that  your  great 
Predecessor,  whose  illustrious  Example,  I  don't  doubt  but 
you  follow,  never  imploy*d  his  time  about  things  ordinary. 
I  have  no  ^Mathematical  intelligence  to  send  you ;  M^  Keil 


*  The  tract  of  which  Cotes  sends  a  specimen  to  Jones  will  be  found  among  his 
Opera  Miscellanea,  pp.  36 — 71.  The  title  of  it  is  •*  Canonotitchnia  sive  Constructio 
Tabularum  per  Differentias."  lie  has  not  copied  out  the  proposition  in  this  draught  of 
his  letter,  and  therefore  it  will  be  sufficient  to  refer  the  curious  reader  to  p.  36  of  tha 
work  just  cited. 

t  Here  also  Cotes  has  not  taken  the  trouble  to  transcribe  the  proposition.  It  may 
be  seen  in  his  Logometria,  {[larmonia  Men$urarum^  pp.  23,  24.) 


222  CORRESPONDENCE  OF 

thinks  he  has  discover  d  a  very  easy  and  Practical  solution 
of  the  Keplerian  Problem  ♦ :  the  Problem  of  the  Refrac- 
tion, or  that  concerning  y®.  description  of  the  Curve  de- 
scribed by  a  Ray  of  Light  in  passing  thro  the  Atmosphere, 
is  here  done  by  two  diflerent  hands ;  one  of  them  endea- 
vours to  apply  it  to  Astronomical  uses,  \v^^.  I  suppose  he 
has  pretty  well  compassed. 

«  •  *  •  •  « 

I  am  extremely  pleas'd  to  find  that  S"^.  Isaac's  Book  is 
so  near  being  finish'd:  his  general  Scholium  I  presume 
he'l  soon  send  you,  if  'tis  not  already  done :  and  'tis  not 
less  agreeable  to  me,  to  hear  that  your  own  Book  is  in 
such  forwardness. 


P.S.  I  have  sent  to  you  four  of  the  Comercium  Epis- 
tolic.  that  is,  one  for  your  self,  and  y®  other  three  as 
before  mention'd  which  I  desire  you  wou'd  deliver,  as  from 
the  Royal  Society  of  London. 


LETTER  CXII. 
COTES  TO  JONES. 


[Extract.] 
S'.  Cambridge  Feb',  n^^ 

I  have  received  Your  obliging  Letter  together  with 
the  very  agreeable  gift  of  the  Commercium  Epistolicu.  I 
have  delivered  one  Copy  to  the  University  Library  Keeper 
another  to  the  Library-keeper  of  Our  College  and  the 
third  to  M'  Sanderson  as  from  the  Royal  Society.  You 
may  be  pleas'd  to  return  our  acknowledgments  of  the 
Favour. 


•  Phil.  Trans,  for  1713,  Vol.  xxviii.  pp.  1—10. 


COTES  AND  JONES.  223 

I  am  very  glad  to  see  this  Peice  at  length  made  pub- 
lick  in  which  quicquam  cuiquam  detractum  non  reperio,  sed 
potius  passim  suum  cuique  tributum*. 


LETTER  CXIII. 
JONES  TO  COTES. 
S'.  London  Aprill  29*\  1713. 

Ever  since  I  received  your  very  kind  Letter,  and 
Mouton's  Book,  I  waited  for  an  opportunity  of  sending 
you  some  old  Manuscripts  I  had  by  me,  and  at  last  am 
oblig'd  to  Venture  them  by  the  Carrier ;  They  relate,  in 
some  measure,  to  the  Method  of  Differences ;  The  folio 
one,  I  find,  was  "writ  by  one  Nath.  Torpcrleyf,  a  Shrop- 
shire man,  who  when  young  was  Amanuensis  to  Vieta,  but 
afterwards  writ  against  him;  he  was  contemporary  with 
Briggs  and  Harriot,  and  intimately  acquainted  with  them ; 
The  Book,  I  think,  can  be  of  no  other  use  to  you,  than  in 
what  relates  to  the  History  of  that  Method,  and  in  having 
y®  Satisfaction  of  seeing  what  has  bin  formerly  done  on 
that  Subject.  The  other  Small  4*°  I\LS.  is  a  piece  of  Mer- 
cator's  about  Differences,  it  seems  to  contain  no  great 
matter ;  nor  indeed,  can  I  be  satisfied,  any  thing  that  he 
has  done,  or  any  one  else,  so  very  considerable,  as  to 
deserve  to  accompany  any  piece  of  yours ;  Therefore  pray 
let  us  have  your  things  entire,  and  as  soon  as  conveniently 
you  can. 

I  am  mightily  pleas'd  to  see  the  end  of  the  Principia, 
and  return  you  many  thanks  for  the  very  Instructive  Index, 


•  Commerc.  Epistol.  p.  119,  (p.  239,  2ntl.  Ed.)  These  are  I^ibniz's  words  in  his 
Letter  to  Sloane,  Dec.  29,  1711,  by  which  he  unfortunately  mode  himself  a  party  to 
the  obnoxious  languag-e  of  the  Leipsic  review  of  Newton's  tract,  "De  Quadrature  Cur- 
varum,"  Leips.  Acts,  Jan.  1705. 

t  Compare  Afacclesfield  Corresp,  ii.  5,  note. 


224  CORRESPONDENCE  OF 

that  you  have  taken  the  pains  to  add,  and  hope  'twill  not 
be  long  before  we  shall  see  the  Beginning  of  that  Noble 
Book. 

I  shall  be  in  some  pain  till  I  hear  that  you  have  re- 
ceiv'd  my  old  M:S.  it  being  a  favorite  one,  purely  upon  the 
account  of  some  extravagancys  in  it,  So  very  uncommon : 
But  I  shall  think  it  safe  when  in  your  hands;  I  am  S'. 
without  reserve,  your  very  affectionate  friend  and  most 
humble  Servant 

W:  Jones. 


LETTER  CXIV. 
COTES  TO  JONES. 


Dear  S"^ 

I  know  not  how  to  return  You  my  thanks  as  I  ought 
for  Your  readiness  to  assist  me.  The  two  Manuscripts  of 
Torperly  &  Mercator  are  come  very  safe  to  my  hands; 
I  hope  I  shall  return  'em  to  You  without  any  damage.  I 
have  been  lately,  and  am  at  present  taken  up  with  some 
College  buisness,  so  that  I  have  scarce  yet  had  any  time 
to  look  into  'em.  If  I  find  any  thing  in  them  of  Moment, 
I  believe  I  shall  request  You  to  let  me  print  it  .with  my 
own,  for  I  would  not  willingly  have  any  one  lose  the  Credit 
due  to  him. 

I  am  glad  You  can  approve  of  the  Index  to  the  Prin- 
cipia.  It  was  not  design'd  to  be  of  any  use  to  such 
Headers  as  Your  self,  but  to  those  of  ordinary  capacity. 
I  hope  the  whole  Book  may  be  finished  in  a  fortnight  or 
three  Weeks.  I  have  lately  been  out  of  Order,  or  it  might 
have  been  done  by  this  time 

I  am  S"" 

Your  most  Obliged  Freind 

May  3**.  1713.  and  Servant  B.  Cotes. 


COTES  AND  JONES  225 


LETTER  CXV. 


JONES  TO  COTES. 
Dear  S'  juiy  ii*>»  ins 

Tis  impossible  to  represent  to  you,  with  what  pleasure 
I  receiv'd  your  inestimable  Present  of  the  Principia,  and 
am  mucli  concern'd  to  find  my  self  so  deeply  charg'd  with 
Obligations  to  you ;  and  such,  I  fear,  as  all  my  future  en- 
deavours will  never  be  able  to  requite.  This  Edition  is 
indeed  exceeding  beautifull,  and  interspersed  with  great 
variety  of  admirable  discoverys,  so  very  natural  to  its  great 
Author ;  but  is  much  more  so,  from  the  additional  advan- 
tage of  your  excellent  Preface  prefix'd ;  which  I  wish  might 
be  got  publish'd  in  some  of  the  foreign  Journals;  and 
since  a  better  account  of  this  Book  cannot  be  given,  I 
suppose  it  will  not  be  difficult  to  get  it  done. 

Now  this  great  Task  being  well  over,  I  hope  you'l 
think  of  publishing  your  own  Papers,  &  not  let  such  valu- 
able pieces  lye  by : 

As  to  w^  you  mention*d  in  your  last  concerning  my 
Old  manuscripts,  tho,  for  my  part,  I  know  of  nothing  worth 
your  notice  publickly  in  them,  buit  if  you  do  find  any,  it  the 
more  answers  the  end  of  my  sending  it,  and  you  know 
that  you  may  do  as  you  please  ; 

S"-  I  am 

your  most  obedient 

humble  Serv' 

W:  Jones 


LETTER  CXVL 
COTES  TO  {WHISTON.} 
Dear  Sir  {March  1715} 

I  have  lately  seen  two  Schemes  of  the  great  Eclipse 
the  one  done  by  Your  self,  the  other  by  D""  Halley.  Yours 
being  to  be  understood  by  those  only  who  are  acquainted 
15 


226  LETTER  OF 

with  Astronomy,  has  upon  that  account  much  the  disad- 
vantage of  the  D"*"  with  most  People.  I  take  the  Liberty 
to  propose  another  Scheme  to  You,  which  I  beleive  would 
give  a  more  general  satisfaction  than  either  of  the  other : 
I  mean  a  Map  of  that  part  of  the  Heavens  in  Avhich  the 
Sun  will  be  at  that  time.  If  the  sky  be  clear  it  will  un- 
doubtedly be  a  great  surprize  to  see  the  Stars,  but  twill  be 
much  more  so  to  the  Vulgar  that  You  should  be  able  to 
describe  the  Positions  of  'em  beforehand:  this  I  am  apt 
to  think  they  will  look  upon  as  a  greater  peice  of  art,  than 
to  predict  the  Eclipse  itself.  By  comparing  the  Ephemeris 
&  Globe  together  I  find  there  will  be  three  Planets  visible 
on  the  West  of  the  Sun,  Jupiter  will  be  very  near  him, 
Venus  will  be  about  the  Meridian,  Mercury  will  lye  between 
them.  You  have  already  spoken  of  the  Moons  Atmo- 
sphere, I  think  it  would  not  be  amiss  if  You  desired  Peo- 
ple to  look  if  tlicy  can  observe  the  Suns  also,  I  mean  that 
light  in  the  Heavens  which  D^  Gregory  describes  pretty 
largely  in  the  Scholium  to  Prop.  8.  Lib.  2  of  his  Astro- 
nomy. A  representation  of  this  may  be  inserted  in  the 
Map  if  You  think  fit,  that  it  may  be  known  beforehand 
how  tis  likely  to  appear.  You  may  caution  those  who  are 
desirous  to  see  this  faint  light,  that  they  prepare  their  eyes 
beforehand  for  it,  by  staying  in  some  dark  place  for  about 
a  quarter  of  an  hour  before  the  Sun  be  totally  obscur'd ; 
You  know  it  requires  about  that  time  to  bring  our  Eyes  to 
the  disposition  they  usually  have  in  the  night  time  for  see- 
ing faint  Lights.  I  would  further  advise,  if  You  think  fit 
to  set  about  this  Project,  that  You  do  it  with  exactness 
that  Mathematicians  may  not  dislike  it,  &  that  Your  Ex- 
plications be  written  in  a  Popular  way  &  as  free  as  may  be 
from  Mathematical  Terms  that  others  may  not  dislike  it. 
I  suppose  You  have  seen  Cassini's  Map  &  Reflections  upon 
the  Eclipse  of  1699,  printed  in  the  Memoires  of  the  Royal 
Academy  of  Sciences  for  that  Year.     If  You  have  not  yet 


COTES  TO  WIIISTON. 


227 


seen  it,  tis  possible  it  may  suggest  something  further  to 
You.  1  shall  not  trouble  You  any  longer  upon  this  sub- 
ject. 

My  Cozen  Smith  was  chosen  Fellow  the  last  Elen.  He 
takes  his  Master's  Degree  this  next  Commencement.  Ho 
has  already  two  Pupils  &  expects  one  or  two  more  in  a 
short  time.  He  presents  his  humbl  service  to  You ;  both 
He  &  my  self  shall  be  obliged  to  You,  if  You  can  assist 
Bim  by  Your  recommendation.  I  need  not  tell  You,  that 
as  he  is  in  all  other  respects  well  qualified  for  that  Buiss- 
ness  so  he  is  very  capable  of  instructing  his  Pupils  in  some 
parts  of  Knowledge  which  You  &  I  esteem,  &  which  very 
few  Tutors  in  the  University  do  at  all  pretend  to. 

This  letter  was  evidently  written  to  AVhiston,  who  "  a  little  before 
the  famous  total  eclipse  of  tlie  Sun,  April  22,  this  year,  1715,  pub- 
lished two  schemes*  of  tliat  eclipse,"  in  the  latter  of  which  lie  adopted 
Cotes's  suj^gcstions,  though  ho  makes  no  mention  of  his  receiving  any 
such  assistance.     "N.B.  This  most  eminent  eclipse,  1715,  was  exactly 

foretold  by  ]\r  Flamstccd,  D'  Ilalley,  &  myself I  myself  by  my 

lectures  before ;  by  the  sale  of  my  schemes  before  &  after ;  by  tho 
generous  presents  of  my  numerous  &  noble  audience;  wlio,  at  tho 
recommendation  of  my  great  friend,  the  lord  Stanhope,  then  secretary 
of  state,  gave  me  a  guinea  apiece ;  by  the  very  uncommon  present  of 
twenty  guineas  from  another  of  my  great  benefactors,  tho  duke  of 
Newcastle;  and  of  five  guineas  at  night  from  tho  lord  Godolphin; 
gained  in  all  about  £120.  by  it."     See  Winston's  Memoirs  i.  204,  5. 


•  The  title  of  the  first  is  "A  Calculation  of  the  prreat  EcMpse  of  the  Sun,  Apr.  22. 
1715,  in  y*  morning,  from  Mr  Flamsteed's  Tables,  as  corrected  according  to  S'  Isaac 

Newton's  Theory  of  y«  Moon  in  y^  Astronomical  Lectures "    In  the  2nd,  which  is 

larger  and  fuller  than  the  1st,  the  Eclipse  is  calculated  "  from  S'  I.  Newton's  last  im- 
provements to  his  Theory  of  y«;  INIoon."  (It  is  dated,  April  2,  1715).  In  the  1st 
Whistou  had  neglected  to  avail  himself  of  the  2nd  Ed.  of  the  Principiuj  a  fact  to 
which  Cotes  in  the  Letter  of  which  we  have  here  only  the  draught,  may  possibly  have 
drawn  his  attention. 

Time  of  Eclipse  at  London. 


Whiston's  Ist  Scheme. 

His  2nd 

Halley. 

Flamstced. 

Observed  Time. 

Beginning       Q^ .  18' 

8h.    7i' 

BV    7' 

8h.    8' 

8»>.6' 

Middle            9  .  24 

9  .  14 

9  .  13 

9  .  13.i 

9  .  lO'.  45'' 

End               10  .  35 

10  .  24i 

10  .24 

10.  24 

10  >  20 

15—2 


228  LETTER  OF 

LETTER  CXVIL 
COTES  TO  LORD  TREVOR. 
My  Lord,  Trinity  College  Cambr.  Jan.  10***  1716 

When  I  waited  upon  Your  Lordship  with  S"^  Isaac 
Newton,  I  remember  my  Lady  Trevor  was  saying,  that 
S'  John  Bernard  was  design'd  for  our  College :  I  have  since 
heard  that  He  will  come  to  us  very  soon.  I  have  not  been 
informed  whether  any  Tutor  is  already  provided  for  Him. 
If  Your  Lordship  is  not  yet  determin'd,  I  beg  leave  to 
propose  one  to  You,  His  name  is  Smith,  a  Junior  Fellow  of 
the  College,  I  have  had  the  oportunity  of  an  intimate 
knowledge  of  His  Temper  Behaviour  &  Learning,  as  He 
has  been  my  Chamber-fellow  for  some  yeares  &  as  He  is 
my  Kinsman.  I  can  therefore  be  bold  to  recommend  Him 
to  You  as  a  person  whom  I  think  to  be  extraordinarily  well 
qualified  to  satisfie  Your  expectation  in  all  respects.  If 
You  desire  to  have  S*"  John  instructed  in  the  Mathematicks 
&  the  new  Philosophy  :  I  do  assure  Your  Lordship,  I  know 
no  one  more  capable  of  doing  it  with  good  success,  both 
on  account  of  His  very  great  skill  in  those  things  &  His 
easy  way  of  teaching.  Your  Lordship  was  formerly  pleas'd 
to  desire  me  to  assist  M"*  Trevor  *  that  way :  I  was  very 
sorry  I  might  not  do  Your  Lordship  that  service,  for  it 
was  not  my  fault  that  I  did  not.  The  remembrance  of  it 
makes  me  beleive  You  have  the  same  views  for  S*"  John : 
I  therefore  thought  it  my  duty  as  Avell  to  Your  Lordship 
as  to  m}'  Kinsman  to  write  thus  to  You.  If  the  appoint- 
ment of  a  Tutor  shall  be  left  to  D"^  Bentlcy ;  I  know  His 
opinion  of  M*"  Smith  is  such,  that  He  will  think  He  cannot 
serve  Your  Lordship  more,  than  by  naming  Him  to  You 

I  am  &c. 
'  E  C 

•  Lord  Trevor's  eldest  son  and  successor  in  the  title.  He  was  entered  a  fellow- 
commoner  at  Trinity  College,  June  19,  1708,  his  tutor  being  Mr  Nic.  Clagett,  Libra- 
rian of  the  Colleg-e,  afterwards  Dean  of  Rochester,  and  Bishop  of  St  David's,  from 
whence  he  was  translated  to  Exeter. 


.   COTES  TO  LORD  TREVOR.  229 

The  application  made  in  this  letter  was  successful,  but  before  tho 
formal  result  of  it  was  realised,  tlio  wann  heart  that  dictated  it  had 
ceased  to  beat,  and  tho  grave  had  parted  tho  two  chamber-fellows. 
Sir  John  Bernard  was  entered  a  Nobleman  under  Smith,  July  6,  \716, 
Cotes  breathed  his  last  on  Juno  5. 

Lord  Trevor  was  one  of  the  twelve  peers  created  by  Queen  Anno  in 
order  to  turn  tho  balance  in  tho  House  of  Lords  in  favour  of  tho  peace 
of  Utrecht.  Ho  was  Chief  Justice  of  tho  Common  Pleas  in  her  reign, 
but  shortly  after  tho  acoesaion  of  George  I.  (Oct.  1714)  ho  was  super- 
seded at^tho  suggestion  of  Lord  Chancellor  Cowper,  and  tho  appoint- 
ment was  bestowed  on  Sir  Peter  King.  See  Lord  Campbell's  Chan- 
cellors IV.  349  note.  592.  .593. 

Ho  married  for  his  second  wife  tho  widow  of  Sir  Robert  Bernard, 
a  brother  of  ^Irs  Bcntley,  and  thus  became  step-father  to  the  young 
baronet  Sir  John. 

On  tho  publication  of  the  2d  Ed.  of  tho  Principia,  Bentley  presented 
him  with  a  copy  of  it.     Bentloy's  Correspondence,  p.  4G5. 


LETl'ER  CXVIIL 
COTES   TO   ROBERT  DANNYE. 

Containing  an  account  of  the  meteor  of  the  (Jth  of  March  171  j . 

Tho  following  is  an  extract  from  tho  Journal  Book  of  the  Royal 

Society.     "  March  7-  ^m*    Tho  President  in  the  Chair A  letter 

of  the  lato  M'  Roger  Cotes  Math.  Professor  at  Cambridge  to  the 
Reverend  Mr  Robert  Dannye  [dated  March  15,  1716}  was  produced 
as  communicated  by  M'  Jurin  of  Trinity  Coll.  Cambridge.  It  contain'd 
some  very  remarkable  circumstances  seen  by  him  in  tho  lato  wonderful 
phoenomcnon  seen  about  a  twelve  month  since,  as  that  about  \  after 
seven  there  was  a  perfect  Canopy  of  Rays  ascending  from  all  parts  round 
tho  Horizon,  but  no  where  reaching  to  it  being  about  10  or  15  degrees 
high  on  tho  North  Side  &  near  forty  on  tho  South,  continuing  in  this 
state  not  above  two  minutes  during  w'*"  interval  several  Colours  appeared, 
some  fainter  &  more  permanent,  others  brighter  but  quickly  vanishing, 
with  several  other  curious  remarks.  This  description  being  better  circum- 
stanced than  w*  had  before  been  communicated  by  most  other  obser>'ers, 
was  thought  worthy  to  bo  preserv'd  in  the  Transactions."  It  will  bo 
found  in  tho  Transactions  for  ^lay — August  1720.  pp.  60-70,  and  in 
Smith's  Optics  (1738)  Vol.  i.  pp.  07-70,  and  therefore  it  has  not  been 
thought  necessary  to  reproduce  it  here. 


230  CORRESPONDENCE  OF  COTES. 

This  letter  closes  Cotos's  corrospoiulcnco  in  the  Trinity  College 
Collection.  Among  the  ]\Iacolesfiol(l  Letters,  however,  there  is  one  of 
a  later  date,  addressed  to  his  friend  Jones  only  a  month  before  his 
death,  in  answer  to  some  inquiries  respecting  the  progress  of  his  tables 
of  integrals  upon  which  he  was  employed.  At  the  beginning  of  the 
year  he  had  returned  to  the  subject  of  the  integration  of  rational  frac- 
tions, and  in  this  letter  ho  refers  exultingly  to  the  success  of  his  re- 
searches, animadverting  upon  a  paper  of  Leibniz,  (Lcips.  Acts,  1702, 

p.  210)  who  was  unable  to  integrate  -j « .     The  letter  is  quoted  by 

Smith  (Harmon.  Mcnsur.  p.  113),  and  an  extract  from  it  is  given  by 
him  in  his  account  of  that  work  printed  in  the  Phil.  Trans,  for  June — 
August  1722,  pp.  140-148.  Leips.  Acts,  April  1723,  pp.  103,  104. 
One  of  the  expressions  which  Cotes  mentions  in  this  letter  as  yielding 

to  hb  method  f .— ,; -^^ ,  where  q  is  some  power  of  2  j ,   Taylor 

sent  to  Monmort  as  a  challenge  from  himself  to  the  mathematicians 
of  the  continent,  without  dropping  any  allusion  to  the  source  to  which 
he  was  indebted  for  the  problem.  Monmort  transmitted  the  question 
to  John  Bernoulli  and  Ilermaim,  tiie  former  of  whom  replied  (Jan.  171^) 
by  oflering  to  lay  Taylor  a  wager  of  50  guineas  that  he  would  produce 
a  solution  within  a  stipulated  time,  but  upon  condition  that  he  should 
in  his  turn  propose  a  problem  to  Taylor  upon  the  same  terms.  Taylor 
at  once  declined  tiie  proposal  in  a  lengthy  reply,  {Contempl,  Philosoph. 
p.  109),  but  before  it  came  to  Bernoulli's  hands,  that  mathematician 
apprehensive,  ho  says,  lest  his  silence  should  be  construed  by  some 
austere  Englishmen  (quidam  ex  severioribus  Anglis)  into  an  acknow- 
ledgment that  the  problem  was  beyond  the  strengtli  of  foreign  analysts, 
had  sent  his  solution,  wliich  ho  had  soon  hit  upon,  for  insertion  in 
the  Leipsic  Acts  (Leips.  Acts,  June  1710,  p.  250.  Bernoull.  0pp.  ii. 
402).     Hermann's  solution  appeared  in  the  Acts  for  August,  p.  351. 

If  an  early  death  had  not  put  an  abrupt  stop  to  his  investigations. 
Cotes  would  no  doubt  have  removed  the  restriction  with  respect  to  the 
value  of  q  in  the  exjiression  given  above.  His  example,  however, 
stimulated  Demoivre  to  make  the  attempt,  which  was  at  last  crowned 
with  success.  See  Miscellanea  Analt/tica^  Loud.  1730.  Taylor  says, 
(see  Letter  cxx,  and  Contempl.  Philos.  p.  113.)  that  ho  himself  could 
prove  the  possibility^  of  the  integration. 


END  OF  COTES'S  CORRESPONDENCE 


LETTERS  OF  TAYLOR  TO  PROF.  SMITH. 


Brook  Taylor  (bom  1685,  died  1731)  was  entered  a  foUow-com- 
moner  at  St  John's  Collcf^e,  Cambridge,  in  1701,  and  took  the  degree  of 
LL.B.  in  1709,  LL.D.  in  1714.  Treatises  on  the  Differential  Calculus 
have  made  his  name  familiar  to  many  who  can  write  out  his  Theorem 
without  having  any  very  precise  idea  of  tlie  personality  of  the  dis- 
coverer of  it.  A  life  of  him,  prefixed  to  his  tract  Contcmplatio  Philo' 
sop/iicay  was  printed  in  1703  by  liis  grandson  Sir  W.  Young.  At  the 
time  when  ho  wrote  the  following  letter  lie  was  Secretary  of  the  Royal 
Society,  though,  about  a  month  before,  ho  had  sent  in  his  resignation 
of  the  office  to  his  brother-secretary  Halley  {ContempL  Philosoph.  p.  103). 
On  Dec.  1,  3Iachin  was  appointed  to  succeed  him.  Before  the  letter 
was  sent  off,  it  was  read  at  the  weekly  meeting  of  the  Society.  "  Nov, 
27,  1718.  Tlie  President  in  the  chair.  D'  Taylor  read  a  letter  he  had 
drawn  up  for  ]\r  Smith,  Professor  of  Astronomy  ic  Cambridge,  re- 
questing him  to  communicate  some  curious  discoveries  in  Geometry 
made  by  the  late  M'  Cotes  his  predecessor  &  kinsman."  Journal 
Book, 

LETTER  CXIX. 
BROOK  TAYLOR  TO  PROP.  SMITH. 
Sir 
When  I  last  saw  your  most  excellent  Predecessor 
M"^  Cotes  I  was  so  very  much  pleased  with  the  account  he 
gave  me  of  some  Mathematical  Tracts  he  had  thoughts  of 
obliging  the  Publick  with,  particularly  a  Sett  of  Tables  for 
the  Squaring  of  Curves  by  the  Measures  of  Ratio's  & 
Angles,  that  I  have  not  been  able  to  forbear  very  fre- 
quently mentioning  of  them,  and  expressing  my  wishes 
that  I  might  soon  see  them  made  publick.  All  Lovers  of 
Mathematical  Learning  do  heartily  joyn  with  me  in  this, 
particularly  the  Royal  Society  is  so  sensible  of  the  great 
usefulness  of  those  Tables,  that  they  have  been  pleased  to 
order  me  to  take  this  occasion  to  let  you  know  that  they 
shall  think  themselves  very  much  obliged  to  you  by  the 


232  LETTERS  OP 

speedy  publication  of  them,  and  shall  be  very  glad  to  give 
you  any  assistance  you  may  have  occasion  for  in  the  doing 
of  it. 

I  myself,  upon  the  memory  of  what  M'  Cotes  sliew'd 
me,  have  made  some  Tables  of  the  same  nature,  and  am 
presst  by  some  friends  to  publish  them,  as  a  thing  they 
say  will  make  amends  for  the  injury  you  do  the  Publick 
and  the  memory  of  M'  Cotes  in  so  long  suppressing  his 
Papers.  But  I  can  by  no  means  prevail  upon  myself  to  do 
this,  being  much  more  desireous  to  see  M""  Cotes's  own 
Tables  publisht  by  you.  And  I  shall  be  very  glad  in  any 
manner  to  assist  you  in  looking  over  the  Papers  them- 
selves, and  in  taking  care  of  the  Press,  if  the  convenience 
of  Types  should  make  you  think  it  proper  to  print  them 
here,  and  your  own  affairs  should  make  it  inconvenient  to 
you  to  attend  this  work  wholly  your  self. 
I  am 

Sir 
Your  most  humble  Servant 
Norfolk  Street  Brook  Taylor 

27*^  Nov';  1718  Seer 

P.  S.  If  there  be  any  other  Papers  of  M*^  Cotes  be- 
sides the  Tables  that  are  fit  to  be  publisht  and  cannot  be 
conveniently  done  so  soon,  the  Tables,  being  a  particular 
thing  by  themselves,  may  be  printed  seperate,  leaving  those 
other  Papers  to  a  more  convenient  opportunity. 


The  purport  of  Smith'a  answer  may  be  gathered  from  the  following 
extract  from  the  Journal  Book  of  the  Royal  Society. 

"  Dec.  11.  171 8.  There  was  read  a  letter  from  M'  Smith,  in  answer 
to  a  letter  of  D'  Taylor  written  to  desire  the  hastening  of  the  Edition 
of  M'  Cotes  his  Posthumous  papers  upon  the  Quadrature  of  Curves. 

M'  Smith  informs  the  Doctor  that  those  papers  are  preparing  with 
all  convenient  speed  to  be  put  in  the  press,  &  are  designed  to  be 
printed  by  Subscription;  that  the  Title  of  the  Book  is  as  follows: 
Ilarmonia  Mensurarum,  sivo  Analysis  et  Synthesis  per  Kationcm  et 
Angulorum  raensuras  promotce." 


BROOK  TAYLOR  TO  PROF.  SMITH.  233 

LETTER  CXX. 

BROOK  TAYLOR  TO  PROP.  SMITH. 
Sir 

I  am  very  much  obliged  to  you  for  the  account  you 
give  me  of  your  design  to  publish  M'  Cotes's  Papers,  and 
I  am  not  only  most  ready  myself,  but  all  my  acquaintance 
will  do  what  is  in  their  power  to  assist  you  in  it.  I  have 
given  your  letter  to  D""  Halley,  and  I  dont  doubt  but  he 
will  acquaint  you  with  the  thoughts  of  the  Royal  Society 
upon  it. 

The  great  impatience  I  am  in  to  see  your  Book 
publisht  makes  me  a  little  concerned  that  it  must  depend 
upon  a  Subscription.  For  tho  such  a  Book  as  this  when 
publisht  cannot  want  purchasers ;  yet  it  will  be  very  hard 
to  find  a  sufficient  number  of  Persons,  who  have  knowledge 
enough  in  these  studies  to  think  it  worth  while  to  interest 
themselves  in  a  Subscription  that  may  turn  to  any  account. 
And  tho  what  you  propose  of  having  no  money  paid  down, 
&  the  price  being  sett  by  the  Vice  chancellor,  be  very 
fair  and  easy  to  the  Subscribers;  yet  there  are  a  great 
many  Persons  who  will  not  care  to  subscribe  without 
knowing  beforehand  what  will  be  the  charge.  In  this  I 
dont  only  \vrite  my  own  sentiments,  but  also  those  of  M' 
Jones,  who  is  the  best  acquainted  with  affairs  of  this 
nature  of  any  one  I  know,  &  whose  character  you  can 
be  no  stranger  to.  He  had  a  correspondance  with  M' 
Cotes  upon  this  Subject,  and  would  particularly  be  glad  to 
do  you  any  service  in  this  matter.  Upon  account  of  what 
I  have  said  I  wish  you  could  rather  think  of  getting  the 
Book  publisht  at  the  Charge  of  the  University,  or  some 
other  way.  Perhaps  the  Royal  Society  would  be  inclined 
to  do  it.  And  it  may  be  tried  whether  there  may  not  be 
some  encouragement  got  from  the  E.  of  Caernarvan. 
Wliat  ever  be  your  resolution  I  will  do  you  all  the  Service 


234  LETTERS  OF 

I  can  in  it.  Particularly  I  will  endeavor  to  get  en- 
couragement from  abroad  by  the  Correspondance  I  have. 
Tho'  I  must  be  so  just  as  tx)  tell  you  that  M'  Cotes  is  but 
little  known  among  the  Foreigners.  His  Logometria  is 
out  of  their  Tast,  (in  short  none  of  them  have  judgement 
enough  to  know  how  to  esteem  it,)  &  his  Preface  to  the 
Principia  is  a  prejudice  to  his  disadvantage  with  them. 
Yet  I  dont  doubt  but  the  newness  of  the  design  will  make 
them  purchase  the  Book  when  it  is  out. 

I  believe  I  can  do  all  that  M"^  Cotes  has  done  in  his 
Tables;  for  I  can  demonstrate  that  any  Curve  may  be 
squared  by  Measures  of  Ratio's  and  Angles,  whose  Absciss 


beinff  «,  the  Ordinate  is  in  this  form ■   ^         .        -     -^    , 

where  t)  is  any  index,  &  ^  &  \  are  any  whole  numbers  affirma- 
tive or  negative,  &  the  denominator  e-i-fz^+gz^'^+hz^^'SiC 
consists  of  any  number  of  terms.  You  know  very  well 
that  the  irrational  forms  depend  upon  the  rational  ones. 
I  have  a  different  way  from  M'  Cotes's*,  and  something 
more  simple,  of  supplying  the  defect  in  Sir  Is :  Newton's 
6"*  form.  I  shall  be  very  ready  and  glad  to  communicate 
to  you  any  thing  that  I  know  in  these  matters  that  may 
render  your  Book  the  more  compleat.  I  believed  it  might 
be  some  Service  to  the  general  design  of  it  to  have  Tables 
of  Natural  Logarithms  and  Arcs  answering  to  the  Tangents, 
when  the  Radius  is  unite ;  wherefore  I  have  wrote  to  M^ 
Sharp  at  Little  Horton  near  Bradford  in  Yorkshire,  to 
know  if  he  will  undertake  to  make  them. 

I  desire  you  will  direct  to  me  in  Norfolk  Street,  and 


•  Given  in  his  letter  of  May  5,  1716  to  Jones,  quoted  p.  230  antea,  which  Taylor 
appears  to  have  seen  since  writing  the  letter  of  Nov.  27,  Smith  having  probably  alluded 
to  it  in  his  answer.  Newton's  6fh  form  (in  his  De  Quadratura  Curvanim)  com- 
prises the  integrals  of  two  expressions  equivalent  to ^—r, r  and  ^ ,  in 

_  a  +  bx^  +  cx*  a  +  bx'+cx** 

the  case  where  ft>2N/ur  and  a,  6,  <?  have  all  the  same  sig^n. 


BROOK  TAYLOR  TO  PROF.  SMITII.  235 

not  to  Crane  Court,  because  the  Servants  there  neglect 
bringing  me  letters,  and  I  am  very  seldom  there. 
I  am 
Sir 
Your  most  humble  Servant 
Norfolk_Street  Brook  Taylor 

11  Dec'  1718 

The  following  extracts  from  the  Journal  Book  of  the  Royal  Society 
will  contribute  to  complete  the  history  of  the  publication  of  the  //rir- 
monia  Mcnsuranim. 

"Dec.  la  1718.  The  Presitlent  acquainted  the  Society  that  D' 
Bentley  informed  him  that  100  Subscriptions  were  already  procured 
for  printing  W  Cotes's  Posthumous  Works." 

"Apr.  26.  1722. ..M'  Smith.. .made  the  Society  a  present  of  his 
Edition  of  the  Mathematical  Works  of  the  late  M'  Cotes... M'  Smith 
was  ordered  thanks  for  this  present." 


Among  the  Lucasian  MSS.  there  are  three  letters  from  Taylor  to 
Keill  (packet  No.  3).  The  1st  dated  17  July,  1717*  contains  a  critique 
upon  Stirling's  Lineco  Tertii  0rdini8  Neutonianw.  The  following  Post- 
script is  added.  "  Pray  do  me  the  favor  to  put  M'  Innys  in  mind  to 
send  me  the  Leipsic  Acts,  &  two  copies  of  Sir  Is:  Newton's  Opticks, 
as  soon  as  it  is  out,  one  bound,  &  another  in  sheets,  which  I  must 
send  to  M:  Monmort." 

The  2nd  (26  Apr.  1719)  contains  the  answer  of  Nic.  Bernoulli  of 
Padua  (John's  nephew)  to  a  message  which  Keill  had  sent  to  him 
through  Taylor  and  IMonmort.  Taylor  says  he  can  hardly  prevail  upon 
himself  to  forward  it,  "it  is  so  disagreable."  As  two  of  the  points 
referred  to  in  it  relate  more  or  less  to  our  philosopher,  we  may  possibly 
be  excused  for  giving  it  a  place  here.  It  is  couched  in  the  following 
language.  "  J'accepte  la  promesso  de  IM.  Keil  qui  est  do  mo  donner 
5  pistolles  pour  chaque  mensonge  dont  je  le  pourrai  convaincre.  Si 
done  ^I.  Keil  tient  sa  parole  jo  gagnerai  au  moins  20  pistolles  car  je 
soutiens  qu'il  ne  pas  dit  la  verite  1°.  lorsqu'il  a  dit  que  dcpuis  men 
sejour  a  Londres  J'avois  public  le  contraire  de  ce  quo  31.  Newton 
m'avoit  demontre  {Cf.  p.  142,  note}.  2".  lorsqu'il  a  dit  qu'on  a  oubliC* 
par  une  faute  d'Impression  le  mot  ut  dans  le  Scholium  qui  est  a  la  fin 
du  traitte  de  quadraturis.  3".  lorsqu'il  a  dit  que  men  oncle  (jo  passe 
sous  silence  ce  qu'il  dit  do  moy  dans  le  memo  endroit)  n'entcnd  pas  lo 
calcul  differenticl.  4°.  lorsqu'il  a  dit  nouvcllement  dans  sa  lettro  a 
M.  Taylor  that  he  can  shew  me  lyes  I  have  made  for  nothing.   Je  vous 


236  LETTER  OF 

prio  do  luy  fairo  notifior  ces  pretentions,  &  d'en  demander  sa  reponso." 
Tho  last  paragraph  of  the  letter  opens  with  the  words  "  Since  I  have 
heard  nothing  from  you  in  answer  to  my  proposal  of  joyning  with  you 
against  Bernoulli  I  have  dra>vn  up  a  paper*,  which  I  think  soon  to 
publish  by  itself." 

The  3rd  (26  Aug.  1721)  begins  thus:  "Tlio  enclosed  is  just  como 
to  me  from  Abbe  Conti,  who  desires  me  to  convey  it  to  you.  He  tells + 
me  that  he  disputes  continually  with  the  French  in  favor  of  Sir  Isaac 
Neuton  and  the  English  Mathematicians ;  but  that  he  can  by  no  means 
make  them  sensible  of  the  true  nature  of  Sir  Isaac's  method,  they  not 
yet  rightly  understanding  what  ho  means  by  first  and  last  ratios  of 

nascent  and  evanescent  quantities I  shall  trouble  you  with  no  more 

at  present,  not  knowing  how  unwelcome  this  little  may  be  to  you  from 
me,  upon  account  of  what  Bernoulli  has  publislit  f  out  of  my  letters  to 
Monmort  in  hopes  to  provoke  your  resentments  against  me."  Taylor 
then  enters  into  an  elaborate  explanation  of  tho  offensive  expression,  in 
the  course  of  which  he  lashes  Monmort  for  "  betraying  so  private  a 
letter  as  that  was,"  and  Bernoulli  for  publishing  it  The  apology 
seems  to  have  come  too  late.  The  letter  bears  tho  London  post-mark 
of  Aug.  28,  and  would  therefore  reach  Oxford  on  the  29th,  the  day  on 
which  poor  Keill  died.    The  address  is  crossed. 


Fran9oi3-Mario  Arouet  (Voltaire)  born  1694,  died  1778. 

LETTER  CXXI. 

VOLTAIRE  TO  PROF.   SMITH. 
S' 
I  have  perus'd   y'  book   of  optics,   I   cannot   be    so 
mightily  pleas'd  with  a  book,  without  Loving  the  author, 


•  Apologia  D.  Brook  Taylor... contra... J,  Bernonllium.  (It  is  a  reply  to  the  charg-e 
of  pla^arism  brought  against  him  in  the  "Epistola  pro  Eminente  Mathematico" 
Leipfiic  Acts,  July  1716).  Philosaph.  Trans.  March— May  1719,  p.  955.  Jo.  Bernoulli 
Opp.  II.  478.  It  was  shewa  by  Jones  to  Newton  before  publication.  See  Taylor's 
letter  to  Jones,  Mace.  Corr.  i.  279.  Keill  was  already  employed  on  his  own  account 
on  his  Epistola  ad. ..Jo.  Bernoulli.    See  p.  187,  antea. 

t  See  Conti's letter  to  Taylor,  (May  22, 1721),  Contempl.  Philos.  p.  124. 

:J:  In  Jo.  Burchardi... Epistola  ad. ..Taylor  (Leipsio  Acts  for  May  1721,  pp.  195 — ^228. 
Jo.  Bernoulli  Opp.  ii,  pp.  483 — 512),  a  reply  to  Taylor's  Apologia.  The  words  more 
especially  referred  to  are  as  follows :  "  Entre  nous,  je  suis  un  peu  de  I'avis  de  Mr. 
Bernoulli  que  Mr.  Keill  is  better  qualified  for  a  Champion  than  for  an  Analyste." 


VOLTAIRE  TO  PROF.  SMITH.  237 

give  me  leave  to  submitt  to  y'  judgement  these  little 
answer  of  mine,  w^'ich  I  have  writ  against  some  ignorant 
ennemies  of  S'  Isaae,  Neiiton,  whom  you  follow  so  closely 
in  the  path  of  truth  and  glory, 
I  am 
S' 
Y'  most  humble  obed 
{Hotel  de  Brie,  rue  Cloche-  Servant  Voltairb. 

Ferche]  Farts  the  10***  of 
October  {1739}  new  stile, 
M^  Smith 

This  letter  was  written  during  a  short  visit  which  Volta.iro  made  to 
Paris.  Ho  had  run  up  from  Bnissels  in  September,  purposing  to  stay 
about  a  month  in  what  he  calls  the  worse  than  Cartesian  tourUlloni  of 
the  French  capital,  but  on  tho  day  of  his  intended  departure  he  had  an 
attack  of  illness  which  detained  him  until  the  end  of  November,  In 
a  letter,  written  the  day  after  the  date  of  the  one  before  us,  he  describes 
the  plight  ho  was  in  between  his  two  medical  attendants  ("on  me 
saigne,  on  mo  baigne").  Under  these  circumstances,  added  to  long 
disuse  of  the  language,  we  need  not  be  surprised  to  find  his  English 
not  quite  so  good  as  when  he  wrote  a  dozen  years  before  during  his  resi- 
dence in  this  country. 

The  "  little  answer "  is  his  "  Reponse  aux  objections  principales 
qu*  on  a  faites  en  France  centre  la  philosophie  de  Newton,"  8vo.  Am- 
sterdam, 1739  (a  defence  against  the  attacks  that  had  been  made  upon 
his  EUmens  de  la  Philosophie  de  Newton...  1738,  and  against  miscon- 
ceptions on  some  points  in  the  Newtonian  philosophy).  The  following 
allusions  to  this  tract  occur  in  his  Correspondence.  Writing  to  Prince 
Frederic  of  Prussia,  "  the  Solomon  of  the  North/'  in  September,  shortly 
after  his  arrival  in  Paris,  he  says,  "  II  a  fallu  d'abord,  en  arrivant,  re- 
pondre  a  beaucoup  d'objections  que  j'ai  trouveos  repandues  a  Paris 
centre  les  d^couvertes  de  Newton.  Mais  ce  petit  devoir  dont  je  me  suis 
acquitte  ne  m'a  point  fait  perdre  de  vue  ce  Mahomet  {his  tragedy}  dont 
j'ai  d6ja  eu  I'honneur  d'envoyer  les  premices  a  votre  altesso  royale. 
Voici  deux  actes  tl-la-fois."  In  a  letter  to  Ilelvetius,  dated  a  week 
previous  to  this  letter  to  Smith,  ho  writes,  *'  Je  ne  sais  comment  je  m*y 
prendrai  pour  envoyer  une  courte  et  modeste  reponse  que  j'ai  faite  aux 
anti-newtoniens.  Je  suis  I'enfant  perdu  d'un  parti  dont  ]M.  de  Buffon 
est  le  chef,  et  je  suis  assez  comme  les  soldats  qui  so  battcnt  de  bon  coeur 
sans  trop  entendre  les  interets  de  leur  prince." 

Voltaire's  "  Elemens  do  la  Philosophic  de  Newton,  mis  d  la  port^ 


238  NOTE  OF  DUKE  OF  CUMBERLAND 

do  tout  le  mondo  "  (the  eight  last  words  were  added  by  the  booksellers) 
issued  from  the  press  at  Amsterdam  in  April,  1738,  without  his  know- 
ledge. The  impati«nco  of  the  booksellers  could  not  wait  for  his  recovery 
from  a  fit  of  sickness,  or  for  the  alterations  that  he  wished  to  make  in 
the  work,  and  they  employed  another  hand  to  complete  it  by  finishing 
the  23rd  chapter,  and  writing  two  additional  chapters  (the  24th  and 
2oth).  The  book  was  reprinted  at  Paris  (with  a  London  title-page) 
the  following  July,  accompanied  with  "  eclaircissements"  and  a  2Gth 
chapter  on  the  tides,  supplied  by  Voltaire :  these  he  also  sent  to  the 
Dutch  corsairs  (as  he  denominates  the  booksellers)  to  be  circulated  with 
their  edition.  Before  leaving  Paris,  in  November,  1739,  ho  tells  Fre- 
deric that  a  now  edition  was  called  for,  and  ho  republished  the  work  in 
an  enlarged  and  otherwise  altered  form  (1741)*,  with  flattering  re- 
ferences to  Smith's  Optics  (see,  for  example,  the  explanation  of  the 
sun  or  moon  appearing  larger  on  the  horizon  than  on  the  meridian, 
Part  2,  ch.  viii.  "  le  docteur  Smith  a  la  gloire  d'avoir  enfin  trouve  la 
solution  complete  d'un  probleme  sur  lequel  les  plus  grands  g^nies  avai- 
ent  fait  des  systemes  inutiles").  Journal  des  Savants y  1738.  Billio- 
thi'que  Franfaise,  1738,  1739.  Voltaire's  Corrcspondance.  His  Life 
in  Blogr*  Univ.  (Beuchot's  note).    Bcuchot's  Voltaire^  tom.  38. 

In  a  letter,  written  from  Leyden  in  Feb.  1737,  Voltaire  says,  "  Je 
pars  incessamment  pour  achever  a  Cambridge  mon  petit  cours  do  new- 
tonisme :"  (he  had  been  studying  the  Newtonian  philosophy  for  some 
weeks  under  's  Gravesande  at  Leyden,  where  he  had  taken  shelter  from 
the  storm  that  burst  upon  him  on  the  appearance  of  "  Le  Mondain  "). 
But  the  announcement  was  intended  only  as  a  blind  to  his  enemies. 
Ho  in  reality  returned  to  his  retreat  at  Ciroy,  in  Champagne.  Some 
of  his  biographers  state  that  his  letters  at  this  time  were  dated  from 
Cambridge,  but  there  are  no  letters  so  dated  in  his  published  Corre- 
spondence. 


William  Augustus,  son  of  George  IL  bom  1721,  died  1705. 
LETTER  CXXII. 
DUKE  OF  CUMBERLAND  TO  PROF.   SMITH. 

{July  3.  1740}. 
Doctor  Smith  I  desire  you  would  lose  no  time  in  pro- 
viding a  Sea  Quadrant  and  Telescope  for  to  fit  my  eye ; 


Lalande  also  mentions  an  edition  in  the  followiniif  year. 


TO  PROF.  SMITH.  239 

my  baggage  goes  at  five  this  afternoon;   I  shall  be  ox- 
trcamly  obliged  to  you. 

William 
Endorsed  by  D'  Smith.     "  Tho  Dulco  of  Cumberlanda  Note  to  mo." 

This  note  was  probably  written  by  tho  fnturo  •'butcher,**  wlien  ho 
was  on  the  point  of  setting  out  to  join  the  squadron  under  Sir  John 
Norris,  which  was  supposed  to  be  destined  for  an  attack  upon  the 
Spanish  fleet  in  Ferrol.  "  Friday,  July  4.  1740.  The  Duke  of  Cum- 
berland who  had  been  some  time  at  his  post  in  tho  camp  at  Ilounslow 
{he  was  Colonel  of  the  Coldstream  Guards}  left  it  on  a  sudden,  and 
arrived  at  Portsmouth  unexpected,"  where  he  "  went  aboard  tho  Victory 
Man  of  AVar  as  a  Volunteer." — Gentleman* 8  MaffazinCy  July  1740.  The 
London  Evening  Post  states  that  ho  set  out  from  St  James's  for  Ports- 
mouth at  4  in  the  morning.  Tho  weather  proving  unfavourable,  tho 
Admiral  and  the  young  Volunteer  returned  to  London  in  September. 

The  Duke  was  now  turned  19.  Smith  had  been  in  attendance 
upon  him  since  June,  1739  (Conclusion  Book,  June  11). 


APPENDIX. 


Henry  Oldenburg,  born  1626  at  Bremen,  died  1677*  Secretary  of 
the  Royal  Society.     He  was  a  friend  of  Milton's. 

No.   I. 

OLDENBURG   TO  NEWTON. 

Beginning  of  their  Correspondence. 

Accompanying  this  letter  were  a  figure  and  description  in  Latin  of 
the  reflecting  telescope  made  by  Newton  the  preceding  autumn  and 
sent  up  "for  the  King's  perusal"  in  December.  See  Syn.  View  of 
Newton's  Life  under  the  year  167L 

S' 
Your  Ingenuity  is  the  occaon  of  this  addresse  by  a 
hand  unknowne  to  you.  You  have  been  so  generous,  as 
to  impart  to  the  Philosophers  here,  your  Invention  of 
contracting  Telescopes.  It  having  been  considered,  and 
examined  here  by  some  of  y*  most  eminent  in  Opticall 
Science  and  practise,  and  applauded  by  them,  they  think 
it  necessary  to  use  some  meanes  to  secure  this  Invention 
from  y*  Usurpaon  of  forreiners  ;  And  therefore  have  taken 
care  to  represent  by  a  scheme  that  first  Specimen,  sent 
hither  by  you,  and  to  describe  all  y®  parts  of  y®  Instru- 
ment, together  w*^  its  effect,  compared  w*^  an  ordinary, 
but  much  larger,  Glasse;  and  to  send  this  figure,  and 
description  by  y®  Secretary  of  y®  B.  Soc.  (where  you  were 
lately  by  y*  L^  B^.  of  Sarum  proposed  Candidat)  in  a 
solemn  letter  to  Paris  to  M.  Hugens*,  thereby  to  prevent 
the  arrogation  of  such  strangers,  as  may  perhaps  have  seen 
it  here,  or  even  w*^  you  at  Cambridge;  it  being  too 
frequent,  y*  new  Inventions  and  contrivances  are  snatched 
away  from  their  true  Authors  by  pretending  bystanders  ; 


•  As  OldeDburg  had  promised  in  a  letter  to  JIuygens,  Jan.  1.    Letter  Bk.  Roy.  Soc.  v.  92. 


Append.]  NEWTON  AND  OLDENBURG.  241 

But  yet  it  was  not  thought  fit  to  send  this  away  w***  out 
first  giving  you  notice  of  it,  and  sending  to  you  y*  very 
figure  and  description,  as  it  was  here  drawne  up*;  y*  so 
you  might  adde,  &  alter,  as  you  shall  see  cause ;  w***  being 
done  here  w***,  I  shall  desire  your  favour  of  returning  it 
w***  all  convenient  speed,  together  w*^  such  alterations,  as 
you  shall  think  fit  to  make  therein. 

Though  divers  of  y*  most  skillfull  examiners  agreed 
y*  your  Tube  magnifyed,  by  measure,  y*  object  here  repre- 
sented by  A-ft  so  much,  as  you  see,  above  w*  a  much 
greater  Telescope  did;  yet  there  were  others,  well  versed 
also  in  Optic  glasses,  yS  though  they  could  not  disprove 
that  mensuraon,  yet  were  positive  to  affirm,  y*  y*  excesse 
of  magnitude  did  not  appeare  such  to  their  eye. 

Besides  it  was  discoursed,  y'  by  this  way  of  yours  it 
was  longsome,  &  difficult  to  find  y®  Object:  w^^  incon- 
venience yet  they  looked  upon  as  possible  to  be  remedied, 
I  shall  be  glad,  S^  to  receive  your  speedy  answer  to  these 
lines,  and  embrace  all  occasions  to  expresse  my  singular 
respects  to  your  merit,  as  becomes 

S' 
Your  humble  Ser>'ant 
Jan.  2.  167^.  Oldenburg  J. 

Newton's  answer,  dated  Jan.  6,  will  bo  found  in  Mace.  Corr.  ii.  31 1, 
and  (not  complete)  in  Birch,  ni.  2,  Ilorsley,  iv.  271.  Comp.  Syn.  View 
under  that  date. 


No.  II. 
NEWTON  TO   OLDENBURG. 
S'  Cambridg  March  16*''  1671  }2 1 

The  book  w*^^  my  Carrier  by  forgetfulncsse  disappointed 
me  of  the  last  week   I  have  now  received  &  thank  you 


•  Orig.  Lett.  Bk.  Roy.  Soc.  N.  1.  37.  Horsley  iv.  270. 

t  This  is  fig.  2.  Tab.  I.  Phil.  Tram.  March  25,  1672.     Or,  see  Horsley  iv.  fig. 
facing  p.  280. 

^  From  a  copy  corrected  by  Oldenburg  (Orig.  Lett.  Bk.  Roy.  Soc,  0. 2. 64). 

16 


242  CORRESPONDENCE  OF  [Append. 

for  it.  With  the  Telescope  w*^^  I  made  I  have  sometimes 
seen  remote  objects  &  particularly  the  Moon  very  distinct 
in  those  p*'  of  it  w*^^  were  neare  the  sides  of  the  visible 
angle.  And  at  other  times  when  it  hath  been  otherwise 
put  together  it  hath  exhibited  things  not  w^^out  some  con- 
fusion. W^^  difference  I  attributed  chicfely  to  some  imper- 
fection that  might  possibly  be  either  in  the  figures  of 
y*  metalls  or  eye  glasse,  &  once  I  found  it  caused  by 
a  little  tarnishing  of  the  Metall  in  4  or  5  days  of  moist 
weather. 

One  of  the  ffellows  of  o*^  College  is  making  such 
another  Telescope  w*'*  w*^^  last  night  I  looked  on  Jupiter  & 
he  seemed  as  distinct  &  sharply  defined  as  I  have  seen 
him  in  other  Telescopes.  When  he  hath  finished  it  I  will 
examin  more  strictly  &  send  you  an  account  of  its  per- 
formances, ffor  it  seemes  to  be  something  better  then  that 
^ych  J  made. 

Yo'  humble  servant 

Tliese  I.  Newton* 

To  Henry  OLDENBUiia  Esq :  at  his  house 
abotit  the  middle  of  the  old  Pall-mail 
in  Westminster.  London 

"rec.  March  18.  71" 

In  Oldenburg's  hand. 


No.  III. 

NEWTON  TO  OLDENliURG. 

March  19.  1671 12|. 
AJler  describing  the  performances  of  the  instrument  mentioned  in  the  last 
letter  he  proceeds : 

This  may  be  of  some  use  to  those  that  shall  endeavour 
any  thing  in  Reflexions ;  for  hereby  they  will  in  some 
measure  be  enabled  to  judge  of  the  goodness  of  their 
Instruments.  And  for  this  end  you  may  annex  these 
observations  made   with  this  last  instrument  to  the  de- 


Orig.  Lett.  Bk.  Roy.  Soc.  N.  1.  35. 


ArrEXD.]  NEWTON  AND  OLDENBURG.  243 

scription  of  it  in  the  Transactions  of  this  month.  But  my 
answer  to  M'  Hooks  observations  will  not  be  ready  for 
them,  because  I  intend  to  annex  to  that  answer  some 
further  explications  of  the  Theory  which  I  shall  not  have 
leisure  to  do  this  week  or  fourtnight. 

I  am  in  hast 

Yo'  faithfull  Serv« 

Endorsed  by  Oldenburg:  I.  New  TON  • 

"  Rcc**.  20.  Ans'*.  23  comiii{unicating} 
v"  Comet  and  «  sub  cap.  Cy^iii  from 
Hevel."  See  Phil,  Trans.  March  25, 
1072,  p.  4017. 


No.   IV. 

NEWTON    TO  OLDENBURG. 
S*"  March  26*''.  1672 

About  10  days  since  at  night  I  saw  a  dull  starr  south 
west  of  Perseus,  which  I  now  take  to  have  beene  that 
Comet  of  which  you  give  me  information ;  But  it  was  very 
small  &  had  not  any  visible  tayle  which  made  me  regard 
it  noe  further,  &  I  feare  it  will  now  bee  difficult  to 
find  itf. 

Since  my  last  letter  I  have  further  compared  the  two 

telescopes  &c.    {See  Phil  Trans,  Apr.  22.  1672  p,  4032.) 

*  *  *  #  ♦ 

Thus  much  of  these  Telescopes,  &  at  present  I  shiJI 
trouble  you  no  further  then  to  thanke  you  for  your  last 
intelligence,  by  which  you  have  obliged 

Your  faithfull  servant 
I.  Newton  J. 

•  Orig.  Lett.  JDk.  Roy.  Soc.  N.  1.  36.  For  the  first  part  of  the  letter  see  Phil 
Tram.  March  25,  1672,  p.  4009,  where  "considerable"  is  printed  by  mistake  for 
"  insensible.*' 

t  Phil.  Tram.  March  25, 1672,  p.  4018. 

X  Lett.  Bk.  Roy.  Soc.  v.  187.  Horsley  iv.  275. 

16—2 


244  CORRESPONDENCE  OF  [Append. 

No.  V. 

NEWTON  TO  OLDENBURG. 
**  Asserting  the  advanta<yo  of  reflecting  telescopes  above  refracting 
ones,  Sc  endeavouring  to  remove  some  inconveniences  in  the  former." 
f^tt,  Dh  Roy.  Soc.  v.  193. 

S'  March  30.  1672 

I  doubt  not  but  Mons'.  Auzout  &c.     (See  Phil,  Tr.  Apr. 
22.  1672.  p.  4034). 

»  «  «  «  * 

In  the  meane  time  to  remedy  in  some  measure  these  incon- 
veniences, I  shall  propound  a  way  *  of  using,  instead  of  the 
little  ovall  metall,  a  glass  or  crystall  figured  like  a  triangu- 
lar Prism,  as  you  see  it  rejpresented  in  the  first  scheme  by 
the  figure  ABc,    It's  side 

ABha  I  suppose  to  per-          ^  j)T\ 

forme  the  office  of  that       ^^^-^ '^  ^ 


metall  by   reflecting   to- 
wards the  eyeglasse  the 

light   which  comes  from  ....^^ 

................■^•11 

the  concave  DE:   which  '^ 

light  I  suppose  to  enter  into  this  Prism  at  its  side  CBhc, 
&  after  reflexion  to  emerge  at  the  side  ACca  before  it  con- 
vene at  F,  the  focus  of  the  glasse.  The  axes  of  the  eye- 
glasse and  concave  metall  must  be  perpendicular  to  the  midle 
of  the  planes  ACca  and  CBbc.  And  least  any  colours 
should  be  produced  by  the  refraction  of  those  planes,  'tis 
requisite  that  the  angles  of  the  Prism  at  J  a  Sz  Bh  bee  pre- 
cisely equall :  which  may  most  conveniently  be  performed 
by  making  them  halfe  right  angles  &  consequently  the  third 
angle  at  Cc  a  right  one.  The  plane  ABha  without  being 
foliated  will  reflect  all  the  light  incident  on  it ;  Especially 
if  the  Prism  be  made  of  Crystall.  But  to  exclude  all  un- 
necessary light,  'tis  convenient  that  it  bee  all  over  covered 
with  some  blackc  substance,  excepting  two  circular  spaces 


•  Comp.  Optics,  Book  i.  Part  1.  Prop.  viii. 


Append.]  NEWTON  AND  OLDENBURG.  245 

of  the  planes  AcSc  Boi^or  the  usefull  light  to  posse  through, 

as  you  see  it  designed  in  the  2<i  scheme.     The  length  of 

this  Prism  should  bee  such,  that  its 

sides  Ac  8c  Be  may  be  four-square, 

and  so  much  of  the  angles  B  Sib,  q.3 

are  superfluous,  ought  to  bee  ground 

ofFj  to  give  passage  to  as  much  liglit 

as  is  possible  from  the  object  to  the 

concave. 

There  is  one  very  considerable 
advantage  of  this  Prism,  which  the 
ovall  metall  is  not  capable  of,  without  using  two  eye- 
glasses, and  it  is,  that  if  its  sides  ACca  &  BCcb  bee 
ground  convex,  it  >vill  erect  the  object  by  performing  the 
office  of  a  double  convex  lens.  The  manner  you  have  ex- 
pressed in  the  3*^  scheme ;  where  suppose  G  to  be  the  focus 

of  the  concave,  and  F  -.  : 

tip 
of  the     eye-glasse    at  \j!^ 

which  the  rays   crosse 

tmce  before  their  arri- 

vall  at  the  eye.     But  it 

is  convenient,  that  the 

first  tryalls  bee    made  * 

with  Prisms  whose  sides 

are  all  of  them  plane.     And  thus  much  concerning  Mons' 

Auzout*s  considerations. 

To  the  queries  of  Mons'  Denys  I  answer,  1.  That  a  Tube 
of  six  inches  is  capable  of  bearing  an  aperture  (limited  next 
the  eye)  so  large,  that  an  obstacle  of  1;^  or  1^  of  an  inch 
in  breadth  shall  be  requisite  to  intercept  all  the  light  com- 
ing from  one  point  of  the  object  towards  the  concave  metall : 
But  it  is  convenient,  that  the  Tube  bee  a  little  wider  than 
that  aperture  precisely  requires,  suppose  1^  or  l§-  of  an 
inch,  &  not  more;  And  the  whole  breadth  of  the  metall 
should  not  bee  lesse  than  two  inches,  because  its  figure  to- 


-n— 


A 


O 


246 


CORRESPONDENCE  OF 


[AlTEND. 


wards  the  edges  will  scarcely  bee  so  true  as  to  bee  usefull. 
And  by  that  meanes  it  may  also  bee  conveniently  fastened 
to  the  end  of  the  Tube  on  the  outside,  so  as  at  pleasure  to 
bee  taken  off  &  layd  up  close  from  the  Air,  to  preserve  it 
from  tarnishing. 

How  the  Diameter  of  the  Tube  is  to  bee  enlarged  ac- 
cording to  its  length,  will  appeare  by  the  Table  of  Aperturs 
and  charges  which  I  sent  you  in  my  last  letter  of  March 
the  2(i*^  Namely  the  Cube  of  its  length  should  be  propor- 
tionable to  the  square-square  of  its  diameter  or  aperture  at 
the  me  tall ;  so  that  the  advantage  of  augmenting  the  length 
of  Tubes  is  by  this  way  far  greater  than  by  refractions, 
where  their  length  ought  to  bee  proportionall  to  the  square 
of  the  diameter  of  the  aperture. 

2.  The  breadth  or  shortest  diameter  of  the  little  ovall- 
metall  for  a  Tube  of  six  inches  should  not  bee  greater  than 
i,  nor  lesse  than  J  of  an  inch ;  And  the  longest  Diameter 
should  bee  to  the  shortest  as  about  10  to  7.  But  you  may 
more  exactly  determine  these  diameters  for  Tubes  of  all 
lengths  after  this  manner ;  In  the  4*^  figure  let  AB  repre- 
sent the  ovall  sett  edg- 
wise ;  DE  the  concave ; 
FG  its  axis;  Gp  the 
reflex  of  that  axis;  st 
the  Diameter  of  the 
hole  through  which  the 
light  is  transmitted  to 
the  eye ;  &  P  the  cen- 
ter of  that  hole.  Pro- 
duce FG  to  TT,  so  that 
Gir  may  bee  cquall  to  Gp;  erect  Tra-  &  ^t  equall  to  ps  &pt, 
&  from  a-Si  T  draw  two  feies,  aD  &  tF,  to  4;he  utmost  parts 
of  the  concave,  w^^in  the  Tube  intersecting  AB  in  ^  &  B;  & 
AB  shall  bee  the  long  djameter  of  the  ovall ;  which  bisect 
in  a?,  &  perpendicular  to  Fx  erect  xy  &  xz  occurring  ^vith  <rD 


sPr 


A^'PEND.]  XE\VTON  AND  OLDENBURG.  247 

&  tE  in  y  &  z,  &  a  meane  proportionall  between  xy  &  xz 
doubled  shal  be  the  other  short  diameter :  fibr,  by  viewing 
y®  scheme  you  will  easily  perceive,  that  an  ovall,  described 
with  those  rectangular  conjugate  diameters,  is  of  sufficient 
bignesse  to  reflect  all  the  uscfuU  light  towards  the  eye,  if  it 
be  rightly  placed  in  the  Tube ;  &  a  broader  metall  would 
not  onely  intercept  too  many  of  the  best  rays,  but  some  of 
the  scattering  light,  reflected  every  way  from  its  superfluous 
parts,  woidd  fall  on  the  eye-glasse  &  make  the  object  ap- 
peare  something  confused  &  as  it  were  in  a  mist.  This,  S^ 
is  that,  which  in  answer  to  your  letter  my  present  thoughts 
suggest  to 

Your  faithfull  Servant 

I.  Newton*. 


No.  VI. 

NEWTON  TO  OLDENBURG. 

S"^  Cambridgo  April  13.  1672. 

I  herewith  send  you  an  answer  to  the  Jesuite  Pardies 
Considerations ;  in  the  conclusion  of  which  you  may  pos- 
sibly apprehend  me  a  little  too  positive,  but  I  speake  only 
for  myselfe.  I  am  highly  sensible  of  your  good  >vill  in  com- 
municating to  me  such  observations  as  occurr  concerning 
my  Theories  or  Catadioptricall  instruments,  and  I  desire 
you  to  continue  that  favour  to  me.  I  shall  immediately 
proceed  to  add  what  I  promised  to  my  answer  to  Mr,  Hooks 
observations,  &  then  send  it  you.  Mons"*  Hugens  has  very 
well  observed  the  confusion  of  refractions  near  the  edges 
of  a  lens,  where  its  two  superficies's  are  inclined  much  like 


•  Lett.  Sk,  Roy.  Soc.  v.  193.  This  and  some  other  letters  have  been  printed  by 
Horsley  (Vol.  IV.)  from  the  MSS.  at  the  Royal  Society,  but  not  so  as  altogether  to 
supersede  the  necessity  of  their  reappearance  here  in  a  more  complete  and  accurate 
form. 


248  CORRESPONDENCE  OF  [Append. 

the  planes  of  a  prisme  whose  refractions  are  in  like  manner 
confused.  But  it  is  not  from  the  inclhiation  of  those  super- 
ficies so  much  as  from  the  heterogeneity  of  light  that  that 
confusion  is  caused:  ffor  by  illuminating  an  object  with 
homogeneall  light,  I  have  seen  it  far  distincter  through  a 
Prism  than  I  could  by  light  that  was  heterogeneal. 

I  suppose,  the  designe  of  S'^  Robt  Moray's  experiments 

is  &c.     {See  Phil  Tr.  May  2^.  1672.  p.  4060). 

«  «  «  «  « 

Thus  far  concerning  S""  E,*  Morays  proposalls.  I  have 
nothing  more  at  present  unlesse  to  desire  you,  that  in  y® 
letter  wherein  I  sent  you  the  Table  of  apertures  and  charges 
you  would  change  an  expression  concerning  the  six  foot 
Tube  where  I  intimated  that  it  was  none  of  the  best  in  its 
kind,  ffor  least  the  friend,  of  whom  it  was  borrowed,  should 
thinke  I  depreciate  it,  I  had  rather  that  the  expression 
should  be  a  little  intimated  after  this  manner ;  that  I  am 
not  very  well  assured  of  its  goodnesse,  &  therefore  desire, 
that  the  other  experiment  of  reading  at  100  foot  distances 
should  rather  be  confided  in.  You  will  do  me  a  favour  to 
peruse  the  rest  of  that  letter  also  before  you  commit  it  to 
the  presse.  ffor  I  writ  it  in  so  much  hast,  that  I  had  no 
time  to  review  it :  And  by  rendring  my  expressions  more 
perspicuous  or  lesse  ambiguous  you  will  still  oblige 

Your  faithfull  Servant 

I.  Newton*. 


No.  VII. 

NEWTON  TO  OLDENBURG. 
S'  June  11***  1672. 

I  have  sent  you  my  Answers  to  M'  Hook  &  P.  Pardies, 
w*^**  I  hope  will  bring  with  y™  y'  satisfaction  w*^^  I  promised. 


Lett.  Bk.  Roy.  Soc.  v.  222. 


Append.]  NEWTON  AND  OLDENBURG.  249 

And  as  there  is  nothing  in  W,  Hooks  Considerations  w*** 
w*^''  I  am  not  well  contented,  so  I  presume  there  is  as  little 
in  mine  w^^  he  ean  exeep{t}  against,  since  you  will  easily 
see  that  I  have  industriously  avoyded  y*  intermixing  of 
oblique  &  glancing  expressions  in  my  discourse.  So  y*  I 
hope  it  will  be  needlesse  to  trouble  the  R.  Society  to 
adjust  matters.  However  if  there  should  possibly  be  any 
thing  esteemed  of  y*  kind,  I  desire  it  may  be  interpreted 
candidly  &  with  respect  to  the  contents  of  M''  Hooks 
Considerations,  &  I  shall  readily  give  way  to  y*  mitigation 
of  whatsoever  y®  Heads  of  y*  11.  Society  shall  esteem 
personall.  And  concerning  my  former  Answer  to  P. 
Pardies,  I  resigne  to  you  y*'  same  liberty  w*^^  he  hath  done 
for  his  Objections,  of  mollifying  any  expressions  that  may 
have  a  shew  of  harshnesse. 

Yo"^  Servant 
TJiese  I.  Newton •. 

To  Henuy  Oldenburg  Esq :  at  his  house 
abcut  y"  middle  of  y^  old  Pall-maile 
in  Westminlsjter  London. 


No.  vni. 

NEWTON  TO   OLDENBURG. 

Cambridg 
S'  July  30*^  1672 

The  last  week  I  wrote  to  you  that  y*  ^Metall  w*^^  you 
sent  me  was  well  for  closenesse  &  hardnesse  but  yet  of  a 
colour  not  very  brisque  &  inclining  to  red.  However  if  it 
be  less  apt  to  tarnish  then  any  other  mixture  yet  known, 
that  will  sufficiently  recompense  y*  other  imperfections. 
Yo"  of  July  IG***  directed  to  Stoake  is  not  yet  come  to  my 
hands.  I  feare  it  is  miscarried,  and  desire  therefore  you 
would  favour  me  w^^  y*  particulars  w*^^  were  in  answer  to 


Orig.  Lett.  Bk.  Roy.  Soc.  N.  1.  39. 


250  CORRESPONDENCE  OF  [Append. 

y'  troublesome  letter  ^^  written  last  from  Stoake,  for  w*^**  I 
hegg  yo'  pardon.  I  send  you  by  John  Stiles  13'  for  the 
last  quarter. 

Yo'  humble  Servant 

These  Newton  t 

To  Henry  Oldenburg  Esq :  at  his  house 

about  the  middle  o/the  old  Pall-Maile 

in  Westminster  London 

^9'^  13*. 

"Reo.  July  31.  72  Answ.  codem.  and  repeated  y®  contents  of  my 
letter  of  July  16."     Mem.  by  Oldenburg. 


No.  VIII.  (bis). 

OLDENBURG   TO  NEWTON. 
[Extract.] 
Sent  in  conformity  with  the  wish  expressed  in  the  preceding  letter. 
S'  Lond.  July  16.  1672. 

I  have  spoken  with  Mr  Cock  about  the  four  foot  Tube, 
which  hath  been  ready  a  pretty  while.  He  saith  that  the 
object-speculum  (being  a  compound  of  copper,  tin,  tin- 
glasse,  antimony  and  a  little  arsenick)  is  of  about  6  inches 
diameter,  wrought  upon  a  tool  of  about  14  or  15  foot,  and 
drawing  4  foot,  more  or  less.  He  adds,  that  tis  very  good 
mettall,  shewing  the  moon  very  well,  but  other  objects 
faint ;  perhaps  for  want  of  giving  it  its  due  charge.  Tis 
lodged  in  a  square  box,  with  a  lid  at  the  end  of  it,  for 
placing  the  speculum-plate,  lodged  in  it,  at  such  a  dis- 
tance as  shall  be  requisite.  He  offers  to  unpolish  this 
plate  again,  and  to  send  you  this  very  Instrument  for  5^^ ; 
and  what  alterations  or  emendations  you  shall  direct  to  bee 
made  herein  upon  triall,  hee  will  make,  without  demanding 


•  Dated  July  13.   It  is  printed  in  Gen.  Diet.  vir.  782.    Mace.  Corr.  ir.  332. 
t  Or'ig.  Lett.Bk.  Roy.Soc.  N.  1.  41. 


Append.]  NEWTON  AND  OLDENBURG.  25J 

any  more  money  for  that  labour.  I  intend,  god  permitting, 
to  send  by  the  next  conveniency  of  your  Cambridge 
Carrier,  J.  Stiles,  a  piece  of  that  very  mettal,  with  the 
s**  objeet-speeulum,  w*^**  the  4  foot  Teleseope  is  com- 
pounded off. 

As  to  the  steely  Speculum,  he  saith,  tis  a  pure  Venice- 
Steel,  forged  with  much  care;  not  melted,  nor  com- 
pounded with  any  thing  ;  of  3  inches  diameter,  but  bearing 
not  so  good  a  polish.  And  this  he  is  not  unwilling  to  send 
also  to  you  to  Cambridge  for  your  examination,  and 
further  directions  about  it.  Hee  saith,  that  tis  very  hard 
&  tedious  to  grind  this  steely  matter  true*. 


No.  IX. 

NEWTON  TO  OLDENBURG. 

For  the  first  part  of  the  letter  see  Rlgaud^s  Appendix  to  his  Essay,  No, 
VIII,  pp.  42,  44,  aiid  the  Phil,  Trans,  for  Juli/  21,  1C73,  p.  6087. 

i|(  «  «  «  « 

Pray  w*^  these  Notes  return  my  thanks  to  M.  Hugens 
for  his  book. 

By  a  former  letter  of  yo"  I  was  a  little  dubious 
whether  M.  Slusius  might  not  apprehend,  by  w*  you  wrote 
to  him  concerning  me,  y^  I  pretended  to  his  Method 
of  drawing  tangents ;  untill  I  understood  by  M.  Collins  y* 
you  signified  to  him  y'  you  thought  it  here  of  a  later  date, 
ffor  it  seems  to  me  that  he  was  acquainted  w***  it  some 
yeares  before  he  printed  his  Mesolabum  &  consequently 
before  I  understood  it.  But  if  it  had  been  otherwise 
yet  since  he  first  imparted  it  to  his  friends  &  y*  world,  it 
ought  deservedly  to  be  accounted  his.  As  for  y®  Methods 
they  are  y®  same,  though  I  beleive  derived  from  different 
principles.     But  I  know  not  whether  his  Principles  afford 


•  Orig.  Lett.  Bk.  Hoy.  Soc.  0. 2.  92. 


252  ALGEBRAICAL  PAPER  [Append. 

it  so  general!  as  mine  w^^  extend  to  Equations  affected  w*'* 
surd  terms,  w*^out  reducing  them  to  another  form.  But  if 
you  please  let  this  pass. 

The  incongruities  you  speak  of,  I  pass  by.  But  I 
must,  as  formerly,  signify  to  you  y*  I  intend  to  be  no 
further  soUicitous  about  matters  of  Philosophy.  And  there- 
fore I  hope  you  will  not  take  it  ill  if  you  find  me  cease 
from  doing  any  thing  more  in  y*  kind,  or  rather  y*  you 
will  favour  me  in  my  determination  by  preventing  so  far 
as  you  can  conveniently  any  objections  or  other  philoso- 
phical letters  that  may  concern  me.  For  your  profer 
about  my  Quarterly  payments  I  thank  you.  But  I  would 
not  have  you  trouble  yo'^self  to  get  them  excused  if  you 
have  not  done  it  already.  And  now  being  tired  w*^  this 
long  letter,  I  must  in  hast  write  myself 

Yo^  humble  Servant 
Cambridg.  June  23.  73.  I.  Newton*. 


No.   X. 

Paper  given  by  Newton  to  Flamsteed  at  lecture  in  1674.  It  is  printed 
here  as  exhibiting  to  us,  perhaps  in  a  more  vivid  manner  than  his 
actual  lectures,  the  philosopher  descending  to  the  level  of  an  elementary 
teacher. 

I.     (a)    +  6  =  0?  per  reductionem  fit  ao?  +  a6  -  6.^' 
wa  ax  '-  wx   seu    wx  =  ox  -  ab,      («)    «  w  -  c   fit 

^    ^    2cy-cc 

a^  -  abb  a?  ^  abb  —  (? 

'^yy  -  3cy  +  cc    seu 

c 

(y)  —  —  o  =a  a?  fit  aa  -  ao?  «  xx.     .  ^ . 

X  ^  ^    cxx       a  +  b  -  X 

a^bb  +  aab^  —  aabbx 


•  Orig.  Lett.  Bk.  Roy.  Soc.  N.  1.  47.  The  date  is  in  Oldenburg's  hand.  The 
part  of  the  letter  which  we  have  given  here  is  crossed  out  in  the  MS.  probably  by 
Oldenburg.    The  whole  of  the  letter  is  printed  in  Ilorsley  iv.  342. 


ArPEPm.]  GIVEN  TO  FLAMSTEED.  253 

TT       /  \    oa-'^**?  «.  o       +  2aa 

II.  (a)    —  +  a  a  or   nt   «rd7 -i       .or 

a  +  6  ^  +  6       +  a6 

y'  -  rt^y  

(i3)   7====f+a  cv/afl-fefefity*- ofty  +  fl66  -  0. 

III.  (a)  \/«rt-fl.t?  +  «afl?  fit  aa^axmrnaw-^Qaof-haa 
seu  tV  =  <7.  (/3)  v/s  :  aaa?  +  2axx  -  a?^-  o  +  a?  =  0  fit  aax 
+  Sa.f.r  —  ,r'  «  «'  —  i^oa.r  +  Saa'.r  -  .ir*  sen  .rar  —  4rt.'«?  -  aa, 

(7)    y  =  V  «y  +  yy  -a  \/ay  -  yy  primo  fit  y  «  \/ay  -  yy 
d{e}in2y  =  «. 

IV.  (a)    2y  =  a  fit  y  -  ^a.      (/3)    ~  -  a   fit  ,r  -  ^  . 

(7)  aat-cx^ac   fit    u;  « .      (d)  it?  xat 

a "  c  —  cc  A-  aac 

a?-a'cc  =  0,  tit  .1'  + a?j?  — aaj7  — =  0.* 

+  aacc  2ac  —  cc  2fls-c 


No.   XI. 

NEWTON  TO   OLDENBURG. 

Not.  13.  1676. 

TJie  principal  part  of  the  letter  is  printed  in  the  Transaetiona  for  January 
24,  1C76:  the  remainder  is  as  follows : 

I  have  returnd  you  Mr  Line's  letter.  It  came  to  my 
hands  but  this  week;  the  Gentleman  by  whom  you  sent  it 
having  not  yet  been  at  Cambridge  but  transmitting  it  to 
me  from  Oxford. 


•  From  the  ori^nal  paper  in  Newton'ti  hand,  pasted  in  at  the  beginning  of  Vol.  42 
of  Flamsteed's  MSS.  at  Greenwich :  at  the  l)ottom  are  the  words  •*  Mr  Newton's  paper 
given  at  one  of  his  lectures,  Midsummer,  1674.'*  Flanisteed  wa?  at  Cambridge,  from  the 
end  of  May  until  July  13.  He  brought  with  him  a  Royal  Mandate  for  tlie  degree  of 
M.A.  which  was  conferred  upon  him  on  June  5.  He  had  been  admitted  a  pensioner  nt 
Jesus  College  Dec.  21. 1670,  durinjc^  a  short  stay  he  made  at  Cambridge  on  his  return 
from  London  to  Derby,  when  he  also  took  the  op|)ortunity  of  eiUing  upon  Barrow  and 
Newton.    Comp.  Ibily,  p.  29. 

I,  III  and  IV  (except  y)  will  be  found  in  the  published  Algebra  Leeture$  {L^cU 
6  and  7 ).  Re^if.  3,  4,  5  pp.  65—67. 


254  CORRESPONDENCE  OF  [Append. 

I  had  some  thoughts  of  writing  a  further  discours 
about  colours  to  be  read  at  one  of  yo'  Assemblies,  but 
find  it  yet  against  y*  grain  to  put  pen  to  paper  any  more 
on  y*  subject.  But  however  I  have  one  discourse  by  me 
of  y*  subject  written  when  I  sent  my  first  letters  to  you 
about  colours  &  of  w*^**  I  then  gave  you  notice.  This  you 
may  command  w"  you  think  it  will  be  convenient  if  y® 
custome  of  reading  weekly  discourses  still  continue*.  In 
y*  meane  while  I  am  S'" 

Yo"'  humble  Serv"* 

Is.  Newton f. 


No.  XII. 
NEWTON  TO  OLDENBURG. 

S'  Cambr.  Novemb  30  1675. 

I  intended  to  have  sent  you  y®  papers  this  week  but 
upon  reviewing  them  it  came  into  my  mind  to  write  ano- 
ther little  scrible  J  to  accompany  them :  You  may  expect 
*em  y*  next  week.  An  ancient  Gentleman  I  met  at  yo' 
Assemblies  (whose  name  I  cannot  recollect,)  being  thick  of 
hearing  desired  me  to  inquire  after  y®  form  of  Mr  Mace's 
Otocousticon  a  INIusitian  here ;  but  he  has  not  been  in 
town  since  I  came  from  London,  but  is  somewhere  in 
London  about  printing  a  book  of  Mu3iq:||.  Yet  y^  last 
week  1  had  opportunity  to  inquire  after  it  of  his  son  &  he 


'  **  Mr  Oldenburg  was  ordered  to  thank  him  for  this  offer,  and  to  desire  him  to  send 
the  said  discourse  ns  soon  as  he  pleased."   Birch,  iii.  232. 

t  OrifT,  leit,  Jik.  Roy.  Soc.  N.  1.  48. 

t  "An  Hypothesis  explaininfr  the  properties  of  light,  discoursed  of  in  my  several 
papers."     liirch,  iii.  2-18. 

II  "  Mustek's  Monument,"  &c.  &:c.  Lond.  1676.  Newton*8  name  appears  in  the  list 
of  subscribers  to  the  work.  Thomas  Mace  was  one  of  the  '  Clerici'  or  Singing  Men  of 
Trinity  College  for  more  than  70  years  (1635— 1706).  Comp.  Burney's  Hist.of  Muf. 
Vol.  3.  Southey's  Doctor,  chapters  193-196.  Cooper's  Annals  of  Camb.  under  year  1690. 


Append.] 


NEWTON  AND  OLDENBURG. 


255 


tells 'me  the  form  is  this.  A  y*  smal  end  to  put  into  y*  ear 
BC  y""  length  sup- 
pose two  foot  CD  y^ 
>vide  end  suppose 
about  eight  inches 
over.  Thetubei?Z>C 
tapers  all  y®  way 
almost  eavenly  like  a  cone  only  at  y*  great  Orifiee  CD 
widens  more,  like  y®  end  of  a  Trumpet.  He  has  of 
several  sizes.  The  biggest  do  y*  best.  If  you  can't 
recollect  who  y®  Gentleman  may  be  I  suppose  M'  Hill  can 
tell  you,  for  I  think  M""  Hill  was  by  when  y"  Gentleman 
spake  to  me,  &  y^  Gentleman  dcsird  me  to  write  to  either 
M*"  Hill  or  you  about  it. 


For  IIenry  Oldenijuro  Esq :  at  his 
house  about  y*  middle  of  y'  Old 
Pal-inel  in  Westminster  London, 


Yo"  in  hast 

Is.  Newton' 


No.  XIII. 
NEWTON  TO  OLDENBURG. 


I  hope  M"*  Linus's  ffriends  will  acquiesce  in  y*  late  tryall 
of  y'^  Exp*  in  debaitf,  for  y*  procurement  of  w*^^  &  for  send- 
ing  them  notice  of  y®  event,  I  return  you  my  hearty  thanks, 
as  I  have  reason.     I  perceive  I  went  upon  a  wrong  suppo- 


•  Orig.  Lett.  Dk,  Roy.  Soc.  N.  1.49. 

t  i.e.  The  Experiment  on  the  Solar  spectrum.  •*  Apr.  27.  The  Experiment  of  Mr 
Newtou  which  had  been  contested  by  Mr  Linus  and  his  fellows  at  Lienro,  wan  tried 
before  the  Society,  according  to  Mr  Newton's  directions,  and  succeeded,  as  lie  all  along 
asserted  it  would  do :  and  it  was  ordered,  that  Mr  Oldenburg- should  signify  this  success 
to  those  of  Liege,  who  had  formerly  certified,  {  by  a  letter,  Dec.  15, 1675  |  that  if  the 
experiment  were  made  before  the  Society,  and  succeeded  according  to  Mr.  Newton's 
assertions,  they  would  acquiesce."  Birch,  in.  313.  Linus  had  maintained  that  the 
sun's  image  was  round,  and  the  colours  arranged  parallel  to  the  axis  of  the  prism. 


266  CORRESPONDENCE  OF  [Append. 

sition  in  what  1  wrote  concerning  M'  Boyles  Exp*.  The 
Papers  in  yo*"  hand  I  have  no  present  need  of:  You  may 
send  them  at  yo"^  best  leisure. .  Sometime  this  Sommer  it's 
possible  I  may  make  use  of  them,  if  I  can  but  get  some 
time  to  write  y*  other  discourse  about  y®  colours  of  y^  Prism 
w*^^  I  have  long  intended.  S*"  I  am 

Yo^  humble  &  obliged 
Servant 
Cambridge.  May  ll*^  1G76.  Is.  Newton*. 

For  Henry  Oldenburo  Esq :  at  his  house 
about  y'  middle  of  y'  old  Pal-mall  in 
Westminster  London. 

Endorsed  by  Oldenburg : 

"RccM2May. 

Answ.  by  D'  Sidnamt  May  l.'i.  and  sent  by  him  liis  Hypothesis 
explaining  y*  properties  of  light ;  as  also  his  discourse  about  y'  various 
colors  exhibited  by  transparent  substances  made  very  thin  by  being 
blown  into  bubles  or  otherwise  form'd  into  plates,  altho  at  a  greater 
thicknes  they  appear  very  clear  and  colorlesse. 

In  my  letter  accompanying  these  papers  I  imparted  to  M'  Newton 
y*  particulars  contain'd  in  M.  Leibniz  his  letter  to  me  of  May  12 
1 67(3.  from  Paris  st.  n."  In  the  letter  just  mentioned  Leibniz  desired 
information  on  the  subject  of  the  analytical  discoveries  recently  made  in 
England,  and  it  was  in  compliance  with  this  request  that  Newton,  at 
the  pressing  solicitation  of  Collins  and  Oldenburg,  drew  up  his  celebrated 
letter  of  June  13.  One  of  the  questions  in  Leibniz's  letter,  of  which  an 
extract  is  printed  in  the  Commercium  Epistolicum.,  will  probably  sur- 

prise  the  modern  student.  The  series  (sin  6  =)0-  jt+....  and  its  converse 

had  been  sent  to  him  from  this  country,  and  he  begs  the  favour  of  a 
demonstration  of  them. 


•  Oris;.  Lett.  Bk.  Roy.Soc.  N.  1.  62. 

•f-  Sydenham  was  going  to  Cambridge  to  take  his  M.  D.  degree.  He  was  admitted 
at  Pembroke,  May  17  (from  Alag-dalcn  Hall,  Oxford) and  was  made  Doctor  the  follow- 
ing day. 


Append.]  NEWTON  AND  OLDENBURO.  257 

No.   XIV. 

NEWTON    TO  OLDENBURO. 

Accompanying  hia  answer  to  Lucas  (dated  Aug.  18,  and  printed  in  the 
Trans,  for  Sept.  25). 

I  have  been  stayed  from  writing  to  you  longer  then  I 
intended  by  reason  that  I  could  not  till  of  late  meet  w*^  a 
day  clear  enough  at  noon-time  to  try  some  of  y*  experi- 
ments herein  set  down.  And  now  1  have  not  sent  you  an 
answer  so  full  as  I  intended  at  first  but  perhaps  more  to  y* 
purpose  considering  who  I  have  to  deale  w*^  whose  buisiness 
it  is  to  cavill.  The  other  buisiness  you  wrote  to  me  about 
viz:  about  stocking  us  w*^  fruit  trees  I  shall  be  glad  to  pro- 
mote. Some  inquiry  I  have  made  about  it,  &  w^^'in  a  few 
days,  when  I  have  got  some  further  information  &  dis- 
coursed it  w*^  some  that  are  most  like  to  entertein  y*  pro- 
posall,  I  hope  to  give  you  a  further  account  of  it.  In  y* 
mean  time  I  rest 

Yo'^  humble  Servant 
Cambridge  Aug:  22.  1676.  Is.  Newton* 

For  Henry  Oldenburg  Esq :  at  his  house 
about  tlie  middle  of  y^  old  Pal-maill  in 


Westmimter  London. 


w***  care. 


No.  XV. 

NEWTON  TO  OLDENBURO. 

S"  Octob  26.  1676. 

Two  days  since,  I  sent  you  an  answer  to  M.  Leibnitz's 

excellent  Letter.     After  it  was  gone,  running  my  eyes  over 

a  transcript  that  I  had  made  to  be  taken  of  it,  I  found 

some  things  w*^^  I  could  wish  altered,  &  since  I  cannot  now 


•  Orig.  TMt.  m.  Roy.Soc.  N.  1.  54. 

17 


258  CORRESPONDENCE  OF  [Append. 

do  it  my  self,  I  desire  you  would  do  it  for  me,  before  you 
send  it  away. 

In  pag :  3.  Sect :  Pudet  dicere.]  ffor  a  Z>.  Barrow  tunc 
Matheseoa  Professare  write  only  per  amicum 

Pag :  5.  Sect :  At  quando.]  After  quibuscum  potest  com- 
parari ;  write  ad  quod  sufficit  etiam  hoc  ipsum  unicum  jam 
descriptum  Theorema  si  dehite  concinnetur.  Pro  Trinomiis 
etiam  et  aliis  quihusdam  Regulas  quasdem  concinnavi  &c. 

Pag :  6.  Sect :  Quamvis  multa.]  Where  you  find  y® 
words  Gregorianis  ad  Circulum  et  Hyperholam  editis  p^rsi- 
miles,  for  persimiles  write  ajffines 

Pag:  9  or  10.   Sect:   Theorema  de.]     ffor  error  erit 

—  +  —  +  &c.  write  error  erit  — h  —  +  &c. 
90      140  90      19^ 

Pag:  6  vel  7.  Sect :  Quamvis  multa.]  about  y*  end  of  y* 
section  turn  plenariam  into  plenam  or  rather  blot  y®  word 
quite  out. 

Pag :  ult.  vel  penult.  Sect :  Ubi  dixi].  write  solutilia  for 
solubilia.  And  if  you  observe  any  other  such  scapes  pray 
do  me  y®  favour  to  mend  them.  So  in  pag  5  or  6.  Sect. 
Quamvis  multa.]  It  may  be  perhaps  more  intellig  { ib }  le  to 
"write  cvOvvarei  for  euthunsi. 

Pag  8  or  9.  Sect :  Per  seriem.]  After  y®  words  product 
ad  multas  figuras :  you  may  if  you  please  add  these  words, 
ut  et  ponendo  summam  terminorum  1  -■f  +  ^-iV'^'TV 
"  ife  "**  '^  "  :3V  +  ^V  ^^  ^^^®  ^^  totam  seriem  1  -  ;J^  +  -J- 
- 1  +  ^  -  ^  +  &c  ut  1  +  \/2  ad  2.  Sed  optimus  ejus 
usus  &c 

I  feare  I  have  been  something  too  severe  in  taking 
notice  of  some  oversights  in  M.  Leibnitz  letter  considering 
y®  goodnes  &  ingenuity  of  y®  Author  &  y*  it  might  have 
been  my  own  fate  in  writing  hastily  to  have  committed  y® 
like  oversights.  But  yet  they  being  I  think  real  oversights 
I  suppose  he  cannot  be  offended  at  it.  If  you  think  any 
thing  be  exprest  too  severely  pray  give  me  notice  &  I'le 


Atpe^nd.]  NEWTON  AND  OLDENBURG.  259 

endeavour  to  molliiy  it,  unless  you  will  do  it  w***  a  word  or 
two  of  your  own.  I  beleive  INI.  Leibnitz  will  not  dislike 
y*  Theorem  towards  y®  beginning  of  my  letter  pag.  4  for 
squaring  Curve  lines  Geometrically.  Sometime  when  I 
have  more  leisure  it's  possible  I  may  send  him  a  fuller 
account  of  it :  explaining  how  it  is  to  be  ordered  for  com- 
paring curvilinear  figures  w*^  one  another,  &  how  y*  simplest 
figure  is  to  be  found  w*^  w*^^  a  propounded  Curve  may  be 
compared.  S"*  I  am 

Yo*"  humble  Servant 

Is.  Newton*. 

Pray  let  none  of  my  mathematical  papers  be  printed 
w^^out  my  special  licence. 

Some  other  things  in  M.  Leibnitz  letter  I  once  thought 
to  have  touched  upon,  as  y®  resolution  of  affected  aqua- 
tions, &  y®  impossibility  of  a  geometric  Quadrature  of  y* 
Circle  in  w*^^  M.  Gregory  seems  to  have  tripped.  But  I 
shall  add  one  thing  here.  That  y®  series  of  a?quations  for 
y®  sections  of  an  angle  by  whole  numbers,  w*^^  M.  Tschum- 
hause  saith  he  can  derive  by  an  easy  method  one  from  an 
other,  is  conteined  in  y^  one  aequation  w*^*  I  put  in  y"  3** 
section  of  y®  Problems  in  my  former  letter  for  cutting  an 
angle  in  a  given  ratio,  and  in  another  aequation  like  that. 
Also  y*  coefficients  of  those  equations  may  be  all  obteined 

n-Oxn-l       n— 2x11—3      w-4xn— 5 
by  this  progression  1  x     ^^^_^      x  __^  x  -^— ^ 


X  ^^    ^^^"-7  ^  ^^     rJ^Y^^  ^^g^  coefficient  being   l,  y*    2^ 
4xn-4 

n-Oxn-1  ^  „rt  ?i-Oxn-l       ri-2xn-3 

1  X .      V®  3°  1  X  X .  &c.  &  n 

lxn-1  Ixw-l  2x71-2 

being  y®  number  by  w*^^  y*  angle  is  to  be  cut.  as  if  n  be  5. 


•  MSS.  Birch,  Brit.  Mus.  4294.  The  signature  which  waa  cut  out  by  some  felonioua 
hand  in  1833,  has  been  recently  r&stored. 

17—2 


260  CORRESPONDENCE  OF  [Append. 

5  y^4t      3x2       1x0 

then  y*  series  is  l  x x  —   x  that  is  1  x  5  x  1  x  0 

^  1x4      2x3       8x2 

&  consequently  y*  coefficients  1.5.5.     So  if  n  be  6  y® 

6x5       4x3      2x1 

series  is  l  x  x  —  x  - —  x  o  that  isl  x6x|x|^xO 

1x5       2x4       3x3  ^ 

&  consequently  y*  coefficients  1.6.9.2.     This  scrible  is 

not  fit  to  be  seen  by  any  body  nor  scarce  my  other  letter 

in  y*  blotted  form  I  sent  it,  unless  it  be  by  a  friend. 

For  Henry  Oldenburg  Esq:  at  his  house 
ahwit  y*  middle  of  y'  old  Pal-mall  in 
Westminster  London 


No.  XVI. 
NEWTON  TO  OLDENBURG. 
S' 

I  am  desired  to  write  to  you  about  procuring  a  recom- 
mendation of  us  to  M'  Austin  y®  Oxonian  planter.  We 
hope  yo'  correspondent*  will  be  pleased  to  do  us  y*  favour 
as  SLs{8ic}  to  recommend  us  to  him,  y*  we  may  be  furnished 
w*^  y*  best  sorts  of  Cider-fruit-trees.  We  desire  only  about 
30  or  40  Graffs  for  y®  first  essay,  &  if  those  prove  for  o'  pur- 
pose they  will  be  desired  in  greater  numbers.  We  desire 
graffs  rather  then  sprags  that  we  may  y®  sooner  see  what 
they  will  prove.  They  are  not  for  M""  Blackley  but  some 
other  persons  about  Cambridge,  But  M*"  Austin  need  only 
direct  his  letters  to  me  or  to  M"^  Bainbrigg  ffellow  of  o' 
College.  In  y®  mean  time  we  return  o^  thanks  to  you  & 
your  ffriend  for  y®  good  will  you  have  already  shewn  us. 

M*"  Lucas  letter t  I  have  received,  &  hope  to  send  you 
an  answer  y**  next  Tuesday  Post.  I  thank  you  for  your 
care  to  prevent  their  prejudicing  me  in  y®  Society,  as  also 


•  Dr  John  Deal,  rector  5f  Yeovil,  wlio  inherited  a"  zeal  for  the  plantation  of  orchards 
for  the  making  of  cider."    See  liirch,  iv.  235. 
t    UateU  Oct.  23. 


Append.]  NEWTON  AND  OLDENBURG.  261 

for  giving  me  notice  of  y*  things  miswritten  in  my  late 

letter.     In  pag  3  y*  words  you  cite  should  run  thus.   Cujus 

2x2x2 
triplo  adde  Log.  0.8,  siquidem  sit  - — - —  -  10.     But  in 

0.8 

pag  8  y®  signes   of  y®  series  i  +i-^-4-  +  ^  +  &c  arc 

rightly  put  two  +  &  two  -  after  one  another,  it  being  a 

different  series  from  y*  of  M.  Leibnitz.    But  in  y*  next  two 

or  3  lines,  to  prevent  future  mistake  you  may  if  you  think 

fit,  after  y^  words  res  tardius  ohtineretur  per  tangentem  45^, 

add  these  words  jxixta  seriem  nobis  commtmicatam, 

Seing  y®  letter  is  still  in  yo""  hands,  you  will  do  me  y* 
favour  to  make  these  further  amendments 

Pag.  3  Sect  [Pudet  dicere]  cum  D.  Collinsio  for  ad  D, 
Collinsium 

pag.  5.  Exempl.  4  after  y®  words  vel  quibus  libet  dujiiU 
tatibus  binomii  cujuscunq :  add  licet  non  direct^  ubi  index  duj* 
nitatis  est  numerus  integer. 

pag  6  or  7  in  y®  end  of  y*  section  quamvis  multa  I  desire 
you  would  cross  out  y®  words  adeo  ut  in  potestate  habeam 
descriptionem  omnium  curvarum  istius  ordinis  quce  per  8  data  ^'^ 
puncta  determinantur.  And  in  y®  2^  sentence  of  y®  next 
section  I  could  wish  these  words  also  numero  infinite  multaa 
were  put  out. 

pag  9f.  Sect  [Prceterea  quae,']  for  mihi  quidem  haud  ita 
clara  sunt  put  nondum  percipio.  And  after  a  line  or  two 
where  you  see  y®  words  et  cert^  minor  est  labor,  put  out  certe» 

By  these  alterations  S*"  you  will  oblige 

Yo'  humble  Servant 
{Tuesday}  Nov.  14  1676.  Is.  Newton |. 


•  "data"  is  written  by  mistake  for  "tantum."  The  words  here  ordered  to  be 
crossed  out  are  inclosed  within  parentheses  in  the  letter  as  printed  in  Wallis's  3rd 
Volume,  and  the  Commercium  Epistolicuni,  wliere  also  leptem  appears  instead  of  octo. 
One  of  the  points  is  supposed  to  be  a  double  point.  See  Newton's  Enumeralio  Lin, 
Tert.  Ord. 

t  The  place  referred  to  is  in  p.  10. 

:  MSS.  Birch,  Brit.  Mus.  4294. 


262  LETTER  TO  MADDOCK.  [Append. 

•  Just  now  I  received  Yo'  packet  conteining  two  books 
from  M'  Boyle.  That  for  D""  Moor  shall  be  conveyed  to 
him.  For  the  other  I  shall  return  my  thanks  to  y®  noble 
Author. 

For  Henuy  Oldenburg  Esq :  at  his  house 
about  y*  middle  of  y*  old  Palmail  in 
Westminster 

Endorsed  by  Oldenburg:  "Rec.  Nov.  15.  76. 
written  to  D'  Beall  about  part  of  y'  contents 
of  this  letter.  Nov.  16.  76.  Anew.  Nov. 
25.  76." 

In  another  letter  to  Oldenburg  written  on  the  following  Saturday, 
he  saya :  "  I  promised  to  send  you  an  answer  to  Mr  Lucas  this  next 
Tuesday,  but  I  find  I  shall  scarce  finish  what  I  have  designed,  so  as  to 
get  a  copy  taken  of  it  by  that  time,  and  therefore  I  beg  your  patience  a 
week  longer."  Mace.  Corr.  ii.  405.  The  answer  was  accordingly  sent 
on  the  28th.  All  that  is  known  respecting  it  is  derived  from  Lucas's 
rejoinder.  See  Syn.  View  of  Newton's  Life,  under  Nov.  28,  1676, 
note. 


No.    XVIL 
NEWTON  TO  DR  JOSHUA  MADDOCK. 

Maddock  had  sent  Newton  some  specimens  of  a  new  branch  of 
optics,  devoted  to  the  consideration  of  the  properties  of  dark  rays. 
Such  a  system  would  afibrd  relief  to  those  commentators  who  are  em- 
barrassed by  expressions  like  /ue\a/u0a€«  €/)e/3o<f,  fxeKaiva  alyXrj,  and 
atrum  lumen*  There  was  a  person  of  that  name  at  Jesus  College,  who 
took  the  degree  of  B.A.  in  1661. 

Vir  dignissime, 
Spccimina  ilia  optica,  qu89  pro  humanitate  tua  ad  me 
nuper  misisti,  tantam  in  his  rebus  peritiam  ostendunt,  ut 
non  possum  quin  doleam  incertitudinem  principiorum  qui- 
bus  omnia  innituntur.  Etenim  qua)ri  potest,  an  sint  in 
rerum  natura  radii  tenebrosi,  et,  si  sint,  an  radii  illi, 
secundum  aliam  legem  refringi  debeant,  quam  radii  lucis. 
Defectu  experientiae,  nescio  prorsus  quid  de  his  principiis 
sentiendum  sit.     Neque  huic  difficultati  tollendi©,  quam  et 


Append.]  LETTER  TO  HOOKE.  263 

tute  ipse  indigitasti  facile  adfuerit  Tiberius*.  At  positis 
cjusmodi  radiis,  una  cum  lege  refractionis  quam  tu  assu- 
mis,  ca)tera  recto  se  habent ;  neque  propositiones  tantum 
utiles  sunt  ac  demonstrationes  artificiosro,  sed,  ct  quod 
majus  est,  omnia  nova  proponis,  quro  opticam,  altera  sui 
parte,  auctura  sunt,  si  modo  defectus  experientiro  in  stabi- 
liendis  principiis  tuis  a!iquo  demum  modo  suppleri  possit. 
Interim,  quod  me  meditationum  tuarum  pcrquam  subtilium 
participem  fieri  dignatus  sis,  gratias  ago.     Vale  I 

Tui  studiosissimus, 
Trin.  Coll.  Cant.  Feb.  7,  167f.  I.  NEWTONf. 

For  his  honoured  friend  Joshua  Maddock, 

Doctor  of  Physic  at  his  house  in   Whit' 

church  in  Shropshire. 


No.   XVIII. 

NEWTON   TO  IIOOKE. 

S' 

One  Dominico  Casparini  an  Italian  Doctor  of  Physick 

of  the   City   of  Lucca  has  composed   a   Treatise   of  the 

Method  of  administring  the  Cortex  Peruviana  in  Fevers, 

in  which  he  particularly  discusses  whether  it  may  be  admi- 

nistred  in  Malignant  fevers  and  also  whether  in  any  fevers 

before  the  fourteenth  day  of  the    Sickness.     Upon   the 

fame  of  the  Eoyal  Society  spread  every  where  abroad,  he 

is  ambitious  to  submit    his    discourse    to   so    great   and 

Authentick    a    Judgment    as    theirs    is,    and   thereupon 

desired  another   D^  of  Physick  of  his    Acquaintance  in 

Italy  to  >vrite  to  his  Correspondent  an  Italian  in  London, 

to  move  that  the  Society  would  give  him  leave  to  dedicate 


•  Allusion  to  Tiberius's  peculiarity  of  vision.  *'  Cum  praegrandibus  oculis,  et  qui, 
quod  mirum  esset,  noctu  etiam  et  in  tenebris  viderent,  sed  ad  breve."  Sueton.  Tib.  68. 
Comp.  Plin.  Nat.  Hist.  xi.  54. 

t  Printed  at  the  end  of  a  Funeral  Sermon  on  the  death  of  Daniel  Maddock  by 
E.  Latliam,  M.D.  Lond.  1745  :  and  Gentleman's  Mag.  Aug.  1782. 


264  CORRESPONDENCE  OF  [Append. 

his  Book  to  them.  The  said  Italian  being  come  from 
London  hither  before  the  Arrival  of  the  Letters,  upon 
the  receipt  of  them  applied  himself  to  me  and  I  promised 
him  I  would  desire  you  to  acquaint  the  Society  with  his 
Request.  If  you  please  to  send  their  Answer  to  me,  the 
Italian  here  will  convey  it  into  Italy. 

For  the  trials  you  made  of  an  Experiment  suggested 
by  me  about  falling  bodies*,  I  am  indebted  to  you  thanks 
which  I  thought  to  have  returned  by  word  of  mouth,  but 
not  having  yet  the  Opportunity  must  be  content  to  do  it 
by  Letter  &cf. 

Trinity  College  Decemb.  3'*  1680. 


William  Briggs,  bom  about  1650,  succeeded  Tenison  in  his  fellow- 
ship at  Corpus  Christi  College,  16()8.  A.M.  1670.  M.D.  1677.  See 
Mrtsters's  Hist,  of  Corp.  p.  249. 

No.   XIX. 

NEWTON  TO  BRIGGS. 
S' 

I  have  perused  yo'  very  ingenious  Theory  of  Vision  J 
in  w*'**  (to  be  free  w*^  you  as  a  friend  should  be)  there 
seems  to  be  some  things  more  solid  &  satisfactory,  others 
more  disputable  but  yet  plausibly  suggested  &  well  de- 
serving y®  consideration  of  y®  ingenious.  The  more  satis- 
factory I  take  to  be  your  asserting  y*  we  see  w*^  both 
eyes  at  once,  yo*"  speculation  about  y®  use  of  y®  musculua 
ohliquus  inferior t  yo'  assigning  every  fibre  in  y®  optick  nerve 
of  one  eye  to  have  its  correspondent  in  y*  of  y®  other, 


•  See  Synoptical  View  of  Newton's  Life  under  the  year  1679  (note). 

•f-  Roy.  Soc.  Letter  Booky  vni.  139.  Hooke's  Answer,  dated  Dec.  18,  is  given  lb. 
140.    Compare  Birch,  IV.  61. 

J  ••  A  New  Theory  of  Vision  "  read  at  the  meeting  of  the  Royal  Society,  March  15, 
1682,  and  printe«l  in  Ilooke's  Philosophical  Collections  for  tliat  month.  A  paper  in  con- 
tinuation of  it,  *'  with  an  examination  of  some  late  objections,"  appeared  in  the  Phil. 
Tram,  for  May  1683. 


Append.]  NEWTON  AND  BRIOGS.  .  265 

both  w^**  make  all  things  appear  to  both  eyes  in  one  & 
y*  same  place  &  yo'  solving  hereby  y*  duplicity  of  y*  object 
in  distorted  eyes  &  confuting  y*  childish  opinion  about  y* 
splitting  y*  optick  cone.  The  more  disputable  seems  yo' 
notion  about  every  pair  of  fellow  fibres  being  unisons  to 
one  another,  discords  to  y®  rest,  &  this  consonance  making 
y*  object  seen  w*^  two  eyes  appear  but  one  for  y*  same 
reason  that  unison  sounds,  seem  but  one  sound.  I  did 
think  to  have  sent  you  what  I  fancy  may  be  objected 
against  this  notion  &  so  staid  for  time  to  write  it  down, 
but  upon  second  thoughts  I  had  rather  reserve  it  for  dis- 
course at  o^  next  meeting :  and  therefore  shall  add  only 
my  thanks  for  yo*"  kind  letter  &  p'sent. 

S'  I  am 
Yo'  much  obliged  &  humble  servant 
Trin.  Coll.  Cambridge  June  20*^  1G82,  Is.  Newton*. 

For  his  honoured  friend  D'  Wiluam  BiiiGos 
at  his  hotise  in  Suffolk  Street  in  London. 


No.  XX. 

NEWTON  TO  BRIGOS. 

For  his  Hon'*  ffriond  D'  W"  Bnwos. 
S' 
Though  I  am  of  all  men  grown  y*  most  shy  of  setting 
pen  to  paper  about  any  thing  that  may  lead  into  disputes 
yet  yo'  friendship  overcomes  me  so  far  as  y*  I  shall  set 
down  my  suspicions  about  yo*"  Theory,  yet  on  this  con- 
dition, that  if  I  can  Avrite  but  plain  enough  to  make  you 
understand  me,  I  may  leave  all  to  yo'^  use  w^^'out  pressing 
it  further  on.     For  I  designe  not  to  confute  or  convince 


•  From  the  original  in  the  British  Museum,  Add.  MSS.  4237.  fol.  32.    Part  of  thiu 
letter  is  lithographed  in  C.  J.  Smith's  Hitt.  and  TJt.  Cunoutie$,  Lond.  1840. 


266  CORRESPONDENCE  OF  [Append. 

you  but  only  to  present  &  submit  my  thoughts  to  yo' 
consideration  &  judgment. 

First  then  it  seems  not  necessary  that  the  bending  of 
y*  nerves  in  y*  Thalamus  opticus  should  cause  a  differing 
tension  of  y*  ffibres.  ffor  those  w*^^  have  y®  further  way 
about,  will  be  apt  by  nature  to  grow  the  longer.  If 
y*  arm  of  a  tree  be  grown  bent  it  follows  not  that  the 
fibres  on  y*  elbow  are  more  stretcht  then  those  on  the 
concave  side,  but  that  they  are  longer.  And  if  a  straight 
arm  of  a  tree  be  bent  by  force  for  some  time,  the  fibres  on 
y*  elbow  vi'''^  were  at  first  on  y*  stretch  will  by  degrees 
grow  longer  &  longer  till  at  length  the  arm  stand  of  it's 
self  in  y®  bended  figure  it  was  at  first  by  force  put  into, 
that  is  till  y*  fibres  on  y®  elbow  be  grown  as  much  longer 
then  y*  rest  as  they  go  further  about,  &  so  have  but  the 
same  degree  of  tension  w*'*  them.  The  observation  is 
ordinary  in  twisted  Codling  hedges,  fruit  trees  nailed  up 
against  a  wall  &c.  And  y®  younger  &  more  tender  a  tree 
is  the  sooner  will  it  stand  bent.  How  much  more  there- 
fore ought  it  to  be  so  in  that  most  tender  substance  of  y® 
Optick  nerves  w*^^  grew  bent  from  y*  very  beginning  ?  And 
whether  if  those  nerves  were  carefully  cut  out  of  y®  brain 
&  outward  coat  &  put  into  brine  made  as  neare  as  could 
be  of  the  same  specific  gravity  w*^  y*  nerves,  they  would 
unbend  or  exactly  keep  the  same  bent  they  had  in  y® 
brain  may  be  worth  considering,  ffor  though  y®  strength 
of  a  single  fibre  upon  the  stretch  be  inconsiderably  little, 
yet  all  together  ought  to  have  as  much  strength  to  unbend 
y*  nerve,  as  would  suffice  by  outward  application  of  y* 
hand  to  bend  a  straight  nerve  of  y®  same  thickness,  the 
dura  Mater  being  taken  off. 

M'  Sheldrake*  further  suggests  wittily  that  an  object 
whether  the  axis  opticus  be  directed  above  it,  under  it,  or 


•  A  Fellow  of  Corpus,  7  years  senior  to  Newton. 


Append.]  NEWTON  AND  BRIOGS.  267 

directly  towards  it,  appears  in  all  cases  alike  as  to  figure  & 
colour  excepting  that  in  y*  3^  case  tis  distincter,  w^^  pro- 
ceeds not  from  y*  frame  of  y*  nerves  but  from  y*  dis- 
tinctness of  y*  picture  made  in  y*  Retina  in  that  case. 
But  in  y*  first  case  where  y'  vision  is  made  by  y*  fibres 
above  &  second  where  tis  made  by  those  below,  the  object 
appearing  alike  :  he  thinks  it  argues  that  the  fibres  above 
&  below  are  of  y**  same  constitution  &  tension,  or  at  least 
if  they  be  of  a  differing  tension,  that  that  tension  has  no 
efiect  on  y®  mode  of  vision,  but  I  understand  you  are 
already  made  acquainted  w*^  his  thoughts. 

It  may  be  further  considered  that  the  cause  of  an 
objects  appearing  one  to  both  eyes  is  not  its  appearing  of 
y®  same  colour  form  &  bigness  to  both,  but  in  y*  same 
situation  or  place.  Distort  one  eye  &  you  will  see  y* 
coincident  images  of  y®  object  divide  from  one  another  & 
one  of  them  remove  from  y®  other  upwards  downwards  or 
sideways  to  a  greater  or  less  distance  according  as  y* 
distortion  is;  &  when  the  eyes  are  let  return  to  their 
natural  posture  the  two  images  advance  towards  one 
another  till  they  become  coincident  &  by  that  coincidence 
appear  but  one.  If  we  would  then  know  why  they  appear 
but  one,  we  must  e{nj  quire  why  they  appear  in  one  &  y* 
same  place  &  if  we  would  know  y®  cause  of  that  we  must 
enquire  why  in  other  cases  they  appear  in  divers  places 
variously  situate  &  distant  one  from  another,  ffor  that 
w*^^  can  make  their  distance  greater  or  less  can  make 
it  none  at  all.  Consider  whats  the  cause  of  f  heir  being  in 
y*  same  altitude  when  one  is  directly  to  y*  right  hand  y* 
other  to  y®  left  &  what  of  their  being  in  y'*  same  coast  or 
point  of  y®  compas,  when  one  is  directly  over  y®  other : 
these  two  causes  joyned  will  make  them  in  y®  same  altitude 
&  coast  at  once  that  is  in  y*  same  place.  The  cause 
of  situations  is  therefore  to  be  enquired  into.  Now  for 
finding  out  this  y®  analogy  will  stand  between  y*  situations 


268  CORRESPONDENCE  OP  [Append. 

of  sounds  &  the  situifCiDns  of  visible  things,  if  we  will 
compare  these  two  senses.  But  the  situations  of  sounds 
depend  not  on  their  tones.  I  can  judge  from  whence  an 
echo  or  other  sound  comes  tho  I  see  not  y*  sounding 
body,  &  this  judgment  depends  not  at  all  on  y*  tone.  I 
judge  it  not  from  east  because  acute,  from  west  because 
grave :  but  be  y*  tone  what  it  will  I  judge  it  from  hence  or 
thence  by  some  other  principle.  And  by  that  principle  I 
am  apt  to  think  a  blind  man  may  distinguish  unisons  one 
from  another  when  y®  one  is  on  his  right  hand  y®  other  on 
his  left.  And  were  our  ears  as  good  &  accurate  at  distin- 
guishing y®  coasts  of  audibles  as  our  eyes  are  at  distin- 
guishing y®  coasts  of  visibles  I  conceive  we  shoidd  judge  no 
two  sounds  the  same  for  being  unisons  unless  they  came  so 
exactly  from  y®  same  coast  as  not  to  vary  from  one  another 
a  sensible  point  in  situation  to  any  side.  Suppose  then  you 
had  to  do  with  one  of  so  accurate  an  ear  in  distinguishing 
y®  situation  of  sounds,  how  would  you  deale  with  him  ? 
Would  you  tell  him  that  you  heard  all  unisons  as  but  one 
sound  ?  He  would  tell  you  he  had  a  better  ear  then  so. 
He  accounted  no  sounds  y®  same  w*^^  differed  in  situation : 
&  if  your  eyes  were  no  better  at  y®  situation  of  things  then 
your  ears,  you  would  perhaps  think  all  objects  y®  same,  w^^ 
were  of  y®  same  colour.  But  for  his  part  he  found  y*  y*' 
like  tension  of  strings  &  other  sounding  bodies  did  not 
make  sounds  one,  but  only  of  y®  same  tone :  &  therefore 
not  allo>ving  the  supposition  that  it  does  make  them  one, 
the  inference  from  thence  that  y®  like  tension  of  y®  optick 
fibres  made  y®  object  to  y®  two  eyes  appeare  one,  he  did  not 
think  himself  obliged  to  admit.  As  he  found  y*  tones 
depended  on  those  tensions  so  perhaps  might  colours,  but 
the  situation  of  audibles  depended  not  on  those  tensions,  & 
therefore  if  the  two  senses  hold  analogy  with  one  another, 
that  of  visibles  does  not,  *  &  consequently  the  union  of 
visibles  as  well  as  audibles  which  depends  on  the  agreement 


AprEND.]  NEWTON  AND  BRIOOS.  269 

of  situation  as  well  as  of  colour  or  tone  must  have  some 
other  cause. 

But  to  leave  this  imaginary  disputant,  let  us  now 
consider  what  may  be  y*  cause  of  y*  various  situations 
of  things  to  y*  eyes.  If  when  we  look  but  w***  one  eye  it 
be  asked  why  objects  appear  thus  &  thus  situated  one  to 
another  the  answer  would  be  because  they  are  really  so 
situated  among  themselves  &  make  their  coloured  pictures 
in  y''  Retina  so  situated  one  to  another  as  they  are  & 
those  pictures  transmit  motional  pictures  into  y*  scnsorium 
in  y®  same  situation  &  by  the  situation  of  those  motional 
pictures  one  to  another  the  soul  judges  of  y*  situation  of 
things  without.  In  like  manner  when  we  look  with  two 
eyes  distorted  so  as  to  see  y*  same  object  double  if  it 
be  asked  why  those  objects  appear e  in  this  or  that  situation 
&  distance  one  from  another,  the  answer  shoiUdbe  because 
through  y®  two  eyes  are  transmitted  into  y®  sensorium  two 
motional  pictures  by  whose  situation  &  distance  then  from 
one  another  the  soule  judges  she  sees  two  things  so  situate 
&  distant.  And  if  this  be  true  then  the  reason  why  when 
the  distortion  ceases  &  y®  eyes  return  to  their  natural 
posture  the  doubled  object  grows  a  single  one  is  that  the 
two  motional  pictures  in  y*  sensorium  come  together  & 
become  coincident. 

But  you  will  say,  how  is  this  coincidence  made?  I 
answer,  what  if  I  know  not?  Perhaps  in  y®  sensorium, 
after  some  such  way  as  y*  Cartesians  would  have  beleived 
or  by  some  other  way.  Perhaps  by  y®  mixing  of  y*  marrow 
of  y®  nerves  in  their  juncture  before  they  enter  the  brain, 
the  fibres  on  y*  right  side  of  each  eye  going  to  y*  right 
side  of  y*  head  those  on  y®  left  side  to  y®  left.  If  you 
mention  y*  experim*  of  y®  nerve  shrunck  all  y®  way  on  one 
side  y®  head,  that  might  be  either  by  some  unkind  juyce 
abounding  more  on  one  side  y®  head  y"  on  y*  otlier,  or  by 
y®  shrinking  of  y®  coate  of  y®  nene  whose  fibres  &  vessels 


570  CORRESPONDENCE  OP  [Append. 

for  nourishment  perhaps  do  not  cross  in  y*  juncture  as  y* 
fibres  of  y*  marrow  may  do.  And  its  more  probable  y*  y® 
stubborn  coate  being  vitiated  or  wanting  due  nourishment 
shrank  &  made  y®  tender  marrow  yeild  to  its  capacity, 
then  that  y®  tender  marrow  by  shrinking  should  make 
y*  coate  yeild.  I  know  not  whether  you  would  have  y® 
succus  nutricius  run  along  y®  marrow.  If  you  would,  'tis  an 
opinion  not  yet  proved  &  so  not  fit  to  ground  an  argument 
on.  If  you  say  y*  in  y®  Camgelion  &  ffishcs  y®  nerves  only 
touch  one  another  without  mixture  &  sometimes  do  not  so 
much  as  touch ;  *Tis  true,  but  makes  altogether  against  you. 
ffishcs  looke  one  way  with  one  eye  y®  other  way  with  y® 
other:  the  Chamajlion  looks  up  w*^  one  eye,  down  w*^ 
t'other,  to  y®  right  hand  w*^  this,  to  y®  left  w*^  y*,  twisting 
his  eyes  severally  this  way  or  that  way  as  he  pleases.  And 
in  these  Animals  wliich  do  not  look  y®  same  way  w**'  both 
eyes  what  wonder  if  y®  nerves  do  not  joyn?  To  make 
them  joyn  would  have  been  to  no  purpose  &  nature  does 
nothing  in  vain.  But  then  whilst  in  these  animals  where 
tis  not  necessary  they  are  not  joyned,  in  all  others  w*^^  look 
y®  same  way  w*^  both  eyes,  so  far  as  I  can  yet  learn,  they 
are  joyned.  Consider  therefore  for  what  reason  they  are 
joyned  in  y®  one  &  not  in  the  other,  ffor  God  in  y®  frame 
of  animals  has  done  nothing  w%ut  reason. 

There  is  one  thing  more  comes  into  my  mind  to  object. 
Let  y®  circle  D  J  represent 
the  Retina,  or  if  you  will  the 
end  of  y®  optick  nerve  cut 
cross.  A  the  end  of  a  fibre 
above  of  most  tension,  C  y® 
end  of  one  below  of  least 
tension.  D  Sc  G  y^  ends  of 
fibres  above  on  either  hand 
almost  of  as  much  tension  as 
At  F  Si  J  the  ends  of  others 


Append.]  NEWTON  AND  BRIGGS.  271 

below  almost  of  as  little  tension  as  C  E  y*  end  of  a  fibre 
of  less  tension  then  ^  or  G  &  of  more  then  C  or.  J,  And 
between  A  &  Q  G  &  J  there  will  {be}  fibres  of  equal  tension 
w*^  E  because  between  them  there  are  in  a  continual 
series  fibres  of  all  degrees  of  tension  between  y*  most 
tended  at  A  Sc  G  8c  least  tended  at  C  &  ./.  And  by  the 
same  argument  that  3  fibres  E,  D  k  II  of  like  tension 
are  noted  let  the  whole  line  of  fibres  of  the  same  Degree 
of  tension  running  from  E  to  //  be  noted.  Do  you  now 
say  y*  y^  reason  why  an  object  seen  w***  two  eyes  appears 
but  one  is  that  y*'  fibres  in  y®  two  eyes  by  w*^^  'tis  seen  are 
unisons?  then  all  objects  seen  by  unison  fibres  must  for 
y®  same  reason  appear  in  one  &  y®  same  place  that  is  all  y' 
objects  seen  by  the  line  of  fibres  E  B  II  running  from  one 
side  of  the  eye  to  y®  other.  fTor  instance  two  stars  one  to 
y®  right  hand  seen  by  y®  fibres  about  //,  the  other  to  y®  left 
seen  by  y®  fibres  about  E  ought  to  appear  but  one  starr, 
&  so  of  other  objects,  ffor  if  consonance  unite  objects 
seen  w*^  the  fibres  of  two  eyes  much  more  will  it  unite 
those  seen  w*^  those  of  y®  same  eye.  And  yet  we  find  it 
much  otherwise.  AVhat  soever  it  is  that  causes  the  two 
images  of  an  object  seen  with  both  eyes  to  appear  in  y* 
same  place  so  as  to  seem  but  one  can  make  them  upon 
distorting  y®  eyes  separate  one  from  y®  other  &  go  as 
readily  &  as  far  asunder  to  y®  right  hand  &  to  y®  left  as 
upwards  &  downwards. 

You  have  now  y®  summ  of  what  I  can  think  of  worth 
objecting  set  down  in  a  tumultuary  way  as  I  could  get 
time  from  my  Sturbridge  ffair  friends.  If  I  have  any 
where  exprest  myself  in  a  more  peremptory  way  tlien 
becomes  y®  weaknes  of  y®  argument  pray  look  on  that  as 
done  not  in  earnestness  but  for  y®  mode  of  discoursing. 
Whether  any  thing  be  so  material  as  y*  it  may  prove  any 
way  useful  to  you  I  cannot  tell.  But  pray  accept  of  it 
as  written  for  that  end.     ffor  having  laid  Philosophical 


272  CORRESPONDENCE  WITH  BRIG  OS.  [Append. 

speculations  aside  nothing  but  y^  gratification  of  a  friend 
would  easily  invite  me  to  so  large  a  scribble  about  things 
of  this  nature. 

S*"  I  am 
Yo'  humble  Servant 
Trin.  Coll.  Cambr.  Sept.  12*^.  1682.  Is.  Newton*. 


No.  XXI. 
NEWTON  TO  BRIGGS. 

Isaacus  Newtonus  Doctori  Gulielmo  Briggio. 
Vir  Clarissime, 
Hisce  tuis  Tractatibusj  duas  magni  nominis  scientias 
uno  opere  promoves,  Anatomiam  dico  &  Opticam.  Or- 
gani  enim  (in  quo  utraque  versatur)  artificio  sumrao  con- 
structi  diligenter  perscrutaris  mysteria.  In  hujus  dis- 
sectione  peritiam  &  dexteritatem  tuam  non  exiguo  olim 
mihi  oblectamento  fuisse  recordor.  Musculis  motoriis 
secundum  situm  suum  naturalem  eleganter  a  te  expansis, 
caaterisque  partibus  coram  expositis,  sic  ut  singularum  usus 
&  ministcria  non  tarn  intelligere  liceret  quam  cernere, 
effecerat  dudiim  ut  ex  cultro  tuo  nihil  non  accuratum 
sperarem.  Nee  spem  fallebat  eximius  ille  Tractatus  Ana- 
tomicus,  quem  postmoddm  edidisti.  Jam  praxeos  hujus 
uKpljieiav  pcrgis  ingeniosissima  Theoria  instruere  &  exor- 
nare.  Et  quis  Theoriis  condendis  aptior  extiterit,  quam 
qui  phienomenis  accurate  observandis  navfirit  operam? 
Nervos  opticoa  ex  capillamentis  varie  tensis  constare  sup- 
ponis,  eaque  magis  esse  tensa  qua)  per  iter  longius  porri- 
guntur;  ex  diversa  autem  tensione  fieri  ut  objectorum 
partes  singulaB  non  coincidant  &  confundantur  inter  se,  sed 


•  From  the  original  in  the  British  Museum,  Add.  MSS.  4237.  lol.  34. 
t  i.  e.  Opthalmographiaf  Cantab.  1676  (2nd  Ed.  Lond.  1687)  and  his  Theory  of 
Viiion. 


Append.]  DIRECTIONS  TO  BENTLEY.  278 

pro  situ  8110  natural!  diversis  in  locis  nppareant :  &  capilla- 
mentis  amborum  oculorum  a;quali  tcnsione  factis  coiiCordU 
bus,  geminas  objectorum  species  uniri.  Sic  ex  tcnsione 
cliordarum,  qud  soni  vel  variantur  vel  concordant  in  Mu- 
sicji,  colligere  videris  quid  fieri  debet  in  Opticii.  Simplex 
etenim  est  Natura,  &  codem  operandi  tenore  in  immensA. 
eflectuum  varietate  sibi  ipsa  constare  solet.  Quanto  vero 
magis  in  sensuum  cognatorum  causis  ?  Et  quamvis  aliam 
ctiam  horum  sensuum  analogiam  suspicari  possim,  ingcnio- 
sam  tamen  esse  quam  tute  excogitasti,  cert^  nemo  non 
lubenter  fatebitur.  Nee  inutilem  censeo  Dissertationem 
ultimam  qua  diluis  objectiones.  Inde  Lector  attentus  & 
pleniCis  intelliget  Hypothesin  totam,  &  in  qua)stiones  incidet 
vel  tuis  Meditationibus  illustratas,  vel  novis  experimentis 
&  disquisitionibus  posthac  dirimendas.  Id  quod  in  usum 
cedet  juventuti  AcademicjB,  &  provectiores  ad  ulteriores  in 
Philosophia  progressus  manuducet.  Pergas  itaque,  vir 
ornatissime,  scientias  hasce  pra)claris  inventis,  uti  facis, 
excolere ;  doceasque  difficultatcs  causarum  naturalium  tam 
facile  solertia  vinci  posse,  quiim  solent  conatibus  vulgari- 
bus  difficulter  cedere. 

Daham  Cantahrigice  7  Kal.  Mail.  Vale  . 

1686. 


No.  XXII. 

Paper-of  Difections  given  by  Newton  to  Bentley,  respecting  the  Books 
to  bo  read  before  entering  upon  the  Principia.  Date  probably 
about  July  1691. 

In  1691  the  vigorous  mind  of  Richard  Bentley,  who  was  then  in  his 
30th  year,  was  attracted  to  the  revelations  which  the  Principia  had 
announced  to  the  philosophical  world  some  four  years  before,  and  with 


•  This  letter  is  prefixed  to  the  Latin  Version  of  Brigprs's  Theory  of  Vhion  (mnde  at 
Newton's  request)  Load.  1685. 

18 


274  DIRECTIONS  TO  BENTLEY.  [Append. 

the  view  of  making  himself  acquainted  with  the  "Great  Charter  of 
Modem  Science/*  as  that  immortal  work  is  called  by  Dr  Whewell,  ho 
applied  through  his  friend  W.  Wotton  to  John  Craigo,  a  mathemati- 
cian of  some  eminence,  for  advice  as  to  the  course  of  reading  to  bo  fol- 
lowed preparatory  to  the  study  of  the  volume  itself.  The  appalling 
list  of  authors  which  Craige  sent  him  (Juno  24,  1691 ;  see  Bentloy's 
Corresp.  p.  736)  probably  induced  him  to  repair  to  the  fountain  head, 
and  the  paper  now  before  us  was  the  result  of  that  step. 

That  Bcntley  studied  the  Principia  to  some  purpose  was  shewn  by 
his  two  last  sermons  at  the  Boyle  Lecture  (founded  by  the  "Will  of 
Robert  Boyle,  who  died  Dec.  31,  1691)  in  November  and  December  of 
the  following  year,  which  led  him  to  consult  our  philosopher  again 
upon  some  points  that  arose  in  tliem  requiring  elucidation.  See  New- 
ton's four  Letters  to  Bentley  in  1(592  and  1693.   (Bcntley 's  Corresp.) 

Next  after  Euclid's  Elements  the  Elements  of  y*  Conic 
sections  are  to  be  understood.  And  for  this  end  you  may 
read  either  the  first  part  of  y*  Elementa  Curvarum  of  John 
De  Witt,  or  De  la  Hire's  late  treatise  of  y-  conick  sec- 
tions, or  D'  Barrow's  epitome  of  Apollonius. 

For  Algebra  read  first  Barth{ol}in's  introduction  & 
then  peruse  such  Problems  as  you  will  find  scattered  up  & 
down  in  y*  Commentaries  on  Cartes's  Geometry  &  other 
Alegraical  {sic]  writings  of  Francis  Scliooten.  I  do  not 
mean  y*  you  should  read  over  all  those  Commentaries,  but 
only  y®  solutions  of  such  Problems  as  you  will  here  &  there 
meet  with.  You  may  meet  with  De  AVitt's  Elementa 
curvarum  &  Bartholin's  introduction  bound  up  together 
w*^  Carte's  Geometry  &  Schooten's  commentaries. 

For  Astronomy  read  first  y®  short  account  of  y® 
Copernican  System  in  the  end  of  Gassendus's  Astronomy 
&  then  so  much  of  Mercator's  Astronomy  as  concerns  y® 
same  system  &  the  new  discoveries  made  in  the  heavens 
by  Telescopes  in  the  Appendix. 

These  are  sufficient  for  understanding  my  book  :  but  if 
you  can  procure  Ilugenius's  Horologium  oscillatorium,  the 
perusal  of  that  will  make  you  much  more  ready. 

At  y®  first  perusal  of  my  Book  it's  enough  if  you  under- 


Append.]  LETTER  TO  LOCKE.  275 

stand  y"  Propositions  w*^  some  of  y*  Demonstrations  w*^** 
are  easier  then  the  rest.  For  when  you  understand  y* 
easier  they  will  afterwards  give  you  light  into  y*  harder. 
When  you  have  read  y*  first  GO  pages,  pass  on  to  y*  3** 
Book  &  when  you  see  the  design  of  that  you  may  turn 
back  to  such  Propositions  as  you  shall  have  a  desire  to 
know,  or  peruse  the  whole  in  order  if  you  think  fit*. 

Memorandum 
by  Bcntlcy. 
"Directions  from  ^I'  Newton  by  his  own  Hand"  . 


No.  XXIIi. 
NEWTON  TO  LOCKE. 


The  first  few  lines  of  the  letter  are  wanting.  Locke  had  sent  him 
some  of  Boyle's  red  earth,  which  tliat  philosopher  had  a  recipe  for 
combining  with  mercury  so  as  to  "  multiply"  gold.  In  a  letter  >Nritten 
on  the  2nd  of  tlie  folloNving  month,  Newton  "  dissuadOs"  Locke  "  from 
too  hasty  a  trial  of  this  recipe,"  whicli  he  states  to  bo  "  imperfect  and 
useless."  Lord  King's  Life  of  Lochy  i.  412. 


as  I  can.  You  have  sent  much  more  earth  then  I  ex- 
pected. For  I  desired  only  a  specimen,  having  no  inclination 
to  prosecute  the  process.  For  in  good  earnest  I  have  no 
opinion  of  it.  But  since  you  have  a  mind  to  prosecute  it 
I  should  be  glad  to  assist  you  all  I  can,  having  a  liberty  of 
communication  allowed  me  by  INIr  B.  in  one  case  which 
reaches  to  you  if  it  be  done  under  y®  same  conditions  in  w^ 
I  stand  obliged  to  Mr  B.  ffor  I  presume  you  are  already 
under  the  same  obligations  to  him.  But  I  feare  I  have  lost 
y®  first  &  third  part  out  of  my  pockett.     I  thank  you  for 


•  From  the  original,  given,  with  Newton's  four  letters  to  Bentley,  by  Cumberland  to 
Trinity  College. 

18—2 


276  LETTER  [Append. 

what  you  communicated  to  me  out  of  yo'  own  notes  about 
it.     S'  I  am 

Yo'  most  humble  Serv^ 
Cambridge  Jul  7***  Is  Newton 

1692. 
When  the  hot  weather  is  over  I  intend  to  try  the  begin- 
ning* tho  y*  success  seems  improbable  f. 

For  Zovfin  Look,  Esq.  at  M'.  Paulen*s  in  Dorset  Court 
in  Channel  Row  in  Westminster, 


No.  XXIV. 
NEWTON  TO  LEIBNIZ. 


In  answer  to  a  letter  of  Leibniz  dated  ^  Mart.  1693,  printed  in  Rapbson's 
Hist,  of  Fluxions,  p.  119.     Leibn,  0pp.  in.  484. 

Ccleberrimo  Viro 

Godefrido  Gulielmo  Leibnitio 

IsAAcus  Newton  S.P.D. 

Literse  tuie,  cilm.  non  statim  acceptis  responderem, 
e  manibus  elapsa?  inter  schedas  meas  diu  latuere,  nee  in 
eas  ante  hesternum  diem  incidere  potui.  Id  quod  me 
moleste  habuit,  ctim  amicitiam  tuam  maximi  faciam,  teq : 
inter  summos  hujus  soeculi  Geometras  a  multis  retro  annis 
habuerim,  quemadmodum  etiam  data  omni  occasione  tes- 
tatus  sim  Nam  quamvis  commercia  philosophica  &  mathe- 
matica  quammaxim^  fugiam,  tamen  metuebam  ne  amicitia 
nostra  ex  silentio  decrementum  acciperet,  idq :  maxime 
cum  Wallisius  noster  Historiam  Algebrae  in  lucem  denuo 


•  i.  e.  the  first  of  tlic  three  parts  of  tho  recipe,  tlie  effect  of  wliich,  accordiner  to 
Boyle,  was  the  production  of  a  mercury  which  would  grow  hot  with  gold, 
t  From  a  transcript  oblipringly  made  for  me  by  I^ord  Lovelace. 


Append.]  TO  LEIBNIZ.  277 

missurus  nova  aliqua  c  litcris  inscriiit  qua8  olim  per 
manus  D°*  Oldenburgi  ad  te  conscripsi,  &  sic  ansam  mihi 
dedit  ea  etiam  de  re  ad  te  scribendi.  Postulavit  enim  ut 
methodum  quandam  duplicem  aperirem  quam  litcris  trans- 
positis  ibi  celaveram.  Qiiocirca  coactus  sum  qua  potui 
brevitate  cxponere  methodum  meam.  fluxionum  quam  hac 
celaveram  sententia.  Data  a;quatione  quantitatea  quotcunque 
fiueiites  involvente  invenire  Jluxiones,  Sf  vice  versa,  Spero 
autem  me  nihil  scripsisse  quod  tibi  non  placeat,  et  siquid 
sit  quod  reprehensione  dignum  censeas,  ut  litcris  id  mihi 
significes  quoniam  amicos  pluris  facio  quam  inventa  ma- 
thematica. 

Eeductioncm  quadraturarum  ad  Curvarum  rectifica- 
tiones*  quam  desiderare  videris,  inveni  talem.  Sit  Curvro 
cujusvis  abscissa  w,  ordinata  y,  et  area  a«f,  posito  quod 
a  sit  data  quantitas  fiuat  .2?  uniformiter,  sitque  ejus  fluxio 
ih  «  a,  et  ipsius  y  sit  fluxio  y.  A  dato  puncto  D  in  rectA 
positione  data  DE  sumatur  DD^x^  et  agatur  indcfinita 
DCG  ea  lege  ut  cosinus  anguli  DBG  sit  ad  Radium  ut 
fluxio  yf  ad  fluxionem  .r  «  a,  et  inveniatur  curva  FG  quam 
recta  BG  perpetuo  taugit.     Id  enim  semper  fieri  potest 


•  Twenty-six  years  later  this  problem,  which  Euler  calls  **  celebre  illud  prolhsma 
multutn  inter  Geomctras  aKitatum,"  wos  proposed  by  Hermann  in  the  Leipsic  Acts 
(Aug.  1719),  and  was  solved  by  him  in  the  number  for  Apr.  1723,  and  by  J.  Bernoulli 
in  the  number  for  Aug.  1724.  The  latter  shews  how  to  obtain  a  moie  general  solution. 
See  also  Newton's  Geometria  Anulytica  (Horsley,  i.  508),  his  letter  to  Oldenburg, 
Jun.  23,  1673,  and  Euler,  Conunent.  Petrop.  Tom.  v.  p.  171,  We  find  no  allusion  to 
Newton's  solution  in  any  of  Leibniz's  published  letters  or  papers.  In  the  fig"ure  FD 
should  be  a  straight  line. 

The  following  may  assist  some  readers  in  verifying  Newton's  construction.  Let 
X,  Y  be  the  co-ordinates  of  the  required  curve,  on  the  length  of  whose  arc  (S)  the 
area  of  the  proposed  curve  is  to  be  made  to  depend.  Then  S  =J'd  Y  Vl  +  P",  (dX^PdY) 

, pYPdP 

=  rv/i^i^«./^--. 

Assume  r  d  P  -  Jx  and  -===-,  =  2  ;  and  X  will  be  found  -  ^i^lril^  ^  -  ,, 
Vl+P'     a  a*         dy       * 

"  dy 

t  fluxio  y.]    This  should  be  either  •'  fluxio  x"  or  ito  equal  "y," 


278  LETTER  [Append. 

Geometrice  ubi  fluxionum  .r  &  y  relatio  gcometrica  est. 

Sit    G   punctum  contactAs,  et   ubi  ^ 

punctum  B  incidit  in  punctum  D, 

incidat  punctum   G  in  punctum  F, 

In  tangente  BG  sumatur  GO  wqualis 

curva;  GF^  et  CII  ajqualis  rcctso  FD,  \  \ji 

et  erit  BH  «  ^r.     Qua  inventa  habe- 

tur  area  qucesita  ax.  ^T b — E 

Quae  vir  summus  Hugenius  in  mea  notavit  ingeniosa 
sunt*.  Parallaxis  soils  niiagr  videtur^uam  ipse  statueram, 
et  motus  sonorum  forte  magis  rectilineus  est,  at  cajlos 
materia  aliqua  subtili  nimis  implere  videtur.  Nam  cum 
motus  coolestes  sint  magls  regulares  quam  si  a  vorticibus 
orirentur,  et  leges  alias  observent,  adeo  ut  vortices  non  ad 
regendos  sed  ad  perturbandos  Planetarum  et  Cometarum 
motus  conducant,  cumque  omnia  caelorum  et  maris  phae- 
nomena  ex  gravitate  sola  secundum  leges  a  me  descripttas 
agente  accurate  quantum  scntio  sequantur,  et  natura  sim- 
plici-ssima  sit,  ipse  causas  alias  omnes  abdicandas  judicavi 
et  cajlos  materia  omni  quantum  fieri  licet  privandos,  nc 
motus  Planetarum  et  Cometarum  impediantur  aut  reddan- 
tur  irregulares.  At  interea  si  quis  gravitatem  una  cum 
omnibus  ejus  legibus  per  actionem  materioe  alicujus  subtilis 
explicuerit,  et  motus  Planetarum  et  Cometarum  ab  liac 
materia  non  perturbatos  \sic\  iri  ostenderit,  ego  minime 
adversabor.  Colorum  phsonomena  tam  apparentium  ut 
loquuntur  quam  fixorum  rationes  certissimas  me  invenisse 
puto,  sed  a  libris  edendis  manum  abstineo  ne  mihi  lites  ab 
imperitis   intententur   et    controversiaa.      Alius    est   New- 


•  In  an  **  Addition"  to  his  "  Discours  de  la  Cause  de  la  Pesanteur."    Leid.  1690. 

Nic.  Fatio  writing  to  Iluygens  from  London,  Feb.  24,  1690,  says:  "Mr.  Newton, 
Mr.,  recevra  parfaitement  bien  tout  ce  que  vous  avez  dit.  Je  I'ai  trouv6  tantde  fois 
pret  a  corrigcr  son  livre  sur  des  choses  que  je  lui  disois,  que  je  n'ai  pil  assez  admirer  sa 
facility,  et  particulierement  sur  les  endroits  que  vous  attaqtiez.  II  a  quelquo  peine  n 
entendre  le  Francois,  mais  il  s'en  tire  pourtant  avec  un  dictionaire."  Again,  Apr.  11  : 
"  Mr.  Newton,  Mr.,  m'a  assure  qu'il  prenit  en  fort  bonne  part  tout  ce  qui  est  dans 
le  traitt^  de  la  cause  de  la  pesanteur.** 


AppENn.]  TO  LEIBNIZ.  279 

tonus*,  cujus  opera  in  Hbrorum  cditorum  indicibus  tibi 
occurrunt.  His  contestari  volui  me  tibi  araicum  intcger- 
rimiim  esse  &  amicitiam  tiiam  maxinie  faccre.  Vale. 
Dabam  Cantabrigiro,  Octob.  J  J.  IGOSf. 

Utinam  rectificationeni  Hyperbola)  quam  te  invenisse 
dudum  significasti  in  lucem  cmitteres. 


No.   XXV. 

NEWTON   TO  IIAWES. 


^fr  Edward  Paget,  Fellow  of  Trinity  College,  and  Mathematical 
Master  at  Christ's  Hospital,  drew  up  in  1()94  a  scheme  of  reading  for 


•  This  refers  to  the  following  passage  in  I^ibnii's  letter :  **  In  Hbrorum  apud  Anglo« 
ejitorum  Indicibus  occurrere  mihi  aliquoties  libri  Alathematici  autore  Neutono,  scd 
dubitavi  a  Te  cssent,  quod  velleni,  an  nb  alio  homonymo."  The  author  in  question 
was  John  Newton,  D.D.,  a  writer  of  mathematical  school-books.  Morhof  was  probably 
thinking  of  this  same  "  Doctor,"  when  he  called  our  pliilosopher  **  Mcdicus  Anglus." 
(The  passajje  alluded  to  occurs  in  n  posthumous  part  of  the  Volyhistor,  but  was  written, 
apparently,  not  long  after  the  publication  of  Newton's  Analysis  of  Solar  Light.  The 
expression  is  retained  in  Fabricius's  editions  of  fhc  work  1732  and  1747.  Morhof  died 
in  1691).  'I'he  title  is  retailed  by  Saxius  Onomatt.  v.  120:  '♦  Isaacus  Newtonus  Wool- 
stropensis  Anglus,  Mcdicus,  Mathematicus  et  Philosophus  Ix)ndinensis..."  Compare 
Report  of  Committee  of  House  of  Commons  on  abuses  in  the  Mint  (Apr.  8,  1697),  in 
which,  on  the  Moneyers  alleffini?  themselves  to  be  a  Corporation,  it  is  stated  that  **  D' 
Newton,  present  Warden  of  the  Mint,  declared  that  he  had  never  seen  any  such  Grant 
or  Patent  to  the  Moneyers ;  and  believed  they  had  no  other  Charter,  but  the  general 
Charter  of  the  Mint,  which  he  had  in  his  possession,"  6ic.  lluding*s  Annals  of  the 
Coinage,  iii.  536,  540.    (London,  1817). 

D«"  Henry  Newton,  Envoy  Extraordinary  (1704—1710)  to  the  Grand  Duke  of 
Tuscany  and  Republic  of  Genoa,  tells  us  that  he  occasionally  received  compliment* 
that  were  intended  for  his  illustrious  namesake:  **  Et  multa  adhuc  expectant  [Itali]  a 
Summo  Mathematico  ejusdem  mecum  Cognominis,  (indc  aliquoties  contigit  ex  errore 
nominis,  me  quoque  non  mcis  laudibus  oruari)  prxseitim  vero  Mundum  qualem  Deus, 
ipsi  quoque  Ilobbesio,  il-Ucrnus  Geometra,  ab  initio  formaverat,  atque  sapientissimus 
Creator  in  mensura,  &  numero  fie  pondere  disposuerat ;  sed  intellectu  facilem,  non 
solum  Mathematicis,  nee  quidem  illis  ex  plcbe,  Icf^endum  intucndumque;  sicque  tile 
demum  optimu,  sibi,  Patria;,  omni  denique  Posteritati  consulat.*'  Inciter  from  Florence, 
Oct.  1,  1705,  in  his  Kpistolt£...L\xcait  1710.  As  a  sort  of  compensation  a  Ictttr  of 
thanks  from  Lord  CowjMjr  has  been  recently  published  as  addressed  to  our  pliiloso- 
pher, which  1  strongly  suspect  was  intended  for  the  author  of  the  work  just  quoted. 
(Lord  Campbell's  Chancellors^  iv.  341.) 

f  Partly  from  Crelle's  Journal,  Band  xxxii.  (where  a  portion  of  the  letter  is 
lithographed  from  the  original  in  the  Royal  Library  at  Hanover),  and  partly  from  a 
copy  in  the  British  Museum,  Add.  MSS.  6399.  fol.  56,  which  seems  to  have  come  to 
the  Museum  with  Cole's  Collections.  The  letter  has  been  recently  printed  in  the 
edition  of  Leibniz's  works  now  in  course  of  publication  at  Berlin. 


280  CORRESPONDENCE  OF  [Append. 

the  boys  under  hia  care.  At  a  meeting  of  the  Committee  of  the  Schools 
of  the  Hospital  on  the  9th  of  May,  Mr  IJawes,  the  Treasurer,  was 
**  desired  when  he  goes  to  Cambridge  on  Friday  next  to  take  with  him 
a  copy  of  the  old  and  new  schemes,  and  advise  with  the  Professor  and 
other  Mathematicians  in  the  University  concerning  them,  and  get  their 
opinions  in  writing  which  of  the  two  schemes  they  judge  best."  New- 
ton's opinion  of  their  respective  merits  is  conveyed  in  this  letter,  which 
was  sent  enclosed  in  another  to  Paget. 

ffiyr  Nathanael  Hawes,  Esq. 

S' 
I  now  returne  you  the  papers  you  left  in  my  hands. 
The  two  Schemes  of  learning  I  have  compared,  and  find 
that  the  old  one  wants  methodizing  &  enlarging ;  the  want 
of  method  you  may  perceive  by  these  instances. 

1.  Arithmetick  is  set  down  preposterously  in  the  12*^ 
Article  after  almost  all  the  rest  of  Mathematicks.  ffor  a 
man  may  understand  and  teach  Arithmetick  w^^'out  any 
other  skill  in  Mathematicks,  as  writing  Masters  usually  doe, 
but  w*^out  Arithmetick  he  can  be  skilled  in  noe  other 
parte  of  Mathematicks,  &  therefore  Arithmetick  ought  to 
have  been  set  downe  in  the  very  first  place  as  the  ffounda- 
tion  of  all  the  rest. 

2.  The  parts  of  Arithmetick  are  set  downe  in  severall 
Articles  preposterously,  ffor  Decimal  Arithmetick  and  the 
Extraction  of  roots  are  enjoined  in  the  3**  Article  before 
the  boyes  have  learnt  Arithmetick  in  integers  &  vulgar 
fractions.  Then  in  the  4*^  &  S*'*  Articles  they  are  enjoined 
Logarithms.  And  after  all  this  tliey  are  required  in  the 
12*^  Article  to  learn  Arithmetick  in  generall,  as  if  they  had 
learnt  nothing  of  it  before. 

3.  Geometry  and  Trigonometry  are  confounded  to- 
gether in  the  first  Article,  and  again  in  the  4***.  Whereas 
Geometry  ought  to  have  made  one  Article  and  Trigono- 
metry another,  ffor  these  are  accounted  distinct  sciences. 

4.  The  use  of  Logarithms  w*^**  is  set  downe  in  the  8*^ 
Article,  ought  to  have  preceded  that  of  Artificial  Sines  & 


ArrEND.]  NEWTON  AND  IIAWES.  281 

Tangents  vf^^  is  in  the  4*^  ffor  how  can  any  man  under- 
stand the  Logarithms  of  Sines  and  Tangents,  before  he 
understands  the  Logarithms  of  Numbers  in  generall. 

5.  The  doctrine  of  the  Globes  is  set  down  in  the  11*** 
Article  and  the  projection  of  the  Sphere  or  globe  and 
making  of  Maps  is  set  down  in  the  10^**.  whereas  the  doc- 
trine of  the  globes  ought  to  precede  the  projection  of 
the  sphere  &  making  of  Maps.  Ifor  how  can  any  man  pro- 
ject the  lines  on  a  sphere  or  globe  into  Maps,  before  he  is 
taught  what  those  lines  are  ? 

6.  The  10^**  Article  is  worded  improperly.  fFor  instead 
of  saying,  The  projection  of  the  Sphere  in  circles  or  globe  in 
a  plain  divers  wai/e^,  It  should  have  been  said  The  projec- 
tion of  the  Sphere  or  globe  in  circles  on  a  plain  divers  wayes, 
ffor  the  projection  of  a  spliere  in  circles  and  that  of  a  Globe 
in  a  plain  are  neither  equipollent  phrases,  nor  branches  of 
a  distinction,  &  therefore  cannot  be  put  together  disjunc- 
tively (as  they  are  in  this  Article)  w^^out  an  impropriety  of 
speech. 

7.  The  Rule  for  finding  the  Latitude  by  the  Sun  or 
Starrs  in  the  sixth  Article,  and  the  questions  of  plane  Sail- 
ing w*^  the  use  of  the  plane  Sea  Chart  in  the  seventh, 
ought  to  have  come  after  the  Doctrine  of  the  globes,  &  the 
making  of  Maps  or  Charts ;  &  yet  these  are  set  after  the 
other  in  the  10^^  and  11*^  Articles.  Soc  alsoe  in  the  second 
Article,  the  making  of  the  Scale  of  hours,  Rumbs  and 
Longitude,  is  improperly  joyned  with  the  Rule  of  three,  & 
more  improperly  set  before  the  doctrine  of  y*  Globes.  And 
in  generall  the  whole  scheme  is  soe  confused  &  imme- 
thodical,  as  mj  es  me  think  that  they  who  drew  it  up, 
had  noe  regard  to  the  order  of  the  things,  but  set  them 
downe  by  chance  as  they  first  thought  upon  them,  w^^out 
giving  themselves  the  trouble  to  digest  and  methodise  the 
heap  of  things  they  had  collected  together ;  w*^**  makes  me 
of  ^opinion,  that  it  will  not  be  for  the  reputation  of  the 


282  CORRESPONDENCE  OP  [Append. 

fibundation  to  continue  this  scheme  any  longer  w%ut  put- 
ting it  at  least  into  a  new  forme. 

But  then  for  the  things  it  conteins  I  account  it  but 
mean  and  of  small  extent.  It  seems  to  comprehend  little 
more  then  the  use  of  Instruments,  and  the  bare  practise  of 
Seamen  in  their  beaten  road,  \\^^  a  child  may  easily  learn 
by  imitiition,  as  a  Parrot  does  to  speak,  w%ut  understand- 
ing in  many  cases  the  reason  of  what  he  does ;  &  w^^  an 
industrious  blockhead,  who  <}an  but  remember  what  he 
has  seen  done,  may  attain  to  almost  as  soon  as  a  child 
of  parts,  and  he  that  knows  it  is  not  assisted  thereby 
in  inventing  new  things  &  practises,  and  correcting  old 
ones,  or  in  judging  of  what  comes  before  him  :  Whereas 
the  INIathematicall  children,  being  the  flower  of  the  Hos- 
pitall,  are  capable  of  much  better  learning,  &  when  well 
instructed  and  bound  out  to  skilful!  Masters,  may  in  time 
furnish  the  Nation  w^''  a  more  skilfuU  sort  of  Sailors, 
Builders  of  Ships,  Architects,  Engineers  and  ]Mathematicall 
Artists  of  all  sorts,  both  by  Sea  and  Land,  then  llrance  can 
at  present  boast  of.  The  defects  of  the  old  scheme  you 
may  understand  by  these  instances. 

1.  It  conteins  nothing  more  of  Geometry  than  what 
Euclid  has  in  the  beginning  of  his  first  book,  and  in  the 
10***  and  12*^  propositions  of  his  sixth  booke.  All  which  is 
next  to  nothing. 

2.  There  is  nothing  at  all  of  symbolical  Arithmetick, 
w*^  tho'  not  requisite  in  the  vulgar  road  of  Seamen,  yet  to 
an  inventive  Artist  may  be  of  good  use. 

3.  The  taking  of  heights  and  distances,  and  measuring 
of  planes  and  solids  is  alsoe  wanting,  tho  of  frequent  use, 

4.  Nor  is  there  any  thing  of  spherical  Trigonom  ?*y, 
tho  the  foundacon  of  a  great  many  usefull  Problei-  in 
Astronomy,  Geography  and  Navigation. 

5.  Neither  is  there  any  thing  of  Sayling  according  to 
the  severall  Hypotheses,  nor  of  Mercators   Chart,  nor  of 


Append.]  NEWTON  AND  HAWES;  283 

computing  the  way  of  Ships  tho  things  w***  a  Sailor  ought 
not  to  be  ignorant  of. 

6.  The  finding  the  difference  of  Longitude,  Amplitude, 
Azimuts  and  variation  of  the  compass  is  alsoe  omitted, 
tho  these  things  are  very  useful!  in  long  voyages,  such  as 
are  those  to  the  East  Indies,  and  a  INIariner  who  knows 
them  not  is  an  ignorant. 

7.  Nor  is  there  one  word  of  reasoning  about  force 
and  motion,  tho  it  be  the  very  life  and  Soul  of  Mechanical 
skill  and  manual  operations  and  there  is  nothing  soe  IMe- 
chanical  as  the  frame  &  managem*  of  a  ship.  By  these 
defects  it's  plain  that  the  old  scheme  wants  not  onely 
methodizing,  but  alsoe  an  enlargem*  of  the  learning,  ffor 
some  of  the  things  here  mentioned  to  be  wanting,  are 
requisite  to  make  a  Mariner  skilfuU  in  the  ordinary  road, 
and  the  rest  may  be  often  found  useful!  to  such  as  shall 
become  eminent  for  skill  &  ingenuity,  either  in  Sea  affaires, 
or  such  other  mechanicall  offices  and  imploymcnts,  as  the 
King  may  have  occasion  in  his  Yards,  Docks,  fforts,  and 
other  places,  to  intrust  them  with. 

Now  the  imperfections  of  the  whole  scheme  are  pretty 
well  supplyed  in  that  new  one  w*^^'  is  proposed  to  be  esta- 
blished, ffor  this  is  methodical,  short  &  comprehensive.  It 
excells  the  old  one  beyond  comparison  ;  I  have  returned  it 
to  you,  w*^  some  few  alterations  for  making  the  affinity, 
coherence  and  good  order  of  the  several!  parts  of  the 
learning,  more  cleare  and  conspicuous,  &  supplying  6ome 
defects.  The  alterations  arc  of  noe  very  great  moment, 
excepting  the  addition  of  the  last  Article,  w*^^  conteins  the 
science  of  Mechanicks.  The  rest  is  as  perfect  as  I  can 
make  it  without  this  Article,  whether  this  should  be  added 
may  be  a  question,  but  since  you  concur  w*^  me  in  the 
affirmative,  I'le  set  downe  my  reasons  for  the  addition, 
ffor  w^'^out  the  learning  in  this  Article,  a  Man  cannot  be 
an  able  and  Judicious  Mechanick,  &  yet  the  contrivance  & 


284  CORRESPONDENCE  OP  [Append. 

managem*  of  Ships  is  almost  wholly  Mechanical.  Tis 
true  that  by  good  naturall  parts  some  men  have  a  much 
better  knack  at  Mechanical  things  then  others,  and  on  that 
acco*  are  sometimes  reputed  good  Mechanicks,  but  yet 
w**'out  the  learning  of  this  Article,  they  are  soe  ffarr  from 
being  soe,  as  a  Man  of  a  good  Geometrical  head  who  never 
learnt  the  Principles  of  Geometry,  is  from  being  a  good 
Geometer,  ffor  whilst  Mechanicks  consist  in  the  Doctrine 
of  force  and  motion,  and  Geometry  in  that  of  magnitude 
and  figure :  he  that  can't  reason  about  force  and  motion, 
is  as  far  from  being  a  true  Mechanick,  as  he  that  can't 
reason  about  magnitude  and  figure  from  be^ng  a  Geometer. 
A  Vulgar  Mechanick  can  practice  what  he  has  been  taught 
or  seen  done,  but  if  he  is  in  an  error  he  knows  not  how 
to  find  it  out  and  correct  it,  and  if  you  put  him  out  of 
his  road,  he  is  at  a  stand;  Whereas  he  that  is  able  to 
reason  nimbly  and  judiciously  about  figure,  force  and 
motion,  is  never  at  rest  till  he  gets  over  every  rub.  Expe- 
rience is  necessary,  but  yet  there  is  the  same  difference 
between  a  mere  practical  Mechanick  and  a  rational  one,  as 
between  a  mere  practical  Surveyer  or  Guager  and  a  good 
Geometer,  or  between  an  Empirick  in  Physick  and  a 
learned  and  a  rational  Physitian.  Let  it  be  therefore 
onely  considered  how  Mechanical  the  frame  of  a  Ship  is, 
and  on  what  a  multitude  of  forces  and  motions  the  whole 
business  and  managem*  of  it  depends.  And  then  let  it  be 
further  considered  whether  it  be  most  for  the  advantage  of 
Sea  affaires  that  the  ablest  of  our  Marriners  should  be  but 
mere  Empiricks  in  Navigation,  or  that  they  should  be  alsoe 
able  to  reason  well  about  those  figures,  forces,  and  motions 
they  are  hourly  concerned  in.  And  the  same  may  be  said 
in  a  great  measure  of  divers  others  Mechanical  employ- 
ments, as  buildings  of  ships.  Architecture,  ffortification. 
Engineering,  ffor  of  what  consequence  Mechanical  skill  is 
in  such  Mechanical  employments  may  be  known  both  by 


Append.]  NEWTON  AND  HAWES.  285 

the  advantage  it  gave  of  old  to  Archimedes  in  defending 
his  City  against  the  Romans,  by  w*^**  he  made  himself  soe 
famous  to  all  future  ages,  and  by  the  advantage  the  ffrench 
at  present  have  above  all  other  Nations  in  the  goodness  of 
their  Engineers,  ffor  it  was  by  skill  in  this  Article  of  learn- 
ing that  Archimedes  defended  his  City,  And  tho  the 
ffrench  Engineers  are  short  of  that  great  Mechanick,  yet 
by  coming  nearer  to  him  then  our  Artificers  doe,  we  see 
how  well  they  fortify  and  defend  their  owne  Cities,  and  how 
readily  they  force  and  conquer  those  of  their  Enemies*. 
You  may  consider  to  what  perfection  that  Nation  by  their 
Schooles  for  Sea-Officers  had  lately  brought  their  Navall 
strength,  even  against  all  the  disadvantages  of  nature,  and 
yet  your  schoole  is  capable  of  out-doeing  them,  ffor  their's 
are  a  mixture  of  all  sorts  of  capacities,  your's  children  of 
the  best  parts  selected  out  of  a  great  multitude. 

Their's  are  young  men  whose  faculties  for  learning 
begin  to  be  as  stiff  and  inflexible  as  their  bones,  and  whose 
minds  are  prepossest  &  diverted  with  other  tilings,  yours 
are  children  whose  parts  are  Limber  and  pliable  and  free 
to  receive  all  impressions.  A  great  part  of  their  schools 
are  scarce  capable  of  much  better  learning  than  that  in 
your  old  scheme,  your's  have  already  shewn  by  experience 
that  they  are  capable  of  all  the  learning  in  the  new  one, 
except  the  last  Article,  w*^^  has  not  yet  been  taught  them, 
and  yet  after  they  have  learnt  the  rest,  will  prove  noo 
harder  then  that  w*^^'  they  had  learnt  before.  And  as  your 
children  are  a  select  Number  for  parts,  and  capable  of  all 
the  learning  here  proposed,  and  it  will  be  for  the  Honour 
&  advantage  of  the  Nation  to  introduce  a  new  spirit  of 


•  The  capture  of  Mons  in  1691,  that  of  Namur  in  1692,  and  of  Charleioi  in  1693, 
were  among-  Vauban*8  recent  triumphs.  When  Newton  wrote  the  above  remarks  he 
probably  little  anticipated  the  example  that  wquld  be  set  by  "  tlmt  nation  '  lo  his  own 
country  in  paying  a  tribute  to  hie  genius.  The  "  Newton  "  in  the  French  sfeani  navy 
is  a  corvette  of  26  guns,  220  horse  power. 


286  CORRESPONDENCE  OP  [Append. 

useful!  learning  among  the  Seamen,  soe  it  will  give  your 
children  a  higher  reputation  for  preferrment.  And  1  take 
it  to  be  for  the  Honour  of  both  King  Charles  his  memory 
and  of  the  foundation,  that  this  School  should  be  as  learned 
for  Sea  affaires  as  you  can  well  make  it ;  and  probably  it 
was  his  designe  and  will,  it  should  be  soe,  tho  all  this  learn- 
ing was  not  started  when  he  founded  it.  If  you  admit  this 
learning,  your  school  will  certainly  grow  into  greater  repu- 
tation, &  may  be  thereby  more  apt  to  stir  up  new  Bene- 
factors and  set  a  Precedent  of  good  learning  to  all  future 
foundations  of  the  same  kind,  and  if  you  admit  it  not,  your 
scheme  of  learning  will  be  imperfect  and  leave  roome  for 
future  foundations  to  outstrip  yours,  w*^^  I  beleive  would 
not  be  for  it's  honour.  fFor  the  scheme  of  learning,  as  I 
now  returne  it  to  you  is  an  entire  thing  w*^**  cannot  well 
want  any  of  it's  members,  ffor  'tis  nothing  but  a  combina- 
tion of  Arithmetick,  Geometry,  Perspective  and  Mecha- 
nicks,  I  mean  Geometry  as  well  in  sphericall  surfaces  as 
in  plane  ones.  Geometry  is  the  foundation  of  Me- 
chanicks,  &  Mechanicks  the  accomplishm*  &  Crown  of 
Geometry,  &  both  are  assisted  by  Arithmetick  for  com- 
puting and  perspective  for  drawing  figures :  Soe  that  any 
part  of  this  Systeme  being  taken  away  the  rest  remaines 
imperfect.  These  considerations  have  moved  me  to  pro- 
pose this  Article  to  you,  but  perhaps  the  Governors  may 
see  reasons  against  it  of  greater  weight  w*^'*  I  am  not  yet 
acq*^**  with,  &  therefore  I  onely  propose  this  business  and 
leave  it  wholly  to  their  prudence. 

The  INIain  difficulty  that  I  can  think  of,  is,  that  the 
learning  of  this  Article  may  take  up  too  much  of  the 
childrens  time.  And  yet  if  for  all  the  rest  of  their  learn- 
ing they  are  allowed  (as  you  tell  me)  but  two  yeares  & 
halfe  I  question  not  but  another  halfe  yeare  would  be 
abundantly  sufficient  for  this  addition,  and  then  they  would 
goe  to  sea  w^*'  a  complete  Systeme  of  Mathematical!  learn- 


Append.]  NEWTON  AND  IIAWES.  287 

ing.  And  perhaps  it  may  be  more  for  their  advantage  to 
spend  this  halfe  yeare  at  schoolc  in  an  important  part  of 
learning  w*^^  they  cannot  get  at  Sea,  then  at  Sea  in  learn- 
ing what  they  will  afterwards  learn  there  more  readily  if 
well  instructed  at  School,  before  they  goe  thither. 

If  two  yeares  were  not  at  first  thought  too  much  for  the 
old  scheme  of  learning  w'^'*  (before  the  addition  of  the 
Article  of  taking  prospects)  was  very  meane  and  narrow ; 
four  or  five  yeares  for  this  new  scheme  would  be  but  a 
moderate  allowance  at  that  reckoning,  &  therefore  tis  very 
much  if  they  can  learn  it  in  three.  And  yet  perhaps 
they  may  run  through  all  the  parts  of  it  in  two  yeares  and 
an  halfe  ;  but  not  soe  well :  And  I  Would  advise  that  they 
should  rather  be  allowed  three  full  yeares,  then  be  sent 
away  smatterers  in  their  learning. 

But  whether  they  be  allowed  two  yeares  &  an  halfe  or 
three  yeares,  1  conceive  the  time  of  their  examination 
ought  to  be  stated,  ffor  the  liberty  w*^**  the  Masters  of 
Ships  have  had  of  taking  away  the  boys  sometimes  before 
they  had  gone  through  the  whole  course  of  their  !Mathe- 
maticall  learning,  seems  to  me  a  mischief  w*^**  may  deserve 
a  reformation,  ffor  the  sending  abroad  unripe  boys  can  be 
neither  a  reputation  to  the  School,  nor  advantage  to  the 
Nation  ;  Such  boyes  being  not  onely  less  knowing  then 
others,  but  alsoe  less  able  to  make  use  of  what  they  have 
learnt,  &  more  apt  to  forget  it,  as  smatterers  in  a  Grainar 
school  doe  their  Latine. 

Nor  doe  I  see  how  the  genius  &  method  of  the  School 
in  goeing  through  the  whole  course  of  the  Mathematicall 
learning  can  be  carried  on  soe  evenly  and  advantagiously, 
as  when  y*  Mathem"  Master  shall  be  at  a  certainty  in  the 
Number  of  Scholars,  &  in  the  time  against  which  he  is  to 
make  them  fit.  As  the  constitution  now  is  you  leave  a  bad 
Mathematicall  Master  a  liberty  of  making  excuses  when 
ever  he  shall  prove  negligent,  &  discourage  a  good  one 


288  CORRESPONDENCE  OF  [ArrENi>. 

by  the  uncertainty  of  his  business  &  method  &  of  the  satis- 
faction &  reputation  of  bringing  his  Schollars  to  perfection, 
&  alsoe  by  leaving  him  exposed  to  such  humours  as  may 
desire  by  that  meanes  to  take  opportunity  of  hurting  him 
in  his  business  or  reputation  :  whereas  it's  your  interest 
to  make  the  place  as  desirable  as  you  can,  that  when  it 
becomes  void  you  may  have  the  greater  choice  of  such  men 
as  are  fittest  for  it,  &  encourage  them  to  goe  on  cheerfull}'' 
with  their  duty.  And  if  it  may  be  for  the  credit  &  interest 
of  y®  foundacon  not  onely  that  the  boyes  should  be  well 
learned,  but  alsoe  that  they  should  be  placed  abroad  w*^ 
the  best  Masters,  &  the  appointing  two  solemn  times  every 
yeare  for  examining  five  boyes  at  a  time  &  binding  them 
out  apprentices  may  draw  together  a  greater  choice  of 
good  Masters  then  in  the  petty  examinations  at  present. 
As  a  ffair  draws  together  a  greater  Number  of  Chapmen 
than  little  markets  doe:  If  the  giving  publick  notice  of 
those  times  may  alsoe  make  the  thing  more  solemn  & 
more  known  to  the  Nation^  &  thereby  conduce  to  the 
honour  of  the  foundation,  &  probably  to  the  stirring  up  of 
new  Benefactors :  I  should  think  the  conjunction  of  soe 
many  advantages  may  well  deserve  an  establishment,  unless 
there  should  be  some  great  objection  against  it  w^^  I  am 
not  yet  aware  of.  ffor  you  have  told  me  that  when  the 
boyes  have  run  through  their  course  of  learning  there  will 
be  noe  danger  of  their  not  meeting  with  Masters  at  the 
next  publick  examination,  and  if  any  of  them,  should  then 
happen  to  fail  of  Masters,  they  would  at  all  times  after 
that  be  at  liberty  to  goe  with  such  Masters  as  could  be 
met  with.  And  as  for  the  Examinations  I  should  think 
that  the  more  publick  they  are,  the  more  the  School  will 
be  concerned  for  its  reputation,  &  the  greater  will  be  the 
reputation  w*^^  it  may  get  by  the  good  performance  of  the 
boyes.  If  there  be  any  advantage  in  publick  Examinations, 
the  more  publick  they  are  the  greater  the  advantage  :  if  in 


Append.]  NEWTON  AND  HAWES.  289 

private  ones  the  Governors  may  have  it  at  their  Visitations 
by  able  and  diligent  Examiners  w*^  as  much  privacy  and 
severity  as  they  please :  And  if  more  such  examinations 
shall  upon  any  occasion  be  found  requisite,  yet  I  con« 
ceive  they  should  be  made  onely  by  Examiners  appointed 
by  the  Governors,  &  obliged,  soe  soon  as  the^  Examination 
is  over  to  give  an  account  to  the  Governors,  &  to  noe  body 
else  \v*^out  their  permission,  of  what  ever  they  find  amiss. 

When  the  boyes  are  sent  to  Trinity  house  to  be  pub- 
lickly  examined  perhaps  it  would  not  be  amiss  that  the 
Mathematical!  Master  send  along  w**^  them  a  larger  &  more 
particular  draught  of  the  things  they  have  been  taught,  & 
are  prepared  to  be  examined  in,  then  that  scheme  of  leam^ 
ing  w*^^  you  establish,  and  that  the  draught  of  every  Master 
with  the  alterations  from  time  to  time  made  in  it  and  the 
Number  of  the  boyes  who  at  every  examination  answer 
well  and  readily  to  the  things  therein,  be  kept  upon  record 
in  the  school  as  a  standard  of  the  learning  w*^^  the  boyes 
are  capable  of  w*4n  the  time  allowed  them. 

And  when  the  boyes  are  put  out  apprentices,  they  may 
be  exhorted  or  obliged  by  the  Governors  to  communicate 
to  the  School  (in  gratitude  to  the  place  of  their  education) 
such  accurate  observations,  curious  discoveries  and  select 
draughts  as  they  shall  make  abroad  in  their  Voyages  and 
ffactories  for  rectifying  the  longitudes  and  situation  of 
places  in  the  Maps,  or  otherwise  improving  Geography, 
Hydrography,  Navigation,  the  building  of  Ships,  Trade  or 
any  valuable  knowledge  of  remote  Nations  or  Regions.  And 
these  or  other  curiosities  communicated  by  them  may  be 
kept  together  in  a  convenient  place  as  an  Ornament  of  the 
Schoole  to  be  consulted  upon  occasions.  I  have  hitherto 
considered  onely  the  Kings  ffoundation,  and  herein  I 
have  been  free  in  comparing  the  old  &  new  schemes  of 
learning,  and  speaking  my  thoughts  about  them,  because, 
as  you  told  me,  it  was  desired.  I  hope  it  will  give  noe 
19 


290  CORRESPONDENCE  OF  [Append. 

offence  to  any  body,  ffor  at  the  first  founding  of  the 
Schoole,  the  old  scheme  might  serve  very  well  for  a  tryall, 
till  it  was  known  what  learning  such  young  children  might 
be  capable  of.  And  I  presume  that  the  Mathematicians 
who  drew  it  up,  intended  for  them  nothing  more  then  part 
of  that  learning  which  is  taught  to  persons  of  riper  age  in 
the  ffrench  Schools,  and  thought  it  more  advisable  to  leave 
the  method  of  the  things  to  the  Mathematical  Master, 
then  to  be  accurate  in  what  could  not  be  made  perfect. 
The  conjunction  of  Mr  Stones  fToundation*  with  the  Kings 
seems  to  be  well  designed :  ffor  as  both  the  Honour  and 
Interest  of  the  Kings  fToundation  is  consulted  by  making 
Mr.  Stone's  subservient  &  usefuU  to  it :  Soe  it  is  both  for 
the  Honour  of  Mr.  Stone's  fToundation  to  have  this  relation 
to  the  King's,  and  for  the  Interest  of  it,  that  his  boyes 
may  be  preferred  to  the  King's,  where  they  will  be  bound 
out  Apprentices  w*^  a  better  allowance.  But  care  should 
be  taken  that  the  Kings  boyes  be  not  retarded  in  their 
learning,  by  joyning  w*^  them  too  great  a  Number  of  other 
boyes  of  inferior  parts,  soe  as  to  hinder  them  from  getting 
through  their  scheme  of  learning  within  the  time  limited. 

I  like  well  the  designe  of  establishing  some  Latin 
Authors  to  be  read  in  the  Schoole,  because  the  best 
Mathematicall  books  are  in  that  language,  &  by  useing 
the  boys  to  Mathematicall  Latin,  they  will  be  enabled  to 
understand  them.  The  Synopsis  Algehraica  and  Wards 
Trigonometry  are  well  chosen  and  soe  is  Eticlides  nova 
methodo  in  regard  of  the  short  time  allowed  the  boyes. 
Yet  Euclid  himself  (suppose  in  Barrow's  edition)  would  doe 
them  more  good  if  it  could  be  compassed  within  the  time, 
and  would  be  more  usefull  to  them  in  reading  other 
Authors  afterwards.      And  therefore  the  Governors  may 


•  Henry  Stone  had,  in  1693,  bequeathed  the  bulk  of  his  property  to  the  Hospital, 
of  which  at  least  £50  a  year  was  to  be  devoted  to  the  improvement  of  the  mathematical 
department  of  the  school. 


Append.]  NEWTON  AND  HAWES.  291 

establish,  if  they  think  fit,  that  the  Boyes  read  either 
Euclidea  nova  methodo  or  else  at  the  discretion  of  the  Ma- 
thematical! Master  the  first  six  books  of  Euclides  Elements 
in  Barrows  edition  for  plane  Geometry  and  the  ll***  and 
12*^  books  thereof  for  Solids.  fTor  soe  the  Mathematical! 
Master  will  be  at  liberty  to  read  the  Elements  themselves 
soe  soon  as  he  finds  he  can  compass  it  and  the  rest  of  the 
scheme  w*Mn  the  time  limited.  As  for  the  Doctrine  of  the 
Sphere  the  first  book  of  Mercator*8  Astronomy  is  brief  and 
well  adapted  to  the  use  of  the  Schoole  and  therefore  may 
be  appointed. 

And  now  I  have  told  you  my  opinion  in  these  things,  I 
will  give  you  Mr.  Oughtred's,  a  Man  whose  judgment  (if 
any  man's)  may  be  safely  relyed  upon,  ffbr  he  in  his  book 
of  the  circles  of  proportion,  in  the  end  of  what  ho  writes 
about  Navigation  (page  184)  lias  this  exhortation  to  Seamen 
"And  if,  saith  he,  the  Masters  of  Ships  and  Pilots  will  take 
the  pains  in  the  Journals  of  their  Voyages  diligently  & 
faithfully  to  set  down  in  several!  columns,  not  oncly  the 
Rumb  they  goe  on  and  the  measure  of  the  Ships  way  in 
degrees,  &  the  observation  of  Latitude  and  variation  of 
their  compass ;  but  alsoe  their  conjectures  and  reason  of 
their  correction  they  make  of  the  aberrations  they  shall 
find,  and  the  qualities  &  condition  of  their  ship,  and  the 
diversities  and  seasons  of  the  winds,  and  the  secret  motions 
or  agitations  of  the  Seas,  when  they  begin,  and  how  long 
they  continue,  how  farr  they  extend  &  w^^  what  inequality  ; 
and  what  else  they  shall  observe  at  Sea  worthy  consideration, 
&  will  be  pleased  freely  to  communicate  the  same  with 
Artists,  such  as  are  indeed  skilful!  in  the  Mathematicks 
and  lovers  &  enquirers  of  the  truth :  I  doubt  not  but  that 
there  shall  be  in  convenient  time,  brought  to  light  many 
necessary  precepts  w*^^  may  tend  to  y"  perfecting  of  Navi- 
gation, and  the  help  and  safety  of  such  whose  Vocations 
doe  inforce  them  to  commit  their  lives  and  estates  in  the 

19—2 


292  CORRESPONDENCE  OP  [Append. 

vast  Ocean  to  the  providence  of  God."  Thu8  farr  that 
very  good  and  judicious  man  Mr.  Oughtred.  I  will  add, 
that  if  instead  of  sending  the  Observations  of  Seamen  to 
able  Mathematicians  at  Land,  the  Land  would  send  able 
Mathematicians  to  Sea,  it  would  signify  much  more  to  the 
improvem*  of  Navigation  and  safety  of  Mens  lives  and 
estates  on  that  element. 

I  hope  S*^  You  will  all  interpret  my  freedome  in  this 
Letter  candidly  and  pardon  what  you  may  therein  think 
amiss,  because  I  have  written  it  with  a  good  will  to  your 
ffoundation,  and  now  I  have  spoke  my  thoughts  I  leave 
the  whole  business  to  the  wisdome  of  your  selfe  and  the 
Governors.     I  am, 

Hon'**  S^ 

Your  most  humble  &  most 

obedient  Servant, 
Cam^  idge  May  25*^,  1694.  Is.  Newton. 


[Accompanying  the  above.] 

A  'Sew  Scheme  of  Learning  proposed  for  the  Mathematical  Boys  in 
Christ's  Hospital.  {Paget's  scheme  with  a  few  alterations  by 
Newton  who  has  also  added  the  10th  article.} 

1.  Arithmetick  in  Integers,  Vulgar  fractions  8e  Deci- 
mals, in  Proportional  numbers  natural  and  Artificial,  in 
Symbols  of  unknown  Numbers  &  in  Equations. 

2.  Geometry  speculative  and  practical  in  planes  and 
Solids. 

3.  The  Application  of  Arithmetick  to  Geometry  in 
determining  and  protracting  Lines,  Angles  and  figures  by 
Numbers  natural  and  Artificial,  Symbols  of  Numbers  and 
tables  of  Sines  &  Tangents. 

4.  The  description  and  properties  of  figures  in  per- 
spective with  the  Arts  of  drawing  and  designing. 


Append.]  NE^VTON  AND  HAWES.  293 

5.  The  use  of  the  best  Instruments  in  working  by 
proportionals  taking  Angles,  heights  and  distances,  and 
measuring  planes  and  solids. 

6.  The  Doctrine  of  the  Globes  and  the  Rudiments  of 
Geography  Hydrography  and  Astronomy. 

7.  The  descriptions  of  the  Globe  in  perspective  com- 
monly called  Projections  and  the  Art  of  making  Charts 
and  Maps. 

8.  The  Doctrine  of  Spherical  Triangles  w*^  their  ap- 
plication in  projecting  and  computing  all  the  usefull  Pro- 
blems in  Geography,  Astronomy  and  Navigation. 

9.  A  full  application  of  the  learning  aforesaid  to 
Navigation  particularly  to  the  severall  Hypotheses  thereof, 
commonly  called  Plane,  Great  circle  and  Mercators  sail- 
ing. As  alsoe  the  use  of  Charts  and  Sea  Instruments  for 
observation  and  their  application  to  the  finding  of  the 
Latitude,  difference  of  Longitude,  Amplitudes,  Azimuths 
and  variation  of  the  compass  by  the  Sun  or  Starrs,  w^**  the 
knowledge  of  the  Tides  and  Roman  Calender,  and  the 
method  of  keeping  Journals  and  of  finding  the  difference 
of  the  Longitudes  of  Shores  by  the  Eclipses  of  Jupiters 
Satellites. 

10.  The  principles  of  reasoning  about  force  &  motion, 
particularly  about  the  five  mechanical  powers,  the  stress  of 
ropes  and  timber,  the  power  of  winds,  tides,  bullets  and 
bombs,  according  to  their  velocity  and  direction  against 
any  plane,  the  line  w*^^  a  bullet  describes,  the  force  of 
weights  and  springs  and  the  power  of  fluids  to  press 
against  immersed  bodies,  and  bear  them  up,  and  to  resist 
their  motions  ;  w*^  the  application  of  this  learning  to  Sea 
affaires,  for  contriving  well  and  managing  easily,  speedily 
&  dextrously.  Levers,  Pulleys,  Skrews,  Anchors,  Pumps, 
Rudders,  Guns,  Sails  and  other  Tackle,  judging  truly  of  the 
advantages  &  disadvantages  of  VesscUs,  Havens,  fforts, 
Engins  and  new  Projects,  &  observing  or  discovering  what 


294  CORRESPONDENCE  OF  [Append. 

ever  tends  to  make  a  Ship  endure  and  Sail  well,  or  other- 
wise to  correct  or  improve  Navigation. 

Is.  Newton  ♦, 


No.  XXVI. 
NEWTON  TO   IIAWES. 
jfor  Nathan".  Ilawes,  Esq. 
S'  Yesterday  I  sent  by  the  Carryer  a  Letter  to  you 
w***  the  papers  you  left  in  my  hands,  inclosed  in  another  to 
M^  Paget.      In  that  I  wrote  to  you,  you  mil  find  my 
thoughts  set  downe  at  large  about  the  old  and  new  schemes 
of  learning.     Looking  this  morning  into  S*"  Jonas  Moore''s 
Systcme  of  Mathematicks  w*^^  he  composed  about  15  or  16 
yeares  agoe  for  the  use  of  your  schoole,  I  find  by  the  title 
page  and  preface  to  that  book,  that  the  new  Scheme  was 
for  the  most  part  composed  at  that  time  by  S"*  Jonas,  ifor 
there  (as  is  mentioned  in  the  preface)  he  proposes  to  teach 
in  order  these  sciences. 

1.  Arithmetick  vulgar,  decimal  and  Logarithmical. 

2.  Practical  Geometry. 

3.  Trigonometry  plane  and  spherical, 

4.  Cosmography  w^^  includes   the  Doctrine  of  the 

Globes  with  Astronomy  and  Geography. 

5.  Navigation  with  the  making  of  Maps. 

After  these  and  many  Tables  &  Geographical  Maps  follow 
Algebra  &  speculative  Geometry  conteined  in  the  first,  6*^ 
&  11*^  &  12*^  books  of  Euclid's  Elements.  The  difference 
between  this  method  and  the  new  Scheme  of  learning  now 
proposed  lies  in  these  things. 

1.  In  the  new  scheme  (as  alsoe  in  the  title  page  to 
S*"  Jonas  Moores  book)  Algebra  is  joyned  w*^  Arithmetick, 
&  speculative  Geometry  w**^  the  practical ;  w^^  certainly  is 


•  This  and  the  two  following  letters  are  from  ilie  official  copies  in  the  Christ's 
Hospital  Court  Book. 


Append.]  NEWTON  AND  IIAWES.  295 

the  best  method  for  Schollars  of  good  parts  who  are  to 
learn  both.  But  in  the  preface  to  S"^  Jonas  Moores  book 
Algebra  &  speculative  Geometry  are  separated  &  taught 
apart  after  all  the  other  Sciences  ;  w*^**  is  best  for  a  mixturo 
of  Schollars  of  all  degrees  of  parts,  some  of  w***  are  not 
capable  of  learning  the  whole  Scheme. 

2.  S^  Jonas  joyns  plane  &  spherical  Trigonometry  to- 
gether, but  in  the  new  scheme  spherical  Trigonometry  ia 
set  after  the  Doctrine  of  the  Sphere  w*^**  is  more  proper 
for  a  learner. 

3.  S"^  Jonas  omits  perspective  and  Mechanicks  & 
referrs  the  taking  heights  and  distances  &  mensuration  of 
planes  &  solids  to  the  end  of  practical  Geometry  and  plane 
Trigonometry  :  whereas  in  the  new  scheme  perspective  is 
inserted  between  them  for  delineating  the  heights,  dis- 
tances and  solids  w*^^  are  to  be  measured,  &  again  after 
y*^  doctrine  of  the  globes  for  the  making  of  Maps. 

This  I  thought  proper  to  signify  to  you,  that  the 
Governors  of  the  Hospitall  might  have  the  judgment  of 
S""  Jonas  in  this  matter.  fFor  he  follows  not  y®  old  scheme 
in  any  thing,  but  agrees  well  enough  w*^  the  new  one,  both 
in  y*  substance  of  the  things  he  teaches,  &  in  the  order 
of  them,  if  perspective  &  Mechanicks  be  inserted  into  his 
Systeme  in  their  proper  places.  By  S"^  Jonas  his  departing 
soe  much  from  y®  method  of  the  whole  scheme,  and  supply- 
ing some  things  w^^  were  wanting  in  it  &  coming  soe  neare 
to  the  new  one,  you  may  gather  that  the  old  one  in  his 
judgm*  wanted  information,  &  that  the  new  one  is  not 
much  amiss.     S"^  I  am, 

Yo'  most  humble  &  most  obed'  Serv' 
Is.  Newton. 

The  new  scheme  with  Newton's  modifications  was  bent  to  "Wallia 
and  David  Gregory  at  Oxford,  who  gave  their  *' opinion  and  advice" 
respecting  it  in  a  joint  paper,  dated  Juno  13,  1C94.  "After  a  very 
large  debate"  on  June  25,  it  was  agreed  to  adopt  the  new  scheme.    The 


296  CORRESPONDENCE  OP  [Append. 

Committee  also  stated  it  as  their  "  opinion  that  the  10""  Art.  in  the 
now  scheme  about  the  5  Mechanical  powers  cannot  be  taught  under  6 
months  longer  time  than  is  allowed  for  their  instruction  in  Mathema- 
tics. Also  that  the  Court  be  desired  to  request  Mr  Newton  to  enlarge 
himself  upon  the  aforesaid  lO*^  Art.  that  so  M'  Paget  may  be  the  bettor 
qualified  for  their  instruction  therein,  being  very  advantageous  to  the 
improvement  of  Navigation."  It  was  at  the  same  time  ordered  that 
"humble  &  hearty  thanks  be  returned  to  Mr  Newton,  D"  Wallis  & 
Gregory  for  their  extraordinary  pains  &  kindness  in  this  affair."  A 
letter  of  thanks  was  accordingly  sent  Aug.  9,  in  which  it  is  'observed 
that  "  the  plan  requiring  long  &  serious  consideration,  we  have  chosen 
a  committee  to  consider  tliereof,  but  being  unwilling  to  defer  our 
acknowledgments"  &c. 


No.    XXVII. 
NEWTON  TO  HAWES. 

S'.  Cambridge  June  14. 1696. 

I  should  have  writ  to  you  by  Mr.  Newton*  but  that  I 
stay'd  to  consider  further  of  y*'  scheme  of  Mathematical 
learning  before  it  be  established,  ffor  the  last  Article 
seemed  too  indefinite  to  be  subscribed,  and  in  the  forme  it 
is  there  set  downe,  has  noe  books  written  of  it„  &  therefore 
I  have  changed  it  into  the  last  A  rticle  of  the  scheme  I 
now  send  you  enclosed  in  this  Letter,  ffor  this  last  Article 
eonteins  as  much  of  the  other,  as  has  been  hitherto  re- 
duced to  a  certain  science  and  something  more,  and  is 
definite,  soe  that  the  Master  may  know  what  he  subscribes, 
and  the  Governors  what  the  Master  is  obliged  to  by  his  sub- 
scription. It  has  alsoe  books  written  upon  every  parte  of 
it  to  make  it  more  fit  for  the  school.  As  for  M^  Newton  I 
never  took  him  for  a  deep  Mathematician,  but  recom- 
mended him  as  one  who  had  Mathematicks  enough  for 
your  business,  w*^  such  other  qualifications  as  fitted  him  for 


•  Mr  Samuel  Newton,  who  had  been  recently  appointed  to  the  IVIathcmatical 
Mastership  at  Christ's  Hospital,  vacant  by  Paget's  resignation.  Compare  Newton's 
letters  in  Baily's  Flamsteedf  pp.  153,  154, 156. 


AprENn.]  NEWTON  AND  IIAWES.  297 

a  Master  in  respect  of  temper  and  conduct  as  well  as 
learning.  But  I  reckon  two  yeares  too  short  a  time  for 
this  scheme  of  learning,  and  D'.  Gregory  who  taught 
Mathematicks  in  Scotland  w***  very  good  success,  and  was 
here  last  weeke,  tells  me  that  by  the  time  he  spent  in 
teaching  he  should  reckon  three  yeares  little  enough  for 
this  scheme.  M'.  Newton  may  try  if  he  can  compass  it 
sufficiently  in  two  yeares  but  if  that  be  found  too  little, 
perhaps  the  wisdome  of  the  Governors  may  soe  order 
things  as  to  allow  him  halfe  a  yeare  more  from  the  other 
schooles.  ffor  were  it  not  for  some  Mathematicall  bookes 
in  Latine,  I  should  think  that  language  of  soe  little  use  to 
a  Seaman  as  not  to  deserve  four  or  five  yeares  of  the  chil- 
drens  time,  while  Mathematicks  are  allowed  but  two ;  I 
thank  you  for  your  concerne  and  pains  in  behalfe  of 
M'.  Newton,  and  am  very  glad  to  understand  that  he 
behaves  himselfe  so  well,  ffor  tho*  I  was  almost  a  stranger 
to  him  when  I  recommended  him,  yet  since  he  was  elected, 
I  reckon  myselfe  concerned  that  he  should  answer  my 
recommendation.  The  ill  will  you  may  have  got  by  your 
acting  for  him  I  perceive  is  but  of  little  extent  and  cannot 
hurt  you.  M*".  CaswePs  freinds  at  Oxford  blame  his 
freind*  neere  London,  and  some  of  them  think  the  place 
would  not  have  suited  with  his  humour,  soe  that  I  am 
satisfyed  you  made  the  best  choice.  S*".  Your  most  humble 
&  most  obedient  Servant. 

Is:  Newton. 


•  Flamsteed,  who  had  recommended  Caswell  as  Paget's  Buccessor. 


298  CORRESPONDENCE  OP  [Append. 

[Enclosed  in  the  above.] 
A  Now  Scheme  of  Learning  proposed  for  the  Mathematical  boyes  in 

Christ's  Hospitall.* 

1.  Arithmetick  in  Integers,  Vulgar  fractions  &  Deci- 
mals, in  Proportional  numbers  natural  and  Artificial,  in 
Symbols  of  unknown  Numbers  &  in  Equations. 

2.  Geometry  in  Planes  &  Solids,  with  the  Demon- 
strations thereof  &  y*  practise  by  the  Rule  &  Compass. 

3.  The  application  of  Arithmetick  to  Geometry  in 
determining  &  protracting  lines,  angles,  and  plane  Trian- 
gles f,  by  numbers  natural  and  artificial,  Symbols  of  Num- 
bers, &  Tables  of  Sines  &  Tangents. 

4.  The  description  &  properties  of  ffigures  (rectilinear 
&  circular)  in  Perspective,  w*^  the  Art  of  Designing  J  & 
Drawing}. 

5.  The  construction  &  use  of  the  best  Instruments  in 
working  by  Proportionals,  taking  Angles,  Heights  &  Dis- 
tances, &  Surveying,  Guaging,  or  otherwise  measuring 
Planes  &  Solids. 

6.  Cosmo^Vi'p.^y,  or  the  rudiments  of  Astronomy,  Geo- 
graphy &  Hydrography,  with  the  Projections  of  the  globe 
in  Perspective,  &  the  art  of  making  Maps  &  Charts. 

7.  The  doctrine  of  Spherical  Triangles,  with  their 
application  in  projecting  &  computing  all  the  useful  Pro- 
blems in  Astronomy,  Geography  &  Navigation. 

8.  A  full  application  of  the  Learning  aforesaid  to 
Navigation  particularly  to  the  several  Hypotheses  thereof 
commonly  called  Plane,  Great  circle.  Parallel  &  Mercator's 
sailing.  As  also  the  use  of  Charts  &  Sea  Instruments  for 
Observation,  &  their  application  to  the  finding  of  the  Lati- 
tude, difference   of  Longitude,   Amplitudes,  Azimuths  & 


•  There  is  a  copy  of  this  paper  in  Newton's  handwriting  in  Trin.  Coll.  Library 
in  a  folio  volume  marked  II.  5.  4. 

t  In  Newton's  MS.  it  is  "  plane  triangles  &  other  figures." 
X  These  words  change  places  in  Newton's  MS. 


Append.]  NEWTON  AND  HAWES.  299 

Variation  of  the  Compas  by  y*  Sun  or  stars,  with  the  know- 
ledge of  Tides,  Currents  &  the  Roman  Calendar  &  the 
method  of  keeping  Journals,  &  of  finding  the  longitudes  of 
shores  by  the  Eclipses  of  Jupiters  Satellites. 

9.  The  mechanical  Arts  or  Sciences  of  the  five  Powers, 
The  laws  of  motion,  Hydrostaticks,  Gunnery  &  ffortifica- 
tion. 

A  minute  dated  19  July,  1695,  states  that  **the  consideration  of 
the  new  sclieme...  drawn  up  by  Mr  Newton...  which  was  referred  by  the 
last  Court  to  this  Committee  is  for  several  reasons  postponed  until 
another  time." 

The  master  seems  to  have  found  the  scheme  difficult  to  carry  into 
practice,  and  a  course  of  study  formed  by  a  fusion  of  the  old  and  new 
schemes,  and  excluding  Mechanics  except  "  so  much  of  gimncry  as  is 
necessary  for  sea  service"  was  afterwards  adopted.  (Minutes  of  Apr.  0 
and  June  10,  169(5.) 

A  few  notices  of  our  philosopher,  taken  from  the  same  source  to 
which  we  are  indebted  for  the  three  preccdinfl;  letters,  and  exhibiting 
him  in  connexion  with  Christ's  Hospital,  may  be  given  hero. 

"  March  25, 1696.  The  Committee  being  informed  that  Mr  Newton 
is  in  town  {summoned  by  Charles  Montagu's  letter  offering  him  the 
Wardenship  of  the  ^lint}  and  will  stay  some  days,  desired  the  Treasurt^r 
to  request  him  to  examine  and  consider  of  the  Library  belonging  to  the 
Mathematical  School.... and  give  his  opinion  what  books  are  wanting 
that  may  be  most  useful  and  necessary. 

July  13,  I6O7.  The  Committee  did  desire  Mr  Isaac  Newton  now 
present  to  deliver  his  opinion  concerning  the  said  jfive}  boys,  who  was 
pleased  to  say  that  about  10  or  14  days  since  ho  examined  them  and 
then  found  them  perfected,  except  in  a  very  few  particulars,  which  by 
this  time  he  don't  question  but  they  are  masters  of,  and  therefore  is  of 
opinion  they  are  well  qualified  to  be  placed  forth  to  sea  as  apprentices... 
And  this  Committee  returned  their  unanimous  thanks  to  Mr  Professor 
Newton  for  his  great  kindness  and  pains  taken  herein." 

He  is  also  mentioned  as  present  at  the  Hospital  meetings  on  Sept. 
23,  (visitation  of  all  the  schools  in  the  hospital)  and  Dec.  16,  1697, 
on  which  latter  day  he  was  appointed  one  of  a  committee  to  consider 
how  £100  might  best  be  laid  out  for  the  improvement  of  the  mathe- 
matical library. 


300  LETTER  OP  WALLIS.  [Aitend. 

No.  XXVIII. 
WALLIS  TO  NEWTON. 

Sip,  Oxford,  Apr.  10, 1696. 

I  was  in  hopes  of  seeing  you  in  Oxford  last  summer ; 
which  made  me  neglect  sending  you  (by  the  Carrier)  two 
Cuts  which  belonged,  to  the  Volume  you  had  before.  They 
were  not  >\TOught  oft  at  y®  Rolling-Press  when  you  had  th« 
rest ;  but  are  easy  to  be  inserted  in  their  proper  places. 
I  send  them  now,  with  the  other  Volume ;  which  I  desire 
you  to  accept. 

I  understand  (from  Mr  Caswell)  you  have  finished  a 
Treatise  about  Light,  Refraction  and  Colours;  which  I 
should  be  glad  to  see  abroad.  'Tis  pitty  it  was  not  out 
long  since.  If  it  be  in  English  (as  I  hear  it  is)  let  it,  how- 
ever, come  out  as  it  is ;  &  let  those  who  desire  to  read 
it,  learn  English.  I  wish  you  would  also  print  the  two 
large  Letters  of  June  and  August  { October  {  1676.  I  had 
intimation  from  Holland,  as  desired  there  by  your  friends, 
that  somewhat  of  that  kind  were  done;  because  your 
Notions  (of  Fluxions)  pass  there  with  great  applause,  by 
the  name  of  Leibnitz's  Calculus  Differ entialis,  I  had  this 
intimation  when  all  but  (part  of)  the  Preface  to  this  Volume 
was  Printed-off';  so  that  I  could  onely  insert  (while  the 
Press  stay'd)  that  short  intimation  thereof  which  you  there 
find.  You  are  not  so  kind  to  your  Reputation  (&  that 
of  the  Nation)  as  you  might  be,  when  you  let  things 
of  worth  ly  by  you  so  long,  till  others  carry  away  the 
Reputation  that  is  due  to  you.  I  have  endeavoured  to 
do  you  justice  in  that  point ;  and  am  now  sorry  that  I  did 
not  print  those  two  letters  verbatim, 

I  understand  you  are  now  about  adjusting  the  Moons 
Motions ;  and,  amongst  the  rest,  take  notice  of  that  of  the 
Comon  Center  of  Gravity  of  the  Earth  &  Moon  as  a 
conjunct  body :  (a   notion  which,  I  think,  was   first  started 


Append.]  LETTER  OP  WALLIS.  301 

by  me,  in  my  Discourse  of  the  Flux  and  Reflux  of  the  Sea.) 
And  it  must  needs  be  of  a  like  consideration  in  that 
of  Jupiter  with  his  Satellites,  &  of  Saturn  with  his.  (And 
I  wonder  we  have  not  yet  heard  of  any  about  ^loon.)  But 
Saturn  and  Jupiter  being  so  far  off,  the  efTects  thereof  are 
less  observable  by  us  than  that  of  the  ^Moon.  My  advise 
upon  the  whole,  is,  that  you  would  not  be  too  slow  in 
publishing  what  you  do. 

I  am  S-^ 
Your  very  humble  Servant, 

John  Wallis*. 

For  Mr  Isaac  Newton, 

Fellow  of  Trinity  College,  ^ 
Professor  of  Mathematick, 
in  Cambridge. 

With  a  Book  {the  1  at  Vol.  of  Wallis's  Works.} 

"NVallis  was  a  strong  advocate  for  the  immediate  publication  of  di»* 
coveries.  In  a  letter  to  Waller  (Sec.  to  Royal  Soc),  April  30,  ho 
dwells  upon  the  same  topics,  and  speaks  of  Newton's  Treatise  as 
"  finished  &  fairly  transcribed  some  while  since.  I  wisli  he  were 
called  upon  to  print  it  without  farther  delay.  Perhaps  Mr  Ilalley  may 
prevail  with  him  so  to  do,  &c.'* 

Waller  writes  back  May  15  "  Mr  Ilalley  has  promised  to  write  to 
Mr  Newton  concerning  those  letters  {to  Leibniz}  you  mention.  I 
hope  they  may  be  procured  from  him  &  thank  you  for  the  intimation 
thereof." 

Wallis  writing  to  Ilalley  Nov.  11,  says :  '*  I  have  written  several  let- 
ters to  Mr  Newton  about  it  {i.  e.  printing  the  two  letters}  pressing  with 
some  importunity  the  printing  of  them,  and  of  his  Treatise  about  Light 
and  Colours  (as  being  neither  just  to  himself  nor  kind  to  the  publick  to 
delay  it  so  long.  As  to  the  Letters  I  sent  him  a  fair  transcript  ready 
for  the  press  {Newton's  copies  of  them  may  have  perished  in  the 
fire  which  destroyed  a  mass  of  other  papers,  and,  as  Wallis  supposed, 
Leibniz's  answers  among  them ;  see  Wallis's  Works  III.  654  or  CW- 
merc.  Epistol.  110  or  211  ed.  2},  which  if  he  would  print,  it  might 

best  be  done  here,  (and  I  would  take  the  care  of  it) But  he  did 

not  seem  forward  for  either As  to  that  about  laght  &  Colours 


•  Orig.  Lett.  Bk,  Roy.  Soc.  W.  2.  48.    Part  of  it  is  printed  in  Raphson's  Hist,  of 
Fluxions,  p.  120, 


802  LETTER  [Append. 

(for  which  I  am  more  solicitous)  your  interest  may  possibly  prevail 
with  him  better  than  mine  to  get  it  published." 

"  In  pursuance  of"  a  letter  from  Ilalley  dated  Nov.  21,  Wallis  sent 
him  copies  of  the  two  letters  on  the  20th,  observing:  "I  am  glad 
Mr  Newton  is  inclinable  to  print  some  of  the  things  he  hath  by  him. 
So  many  as  he  hath  on  his  hands  at  once  do  hinder  one  another.  I  am 
most  fond  of  his  Book  of  Light  and  Colours.  His  fear  of  disputes 
and  cavils  need  not  trouble  him.  It  will  be  at  his  choice  whether  or 
not  to  answer  them.  His  Hypothesis  will  defend  itself.  "We  are  told 
here  that  he  is  made  Master  of  the  Mint"  &c.  Orig,  Lett.  Bk.  Roy.  Soc, 
W.  2.  56. 


No.   XXIX. 
NEWTON  TO  HAUINGTON. 

Mr  John  Harington  (of  the  family  of  "Ariosto"  Ilarington  and 
**  Oceana"  Harington),  an  undergraduate  of  Oxford,  seems  to  have  had 
some  conversation  with  Newton  upon  a  method  which  had  occurred  to 
him  of  representing  musical  intervals  by  the  additions  of  the  sides 
(3,  4, 5)  of  a  right-angled  triangle,  and  to  have  tlluded  to  the  bearing  of 
the  subject  upon  the  principles  of  architectural  beauty.  At  Newton's 
request  he  sent  the  details  of  his  method  with  remarks  upon  the  appli- 
cation of  harmonical  ratios  to  architecture,  in  a  letter  dated  Wadhain 
College  May  22.  1698.  The  receipt  of  this  letter  Newton  acknow- 
ledged in  the  following  kind  and  encouraging  terms. 

Sir, 
By  the  hands  of  your  friend,  Mr.  Conset,  I  was  favoured 
with  your  Demonstration  of  the  Harmonic  Ratios,  from  the 
Ordinances  of  the  47th  of  Euclid.  I  think  it  very  explicit 
and  more  perfect  than  the  Helicon  of  Ptolemy,  as  given  by 
the  learned  Doctor  Wallis.  Your  observations  hereon  are 
very  just,  and  afford  me  some  hints  which,  when  time 
allows,  I  would  pursue,  and  gladly  assist  you  with  any 
thing  I  can,  to  encourage  your  curiosity  and  labours  in 
these  matters.  I  see  you  have  reduced,  from  this  wonder- 
ful proposition,  the  inharmonics  as  well  as  the  coincidences 
of  agreement,  all  resulting  from  the  given  lines  three,  four, 
and  five.  You  observe  that  the  multiples  hereof  furnish 
those  ratios  that  afford  pleasure  to  the  eye  in  architectural 


Append.]  *  TO  IIARINOTON.  803 

designs :  I  have,  in  former  considerations,  examined  these 
things,  and  wish  my  other  employments  would  permit  my 
further  noticing  thereon,  as  it  deserves  much  our  strict 
scrutiny,  and  tends  to  exemplify  the  simplicity  in  all  the 
works  of  the  Creator ;  however,  I  shall  not  cease  to  give 
my  thoughts  towards  this  subject  at  my  leisure.  I  beg  you 
to  pursue  these  ingenious  speculations,  as  your  genius 
seems  to  incline  you  to  mathematical  researches.  You 
remark  that  the  ideas  of  beauty  in  surveying  objects  arises 
from  their  respective  approximations  to  the  simple  con- 
structions,  and  that  the  pleasure  is  more  or  less,  as  the 
approaches  are  nearer  to  the  harmonic  ratios*.  I  believe 
you  are  right;  portions  of  circles  are  more  or  less 
agreeable,  as  the  segments  give  the  idea  of  the  perfect 
figure  from  whence  they  are  derived.  Your  examinations 
of  the  sides  of  polygons  with  rectangles  certainly  quadrate 
with  the  harmonic  ratios.  I  doubt  some  of  them  do  not ; 
but  then  they  are  not  such  as  give  pleasure  in  the  for- 
mation or  use.  These  matters  you  must  excuse  my  being 
exact  in,  during  your  inquiries,  till  more  leisure  gives  me 
room  to  say  with  more  certainty  hereon.  I  presume  you 
have  consulted  Kepler,  !Mersenne,  and  other  writers  on  the 
construction  of  figures.  What  you  observe  of  the  ancients 
not  being  acquainted  with  a  division  of  the  sesquialteral 
ratio  is  very  right ;  it  is  very  strange  that  geniuses  of  their 
great  talents,  especially  in  such  mathematical  considera- 
tions, should  not  consider  that,  although  the  ratio  of  three 
to  two  was  not  divisible  under  that  very  denomination,  yet 
its  duple  members  six  to  four  easily  pointed  out  the  ditone 
four  to  five,  and  the  minor  tierce  six  to  five,  which  are  the 
chief  perfections  of  the  diatonic  system,  and  without  which 
the   ancient   system   was   doubtless    very   imperfect.      It 


•  Comp.  Kepler,  Harmon.  Munrfi,  p.  126.  In  Architectonica  quaecuiw^ue  propor- 
tiones  long^itudinis  ad  latitudinem  vel  crassitiem  plurimum  probantur,  etiam  a  non 
Mathemnticis  spectatoribus,  os  quum  proximtc  harmonicis  inveniuntur. 


304  REFORMATION   OF  CALENDAR.  [Append. 

appears  strange,  that  those  whose  nice  scrutinies  carried 
them  so  far  as  to  produce  the  small  limmas,  should  not 
have  been  more  particular  in  examining  the  greater 
intervals,  as  they  now  appear  so  serviceable  when  thus 
divided.  In  fine,  I  am  inclined  to  believe  some  general 
laws  of  the  Creator  prevailed  with  respect  to  the  agreeable 
or  unpleasing  affections  of  all  our  senses ;  at  least  the  sup- 
position does  not  derogate  from  the  wisdom  or  power  of 
God,  and  seems  highly  consonant  to  the  macrocosm  in 
general.  Whatever  else  your  ingenious  labours  may  pro- 
duce I  shall  attentively  consider,  but  have  such  matters  on 
my  mind,  that  I  am  unable  to  give  you  more  satisfaction 
at  this  time ;  however,  I  beg  your  modesty  will  not  be  a 
means  of  preventing  my  hearing  from  you,  as  you  proceed 
in  these  curious  researches ;  and  be  assured  of  the  best 
services  in  the  power  of 

Your  humble  Servant, 
{Jermyn  Street}  May  30,  1698.  Is.  Newton*. 


No.    XXX. 

The  decree  of  the  German  Diet  (Ratisbon,  Sept.  23.  1699,  see  Mim^ 
tucla^  Hist,  des  Math.  iv.  325,)  refonning  the  Julian  Calendar  and  order- 
ing (1)  that  the  day  after  Febr.  18.  1700  should  bo  March  1,  and  (2) 
that  Easter  should  bo  determined  by  astronomical  calculation  (viz.  of  the 
exact  time  of  the  vernal  equinox  and  the  full  moon  following  it),  gave 
rise  to  considerable  discussion  among  the  theologians  and  scientific  men 
of  the  Empire.  In  Leibniz's  "Works  (iv.  pars  ii.  115 — 137)  will  bo 
found  the  correspondence  which  he  had  with  Roemcr  upon  the  subject. 
Leibniz  also  consulted  the  French  Academy  (lb.  143)  and  the  Royal 
Society  on  the  second  of  the  two  Articles  of  the  Ratisbon  conclusum  : 
his  application  to  the  latter  body  was  laid  before  Newton,  whose 
answer  is  contained  in  the  paper  now  presented  to  the  reader. 


•  H.  Harington's  Nug<e  Antiqnce,  Lond.  1779.  (U.  107),  where  Harington's  letter 
and  Newton's  answer  are  dated  1693,  but  as  Hanngton  was  not  admitted  at  Wadliam 
until  .June  1696,  being  then  in  his  17th  year,  I  have  ventured  to  suppose  that  the  3  has 
been  printed  by  mistake  for  8. 


ArrEm]  tBIE  OF  VERNAL  EQUINOX.  305 

Subjoined  aro  some  notices,  bcarin<:f  upon  the  subject,  extracted  from 
the  Journal  Book  of  tlio  Royal  Society. 

Febr.  21.1700.  A  letter  from  Leibniz  to  Sloano  {Jan.  30.  Letter 
Bk\  27()}  was  read  conceruinpr  the  dianrjo  of  style,  {in  which  tho  writer 
desires  tlio  opinion  of  tlie  Society  upon  tho  point}. 

Sloano  said  ho  heard  IMr  Newton  had  made  a  very  jjood  calculation 
of  tho  year,  and  that  the  settlinf;  that  affiiir  mi^^ht  be  helped  by  it. 
Sloano  was  ordered  to  wait  on  Mr  Newton  about  it. 

Apr.  25.  Sloano  read  an  answer  to  Tioibniz's  letter  containing  Mr, 
Newton's  opinion  concerning  tho  alteration  of  tho  stylo,  {the  paper  hero 
printed}. 

Tho  Vice-President  (Sir  Robt.  Southwell)  said  his  opinion  was  that 
this  paper  bo  sent  to  Air.  Leibniz,  and  in  tho  nieantimo  that  ho  procure 
Mr  Flamstced's  and  Dr  AVallis's  opinion,  and  send  them  to  him :  also 
that  a  copy  of  this  be  kept. 

]\Lay  I.  Copy  of  Leibniz's  letter  and  Newton's  answer  ordered  to 
bo  sent  to  Flamsteed,  and  an  answer  rec]  nested. 

May  22.  A  letter  from  Wallis  read  (returning  Newton's  paper) 
concerning  tho  Julian  account.  (Orif/.  Lett.  Bk.  "W,  2.  06).  Copy 
ordered  to  bo  sent  to  Leibniz. 

May  21).  Flamstced's  opinion  of  Leibniz's  letter  read  (dated  IVIay 
22.    Ze«. /?^^  XII.  320). 

Jun.  5.  Sloano  read  a  letter  from  Flamsteed  against  Leibniz's  rea- 
sons for  changing  tho  style. 

Among  Flamstced's  MSS.  at  Greenwich  (Vol.  33)  are  copies  in  his 
hand  of  Leibniz's  letter  and  Newton's  answer,  to  tho  latter  of  which 
ho  has  added  remarks  redolent  as  usual  of  amour-propre.  Of  tho 
former  ho  observes  "This  letter  imparted  to  mo  by  Dr  Sloano,  May 
2.  2;  1700,  but  tho  schedule  of  Mr  Newton  was  sent  away  without 
expecting  my  answer."  The  paper  as  revised  by  I'lam steed  was  sent 
to  Leibniz  with  Newton's  approval  in  a  letter,  dated  July  4.  "  IIo 
(Newton)  docs  not  say  tis  his  own,  but  what  ho  approves  of  from  the 
best  observations  ho  thinks  have  been  made  in  England  by  Mr  Flam- 
steed, Halley,"  &c.     (firif/.  Lett.  Bk.  S.  2.  14.) 

Elementa  motuum  Solis  et  Lunro 
ab  iEquinoctio  verno. 
Tempus  rcquabile,  quod  vcrum  dici  solet  diurnro  non 
solis   sed  flixarum    revolution!   proportionalc  est   et   inde 
condenda3  sunt  Tabulae  pro  aquatione  Temporis. 
20 


306         .  TIME  OF  VERNAL  EQUINOX.  [ArrESD. 

In  Observatorio  Rcgio  Grenovicensi,  Anno  Cliristi 
1701  ineunte  ad  meridiem  Kal.  Jan.  stylo  veteri,  erit 
medius  motus  Solis  9*.  2l^^  42'.  38".  Apogsei  ejus  3".  07^^ 
44'.  30",  Lunaj  10*.  28^\  30\  12"  &  Apogooi  ejus  11*.  085^ 
25'.   14". 

Uraniburgum  est  orientalius  Observatorio  Regio 
Parisiensi  00^.  42'.  lo"  &  hoc  Observatorium  est  orientalius 
Grenovieensi  00^.  09'.  15",  et  inde  per  reductionem  lia- 
bentur  motus  illi  medii  eodem  die  et  liora  ad  meridianum 
Uraniburgi,  viz*.  Solis  9*.  21^.  40'.  32"  Apogaei  ejus  3*.  O?^*". 
44'.  30"  Lunffi  10*.  28^''.  01'  58"  &  Apogaji  ejus  11*.  8^'.  25'. 
00".  Et  ante  undeeim  dies  seu  meridie  Kal.  Jan.  stylo 
novo  erit  motus  medius  Solis  9^  H^"".  50'.  00"  Apogaei  ejus 
3*.  7^.  44'.  32"-^,  Luna)  6*.  03^.  05'.  33''  &  Apogoci  ejus 
11".  O?"^.    11'.  28". 

Maxima  Solis  Prost{h}aphsBresi8  qua?  Keplero  est 
plusquam  2^',  3'  debet  esse  tan  turn  l^^  5&,  20". 

Ubi  haec  ajquatio  additur  vel  subducitur  medio  motui 
Solis  debet  ejus  pars  decima  e  eontra  subduci  vel  addi 
medio  motui  Luna).  Nam  medius  motus  Lunae  non  est 
uniformis  sed  per  vices  tardescit  et  acceleratur  propterea 
quod  orbis  Lunae  dilatatur  in  perigaeo  Solis  et  contrahitur 
in  ejus  Apogseo. 

Postquam  motus  medius  Lunae  sic  correetus  habetur, 
reliqua  peragenda  sunt  per  Tabulas  Kepleri:  et  JSqui- 
noctium  vernum  incidet  semper  in  diem  horam  et  minu- 
tum  ubi  longitudo  Solis  per  hoc  computum  prodit 
00*.  00^.  00'.  00"  f. 


*  Gregory  informed  Wallis  that  the  "32"  is  miswritten  for  28.     Wallis'e  letter. 
May  11,  1700. 

t  Orig.  Lett.  Bk,  Uoyal  Soc.  N.  1.  63. 


Append.]  LKTTER  TO  SIR  JOHN  NKWTON.  307 

No.   XXXI. 
NEWTON  TO  SIR  JOHN  NEWTON. 
Sir  John 
I  was  very  much  surprized  at  the  notic  of  IVF.  Cook's* 
death  brought  me  this  morning  by  the  bearer  who  being 
an  undertaker  came  to  me  to  desire  that  I  would  speak 
to  you  that  he  might  be  employed  in  furnishing  things 
for  y®  funeral.     lie  having  married  a  near  kinswoman  of 
mine  I  could  not  refuse  troubling  you  with  this  letter  in 
his  behalf  beleeving  that  he  will  do  it  well  if  you  are  not 
otherwise  provided.     I  had  an  opinion  that  my  Cousin  was 
not  in  danger  tho  weak,  w*^^  makes  my  concern  the  greater 
for  the  loss.      I  am 

Yo""  aflcctionate  Kinsman 
and  most  humble  Servant 
{.Termyn  Street,  Apr.  1707}.  P  Newtonj. 

For  S'  John  Newton,  Baron* 
{at  his  house  in  Soho  Square.} 


No.  XXXII. 


This  is  the  rough  draught  of  a  critique  on  three  papers  of  Leibniz's 
in  the  Leipsic  Acts  for  Jan.  and  Febr,  1089  (pp.  36,  38,  82),  and  was 
probably  written  in  1712,  after  tho  receipt  of  Leibniz's  second  letter  to 
Sloano  (see  p.  55,  ante).  It  is  copied  from  a  folio  sheet  in  Newton's 
liand  which  formerly  belonged  to  Keill  and  is  now  preserved  among  tho 
Lucasian  papers  (packet  No.  8.)  Several  expressions  in  the  introduc- 
tory sentences,  as  Newton  had  first  written  them,  coincide  witli  some  of 
those  in  the  second  of  the  two  statements  published  in  Rigaud's  Essay 
on  the  First  Publication  of  the  PHwc//)k  (Appendix,  p.  07) :  but  New- 
ton afterwards  crossed  them  out  and  substituted  others  for  them. 
These  alterations  (with  one  or  two  others)  bring  tho  language  of  this 
document  into  still  closer  agreement  with  that  used  in  the  Commercium 


•  Fossibly  Edw.  Coke,  Esq.,  of  Ilolkham,  (great -great -grandson  of  the  Chief 
Justice),  wlio  married  Cary,  daughter  of  Sir  John  Newton,  and  died  Apr.  13,  1707. 
His  son  Thomas  was  created  Earl  of  I^icoster  in  1744. 

t  The  original  is  in  the  possession  of  P.  O'Callaghan,  Esq.,  to  whom  I  am  indebted 
for  a  copy  of  it 


808  "  STRICTURES  ON  [Append. 

EpittoHeum  (p.  97,  cd.  1  ;  p.  206,  ed.  2),  tho  editors  of  whicli  work 
must  therefore  havo  seen  either  tho  dociinient  itself  or  a  copy  of  it,  or 
perhaps  a  still  lator  corrected  fonn  of  it.  The  opening  sentence  of  this 
paper  seems  to  havo  passed  through  the  following  stages  : 

1.  Ncwtonus  anno  1(>84  Propositioncs  princi pales  earum  quaB  in 
Philosophia*  Principiis  Mathematicis  hahentur  cum  Socictato  Regia 
conimunicaro  coopit,  &c. 

2.  Ineunto  anno  1(504  Ncwtonus  Propositioncs cum  Socictato 

Regia  communicavit,  &c. 

3.  Anno  J  083  Ncwtonus  Propositioncs 

4.  Anno  1083  ad  finem  vergento  Nowtonus  Propositioncs  princi- 
pales  carum...habontur  Londinum  misit  eaodemque  cum  Socictato  Regia 
mox  communicata)  sunt,  &c. 

Newton  first  of  all  clearly  wrote  1G84,  then  altered  tho  4  to  a  3, 
afterwards  crossed  all  tho  figures  out  and  wrote  distinctly  1683.  I 
mention  this  the  more  particularly,  because  Mr  Rigaud  says  (Essay, 
p.  20)  that  in  tho  MS.  of  tho  latter  of  the  two  fragments  which  ho  has 
published  from  th.e  Macclesfield  Collection,  tho  year  1683  was  at  first 
written,  "  tho  last  figure  having  been  evidently  altered  to  a  4."  Newton 
therefore  after  endeavouring  to  recollect  tlie  exact  year  in  which  ho  sent 
up  tho  fundamental  propositions  of  tho  Prindpia  to  London,  antedated 
the  event  by  a  twelvemonth.  Sec  Syn.  View  of  his  Life,  under  date 
Nov.  1684. 

Ex  Epistola  cujusdam  ad  Amicum. 

Anno  1683  ad  fmem  vergente  Newtonus  Propositiones 
principales  earum  qurs  in  Philosophis9  Principiis  Mathe- 
maticis habentur  Londinum  misit  e^edemq :  cum  Socic- 
tato Regia  mox  communicatae  sunt,  annoq :  1686  Liber 
ille  ad  Societatem  missus  est  ut  imprimeretur,  et  pr» 
irao  anno  lucem  vidit.  Deinde  anno  1688  epitome  ejus 
in  Actis  Lipsicis  impressa  est,  qua  lecta  D.  Lcibnitius 
Epistolam  de  lineis  opticis,  Schediasma  de  res{is}tentia 
Medii  &  motu  projcctilium  gravium  in  Medio  resistente, 
&  Tentamen  de  motuum  ccelcstium  causis  composuit  & 
in  Actis  Lipsicis  ineuntc^anno  1689  imprimi  curavit,  quasi 
Ipse  quoque  prtecipuas  Newtoni  de  his  rebus  Proposi- 
tioncs invenissct  idque  methodo  diversa  ct  Librum  New- 
toni nondum  vidissct.^  Qua  licentia  concessa  Authorcs 
quilibet  inventis  suis  facile  privari  possunt.     Quam  primum 


Append.]  LEIBNIZ'S  PAPERS.  309 

Liber  Newtoni  lucem  vidit  exemplar  ejus  D.  Nicolno 
Fatio  datum  est  ut  ad  Leibnitium  mitteretur.  Vidcrat 
Leibnitius  Epitomen  ejus  in  Actis  Lipsicis.  Per  commer- 
cium  epistolicum  quod  cum  viris  doctis  passim  habcbat, 
cognoscere  potuit  Propositiones  principalcs  m  libro  illo 
contentas  imo  &  librura  ipsum  procurarc.  Sin  Librum 
ipsura  non  vidisset  videre  tamon  dcbuissct  antequara  sua 
de  iisdem  rebus  cogitata  publicaret,  idq:  ne  festinando 
errarct  in  sub  { j  \  ecto  novo  ac  difficili  et  Newtono  injurius 
esset  auferendo  inventa  ejus,  et  Lectori  molestus  repe- 
tendo  qujB  Xewtonus  antea  dixerat,  &  contentioncs  de 
inventis  excitaret,  ut  antea  fecerat  in  causa  ^Moutoni. 
Dicit  enim  in  fine  Scbediasmatis  de  resistentia  Medii : 
Nobis  nunc  fundamenta  Geometrica  jecisse  suffccerit  in 
quibus* 

Qua3  de  Lineis  Opticis  habet,  primo  intuitu  ex  New- 
tonianis  consequuntur,  positis  sinubus  incidentiaj  et  reflexi- 
onis  a?qualibus. 

In  scliediasmate  de  llesistentia  Medii,  Resistentiam 
cum  Xcwtono  duplicem  facit,  unam  qua3  a  Medii  glutino- 
sitate  et  frictione  oritur,  alteram  quae  a  Medii  densitate. 
Priorem  vocat  absolutam,  posteriorem  relativam.  Prio- 
rem  facit  velocitati  proportionalem  posteriorem  cum  Xcw- 
tono facit  in  duplicata  ratione  vclocitatis.  Priorem  tractat 
in  tribus  Articulis,  eaq :  sola  tradit  qua;  Xewtonus  in  Libri 
secundi  Propositionibus  quatuor  primis  de  hujusmodi 
resistentia  prius  dixerat.  Posteriorem  tractat  in  Articulo 
quarto  quinto  et  sexto.  Et  qua)  in  articulo  quarto  habet 
Xewtoniana  sunt.  In  quinto  Propositiones  quatuor  (tertia 
quarta  sexta  et  septima)  sunt  falsa)  f.     In   sexto  Propo- 


•  There  not  being  room  for  the  remainder  of  the  quotation  in  the  MS.,  there  is  a 
mark  after  "quibus"  apparently  referring  to  anotlier  paper  which  is  lost.  The  whole 
of  the  i):u^sii!,'e  will,  however,  be  found  quoted  afterwards,  p.  313,  lin.  11. 

+  Newton  does  not  seem  to  have  decided  whetlier  to  write  "non  sunt  veras"  or 
"sunt  falsaj."  He  first  of  all  used  the  latter  phrase,  then  crossed  it  out  and  wrote  the 
former  above  it,  but  afterwards  restored  tiie  old  phrase  underneath  its  oriffinal  place. 


310  STRICTURES  ON  [Append. 

sitiones  sunt  tantum  diKOj,  et  iitraq :  falsa  est.  Corpus  enim, 
ubi  resistentia  est  in  duplicata  ratione  velocitatis,  non 
fertur  motu  composito  ea  motibus  duorum  Articulorum 
praecedentium.  Demonstret  Leibnitius  hasce  sex  Propo- 
sitiones  si  pro  veris  haberi  velit. 

In  tentamine  de  motuum  ccelestium  causis*,  Leibnitius 
deducit  circulationem  harmonicam  Planetarum  a  circu- 
latione  harmonica  Vorticum,  &  ascensum  et  descensum 
Planetarum  ab  eorum  gravitate,  dicitq :  (in  Propositione 
tertia)  nihil  referre  quia  sit  motus  rectilineus  quo  ad  centrum 
acceditur  vel  ah  ij^so  receditur  (quern  motum  vocat  paracen- 
tricu)  modo  circulationes  sunt  harmoniccc.  Imo  multum 
refert.  Nam  si  motus  paracentricus  si  paulo  velocior  vel 
paulo  tardier  Apsides  Planetarum  non  manebunt  in  locis 
suis,  &  propterea  Sectiones  oonicas  non  describentur.  Co- 
nicas  igitur  Seetiones  deseribi  Leibnitius  non  demon- 
stravit. 

In  sexta  Tentaminis  Propositione  docet  ex  Phrenomenis 
Planetas  motu  harmonico  ferri,  in  septima  deducit  inde 
motum  liarmonicum  vorticum.  Qua?  de  Vorticibus  dicuntur 
sunt  mere  hypothetica,  &  cum  motu  Cometarum  conciliari 
non  possunt,  neque  quadrant  cum  Planetarum  temporibus 
periodieis  qua3  sunt  in  ratione  sesquiplicata  distantiarum 
ab  orbium  centre  communi.  Hoc  notavit  Gregoriusf,  et 
Pespondit  Leibnitius  Vortices  non  moveri  motu  harmonico 
nisi  in  singulis  Planetarum  orbibus  seorsim  spectatis ;  in 
intervallis  orbium  vortices  alia  ratione  moveri;  id  est, 
partes  vorticum  alternis  vicibus  harmonice  et  non  harmo- 
nice  per  multa  orbium  intervalla  revolvi.  Miraculis  plena 
est  hsec  hypothesis   motumq :     Cometarum  adhue  magis 


•  Among  the  Lucasian  MSS.  there  is  a  paper  in  Keill's  handwriting  entitled  "  Notae 
in  Acta  Erud...Anno  1689,  Pag.  84  et  scq."  in  which  the  errors  of  this  essay  of  Leibniz's 
are  briefly  exposed,  li  seems  to  be  the  same  as  tliat  mentioned  by  Wilson  (Robins's 
Tractit  II.  351)  and  apparently  attributed  by  him  to  Newton. 

t  Astron.  Element,  p.  102. 


Append.]  LEIBNIZ'S  PATERS;  311 

perturbat  &  cum  Vorticibiis  Satellitum  Planetarum  minimc 
consistit.  JMotiis  Satellitum  Jovis  sunt  summe  regulares 
&  Vorticem  summe  regularem  circum  Jovem  requirunt :  et 
hujusmodi  Vortex  impediet  motum  harmonicum  Vorticis 
Solaris  intra  Orbem  Jovis.  Et  pra3terea  si  Planetoe  a 
Vorticibus  deferuntur  &  gravitant  etiam  in  Solem  ut  vult 
Leibnitius,  ut  ha)  dua;  vires  seinvicem  non  perturbent, 
necesse  est  ut  vis  ilia  qua  Planetae  deferuntur  a  vorticibus 
in  Orbem  &  versus  Solem  incurvantur  sit  ipsa  gravitas:  cum 
tamen  gravitas  non  minor  sit  ad  polos  Solis  et  Planetarum 
quam  ad  eorum  ajquatores,  vortices  vero  non  agant  ad 
polos,  ad  luce  vis  centripeta  a  motu  harmonico  oriunda 
debet  esse  reciproce  non  ut  quadratum  sed  ut  cubus 
distantia)  Planeta)  a  Sole  per  Corol.  1  Prop.  4  Lib.  1 
Principiorum  Mathemat.  Dcniq :  Leibnitius  nullam  reddit 
causam  motus  harmonic!  vorticum  sed  hunc  motum  sup- 
ponit  tantum  ut  motibus  Phmetarum  a  Kcplcro  detectis 
consentaneum,  ideoq:  non  demonstravit  Planetas  in  Or- 
bibus  EUipticis  harmonice  ferri.  Et  hoc  non  demonstrate 
nihil  demonstravit  quod  alicujus  sit  momenti. 

Undecima  Tentaminis  Propositi©  est  ha;c.  Conatus 
centrifugus  exprimi  potest  per  sinum  versum  anguli  circula- 
tionis,  Et  vera  quidem  est  hajc  Propositi©  ubi  circulatio 
fit  in  circulo  sine  motu  paracentrico.  Sed  ubi  fit  in  Orbe 
excentrico  Propositi©  vera  non  est.  Conatus  centrifugus 
semper  ajqualis  est  vi  gravitatis  &  in  contrarias  partes 
dirigitur  per  tertiam  motus  Legem  in  Principiis  IMathe- 
maticis  Newtoni,  et  vis  gravitatis  exprimi  non  potest  per 
sinum  versum  anguli  circulationis,  sed  est  reciproce  ut 
quadratum  Radii. 

Duodecima  Tentaminis  Propositi©  ha;c  est.  Conatua 
centrifwji  harmonice  circulantis  aunt  in  ratione  radiorum 
reciproce  triplicata.  Pectins  dixisset  quod  sunt  in  ratione 
radiorum  reciproca  diiplicata.  Sunt  qnim  viribus  gravitatis 
ajquales  ut  supra  dictum  est;  et  gravitas  est  in  raticne 
radiorum  reciproca  duplicata. 


312  STRICTURES   ON  [ArpoD. 

Decima  quinta  Tentaminis  Propositio  ha?c  est.  In  omni 
circulatione  harmonica  elementum  impetus  paracentrici  {hoc 
est  incrementum  aut  decrementum  velocitatis  descendendi  versus 
centrum  vel  ascendendi  a  centra)  est  differentia  vel  summa 
sollicitationis  paracentricce,  {hoc  est  impressionis  a  gravitate 
vel  levitate  aut  causa  simili  factop)  et  dujili  conatus  centrifiuji 
ah  ipsa  circulatione  harmonica  orti.  Summa  quidem  si  levitas 
adsitt  differentia  si  gravltas,  Errorem  eiiormem  in  liac 
Propositione  Leibnitius  postea  correxit  &  pro  cluplo  conatu 
ccntrifugo  conatum  simplum  scripsit  (Vide  Acta  Lips.  Anno 
1706  pag.  447.)  Sed  Propositio  tamen  etiamnum  falsa 
manet.  Ob  sollicitationem  paracentricam  &  eonatiini  cen- 
trifugum  inter  se  aquales,  differentia  eorum  nulla  est, 
ideoq :  elementum  impetus  paracentrici  per  banc  Proposi- 
tionem  semper  debet  esse  nullum,  et  velocitas  descendendi 
versus  centrum  vel  ascendendi  a  centro  semper  debet  esse 
uniformis.  Quod  verum  esse  non  potest.  Prajterca  in  De- 
monstratione  hujus  Propositionis  error  admittitur  his  verbis. 
Jam  F2M  cequ.  {N2M  seu)  G^D^NP.  Pro  N2M  Lie 
scribitur  G^D  quamvis  G 2D  sit  major  quam  N2M  excessu 
G^M, 

Tandem  ex  falsis  hisce  Propositionibus  Leibnitius  co- 
natur  demonstrare,  Quod  Planeta?  circa  Solem  in  Ellipsi 
harmonice  circulantis  gravitas  in  Solem  sit  reciproce  ut 
quadratum  distantioe  Planetae  a  Sole.  Et  ha3C  est  Leibnitii 
Propositio  decima  nona.  Errat  vero  in  Demonstratione 
citando  duas  falsas  Propositiones  duodecimam  scilicet  et 
decimam  quintam  quarum  errores  se  mutuo  corrigunt :  Et 
errando  Propositionem  minime  invenit  minime  demonstravit 
sed  a  Newtono  invcntam  et  demonstratam  conatus  est 
aliter  invenire  et  demonstrare  ut  suam  faceret.  Per  duos 
errores  se  invicem  corrigentes  calculum  aptare  potuit  ad 
eonclusionem  propositam,  veritatem  invenire  ac  demon- 
strare non  potuit. 

Propositio  vigesima  deducitur  a  decima  nona  ideoq ; 
non  demonstratur. 


Append.]  LEIBNIZ'S  PAPERS.  3J3 

Propositio  vegesima  [sic]  prima  et  Propositio  vigesima 
quiiita,  niinorcm  cxliibcnt  vim  ccntrifugam  quam  gravita- 
tem  Plane toD  in  Solem  ideoq :  falsa)  sunt.  Motus  Planetm 
in  orbe  non  pendet  ab  cxcessu  gravitatis  supra  vim  centri- 
fugam  (uti  credit  Leibnitius)  sed  Orbis  incur vatur  a  gravi- 
tatis actionc  sola,  cui  vis  centrifuga  (ut  reactio  vel  resistentia) 
semper  est  ajqualis  &  contraria  per  motus  Legem  tertiara 
a  Newtono  positam. 

In  fine  Scliediasmatis  dc  resistentia  Medii  Leibnitius 
subjungit.  Miilta  ex  his  decluci  possent  praxi  accommodata, 
sed  nobis  nimc  fandameiita  Geometrica  jecisse  stiff  {ec]erit  in 
quihus  maxima  consistcbat  difficidtas,  Et  fortassis  attente  con* 
sideranti  vias  quasdam  novas  vel  certe  satis  antea  impeditas 
apendsse  videhimur.  Omnia  autem  respondent  nostrce  Analyst 
infinitorumf  hoc  est  calculo  summarum  et  differentiarum. 

Analysim  banc  per  annos  undecim  vel  duodecim  Leib- 
nitius in  differentiis  primis  jam  exercuerat  et  notaverat 
differentias  differentiarum  per  dd  easq:  ad  inventionem 
puncti  flexus  contrarii  applicuerat,  sed  problemata  difficiliora 
per  differentias  differentiarum  soluta  nondum  dederat.  Jam 
vero  per  opus  Newtonianum  excitatus  hroc  aggreditur  ac 
gloriatur  se  nunc  fundamenta  Geometrica  jecisse  in  quibus 
maxima  consistcbat  difficultas  et  vias  quasdam  novas  vel 
certe  satis  antea  impeditas  aperuisse  &  htec  fecisse  per 
Analysin  suam  infinitorum  quam  differentialcm  vocat.  Sed 
primo  tamen  conatu  multipliciter  erravit  &  per  errorcs 
suos  prodidit  se  methodum  illam  in  difficilioribus  hisce 
nondum  probe  intellexisse,  prodidit  se  Propositioncs  New- 
toni  minime  invenisse  sed  calculum  tantum  ad  conclusioncs 
aptasse.  Noverat  methodum  infinitorum  Newtono  prius 
cognitam  fuisse  ut  ex  ejus  Epistolis  manifestum  est*.  Nove- 
rat Propositioncs  de  resistentia  mediorum  deq:    motibus 


•  In  the  margin  Newton  has  wrillen  "  vide  pag,"  intending  prohably  to  refer  to 
Leibniz's  letter  to  Wallis  (I\Iuy  28,  l()97)and  his  answer  to  Fatio,  which  are  printed 
in  the  Comniercimn  Epistolicum  (pp.  104,  107). 

21 


314  ANCIENT  FORMS  [Appexd. 

corporum  ccBlestium  a  Newtono  primum  inventas  fuisse 
idq:  per  meth{od|um  illam  infinitorum,  et  omnia  tamen 
sibi  arrogat,  &  passim  novis  vcstit  nominibus  ne  Newtonum 
sequi  videatur.  Quod  prius  feccrat  cum  Moutono  hoc 
postea  facere  cum  Newtono  non  dubitavit.  Noverat  etiam 
mcthodum  serierum  infinitarum  a  Newtono  inventam  fuisse 
et  hujus  methodi  adminiculo  Gregorium  ineunte  anno  1671 
in  sericm  pro  arcu  ex  tangente  incidisse  et  tamen'  banc 
seriem  ut  suam  in  Actis  Lipsicis  Anno  1682  magnifice  in 
lucem  edidit. 


•     No.  XXXIII. 

Seo  Synoptical  View  of  No-ytni's  Life  under  date  1713  Nov. 

S"^  Isaac  Newton  represents  that  he  did  formerly  dis- 
course w*^  your  LordP  about  the  ancient  year  of  360  days, 
&  represented  to  yo'^  Lord^  that  it  was  the  Kalendar  of  the 
ancient  Lunisolar  year  composed  of  the  nearest  round  num- 
ber of  Lunar  months  in  a  year  &  days  in  a  Lunar  month : 
that  the  ancients  corrected  this  Kalendar  monthly  by  the  new 
moons  &  yearly  by  the  returns  of  the  four  seasons,  drop- 
ping a  day  or  two  when  they  found  the  Kalendar  month  of 
30  days  too  long  for  the  return  of  the  Moon,  and  adding 
a  month  to  the  end  of  the  year  when  they  found  the  year 
of  12  Lunar  months  too  shor£  for  the  return  of  the  seasons 
&  fruits  of  the  earth :  that  Moses  in  describing  the  flood 
uses  the  Kalendar  months  not  corrected  by  the  course  of 
the  INIoon,  the  cloudy  rainy  weather  not  suffering  her  then 
to  appear  to  Noah :  that  when  Herodotus  or  any  other 
author  reccons  30  days  to  the  months  &  360  days  to 
y®  year,  he  understands  the  Kalendar  month  &  year  with- 
out correcting  them  by  the  courses  of  the  Sun  and  Moon : 
that  when  Herodotus  reccons  by  years  of  12  &  13  months 
alternately  for  70  years  together,  he  understands  the  Diet- 
eris  of  the  ancients  continued  70  years  without  correcting 


Append.]  OF  THE  YEAR.  315 

it  by  the  Luminaries :  &  that  when  we  read  of  a  week  or 
a  month  or  a  year  consisting  of  any  other  days  then  the 
natural,  we  are  to  reecon  7  days  or  30  days  or  3G0  days 
aecording  to  the  Kalendar  because  where  the  days  are  not 
natural  ones  the  Kalendar  cannot  be  corrected  by  the 
courses  of  the  Sun  &  Moon;  and  if  the  days  be  taken 
mystically  for  the  years  of  any  nation,  we  are  to  take  these 
years  in  the  vulgar  sense  for  7  or  30  or  3G0  practical  years 
of  that  nation  such  as  they  commonly  use  in  their  civil 
affairs.  S'^  Isaac  saith  further  that  he  meets  w*^  nothing  in 
yo'^  LordP^  paper  w*^'*  in  his  opinion  makes  against  what  he 
tlreu  represented  to  y^""  Lord*",  that  Suidas  (in  Sa^i)  tells 
us  that  y®  months  of  the  Chaldees  were  Lunar,  their  cv 
dinary  years  composed  of  12  Lunar  months,  and  the 
Sarus  composed  of  18  such  years  &  six  months,  w*^**  months 
he  takes  to  be  intercalary  (the  end  of  all  cycles  of  years 
being  to  know  when  to  intercale  the  months  of  y*  Luni- 
solar  year  for  keeping  the  year  to  the  seasons;)  &  that 
Censorinus  mentions  a  Chaldean  cycle  of  12  years,  &  y*  the 
Jews  in  returning  from  captivity  called  their  own  months 
by  the  names  of  the  Chaldean,  &  that  the  feast  Sacca*  of 
the  Babylonians  was  celebrated  on  y*  16*^  day  of  a  Lunar 
month  &  kept  to  the  same  season  of  y®  year,  &  that  in  all 
antiquity  he  meets  w*^  no  other  sorts  of  years  than  the 
Luni-solar  the  Solar  &  the  Lunar,  &  their  Calendars  & 
cycles  f.    • 


■  AthenaEus  xiv.  639. 

t  From  the  original  in  Brit.  Mus.  Add.  MSS.  6489.  fol.  < 


316  LETTER  TO  LORD  TOWNSHEND.  [Append. 

No.  XXXIV. 

NEWTON  TO  LORD  TOWNSHEND. 

Containing  an  opinion  of  Bomo  value  in  connexion  with  the  subject  of 
capital  punishments. 

My  Lord 
I  know  nothing  of  Edmund  Metcalf  convicted  at  Derby 
assizes  of  counterfeiting  the  coyne ;  but  since  he  is  very 
evidently  convicted,  I  am  humbly  of  opinion  that  its  better 
to  let  him  suffer,  than  to  venture  his  going  on  to  counter- 
feit the  coin  &  teach  others  to  do  so  untill  he  can  be  con- 
victed again,  ffor  these  people  very  seldom  leave  off.  And 
its  difficult  to  detect  them.  I  say  this  with  most  humble 
submission  to  his  Maj^  pleasure  &  remain 

My  Lord 
your  LordP''  most  humble  &  obedient  Servant 
Mint  office  Aug.  25,  1724.  Is.  Newton*- 

i**.  Tawnshend  {Secretary  of  State}, 


*  From  a  copy  communicated  by  P.  O'Callaghan,  Esq.    The  origiDal  is  in  the 
possession  of  I\l.  A.  Donnadieu. 


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2  Select  List,  No.  II.,  of  Works 


History  of  the  Royal  Society.     Compiled  from  Original  Authentic 

icuments.    By  C.  R.  Weu   "      "^       ^*  A-f-x.-iO       ^       x.  al.  t»-_.i  c.-.:.*_ 

Two  Volumes,  Octavo.    30a. 


Documents.    By  C.  R.  Weld,  Barrister-at-law ;  Assistant  Secretary  to  the  Royul  Society. 
I.  Oc 


Elements  of  Meteorology.     By  the  late  Professor  Daniell,  D.C.L., 

For.  Sec.  R.S.    Two  Volumes,  Octavo,  with  Plates  and  coloured  Charts.    32*. 

Daniell's  Introduction  to  the   Study  of  Chemical  Philosophy. 

With  numerous  Illustrations.    Second  Edition,  much  Enlarged,  21». 

A  Cycle  of  Celestial  Objects.     By  Captain  W.  H.  Smyth,  R.N., 

K.S.F.,  D.C.L.,  F.U.S.,  President  of  the  Astronomical  Society.    Two  Volumes,  Octavo, 
with  numerous  Illustrations.    21.  28. 


Views  of  the  Architecture  of  the  Heavens.    By  J.  P.  Nichol,  LL.D., 

Professor  of  Practical  Astronomy  in  the  University  of  Glasgow.  A  New  Edition.  Illus- 
trated with  large  Plates  of  Clusters  and  Nebulee,  recently  discovered  by  means  of  the  great 
Telescope  of  Lord  Rosse,  and  Sketches  by  the  late  David  Scott.    Royal  Octavo,  21«. 


A  Manual  of  Chemistry.     By  W.  T.  Brande,  F.R.S.,  Professor 

of  Chemistry  at  the  Royal  Institution.     Sixth  Edition,  greatly  Enlarged,  and  Embracing 
all  New  Facts  of  the  Science.    Two  large  and  closely-printed  Volumes.    45*. 

Vol.  I.  contains.  History  of  Chemical  Philosophy — the  Powers  of  Matter  and  the 

Non-Metallic  Elements — Chemistry  of  the  Metals. 
Vol.  II.  Organic  Chemistry— Appendix  of  Weights  and  Measures — and  a  complete 

Index,  giving  more  than  11,000  references. 

Brande's  Dictionary  of  Materia  Medica  and  Pharmacy.  8vo,  155. 
Brande*s  Tables  of  Chemical  Equivalents,  Weights,  Measures,  &c. 

On  four  large  sheets,  3*.  6d. 

Lectures  on  the  Principles  and  Practice  of  Physic  ;  delivered  at 

King's  College,  London.    By  T.  Watson,  M.D.,  Fellow  of  the  Royal  College  of  Physi- 
cians.   Two  Volumes.    Octavo.     Third  Edition,  revised.    34*. 


Lectures  on  Dental  Physiology  and   Surgery,   delivered  at  the 

Middlesex  Hospital  Medical  School.  13y  John  Tomes,  Surgeon-Dentist  to  the  Middlesex 
Hospital.    Octavo,  with  upwards  of  One  Hundred  Illustrations,  I2s. 

Physiological  Anatomy  and  Physiology  of  Man.     By  K.  B.  Todd, 

M.D.,  F.K.S.,  and  W.  Bowman,  F.K.S.,  of  King's  College,  London.  With  numerous 
Original  Illustrations.  Part  III.,  Octavo,  7*. ;  also  the  First  Volume,  ISs.  cloth.  To  be 
completed  in  Four  Parts,  forming  Two  Volumes. 


The  Nervous  System  and  its  Functions.     By  Herbert  Mayo, 

F.R.S.,  late  Senior  Surgeon  of  the  Middlesex  Hospital.    Post  Octavo,  6*.  6d.  • 

Mayo  on  the  Management  of  the  Organs  of  Digestion  in  Health 

and  in  Disease.    Second  Edition,  6*.  6d, 

On  Spasm,  Languor,  and  Palsy.     By  A.  J.  Wilson,  M.D.,  Phy- 
sician to  St.  George's  Hospital.    Post  Octavo*  7*. 

On   Gout,    llheumatic   Fever,    and    Chronic   Kheumatism.      By 

11.  B.  Toi>D,  M.D.,  F.R.S.,  Professor  of  Physiology  in  King's  College,  London.    7*.  Gd. 


Published  by  John  W.  Parker.     .  3 

Lunacy  and  Lunatic  Life :  with  Hints  on  the  Personal  Caro  and 

Management  of  those  afflicted  with  Temporary  or  Permanent  Derangement.    By  tho  late 
Medical  Supsrintendemt  or  an  Asylum  tor  tub  Imsans.    3i.  Gd. 


On  the  Nature  of  Thunderstorms,  and  on  the  Means  of  Pro- 
tecting Churches  an'l  other  Buildings,  and  Shipping,  against  tho  destructifo  effects  of 
Lightning.    By  Sir  W.  Snow  IIahuis,  F.ll.S.    Octavo,  10«.  Gd, 

Practical  Geology  and  Mineralogy,  and  the  Chemistry  of  Metals. 

By  JosuuA  Tkimmkr,  F.G.S.    Octavo,  with  200  Illustrations,  12*. 

Trimmer's  Practical   Chemistry  for  Farmers  and  Landowners. 

Post  Octavo,  5s. 

Minerals  and  their  Uses;  in  a  Series  of  Letters  to  a  Lady.     By 

J.  R.  Jackson,  F.ll.S.    AVith  a  Coloured  Frontispiece.    7*.  Gd. 

Practical    Geodesy,   comprising    Chain   Sui  .eying,   the   Use    of 

Surveying  Instruments,  Levelling,  Trigonometrical,  Mining,  and  Maritime  Siirvcying.  By 
Butler  Williams,  C.E.,  F.G.S.  New  Edition,  with  Additional  Chapters  on  Estate, 
Parochial  and  llailroad  Surveying.    Octavo,  with  Illustrations.     12«.  Gd. 

Williams'.  Manual  of  Model-Drawing  from  Solid  Forms;  with  a 

Popular  View  of  Perspective.  Octavo,  with  Shaded  Engravings  of  tho  !Mode)s,  and  Wood 
Cuts.    I5s.    Published  under  the  Sanction  of  the  Committee  of  the  Council  on  Education. 


Principles  of  Mechanism.     By  R.  Willis,  M.A.,  F.R.S.,  Jack- 

sonian  Professor  of  Natural  Philosophy,  Cambridge.    With  250  Wood  Cuts.    Octavo,  15*. 

WiLLis/s  Architectural  History  of  the  Holy  Sepulchre.    Reprinted 

from  Williams'  Hol^  City.     Illustrated  from  original  drawings.     Octavo,  9*. 


An  Elementary  Treatise  on  the  Differential  and  Integral  Calculus. 

By  T.  G.  Hall,  M.A.,  Professor  of  Mathematics  in  King's  College,  London.    Fourth 
Edition,  enlarged.    Octavo,  12*.  Gd. 

IIall*s  Elements  of  Descriptive   Geometry,  chiefly  intended  for 

Students  in  Engineering.    With  80  Illustrations.    6*.  Gd. 

A  Treatise  on  the  Motion  of  a  Rigid  Body.     By  W.  N.  Griffin, 

B.D.,  Fellow  of  St.  John's  College,  Cambridge.     Octavo,  Gs.  Gd. 

Elements  of  Euclid,  from  the  Text  of  Simson.  With  Notes,  Geo- 
metrical Exercises,  Examination  Papers,  and  a  History  of  Geometry.  By  U.  Potts,  M.A. 
The  College  Edition,  Octavo,  10*. 

Mechanics  applied  to  the  Arts.     By  H.  Moseley,  M.A.,  F.R.S., 

one  of  Her  Majesty's  Inspectors  of  Schools.    Third  Edition,  Gs.  Gd. 

Moseley's  Lectures  on  Astronomy,  delivered  at  King's  College, 

London.    Third  Edition,  5s.  Gd. 

Mathematical  Tracts.     By  G.  B.  Airy,  M.A.,  F.R.S.,  Astronomer 

Royal.     Octavo,  Third  Edition,  15.v. 

Mathematical  Tracts.     By  M.  O'Bkien,  M.A.,  F.R.S.,  Professor 

of  Natural  Pliilosophy  in  King's  College,  London.  4*.  Gd, 


4         Select  Works  published  by  John  W.  Parker. 

By  Direction  of  (he  Lords  Commissioners  of  the  Admiralty, 

Descriptive  Geometry;   and   its   Application   to   Ship   Building. 

Arranged  for  the  School  of  Naval  Construction,  Portsmouth  Dockyard,  and  adapted  for 
the  Use  of  Students  in  the  Universities  and  Civil  Engineers.  By  Joseph  Woolley,  M.A., 
LL.D.,  Principal.  Part  I. :  Elements  of  Descriptive  Geometry.  Octavo,  with  Numerous 
Large  Plates,  20j. — Part  II. :  The  Application  of  Descripie  Geometry  to  the  Laying-off 
of  Ships.  

By  W.  WHEWiELL,  D.D.,  Master  of  Trinity  College,  Cambridge. 

Of  a  Liberal  Education  in  general,  and  with  particular  reference 

to  the  leading  Studie  of  the  University  of  Cambridge.    Octavo,  9«. 

The  Principles  of  English  University  Education.  Second  Edition,  bs. 
Architectural  Notes  en  German  Churches.     Third  Edition,  with 

Additions,  12«. 

The  Doctrine  of  Limits,  with  its  Applications ;  namely,  the  First 

Three  Sections  of  Newton — Conic  Sections — the  Differential  Calculus.    Octavo,  9«. 

Mechanics  of  Engineering.    Octavo,  9^. 

^lechanical  Euclid,  containing  the  Elements  of  Mechanics   and 

Hydrostatics  demonstrated  after  the  manner  of  the  Elements  of  Geometry,  with  Remarks 
on  Mathematical  Ileasoning.  Fifth  Edition,  (ca,refully  adapted  to  the  ordinary  B.A.  Exa- 
mination.)   5*. 

Conic  Sections:  their  principal  Properties  proved  Geometrically. 

Second  Edition.    Is.  Qd. 

The  Propositions    in  Mechanics  and  Hydrostatics   required,    at 

Cambridge,  of  Questionists,  not  Candidates  for  Honours ;  with  Illustrations  and  Examples. 
By  A.  C.  Babbett,  M.A.     Octavo,  7s. 

Solutions  of  Geometrical  Problems  proposed  at  St.  John's  College, 

Cambridge,  consisting  chiefly  of  Examples  in  Plane  Co-ordinate  Geometry.  By  T. 
Gaskjn,  M.A.,  late  Fellow  and  Tutor  of  Jesus  College,  Cambridge.     12*. 

Newton's  Principia.     Books  I.  IL  III.     In  Latin ;  with  Notes 

and  References.    Edited  by  Professor  Wiieweix,  D.D.     2s.  (yd. 

Lectures  upon  Trigonometry,  and  the  Application  of  Algebra  to 

Geometry.     Second  Edition,  corrected.     7«.  6^?. 

An  Analytical  System  of  Conic  Sections.     By  H.  P.  Hamilton, 

M.A.,  F.RS.,  Dean  of  Salisbury.    Fourth  Edition,  revised,  10*. 


The  Undulatory  Theory,  as  applied  to  the  dispersion  of  Light. 

By  B.Powell,  M.A.,  F.R.S.,  Savilian  Professor,  Oxford.     Octavo,  9*.,  with  a  Coloured 
Chart  of  the  Prismatic  Spectra. 

Theory  of  Heat.     By  Professor  Kelland,  M.A.,  F.R.S.,  of  the 

University  of  Edinburgh,  9*.    

The  Dangers  of  Superficial  Knowledge.     An  Introductory  Lecture. 

By  James  D.  Fobbes,   F.ll.S.,    Professor  of  Natural  Philosophy  in  the  University  of 
Edinburgh.     2*.  ^/^  ^ 

Davies*s  Estimate  of  the  Human  Mind :  a  Philosophical  Inquiry 

into  the  legitimate  Application  and  Extent  of  its  leading  Faculties,  as  connected  with  the  Prin- 
ciples and  Obligations  of  the  Christian  Keligion.  Second  Edition,  with  Additions.   8vo,  14*. 


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