340 A. ©. Twining—Fuclid’s Doctrine of Parallels. 
the third angle. Therefore, it was argued, this third angle is a 
function of the two angles and the side—but a function into 
which the side cannot in fact enter, because a (ine cannot enter 
into the composition of an angle,—on which account the two 
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point, whatever the bases, because they so meet with the assumed 
base. On this assumption the entire doctrine would follow 
apace from the ordinary rudiments of geometry. 
Mr. T. P. Thompson, the author before alluded to, has 
himself introduced into his above-mentioned modification of 
W X, of unlimited length both ways, travel along the axis from 
the vertex A toward Z, till it cuts the axis in M; and it has 
been shown (28p, 1 Cor.) that during such travel it cannot cease 
to cut the series, &¢.” The fatal objection is that W X 1s 80 
restricted by the conditions of its cutting that, although ever 
approaching the point M, it cannot be proved capable of reach- 
e grounds of this objection will be made yet more 
clear in the recasting of Mr. Thompson’s proof, which follows. 
