ON MATHEMATICAL TABLES. 89 



There is a short preface of 2 pp. by Dr. Eosenberg, who edited the work, 

 which was left nearly complete by Dase. 



Dase and Rosenberg (Ninth Million), 1865. The arrangement is the 

 same as in the pre\doiis two millions ; and the table extends from 8,010,000 

 to 9,000,000. The work left incomplete by Dase at his death was finished 

 by Dr. Rosenberg ; the paging runs from 225-334. 



It is stated in the preface that the tenth million (the last which the tables 

 were intended to include) was nearly completed ; but we believe it has not 

 yet appeared. 



It will have been seen from the above accounts that CnERNAc's, Btjeck- 

 haedt's, and Dase's tables together contain all the published results with re- 

 gard to factors of numbers ; and by means of them we can find aU the 

 simple divisors of numbers between one million and three millions and 

 between six millions and nine millions easily, and between unity and one 

 million at sight. There is, however, the gap from three millions to six 

 millions ; and it is very much to be regretted that this is not filled up. 

 Gauss states a table of divisors from three millions to six millions exists in 

 manuscript at Berlin ; and Eurckhardt also formed a similar table ; so that 

 this portion has apparently been twice calculated (by Crelle ? and Eurck- 

 hardt). 



Gauss's letter is dated 1850 ; and it is a calamity that the anticipations con- 

 tained in it have not been realized, as a manuscript unpublished does more 

 liarm than if it were non-existent, by checking others from attempting the 

 task. The completion of Gauss's scheme (viz. the publication of tables to ten 

 millions) is very desirable, as these tables may be regarded as data in regard 

 to investigations in the theory of numbers (see references to memoirs of Euler 

 and Gauss in Cheenac, and Gauss's letter). The tenth million also seems to 

 be still unpublished, though seven years ago we had Dr. Eosenberg's assurance 

 that it was nearly completed. If the whole ten millions were published, we 

 should much like to see a list of all the primes up to this point published 

 separately. 



Oakes, 1865 (Machine table). The object is to find the prime or least 

 factors of numbers less than 100,000 ; and for this purpose there are three 

 tables, A (1 page large 8vo), E (4 pp. folio), and C (1 page obi. folio), and 

 nine perforated cards, the one to be employed depending on the group of 

 10,000 that contains the argument. The mode of entry is somewhat compli- 

 cated ; and the table can only be regarded as a matter of curiosity ; for in the 

 method of arrangement of Eurckhasdt or Dase the least factors of aU 

 numbers under 100,000 only occupy a little over 11 pp. or six leaves 

 of small folio or large 8vo size — while the present apparatus consists of six 

 leaves of large and diiferent sizes, and nine cards, besides requiring an 

 involved course of procedure. Col. Oakes does not explain the principle 

 on which his method depends. 



The following is a list of tables contained in works that are described in 



§4. 



TaUes of Divisors.— Dosso:s, 1747, T. XVII. (to 10,000) ; Maseres, 1795 

 (to 100,000) ; Yega, 1797, Vol. II. T. I. (to 102,000) ; Lambert, 1798, 

 T. I. (to 102,000) ; Earlow, 1814, T. I. (to 10,000) ; Hantschl, 1827, 

 T. VII. (to 18,277): *Salomon, 1827, T. II. (to 102,011); Hulsse's Vega, 

 1840, T. V. ; KoHLEE, 1848, T. VIII. (to 21,524) ; Houel, 1858, T. VII. (to 

 10,841); Eanktne, 1866 (to 256). See also Gruson, 1798, § 3, art. 1. 



List of Prime Numhers.—JiomoTi, 1747, T. XVIII. (10,000 to 15,000) ; 

 Vega, 1797, Vol. II. T. I. (102,000 to 400,000) ; Lambert, 1798, T. II. 



