4 History of the Theory of Numbers. [Chap. I 
lamblichus* (about 283-330) repeated in effect the remarks by Nico- 
machus on perfect, abundant, and deficient numbers, but made erroneous 
additions. He stated that there is one and but one perfect number in the 
successive intervals between 1, 10, 100,..., 100000, etc., to infinity. 
"Examples of a perfect number are 6, and 28, and 496, and 8128, and the like 
numbers, alternately ending in 6 and 8." He remarked that the Pythag- 
oreans called the perfect number 6 marriage, and also health and beauty 
(on account of the integrity of its parts and the agreement existing in it). 
Aurelius Augustinus^ (354-430) remarked that, 6 being the first perfect 
number, God effected the creation in 6 daj's rather than at once, since the 
perfection of the work is signified by the number 6. The sum of the aUquot 
parts of 9 falls short of it; likewise for 10. But the sum of the aliquot 
parts of 12 exceeds it. 
Anicius Manhus Severinus Boethius^ (about 481-524), in a Latin exposi- 
tion of the arithmetic of Nicomachus, stated that perfect numbers are rare, 
easily counted, and generated in a very regular order, while abundant 
(superfluos) and deficient (diminutos) numbers are found to an unlimited 
extent and not in regular order. The perfect numbers below 10000 are 
6, 28, 496, 8128. And these numbers alwaj^s end alternately in 6 and 8. 
Munyos^ stated that Boethius added to EucUd's idea of perfect number 
that of deficient (diminute) and abundant (redundantem) numbers. 
Isidorus of Seville^ (570-636) distinguished even and odd numbers, 
perfect and abundant numbers, linear, flachen and Korper Zahlen (primes, 
products of two, products of three factors). 
Alcuin^ (735-804) , of York and Tours, explained the occurrence of the 
number 6 in the creation of the universe on the ground that 6 is a perfect 
number. The second origin of the human race arose from the deficient 
number 8; indeed, in Noah's ark there were 8 souls from which sprung the 
entire human race, showing that the second origin was more imperfect than 
the first, which was made according to the number 6. 
^lamblichus Chalcidensis ex Coele-Syria in Nicomachi Geraseni arithmeticam introduc- 
tionem, et de Fato. Accedit Joachimi Camerarii explicatio in duos libros Nicomachi. 
Ed., Samuel Tennulius. Amhemiae, 1668, pp. 43-47. (Greek text and Latin translation 
in parallel columns.) 
lamblichi in Nicomachi arithmeticam introductionem Uber ad fidem codicis Florentini. 
Ed., H. Pistelli. Lipsiae, 1894. (Greek.) 
*De Civitate Dei, hber XI, cap. XXX, ed., B. Dombart, Lipsiae, 1877, 1, p. 504. The reference 
by Frizzo"' i' to lib. II, cap. 39. 
"Arithmetica boetij, Augsburg, 1488; Cologne, 1489; Leipzig, 1490; Venice, 1491-2, 1499; 
Paris, [1496, 1501], 1503, etc.; lib. 1, cap. 20. "De generatione numeri perfecti." 
Opera Boetii, Venice, 1491-2, etc.; ed., Friedlein, Leipzig, 1867. 
"Institvtiones arithmeticae ad percipiendam astrologiam et mathematicas facultates neces- 
sariae. Auctore Hieronymo Mimj'os, Valentiae, 1566, f. 5, verso. 
*Incipit epistola Isidori iunioris hispalensis . . . Finit Uber etymologiarum . . . [Augsburg, 
1472]; Venice, 1483, etc. In this book of etymologies, arithmetic is treated very briefly 
in Book 3, beginning f . 15. 
•Bibhotheca Rerum Germanicarum, tomus sextus: Monumenta Alcuiniana, Berlin, 1873, 
epistolae 259, pp. 818-821. Cf. Migne, Patrologiae, vol. 100, 1851, p. 665; Hankel, 
Geschichte Math., p. 311. 
