70 
It is easy to see that the multiplical over MBF is: 
P(BMF) = P(BMC): P(FMCO)=v = =u 
1 
The rational angle comprised between two rational radii through 
M is the second power of the rational area of the figure enclosed 
by the radii and the logarithmic circle with radius e and centre 
M. For the rational area of a logarithmic circle holds: 
(nr)? 
For rational length of chord and circumference we find: 
r, u and r 7 
By two rationals of centre sectors are cut out of concentric 
logarithmic circles whose rational areae form with the second 
gradations of the radii a logarithmic proportion. Such figures are in 
rational sense congruent. 
§ 28. Besides the rational circle functions the rational hyperbolic 
functions are of importance. Just as in difference geometry they 
appear in the simplest way by the consideration of areae of the 
logarithmic equilateral hyperbola with equation : 
*(x) : *(y) =*(@)- 
Side by side with the current notation of the ordinary functions 
we can write: 
x % » 
1 1 
hin=a(u+—) 7 dum t(e) 
u u 
If a is again the parameter the area of a sector is given by: 
