A. //. Church. 
54 
data afforded by the transition of the morphological standpoint from 
that of adult construction to that of growing systems. 
There can clearly be no objection to the application of a 
mathematical conception in itself; everything in nature is capable 
of mathematical expression if the conditions are only sufficiently 
well known; the difficulty is to select the fundamental conception 
which may afterwards be modified as new factors are introduced 
into the equation. 
The method of application of such a view is again clearly 
indicated: just as Bonnet and Calandrini postulated an ideal adult 
structure, so it becomes necessary, now that growth and develop¬ 
ment are assumed to he the foundation of all views of morphology, 
to postulate a certain ideal condition for a growing system. 
A given plant need not show ideal growth, any more than it may 
be expected to show an ideal adult phase; but a definite standpoint 
is thus taken up, which comes into line with modern conceptions of 
the structure and development of living protoplasm, and this admits 
of future modification as our knowledge becomes increasingly exact. 
The value of a mathematical proposition depends entirely on the 
initial premisses; the point therefore remains to determine what 
mathematical premisses can he established as the basis of a growing 
system composed of axis and lateral members. Bonnet’s system 
was perfect for the adult construction it was alone intended to 
summarise; the case of a growing shoot presents a much greater 
degree of complexity, and a certain ideal condition may therefore 
be deduced as a starting-point. 
Just as in the consideration of the Newtonian laws of motion, 
the purely abstract and mathematical conception of uniform motion 
precedes that of varying motion, so the growth of a mass of 
protoplasm by interstitial development throughout its whole 
substance may, in the simplest case, be conceived as a uniform 
growth expansion taking place around a hypothetical central point, 
the “growth-centre,” and proceeding at a uniform rate radially in 
all directions, with the result that the protoplasmic mass presents 
the phenomenon of a uniformly expanding sphere. Resolving such 
a solid sphere along planes passing through the centre, the plane 
projection of a section would be plotted geometrically as a circle the 
area of which may be divided indefinitely into a series of similar 
figures, in the simplest case small quasi-squares formed by the 
intersection of radii and concentric circles. In such a circular 
mesh-work of squares, any curve drawn in a constant manner 
