¹ Murray, Alexander, Reason and Society in the Middle Ages (Oxford, 1978), 142. This essay is indebted to several individuals and institutions, who have made its combination of medieval and mathematical interests possible: Lyman Briggs College, for an undergraduate education in interdisciplinary science that fostered both mathematical and literary study; Professor Marvin Tomber, for the initial introduction to number theory; MacCluer, Professor C. R., for requiring his abstract algebra classes to explore the historical origins and proofs of modern theorems; and the National Humanities Center, for providing the time to read Dives and Pauper all the way through.Google Scholar

² Priscilla Heath Barnum, ed., Dives and Pauper, 2 vols., EETS 275, 280 (London, 1976; Oxford, 1980), Precept 3.12 (1:284). All references are to this edition.Google Scholar

³ For this tradition, see Heinz Meyer and Rudolf Suntrup, Lexikon der mittelalterlichen Zahlenbedeutungen (Munich, 1987), s.v. “Sechs,” 442–50.Google Scholar

⁴ Three manuscripts (of eight) omit this sentence, perhaps finding its theoretical content unnecessary for the chapter.Google Scholar

⁵ Meyer and Suntrup, Lexikon, 445–48 (also quoting similar remarks by Bede, Rabanus, Alcuin, Isidore, Rupert of Deutz, and others; emphasis mine). One important voice of semi-dissent is that of Gregory the Great, who sees the mathematical perfection of six as mere “wisdom of the world,” and insists that from a spiritual point of view, six should be seen as perfect because God created the world in six days, not for some mathematical reason (ibid., 443–44). However, Augustine's Platonic reverence for the transcendent reality of number was more influential than Gregory's general distrust of secular mathematics.Google Scholar

⁶ Shanks, Daniel, Solved and Unsolved Problems in Number Theory, 3rd ed. (New York, 1985), 3. Both Shanks and Leonard Eugene Dickson, the author of the still-classic History of the Theory of Numbers, 3 vols. (Washington, 1919–23; repr. New York, 1966), indicate the importance of perfect numbers in the development of number theory by making them the central topic of their first chapters. Dickson identifies over 150 classical, medieval, and early modern writers on perfect numbers, including such historically significant figures as Nicomachus, Boethius, Fibonacci, Cataldi, Fermat, Descartes, Mersenne, Euler, and others (1:3–33). Although some of the just-named authors were mathematically more rigorous than others in their discussion of perfect numbers, all of them were interested in genuinely mathematical properties of numbers and not—or not primarily—in the numerological symbolism common in many literary and other artistic uses of number in pre-modern times. The discussion that follows here draws extensively on Dickson and Shanks. For the importance and treatment of perfect numbers in the work of Fermat, one of the principal founders of modern number theory, see André Weil, Number Theory: An Approach Through History from Hammurapi to Legendre (Boston, 1984), 51–55.Google Scholar

⁷ The Thirteen Books of Euclid's Elements. trans. Heath, Thomas L. 2d ed., 3 vols. (Cambridge, 1926; repr. New York, 1956), 2:421. Although book 9 of Adelard of Bath's Latin translation of the Elements is not extant, one can find a medieval Latin version of the theorem in the translation ascribed to Hermann of Carinthia. See The First Latin Translation of Euclid's Elements Commonly Ascribed to Adelard of Bath, ed. Busard, H. L. L. (Toronto, 1983), 248; and The Translation of the Elements of Euclid from the Arabic into Latin by Hermann of Carinthia (?), Books VII–XII, ed. Busard, H. L. L. (Amsterdam, 1977), 74–77.Google Scholar

⁸ It can also be shown that 2^p−1 can be prime only if p is itself prime (i.e., if 2^p−1 is prime, then p must be prime), though a prime p does not guarantee the primality of 2^p−1. See Shanks, , Solved and Unsolved Problems, 3–4, 14.Google Scholar

⁹ Nicomachus of Gerasa, Introduction to Arithmetic 1.16, trans. Martin Luther D'Ooge (New York, 1926), 209.Google Scholar

¹⁰ Ibid., 211; emphasis mine.Google Scholar

¹¹ Ibid., 210.Google Scholar

¹² Boethian Number Theory: A Translation of the De Institutione Arithmetica, trans. Michael Masi (Amsterdam, 1983), 98; emphasis mine.Google Scholar

¹³ Scritti di Leonardo Pisano, ed. Baldassarre Boncompagni, vol. 1: Il Liber Abbaci di Leonardo Pisano (Rome, 1857), 283.Google Scholar

¹⁴ Arithmetica speculatiua (Paris, 1502), sig. A5r (my translation): “Numerorum autem perfectorum in limite primo solum vnicum reperimus vt infra .10.96 [read 6]. infra .100.28. infra .1000.496. & infra .10000.8128…. Erit autem generatio eius talis: …. Sic quamdiu per hunc modum progressi fuerimus non cessabit generatio perfectorum.”Google Scholar

¹⁵ Dickson mentions, among others who made the assertion, Cardan (1537), Noviomagus (1539), Henischiib (1605), and Puteanus (1637); History, 8–12.Google Scholar

¹⁶ On the daunting complexity of medieval European division techniques, see Robert Steele, ed., The Earliest Arithmetics in English, EETS e.s. 118 (London, 1922), x–xiii, 43–45, 61–63; and Leonardo di Pisano, Liber Abbaci, chaps. 5 (pp. 23–47) and 7 (pp. 63–83). Alexander Murray provides an excellent historical analysis of the development of the “arithmetical mentality” by way of the abacus and Arabic numerals, under the influence of both commerce and government, in Reason and Society (n. 1 above), chaps. 7–8 (pp. 162–210).Google Scholar

¹⁷ Arabic mathematicians had found larger perfect numbers somewhat sooner; the earliest known Arabic writer to give correct perfect numbers above 8128 is one Ibn Fallūs (1194–1252), who correctly identified the fifth, sixth (8,589,869,056), and seventh (137,438,691,328) perfect numbers. Knowledge of perfect numbers above 8128 is found elsewhere in the Islamic tradition, as is a recognition that there is not always a perfect number for every power of ten. See Sonja Brentjes, “Untersuchungen zum Nicomachus Arabus,” Centaurus 30 (1987): 217–18, 222–23, and nn. 39–40. Heath speculates that the Greek mathematician Iamblichus (who was unread in medieval Western Europe) may have known the fifth perfect number, though Iamblichus does not explicitly identify it (Euclid's Elements [n. 7 above], 2:426).Google Scholar

¹⁸ Dickson gives many partially incorrect lists of perfect numbers proposed by European mathematicians from the late fifteenth century on (History, 6–18). In fairness to the Western tradition, it must be noted that although Arabic mathematicians found some of the larger perfect numbers sooner than their European counterparts, they too wrongly identified some large numbers as perfect in their lists (Brentjes, “Nicomachus Arabus,” 223).Google Scholar

¹⁹ Dickson, , History, 13.Google Scholar

²⁰ See, for instance, Alexander Hurwitz, “New Mersenne Primes,” Mathematics of Computation 16 (1962): 249–51; Curt Noll and Laura Nickel, “The 25th and 26th Mersenne Primes,” Mathematics of Computation 35 (1980): 1387–90; David Slowinski, “Searching for the 27th Mersenne Prime,” Journal of Recreational Mathematics 11 (1979): 258–61; and generally Shanks, Solved and Unsolved Problems (n. 6 above), 195–96, 237, 265–66.Google Scholar

²¹ Editors’ announcement, “The Latest Mersenne Prime,” American Mathematical Monthly 99/4 (April 1992): 360, and corrigendum in AMM 99/7 (August-September 1992): 617; “2⁸⁵⁸⁴³³−1 is Prime,” AMM 101/4 (April 1994): 338. I am grateful to Professor Edward Ingraham for these references.Google Scholar

²² This comment in the “Progress” chapter of the 3rd edition of Solved and Unsolved Problems (266) may reflect a change of mind from Shanks's 1962 opinion, still visible in chapter 1: “For Conjecture 1 [‘There are infinitely many perfect numbers,’ a logical consequence of an infinitude of Mersenne primes], the evidence is, in fact, not very compelling” (2). Or perhaps the conviction is based on mathematical hunch or hope, rather than the compulsion of logic.Google Scholar

²³ Anne Hudson and Spencer, H. L., “Old Author, New Work: The Sermons of MS Longleat 4,” Medium Ævum 53 (1984): 220–38. Hudson and Spencer note that the word “postils” might better characterize the collection than “sermons,” since it seems to have been written more for private reading than public preaching (226–27); they continue to refer to the collection by the more familiar term “sermons,” however, and I will follow their example in this essay.Google Scholar

²⁴ Ibid., 231–32 and n. 87, quoting Longleat 4, fols. 1r–v and 90v, and Dives and Pauper, Prec. 4.11 (1:327) and Prec. 6.3 (2:64).Google Scholar

²⁵ Criticism of the clergy is noted by Anne Slater, “Dives and Pauper: Orthodoxy and Liberalism,” Journal of the Rutgers University Library 31 (1967): 5; Pfander, H. G., “Dives et Pauper,” The Library 4th ser., 14 (1933): 304–05; and Hudson and Spencer, “Old Author,” 230–31, 233.Google Scholar

²⁶ Hudson and Spencer, “Old Author,” 231.Google Scholar

²⁷ As Priscilla Barnum observes, “his story is a precursor to that of Reginald Pecock a generation or so later, with the difference that, by Pecock's time, positions had hardened. By the 1450s vernacular discussion of church doctrine and practices, and appeals to the lay conscience, had become prosecutable offenses,” even when undertaken by a bishop. See Medieval England: An Encyclopedia, ed. Paul Szarmach et al. (New York, forthcoming), s.v. “Dives and Pauper.” In a still later age, with somewhat different institutions, our author might have followed a path similar to that of Mersenne himself, who became a noted religious apologist and rationalist, seeking to reconcile science, philosophy, and religion in his prolific writings and wide-ranging correspondence with all manner of scholars; for Mersenne's place in seventeenth-century intellectual history, see Peter Dear, Mersenne and the Learning of the Schools (Ithaca, N.Y. 1988).Google Scholar

²⁸ In addition to the examples given here, see Dickison, Roland B., “Superstitions and Some Common Sense Refutations in Fifteenth Century England,” Southern Folklore Quarterly 24 (1960): 164–66, 169 (soot falling in chimneys is not a marvelous omen of bad weather, but caused by naturally moist air before a storm; prognostications based on the weather of the twelve days of Christmas or the day of the week on which New Year falls are not borne out by reason or experience; the foolishness of following bird omens and dream books); and Slater, “Orthodoxy,” 5–6 (falling soot; the natural causes of eclipses and other astronomical phenomena; natural reasons for the medical efficacy of certain charms; etc.).Google Scholar

²⁹ Note also the author's willingness both to admit uncertainty and to speculate about the nature of the star of Bethlehem, after he has given arguments of “skyl and resoun” to show that it was “noo planete ne sterre of þe firmament” (Prec. 1.23; 1:133–34).Google Scholar

³⁰ For parallel comments about lions, pelicans, storks, and chameleons, see On the Properties of Things: John Trevisa's Translation of Bartholomaeus Anglicus De Proprietatibus Rerum, 3 vols., gen. ed. Seymour, M. C. (Oxford, 1975–88), 18.45, 12.30, 12.9, and 18.21 (2:1214–17, 1:636–37, 1:619–20, 2:1159–61).Google Scholar

³¹ On these friars and their interests, see Beryl Smalley, The Study of the Bible in the Middle Ages, 2d ed. (Oxford, 1952; rpt. Notre Dame, 1964), 308–28, 371–73, and Judson Boyce Allen, The Friar as Critic: Literary Attitudes in the Later Middle Ages (Nashville, Tenn. 1971), 34–51. Other characterizations of the mind behind Dives and Pauper include Hudson and Spencer's “radical but not heterodox” (“Old Author,” 233); and Slater's “personal expansiveness of spirit within the framework of accepted attitudes,” “not a particularly original thinker, although he often articulates common sense conclusions drawn from personal observation,” and “within [his orthodox] context he almost always chooses the gentler path” (“Orthodoxy,” 9–10).Google Scholar

³² See Weisheipl, James A., “Curriculum of the Faculty of Arts at Oxford in the early Fourteenth Century,” Mediaeval Studies 26 (1964): 170. For discussion of the importance of Boethius's treatise at both the elementary level and as a starting point for much more complex mathematical developments by such figures as Roger Bacon, Thomas Bradwardine, and Nicholas of Oresme, see Pearl Kibre, “The Boethian De institutione arithmetica and the Quadrivium in the Thirteenth Century University Milieu at Paris,” and Michael Masi, “The Influence of Boethius’ De arithmetica on Late Medieval Mathematics,” in Boethius and the Liberal Arts, ed. Michael Masi (Berne, 1981), 67–80, 81–95.Google Scholar

³³ The text is quite brief and simple; Weisheipl reports that it “seems to have been used later as a text, at least on the continent” (“Curriculum,” 170).Google Scholar

³⁴ Ibid., 167–76.Google Scholar

³⁵ Pauper's list of associations reproduces nearly all the major topics identified by Meyer and Suntrup as the fundamental numerological significations for six: Creation in general; the Ages of the World; the creation of humanity on the sixth day; the Passion and Redemption on the sixth day of the week; human good works, including the works of mercy (Meyer and Suntrup, Lexikon [n. 5 above], s.v. “Sechs,” 442, 444).Google Scholar

³⁶ As Slater notes, Dives and Pauper places the blame for men's sexual sins squarely on men's own shoulders, rejecting a view of women as being automatically at fault as sources of temptation (“Orthodoxy,” 7–8). Interestingly, the treatise also clearly rates lechery as a lesser sin than suicide—a logical and orthodox moral consequence of the nature of the two sins (5.22), but a conclusion in some conflict with the familiar misogynistic characterization of female virtue that praised such “martyrs” as Lucretia and other classical women named in Jerome's Adversus Jovinianum. Google Scholar

³⁷ His language at the end of Prec. 3.12—“we schuldyn for no coueytise don to mychil ne for non slauþe don to lytil but alwey holdyn us in a mene and in efnehed”—reflects the common medieval and classical notion of true virtue as a mean, neither excessive nor defective in any way. Perfect numbers were considered the ideal mathematical image of such balance, and all other numbers were considered to be “excessive” (if their factors added up to more than themselves) or “defective” (if their factors added up to less than themselves). Thus Boethius says, “There is in these [numbers] a great similarity to the virtues and vices. You find the perfect numbers rarely, you may enumerate them more easily, and they are produced in a very regular order. But you find superfluous or diminished numbers to be many and infinite and not disposed in any order, but arranged randomly and illogically, not generated from a certain point” (Boethian Number Theory [n. 12 above], 98; cf. Nicomachus, Introduction to Arithmetic 1.16 [n. 9 above], 209). The rarity of perfect numbers, and difficulty of finding them, turned out to be even greater than medieval mathematicians thought—according to Colquitt, W. N. and Welsh, L., there are only 30 Mersenne primes less than (2¹³⁹²⁶⁸−1), and thus only 30 even perfect numbers less than 2¹³⁹²⁶⁷(2¹³⁹²⁶⁸−1), a number of nearly 84,000 digits; see “A New Mersenne Prime,” Mathematics of Computation 56 (1991): 867. Had the medieval conjecture about powers of ten been right, there would be almost 84,000 even perfect numbers in that range. Colquitt and Welsh also state that it is unknown whether there are any still-undiscovered Mersenne primes between 2¹³⁹²⁶⁸−1 and 2²¹⁶⁰⁹⁰−1, or how many there are beyond 2³⁵³⁶²⁰−1; no odd perfects (of any size) have ever been discovered, though their possible existence remains an open question.Google Scholar

³⁸ Barlow, Peter, A New Mathematical and Philosophical Dictionary (London, 1814), s.v. “Perfect Number”; quoted in Shanks, Solved and Unsolved Problems (n. 6 above), 195n.Google Scholar

³⁹ The mathematical and scientific excitement attending the possible discovery of such a proof, announced by Andrew Wiles in June 1993, even managed to reach the popular press, being reported in such sources as the New York Times (24 June 1993: A1, D22), USA Today (28 June 1993: A12), the Boston Globe (2 July 1993: 15), and the Washington Post (2 July 1993: A19). See also Science 261 (2 July 1993): 32–33. For a brief report on current thinking about Wiles's proof, see Keith Devlin, “Fermat's Last Theorem,” Math Horizons (Spring 1994): 4–5.Google Scholar

⁴⁰ Colquitt and Welsh, “A New Mersenne Prime,” 867; compare the language of the editors of the American Mathematical Monthly: “a good test of reliability for both hardware and software on new machines” (“2⁸⁵⁸⁴³³−1 is Prime” [n. 21 above], 338).Google Scholar

⁴¹ I am indebted to an anonymous mathematical reader of the present article for this observation.Google Scholar

⁴² Solved and Unsolved Problems, 25; emphasis original. Visually, Shanks's point looks like this for 6 and 28:Google Scholar

Note that every column contains only a single 1 in it, with all other digits in the column being zero, hence a sum “without a single carry!”Google Scholar

⁴³ At the higher reaches of fourteenth-century philosophy and theology, questions of infinity and perfection were also at stake, and in some ways their discussion grew out of mathematical notions of those concepts. See Murdoch, John E., “Mathesis in philosophiam scholasticam introducta: The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology,” in Arts libéraux et philosophie au moyen âge, Actes du quatrième Congrès international de philosophie médiévale, Montréal, 1967 (Montréal, 1969), 221–24 (the infinite) and 238–46 (perfection).Google Scholar