42 



terial beings, may not the imagination be regarded as 

 having added a new element to the capacity of space, a 

 fourth dimension of which there is no evidence in experi- 

 mental fact ? 

 Non-Euclid The third method proposed for special remark is that 

 geometry. ^yhich has been termed Non-Euclidean Geometry ; and 

 the train of reasoning which has led to it may be 

 described in general terms as follows : some of the pro- 

 perties of space which on account of their simplicity, 

 theoretical as well as practical, have, in constructing the 

 ordinary system of geometry, been considered as funda- 

 mental, are now seen to be particular cases of more 

 general properties. Thus a plane surface, and a straight 

 line, may be regarded as special instances of surfaces and 

 lines whose curvature is everywhere uniform or con- 

 stant. And it is perhaps not difficult to see that, when 

 the special notions of flatness and straightness are 

 abandoned, many properties of geometrical figures which 

 we are in the habit of regarding as fundamental will 

 undergo profound modification. Thus a plane may be 

 considered as a special case of the sphere, viz., the limit 

 to which a sphere approaches when its radius is increased 

 without limit. But even this consideration trenches 

 upon an elementary proposition relating to one of the 

 simplest of geometrical figures. In plane triangles the 

 interior angles are together equal to two right angles ; 

 but in triangles traced on the surface of a sphere this 

 proposition does not hold good. To this, other instances 

 might be added. 



Further, these modifications may affect not only 

 our ideas of particular geometrical figures, but the 

 very axioms of the Science itself. Thus, the idea 

 which, in fact, lies at the foundation of Euclid's method 

 that a geometrical figure may be moved in space 



