Elements (Euclid)

Introduction

The "Elements" is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria, Egypt c. 300 BCE. It is a collection of definitions, postulates (propositions requiring no proof), and mathematical proofs of the propositions. The books cover Euclidean geometry and the ancient Greek version of elementary number theory. With the exception of Autolycus' "On the Moving Sphere", the "Elements" is one of the oldest Greek mathematical treatises to have survived, with the earliest copies dating from the time of Ptolemy (c. AD 100).

Structure and Content

The "Elements" begins with a set of definitions—some of which are primitive terms, not defined—and continues with five postulates, five "common notions", and then follows with propositions, each proved logically from the definitions and postulates. The first three books cover plane geometry, and the remainder cover the theory of numbers, in particular prime numbers and their properties, and solid geometry.

Definitions

The "Elements" begins with 23 definitions, including the definitions of a point, line, plane, and angle, as well as the notions of equality of angles and segments, and the concept of proportionality.

Postulates

Euclid's five postulates are:

1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Common Notions

The five common notions are:

1. Things which are equal to the same thing are also equal to one another (the Transitive property of equality). 2. If equals are added to equals, then the wholes are equal (Addition property of equality). 3. If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality). 4. Things which coincide with one another are equal to one another (Reflexive property). 5. The whole is greater than the part.

Propositions

The propositions each consists of a problem or a theorem, with the problem being something to be done (such as 'to draw a straight line from A to B') and the theorem being something to be proved. The "Elements" contains around 465 propositions, or theorems, across its 13 books.

Influence and Legacy

The "Elements" has been referred to as the most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published.