208 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



metrical growth was founded, as in the case of the three- sided apical- 

 cell of the Fern, the genetic spiral is present and apparently actually 

 represents the asymmetrical formation of new growth-centres, one 

 at a time. To what extent this can be regarded as holding for the 

 more complicated production of the growth-centres of more massive 

 primordia must necessarily be obscure, until more is known as to 

 what is really implied by the convention " growth-centre," and how 

 far such a centre has any material existence, or possesses a finite 

 character. 



It is meanwhile interesting to note that the genetic spiral as a 

 single determining path was the creation of Schimper, and that the 

 older writers, including Bonnet, were content with the expressions 

 " Multiple Spirals," " Parallel Spires," for even slightly compUcated 

 constructions. The deduction of a single genetic spiral is, in fact, 

 the result of the assumption of a spiral of Archimedes as the funda- 

 mental growth spiral. The utilisation of such a spiral, passing 

 through equidistant points on the radii vectores, is clearly the 

 simplest mode of expressing such a construction ; and Sachs is so 

 far correct in stating that the orthostichy system of Schimper and 

 Braun is preferable to the parastichy system of the Bravais : if a 

 given set of points can be defined in terms of two sets of spirals, but 

 also in terms of one spiral and definitely straight lines, the latter is 

 certainly preferable. But with the elimination of spirals of Archi- 

 medes straight lines vanish (for practical purposes), and the points of 

 intersection of log. spirals can only be defined in terms of two of the 

 orthogonally intersecting curves ; the genetic spiral thus becomes 

 useless theoretically, since its complementary orthogonal path is not 

 obvious, while the parastichy ratios are simple and readily observed 

 and tabulated. The genetic spiral thus tends to vanish as the log. 

 spiral theory replaces that of Schimper and Braun, but at the same 

 time the " orthostichy " curves are often so nearly straight that the 

 Schimper-Braun formulae will remain very useful in a large number 

 of cases for descriptive purposes ; nor can there be any objection to 

 such a proceeding so long as the convention is recognised. 



The error of the older phyllotaxis systems which postulate spirals 

 of Archimedes is, however, more deeply seated than appears at first 

 sight; it now becomes evident that its introduction into Botany 



