162 KEASONING. 



one within which the case evidently falls, to bring it within others in which 

 it can not be directly seen to be included. 



When the more obvious of the inductions which can be made in any 

 science from direct observations, have been made, and general formulas 

 have been framed, determining the limits within which these inductions 

 are applicable ; as often as a new case can be at once seen to come within 

 one of the formulas, the induction is applied to the new case, and the busi- 

 ness is ended. But new cases are continually arising, which do not obvi- 

 ously come within any formula whereby the question we want solved in 

 respect of them could be answered. Let us take an instance from geome- 

 try : and as it is taken only for illustration, let the reader concede to us 

 for the present, what we shall endeavor 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 formu- 

 lae 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 isos- 

 celes triangle do not obviously come within any of these. The formulae 

 specify certain marks of equality and of inequality, but the angles can not 

 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 recognizing these angles as the differences of equal things? 

 Because each of them is the difference not of one pair only, but of innu- 

 merable 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 part of the first inveptor, deserves to be re- 

 garded as considerable, two pairs of angles were hit upon, which united 

 these requisites. First, it could be perceived intuitively that their differ- 

 ences were the angles at the base ; and, secondly, they possessed one of the 

 marks of equality, namely, coincidence when applied to one another. This 

 ^ coincidence, however, was not perceived intui- 



tively, but inferred, in conformity to another 

 formula. 



For greater clearness, I subjoin an analysis 

 of the demonstration. Euclid, it will be re- 

 membered, demonstrates his fifth proposition 

 by means of the fourth. This it is not allow- 

 able for us to do, because we are undertaking 

 to trace deductive truths not to prior deduc- 

 tions, but to their original inductive founda- 

 ^ tion. We must, therefore, use the premises of 

 the fourth proposition instead of its conclusion, and prove the fifth dii-ectly 

 from first principles. To do so requires six formulas. (We must begin, as 



