IDEAL ANGLES. 



73 



and this is borne out by the observation that these higher numbers 

 are found, intermingled with the normal series, in such inflorescences 

 as those of Helianthus, and especially of Dipsacus fullonum. 

 Others are met with in the leafless Cactaceae, bractless spadices of 

 Aroids, and xerophytic types, species of Lycopodium, Sedum, 

 Euphorbia, etc. 



Only in some of the lower terms is any marked advance toward 

 symmetrical curve-construction exhibited ; e.g. (3 : 4), (4 : 5), and 

 (5 : 6). Cases in which such types occur are again no commoner 

 than even closer approximations to equality, for which the necessity 

 of constructing continuous fractional expressions is still less apparent. 

 Thus :— 



