6 History of the Theory of Numbers. [Chap. I 
In the manuscripts^ Codex lat. Monac. 14908, a part dated 1456 and a 
part 1461, the first four perfect numbers are given (J. 33') as usual and the 
fifth perfect number is stated correctly to be 33550336. 
Nicolas Chuquet^^ defined perfect, deficient, and abundant numbers, 
indicated a proof of EucHd's rule and stated incorrectly that perfect num- 
bers end alternately in 6 and 8. 
Luca Paciuolo, de Borgo San Sepolcro,^^ gave (f. 6) Euclid's rule, saying 
one must find by experiment whether or not the factor 1+2+4+. . . is 
prime, stated (f. 7) that the perfect numbers end alternately in 6 and 8, as 
6, 28, 496, etc., to mfinity. In the fifth article (ff. 7, 8), he illustrated the 
finding of the aliquot divisors of a perfect number by taking the case of the 
fourteenth perfect number 9007199187632128. He gave its half, then the 
half of the quotient, etc., until after 26 divisions by 2, the odd number 
134217727, marked " Indi^dsibilis " [prime]. Dividing the initial number 
by these quotients, he obtained further factors [1,2,..., 2'^, but written at 
length]. The proposed number is said to be evidently perfect, since it is the 
sum of these factors [but he has not employed all the factors, since the above 
odd number equals 2'-'^ — 1 and has the factor 2^ — 1 = 7] . Although Paciuolo 
did not list the perfect numbers between 8128 and 90 . . .8, the fact that he 
called the latter the fourteenth perfect number imphes the error expressly 
committed bj^ Bo^illus.^" 
Thomas Bradwardin^" (1290-1349) stated that there is only one perfect 
number (6) between 1 and 10, one (28) up to 100, 496 up to 1000, 8128 up 
to 10000, from which these numbers, taken in order, end alternately in 6 
and 8. He then gave EucUd's rule. 
Faber Stapulensis^^ or Jacques Lefevre (born at Etaples 1455, tl537) 
stated that all perfect numbers end alternately in 6 and 8, and that Euclid's 
rule gives all perfect numbers. 
Georgius Valla^^ gave the first four perfect numbers and observed that 
"The manuscript is briefly described by Gerhardt, Monatsber. Berlin Ak., 1870, 141-2. 
See Catalogus codicum latinorum bibliothecae regiae Monacensis, Tomi II, pars II, 
codices nuna. 11001-15028 complectus, Munich, 1876, p. 250. An extract of ff. 32-34 
on perfect numbers was published by MaximiUan Curtze, BibUotheca Mathematica, 
(2), 9, 1895, 39-42. 
"Triparty en la science des nombres, manuscript No. 1436, Fonds Fran^ais, BibliothSque 
Nationale de Paris, written at Lyons. 1484. Published by Aristide Marre, Bull. Bibl. 
Storia Sc. Mat. et Fis.. 13 (1880), 593-659, 693-814; 14 (1881), 417-460. See Part 1, 
Ch. Ill, 3, 619-621, manuscript, ff. 20-21. 
"Summa de Arithmetica geometria proportioni et proportionalita. [Suma . . , Venice, 1494.] 
Toscolano, 1523 (two editions substantially the same). 
"Arithmetica thome brauardini. Tractatus perutilis. In arithmetica speculativa a magistro 
thoma Brauardini ex libris eucUdis boecij & ahorum qua optimne excerptus. Parisiis, 
1495, 7th unnumbered page. 
Arithmetica Speculativa Thome Brauardini nuper mendis Plusculis tersa et diligenter Impressa, 
Parisiis [1502], 6th and 7th unnumbered pages. Also undated edition [1510], 3d page. 
"Epitome (iii) of the arithmetic of Boethius in Faber's edition of Jordanus," 1496, etc. 
Also in Introductio Jacobi fabri Stapulesis in Arithmecam diui Seuerini Boetij pariter 
Jordani, Paris, 1503, 1507. Also in Stapulensis, Jacobi Fabri, Arithmetica Boethi 
epitome, Basileae, 1553, 40. 
"De expetendis et fvgiendis rebvs opvs, Aldus, 1501. Liber I ( = Arithmeticae I), Cap. 12. 
