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2 弗朗西斯 波尔塞 Francis Borceux 公理 An Axiomatic 几何学方法 Approach to Geometry 几何三部曲 I Geometric Trilogy I

3 几何的公理化方法 An Axiomatic Approach to Geometry

4 弗朗西斯 波尔塞 Francis Borceux 几何的公理化方法 An Axiomatic Approach to Geometry 几何三部曲 I Geometric Trilogy I

5 比利时卢浮宫天主教大学 Francis Borceux Université catholique de Louvain Louvain-la-Neuve, Belgium ISBN ( 电子书 ) ISBN ISBN (ebook) DOI / 斯普林格查姆海德堡 DOI / 纽约多德雷赫特伦敦 Springer Cham Heidelberg New York Dordrecht London 国会图书馆编号 : Library of Congress Control Number: 数学学科分类 (2010): A05 51A05 51A15 51A30 51A35 51B05 51B20 Mathematics Subject Classification (2010): 51-01, 51-03, 01A05, 51A05, 51A15, 51A30, 51A35, 51B05, 51B 年瑞士斯普林格国际出版公司 Springer International Publishing Switzerland 2014 这件作品受版权保护. 出版方保留所有权利, 无论是全部或部分相关材料, 特别是翻译 重印 再利用插图 背诵 广播 缩微胶片或 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of 任何其他物理方式的复制 传输或信息存储和检索 电子改编 计算机软件的权利, 或通过现在已知或以后发展的类似或不同的方法. the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology 免除本法律保留的是与审查或学术分析有关的简短摘录, 或专门为在计算机系统上输入和执行而提供的材料, now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection 供买方独家使用. with reviews or scholarly analysis or material supplied specifically for the purpose of being entered 本出版物或其部分内容 and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of 的复制, 只有在出版商所在地的版权法 初始版本和使用许可的规定下, 方可从斯普林格获得. this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. 使用许可可以通过版权许可中心的 RightsLink 获得. 违反版权法的行 Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations 为将受到相应版权法的起诉. are liable to prosecution under the respective Copyright Law. 通用名称 注册名称 商标 服务标志的使用, 等并不意味着, 即使在没有具体说明的情况下, 这些名称也不受有关保护性法律和条例的限制, 因此可供一般使用. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 尽管本书中的建议和信息在出版之日被认为是真实和准确的, 但作者 编辑和出版商均不对可能出现的任何错误或遗漏承担任何法律责任. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any 出版商对本文所含材料不作任何明示或暗示的保证. errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. 封面图片 : 安德烈 塞维特的欧几里德 玛格丽安 ( ) Cover image: Euclide Margarean by André Thévet ( ) 印在无酸纸上 Printed on acid-free paper Springer 是 Springer Science+ 商业媒体的一部分 ( 网站 ) Springer is part of Springer Science+Business Media (

6 致克里斯蒂娜 To Christiane

7 前言 Preface 读者被邀请沉浸在一个已经展开 35 个世纪的爱情故事中 : 数学家和几何学之间的爱情故事. The reader is invited to immerse himself in a love story which has been unfolding 除了陪伴读者了解 for 35 centuries: the love story between mathematicians and geometry. In addition 当前的艺术现状外, 这部三部曲的目的正是讲述这个故事. to accompanying the reader up to the present state of the art, the purpose of this Trilogy is precisely to tell this story. The Geometric Trilogy will introduce the reader to 几何学三部曲将向读者介绍几何学的多个互补方面, 首先赞扬它所依据的历史著作, 然后转向更现代的处理方式, 充分利用现代逻辑 代数和分析. the multiple complementary aspects of geometry, first paying tribute to the historical work on which it is based and then switching to a more contemporary treatment, 在这三部曲中, 几何学绝对被看作是一门 making full use of modern logic, algebra and analysis. In this Trilogy, Geometry 独立的学科, 而不是代数或分析的子产品. is definitely viewed as an autonomous discipline, never as a sub-product of algebra 三部曲的三卷本被写成了三本相互独立但互为补充的书, 分别侧重于几何的公理 代数和微分方法. or analysis. The three volumes of the Trilogy have been written as three independent but complementary books, focusing respectively on the axiomatic, algebraic 根据教授的选择, 它们包含了各种可能非常不同的本科几何课程的 and differential approaches to geometry. They contain all the useful material for a 所有有用材料. wide range of possibly very different undergraduate geometry courses, depending 它们也为其他学科的研究人员提供了必要的几何背景, 他们需要掌握几何 on the choices made by the professor. They also provide the necessary geometrical 技术. background for researchers in other disciplines who need to master the geometric techniques. 本书带领读者走过 35 个世纪的几何学 : 从 16 个世纪前埃及文士艾哈迈斯的纸莎草, 到 19 个世纪后希尔伯特著名的几何学公理化. The present book leads the reader on a walk through 35 centuries of geometry: from the papyrus of the Egyptian scribe Ahmes, 16 centuries before Christ, to 我们一步一步地发 Hilbert s famous axiomatization of geometry, 19 centuries after Christ. We discover 现, 现代几何学的所有成分都是如何慢慢地获得最终形式的. step by step how all the ingredients of contemporary geometry have slowly acquired their final form. 事实上 : 三千年来, 几何学基本上是通过综合的方法来研究的, 即从一个给定的公理系统. It is a matter of fact: for three millennia, geometry has essentially been studied 直到 17 世纪, 代数和微分方 via synthetic methods, that is, from a given system of axioms. It was only during 法才被认真地考虑, 尽管它们从古代就一直以伪装的形式存在. the 17th century that algebraic and differential methods were considered seriously, even though they had always been present, in a disguised form, since antiquity. 在快速回顾了埃及人和巴比伦人经验性地知道的一些结果之后, 我们展示了古希腊几何学家如何慢慢地, 有时遇到巨大的困难, After rapidly reviewing some results that had been known empirically by the 得出一个连贯有力的演绎理论, 允许对所有这些经验性结果和许多其他结果进行严格的证明. Egyptians and the Babylonians, we show how Greek geometers of antiquity, slowly, sometimes encountering great difficulties, arrived at a coherent and powerful deductive theory allowing the rigorous proof of all of these empirical results, and many 一些著名的问题, 如使圆成直角, 引起了复杂方法的发展. others. Famous problems such as squaring the circle induced the development 特别是在公元前 4 世纪, 尤多克斯用一种方法克服了不可通约量这一令人费解的困难, 这种方 of sophisticated methods. In particular, during the fourth century BC, Eudoxus overcame the puzzling difficulty of incommensurable quantities by a method which 法是 is 七 vii

8 viii Preface 基本上是因为 rhandlingrealnumbers.eudoxuso 已批准极限过程 ( 穷竭定理 ) 的有效性, 它允许他回答与各种曲线或曲面有关的 essentially that of Dedekind cuts for handling real numbers. Eudoxus also proved the 或体积等问题. validity of a limit process (the Exhaustion theorem) which allowed him to answer questions concerning, among other things, the lengths, areas or volumes related to various curves or surfaces. 我们首先通过介绍欧几里得元素的主要方面来总结当时希腊几何学家的知识. We first summarize the knowledge of the Greek geometers of the time by presenting the main aspects of Euclid s Elements. We then switch to further work by 然后我们转到阿基米德 ( 圆 球面 螺旋等 ) 阿波罗牛( 电子 ) 墨涅劳斯和托勒密( 三角学的诞生 ) 帕普斯( 射影几何学的祖先 ) 等的进一步研究. Archimedes (the circle, the sphere, the spiral,...),apollonius (the conics), Menelaus and Ptolemy (the birth of trigonometry), Pappus (ancestor of projective geometry), and so on. 我们还回顾了欧几里得之后几个世纪才研究的经典欧几里得方法的一些相关结果, 如三角形和二次曲线的附加性质 Ceva 定理 三 We also review some relevant results of classical Euclidean geometry which were 角形的三分线 赤平投影等. only studied several centuries after Euclid, such as additional properties of triangles and conics, Ceva s theorem, the trisectors of a triangle, stereographic projection, and 然而, 这种精神中最重要的新方面可能是庞赛莱在 19 世纪发展的逆理论 ( 保角映射的一个特例 ). so on. However, the most important new aspect in this spirit is probably the theory of inversions (a special case of a conformal mapping) developed by Poncelet during the 19th century. 我们继续研究几何中的射影方法. 它们出现在 17 世纪, 起源于 We proceed with the study of projective methods in geometry. These appeared in 一些画家理解透视法则的努力. the 17th century and had their origins in the efforts of some painters to understand 在一个好的透视图中, 平行线似乎在地平线上相交. the rules of perspective. In a good perspective representation, parallel lines seem to 由此产生了在欧几里德平面无穷远处添加点的想法, 即平行线最终相交的点. meet at the horizon. From this comes the idea of adding points at infinity to the 有一段时间, 投影方法被认为是处 Euclidean plane, points where parallel line eventually meet. For a while, projective 理一些欧几里德问题的简便方法. methods were considered simply as a convenient way to handle some Euclidean 把射影几何学作为一种受人尊敬的几何学理论, 仅在二十世纪初, 从平面上的两条直线的自年龄测量中得到承认. problems. The recognition of projective geometry as a respectable geometric theory in itself a geometry where two lines in the plane always intersect only came later. 在讨论了导致投影几何的基本原理之后, 我们重点讨论了令人惊奇的希尔伯特定理. After having discussed the fundamental ideas which led to projective geometry we 这些定理表明射影平面的非常简单的经典公理表示强制存在一个潜在 focus on the amazing Hilbert theorems. These theorems show that the very simple 的坐标场. classical axiomatic presentation of the projective plane forces the existence of an 感兴趣的读者将在本三部曲第二卷的 [5] 中找到, 一个在任意维上, 充分利用当 underlying field of coordinates. The interested reader will find in [5], Vol. II of 代线性代数技术的射影空间的系统研究. this Trilogy, a systematic study of the projective spaces over a field, in arbitrary dimension, fully using the contemporary techniques of linear algebra. 19 世纪另一种截然不同的几何学方法 : 非欧几里德几何学. Another strikingly different approach to geometry imposed itself during the 19th 欧几里得的平面公理化首先指的是四个没有人认为可以反驳的高度自然的假设 century: non-euclidean geometry. Euclid s axiomatization of the plane refers. 但它也包含了著 first to four highly natural postulates that nobody thought to contest. But it also 名但更为复杂的第五个假设, 迫使平行线通过给定点的唯一性. contains the famous but more involved fifth postulate, forcing the uniqueness 几个世纪以来, 许多数学家努力从其他公理中证明 of the parallel to a given line through a given point. Over the centuries many mathematicians made considerable efforts to prove Euclid s parallel postulate from the 欧几里德的平行假设. 试图获得这样一个证明的一种方法是通过一个荒谬的还原 : 假设一条给定的线通过一个给定的点有几个平行线, other axioms. One way of trying to obtain such a proof was by a reductio ad absurdum: assume that there are several parallels to a given line through a given point, 然后从这个假设中尽可能地推断出更多的结果, 直到你达到一个矛盾的时刻. then infer as many consequences as possible from this assumption, up to the moment when you reach a contradiction. But very unexpectedly, rather than leading to 但出乎意料的是, 这些努力并没有导致矛盾, 反而一步一步地导致了一个惊人的新几何理论. a contradiction, these efforts instead led step by step to an amazing new geometric theory. When actual models of this theory were constructed, no doubt was left: 当这个理论的实际模型被构造出来时, 毫无疑问 : 从数学上讲, 这个非欧几里德几何和欧几里德几何一样连贯. mathematically, this non-euclidean geometry was as coherent as Euclidean geometry. We recall first some attempts at proving Euclid s fifth postulate, and then 我们首先回顾了证明欧几里德第五公设的一些尝试, 然后发展了非欧几里德平面的主要特征 : 极限平行和三角形的一些性质. develop the main characteristics of the non-euclidean plane: the limit parallels and 接下来, 我们详细描述了两个著名的非欧几里德几何模型 :Beltrami-Klein 盘和 Poincar some properties of triangles. Next we describe in full detail two famous models of

9 Preface ix é 盘. 另一个模型 non-euclidean geometry: the Beltrami Klein disc and the Poincaré disc. Another 著名的庞加莱半平面将在这三部曲的第三卷 [6] 中给予充分关注, 使用黎曼几何的技术. model the famous Poincaré half plane will be given full attention in [6], Vol. III of this Trilogy, using the techniques of Riemannian geometry. 我们用 Hilbert 著名的平面公理化来总结综合几何的这一概述. We conclude this overview of synthetic geometry with Hilbert s famous axiomatization of the plane. Hilbert has first filled in the small gaps existing in Euclid s ax- 希尔伯特首先填补了欧几里德公理化中存在的小空白 : 本质上, 与点和线的相对位置 ( 左, 右, 中间, ) 有关的问题, 这些问题是希腊几何学家认为是直观的或从图片中可以看出的. iomatization: essentially, the questions related to the relative positions of points and lines (on the left, on the right, between,...), aspects that Greek geometers considered as being intuitive or evident from the picture. A consequence of Hilbert s Hilbert 公理化欧几里德平面的一个结果是该平面与欧几里德平面 R2 之间的同构 : 这与本书第二卷的 [5] 形成了联系三部曲. 但是首先, 希尔伯特认为, 只要用一点对一条给定直线存 axiomatization of the Euclidean plane is the isomorphism between that plane and 在多条平行线的要求来代替关于平行线唯一性的公理 the Euclidean plane R 2, 就可以得到非欧几里德平面的公理化, 如前一章所研究的. : this forms the link with [5], Vol. II of this Trilogy. But above all, Hilbert observes that just replacing the axiom on the uniqueness of the parallel by the requirement that there exist several parallels to a given line through a same point, one obtains an axiomatization of the non-euclidean plane, as studied in the preceding chapter. 总而言之, 我们还记得古希腊的几何学家在早期就提出了许多众所周知的问题, 但这些问题是他们无法解决的. To conclude, we recall that there are various well-known problems, introduced 最著 early in antiquity by the Greek geometers, and which they could not solve. The 名的例子是 : 平方圆, 三等分一个角, 复制一个立方体, 构造一个 n 边的正多边形. most famous examples are: squaring a circle, trisecting an angle, duplicating a cube, 直到 19 世纪, 随着代数的新发展, 这些尺子和指南针的结 constructing a regular polygon with n sides. It was only during the 19th century, 构才被证明是不可能的. with new developments in algebra, that these ruler and compass constructions were 通过场理论和多项式理论, 我们给出了这些不可能结果的显式证明. proved to be impossible. We give explicit proofs of these impossibility results, via 特别地, 我们证明了 Gauss-Wantzel 定理的超越性, 同时也证明 field theory and the theory of polynomials. In particular we prove the transcendence 了 Gauss-Wantzel 定理, 刻画了那些用标尺和指南针构成的规则多边形. of π and also the Gauss Wantzel theorem, characterizing those regular polygons 由于所涉及的方法完全超出了本书所致力于的几何综 which are constructible with ruler and compass. Since the methods involved are 合方法, 我们在几个附录中给出了这些不同的代数证明. completely outside the synthetic approach to geometry, to which this book is dedicated, we present these various algebraic proofs in several appendices. 每章以一节问题和另一节练习结束. 问题通常包括书中未涉及但仍具有理论意义的陈述, 而练习是为读者实践技巧和进一步研究正文 Each chapter ends with a section of problems and another section of exercises. Problems generally cover statements which are not treated in the book, but 中包含的概念而设计的. which nevertheless are of theoretical interest, while the exercises are designed for the reader to practice the techniques and further study the notions contained in the main text. 提供了本书所讨论主题的选择性参考书目. 书中未提及 A selective bibliography for the topics discussed in this book is provided. Certain 的某些项目已列入供进一步阅读. items, not otherwise mentioned in the book, have been included for further reading. 作者希望通过多年的合作, 帮助他提高几何课程和本书的质量. The author thanks the numerous collaborators who helped him, through the years, 其中, 作者特别要感 to improve the quality of his geometry courses and thus of this book. Among them, 谢帕斯卡杜邦, 他也给了一些有用的提示, 绘制了一些插图, 实现了与 Mathematica 和 Tikz. the author particularly wishes to thank Pascal Dupont, who also gave useful hints for drawing some of the illustrations, realized with Mathematica and Tikz.

10 几何三部曲 The Geometric Trilogy 几何的公理化方法 I. An Axiomatic Approach to Geometry 一. 前希腊时代 1. Pre-Hellenic Antiquity 希腊几何学的一些先驱 2. Some Pioneers of Greek Geometry 三. 欧几里得 s 元素 3. Euclid s Elements 一些希腊几何大师 5. 后希腊欧几里得几何学 4. Some Masters of Greek Geometry 5. Post-Hellenic Euclidean Geometry 6. 射影几何 6. Projective Geometry 7 号. 非欧几里得几何 7. Non-Euclidean Geometry 8 个. 平面的希尔伯特公理化 8. Hilbert s Axiomatization of the Plane 附录 Appendices A. 可施工性 B. A. Constructibility 三个经典问题 C. 正则多边形 B. The Three Classical Problems C. Regular Polygons 几何学的代数方法 II. An Algebraic Approach to Geometry 解析几何的诞生 2. 仿射几何 1. The Birth of Analytic Geometry 2. Affine Geometry 三. 更多关于实仿射空间 4. 欧几里得几何 3. More on Real Affine Spaces 4. Euclidean Geometry 5 个. 厄米空间 5. Hermitian Spaces 6. 射影几何 6. Projective Geometry 7 号. 代数曲线 7. Algebraic Curves 附录 Appendices A. 域上的多项式 A. Polynomials over a Field B. 多元多项式 B. Polynomials in Several Variables C. 齐次多项式 D. 结果 C. Homogeneous Polynomials D. Resultants E. 对称多项式 F. 复数 E. Symmetric Polynomials F. Complex Numbers 席 xi

11 xii The Geometric Trilogy G. 二次型 G. Quadratic Forms H. 对偶空间 H. Dual Spaces 几何学的微分方法 III. A Differential Approach to Geometry 微分方法的起源 1. The Genesis of Differential Methods 2. 平面曲线 2. Plane Curves 曲线博物馆 3. A Museum of Curves 四. 斜曲线 4. Skew Curves 曲面的局部理论 The Local Theory of Surfaces 朝向黎曼几何 7. 曲面整体理论要素 6. Towards Riemannian Geometry 7. Elements of the Global Theory of Surfaces Appendices A. 拓扑结构 A. Topology B. 微分方程 B. Differential Equations

12 目录 Contents 前希腊时代 1 Pre-Hellenic Antiquity... 1 史前 1.1 Prehistory... 1 埃及 1.2 Egypt... 3 美索不达米亚 1.3 Mesopotamia... 5 问题 1.4 Problems... 6 练习 1.5 Exercises... 7 希腊几何学的一些先驱 2 Some Pioneers of Greek Geometry... 9 泰勒斯 2.1 ThalesofMiletus 毕达哥拉斯与黄金分割率 2.2 Pythagoras and the Golden Ratio 三等分角 2.3 TrisectingtheAngle 使圆成直角 2.4 Squaring the Circle 复制多维数据集 2.5 DuplicatingtheCube 不可公度 2.6 Incommensurable Magnitudes 穷竭法 2.7 The Method of Exhaustion 论空间的连续性 2.8 On the Continuity of Space 问题 2.9 Problems 练习 2.10Exercises 欧几里得 s 元素 3 Euclid s Elements 第一册直线 3.1 Book1:StraightLines 第二册 : 几何代数 3.2 Book 2: Geometric Algebra 第三册 : 圆圈 3.3 Book3:Circles 第 4 册 : 多边形 3.4 Book 4: Polygons 第 5 册 : 比率 3.5 Book5:Ratios 第六册 : 相似之处 3.6 Book 6: Similarities 第七册 : 算术的可分性 3.7 Book 7: Divisibility in Arithmetic 第 8 册 : 几何级数 3.8 Book8:GeometricProgressions 第九册 : 关于数字的更多内容 3.9 Book9:MoreonNumbers 第 10 册 : 不可比较的震级 3.10 Book 10: Incommensurable Magnitudes 第 11 册 : 立体几何 3.11Book11:SolidGeometry 第十二册 : 穷竭法 3.12 Book 12: The Method of Exhaustion 十三 xiii

13 xiv Contents 第 13 册 : 正多面体 3.13 Book 13: Regular Polyhedrons 问题 3.14Problems 练习 3.15Exercises 一些希腊几何大师 4 Some Masters of Greek Geometry 圆上的阿基米德 4.1 Archimedes on the Circle 阿基米德的号码 4.2 Archimedes on the Number π 球面上的阿基米德 4.3 Archimedes on the Sphere 抛物线上的阿基米德 4.4 Archimedes on the Parabola 阿基米德螺旋 4.5 Archimedes on the Spiral 圆锥形截面上的阿波罗 4.6 ApolloniusonConicalSections 共轭方向上的阿波罗 4.7 ApolloniusonConjugateDirections 切线上的阿波罗 4.8 Apollonius on Tangents 北极和极地线上的阿波罗 4.9 ApolloniusonPolesandPolarLines 阿波罗尼乌斯聚焦 4.10ApolloniusonFoci 三角上的苍鹭 4.11 Heron on the Triangle 三角墨涅劳斯 4.12 Menelaus on Trigonometry 托勒密三角学 4.13 Ptolemy on Trigonometry 关于非谐比的 Pappus 4.14 Pappus on Anharmonic Ratios 问题 4.15Problems 练习 4.16Exercises 后希腊欧几里得几何学 5 Post-Hellenic Euclidean Geometry 还在追查数字 5.1 Still Chasing the Number π 三角形的中间 5.2 The Medians of a Triangle 三角形的高度 5.3 The Altitudes of a Triangle 塞瓦定理 5.4 Ceva s Theorem 三角形的三色性 5.5 The Trisectrices of a Triangle 再看圆锥曲线的焦点 5.6 AnotherLookattheFociofConics 平面上的倒转 5.7 InversionsinthePlane 固体空间中的反演 5.8 Inversions in Solid Space 赤平投影 5.9 The Stereographic Projection 让我们烧掉我们的统治者 5.10LetusBurnourRulers! 问题 5.11Problems 练习 5.12Exercises 射影几何 6 Projective Geometry 透视表示 6.1 Perspective Representation 射影与欧几里德 6.2 Projective Versus Euclidean 非谐比 6.3 Anharmonic Ratio Desargues 与 Pappus 定理 6.4 The Desargues and the Pappus Theorems 公理射影几何 6.5 AxiomaticProjectiveGeometry 阿圭斯和帕皮安平面 6.6 Arguesian and Pappian Planes 斜场上的射影平面 6.7 TheProjectivePlaneoveraSkewField 希尔伯特定理 6.8 The Hilbert Theorems 问题 6.9 Problems 练习 6.10Exercises

14 Contents xv 非欧几里得几何 7 Non-Euclidean Geometry 追寻欧几里得第五公设 7.1 ChasingEuclid sfifthpostulate Saccheri 四边形 7.2 The Saccheri Quadrilaterals 三角形的角 7.3 The Angles of a Triangle 极限平行 7.4 TheLimitParallels 三角形的面积 7.5 The Area of a Triangle Beltrami Klein 和 PoincaréDisks 7.6 The Beltrami Klein and Poincaré Disks 问题 7.7 Problems 练习 7.8 Exercises 平面的希尔伯特公理化 8 Hilbert s Axiomatization of the Plane 关联公理 8.1 The Axioms of Incidence 秩序公理 8.2 TheAxiomsofOrder 同余公理 8.3 The Axioms of Congruence 连续性公理 8.4 TheAxiomofContinuity 平行公理 8.5 TheAxiomsofParallelism 问题 8.6 Problems 练习 8.7 Exercises 附录 A 可施工性 Appendix A Constructibility 极小多项式 A.1 The Minimal Polynomial 艾森斯坦准则 A.2 TheEisensteinCriterion 尺规结构 A.3 Ruler and Compass Constructibility 可构造性与场论 A.4 Constructibility Versus Field Theory 附录 B 经典问题 Appendix B The Classical Problems 复制多维数据集 B.1 DuplicatingtheCube 三等分角 B.2 TrisectingtheAngle 使圆成直角 B.3 Squaring the Circle 附录 C 规则多边形 Appendix C Regular Polygons 希腊几何学家所知道的 C.1 WhattheGreekGeometersKnew 代数术语中的问题 C.2 The Problem in Algebraic Terms 费马素数 C.3 FermatPrimes 模运算元素 C.4 Elements of Modular Arithmetic 伽罗瓦理论的味道 C.5 A Flavour of Galois Theory Gauss-Wantzel 定理 C.6 The Gauss Wantzel Theorem 参考文献和进一步阅读 References and Further Reading 索引 Index

15 第一章 Chapter 1 前希腊时代 Pre-Hellenic Antiquity 这很短的一章只想概述在希腊几何学家系统工作的影响之前, 在各种文明中出现的一些最初的几何思想. This very short chapter is intended only to give an overview of some of the first geometric ideas which arose in various civilizations before the influence of the systematic work of the Greek geometers. So pre-hellenic should be understood here as 所以在这里, 前希腊语应该和希腊影响之前一样理解. before the Greek influence. 从这个前希腊的古代, 我们知道各种各样的作品, 由于埃及人和巴比伦人. From this pre-hellenic antiquity, we know of various works due to the Egyptians and the Babylonians. Indeed, these are the only pre-hellenic civilizations 事实上, 这些是唯一的前希腊文明, 产生了书面几何文件, 并存活到今天. which have produced written geometric documents that have survived to the present day. 不过, 这里应该提到的是, 在希腊几何学黄金时代之后, 中国和印度的一些作品被一些历史学家认为是希腊之前的作品, 因为没 It should nevertheless be mentioned here that some works in China and India 有希腊的影响. posterior to the golden age of geometry in Greece are considered by some historians as pre-hellenic in the sense of being absent of Greek influence. But not all 但并非所有专家都同意这一点. 因此, 我们选择在这本书中提到这些发展的时间顺序的地方, 希腊时期之后. specialists agree on this point. Therefore we choose in this book to mention these developments at their chronological place, after the Greek period. 1.1 史前史 1.1 Prehistory 史前时期的特点是没有文字. 在那些日子里, 知识的传播基本上是口头的. Prehistory is characterized by the absence of writing. In those days, the transmission 但现在, 我们再也听不到这些声音了. of knowledge was essentially oral. But nowadays, we no longer hear those voices. 因此, 史前对于几何学和人类生活的所有其他方面都保持沉默. Therefore prehistory remains as silent about geometry as it is about all other aspects 我们所能做的最好的事情就是依靠考古发现, 试着解读各种洞穴图片和发现的物体. of human life. The best that we can do is to rely on archaeological discoveries and try to interpret the various cave pictures and objects that have been found. 第一幅几何图形可追溯到公元前 年. 它们已经表明了对对称和数字同余概念的一些 The first geometric pictures date from BC. They already indicate some 掌握. 同一时期的其他一些物体显示 mastering of the notions of symmetry and congruence of figures. Some other objects 了第一次算术发展的证据, 例如计数的思想. of the same period show evidence of the first arithmetical developments, such as the idea of counting. 特别有趣的是图 1.1 中的图片 : 很明显, 史前艺术家并不只是想画一幅图画头晕 : 他 / 她想要强调一些数学发现. Particularly intriguing is the picture in Fig. 1.1: it seems to be evident that the prehistoric artist did not just want to draw a nice picture: he/she wanted to emphasize 事实上, 看看这张照片, 我们立刻注意到 : some mathematical discovery. Indeed, looking at this picture, we notice at once that: 将三角形的边加倍乘以 4; 将三角形的边加倍乘以 9; doubling the side of the triangle multiplies the area by 4; tripling the side of the triangle multiplies the area by 9; 波尔图, 几何的公理方法, DOI / , F. Borceux, An Axiomatic Approach to Geometry, DOI / _1, 2014 年瑞士斯普林格国际出版公司 Springer International Publishing Switzerland

16 2 1 Pre-Hellenic Antiquity 图 1.1 Fig. 1.1 计算每条线上的小三角形的数量我们观察到 counting the number of small triangles on each line we observe that 1 = 1, = 4, = 9. 我们所知道的最早的关于几何学的书面文件已经提到了相应的一般结果 : The oldest written documents that we know concerning geometry already mention the corresponding general results: 长度乘以 n, 面积乘以 n2; multiplying the lengths by n results in multiplying the areas by n 2 ; n 个第一奇数之和等于 n2. the sum of the n first odd numbers is equal to n 2. 史前艺术家在多大程度上了解这些定理? 我们可能永远不会知道. To what extent was the prehistoric artist aware of these theorems? We shall probably never know. 一种传统认为算术和几何学的起源可以在我们祖先的宗教仪式中找到 : 他们被一些形式和数字的特性所吸引, 并将其归因于魔法的 A tradition claims that the origin of arithmetic and geometry is to be found in 力量. the religious rituals of our ancestors: they were fascinated by the properties of some 通过在他们的仪式中 forms and some numbers, to which they attributed magical powers. By introducing such magical forms and numbers into their rituals they might perhaps draw the 引入这种神奇的形式和数字, 他们也许可以得到他们的神的祝福. benediction of their gods. 另一个传统, 由希罗多德报道 (c. 公元前 425 年 ) 作为尼罗河摩羯座的宝贝女 Another tradition, reported by Herodotus (c. 484 BC c. 425 BC) presents geometry as the precious daughter of the caprices of the Nile. Legendary Pharaoh Sesostris 儿出现在几何上. 传说中的法老塞索斯特里斯 ( 约公元前 1300 年 ; 但可能是塞提和拉美西斯二世的混合物 ) 声称, 希罗多德将埃及土地分布在 ( 据我们所知, 只有少数特权 ) 居民之间. (around 1300 BC; but probably a compound of Seti and Ramesses II) had, claims Herodotus, distributed the Egyptian ground between the (by which we understand 尼罗河流域每年的洪水是其肥沃的起源, 也是许多戏剧性事件的根源, 因此 some few privileged ) inhabitants. The annual floods of the Nile valley, the origin 有必要设计出在每次洪水之后追溯每个庄园界限的实用方法. of its fertility but also of many dramatic events, made it necessary to devise practical 这些方法是基于三角剖分的, 可 methods of retracing the limits of each estate after each flood. These methods were 能利用毕达哥拉斯定理的一些特殊例子来构造直角. based on triangulation and probably made use of some special instances of Pythagoras theorem for constructing right angles. For example, the fact that a triangle with 例如, 一个边为 3, 4, 5 的三角形有一个直角的事实似乎至少在公元前 2000 年就已经知道了. sides 3, 4, 5 has a right angle seems to have been known at least since 2000 BC. 但尼罗河流域当然没有早期数学发展的霸权, 甚至在非洲也没有 :1950 年在刚果发现的伊尚戈骨, 可追溯到公元前 年, 是一 But the Nile valley certainly does not have the hegemony of early developments 些数学活动最古老的见证之一. of mathematics, not even in Africa: the discovery in 1950 of the Ishango bone in Congo, dating from BC, is one of the oldest testimonies of some mathematical activity. And various discoveries in Europe, India, China, Mesopotamia, and 而在欧洲 印度 中国 美索不达米亚等地的各种发现表明, 在不同的发展水平上, 古代世界上许多地方都存在着数学思想. so on, indicate that at different levels of development mathematical thought was present in many places in the world during antiquity. 然而, 到目前为止, 谦逊的史前历史很少揭示它与几何学的个人关系. However, up to now, modest prehistory has unveiled very little of its personal relations with geometry.

17 1.2 Egypt 3 图 1.2 Fig. 1.2 图 1.3 Fig 埃及 1.2 Egypt 最古老的数学纸莎草已经到达我们是所谓的莫斯科纸莎草, 最有可能写在公元前 1850 年左右. The oldest mathematical papyrus which has reached us is the so-called Moscow papyrus, most likely written around 1850 BC. But our main knowledge of Egyptian 但我们对古埃及数学的主要知识来自于公元前 1650 年左右抄写员艾哈迈斯抄写的一份更为广泛的纸莎草. mathematics during high antiquity comes from a more extended papyrus copied by 这些纸莎草包含了许多算术和几何问题的解决方案, 根据艾哈迈斯的说法, 这 the scribe Ahmes around 1650 BC. These papyri contain the solutions to many arithmetical and geometrical problems whose elaboration, according to Ahmes, dates 些问题的阐述可以追溯到公元前 2000 年. back to 2000 BC. 莫斯科纸莎草也被称为 Golenischev 数学纸莎草 The Moscow papyrus is also called the Golenischev Mathematical Papyrus,after 1893 年左右在底比斯买纸莎草的埃及古物学家弗拉基米尔 戈勒尼什切夫. the Egyptologist Vladimir Golenishchev who bought the papyrus in Thebes around 这些纸莎草后来进入莫斯科普希金国家美术馆收藏, 至今仍保存在那里 The papyrus later entered the collection of the Pushkin State Museum of Fine 艾哈迈斯纸莎草通常被称为莱茵纸莎草, 以 1858 年在埃及卢克索购 Arts in Moscow, where it remains today. The Ahmes papyrus is often referred to 买该纸莎草的苏格兰古董学家亚历山大 亨利 莱茵的名字命名. as the Rhind papyrus, so named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt. The papyrus was appar- 这些纸莎草显然是在法老拉美西斯二世太平间神庙遗址的非法挖掘中发现的. ently found during illegal excavations on the site of the mortuary temple of Pharaoh 它保存在伦敦大英博物馆. Ramesses II. It is kept at the British Museum in London. 例如, 艾哈迈斯纸莎草的问题 51 表明 For example, Problem 51 of the Ahmes papyrus shows that 等腰三角形的面积等于高度乘以底面的一半. The area of an isosceles triangle is equal to the height multiplied by half of the base. 解释是如图 1.2 所示的剪切和粘贴参数. 沿着三角形的高度剪下它 ; 把一块倒过来, 把它翻过来, 把两块粘在一起 ; 矩形就在右边. The explanation is a cut and paste argument as in Fig Cut the triangle along its height; reverse one piece, turn it upside down and glue both pieces together; you get the rectangle on the right. 问题 52 中使用了类似的论点来说明 An analogous argument is used in Problem 52 to show that 等腰梯形的面积等于高度乘以底面的一半. The area of an isosceles trapezium is equal to the height multiplied by half the sum of the bases. 见图 1.3, 这本身也是一个证明. 至少, 这是时代精神的证明 : 在任何情况下, 都是基于 See Fig. 1.3, which is again a proof in itself. At least, it is a proof in the spirit 数字一致性的论点. of the time: in any case, an argument based on congruences of figures. 然而, 让我们强调, 埃及人并不知道什么是定理或形式证明, 在数学意义上的术语. However, let us stress that the Egyptians did not have a notion of what a theorem or a formal proof is, in the mathematical sense of the term. In particular, 特别是, 他们没有明确区分精确结果和近似结果. they did not make a clear distinction between a precise result and an approximative

18 4 1 Pre-Hellenic Antiquity 例如, 人们在埃及文献中发现了计算四边形面积的下列奇怪规则 : one. For example, one finds in Egyptian documents the following strange rule to compute the area of a quadrilateral: 任意四边形的面积是由对边对的一半和相乘得到的. The area of an arbitrary quadrilateral is obtained by multiplying the half-sums of the pairs of opposite sides. 这显然与艾哈迈斯纸莎草中的问题 52 矛盾 : 等腰梯形的区域. This is in clear contradiction with Problem 52 in the Ahmes papyrus: the area of an 但这似乎没有打扰任何人. 更令人惊奇的是, 根据这条一般 isosceles trapezium. But this did not seem to disturb anybody. Even more amazing 规则推断出的推论 ( 表现为这样 ): is the corollary (presented as such) inferred from this general rule : 三角形的面积等于一条边的一半乘以另两条边的一半和. The area of a triangle is equal to the half of one side multiplied by the half sum of the other two sides. 这再次与艾哈迈斯纸莎草中的问题 51 相矛盾, 但最大的意外是其他地方 : This is again in contradiction with Problem 51 in the Ahmes papyrus, but the biggest surprise is elsewhere: 埃及人可以把三角形看作是四边形的一种特殊情况 : 四边形的一边长度为零. The Egyptians were able to consider a triangle as a special case of quadrilateral: a quadrilateral having one side of length zero. 这是一个抽象的过程, 在那些日子里我们是不会想到的. This is an abstraction process which we would not have expected in those days. 这两种纸莎草也考虑一个圆圈的面积. 在艾哈迈斯纸莎草的问题 50 中, 据称 Both papyri also consider the area of a circle. In Problem 50 of the Ahmes papyrus, it is claimed that 直径为 9 个单位的圆与边为 8 个单位的正方形面积相同. A circle whose diameter is 9 units has the same area as a square whose side is 8 units. 鉴于目前已知的圆面积公式, 这将产生 In view of the now well-known formula for the area of a circle, this yields the value π = 这还不错! 作为一个比较, 圣经 ( 在国王的第一本书, VII-23) 给出了一个直径为 10 的圆周长的值 which is not that bad! As a matter of comparison, the Bible (in the first book 这是一个值 =3. of Kings, VII-23) gives the value 30 for the circumference of a circle of diameter 埃及人似乎没有意识到存在着一种独特的数量, 可以用于所有的圈子, 无论其大小 that is a value π = 3. It does not seem that the Egyptians were aware of the existence of a unique quantity π to be used for all circles, whatever their size. 稍后我们将讨论这种数量的性质在古代可能是什么 ( 见第四节 ). Later we shall discuss what the nature of such a quantity π could have been in antiquity (see Sect. 2.6). 另一方面, 埃及人发现了圆的长度和面积之间的关系 ( 这次是一个精确的关系 ). On the other hand the Egyptians had discovered the relation (an exact relation, this time) between the length and the area of a circle. 圆的面积等于它的长度, 正如在直径上构造的正方形的面积等于它的周长一样. The area of a circle is to its length as the area of the square constructed on the diameter is to its perimeter. 在现代代数记数法中, 如果 R 是圆的半径, 这意味着 In modern algebraic notation, if R is the radius of the circle, this means πr 2 2πR = (2R) 2 4(2R). 埃及人也知道如何计算金字塔的体积 : The Egyptians also knew how to compute the volume of a pyramid: 3 1 底座 高度. base height. 3 我们不知道他们是如何发现这个公式的, 但我们可以很容易地想象他们会如何利用这个公式. We do not know how they discovered this formula, but we can easily imagine how they would have made use of it. 问题 56 的艾哈迈斯纸莎草也调查相似的三角形. Problem 56 of the Ahmes papyrus also investigates the similarity of triangles.

19 1.3 Mesopotamia 5 图 1.4 Fig. 1.4 两个直角分别为两个直角, 相对应的角应相等. Two right angled triangles having their respective sides proportional have their corresponding angles equal. 在这个问题中 ( 见图 1.4), 角度是用我们今天所说的它们的余切来测量的. 这样的结果对于金字塔的建造是很重要的, 以便保持坡度恒定 In this problem (see Fig. 1.4), the angles are measured by what we call today their. cotangent. Such a result was important for the construction of pyramids, in order to keep the slope constant. 1.3 美索不达米亚 1.3 Mesopotamia 现在让我们离开尼罗河流域, 跳到美索不达米亚的幼发拉底河和底格里斯河流域. Let us now leave the Nile valley and jump to the valleys of the Euphrates and 因此, 我们放弃了象形文字, 转而使用楔形文字 ( 最早在公元前 3000 年 ), 通常是刻在 the Tigris, in Mesopotamia. Thus we leave behind the hieroglyphs and switch to 泥板上, 而不是纸莎草上 : 这确保了几个世纪以来更好的保护. cuneiform writing (as early as 3000 BC), most often carved on clay tablets instead of papyrus: and this ensured a much better conservation through the centuries. 巴比伦人是杰出的代数学家和天文学家. 他们能够解一 二阶方程 The Babylonians were exceptional algebraists and astronomers. They were able, 甚至一些更高阶的方程. to solve the equations of degree one or two, and even some equations of higher 我们从他们那里继承了时间和角度的性别划分. 有些药片 degrees. We inherited from them the sexagesimal division of time and angles. Some 也是三角表, 给出了角度的割线值. tablets are also trigonometric tables, giving the values of the secants of the angles. 但美索不达米亚和埃及一样, 并没有真正区分精确和近似结果. But Mesopotamia, like Egypt, did not really distinguish between exact and approximate results. A tablet (see [1]) gives the approximate areas of the first seven 平板 ( 见 [1]) 给出了前七个规则多边形的近似面积. 就圆而言, 据称圆内接的正六边形的周长 (= 半径的六倍 ) 等于周长的 regular polygons. As far as the circle is concerned, it is claimed that the perimeter of the regular hexagon inscribed in a circle (= six times the radius) is equal to of 现代记谱法 the circumference. In modern notation 从而产生价值 which yields the value 6R = πR π = 25 8 = 另一块平板声称截锥或棱锥的体积是由底部的一半 (sic) 和乘以高度得到的! Another tablet claims that the volume of a truncated cone or pyramid is obtained 埃及人有正确的公式 ( 三分之一而不是一半 as the half (sic) sum of the base multiplied by the height! Egyptians had the correct ). 就几何学的发展而言, 我们可以假设两个文明之间几乎没有交流. formula (one third instead of one half). As far as the development of geometry is concerned we can assume there was little communication between the two civilizations. 事实上, 巴比伦人正在广泛使用毕达哥拉斯定理, 至少在毕达哥拉斯诞生前一千年. As a matter of fact, the Babylonians were making extensive use of Pythagoras 在一个平板电脑上, 会发现以 theorem, at least one millennium before Pythagoras was born. On one tablet, one 下问题 : finds the following problem:

20 6 1 Pre-Hellenic Antiquity 图 1.5 Fig. 1.5 阿拉德岛和塔龙 a 墙. 柱头当梯子的脚在地面上移动九个单位时, 加达滑翔机可以在墙上监视. A ladder is leant along a wall. The top of the ladder glides of three units along the wall while 梯子的长度是多少? the foot of the ladder moves nine units on the ground. What is the length of the ladder? 另一块石板解释了如何利用毕达哥拉斯定理找到一个圆内弦的顶点. Another tablet explains how to find, using Pythagoras theorem, the apothem of a chord inscribed in a circle. 与人们长期以来的想法相反, 美索不达米亚几何学的发展至少可以与古埃及几何学的发展相媲美, 甚至可能更为优越. Contrary to what was thought for a long time, the development of geometry in Mesopotamia was at least comparable, and maybe even superior, to the development 特别是, 毕达哥拉斯定理的系统应用与埃及的纸莎草书中的明显缺失形 of geometry in Egypt during antiquity. In particular, the systematic use of Pythagoras theorem contrasts with its explicit absence in the Egyptian papyri. 成了鲜明对比. 巴比伦人也知道 The Babylonians also knew that 定理内接在半圆上的角必然是直角. Theorem An angle inscribed in a half circle is necessarily a right angle. 埃及人没有意识到这一事实, 这通常归因于泰勒斯, 谁生活了一千年后. The Egyptians were unaware of this fact, which is generally attributed to Thales, 巴比伦人是如何发现并证明这个结果的? who lived a millennium later. How did the Babylonians find and justify this result? 我们不知道. 或许他们做了以下经验观察 ( 见图 1.5): We do not know. Perhaps they made empirical observations of the following kind (see Fig. 1.5): 设 ABC 为内接在半圆上的角. Let ABC be the angle inscribed in a half circle. 构造 B, 相对于圆的中心 Construct B O 与 B 对称的点., the point symmetric to B with respect to the centre O of the circle. 这产生四个等腰三角形, 形成两个相等的对. This yields four isosceles triangles, forming two equal pairs. 因此 A, B, C, B 的四个角是相等的. Thus the four angles at A, B, C, B are equal. 根据对称性, 四边形 ABCB 的对边是相等的. By symmetry, the opposite sides of the quadrilateral ABCB are equal. 四边形 ABCB 的对角线也相等 ( 并且等于圆的直径 ). The diagonals of this quadrilateral ABCB are equal as well (and equal the diameter of the circle). 这无疑是使他们相信四边形是矩形的充分理由. This was certainly sufficient reason to convince them that the quadrilateral is a rectangle. 1.4 问题 1.4 Problems 证明立方体是三个相等金字塔的并集. 这立刻给出了其中一个金字塔的体积 Show that a cube is the union of three equal pyramids. This yields at once the 公式. formula for the volume of one of these pyramids.

21 1.5 Exercises 考虑一个金字塔, 其底部是一个正方形, 其顶部垂直于底部的中心 ( 如埃及人建造的金字塔 ) Consider a pyramid whose base is a square and whose summit projects orthogonally on the center of the base (like the pyramids constructed by the Egyptians). By a cut and paste argument, infer the formula giving the volume of such 通过剪切和粘贴参数, 推断出给出这种金字塔体积的公式. a pyramid 证明巴比伦人的梯子问题 ( 见. 1.3) 有无限多的解 Prove that the ladder problem of the Babylonians (see Sect. 1.3) has infinitely many solutions. 1.5 练习 1.5 Exercises 通过剪切粘贴参数, 推导平行四边形面积的公式 By a cut and paste argument, infer the formula for the area of a parallelogram 通过剪切粘贴参数, 推导任意三角形面积的公式 By a cut and paste argument, infer the formula for the area of an arbitrary triangle 通过剪切和粘贴参数, 推导任意梯形的面积公式 By a cut and paste argument, infer the formula for the area of an arbitrary trapezium.

22 第二章 Chapter 2 希腊几何学的一些先驱 Some Pioneers of Greek Geometry 本章开始研究数学史上的黄金页. This chapter begins our study of the golden pages of the history of mathematics. 这几页是数学的真正起源, 因为比它们所包含的结果更重要的是, 它们告诉我们数学中演绎方法的诞生. These pages are the true genesis of mathematics, because more importantly than the results that they contain they tell us of the birth of the deductive approach in mathematics. 欧几里得的著名元素 ( 约公元前 300 年 ) 构成了研究希腊几何学的经典参考. The famous Elements of Euclid (around 300 BC) constitute the classical reference for the study of Greek geometry. These books are a kind of achievement and 这些书是一种成就, 将在下一章给予充分的关注. 欧几里德在一个系统的 连贯的理论中组织的许多结 will be devoted full attention in the next chapter. The origins of the many results 果的起源通常不太为人所知, 或者根本不为人所知. organized by Euclid in a systematic and coherent theory are generally less known, or are not known at all. 在本章中, 我们将集中讨论一些希腊几何学家在欧几里得之前所做的努力, 他们在建立几何基础的过程中遇到了各种挑战性的问题 In the present chapter, we focus on the efforts of some Greek geometers, anterior. to Euclid, who encountered various challenging problems on their journey towards 在这样做的过程中, 他们慢慢地收集了所有必要的成分, 最终导致了欧几 establishing the bases of geometry. In doing so, they slowly gathered all the necessary ingredients, eventually leading to the marvelous synthesis written by Euclid, 里德所写的奇妙的合成, 同时也为阿基米德 阿波罗 帕普斯和其他人的进一步工作开辟了道路. but also opening the road to further work by Archimedes, Apollonius, Pappus and 这些开创性的几何学家从他们的前辈那里学到了关于三角形和圆的各种基本结果. others. These pioneering geometers had learned from their predecessors various basic results on triangles and on the circle. What did they know exactly? Who gave 他们到底知道什么? 谁第一次给出了这些经验已知结果的正式证明? 这样的证据是基 for the first time a formal proof of these empirically known results? On which 于哪些论据和哪些证据? 我们不知道. 然而, arguments and which evidences was such a proof based? We do not know. Nevertheless, when necessary, and to avoid repetition, we shall freely use these basic 在必要时, 为了避免重复, 我们将在本章中自由使用这些基本结果, 并将读者引向下一章, 在这一章中, 欧几里德对这些问题进行了系统的论述. results in this chapter and refer the reader to the next chapter where Euclid s systematic treatment of these matters is presented. 在希腊几何学的最大挑战中, 每个人都听说过著名的圆平方问题, 可能还听说过立方体和角的三等分 ( 否定 ) 对两千多年后出现的这 Among the greatest challenges of Greek geometry, everybody has heard of the 些问题的答案! famous circle squaring problem, and probably also of the duplication of the cube and the trisection of the angle. A final (negative) answer to these problems was found 但是, 为解决这些问题所做的许多不成功的努力导致了许多技术和结果的发现, more than two millenniums later! But the many unsuccessful efforts developed to 这些技术和结果比引起它们的问题更为重要, 例如圆锥曲线理论. try to solve these problems resulted in the discovery of a host of techniques and results which are much more important than the problems that gave rise to them, such as, for example, the theory of conics. 对于几何学和数学的发展来说, 不太流行但更基本的是不可公度和穷竭法. Less popular but much more fundamental for the development of geometry and eventually, of mathematics are the incommensurable magnitudes and the method 从现代意义上讲, 非有理数和实数的发现 of exhaustion. In modern terms, the discovery of irrational and real numbers from 波尔图, 几何的公理方法, DOI / , F. Borceux, An Axiomatic Approach to Geometry, DOI / _2, 2014 年瑞士斯普林格国际出版公司 Springer International Publishing Switzerland

23 10 2 Some Pioneers of Greek Geometry 一种几何观点和使用极限来处理曲线图形. 值得注 a geometric point of view and the use of limits to handle curved figures. It is 意的是, 在德德金诞生前 23 个世纪, 希腊几何学家就已经考虑并使用了德德金切割法 : 区别很简单, 希腊几何学家并没有把它们称为数 remarkable that Dedekind cuts had already been considered and used by the Greek 字. geometers 23 centuries before Dedekind was born: the difference was simply that, the Greek geometers did not call them numbers. 2.1 米利都的泰勒斯 2.1 Thales of Miletus 米利图斯的泰勒斯 (c. 公元前 546 年 ) 是第一个希腊几何明确提到的文件中, 我们已经到达. Thales of Miletus (c. 620 BC c. 546 BC) is the first Greek geometer explicitly mentioned in the documents that have reached us. However these documents rely only 然而, 这些文件仅仅依赖于传统, 因为它们是在泰利斯死后几个世纪写的. 因此, 关于 on tradition, since they were written several centuries after Thales death. For that 泰勒斯及其作品存在着一些争议. reason, some controversy exists concerning Thales and his work. 泰利斯游历了埃及和美索不达米亚, 在那里他接触到了大量重要的科学知识. Thales travelled through Egypt and Mesopotamia, where he came into contact 这大量的信息在泰尔聪明的头脑中找到了一块肥沃的土 with a wealth of important scientific knowledge. This mass of information found in 地, 在上面它能够生长和繁荣. Thale s brilliant mind a fertile ground upon which it was able to grow and flourish. 由于底比斯被授予被公认为第一位希腊几何学家的荣誉, 他也被授予许多他在旅行中仔细收集的结果的父权. As Thebes is given the honor of being recognized as the first Greek geometer, he has also been granted the paternity of many results that he carefully gathered during 例如 : his trips. For example: 内接在半圆上的角是直角. An angle inscribed in a half circle is a right angle. 等腰三角形底部的角度相等. The angles at the base of an isosceles triangle are equal. 当两条线相交时, 相反的角度相等. When two lines intersect, the opposite angles are equal. 当然, 这些结果早在泰勒斯之前就已经知道了, 但泰勒斯可能是第一个提供了正式证据的人. Of course, these results were known long before Thales, but Thales may have been the first to have provided a formal proof of them. 事实上, 传统认为泰勒斯是第一个用逻辑和演绎论证证明其定理的人. Indeed, the tradition attributes to Thales the merit of having been the first one to 例如, 他所取得的非常基本的成果 prove his theorems using logical and deductive arguments. The kind of very basic results attributed to him, for example 直径把圆切成两等分. A diameter cuts a circle in two equal pieces. 当第一个三角形的一条边等于第二个三角形的一条边时, 两个三角形就相等, 并且这两条边末端的对应角也相等. Two triangles are equal as soon as a side of the first triangle is equal to a side of the second triangle, and the corresponding angles at the extremities of these two sides are mutually equal as well. 似乎表明他可能试图从这些基本结果中推断出更复杂的结果. Seem to indicate that he probably tried to infer more sophisticated results from these 不幸的是, 没有一份关于泰利斯工作的幸存文件能证实这一点. elementary ones. Unfortunately, no surviving document relating to Thales work 因此, 我们必须非常小心地把真理从各种传统和传说中分离出来, 这些传统和传说把他描绘成数学家 商人 can confirm this. Thus it is with the greatest care that we have to separate truth from 政治家 天文学家... 还有单身汉. various traditions and legends which depict him as a mathematician, a tradesman, a politician, an astronomer... and a champion of bachelorhood. 亚里士多德 ( 公元前 384 年, 公元前 322 年 ) 写道 : Aristotle (384 BC 322 BC) wrote: 对泰利斯来说, 根本的问题不是我们知道什么? For Thales, the fundamental question was not What do we know? 但我们怎么知道呢? but How do we know it?. 这种完美的理想使希腊的几何学家们创造了一部令人难忘的作品, 两千年来, 它一直被认为是数学思想的最高成就. This ideal of perfection led Greek geometers to create an unforgettable work which, for two millennia, has been considered as the highest achievement of mathematical thought. 然而, 今天泰勒斯的名字首先与以下结果联系在一起 : Today, however, the name of Thales is above all attached to the following result:

24 2.1 Thales of Miletus 11 图 2.1 Fig. 2.1 Thales 截获定理包含两条任意线性 d 和四条平行线 d1 d2 d3 d4, 如图 2.1 的左图所示 Thales intercept theorem Consider two arbitrary lines d and d., and four parallel lines d 1, d 2, d 3, d 4 cutting d and d 一个人的比率 as in the left hand diagram of Fig One has 是相等的 then the equality of ratios AB A B = CD C D. 泰勒斯证明了这个结果吗? 如果是, 他是怎么证明的? 我们不知道. 传统的说法 Did Thales prove this result? If so, how did he prove it? We do not know. The tradition claims that Thales used this result to measure the height of the Cheops pyra- 是泰利斯用这个结果来测量基奥普斯金字塔的高度. 关于这一结果, 我们所听到的最古老的论据有时被称为石器时代的证据, 因为它的灵感来自于公元前 2500 年左右的新石器时代 mid. The oldest argument which has reached us concerning this result is sometimes 石碑上的一幅图片, 但它仍然出现在 20 世纪的一些教科书中! called the stone age proof, since it is inspired by a picture found on a neolithic stele of around 2500 BC, but it still appears in some textbooks of the twentieth century! 首先考虑图 2.1 的右图中描述的特殊情况, 其中我们选择了 AB=CD. Consider first the special case depicted in the right hand diagram of Fig. 2.1, where we have chosen AB = CD. 把 AX, CY 和 d 画平行. Draw AX, CY parallel to d. 三角形 AXB 和 CYD 相等, 因为 AB=CD, 而按平行度计算, 它们对应的角度相等. The triangles AXB and CYD are equal, because AB = CD, while by parallelism, their corresponding angles are equal. 因此 AX=CY, 由于 AXB A 和 CYD C 是平行四边形 Thus AX = CY and since AXB, A AX= and CYD C are parallelograms, AX = A A B, B CY=C D,, CY = C D 因此 A B=C, thus A B D. = C D. 这证明了 This proves that AB = CD = A B = C D. 现在我们回到一般情况. 在 d 线上选择一个单位长度, 足够小, 可以用这个单位测量 AB 和 CD. Now we return to the general case. Choose a unit length ε on the line d, sufficiently 我们假设 AB 的长度是 n, BC 的长度是 small to be able to measure both AB and CD with this unit. Let us say that AB has m, 其中 n 和 m 是两个整数. 在第一种情况下, d 上的所 length nε and BC has length mε, where n and m are two integers. By the first case, 有单位长度都以相同长度的段投射到 d 上, 比如说. all unit lengths on d project on d in segments of the same length, let us say, ε 因此,. Thus A A B 和 B B C 具有各自的长度 and B C n 和 m. have respective lengths nε and mε 这最终会产生. This yields eventually 结果证明了这一点. and so the result is proved. AB A B = nε nε = ε ε = mε mε = A B C D

25 12 2 Some Pioneers of Greek Geometry 图 2.2 Fig. 2.2 当然, 我们今天知道, 这个证明有一个很大的差距 : 选择一个单位来测量 AB 段和 CD 段的可能性. Of course we know today that there is a big gap in this proof : the possibility 说有这样一个单位存在, 恰恰意味 of choosing a unit ε to measure both segments AB and CD. Saying that such a unit 着两个长度之比是一个有理数. 我们 ε exists means precisely that the ratio of the two lengths is a rational number. We 将在图 2.16 中回到这个困难. shall come back to this difficulty in Fig 泰勒斯定理的主要应用是相似三角形理论 ( 见图 2.2): The main application of Thales theorem is the theory of similar triangles (see Fig. 2.2): 推论如果两个三角形对应的角成对相等, 则它们对应的边的比率相同. Corollary If two triangles have their corresponding angles pairwise equal, then their corresponding sides are in the same ratio. 实际上, 如果两个三角形 A B C 和 ABC 分别在 A 和 A B Indeed if the two triangles ABC and A 和 B B C C 和 C 处具有相等的角度, 则将三角形 BAC 转换到三角形 BAC 上, 迫使 A 和 have equal angles, respectively at A A 处的角度重合 and A., B and B, C and C, translate the triangle B A C onto the triangle BAC, forcing the angles at A and A 由于 B 和 B 处的角度也相等, 因此 B C 和 BC to coincide. Since the angles at B and B 线平行, 因此泰雷兹 are equal 定理适用 : as well, the lines BC and B C are parallel and therefore Thales theorem applies: AB A B = AC A C. 一个类似的论点, 迫使 B 和 B 的角度重合, 进一步产生 An analogous argument, forcing the angles at B and B to coincide, yields further 所以 and so BA B A = BC B C AB A B = AC A C = BC B C. 相似三角形上的这一结果对希腊几何学的发展起到了至关重要的作用. This result on similar triangles played an essential role in the development of Greek geometry.

26 2.2 Pythagoras and the Golden Ratio 毕达哥拉斯与黄金比例 图 2.3 Fig Pythagoras and the Golden Ratio 毕达哥拉斯. 公元前 500 年 ), 和泰利斯一样, 在穿越埃及和美索不达米亚的途中学会了几何学. Pythagoras (c. 580 BC c. 500 BC), like Thales, learned geometry while travelling 他特别学习了现在被称为毕达哥拉斯定理的东西, 一千年前在埃及就知道了. through Egypt and Mesopotamia. He learned in particular what is now known as Pythagoras theorem, known in Egypt one millennium earlier. 毕达哥拉斯定理给出一个直角三角形, 直角边上的正方形面积之和等于斜边上的正方形面积 ( 见图 2.3). Pythagoras theorem Given a right angled triangle, the sum of the areas of the squares constructed on the sides of the right angle equals the area of the square constructed on the hypotenuse (see Fig. 2.3). 当然, 毕达哥拉斯知道或者可能给自己一个正式的演绎证明. Quite certainly, Pythagoras knew or maybe gave himself a formal deductive 下面的证明, 基于类似三角形的理论, 因此基于泰勒斯定理 ( 见第. proof of this result. The following proof, based on the theory of similar triangles 2.1) 一直很受欢迎, 毕达哥拉斯可能也知道. thus on Thales theorem (see Sect. 2.1) has always been very popular and was probably known to Pythagoras. 直角三角形 ABC 和 ADC 在 a 处有一个公共角度, 因此是相似的 ; 类似地, 三角形 ABC 和 BDC 是相似的. The right angled triangles ABC and ADC have a common angle at A, thus are 这就产生了 similar; analogously the triangles ABC and BDC are similar. This yields AB AD = AC AC AB 和 BC AB and AC= BC AC = DC BC. 因此 Therefore AB 2 + BC 2 = AD AC + DC AC = (AD + DC)AC = AC AC = AC 2. 早在希腊, 毕达哥拉斯就建立了一个宗教团体, 专门研究哲学和数学问题. Back in Greece, Pythagoras founded a religious society involved in philosophical 像以前那样频繁 ( 有时还是今天? ) 追随 and mathematical questions. As often in those days (and still sometimes today?) the 者们发现的结果都归功于领袖, 甚至在他死后! results discovered by the followers were attributed to the leader, even after his death! 因此, 当我们提到毕达哥拉斯的作品时, 将其解读为毕达哥拉斯学派的作品可能更为明智. So when we mention the work of Pythagoras, it would probably be more sensible to interpret this as the work of his school. 毕达哥拉斯人把神奇的美德归因于一些数字和一些几何形式, 特别是规则形式. The Pythagoreans attributed magical virtues to some numbers and some geometric forms, in particular the regular forms. Some historians claim that they already 一些历史学家声称他们已经知道五个正多面体 : 四面体 ( 四个三角形 ) 立方体( 六个正方形 ) 八面体( 八个三角形 ) 十二面体( 十二个五角体 ) 和二十面体 ( 二十个三角形 ). knew the five regular polyhedrons: the tetrahedron (four triangles), the cube (six squares), the octahedron (eighttriangles), thedodecahedron (twelvepentagons)and 但没有证据证明这一事实. the icosahedron (twenty triangles). But there is no evidence of this fact. 另一方面, 事实上, 毕达哥拉斯派的标志是五角大楼, 巴比伦人已经知道了 ( 见图 2.4). On the other hand, it is a matter of fact that the emblem of the Pythagorian sect was the star (regular) pentagon, already known to the Babylonians (see Fig. 2.4). 它可能是由几何学上的轮廓学家发现了所谓的黄金地段. It was perhaps by scrutinizing this geometric figure that Greek geometers discovered 让我们更明确地说明这一点, 自由地使用圆内接角的性质, 这将在下一章中得到证明. the so-called Golden section. Let us make this point more explicit, freely using the properties of angles inscribed in a circle, which will be proved in the next chapter. 我们不知道毕达哥拉斯是否知道这些结果, 也不知道他是否用其他论据来达到他的最终目标. We do not know if Pythagoras knew these results or if he used other arguments to reach his final goal.

27 14 2 Some Pioneers of Greek Geometry 图 2.4 Fig. 2.4 两个内接在一个圆上的角与截取同一和弦的角相等 ( 见命题 3.3.5). Two angles inscribed in a circle and intercepting the same chord are equal 因此 (BCA)=(BDA). 加上图形的五边形对称性, 这就产生了 (see Proposition 3.3.5). Thus (BCA) = (BDA). Together with the pentagonal symmetry of the figure, this yields ( BCD ) = (BCA) = (BDA) = (CEB). 当然, And of course, ( CBD ) = (CBE). 所以两个三角形 BCD 和 CEB 是相似的, So the two triangles BCD 特别是因为 CEB 是等腰的, 所以三角形 BCD 也是等腰的. and CEB are similar; in particular since CEB is isosceles, so is the triangle BCD 相似三角形边的比例进一步告诉我们. The proportionality of sides in similar triangles tells us further that BE CD = BC BD 因此, 通过五边形对称, 由于三角形是等腰的 thus also, by pentagonal symmetry and since the triangles are isosceles AC BC = AC CD = CD AD. 第二个等式可以重新表述为 The second equality can be rephrased as 点 D 将段 AC 切成两段 The point D, 使得整个段是最长的部分, 而最长的部分是最小的部分. cuts the segment AC into two pieces, in such a way that the whole segment is to the longest part as the longest part is to the smallest one. 此外, 第一等式表明, 这个比例正是恒星正五边形的一面和相应凸正五边形的一面之间的比例. Furthermore, the first equality shows that, this proportion is precisely the proportion between a side of the star regular pentagon and a side of the corresponding convex regular pentagon. 黄金比率黄金比率是恒星五边形侧面和相应的正五边形侧面之间的比率. The Golden ratio The Golden ratio is that between the side of the star pentagon and the side of the corresponding regular pentagon. 因此, 一个片段的黄金部分的特点是它无限地重复自己 : 当你意识到一个片段的黄金部分时, 只要把小部分放在长部分上, 你就 The Golden section of a segment is thus characterized by the fact that it repeats 得到了长部分的黄金部分 ; 你可以随心所欲地重复这个过程. itself infinitely: when you have realized the golden section of a segment, simply put the small part on the long one and you get a golden section of the long part; 这个神奇的属性 and you can repeat the process as many times as you want. This magic property

28 2.2 Pythagoras and the Golden Ratio 15 图 2.5 Fig. 2.5 图 2.6 Fig. 2.6 希腊的几何学家认为黄金分割是最理想的, 而希腊的建筑师推断, 最美观的矩形是其边的黄金比例 ( 见图 2.5): 他们所有的神庙都是按照 led Greek geometers to consider the golden section as the ideal one, and Greek 这一规则建造的. architects inferred that the most aesthetic rectangle is that whose sides are in the golden ratio (see Fig. 2.5): all their temples were constructed following this rule. 黄金比率的值很容易计算. 在正规五角大厦的边上写 1, 在对应的星形五 The value of the golden ratio is easy to compute. Write 1 for the side of the 角大厦的边上写 x. 因此黄金比率是 x, regular pentagon and x for the side of the corresponding star pentagon. The golden 其特征是 ratio is thus x and is characterized by x 1 = 1 x 1 从而得出方程 which yields the equation x2 x 2 固定 1=0 x 1 = 0 其正解由 whose positive solution is given by x = AC 段的黄金部分可以很容易地用尺子和罗盘建造 ( 见图 2.6). The golden section of the segment AC can easily be constructed with ruler and 用长度 12 画垂直于 AC 的轴 AC. 图纸中心 X 和半径 X a 的圆, 在 Y compass (see Fig. 2.6). Draw AX perpendicular to AC with length 1 处切割 X C. 然后绘制中心 C 和半径 AC. Drawa CY 的圆, 在 D 处切割 AC. 2 circle of center X and radius XA, cutting XC at Y. Next draw a circle of center C and radius CY, cutting AC at D 这一点 D 实现了 AC 段的黄金分割. This point D. realizes the golden section of the 事实上毕达哥拉斯定理表明 segment AC. Indeed Pythagoras theorem shows that XC 2 = AX 2 + AC 2 = AX 2 + 4AX 2 = 5AX 2. 因此 It follows that AC D C = 2AX YC = 2AX XC XY = 2AX 2 = = AX AX 毕达哥拉斯人很可能知道黄金地段的几何结构. This geometric construction of the golden section was most likely known to the Pythagoreans.

29 16 2 Some Pioneers of Greek Geometry 注意, 上面的构造也产生了一种构造规则五边形的方法, 因为星形五边形的角度是等腰三角形的角度, 它的边和底是黄金比例. Notice that the construction above also yields a way of constructing the regular pentagon, since the angle of the star pentagon is that of an isosceles triangle whose 因此, 在毕达哥拉斯时代, 希腊的几何学家已经能够用尺子和指南针来 side and base are in the golden ratio. So, already at the time of Pythagoras, Greek 构造等边三角形 正方形 正五边形, 当然, 所有的多边形都是通过不断地将边的数量加倍而得到的. geometers were able to construct with ruler and compass the equilateral triangle, the square, the regular pentagon and of course, all polygons obtained from these 后来, 他们还发现了如何构造 15 条边的正多边形. by successively doubling the number of sides. Later, they also discovered how to 自从 construct the regular polygon with 15 sides. Since (2 3) 5 = 1 一下子就得到 one gets at once 1 15 = = 因此, 若要将圆分成 15 等分, 则只需先在直角三角形中画出两个连续的边, 然后再从直角三角形的角度画回. Thus to divide the circle into 15 equal parts, it suffices to first draw two consecutive sides of the regular pentagon, and next to come back along one side of the equilateral triangle. 就规则多边形而言, 这是所有希腊几何学家所能建造的. As far as regular polygons are concerned, this is all the Greek geometers were 在这个问题上的进一步进展需要等待两千年, 当时高斯描述了 17 条边的正多边形的构造. able to construct. Further progress on this problem had to wait for two thousand years, when Gauss described the construction of the regular polygon with 17 sides. 在同一时期, 伽罗瓦等人关于用根解方程的工作提供了用尺和罗盘构造正多边形的一个 if-an-if 条件 ( 见定理 C.6.4). During the same period, the work of Galois and others on the solution of equations by radicals provided an if an only if condition for a regular polygon to be constructible with ruler and compass (see Theorem C.6.4). Nowadays, however, we still 然而, 现在我们仍然只知道六个基本的这样的规则多边形 : 那些有 3, 4, 5, 17, 257 和 边的多边形! only know six basic such regular polygons: those with 3, 4, 5, 17, 257 and sides! 2.3 三等分角度 2.3 Trisecting the Angle 泰勒斯定理 ( 见第. 2.1) 特别告诉我们如何将任意线段 AB 分成 n 个相等的部分 ( 见图 2.7): Thales theorem (see Sect. 2.1) tells us in particular how to divide an arbitrary segment AB into n equal parts (see Fig. 2.7): 通过 A 画另一条线 ; 从 A 开始, 在该线上画任意长度 draw another line through A; 的 n 个连续相等的线段 ; from A, on that line, draw n consecutive equal segments of arbitrary length; 画线 XnB; 通过点 Xi, 画出与 XnB 的 draw the line X n B; 平行线 ; 各个交点 Yi 将线段 AB 等分为 n 个部分. through the points X i, draw the parallels to X n B; the various intersection points Y i divide the segment AB into n equal parts. 所以下一个自然的问题是 So the next natural question is 如何用 n 等分来划分任意角度? How to divide an arbitrary angle with n equal parts? 当然, 希腊几何学家试图解决这个问题. Of course, Greek geometers tried to solve this problem. 将角度 BAC 等分为两部分很容易 ( 见图 2.8): Dividing an angle BAC in two equal parts is easy (see Fig. 2.8): 画一个中心为 A 的任意圆, 在 X 处切 AB, 在 Y 处切 AC; draw an arbitrary circle with center A, cutting AB at X and AC at Y ; 画两个圆心分别为 X 和 Y, 穿过 A, 在 Z 相交的圆 ; draw two circles with respective centers X and Y, passing through A and intersecting at Z;

30 2.3 Trisecting the Angle 17 图 2.7 Fig. 2.7 图 2.8 Fig. 2.8 图 2.9 Fig. 2.9 一个得到一个菱形轴, 它的对角线将角 XAY 分成两个相等的部分. one obtains a rhombus AXZY and its diagonal AZ divides the angle XAY into two equal parts. 所以下一步是把角 BAC 分成三等分 : 所谓的角的三等分. 同样, 只有在 19 世纪, 才证明仅使用尺子和罗盘是不可能解决这个问题的 ( 见命题 So the next step is to divide the angle BAC into three equal parts: the so-called trisection of the angle. Again, it was only during the nineteenth century that the im- B.2.1). possibility of this problem using only ruler and compass was proved (see Proposition B.2.1). 公元前 420 年左右, 希庇亚对把一个角分成 n 个等分的问题提出了以下解决方案 ( 见图 2.9 的左图 ). Around 420 BC, Hippias proposed the following solution to the problem of dividing an angle into n equal parts (see the left hand diagram of Fig. 2.9). 考虑一个正方形 ABCD. Consider a square ABCD. 让侧 DA 以恒定的角速度围绕中心 D 旋转, 直到到达最终位置 DC. Let the side DA rotate around the centre D, at constant angular speed, until it reaches the final position DC. 同时, 让 AB 侧以恒定的线速度向下移动, 始终保持与 DC 平行, 直到到达相同的最终位置 DC. At the same time, let the side AB move down at constant linear speed, remaining all the time parallel to DC, until it reaches this same final position DC.

31 18 2 Some Pioneers of Greek Geometry 考虑曲线 ( 所谓的希庇亚三色线 ), 它是两个运动段交点的轨迹. Consider the curve (the so-called trisectrix of Hippias) which is the locus of the intersection point of the two moving segments. 这条曲线因此被完美地定义了, 除了它的终点 : 移动段的两个最终位置, 即 DC 和 DC 的交点! This curve is thus perfectly defined, except for its final point: the intersection of 但是这个对 the two final positions of the moving segments, that is, of DC with DC! But this 应于零角的点在这个问题中是不相关的. point corresponding to the zero angle is irrelevant in the present problem. 简单地说, 这个三等分线立刻解决了将任意角度 XDC 分成 n 个相等部分的问题 ( 参见图 2.9 的右图, 其中 n=3) Trivially, this trisectrix solves at once the problem of dividing an arbitrary angle XDC into n equal parts (see the right hand figure of Fig. 2.9, where n = 3) 写 Y 表示 X 在 DA 上的正交投影 ; 通过点 Yi 将线段 DY 分成 n 个等分 ; 通过每个点 Yi 画出与 DC 平 write Y for the orthogonal projection of X on DA; 行的线, 在点 Xi 处切割三等分线 ; 因此, 线 DXi 将角 XDC 分成 n 个等分. divide the segment DY into n equal parts via points Y i ; through each point Y i draw the parallel to DC, cutting the trisectrix at a point X i ; the lines DX i thus divide the angle XDC into n equal parts. 但当然, 绘制三等分线的所有点不可能通过尺子和罗盘结构来实现. But of course, drawing all the points of the trisectrix cannot possibly be achieved 第一, 有无限多的点, 第二, 并非所有的点都可以用尺子和罗盘来构 via a ruler and compass construction. First, there are infinitely many points, and 造 : 例如, 我们将在章节中看到. second, not all points are constructible with ruler and compass: for example, as we B.2. 与 20 度角相对应的点无法构造. shall see in Sect. B.2, the point corresponding to an angle of 20 degrees cannot be constructed. 公元前四世纪, 希庇亚的这种结构, 似乎是第一个已知的对曲线而不是直线或圆的数学描述. This construction of Hippias, in the fourth century BC, seems to be the first known mathematical description of a curve other than a straight line or a circle. 此外, 曲线是由一个动力学参数定义的. Moreover, the curve is defined by a dynamical argument. 2.4 方圆 2.4 Squaring the Circle 如今, 圆的平方这个短语被用作试图解决一个不可能解决的问题的同义词 : 一个只有疯狂的数学家才会感兴趣的问题. The phrase to square the circle is nowadays used synonymously with trying to solve an impossible problem: a problem that only a crazy mathematicians would be interested in. Indeed, every sensible person knows that a circle is not square shaped. But 事实上, 每个明智的人都知道圆不是正方形的. 但这真的是重点吗? 一点也不! is that really the point? Not at all! 手稿 ( 见章节. 1.2) 据说公元前 2000 年左右, 吉普赛人已经可以计算出三角形的面积当然是 The Ahmes manuscript (see Sect. 1.2) tells us that around 2000 BC, the Egyptians 梯形的任何多边形的面积也可以计算出来, 只要把它分成三角形就行了. were already able to compute the area of a triangle or a trapezium. Of course the area 所 of any polygon could be computed as well, simply by dividing it into triangles. So 以下一步显然是计算圆的面积. 更普遍地说, 人们可能对计算使用圆的弧 the next step was clearly to compute the area of a circle. More generally, one could 构造的任意图形的面积感兴趣, 甚至可以同时使用圆的段和弧. be interested in computing the area of an arbitrary figure constructed using arcs of 如前所述. circles, or even, using both segments and arcs of circles. As already mentioned in 1.2 埃及人知道一些实用的计算圆面积的公式. Sect. 1.2, the Egyptians knew some pragmatic formulas to compute the area of some 但这些是近似公式, 可能取决于圆的大小! circles. But these were approximate formulas, possibly depending on the size of 对于希腊几何学家来说, 计算一个圆的面积意味着用尺子和罗盘精确地构造一个面积与这个圆相同的正方形 ( 从中我们 the circle! For Greek geometers, computing the area of a circle meant constructing 得到了圆平方这个术语 ). precisely, with ruler and compass, a square having the same area as this circle (from which we get the term circle squaring ). 一种传统, 被普鲁塔克引用 ( 约 ) 在他关于流亡的著作中, 将其归因于阿纳克萨戈拉斯 (?)? A tradition, cited by Plutarch (ca ) in his work On Exile, attributes to ( 公元前 428 年 ) 圆平方问题的父系. 阿纳克萨格拉斯曾 Anaxagoras (? 428 BC) the paternity of the circle squaring problem. Anaxagoras 因公开宣称太阳不是上帝而被监禁在雅典, 因为它只是一块巨大的石头, 由于酷热而发光, 而且月亮只是反射太阳的光. had been jailed in Athens for publicly claiming that the Sun is not a god, but merely a very large stone that was glowing due to its extreme heat, and furthermore the 为了在监狱里呆上一段时间, 阿纳克萨格拉斯转向了一个问题 : Moon is simply reflecting the Sun s light. To occupy his time in jail, Anaxagoras 试图建造一个面积与给定圆相同的广场. turned to the problem of trying to construct a square having the same area as a given circle.

32 2.4 Squaring the Circle 19 图 2.10 Fig 传统还归因于希波克拉底, 大约公元前 430 年, 以下定理, 这是研究圆的必要条件 ( 见图 2.10): The tradition also attributes to Hippocrates, around 430 BC, the following theorem, which is essential for studying the circle (see Fig. 2.10): 两个相似圆段的面积与其基的平方成相同的比例. The areas of two similar circular segments are in the same ratio as the squares of their bases. 鉴于当时的数学知识, 希波克拉底不太可能知道这个结果的正式证明. In view of the mathematical knowledge of the time, it is unlikely that Hippocrates knew a formal proof of this result. 当然, 这个结果特别适用于半圆的情况, 在这种情况下, 底是直径, 我们从中可以马上得到 : Of course the result applies in particular to the case of half circles, in which case the base is the diameter, from which we obtain at once: 两个圆的面积与其直径的平方成相同的比例. The areas of two circles are in the same ratio as the squares of their diameters. 在当代, 考虑半径 R 和 R 的两个圆以及各自的区域 In contemporary terms, considering two circles of radii R and r and respective areas A 和 A, 这个结果告诉我们 A and a, this result tells us that A a = (2R) 2 (2r)2 因此, R2=R2. thus A (2r) 2 R = a 2 r. 2 写下最后一个比率, 因此与圆的大小无关, 我们得到了众所周知的公式 Writing π for this last ratio, thus independent of the size of the circle, we get the well-known formula A = πr 2. 但是, 让我们清楚地表明, 对于希腊几何学家来说, 上面的论证中根本没有数字 : 只是两个面积的比率, 这对所有的圆都是一样的. But let us make clear that, for the Greek geometers, there was no number π at all in the argument above: just a ratio of two areas, which was the same for all circles. 两千年后才正式定义实数! 然而, 观察到同样的 Real numbers were only formally defined two thousand years later! Nevertheless, 比率被用来计算每一个圆的面积, 不管它的大小, 无疑是一个很大的成就. observing that this same ratio is to be used to compute the area of every circle, whatever its size, was certainly a very big achievement. 因此, 知道半径 R 的圆的面积, 就等于知道这个比值的值, 也就是说, 一旦单位长度固定下来, 就可以构造一段长度. 希腊几何 So knowing the area of a circle of radius R reduces to knowing the value of this 学家已经知道, 用加减乘除法和平方根由给定单位构成的所有长度, 都可以用尺子和指南针构成 ( 见第四节 ). ratio π, that is once a unit length is fixed to constructing a segment of length π. Greek geometers knew already that all lengths constructed from a given unit using additions, subtractions, multiplications, divisions and square roots, are constructible 是那种长度吗? 这对数学家来 with ruler and compass (see Sect. 3.2). Was the length π of that kind? This was 说无疑是个好问题... 不仅仅是那些疯子! 1882 certainly a good question for mathematicians... and not only for crazy ones! The 年林德曼对超越性的证明为希腊圆的平方问题提供了一个最终的否定答案 ( 见第三节 ). proof of the transcendence of π by Lindemann in 1882 provided a final negative answer to the Greek circle squaring problem (See Sect. B.3). 让我们强调这样一个事实 : 希腊的几何学家有充分的理由相信, 用尺子和指南针将一个圆平方是可能的, 毕竟他们能够将由圆的弧 Let us stress the fact that Greek geometers had some good reasons to believe 线界定的更复杂的数字平方. that squaring a circle with ruler and compass could be possible, after all they were 例如, able to square more complicated figures delimited by arcs of circles. For example,

33 20 2 Some Pioneers of Greek Geometry 图 2.11 Fig 图 2.12 Fig 一些卫星. 一个卫星是由两个不同半径的相交圆弧所分隔的图形, 位于它们共同和弦的同一侧 ( 见图 2.11). some moons. Amoon is the figure delimited by two intersecting arcs of circles, of 让我 different radii, situated on the same side of their common chord (see Fig. 2.11). Let 们回顾一下希腊关于月球的两个 ( 许多 ) 结果. us review two of the (many) Greek results concerning moons. 命题考虑图 2.12 的等腰三角形. 画出直径 A C 的半圆和 A 处与 AB 相切, C 处与 CB 相切的圆弧. Proposition Consider the isosceles triangle of Fig Draw the half circle of 对应的月亮与原 diameter AC and the circular arc tangent to AB at A and to CB at C. The corresponding moon has the same area as the original 来的三角形面积相同. triangle. 半圆的中心是线段 AC 的中点 E. 由于角度 ABC 是 The centre of the half circle is the midpoint E of the segment AC. Since the 对的, 它包含在直径 AC 的半圆中, 因此 B 在刚才提到的半圆上. angle ABC is right, it is contained in the half circle of diameter AC, thus B is on 完成正方形 ABCD 后, 点 D 是与 AB 和 CB 相切的圆弧的中心. the half circle just mentioned. Completing the square ABCD, the point D is the 紧接着, 基地 AC 和中心 D 的圆形部分与基地 AB 和中心 E 的 centre of the circular arc tangent to AB and CB. It follows at once that the circular 圆形部分相似. segment of base AC and centre D is similar to the circular segment of base AB and 根据希波克拉底定理, 两个圆段的面积在 centre E. By Hippocrates theorem, the areas of the two circular segments are in the 比率 AC2 ratio AC2 AB2. 但是根据毕达哥拉斯定理,. But by Pythagoras theorem, AB2 AC 2 = AB 2 + BC 2 = 2 AB 2. 因此, 具有基 AC 的圆段的面积是具有基 AB 的圆段面积的两倍, 即精确地说, 具有基 AB 和 BC 的两个相等圆段的面积之和. Thus the area of the circular segment with base AC is twice the area of the circular segment with base AB, that is, precisely the sum of the areas of the two equal 这立刻证明了结果. circular segments with bases AB and BC. This proves at once the result. 这个命题考虑一个内接在半径 R 的圆上的正六边形. 在六边形的每 Proposition Consider a regular hexagon inscribed in a circle of radius R. On each 一侧, 构造一个半圆. 这样得到的六个卫星的面积之和等于六边形的面积减去一个直径 side of the hexagon, construct a half circle. The sum of the areas of the six moons 圆的面积 - so obtained is equal to the area of the hexagon, minus the area of a circle of diameter R.(See Fig. 后 ( 见图 2.13) 2.13.)

34 2.4 Squaring the Circle 21 图 2.13 Fig 图 2.14 Fig 六边形的边的长度为 R, 因此在这些边上构造的半个小圆的半径为 R 2. A side of the hexagon has length R, thus the half small circles constructed on these sides have radius R 根据希波克拉底定理, 每个小圆的大小等于大圆的大小的 14.. By Hippocrates theorem, each small circle thus has size 2 equal to 1 六边形的面积加上六个半小圆的面积, 等于大圆的面积加上六个月亮的面 of that of the big circle. The area of the hexagon, plus that of the six 积. 4 也就是 half small circles, is equal to the area the big circle, plus that of the six moons. That 说, 六边形的面积加上三个小圆的面积等于四个小圆的面积加上六个卫星的面积. is, the area of the hexagon plus that of three small circles is equal to the area of four 这证明了结果. small circles plus the areas of the six moons. This proves the result. 计算六边形的面积是一件容易的事 : 它等于 R 边等边三角形面积的六倍. 因此刚才提到的第二个命题将直径 R 的圆的面积计算简化为月 Computing the area of the hexagon is an easy matter: it is equal to six times the 球的面积. area of an equilateral triangle of side R. So the second proposition just mentioned 正如我 reduces the computation of the area of a circle of diameter R to that of a moon. And 们在上面的第一个命题中所看到的, 由于希腊的几何学家能够计算出一些卫星的面积, 他们有理由希望最终能够计算出一个圆的面积. since Greek geometers, as we have seen in the first proposition above, were able to compute the area of some moons, they could reasonably hope to be able to eventually compute the area of a circle. Unfortunately for them, as the transcendence of π 不幸的是, 正如超越性所显示的那样, 六边形问题所涉及的卫星不能用尺子和罗盘来平方, 因为正如上面的第二个问题所显示的那样, 平方这样的卫星相当于平方圆. shows, the moons involved in the hexagon problem cannot be squared with ruler and compass, since as the second problem above shows squaring such moons is equivalent to squaring the circle. 另一个试图使圆圈方正的尝试值得一提. 这是由于公元前 350 年左右的狄 Another attempt at squaring the circle is worth mentioning. It is due to Dinostrates, around 350 BC, who proved the following property of the trisectrix of Hip- 诺斯特拉特证明了希庇亚三色线的以下性质 ( 见第三节 ). 2.3 和图. pias (see Sect. 2.3 and Fig. 2.14). 命题考虑希庇亚的三等分线, 用中心 D 和半径 DC 画出圆 AC 的弧. Proposition Consider the trisectrix of Hippias and draw the arc of circle ÂC with 如果 Q 是三等分线与段 DC 的交点, 则 centre D and radius DC. If Q is the intersection point of the trisectrix with the segment DC, then 电弧交流 arc ÂC DC = DC DQ. 这是证据, 而不是财产本身, 这很有趣. It is the proof, more than the property itself, which is interesting.

35 22 2 Some Pioneers of Greek Geometry 首先, Dinostrates 考虑了点 Q, 正如我们在 Sect 中所观察到的. 2.3 是 First, Dinostrates considers the point Q which, as we have observed in Sect. 2.3, 希庇亚三色线的唯一一个点, 希庇亚的结构并没有这样定义. is the only point of the trisectrix of Hippias which is not defined as such by the 这一点实际上是希庇亚三色线其他点的极限. Hippias construction. This point is in fact the limit of the other points of the Hippias 当然, 当时不存在极限的概念, 但这并不能阻止恐龙解决这个问题. trisectrix. Of course the notion of limit did not exist in those days, but this did not prevent Dinostrates from solving the problem. 第二, 我们所知道的, 由希腊几何学家给出的所有前面的证明, 都是建设性的证明 : 一个人所做的各种构造, 提供了结果是真实 Second, all the anterior proofs that we know, given by the Greek geometers, are 的证据. constructive proofs : one performs various constructions which provide evidence 但狄诺斯特拉蒂用一种荒谬的还原来证明他的结果 : 他没有给出任何直接的论据来解释为什么结果是 that the result is true. But Dinostrates proves his result by a reductio ad absurdum: 真的, 他只是证明了其他的可能性必然是假的. he does not give any direct argument explaining why the result is true, he simply proves that the other possibilities are necessarily false. 证据如下. 考虑 DC 的 R 点 Dinostrates proof is as follows. Consider the point R of DC such that 电弧交流 arc ÂC DC = DC DR. 我们必须证明 R=Q. 如果不是这样, R 是位于 Q 的左边或右边的 DC 段的另一个点. 让我们证明这两种情况都必须排 We must prove that R = Q. If this is not the case, R is another point of the segment 除. DC, situated on the left or on the right of Q. Let us show that both cases must be excluded. 如果 R 在 Q 的右侧, 让我们画一个中心 D 通过 R 的圆 ( 见图 2.14 的右侧图 ), 在 S 处切割三等分线, 在 T 处切割线段 D a. 在 DC 上写出 S 的 If R is on the right hand side of Q, let us draw a circle with center D passing 正交投影 U. through R (see the right hand diagram of Fig. 2.14), cutting the trisectrix at S and 由于两个周长与 the segment DA at T. Write U for the orthogonal projection of S on DC. Since two 其直径 ( 或半径 ) 的比率相同, 因此可以得到 circumferences are in the same ratio as their diameters (or radii), one gets 电弧交流 arc ÂC 直流 arc TR = DC - 直流 - 交流弧形 arc ÂC thus DR DC = arc TR TR 弧形 TR= DR. 与 R 点定义中的比率相比, 我们得到 Comparing with the ratio in the definition of the point R, we obtain 电弧 TR=DC. arc TR= DC. 根据三色线的定义, 考虑到角 (CDT)=(CDA) 和 (CDS), 我们进一步得到 By definition of the trisectrix, considering the angles (CDT ) = (CDA) and (CDS), we obtain further 因此 Therefore 弧形 TR arc TR = AD arc ŜR SU = DC 弧形 SU = arc TR TR 弧形 SR= SU. 弧 SR=SU arc ŜR = SU 这是一个矛盾, 因为垂直 SU 比连接 S 和 DC 点的任何其他路径都短. which is a contradiction, since the perpendicular SU is shorter than any other path joining S and a point of DC. 当 R 在 Q 的左手边时, 图 2.14 的左手图上的一个类似的参数表示, 弧 TR 与段 DC 具有相同的长度 When R is on the left hand side of Q, an analogous argument on the left hand diagram of Fig shows that the arc TRhas the same length as the segment DC 弧 UR 的长度与 SR 段的长度相同. 一个额外的小技巧是必要的, 以完成证 and the arc ÛR has the same length as the segment SR. An additional small trick 明 : 画紫外垂直于 DS. 然后 is then necessary to complete the proof: draw UV perpendicular to DS. Then by

36 2.5 Duplicating the Cube 23 毕达哥拉斯定理应用于三角 SUV, 边的紫外比斜边的 SV 短. Pythagoras theorem applied to the triangle SUV, the side UV is shorter than the 因此 hypotenuse SV. Therefore 弧 UR<UV+VR<SV+VR<SR arc ÛR <UV + VR<SV + VR<SR 这是一个矛盾. which is a contradiction. 当然, 今天, 我们会说, 在这个证据中有漏洞. 例如, 在证明的第二 Of course, today, we would claim that there are gaps in this proof. For example, 部分中, Dinostrates 没有正式证明弧 UR 比多边形线 UVR 短的事实. in the second part of the proof, Dinostrates does not formally justify the fact that the 然而, 做出这样的批评是忽略了一点 : 这个证据是在公 arc ÛR is shorter than the polygonal line UVR. However, to make this criticism is 元前 350 年写的. to miss the point: this proof was written in 350 BC. 一个更明智的评论是强调这样一个事实 : 当时的数学家已经能够认识到形式论证的有效性, 独立于支持推理的图片. A more sensible comment is to underline the fact that the mathematicians of the time were already able to recognize the validity of a formal argument, independently 图 2.19 的两部分都是错误的, 因为不存在与 Q 不同的 R 点, 且该点 of the picture supporting the reasoning. Both parts of Fig are indeed false, 具有所示的特性. 因此, since, a point R distinct from Q, with the property indicated, does not exist. Thus 希腊几何学家已经发现, 人们可以写下一个正确的证明 Greek geometers had already discovered that one can write down a correct proof 用假照片. using a false picture. 我们不知道恐龙是如何使用这个结果的. 他当然知道计算周长的公式 2r. We do not know precisely how Dinostrates used this result. Of course he knew 因此, 以直流电为单位段, 由 the formula 2πR for computing the length of a circumference. Thus taking DC as 于电弧交流是半径 1 周长的四分之一, 他的结果产生 unit segment, since the arc ÂC is a quarter of the circumference of radius 1, his result yields 那就是 that is 1 4 2π1 1 = 1 DQ π = 2 DQ. 因此, 能够用尺子和指南针来构造点 Q, 就可以给出值, 圆就可以平方了. So being able to construct the point Q with ruler and compass would have given the value of π and the circle would have been squared. 2.5 复制立方体 2.5 Duplicating the Cube 公元前 429 年, 雅典遭受了一场严重的瘟疫. 这个传说告诉我们, 一个代表团去问 In 429 BC, Athens was suffering a severe plague epidemic. The legend tells us that 阿波罗神谕, 他是否可以阻止这种流行病. a delegation went to ask the Apollo oracle if he could possibly stop this epidemic. 神谕回答说 : 是的, 如果你把阿波罗的立方祭坛翻一番坦普尔 The oracle answered: Yes, if you double the cubic altar in the Apollo temple. The 雅典人立即建造了一座更大的祭坛, 但瘟疫并没有停止. 于 Athenians immediately constructed a larger altar, but the epidemic did not stop. So 是他们回到神谕那里, 提醒他, 他还得履行合同中的义务. they went back to the oracle to remind him that he still had to meet his part of 但神谕回答说 : 但你没有达到我的要求! 你 the contract. But the oracle answered: But you did not meet my requirement! You 把祭坛的所有尺寸都加倍, 这样你就把它的体积乘以 8, 而不是 doubled all the dimensions of the altar, thus you multiplied its volume by 8, not 到 2 点. 雅典人称当时最好的几何学家试图解决这个新问题, 但没有人能! by 2. The Athenians called the best geometers of that time to try to solve the new 尽管如此, 这种流行病最终还是停止了. 这至少 problem, but no one could! Nevertheless, eventually, the epidemic stopped. This 证明了阿波罗的仁慈. proves at least the clemency of Apollo. 如果将原始立方体的边作为单位长度, 那么问题是构造一个体积为 2 的立方体, 即边为 32 的立方体. If you take as unit length the side of the original cube, the problem is thus to construct a cube with volume 2, that is, a cube whose side is 3 全世界的问题 2. The whole problem

37 24 2 Some Pioneers of Greek Geometry 图 2.15 Fig 因此减少到 32 的结构. thus reduces to the construction of 3 与我们在埃及和美索不达米亚所观察到的情况相反, 希腊的地理学 2. In contrast to what we have observed in 家并不接受近似的答案 : 只有他们能够证明形式上精确的解决方案. Egypt and Mesopotamia, Greek geometers did not accept approximate answers: 在几何学中, 他们唯一能使用的两种精确 only solutions that they could prove to be formally exact. In geometry, the only two 仪器是尺子和罗盘. 因此, 解决一个几 precise instruments that they could use were the ruler and the compass. Thus solving 何问题意味着只用尺子和指南针来解决它. a geometric problem meant solving it using only ruler and compass constructions. 在复制立方体的情况下 : 给定一段长度为 1 的线段, 用尺子和指南针构造一段长度为 32 的线段. In the case of the duplication of the cube: given a segment of length 1, construct with ruler and compass a segment of length 3 再次有必要等到 19 世纪才知道这是不可能的 ( 见第. 2. Once more it was necessary to wait until the 19th century to learn that this is impossible (see Sect. B.1). 然而, 为解决这一问题所作的各种努力值得注意, 因为它们产生了许多重要的几何概念和方法. Nevertheless, various efforts made to solve the problem are worth some attention, because they gave rise to a number of important notions and methods in geometry. For example, here is the solution proposed by Archytas, around 380 BC (see 例如, 这是公元前 380 年左右阿奇塔斯提出的解决方案 ( 见图 2.15). 为了简单起见, 我们用现代术语解释. 考虑三维空间及其通常的直角笛卡尔坐标 Fig. 2.15). For simplicity, we explain it in modern terms. Consider the three dimensional space and its usual orthogonal system of Cartesian coordinates (x,y,z). 系 (x, y, z). 首先构造一个顶点 (0,0,0) 的右圆锥体, 其规则穿过半径为 1 的圆 ( 以 (1,0,0) 为中心 ), 在垂直于 X 轴的平面上. Construct first a right circular cone of vertex (0, 0, 0), whose rulings pass through the circle of radius 1, centered at (1, 0, 0), in the plane perpendicular to the x- 因此, 该圆锥体在平面 x=x0 处的截面为半径 x0 的圆 ; 因此, 圆锥体的方程式为 axis. The section of this cone by the plane x = x 0 is thus a circle of radius x 0 ;the equation of the cone is therefore y 2 + z 2 = x 2. 下一个结构圆柱, 其平行于 z 轴, 并在垂直于 z 轴的平面上通过以 (1,0,0) 为中心的半径为 1 的圆. Next construct a circular cylinder, whose rulings are parallel to the z-axis and pass through the circle of radius 1, centered at (1, 0, 0), in the plane perpendicular to 这个圆柱体的方程是这样的 the z-axis. The equation of this cylinder is thus (x 1) 2 + y 2 = 1. 最后, 在垂直于 y 轴的平面上, 围绕半径为 1 的圆 ( 以 (1,0,0) 为圆心 ) 的 z 轴旋转生成圆环. Finally, construct the torus generated by the rotation, around the z-axis, of the circle of radius 1, centred at (1, 0, 0), in the plane perpendicular to the y-axis.