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A year or more ago while the author was engaged in extending the work of J. C. Adams' by computing the Napierian logarithms of 11, 13, 17, 19, 23, 29 and 31 to a fairly large number of decimal places, but before the calculations had been completed, he became convinced that it would not be justifiable to claim for the new values complete absence of error unless the basic data, namely, the logarithms of 2, 3, 5 and 7, were proved to be correct as printed. This conclusion was reached after a careful search through the available literature failed to show that in the interim any other mathematician had repeated or independently verified the values found by Adams. In order to contribute more than the mere checking of the classical data the goal of nearly 330 decimal places was set for the present investigation. Since Adams' approximations were claimed by him to be valid to about 273 decimal places the gain proposed would be 55 figures. The notation used here is the same as that employed by me in an earlier paper.2 The actual work involved the following steps in the order stated: (a) the selection of suitable pairs of numbers (p, q) which satisfy the condition p q = 1; (b) the calculation of terms of the simple geometrical series 2(2p )(2m + ) m = 0, 1, 2, 3, ...; (c) copying from the Monroe computing machine (in black ink) the numerical values of these terms on alternate horizontal lines of rectangularly ruled paper; (d) forming the sum of these data and comparing it with the previously computed value of S that is, the limit (2p 1)/4p(p 1); (e) dividing each value of (2p 1) (2m + ') by 2m + 1 and entering the quotients (in red ink) in the blank lines directly below the corresponding terms of the geometrical series; (f) summing separately the lines which represent the even and odd terms of the final series, s(+) = 2(2p 1)-(4m+ 1)/(4m + 1) and s(-)

Recalculation and Extension of the Modulus and of the Logarithms of 2, 3, 5, 7 and 17