ON MATHEMATICAL TABLES. 87 



These tables occupy 256 pp., and are followed by 78 pp. of formulae, weights, 

 and measures, &c. 



There is a full introduction, stating whence the tables were derived, or, if 

 computed, from what formulae, ifec. The hyperbolic logarithms were taken 

 from "Wolfeam's table in Schulze ; and the reciprocals, factors, square and 

 cube roots, and several other ^tables were the result of independent cal- 

 culations. 



The squares, cubes, square and cube roots, and reciprocals from this table 

 were reprinted and stereotyped, at the suggestion of De Morgan, in 1840 (see 

 Barloav's tables, 1840, in § 3, art. 4). The reprint thus gives T. I., the 

 column of factors being omitted. A list of 90 errors in T. I. of the original 

 work is given in the reprint ; and 25 errors in T. YI. are given by Prof. 

 Wackerbarth in the ' Monthly Notices of the Eoyal Astronomical Society ' for 

 April 1867. 



Bates, 1781. [T. I.] Five-figure logarithms to 10,000, without dif- 

 ferences. 



[T. n.] Log sines and tangents (to 5 places), and natural sines and tan- 

 gents (to 7 places), for every minute of the quadrant, semiquadrantally 

 arranged: no differences. 



The tables (which have a separate titlepage, bearing the date 1779) are 

 preceded by 211 pp. of trigonometry, and followed by an Appendix on the 

 motion of projectiles in a non-resisting medium. The work was intended for 

 use in the Military Academy, Belmont, near Dublin. 



Beardmore, 1862. Only 23 pages (pp. 84-106) of this work contain 

 tables that come within the scope of this Beport. 



T. 34. Areas and circumferences of circles, to 3 places, for diameters 

 •1, -2, -9, and from 1-00 to 100, at intervals of -25. 



T. 35. Squares, cubes, fifth powers, square and cube roots (to 3 places), 

 and reciprocals (to 9 places) for numbers from 1 to 100, the squares and 

 square and cube roots being given as far as 1100. 



T. 36. Six-figure logarithms of numbers from 100 to 1000. 



T. 37. Log sines from 0° to 4-5° 50', at intervals of 10', to 6 places. 



T. 38. Natural sines, tangents, and secants for 1°, 2°, . . . . 90"^, to 6 places. 

 The other tables relate to hydraulics, rainfall, &c. 



The work was first published in 1850 ; and a second edition, in an extended 

 form, was issued in 1851. 



Beverley [1833?] T. VI. (p. 127). Any number of minutes less than 

 12'' expressed as a decimal of 12'', to 4 places. 



T. VI. (pp. 232-243). Sexagesimal cosecants and cotangents for every 

 minute from 20° to 90°. A sexagesimal cotangent is the cotangent when 

 the radius is taken = 60' (or 1°); viz. it bears to 60' the same ratio that the 

 ordinary cotangent does to unity, and is usually expressed in minutes, seconds, 

 and decimals of a second. The same, of course, holds for sines, cosines, &c. 

 Thus the sexagesimal sine of 30° is 30', cosecant 30°= 120', &c. 



In this table the quantities tabulated are not sexagesimal functions, but 

 sexagesimal functions divided by 3 (and are therefore to radius 20') : we thus 

 have cosec 30°=40'. The table is given to two decimal places of a second. 



T. XV. Sexagesimal sines, tangents, secants, and versed sines (viz. to rad. 

 60') to every degree to 90°, to one decimal place of a second, with differences. 



T. XVII. Log sines and tangents, from 18° to 90°, at intervals of 1', to 

 4 places. 



T. XVIII. Proportional logarithms for every second to 3°, to 4 places ; 

 same as T. 74 of Rapek. 



