CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 7 "these happen to end in 6 or 8.. .and these terminal numbers will always be found alternately." Carolus Bovillus20 or Charles de Bouvelles (1470-1553) stated that every perfect number is even, but his proof applies only to those of Euclid's type. He corrected the statement of Jordanus12 that every abundant number is even, by citing 45045 [ = 5-9-74M3] and its multiples. He stated that 2n—1 is a prime if n is odd, explicitly citing 511 [ = 7-73] as a prime. He listed as perfect numbers 2n""1(2n—l), n ranging over all the odd numbers ^39 [Cataldi44 later indicated that 8 of these are not perfect]. He repeated the error that all perfect numbers end alternately in 6 and 8. He stated (f. 175, No. 25) that if the sum of the digits of a perfect number > 6 be divided by 9, the remainder is unity [proved for perfect numbers of Euclid's type by Cataldi,44 p. 43]. He noted (f. 178) that any divisor of a perfect number is deficient, any multiple abundant. He stated (No. 29) that one or both of 6n=*= 1 are primes and (No. 30) conversely any prime is of the form 6n±l [Cataldi,44 p. 45, corrects the first statement and proves the second]. He stated (f. 174) that every perfect number is triangular, being 2n(2n—l)/2. Martinus21 gave the first four perfect numbers and remarked that they end alternately in 6 and 8. Gasper Lax22 stated that the perfect numbers end alternately in 6 and 8. V. Rodulphus Spoletanus23 was cited by Cataldi,44 with the implication of errors on perfect numbers. [Copy not seen.] Girardus Ruffus24 stated that every perfect number is even, that most odd numbers are deficient, that, contrary to Jordanus,12 the odd number 45045 is abundant, and that for n odd 2n —1 always leads to a perfect number, citing 7, 31, 127, 511, 2047, 8191 as primes [the fourth and fifth are composite]. Feliciano25 stated that all perfect numbers end alternately in 6 and 8. Regius26 defined a perfect number to be an even number equal to the sum of its aliquot divisors, indicated that 511 and 2047 are composite, gave correctly 33550336 as the fifth perfect number, but said the perfect numbers 20Caroh' Bouilli Samarobrini Liber De Perfectis Numeris (dated 1509 at end), one (ff. 172-180) of 13 tracts in his work, Quc hoc volumine continetur: Liber de intellects, . . . De Numeris Perfectis, . . . , dated on last page, 1510, Paris, ex officina Henrici Stephani. Biography in G. Maupin, Opinions et Curiosit6s touchant la Math., Paris, 1,1901,186-94. "Are Arithmetica loannis Martini, Silicei: in theoricen & praxim. 1513, 1514. Arithmetics loannis Martini, Scilicci, Paris, 1519. "Arithmetica speculatiua magistri Gasparis Lax. Paris, 1515, Liber VII, No. 87 (end). MDe proportione proportionvrn dispvtatio, Rome, 1515. J4Divi Severini Boetii Arithmetica, dvobvs discreta libria, Paris, 1521; ff. 40-44 of the commentary by G. RufFus. "Libro di Arithmetica & Geometria speculatiua & praticale: Compoato per maestro Francesco Feliciano da Lazisio Veronese Intitulato Scala Grimaldelli: Nouamente etampato. Venice, 1526 (p. 3), 1527, 1536 (p. 4), 1545,1550, 1560, 1570,1669, Padoua, 1629, Verona, 1563, 1602. MVtrivsqve Arithrnetices, epitome ex uariis authoribus concinnata per Hvdalrichum Regium. Strasburg, 1536. Lib. I, Cap. VI: De Perfecto. Hvdalrichvs Regius, Vtriveque. . . ex variie . . . , Friburgi, 1550 [and 1543], Cap. VI, fol. 17-18.