Dictionary of Scientific Biography Dictionary of Scientific Biography

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# JORDANUS DE NEMORE (fl. ca. 1220), mechanics, mathematics.

The general pattern of every proposition is thus (1) formal enunciation of the proposition; (2) proof; and (3) a numerical example, which is certainly non-Euclidean and was perhaps patterned after Arabic algebraic treatises. In this wholly rhetorical treatise, Jordanus used letters of the alphabet to represent numbers. An unknown number might be represented as ab or abc, which signify a + b and a + b + c respectively. Occasionally, when two unknown numbers are involved, one would be represented as ab, the other as c.

Typical of the propositions in the De numeris datis are Book I, Proposition 4, and Book IV, Proposition 7. In the first of these, a given number, say s,21 is divided into numbers x and y, whose values are to be determined. It is assumed that g, the sum of x2 and y2, is also known. Now s2 - g = 2xy = e and g - e = h = (x - y)2. Therefore (x - y) = ?h = d. Since d is the difference between the unknown numbers x and y, their values can be determined by Book I, Proposition 1, where Jordanus demonstrated that “if a given number is divided in two and their difference is given, each of them will be given.” The numbers are found from their sum and difference. Since x + y = s and x - y = d, it follows that y + d = x and, therefore, 2y + d = x + y = s; hence 2y = s - d, y = (s - d)/2, and x = s - y. Should we be given the ratio between x and y, say r, and the product of their sum and difference, say p, the values of x and y are determinable by Book IV, Proposition 7, as follows: since x/y = r, x2/y2 = r2; moreover, since (x + y) (x - y) = p, therefore x2 - y2 = p. Now x2 = r2y2, so that (r2 - 1)y2 = p and y = ?p/(r2 - 1). In the numerical example r = 3 and p = 32, which yields y = 2 and x = 6.

In the Arithmetica, the third and probably most widely known of his three major mathematical works, Jordanus included more than 400 propositions in ten books which became the standard source of theoretical arithmetic in the Middle Ages. Proceeding by definitions, postulates, and axioms, the Arithmetica was modeled after the arithmetic books of Euclid's Elements, a treatise which Jordanus undoubtedly used, although the proofs frequently differ. Jordanus' Arithmetica contrasts sharply with the popular, nonformal, and often philosophical Arithmetica of Boethius. A typical proposition, which has no counterpart in Euclid's Elements, is Book I, Proposition 9:

The [total sum or] result of the multiplication of any number by however many numbers you please is equal to [est quantum] the result of the multiplication of the same number by the number composed of all the others.

Let A be the number multiplied by B and C to produce D and E [respectively]. I say that the composite [or sum] of D and E is produced by multiplying A by the composite of B and C. For it is obvious by Definition  that B measures [numerat] D A times and that C measures E by the same number, namely, A times. By the sixth proposition of this book, you will easily be able to argue this.22

Thus Jordanus proves that if A ? B = D and A ? C = E, then D + E = A(B + C). By Definition 7, D/B = A and E/C = A. And since D and E are equimultiples of B and C, respectively, then, by Proposition 6, it follows that B + C = (1/n) (D + E) and, assuming n = A, we obtain A(B + C) = D + E.

The Arabic number system also attracted Jordanus' attention--if the Demonstratio Jordani de algorismo and a possible earlier and shorter version of it, the Opus numerorum, are actually by Jordanus. Once again Jordanus proceeded by definitions and propositions in a manner that differed radically from Johannes Sacrobosco's Algorismus vulgaris, or Common Algorism. Unlike Sacrobosco, Jordanus described the arithmetic operations and extraction of roots succinctly and formally and without examples. Among the twenty-one definitions of the Demonstratio Jordani are those for addition, doubling, halving, multiplication, division, extraction of a root (these definitions are illustrated as propositions), simple number, composite number, digit, and article (which is ten or consists of tens). Propositions equivalent to the following are included:

3. If a : b = c : d, then a ? 10n : b = c ? 10n : d

12. 1 ? 10n + 9 ? 10n = 1 ? 10n+1

19. a ? 10n + b ? 10n = (a + b)10n

32. If a1 = a ? 10, a2 = a ? 100, a3 = a ? 1,000, then (a ? a1)/a2 = (a1 ? a2)/a21 = (a2 ? a3)/a22 = ???.

An algorithm of fractions, called Liber or Demonstratio de minutiis in some manuscripts, may also have been written by Jordanus. It describes in general terms arithmetic operations with fractions alone and with fractions and integers. He also composed a Liber de proportionibus, a brief treatise containing propositions akin to those in Book V of Euclid's Elements.

## NOTES

1. Marshall Clagett, The Science of Mechanics in the Middle Ages, pp. 72-73.
2. Leopold Delisle, Le cabinet des manuscrits de la Bibliothèque nationale, II (Paris, 1874), 526, 527.
3. This has been suggested by O. Klein, “Who Was Jordanus Nemorarius?,” in Nuclear Physics, 57 (1964), 347.
4. Maximilian Curtze, “Jordani Nemorarii Geometria vel De triangulis libri IV,” in Mitteilungen des Coppernicus-Vereins, 6 (1887), iv, n. 2.
5. Ibid., p. vi.
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