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NEWTON, ISAAC (b. Woolsthorpe, England,
25 December 1642; d. London, England, 20 March
1727), mathematics, dynamics, celestial mechanics,
astronomy, optics, natural philosophy.
NOTES
16. L. T. More, Isaac Newton, p. 368.
17. See J. Edleston, ed., Correspondence . . . Newton and
. . . Cotes, pp. xxxvi, esp. n. 142.
18. Mathematical Papers of Isaac Newton, D. T. Whiteside,
ed., in progress, to be completed in 8 vols. (Cambridge,
1967- ); these will contain edited versions of Newton's
mathematical writings with translations and explanatory
notes, as well as introductions and commentaries that
constitute a guide to Newton's mathematics and scientific
life, and to the main currents in the mathematics of the
seventeenth century. Five volumes have been published
(1973).
19. See D. T. Whiteside, “Newton's Discovery of the General
Binomial Theorem,” in Mathematical Gazette, 45 (1961),
175.
20. Especially because of Whiteside's researches.
21. Whiteside, ed., Mathematical Papers, I, 1-142. Whiteside
concludes: “By and large Newton took his arithmetical
symbolisms from Oughtred and his algebraical
from Descartes, and onto them . . . he grafted new modifications
of his own” (I, 11).
22. Ca. 1714; see University Library, Cambridge, MS Add.
3968, fol. 21. On this often debated point, see D. T.
Whiteside, “Isaac Newton: Birth of a Mathematician,”
in Notes and Records. Royal Society of London, 19 (1964),
n. 25; but compare n. 48, below.
23. University Library, Cambridge, MS Add. 3968. 41, fol. 85.
This sentence occurs in a passage canceled by Newton.
24. Ibid., fol. 72. This accords with De Moivre's later statement
(in the Newton manuscripts recently bequeathed the
University of Chicago by J. H. Schaffner) that after
reading Wallis' book, Newton “on the occasion of a
certain interpolation for the quadrature of the circle,
found that admirable theorem for raising a Binomial to
a power given.”
25. Translated from the Latin in the Royal Society ed. of the
Correspondence, II, 20 ff. and 32 ff.; see the comments by
Whiteside in Mathematical Papers, IV, 666 ff. In the second
term, A stands for Pm/n (the first term), while in
the third
term B stands for (m/n) AQ (the second term), and so
on.
This letter and its sequel came into Wallis' hands and he
twice published summaries of them, the second time with
Newton's own emendations and grudging approval.
Newton listed some results of series expansion—coupled
with quadratures as needed—for z = r sin-1
[x/r] and the
inverse x = r sin[z/r]; the versed sine r(1 -
cos[z/r]); and
x = ez/b - 1, the inverse of z = b
log(1 + x), the Mercator
series (see Whiteside, ed., Mathematical Papers, IV,
668).
26. Translated from the Latin in the Royal Society ed. of the
Correspondence, II, 110 ff., 130 ff.; see the comments by
Whiteside in Mathematical Papers, IV, 672 ff.
27. See Whiteside, Mathematical Papers, I, 106.
28. Ibid., I, 112 and n. 81.
29. The Boothby referred to may be presumed to be Boothby
Pagnell (about three miles northeast of Woolsthorpe),
whose rector, H. Babington, was senior fellow of Trinity
and had a good library. See further Whiteside, Mathematical
Papers, I, 8, n. 21; and n. 8, above.
30. The Mathematical Works of Isaac Newton, I, x.
31. Ibid., I, xi.
32. Here the “little zero” o is not, as formerly, the
“indefinitely
small” increment in the variable t, which “ultimately
vanishes.” In the Principia, bk. II, sec. 2, Newton
used an alternative system of notation in which a, b,
c, ...
are the “moments of any quantities A, B, C,
&c.,” increasing
by a continual flux or “the velocities of the
mutations which are proportional” to those moments,
that is, their fluxions.
33. See Whiteside, Mathematical Works, I, x.
34. See A. R. and M. B. Hall, eds., Unpublished Scientific
Papers of Isaac Newton (Cambridge, 1962).
35. Mathematical Works, I, xi.
36. Ibid., xii.
37. University Library, Cambridge, MS Add. 3968.41,
fol. 86, v.
38. Whiteside, Mathematical Papers, II, 166.
39. Ibid., 166-167.
40. Ibid., I, 11, n. 27. where Whiteside lists those “known
to
have seen substantial portions of Newton's mathematical
papers during his lifetime” as including Collins, John
Craig, Fatio de Duillier, Raphson, Halley, De Moivre,
David Gregory, and William Jones, “but not, significantly,
John Wallis,” who did, however, see the “Epistola prior”
and “Epistola posterior” (see n. 25, above); and II, 168.
Isaac Barrow “probably saw only the De analysi.”
41. The Methodus fluxionum also contained an amplified
version of the tract of October 1666; it was published in
English in 1736, translated by John Colson, but was not
properly printed in its original Latin until 1779, when
Horsley brought out Analysis per quantitatum series,
fluxiones, ac differentias, incorporating William Jones's
transcript, which he collated with an autograph manuscript
by Newton. Various MS copies of the Methodus fluxionum
had, however, been in circulation many years before 1693,
when David Gregory wrote out an abridged version.
Buffon translated it into French (1740) and Castillon
used Colson's English version as the basis of a retranslation
into Latin (Opuscula mathematica, I, 295 ff.). In all these
versions, Newton's equivalent notation was transcribed
into dotted letters. Horsley (Opera, I) entitled his version
Artis analyticae specimina vel geometria analytica. The
full text was first printed by Whiteside in Mathematical
Papers, vol. III.
42. Mathematical Papers, II, 170.
43. P. xi; and see n. 41, above.
44. The reader may observe the confusion inherent in using
both “indefinitely small portions of time” and “infinitely
little” in relation to o; the use of index notation for powers
(x3, x2, o2) together
with the doubling of letters (oo) in the
same equation occurs in the original. These quotations
are from the anonymous English version of 1737, reproduced
in facsimile in Whiteside, ed., Mathematical
Works. See n. 46.
45. In this example, I have (following the tradition of more
than two centuries) introduced ? and ? where Newton
in
his MS used m and n. In his notation, too, r stood for
the
later ż.
46. Mathematical Papers, III, 80, n. 96. In the anonymous
English version of 1737, as in Colson's translation of
1736, the word “indefinitely” appears; Castillon followed
these (see n. 41). Horsley first introduced
“infinité.”
47. Ibid., pp. 16-17.
48. See Whiteside, ibid., p. 17; on Barrow's influence, see
further pp. 71-74, notes 81, 82, 84.
49. Ibid., pp. 328-352. On p. 329, n. 1, Whiteside agrees with a
brief note by Alexander Witting (1911), in which the
“source of the celebrated ‘fluxional’ Lemma II of the
second Book of Newton's Principia” was accurately
found in the first theorem of this addendum; see also
p. 331, n. 11, and p. 334, n. 16.
50. On this topic, see the collection of statements by Newton
assembled in supp. I to I. B. Cohen, Introduction to
Newton's Principia.
51. This and the following quotations of the De quadratura are
from John Stewart's translation of 1745.
52. As C. B. Boyer points out, in Concepts of the Calculus,
p. 201, Newton was thus showing that one should not
reach the conclusion “by simply neglecting infinitely small
terms, but by finding the ultimate ratio as these terms
become evanescent.” Newton unfortunately compounded
the confusion, however, by not wholly abjuring infinitesimals
thereafter; in bk. II, lemma 2, of the Principia he
warned the reader that his “moments” were not finite
