ON MATHEMATICAL TABLES. 11:^7 



[T, IV.] Logistic logarithms to every second to one degree, to four places. 

 The pages in [T. III.] and [T. IV.] are not numbered. 



[T. v.] is the first table in the second volume. It contains : — natural sines, 

 tangents, and secants to seven places, Avith differences ; log sines and tangents 

 to seven places, with differences (from 0° to 4° the simple difference, and from 

 4° to 45° one sixth part of the difference, is given) ; and Napisrian (see § 3, 

 art. 17) log sines and tangents to eight places, without differences ; all for 

 every ten seconds for the first four degrees, and thence for every minute to 45°. 

 The Napierian logarithms (see first page of Preface to the second volume) are 

 taken from the ' Canon Mirificus ' of Napier, augmented by Ursinus. The 

 arrangement of the table is not very convenient, but perhaps the best 

 possible. 



[T. VI.] (pp. 262, 263). First nine multiples of the sines of 1°, 2°, .3° 

 .... 90°. One or two constants are given on p. 264. 



[T. VII.] Circular measiire of all angles from 1° to 360° at intervals of 

 ]°. This is followed by similar tables for minutes from 1' to 60' at intervals 

 of 1', and for seconds from 1" to 60" at intervals of 1", all to 27 places. 



[T. VIII.] Powers, as far as the eleventh, of decimal fractions from '0 to 

 J -00 at intervals of -01, to eight places. 



[T. IX.] Squares of numbexs to 1000, 



[T. X.] Cubes of numbers to 1000. 



[T. XI.] Square roots of numbers to 1000, to seven places. 



[T. XII.] Cube roots of numbers to 1000, to seven places. 



[T. XIII.] The first six binomial-theorem coefficients, viz. ,r, ^' — '-, .... 



'—- — -,,L''^ -5 for X = -01 to cc = 1-00, at intervals of -01, to seven 



places. 



The other tables connect the height and velocity of falling bodies, and 

 contain specific gravities &c. A table on the last page is for the conversion 

 of minutes and seconds of arc into decimals of an hour. 



A table headed Jiaiionale Trigonometrie occupies pp. 308-311 , and is very 

 interesting. It gives right-angled triangles whose sides are rational and 

 such that tan |w (w being one of the acute angles of the triangle) is 

 greater than J^. Such triangles (though not so called here) are often known 

 as Pythagorean. Those with sides 3, 4, and 5 ; and 5, 12, and 1.3 are the 

 1)est-known cases; and 8, 15, and 17, 9, 40, and 41, 20, 21, and 29, &c. are 

 among the next in point of simplicity. This table contains 100 such tri- 

 angles ; but some occur twice. It gives in fact a table of integer values of 

 a, b, c, satisfying (r-\-h'-=c-, subject to the condition mentioned above: 

 tan iio, expressed both as a vulgar fraction and as a decimal, is given, as also 

 are w and 90° — w. For a larger table of the same kind, see Sang, 'Edinburgh 

 Transactions,' t. xxiii. p. 757, 1864. On the whole, this collection of tables 

 is very useful and valuable. 



[Schi^macher, 1822 ?]. T. V. Five-figure logarithms of numbers for 

 every second to 10,800" (3°), arguments expressed in degrees, minutes, and 

 seconds. 



T. VI. Log sines for every second to 3°, to five places. There is no name 

 at all on the table ; but it is assigned (and no doubt correctly) to Schumacher 

 in the Royal Society's Librarj- ; and De Morgan, speaking of Waexstoeff's 

 ScnuMACHER (1845), says that the original publication was Altona, 1822; 

 but there was an earlier edition, we believe, at Copenhagen, in 1820. 



Shanks, 1853. The bulk of this work _([T. I.] jjp. 2-85) consists of the 

 values of the terms in Mr. Shanks's calculation of the value of n by Machin'a 



