167 



(Pinus, Ananassa, Helianthus, etc.) a change of the ratio between 

 longitudinal and transversal growth does not occur; hence, the laws 

 of phyllotaxis are best studied in the case of such multiple fruits 

 and inflorescences, in the capitulum of the sunflower, in the cones 

 of Pinus, in terminal buds, etc. It is, therefore, absolutely necessary 

 to draw attention more exclusively to the study of the growing 

 apex of the plant : the first zone of growth in the terminal bud must, 

 in the opinion of this author, reveal the phenomena of phyllotaxis 

 in their most pure and undisturbed form. The explanation to be 

 given to it must, moreover, really satisfy the requirements of onto- 

 genetic observation. Thus, starting from Sachs' theory of cell-for- 

 mation and of the orthogonal intersection of cell-walls x ) in the 

 terminal bud, he comes to the conclusion that the parastichies must 

 have the shape of logarithmic spirals (p = a s ), intersecting every- 



a. 



Fig. 131 a and b. 



where at right angles. Indeed, in two-dimensional space the loga- 

 rithmic spiral is the only curve, in which one part differs from the 

 other only in size, but not in shape, - - a property which brings 

 out very 'strikingly the essential character of such curves as lines 

 of growth. 



If it be kept in mind that the primordial cells will be greater as 

 they are older, Church gives the following constructions of the spiral 

 arrangements in the first zone of growth. As symbols of the emer- 

 gences based on lateral members of cell-aggregates, he takes, like 

 De Candolle (loco cit. p. 52), circles of different diameter packed 

 closely together in the way of the most closely packed "cubic" arrange- 

 ments (fig. /j/ a , 13 i b ), and in both principal directions (see fig. 123), 

 as well laterally (fig. iji a ), as diagonally (fig. 13 i b ) oriented along 



*) J. Sachs, loco cit. p. 253; Cf. also: L. Errera, Bull, de la Soc. beige de 

 Microscopic, 13, 1, (1886). 



