TRAINS OF REASONING. 143 



First Formula. The sums of equals arc equal. 

 A D and A E arc sums of equals by the supposition. Having that 

 niai-k of equahty, they are concluded by this formula to be equal. 



Second Formula. Equal straight Inics being aj)j)licd 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 

 sti'aight lines have the property of e(}uality ; which, according to the 

 second formula, is a mark that, if applied to each otlier, they will coin- 

 cide. Coinciding altogether means coinciding in every part, and of 

 coui'se at their extremities, DE and BC. 



Third Formula. Straight lines, having their extremities coincident, 

 coincide. 

 BE and DC have been brought within this formula by the preceding 

 induction ; they will therefore coincide. 



Fourth Formula. Angles, having their sides coincident, coincide. 



The two previous inductions having shown that BE and DC coin- 

 cide, and'that AD, AE, coincide, the angles ABE and ACD are 

 thereby brought within the fourth formula, and accordingly coincide. 



Fifth Formula. Things which coincide are equal. 



The angles ABE and ACD are brought within this formula by the 

 induction immediately preceding. This train of reasoning being also 

 applicable, mutatis mutandis, to the angles E BC, D CB, 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 BE, and the angle 

 ACB being the difference of ACD, DCB; which have been proved 

 to be equals ; ABC and ACB are brought within the last formula 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 pair of angles out of another, while each pair shall be 

 con-esponding angles of ti'iangles which have two sides and the inter- 

 vening angle equal. It is by this happy contrivance that so many dif- 

 ferent inductions are brought to bear upon the same particular case. 

 And this not being at all an obvious idea, it may be seen from an 

 example so near the threshold of mathematics, how much scope there 

 may well be for scientific dexterity in the higher branches of that and 

 other sciences, in order so to combine a few simple inductions, as to 

 bring within each of them innumerable cases which arc not obyiou,sly 

 included in it ; and how long, and numerous, and complicated, may be 

 the processes, necessary for bringing the inductions together, even wheu 

 each induction may itself be very easy and simple. All the inductions 

 involved in all geometry are comprised in those simj^le ones, the for- 

 mula of which are the Axioms, and a few of the so-called Definitions. 

 The remainder of the science is made up of the processes employed 

 for bringing unforeseen cases within these inductions ; or (in syllogistic 



