¹ Europiiische Literatur und lateinisches Miltdaller (Bern, 1948), tr. Willard R. Trask, European Literature and the Latin Middle Ages (New York, 1953), p. 509.

² “Zur Interpretation des Alexiusliedes,” ZKPh, LVI (1936), 113–137.

³ “The Old-French Poem St. Alexis: A Mathematical Demonstration,” Traditio, vin (1952), 112–158.

⁴ I have been assisted in the formulation of my arithmetic statement by my friend Marcel Riesz, who is Visiting Research Professor in Mathematics at the University of Mary land, and who found himself with some surprise applying his formidable knowledge of mathematics to the field of Old-French Literature.

⁵ The symbolic meaning of the key numbers in this structure is the subject of a study in documentation now being conducted by my student Mr. Paul Imhoff for a master's thesis.

⁶ This is the arrangement of the twelfth-century Lam-springe manuscript (usually designated L) which is now in the library of the Church of St. Godoard at Ffildesheim in Hanover. Of the seven manuscripts extant, L is one of the three oldest, the others, also of the twelfth century, being: A, from Lord Ashburnham's collection but now in France, and V (Vatican, Cod. Vat. lal., fol. 125), a fragment of 178 lines of the last part of the poem, which was published by Pio Rajna in Archimm Romanicum, xiii (1929), 1–86. Manuscript L is the most complete, having only 5 lines missing: 255, 274, 275, 349, and 472. In the V fragment stanzas 109 and 110 of L (along with two others: 87 and 108) do not appear. They do appear in A, where, as stanzas 107 and 108, they comprise the conclusion of the poem, since the final section is missing in this version. Stanza 109 also appears in a thirteenth-century manuscript P (Cod. Paris, lat., 19525) which, although lacking nine complete stanzas and eight isolated lines, gives a very good reading. Thus, although stanzas 109 and 110 of L do appear in two of the three twelfth-century manuscripts, and although stanza 109 appears in a good thirteenth-century manuscript, Rajna maintains that these two stanzas are spurious, since they do not appear in the V fragment and since they have somewhat the character of interpolations. Hatcher does not concur in this but Curtius (“Zur Interpretation . . . ”) does. The problem of the interpolations in the Alexis text is discussed in n. 12 below. It suffices to say at this point that I am in accord with Hatcher in that stanzas 109 and 110 be retained. My reasons are, first, that Rajna's arguments for excluding them are by no means conclusive, and secondly, that the complex and meaningful arithmetic pattern which their retention makes possible mitigates against the probability that they were not a part of the original composition.

⁷ In my citations I follow the reading of manuscript L as given by C. Storey, Saint Alexis, étude de la langue du manuscrit de Httdesheim, suivie d'une édition critique du texte (Paris, 1934).

⁸ The reference works which I have consulted for this discussion of medieval arithmetic are: Florian Cajori, A History of Mathematics (New York and London, 1894), pp. 84–137; and A History of Mathematical Notations, i (Chicago, 1928), passim; Jacques Boyer, Histoire des mathématiques (Paris, 1900), pp. 63–87; and Bartel Leendert van der Waerden, Science Awakening, tr. Arnold Dresden (Groningen, 1954), pp. 37–61, 82–104, et passim.

⁹ (New York, 1938), pp. 9, 10.

¹⁰ The medieval tendency to attribute special mystical properties to the number 10 and its multiples was more or less inherited from the Pythagoreans. Hopper (p. 42) explains that in the Pythagorean view “the particular glory of the archetypal numbers [i.e., the tetrad or first four digits] is that they produce the decad, either as a sum (1+2+3+4 = 10) or in the figured representation of 10 as a triangular number. This figure was known as the tetraktys, the legendary oath of the Pythagoreans. Lucian in the Sale of Philosophers represents the Pythagorean as asking the prospective buyer to count. When he has counted to 4, the philosopher interrupts, 'Lo! what thou thinkest four is ten, and a perfect triangle, and our oath.' Philo adds that there are 4 boundaries of number, the unit, decad, century, and thousand, all of which are measured by the tetraktys. For just as 1+2+3+4 = 10, so 10+20+30+40=100, and 100+200+300+400 = 1,000.” Hopper continues (pp. 44 and 45) : “Ten and 1 are mystically the same, as are also 100 and 1,000, the 'boundaries' of number. In the decad, multiplicity returns again to unity. Ten are the categories … for 10 is the total of all things, embracing the entire world. It is the most perfect of all ‘perfect’ numbers and is called by Porphyry ‘Comprehension,‘ as comprehending all differences of numbers, reasons, species and proportions.”

¹¹ According to Cajori, A History of Mathematics, pp. 159, 160, the idea of decimal fractions made its first appearance in methods practiced in Europe as early as the mid-twelfth century for approximating to the square roots of numbers. However, it is Simon Stevin (1548-1620) of Bruges in Belgium who is generally credited with the development of the first systematic treatment of decimal fractions, although his method of notation, described in his work La Disme (1585), differs somewhat from that in use today.

¹² “How nappy those who honored him by faith!” It will be noted that here again, as in stanza 67 discussed above, the poet rounds out his stanza in line 500 with an exclamatory intercalation more or less extraneous to the narrative. In the occurrence of these insertions at two such crucial points an explanation is offered for the interpolatory character of the stanzas 109 and 110 which Rajna wished to delete (see n. 6 above). They were simply needed to fill out the number scheme which the poet had devised.

¹³ We are reminded in this of that other more famous division of 100 by the number of years of Christ's life, where Dante conducts us through the structure 1:33:33:33 of the Divina Commedia, as, to use the words of Curtius, “triads and decads intertwine into unity,” across the three realms of the Christian cosmos to the tenth heaven at its apex.

¹⁴ One also has in memory that other manipulation of the powers of numbers whereby the miracle of Beatrice is explained as the number nine, because “lo numéro del tre è la radice del nove, perô che sanza numéro altro alcuno, per se medesimo fa nove, si come vedemo manifestamente che tre via tre fa nove. Dunque se lo tre è fattore per se medesimo del nove, e lo fattore per se medesimo de la miracoli è tre, cioè Padre e Figlio e Spirito Santo, li quali sono tre e uno” (Dante, Vita Nuova, xxrx, ed. Kenneth McKenzie, La Vita Nuova di Dante Alighieri [New York, 1922], pp. 51, 52).

¹⁵ See notes 2 and 3 above.

¹⁶ An interpretation of the Contra Celsum by Jean Daniélou, Origen, tr. Walter Mitchell (New York, 1955), p. 35.

¹⁷ Cf., of course, the frequently quoted verse from Wisdom xi, 2: “Sed omnia in mensura et nomine et pondère dis-posuisti.”

¹⁸ As cited by Hopper, p. 38, from the translation of Martin Luther d'Oge (New York, 1926). Cf., on the one hand Plato, Republic, vii, tr. B. Jowett, The Works of Plato, 4 vol. complete in one, ed. Tudor Publishing Co. (New York, n.d.), ii, 281: “We must endeavor to persuade the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only. . . . Arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible and tangible objects into the argument”; p. 290: “For they [the Pythagoreans] too are in error, like the astronomers; they investigate the numbers of the harmonies which are heard, but they never attain to problems—that is to say, they never reach the natural harmonies of number, or reflect why some numbers are harmonious and others not.” On the other hand, cf. St. Augustine, On Christian Doctrine, n, 38, 56, tr. Marcus Dods, The Works of Aurelius Augustine (Edinburgh, 1771–76), ix, 73: “Concerning now the Science of number, it is clear to the dullest apprehension that this was not created by man, but was discovered by investigation. … It is not in any man's power to determine at his pleasure that 3X3 are not 9, or do not make a square or are not the triple of 3”; also The City of God, xi, 30 (Dods' trans. I, 475): “we must not despise the science of numbers, which, in many passages of Holy Scripture, is found to be of eminent service to the careful interpreter. Neither has it been without reason numbered among God's praises, ‘thou has ordered all things in number, and measure and weight.‘ ” (The passages from Augustine are cited by Hopper, p. 78.)