Euclidean Geometry - The Elements

The Elements

The Elements are mainly a systematization of earlier knowledge of geometry. Its superiority over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost.

Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, e.g., If a triangle has two equal angles, then the sides subtended by the angles are equal. The Pythagorean theorem is proved.

Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as prime numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved.

Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base.

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