Church.'-^The Principles of Phyllotaxis. 241 



be taken as established; and attention may now be drawn to anotHer 

 feature of the mathematical proposition. 



It follows from the form of the equation ascribed to the quasi-circle 

 that whatever value be given to m. and «, the curve itself is bilaterally 

 symmetrical about a radius of the whole system drawn through its centre 

 of construction. That it should be so when »« = », i.e. in a symmetrical 

 (wkorled) leaf-arrangement, would excite no surprise ; but that the primor- 

 dium should be bilaterally symmetrical about a radius drawn through 

 its centre of construction, even when the system is wholly asymmetrical 

 and spiral, is little short of marvellous, since it implies that identity of 

 leaf-structure in both spiral and whorled systems, which is not only their 

 distinguishing feature, but one so usually taken for granted that it is 

 not considered to present any difficulty whatever. Thus, in any system of 

 spiral phyllotaxis, the orientation of the rhomboidal leaf-base is obviously 

 oblique, and as the members come into lateral contact they necessarily 

 become not only oblique but asymmetrical, since they must under mutual 

 pressure take the form of the full space available to each primordium, 

 the quasi-square area which appears in a spiral system as an oblique 

 unequal-sided rhomb (Fig. 35). Now the base of a leaf (in a spiral system) 

 is always such an oblique, anisophyllous structure, although the free appen- 

 dage is isophyllous, bilaterally symmetrical, and flattened in a horizontal 

 plane ^. The quasi-circle hypothesis thus not only explains the inherent 

 bilaterality of a lateral appendage, but also that peculiar additional attri- 

 bute which was called by Sachs its ' dorsiventralityl or the possession 

 of different upper and lower sides, and what is more remarkable, since 

 it cannot be accounted for by any other mathematical construction, the 

 isophylly of the leaves produced in a spiral phyllotaxis system ^- 



It has been the custom so frequently to assume that a leaf-primordium 

 takes on these fundamental characters as a consequence of biological 

 adaptation to the action of such external agencies as light and gravity, that 

 it is even now not immaterial to point out that adaptation is not creation, 

 and that these fundamental features of leaf-structure must be present in the 

 original primordium, however much or little the action of environment may 



' These relations are beautifully exhibited in the massive insertions of the huge succulent leaves 

 of large forms of Agave : the modelling of the oblique leaf-bases with tendency to rhomboid section, 

 •as opposed to that of the horizontal symmetrical portion of the upper free region of the appendage, 

 may be followed by the hand, yet only differs in bulk from the case of the leaves of Sempervivum or 

 the still smaller case of the bud oiPinus. 



". Anisofhylly is equally a mathematical necessity of all eccentric shoot systems. 



It will also be noted that the adjustment required in the growing bud, as the free portions of 

 such spirally placed priinordia tend to orientate their bilaterally symmetrical lamina in a radial and 

 not spiral plane, gives the clue to those peculiar movements in the case of spired growth systems, which, 

 in that they could be with difficulty accounted for, although as facts of observation perfectly obvious, 

 has resulted in the partial acceptance of Schwendener's Dachstuhl Theory. This theory was in fact 

 mainly based on the necessity for explaining this ' slipping ' of the members, but in the logarithmic 

 spiral theory it follows as a mathematical property of the construction. 



