136 PROCEEDINGS OF THE AMERICAN ACADEMY 



KM must fall without the triangle jff/J?'(orthat it will meet H I^ 

 produced in the direction HI), for if KM does not fall without the 

 triangle KIH, but coincides with K I, or falls at some point between 

 H and I, then we shall have the triangle H K L such that the sum of 

 its angles is less than V, and as V is any finite angle taken as small as 

 we please, .•. the sum of the angles of the triangle H K L is less than 

 any finite angle, or it is infinitesimal, which is impossible. Hence the 

 perpendicular KM falls oa HI produced in the direction HI, so as to 

 make the angle HKM equal to some finite angle ; and it is evident 

 that the perpendicular cannot intersect HI produced in the direction 

 IH, for if it could, a triangle would be formed having the sum of two 

 of its angles greater than two right angles, which is impossible. Hence, 

 since the angle H IK is the exterior angle of the triangle KIM, it is 

 greater than the right angle KM I (Sim., p. 16, B. I.) ; hence the 

 sum of the angles of any triangle is greater than a right angle, as re- 

 quired. 



'■'■Prop. 4'. The sum of the angles of any triangle is not less than 

 two right angles. 



" We shall use the figure to Prop. 3'. It is evident that we may 

 suppose the sum of the angles of the triangle KHL not less than that 

 of the triangle KIH, or, since the sum of the angles IHK, IKH is 



A A 



not greater than 7^, we shall have 2 IKM-\--^, not less than the an- 

 gle KIH. Hence, if we denote a right angle by R, since the sum 

 of the angles HIK, KIM is equal to 2i? (Sim., p. 13, B. I.), and that 

 the sum of the angles K I M, I K M is not greater than R (see our 

 Prop. 2, and observe that the angle IMK=^ R), we get I KM 

 not less than R — ^, or ^„ is not less than the angle KIM. But ^sr 

 is less than any given angle, .•. the angle KIM is infinitesimal, con- 

 sequently the angle KIH differs from 2R by an infinitesimal angle, 

 and of course the sum of the angles of the triangle KIH or A B C is 

 not less than 2R, as required. 



" Cor. Hence, since the sum of the angles of any triangle (ABC), 

 is neither greater nor less than 2 R, it is equal to2R, = two right an- 

 gles." 



II. " ^n attempt to show {analytically) that the sum of the angles 

 of any rectilineal triangle is equal to two right angles. 



" Ax. The angle formed by two (right) lines is independent of the 

 lengths of the lines. 



