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

As yet there is no ed. of the ten-book Arithmetica, although the enunciations of the propositions were published by Jacques Lefèvre d'Étaples (Jacobus Faber Stapulensis), who supplied his own demonstrations and comments in Arithmetica (Iordani Nemorarii) decem libris demonstrata (Paris, 1496, 1503, 1507, 1510, 1514). At least sixteen complete or partial MSS of it are presently known, among which are two excellent and complete thirteenth-century MSS: Paris, BN 16644, 2r-93v; and Vat. lat., Ottoboni MS 2069, 1r-51v.

The Latin text of the definitions and enunciations of the 34 propositions of the Demonstratio Jordani de algorismo were published by G. Eneström, “Über die ‘Demonstratio Jordani de algorismo,’ ” in Bibliotheca mathematica, 3rd ser., 7 (1906-1907), 24-37, from MSS Berlin, lat. 4? 510, 72v-77r (Königliche Bibliothek, renamed Preussische Staatsbibliothek in 1918; the fate of this codex after World War II, when the basic collection was divided between East and West Germany, is unknown to me) and Dresden Db 86, 169r-175r. The Demonstratio appears to be an altered version of a similar and earlier work beginning with the words “Communis et consuetus ...,” which Eneström called Opus numerorum. The Latin text of its introduction and a comparison of its propositions with those of the Demonstratio Jordani were published by Eneström as “Über eine dem Jordanus Nemorarius zugeschriebene kurze Algorismusschrift,” in Bibliotheca mathematica, 3rd ser., 8 (1907-1908), 135-153. He relied primarily on MS Vat. lat. Ottob. 309, 114r-117r, supplemented by MSS Vat. lat. Reg. Suev. 1268, 69r-71r; Florence, Bibl. Naz. Centr., Conv. Soppr. J.V. 18 (cited by Eneström as San Marco 216, its previous designation), 37r-39r; and Paris, Mazarin 3642, 96r and 105r. Since the Demonstratio Jordani was definitely ascribed to Jordanus, and the Opus numerorum seemed an earlier version of it, Eneström conjectured that the Opus was a more likely candidate for Jordanus' original work, while the Demonstratio Jordani, which omits most of the introduction but expands the text itself, may have been revised by Jordanus or someone else.

Each of these two treatises has associated with it a brief work, attributed in some MSS to Jordanus, on arithmetic operations with fractions. The treatise associated with the Opus numerorum, which Eneström calls Tractatus minutiarum, contains an introduction and 26 highly abbreviated propositions; the work on fractions associated with the Demonstratio Jordani de algorismo, which Eneström calls Demonstratio de minutiis, consists of an introduction and 35 propositions. Although the introductions differ, all 26 propositions of the Tractatus minutiarum have, according to Eneström, identical counterparts in the longer Demonstratio de minutiis. In “Das Bruchrechnen des Jordanus Nemorarius,” in Bibliotheca mathematica, 3rd ser., 14 (1913-1914), 41-54, Eneström includes a list of MSS for both treatises (pp. 41-42), the Latin texts of the introductions, the texts of the enunciations of the propositions, and analytic representations of the propositions. By analogy with his reasoning about the relations obtaining between the Opus numerorum and Demonstratio Jordani de algorismo, Eneström conjectures that Jordanus is the author of the Tractatus minutiarum, the briefer treatise associated with the Opus numerorum. One of the MSS is Bibl. Naz. Centr., Conv. Soppr. J.V. 18, 39r-42v, which follows immediately after the Opus numerorum in the same codex cited above; correspondingly, MS Berlin, lat. 4? 510, 72v-77r, of the Demonstratio Jordani de algorismo is followed immediately by a version of the Demonstratio de minutiis on fols. 77r-81v, a relation which also seems to obtain in Bibl. Naz. Centr., Conv. Soppr. J.I. 32, 113r-118v, 118v-124r. Whether the two algorithm treatises and the two associated treatises on fractions bear any relation to works (5), (6), (7), or (10), cited above from the Biblionomia, has yet to be determined and may, indeed, be impossible to determine. The Algorismus demonstratus published in 1534 by J. Schöner and formerly ascribed to Jordanus, was composed by a Master Gernardus, who is perhaps identical with Gerard of Brussels.

The Liber de proportionibus, mentioned in the Biblionomia, is probably a brief work by Jordanus beginning with the words “Proportio est rei ad rem determinata secundum quantitatem habitudo ....” A seemingly complete MS of it is Florence, Bibl. Naz. Centr., Conv. Soppr. J.V. 30, 8r-9v. Other MSS are listed in L. Thorndike and P. Kibre, A Catalogue of Incipits of Mediaeval Scientific Writings in Latin, rev. ed. (Cambridge, Mass., 1963), col. 1139. The Suppletiones plane spere of the Biblionomia is probably a commentary on Ptolemy's Planisphaerium. According to G. Sarton, Introduction to the History of Science, 3 vols. in 5 pts., II, pt. 2 (Baltimore, 1931), 614, it is “a treatise on mathematical astronomy, which contains the first general demonstration of the fundamental property of stereographic projection--i.e., that circles are projected as circles (Ptolemy had proved it only in special cases).” In Thorndike and Kibre, op. cit., Jordanus' Planisphaerium is listed under three separate and different incipits (see cols. 1119, 1524, and 1525, where MSS are listed for each). An edition appeared at Venice in 1558, under the title Ptolemaei Planisphaerium: Iordani Planisphaerium; Federici Commandi Urbinatis in Ptolemaei Planisphaerium commentarius. A work on isoperimetric figures, De figuris ysoperimetris, is also attributed to Jordanus: MSS Florence, Bibl. Naz. Centr., Conv. Soppr. J.V. 30, 12v (a fragment) and Vienna 5203, 142r-146r, the latter actually copied by Regiomontanus, who was also acquainted with Jordanus' De triangulis, Planisphaerium, Arithmetica, De numeris datis, and De proportionibus; the enunciations of the eight propositions in the Vienna MS were published by Maximilian Curtze, “Eine Studienreise,” in Zentralblatt für Bibliothekswesen, 16 (1899), 264-265.

II. SECONDARY LITERATURE.

The most significant studies on Jordanus are monographic in character and have been cited above, since they are associated with editions and translations of his works. No general appraisal and evaluation of his scientific works has yet been published. To what has already been cited, the following may be added: O. Klein, “Who Was Jordanus Nemorarius?,” in Nuclear Physics, 57 (1964), 345-350; Benjamin Ginzberg, “Duhem and Jordanus Nemorarius,” in Isis, 25 (1936), 341-362,