ADDRESS. 23 



it is a mode of representation, or a method which, having served its pur- 

 pose, vanishes from the scene. Like a rainbow, if we try to grasp it, it 

 eludes our very touch ; but, like a rainbow, it arises out of real conditions 

 of known and tangible quantities, and if rightly apprehended it is a true 

 and valuable expression of natural laws, and serves a definite purpose in 

 the science of which it forms part. 



Again, if we seek a counterpart of this in common life, I might remind 

 you that perspective in drawing is itself a method not altogether dissimilar 

 to that of which I have been speaking ; and that the third dimension of 

 space, as represented in a picture, has its origin in the painter's mind, and 

 is due to his skill, but has no real existence upon the canvas which is the 

 groundwork of his art. Or again, turning to literature, when in legendary 

 tales, or in works of fiction, things past and future are pictured as present, 

 has not the poetic fancy correlated time with the three dimensions of 

 space, and brought all alike to a common focus ? Or once more, when 

 space already filled with material substances is mentally peopled with 

 immaterial 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 experimental fact? 



The third method proposed for special remark is that which 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 con- 

 sidered as fundamental, 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 uni- 

 form or constant. 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 fundamen- 

 tal 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, viz., that a geo- 

 metrical figure may be moved in space without change of size or alteration 

 of form, entirely falls away, or becomes only approximate in a space 

 wherein dimension and form are dependent upon position. For instance, 

 if we consider merely the case of figures traced on a flattened globe like 



