218 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



reading the pattern — that is, as a complex of intersecting spiral 

 curves. The mathematical properties of such intersecting spirals 

 are readily deduced mathematically, and still more obviously by 

 simple geometrical constructions, of which several examples have 

 been previously given (figs. 25, 26, 28, 55, 63, 70). From these it 

 becomes clear that, in dealing with such intersecting curves, three 

 cases are mathematically possible, and all occur widely distributed 

 in the plant-kingdom.* 



First, if the two integers which express the spiral curves in 

 either direction are divisible by unity only, one spiral of the same 

 class can be drawn through the entire series of intersections. A 

 numerical value can be given to all the points of intersection by 

 counting along the spirals in either direction numerals differing 

 by the number of the same spirals in the set. The fact that such 

 a numerical value can be given is a mathematical consequence of 

 the peculiar curve construction ; and in this case, since one spiral 

 passes through the entire series of points, the numerals utilised 

 are successive numbers (Braun's method). 



Secondly, if the two integers are divisible by a common factor 



(w), n spirals of the same class can be each drawn through — of 



n 



the points of intersection (fig. 70) ; the same method of numbering 



up does not utilise successive numerals, but gives n sets. 



Lastly, as a special case of the preceding, equality of the 

 integers results in the same number of spirals passing each 

 through its own share of the points ; but each set of points lies 

 on a common and readily observed circular path. 



These three sets of mathematical phenomena are properties not of 

 plants hut of intersecting spiral curves. They follow in the plant 

 because the rhythmic expression of phyllotaxis takes this 

 particular form of distribution. Why it should take this form, 

 is of course the next fundamental question. But so far it will 

 be seen that the first case constitutes the condition of normal 

 spiral phyllotaxis, extremely general because the Fibonacci ratios 



* For the general proof of these statements in mathematical form I am 

 indebted to Mr H. Hilton ; for log. spirals or spirals of Archimedes it can be 

 shown geometrically on the diagrams, 



