284 REASONING. 



results of induction. Our example shall be the fifth 

 proposition of the first book of Euclid. The inquiry 

 is, Are the angles at the base of an isosceles triangle 

 equal or unequal ? The first thing to be considered 

 is, what inductions we have, from which we can infer 

 equality or inequality. For inferring equality we 

 have the following formulae : Things which being 

 applied 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 differences of equal 

 things are equals. There are no other formula? to 

 prove equality. For inferring inequality 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 formulae. 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 inequality, but the angles 

 cannot be perceived intuitively to have any of those 

 marks. We can, however, examine whether they 

 have properties which, in any other formulae, are set 

 down as marks of those marks. On examination it 

 appears that they have ; and we ultimately suc- 

 ceed in bringing them within this formula, " The 

 differences of equal things are equal." Whence 

 comes the difficulty in 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 possessed some 

 of the marks of equality set down in the various 

 formulae. By an exercise of ingenuity, which, on the 



