Page:The Construction of the Wonderful Canon of Logarithms.djvu/117

Notes. 93 adapted to meet any requirements of accuracy. In sec. 60, Napier, in suggesting the construction of a table of logarithms to a greater number of places, proposes to take 100000000 as radius. The effect of this would be to throw still further back the error involved in taking the arithmetical mean of the limits for the true logarithm. Thus, using the formula given, substituting 100000000 for n, and multiplying the result by that amount as already explained, we should have for the true logarithm of 99999999, the first proportional after radius in the new First table,

If we take the arithmetical mean of the limits, we have

This brings out a difference of

or a sixty thousand billionth part of the logarithm. We see that the logarithms only begin to differ in the 18th place, and that thus to however many places the radius is taken, the logarithms of proportionals deduced from it will be given with absolute accuracy to a very much greater number of places.

To ensure accuracy in the figures given above, the three preparatory tables were recomputed strictly according to the methods described in the Constructio, fourth proportionals being found in all the preceding tables, and both limits of their logarithms being calculated, the work being carried to the 27th place after the decimal point.

As logarithms to base $$\scriptstyle e^{-1}$$ are now quite superseded, it is not worth while printing these preparatory tables. The following values (pp. 94-95), however, may be of service for comparison, and as a check to any one who may desire to work out for himself the tables and examples in the Constructio. The values given are the first proportional after radius, and the last proportional in each of the three tables, and also in the Third table, the last proportional in col. 1, and the first proportionals in col. 2 and 69. Opposite these are given their logarithms to base $$\scriptstyle e^{-1}$$, computed, first, according to Napier’s method, and second, by the present method of series which gives the value true to the last place, which is increased by unit when the next figure is 5 or more. The proportionals and logarithms are each multiplied by 10000000, as explained above.

Though the logarithms in the Canon of 1614 were affected by the mistake