34 ORIGIN AND SIGNIFICANCE OF 



suppositions we make, in employing the common con- 

 structive method. We evade them when we apply, to 

 the investigation of principles, the analytical method 

 of modern algebraical geometry. The whole process 

 of algebraical calculation is a purely logical operation ; 

 it can yield no relation between the quantities sub- 

 mitted to it that is not already contained in the equa- 

 tions which give occasion for its being applied. The 

 recent investigations in question have accordingly been 

 conducted almost exclusively by means of the purely 

 abstract methods of analytical geometry. 



However, after discovering by the abstract method 

 what are the points in question, we shall best get a 

 distinct view of them by taking a region of narrower 

 limits than our own world of space. Let us, as we 

 logically may, suppose reasoning beings of only two 

 dimensions to live and move on the surface of some 

 solid body. We will assume that they have not the 

 power of perceiving anything outside this surface, but 

 that upon it they have perceptions similar to ours. If 

 such beings worked out a geometry, they would of 

 course assign only two dimensions to their space. 

 They would ascertain that a point in moving describes 

 a, line, and that a line in moving describes a surface. 

 But they could as little represent to themselves what 

 further spatial construction would be generated by a 

 surface moving out of itself, as we can represent what 



