242 REASONING. 



equality, namely, coincidence when applied to one another. 

 This coincidence, however, was not perceived intuitively, but 

 inferred, in conformity to another formula. 



For greater clearness, I subjoin an analysis of the de- 

 monstration. Euclid, it will be remembered, demonstrates 

 his fifth proposition by means of the fourth. This it is not 

 allowable for us to do, because we are undertaking to trace 

 deductive truths not to prior deductions, but to their original 

 inductive foundation. We must therefore use the premises 

 of the fourth proposition instead 

 of its conclusion, and prove the 

 fifth directly from first principles. 

 To do so requires six formulas. 

 (We must begin, as in Euclid, 

 by prolonging the equal sides 

 AB, AC, to equal distances, and 

 joining the extremities BE, 

 DC.) 



FIRST FORMULA. The sums of equals are equal. 



AD and AE are sums of equals by the supposition. Hav- 

 ing that mark of equality, they are concluded by this formula 

 to be equal. 



SECOND FORMULA. Equal straight lines being applied 

 to one another coincide. 



AC, AB, are within this formula by supposition; AD, 

 AE, have been brought within it by the preceding step. 

 Both these pairs of straight lines have the property of equality; 

 which, according to the second formula, is a mark that, if ap- 

 plied to each other, they will coincide. Coinciding altogether 

 means coinciding in every part, and of course at their extremi- 

 ties, D, E, and B, C. 



THIRD FORMULA. Straight lines, having their extremities 

 coincident, coincide. 



B E and C D have been brought within this formula by 

 the preceding induction ; they will, therefore, coincide. 



