For greater clearness, I 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 foundation. 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 requires six formulas. (We presuppose an equilateral triangle, whose vertices are A, D, E, with point B on the side AD, and point C on the side AE, such that BC is parallel to DE. 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._
A D and A E are sums of equals by the supposition. Having that mark of equality, they are concluded 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; A D, A E, 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 applied to each other, they will coincide. Coinciding altogether means coinciding in every part, and of course at their extremities, 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.
FOURTH FORMULA. _Angles, having their sides coincident, coincide_.
The third induction having shown that B E and C D coincide, and the second that A B, A C, coincide, the angles A B E and A C D are thereby brought within the fourth formula, and accordingly coincide.
FIFTH FORMULA. _Things which coincide are equal_.
The angles A B E and A C D are brought within this formula by the induction immediately preceding. This train of reasoning being also applicable, _mutatis mutandis_, to the angles E B C, D C B, these also are brought within the fifth formula. And, finally,
SIXTH FORMULA. _The differences of equals are equal_.