Descriptive Morphology. 53 
theory as they do in practice to ever newly developing members. 
The whole attitude of the botanist with regard to the plant world 
thus became altered and entirely revolutionised; but the corres¬ 
ponding change was not simultaneously effected in the descriptive 
nomenclature previously built up under the old regime. 
To Hofmeister and Sachs, the phyllotaxis formulae of Schimper 
and Braun, when approached from the standpoint of the development 
of lateral members, were naturally found wanting. They did not 
explain or agree precisely with the facts of ontogeny, and Sachs 
expressly denounced the Spiral Theory as a mode of view 
gratuitously introduced into the plant, and phyllotaxis formulae as 
mere playing with the mathematical properties of numbers. That 
Bonnet’s mathematical conception was, from the modern standpoint^ 
gratuitously introduced, is obvious. Every mathematical conception 
as applied to the plant must necessarily be equally gratuitous ; but 
it was never intended to do more than represent a theoretically 
ideal adult condition. The fallacy appears to have been introduced 
by Schimper himself, and remainded unnoticed by his followers, in 
that they did not grasp the fact that having accepted Bonnet’s 
spiral, they were now including the case of all visible systems, 
growing or not; while Bonnet, in equal ignorance of protoplasmic 
growth, dealt solely with adult structures which had ceased growing 
on the attainment of a uniform specific bulk. 
That a mathematical conception which was based on the 
postulate that the lateral members were equal in volume or equally 
spaced would not hold for developing members which present a 
gradated sequence in bulk, space and time, appears to have been 
entirely overlooked. Similarly any view which attempts to adopt 
the Schimper-Braun conventions to explain the facts of develop¬ 
ment is open to a fundamental error: a remarkable example being 
seen in Schwendener’s “ Dachstuhl” Theory, in which somewhat 
hypothetical forces of Contact-pressure were supposed to alter 
phyllotaxis formulae in all cases planned after the helix theory 
of Bonnet. 
The error lying at the root of all theories of spiral phyllotaxis 
is so far clear; it was the old story of putting new wine into old 
bottles,' the Schimper-Braun theory collapsed without being 
replaced by a new one, and all attempts at patcning on the part of 
Schwendener have proved futile. What is really required is, after 
all, only what should have been put forward fifty years ago, namely 
a new mathematical conception founded on the new mathematical 
