342 RELATION BETWEEN THE AREA OF A PLANE 
numbers of the Journal at hand, the substance of my refutation of 
Legendre is given in an Appendix to the present paper. 
Proposition I. 
The sum of the angles of a triangle AHE (Fig. 1) is not greater 
than two right angles. 
FIG | 
For, produce HE to F. Bisect AKin M. Draw HMB, making 
MB = HM; and join BE. In like manner construct the triangie 
CHE; N being the middle point of BE ; and CN being equal to 
HN. In like manner construct the triangle DHE; P being the 
middle point of CE; and DP being equal to PH. And so on in- 
definitely. Denote by 8, 8,,S., &c., the sum of the angles of the 
triangles AHE, BHE, CHE, &c., respectively ; and by A,, A,, A,, 
&e., the angles HBE, HCE, HDE, Xc., respectively. Then it is 
plain that the quantities 8, S,, S,, &c., are all equal to one an- 
other. Also, as the number 2 becomes indefinitely great, the angle 
A,, becomes indefinitely small. For, the sum of all the angles in the 
series, A, A,, Ay, &e., is less than ANF; and, since the series, A, Aj, 
&c., may be made to contain an indefinite number of terms, those 
terms which are ultimately obtained must be indefinitely smail, in 
order that AEF may be a finite angle. But, the exterior angle DEF 
being greater than the interior and opposite angle DHE, S, cannot 
exceed two right angles by D. And 8S, =S. Therefore S cannot 
exceed two right angles by D or A,. In like manner it may be 
proved that S cannot exceed two right angles by A,, whatever n be. 
And A, is ultimately less than any assignable angle. Therefore S 
cannot exceed two right angles by any finite angle whatsoever. 
Cor. 1.—If a line AE (Fig. 2) be drawn from A, an angle of a 
triangle ADF, to a point in the opposite side; and if the sum of the 
