CHAP. I] PERFECT, MULTIPLY PEKFECT, AND AMICABLE NUMBERS. 9 Postello35 stated erroneously that 130816 [=256-511] is perfect. Lodoico Baeza36 stated that Euclid's rule gives all perfect numbers. Pierre ForcadeF (fl574) gave 130816 as the fifth perfect number, implying incorrectly that 511 is a prime. Tartaglia38 (1506-1559) gave an erroneous [Kraft86] list of the first twenty perfect numbers, viz., the expanded forms of 2n~1(2rió1), for n = 2 and the successive odd numbers as far as n = 39. He stated that the sums 1+2+4, 1+2+4+8,.. .are alternately prune and composite; and that the perfect numbers end alternately in 6 and 8. The third "notable property" mentioned is that any perfect number except 6 yields the remainder 1 when divided by 9. Robert Recorde39 (about 1510-1558) stated that all the perfect numbers under 6-109 are 6, 28, 496, 8128,130816, 2096128, 33550336, 536854528 [the fifth, sixth, eighth of these are not perfect]. Petrus Ramus40 (1515-1572) stated that in no interval between successive powers of 10 can you find more than one perfect number, while in many intervals you will find none. At the end of Book I (p. 29) of his Arith-meticae libri tres, Paris, 1555, Ramus had stated that 6, 28, 496, 8128 are the only perfect numbers less than 100000. Franciscus Maurolycus41 (1494-1575) gave an argument to show that every perfect number is hexagonal %d hence triangular. Peter Bungus42 (fl601) gave (1584, pars altera, p. 68) a table of 20 numbers stated erroneously to be the perfect numbers with 24 or fewer digits [the same numbers had been given by Tartaglia38]. In the editions of 1591, etc., p. 468, the table is extended to include a perfect number of 25 digits, one of 26, one of 27, and one of 28. He stated (1584, pp. 70-71; 1591, pp. 471-2) that all perfect numbers end alternately in 6 and 28; employing Euclid's formula, he observed that the product of a power of 2 ending in 4 by a number ending in 7 itself ends in 28, while the product of one ending in 6 by one ending in 1 ends in 6. He verified (1585, pars Theoricae Arithineticea Compendium a Guilielmo Postello, Lutetiae, 1552, a syllabus on one large sheet of arithmetic definitions. 36Nvmerandi Doctrina, Lvtetiae, 1555, fol. 27-28. 87L'Arithmeticqve de P. Forcadel de Beziers, Paris, 1556-7. Livre I (1556), fol. 12 verso. 38La seconda Parte del General Trattato di Nvmeri, et Misvre di Nicolo Tartaglia, Vinegia, 1556, f. 146 verso. L' Arithmetiqve de Nicolas Tartaglia Brescian .... Recueillie, & traduite d'ltalien en Francois, par Gvillavme Gosselin de Caen, .... Paris, 1578, f. 98 verso, f. 99. 39The Whetstone of witte, whiche is the seconde parte of Arithmetike, London, 1557, eighth unnumbered page. 40Petri Kami Scholarum Mathematicarum, Libri unus et triginta, a Lazaro Schonero recog- niti & emendati, Francofvrti, 1599, Libr. IV (Arith.), p. 127, and Basel, 1578. "Arithmeticorvm libri dvo, Venetiis, 1575, p. 10; 1580. Published with separate paging, at end of Opuscula inathematica. 42Mysticae nvmerorvm significationis liber in dvas divisvs partes, R. D. Petro Bongo Canonico Bergomate avctore. Bergomi. Pars prior, 1583, 1585. Pars altera, 1584. Petri Bungi Bergomatis Numerorurn mysteria, Bergomi, 1591, 1599, 1614, Lutetiae Parisio- rum, 1618, all four with the same text and paging. Classical and biblical citations on numbers (400 pages on 1, 2, . . , 12). On the 1618 edition, see Font6s, M6m. Acad. Sc. Toulouse, (9), 5, 1893, 371-380.