134 PROCEEDINGS OF THE AMERICAN ACADEMY 



by a right line ; also extend EH to /, so that I H= E H, and draw 

 a right line from / to G. Since DK= GK, EK=HK, and the 

 angles D K E, GKH are equal (Sim., Prop. 15, B. I.), the triangles 

 DKE, GKH are identical (Sim., p. 4, B. I.), and DE=GH, 

 the angle G H K=the angle D E K= R, and the angle H G K=^ 

 the angle ED K; hence the triangles E H G, IHG are identical, 

 since they have E H=^ I H, HG common, and that the angles at 

 if are right (Sim., p. 4, B. I.) ; hence the sides G E, G I are equal, 

 the angle G I H equals the angle G E H, and the angle IG H equals 

 the angle E G H. Since the angle D F E is greater than either of 

 the angles F G E, FE G (Sim., p. 16, B. L), it follows from what 

 has been shown that the angle E G H is not greater than the sum of 

 the angles E D F^ E F D, and of course the sum of the angles E G H, 

 KEF is not greater than F, and since G E F is less than E F D, 

 G E F is not greater than g^,, .•. the sum of the angles E G H, HE G 

 is not greater than V-\- n^„ consequently the sum of the angles of the 

 triangle E G I is not greater than 2V + ^,. But by hypothesis the 

 sum of the angles of the triangle ABC, which equals the sum of the 

 angles of the triangle D E F, is not greater than the sum of the angles 

 of the triangle E G I; .-. 2F-f- |^^ is not less than R+V,oy V is 

 not less than R — |^^. Hence V cannot differ from R by any given 

 angle, as a, so that V= R — a, a being a positive finite angle ; for 

 by taking a sufficiently great positive integer for m (which is evi- 

 dently arbitrary), we shall make ~ less than a, which is absurd ; .•. V 

 is not less than jR. Hence the sum of the angles of the triangle 

 A B C is not less than 2 R. 



" Cor. Since by Prop. 2 the sum of the angles of any triangle is 

 not greater than 2 R, and from what has been shown in this Prop, it 

 is not less than 2 R, it follows that the sum of the angles of any trian- 

 gle = 2 jR = two right angles, as required. 



" Appendix to Propositions 3 and 4. 



" Lem. No triangle can exist such that the sum of its angles shall 

 be less than any given angle ; or such that the sum of its angles shall 

 equal an infinitesimal angle. For, if possible, let A B C be such a tri- 

 angle ; then, since the sum ^ 

 of its angles is less than 

 any given angle, each of its 



angles is of course less than (L^::—— '^Ji 



