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On a Table for the Formation of Logarithms and Anti-Logarithms to Twelve Places (Part III)
Published online by Cambridge University Press: 18 August 2016
Extract
The twelve-figure table which accompanied Part I. was extracted from a table extending to twenty-four places, which is still in manuscript. Hence the details I am about to give of the methods of construction and verification made use of, will necessarily have reference to the more extensive table.
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- Copyright © Institute and Faculty of Actuaries 1866
References
page 212 note * A definite relation subsists between t, the number of places in the tabulated logarithms, m, the number by which the process is designated, and c, the number of columns which it is necessary to exhibit. The number of tabular entries requisite, irrespective of the auxiliary table, is t: m. The first half of these require a column each, while for the second half one additional column suffices—one period of m figures being dropped at the second entry in it, two at the third, and so on. Hence, generally,
When m = 3 this gives, for t=12, c=3; and for t = 24, c = 5.
The property of which advantage is thus taken to restrict the number of columns is an obvious consequence of the relation
the first significant figure of the greatest value of n, in Col. V., being in the thirteenth decimal place.
page 213 note * This work is very valuable, and is also, I believe, very scarce. The copy I possess belonged to Mr. Baily, at the sale of whose library it was bought by Mr. Woollgar, who bequeathed it to me.
page 214 note * See in particular Morgan's, De Differential and Integral Calculus, p. 312,Google Scholar and , Boole's Calculus of Finite Differences, pp. 86, 87.Google Scholar
page 214 note † It is hardly necessary to mention, that in applying the theorem between limits, the constant, log¬ (2π), disappears
page 215 note * The tables in which the errors were found are Hutton's and Callet's tables to twenty-places, and Wolfram's table to forty-eight places. The logarithms being in all the tables arranged in periods, it is sufficient to give the erroneous period and its correction.
As the whole of the above errors may not be found in all the editions of Hutton and Callet, it is proper that I should mention that my copies of these works are of the editions following:—Hutton, 7th edition, 1838; and Callet, “(1795) An 3e.” I find also that the two twenty-figure tables referred to, are very far from being universally true to the nearest figure in the twentieth place.
page 216 note * I carefully preserve these sums, as well as the corresponding sums of Col. I., for use in verifying the several columns, in case the large table should be hereafter put in type.
page 216 note † There is another theorem that would seem more directly to answer the end in view, namely,
which, on making n = 2, and taking the values of Δ2, Δ03, Δ204, &c, from a table of the Differences of Nothing, becomes
But the exponential theorem is rather more easily used and as it is the logarithm of that is wanted, the theorem (100001-1) that is wanted, the theorem last named answers the purpose just as well.