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 witnesses: this mark, the government under discussion was recognized by the senses as possessing. Hence that government fell within the last induction, and by it was brought within all the others. The perceived resemblance 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 consist, as in the examples hitherto exhibited, of a single chain, a a mark of b, b of 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 of e, c of f, d e f of n, therefore _a b c a mark of n_. Suppose, for example, the following combination of circumstances: 1st, rays of light impinging 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 a reflecting surface are a mark that those rays will be reflected at an angle equal to the angle of incidence. The parabolic form 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 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 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 inductions 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 Mathematics, requiring the highest scientific genius in those who contributed to its creation, 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 considerations more recently adduced remove the mystery, by showing, that even when the inductions themselves are obvious, there may be much difficulty in finding whether the particular case which is the subject of inquiry comes within them; and ample room for scientific ingenuity in so combining various inductions, as, by means of
