There are such certain and universal inductions; and it is because there are such, that a Logic of Induction is possible.
CHAPTER V. OF THE LAW OF UNIVERSAL CAUSATION.
§ 1. The phenomena of nature exist in two distinct relations to one another; that of simultaneity, and that of succession. Every phenomenon is related, in an uniform manner, to some phenomena that coexist with it, and to some that have preceded or will follow it.
Of the uniformities which exist among synchronous phenomena, the most important, on every account, are the laws of number; and next to them those of space, or in other words, of extension and figure. The laws of number are common to synchronous and successive phenomena. That two and two make four, is equally true whether the second two follow the first two or accompany them. It is as true of days and years as of feet and inches. The laws of extension and figure, (in other words, the theorems of geometry, from its lowest to its highest branches,) are, on the contrary, laws of simultaneous phenomena only. The various parts of space, and of the objects which are said to fill space, coexist; and the unvarying laws which are the subject of the science of geometry, are an expression of the mode of their coexistence.
This is a class of laws, or in other words, of uniformities, for the comprehension and proof of which it is not necessary to suppose any lapse of time, any variety of facts or events succeeding one another. If all the objects in the universe were unchangeably fixed, and had remained in that condition from eternity, the propositions of geometry would still be true of those objects. All things which possess extension, or in other words, which fill space, are subject to geometrical laws. Possessing extension, they possess figure; possessing figure, they must possess some figure in particular, and have all the properties which geometry assigns to that figure. If one body be a sphere and another a cylinder, of equal height and diameter, the one will be exactly two-thirds of the other, let the nature and quality of the material be what it will. Again, each body, and each point of a body, must occupy some place or position among other bodies; and the position of two bodies relatively to each other, of whatever nature the bodies be, may be unerringly inferred from the position of each of them relatively to any third body.