354 History of the Theory of Numbers. [Chap, xiii 
C. A. Laisant'^^" would exhibit a factor table by use of shaded and un- 
shaded squares on square-ruled paper without using numbers for entries. 
G. Speckmann'^'' made tri\aal remarks on the construction of a list of 
primes. 
P. Valerio'® 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 primes 7, 11, etc., instead of 3. 
J. P. Gram" pubhshed 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 Burckhardt^^ and Dase." 
G. L. Bourgerel"^ gave a table with 0, 1, . . . , 9 in the first row, 10, . . . , 19 
in 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 Une 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. Suchanek" continued to 100 000 Simony's^^ table of primes to base 2. 
D. von Sterneck^" 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. Vollprecht^^ discussed the determination of the number of primes <N 
by use of the primes < v^- 
A. Cunningham and H. J. WoodalP^ discussed the problem to find all 
the primes in a given range and gave many successive primes >9 million. 
They^^a ^^^^^^ jj^y primes between 224±1020. 
H. Schapira^^'' 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 Tennant^ wrote numbers to the base 900 and used auxiliary tables. 
A. Cunningham^" gave long lists of primes between 9-10^ and 10^^ 
Ph. Jolivald^ 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. 
^'"Assoc. frang., 1891, II, 165-8. 7**Archiv Math. Phys., (2), 11, 1892, 439-441. 
'«La 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. 
^*La revue scientifique de France, (4), 1, 1894, 411-2. 
"Sitzunpsber. 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, 6.5-83; 41, 1911, 1-16. s^^Report British Assoc, 1900, 646. 
»«'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. 
•♦L'intermfidiaire des math., 11, 1904, 97-98. 
