直观的，教学上有用的，历史上可能的证明两个埃及金字塔体积公式（1850年BCE）。关于数学史与数学教学论边界的思考
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
摘要：
1930年前后的数学家和数学史学家做了一项令人钦佩的工作，证明了公元前1850年左右新发现的莫斯科纸莎草纸的第14个问题相当于对截头金字塔（平截头体）体积的一般和精确计算。他们不太成功地给尝试性的解释是什么说服埃及人的正确性。特别是，他们从来没有考虑过将三个相同的截头锥体复制品解剖并将其重新组合成三个大小不同的盒子的可能性，这些盒子的体积可以很容易地计算出来。这是令人惊讶的，因为历史学家得出的公式似乎暗示了这一程序。在埃及人之后大约2000年，中国学者刘晖正是这样做的，因为九章中几乎相同的问题。如果这些数学史学家在1930年左右已经知道刘晖的算法，他们可以很容易地得出初步结论，也为埃及的情况。本文认为，这是他们的知识，严格的欧几里德几何和相对现代的代数扭曲了这些历史学家的判断，这不可能是刘晖的情况。他们的失败似乎使后来的考古学家不愿寻找解释，甚至在刘徽的工作为人所知时也不愿指出他。因此，一个机会被错过了使用伟大的直觉和教育潜力的一个显着的一块埃及数学。本文不是对埃及数学史学的贡献，作者不是专家，但主要是在方法论层面上使用次要历史资料。这篇论文的部分灵感来自于Paul Shutler（2009）最近发表的一篇文章，他提出了一个直观的和历史上可能的证明，同样适用于第二个公式的特殊和数学上至关重要的情况，这与完整的金字塔有关。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
