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JORDANUS DE NEMORE (fl. ca. 1220), mechanics,
6. Clagett, op. cit., p. 73.
7. Translated by E. A. Moody and M. Clagett, The Medieval
Science of Weights, p. 129.
8. Ibid., p. 131.
9. 850b.4-6 in the translation of E. S. Forster (Oxford, 1913).
10. Moody and Clagett, trans., op. cit., pp. 139, 141.
11. A position adopted by Moody, ibid., pp. 171-172.
12. Joseph E. Brown, The Scientia de ponderibus in the Later
Middle Ages, pp. 64-66.
13. Clagett, op. cit., pp. 76-77.
14. Moody and Clagett, trans., op. cit., p. 191.
15. Ibid., pp. 185, 187.
16. Clagett, op. cit., p. 82.
17. Translated by Stillman Drake in Mechanics in Sixteenth-Century
Italy, trans. and annotated by Stillman Drake and
I. E. Drabkin (Madison, Wis., 1969), p. 295.
18. Marshall Clagett, Archimedes in the Middle Ages, I, 675.
19. Trans. by Clagett, ibid., pp. 573-575.
20. Barnabas B. Hughes (ed. and trans.), The De numeris
datis of Jordanus de Nemore, pp. 50-52.
21. In his ed. of the De numeris datis, Curtze altered the letters
in presenting the analytic summaries of the propositions;
in a few instances I have altered Curtze's letters.
22. My translation from Edward Grant, ed., A Source Book in
Medieval Science (in press).
I. ORIGINAL WORKS.
In Richard Fournival's Biblionomia
twelve works are attributed to Jordanus. In codex 43 we
find (1) Philotegni, or De triangulis; (2) De ratione
(3) De ponderum proportione; and (4) De quadratura
circuli. In codex 45 three works are listed: (5) Practica, or
Algorismus; (6) Practica de minutiis; and (7) Experimenta
super algebra. Codex 47 contains the lengthy (8) Arithmetica.
Codex 48 includes (9) De numeris datis; (10) Quedam
experimenta super progressione numerorum; and (11) Liber
de proportionibus. Codex 59 includes a treatise called
(12) Suppletiones plane spere.
Numbers (2) and (3) are obviously statical treatises. The
De ratione ponderum is probably the De ratione ponderis
edited and translated by E. A. Moody in E. A. Moody and
M. Clagett, The Medieval Science of Weights (Scientia de
ponderibus) (Madison, Wis., 1952); its attribution to Jordanus
has been questioned by Joseph E. Brown, The
Scientia de ponderibus in the Later Middle Ages (Ph.D.
diss., University of Wis., 1967), pp. 64-66. In the same
volume Moody has also edited and translated the Elementa
Jordani super demonstrationem ponderum, a genuine work
of Jordanus, which perhaps corresponds to the De ponderum
proportione of the Biblionomia.
The Philotegni, or De triangulis, exists in two versions.
The longer, and apparently later, version was published by
Maximilian Curtze, “Jordani Nemorarii Geometria vel De
triangulis libri IV,” in Mitteilungen des Coppernicus-Vereins
für Wissenschaft und Kunst zu Thorn, 6 (1887), from MS
Dresden, Sächsische Landesbibliothek, Db 86, fols. 50r-61v.
Utilizing additional MSS, Marshall Clagett reedited
and translated Props. IV.16 (quadrature of the circle),
IV.20 (trisection of an angle), and IV.22 (finding of two
mean proportionals) in his Archimedes in the Middle Ages,
I, The Arabo-Latin Tradition (Madison, Wis., 1964), 572-575,
672-677, and 662-663, respectively. A shorter version,
which lacks Props. II.9-12, 14-16, and IV.10 and terminates
at IV.9 or IV.11, has been identified by Clagett. Both
versions will be reedited by Clagett in vol. IV of his
Archimedes in the Middle Ages. Of the 17 MSS of the two
versions which Clagett has found thus far, we may note,
in addition to the Dresden MS used by Curtze, the following:
Paris, BN lat. 7378A, 29r-36r; London, Brit. Mus.,
Sloane 285, 80r-92v; Florence, Bibl. Naz. Centr., Conv.
Soppr. J. V. 18, 17r-29v; and London, Brit. Mus., Harley
The De quadratura circuli attributed to Jordanus as a
separate treatise in Fournival's Biblionomia may be identical
with Bk. IV, Prop. 16 of the De triangulis, which bears
the title “To Form a Square Equal to a Given Circle”
(quoted above; for the Latin text, see Clagett, Archimedes
in the Middle Ages, I, 572, 574). In at least one thirteenth-century
MS (Oxford, Corpus Christi College 251, 84v) the
proposition stands by itself, completely separated from the
rest of the De triangulis, an indication that it may have
circulated independently (for other MSS, see Clagett,
Archimedes, I, 569).
The De numeris datis has been edited three times. It was
first published on the basis of a single fourteenth-century
MS, Basel F.II.33, 138v-145v, by P. Treutlein, “Der Traktat
des Jordanus Nemorarius ‘De numeris datis,’ ” in
Abhandlungen zur Geschichte der Mathematik, no. 2 (Leipzig,
1879), pp. 125-166. Relying on MS Dresden Db86,
supplemented by MS Dresden C80, Maximilian Curtze
reedited the De numeris datis and subdivided it into four
books in “Commentar zu dem ‘Tractatus de numeris datis’
des Jordanus Nemorarius,” in Zeitschrift für Mathematik
und Physik, hist.-lit. Abt., 36 (1891), 1-23, 41-63, 81-95,
121-138. In MS Dresden C80 Curtze found additional
propositions (IV.16-IV.35) beyond the concluding proposition
in Treutlein's ed. The additional propositions included
no proofs but only the enunciations of the propositions
followed immediately by a single numerical example for
each. That these extra propositions formed a genuine part
of the De numeris datis was verified by MSS Vienna 4770
and 5303, which included not only the additional propositions
but also their proofs. Using MS Vienna 4770, from
which 5303 was copied, R. Daublebsky von Sterneck published
complete versions of Props. IV.15-IV.35 and also
supplied corrections and additions to a few propositions
in Bk. I in “Zur Vervollständigung der Ausgaben der
Schrift des Jordanus Nemorarius: ‘Tractatus de numeris
datis,’ ” in Monatshefte für Mathematik und Physik,
(1896), 165-179. A third ed., with the first English trans.,
has been completed by Barnabas Hughes: The De numeris
datis of Jordanus de Nemore, a Critical Edition, Analysis,
Evaluation and Translation (Ph.D. diss., Stanford University,
1970). Hughes's thorough study also includes (pp. 104-126)
a history of previous editions, as well as a description
of twelve MSS, whose relationships are discussed in detail.
A Russian translation from Curtze's edition was made
by S. N. Šre?der, “The Beginnings of Algebra in Medieval
Europe in the Treatise De numeris datis of Jordanus de
Nemore,” in Istoriko-Matematicheskie issledovaniya, 12