XIV] OR PHYLLOTAXIS 917 



Another, and a younger leaf (B) will be found standing at a certain 

 distance around the stem, and a certain distance along the stem, 

 from the first. The former distance may be expressed as a fractional 

 "divergence" (such as two-fifths of the circumference of the stem) 

 as the botanists describe it, or by an "angle of azimuth" (such as 

 (f) = 144°) as the mathematician would be more likely to state it. 

 The position of B relatively to A may be determined, not only by 

 this angle (/>, in the horizontal plane, but also by an angle of slope {6), 

 or merely by linear distance from its basal plane; for the height of 

 B above the level of ^, in comparison with the diameter of the 

 cylinder, will obviously make a great difference in the appearance of 

 the whole system. But this matter botanical students have not 

 concerned themselves with ; in other words, their studies have been 

 limited (or mainly limited) to the relation of the leaves to one ailother 

 in azimuth — in other words, to the angle ^ and its multiples. 



Whatever relation we have found between A and B, let precisely 

 the same relation subsist between B and C: and so on. Let the 

 growth of the system, that is to say, be continuous and uniform; 

 it is then evident that we have the elementary conditions for the 

 development of a simple cyhndrical helix; and this "primary helix" 

 or "genetic spiral" we can now trace, winding round and round the 

 stem, through A, B,C, etc. But if we can trace such a helix through 

 A, B, C, it follows from the symmetry of the system, that we have 

 only to join A to some other leaf to trace another spiral helix, such, 

 for instance, as ^, C, E, etc. ; parallel to which will run another and 

 similar one, namely in this case B, D, F, etc. And these spirals 

 will run in the opposite direction to the spiral ABC*. 



In short, the existence of one helical arrangement of points im- 

 plies and involves the existence of another and then another helical 

 pattern, just as, in the pattern of a wall-paper, our eye travels from 

 one linear series to another. 



A modification of the helical system will be introduced when, 

 instead of the leaves appearing, or standing, in singular succession, 

 w^ get two or more appearing simultaneously upon the same level. 

 If there be two such, then we shall have two generating spirals 



* For the spiral ACE to be different from ABC, the angle of divergence, or angle 

 of azimuth for one step, must exceed 90°, so that the nearer way from ^ to C is 

 backwards; otherwise the spiral ACE is ABCDE, or ABC over again. 



