TRAINS Of REASONING. 243 



FOURTH FORMULA. Angles, having their sides coincident, 

 coincide. 



The third induction having shown that BE and CD co- 

 incide, and the second that AB, AC, coincide, the angles 

 ABE and ACD are therehy brought within the fourth for- 

 mula, 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 EBC, DCB, 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, CBE, 

 and the angle CB 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 corresponding angles of 

 triangles which have two sides and the intervening angle 

 equal. It is by this happy contrivance that so many different 

 inductions are brought to bear upon the same particular case. 

 And this not being at all an obvious thought, 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 com- 

 bine a few simple inductions, as to bring within each of them 

 innumerable cases which are not obviously included in it ; and 

 how long, and numerous, and complicated may be the processes 

 necessary for bringing the inductions together, even when each 

 induction may itself be very easy and simple. All the induc- 

 tions involved in all geometry are comprised in those simple 

 ones, the formulae of which are the Axioms, and a few of the 



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