SCIENCE ABSOLUTE OF SPACE. 31 



areas bounded by these lines could also be ex- 

 pressed. 



We know, that a surface t, \\\ to a plane fig- 

 ure/* (at the distance q], is to/> in the ratio of 

 the second powers of homologous lines, or as 

 f 4 -\ 2 



I^ |^i_?i J : 1. 



It is easy to see, moreover, that the calcula- 

 tion of volume, treated in the same manner, 

 requires two integrations (since the differen- 

 tial itself here is determined only by integra- 

 tion) ; and before all must be investigated the [in 

 volume contained between p and /, and the ag- 

 gregate of all the straights A-p and joining 

 the boundaries of p and t. 



We find for the volume of this solid (whether 

 by integration or without it) 



f 2q ^ 



The surfaces of bodies may also be deter- 

 mined in S, as well as the curvatures, the 

 involutes, and evolutes of any lines, etc. 



As to curvature; this in S either is the curv- 

 ature of L, or is determined either by the 

 radius of a circle, or by the distance to a 

 straight from the curve ||| to this straight; since 

 from what precedes, it may easily be shown, 

 that in a plane there are no uniform lines other 

 than L-lines, circles and curves ||| to a straight. 



