216 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



curiously academic view of a plant thus survives very generally 

 in text-books which bears little reference to the facts of ontogeny 

 and the manner in which a leafy shoot is actually constructed. 

 The fact that all internodes are secondary and subsidiary growths, 

 and that the elongation of a typical shoot is a secondary and 

 extremely complicated phenomenon, is often forgotten or unex- 

 pressed. The fact that the arrangement of leaves on such shoots 

 produces the subjective effect of circles or winding spirals is also 

 entirely secondary, the primary construction system only being 

 observed at the apex of the shoot, or on shoots which exhibit no 

 secondary elongation whatever. 



Leaving on one side, therefore, all academic prejudices in favour 

 of whorls and a single genetic-spiral traced on an elongated leafy 

 axis by drawing a subjective line through successive members, 

 the actual data of the rhythm exhibited by the plant in building 

 its leafy shoot system reduce merely to the enumeration of a 

 certain number of curves which intersect in either direction. No 

 further data can be obtained from the living organism than such 

 observation of these intersecting curves, the contact-parastichies. 

 These are therefore simple numerical expressions involving two 

 whole numbers only ; and not only so, but every additional 

 factor read into the subject comprises, to use Sachs' expression, 

 " gratuitously introduced mathematics." 



There can be, however, no objection to the introduction of the 

 mathematical properties of the numbers, since the numbers are 

 given ; and the fact that mathematics may be introduced follows 

 directly from the presence of continued rhythmic phenomena. 

 But error creeps in as soon as the bare numerical data afforded 

 by the plant are combined to constitute a mathematical expression 

 or formula. The facts of observation supply an intersecting 

 system of equally distributed spiral curves, the number of which 

 must be integral and can usually be readily checked. The only 

 additional mathematical data that can be introduced, therefore, 

 are the mathematical properties of such intersecting curves. But 

 in expressing the relation of the numbers of these intersecting 

 curves, care must be taken to render the resultant expression 

 mathematically harmless. To this end, the notation has been 



