4 HISTORY OF THE THEORY OF NUMBERS. [CHAP. I lamblichus4 (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 Pythagoreans called the perfect number 6 marriage, and also health and beauty (on account of the integrity of its parts and the agreement existing hi it). Auxelius Augustinus5 (354r430) remarked that, 6 being the first perfect number, God effected the creation in 6 days rather than at once, since the perfection of the work is signified by the number 6. The sum of the aliquot parts of 9 falls short of it; likewise for 10. But the sum of the aliquot parts of 12 exceeds it. Anicius Manlius Severinus Boethius6 (about 481-524), in a Latin exposition 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 always end alternately in 6 and 8. Munyos7 stated that Boethius added to Euclid's idea of perfect number that of deficient (diminute) and abundant (redundantem) numbers. Isidorus of Seville8 (570-636) distinguished even and odd numbers, perfect and abundant numbers, linear, flachen and Korper Zahlen (primes, products of twQj products of three factors). Alcuin9 (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 numbef. 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. 4Iamblichus Chalcidensis ex Coele-Syrla in Nicomachi Geraseni arithmeticam introduc- tionem, et de Fato. Accedit Joachimi Camerarii explicatio in duos libros Nicomachi. Ed., Samuel Tennulius. Arnhemiae, 1668, pp. 43-47. (Greek text and Latin translation in parallel columns.) lamblichi in Nicomachi arithmeticam introductionem liber ad fidem codicia Florentini. Ed., H. Pistelli. Lipsiae, 1894. (Greek.) BDe Civitate Dei, liber XI, cap. XXX, ed., B. Dombart, Lipsiae, 1877,1, p. 504. The reference by Frizzo29 r> to lib. II, cap. 39. "Arithraetica 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. 7Institvtiones arithmeticae ad percipiendam astrologiam et mathematicas facultates neces- sariae. Auctore Hieronymo Munyos, Valentiae, 1566, f. 5, verso. 8Incipit epistola Isidori iunioris hispalensis . . . Finit liber etymologiarum . . . [Augsburg, 1472]; Venice, 1483, etc. In this book of etymologies, arithmetic is treated very briefly in Book 3, beginning f. 15. •Bibliotheca Rerum Germamcarum, tomus eextus: Monumenta Alcuiniana, Berlin, 1873, epistolae 259, pp. 818-821. Cf. Migne, Patrologiae, vol. 100, 1851, p. 665; Hankel, Geschichte Math., p. 311.