10 HlSTOEY OF THE THEOEY OF NtJMBEES. [CHAP. I first seven numbers of his table [two im of the digits of a perfect number Every perfect number is triangular A a perfect number is abundant, every Bungus and repeated his error that odd and that all perfect numbers end Cataldi44 (1548-1626) noted in his Preface that Paciuolo's16 fourteenth perfect number 90.. .8 is in fact abundant since it arose from 1+2+4+... +226 = 134217727,which is divisible by 7,whereas Paciuolo said it was prime. Citing the error of the latter, Bovillus,20 and others, that all perfect numbers end alternately in 6 and 8, Cataldi observed (p. 42) that the fifth perfect number is 33550336 and the sixth is 8589869056, from 8191 =213 -1 and 131071 = 2171, 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 219 1 to be prime. He stated (p. 11) that 2n-1 is a prime for n = 2, 3, 5, 7,13,17,19, 23, 29,31, 37, remarking that the prune n = ll does not yield a perfect number since (p. 5) 2U-1 = 2047=23-89, while it is composite if n is composite. He proved (p. 8) that the perfect numbergtgiven 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 Henischiib45 (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 2n1 to primes. Johan Rudolff von Graffenried46 stated that all perfect numbers are given by Euclid's rule, which he applied without restricting 2n 1 to primes, expressly citing 256X511 as the fifth perfect number. Every perfect number is triangular. Bachet de MSzirac47 (1581-1638) gave (f. 102) a lengthy proof of Euclid's theorem that 2np is perfect if p = l+2+.. .+2n is a prime, but «De 1'arithmetica vniversale del Sig. loseppo Vnicorno, Venetia, 1598, f. 57. "Trattato de nvmeri perfetti di Pierto Antonio Cataldo, Bologna, 1603. According to the Preface, this work was composed in 1588. Cataldi founded at Bologna the Academia Erigeade, 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 nelT Academia di Perugia" (G. Libri, Hist. Sc. Math, en Italie, 2d ed., vol. 4, Halle, 1865, p. 91). G. Wertheim, Bibliotheca Math., (3), 3,1902,76-83, gave a summary of the Trattato. 46Arithmetica Perfecta et Demonstrata, Georgii Henischiib, Augsburg , 1609, pp. 63-64. 48Arithmeticae Logistica- Popularis Librii IIII. In welchen der Algorithmus in gantzen Zahlen u. Fracturen , . . . , Bern, 1618, 1619, pp. 236-7. 47Elementorum arithmeticorum libri XIII auctori D . . . , a Latin manuscript in the Biblio-theque de 1'Institut de France. On the inside of the front cover is a comment on the sale of the manuscript by the son of Bachet to Dalibert, treasurer of France. A general account of the contents of the manuscript was given by Henry, Bull. Bibl. Storia Sc. Mat. e Ks., 12,1879, pp. 619-641. The present detailed account of Book 4, on perfect numbers, was taken from the manuscript.