46 



SCIENCE ABSOLUTE OF SPACE. 



M 



A 



FIG. 40. 



Designating henceforth, for brevity, any tri- 

 angle the supplement of whose angle-sum is z 

 by A, we will therefore have A z.i^ 1 . 



Hence it readily flows 

 that, if OR II AM and 

 RO||AB, the area com- 

 prehended between the 

 straights OR, ST, BC D: 



(which is manifestly the 

 absolute limit of the area of rectilineal tri- 

 angles increasing without bound, or of A for 

 ^=st.^), is T,^ area 0^, in F. 



This limit being denoted by n , moreover 

 (by 30) 7rr 2 =tan 2 ^.n = area >r in F ( 21) = 

 area s (by 32, VI) if the chord CD is called s. 

 If now, bisecting at right angles the given 

 radius 5 of the circle in a plane (or the Iy form 

 radius of the circle in F), we construct (by 

 34) DB|^CN; by dropping CA 1 DB, and 

 erecting CM 1 CA, we shall 

 1 get z; whence (by 37), assum- 

 ing at pleasure an L form 

 radius for unity, tan 2 ^ can be 

 determined geometrically by 

 means of two uniform lines 

 A_ of the same curvature (which, 

 3 their extremities alone being 

 given and their axes con- 



FIG. 41. 



