t have come in somehow, though at what stage or under
what name must have depended upon the casualties of algebraical invention.
This will readily be seen when it is stated that [pi] is nothing but four
times the series

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...

_ad infinitum_.[616] It would be wonderful if so simple a series {285} had
but one kind of occurrence. As it is, our trigonometry being founded on the
circle, [pi] first appears as the ratio stated. If, for instance, a deep
study of probable fluctuation from average had preceded, [pi] might have
emerged as a number perfectly indispensable in such problems as: What is
the chance of the number of aces lying between a million + x and a million
- x, when six million of throws are made with a die? I have not gone into
any detail of all those cases in which the paradoxer finds out, by his
unassisted acumen, that results of mathematical investigation _cannot be_:
in fact, this discovery is only an accompaniment, though a necessary one,
of his paradoxical statement of that which _must be_. Logicians are
beginning to see that the notion of _horse_ is inseparably connected with
that of _non-horse_: that the first without the second would be no notion
at all. And it is clear that the positive affirmation of that which
contradicts mathematical demonstration cannot but be accompanied by a
declaration, mostly overtly made, that demonstration is false. If the
mathematician were interested in punishing this indiscretion, he could make
his denier ridiculous by inventing asserted results which would completely
take him in.

More than thirty years ago I had a friend, now long gone, who was a
mathematician, but not of the higher branches: he was, _inter alia_,
thoroughly up in all that relates to mortality, life assurance, &c. One
day, explaining to him how it should be ascertained what the chance is of
the survivors of a large number of persons now alive lying between given
limits of number at the end of a certain time, I came, of course upon the
introduction of [pi], which I could only describe as the ratio of the
circumference of a circle to its diameter. "Oh, my dear friend! that must
be a delusion; what can the circle have to do with the numbers alive at the
end of a given time?"--"I cannot demonstrate it to you; but it is
demonstrated."--"Oh! stuff! I think you can prove anything with your
differential calculus: figment, {286} depend upon it." I said no more; but,
a few day