228 Church . — The Principles of Phyllotaxis . 
of the subject, the geometry of the system is based solely on a mathematical 
conception put forward by Bonnet and Calandrini in 1754; and this 
mathematical conception applied only to adult shoots and adult members 
of equal volume arranged in spiral sequence, and thus involved a system 
of intersecting helices of equal screw-thread, or, reduced to a plane 
expression, of spirals of Archimedes, also with equal screw-thread. A 
system of helical mathematics was thus interpolated into botanical science, 
and these helical systems were correctly tabulated by * orthostichies ’ and 
‘ divergence angles * obtained from simple fractional expressions themselves 
deduced from the observation of orthostichies. 
But in transferring the study of phyllotaxis to the ontogenetic sequence 
of successively younger, and therefore gradated , primordia at the apex of 
a growing plant-shoot which was not cylindrical, these mathematical 
expressions were retained, although the helices originally postulated have 
absolutely vanished ; and it is somewhat to the discredit of botanical science 
that this simple error should have remained so long undetected and 
unexpressed. As soon as one has to deal with spirals which have not an 
equal screw-thread, the postulated orthostichies vanish as straight lines ; 
the fractional expressions therefore no longer present an accurate statement 
of the facts ; and the divergence angles, calculated to minutes and seconds, 
are hopelessly out of the question altogether ; while any contribution 
to the study of phyllotaxis phenomena which continues the use of such 
expressions must only serve to obscure rather than elucidate the inter- 
pretation of the phenomena observed. That the required orthostichies 
were really non-existent at the growing point, a feature well known to 
Bonnet himself, has thus formed the starting-point for new theories of 
displacement of hypothetically perfect helical systems, as, for example, 
in the contact-pressure theory of Schwendener. But once it is grasped 
that the practice of applying helical mathematics to spiral curves which, 
whatever they are, cannot be helices, is entirely beside the mark, it is 
clear that the sooner all these views and expressions are eliminated the 
better, and the subject requires to be approached without prejudice from 
an entirely new standpoint. r 
The first thing to settle therefore is what this new standpoint is to be ; 
and how can such a remarkable series of phenomena be approached on 
any general physical or mathematical principles ? 
Now in a transverse section of a leaf-producing shoot, at the level 
of the growing point, the lateral appendages termed leaves are observed 
to arrange themselves in a gradated sequence as the expression of a 
rhythmic production of similar protuberances, which takes the form of 
a pattern in which the main construction lines appear as a grouping 
of intersecting curves winding to the centre of the field, which is occupied 
by the growing point of the shoot itself. As the mathematical properties 
