6 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. I In the manuscript14 Ccidex lat. Monac. 14908, a part dated 1456 and a part 1461, the first four perfect numbers are given (/. 33') as usual and the fifth perfect number is stated correctly to be 33550336. Nicolas Chuquet15 defined perfect, deficient, and abundant numbers, indicated a proof of Euclid's rule and stated incorrectly that perfect numbers end alternately in 6 and 8. Luca Paciuolo, de Borgo San Sepolcro,16 gave (f. 6) Euclid's rule, saying one must find by experiment whether or not the factor 1+2+4+... is prune, stated (f. 7) that the perfect numbers end alternately in 6 and 8, as 6, 28, 496, etc., to infinity. 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 "Indivisibilis" [prime]. Dividing the initial number by these quotients, he obtained further factors [1,2,..., 226, 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 227 — 1 and has the factor 23—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 implies the error expressly committed by Bovillus.20 Thomas Bradwardin17 (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 Euclid's rule. Faber Stapulensis18 or Jacques Lef£vre (born at fitaples 1455, f!537) stated that all perfect numbers end alternately in 6 and 8, and that Euclid's rule gives all perfect numbers. Georgius Valla19 gave the first four perfect numbers and observed that "The manuscript is briefly described by Gerhardt, Monatsber. Berlin Ak., 1870, 141-2. See Catalogue codicum latinorum bibliothecae regiae Monacensis, Tomi II, pars II, codices num. 11001-15028 complectus, Munich, 1876, p. 250. An extract of ff. 32'-34 on perfect numbers was published by Maximilian Curtze, Bibliotheca Mathematica, (2), 9, 1895, 39-42. "Triparty en la science des nombres, manuscript No. 1436, Fonds Francais, Bibliotheque Nationale de Paris, written at Lyons, 1484. Published by Aristide Marre, Bull. Bibl. Storia Sc. Mat. et Fia.. 13 (1880), 593-659, 693-814; 14 (1881), 417-460. See Part 1, Ch. Ill, 3, 619-621, manuscript, ff. 20-21. 18Summa de Aiithmetica geometria proportion! et proportionalita. [Surna . . . , Venice, 1494.] Toscolano, 1523 (two editions substantially the same). 17Arithmetica thome brauardini. Tractatus perutilis. In arithmetica spcculutiva a magiatro thoma Brauardini ex libris euclidis boecij & aliorum qua optirnne excerptus. Parisiis, 1495, 7th unnumbered page. Arithmetica SpeculativaThome Brauardini nuper mendis Plusculis tersa et diligenter Impressa, Parisiis , 6th and 7th unnumbered pages. Also undated edition , 3d page. 18Epitome (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. 19De expetendis et fvgiendis rebvs opvs, Aldus, 1501. Liber I (=Arithmeticae I), Cap. 12.