Instruct: Given a question, retrieve relevant passages that answer the question.
Query: What was Euclid's most important mathematical contribution?
Euclid (/ˈjuːklɪd/; Ancient Greek: Εὐκλείδης – Eukleídēs, pronounced[eu̯.klěː.dɛːs]; fl. 300 BC), sometimes given the name Euclid of Alexandria[1] to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry"[1] or the "father of geometry". He was active in Alexandria during the reign of PtolemyI (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.[2][3][4] In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and rigor.
In the late 11th century, Omar Khayyam wrote Discussions of the Difficulties in Euclid, a book about what he perceived as flaws in Euclid's Elements, especially the parallel postulate. He was also the first to find the general geometric solution to cubic equations. He was also very influential in calendar reform.[151]
The success of the "Elements" is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are supposedly his. Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the "Elements", encouraged its use as a textbook for about 2,000 years. The "Elements" still influences modern geometry books. Further, its logical axiomatic approach and rigorous proofs remain the cornerstone of mathematics.
Writers before Euclid had developed theories of vision. However, their works were mostly philosophical in nature and lacked the mathematics that Euclid introduced in his "Optics". Efforts by the Greeks prior to Euclid were concerned primarily with the physical dimension of vision. Whereas Plato and Empedocles thought of the visual ray as "luminous and ethereal emanation", Euclid’s treatment of vision in a mathematical way was part of the larger Hellenistic trend to quantify a whole range of scientific fields.
Becker also made important contributions to the history and interpretation of ancient Greek mathematics. Becker, as did several others, emphasized the "crisis" in Greek mathematics occasioned by the discovery of incommensurability of the side of the pentagon (or in the later, simpler proofs, the triangle) by Hippasus of Metapontum, and the threat of (literally) "irrational" numbers. To German theorists of the "crisis", the Pythagorean diagonal of the square was similar in its impact to Cantor's diagonalization method of generating higher order infinities, and Gödel's diagonalization method in Gödel's proof of incompleteness of formalized arithmetic. Becker, like several earlier historians, suggests that the avoidance of arithmetic statement of geometrical magnitude in Euclid is avoided for ratios and proportions, as a consequence of recoil from the shock of incommensurability. Becker also showed that all the theorems of Euclidean proportion theory could be proved using an earlier alternative to the Eudoxus technique which Becker found stated in "Aristotle's Topics", and which Becker attributes to Theaetetus. Becker also showed how a constructive logic that denied unrestricted excluded middle could be used to reconstruct most of Euclid's proofs.
This is often regarded as not only the most important work in geometry but one of the most important works in mathematics. It contains many important results in plane and solid geometry, algebra (books II and V), and number theory (book VII, VIII, and IX). More than any specific result in the publication, it seems that the major achievement of this publication is the promotion of an axiomatic approach as a means for proving results. Euclid's "Elements" has been referred to as the most successful and influential textbook ever written.
Mayberry's position is that all of this, right from Book V of Euclid, constitutes an aberration from the true spirit of mathematics as exemplified in Euclid Books I-IV. The central purpose of his book is to explain his position and to show that it is not corrosive of the essential content or the modern practice of mathematics, but, in his recommendation of a clearer Aristotelian understanding of what mathematics is about and the standard of rigour appropriate to his more exigent understanding of meaning, he is following in a tradition initiated by Cantor of restoring meaning to mathematics after three centuries of formalism. However, in Mayberry's eyes a modern platonically inspired doctrine that holds that say, proper classes, objectively exist is as much a departure from good sense and probable truthfulness as, say, the early 19th century formalistically inspired doctrine, Peacock's “Principle of the equivalence of Permanent Forms”.
As was common in ancient mathematical texts, when a proposition needed proof in several different cases, Euclid often proved only one of them (often the most difficult), leaving the others to the reader. Later editors such as Theon often interpolated their own proofs of these cases.
In 1570, Billingsley published his translation of Euclid's Elements "The elements of geometrie of the most ancient philosopher Euclide of Megara". (Actually, it should have been Euclid of Alexandria; the two Euclids were frequently confused in the Renaissance.) The work included a lengthy preface by John Dee, which surveyed all the existing branches of pure and applied mathematics. Dee also provided copious notes and other supplementary material. The work was printed in folio by John Day, and included several three-dimensional fold-up diagrams illustrating solid geometry. Though not the very first, it was one of the first books to include such a feature.
Recently the most important contributions to the development of the mathematical models of long-term ("secular") sociodemographic cycles have been made by Sergey Nefedov, Peter Turchin, Andrey Korotayev, and Sergey Malkov. What is important is that on the basis of their models Nefedov, Turchin and Malkov have managed to demonstrate that sociodemographic cycles were a basic feature of complex agrarian systems (and not a specifically Chinese or European phenomenon).
[Their work] takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness [of Indian and Arabic mathematics] is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilisations – most notably China and India, are perceived either as borrowers from Greek sources or having made only minor contributions to mainstream mathematical development. An openness to more recent research findings, especially in the case of Indian and Chinese mathematics, is sadly missing"
Ancient Greek mathematics contributed many important developments to the field of mathematics, including the basic rules of geometry, the idea of formal mathematical proof, and discoveries in number theory, mathematical analysis, applied mathematics, and approached close to establishing integral calculus. The discoveries of several Greek mathematicians, including Pythagoras, Euclid, and Archimedes, are still used in mathematical teaching today.
The scientific exploitation of this enormous data set will be carried out by a European-led consortium of more than 1200 people in over 100 laboratories in 15 countries (Austria, Belgium, Denmark, Finland, France, Germany, Italy, the Netherlands, Norway, Portugal, Romania, Spain, Switzerland, UK, Canada and the US). The Euclid Consortium is also responsible for the construction of the Euclid instrument payload and for the development and implementation of the Euclid ground segment which will process all data collected by the satellite. The laboratories contributing to the Euclid Consortium are funded and supported by their national space agencies, which also have the programmatic responsibilities of their national contribution, and by their national research structures (research agencies, observatories, universities). Overall, the Euclid Consortium contributes to about 30% of the total budget cost of the mission until completion.
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the "Elements". Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The "Elements" begins with plane geometry, still taught in secondary school (High School) as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the "Elements" states results of what are now called algebra and number theory, explained in geometrical language.
In 2011, Project Euclid received the 2011 Division Award from the Physics-Astronomy-Mathematics Division of the Special Libraries Association. Given annually, this award recognizes significant contributions to the literature of physics, mathematics, or astronomy, and honors work that demonstrably improves the exchange of information within these three disciplines. The award also takes into consideration projects that benefit libraries.
In mathematics, Euclid numbers are integers of the form , where "p"# is the "n"th primorial, i.e. the product of the first "n" prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers.
Euclid's list of axioms in the "Elements" was not exhaustive, but represented the principles that were the most important. His proofs often invoke axiomatic notions which were not originally presented in his list of axioms. Later editors have interpolated Euclid's implicit axiomatic assumptions in the list of formal axioms.
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra (named for "The Compendious Book on Calculation by Completion and Balancing" by scholar Al-Khwarizmi), and advances in geometry and trigonometry.
For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the "elegance" of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematician's Apology" expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic. Mathematical research often seeks critical features of a mathematical object. A theorem expressed as a characterization of the object by these features is the prize. Examples of particularly succinct and revelatory mathematical arguments has been published in "Proofs from THE BOOK".
Definition 3 describes what a ratio is in a general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself. Euclid defines a ratio as between two quantities "of the same type", so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists when there is a multiple of each that exceeds the other. In modern notation, a ratio exists between quantities "p" and "q" if there exist integers "m" and "n" so that "mp">"q" and "nq">"p". This condition is known as the Archimedes property.
In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria.[53] It was there that Euclid (c. 300 BC) taught, and wrote the Elements, widely considered the most successful and influential textbook of all time.[1] The Elements introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.[54] The Elements was known to all educated people in the West up through the middle of the 20th century and its contents are still taught in geometry classes today.[55] In addition to the familiar theorems of Euclidean geometry, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry,[54] including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.[56]
Hovhannes Imastaser’s work in mathematics is represented by the volume "Haghaks Ankiunavor Tvots" (Concerning Polygonal Numbers). This work indicates a profound knowledge of all important ancient and medieval mathematicians, including Pythagoras, Euclid and Aristotle. Hovhannes Imastaser translated into Armenian the works of the following classical scholars: Philo of Alexandria, Dionysius the Areopagite, Gregory of Nyssa, Porphyry, and, as mentioned, Aristotle and Euclid.
Because "Optics" contributed a new dimension to the study of vision, it influenced later scientists. In particular, Ptolemy used Euclid's mathematical treatment of vision and his idea of a visual cone in combination with physical theories in Ptolemy's "Optics", which has been called "one of the most important works on optics written before Newton". Renaissance artists such as Brunelleschi, Alberti, and Dürer used Euclid's "Optics" in their own work on linear perspective.
"Euclidean geometry" is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described (although non-rigorously by modern standards) in his textbook on geometry: the "Elements". Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The "Elements" begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the "Elements" states results of what are now called algebra and number theory, explained in geometrical language.
In ancient Greek mathematics, "space" was a geometric abstraction of the three-dimensional reality observed in everyday life. About 300 BC, Euclid gave axioms for the properties of space. Euclid built all of mathematics on these geometric foundations, going so far as to define numbers by comparing the lengths of line segments to the length of a chosen reference segment.