364
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
for integer n, and observe the simple relations between the neighbouring
terms, then to interpolate the cases n = 1/2, 3/2, etc. starting from these
relations. A natural idea to him: during the English Civil War Wallis rendered
eminent service to Cromwell's party by decrypting a quantity of ciphered
messages exchanged by the other camp; now, when one speaks of "ciphered"
messages, one is almost always dealing, then as now, with replacing letters or
groups of letters by numbers according to generally very simple rules, and one
needs to discover them; "interpolation" and "extrapolation" are operations
to which the decrypters have constant recourse. This does not prove that the
calculation of the integrals which interested Wallis will obey the same rules,
but he tried, and, though not succeeding, discovered his famous formula for 71";
in this way he opened Newton's path to the binomial formula which enabled
him to calculate the integral of (1 - x 2 )1/2 trivially, the same formula that
Newton also obtained by clever interpolations starting from the binomial
coefficients for integer exponents.
Wallis also sought to calculate the area of a segment of hyperbola, but,
instead of the curve xy = a 2 , chose the equation y2 - x 2 = a 2 which so closely
resembles that of a circle - except for an inoffensive change of sign - that
one might hope to resolve the problem; it resisted him. Wallis spoke of this
in 1655 to one of his friends, Lord Brouncker, who had studied mathematics
at Oxford, occupied very high positions in government, was interested in the
sciences, and became the first president of the Royal Society in 1662. The
latter calculated the area of the hyperbola xy = 1 contained between x = 1
and x = 2 by the usual method, but in a markedly more ingenious way: he
decomposed [1,2] into equal intervals of length 1/2 n and grouped the partial
areas obtained to conclude that log 2 = 1/1.2 + 1/3.4 + 1/5.6 + ....
For his part, the Italian Pietro Mengoli apparently published the series
in Bologna, in a book of 1659 where he gives an almost rigorous definition of
the log of a rational number, inspired by the area of the hyperbola, though
he did not say this explicitly. To define log(a/b) where a, b are integers such
that b < a, he introduced, up to notation, the sums
L:;'(a/b) = L l/p, L~(a/b) = L l/p
bn
pin with p integer, restricts to those which lie between a and b, i.e. such that
bn ~ p ~ an, and then approximates the area of the hyperbola y = l/x lying
between the verticals at a and b by the vertical rectangles having for bases
the intervals of length l/n in question, etc.
This done, Mengoli proved that these sums possess the following properties: (i) as n increases, the sums L - increase and the sums L + decrease,
(ii) the first are smaller than the second (obvious), (iii) the difference between
the second and the first tends to 0 as n increases (obvious). He then defined
log (a/b) by imposing the condition that it satisfy L;.(a/b) ~ log(a/b) ~
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
for integer n, and observe the simple relations between the neighbouring
terms, then to interpolate the cases n = 1/2, 3/2, etc. starting from these
relations. A natural idea to him: during the English Civil War Wallis rendered
eminent service to Cromwell's party by decrypting a quantity of ciphered
messages exchanged by the other camp; now, when one speaks of "ciphered"
messages, one is almost always dealing, then as now, with replacing letters or
groups of letters by numbers according to generally very simple rules, and one
needs to discover them; "interpolation" and "extrapolation" are operations
to which the decrypters have constant recourse. This does not prove that the
calculation of the integrals which interested Wallis will obey the same rules,
but he tried, and, though not succeeding, discovered his famous formula for 71";
in this way he opened Newton's path to the binomial formula which enabled
him to calculate the integral of (1 - x 2 )1/2 trivially, the same formula that
Newton also obtained by clever interpolations starting from the binomial
coefficients for integer exponents.
Wallis also sought to calculate the area of a segment of hyperbola, but,
instead of the curve xy = a 2 , chose the equation y2 - x 2 = a 2 which so closely
resembles that of a circle - except for an inoffensive change of sign - that
one might hope to resolve the problem; it resisted him. Wallis spoke of this
in 1655 to one of his friends, Lord Brouncker, who had studied mathematics
at Oxford, occupied very high positions in government, was interested in the
sciences, and became the first president of the Royal Society in 1662. The
latter calculated the area of the hyperbola xy = 1 contained between x = 1
and x = 2 by the usual method, but in a markedly more ingenious way: he
decomposed [1,2] into equal intervals of length 1/2 n and grouped the partial
areas obtained to conclude that log 2 = 1/1.2 + 1/3.4 + 1/5.6 + ....
For his part, the Italian Pietro Mengoli apparently published the series
in Bologna, in a book of 1659 where he gives an almost rigorous definition of
the log of a rational number, inspired by the area of the hyperbola, though
he did not say this explicitly. To define log(a/b) where a, b are integers such
that b < a, he introduced, up to notation, the sums
L:;'(a/b) = L l/p, L~(a/b) = L l/p
bn
pin with p integer, restricts to those which lie between a and b, i.e. such that
bn ~ p ~ an, and then approximates the area of the hyperbola y = l/x lying
between the verticals at a and b by the vertical rectangles having for bases
the intervals of length l/n in question, etc.
This done, Mengoli proved that these sums possess the following properties: (i) as n increases, the sums L - increase and the sums L + decrease,
(ii) the first are smaller than the second (obvious), (iii) the difference between
the second and the first tends to 0 as n increases (obvious). He then defined
log (a/b) by imposing the condition that it satisfy L;.(a/b) ~ log(a/b) ~