92 
THE ESSEX NATURALIST. 
comparable parts in one complete turn of the spiral apart, e„g., the widths 
of the openings on one side or the other, when a section is made in the 
plane of the columella, will also give the same result. The ratio, so deter 
mined, by whatever method, is usually a specific character, 
f** Naturally there is a very considerable range of ratios to be found among 
the widely different types of univalve shells. For example, in some 
species of Planorbis the ratio is very small so that the appearance is more 
that of the non-expanding spiral of Archimedes than that of a logarithmic 
spiral, whereas in such a form as the Ear-shell, Haliotis tu erculata, the 
ratio is very large, probably as much as io : 1. 
In view of the many factors involved in the growth of an organism, 
it is not surprising that measurements show that the ratio is not always 
constant throughout life. There is very often a falling-off in the rate of . 
increase (e.g., some species of Planorbis, Pupa,Clausilia, etc.), and it some¬ 
times happens that even the spiral form is lost in the later stages of growth 
(e.g., the ammonitoid fossil Lituites). On the other hand, cases do occur 
in which the rate of expansion of the spiral is actually accelerated as time 
goes on (e.g., Succinea, etc.). 
It is not only the shells themselves, however, which show more or less 
close approximations to logarithmic spirals, but sometimes also their 
opercula. This is very well shown in species of Turbo and can also be 
seen in the land snail, Cyclostoma elegans, and in the common periwinkle 
(Littorina ), etc. In the case of the pearly Nautilus the septa dividing the 
shell into chambers are, in median section, also parts of a logarithmic 
spiral exactly similar to that of the shell itself. 
Another group of animals in which many of the forms embody the 
logarithmic spiral is the Foraminifera. In these, for the most part 
microscopic organisms, the effect is produced by the addition of new 
chambers, each of which is a “ gnomon ” to the pre-existing group of 
chambers. Some genera, e.g., Rotalia, Cornuspira, Cyclammiva, etc., 
are discoid, having the chambers arranged in a plane, and they look 
remarkably like diminutive forms of Nautilus and other Cephalopods. 
In other genera, e.g., Globigerina, the chambers are pushed more or less 
out of a plane into a screw pattern. As in the Mollusca, the ratio of in¬ 
crease is not always maintained throughout life. In Nummulites the 
spiral in its central part seems to be roughly logarithmic although the 
rate of increase is very small. Later turns of the spiral, however, keep 
at almost exactly the same distance apart and therefore form an 
Archimedean spiral. In Peneroplis and other forms the early spiral 
arrangement is lost in later stages, producing various fantastic effects. 
The third principal group of animals exhibiting the logarithmic spiral, 
though only as a small part of their structure, is the group comprising 
cattle, sheep, goats, etc. In these vertebrates the horns are sometimes 
twisted round more than once into very close approximations to logarith¬ 
mic spirals, though not in one plane (e.g., rams of various breeds of sheep), 
and in other cases, although the spiral is only partial, its logarithmic 
nature is quite evident. 
Yet another example of the logarithmic spiral among animals may 
be mentioned, namely, the cochlea of the mammalian ear. 
Generally speaking it will be seen that, with the exception of the Fora- 
