SSG. J. R. Hind. Natural versed sines .. used in computing lunar distances for the Nautical Almanac.' This table gives natural and logarithmic versed sines to seven decimals : natural, 0(10") 125°, with the proportional parts for each second ; logarithmic, Oh (1') with the values of the angles in space annexed. This table, though bearing in its title only a limited notion of application, will be ex ceedingly valuable, especially to those who want sines, cosines, and their squares.
1857. George and Edward Scheutz, Specimens of tables calculated, stereomoulded, and printed by machinery,' London. The first pro. Auction of the machine which two Swedes, father and son, constructed on Mr. Babbago's principles, as suggested by Dr. Lardner's article in the Edinburgh Review' (1834), with their own details ; five-figure logmithins 0 (1) 10000; with some specimens of other tables. This work was reproduced at Paris, with a French preface, in 1858. In 1859, was published Mountain Barometer Tables,' calculated by the same machine.
1S59. Edward Sang,' Five-place logarithms,' 0 (1) 10000.
1860. Ludwig Sehr6n, Schron's Logarithms,' Tafel I., II., III. We had just, as we thought, put the finishing hand to this article, when the table above-named reached us. Should it turn out to possess the requisite accuracy of printing, it will have decided success. It is a large octavo volume of 550 pages of tables. The type, though without heads and tails, is all as nearly as possible of ono thickness, and that thickness not too great, so that it might be called thin Egyptian ; and it is very legible. The contents are all to seven decimals. Logarithms of numbers 0 (1)10S000, with subsidiary tables at the bottom of the page, which by addition of two logarithms in the page give the loga rithms of sines and tangents, 0 (0'001) 1" (O".01) 10" (0'1) 1'40" and 1'40" (0'01) 16' 40" 3°. Logarithms of sines and tangents, 0 (10") 45°. The differences begin to be inserted from 3° ; and the first nine multiples complete, that is, one figure more than in the common table of proportional parts, are given in the same page ; first for every fifth number, then for every three, &c., as the page will bear it. Under 3° multiples of subsidiary numbers are given, as explained. Then follows a table of proportional parts to complete hundredths, for numbers from 40 to 409.
1860. Galbraith and Haughton, 'Manual of Mathematical Tables.' The usual five-figure table, with Gauss's table. A 0 (•001) 2 (.1) 5. The first British five-figure table, we believe, which gives Gauss's table.
1861. A. De Morgan. Three-figure logarithms : three figures of number to three of logarithm, complete,, on a sheet of 7) by 6 inches. The third figure of the number in every case by the side of the loga rithm : all numbers in red, logarithms in black. The quarter of a
unit in which the logarithm lies, shown by use of the four common punctuating stops. Intended for the earliest instruction in logarithms, and ns a substitute for the sliding rule in ceitaM cases.
§ 7. The next tables which we shall mention are those which are wanted in the higher mathematics.
Extensive tables of elliptic functions are in Legendre's "frait6 des Fonetions Elliptiques; 2 vols. 4to., 1825 and 1826. The factorial function, Tx, is tabulated in the same work ; and also in the 'Exer cicea du Calcul Integral' of the same author, Paris. 1817, in which several other definite integrals are also tabulated. An abridgment of this table (with ready means of restoring it fully) is in the treatise on the Differential Calculus (' Lib. Use. Know?), p. 587. Tables of the integer form of I's, or 1.2.3 .... (x— 1), or rather of the logarithms of the values, are given by C. F. Degen, Tabularnin Enneas,' Copen hagen. 1824, up to .r. 1201, to IS decimal places: this table is re printed to six decimal places at the end of the article Theory of Probabilities' in the Encyclopaedia Metropolitaua. Tables of the were first given by Kramp, with logarithms of the values, Analyse des Refractions Astronoiniques; Strasburg, 1799. This table is reprinted in the Encyc. Metrop., art. Theory of Probabilities.' The form in which this integral more usually occurs in the theory of probabilities (with the factor 2 : Vs) was given (by Profess( r Encke, we believe) in the Berlin Astrono miaches Jahrbuch ' for 1834, from whence it was copied into the article in the Encyc. 3letrop., above noticed; and (with extensions) into the Essay on Probabilities and Life Contingencies' in the Cabi net Cycloptedia, aud into the article on Probability in the new edition of the Encyclopaedia Britannica.' A few other definite integrals have been tabulated : one very useful one...frx ; log a., by Soldner, Nouvelle Fonction Transeendente.' 3Iiinich, 1809, copied into the Differential Calculus' (' Lib. Use. Know.'), p. 662. The integrals known by thee name of Spence's Logarithmic Transcendunts, aro in the work with that title (Edinburgh, 1809; Sir J. Herachel'a editiOn, London, 1S20). There are a few of the integrals in optics scattered through the Me — • — rmiirs of the Institute and of the Cambridge and Philosophical Society (in memoirs by Fresnel and 3Ir. Airy). Perhaps we should also mention the tables for the solution of indeterminate equations of the second degree. Of these there is one in Legcndre's Theorie des Nombres r another has been given by Jacobi; and a third, by Degen, called ' Canon Pellianus,' Copenhagen, 18 I 7.