TRAINS OF REASONING. 161 



was drawn according to a formula in which desire of the public good was 

 set down as a mark of not being likely to be overthrown ; a mark of this 

 mark was, acting in a particular manner ; and a mark of acting in that 

 manner was, being asserted to do so by intelligent and disinterested wit- 

 nesses : this mark, the government under discussion was recognized by the 

 senses as possessing. Hence that government fell within the last induc- 

 tion, and by it was brought within all the others. The perceived resem- 

 blance of the case to one set of observed particular cases, brought it into 

 known resemblance with another set, and that with a third. 



In the more complex branches of knowledge, the deductions seldom con- 

 sist, as in the examples hitherto exhibited, of a single chain, a a mark of b, 

 b oi c,c of d, therefore a a mark of d. They consist (to carry on the same 

 metaphor) of several chains united at the extremity, as thus : a a mark of 

 d, b oi e,c of f, d ef oi n, therefore a b c a mark of n. Suppose, for exam- 

 ple, the following combination of circumstances: 1st, rays of light impin- 

 ging on a reflecting surface ; 2d, that surface parabolic ; 3d, those rays 

 parallel to each other 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 circumstances is singly a mark of something material to the case. 

 Rays of light impinging on r. reflecting surface are a mark that those rays 

 will be reflected at an angle equal to the angle of incidence. The parabol- 

 ic form of the surface, is a mark that, from any point of it, a lino drawn to 

 the focus and a line pai-allel 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 re- 

 flection 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 ma. 2; that the reflected rays pass through 

 the focus. Most chains of physical deduction are of this more complicated 

 type ; and even in mathematics such are abundant, as in all propositions 

 where the hypothesis includes numerous conditions: "If a circle be taken, 

 and if within that circle a point be taken, not the centre, and if straight 

 lines be drawn from that point to the circumference, then," etc. 



§ 4. The considerations now stated remove a serious diflSculty 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 induc- 

 tion, that the difficulties of philosophical investigation must lie in the in- 

 ductions exclusively, 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 extensive Science of Mathemat- 

 ics, requiring the highest scientific genius in those who contributed to its 

 ireation, and calling for a most continued and vigorous exertion of intel- 

 ect in order to appropriate it when created, may seem hard to be accounted 

 'or on the foregoing theory. But the considerations more recently adduced 

 •emove the mystery, by showing, that even when the inductions themselves 

 ire obvious, there may be much difficulty in finding whether the particular 

 •ase which is the subject of inquiry comes within them ; and ample room 

 or scientific ingenuity in so combining various inductions, as, by means of 



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