THE FOURTH DIMENSION. 83 



which we regarded as substitutes for the eleventh axiom, though 

 valid for the plane, is not true for the surface of a sphere; namely, 

 the postulate that defines the sum of the angles of a triangle. This 

 sum in a plane triangle is two right angles ; in a spherical tri- 

 angle it is more than two right angles, the spherical triangle be- 

 ing greater, the greater the excess the sum of its angles is above 

 two right angles. It will be seen, from these considerations, that 

 in geometries in which curved surfaces and not fixed planes are 

 studied, the axioms of Euclid are either all or partially false. 



The axioms of geometry thus having been revealed as facts of 

 experience, the question suggested itself whether in the same way 

 in which it was shown that different two-dimensional geometries 

 were possible, also different three-dimensional systems of geometry 

 might not be developed ; and consequently what the relations were 

 in which these might stand to the geometry of the space given by 

 our senses and representable to our mind. As a fact, a three-dimen- 

 sional geometry can be developed, which like the geometry of the 

 surface of an egg will exclude the axiom that a figure or body can 

 be transferred from any one part of space to any other and yet re- 

 main congruent to itself. Of a three-dimensional space in which 

 such a geometry can be developed we say, that it has no constant 

 measure of curvature. 



The space which is representable to us, and which we shall 

 henceforth call the space of experience, possesses, as our experiences 

 without exception confirm, the especial property that every bodily 

 thing can be transferred from any one part of it to any other with- 

 out suffering in the transference any distension or any contraction. 

 The space of experience, therefore, Ijas a constant measure of cur- 

 vature. The question, however, whether this measure of curvature 

 is zero or positive, that is, whether the space of experience possesses 

 the properties which in two-dimensional structures a plane pos- 

 sesses, or whether it is the three dimensional analogue of the surface 

 of a sphere is one which future experience alone can answer. If the 

 space of experience has a constant positive measure of curvature 

 which is different from zero, be the difference ever so slight, a point 

 which should move forever onward in a straight line, or, more a c- 



