32 



REPORT — 1873. 



1,000,000, arranged in order of magnitude, with the values of n and m, and also 

 the values of the reciprocals of the numbers (expressed as decimals) and the 

 total number of the proper vulgar fractions in their lowest terms which can 

 arise for any of the arguments as denominator. An example of the use of 

 the tables is given at the end of the book. 



The First Centenary Sfc. [1816] contains the factors of all numbers to 100, 

 and the complete periods of their reciprocals or multiples of theii- reciprocals, 

 also the first six figures of every decimal fraction equivalent to a vulgar frac- 

 tion whose denominator is equal to the argument. The following is a spe- 

 cimen of one of the tables : 



The explanation is veiy simple: we have || = -970588, and the other 

 figiires of the period are 23529411764; fj- = •911764, and the other figures 

 are 70588235294, &c. If the numerator is in the third column m-c take the 

 complement of the result {i. c. subtract each figure from 9) ; thus J^ = 

 •029411, and the other figures of the period are 76470588235'. The even 

 numbers are omitted, as the fractions are not in their lowest terms ; thus ^^ 

 = -}--f-, and must be sought under argument 17. [This table was published 

 separately by Goodwyu for private circulation. There is no date on the title- 

 page*; but the address is Avritten from Blaekheath, and dated March 5, 1816.] 

 There is added a tabular series of complete decimal quotients of fractions 

 whose numerator is not greater than 50 and denominater not greater than 

 100 (the heading of the table incorrectl)" says, " iieither numerator nor de- 

 nominator greater than 100 "), arranged as in the ' Tabular Series' &c.,1823 ; 

 it is followed by an auxiliary table for completing such quotients as consist 

 of too many places to allow all the digits of their periods to appear in the 

 principal table. There is an appendix on Circulates &c. The ' Tabular Series' 

 (1816 and 1823) are interesting as exhibiting in the order of magnitude all 

 fractions whose numerators and denominators are both less than 100 up to ^, 

 and whose numerators and denominators are both less than 1000 up to Afl_. 

 In the preface to the latter table the author gives as a fact he has observed, that 



* It IS by no means improbable that the titlepage has been torn out from the onlv copy 

 we have seen, viz. that in the Eoyal Society's Library. 



