1878.] Ilr Glaisher, On circulating decimals. 191 



141 



•0070921985815602836879432624113475177304964589 

 •9929078014184397163120567375886524822695035460 



In this manner all the periods of the numbers prime to 10 up to 

 1024 are given ; the digits in the complementary periods or half 

 periods being always printed under the digits to which they are 

 complementary. Thus in the case of 41 and 141 the comple- 

 mentary periods are printed in the second line, and in the case of 

 127 the second half of each period is printed under the first half. 



The ' tabular series ' contains the first eight (or in special cases 

 nine or ten) digits of the decimal equivalent to every vulgar frac- 

 tion, in its lowest terms, whose numerator and denominator are 

 both not greater than 1000 from jJ^y^ to -^-^j, arranged in order of 

 magnitude. It was Mr Goodwyn's intention^ to have extended the 

 table as far as | ; this would have included all fractions which, in 

 their lowest terms, have numerators and denominators both not 

 greater than 1000, as the decimal values of the vulgar fractions 

 between ^ and 1 are obtained at once from those between and |, 

 by replacing each digit by its complement. 



There is also at the end of the ' table of circles ' a small table 

 of two pages entitled, " A table shewing under what divisor in the 

 preceding table of circles those are to be found that arise from 

 any given divisor between 1 and 1024." Adopting the notation of 

 § 3, it gives the value of s corresponding to ever}^ value of q from 

 1 to 1024, i.e. corresponding to q the table gives the resulting 

 factor when powers of 2 and 5 have been thrown out. If q is of 

 the form 2™5" the letter T is given as tabular result, indicating 

 that the decimal terminates. 



Taking the examples in § 3, if we require the complete decimal 

 fraction equivalent to y^^\ we enter the ' tabular series ' with this 

 number and find as tabular result •09311224. Entering the small 

 table with 784 we obtain as tabular result 49, and entering the 

 ' table of circles ' with 49 we find the single period of 49 which 

 has been quoted in § 1 (vi), and beginning this with the digits 1224 



1 It is stated in the introduction that "the table which is now submitted to 

 public inspection is the iirst part of one which is intended to exhibit the decimal 

 value of every proper fraction whose denominator is less than 1000," and at the end 

 is printed "end of part I." This part contains all fractions whose decimal values 

 begin with -0, and the author's intention was that the second part should contain 

 those whose decimal values begin with •! ; the third those whose decimals begin with 

 •2 ; the fourth with "3, and the fifth with -4. The publication of these other parts 

 is stated to be dependent upon the reception which the present work met with : and 

 no more was ever published. De Morgan, in his article on Tables in the English 

 Cyclopedia, states that "Mr Goodwyn's manuscripts, an enormous mass of similar 

 calculations, came into the jJossession of Dr Olinthus Gregory, and were purchased 

 by the Eoyal Society at the sale of his books in 1812." Nothing however is known 

 nf them at the Eoyal Society. 



