GENEEAL MORPHOLOGY OF THE PLANT 



85 



afforded by the Holly and Aconite. In this the ninth leaf is over 

 the first, the tenth over the second, the eleventh over the third, 

 and so on ; thus eight leaves -will complete the cycle ; and, as 

 the spiral line in this distance makes three tvirns round the stem, 

 the angular divergence will be measured by f of the circum- 

 ference, that is, it will be f of 360° = 135°. 



We can arrange the fractions representing these divergencies 

 in a series thus : J, J, f , |, and we see that a certain relation 

 exists between them ; for the numerator of each fraction after 

 the second is equal to the sum of the numerators, and the de- 



FlG. 167. 



Fig. 16H. 



Fig. 167. Pineapple fruit (Sorosis), snnnountea by a ojown of empty bracts. 

 rig. 168. Cone of the Sootoh Fir. Both show a spiral arrangement of the 

 leaves. 



nominator to the sum of the denominators of the two preceding 

 fractions. We may continue the series in the same way, when we 

 get the fractions ^^, /t, &c., and we find that instances of the 

 occurrence of these more complex divergencies are met with. 

 We find also that other series exist. We can meet with oases 

 which taken together form the series J, |, f, ^, &c., and others 

 again which may be arranged in the order ^, |, f , g, ^"j, &c. 



In computing these fractions we have assumed that the 

 genetic spiral is imiform in its direction all along the axis. It 

 may pass either from right to left or in the opposite direction. 

 Occasionally cases are found, however, where the direction of 



