142 REASONING. 



formula whereby the questions 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 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 induc- 

 tions we have, from which we can infer equality or inequalit}'. For 

 iufen-ing equality we have the following fonnulae : — 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 dif- 

 ferences of equal things are equals. There are no other fonnulse 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 ob\-iously come within any of 

 these. The formulas specify certain marks of equality and of in- 

 equality, 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 succeed in bringing them within this formula, " The 

 differences of equal things are equal." Whence comes the difficulty 

 in recognizing these angles as the differences of equal things'? Be- 

 cause each of them is the difference not of one pair only, but of in- 

 numerable 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 

 formulcB. By an exercise of ingenuity, which, on the part of the first 

 inventoi, deserves to be regarded as considerable, two pairs of angles 

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

 ceived 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 coincidence, how- 

 ever, was not peiceived intuitively, but infeired, in conformity to 

 another formula. 



To make all clear, we subjoin an analysis of the demonstration. 

 Euclid, it ^^■^.\l be remembered, demon- 

 strates his fifth proposition by means of 

 the fourth. This it is not allov.'able for us 

 to do, because we are undertaking to ti-ace 

 deductive truths not to prior deductions, 

 but to their original inductive foundation. 

 We must therefore use the premisses of 

 the fourth proposition instead of its con- 

 clusion, and prove the fifth directly from 

 first principles. To do so requires six for- 

 mulas. (We must begin, as in Euclid, 

 by prolonging the equal sides A B, AC, to equal distances, and join- 

 ing the extremities B E, D C.) 



