228 THE POPULAR SCIENCE MONTHLY. 



tinue 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, w r e 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 di- 

 vision 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 meta- 

 physical point was hardly the proper way to build up the science of 

 geometry. Of course, it being nothing, the geometry that he con- 

 structed out of it, no matter how many intermediate propositions in- 

 tervened, 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 propo- 

 sitions, or there can be no geometry, namely, that " space is infinite 

 in extent, that it is infinitely divisible, and that it is infinitely con- 

 tinuous." 



Now, I deny that geometry has anything to do with infinity ; in- 

 deed, 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. Indefi- 

 nite 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. Ex- 

 perience 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 



