286 REASONING. 



ceding step. Both these pairs of straight lines have 

 the property of equality; which, according to the 

 second formula, is a mark that, if applied to each 

 other, they will coincide. Coinciding altogether 

 means coinciding in every part, and of course at their 

 extremities, DE and B C. 



THIRD FORMULA. Straight lines, having their extremi- 

 ties coincident, coincide. 



B E and D C have been brought within this for- 

 mula by the preceding induction ; they will, therefore, 

 coincide. 



FOURTH FORMULA, Angles, having their sides coinci- 

 dent, coincide. 



The two previous inductions having shown that 

 BE and DC coincide, and that AD, AE, coincide, 

 the angles ABE and A C D are thereby brought 

 within the fourth formula, and accordingly coincide. 



FIFTH FORMULA. Things which coincide are equal. 



The angles A BE and A CD are brought within 

 this formula by the induction immediately preceding. 

 This train of reasoning being also applicable, mutatis 

 mutandis, to the angles EBC, D C B, these also are 

 brought within the fifth formula. And, finally, 



SIXTH FORMULA. The differences of equals are equal. 



The angle ABC being the difference of ABE, 

 C B E, and the angle A C B being the difference of 

 A C B,, D C B ; which have been proved to be equals ; 

 ABC and ACB are brought within the last for- 

 mula by the whole of the previous process. 



The difficulty here encountered is chiefly that of 

 figuring to ourselves the two angles at the base of the 

 triangle ABC, as remainders made by cutting one 



