The following article by Professors Menso Folkerts and Richard Lorch, from Munich University in Germany, describes the influences of Arabic sciences in the works of Jordanus de Nemore, a scholar who flourished in Western Europe in the 13th century.
The article is devoted to describing the works of Jordanus de Nemore, a European scholar of the 13th century, to whom various scientific works are ascribed in the fields of mathematics, astronomy and theoretical mechanics. Several of these texts reveal clear and strong influences from Arabic scientific works, many of which were translated into Latin in the 12th century.
Historians of mediaeval mathematics agree that Jordanus de Nemore was one of the most important writers on mechanics and mathematics in the Latin West – to be compared only with Leonardo Fibonacci and Nicole Oresme. But almost nothing is known about his life. He must have lived before the middle of the 13th century, because Jordanus’ works are mentioned in Richard de Fournival’s Biblionomia, a catalogue of books compiled towards 1250, and because Campanus cites Jordanus in his redaction of Euclid’s Elements which must have been written before 1259.
Jordanus de Nemore is the name given in manuscripts of the 13th and 14th centuries to a mathematician who in the Renaissance period was called Jordanus Nemorarius. A number of his works are extant, but nothing is known of his life. It is customary to place him early in the 13th century. Michel Chasles, the geometer, concluded from a study of the Algorismus Jordani that its author lived not later than the 12th century. In the 14th century the English Dominican Nicholas Triveth, in a chronicle of his order, attributed the De ponderibus Jordani and the De lineis datis Jordani to Jordanus Saxo, who, in 1222, succeeded St. Dominic as master general of the Friars Preachers. Since then, the identity of Jordanus Saxo with Jordanus Nemorarius has been accepted by a great many authors; it seems difficult to maintain this opinion, however, as the Dominican superior general never adds de Nemore to his name, and the mathematician never calls himself Saxo.
The treatises attributed to Jordanus de Nemore are:
(1) Liber philotegni, is an advanced textbook on geometry that appears to be a genuine work by Jordanus; it was reworked under the name Liber de triangulis Iordani.
(2) Elementa de ponderibus: There are some Latin treatises on statics in the manuscripts attributed to Jordanus, in which the dynamical approach of Aristotelian physics is combined with the abstract mathematical physics of Archimedes, the proofs being presented in the Euclidean way. But only one treatise, the Elementa super demonstrationem ponderum or Elementa de ponderibus, may be definitely assigned to him; other treatises are based upon Greek works that were mostly transmitted through the Arabic and upon Arabic works – for instance, the Liber karastonis, which is the Latin translation by Gerard of Cremona of Kitab fi’l-qarastun by Thabit ibn Qurra.
(3) The algorismus treatises are various algorismus treatises ascribed to Jordanus, not yet edited in their entirety, and only two of them might have been written by Jordanus. The treatises that seem to have been written by Jordanus teach the six basic operations with integers (including duplation and mediation) and the extraction of the square root within the Arabic number system, but without examples and in a more formal way than in the common algorismus treatises of the 13th century (Johannes de Sacrobosco, Alexander de Villa Dei). All this is strongly reminiscent of Arabic texts (which begin with al-Khwarizmi’s Arithmetic).
4. De numeris datis: In this treatise, Jordanus solved algebraic problems in a way different from those found in Arabic texts. He formulated problems by saying what is known and what has to be found, and then transformed the initial equation into canonical form by using letters to represent numbers. At the end of every problem he gives a numerical example. Although some bits and pieces can be found in other works, the whole is not a compilation, but a unique tract in advanced algebra. As for the sources, the approach is too different from that of al-Khwarizmi for the latter’s Algebra to have been the decisive influence – and in general, we have found no telling evidence of any Arabic source for this work.
5. De plana spera: This treatise, existing in three versions, may be compared with Ptolemy’s Planisphaerium. It treats the principles of stereographic projection – the central concept used in constructing the astrolabe – and gives inter alia a general demonstration of its fundamental property, i.e. that circles are projected as circles.
6. De elementis arismetice artis: This text, the most widely known mathematical work of Jordanus, was edited in its original form only in 1991. It is divided into ten books and comprises more than 400 propositions. As in Euclid’s Elements – and, it seems, derived from it – Jordanus starts with definitions and postulates and then proceeds to the enunciations.
Finally, (7) Liber de proportionibus is dubiously ascribed to Jordanus and it is not clear whether it is an original work or a translation from an original Arabic text by Thabit ibn Qurra. At any rate, it reflects a clear Arabic influence.
Figure 1: Extract from Jordanus’ De planisphaeri figurationei (Source).
Instances of the scientific Arabic influence on Jordanus
In some of his quotation of Euclid’s Elements in Liber philotegni, it is evident that Jordanus’ source was a text that came from the Arabic, because he mentions twice the word mutekefia (= reciprocally proportional), which is also given, with the same meaning, in propositions VI.13 and 14, by some Latin texts of the Elements based on Arabic sources such as the Latin translation of the Elements from Arabic by Adelard of Bath and Hermann of Carinthia. It should also be mentioned that one of the earliest manuscripts of the Liber philotegni cites (prop. 28) the Pythagorean theorem by per dulk. The term dulcarnon (= the two-horned) for this theorem came from Arabic texts. It was first used in the West in some manuscripts of the Elements and became later very common.
Likewise, it seems that Jordanus also used in Liber philotegni another treatise by Euclid: the Liber divisionum. Today this text is available only in Arabic. In the 12th century it was translated into Latin by Gerard of Cremona, but his translation is lost. In propositions 21-23 of the Liber philotegni Jordanus presents problems on the division of triangles, and it is very likely that he used Gerard’s translation of Euclid’s Liber divisionum.
Furthermore, Jordanus cites in his Liber philotegni a treatise, Liber de similibus arcubus, whose author was Ahmad b. Yusuf b. Ibrahim ibn al-Daya, who lived in the second half of the 9th century in Egypt.
De plana spera offers other instances of the influence of the Arabic scientific heritage on Jordanus’ work, as it makes use of properties found in Ptolemy’s Planisphaerium as well as in a text written by Maslama al-Majriti, the well-known Andalusian astronomer and scholar. This text is extant in Arabic, and it was translated into Latin in the 12th century. It is almost certainly the ultimate source, if not the immediate source, of Jordanus’ treatise.
Figure 2: Jordanus de Nemore, Liber de ratione ponderis in the edition of Nicolo Tartaglia (Venice, 1565) (Source).
In the same vein, the sources of De elementis arismetice artis include Arabic texts. Because Jordanus does not cite any author in his Arithmetica – except Boethius’ Arithmetica –, we are only able to list the propositions of Jordanus that can also be found in earlier texts and therefore might have been taken from them. Among these we can quote Al-Nayrizi’s commentary of Euclid and Ahmad b. Yusuf’s De proportione et proportionalitate, which were translated into Latin by Gerard of Cremona.
Jordanus was one of the few mathematicians of the Latin Middle Ages who showed any originality. He had also a strong inclination to rework the material that came to hand. None the less, it is possible to trace many of the ideas in his works to his predecessors, in particular to the translations from the Arabic in the 12th century. All his major works were reworked, often more than once. It is remarkable that in many cases yet more material derived from the Arabic finds its way into the reworked texts, and this material is often more easily recognized, because it is more often supplied with the name of the source or because the style is less transformed. In the works by, or attributed to, Jordanus, which formed a large part of mathematics in the West from the founding of the universities until the Renaissance, we find a wonderful repository of mathematical learning transmitted from the rich Arabic heritage.