EUCLIDS ELEMENTS OF GEOMETRY

The Greek text of J.L. Heiberg (18831885)

from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibusB.G. Teubneri, 18831885

edited, and provided with a modern English translation, by

Richard Fitzpatrick

Introduction

Euclids Elements is by far the most famous mathematical work of classical antiquity, and also has the distinctionof being the worlds oldest continuously used mathematical textbook. Little is known about the author, beyondthe fact that he lived in Alexandria around 300 BCE. The main subjects of the work are geometry, proportion, andnumber theory.

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

The geometrical constructions employed in the Elements are restricted to those which can be achieved using astraight-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 greaterthan the other.

The Elements consists of thirteen books. Book 1 outlines the fundamental propositions of plane geometry, includ-ing the three cases in which triangles are congruent, various theorems involving parallel lines, the theorem regardingthe sum of the angles in a triangle, and the Pythagorean theorem. Book 2 is commonly said to deal with geometricalgebra, since most of the theorems contained within it have simple algebraic interpretations. Book 3 investigatescircles and their properties, and includes theorems on tangents and inscribed angles. Book 4 is concerned with reg-ular 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 figures. Book 7 dealswith elementary number theory: e.g., prime numbers, greatest common denominators, etc. Book 8 is concerned withgeometric series. Book 9 contains various applications of results in the previous two books, and includes theoremson the infinitude of prime numbers, as well as the sum of a geometric series. Book 10 attempts to classify incommen-surable (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 relativevolumes of cones, pyramids, cylinders, and spheres using the method of exhaustion. Finally, Book 13 investigates thefive so-called Platonic solids.

This edition of Euclids Elements presents the definitive Greek texti.e., that edited by J.L. Heiberg (18831885)accompanied by a modern English translation, as well as a Greek-English lexicon. Neither the spuriousbooks 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 stilladhering closely to the meaning of the original Greek. Text within square parenthesis (in both Greek and English)indicates material identified by Heiberg as being later interpolations to the original text (some particularly obvious orunhelpful interpolations have been omitted altogether). Text within round parenthesis (in English) indicates materialwhich is implied, but not actually present, in the Greek text.

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ELEMENTS BOOK 1

Fundamentals of plane geometry involvingstraight-lines

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. ELEMENTS BOOK 1

+. Definitions. 1 4, 7 :. 1. A point is that of which there is no part.. > @ A C. 2. And a line is a length without breadth.. A @ 1. 3. And the extremities of a line are points.. :1 4, H 4 J 1 4M NA 4. A straight-line is whatever lies evenly with points

1. upon itself.. M 4, Q A R 5. And a surface is that which has length and breadth

S. alone.U. M @ . 6. And the extremities of a surface are lines.. M 4 4, H 4 J 1 4M 7. A plane surface is whatever lies evenly with

NA : 1. straight-lines upon itself.. M @ 4R X 4 4Y 8. And a plane angle is the inclination of the lines,

[ \ C R > 4M : when two lines in a plane meet one another, and are not ] C [ [ . laid down straight-on with respect to one another.

. + @ ^ > R 9. And when the lines containing the angle are:1 _, : 1 X . straight then the angle is called rectilinear.

. + @ :1 4M :1 1 ` 4A 10. And when a straight-line stood upon (another) J C a, b> N [ J straight-line makes adjacent angles (which are) equal to[ 4, R X 41 :1 1, one another, each of the equal angles is a right-angle, and4M c 4. the former straight-line is called perpendicular to that

. M1 4R X bA. upon which it stands.. M1 @ X 4 bA. 11. An obtuse angle is greater than a right-angle.. + 4, e 4 . 12. And an acute angle is less than a right-angle.. A 4 ] f g e - 13. A boundary is that which is the extremity of some-

. thing.. 4R A 4 f] i A 14. A figure is that which is contained by some bound-

[c 1 ], ] c CM ary or boundaries.N] [ 4] j i 15. A circle is a plane figure contained by a single^ :1 [] > j - line [which is called a circumference], (such that) all of] J C k. the straight-lines radiating towards [the circumference]

U. @ j ] 1 1. from a single point lying inside the figure are equal to. @ j 4R :1 ` j one another.

m R 4M N ` 16. And the point is called the center of the circle.f] A j , H R 17. And a diameter of the circle is any straight-line,] . being drawn through the center, which is brought to an

. n 4 ] A f end in each direction by the circumference of the circle. A R A C fM :A And any such (straight-line) cuts the circle in half.

. @ j X ] :, Q R 18. And a semi-circle is the figure contained by thej 4. diameter and the circumference it cuts off. And the center

. : 4 ` f] :[ - of the semi-circle is the same (point) as (the center of) the, @ ` f] [, circle.@ ` f] , @ ` f] p 19. Rectilinear figures are those figures contained by :[ . straight-lines: trilateral figures being contained by three

. [ @ k @ straight-lines, quadrilateral by four, and multilateral by 4 ] ` 1 J S , k@ more than four.@ ] ` J S , ] @ ] 20. And of the trilateral figures: an equilateral trian-` 1 C S . gle is that having three equal sides, an isosceles (triangle)

q @ [ b that having only two equal sides, and a scalene (triangle)@ 4 ] S b> , C that having three unequal sides.

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. ELEMENTS BOOK 1

@ ] S C1 , b @ ] ` 1 21. And further of the trilateral figures: a right-angledb S . triangle is that having a right-angle, an obtuse-angled

. s @ (triangle) that having an obtuse angle, and an acute-4, Q k 4 R b, N angled (triangle) that having three acute angles., Q b , : k , t , 22. And of the quadrilateral figures: a square is thatQ k , : b , t@ @ which is right-angled and equilateral, a rectangle that] ` C R J C which is right-angled but not equilateral, a rhombus thatS, Q u k 4 u b ` @ which is equilateral but not right-angled, and a rhomboid` j . that having opposite sides and angles equal to one an-

. k :1, x 4 y :y other which is neither right-angled nor equilateral. And4Y z R 4 k { 4M N let quadrilateral figures besides these be called trapezia.` 4R C. 23. Parallel lines are straight-lines which, being in the

same plane, and being produced to infinity in each direc-tion, meet with one another in neither (of these direc-tions).

This should really be counted as a postulate, rather than as part of a definition.

k. Postulates. M C] ] 4R i 1 1. Let it have been postulated to draw a straight-line

:1 > C1. from any point to any point.. R :1 ` ] @ 4M 2. And to produce a finite straight-line continuously

: 41. in a straight-line.. R R Y R - 3. And to draw a circle with any center and radius.

. 4. And that all right-angles are equal to one another.. R ` b` J C |. 5. And that if a straight-line falling across two (other). R 4` k : :1 4 straight-lines makes internal angles on the same side (of

` 4] R 4R ` :` b[ itself whose sum is) less than two right-angles, then, be-4 a, 4 ` : 4M {- ing produced to infinity, the two (other) straight-lines , 4M } kR ^ [ b[ meet on that side (of the original straight-line) that the4. (sum of the internal angles) is less than two right-angles

(and do not meet on the other side).

This postulate effectively specifies that we are dealing with the geometry of flat, rather than curved, space.

R S. Common Notions

. ` y :y J R C 4R J. 1. Things equal to the same thing are also equal to. R 4` J J a, ` e 4R J. one another.. R 4` C] J ~ Ca, ` 2. And if equal things are added to equal things then

4 J. the wholes are equal.. R ` 4 4M C J C 3. And if equal things are subtracted from equal things

4. then the remainders are equal.

. R ] e j 1 [4]. 4. And things coinciding with one another are equalto one another.

5. And the whole [is] greater than the part.

As an obvious extension of C.N.s 2 & 3if equal things are added or subtracted from the two sides of an inequality then the inequality remains

an inequality of the same type.

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. ELEMENTS BOOK 1