XIV] OR PHYLLOTAXIS 637 



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

 fractio*ial "divergence" (such as two-fifths of the circumference 

 of the stem) as the botanists describe it, or by an "angle of 

 azimuth" (such as ^ = 144°) as the mathematician would be more 

 likely to state it. The position of B relatively to A must be 

 determined, not only by this angle ^, in the horizontal plane, but 

 also by an angle (6) in the vertical plane ; for the height of B above 

 the level of A, in comparison with the diameter of the cylinder, 

 will obviously make a great difference in the appearance of the 

 whole system, in short the position of each leaf must be expressed 

 by F{<^ . sin 6). 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 another in azimuth. 



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 cylindrical 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 A, 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 

 implies 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, 

 we get two or more appearing simultaneously upon the same level. 

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

 precisely equivalent to one another; and we may call them 

 A, B, C, etc., and A', B', C, and so on. These are the cases 

 which we call "whorled" leaves, or in the simplest case, where 



