欧几里得《几何原本》电子版

    坦率地说，传统微积分源自欧几里得《几何原本》，而现代微积分源自希尔伯特《几何基础》。这话是有根据的。为什么？

历史上，直到2007年，欧几里得《几何原本》电子版（英文PDF版本）才上网，请查看本文附件（缺图）。

注意：欧几里得《几何原本》电子版的英文原文是：EULID’S ELEMENTS OF GEOMETRY，其PDF版本有示意图。

希尔伯特《几何基础》与欧几里得《几何原本》的对比研究正在进行中。

袁萌  陈启清   3月14日

附件：

EULID’S ELEMENTS OF GEOMETRY

The Greek text of J.L. Heiberg (1883–1885)

from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri, 1883–1885

edited, and provided with a modern English translation, by

Richard Fitzpatrick

First edition - 2007 Revised and corrected - 2008

ISBN 978-0-6151-7984-1

Contents

Introduction 4

Book 1 5

Book 2 49

Book 3 69

Book 4 109

Book 5 129

Book 6 155

Book 7 193

Book 8 227

Book 9 253

Book 10 281

Book 11 423

Book 12 471

Book 13 505

Greek-English Lexicon 539

Introduction

Euclid’s Elements is by far the most famous mathematical work of classical antiquity, and also has the distinction of being the world’s oldest continuously used mathematical textbook. Little is known about the author, beyond the fact that he lived in Alexandria around 300 BCE. The main subjects of the work are geometry, proportion, and number theory.

Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from ve simple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of previously discovered theorems: e.g., Theorem 48 in Book 1.

The geometrical constructions employed in the Elements are restricted to those which can be achieved using a straight-rule and a compass. Furthermore, empirical proofs by means of measurement are strictly forbidden: i.e., any comparison of two magnitudes is restricted to saying that the magnitudes are either equal, or that one is greater than the other.

The Elements consists of thirteen books. Book 1 outlines the fundamental propositions of plane geometry, including the three cases in which triangles are congruent, various theorems involving parallel lines, the theorem regarding the sum of the angles in a triangle, and the Pythagorean theorem. Book 2 is commonly said to deal with “geometric algebra”, since most of the theorems contained within it have simple algebraic interpretations. Book 3 investigates circles and their properties, and includes theorems on tangents and inscribed angles. Book 4 is concerned with regular polygons inscribed in, and circumscribed around, circles. Book 5 develops the arithmetic theory of proportion. Book 6 applies the theory of proportion to plane geometry, and contains theorems on similar gures. Book 7 deals with elementary number theory: e.g., prime numbers, greatest common denominators, etc. Book 8 is concerned with geometric series. Book 9 contains various applications of results in the previous two books, and includes theorems on the innitude of prime numbers, as well as the sum of a geometric series. Book 10 attempts to classify incommensurable (i.e., irrational) magnitudes using the so-called “method of exhaustion”, an ancient precursor to integration. Book 11 deals with the fundamental propositions of three-dimensional geometry. Book 12 calculates the relative volumes of cones, pyramids, cylinders, and spheres using the method of exhaustion. Finally, Book 13 investigates the ve so-called Platonic solids.

This edition of Euclid’s Elements presents the denitive Greek text—i.e., that edited by J.L. Heiberg (1883– 1885)—accompanied by a modern English translation, as well as a Greek-English lexicon. Neither the spurious books 14 and 15, nor the extensive scholia which have been added to the Elements over the centuries, are included. The aim of the translation is to make the mathematical argument as clear and unambiguous as possible, whilst still adhering closely to the meaning of the original Greek. Text within square parenthesis (in both Greek and English) indicates material identied by Heiberg as being later interpolations to the original text (some particularly obvious or unhelpful interpolations have been omitted altogether). Text within round parenthesis (in English) indicates material which is implied, but not actually present, in the Greek text.

My thanks to Mariusz Wodzicki (Berkeley) for typesetting advice, and to Sam Watson & Jonathan Fenno (U. Mississippi), and Gregory Wong (UCSD) for pointing out a number of errors in Book 1.

4

ELEMENTS BOOK 1

Fundamentals of Plane Geometry Involving Straight-Lines

5

STOIQEIWNaþ. ELEMENTS BOOK 1 VOroi. Denitions α. Σημεν στιν, ο μρος οθν. 1. A point is that of which there is no part. β. Γραμμ δ μκος πλατς. 2. And a line is a length without breadth. γ. Γραμμς δ πρατα σημεα. 3. And the extremities of a line are points. δ. Εθεα γραμμ στιν, τις ξ σου τος φ αυτς 4. A straight-line is (any) one which lies evenly with σημεοις κεται. points on itself. ε. Επιφνεια δ στιν, μκος κα πλτος μνον χει. 5. And a surface is that which has length and breadth . Επιφανεας δ πρατα γραμμα. only. ζ. Εππεδος πιφνει στιν, τις ξ σου τας φ 6. And the extremities of a surface are lines. αυτς εθεαις κεται. 7. A plane surface is (any) one which lies evenly with η. Εππεδος δ γωνα στν ν πιπδ δο γραμμν the straight-lines on itself. πτομνων λλλων κα μ π εθεας κειμνων πρς 8. And a plane angle is the inclination of the lines to λλλας τν γραμμν κλσις. one another, when two lines in a plane meet one another, θ. Οταν δ α περιχουσαι τν γωναν γραμμα εθεαι and are not lying in a straight-line. σιν, εθγραμμος καλεται γωνα. 9. And when the lines containing the angle are ι. Οταν δ εθεα π εθεαν σταθεσα τς φεξς straight then the angle is called rectilinear. γωνας σας λλλαις ποι, ρθ κατρα τν σων γωνιν 10. And when a straight-line stood upon (another) στι, κα φεστηκυα εθεα κθετος καλεται, φ ν straight-line makes adjacent angles (which are) equal to φστηκεν. one another, each of the equal angles is a right-angle, and ια. Αμβλεα γωνα στν μεζων ρθς. the former straight-line is called a perpendicular to that ιβ. Οξεα δ λσσων ρθς. upon which it stands. ιγ. Ορος στν, τινς στι πρας. 11. An obtuse angle is one greater than a right-angle. ιδ. Σχμ στι τ π τινος τινων ρων περιεχμενον. 12. And an acute angle (is) one less than a right-angle. ιε. Κκλος στ σχμα ππεδον π μις γραμμς 13. A boundary is that which is the extremity of someπεριεχμενον [ καλεται περιφρεια], πρς ν φ νς thing. σημεου τν ντς το σχματος κειμνων πσαι α 14. A gure is that which is contained by some boundπροσππτουσαι εθεαι [πρς τν το κκλου περιφρειαν] ary or boundaries. σαι λλλαις εσν. 15. A circle is a plane gure contained by a single line ι. Κντρον δ το κκλου τ σημεον καλεται. [which is called a circumference], (such that) all of the ιζ. Διμετρος δ το κκλου στν εθε τις δι το straight-lines radiating towards [the circumference] from κντρου γμνη κα περατουμνη φ κτερα τ μρη one point amongst those lying inside the gure are equal π τς το κκλου περιφερεας, τις κα δχα τμνει τν to one another. κκλον. 16. And the point is called the center of the circle. ιη. Ημικκλιον δ στι τ περιεχμενον σχμα π τε 17. And a diameter of the circle is any straight-line, τς διαμτρου κα τς πολαμβανομνης π ατς περι- being drawn through the center, and terminated in each φερεας. κντρον δ το μικυκλου τ ατ, κα το direction by the circumference of the circle. (And) any κκλου στν. such (straight-line) also cuts the circle in half.† ιθ. Σχματα εθγραμμ στι τ π εθειν πε- 18. And a semi-circle is the gure contained by the ριεχμενα, τρπλευρα μν τ π τριν, τετρπλευρα δ τ diameter and the circumference cuts off by it. And the π τεσσρων, πολπλευρα δ τ π πλεινων τεσσρων center of the semi-circle is the same (point) as (the center εθειν περιεχμενα. of) the circle. κ. Τν δ τριπλερων σχημτων σπλευρον μν 19. Rectilinear gures are those (gures) contained τργωνν στι τ τς τρες σας χον πλευρς, σοσκελς by straight-lines: trilateral gures being those contained δ τ τς δο μνας σας χον πλευρς, σκαληνν δ τ by three straight-lines, quadrilateral by four, and multiτς τρες νσους χον πλευρς. lateral by more than four. κα Ετι δ τν τριπλερων σχημτων ρθογνιον μν 20. And of the trilateral gures: an equilateral trianτργωνν στι τ χον ρθν γωναν, μβλυγνιον δ τ gle is that having three equal sides, an isosceles (triangle) χον μβλεαν γωναν, ξυγνιον δ τ τς τρες ξεας that having only two equal sides, and a scalene (triangle) χον γωνας. that having three unequal sides.