238 Church. — The Principles of Phyllotaxis. 



since it is more convenient to trace a solid in separates planes, it will be 

 illustrated by a diagram in which a system of concentric circles encloses 

 a series of similar figures, which represent a uniform growth increment 

 in equal intervals of time. Such a circular figure, in which the expanding 

 system is subdivided into an indefinite number of small squares repre- 

 senting equal time-units, is shown in Fig. 40, and presents the general 

 theory of mathematical growth, in that in equal times the area represented 

 by one ' square ' grows to the size of the one immediately external 

 to it^ 



Now it is clear that while these small areas would approach true 

 squares if taken ^sufficiently small, at present they are in part bounded 

 by circular lines which intersect the radii orthogonally; they may there- 

 fore be termed quasi-squares : and while a true square would contain 

 a true inscribed circle, the homologous curve similarly inscribed in a quasi- 

 square will be a quasi-circle. 



It is to this quasi-circle that future interest attaches; because, just 

 as the section of the whole shoot was conceived as containing a centric 

 growth-centre, so the lateral, i. e. secondary, appendages of such a shoot 

 may be also conceived as being initiated from a point and presenting a 

 centric growth of their own. These lateral growth-centres, however, are 

 component parts of a system which is growing as a whole. The con- 

 ception thus holds that the plane representation of the primary centric 

 shoot-centre is a circular system enclosing quasi-circles as the representatives 

 of the initiated appendages. 



To this may now be added certain mathematical and botanical facts 

 which are definitely established. 



I. Any such growth-construction involving similar figures (and quasi- 

 circles would be similar) implies a construction by logarithmic spirals. 



II. A growth-construction by intersecting logarithmic spirals, and 

 only by curves drawn in the manner utilized in constructing these diagrams 

 (Figs. 35-38), is the only possible mathematical case of continued orthogonal 

 intersection ^. 



III. The primordia of the lateral appendages of a plant only make 

 contact with adjacent ones in a definite manner, which is so clearly that 

 of the contacts exhibited by quasi-circles in a quasi-square meshwork, 

 that Schwendener assumed both a circular form and the orthogonal 

 arrangement as the basis of his Dachstuhl Theory : these two points being 

 here just the factors for which a rigid proof is required, since given these 

 the logarithmic spiral theory necessarily follows. 



A construction in terms of quasi-circles would thus satisfy all theo- 



^ The same figure may also be used to illustrate a simple geometrical method of drawing any 

 required pair of orthogonally intersecting logarithmic spirals. 



' For the formal proof of this statement I am indebted to Mr. H. Hilton. 



