TRAINS OF REASONING. 



which do not obviously come within any formula whereby the 

 question we want solved in respect of them could be answered. 

 Let us take an instance from geometry : and as it is taken 

 only for illustration, let the reader concede to us for the pre- 

 sent, what we shall endeavour to prove in the next chapter, 

 that the first principles of geometry are 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 original for- 

 mulae 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 differ- 

 ences of equal things and unequal things are unequals. In 

 all, eight formulas. 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. On examination it appears that they have ; and 

 we ultimately 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 possessed some of the 

 marks of equality set down in the various formulae. By an ex- 

 ercise of ingenuity, which, on the part of the first inventor, 

 deserves to be regarded as considerable, two pairs of angles 

 were hit upon, which united these requisites. First, it could be 

 perceived intuitively that their differences were the angles at 

 the base ; and, secondly, they possessed one of the marks of 

 VOL. i. 16 



