658 PSYCHOLOGY. 



a new relation to the parts that pre-exist ; and assuming a 

 new relation means ceasing to be straight or plane. If we 

 mean by a parallel a line that will never meet a second 

 line ; and if we have one such line drawn through a point, 

 any new line drawn through that point which does not 

 coalesce with the first must be inclined to it, and if inclined 

 to it must approach the second, i.e., cease to be parallel 

 with it. No properties of outlying space need come m 

 here : only a definite conception of uniform direction, and 

 constancy in sticking to one's point. 



The other two axioms peculiar to geometry are that 

 figures can be moved in space without change, and that no 

 Tariation in the way of subdividing a given amount of space 

 alters its total quantity.* This last axiom is similar to 

 what we found to obtain in numbers. ' The whole is equal 

 to its parts ' is an abridged way of expressing it. A man is 

 not the same biological whole if we cut him in two at the 

 neck as if we divide him at the ankles ; but geometrically 

 he is the same whole, no matter in which place we cut him. 

 The axiom about figures being movable in space is rather 

 a postulate than an axiom. So far as they are so movable, 

 then certain fixed equalities and diflerences obtain between 

 forms, no matter ivhere placed. But if translation through 

 space warped or magnified forms, then the relations of 

 equality, etc., would always have to be expressed Avith a 

 position-qualification added. A geometry as absolutely 

 certain as ours could be invented on the supposition of 

 such a space, if the laws of its warping and deformation 

 were fixed. It would, however, be much more complicated 

 than our geometry, which makes the simplest possible sup- 

 position ; and finds, luckily enough, that it is a supposition 

 with which the space of our experience seems to agree. 



By means of these principles, all playing into each 

 other's hands, the mutual equivalences of an immense num- 

 ber of forms can be traced, even of such as at first sight 

 bear hardly any resemblance to each other. We move and 



* The subdivision itself consumes none of the space. In all practical 

 experience our subdivisions do consume space. They consume it in our 

 geometrical figures. But for simplicity's sake, in geometry we postulate 

 subdivisions which violate experience and consume none of it. 



