Euclid's Elements
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Euclid's Elements (Greek: Στοιχεῖα) 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. It is considered the most successful textbook ever written: the Elements was one of the very first books to go to press, and is second only to the Bible in number of editions published (well over 1000). For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century did it cease to be considered something all educated people had read. It is still (though rarely) used as a basic introduction to geometry today. Image:Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570 (560x900).jpg The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570
First principlesEuclid based his work in Book I on 23 definitions, such as point, line and surface, five postulates and five "common notions" (both of which are today called axioms). Postulates in Book I:
Common notions in Book I:
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. 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. 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. Throughout history there have been controversies surrounding many of Euclid's axioms and proofs. Nevertheless, the Elements has withstood the test of time and is still considered a masterpiece in the application of logic to mathematics, and, historically, it has been enormously influential in many areas of science. 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. Parallel postulateOf 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. 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, 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). It is a testament to Euclid's dedication to a logical development from as few assumptions as possible that he recognized the independence of the parallel postulate. His statement of it as a fifth separate axiom predates by two millenia its acceptance as such by other mathematicians. HistoryElements was written in approximately 300 BC by Euclid, an ancient Greek mathematician who probably studied under the pupils of Plato. Although most of the theorems had been developed earlier, Elements was so impressive and comprehensive that the Greeks had no use for the older books, and little is known about earlier geometers today. It was translated later into Arabic after being gifted to the Arabs by Byzantium and from those secondary translations into Latin. The first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Copies of the Greek text also exist, e.g. 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 editions of the text. Also of importance are the scholia, or footnotes to the text. These additions, which 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. ContentsAlthough 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:
Books 5 through 10 introduce ratios and proportions:
Books 11 through 13 deal with spatial geometry:
External links
Complete and fragmentary manuscripts of versions of Euclid's elements: bg:Елементи ca:Elements d'Euclides de:Euklids Elemente es:Los elementos fr:Éléments d'Euclide it:Elementi (Euclide) ja:ユークリッド原論 no:Euklids Elementene pl:Elementy sv:Elementa zh:几何原本 |
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