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 that i/k, l/m,... are all less than unity.
Then the fraction i/k+ l/m+ ... is incommensurable, as proved: let it be
[kappa]. Then g/(h + [kappa]) is incommensurable, say [lambda]; e/(f +
[lambda]) is the same, say [mu]; also c/(d + [mu]), say [nu], and a/(b +
[nu]), say [rho]. But [rho] is the fraction a/b+ c/d+ ... itself; which is
therefore incommensurable.