Euclid's Elements

Euclid's Elements (Greek Στοιχεία) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. Euclid's books are in the fields of Euclidean geometry, as well as the ancient Greek version of number theory. The Elements is one of the oldest extant axiomatic deductive treatments of geometry, and has proved instrumental in the development of logic and modern science.

First principles

Euclid based his work in Book I on 23 definitions, such as point, line and surface, five postulates and five "common notions" (both of which are today called axioms).

Related Topics:
Point - Line - Surface - Postulate - Axiom

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Postulates in Book I:

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• A straight line segment can be drawn by joining any two points.
• A straight line segment can be extended indefinitely in a straight line.
• Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center.
• All right angles are congruent.
• 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 in Book I:

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• Things which equal the same thing are equal to one another.
• If equals are added to equals, then the sums are equal.
• If equals are subtracted from equals, then the remainders are equal.
• Things which coincide with one another are equal to one another.
• The whole is greater than the part.

These basic principles reflect the constructive geometry Euclid, along with his contemporary Greeks, was interested in.

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The first three postulates basically describe the constructions one can carry out with a compass and an unmarked straightedge or ruler.

Related Topics:
Constructions - Compass - Straightedge - Ruler

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