164 RELATION OP PHYLLOTAXlS TO MECHANICAL LAWS. 



between superposition which is mathematically accurate and that 

 which is only apparent to the eye. 



The determination of phyllotaxis systems in flower-shoots 

 in which the construction is not continued for a sufiScient 

 number of members to judge whether the apparent orthostichies 

 are truly vertical or really spiral, may present a difficulty. 



Thus, spiral flowers may be constructed in the systems (1 + 2), 

 (2 + 3), (3 + 5), giving respectively cycles of 3, 5, or 8 apparently 

 superposed members, on the lines of the three-spired Gyperus, 

 five-spired Apicra, or an eight-ridged Cactus or EwphorKa melo- 

 formis. 



If the number of members is few, and their relative bulk 

 very nearly equal, superposition may be sufficiently accurate 

 to the eye, or may actually become so by secondary growth 

 changes, as possibly in the flowers of Beta and ATnaranthus with 

 superposed perianth and androecium. 



Thus the five-spired terminal flower of Berheris vulgaris presents 

 cycles sufficiently superposed to the eye in the expanded flower, 

 but in development the spires are better marked, so that the first- 

 formed sepals would not be said to be at all superposed to the 

 petals : the construction being, in fact, as markedly (3 + 5) as 

 in the case of Delphinium Ajacis. Similarly in Nigella damasccTva, 

 in which the androecium is constructed in a (5 -|- 8) system, the 

 eight shorter curves, which are well marked in the expanded 

 flower, have been interpreted as "obhque orthostichies." 



On the other hand, mathematical superposition can only 

 be produced in a symmetrical construction in which circles 

 and straight lines really are present as the orthogonal construction 

 paths of the system, and in such cases the superposition takes place 

 between members of alternate whorls, the construction being that 

 of a concentrated system. 



Again, when some of the floral members show true alternation 

 and others do not (Buta, Primula), some secondary change must 

 be implied; the presence of (5 -|- 5) formation in part shows that 

 at one time the essential organs of the flower must have attained 

 this symmetrical construction throughout, and a definite stand- 

 point is thus opened up for the consideration of obdiplostemony, 



