This article presents the Arabic text of al-Q's short geometric treatise, The Book of the centres of the tangent circles on lines, by the method of analysis, with translation and a mathematical commentary. In this treatise al-Q solves by analysis an ordered set of eight problems where the goal is to locate, on a given line, the centre of a circle which is tangent to two given elements, which may be points, straight lines or circles. For example, in the first (...) problem he wanted to find the centre of a circle which contains two given points. In fact, al-Q establishes the locus of the solutions, and this locus can be a straight line or a conic. The treatise is reminiscent of one of the purposes of al-Q's work on the perfect compass, namely to assign to the conics the status of regular lines. (shrink)

RésuméDans son traité intituléLe Planisphère, Ptolémée présente une méthode pour représenter une sphère sur un plan, selon des principes compatibles avec ce qu'on nomme aujourd'hui la projection stéréographique. Mais cette dernière ne sera traitée mathématiquement, en tant que telle, que bien plus tard, au IXẹsiècle, dans l'œuvre d'al-Farghānī qui démontrera notamment la propriété fondamentale de cette projection. Ce n'est qu'au Xesiècle qu'al-Qūhī et Ibn Sahl écriront une première théorie générale des projections de la sphère. Cet article analyse les raisons qui (...) permettent d'affirmer que Ptolémée n'a pas conçu mathématiquement de projection, et que cette conception n'a pu être possible qu'avec l'aide de la définition de la surface conique telle qu'Apollonius l'a exposée dans sesConiques. (shrink)

The present article is devoted to two issues. The first is the identification of lead and tin in medieval Arabic alchemy. The second is the investigation of whether Arabic alchemists differentiate between these problematic substances or not. These two issues are investigated in the light of a comparison which is made between the facts that are stated about the two problematic substances in the original Arabic alchemical works and those stated in modern chemical literature. It is proved that Arabic alchemists (...) made a sharp distinction between lead and tin. Also, it becomes clear that these two metals were used in a satisfactory purity in the era of medieval Arabic alchemy. As a consequence of the present study, some conclusions are drawn about the existence of some categories of ‘derivatives’ of fusible bodies in Arabic alchemy which are degenerate to modern categories of oxides, carbonates, etc. (shrink)

The development of geometrical analysis in the 10th century was partly inspired by the reception of the works of Apollonius, which Arab mathematicians translated as early as the preceding century. Al-Qūhī contributed to this development by writing several collections of problems dealing with Apollonian themes and solved by the method of analysis; however, it seems that they do not all occupy the same place in his work. The author gives here the edition, translation, and mathematical commentary of a short work, (...) entitled The determination of two straight lines from a point along a known angle, which presents the particularity of providing problems as geometrical lemmas to other studies. Indeed, al-Qūhī uses two of these lemmas in more complex constructions which belong to his Treatise on the art of the astrolabe. In this same treatise on the astrolabe, as well as in his Treatise on the perfect compass, this scholar also uses as lemmas several problems solved in another of his works, entitled The generation of points on straight lines according to ratios of which the terms are surfaces. This work is unfortunately lost, and all that remains of it are the traces which subsist in other treatises in which it has been used. This study seems to be necessary if one wishes to understand the organization of the work of al-Qūhī, a mathematician of the first rank who was representative of his time. (shrink)

The development of geometrical analysis in the 10th century was partly inspired by the reception of the works of Apollonius, which Arab mathematicians translated as early as the preceding century. Al-Qūhī contributed to this development by writing several collections of problems dealing with Apollonian themes and solved by the method of analysis; however, it seems that they do not all occupy the same place in his work. The author gives here the edition, translation, and mathematical commentary of a short work, (...) entitled The determination of two straight lines from a point along a known angle, which presents the particularity of providing problems as geometrical lemmas to other studies. Indeed, al-Qūhī uses two of these lemmas in more complex constructions which belong to his Treatise on the art of the astrolabe. In this same treatise on the astrolabe, as well as in his Treatise on the perfect compass, this scholar also uses as lemmas several problems solved in another of his works, entitled The generation of points on straight lines according to ratios of which the terms are surfaces. This work is unfortunately lost, and all that remains of it are the traces which subsist in other treatises in which it has been used. This study seems to be necessary if one wishes to understand the organization of the work of al-Qūhī, a mathematician of the first rank who was representative of his time. (shrink)