10 History of the Theory of Numbers. [Chap, i 
prior, p. 238; 1591, p. 343) for the first seven numbers of his table [two 
being imperfect, however] that the sum of the digits of a perfect number 
exceeds by unity a multiple of 9. Every perfect number is triangular 
(1591, p. 270). Every multiple of a perfect number is abundant, every 
divisor deficient (1591, p. 464). 
Unicornus^^ (1523-1610) cited Bungus and repeated his error that 
2"- 1 (2^ — 1) is always perfect for n odd and that all perfect numbers end 
alternately in 6 and 8. 
Cataldi"^ (1548-1626) noted in his Preface that Paciuolo's^^ fourteenth per- 
fect number 90. . .8 is in fact abundant since it arose from 1+2+4+ • ■ • 
+2^^ = 134217727, which is divisible by 7,whereas Paciuolo said it was prime. 
Citing the error of the latter, Bovillus,^° and others, that all perfect num- 
bers end alternately in 6 and 8, Cataldi observed (p. 42) that the fifth per- 
fect number is 33550336 and the sixth is 8589869056, from 8191 =2'^- 1 and 
131071=2^^ — 1, respectively, proved to be primes (pp. 12-17) by actually 
trying as possible divisor every prime less than their respective square roots. 
He gave (pp. 17-22) the corresponding work showing 2^^ — 1 to be prime. He 
stated (p. 11) that 2'*-! is a prime forn = 2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 
remarking that the prime n = ll does not yield a perfect number since 
(p. 5) 2^^ — 1=2047 = 23*89, while it is composite if n is composite. He 
proved (p. 8) that the perfect numbers given by Euclid's rule end in 6 or 8. 
He gave (pp. 28-40, 48) a table of all divisors of all even and odd numbers 
^ 800, and a table of primes < 750. 
Georgius Henischiib^^ (1549-1618) stated that the perfect numbers end 
alternately in 6 and 8, and that one occurs between any two successive 
powers of 10. He applied Euclid's formula without restricting the factor 
2"—! to primes. 
Johan Rudolff von Graff enried"*^ stated that all perfect numbers are 
given by Euclid's rule, which he applied without restricting 2" — 1 to primes, 
expressly citing 256X511 as the fifth perfect number. Every perfect 
number is triangular. 
Bachet de Mezirac''^ (1581-1638) gave (f. 102) a lengthy proof of 
Euclid's theorem that 2'*p is perfect if p = l+2+ . . . +2^* is a prime, but 
"De I'arithmetica vniversale del Sig. loseppo Vnicorno, Venetia, 1598, f. 57. 
"Trattato de nvmeri perfetti di Pierto Antonio Cataldo, Bologna, 16C3. According to the 
Preface, this work was composed in 1588. Cataldi founded at Bologna the Academia 
Erigende, the most ancient known academy of mathematics; his interest in perfect 
numbers from early youth is shown by the end of the first of his "due lettioni fatte nell' 
Academia di Perugia" (G. Libri, Hist. Sc. Math, en ItaUe, 2d ed., vol. 4, Halle, 1865, p. 
91). G. Wertheim, BibHotheca Math., (3), 3, 1902, 76-83, gave a summary of the Trattato. 
"Arithmetica Perfecta et Demonstrata, Georgii Henischiib, Augsburg [1605], 1609, pp. 63-64. 
"Arithmeticae Logistica Popularis Librii IIII. Jn welchen der Algorithmus in gantzen Zahlen 
u. Fracturen . . . . , Bern, 1618, 1619, pp. 236-7. 
*^Elementorum arithmeticorum libri XIII auctori D . . . , a Latin manuscript in the Biblio- 
thfeque de I'lnstitut de France. On the inside of the front cover is a comment on the 
sale of the manuscript by the son of Bachet to DaUbert, treasurer of France. A general 
account of the contents of the manuscript was given by Henry, Bull. Bibl. Storia Sc. Mat. 
e Fis., 12, 1879, pp. 619-641. The present detailed account of Book 4, on perfect numbers, 
was taken from the manuscript. 
