RELATIVE POSITIONS OF LATERAL MEMBERS. 



17.3 



the rosette of leaves of the older shoot radiating on all sides. Similar relation- 

 ships appear to exist in Dracaena and in some Aroideoe. At first sight such kinds 

 of phyllotaxis appear as if the leaves were placed in two lines which have become 

 changed by the torsion of the stem ; but this hypothesis seems in this case scarcely 

 admissible. 



If we now turn, in conclusion, to those cases which clearly gave rise in times 

 past to the erroneous hypothesis that the primary law of phyllotaxis is a universal 

 spiral arrangement, we find the leaves placed singly, their divergences almost or 

 quite equal or gradually passing over into some other value corresponding to the 

 second case named above of spiral arrangement. In these cases the spiral con- 

 struction affords a simple expression of the law of phyllotaxis; the only thing 

 required is to name the constant angle of divergence ; — according as this is ^, 

 ], i, f, I, &c., the phvllotaxis is termed simply one of ^, i, i, and so on. 

 It is usual in such cases for the divergence not to remain constant for all the 

 members of an axis ; shoots which form numerous leaves mostly begin with more 

 simple arrangements, as .J, and then pass over into more complicated ones, an 

 arrangement being considered more complicated when the numerator and denomi- 

 nator of the fraction of divergence are larger. When the divergences between 

 lateral members placed solitarily with spiral arrangement are equal, they must then 

 also stand in straight rows, the number of which is expressed by the denominator of 



the angle of divergence. If, for instance, 

 the divergence is a constant one of f , as 

 in Fig. 145, there are eight orthostichies, 

 the 9th member standing on the same 

 median plane as the ist, the loth as the 

 2nd, the nth as the 3rd, and so on. In 

 a f phyllotaxis, in the same manner, the 

 6th member stands over the ist, the 7th 

 over the 2nd, and so on. In some cases 

 the orthostichies are very obvious, as, for 

 instance, in Cacti with prominent angles, 

 the angles corresponding to the ortho- 

 stichies of the spirally arranged leaves, 

 which, however, in this case mostly re- 

 main undeveloped. In verticillate leaves 

 also the straight rows are mostly conspicuous if the shoot is looked at from above, 

 as, for instance, in the decussate two-leaved whorls oi Etiphorbia Lathyris, and the 

 cactus-like E. canarieiisis. 



When the members of a spiral phyllotaxis with a constant angle of divergence 

 stand sufficiently close to one another, spiral arrangements are easily seen and 

 followed to the right and left which more or less conceal the genetic spiral. These 

 rows are called Parastichies, and are particularly clear in the cones of species 

 of Pinus, the leaf-rosettes of Crassulaceae, the flowers of the sunflower and other 

 Compo^itee, and in the spadices of Aroide^. They may be seen in every spiral 

 phyllotaxis with a constant divergence, and can always be made clear in the dia- 

 gram, or when the arrangement is represented on an unrolled cylindrical surface. 



Fig. 145.— Diagram of a shoot in which the leaves have 



a cuiistant phyllotaxis of tj. 



