The Epeira complies to the best of her ability with this law of the endless volute.  The spiral revolutions come closer together as they approach the pole.  At a given distance, they stop abruptly; but, at this point, the auxiliary spiral, which is not destroyed in the central region, takes up the thread; and we see it, not without some surprise, draw nearer to the pole in ever-narrowing and scarcely perceptible circles.  There is not, of course, absolute mathematical accuracy, but a very close approximation to that accuracy.  The Epeira winds nearer and nearer round her pole, so far as her equipment, which, like our own, is defective, will allow her.  One would believe her to be thoroughly versed in the laws of the spiral.

I will continue to set forth, without explanations, some of the properties of this curious curve.  Picture a flexible thread wound round a logarithmic spiral.  If we then unwind it, keeping it taut the while, its free extremity will describe a spiral similar at all points to the original.  The curve will merely have changed places.

Jacques Bernouilli, {42} to whom geometry owes this magnificent theorem, had engraved on his tomb, as one of his proudest titles to fame, the generating spiral and its double, begotten of the unwinding of the thread.  An inscription proclaimed, ’Eadem mutata resurgo:  I rise again like unto myself.’  Geometry would find it difficult to better this splendid flight of fancy towards the great problem of the hereafter.

There is another geometrical epitaph no less famous.  Cicero, when quaestor in Sicily, searching for the tomb of Archimedes amid the thorns and brambles that cover us with oblivion, recognized it, among the ruins, by the geometrical figure engraved upon the stone:  the cylinder circumscribing the sphere.  Archimedes, in fact, was the first to know the approximate relation of circumference to diameter; from it he deduced the perimeter and surface of the circle, as well as the surface and volume of the sphere.  He showed that the surface and volume of the last-named equal two-thirds of the surface and volume of the circumscribing cylinder.  Disdaining all pompous inscription, the learned Syracusan honoured himself with his theorem as his sole epitaph.  The geometrical figure proclaimed the individual’s name as plainly as would any alphabetical characters.

To have done with this part of our subject, here is another property of the logarithmic spiral.  Roll the curve along an indefinite straight line.  Its pole will become displaced while still keeping on one straight line.  The endless scroll leads to rectilinear progression; the perpetually varied begets uniformity.

Now is this logarithmic spiral, with its curious properties, merely a conception of the geometers, combining number and extent, at will, so as to imagine a tenebrous abyss wherein to practise their analytical methods afterwards?  Is it a mere dream in the night of the intricate, an abstract riddle flung out for our understanding to browse upon?