194 



MORPHOLOGY OF MEMBERS, 



may be followed in this manner up to the terminal rosette of leaves. Similar 

 phyllotaxes appear to occur in Dracccna and in some Aroidea^ ; and at first sight 

 present the appearance as if the leaves were placed in two rows which have become 

 changed into spirals by the torsion of the stem. 



If we now turn to those cases which clearly gave rise to the erroneous 

 hypothesis that the primary law of phyllotaxis is a universal spiral arrangement, 

 we find the leaves placed singly, and their divergences almost or quite equal or 

 gradually passing over into some other value, thus corresponding to the second 

 case named above of spiral arrangement. In these cases the spiral construction 

 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 J, j, -g^, f , |, 

 &c., the phyllotaxis is termed simply one of \, \, \, and so on. It is usual in 

 such cases for the divergence not to remain constant for all the lateral members 

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

 arrangements, as ^, and then pass over into more complicated ones, an arrange- 

 ^ ment being considered more complicated 



when the numerator and denominator 

 of the fraction of divergence are larger. 

 When the divergences between lateral 

 members placed solitarily with a spiral 

 arrangement are equal, they must also 

 stand in straight rows, the number 

 of which is expressed by the denom- 

 inator of the angle of divergence. If, 

 for instance, the divergence is a constant 

 one of I, as in Fig. 153, 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 to the stem, the angles corresponding to the orthostichies of the 

 spirally arranged leaves, which, however, in this case mostly remain undeveloped. 

 In verticillate leaves also the straight rows are mostly conspicuous if the shoot be 

 looked at from above, as, for instance, in the decussate two-leaved whorls of Eu- 

 phorbia Lathy r is ^ and the cactus -like E. canarihtsis. 



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

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

 one of which may be followed to the right and the other to the left, and more or less 

 completely concealing the genetic spiral. These rows are called Parastichies, and 

 are particularly clear in fir-cones, the leaf-rosettes of Crassulacese, the flower-heads 

 of the sunflower and other Compositse, and the spadices of Aroideae. They may be 

 seen in ev-ery spiral phyllotaxis with a constant divergence, and can always be made 

 clear in the diagram, or when the arrangement is represented on an unrolled 

 cylindrical surface. The consideration of these constructions leads to definite 



Fig. 153.— Diagram of a shoot in which the leaves have 

 3 constant phyllotaxis of S. 



