内容概述
故事取材于莫斯科纸草卷（约公元前1850年）中的第14题：已知一平顶（截顶）金字塔的垂直高为6，下底边长为4，上底边长为2，按古埃及书吏给出的步骤计算其体积，结果为56。作者指出，这实际上等价于现代截头方锥体积公式 V = (1/3)h(a² + ab + b²)。E. T. Bell 因此称这一发现为“最伟大的埃及金字塔”，因为其归纳发现可能比任何实体金字塔的建造更令人惊叹。
中文正文
莫斯科纸草卷是公元前1850年的古埃及文书，包括25个数学问题，我们发现下面一个计算的例子：
告诉你：兹有一平顶金字塔，高为6，底边为4，顶边为2。你须将4平方，得16；将4加倍，得8；将2平方，得4；将这些数字加起来，结果28。你须将6三分，得2。然后将28加倍，得56。看，结果是56。你可以发现它是正确的。
现在我们知道，古埃及人造的都是四方的金字塔，这个例子当然说明了如下公式的应用：
V = (1/3)h(a² + ab + b²)
它计算一个高为h、底边分别为a和b的平顶金字塔的体积。在上面的例子中，h=6，a=4，b=2。问题里的步骤，等于把这些数值逐个带入那公式，然后得到正确的答案，体积为56。假如
（中文 PDF 第33页末尾于此处截断，下文据英文原文补齐来源信息，不补入中文正文。）
英文正文
15° The greatest Egyptian pyramid. In the Moscow papyrus, an ancient Egyptian text dating from about 1850 B.C. and consisting of twenty-five mathematical problems, we find the following numerical example:
If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. You are to take 28 twice, result 56. See, it is 56. You will find it right.
Now if we recall that the ancient Egyptians concerned themselves with only square pyramids, this certainly seems to be a numerical example illustrating the use of the formula
V = (1/3)h(a² + ab + b²)
for computing the volume of a frustum of a square pyramid of height h and with bases of sides a and b. In the example, h = 6, a = 4, b = 2. The instructions of the problem carry us step by step through the substitution of these values for a, b, h in the above formula, resulting in the correct answer of 56 for the volume of the frustum. If this interpretation is the right one (and it is difficult to imagine that it is not), then we must concede to the ancient Egyptians the extraordinary achievement of the discovery of the formula. Now any rigorous derivation of the formula requires some form of the integral calculus, and so the discovery must have been an inductive, or empirical, one. No other unquestionably genuine example of this formula has been found in pre-Hellenic mathematics, and several involved conjectures have been furnished to explain how it might have been empirically discovered. It appears to be such a magnificent piece of induction that the historian of mathematics E. T. Bell has aptly referred to this early Egyptian example as "the greatest Egyptian pyramid," since the inductive discovery involved is perhaps more remarkable than the physical construction of any of the great stone pyramids of antiquity.
英语学习拓展
常用词汇
truncated音标/ˈtrʌŋkeɪtɪd/
截顶的，截断的。
原文用法：A truncated pyramid of 6 for the vertical height。
vertical音标/ˈvɜːtɪkl/
垂直的。
原文用法：the vertical height。
base音标/beɪs/
底边，底面。
原文用法：4 on the base by 2 on the top。
square音标/skweə(r)/
平方；正方形。
原文用法：You are to square this 4, result 16。
double音标/ˈdʌbl/
加倍。
原文用法：You are to double 4, result 8。
frustum音标/ˈfrʌstəm/
截头锥体。
原文用法：the volume of a frustum of a square pyramid。
substitution音标/ˌsʌbstɪˈtjuːʃn/
代入。
原文用法：the substitution of these values for a, b, h in the above formula。
interpretation音标/ɪnˌtɜːprɪˈteɪʃn/
解释，解读。
原文用法：If this interpretation is the right one。
concede音标/kənˈsiːd/
承认，让步。
原文用法：we must concede to the ancient Egyptians the extraordinary achievement。
rigorous音标/ˈrɪɡərəs/
严格的。
原文用法：any rigorous derivation of the formula。
derivation音标/ˌderɪˈveɪʃn/
推导。
原文用法：any rigorous derivation of the formula。
inductive音标/ɪnˈdʌktɪv/
归纳的。
原文用法：an inductive, or empirical, one。
empirical音标/ɪmˈpɪrɪkl/
经验的。
原文用法：an inductive, or empirical, one。
magnificent音标/mæɡˈnɪfɪsnt/
壮丽的，了不起的。
原文用法：such a magnificent piece of induction。
aptly音标/ˈæptli/
恰当地。
原文用法：has aptly referred to this early Egyptian example as ...
同义词与近义词
对应核心词汇：truncated
对应核心词汇：truncated。
truncated 是数学/几何专用词，指“截去顶部”；cut off 更口语；shortened 仅指缩短。
对应核心词汇：inductive、empirical
对应核心词汇：inductive。
inductive 强调从特殊到一般的推理；empirical 强调基于观察和经验。
原文把两者并列，说明古代发现兼具这两种性质。
对应核心词汇：rigorous
对应核心词汇：rigorous。
rigorous 常用于数学和逻辑，指“严格、严密”；strict 强调规则严格；exact 强调精确。
常用短语与固定搭配
- be concerned with：关注，研究。原文用法：the ancient Egyptians concerned themselves with only square pyramids。
- carry sb. through：引导某人完成。原文用法：The instructions of the problem carry us step by step through the substitution。
- step by step：逐步地。原文用法：carry us step by step through the substitution。
- it is difficult to imagine that：难以想象。原文用法：it is difficult to imagine that it is not。
- refer to ... as ...：把……称为……。原文用法：referred to this early Egyptian example as "the greatest Egyptian pyramid"。
常用句式
- Pattern: If you are told ..., you are to ...：如果告诉你……，你应当……。原文例句：If you are told: A truncated pyramid of 6 ..., You are to square this 4, result 16.
- Pattern: If this ... is the right one (and it is difficult to imagine that it is not), then ...：如果这个解释是正确的（而且很难想象它不是），那么……。原文例句：If this interpretation is the right one ..., then we must concede to the ancient Egyptians ...
- Pattern: ... such ... that ...：如此……以至于……。原文例句：It appears to be such a magnificent piece of induction that ...
长难句解析
Now any rigorous derivation of the formula requires some form of the integral calculus, and so the discovery must have been an inductive, or empirical, one.
- 主干：any rigorous derivation requires some form of the integral calculus, and so the discovery must have been an inductive, or empirical, one（任何严格推导都需要某种形式的积分学，因此这一发现必然是归纳的或经验的）。
- 修饰与从句：of the formula 修饰 derivation；some form of the integral calculus 为宾语；and so 引导结果分句；must have been 表示对过去情况的推测。
- 逻辑关系：前一句说明“严格推导需要微积分”，后一句用 and so 引出结论——古人没有微积分，因此只能是归纳/经验发现。
语言使用特点
- 文本大量使用数学史评价的措辞，如 extraordinary achievement, rigorous derivation, magnificent piece of induction。
- 使用 must have been 等情态动词表示对历史事实的推测。
- 引语和公式并用，体现学术说明的风格。
迁移练习
- 用 step by step 描述解一道数学题的过程。
- 用 refer to ... as ... 介绍一个你熟悉的数学家或数学定理。
- 将 Any rigorous derivation of the formula requires some form of the integral calculus. 改写成 It is ... that ... 强调句。
点击展开参考答案
- （示例）To solve the equation, I followed the instructions step by step: first I moved the constant term, then I divided both sides by the coefficient.
- （示例）Many people refer to Archimedes as one of the greatest mathematicians of antiquity.
- （示例）It is some form of the integral calculus that any rigorous derivation of the formula requires.