DISTRIBUTION OF LEAVES ON THE STEM. 129 



It is further noticeable that the smaller number of parallel sec- 

 ondary spirals, 2, agrees with the numerator of the fraction in 

 this the f arrangement ; and that this number, added to that of 

 the parallel secondary spirals which wind in the opposite direction, 

 viz. 3, gives the denominator of the fraction. This holds good 

 throughout ; so that we have only to count the number of par- 

 allel secondary spirals in the two directions, and assume the 

 smaller number as the numerator, and the sum of this and the 

 larger number as the denominator, of the fraction which ex- 

 presses the angular divergence sought. For this, we must, how- 

 ever, take the order of secondary spirals nearest the vertical 

 rank in each direction, when there are more than two, as in all 

 the higher forms. But, in all, it is necessary to count only the 

 most manifest secondary spiral of each direction in order to 

 lay down the proper number on the leaves or scales, and so deter- 

 mine the phyllotaxy. 1 In a rosette of the leaves of Houseleek 

 (Fig. 245) and a cone of Pinus Strobus (Fig. 246), the num- 

 bers which can be seen at one view are appended, and in the 

 latter the conspicuous secondary spirals are indicated : one to 

 left with a common difference of 5 ; and two to the right, of 

 which the most depressed and prominent has the common dif- 

 ference of 3, the other, nearest the vertical, the common differ- 

 ence 8. The 14th leaf is superposed to the first, indicating the 

 ^ arrangement. The same conclusion is derived from the num- 

 ber of the higher spirals, the smaller 5 for the numerator, and 

 this added to 8 for the denominator. The mathematical discus- 

 sion of these relations, and of the whole subject of phyllotaxy, 

 leads into interesting fields. But this sketch may suffice for 

 botanical uses. 



246. Relations of Whorls to Spirals. Verticillate and alternate 

 phyllotaxy, or whorls and spirals, in all complete exemplifica- 

 tions, are to be considered morphologically as distinct modes, 

 not to be practical!}' homologized into one. Nevertheless, transi- 

 tions between the two, and abrupt changes from one to the other 

 on the same axis are not uncommon, the former especially in 

 the foliage, the latter in the blossom. If the spiral be assumed 

 as the fundamental order, it is not difficult to form a clear con- 

 ception as to how such changes come to pass. A single whorl 



1 In applying this method to the determination of the phyllotaxy of a 

 cone, or any such assemblage of leaves, the student should be warned that, 

 although the cones of Pines and Firs are all normally on the alternate plan 

 (while those of Cypresses are on the verticillate), yet in individual cases 

 (common in Norway Spruce) the cone is plainly made up of pairs of oppo- 

 site scales which are spirally arranged. See note under 236. 



