内容概述
毕达哥拉斯学派坚信万物皆依赖于整数。当一位门徒发现 √2 不是有理数时，正方形的边与对角线不可公度的几何事实动摇了学派的比例理论和相似图形理论。这一“逻辑丑闻”甚至导致泄密者被兄弟会放逐，并为其修建坟墓。
中文正文
55° 毕达哥拉斯的哲学和几何的灾难
毕达哥拉斯怀着萨沃纳罗拉（Savonarola）的激情宣扬天下万物——没有例外——都依赖于整数。于是我们可以想象，当一个兄弟发现 √2 不是有理数——不能表达为两个整数之比，因而似乎也就与整数无关——在毕达哥拉斯的信徒们看来，该是多么恐慌的事情！这个可怕发现对应的几何事实也同样令人惊讶。因为，假如 √2 不是有理数，那么正方形的边与对角线将没有共同的单位以整数度量它们的长度。而人们凭直觉坚定地相信，任何两个线段都有共同的度量单位（哪怕很小、很小的单位），这就产生了矛盾。而整个毕达哥拉斯的比例理论和相似图形的理论都建立在这似乎显而易见的假设基础上。在这致命的打击下，毕达哥拉斯的哲学基础和许多几何功绩都摇摇欲坠了。据说，这个“逻辑丑闻”太可怕，有人试图把它遮掩起来；还有传说，第一个把秘密泄露出去的门徒，被兄弟会驱逐，他们还为他修了坟墓，就当他已经死了。
英文正文
55° Pythagorean philosophy and geometry at stake. Pythagoras preached, with all the fervor of a Savonarola, that everything—simply everything—depends upon the whole numbers. Imagine the consternation, then, within the Pythagorean ranks when some Brother discovered the devastating fact that √2 is not a rational number, that is, cannot be expressed as the ratio of two whole numbers, and therefore appears to be independent of the whole numbers. The geometrical counterpart of this discovery was equally startling, for if √2 is not a rational number, it follows that a side and diagonal of a square possess no common unit of measure that can be stepped off exactly a whole number of times into each. This contradicted the firm intuitive belief that any two line segments must possess some common unit of measure, though perhaps very, very small. Now the entire Pythagorean theory of proportion and similar figures was built upon this seemingly obvious assumption. In one fell stroke, both the basic Pythagorean philosophy and much of Pythagorean accomplishment in geometry were threatened. It is reported that so great was the "logical scandal" that efforts were made for a while to keep the matter secret, and one legend has it that the first Pythagorean to divulge the secret to outsiders was banished from the Brotherhood and a tomb was erected for him as though he were dead.
英语学习拓展
常用词汇
fervor音标/ˈfɜːvə(r)/
激情；热忱。
原文用法：with all the fervor of a Savonarola.
preach音标/priːtʃ/
宣扬；布道。
原文用法：Pythagoras preached ... that everything ... depends upon the whole numbers.
depend upon：依赖于。原文用法：同上。
consternation音标/ˌkɒnstəˈneɪʃn/
惊愕；恐慌。
原文用法：Imagine the consternation, then, within the Pythagorean ranks...
devastating音标/ˈdevəsteɪtɪŋ/
毁灭性的；令人震惊的。
原文用法：the devastating fact that √2 is not a rational number.
rational number音标/ˈræʃnəl ˈnʌmbə(r)/
有理数。
原文用法：同上。
ratio音标/ˈreɪʃiəʊ/
比；比率。
原文用法：cannot be expressed as the ratio of two whole numbers.
independent音标/ˌɪndɪˈpendənt/
独立的；无关的。
原文用法：therefore appears to be independent of the whole numbers.
geometrical音标/ˌdʒiːəˈmetrɪkl/
几何的。
原文用法：The geometrical counterpart of this discovery...
counterpart音标/ˈkaʊntəpɑːt/
对应物；对应方面。
原文用法：同上。
startling音标/ˈstɑːtlɪŋ/
令人震惊的。
原文用法：was equally startling.
diagonal音标/daɪˈæɡənl/
对角线。
原文用法：a side and diagonal of a square.
square音标/skweə(r)/
正方形；平方。
原文用法：同上。
common unit of measure：共同度量单位。原文用法：possess no common unit of measure.
step off：用步测；量出。原文用法：that can be stepped off exactly a whole number of times into each.
contradict音标/ˌkɒntrəˈdɪkt/
与……矛盾。
原文用法：This contradicted the firm intuitive belief...
intuitive音标/ɪnˈtjuːɪtɪv/
直觉的。
原文用法：同上。
line segment音标/ˈlaɪn ˈseɡmənt/
线段。
原文用法：any two line segments.
assumption音标/əˈsʌmpʃn/
假设。
原文用法：this seemingly obvious assumption.
theory of proportion：比例理论。原文用法：the entire Pythagorean theory of proportion.
similar figures：相似图形。原文用法：and similar figures.
seemingly音标/ˈsiːmɪŋli/
表面上； seemingly。
原文用法：this seemingly obvious assumption.
in one fell stroke：一举；一下子。原文用法：In one fell stroke, both ... were threatened.
threaten音标/ˈθretn/
威胁。
原文用法：同上。
scandal音标/ˈskændl/
丑闻。
原文用法：the "logical scandal".
divulge音标/daɪˈvʌldʒ/
泄露。
原文用法：the first Pythagorean to divulge the secret.
banish音标/ˈbænɪʃ/
放逐。
原文用法：was banished from the Brotherhood.
Brotherhood音标/ˈbrʌðəhʊd/
兄弟会。
原文用法：同上。
tomb音标/tuːm/
坟墓。
原文用法：a tomb was erected for him.
erect音标/ɪˈrekt/
竖立；修建。
原文用法：同上。
同义词与近义词
对应核心词汇：fervor
对应核心词汇：fervor。
辨析：fervor 强调热烈、狂热的情感；passion 泛指激情；zeal 强调热忱、狂热。
对应核心词汇：consternation
对应核心词汇：consternation。
辨析：consternation 指因突发坏消息而产生的惊慌；dismay 强调沮丧；shock 强调震惊。
对应核心词汇：devastating
对应核心词汇：devastating。
辨析：devastating 强调毁灭性打击；shattering 强调彻底粉碎；destructive 强调破坏性。
对应核心词汇：ratio
对应核心词汇：ratio。
辨析：ratio 指两个量的比；fraction 指分数；proportion 指比例关系。
对应核心词汇：counterpart
对应核心词汇：counterpart。
辨析：counterpart 指对应的人或物；equivalent 指等价物；parallel 指类似对应物。
对应核心词汇：contradict
对应核心词汇：contradict。
辨析：contradict 指与……矛盾；conflict with 指冲突；go against 指违背。
对应核心词汇：intuitive
对应核心词汇：intuitive。
辨析：intuitive 指直觉的；instinctive 指本能的；gut 为口语，指凭直觉的。
对应核心词汇：divulge
对应核心词汇：divulge。
辨析：divulge 常指泄露秘密；reveal 泛指揭示；disclose 较正式，指公开。
对应核心词汇：banish
对应核心词汇：banish。
辨析：banish 指放逐；exile 也指流放；expel 指开除、驱逐。
常用短语与固定搭配
- with all the fervor of：怀着……的全部激情。原文用法：with all the fervor of a Savonarola.
- depend upon：依赖于。原文用法：everything ... depends upon the whole numbers.
- the whole numbers：整数。原文用法：同上。
- within the ranks：在……内部。原文用法：within the Pythagorean ranks.
- the devastating fact that ...：……的惊人事实。原文用法：the devastating fact that √2 is not a rational number.
- rational number：有理数。原文用法：同上。
- the ratio of two whole numbers：两个整数之比。原文用法：the ratio of two whole numbers.
- geometrical counterpart：几何对应。原文用法：The geometrical counterpart of this discovery.
- common unit of measure：共同度量单位。原文用法：no common unit of measure.
- step off exactly a whole number of times：恰好量出整数次。原文用法：can be stepped off exactly a whole number of times into each.
- firm intuitive belief：坚定的直觉信念。原文用法：the firm intuitive belief that any two line segments must possess some common unit of measure.
- line segments：线段。原文用法：同上。
- built upon：建立在……之上。原文用法：was built upon this seemingly obvious assumption.
- seemingly obvious assumption：看似显而易见的假设。原文用法：同上。
- in one fell stroke：一举。原文用法：In one fell stroke, both ... were threatened.
- logical scandal：逻辑丑闻。原文用法：the "logical scandal".
- keep the matter secret：把事情保密。原文用法：keep the matter secret.
- divulge the secret：泄露秘密。原文用法：to divulge the secret to outsiders.
- banished from：被逐出。原文用法：was banished from the Brotherhood.
- as though：好像。原文用法：as though he were dead.
常用句式
- Pattern: Imagine the consternation, then, ... when ...：想象一下……时的惊慌。原文例句：Imagine the consternation, then, within the Pythagorean ranks when some Brother discovered the devastating fact that √2 is not a rational number...
- Pattern: It follows that ...：由此得出……。原文例句：it follows that a side and diagonal of a square possess no common unit of measure...
- Pattern: so ... that ...：如此……以至于……。原文例句：so great was the "logical scandal" that efforts were made for a while to keep the matter secret.
- Pattern: as though + 虚拟语气：好像……。原文例句：a tomb was erected for him as though he were dead.
长难句解析
The geometrical counterpart of this discovery was equally startling, for if √2 is not a rational number, it follows that a side and diagonal of a square possess no common unit of measure that can be stepped off exactly a whole number of times into each.
- 主干：The geometrical counterpart was equally startling.
- 修饰与从句：
- for 引导原因状语从句； - if √2 is not a rational number 为条件状语从句； - it follows that ... 中 it 为形式主语，that 引导主语从句； - a side and diagonal of a square possess no common unit of measure 为主语从句主干； - that can be stepped off exactly a whole number of times into each 为定语从句，修饰 common unit of measure。
- 逻辑关系：先指出几何对应同样惊人，再用条件推理说明：若 √2 无理，则正方形边与对角线不可公度。
语言使用特点
- 使用正式、抽象的学术语体；
- 频繁使用条件句、因果句表达逻辑推演；
- 用 devastating, startling, consternation, scandal 等强烈词汇渲染历史冲击力；
- as though he were dead 使用虚拟语气，表达与事实相反的假设。
迁移练习
- 用自己的话解释为什么 √2 的无理性会威胁毕达哥拉斯几何。
- 用 in one fell stroke 写一个句子。
- 翻译：据说，第一个把秘密泄露出去的门徒被兄弟会驱逐，他们还为他修了坟墓，就当他已经死了。
点击展开参考答案
- （示例）The Pythagoreans believed that any two line segments could be measured by a common unit. But the diagonal of a square with side 1 has length √2, which cannot be expressed as a ratio of whole numbers. This contradicted their theory of proportion and similar figures.
- （示例）In one fell stroke, the new evidence destroyed the old theory.
- It is reported that the first Pythagorean to divulge the secret to outsiders was banished from the Brotherhood and a tomb was erected for him as though he were dead.