TRAINS OF REASONING. 163 



in Euclid, by prolonging the equal sides AB, AC, to equal distances, and 

 joining the extremities BE, DC.) 



First Formula. — TJie sums of equals are equal. 



AD and AE 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 or angles, 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. The angle at A considered as an 

 angle of the triangle ABE, and the same angle considered as an angle of 

 the triangle ACD, are of course within the formula. All these pairs, there- 

 fore, possess the property which, according to the second formula, is a 

 mark that when applied to one another they will coincide. Conceive 

 them, then, applied to one another, by turning over the triangle ABE, and 

 laying it on the triangle ACD in such a manner that AB of the one shall 

 lie upon AC of the other. Then, by the equality of the angles, AE will lie 

 on AD. But AB and AC, AE and AD are equals ; therefore they will co- 

 incide altogether, and of course at their extremities, D, E, and B, C. 



Third Formula. — Straight lines, having their extremities coincident^ 



coincide. 



BE and CD have been brought within this formula by the preceding in- 

 duction ; they will, therefore, coincide. 



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



The third induction having shown that BE and CD coincide, and the 

 second that AB, AC, 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 in- 

 duction immediately preceding. This train of reasoning being also appli- 

 cable, mutatis mtitandis, 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 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 cut- 

 ting one pair of angles out of another, while each pair shall be correspond- 

 ing 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 

 combine a few simple inductions, as to bring within each of them innumer- 

 able cases which are not obviously included in it ; and how long, and nu- 



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