50 



THE ROOT, OR DESCENDING AXIS. 



etc., the common difference being also five) ; two sets (one of three, the other of 

 eight) turning left ; and still another set, of thirteen, steepest of all, turning right 

 (1 14 27, etc.). Now the sum of the spirals contained in the two steepest sets gives 

 the denominator of the fraction expressing the true formative spiral sought. Thus, 

 8-}-13=21. The numerator corresponding is already known, and the fraction is 

 JL. See also the white pine cone, whose cycle is - 5 r> . 



238. Diagram 97 represents the leaves of a cherry cycle as seen from above, and 

 verified in the aestivation of the flowers in the rose-family. 



MORPHOLOGY OF THE LEAF. 



239. GENERAL CHARACTER. The leaf may be regarded as an expan- 

 sion of the substance of the bark, extended into a broad thin plate by 

 means of a woody frame work or skeleton, issuing from the inner part 

 of the stem. The expanded portion is called the lamina or blade of 

 the leaf, and it is either sessile, that is, attached to the stem by its base, 

 or it is petiolate, attached to the stem by a footstalk called the petiole. 



240. STIPULES. But the regular petiole very often bears at its base 



a pair of leaf-like appendages, more or less ap- 

 parent, called stipules. Leaves so appendaged 

 are said to be stipulate, otherwise they are ex- 

 stipulate. 



241. Therefore a complete leaf consists of 

 three distinct parts ; the lamina or blade, the 

 petiole, and the stipules. 



242. TRANSFORMATIONS. Both the petiole, 

 blade and stipules are subject to numerous mod- 

 ifications of form. Either of them may exist 

 without the others, or they may all be transformed 

 into other organs, as pitchers, spines, tendrils, 

 and even into the organs of the flower, as will 



j hereafter appear. 



OF THE PETIOLE. 



243. THE FORM OF THE DISTINCT PETIOLE is 

 rarely cylindrical, but more generally flattened 

 or channeled on the upper side. When it is 

 flattened in a vertical direction, it is said to be 

 compressed, as in the aspen or poplar. In this 

 case the blade is very unstable, and agitated by 

 the least breath of wind. 



244. THE WINGED PETIOLE is flattened or ex- 



93. Leaf of willow (Saiix 



lucida) ; s, the stipules. The 

 raidvein is 3-lined ; veinlets 



2-iined ;veinuiets single-lined. pan d e d into a margin, but laterally instead of 



99, clover leaves ;, stipules, ^ . o J . 



p petiole i leaflets vertically, as in the asters. Sometimes the 



