Yusuf al-Mu'taman ibn Hud

Ceva's theorem, discovered at the end of the 11th century by al-Mutaman.

The main work of al-Mu'taman was his Kitab al-Istikmal (Book of Perfection). This book was a compendium of the Greek mathematics of Euclid and Archimedes among others, but also contained the teachings of Thabit ibn Qurra, the Banu Musa and Ibn al-Haytham, and included some theorems and proofs not found in earlier extant sources. The book only persists as fragments from several anonymous manuscripts, not including any preface or introduction, but it is clear from the remaining content that the intention was to organize and comprehensively describe the known results in Euclidean geometry in a single self-contained work.[2] It is possible al-Mu'taman listed his sources in a now-lost introductory section, but none of the remaining fragments credit past authors or works for any of the content.[2]

The Kitab al-Istikmal was not completed but it was still seen as an important work from the eleventh-century king. The encyclopedist Ibn al-Akfani said that had the Istikmal been completed, it would have made the existing geometrical literature superfluous.[2] Ibn Aknin suggested that the Istikmal should be read by mathematicians alongside such works as the Elements of Euclid, On the Sphere and Cylinder of Archimedes, and Conics of Apollonius. Because this was not a completed piece of work it was never as widely copied or taught as the works of Euclid or Archimedes. A copy was sent to Egypt by Maimonides, and from there it spread to Baghdad in the 14th century, but did not directly influence later European mathematicians.

The Kitab al-Istikmal deals with irrational numbers, conic sections, quadrature of the parabolic segment, volumes and areas of various geometric objects, and the drawing of the tangent to a circle, among other mathematical problems. In the work appears an attempt to classify mathematics into Aristotelian categories. The classification includes a chapter for arithmetic, two chapters for geometry and two others for stereometry.

The Kitab al-Istikmal contains the first known formulation of Ceva's theorem, which only became known in Europe after 1678 from Italian geometer Giovanni Ceva's treatise De lineis rectis. It is unknown whether Al-Mu'taman discovered this theorem himself or obtained it from another source, and it is also unknown whether Ceva rediscovered the theorem independently.[3] The theorem can be stated as follows: "Let ABC be a triangle and D, E, F points on the sides BC, CA, and AB. We draw the lines AD, BE and CF. These three lines intersect at one point if and only if ${\displaystyle {\frac {AF}{FB}}={\frac {EA}{CE}}{\frac {DC}{BD}}}$.