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Euclid's Elements

Euclid's Elements is a mathematical treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC. The Elements is a collection of definitions, postulates, and proofs from Euclidean geometry, named after Euclid.

Euclid based his work on five postulates (today they are called axioms):

  1. To draw a straight line from any point to any other.
  2. To produce a finite straight line continuously in a straight line.
  3. To describe a circle with any center and radius.
  4. That all right angles are equal to each other.
  5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

These postulates reflect the constructive geometry Euclid, along with his contemporary Greeks, was interested in. The first three postulates basically describe the constructions one can carry out with a compass and an unmarked straightedge or ruler.

The last, so-called "parallel postulate" seems less obvious than the others and many geometers tried in vain to prove it from them. 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. Mathematicians say that the parallel postulate is independent of the other postulates. 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). That other geometries could be logically consistent was one of the most important discoveries in mathematics, with vast implications for science and philosophy. Indeed, Einstein's theory of general relativity shows that the "real" space in which we live can be non-Euclidean. 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.

One criticism that arose as mathematicians investigated Euclid's system is that Euclid's five axioms are incomplete[?], meaning that they are insufficient to produce the results one would like to be true in Euclidean geometry. Euclid made some hidden assumptions, which were made explicit by later mathematicians. For example, one of his theorems is that any line segment is part of a triangle, which he constructs in the usual way, by drawing circles around both endpoints[?] and taking their intersection and them as three corners. However, his axioms do not guarantee that the circles actually do intersect. David Hilbert gave a revised list containing no fewer than 23 separate axioms. As Gödel proved, all axiomatic systems -- excepting the very simplest -- are either incomplete or contradict themselves, and this is no exception.

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:

Books 1 through 4 deal with plane geometry:

  • 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 geometric 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[?]:

Books 11 through 13 deal with spatial geometry:

  • 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.

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