5o. Tables of the Square Roots and Cube Roots of Numbers. A table of the first kind has been given by Mr. Lambert: and a more extended one by Mr. Barlow, in his Collection. The latter writer constructed his table by means of differences; ane advantage which may be applied with greater effect to the table of Cube Roots, on account of the greater convergency of this order of differences.
6o. Tables of the Reciprocals of Numbers. These are amongst the most simple but most useful of arithmetical tables; and are peculiarly valuable in converting various series into numbers, — thus facilitating the calculation of differences for the more ready construction of other tables. In order, however, to be employed in such operations, it is absolutely necessary that they should be infallible. Several tables of this kind have been printed: the most recent and extensive of which are those of Mr. Barlow, and Mr. Goodwin.
7o. Tables of Natural Sines, Cosines, Tangents etc. The utility of tables of this kind is evident from the variety of forms in which they have been, from time to time, printed: and it is needleſs to insist on their importance at the present day, since no seaman dare venture out of sight of land without a knowledge of their use. In ordre to be of any real service however, they should be accurate; et diligently revised from time to time: otherwise they may be worse than useleſs. The labour of computing tables of this kind will vary according to the number of figures contained in the result. It appears desirable that the larger tables of this sort should be printed with their several orders of differences to a much greater extend than formerly, for the purpose of making other tables, and for executing several mathematical operations beneficial to science. It would be difficult to state precisely the quantity of mental labour saved by the machine, in constructing tables of this kind; but, I believe, it may be facily reduced to the two thousandth part of the whole.
8o. Tables of the Logarithms of Numbers. Tables of this kind are in the hands of every person engaged in numerical investigations: and it is needleſs to dwell on their utility et importance. The logarithms of numbers from 1 to 108000 have been already computed, with a greater or leſs number of figures; but this has been the work of various authors, and of several successive years: the labour is so immense that no human being has ventured to undertake the whole. The table which now exist are chiefly copies from those original and partial computations. By the help of the machine, however, this immense labour vanishes, and new tables may be readily computed and re-computed as often as may be required by the public. It is probably that the present tables if extended from 108000 to 1000000 would be of greater utility, than an extension of the present tables to a larger number of figures. The quantity of mental labour saved by the machine may be estimated in the following manner. Suppose a machine constructed, capable of computing with five orders of differences; it would be necessary to calculate those differences for every thousandth logarithm only: consequently, if the table extended from 10000 to 1000000, there would be but ninety sets of differences to compute. Any one of these sets being placed in the machine, with its first five differences, it will deliver the 500 preceding logarithms and also 500 succeding ones; thus producing a thousand logarithms: at the end of which term, another set of differences must be substituted. With five orders of differences, a table of logarithms may be computed to eight places of figures, which shall be true to the last figures, and it would not require more than half an hour to compute each set of differences; particularly as the higher numbers require very little labour, two or three terms of the series being quite sufficient.
9o. Tables of Logarithmic Sines, Cosines, Tangents et Cotangents. The remarks which have been made in the preceding article, will apply with nearly equal propriety to the tables here alluded to. The mental labour required for the construction by the machine is reduced to a very insignificant quantity, when compared with the prodigious labour employed in the usual way.
10o. Tables of Hyperbolic Logarithms. Some small tables of this kind have been printed in several works, and are useful in various integrations: but the most comprehensive set was computed by Mr. Barlow, which contains the hyperbolic logarithms of all numbers from 1 to 10000. The labour of computing them is very great, which is the cause of their not being more extended. From a slight examination of the subject, it would appear that the mental labour may, in this case, be reduced by the machine to about a two hundredth part of what was formerly necessary.
11o. Tables for finding the Logarithms of the sum or difference of two quantities, each of whose logarithms is given. This table, which was first suggested, by Mr. Gauſs, has been printed in at least three different