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.

Contents

Although Elements is a geometric work, it also includes results that today would be classified as number theory. The contents of the work are as follows:

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Books 1 through 4 deal with plane geometry:

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• Book 1 contains the basic properties of geometry: the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area).
• Book 2 is commonly called the "book of geometrical algebra," because the material it contains may easily be interpreted as algebra.
• Book 3 deals with circles and their properties: inscribed angles, tangents, the power of a point.
• Book 4 is concerned with inscribing and circumscribing triangles and regular polygons.

Books 5 through 10 introduce ratios and proportions:

Related Topics:
Ratio - Proportions

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• Book 5 is a treatise on proportions of magnitudes.
• Book 6 applies proportions to geometry: Thales' theorem, similar figures.
• Book 7 deals strictly with number theory: divisibility, prime numbers, greatest common divisor, least common multiple.
• Book 8 deals with proportions in number theory and geometric sequences.
• Book 9 applies the results of the preceding two books: the infinitude of prime numbers, the sum of a geometric series, perfect numbers.
• Book 10 attempts to classify incommensurable (in modern language, irrational) magnitudes by using the method of exhaustion, a precursor to integration.

Books 11 through 13 deal with spatial geometry:

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• Book 11 generalizes the results of Books 1–6 to space: perpendicularity, parallelism, volumes of parallelepipeds.
• Book 12 calculates areas and volumes by using the method of exhaustion: cones, pyramids, cylinders, and the sphere.
• Book 13 generalizes Book 4 to space: golden section, the five regular (or Platonic) solids inscribed in a sphere.