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:— 
• * ' l 
1st 
2nd 
3rd 
revolution. 
revolution. 
revolution. 
inches. 
inches. 
inches. 
Radius 
No. 
1 . 
. 1.00 
... 3.00 .. 
9.00 
1 9 
9 9 
2 . 
••• 3-35 •• 
10.04 
» 9 
9 9 
3 . 
. 1.25 . 
... 3.74 .. 
11.21 
9 9 
9 9 
4 . 
• 1-39 • 
... 4.17 .. 
.. 12.51 
9 9 
9 9 
5 . 
• i *55 • 
... 4.65 .. 
.. 13.96 
9 9 
9 9 
6 . 
• 1-73 • 
... 5.19 .. 
.. I 5-58 
9 9 
9 9 
7 . 
1.93 • 
... 5.80 .. 
.. 17.40 
9 9 
9 9 
8 . 
... 6.47 .. 
.. 19.42 
9 9 
9 9 
9 . 
... 7.22 
.. 21.67 
9 9 
9 9 
10 .. 
... 8.06 
.. 24.19 
9 9 
9 9 
1 . 
. 3.00 . 
... 9.00 
.. 27.00 
These lengths (after the first) correspond to the logarithms of ioth, 
roths, Voths, etc., of the logarithm of 3, i.e., to .0477, .0954, • I 43 I , 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 secoiid 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 ar<b almost 
equally good examples, and this applies not only to the discoid forms, but 
to the mo^e 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 
