ON MATHEMATICAL TABLES. 81 



called a dual logarithm, and is to be considered negative if the ascending 

 dual logarithm is taken positive, and vice versa. 



Byrne, 1867. T. I. contains aU the dual numbers of the ascending 

 branch of dual arithmetic from \|/ 0, 0, 0, 1 to \|/ 7, 3, 1, 9, and their 

 corresjjonding dual numbers and natural numbers. The range of the dual 

 logarithms is from 00000 to 69892175, and of the natural numbers from 

 1-00000000 to 2-01167234. Marginal tables are added, by means of which 

 all dual numbers of 8 digits, and their corresponding dual logarithms and 

 natural numbers, may be derived : the table occupies 74 pp. 



T. II. Dual logarithms and dual numbers of the descending branch of 

 dual arithmetic from '0 '0 '0 '1 '0 '0 '0 '0 /|\ to '3 '6 '9 '9 '0 '0 '0 '0 /|\ with 

 corresponding natural numbers. The range of the dual logarithms is from 

 '10001 to '39633845, and of the natural numbers from -99990000 to 

 •67277805. Marginal tables are added, by means of which all intermediate 

 dual numbers of 8 digits and their corresponding dual logarithms and natural 

 numbers may be derived. This table is printed in red, T. I. and III. being 

 in black. It occupies 38 pp. 



T. III. Natural sines and arcs to 7 places for every minute of the 

 quadrant. The length of the arc is, of course, the circular measure of the 

 angle, so that we have a table of circular measures to minutes : the arrange- 

 ment is quadrantal. Proportional parts are given for 10", 20". . . .90" for 

 each difference ; and these occupy two thirds of the page. There are small 

 proportional-part tables for the arc : the table occupies 90 pp. 



The author claims that his tables are incomparably superior to those of 

 common logarithms, and asserts that " these tables are equal in power to 

 Babbage's and Callet's, and take up less than one eighth of the space " 

 ('Dual Arithmetic,' part ii. p. ix). Bahhage and Callet seems an error 

 (unless the Callex of 1827 (§ 3, art. 15) is meant), as the latter work con- 

 tains the table of the logarithms of numbers which is the sole contents of the 

 former. Mr. Byrne's works on the subject are : — ' Dual Arithmetic : a new 

 Art,' London, 1863, 8vo (pp. 244) ; ' Dual Arithmetic : a new Art. New 

 Issue, with a complete analysis,' 1864 (pp. 83) [this work contains a table 

 of 3 pp., " to facilitate the conversion of dual numbers into common ones, or 

 the converse "] ; ' Dual Arithmetic: a new Art. Part the Second ' (pp. 218), 

 and the work above described. Mr. Byrne has also published ' The Dual 

 Doctrine of Angular Magnitude and Functions, &c.,' and the ' Young Dual 

 Arithmetician,' neither of which we have seen : the latter contains an 

 abridgment to 3 dual digits of the tables in the work described above. 



In spite of the somewhat extravagant claims advanced by the author for 

 his system, dual logarithms have found but little favour as yet either from 

 mathematicians or computers. 



Art. 24. Matliematical Cotistants. 

 In nearly all tables of logarithms there is a page devoted to certa^ 

 frequently used constants and their logarithms, such as n, -, tt-, i/n, aYt:, 



&c., the radius of the circle in degrees, minutes, &c., the modulus &c. 

 There are not generally more than four or five logarithms involving tt given ; 

 and usually half the page is devoted to constants relatiug to the conversion 

 of weights and measures. It is only necessary, therefore, here to refer to 

 works in which tliere is a better collection than usual of constants. 

 1873. s 



