In his investigations respecting cycloidal lines and various spiral curves, his attention was directed to the loxodromic and logarithmic spirals, in the last of which he took particular interest from its remarkable property of reproducing itself under a variety of conditions.
The title of Gunter's book, which is very scarce, is Canon triangulorum, and it contains logarithmic sines and tangents for every minute of the quadrant to 7 places of decimals.
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.
The definitions of the logarithmic and exponential functions may be extended to complex values of x.
Remarkable for the beauty of its logarithmic curves; Iwaki-san (5230 ft.), known as Tsugaru-Fuji, and said by some to be even more imposing than Fuji itself; and the twin mountains Gassan (6447 ft.) and Haguro-san (5600 ft.).
Like another Archimedes, he requested that the logarithmic spiral should be engraven on his tombstone, with these words, Eadem mutata resurgo.
As originally proposed, many of these formulae were cast in exponential form, but the adoption of the logarithmic method of expression throughout the list serves to show more clearly the relationship between the various types.
It now remains to notice briefly a few of the more important events in the history of logarithmic tables subsequent to the original calculations.
The point separating the integers from the decimal fractions seems to be the invention of Bartholomaeus Pitiscus, in whose trigonometrical tables (1612) it occurs and it was accepted by John Napier in his logarithmic papers (1614 and 1619).
The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x.