Another look at the two Egyptian pyramid volume ‘formulas’ of 1850 BCE

Abstract

This paper provides some methodological, didactical, and historiographical reflections on Egyptian pyramid volume formulas, responding to suggestions by Paul Shutler from 2009. These suggestions partly reiterate a historically documented proof by the Chinese Liu Hui (third century CE), although Lui Hui’s contribution was apparently unknown to Shutler. The latter came forward, in addition, with intuitive arguments which might have been used by the Egyptians to convince themselves of the correctness of their formula for the volume of the full pyramid. In a broad sense, the reflections in this paper may contribute to the use of history in the mathematical classroom. As a cautionary note: The paper is an abridged version of a longer manuscript that contains detailed explanations and discussions of historical secondary sources. Since the paper is somewhat outside the usual canon of mathematics historiography, I have deposited the longer manuscript on <arXiv.org>.

2020 Mathematics Subject Classification:

• Primary 00A35
• 01A16
• Secondary 97-03

Acknowledgments

I am grateful to the following colleagues and friends for reading earlier versions of this manuscript and giving helpful advice: June Barrow-Green (London), Reinhard Bölling (Berlin), Karine Chemla (Paris), Joseph Dauben (New York), Simon Goodchild (Kristiansand), Christopher Hollings (Oxford), Jesper Lützen (Copenhagen), Nathan Sidoli (Tokyo), and Donald Wagner (Copenhagen). I thank the editor of BJHM for supporting this publication which is somewhat on the border between history, methodology, and pedagogy.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 I call them in this paper summarily ‘historians around 1930’, being aware that among them were Egyptologists and mathematicians with different aims and competences. Imhausen refers – apparently with some critical intent – to several of these works in footnote 21 of her book (Imhausen 2016, 5).

2 In the full manuscript (Siegmund-Schultze 2022) I revisit Lui Hui’s proof, but I will not do so here. The proof is presented and commented on in detail in (Wagner 1979) and (Chemla and Guo 2004, 387–395, 817–818.)

3 Because the factor h/3 is applied at the end of the algorithm expressed by formula F_T and therefore here written in the beginning, the interpretations as an average of area or volume seem equally plausible. In the Chinese version of the formula, the factor 1/3 is applied separately at the very end of the algorithm. This makes its interpretation as an average of volumes very likely from the outset.

4 Throughout this paper, I consider ‘proof’ generally as a rational argument which contributes to the conviction of correctness of some mathematical claim and is historically variable. See Chemla (2012).

5 Interestingly, Liu Hui derives formula F_TA additionally, but he is content with it and does not use it at all to derive the classical formula F_T once again.

6 In the course of this debate one of the most famous historians of ancient mathematics, Otto Neugebauer, was at times accused of unduly insinuating a hidden algebraic agenda among ancient mathematicians. See the most recent contribution (Blåsjö 2016), which recognizes the dangers of projecting modern algebraic methodology on the past but also acknowledges the relative merits of such procedure in the historical reconstruction of ancient geometry.

7 More details in (Siegmund-Schultze 2022).

8 The famous historian Otto Neugebauer, who started his historical work with Egyptian mathematics before 1930, assumed without any explanation that the Egyptians were in possession of Cavalieri’s principle which would have allowed them to compare the volumes of symmetric and oblique pyramids (Neugebauer 1934, 128). Shutler in his ‘speculation’ assumes much less because his argument is based on two symmetrical (!) pyramids which are in a very intuitive sense ‘similar’.

9 The present author goes as far as accepting even an occasional use of counterfactual history, which is of course not the case in the present example because we cannot rule out that the Egyptians used exactly this argument.