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Updated: May 25, 2025
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.
In the stagnant waters of our grassy ditches, the flat shells, the humble Planorbes, sometimes no bigger than a duckweed, vie with the Ammonite and the Nautilus in matters of higher geometry. At least one of them, Planorbis vortex, for example, is a marvel of logarithmic whorls. In the long-shaped shells, the structure becomes more complex, though remaining subject to the same fundamental laws.
Luckily, none of these injuries seriously affected the craft's safety, while most of them could be at least temporarily patched-up in a few hours; also, very luckily, all the navigating instruments, the chronometer, my sextant, the nautical almanac, and my book of logarithmic and other tables had almost miraculously escaped all injury.
Simon, Fourier, and their respective flocks committed a serious blunder in attempting to unite, the one, inequality and communism; the other, inequality and property: but you, a man of figures, a man of economy, you, who know by heart your LOGARITHMIC tables, how can you make so stupid a mistake?
Cut and polished length-wise, the fossil shows a magnificent logarithmic spiral, the general pattern of the dwelling which was a pearl palace, with numerous chambers traversed by a siphuncular corridor.
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?
He joined the circling universe, which he had hitherto kept away from him by main strength, and whirled about his own body, tracing a logarithmic spiral with infinite velocity leaving his body an infinite distance behind. "Thirty-five." The stars and the fiery pen exploded in a wild coruscation of searing, blinding light and he plunged from his spiral into a black abyss.
The consequence of this result is easily deduced. If projected on a plane perpendicular to the axis of the shell, the generating lines of the cone would become radii; and the groove which winds upwards from the base to the apex would be converted into a plane curve which, meeting those radii at an unvarying angle, would be neither more nor less than a logarithmic spiral.
By this characteristic we recognize the 'logarithmic spiral. Geometricians give this name to the curve which intersects obliquely, at angles of unvarying value, all the straight lines or 'radii vectores' radiating from a centre called the 'Pole. The Epeira's construction, therefore, is a series of chords joining the intersections of a logarithmic spiral with a series of radii.
Both of them are logarithmic spirals. I see no mechanism of the legs, be they long or short, that can account for this alteration. Can it then be a premeditated design on the part of the Epeira? Can there be calculation, measurement of angles, gauging of the parallel by means of the eye or otherwise?
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