ON MATHEMATICAL TABLES. 41 



in seconds wholly, so that the expression of the result in seconds wholly is the 

 chief characteristic of [T. III.]. 



This table is followed by 3 pp. to convert sexagesimals into decimals and 

 vice versa, and numbers into sexagesimals and vice versa. The other tables 

 are weights and measures &c. There are numerous examples given in the 

 introduction. 



[T. IV.]. Another table occupying one page (p. 252) should be noticed ; 

 it gives squares, cubes, fourth, fifth, and sixth powers of any number of 

 minutes up to GO' : thus the square of 3' is 9" ; the cube, 27'" ; the fourth 

 power 1'" 21'" ; the fifth 4'" 3% &c. The words sursolid and square cube are 

 used for the fifth and sixth powers. 



On the present work see also Beverley (1833?) (§ 4). 



It was the author of this table (Taylor) who afterwards calculated the 

 logarithmic trigonometrical canon to every second. 



The following are references to works in § 4 : — - 



Sexagesimal tables :— Lynx, 1827, T. Z ; Bagay, 1829, T. XXIV, (lo- 

 garithms with sexagesimal arguments); Beverley (1833 ?),T. VI. (pp. 232 

 &c.) and T. XV.; Shortkede (Com. log. Tab.), 1844; Gordon, 1849, T. 

 XVII. (half sines, &c., expressed sexagesimally). 



Tables for the conversion of sexagesimals into decimals, and vice versa: — 

 Douglas, 1809, T. III., Supplement ; Ducom, 1820, T. XX. ; HtJLssE'a 

 Vega, 1840, T. IV. 



Art. 10. Tables of natural Trigonometrical Functions. 



A history of trigonometrical tables by Hutton is prefixed to all the editions 

 of his ' Tables of Logarithms ' published during his lifetime * ; and, in his 

 Article on Tables in the ' English Cyclopaedia,' De Morgan has given what 

 is by far the most complete and accurate account of printed tables of this 

 kind that has been published. Information about the earlier tables is also 

 to be found in Montucla and Delambre (see references in De Morgan). For 

 many years, when Mathematics had not passed beyond Trigonometry, 

 the method of construction and calculation of the * Canon Trigonometricus ' 

 formed one of the chief objects of the science, and the works on the subject 

 were comparatively numerous, though now, of course, of purely historic 

 interest only. Prior to the introduction of sines from the Arabians by 

 Albategnius, trigonometrical calculations were always made by chords. The 

 imit-arc was the arc whose chord was equal to the radius (viz. 60°) ; and 

 both arc and radius were divided into 60 equal parts, and these subdivided 

 again into 60 parts, and so on. (It thus appears that it was not the right 

 angle that was divided into 90, 60 and 60 pai-ts, &c., but that the unit-angle 

 was 60°, so that the division was strictly sexagesimal throughout. It is 

 curious that in some modern tables (see Beverley, T. VI. and XV. &c.) the 

 original arrangement has been restored, for convenience of interpolating by 

 Taylor's sexagesimal table). Thus in the earliest existing table, viz. the 

 table of chords in the Syntaxis of Ptolemy (died a.d. 178), the chord of 90° 

 is 84° 51' 10". Purbach (born 1423) and llegiomontanus (born 1436) calcu- 

 lated sines, the former to radius 600,000 and the latter to the same radius 

 and also to radius 1,000,000; but it is not certain whether they were printed. 

 The first known printed table, according to De Morgan, is a table of sines 

 to minutes, without date, but previous to 1500. Peter Apian first published 

 a table with the radius divided decimally (1533). Tangents were first pub- 



* It also forms Tract XIX. vol. i. pp. 278-306 of his ' Mathematical Tracts,' 1812. 



