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.
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 + ω), where ω is between ±1: for, p/q, being < 1, and p and q integer, cannot be brought up to 1, by the value of ω. Hence, if we take any of the fractions