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 -^.i 2 . 



Hence it readily flows 

 that, if OR II AM and 

 RO||AB, the area com 

 prehended between the 

 straights OR, ST, BC [25] 

 (which is manifestly the 

 absolute limit of the area of rectilineal tri 

 angles increasing without bound, or of A for 

 ^=st.^), is r^= area 0/, in F. 



This limit being denoted by n, moreover 

 (by 30) rr 2 -tan 2 ^.n= area Or in F ( 21) = 

 area 0s (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 L form 

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

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

 erecting CM J_ CA, we shall 

 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 



FIG. 41. 



A_ of the same curvature (which, 

 D their extremities alone being 



given 



and their axes con- 



