A. C. Twining—FHuclid’s Doctrine of Parallels. 339 
DIH, or the angle HIG, is greater than DHG less IHG, or 
the angle DH I, or its equal HIP, by construction, the angle 
HIG is greater than HI P, and much more greater than HI K. 
Therefore the line CI produced to G makes HG greater than 
K, if the angles of C DI are together less than two right 
angles—that is, equally, makes HG greater than HK, if OI 
produced does not pass through K. But if CI produced passes 
through K, then is HG by supposition not greater than H K, 
and the angles of CD I accordingly are not less than but (28 a. I) 
equal to two right angles, and Cl meets A B. 
JoinI A. It has been shown that if CI produced meets K, 
or makes with I A the angle AIK, it must meet A B, and also 
that if it does not meef K it makes with I A an angle LAG 
within I A K, and therefore much more must it meet AB. There- 
fore CI produced must meet AB, whether it does or does 
not pass through K—that is, i must meet AB. And the same 
may be proved on the other side of AC. Therefore C R is the 
only line through C which cannot be produced to meet AB. 
Remarks. 
1st. The two principal propositions of the foregoing demon- 
Stration are, no doubt, too difficult for beginners. hat fact, 
in themselves, prime qualities, they do not of necessity counter- 
balance the advantage of a system like Euclid’s, preéminent in 
0 
more than average difficulty, was not on that account refused 
by its author a place in due order among his elementary proposi- 
ons; and, though unsatisfactory in its concluding inference, 
and therefore omitted in subsequent editions, it will ever remain 
Worthy of preservation and of study by reason of the beauty and 
Skill of its conception and conduct. 
- It is quite otherwise, however, with the so-called 
analytic or functional proof by the same author, which has been 
made the subject of earnest controversy. This, it is familiarly 
nown, depended upon the consideration that in any given tri- 
angle the given base and the given angles at the base determine 
