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Euclid's Elements

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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 23 definitions, such as point, line and surface, five postulates and five "common notions" (today they are called axioms).

Postulates:

  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, the two lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Axioms:

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

These basic principles 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.

Table of contents
1 Success
2 History
3 Criticism
4 Contents
5 External link

Success

The success of the 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. (Verify?) Its systematic development from a small set of axioms to deep results encouraged its use as a textbook for hundreds of years, and still influence modern geometry books. The mathematician Eric Temple Bell made an unusual comparison between Euclid and a profession of the American West: "The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry."

Of the five postulates Euclid used, the last, so-called "parallel postulate" seems less obvious than the others. 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.

History

The Elements was written in approximately 300 BC by Euclid, an ancient Greek mathematician who probably studied under the pupils of Plato. It was translated later into Arabic after being gifted to the Arabs by Byzantium and from those secondary translations into Latin. Copies of the Greek text also exist, eg in the Vatican library and the Bodlean library in Oxford. However, the manuscripts available are of very variable quality and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been drawn about the contents of the original text (copies of which are no longer available). Texts which refer to the Elements itself and mathematical theories which were current at the time it was written are also important in this process. Such analyses are conducted by J.L. Heiberg and Sir Thomas L. Heath in their translations of the text.

Also of importance are the scholia, or footnotes to the text. These additions, that often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or elucidation. Some of these are useful and add to the text, but many are not.

It is strongly suspected that book XIII was added to the others at a later date.

Criticism

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.

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:

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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