a
-----
b + c
-----
d + e
-
f + etc.,
be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as
falling down to the side of the preceding sign +. In every such fraction we
may suppose b, d, f, etc. {368} positive; a, c, e, &c. being as required:
and all are supposed integers. If this succession be continued ad
infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive,
the limit of the fraction must be incommensurable with unity; that is,
cannot be A/B, where A and B are integers.
First, whatever this limit may be, it lies between -1 and +1. This is
obviously the case with any fraction p/(q + [omega]), where [omega] is
between +-1: for, p/q, being < 1, and p and q integer, cannot be brought up
to 1, by the value of [omega]. Hence, if we take any of the fractions
a/b, a/b+ c/d, a/b+ c/d+ e/f, etc.
say a/b+ c/d+ e/f+ g/h we have, g/h being between +-1, so is e/f+ g/h, so
therefore is c/d+ e/f+ g/h; and so therefore is a/b+ c/d+ e/f+ g/h.
Now, if possible, let a/b+ c/d+ etc. be A/B at the limit; A and B being
integers. Let
P = A c/d+ e/f+ etc., Q = P e/f+ g/h+ etc., R = Q g/h + i/k + etc.
P, Q, R, etc. being integer or fractional, as may be. It is easily shown
that all must be integer: for
{369}