ON MATHEMATICAL TABLES, 31 



at length — a curious mode of statement at the end of a book occupied with 

 decimal fractions. 



Goodwyn's Tables, 1816-1S23. It is convenient to describe Good- 

 ■wyn's four works (the titles of which are given at length in § 5) together, as 

 they all relate to the same subject. 



The Tahidar Series of Decimal Quotients (1823) forms a handsome table of 

 153 pages, and gives to eight places the decimal corresponding to every vulgar 

 fraction less than ^^j, whose numerator and denominator are both not greater 

 than 1000. The arguments are not arranged according to their numerators or 

 denominators, but according to their magnitude, so that the tabular results 

 exhibit a steady increase from -001 (= xu\rLr)to -09989909 (= JJL). The 

 author intended the table to include all fractions whose numerators and deno- 

 minators were both less than 1000 without restriction ; and at the end of the 

 book is printed " End of Part I. ; " but no more was ever published. 



The arrangement of the arguments in order of magnitude is not very good, 

 as it requires the first two figures of the decimal to be known in order to know 

 where to look for it in the table ; the table would be more useful if it were re- 

 quired to find a vulgar fraction (with not more than three figures in numerator 

 or denominator) nearly equal to a given decimal*; but this is not a trans- 

 formation that is often wanted. "When the decimal circulates and its period 

 is completed within the first eight figures, points are placed over the first and 

 last figures of the period, if not, of course only over the first; and by means 

 of the same author's table of ' Circles ' described boloM", the period can be 

 easily completed, and the whole decimal fraction found. The fractions which 

 form the arguments are given in their lowest terms. 



The Table of Circles (1823) gives all the periods of the circulating decimals 

 that can arise from the division of any integer by another integer less than 

 1024. Thus for 13 we find -076923 and -1 53846, which are the only periods 

 in which the fraction I- can circulate. 



The periods for denominator 2" 5"' x are evidently the same as those for 

 denominator x; and arguments of this form are therefore omitted ; but a table is 

 given at the end (pp. 110 and 111), showing whether for any denominator loss 

 than 1024 the decimal ( 1 ) terminates, and is therefore not included in the table, 

 (2) is in the table as it stands, or (3) is in the table but has to be sought 

 under a different argument (these last being numbers of the form 2» 5'" .r). 

 A third table (p. 112) also gives the' number of places after the sejyaratrix 

 (decimal point) at which the period commences. 



The principal table occupies 107 ])p. Some of the numbers are very long, 

 (e. g., for 1021 there are 1020 figui-es in the period), and are printed in lines of 

 difterent lengths, giving a very odd ai^pearance to many of the pagesf. 



A table at the end contains all numbers of the form 2" 5"' that are less than 



* It is proper to note, however, that the table was no doubt calculated for this purpose ; 

 the author considered his 'Table of Circles ' as giving decimals to vulgar fractions, and in- 

 tended this table to give vulgar fractions to decimals (see the introduction to the second 

 part of the ' Centenary ' 1816) ; the ' Tabular Series ' (181C) is complementary to the ' Cen- 

 tenary ;' but not so the ' Tabular Series' (1823) to the ' Table of Circles ' (1823), as the 

 latter only gives the periods. 



t If the period of a decimal consists of an even number of figures, it is well known 

 that the figures in the last half ai-e the complements to nine of tlie figures in the first 

 lialf; and tlie periods have been printed so that the comiilementary figures should bo under 

 one another. When the period is odd, there is always anollicr period of eoraplementary 

 figures, and the two are printed one under the other ; these facts account for wliat at first 

 siglit appears a capricious arrangement of the figures. 



