This table distinctly involves the principle of logarithms and may be described as a modified table of antilogarithms. It consists of two series of numbers, the one being an arithmetical and the other a geometrical progression: thus 0, 1,0000 0000 10, I,0001 0000 20, 1,0002 000 I 9 90, 1,0099 4967 In the arithmetical column the numbers increase by io, in the geometrical column each number is derived from its predecessor by multiplication by i 0001.
If we divide the numbers in the geometrical column by lo g the correspondence is between lox and (I 000l) x, and the table then becomes one of antilogarithms, the base being (1.0001) 1 / 10, viz.
Napier gives logarithms to base e ', Byrgius gives antilogarithms to base (I.coo')='a.
Decimal or Briggian Antilogarithms. - In the ordinary tables of logarithms the natural numbers are all integers, while the logarithms tabulated are incommensurable.
Antilogarithmic tables are few in number, the only other extensive tables of the same kind that have been published occurring in Shortrede's Logarithmic tables already referred to, and in Filipowski's Table of antilogarithms (1849).
Hyperbolic antilogarithms are simple exponentials, i.e.
I r n is generally known as Weddle's method, having been published by him in The Mathematician for November 1845, and the corresponding method for antilogarithms by means of factors of the form i+( I) r n is known by the name of Hearn, who published it in the same journal for 1847.
This was published in 1876 under the title Tables for the formation of logarithms and antilogarithms to twenty-four or any less number of places, and contains the most complete and useful application of the method, with many improvements in points of detail.
Antilogarithms 8.5 88.
2, as base, and take as indices the successive decimal numbers to any particular number of places of decimals, we get a series of antilogarithms of the indices to this base.
We thus get the numbers 2 0l .02 2.03 which are the antilogarithms of 01, 02, .03,.
As the table of antilogarithms is formed by successive multiplications, so the logarithm of any given number is in theory found by successive divisions.