I think it right to give the proof that the ratio of the circumference to
the diameter is incommensurable. This method of proof was given by
Lambert,[672] in the _Berlin Memoirs_ for 1761, and has been also given in
the notes to Legendre's[673] Geometry, and to the English translation of
the same. Though not elementary algebra, it is within the reach of a
student of ordinary books.[674]

Let a continued fraction, such as

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}

A/B = a/b+ P/A, or, P = aB - bA

P/A = c/d+ Q/P, or, Q = cA - dP

Q/P = e/f+ R/Q, or, R = eP - fQ

etc., etc. Now, since a, B, b, A, are integers, so also is P; and thence Q;
and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. are all between -1
and +1, it follows that the unlimited succession of integers P, Q, R, are
each less in numerical value than the preceding. Now there can be no such
_unlimited_ succession of _descending_ integers: consequently, it is
impossible that a/b+ c/d+, etc. can have a commensurable limit.

It easily follows that the continued fraction is incommensurable if a/b,
c/d, etc., being at first greater than unity, become and continue less than
unity after some one point. Say t