200 



Mr Glaisher, On circulating decimals. [Oct. 28, 



Thus for example if = 410256, if = -SSSOT^, 3^ = -128200, &c. 

 With this arrangement, it would be necessary to search among 

 the columns of arguments for the particular numerator required; 

 but each period would only be printed once, and all the numerators 

 having the same period would be placed together. 



§ 11. The property of the number 487 referred to in § 2 (x), 



viz. that the periods of -^ and (4^)^ both contain 486 digits, was, 



I believe, discovered by Desmarest, who seems to have divided the 



period of each number up to 1000 by the number and found that in 



no other case except that of 3 and 487 was the period a multiple 



of the number. For he enunciates the rule in § 2 (x) as follows^: 



"1° Si on transforme en fractions de I'ordre decimal une fraction 



A 



p2> le norabre P premier absolu ^tant inferieur a 1000; 2° si on 



d^signe par^ le nombre des chiffres de la pdriode donn^e par la 



fraction ^ ; 8** si on excepte les nombres premiers 3, 487 ; le nombre 



de chiffres de la p^riode inconnue, c'est-a-dire de la periode donnee 



par la fraction -p-^ , est le nombre entier j) . P." These words and 



others distinctly imply that 3 and 487 are the only exceptions to 

 the general rule up to 1000. In order to establish this and to 

 find that 487 was an exception, Desmarest must have performed 

 the divisions (or employed some equivalent process); but it seems 

 strange that if he had actually performed this heavy work he 

 should not have stated the fact explicitly. On the other hand, I 

 have been able to find no allusion to the property of the number 

 487 prior to the date of Desmarest's work ; and it is scarcely to be 

 supposed that Desmarest would adopt so important a statement 

 as that quoted above without giving his authority. 



I have verified Desmarest's statement with regard to 487, and I 



\=23Ti6d> ^^ obtained by actual 



53593 

 14579 

 27720 

 69609 

 28131 

 08418 

 45585 

 07802 

 37371 

 31622 



42915 

 05544 

 73921 

 85626 

 41683 

 89117 

 21560 

 87474 

 66324 

 17659 



81108 

 14784 

 97125 

 28336 

 77823 

 04312 

 57494 

 33264 

 43531 

 13757 



82956 

 39425 

 25667 

 75564 

 40862 

 11498 

 86652 

 88706 

 82751 

 7 



878852 

 05133 

 35112 

 68172 

 42299 

 97330 

 97741 

 86550 

 54004 



01026 

 47022 

 93634 

 48459 

 79466 

 59548 

 27310 

 30800 

 10677 



^ Theorie des oiomhres, pp. 294 — 295. 



2 It may be noticed that the remainder following this quotient digit is 5, so that 

 the succeeding figures 01026... to the end of the period are written down at once by 

 halving the part of the period already obtained, beginning with the second digit. 



