SECT. I. OKGANOGRAPHY AND GLOSSOLOGY. 



131 



directions. And always, where the denominator of the 

 fraction is a prime number, 

 there will exist one number 

 less than that of the divergences, 

 according to which the gener- 

 ating spiral may be construct- 

 ed and a similar number of 

 vertical ranges will still be the 

 result. But where -the deno- 

 minator is not a prime number, 

 then some of the fractions which 

 express these different diver- 

 gences, are not in their lowest 

 terms ; and these divergences 

 represent the very same spirals 

 as when such fractions are so 

 reduced. Thus, when there are 

 six vertical ranges (fig. 141.), 

 the divergences may be taken as 



'*// 



.' '','l 



represent the trifarious arrange- 

 ment ; also jj- i, which is the 

 distichous. Hence ^ and -j? are >/' 

 the only divergences which fj 

 represent the hexafarious ar- | | 

 rangement, and even these may ' ' 

 be reduced to one kind, only ^ \ 

 the spiral would be turned in * \ 

 opposite directions in the two 

 cases. 



= f, both of which 



*\ 

 i 



=>/ 



I I 



