Euclid discusses the ex aequali relationship twice in the Elements. The first is in Book V (based on definitions 17 and 18, propositions 22 and 23), during his discussion of arithmetical relations between mathematical magnitudes in general. The second is in Books VII–IX (developed using proposition VII,14), where he focuses on arithmetical relations in the case of numbers only. Although the distinction between mathematical magnitudes in general and numbers in particular often seems somewhat forced to contemporary philosophers, it was apparently very real to Euclid. Because Euclid seemed so conscious of the differences between the subject matter of Book V (magnitudes) and Books VII–IX (numbers), he was not much troubled by the differences between his treatment of ex aequali ratios in these two contexts. Later generations of mathematicians, however, found these differences less acceptable and tried to minimize them in various ways. This paper summarizes Euclid's use of the ex aequali relation in developing his mathematics. The paper then outlines the fate of the post-Theonine Greek attempts to “improve” the Euclidean discussion when the Elements entered the Arabic/Islamic intellectual tradition. The study concludes with the attempts by Ibn al-Hayṯam and Ibn al-Sarī to improve the parallelism between the discussions of ex aequali ratios in Book V and Book VII.

The editing of three anonymous Greek texts preserved in the Parisinus Suppl. gr. 643 allows us to clarify certain ideas on the transmission of knowledge in the Mediterranean during the second half of the 13th century. These texts – an introduction to the Physics of Aristotle, one to De generatione et corruptione and a page of Medical Problems – are in fact translations from Latin probably made at Salerno at the end of the Norman period or at the beginning of the Angevin dynasty. They allow us to establish the influence of the Parisian Faculty of Arts on the Sicilian intellectual milieu of the period and to illustrate how, whilst remaining true to its medical vocation, the University of Salerno evolved nonetheless towards a model of general education in the Arts. Finally these texts reveal the considerable influence – both philological and doctrinal – of Arabic learning on the Aristotelian teaching of their author. This very fact, combined with the presence of the Parisinus in Byzantium, in an environment of advanced philological learning, a few decades after its composition, leads us to question our understanding of the Palaeologan Renaissance as well as its independence with regard to the Arabo-latin scholarly tradition of the 13th century.

There are three principal philosophical-scientific encyclopaedias written in Hebrew during the Middle Ages: Yehudah ha-Cohen's Midrash ha-Ḥokmah (1245–1247), Shem Tov ibn Falaquera's De'ot ha-Filosofim (ca. 1270) and Gershon ben Shlomoh's Sha'ar ha-Shamayin (end of the 13th century). All three include detailed treatments of zoology, and the last two of botany and mineralogy as well. The principal feature of their treatments is their “theoretical” – not merely “descriptive” – approach: these encyclopaedias do not contain only lists of stones, plants and animals (such as other Arabic and Latin Medieval encyclopaedias), but also attempts at systematization and philosophical arrangement of the various available theories in the fields of mineralogy, botany and zoology qua sciences. An examination of the doctrines and the sources of these texts shows that, while the treatment of zoology relies upon Aristotle's zoological works and, above all, their Compendia by Averroes, the treatment of mineralogy and botany reflects the non-Aristotelian theories of the Brethren of Purity (Iḫwān al-Ṣafā'), rather than such texts as pseudo-Aristotle's De lapidibus and Nicolaus Damascenus' De plantis. In particular, Falaquera's encyclopaedia represents the most convincing effort to provide a truly scientific discussion of mineralogy and botany, comparable to that of his contemporary Albert the Great, and based upon the Brethren, Avicenna and, maybe, some lost works by Averroes.