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.

Success

The success of Elements is due primarily to its logical presentation of much of the mathematical knowledge available to Euclid. Most of the material is not original to him, although a few of the proofs are his.

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Its systematic development from a small set of axioms to deep results encouraged its use as a textbook for hundreds of years, and still influences modern geometry books.

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Until the late 19th century, Elements was considered one of the best examples (if not the best) of a complete deductive structure: all of its components were thought to follow logically from previous components. However, the publication of David Hilbert's 'Grundlagen der Geometrie' (Foundations of Geometry) made evident many previously overlooked logical flaws in the Elements. Nevertheless, the Elements continues to be used as an adequate example of the application of logic, and, historically, it has been enormously influential in many areas of science.

Related Topics:
19th century - Deductive - David Hilbert - Logic - Science

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European scientists Nicolaus Copernicus, Johannes Kepler, Galileo Galilei and especially Sir Isaac Newton were all influenced by the Elements, and applied their knowledge of it to their work. Mathematicians (Bertrand Russell, Alfred North Whitehead) and philosophers (Baruch Spinoza) have also attempted to provide their own Elements; that is, axiomatized deductive structures of their own respective disciplines.

Related Topics:
Nicolaus Copernicus - Johannes Kepler - Galileo Galilei - Isaac Newton - Bertrand Russell - Alfred North Whitehead - Baruch Spinoza

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Of the five postulates Euclid used, the last, so-called "parallel postulate" seemed less obvious than the others. Many geometers suspected that it may be provable from the other postulates but all attempts to do this failed. By the mid-19th century, it was shown that no such proof exists, because one can construct non-Euclidean geometries where the parallel postulate is false, while the other postulates remain true.

Related Topics:
Parallel postulate - 19th century - Non-Euclidean geometries

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Mathematicians say that the parallel postulate is independent of the other postulates.

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Two alternatives are possible: either an infinite number of parallel lines can be drawn through a point not on a straight line (hyperbolic geometry, also called Lobachevskian geometry), or none can (elliptic geometry, also called Riemannian geometry).

Related Topics:
Hyperbolic geometry - Lobachevskian - Elliptic geometry - Riemann

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That other geometries could be logically consistent was one of the most important discoveries in mathematics, with vast implications for science and philosophy.

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Indeed, Albert Einstein's theory of general relativity shows that the "real" space in which we live can be non-Euclidean (for example, around black holes and neutron stars).

Related Topics:
Albert Einstein - General relativity - Black holes - Neutron star

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That Euclid recognized the independence of the parallel postulate long before other mathematicians accepted it is a testament to Euclid's dedication to a logical development from as few assumptions as possible.

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