内容概述
欧几里得《原本》第五公设——即著名的平行公设——表述复杂，不如其他公设简洁直观。两千多年来，几何学家或试图从其余公设、公理推出它，或寻找更易接受的等价形式。围绕第五公设的持续探讨，推动了现代数学的发展，促使几何学从传统模式中解放出来，催生了多种与欧氏几何同样和谐的非欧几何，也改变了人们对公设本质与数学真理的认识。该公设“科学史上最著名命题”之称来自凯瑟的《数学哲学》。
中文正文
70° 科学史上最著名的一个命题
命题：假如一条直线与两条直线相交，直线同一侧的两个夹角之和小于两直角，则此二直线，假如无限延伸，将在两夹角小于两直角的一侧相交。
这算什么？这是欧几里得《原本》第五公设，即所谓的平行公设。
为什么出名？这个公设不像欧几里得的其他几个公设那么简洁，也不那么一目了然，于是引出许多纷争。两千多年来，学者们一直不满意公设的这样一种陈述方式，几何学家或者尝试从欧几里得的其他公设和公理导出它，或者试图以更能接受的等价的形式来取代它。对第五公设的关心，刺激了许多现代数学的发展，也激发人们深入探询几何学的哲学和逻辑基础。更重要的是，它将几何从传统模式解放出来，产生出后来的许多不同于欧几里得的同样和谐的几何学。实际上，它也给整个数学带来了自由的气息。它沉重打击了数学的绝对真理的观点，革新了对公设本质的认识。数学的出现，是人类思想的自由创造，而不是我们生活的世界派给我们的什么必然产物。
谁给它命名的？凯瑟（Cassius J. Keyser），见他的《数学哲学》（Mathematical Philosophy, New York: E. P. Dutton and Company, Inc., 1922. p. 113.）
英文正文
70° The most famous single utterance in the history of science. The utterance: If a straight line falling on two straight lines makes the interior angles on the same side together less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than two right angles.
What is it? It is the fifth, or so-called parallel, postulate of Euclid's Elements.
Why is it famous? The lack of terseness and ready comprehensibility of this postulate, when compared with Euclid's other postulates, proved to be the source of much controversy. The dissatisfaction of mathematical scholars with its statement as a postulate is indicated by the fact that many geometers attempted over a period of some twenty centuries either to derive it from Euclid's other postulates and axioms or to replace it by a more acceptable equivalent. This concern over Euclid's fifth postulate furnished the stimulus for the development of a great deal of modern mathematics, and also led to deep and revealing inquiries into the logical and philosophical foundations of the subject. In particular, it led to the liberation of geometry from its traditional mold and to the subsequent creation of many other equally consistent geometries different from that of Euclid. Indeed, it had a similar liberating effect on mathematics as a whole. It dealt a severe blow to the absolute truth viewpoint of mathematics and brought forth a new view of the nature of a postulate. Mathematics emerged as an arbitrary creation of the human mind, not as something essentially dictated to us of necessity by the world in which we live.
Who named it so? Cassius J. Keyser, in his Mathematical Philosophy (New York: E. P. Dutton and Company, Inc., 1922), p. 113.
英语学习拓展
常用词汇
utterance音标/ˈʌtərəns/
话语；言论。
原文用法：The most famous single utterance in the history of science
straight音标/streɪt/
直的。
原文用法：a straight line falling on two straight lines
interior音标/ɪnˈtɪəriə/
内部的。
原文用法：the interior angles on the same side
indefinitely音标/ɪnˈdefɪnətli/
无限地；不确定地。
原文用法：if produced indefinitely
postulate音标/ˈpɒstjʊlət/
公设；假设。
原文用法：the fifth, or so-called parallel, postulate
terseness音标/ˈtɜːsnəs/
简洁；简练。
原文用法：The lack of terseness and ready comprehensibility
comprehensibility音标/kɒmˌpriːhenˈsɪbɪləti/
可理解性。
原文用法：The lack of terseness and ready comprehensibility
controversy音标/ˈkɒntrəvɜːsi/
争论；争议。
原文用法：the source of much controversy
dissatisfaction音标/dɪsˌsætɪsˈfækʃn/
不满。
原文用法：The dissatisfaction of mathematical scholars
derive音标/dɪˈraɪv/
推导；演绎出。
原文用法：either to derive it from Euclid's other postulates and axioms
replace音标/rɪˈpleɪs/
取代；替换。
原文用法：or to replace it by a more acceptable equivalent
equivalent音标/ɪˈkwɪvələnt/
等价物；等价命题。
原文用法：a more acceptable equivalent
stimulus音标/ˈstɪmjʊləs/
刺激；激励。
原文用法：furnished the stimulus for the development
inquiry音标/ˈɪnkwəri/
探究；调查。
原文用法：deep and revealing inquiries into the logical and philosophical foundations
foundation音标/faʊnˈdeɪʃn/
基础。
原文用法：the logical and philosophical foundations of the subject
liberation音标/ˌlɪbəˈreɪʃn/
解放。
原文用法：the liberation of geometry from its traditional mold
subsequent音标/ˈsʌbsɪkwənt/
随后的；后来的。
原文用法：the subsequent creation of many other equally consistent geometries
consistent音标/kənˈsɪstənt/
一致的；相容的。
原文用法：many other equally consistent geometries
viewpoint音标/ˈvjuːpɔɪnt/
观点；看法。
原文用法：the absolute truth viewpoint of mathematics
arbitrary音标/ˈɑːbɪtrəri/
任意的；武断的。
原文用法：Mathematics emerged as an arbitrary creation
dictated音标/dɪkˈteɪtɪd/
命令；强加。
原文用法：not as something essentially dictated to us of necessity
同义词与近义词
对应核心词汇：postulate
对应核心词汇：postulate。
辨析：postulate 和 axiom 都指公理/公设，常可互换；assumption 是更广义的“假设”，未必是体系的基本公理。
对应核心词汇：derive
对应核心词汇：derive。
辨析：derive 强调从已知命题或前提推出；deduce 强调演绎推理；infer 强调根据证据推断。
对应核心词汇：replace
对应核心词汇：replace。
辨析：replace A by B 表示用 B 取代 A；substitute B for A 结构相同但动词不同。
常用短语与固定搭配
- interior angles：内角。原文用法：the interior angles on the same side
- right angles：直角。原文用法：less than two right angles
- parallel postulate：平行公设。原文用法：the fifth, or so-called parallel, postulate
- source of controversy：争议的根源。原文用法：the source of much controversy
- derive from：从……推导出来。原文用法：derive it from Euclid's other postulates and axioms
- replace by：用……取代。原文用法：replace it by a more acceptable equivalent
- concern over：对……的关注。原文用法：This concern over Euclid's fifth postulate
- deal a severe blow to：给……以沉重打击。原文用法：It dealt a severe blow to the absolute truth viewpoint
- bring forth：产生；提出。原文用法：brought forth a new view of the nature of a postulate
- emerge as：显现为；成为。原文用法：Mathematics emerged as an arbitrary creation
常用句式
- If ..., then ... / 主句：条件状语从句表达几何命题。原文例句：If a straight line falling on two straight lines makes the interior angles on the same side together less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than two right angles.
- The lack of ... proved to be the source of ...：缺少……成为……的根源。原文例句：The lack of terseness and ready comprehensibility of this postulate ... proved to be the source of much controversy.
长难句解析
The dissatisfaction of mathematical scholars with its statement as a postulate is indicated by the fact that many geometers attempted over a period of some twenty centuries either to derive it from Euclid's other postulates and axioms or to replace it by a more acceptable equivalent.
- 主干：The dissatisfaction ... is indicated by the fact。
- 修饰与从句：of mathematical scholars with its statement as a postulate 为介词短语修饰 dissatisfaction；that many geometers attempted ... 为同位语从句，说明 fact 的内容；从句中 either to derive it from ... or to replace it by ... 为并列不定式结构，作 attempted 的宾语。
- 逻辑关系：学者们对平行公设表述的不满，体现在两千年来许多几何学家试图证明它或替换它。
语言使用特点
- 科学史叙述中大量使用抽象名词（如 dissatisfaction、controversy、liberation）来表达历史过程和因果关系。
- 用 deal a blow to、bring forth、emerge as 等动词短语增强论述的动态感。
迁移练习
- 用英文写出平行公设的一个等价形式。
- 用 The lack of ... proved to be the source of ... 说明一个你观察到的学习或社会问题。
点击展开参考答案
- 示例：Through a point not on a given line, there is exactly one line parallel to the given line.
- 示例：The lack of clear communication proved to be the source of much misunderstanding in our group project.