parallel Lines in Geometry. 167 
figure C GL M is-divided, constitute the four angles of that figure, 
together with the angles round each of points H, K, &e., and 
the angles, ‘directed into the interior of the figure, at the points 
A,B, E, F,&c. But all the angles round the points H, K, &c., 
of which points the number is m—2, are equal to (m—2) x 4P, 
or to4mP—S8P; and all the angles at the points A, B, E, F, 
&c., are equal tom x2P. Wherefore the sum of all the angles 
of all the triangles into which the quadrilateral CGLM is di- 
vided, is equal to the four angles of that figure together with 
4mP—S8P + 2mP =6mP—SP. 
Again: the three angles of the triangle ABC are, by the hy- 
pothesis, equal to 2P—.2; and, as the number of the triangles 
CAB, BH E, EKF, F LG is equal to m, the sum of all the an- 
gles of all these triangles, will be equal to 2mP—m x x. Upon 
each of the lines AH, HK, KL, there stand two triangles, one 
above and one below ; and, as the three angles of a triangle 
cannot exceed two right angles, it follows that all the angles of 
those triangles, the number of which is equal to 2m—2, can- 
not exceed 4mP—4P. Wherefore the sum of all the angles of 
all the triangles into which the quadrilateral C GM L is divided, 
- cannot exceed 4mP —4P+2mP—m x «x =6mP—S8P + 
4P—mxwe. 
It follows, from what has now been proved, that the four angles 
of the quadrilateral CG L M, together with 6 mP—S8P, cannot 
exceed 6mP—8P+4P—m xa. Wherefore, by taking the 
same thing, viz. 6mP—8P, from the two unequal things, the 
four angles of the quadrilateral CGLM cannot exceed 4 P —m x x. 
But 4P—m x a is less than the sum of the two angles AC B 
and ABC, or than the sum of the two angles AC B and LGF: 
wherefore, a fortiori, the four angles of the quadrilateral cannot 
exceed the suin of the two angles ACB and LGF; that is, a 
whole cannot exceed a part of it; which is absurd. ‘Therefore 
the three angles of the triangle ABC cannot be less than two 
right angles. 
And hecause the three angles of a triangle can neither be greater 
nor less than two right angles, they are equal to two right angles. 
By the help of the proposition just proved, the defect in Eu- 
clid’s Theory of parallel Lines may be removed, as the reader 
will see by consulting the notes to Professor Playfair’s Elements 
of Geometry. 
Legendre has demonstrated the same proposition in a-different 
manner, by means of algebraic functions. The like mode of rea- 
soning has also heen applied to elementary propositions in other 
branches of science ; particularly to the composition of forces in 
mechanics. The evidence of such demonstrations may, on an- 
other occasion, become the subject of inquiry. 
March 6, 1822. : J. Ivory. 
