352 HISTORY OF THE THEOKY OF NUMBERS. [CHAP, xin table to 100 million; the manuscript59 has been hi the library of the Vienna Royal Academy since 1867. Lehmer92 gave an account of the first of the eight volumes of the manuscript, listed 226 errors in the tenth million, and concluded that Kulik's manuscript is certainly not accurate enough to warrant publication, though of inestimable value in checking a newly constructed table. Lehmer95 gave a further account of this manuscript which he examined in Vienna. Volume 2, running from 12 642 600 to 22 852 800 is missing. The eight volumes contained 4,212 pages. B. Goldberg60 gave all factors of numbers prime to 2, 3, 5, to 251 647. Zacharias Dase,61 in the introduction to the table for the seventh million, printed a letter from Gauss, dated 1850, giving a brief history of previous tables and referring to the manuscript factor table for the fourth, fifth and sixth millions presented to the Berlin Academy by A. L. Crelle. Although Gauss was confident this manuscript would be published, and hence urged Dase to undertake the seventh million, etc., the Academy found the manuscript to be so inaccurate that its publication was not advisable. Dase died in 1861 leaving the seventh million complete and remarkably accurate, the eighth nearly complete, and a large part of the factors for the ninth and tenth millions. The work was completed by Rosenberg, but with numerous errors. The table for the tenth million has not been printed; the manuscript was presented to the Berlin Academy hi 1878, but no trace of it was found when Lehmer92 desired to compare it with his table of 1909. C. F. Gauss62 gave a table showing the number of primes in each thousand up to one million and in each ten thousand from one to three million, with a comparison with the approximate formula J dx/log x. V. A. Lebesgue63 discussed the formation of factor tables and gave that to 115500 constructed by Houel. W. H. Oakes64 used a complicated apparatus consisting of three tables on six sheets of various sizes and nine perforated cards (cf. Committee,68 p. 39). W. B. Davis65 considered numbers in the vicinity of 108, and of 1011. E. Meissel66 computed the number of primes in the successive sets of 100 000 numbers to one million and concluded that Burckhardt's45 table gives correctly the primes to one million. "Cited by Kulik, Abh. Bohm. Gesell. Wiss., Prag, (5), 11, 1860, 24, footnote. A report on the manuscript was made by J. Petzval, Sitzungsberichte Ak. Wiss. Wien (Math.), 53, 1866, II, 460. Cited by J. Perott, 1'interme'diaire des math., 2, 1895, 40; 11, 1904, 103. 80Primzahlen- u. Faktortafeln von 1 bis 251 647, Leipzig, 1862. Errata, Cunningham.85 61Factoren-Tafelnfur alle Zahlen dersiebenten Million..., Hamburg, 1862;.. .der achten Million, 1863;.. .der neunten Million (erganzt von H. Rosenberg), 1865. "Posthumous manuscript, Werke, 2, 1863, 435-447. ^Tables diverses pour la d6composition des nombres en leurs facteurs premiers, Me"m. soc. sc. phys. et nat. de Bordeaux, 3, cah. 1, 1864, 1-37. "Machine table for determining primes and the least factors of composite numbers up to 100000, London, 1865. «Jour. de Math., (2), 11, 1866, 188-190; Proc. London Math. Soc., 4, 1873, 416-7. Math. Quest. Educ. Times, 7, 1867, 77; 8, 1868, 30-1. "Math. Annalen, 2, 1870, 63&-G42. Cf. 3, p. 523; 21, 1883, p. 304; 25, 1885, p. 251.