238 Church —The Principles of Phyllotaxis. 
since it is more convenient to trace a solid in separate 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 5 grows to the size of the one immediately external 
to it 1 . 
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 2 . 
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- 
1 The same figure may also be used to illustrate a simple geometrical method of drawing any 
required pair of orthogonally intersecting logarithmic spirals. 
2 For the formal proof of this statement I am indebted to Mr. H. Hilton. 
