Spherical Interpretation of Plane Geometry in Menelaus of Alexandria's Sphaerica

Menelaus’ Sphaerica can be considered as the most important classical text in the tradition of spherics books، written with the aim of the solution of problems arising in spherical astonomy. Euclid’s Elements is the the most important book on plane geometry. This article aims at a comparative study of Menelaus’s Sphaerica and Euclid’s Elements، to show that Book I of Sphaerica is an attempt to reconstruct Book I of the Elements for the case of spherical figures. We mention Menelaus’ achievements as well as the limits of his project. The topic of the congruent triangles is treated with special reference to the differences which exist between the plane and the spherical cases. We also show that the spherical counterpart of Euclid’s important theorm on the sum of the interior angles of a triangle (Euclid، I/32) has been put forward، for the first part، by Naṣīr al-Dīn al-Ṭūsī in his Recension of Menelaus’s Sphaerica.