CHAPTER XIII. FACTOR TABLES, LISTS OF PRIMES. Eratosthenes (third century B.C.) gave a method, called the sieve or crib of Eratosthenes, of determining all the primes under a given limit I, which serves also to construct the prime factors of numbers, <Z. From the series of odd numbers 3, 5, 7,.. ., strike out the square of 3 and every third number after 9, then the square of 5 and every fifth number after 25, etc. Proceed until the first remaining number, directly following that one whose multiples were last cancelled, has its square >l. The remaining numbers are primes. Nicomachus and Boethius1 began with 5 instead of with 52, 7 instead of with 72, etc., and so obtained the prime factors of the numbers <L A table containing all the divisors of each odd number g 113 was printed at the end of an edition of Aratus, Oxford, 1672, and ascribed to Eratosthenes by the editor, who incorrectly considered the table to be the sieve of Eratosthenes. Samuel Horsley2 believed that the table was copied by some monk in a barbarous age either from a Greek commentary on the Arithmetic of Nicomachus or else from a Latin translation of a Greek manuscript, published by Camerarius, in which occurs such a table to 109. Leonardo Pisano3 gave a table of the 21 primes from 11 to 97 and a table giving the factors of composite numbers from 12 to 100; to determine whether n is prime or not, one can restrict attention to divisors ^ -\/n. Ibn Albann& in his-Talkhys4 (end of 13th century) noted that in using the crib of Eratosthenes we may restrict ourselves to numbers g VZ- Cataldi5 gave a table of all the factors of all numbers up to 750, with a separate list of primes to 750, and a supplement extending the factor table from 751 to 800. Frans van Schooten6 gave a table of primes to 9979. J. H. Rahn7 (Rhonius) gave a table of the least factors of numbers, not divisible by 2 or 5, up to 24000. T. Brancker8 constructed a table of the least divisors of numbers, not divisible by 2 or 5, up to 100 000. [Reprinted by Hinkley.55] *Introd. in Arith. Nicomachi; Arith. Boethii, lib. 1, cap. 17 (full titles in the chapter on perfect numbers). Extracts of the parts on the crib, with numerous annotations, were given by Horsley.2 Cf. G. Bernhardy, Eratosthenica, Berlin, 1822, 173-4. 2Phil. Trans. London, 62, 1772, 327-347. 3I1 Liber Abbaci di L. Pisano (1202, revised 1228), Roma, 1852, ch. 5; Scritti, 1, 1857, 38. 4Transl. by A. Marre, Atti Accad. Pont. Nuovi Lincei, 17, 1863-4, 307. 6Trattato de' numeri perfetti, Bologna, 1603. Libri, Histoire des Sciences Math, en Italic, ed. 2, vol. 4, 1865, 91, stated erroneously that the table extended to 1000. "Exercitat. Math., libri 5, cap. 5, p. 394, Leiden, 1657. 'Algebra, Zurich, 1659. Wallis,10 p. 214, attributed this book to John Pell. 8An Introduction to Algebra, translated out of the High-Dutch (of Rahn's7 Algebra] into English by Thomas Brancker, augmented by D. P. [=Dr. Pell], London, 1668. It is cited in Phil. Trans. London, 3, 1668, 688. The Algebra and the translation were described by G. Wertheim, Bibliotheca Math., (3), 3, 1902, 113-126. 347