Page:Popular Science Monthly Volume 9.djvu/248

228 to think that the above propositions differ very much from the two fundamental axioms of mathematics, "Equals added to equals and the sums are equal; and two things each equal to a third are equal each to each." In denying these, we must deny the laws of thought, the powers of the mind in distinguishing a thing from what it is not, or from that which it stands in contrast with, or in opposition to. All the other axioms of geometry, as Bain has shown, are either verbal propositions or can be derived from these, since subtraction is implicated in addition, multiplication derived from addition, and division implicated in multiplication.

The absurd conclusion at which the doctor arrives, namely, "Ex nihilo geometria fit," ought to show him that to begin with a metaphysical point was hardly the proper way to build up the science of geometry. Of course, it being nothing, the geometry that he constructed out of it, no matter how many intermediate propositions intervened, must be nothing. Suppose we try the analytic method of arriving at definitions. But first we are compelled to controvert the assertion that it is necessary to believe the three following propositions, or there can be no geometry, namely, that "space is infinite in extent, that it is infinitely divisible, and that it is infinitely continuous."

Now, I deny that geometry has anything to do with infinity; indeed, the doctor, before he gets through, says even more than this. "Science," says he, "has the finite for its domain, religion the infinite." What we have to do with in geometry is simply the relations of the attributes or propria of definite extension. But as definite extension has for its correlative indefinite extension, we need to understand it in a sort of general way. Experience furnishes us with the mutually-implicated notions of the contained and the containing, the bounded and the bounding. We cannot separate them completely in thought. The assertion of the one implicates the other. What lies without any extension is space—indefinite space. Simply that it is outside of our particular part of space is all that we have to do with it: whether it is infinite or not is none of the business of the geometrician. Indefinite extension, or the notion of space in general, is very different from the notion, if there be such a one, the words infinite space would connote. Indefinite space is comprehensible in the only sense that it needs to be comprehended, namely, as the correlative of extension or definite space.

This brings us to the genesis of the definitions of geometry. Experience makes us at first acquainted with extended bodies. This acquaintance goes no further than a knowledge of their attributes, or propria. All these properties come into the mind as a confused aggregate; it is not clearly perceived as a whole made up of distinct parts. The relation of part to part is perceived only in a vague and general manner. The work of the geometrician is to analyze these