# Euclid's Elements of Geometry

*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 that can be achieved using a straight-rule and a compass.
Furthermore, empirical proofs by means of measurement are strictly forbidden: that is,
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'', because 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 figures. *Book 7* deals
with elementary number theory: for instance, prime numbers,
greatest common denominators, et cetera. *Book 8* is concerned with geometric
series. *Book 9* contains various applications of results in the
previous two books, and includes theorems on the
infinitude 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 five so-called Platonic solids.

I have prepared a new edition of Euclid's Elements that presents the definitive (and completely out-of-print) Greek text - that is, the one 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 added to
the Elements over the centuries, are included in the edition.
The aim of the translation is to make the mathematical argument as clear and unambiguous as possible, while still adhering 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 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 modern edition of Euclid's Elements is available in various formats:
- [PDF]
- A downloadable fully-bookmarked PDF document.
- [Amazon]
- A professionally-bound paperback print-on-demand book.

#### Bibliography:

*Euclidus Opera Ominia*, I.L. Heiberg & H. Menge (editors), Teubner (1883-1916).
*Euclid in Greek, Book 1*, T.L. Heath (translator), Cambridge (1920).
*Euclid's Elements*, T.L. Heath (translator), Dover (1956).
*History of Greek Mathematics*, T.L. Heath, Dover (1981).

Richard Fitzpatrick
Last modified: Tue Aug 26 21:02:48 CDT 2014