55 
D e sc rip live Morphology. 
through the points of intersection of the radii and circles is a spiral* 
the tangent to which makes equal angles with every radius vector, 
that is to say, it is an equiangular or logarithmic spiral. Contem¬ 
plation of such a figure shows at once that log. spiral curves with 
the straight line and circle as limiting cases are the sole curves of 
uniform growth expansion. Any morphological theory of spiral 
growth, including spiral phyllotaxis, which is based on a proposition 
involving growth, should therefore be based on a logarithmic spiral 
on a plane surface; and not on a helix winding on a cylinder, which, 
carried on to a plane as a spiral with equal screw-thread, would 
become a spiral of Archimedes (Bravais). In its application to 
phyllotaxis, therefore, the logarithmic spiral represents a second 
mathematical conception as applied to growth phenomena; and as 
far as the living and admittedly irregular growing plant is concerned, 
it would no doubt be as difficult to prove by actual measurement as 
Bonnet’s theoretical helix. But the standpoint is changed, it now 
becomes possible not only to deal with uniformly growing systems 
and deduce mathematically their special properties, but also to draw 
them and see them, as it were, grow on paper. Similarly, once the 
properties of systems exhibiting uniform growth have been ascer¬ 
tained, it may become possible by adding secondary conditions, to 
study the possibilities of varying rates of growth, and thus it 
becomes conceivable that the irregular growth of a living body may 
ultimately be as closely approximated in terms of mathematical 
formulae, as for example the erratic path of a comet, if the gain to 
botanical science should be at all commensurate with the labour 
expended. 1 
Botanic Garden, Oxford, Feh. 1902. 
1 Cf On the Relation of Phyllotaxis to Mechanical Lavos —A. H. CHURCH. 
Part I. —Construction by Orthogonal Trajectories .— Sept. 1901. 
Part II .—Asymmetry and Symmetry , Jail. 1902. 
210 pp. and over 80 figures. Williams & Norgate, London and Oxford. 
