More Archimedean than Archimedes: A New Trace of Abū Sahl al-Kūhī’s Work in Latin

In 1661, Borelli and Ecchellensis published a Latin translation of a text which they called the Ltmmas of Archimedes. The first fifteen propositions of this translation correspond to the contents of the Arabic Book of Assumptions, which the Arabic tradition attributes to Archimedes. The work is not found in Greek and the attribution is uncertain at best. Nevertheless, the Latin translation of the fifteen propositions was adopted as a work of Archimedes in the standard editions and translations by Heiberg, Heath, Ver Eecke and others. Our paper concerns the remaining two propositions, 16 and 17, in the Latin translation by Borelli and Ecchellensis, which are not found in the Arabic Book of Assumptions. Borelli and Ecchellensis believed that the Arabic Book of Assumptions is a mutilated version of a lost “old book” by Archimedes which is mentioned by Eutodus (ca. A.D. 500) in his commentary to Proposition 4 of Book 2 of Archimedes' On the Sphere and Cylinder. This proposition is about cutting a sphere by a plane in such a way that the volumes of the segments have a given ratio. Because the fifteen propositions in the Arabic Book of Assumptions have no connection whatsoever to this problem, Borelli and Ecchellensis “restored” two more propositions, their 16 and 17. Propositions 16 and 17 concern the problem of cutting a given line segment AG at a point X in such a way that the product $$AX\;\cdot\;XG^2$$ is equal to a given volume K. This problem is mentioned by Archimedes, and although he promised a solution, the solution is not found in On the Sphere and Cylinder. In his commentary, Eutodus presents a solution which he adapted from the “old book” of Archimedes which he had found. Proposition 17 is the synthesis of the problem by means of two conic sections, as adapted by Eutodus. Proposition 16 presents the diorismos: the problem can be solved only if $$K\leqslant AB\;\cdot\;BG^2$$ , where point B is defined on AG such that $$AB\;=\; {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}BG $$ We will show that Borelli and Ecchellensis adapted their Proposition 16 not from the commentary by Eutocius but from the Arabic text On Filling the Gaps in Archimedes’ Sphere and Cylinder which was written by Abū Sahl al-Kūhī in the tenth century, and which was published by Len Berggren. Borelli preferred al-Kūhī’s diorismos (by elementary means) to the diorismos by means of conic sections in the commentary of Eutocius, even though Eutocius says that he had adapted it from the “old book.” Just as some geometers in later Greek antiquity, Borelli and Ecchellensis bdieved that it is a “sin” to use conic sections in the solution of geometrical problems if elementary Euclidean means are possible. They (incorrectly) assumed that Archimedes also subscribed to this opinion, and thus they included their adaptation of al-Kūhī’s proposition in their restoration of the “old book” of Archimedes. Our paper includes the Latin text and an English translation of Propositions 16 and 17 of Borelli and Ecchellensis.