§2. Series expansions
363
segments DE and PQ are in the ratio qm, and the segments HI and KG in
the ratio qn, then the ratio between the areas DEQP and HIKG is equal
to min, which comes down to saying that when the segment EQ takes the
values of a geometric progression, the area DEPQ varies according to an
arithmetic progression, an idea that one will find again in Newton, who does
not however appear to have read Saint-Vincent, in contrast to Leibniz. Napier
would have immediately deduced that this area is proportional to the log of
EQ since this is precisely his method of constructing log.
fig.!.
Saint-Vincent's method is the better since, instead of decomposing the
segment EQ using an arithmetic progression like Archimedes or Cavalieri, he
decomposes into segments (qi, qi+ 1 ); the ordinate of the point of the hyperbola y = 1 I x with abscissa qi is equal to 1 I qi, the area of the slice of the
hyperbola contained between the verticals qi and qi+l is contained between
i.e. between q - 1 and (q - 1) I q; since there are m intervals (qi, qi+l) between
E and Q, the area of hyperbola EQ P D must lie between m( q - 1) and
m(q-l)lq; further, that of IGKH lies between n(q-l) and n(q-l)lq, which
yields the proportionality sought. As we saw at the end of nO 11 of Chap. II,
this calculation can yield much more: it is not difficult, three hundred years
later, to be cleverer than a Jesuit of 1647.
Then one finds the English. In the innocent hope of obtaining a beautiful
explicit formula for calculating 7f, John Wallis calculated total areas, i.e. the
integrals over the interval (0,1) of the curves y = (1 - xl/m)n, where m
and n are natural integers: it is enough to expand and take account of the
fact, which he extrapolates starting from several already known particular
cases, that for r rational the integral of xr on the interval (0, 1) is equal to
I/(r + 1). This procedure unfortunately does not apply to calculating the
area of the circle, since in this case n = 1/2 is not integer. Wallis' idea,
which Newton exploited successfully, was to draw up a table of integrals
363
segments DE and PQ are in the ratio qm, and the segments HI and KG in
the ratio qn, then the ratio between the areas DEQP and HIKG is equal
to min, which comes down to saying that when the segment EQ takes the
values of a geometric progression, the area DEPQ varies according to an
arithmetic progression, an idea that one will find again in Newton, who does
not however appear to have read Saint-Vincent, in contrast to Leibniz. Napier
would have immediately deduced that this area is proportional to the log of
EQ since this is precisely his method of constructing log.
fig.!.
Saint-Vincent's method is the better since, instead of decomposing the
segment EQ using an arithmetic progression like Archimedes or Cavalieri, he
decomposes into segments (qi, qi+ 1 ); the ordinate of the point of the hyperbola y = 1 I x with abscissa qi is equal to 1 I qi, the area of the slice of the
hyperbola contained between the verticals qi and qi+l is contained between
i.e. between q - 1 and (q - 1) I q; since there are m intervals (qi, qi+l) between
E and Q, the area of hyperbola EQ P D must lie between m( q - 1) and
m(q-l)lq; further, that of IGKH lies between n(q-l) and n(q-l)lq, which
yields the proportionality sought. As we saw at the end of nO 11 of Chap. II,
this calculation can yield much more: it is not difficult, three hundred years
later, to be cleverer than a Jesuit of 1647.
Then one finds the English. In the innocent hope of obtaining a beautiful
explicit formula for calculating 7f, John Wallis calculated total areas, i.e. the
integrals over the interval (0,1) of the curves y = (1 - xl/m)n, where m
and n are natural integers: it is enough to expand and take account of the
fact, which he extrapolates starting from several already known particular
cases, that for r rational the integral of xr on the interval (0, 1) is equal to
I/(r + 1). This procedure unfortunately does not apply to calculating the
area of the circle, since in this case n = 1/2 is not integer. Wallis' idea,
which Newton exploited successfully, was to draw up a table of integrals