142 



REASONING. 



plipd to each other coincide are equals. 

 Things which are equal to the same 

 thing are equals. A whole and the 

 sum of its parts are equals. The sums 

 of equal things are equals. The dif- 

 ferences of equal things are equals. 

 There are no other original formulae 

 to prove equality. For inferring in- 

 equality we have the following : — A 

 whole and its parts are unequals. 

 The sums of equal things and unequal 

 things are unequals. The differences 

 of equal things and unequal things 

 are unequals. In all, eight formulEe. 

 The angles at the base of an isosceles 

 triangle do not obviously come within 

 any of these. The formulae specify 

 certain marks of equality and of in- 

 equality, but the angles cannot be 

 perceived intuitively to have any of 

 those marks. On examination it ap- 

 pears that they have ; and we ulti- 

 mately succeed in bringing them 

 within the formula, " The differences 

 of equal things are equal." Whence 

 comes the difficulty of recognising 

 these angles as the differences of 

 equal things? Because each of them 

 is the difference not of one pair only, 

 but of innumerable pairs of angles ; 

 and out of these we had to imagine 

 and select two, which could either 

 be intuitively perceived to be equals, 

 or poasessed some of the marks of 

 equality set down in the various for- 

 mulae. By an exercise of ingenuity, 

 which, on the part of the first inven- 

 tor, deserves to be regarded as con- 

 siderable, two pairs of angles were 

 hit upon which united these requi- 

 sites. 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 coin- 

 cidence, however, was not perceived 

 intuitively, but inferrediu coniormity 

 to another fi)rmula. 



For greater clearness, I subjoin an 

 analysisof Lliedemonstratiou. Kuclid, 

 it will be remembered, demonstrates 

 his fitth proposition by means of the 

 fourth. This is uut aliowabU for U4 



to do, because we are undertaking to 

 trace deductive truths not to prior 

 deductions, but to their original in- 

 ductive foundation. We must there- 

 fore use the premises of the fourth 

 proposition instead of its conclusion, 

 and prove the fifth directly from 

 first principles. To do so requires 

 six formulas. (We must begin, as in 

 Euclid, by prolonging the equal sides 

 AB, AC, to equal distances, and join- 

 ing the extremities BE, DC. ) 



First Formula. — The 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 

 therefore possess the property which, 

 according to the second formula, is a 

 mark that when applied to one ano- 

 ther 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 upoQ AC qf the other. Then, 



