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10,000, and of the sines and tangents for every minute to five places of decimals.
For many astronomical computations and reductions, where only small quantities are concerned, or a rough approximation is required, logarithms to five decimals are sufficient; but in many of the cases of most frequent occurrence in practical mathematics, they do not afford the requisite precision. This inconvenience is felt more particularly in Surveying and Navigation ; and hence the Requisite Tables used in the navy, and works on navigation in general, contain the logarithms to six decimal places. In fact, in all trigonometrical computations in which results are required to be found correct to single seconds, or even two or three seconds, logarithms of six figures are indispensable.
In the present collection the logarithms of the numbers are arranged according to the mode adopted in Callet's table. The three first figures of the number are placed in the first column of the page, headed N., and the last figure at the top and foot of the succeeding columns. The two first figures of the logarithm are placed in the second column, and not repeated; and the remaining four in the column indicated by the fourth figure of the number. A blank space marks the change of the second figure of the logarithm; and when the change takes place, the three first digits of the corresponding number in the first column are repeated (which is not done in Callet's table), by which means the four last figures of the logarithm always stand in the same line with the three first of the corresponding number, and some chances of mistake are avoided in seeking the number corresponding to a given logarithm. The last column of the page, headed D., contains the differences. They are, of course, the mean differences of the ten belonging to the same line of logarithms; but it can very rarely happen that an error will be made in the last figure of a logarithm or number by using the mean instead of the true difference when the two happen not to be the same.
By adopting this arrangement instead of placing each logarithm after its natural number, two advantages are obtained. In the first place, there is only one column of natural numbers on the page; and in the second place, as the page admits of a much greater number of logarithms, without any sacrifice of distinctness, the finding of the logarithm to any number, or vice versâ, is rendered much easier in consequence of the greater number brought under the eye at once. In Lalande's table, the logarithms of numbers occupy 112 pages, in this table they occupy only 22 pages.
At the end of this table is given a page of the logarithms of the numbers from 1000 to 1200, correct to seven figures: they are sometimes required in calculations of compound interest, &c.
In the trigonometrical table the differences are given not for the whole minutes, as in Lalande's table, but for every 100", whereby considerable facility is afforded in the use of the table. To find the difference corresponding to any given number of seconds, it is only necessary to multiply the tabular difference by the number, and strike off the two last figures.
To this table is added another short table, containing the logarithms of the sines for every six seconds (or tenth of a minute) of the two first degrees. This will often be found useful in computations where small angles occur, and precision is required.
The formulæ for the solution of triangles are adapted to logarithmic computation, and include all the cases of plane and spherical triangles. As most persons who have occasion to make trigonometrical calculations frequently find it necessary to refer to the formulæ, it was considered that their insertion would render the work a more useful manual.
The table of constant numbers will be found a useful addition to the mathematical student. A few of those numbers, relative to the circle, were given by Lalande and Callet. A more extensive list was given in Babbage's Logarithms, to which considerable additions were made in the Society's table before mentioned. In the present work the collection is still further augmented.
No trouble has been spared to render the work not only generally correct, but, if possible, absolutely free from error.
For this purpose the publishers placed it in the hands of Mr. Farley, whose experience and accuracy in such matters are well known, and the following is the course of examination that was pursued.
The logarithms of numbers were set up from a copy of Riddle's Navigation, 2d edit.; the proofs were read with Vega’s Tables (which are to 10 places); examined entirely by differences, and then stereotyped; proofs from the stereotype plates were then read with Callet's
Tables (which are to 7 places), and the whole again examined by differences, after which the impression was worked off.
The logarithms of sines and tangents were amined in a similar manner; the differences for 100" were obtained by increasing those for l', as indicated by the six-figure logarithms, by two thirds of themselves; this was done both in the original and stereotype proofs.
With respect to typography, it has been thought advisable to follow the example recently set by the Society for the Diffusion of Useful Knowledge in returning to the use of the old numerals. Figures of this form are used in all the French tables; and there can be no doubt that they are greatly preferable, in point of distinctness, to the numerals of uniform height used in modern English works.