RELATIVE POSITIONS OF LATERAL MEMBERS. 171 



row. Cells of the nodal sections occupying clearly-defined positions produce the 

 leaves in the ord^r stated. This development furnishes no evidence that the 

 leaves are formed in spiral succession ; the bilateral structure of the stem shows 

 rather that a spiral construction is in this case altogether inadmissible. The same 

 may be shown to be the case in Marsilea, where the creeping stem bears on its 

 upper side two rows of leaves, while the under-side forms roots; the leaves borne 

 on the upper side may in this case be united in the order of their age by a zigzag 

 line broken right and left, which does not anywhere touch the leafless under-side 

 of the stem, and corresponds also in its course to the bilateral structure of the 

 stem. The spiral construction appears also to be meaningless in all those cases 

 where it is indifferent whether the spiral be carried right or left. This is the case 

 where the members are placed in two rows, with a constant divergence of |-, 

 and are thus arranged alternately in two orthostichies lying exactly opposite to 

 one another, as is the case with the branchings of many thallomes {e.g. Stypocaulon, 

 Fig. 98), the leaves of Grasses, the lateral shoots of Tilia, Ulmus, Corylus, &c. In 

 all these cases of decidedly bilateral construction the genetic spiral may be imagined 

 just as well and with the same divergence ascending right or left, by which of course 

 it loses its importance for any morphological conclusion as much as if one sup- 

 posed it to change its direction from leaf to leaf. 



In upright free-growing axes with solitary leaves arranged in three, four, five, 

 or more directions, the spiral construction appears especially natural ; and this 

 also agrees with the symmetrical relationships of plants, of which more will be 

 said hereafter, as well as with the fact that the spiral construction proves to be 

 opposed to nature in bilateral structures, especially in creeping or climbing stems, 

 and in lateral branches. 



In those cases in which the spiral construction may be employed natur- 

 ally, I. e. in the least forced manner, to elucidate the relative positions of the 

 members, two cases may be distinguished, according as the divergences are very 

 unequal and change abruptly, or are nearly or quite equal to one another or only 

 change gradually. In the first case the members appear to be arranged irregularly 

 and without order, as the foliage-leaves on the stem of Fritillaria imperialis 

 (Fig. 143), the flowers on the rachis of the raceme of Triglochin palustre, and 

 in many Dicotyledons. When the change of divergence on the same axis is abrupt, 

 it may also appear more natural to represent the position of the leaves by two 

 homodromal spirals instead of one, as in many species of Aloe, where the shoots 

 commence with leaves arranged in two rows and then pass over into complicated 

 divergences, which lead finally to rosettes of leaves radiating on all sides. This 

 occurs, e. g. in Aloe ciliaris, laii/olia, brachyphylla, lingua, nigricans, and Serra. 

 Fig. 144 shows the transverse section of a shoot of the last-named species; the 

 first six leaves are arranged exactly alternately m two rows with a constant 

 divergence \ ; at the 7th leaf this arrangement is suddenly changed ; instead of 

 being placed over 5, its position is between 5 and 6 ; but the 8th leaf ex- 

 hibits the divergence \ from the 7 th ; the 9 th again changes the divergence, 

 instead of being placed over 7, it is between 7 and 6 ; the loth leaf again di- 

 verges about \ from the 9th; and so it goes on. The leaves 7-15 are evidently 

 arranged in pairs, the pairs being 7, 8 ; 9, 10; n, 12; 13, 14; each pair 



