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