180 ADAAIS— THE EINSTEIN THEORY. 



the distance between any two points shall be expressed as the sum 

 of the squares of the relative coordinates of the two points. In the 

 generalized space of Riemann this cannot be done. An analogy will 

 make this distinction clear. A plane in three-dimensional space 

 may be regarded as Euclidean space of two dimensions, for by 

 choosing any rectangular coordinates in it it is possible to express 

 the square of the distance between two points as the sum of the 

 squares of the relative coordinates of the points. A curved surface 

 in three dimensions, however, is non-Euclidean space of two dimen- 

 sions, for the distance between two points on the surface measured 

 along the surface cannot be expressed in the same way as on a plane. 

 The geometry of curved surfaces in three-dimensional space was 

 developed by Gauss, and Riemann's geometry is an extension of the 

 Gaussian methods to surfaces of a greater number of dimensions. 

 In this way the conception of curvature of space arose, as a per- 

 fectly logical development of the easily conceived curvature of a 

 surface. Space of zero curvature is Euclidean space ; if the curva- 

 ture is different from zero, whether constant or varying from point 

 to point, space is non-Euclidean. Measurements on a two-dimen- 

 sional surface will tell whether the surface is plane or curved — that 

 is, whether it is Euclidean space or not. For by measuring the 

 circumference of a circle drawn on the surface with a known 

 radius, if the circumference is 27r times the radius, the surface is 

 plane. If the surface is curved the result will in general be dif- 

 ferent. So it might be thought that measurements in our actual 

 three-dimensional space would tell whether our space is Euclidean 

 or not. In fact. Gauss did attempt to test this question by carefully 

 measuring the angles between three distant points, but needless to 

 say he found no departure from Euclidean space. 



We must now consider the question of time. Until Lorentz in- 

 troduced what he called the " local time " in his theory of electrical 

 and optical phenomena in moving bodies, and thus laid the founda- 

 tion for the theory of relativity, time and space were regarded as 

 wholly independent concepts, at least for the purpose of describing 

 physical phenomena. Our knowledge of the physical universe we 

 obtain by experience, and it is certainly true that no one ever de- 

 termined a position in space except at a definite time, nor noted a 



