354 HISTORY OP THE THEORY OF NUMBERS. [CHAP. XHI C. A. Laisant780 would exhibit a factor table by use of shaded and unshaded squares on square-ruled paper without using numbers for entries. G. Speckmann756 made trivial remarks on the construction of a list of primes. P. Valerio76 arranged the odd numbers prime to 5 in four columns according to the endings 1, 3, 7,9. From the first column cross out the first multiple 21 of 3, then the third following number 51, etc. Similarly for the other columns. Then use the prunes 7, 11, etc., instead of 3. J. P. Gram77 published the computation by N. P. Bertelsen of the number of primes to ten million in intervals of 50000 or less, which led to the detection of numerous errors in the tables of Burckhardt46 and Dase.61 G. L. Bourgerel78 gave a table with 0,1,..., 9 in the first row, 10,..., 19 hi the second row (with 10 under 0), etc. Then all multiples of a chosen number lie in straight lines forming a paralellogram lattice, with one branch through 0. For example, the multiples of 3 appear in the line through 0,12, 24, 36,..., the parallel through 3,15, 27,..., the parallel 21, 33,45,...; also in a second set of parallels 3, 12, 21, 30; 6, 15, 24, 33, 42, 51, 60; etc. E. Suchanek79 continued to 100 000 Simony's73 table of primes to base 2. D. von Sterneck80 counted the number of primes 100 n+1 in each tenth of a million up to 9 million and noted the relatively small variation from one-fortieth of the total number of primes in the interval. H. Vollprecht81 discussed the determination of the number of prunes <N by use of the prunes < vl^ A. Cunningham and H. J. Woodall82 discussed the problem to find all the primes in a given range and gave many successive primes >9 million. They82a listed 117 primes between 224=»=1020. H. Schapira826 discussed algebraic operations equivalent to the sieve of Eratosthenes. *V. Di Girio, Alba, 1901, applied indeterminate analysis of the first degree to define a new sieve of Eratosthenes and to factoring. John Tennant83 wrote numbers to the base 900 and used auxiliary tables. A. Cunningham830 gave long lists of primes between 9-106 and 1011. Ph. Jolivald84 noted that a table of all factors of the first 2n numbers serves to tell readily whether a number <4n-{-2 is prime or not. 75*Assoc. franc., 1891, II, 165-8. 76i>Archiv Math. Phys., (2), 11, 1892, 439-441. 78La revue scientifique de France, (3), 52, 1893, 764-5. "Acta Math,, 17, 1893, 301-314. List of errors reproduced in Sphinx-Oedipe, 5,1910,49-51. 7«La revue scientifique de France, (4), 1, 1894, 411-2. "Sitzungsber. Ak. Wiss. Wien (Math.), 103, II a, 1894, 443-610. "Anzeiger K. Akad. Wiss. Wien (Math.), 31, 1894, 2-4. Cf. Kronecker, p. 416 below. "Zeitschrift Math. Phys., 40, 1895, 118-123. "Report British Assoc., 1901, 553; 1903, 561; Messenger Math., 31, 1901-2, 165; 34, 1904-5, 72, 184; 37, 1907-8, 65-83; 41, 1911, 1-16. 8*aReport British Assoc., 1900, 646. 82&Jahresber. d. Deutschen Math. Verein., 5, 1901, I, 69-72. "Quar. Jour. Math., 32, 1901, 322-342. **<>Ibid., 35, 1903, 10-21; Mess. Math., 36, 1907, 145-174; 38, 1908, 81-104; 38, 1909, 145-175; 39, 1909, 33-63, 97-128; 40, 1910, 1-36; 45, 1915, 49-75; Proc. London Math. Soc., 27, 1896, 327; 28, 1897, 377-9; 29, 1898, 381-438, 518; 34, 1902, 49. ML'intenne"diaire des math., 11, 1904, 97-98.