Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 9 
Postello^^ stated erroneously that 130816 [ = 256-511] is perfect. 
Lodoico Baeza^^ stated that Euchd's rule gives all perfect numbers. 
Pierre Forcadel" (tl574) gave 130816 as the fifth perfect number, 
implying incorrectly that 511 is a prime. 
Tartaglia^^ (1506-1559) gave an erroneous [Kraft ^^J list of the first 
twenty perfect numbers, viz., the expanded forms of 2"~^(2'* — 1), 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 prime and composite; and that 
the perfect numbers end alternately in 6 and 8. The third ''notable prop- 
erty" mentioned is that any perfect number except 6 yields the remainder 1 
when divided by 9. 
Robert Recorde^^ (about 1510-1558) stated that all the perfect numbers 
under 6-10^ are 6, 28, 496, 8128, 130816, 2096128, 33550336, 536854528 [the 
fifth, sixth, eighth of these are not perfect]. 
Petrus Ramus^° (1515-1572) stated that in no interval between succes- 
sive 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 lOOpOO. 
Franciscus Maurolycus*^ (1494-1575) gave an argument to show that 
every perfect number is hexagonal and hence triangular. 
Peter Bungus^^ (fieoi) 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 Tartaglia^^]. 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 Arithmetices Compendium h Guilielmo Postello, Lutetiae, 1552, a syllabus on one 
large sheet of arithmetic definitions. 
"Nvmerandi Doctrina, Lvtetiae, 1555, fol. 27-28. 
''L'Arithmeticqve de P. Forcadel de Beziers, Paris, 1556-7. Livre I (1556), fol. 12 verso. 
3*La seconda Parte del General Trattato di Nvmeri, et Misvre di Nicolo Tartagha, Vinegia, 
1556, f. 146 verso. 
L' Arithmetiqve de Nicolas Tartagha Brescian .... Recueillie, & traduite d'ltalien en 
FranQois, par Gvillavme Gosselin de Caen, .... Paris, 1578, f. 98 verso, f. 99. 
'®The Whetstone of witte, whiche is the seconde parte of Arithmetike, London, 1557, eighth 
unnumbered page. 
^''Petri Rami Scholarum Mathematicarum, Libri unus et triginta, k Lazaro Schonero recog- 
niti & emendati, Francofvrti, 1599, Libr. IV (Arith.), p. 127, and Basel, 1578. 
"Arithmeticorvm hbri dvo, Venetiis, 1575, p. 10; 1580. Published with separate paging, at 
end of Opuscula mathematica. 
*^Mysticae 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 Numerorum 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 Font^s, M6m. Acad. So. 
Toulouse, (9), 5, 1893, 371-380. 
