8 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



genious constructions were adopted by Braun and Eichler to adapt 

 the " obliquely vertical " rows of stamens in several Eanuneulaceous 

 flowers as true orthostichies. But it is clear that no sharp line 

 can be drawn between parastichies and orthostichies when once the 

 latter become curved. 



Hofmeister, who approached the subject with the most open mind, 

 came nearest the truth in formulating the statement that, in the 

 bud, a new member always arises in the widest gap between two 

 older ones. That the logical consequence of this would be '.that no 

 member would ever be vertically superposed to another, nor again 

 would it be so if developed at the "ideal angle," has b^^duly 

 recognized. But such conclusions have always been slurred over by 

 supporters of the spiral theory: either the observations must be 

 imperfect, or the specimens must have suffered from torsions or 

 displacements; the remarkable series of mathematical fractions 

 could not possibly be wrong: the perfect accuracy of the "ideal 

 angle " could not be expected of the plant : the object to be attained 

 namely, the best possible distribution of assimilating surface being 

 sufficiently approximated at a comparatively low divergence.* 



When once phyllotaxis is committed to this series of fractions, 

 expressing actual ratios of angular measurement, all deductions 

 from the mathematical properties of such a series naturally follow. 

 The remarkable superstructure therefore stands or falls according 

 to the correctness of the original series, based, as already noticed, 



* Cf. Bonnet, 1754, p. 160 ; De Oandolle, 1827, Organogrwphic V^gOale, vol. 

 i. p. 331. 



Of. Chauncey Wright, 1871. " On tlie uses and origin of arrangements of 

 leaves in plants" {Mem. Amer. Acad. ix. 387, 390). The continuation of this 

 theory of leaf distribution initiated by Bonnet, aflfords a remarkable example of 

 the method of biological interpretation of phenomena. Because a spiral series 

 gives a scattered arrangement of leaves and is very generally met with it 

 does not at all follow that such a scattered arrangement is beneficial or at aU an 

 aim on the part of the plant : nor again that the " ideal angle " would give 

 the ideal distribution. It is clear that in the intercalary growth of petiole- 

 formation the plant has a means of carrying leaves beyond their successors 

 whatever the phyUotaxis may be ; while if the ideal angle of a spiral phyllo- 

 taxis becomes the ideal angle of leaf-distribution, the formation of whorled 

 series from primitive spirals, to say nothing of secondary dorsiventral systems, 

 becomes curiously involved. 



