Medieval Hebrew texts on the quadrature of the lune (Q1910916)

The problem of the quadrature of the lune, and of course that of the circle, are problems that attracted the attention of mathematicians since antiquity. Langermann documents in this article the persistence of this interest well into medieval and renaissance times in the Hebrew scientific tradition, which was itself widely thought to be derivative from the Arabic tradition. In this particular instance although the author maintains that the said texts that he is studying have no parallels in the Arabic or the Latin tradition, he cautions that they may indeed derive from Arabic texts that are no longer extant. In a tangential way this article also touches upon one crucial aspect of the story of transmission of scientific ideas from the world of Islam to the Latin west, and specifically regarding the story of the transmission of astronomical theorems from Arabic into the works of Copernicus. While discussing the work of some Alfonso, otherwise identified as Abner of Burgos (ca. 1270-1340), on the asymptotes the author notes that this Alfonso does not only exhibit knowledge of Nichomedes' construction but that he proposes a theorem demonstrating a device that has come to be known in the literature as the ``Ṭūsī Couple'', defined quickly by Langermann as being ``two tangential circles, one inside the other, whose motion is such that a point on the smaller one oscillates rectilinearly.'' He goes on to say that ``the device was introduced into late Islamic astronomy in an attempt to meet some of the philosophical objections to the Ptolemaic models and was exploited as well by Copernicus in his De Revolutionibus.'' Current research is still trying to determine how could Copernicus have possibly known about the workings of such a device. In his attempt to shed some light on the problem Langermann notes that Alfonso adduces this theorem in a mathematical context where he was proposing ``to construct a continuous and unending rectilinear motion, back and forth along a finite straight line,without resting when reversing direction''. I would say that this indeed is philosophical context rather than mathematical, and the point being made is whether a body changing direction of motion will have to come to rest at the end of one motion and before the opposite motion begins. Langermann then goes on to assert that the same theorem of Alfonso was known to Mordecai Finzi, ``a Jewish savant active in Mantua and its environs in the mid 15th century''. As far as the contacts with Copernicus is concerned Langermann asserts that almost a full half century before the arrival of Copernicus into northern Italy the savants of that region were already cognizant of the workings of the Ṭūsī Couple through Finzi. Although the present reviewer is sympathetic to this line of argumentation, and is fully convinced, due to many other reasons, that the works of Arabic astronomers were known in Italy towards the end of the fifteenth century and during the time of Copernicus's sojourn in that country towards the beginning of the sixteenth, he still thinks that the origin of the Ṭūsī Couple, as is now well documented, is not to be sought in the philosophical works relating to the reversal of motion mentioned above but in the work on Ptolemy's planetary latitude theory which requires such an oscillatory motion irrespective whether the moving element comes to rest in between the two opposite motions or not. The theorem as was first proposed by Ṭūsī (d. 1274) was specifically designed to answer this specific problem, and only later did commentators on the text of Ṭūsī notice that the same theorem could be used as an illustration to two main Aristotelian tenets, namely that for motion to reverse direction the body has to come to rest and that circular motion is distinct from rectilinear motion and pertains to the heaven only while rectilinear motion pertains to the sublunar region. As was already noted by Willy Hartner almost thirty years ago, the revolutionary character of the Ṭūsī\ Couple lies in the fact that it abolished that Aristotelian dichotomy by demonstrating that circular motion can be generated from rectilinear motion and vice versa.