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



these were decided by the actual performance of the divisions, 

 but the work is still incomplete, though I hope to resume it 

 shortly. It was the special liability to error to which tables of this 

 kind are subject, in connexion with the inaccuracies which I had 

 found to exist, which seemed to make it desirable to publish the 

 table on pp. 204 — 206; this, being the result of actual counting, is 

 free from possibility of errors such as those alluded to, and may be 

 regarded as a table of observed facts. Extended tables in the higher 

 arithmetic are likely to find their chief use in affording the means of 

 verifying (or even discovering) theoretical laws, and it is important 

 that they should be quite free from error, and particularly from 

 error connected with points which may be essential in the theory. 

 For the sake of completeness, I may here mention that tables 

 of primes having a given number n of digits in their periods, i. e. 

 tables of the resolutions of 10" — 1 into factors, and, as far as 

 known, into prime factors, have been given by Loof Mn t. xvi. (1851) 

 pp. 54 — 57 of Grunert's Archiv der Mathematik and by Mr Shanks 

 in the Proceedings of the Royal Society, vol. xxii. pp. 381 — 384. 

 The former extends from n = l to n = 60, and the latter from n=l 

 to n = 100, but there are of course gaps in both. The tract of 

 Reuschle referred to at the beginning of this section also contains 

 resolutions of 10" — L 



§ 9. Among the posthumous tables at the end of the second 

 volume of Gauss's Werke is one entitled Tafel zur Verwandlung 

 gemeiner Bruche mit Nennern aus dem ersten Tausend in Decimal- 

 hruche, which occupies pp. 412 — 434. It consists of two parts. 

 The first contains all the periods of the primes and powers of 

 primes from 3 to 463, and the second the period of the reciprocal of 

 every prime and power of a prime from 467 to 997. Thus up to 

 463 Gauss's table is the same as Mr Goodwyn's except that it 

 includes only primes and powers of primes, while the latter 

 includes all numxbers prime to 10 ; above 463 there is the additional 

 restriction that only one period is given. In Mr G(Jodwyn's table 

 the periods are arranged in order of magnitude, i. e. so that each 

 period as it stands is equal to the least fraction to which it belongs, 

 and that the periods regarded as decimal fractions are in order of 

 magnitude^; but in Gauss's table the periods of q which are 

 marked (1), (2), (3), ... (0) correspond to the periods of 



lOr lOr' lOr' 10 



^ This table was reprinted in the Nouvelles Annales de MatMmatiques, t. xiv. 

 (1855), pp. 115—117. 



2 According to this rule the last period of 73 (quoted in § 2 (v), p. 187) should 

 have been printed -2465 7534. The periods are generally arranged in order of 

 magnitude, and this exception was probably accidental. 



