Church.— Note on Phy Hot axis . 485 
phenomena, it is clearly impossible to found any modem 
scientific generalizations on angles which cannot be measured, 
and lines which cannot be proved to be straight: it thus 
follows that all speculations based on the assumption of the 
Schimper-Braun series must rest on a purely hypothetical 
foundation which may at any time be overturned. Such 
expressions, as Sachs constantly pointed out, attempt to 
imitate the phenomena observed without giving any reason 
for such geometrical construction. 
Again, taking the mathematical interpretation of the 
Schimper-Braun system, that the genetic spiral and the 
parastichies are represented by spirals of Archimedes, while 
the orthostichies are radii vectores, a simple geometrical con- 
struction in terms of these spirals should bring out either the 
truth or error of this hypothetical relationship of the lateral 
members. 
Thus, from the equation to the Archimedean spiral (r — aQ), 
it is easy to construct a pair of spirals whose variable a shall 
have the ratio of the parastichies observed on any given speci- 
men. Take for example the / T system, the primary contact 
parastichies of which are 8 and 13 ; Fig. 2 shows such a system 
geometrically planned for a left-hand genetic spiral : the 
members along the twenty-one orthostichy lines differ by 
twenty-one, and fall on the mathematically straight radii 
vectores of the system. The intersections of these parastichy 
spirals mark the points at which the lateral members are 
inserted, and the views of Schimper and Braun included only 
the consideration of such points. It is clear, however, that if 
the spaces between the spiral planes are regarded as contain- 
ing the members pressed into close lateral contact, as seen in 
the transverse section of a foliage bud, the appearance of the 
progressive dorsiventrality of such lateral members is very 
fairly imitated. The construction, in fact, becomes more and 
more like the appearances seen in the plant as the periphery 
of the system is reached, but the central part which includes 
the actual seat of development is very inadequately repre- 
sented : thus, the areas become so relatively elongated in the 
K k 2 
