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 Bovillus^" 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 Jordanus^^ that every abundant number is 
even, by citing 45045 [ = 5-9-7-ll-13] and its multiples. He stated that 
2" — 1 is a prime if n is odd, expUcitly citing 511 [ = 7-73] as a prime. He 
listed as perfect numbers 2"~^(2'* — 1), n ranging over all the odd numbers 
^ 39 [Cataldi^ 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,^^ 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=i=l are primes and (No. 30) conversely any prime is of the form 
6n=t 1 [Cataldi,^ p. 45, corrects the first statement and proves the second]. 
He stated (f. 174) that every perfect number is triangular, being 2" (2'' — l)/2. 
Martinus^^ gave the first four perfect numbers and remarked that they 
end alternately in 6 and 8. 
Gasper Lax'^^ stated that the perfect numbers end alternately in 6 and 8. 
V. Rodulphus Spoletanus^^ was cited by Cataldi,'*^ with the implication 
of errors on perfect numbers. [Copy not seen.] 
Girardus Ruffus^^ stated that every perfect number is even, that most 
odd numbers are deficient, that, contrary to Jordanus,^^ the odd number 
45045 is abundant, and that for n odd 2^* — 1 always leads to a perfect num- 
ber, citing 7, 31, 127, 511, 2047, 8191 as primes [the fourth and fifth are 
composite]. 
Feliciano^^ stated that all perfect numbers end alternately in 6 and 8. 
Regius^^ 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 
^''Caroli Bouilli Samarobrini Liber De Perfectis Numeris (dated 1509 at end), one (ff. 172-180) 
of 13 tracts in his work, Que hoc volumine continetur: Liber de intellectu, . . . De 
Numeris Perfectis, . . . , dated on last page, 1510, Paris, ex ofEcina Henrici Stephani. 
Biography in G. Maupin, Opinions et Curiosit^s touchant la Math., Paris, 1, 1901, 186-94. 
"Ars Arithmetica loannis Martini, Silicei: in theoricen & praxim. 1513, 1514. Arithmetica 
loannis Martini, Scilicei, Paris, 1519. 
"Arithmetica speculatiua magistri Gasparis Lax. Paris, 1515, Liber VII, No. 87 (end). 
*3De proportione proportionvm dispvtatio, Rome, 1515. 
"Divi Severini Boetii Arithmetica, dvobvs discreta hbris, Paris, 1521; ff. 40-44 of the commen- 
tary by G. Ruffus. 
"Libro di Arithmetica & Geometria speculatiua & praticale: Composto per maestro Fran- 
cesco FeUciano da Lazisio Veronese Intitulato Scala Grimaldelli: Nouamente stampato. 
Venice, 1526 (p. 3), 1527, 1536 (p. 4), 1545, 1550, 1560, 1570, 1669, Padoua, 1629, Verona, 
1563, 1602. 
*Vtrivsqve Arithmetices, epitome ex uariis authoribus concinnata per Hvdalrichum Regium. 
Strasburg, 1536. Lib. I, Cap. VI: De Perfecto. Hvdalrichvs Regius, Vtrivsque. . . 
ex variis . . . , Friburgi, 1550 [and 1543], Cap. VI, fol. 17-18. 
