TRAINS OF REASONING. 



141 



and to the axis of the surface. It is 

 to be proved that the concourse of 

 these three circumstances is a mark 

 that the reflected rays will pass 

 through the focus of the parabolic 

 surface. Now, each of the three cir- 

 cumstances is singly a mark of somek- 

 thing material to the case. Rays of 

 light impinging on a reflecting sur- 

 face are a mark that those rays will 

 be reflected at an angle equal to the 

 angle of incidence. The parabolic 

 fonn of the surface is a mark that, 

 from any point of it a line drawn to 

 the focus and a line parallel to the 

 axis will make equal angles with the 

 surface. And finally, the parallelism 

 of the rays to the axis is a mark that 

 their angle of incidence coincides with 

 one of these equal angles. The three 

 marks taken together are therefore a 

 mark of all these three things united. 

 But the three united are evidently a 

 mark that the angle of reflection must 

 coincide with the other of the two 

 equal angles, that formed by a line 

 drawn to the focus ; and this again, 

 by the fundamental axiom concerning 

 straight lines, is a mark that the re- 

 flected rays pass through the focus. 

 Most chains of physical deduction are 

 of this more complicated type ; and 

 even in mathematics such are abun- 

 dant, as in all propositions where the 

 hypothesis includes numerous condi- 

 tions : " If a, circle be taken, and if 

 Avithin that circle a point be taken, 

 not the centre, and if straight lines be 

 drawn from that point to the circum- 

 ference, then,' &c. 



§ 4. The considerations now stated 

 remove a serious difficulty from the 

 view we have taken of reasoning, 

 which view might otherwise have 

 seemed not easily reconcilable with 

 the fact that there are Deductive or 

 Ratiocinative Sciences. It might seem 

 to follow, if all reasoning be induction, 

 that the difficulties of philosophical 

 investigation must lie in the induc- 

 tions exclusi.vely, and that when these 

 were easy, and susceptible of no doubt 

 or hesitation, there could be no science, 



or, at least, no difficulties in science. 

 The existence, for example, of an ex- 

 tensive Science of Mathematics, re- 

 quiring the highest scientific genius 

 in those who contributed to its crea- 

 tion, and calling for a most continued 

 and vigorous exertion of intellect in 

 order to appropriate it when created, 

 may seem hard to be accounted for on 

 the foregoing theory. But the con- 

 siderations more recently adduced re- 

 move the mystery, by showing that 

 even when the inductions themselves 

 are obvious, there may be much diffi- 

 culty in finding whether the particular 

 case which is the subject of inquiry 

 comes within them ; and ample room 

 for scientific ingenuity in so combin- 

 ing various inductions, as, by means 

 of one within which the case evidently 

 falls, to bring it within others in which 

 it cannot be directly seen to be in- 

 cluded. 



When the more obvious of the in- 

 ductions 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 ap- 

 plicable ; 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 business is 

 ended. But new cases are continually 

 arising, which do not obviously come 

 within any formula whereby the ques- 

 tion 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 endeavour to prove in 

 the next chapter, that the first prin- 

 ciples of geometry are results of in- 

 duction. 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 tri- 

 angle equal or unequal? The first 

 thing to be considered is, what in- 

 ductions we have, from which we can 

 infer equality or inequality. For in- 

 ferring equality we have the following 

 formulae : — Things which being ap- 



