CHAP. I] PEKPECT, MULTIPLY PEEFECT, AND AMICABLE NUMBEKS. 11
(f. 103, verso) is abundant if p is composite. Every multiple of a perfect or abundant number is abundant, every divisor of a perfect number is deficient (ff. 104 verso, 105). The product of two primes, other than 2X3, is deficient (f. 105 verso). The odd number 945 is abundant, the sum of its aliquot divisors being 975 (f. 107). Commenting (f. Ill verso, f. 112) on the statement of Boethius6 and Cardan27 that the perfect numbers end alternately in 6 and 8, he stated that the fourth is 8128 and the fifth is 2096128 [an error], the fifth not being 130816 = 256X511, since 511 =7X73.
Jean Leurechon48 (about 1591-1670) stated that there are only ten perfect numbers between 1 and 1012, listed them (noting the admirable property that they end alternately in 6 and 8) and gave the twentieth perfect number. [They are the same as in Tartaglia's38 list.]
Lantz49 stated that the perfect numbers are 2(4-1), 4(8-1), 16(32-1), 64(128-1), 256(512-1), 1024(2048-1), etc.
Hugo Sempilius50 or Semple (Scotland, 1594-Madrid, 1654) stated that there are only seven perfect numbers up to 40,000,000; they end alternately in 6 and 8.
Casper Ens51 stated^hat there are only seven perfect numbers <4-107, viz., 6, 28,496,8128,130816,1996128 [for 2096128], 33550336, and that they end alternately in 6 and 8.
Daniel Schwenter52 (1585-1636) made the same error as Casper Ens.51
Erycius Puteanus53 quoted from Martiano Capella, lib. VII, De Nuptiis Philologiae, to the effect that the perfect number 6 is attributed to Venus; for it is made by the union of the two sexes, that is, from triad, which is male since it is odd, and from diad, which is feminine since it is even. Puteanus said that the perfect numbers in order are 6, 28, 496, 8128, 130816, 2096128, 33550336, and gave all their divisors [implying that 511, 2047, 8191 are primes], and stated that these seven and all the remaining end alternately in 6 and 8. Between any two successive powers of 10 is one perfect number. That they are all triangular adds perfection to the perfect.
Joannes Broscius54 or Brocki remarked that there is no perfect number between 10000 and 10000000, contrary to Stifel,31 Bungus,42 Sempilius,50 Puteanus,53 and the author of Selectarum Propositionum Mathematicarum, quas propugnavit, Mussiponti, Anno 1622, Maximilianus Willibaldus, Baro
48R6cr6ations math<5matiques, Pont-a-Mousson, 1624; London, 1633, 1653, 1674 (these three English editions by Wm. Oughtred), p. 92. The authorship is often attributed to Leurechon's pupil Henry Van Etten, whose name is signed to the dedicatory epistle. Cf. Poggendorff, Handworterbuch, 1863, 2, p. 250 (under C. Mydorge); Bibliotheque des e*crivains de la cornpagnie de Jdsus, par A. de Backer, 2, 1872, 731; Biographic G6ne>ale, 31, 1872, 10.
"Institutionum Arithmeticarum libri quatuor & loanne Lantz, Coloniae Agrippinae, 1630, p. 54.
60De Mathematicis Diaciplinis libri Duodecim, Antverpiae, 1635, Lib. 2, Cap. 3, N. 10, p. 46. There is (pp. 263-5) -an index of writers on geometry and one for arithmetic.
61Thaumaturgus Math., Munich, 1636, p. 101; Coloniae, 1636, 1651; Venice, 1706.
"Deliciae Physieo-Mathematicae oder Mathemat: vnd Philosophische Erquickstunden, part I (574 pp.), Niirnberg, 1636, p. 108.
B3De Bissexto Liber: nova ternporis facula qua intercalandi arcana .... Lovanii, 1637; 1640, pp. 103-7. Reproduced by J. G. Graevius, Thesaurus Antiquitatum Romanarum (12 vols., 1694-9), Lugduni Batavorum, vol. 8.