TRAINS OF REASONING. 285 



part of the first inventor, deserves to be regarded as 

 considerable, two pairs of angles were hit upon, which 

 united these requisites. First, it could be perceived 

 intuitively that their differences were the angles at 

 the base ; and, secondly, they possessed one of the 

 marks of equality, namely, coincidence when applied 

 to one another. This coincidence, however, was not 

 perceived intuitively, but inferred, in conformity to 

 another formula. 



To make all clear, we subjoin an analysis of 

 the demonstration. 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 foun- 

 dation. We must therefore use the premisses of the 

 fourth proposition instead of 

 its conclusion, and prove 

 the fifth directly from first 

 principles. To do so re- 

 quires six formulas. (We 

 must begin, as in Euclid, by 

 prolonging the equal sides 

 A B, A C, to equal distances, 

 and joining the extremities 

 BE, DC.) 



FIRST FORMULA. The sums of equals are equal. 



A D and A E are sums of equals by the supposi- 

 tion. Having that mark of equality, they are con- 

 cluded by this formula to be equal. 



SECOND FORMULA. Equal straight lines being applied 



to one another coincide. 



A C, A B, are within this formula by supposition ; 

 AD, A E, have been brought within it by the pre- 



