80 KEPORT— 1873. 



Some perhaps may find shorter waj's, as I believed I had myself, till advised 

 otherwise by the truly Honourable the Lord Bruncker, &c." This is perhaps 

 the first tabulation of an elliptic integral. 



Bounycastle, 1831. A table of the areas of segments (pp. 295-300) : 

 the same as T. XIII. of Hantschi. 



Todd, 1853. T. I. Areas (to 6 places) and circumferences (to 5 places) 

 of circles for the diameter as argument, the range being from diameter ^ 

 to diameter 24 at intervals of Jg^; the decimal fractions (to 4 places) 

 equivalent to J^^, ^2_.^ <^c., are printed at the top of each page. 



T. II. The same from diameter 24 to 100 at intervals of ^ (4 places 

 only for the circumferences). 



T. III. The same from diameter 12 to 600 at intervals of unity. Both 

 areas and circumferences are only given to 4 places. 



T. IV. The same from diameter "1 to 100 at intervals of 4. Areas to 6 

 places, circumferences to 5. 



T. V. to VII. stand in exactly the same relation to spheres that T. I. to 

 IV. do to circles, except that T. V. is equivalent to T. I. and II., the 

 intervals being ^ from 1 to 100 ; and T. VI. commences at 1 (not 12), The 

 volumes and superficies are given to 4 places. 



T, VIII. Areas (exact) and diagonals (to 5 places) of squares for side as 

 argument, from g to 100 at intervals of ^. 



In all cases the arguments are given in inches, and the results in square 

 and cubic inches ; but in T. III. and VI. the corresponding numbers of 

 linear, square, and cubic feet are also given. 



The original work, of which this is the second and greatly augmented 

 edition, was published in 1826 ; and the tables were the result of original 

 calculations. There are besides some specific gravities, &c. 

 The following tables are more fully described in § 4. 

 Mensuration tables. — Sharp, 1717 [T. II.], areas of segments of circles ; 

 [T. III.], table for computing the solidity of the upright hyperbolic section 

 of a cone ; Dodson, 1747, T. XXVI., XXVIII., and XXIX. ; Galbeaith, 

 1827, T. XV. and XVI. (Introd.) ; Hantschl, 1827, T. XIII. ; Troxter, 

 1841 [T. v.] and [T. XII.]; Willich, 1853, T. C (circumferences and areas 

 of circles) ; Beaedmore, 1862, T. 34 (circumferences and areas of circles) ; 

 Raneine, 1866, T. 4 and 5. 



Art. 23. Dual Logarithms. 



Dual logarithms were invented, and the tables of them calculated, by Mr. 

 Oliver Byrne, who, besides the work described below, has published ' Dual 

 Arithmetic ' and the ' Young Dual Arithmetician ' on the subject. A dual 

 number of the ascending scale is a continued product of powers of 1*1, l-Ol, 

 I'OOl, &c., taken in order, the powers only being expressed. To distinguish 

 these numbers from ordinary numbers, they are preceded by the sign \|/ : 

 thus, \i/ 6, 9, 7, 6 = (M)»(l-01)' (1-001)^ (l-OOOl)" ; n,]/ 0, 0, 2 = (1-1)° 

 (1-01)° (1-001)^, the numbers following the \j/ being called dual digits. 

 "When all but the last digit of a dual number are zeros, the dual number is 

 called a dual logarithm ; but the dual logarithms used by Mr. Byrne are " of 

 the eighth position," viz. there are 7 ciphers between the \|/ and the 

 logarithm. 



A dual number of the descending branch is a continued product of powers 

 of -9, -99, -999, &c., and the dual number is followed by the symbol /|\ ; 

 thus, (-9)3 (-99)2 = '3 '2 /|\; (-999)= (-999999)2 = '0' 0' 3' 0' 0' 2 /|\. In the 

 descending branch also a dual number reduced to the eighth position is 



