OF ARTS AND SCIENCES. 



131 



than ^. And proceeding in the same way from triangle to triangle, 

 until we obtain the m^*" derived triangle, then the sum of its angles will 

 equal that of the given triangle ABC, and one of its angles will not 

 be greater than g^ ; where m is of course a positive integer. 



" Cor. 2. If we obtain the derived triangle whose number is ffi-}-l,the 

 sum of two of its angles will equal that angle of the m^^ triangle which 

 has been shown not to be greater than j^ ; we hence see how from 

 any given triangle to derive another triangle such that the sum of its 

 angles shall equal that of the given triangle, and such that the sum of 

 two of its angles shall not be greater than ^Tn ; where A denotes one 

 of the angles of the given triangle. 



" Remark. Cor. 1 is substantially the same as Mr. I. Ivory's pro- 

 cess, given at page 189 of the New York edition of J. R. Young's 

 Elements of Geometry. 



" Prop. 2. The sum of the angles of any triangle is not greater 

 than two right angles. 



" Let the triangle A B C, o( Prop. 1, represent any triangle, and de- 

 note a right angle by R, and if possible let the sum of the angles of 

 the triangle equal 2 A-{- V, V being a finite positive angle. Then, 

 using A to represent the angle B A C of the triangle, some positive 

 integer, m, may be found so that the inequality m V^ A shall exist. 

 From m 7> ^, it follows that T^> ^ > ^, or F is greater than g^. 

 By Cor. 2, Prop. 1, we may derive a triangle from ABC, such that 

 the sum of two of its angles shall not be greater than ^ ; hence the 

 sum of these two angles is less than F; consequently the third angle 

 of the triangle must be greater than 2 R, which is impossible. Hence 

 the sum of the angles of the triangle A B C is not greater than two 

 right angles. 



" Prop 3. The sum of the angles of any triangle is greater than 

 a right angle. 



" Let ABC represent any tri- 

 angle, and suppose (for conven- 

 ience) that the angle A B C is 

 not less than either of the other 

 angles of the triangle. Let the 

 base be extended in the direction 

 AC to F, so that C F equals the 

 base (AC), and through C and 

 F draw the right lines C D and 

 F G, each equal to A B, and so 



