THE ESSEX FIELD CLUB. 91
angles about the pole (and therefore ten radii, the first being taken, say,
as 1") then the lengths of the successive radii are approximately as.
follow:-
These lengths (after the first) correspond to the logarithms of 1/10th,
2/10ths, 3/10ths, etc., of the logarithm of 3, i.e., to .0477, .0954, .1431, etc.
It will be seen from the above figures that not only are the distances
of the successive turns of the spiral, measured from the pole along each
radius, in the ratio of 3:1, but that the differences between the first and
second, and between the second and the third turns, (i.e., the widths of
the whorls in the case of a shell) are also in the same ratio. And this is
true of any radii drawn in intermediate positions.
Another peculiarity of the logarithmic spiral is that the angle between
the tangent and the radius is always the same, wherever taken. It is from
this fact that this spiral is often called the "equiangular spiral."
Yet another property of the logarithmic spiral is its constant simi-
larity of form whatever its size. It follows from this that in such an ex-
ample of the spiral as a shell, every increment is what is known mathematic-
ally as a "gnomon" to the pre-existing shell. And this further implies
that every increment has its proportional effect upon the following
increment, which suggests that the logarithmic spiral might also be
called the "compound interest spiral."
Although it is probably true that there is no such thing as a perfect
logarithmic spiral in nature, some natural objects come remarkably
close to it.
The best examples are to be found among the shells of the Mollusca,
especially the Cephalopoda and Gasteropoda. A section of a pearly
Nautilus shows an almost perfect logarithmic spiral, with a threefold
increment for each complete turn. But many other shells are almost
equally good examples, and this applies not only to the discoid forms, but
to the more or less elongated or turbinate types, i.e., to the majority of
marine, land and fresh-water Gastropods, in which the logarithmic spiral
can be quite easily recognised, although no longer in a plane. In these
latter cases a series of measurements of the widths of the whorls, if separ-
ately visible, taken along any straight line drawn from the apex towards
the mouth, will give the same constant ratio of increase as do the widths
of the whorls across a radius in the discoid forms. Any other series of