240 Church, — The Principles of Phyllotaxis. 



section of m spirals crossing n, in the manner required, is given in such 



a form as, 



log?- = logc + 1-36438 /__£__ _ .000030864 0", 



where the logarithm is the tabular logarithm, and 6 is measured in 

 degrees ; or where the logarithm is the natural logarithm and Q in circular 



measure 



+ 



(^O 



m' + n' 



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 (a + a), (i + a) and (i + 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 (i + 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 + a) 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 



