Intuitive, didactically useful, and historically possible proofs for the two Egyptian pyramid volume formulas (1850 BCE). Thoughts on the border between history and didactics of mathematics

Abstract

Egyptologists and historians of mathematics around 1930 did an admirable job in showing that problem 14 of the newly discovered Moscow Papyrus from around 1850 BCE amounts to a general and exact calculation of the volume of a truncated pyramid (frustum). They were less successful in giving tentative explanations of what convinced the Egyptians of its correctness. In particular, they never looked into the possibility of dissecting three identical copies of the frustum and recomposing them into three boxes of differing sizes whose volumes can be easily calculated. This is surprising because the formula at which the historians arrived seems to suggest this procedure. About 2000 years after the Egyptians, the Chinese scholar Liu Hui did exactly this for the almost identical problem from the Nine Chapters. If those historians of mathematics around 1930 had known Liu Hui's algorithm they could have easily drawn tentative conclusions also for the Egyptian case. The present paper suggests that it was their knowledge of rigorous Euclidean geometry and of relatively modern algebra which distorted the judgment of those historians, something which could not have been the case for Liu Hui. Their failure seems to have discouraged later Egyptologists to look for explanations or to even point to Liu Hui when his work had become known. Thus a chance was missed to use the great intuitive and pedagogic potential of a remarkable piece of Egyptian mathematics. This paper is not a contribution to the historiography of Egyptian mathematics for which the author is no specialist but argues primarily on a methodological level using secondary historical sources. The paper is partly inspired by a more recent publication of Paul Shutler (2009) who suggests an intuitive and historically possible proof also for the special, and mathematically crucial case of the second formula, which is related to the full pyramid.