240 
Church — The Principles of Phyllotaxis . 
section of m spirals crossing «, in the manner required, is given in such 
a form as, 
log c + 1-36438 
log r — 
— 5 — .000030864 6 2 , 
m 2 + 0 
where the logarithm is the tabular logarithm, and 0 is measured in 
degrees ; or where the logarithm is the natural logarithm and 6 in circular 
measure : 
From these equations the curve required for any phyllotaxis system 
can be plotted out ; and a series of three such curves is shown in Fig. 41, 
grouped together for convenience of illustration, i. e. those for the lowest 
systems (2 + 2), (1 + 2) and (1 + 1). 
It will be noticed immediately that the peculiar characters of these 
curves are exaggerated as the containing spiral curves become fewer: 
thus with a larger number than 3 and 5, the difference between the shape 
of the curve and that of a circle would not be noticeable to the eye. 
While in the kidney-shaped (1 + 1) curve the quasi-circle would no longer 
be recognized as at all comparable in its geometrical properties with 
a true centric growth-centre. But even these curves, remarkable as they 
are, are not the shape of the primordia as they first become visible at the 
apex of a shoot constructing appendages in any one of these systems. 
The shape of the first formed leaves of a decussate system, for example, 
is never precisely that of the (2 + 2) curve (Fig. 41), but it is evidently of 
the same general type ; and it may at once be said that curves as near 
as possible to those drawn from the plant may be obtained from these 
quasi-circles of uniform growth by taking into consideration the necessity 
of allowing for a growth-retardation. Growth in fact has ceased to be 
uniform even when the first sign of a lateral appendage becomes visible 
at a growing point ; but, as already stated, this does not affect the correct- 
ness of the theory in taking this mathematical construction for the 
starting-point ; and, as has been insisted upon, the conception of the actual 
existence of a state of uniform growth only applies to the hypothetical 
‘ growth-centre.’ 
On the other hand, the mere resemblance of curves copied from the 
plant to others plotted geometrically according to a definite plan which 
is however modified to fit the facts of observation, will afford no strict 
proof of the validity of the hypothesis, although it may add to its general 
probability, since there is obviously no criterion possible as to the actual 
nature of the growth-retardation ; that is to say, whether it may be taken 
as uniform, or whether, as may be argued from analogy, it may exhibit daily 
or even hourly variations. Something more than this is necessary before 
the correctness of the assumption of quasi-circular leaf-homologues can 
