148 THE LEAVES. 



difference 3, and gives the numerator, and 3 -f- 5 the denomina- 

 tor, of the fraction f . The next case, y 5 ^, which is exemplified in 

 the rosettes of the Houseleek (Fig. 174) and in the cone of the 

 White Pine (Fig. 176), introduces a fourth set of secondary spi- 

 rals, eight in number, with the common difference 8, viz. that of 

 which the series 1, 9, 17, 25 is a representative. 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 -f- 8 the 

 denominator, of the fraction T 5 vj-. We may here compare the dia- 

 gram with an actual example (Fig. 176) : 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 vari- 

 ous in allied species. Thus, it is only f in our common American 

 Larch ; in the European species, 8 T . The White Pine is y 5 ^, as is 

 also the White Spruce ; but other Pines with thicker cones exhibit 

 in different species the fractions / r , ^f , 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 spe- 

 cies, yet both directions are not unfrequently met with even in the 

 same individual plant. 



247. But when a branch springs from a stem or parent axis, 

 the spiral is found to be continued directly 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 like course) : when it turns in 

 the opposite direction, it is said to be hcterodromous (or of unlike 

 course ) . 



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

 most stable and certain, as well as the easiest to observe. In the 

 higher forms, the exact order of superposition often becomes un- 

 certain, owing to a slight torsion of the axis, or to the difficulty 

 of observing whether the 9th, 14th, 21st, 35th, or 56th leaf is di- 

 rectly over the first, or a little to the one side or the other of the 

 vertical line. Indeed, if we express the angle of divergence in 

 degrees and minutes, we perceive that the difference is so small a 

 part of the circumference, that a very slight change will substitute 

 one order for another. The divergence in -f-j = 138 24'. In all 



