Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 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 
ahquot divisors being 975 (f. 107). Commenting (f. Ill verso, f. 112) on 
the statement of Boethius^ and Cardan^^ 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 Leurechon^^ (about 1591-1670) stated that there are only ten 
perfect numbers between 1 and 10^^, listed them (noting the admirable 
property that they end alternately in 6 and 8) and gave the twentieth per- 
fect number. [They are the same as in Tartaglia's^^ list.] 
Lantz^^ 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 Sempilius^° 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 Ens^^ stated that there are only seven perfect numbers <4-10'', 
viz., 6, 28, 496, 8128, 130816, 1996128 [for 2096128], 33550336, and that they 
end alternately in 6 and 8. 
Daniel Schwenter^^ (1585-1636) made the same error as Casper Ens.^^ 
Erycius Puteanus^^ 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 Broscius^"^ or Brocki remarked that there is no perfect number 
between 10000 and 10000000, contrary to Stifel,^^ Bungus,^^ Sempilius.^" 
Puteanus,^^ and the author of Selectarum Propositionum Mathematicarum, 
quas propugnavit, Mussiponti, Anno 1622, Maximilianus Willibaldus, Baro 
^^Recreations math^matiques, Pont-^-Mousson, 1624; London, 1633, 1653, 1674 (these three 
EngUsh 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 6crivains de la compagnie de Jesus, par A. de Backer, 2, 1872, 731; Biograpliie 
Generale, 31, 1872, 10. 
"Institutionum Arithmeticarum hbri quatuor h loanne Lantz, Coloniae Agrippinae, 1630, p. 54. 
"De Mathematicis Disciphnis hbri 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. 
"Thaumaturgus Math., Munich, 1636, p. 101; Coloniae. 1636, 1651; Venice, 1706. 
"Dehciae Physico-Mathematicae oder Mathemat: vnd Philosophische Erquickstunden, 
part I (574 pp.), Numberg, 1636, p. 108. 
"De Bissexto Liber: nova temporis 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. 
