140 THE LEAVES. 



take, however, the order of secondaiy spirals nearest the vertical 

 rank in each direction, when there are more than two, as thei'e are 

 in all the succeeding cases. 



245. A similar diagram of the f arrangement introduces a set of 

 secondary spirals, in addition to the two foregoing, ascending in a near- 

 er approach to a vertical line, and with a higher common difference, 

 viz. 6. There are accordingly five of this sort, viz. those indicated 

 in the diagi-am by the series 1, 6, 11, 16 ; 4, 9, 14, 19, 24 ; 2, 7, 12, 

 17, 22 ; 5, 10, 15, 20, 25 ; and 3, 8, 13, 18, 23. The highest obvious 

 spiral in the opposite direction, viz. that of which the series 1, 4, 7, 

 10, 13 is a specimen, has the common difference 3, and gives the 

 numerator, and 3 -)- 5 the denominator, of the fraction f . The next 

 case, y^3, which is exemplified in the rosettes of the Houseleek (Fig. 

 207) and in the cone of the Wliite Pine (Fig. 209), introduces a 

 fourth set of secondary spirals, eight in number, with the common 

 difference eight, viz. that of which the series 1, 9, 17, 25 is a repre- 

 sentative. The set that answers to this in the opposite direction, 

 viz. 1, 6, 11, 16, 21, 26, with the common difference 5, gives the 

 numerator, and 5-1-8 the denominator, of the fraction -fr^. We may 

 here compare the diagram with an actual example (Fig. 209) : a 

 part of the numbers are of course out of sight on the other side of 

 the cone. The same laws equally apply to the still higher modes. 



246. The order is uniform in the same species, but often various 

 in alhed species. Thus, it is only § in our common American Larch ; 

 in the European species, -f^. The White Pine is y^^, as is al?o the 

 Black Spruce ; but other Pines witli thicker cones exhibit in differ- 

 ent species the fractions -^j, ^, and f i-. Sometimes the primitive 

 spiral ascends from left to right, sometimes from right to left. One 

 direction or the other generally prevails in each species, yet both 

 directions are not unfrequently met with, even in different cones of 

 the same tree. 



247. Wlien a branch springs from a stem or parent axis, the spi- 

 ral is continued from the leaves of the stem to those of the branch, so 

 that the leaf from whose axil the branch arises begins the spire 

 of that branch. When the spire of the branch turns in the same 

 direction as that of the parent axis, as it more commonly does, it is 

 said to be homodromous (from two Greek words, signifying lilce 

 course) : when it turns in the opposite direction, it is said to be 

 heterodromous (or of different course). 



248. The cases represented by the fractions ^, ^, and f are the 



