8 History of the Theory 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. 
Cardan^^ (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 Roche-^ stated in effect that 2""^ (2" — 1) is perfect for every odd n, 
citing in particular 130816 and 2096128, given by n = 9, n = ll. This 
erroneous law led him to believe that the successive perfect numbers end 
alternately in 6 and 8. 
Noviomagus-^ 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. 
WilUchius^'^ (tl552) 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 StifeP^ (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, Kraft^°]. He later^- repeated the latter error, 
listing as perfect 
2X3, 4X7, 16X31, 64X127, 256X511, 1024X2047, 
"& so fort an ohn end." Every perfect munber is triangular. 
Peletier^^ (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.^ 
^'Hieronimi C. Cardani Medici Mediolanensis, Practica Arithmetice, & Mensurandi singu- 
laris. Milan, 1537, 1539; Xiirnberg, 1541, 1542, Cap. 42, de proprietatibus numerorum 
mirificis. Opera IV, Lyon, 1663. 
-*Larismetique & Geometrie de maistre Estienne de la Roche diet Ville Franche, Nouuelle- 
ment Imprimee & des fautea corrigee, Lyon, 1538, fol. 2, verso. Ed. 1, 1520. 
'"De 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. 
'°Iodoci Vvillichii Reselliani, Arithmeticae libri tres, Argentorati, 1540, p. 37. 
'^Arithmetica Integra, Norimbergae, 1544, ff. 10, 11. 
"Die Cosa Cbristoffs Rudolffs Die schonen Exempeln der Coss Durch Michael Stifel Gebessert 
vnd sehr gemehrt, Konigsperg in Preussen, 1553, Anbang Cap. I, f. 10 verso, f. 11 (f. 
27 v.), and 1571. 
"L'Arithmetiqve de lacqves Peletier dv Mans, departie en quatre Liures, Poitiers, 1549, 
1550, 1553. . . . , ff. 77 v, 78 r. Reviie e augmentee par 1' Auteur, Lion, 1554 
Troisieme edition, reucue et augmentee, par lean de Tovmes, 1607. 
"Arithmeticae Practicae methodvs facilis, per Gemmam Frisivm, Medicvm, ac Mathematicum 
conscripta .... In eandem loannis Steinii & lacobi Peletarii Annotationes. Antver- 
piae, 1581, p. 10. 
