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I should enjoy working it out in all its details, were I not afraid of wearying the reader's attention. Perhaps I have even gone too far in the little that I have said, in which case I owe him some compensation: 'Would you like me, I will ask him, 'would you like me to tell you how I acquired sufficient algebra to master the logarithmic systems and how I became a surveyor of Spiders' webs?
Thereupon followed the epicycloids, curves which as the cycloid is generated by a point upon the circumference of a circle rolled along a straight line are generated by a similar point when the path of the circle becomes any curve whatever. Caustic curves, spirals without number, succeeded, of which but one shall claim our notice, the logarithmic spiral, first fully discussed by James Bernouilli.
To oversee all the details yourself in person; to be at once pilot and captain, and owner and underwriter; to buy and sell and keep the accounts; to read every letter received, and write or read every letter sent; to superintend the discharge of imports night and day; to be upon many parts of the coast almost at the same time often the richest freight will be discharged upon a Jersey shore; to be your own telegraph, unweariedly sweeping the horizon, speaking all passing vessels bound coastwise; to keep up a steady despatch of commodities, for the supply of such a distant and exorbitant market; to keep yourself informed of the state of the markets, prospects of war and peace everywhere, and anticipate the tendencies of trade and civilization taking advantage of the results of all exploring expeditions, using new passages and all improvements in navigation; charts to be studied, the position of reefs and new lights and buoys to be ascertained, and ever, and ever, the logarithmic tables to be corrected, for by the error of some calculator the vessel often splits upon a rock that should have reached a friendly pier there is the untold fate of La Prouse; universal science to be kept pace with, studying the lives of all great discoverers and navigators, great adventurers and merchants, from Hanno and the Phoenicians down to our day; in fine, account of stock to be taken from time to time, to know how you stand.
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.
A problem is given to the machine, and it solves it by computing a long series of numbers following some given law. In this manner it calculates astronomical, logarithmic, and navigation tables, as well as tables of the powers and products of numbers.
To suggest an explanation why this spiral has so greatly exercised the meditations of science, let us confine ourselves for the present to a few statements of which the reader will find the proof in any treatise on higher geometry. The logarithmic spiral describes an endless number of circuits around its pole, to which it constantly draws nearer without ever being able to reach it.
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.
A line may be straight or broken, and if curved, curving continuously or brokenly, etc. That this quality of line is distinct from form may be shown by the simple experiment of turning a spiral a logarithmic spiral, let us say in different ways about its focus.
The first and third parts of the work contain logarithmic and other auxiliary tables, for the purpose of facilitating astronomical calculations. The second part contains tables of the sun, moon, and planets; and the fourth a catalogue of 1000 stars, as determined by Tycho.
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