§2. Series expansions
365
Lt,(a/b) for any n, an impeccable idea in view of properties (i), (ii) and (iii)
above. Having done this, he proved that if alb, c/d > 1 are two fractions,
then
L~ia/b) + L;;a(c/d) = L;;(ac/bd),
which (for us, with our modern notation ... ) is easy: in the first expression
the index of summation p varies between bnd and and, in the second between
dna = and and cna, and in third between bdn and acn = cna. As n increases,
the sequence L~d(a/b), extracted from the sequence L:;;,(a/b), tends to the
same limit log (a/b) as the latter, with a similar remark for the sequence
relative to c/d. The preceding relation then shows Mengoli that his definition
satisfies the condition log(ac/bd) = log(a/b) + log(c/d). Finally, the limits of
the "Riemann" sums provides him less attractive series expansions, though,
for a = 2, b = 1, they lead to the alternating harmonic series. In this case
one must seek the limit of the sum
clearly equal to 1 - 1/2 + 1/3 - 1/4 + ... - 1/(2n - 2) + 1/(2n - 1), which,
in the limit, yields Brouncker's series.
It is hard not to deduce from these arguments that Mengoli, and those
of his contemporaries who performed the same kinds of calculation, already
had a clear enough idea of what a real number and an integral are.
After Mengoli, we arrive at Newton and Mercator. From 1664, reading the
Geometrie of Descartes in the Latin translation published by a Dutch algebraist, and, even more, reading Wallis' Arithmetica Infinitorum, led Newton
to reflect, and a year later to the first case of his binomial formula. At the
same time, and by the same method - expand in a power series and integrate term-by-term - to attack the hyperbola y = 1/(1 + x), he wrote
y = 1 - x + x 2 - x 3 + ... and deduced, from the rule xm ---+ x m + 1 /(m + 1)
which he found in Wallis but, in contrast to him, applied to an arbitrary abscissa x, that the area contained between the verticals 0 and x is equal to
x - x 2 /~ + x 3 /3 - x4 /4 + ... This result for x = 1 yields Brouncker's formula,
though Newton does not mention this.
The mystery is that none of the authors whom we have mentioned above,
except perhaps Saint-Vincent, made explicit the connection between calculating this area and the log of the tables. Newton contented himself, in a
manuscript of 1667, with affirming in a phrase that "the areas are to their
abscissae as the logarithms {i. e. when the abscissae increase in geometric progression, the areas increase in arithmetic progression)". The idea was doubtless already a part of the English mathematical folklore of the time.
And Newton began to embark on numerical calculations of which, later,
he would be little ashamed in front of Leibniz. Successively he took x =
1/10,2/10, -1/10, -2/10,1/100, etc. in his series and calculated the results
to 52 decimal places, committing several small errors. This is not as difficult as
365
Lt,(a/b) for any n, an impeccable idea in view of properties (i), (ii) and (iii)
above. Having done this, he proved that if alb, c/d > 1 are two fractions,
then
L~ia/b) + L;;a(c/d) = L;;(ac/bd),
which (for us, with our modern notation ... ) is easy: in the first expression
the index of summation p varies between bnd and and, in the second between
dna = and and cna, and in third between bdn and acn = cna. As n increases,
the sequence L~d(a/b), extracted from the sequence L:;;,(a/b), tends to the
same limit log (a/b) as the latter, with a similar remark for the sequence
relative to c/d. The preceding relation then shows Mengoli that his definition
satisfies the condition log(ac/bd) = log(a/b) + log(c/d). Finally, the limits of
the "Riemann" sums provides him less attractive series expansions, though,
for a = 2, b = 1, they lead to the alternating harmonic series. In this case
one must seek the limit of the sum
clearly equal to 1 - 1/2 + 1/3 - 1/4 + ... - 1/(2n - 2) + 1/(2n - 1), which,
in the limit, yields Brouncker's series.
It is hard not to deduce from these arguments that Mengoli, and those
of his contemporaries who performed the same kinds of calculation, already
had a clear enough idea of what a real number and an integral are.
After Mengoli, we arrive at Newton and Mercator. From 1664, reading the
Geometrie of Descartes in the Latin translation published by a Dutch algebraist, and, even more, reading Wallis' Arithmetica Infinitorum, led Newton
to reflect, and a year later to the first case of his binomial formula. At the
same time, and by the same method - expand in a power series and integrate term-by-term - to attack the hyperbola y = 1/(1 + x), he wrote
y = 1 - x + x 2 - x 3 + ... and deduced, from the rule xm ---+ x m + 1 /(m + 1)
which he found in Wallis but, in contrast to him, applied to an arbitrary abscissa x, that the area contained between the verticals 0 and x is equal to
x - x 2 /~ + x 3 /3 - x4 /4 + ... This result for x = 1 yields Brouncker's formula,
though Newton does not mention this.
The mystery is that none of the authors whom we have mentioned above,
except perhaps Saint-Vincent, made explicit the connection between calculating this area and the log of the tables. Newton contented himself, in a
manuscript of 1667, with affirming in a phrase that "the areas are to their
abscissae as the logarithms {i. e. when the abscissae increase in geometric progression, the areas increase in arithmetic progression)". The idea was doubtless already a part of the English mathematical folklore of the time.
And Newton began to embark on numerical calculations of which, later,
he would be little ashamed in front of Leibniz. Successively he took x =
1/10,2/10, -1/10, -2/10,1/100, etc. in his series and calculated the results
to 52 decimal places, committing several small errors. This is not as difficult as