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On a Table for the Formation of Logarithms and Anti-Logarithms to Twelve Places
Published online by Cambridge University Press: 18 August 2016
Extract
So far as I am aware, the earliest method for the formation of logarithms, in which it was proposed to resolve the number whose logarithm is required into factors by a continuous process, is one which was published by Mr. Manning in the Philosophical Transactions for 1806. My acquaintance with Mr. Manning's method is derived, not from the original paper, which I have not seen, but from a reprint of it, “nearly as it stands,” in Young's Elementary Essay on the Computation of Logarithms. Mr. Manning applies his method to the formation of Naperian logarithms only; but it is equally applicable to the formation of common logarithms. I repeat, in accordance with this method, an example I have already more than once given.
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- Copyright © Institute and Faculty of Actuaries 1866
References
page 252 note * Second edition, London, 1835, pp. 67–79.
page 254 note * Mr. Manning (or Mr. Young) says, that the number of subtractions in any series can “never exceed nine.” Here, in an example casually taken, is evidence to the contrary.
page 255 note * I have in my example supposed the data arranged in columns. They were not so arranged by Mr. Manning. And I may mention, that although I have here, as in previous examples, indicated a number of columns equal to that of the possible tabular entries, and shall follow the same course in subsequent examples, yet, in all cases with which we have had or shall have to do, the second half of the entries are obtained from a single column. (See note, p. 212.)
page 257 note * As in Mr. Manning's method, so also here, it is unnecessary formally to carry the resolving process beyond the exhaustion of the first six decimal places; since at that point what would be the result of its formal completion is visible in the remainder.
page 257 note † He arranges them, not in columns, as the indications I have attached to the preceding example would seem to imply, but in an unbroken series. There are other breaches of continuity in the method as delivered by Mr. Weddle, which also I have removed.
page 258 note * See page 219.
page 259 note * I am glad to have an opportunity of placing it upon record that Mr. Waddle, with whom I was not previously acquainted, called upon me and thanked me very cordially for the improvements I had introduced in his method, and for my endeavours to bring it more prominently before the public.
page 261 note * It was singular, to say the least, to find given as original, and without remark, a method which had appeared in the pages of the same periodical sixteen months before. I had some correspondence with Mr. Hearn on the subject of his paper, in the course of which he informed me of the circumstances connected with the preparation and publication of it. The account he gave me was confirmed by a friend of his own, a gentleman of unquestionable veracity; and also, subsequently, so far as they were cognisant of the circumstances, by the Editors of the Mathematician. The circumstances were as follows:—The paper was prepared not later than September, 1845, at least two months before the appearance of Mr. Weddle's paper. In November of the same year it was handed to one of the Editors of the Mathematician; and the identity of the logarithmic methods developed in the two papers having been recognised, Mr. Hearn accompanied it with the requisite explanation. For some reason unexplained the paper did not appear till March, 1847, and the explanation was not given. I t was given subsequently, as already hinted; but the omission of it at the proper time exposed Mr. Hearn to a charge of plagiarism, which is now seen to have been altogether undeserved.
It is right to add that Mr. Weddle's paper is dated August 18, 1845; and that he states in it that his methods “were discovered nearly seven years ago.”
page 264 note * This indirectness would be got rid of if the cologarithms instead of the logarithms of the factors were tabulated. But it would then appear in the converse process, as it does in Mr. Weddle's method. There is no avoiding it, in the one operation or the other, in any method making use of the principle laid down on p. 255.
page 264 note † Further particulars regarding Mr. Orchard are contained in an Introduction, prepared by me, to a re-issue of his work in 1856.