A Mechanical Derivation of the Area of the Sphere

1. INTRODUCTION. In the beginning of the twelfth century CE, an interesting new geometry book appeared: The Book of Mensuration of the Earth and its Division, by Rabbi Abraham Bar Hiya (acronym RABH), a Jewish philosopher and scientist. This book is interesting both historically and mathematically. Its historical aspect is discussed in [3]. In this paper we consider the mathematical aspect. The second part of the book contains a beautiful mechanical derivation of the area of the disk [5, §95]. Roughly speaking, the argument goes as follows (see Figure 1): The disk is viewed as the collection of all the concentric circles it contains. If we cut