342 A. C. Twining—Euclid’s Doctrine of Parallels, 
indefinitely. Again, the perpendiculars C O, E Q, &., and BN, 
DP, &c., to the straight line Q P, can never meet each other or 
the axis,—consequently the cusps must each, as EG F, ; 
be less than the angles EG T, DFS, respectively (that is, than 
half either of the equal angles of the equiangular and equilateral 
polygon IGECAB , &c.); for, if otherwise, G F would 
e one and the same straight line which cuts another straight 
line T'S in two points T and §, or beyond them. 
Again, let W X cut the axis at any intermediate position, as 
Y, within the tessera EH D F G. It cannot cut the line ED 
or GF, since all three are normal to A Z. Therefore it cuts the 
sides EG, DF, of that tessera in V, U, making angles or cusps 
V UD, UVE, which cannot be less than EGF, DFG, respectively, 
because, if so, the four angles of the tessera EH F must exceed four 
right angles. Therefore the cusps formed at V and U are each 
greater than a given angle ACB. And because TG F is less 
than a tight angle, T G W’ is greater, and the half angle T GI of 
the polygon will be within it,—so that W X, after passage 
through any tessera, as ET, of the entire series, may enter and 
traverse another, as IT, and cut the polygon in the sides GI, 
F Therefore W X, as it approaches to Q P, can never cease 
to cut the polygon (and at an angle which can be shown, as 
above, to be greater than a given angle). Let it move on at right 
angles to the axis till it reaches M. It will then coincide with 
the straight line of bases QP, which therefore will somewhere 
cut the polygon. That is, the base of some one of the tesseras 
will cut its side opposite the base—which is impossible; com 
sae teat the original supposition is also impossible. 
his exhibits with fidelity Mr. Thompson’s complete pro- 
cess,—only excluding his supposition that the cusps may at 
length come to equal or exceed half the constant angle of the 
polygonal series by proving such supposition itself to be impos- 
sible. The exceptional point, as already remarked, is the last 
step of the process—that of WX moving forward to coincide 
with M. Indeed, beginning back of that step, and at the close. 
of the one preceding it, the really legitimate conclusion would be 
deduced as follows: “ But W X cannot actually reach M during 
this consecutive intersection of the tesseras. For suppose it to 
advance to M while cutting the polygonal series, as in I, H; 
then IM, which makes the angle IMQ, must also make an 
equal vertical angle (15-1) on the opposite side of Q P, instead 
f HM P on the same side,—which last is impossible.” _ 
It would, no doubt, be urged by our author that if wx 
cuts the polygon when in the position Y, but ceases to cut ID 
the position M, it could not but be that the cuttin ig c 
o 
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