§2. Series expansions
367
Latin, which Newton accomplished in a few days; the manuscript, De Analysi
per aequationes numero terminorum infinitas, was in Barrow's possession at
the beginning of July 1669. Newton, who still lacked confidence, asked him to
send it, without naming the author, to John Collins, an official in the English
government who was greatly interested in science (Collins, not the government) and maintained an extensive correspondence with many scholars of
the age. Collins made a copy - the original was returned to Newton -, and
managed to extract the name of the author from Barrow in August "who,
with an unparalleled genius, has made very great progress in this branch of
mathematics" and showed his copy to Lord Brouncker, president of the Royal
Society, and to other English and continentals. The obvious thing to do to
safeguard Newton's priority was to publish his manuscript in the Philosophical Transactions (PT) of the Royal Society or, at least, to list it in the Reports
of its meetings, and moreover Brouncker published his series for log 2 in 1668,
in the same PT; nothing of kind happened; according to Moritz Cantor (III,
p. 68) neither Collins nor Brouncker had judged that the contents of the paper was worth so protecting ... Others mention Newton's tendency not to
publish work as yet uncompleted; it remained so, since after his first success
Newton distanced himself from mathematics, having judged it too arid 24 .
Nevertheless, the impact of the manuscript on Barrow was such - "J am
only a child beside him" - that, already wanting to return to theology, much
more prestigious, he resigned his post and recommended Cambridge to offer
it to Newton who, in the autumn of 1669, aged 27, obtained a chair and
lodgings for life!
The contents of the De Analysi - fifteen or so pages - is prodigious.
Newton started by stating, with examples, the rules of calculus of areas (i.e.
of integrals) for functions expandable in series of rational powers of x: one
replaces xm by xm+1/(m + 1) and adds the results. In the case where y is
given by a formula involving divisions and square roots one reduces to the
preceding case by standard operations: the quotient of two series of powers
of x, for example y = (2Xl/2 - x 3 / 2 )/(1 + Xl/2 - 3x) is calculated with the
help of the algorithm for decimal division, and, in the example that we have
just mentioned, yields,
y = 2Xl/2 - 2x + 7X 3 / 2 - 13x 2 + 34x 5 / 2 + ... ;
the square root is similarly obtained by applying the decimal algorithm, so
that if, for example, y = (a 2 - x 2 )1/2, one finds
y = a - x 2 /2a - x 4 /8a 3 - x 6 /16a 5 - ..• ,
which explained how Newton guessed - or confirmed - the first, crucial, line
of the table of coefficients of the functions (1 - x2)n/p.
24 On Newton, see the DSB, the large biography by Richard Westfall, Never at
Rest: a Biogmphy of Isaac Newton (CUP, 1980), A. Rupert Hall, Isaac Newton.
Adventurer in Thought (Cambridge, 1992) and Loup Verlet, La malle de Newton
(Gallimard, 1993) already mentioned.
367
Latin, which Newton accomplished in a few days; the manuscript, De Analysi
per aequationes numero terminorum infinitas, was in Barrow's possession at
the beginning of July 1669. Newton, who still lacked confidence, asked him to
send it, without naming the author, to John Collins, an official in the English
government who was greatly interested in science (Collins, not the government) and maintained an extensive correspondence with many scholars of
the age. Collins made a copy - the original was returned to Newton -, and
managed to extract the name of the author from Barrow in August "who,
with an unparalleled genius, has made very great progress in this branch of
mathematics" and showed his copy to Lord Brouncker, president of the Royal
Society, and to other English and continentals. The obvious thing to do to
safeguard Newton's priority was to publish his manuscript in the Philosophical Transactions (PT) of the Royal Society or, at least, to list it in the Reports
of its meetings, and moreover Brouncker published his series for log 2 in 1668,
in the same PT; nothing of kind happened; according to Moritz Cantor (III,
p. 68) neither Collins nor Brouncker had judged that the contents of the paper was worth so protecting ... Others mention Newton's tendency not to
publish work as yet uncompleted; it remained so, since after his first success
Newton distanced himself from mathematics, having judged it too arid 24 .
Nevertheless, the impact of the manuscript on Barrow was such - "J am
only a child beside him" - that, already wanting to return to theology, much
more prestigious, he resigned his post and recommended Cambridge to offer
it to Newton who, in the autumn of 1669, aged 27, obtained a chair and
lodgings for life!
The contents of the De Analysi - fifteen or so pages - is prodigious.
Newton started by stating, with examples, the rules of calculus of areas (i.e.
of integrals) for functions expandable in series of rational powers of x: one
replaces xm by xm+1/(m + 1) and adds the results. In the case where y is
given by a formula involving divisions and square roots one reduces to the
preceding case by standard operations: the quotient of two series of powers
of x, for example y = (2Xl/2 - x 3 / 2 )/(1 + Xl/2 - 3x) is calculated with the
help of the algorithm for decimal division, and, in the example that we have
just mentioned, yields,
y = 2Xl/2 - 2x + 7X 3 / 2 - 13x 2 + 34x 5 / 2 + ... ;
the square root is similarly obtained by applying the decimal algorithm, so
that if, for example, y = (a 2 - x 2 )1/2, one finds
y = a - x 2 /2a - x 4 /8a 3 - x 6 /16a 5 - ..• ,
which explained how Newton guessed - or confirmed - the first, crucial, line
of the table of coefficients of the functions (1 - x2)n/p.
24 On Newton, see the DSB, the large biography by Richard Westfall, Never at
Rest: a Biogmphy of Isaac Newton (CUP, 1980), A. Rupert Hall, Isaac Newton.
Adventurer in Thought (Cambridge, 1992) and Loup Verlet, La malle de Newton
(Gallimard, 1993) already mentioned.