EUCLID’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 five 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, includ-
ing the three cases in which triangles are congruent, various theor