Better still.  Let us imagine a plane perpendicular to the aids of the shell and passing through its summit.  Let us imagine, moreover, a thread wound along the spiral groove.  Let us unroll the thread, holding it taut as we do so.  Its extremity will not leave the plane and will describe a logarithmic spiral within it.  It is, in a more complicated degree, a variant of Bernouilli’s ‘Eadem mutata resurgo:’  the logarithmic conic curve becomes a logarithmic plane curve.

A similar geometry is found in the other shells with elongated cones, Turritellae, Spindle-shells, Cerithia, as well as in the shells with flattened cones, Trochidae, Turbines.  The spherical shells, those whirled into a volute, are no exception to this rule.  All, down to the common Snail-shell, are constructed according to logarithmic laws.  The famous spiral of the geometers is the general plan followed by the Mollusc rolling its stone sheath.

Where do these glairy creatures pick up this science?  We are told that the Mollusc derives from the Worm.  One day, the Worm, rendered frisky by the sun, emancipated itself, brandished its tail and twisted it into a corkscrew for sheer glee.  There and then the plan of the future spiral shell was discovered.

This is what is taught quite seriously, in these days, as the very last word in scientific progress.  It remains to be seen up to what point the explanation is acceptable.  The Spider, for her part, will have none of it.  Unrelated to the appendix-lacking, corkscrew-twirling Worm, she is nevertheless familiar with the logarithmic spiral.  From the celebrated curve she obtains merely a sort of framework; but, elementary though this framework be, it clearly marks the ideal edifice.  The Epeira works on the same principles as the Mollusc of the convoluted shell.

The Mollusc has years wherein to construct its spiral and it uses the utmost finish in the whirling process.  The Epeira, to spread her net, has but an hour’s sitting at the most, wherefore the speed at which she works compels her to rest content with a simpler production.  She shortens the task by confining herself to a skeleton of the curve which the other describes to perfection.

The Epeira, therefore, is versed in the geometric secrets of the Ammonite and the Nautilus pompilus; she uses, in a simpler form, the logarithmic line dear to the Snail.  What guides her?  There is no appeal here to a wriggle of some kind, as in the case of the Worm that ambitiously aspires to become a Mollusc.  The animal must needs carry within itself a virtual diagram of its spiral.  Accident, however fruitful in surprises we may presume it to be, can never have taught it the higher geometry wherein our own intelligence at once goes astray, without a strict preliminary training.

Are we to recognize a mere effect of organic structure in the Epeira’s art?  We readily think of the legs, which, endowed with a very varying power of extension, might serve as compasses.  More or less bent, more or less outstretched, they would mechanically determine the angle whereat the spiral shall intersect the radius; they would maintain the parallel of the chords in each sector.