Euclid

Euclid of Alexandria ( ancient Greek Εὐκλείδης Euclid, Latinized Euclides ) was a Greek mathematician, who probably lived in the third century BC in Alexandria.

The surviving works include all aspects of ancient Greek mathematics: these are the theoretical disciplines arithmetic and geometry ( Elements, Data), Music Theory ( The division of the canon ), a methodological guide for finding planimetric solutions of certain secured starting points ( Porismen ) and the physical and applied works (optics, astronomical phenomena ).

In his most famous work elements ( ancient Greek Στοιχεῖα STOICHEIA, rudiments ', ' principles ',' elements ') he carried the knowledge of Greek mathematics of his time together. He showed in the construction of geometric objects, natural numbers and certain sizes and examined their properties. He used definitions, postulates (after Aristotle principles that can be accepted or rejected ), and axioms (general and indubitable principles according to Aristotle ). Many sets of elements apparently do not originate from Euclid himself His main achievement consists rather in the collection and uniform representation of mathematical knowledge and the strict evidence, which became a model for later mathematics.

Life

About life Euclid almost nothing is known. From a note with the Pappus it has been concluded that he was in Alexandria, Egypt. The survival data are unknown. The assumption that he lived around 300 BC, is based on a directory of mathematicians in Proclus, other evidence can, however, assume that (ca. 285-212 BC) was a little younger than Euclid, Archimedes.

From a point in Proclus has also concluded that he was born around the year 360 BC in Athens, there received his education at Plato's Academy and then at the time of Ptolemy I (ca. 367-283 BC ) worked in Alexandria.

He should not be confused with Euclid of Megara, as often happened to the early modern period, so that the name of Euclid of Megara also appeared on the covers of the issues of the elements.

Geometry - Arithmetic - theory of proportion

The elements were in many places to the 20th century, based on the geometry teaching, especially in the Anglo- Saxon countries.

In addition to the Pythagorean geometry Euclid's Elements in Book VII -IX contain the Pythagorean arithmetic, the beginnings of number theory (which already Archytas knew ) and the concepts of divisibility and the greatest common divisor. For the determination of which he found an algorithm, the Euclidean algorithm. Euclid also proved that there are infinitely many primes, named after him sentence of Euclid. Even Euclid's music theory is based on the arithmetic. In addition, the book contains the theory of proportion V of Eudoxus, a generalization of arithmetic to positive irrationals.

The famous fifth postulate of plane Euclidean geometry ( now called the parallel postulate ) asks: When a line falling on two straight lines g and h causes s that the inside on the same page resulting from s angles are α and β together less than two right angles, then the two straight lines meet g and h on the flat side of s on the lie angles α and β. So cutting two straight lines a distance ( or line) so that on one side enclosed by the track and the two straight two angles are smaller than 180 °, then the two lines intersect on this site and limit together with the distance ( or third line) a triangle.

For the history of science dealing with the parallel axiom is of great importance, because it has done much to clarify mathematical concepts and methods of proof. In the course of the inadequacy of the Euclidean axioms was evident in the 19th century. A formal axiomatics of Euclidean geometry can be found in David Hilbert's work Foundations of Geometry (1899 ), which has led to many subsequent editions and subsequent research. In it for the first time, a complete construction of Euclidean geometry will be paid, up to the realization that each model of Hilbert's axiom system is isomorphic to the three-dimensional real number space with the usual interpretations of the basic geometric concepts (such as point, line, plane, length, angle, congruence similarity, etc.) in analytical geometry. Since ancient times many important mathematicians tried in vain to prove the axiom of parallels with the other axioms and postulates ( it would be unnecessary ). Only in the 19th century, the indispensability of the axiom of parallels with the discovery of non-Euclidean geometry by Bolyai and Lobachevsky became clear. The Poincaré half-plane H ( Henri Poincaré ) is a model for such an axiom system in which the parallel axiom does not apply. Thus the axiom of parallels can not be inferred from the other axioms (see non-Euclidean geometry).

Music Theory

In Euclid's music theory font The division of the canon (Greek Katatomē kanonos, Latin sectio canonis ), which is classified as authentic, he took the music theory of Archytas on and set it on a more solid acoustic base, namely frequencies of vibrations ( he spoke of frequency of movements). He generalized the theorem of Archytas thereby about the irrationality of the square root and proved more generally, the irrationality of any roots. The reason for this brilliant generalization is its antithesis to the harmony of Aristoxenus, the on rational multiples of the tone ( semitone ... n- tel - sound) builds. For in the Pythagorean harmony of the tone ( whole step ) has proportion 9:8, which Euclid to its antithesis "The tone is neither divisible into two or into several equal parts" caused; However, it presupposes commensurable frequencies until the end of the 16th century (Simon Stevin ) were adopted in the Pythagorean harmony. The antithesis " The octave is less than 6 whole tones " it was based on the calculation of the Pythagorean comma. In addition, contains Euclid division of the canon - as its title indicates - the oldest surviving representation of a sound system on the canon, a split string, one Pythagorean reinterpretation of the full diationischen sound system of Aristoxenus. Euclid's sound system was handed down by Boethius; it became the basis of modern sound system in the tone letters notation Odos.

Eponyms

After Euclid following mathematical structures are named:

• Euclidean distance, the length of the direct connection between two points in the plane or in space
• Euclidean algorithm, a method for calculating the greatest common divisor of two integers
• Euclidean geometry, descriptive geometry of the plane or the space
• Euclidean body, a parent body in which each member has a non-negative square root
• Euclidean norm, the length of a vector in the plane or in space
• Euclidean space, the space of intuition, a real affine space with the standard scalar
• Euclidean relation, a relation that applies: there are two members each to a third in relation, then they are also related to each other
• Euclidean ring, a ring in which one division with remainder is possible
• Euclidean tools that allowed actions in the construction by ruler and compass

In addition, according to Euclid following mathematical theorems and proofs are named:

• Euclid's proof of the irrationality of the square root of 2, the first proof by contradiction in the history of mathematics
• Height Type of Euclid: In a right triangle, the square is above the level surface is equal to the square of the hypotenuse
• Kathetensatz of Euclid: In a right triangle the Kathetenquadrate are the same in each case the product of the hypotenuse and the associated Hypotenusenabschnitt
• Lemma of Euclid: Does a prime number, a product of two numbers, then at least one of the two factors
• Set of Euclid: There are infinitely many prime numbers

Next are named after Euclid:

• Euclides (crater ), a crater on the lunar front side
• ( 4354 ) Euclides, an asteroid of the main belt

Works

The edition of the works of Euclid of Johan Ludvig Heiberg, Heinrich amount (ed.):

• Euclidis Opera Omnia, 9 volumes, Teubner, Leipzig 1888-1916 (Greek / Latin ), more precise volumes of Supplement 8 ( a commentary on the elements of Al- Nayrizi translated by Gerard of Cremona edited by Maximilian Curtze )

Translations:

• Euclid: The Elements. Books I- XIII. Edited and transl. Clemens von Thaer. 4th Edition, Harri German, Frankfurt am Main 2003 ( Ostwald Klass d. Exact Wiss. 235), ISBN 3-8171-3413-4
• Euclid: The thirteen books of Euclid 's elements. Edited and transl. v. Thomas Heath, 3 volumes, Cambridge University Press, 1908, reprint Dover, 1956 ( English translation with detailed commentary and introduction to Euclid )
• Euclid: Data. The Data of Euclid, according to Menges text from d Griech. transl. and ed. Clemens von Thaer. Springer, Berlin 1962
• The Medieval Latin Translation of the Data of Euclid, translated by Shuntaro Ito, Tokyo University Press, 1980, Birkhauser, 1998.
• Euclid: cesarean section canonis, newly edited, translated and commented: Oliver Busch: logos syntheseos. The Euclidean Sectio canonis, Aristoxenus, and the role of mathematics in ancient musical theory. Hildesheim 2004, ISBN / ISSN: 3-487-11545- X
• Paul ver Eecke Euclide, L' Optique et la catoptrique, Paris, Bruges 1938 ( French translation of the Optics )

Received writings of Euclid, in addition to the elements, the data and the division of the canon: Optika, over the division of the figures (extract obtained in an Arabic translation ). From other works just the titles are well known: among others Pseudaria ( fallacies ), and Katoptrika Phainomena ( astronomy).

Some of the surviving only in Arabic works:

• January Hogendijk: The Arabic version of Euclid 's 'On divisions ', in: Vestigia mathematica, Amsterdam, 1993, pp. 143-162.