CHAP, i] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 39 In the ninth century the Arab Thabit ben Korrah10 (prop. 10) noted that 2nht and 2ns are amicable numbers if (1) /i = 3-2n-l, ^S-^-1-!, s are primes > 2, literally, if 7i = 24-2n, t = z-2n~\ z = l+2+. . .+2n, s==(2ft+1+2n-"2)2n+1-l. The term used for amicable numbers was se invicem amantes. In the article in which F. Woepcke10 translated this Arabic manuscript into French, he noted that a definition of these numbers, called congeneres, occurs in the 51st treatise (on arithmetic) of Ikhovan Algafa, manuscript 1105, anciens fonds arabes, p. 15, of the National Library of Paris. Among Jacob's presents to Esau were 200 she-goats and 20 he-goats, 200 ewes and 20 rams (Genesis, XXXII, 14). Abraham Azulai349 (1570-1643), in commenting on this passage from the Bible, remarked that he had found written in the name of Rau Nachshon (ninth century A. D.) : Our ancestor Jacob prepared his present in a wise way. This number 220 (of goats) is a hidden secret, being one of a pair of numbers such that the parts of it are equal to the other one 284, and conversely. And Jacob had this in mind; this has been tried by the ancients in securing the love of kings and dignatories. Ibn Khaldoun350 related "that persons who have concerned themselves with talismans affirm that the amicable numbers 220 and 284 have an influence to establish a union or close friendship between two individuals. To this end a theme is prepared for each individual, one during the ascendency of Venus, when that planet is in its exaltation and presents to the moon an aspect of love or benevolence; for the second theme the ascendency should be in the seventh. On each of these themes is written one of the specified numbers, the greater (or that with the greater sum of its aliquot parts?) being attributed to the person whose friendship is sought." The Arab El Madschriti,351 or el-Magrttl, (flOOT) of Madrid related that he had himself put to the test the erotic effect of "giving any one the smaller number 220 to eat, and himself eating the larger number 284. " Ibn el-Hasan351a (fl320) wrote several works, including the "Memory of Friends/' on the explanation of amicable numbers. Ben Kalonymos351** discussed amicable numbers in 1320 in a work written for Robert of Anjou, a fragment of which is in Munich (Hebr. MS. 290, f. 60). A knowledge of amicable numbers was considered necessary by Jochanan Allemanno (fifteenth century) to determine whether an aspect of the planets was friendly or not. 849Baale Brith Abraham [Commentary on the Bible], Wilna, 1873, 22. Quotation supplied by Mr. Ginsburg. 3BOProl6gom6nes hist. d'Ibn Khaldoun, French transl. by De Slane, Notices et Extraits des Manuscrits de la Bibl. Impfriale, Paris, 21, I, 1868, 178-9. 381Manuscript Magriti; Steinechneider, Zur pseudoepigraphischen Literatur inbesondere der geheimcn Wissenschaften des Mittelaltere, Berlin, 1862, p. 37 (cf. p. 41). 35iaH. Suter, Abh. Gesch. Math. Wiss., 10, 1900, 159, §389. J6l6Hebr. Bibl., VII, 91. Steinschneider, Zeitschrift der Morgenlandischen Ges., 24, 1870, 369.