A. C. Twining—Euclid’s Doctrine of Parallels. 341 
Thompson’s Proof transformed and simplified. 
In our here abbreviated process, Mr. Thompson’s chain of 
proof will be presented unbroken (and in some parts even 
a fortiori) but in propositions involving altogether not one-third 
the bulk and labor of Mr. Thompson’s own method. This 
Statement includes our antecedent demonstration (p. 3) that no 
triangle can have the sum of tts angles greater than two right angles ; 
ut, aside from this, the proof is reduced in compass to one-tenth. 
It will be the purport of this our transformed process—equally 
as it was of Mr. Thompson’s process—to show that no triangle 
can have the sum of its angles less than two right angles. 
_¥or—looking back to the figure of our xxviii a—if ABC, 
right angled at A, is supposed to be such a triangle, and the 
quadrilateral (or tessera—so called by Thompson) be constructed 
as before, then, since the angles of BCD cannot exceed two 
right angles, the four angles of the tessera are less than four 
night angles, and the equal angles AC D, BDC, are each less 
than a right angle. Construct, as below, Thompson’s figure 
under his Caption xxviii E—and with the saine! designating 
letters—in which QM P is astraight line of bases on which the 
tesseras A on the 
8a polygon. Join BC, CD, DE; then ABC,=ACB, must 
han ACD,=BDC, even were ABD one line, and 
much more, as easily shown, for the angle at B. But BI 
/ 4 D, is greater than BDO, and much more than ACB. In 
like manner may it be shown by joining E F, FG, that the equal 
cusps HG F, D FG are greater than the cusps CED, BDE. In 
the same manner, also, if A Z is an axis perpendicular to B C— 
and easily shown to be normal also to ED, G F—it may be proved 
that if an indefinite line W X moves from A toward Z, keeping at 
Tight angles to the axis, it shall make the cusps formed by it at I 
and H greater than the preceding cusps at G and F, and so on 
