Euclid
- For other senses of this word, see Euclid (disambiguation).
Euclid also wrote works on perspective, conic sections, Spherical geometry, and possibly quadric surfaces. Neither the year nor place of his birth have been established, nor the circumstances of his death.
The Elements
- Main article: Euclid's Elements
Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework. In addition to providing some missing proofs, Euclid's text also includes sections on Number theory and three-dimensional geometry.
The geometrical system described in Elements was long known simply as "the" geometry. Today, however, it is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries which were discovered in the 19th century. These new geometries grew out of more than two millennia of investigation into Euclid's fifth postulate, one of the most-studied axioms in all of mathematics. Most of these investigations involved attempts to prove the relatively complex and presumably non-intuitive fifth postulate using the other four (a feat which, if successful, would have shown the postulate to be in fact a Theorem).
While the Elements was used well into the 20th century as a geometry textbook and has been considered a fine example of the formally precise axiomatic method, Euclid's treatment does not hold up to modern standards of rigor; some logically necessary axioms are missing, and the definitions of primitive terms appeal to spatial intuition. The first correct axiomatic treatment of geometry by modern standards was provided by David Hilbert in 1899, in his Grundlagen der Geometrie.
Other works
In addition to the Elements, four works of Euclid have survived to the present day.- Data deals with the nature and implications of "given" information in geometrical problems; the subject matter is closely related to the first four books of the Elements.
- On Divisions of Figures, which survives only partially in Arabic translation, concerns the division of geometrical figures into two or more equal parts or into parts in given ratios. It is similar to a third century (AD) work by Heron of Alexandria, except Euclid's work characteristically lacks any numerical calculations.
- Phaenomena concerns the application of Spherical geometry to problems of Astronomy.
- Optics, the earliest surviving Greek treatise on Perspective, contains propositions on the apparent sizes and shapes of objects viewed from different distances and angles.
All of these works follow the basic logical structure of the Elements, containing definitions and proved propositions.
There are four works credibly attributed to Euclid which have been lost
- Conics was a work on conic sections that was later extended by Apollonius of Perga into his famous work on the subject.
- Porisms might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial.
- Pseudaria, or Book of Fallacies, was an elementary text about errors in Reasoning.
- Surface Loci concerned either loci (sets of points) on surfaces or loci which were themselves surfaces; under the latter interpretation, it has been hypothesized that the work might have dealt with quadric surfaces.
Biographical sources
Almost nothing is known about Euclid outside of what is presented in Elements and his few other surviving books. What little biographical information we do have comes largely from commentaries by Proclus and Pappus of Alexandria: he was active at the great library in Alexandria and may have studied at Plato's Academe in Greece, but his exact lifespan and place of birth are unknown.In the Middle Ages, writers sometimes referred to him as Euclid of Megara, confusing him with a Greek Socratic Philosopher who lived approximately one century earlier.