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JORDANUS DE NEMORE (fl. ca. 1220), mechanics,
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
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
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,
(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,
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
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 ?
E, then D + E = A(B + C). By
D/B = A and E/C = A. And since D and
equimultiples of B and C, respectively, then, by
Proposition 6, it follows that B + C = (1/n) (D
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 :
12. 1 ? 10n + 9 ? 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 =
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.
1. Marshall Clagett, The Science of Mechanics in the Middle
Ages, pp. 72-73.
2. Leopold Delisle, Le cabinet des manuscrits de la
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.