Church . — The Principles of Phyllo taxis. 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 n, 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 m—n , i. e. in a symmetrical 
( whorled ) 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 1 . 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 * dorsiventrality] 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 2 . 
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 
1 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, 
us 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 of Pinus. 
2 Anisophylly 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 primordia 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 spiral 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. 
