30 REPORT— 1873. 



Eaneine, 1866, T. 2. See also Sir Jonas Moobe [1650?], § 3, art. 4; 

 Taylor, 1781 [T. IV.] (§ 3, art. 9). 



Tables for the solution of Cubic Equations, Vv/.. +(x — aj^). — Lambert, 1798, 

 T. XXIX. ; Barlow, 1814, T. IV. 



Art. 6. Tables for the expression of vulgar fractions as decimals. 



The only separate tables we have seen are Wuoherer and Goobwyn's 

 works described at length below. The Babbago Catalogue contains the title 

 of an anonymous book, " Tafeln zur Verwandlung aller Briichc von j^-^ bis 

 ■^roli^' ^^^"^ "^°^ Tu'uo ^^^ TtMmro ^^ fiinf- bis siebenziftrige Decimalbriiche, 

 4to, Oldenbiirg, 1842," of which De Morgan says "it gives every fraction 

 less than unity whose denominator does not exceed three figures, nor its nu- 

 merator two, to seven places of decimals. It is arranged by numerators ; 

 that is, all fractions of one numerator are upon one double page." Eecipro- 

 cals would property be included in this article ; but from their more frequent 

 use they have been jjlaccd in an article by themselves (§ 3, art. 7) ; Picabte's 

 table in that article gives multiples of reciprocals. 



We must especially mention the " Tafel zur Verwandlung gemeiner 

 Briiche mit Nennern aus dem ersten Tausend in Decimalbriiche," Avhicli 

 occupies pp. 412-434 of vol. ii. of ' Carl Friedrich Gauss Werke,' Gottingen, 

 4to, 1863, and which somewhat resembles Goodwyn's tables described below. 

 In it, among other things, the reciprocal of every prime less than 1000 is 

 given completeh/ (i. e. till the figures circulate). Had we met with the table 

 eai'lier we should have given a full description ; but we merely confine our- 

 selves hei'e to giving the reference, reserving a more detailed explanation for 

 a future Eeport. 



Wucherer, 1796, The decimal fractions (to five places) for all vulgar 

 fractions, whose numerators and denominators are both less than 50 and 

 prime to one another, arranged according to denominators ; so that all 

 having the same denominator are given together : tlius the order is ... . -jL., 

 -fif' TT'- • • -i?' tV' t\- • • -5 the arguments being only given in their lowest 

 terms. After ^^ the system is changed, and the decimals are given for 

 vulgar fractions whose numerators are less than 11 only; thus we have -Jj,, 

 tV' TiJ- • ■ •'k?)' ftV' 7)"t- • • -^3 consecutive arguments (tlie arguments not being 

 necessarily in their lowest terms) ; and the denominators proceed from 50 to 

 999. 



[T. II.]. Sea-arjesimal-Bi'ikhe, viz. sexagesimal multiplication table to 60 

 X 60 ; thus, as 5 X 29" = 145" = 2' 25", the table gives 2.25 as the tabular 

 result for the joint-entry 5 and 29. There are seven other tables (II.-VIII.) 

 for the conversion of money into decimals of other money, for the coins of 

 different countries ; the English table will serve as an example. There are 



given as arguments ^-^, -jf^, ^„ |fi^ (/. e. Id., 2d., 'M., &c.), and as 



tabular results the corresponding decimal fraction to ten places (/. e. of £1), 

 and also the shillings and pence ; thus for ^4^ there are given -3041666666, 

 and 6s. Id. 



Tlie Ileichs-Geld and Pfennig table is practically the same ; the denomi- 

 nators are in all cases 240, or 960, or submultiples of the latter. Regarded ma- 

 thematically the English table gives nearly as much as all the rest, as for 

 denominators above 240 only a few numerators are taken. There are also tables 

 of interest, present value, &e., to a great many places. The value of tt is given 

 on the last page to 306 places; thus, if the diameter = lOOUO. . . .(306 

 ciphers), then tt = 31415 (307 figures), the ciphers and figures being written 



