8 HISTORY OF THE THEOBY OF NUMBERS. [CHAP. I end alternately in 6 and 8. A multiple of an abundant or perfect number is abundant, a divisor of a perfect number is deficient. Cardan27 (1501-1576) stated that perfect numbers were to be formed by Euclid's rule and always end with 6 or 8; and that there is one between any two successive powers of ten. De la Roche28 stated in effect that 2n~1(2n — 1) is perfect for every odd n, citing hi particular 130816 and 2096128, given by n = 9, n = ll. This erroneous law led him to believe that the successive perfect numbers end alternately hi 6 and 8. Noviomagus29 or Neomagus or Jan Bronckhorst (1494-1570) gave Euclid's rule correctly and stated that among the first 10 numbers, 6 alone is perfect,..., among the first 10000 numbers, 6,28,496,8128 alone are perfect, etc., etc. [implying falsely that there is one and but one perfect number with any prescribed number of digits]. In Lib. II, Cap. IV, is given the sieve (or crib) of Eratosthenes, with a separate column for the multiples of 3, a separate one for the multiples of 5, etc. Willichius30 (fl552) listed the first four perfect numbers and stated that to these are to be added a very few others, whose nature is that they end either in 6 or 8. Michael Stifel31 (1487-1567) stated that all perfect numbers except 6 are multiples of 4, while 4(8-1), 16(32-1), 64(128-1), 256(512-1), etc., to infinity, are perfect [error, Kraft85]. He later*2 repeated the latter error, listing as perfect 2X3, 4X7, 16X31, 64X127, 256X511, 1024X2047, u& so fort an ohn end." Every perfect number is triangular. Peletier33 (1517-1582) stated (1549, V left; 1554, p. 20) that the perfect numbers end in 6 or 8, that there is a single perfect number between any two successive powers of 10, and (1549, C III left; 1554, pp. 270-1) that 4(8-1), 16(32-1), 64(128-1), 256(511),.. .are perfect. The first two statements were also given later by Peletier.34 27Hieronimi C. Cardani Medici Mediolanensis, Practica Arithmetice, & Mensurandi singu- laris. Milan, 1537, 1539; Niirnberg, 1541, 1542, Cap. 42, de proprietatibus numerorum mirificis. Opera IV, Lycra, 1663. 28Larismetique & Geometrie de maistre Eatienne de la Roche diet Ville Franche, Nouuelle- ment Imprimee & des fautes corrigee, Lyon, 1538, fol. 2, verso. Ed. 1, 1520. 29De Nvmeris libri dvo .... authore loanne Nouiomago, Paris, 1539, Lib, II, Cap. III. Reprinted, Cologne, 1544; Deventer, 1551. Edition by G. Frizzo, Verona, 1901, p. 132. 80Iodoci Vvillichii Reselliani, Arithmeticae libri tres, Argentorati, 1540, p. 37. "Arithmetica Integra, Norimbergae, 1544, ff. 10, 11. *2Die Coss Christoffs Rudolffs Die sehonen Exempeln der Coss Durch Michael Stifel Gebessert vnd eehr gemehrt, Kdnigsperg in Preussen, 1553, Anhang Cap. I, f. 10 verso, f. 11 (f. 27 v.), and 1571. "L'Arithmetiqve de lacqvea Peletier dv Mans, departie en quatre Liures, Poitiers, 1549, 1550, 1553. . . . , ff. 77 v, 78 r. Revile e augmentee par 1' Auteur, Lion, 1554..... Troisieme edition, reucue et augmentee, par lean de Tovrnes, 1607. "Arithmeticae Practicae methodvs facilis, per Gemmam Frisivm, Medicvm, ac Mathematicum conscripta .... In eandem loannis Steinii & lacobi Peletarii Annotationes. Antver- piae, 1581, p. 10.