Cambridge University Press

Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the world's oldest publishing house and the second-largest university press in the world, it holds letters patent as the Queen's Printer. The press mission is "to further the University's mission by disseminating knowledge in the pursuit of education and research at the highest international levels of excellence". Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global sales presence, publishing hubs, offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries, its publishing includes academic journals, reference works and English language teaching and learning publications. Cambridge University Press is a charitable enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press.

It originated from letters patent granted to the University of Cambridge by Henry VIII in 1534, has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses. Authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, Stephen Hawking. University printing began in Cambridge when the first practising University Printer, Thomas Thomas, set up a printing house on the site of what became the Senate House lawn – a few yards from where the press's bookshop now stands. In those days, the Stationers' Company in London jealously guarded its monopoly of printing, which explains the delay between the date of the university's letters patent and the printing of the first book. In 1591, Thomas's successor, John Legate, printed the first Cambridge Bible, an octavo edition of the popular Geneva Bible; the London Stationers objected strenuously. The university's response was to point out the provision in its charter to print "all manner of books".

Thus began the press's tradition of publishing the Bible, a tradition that has endured for over four centuries, beginning with the Geneva Bible, continuing with the Authorized Version, the Revised Version, the New English Bible and the Revised English Bible. The restrictions and compromises forced upon Cambridge by the dispute with the London Stationers did not come to an end until the scholar Richard Bentley was given the power to set up a'new-style press' in 1696. In July 1697 the Duke of Somerset made a loan of £200 to the university "towards the printing house and presse" and James Halman, Registrary of the University, lent £100 for the same purpose, it was in Bentley's time, in 1698, that a body of senior scholars was appointed to be responsible to the university for the press's affairs. The Press Syndicate's publishing committee still meets and its role still includes the review and approval of the press's planned output. John Baskerville became University Printer in the mid-eighteenth century.

Baskerville's concern was the production of the finest possible books using his own type-design and printing techniques. Baskerville wrote, "The importance of the work demands all my attention. Caxton would have found nothing to surprise him if he had walked into the press's printing house in the eighteenth century: all the type was still being set by hand. A technological breakthrough was badly needed, it came when Lord Stanhope perfected the making of stereotype plates; this involved making a mould of the whole surface of a page of type and casting plates from that mould. The press was the first to use this technique, in 1805 produced the technically successful and much-reprinted Cambridge Stereotype Bible. By the 1850s the press was using steam-powered machine presses, employing two to three hundred people, occupying several buildings in the Silver Street and Mill Lane area, including the one that the press still occupies, the Pitt Building, built for the press and in honour of William Pitt the Younger.

Under the stewardship of C. J. Clay, University Printer from 1854 to 1882, the press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks. During Clay's administration, the press undertook a sizeable co-publishing venture with Oxford: the Revised Version of the Bible, begun in 1870 and completed in 1885, it was in this period as well that the Syndics of the press turned down what became the Oxford English Dictionary—a proposal for, brought to Cambridge by James Murray before he turned to Oxford. The appointment of R. T. Wright as Secretary of the Press Syndicate in 1892 marked the beginning of the press's development as a modern publishing business with a defined editorial policy and administrative structure, it was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories. The Cambridge Modern History was published

Projective space

In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when V = R2 and V = R3 are the real projective line and the real projective plane where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, R3 denotes ordered triplets of real numbers; the idea of a projective space relates to perspective, more to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a projection line, intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the vector space is R3 with the camera entrance pupil at the origin, the projective space corresponds to the image points. Projective spaces can be studied as a separate field in mathematics, but are used in various applied fields, geometry in particular. Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on homogeneous coordinates.

As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard Euclidean geometry for the plane, two lines always intersect at a point except when parallel. In a projective representation of lines and points, such an intersection point exists for parallel lines, it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, their representation theories; as outlined above, projective space is a geometric object that formalizes statements like "Parallel lines intersect at infinity." For concreteness, we give the construction of the real projective plane P2 in some detail. There are three equivalent definitions: The set of all lines in R3 passing through the origin.

Every such line meets the sphere of radius one centered in the origin twice, say in P = and its antipodal point. P2 can be described as the points on the sphere S2, where every point P and its antipodal point are not distinguished. For example, the point is identified with, etc, yet another equivalent definition is the set of equivalence classes of R3 ∖, i.e. 3-space without the origin, where two points P = and P∗ = are equivalent iff there is a nonzero real number λ such that P = λ⋅P∗, i.e. x = λx∗, y = λy∗, z = λz∗. The usual way to write an element of the projective plane, i.e. the equivalence class corresponding to an honest point in R3, is. The last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with z ≠ 0 is equivalent to. So there are two disjoint subsets of the projective plane: that consisting of the points = for z ≠ 0, that consisting of the remaining points; the latter set can be subdivided into two disjoint subsets, with points and. In the last case, x is nonzero, because the origin was not part of P2.

This last point is equivalent to. Geometrically, the first subset, isomorphic to R2, is in the image the yellow upper hemisphere, or equivalently the lower hemisphere; the second subset, isomorphic to R1, corresponds to the green line, or, equivalently the light green line. We have the red point or the equivalent light red point. We thus have a disjoint decomposition P2 = R2 ⊔ R1 ⊔ point. Intuitively, made precise below, R1 ⊔ point is itself the real projective line P1. Considered as a subset of P2, it is called line at infinity, whereas R2 ⊂ P2 is called affine plane, i.e. just the usual plane. The next objective is to make the saying "parallel lines meet at infinity" precise. A natural bijection between the plane z = 1 and the sphere of the projective plane is accomplished by the gnomonic projection; each point P on this plane is mapped to the two intersection points of the sphere with the line through its center and P. These two points are identified in the projective plane. Lines in the plane are mapped to great circles if one includes one pair of antipodal points on the equator.

Any two great circles intersect in two antipodal points. Great circles corresponding to parallel lines intersect on the equator. So any two lines have one intersection point inside P2; this phenomenon is axiomatized in projective geometry. The real projective space of dimension n or projective n-space, Pn, is the set of the lines in Rn+1 passing through the origin. For defining it as a topological space and as an algebraic variety it is better to define it as the quotient space of Rn+1 by the equivalence relation "to be aligned with the origin". More Pn:= / ~,where ~ is the equivalence relation defined by: ~ if there is a non-zero real number λ such that =; the elements of the projective space are called points. The projective coordinates of a point P are x0... xn, where is any element of the corresponding equivalence class. This is denoted P =, the colons and the brackets emphasizing that the right-hand side is an equivalence class, whic

Mathematics

Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.

The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.

Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.

The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to

Felix Klein

Christian Felix Klein was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, on the associations between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Felix Klein was born on 25 April 1849 to Prussian parents. Klein's mother was Sophie Elise Klein, he attended the Gymnasium in Düsseldorf studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, during 1866, Plücker's interest was geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn during 1868. Plücker died during 1868. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes, thus became acquainted with Alfred Clebsch, who had relocated to Göttingen during 1868.

Klein visited Clebsch the next year, along with visits to Paris. During July 1870, at the beginning of the Franco-Prussian War, he was in Paris and had to leave the country. For a brief time he served as a medical orderly in the Prussian army before being appointed lecturer at Göttingen during early 1871. Erlangen appointed Klein professor during 1872. For this, he was endorsed by Clebsch, who regarded him as to become the best mathematician of his time. Klein did not desire a school at Erlangen where there were few students, so he was pleased to be offered a professorship at Munich's Technische Hochschule during 1875. There he and Alexander von Brill taught advanced courses to many excellent students, Adolf Hurwitz, Walther von Dyck, Karl Rohn, Carl Runge, Max Planck, Luigi Bianchi, Gregorio Ricci-Curbastro. During 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel. After five years at the Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig.

There his colleagues included Walther von Dyck, Eduard Study and Friedrich Engel. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. During 1882, his health collapsed. Nonetheless his research continued. Klein accepted a professorship at the University of Göttingen during 1886. From until his 1913 retirement, he sought to re-establish Göttingen as the world's main mathematics research center, yet he never managed to transfer from Leipzig to Göttingen his own primacy as a developer of geometry. At Göttingen, he taught a variety of courses concerning the interface between mathematics and physics, such as mechanics and potential theory; the research facility Klein established at Göttingen served as a model for the best such facilities throughout the world. He introduced weekly discussion meetings, created a mathematical reading room and library. During 1895, Klein hired David Hilbert away from the University of Königsberg. With Klein's editorship, Mathematische Annalen became one of the best mathematics journals in the world.

Founded by Clebsch, only with Klein's management did it first rival surpass Crelle's Journal based in the University of Berlin. Klein established a small team of editors who met making democratic decisions; the journal specialized in complex analysis, algebraic geometry, invariant theory. It provided an important outlet for real analysis and the new group theory. During 1893 in Chicago, Klein was a major speaker at the International Mathematical Congress held as part of the World's Columbian Exposition. Due to Klein's efforts, Göttingen began admitting women during 1893, he supervised the first Ph. D. thesis in mathematics written at Göttingen by a woman. During 1897 Klein became a foreign member of the Royal Netherlands Academy of Sciences. About 1900, Klein began to become interested in mathematical instruction in schools. During 1905, he was decisive in formulating a plan recommending that analytic geometry, the rudiments of differential and integral calculus, the function concept be taught in secondary schools.

This recommendation was implemented in many countries around the world. During 1908, Klein was elected president of the International Commission on Mathematical Instruction at the Rome International Congress of Mathematicians. With his guidance, the German part of the Commission published many volumes on the teaching of mathematics at all levels in Germany; the London Mathematical Society awarded Klein its De Morgan Medal during 1893. He was elected a member of the Royal Society during 1885, was awarded its Copley Medal during 1912, he retired the next year due to ill health, but continued to teach mathematics at his home for some years more. Klein was one of the 93 signatories of the Manifesto of the Ninety-Three, a document penned in support of the German invasion of Belgium in the early stages of World War I. Klein had the title of Geheimrat, he died in Göttingen during 1925. Klein's dissertation

Arthur Cayley

Arthur Cayley was a British mathematician. He helped; as a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, where he excelled in Greek, French and Italian, as well as mathematics, he worked as a lawyer for 14 years. He postulated the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, verified it for matrices of order 2 and 3, he was the first to define the concept of a group in the modern way—as a set with a binary operation satisfying certain laws. When mathematicians spoke of "groups", they had meant permutation groups. Cayley tables and Cayley graphs as well. Arthur Cayley was born in Richmond, England, on 16 August 1821, his father, Henry Cayley, was a distant cousin of Sir George Cayley, the aeronautics engineer innovator, descended from an ancient Yorkshire family. He settled in Russia, as a merchant, his mother was daughter of William Doughty. According to some writers she was Russian, his brother was the linguist Charles Bagot Cayley.

Arthur spent his first eight years in Saint Petersburg. In 1829 his parents were settled permanently near London. Arthur was sent to a private school. At age 14 he was sent to King's College School; the school's master observed indications of mathematical genius and advised the father to educate his son not for his own business, as he had intended, but to enter the University of Cambridge. At the unusually early age of 17 Cayley began residence at Cambridge; the cause of the Analytical Society had now triumphed, the Cambridge Mathematical Journal had been instituted by Gregory and Robert Leslie Ellis. To this journal, at the age of twenty, Cayley contributed three papers, on subjects, suggested by reading the Mécanique analytique of Lagrange and some of the works of Laplace. Cayley's tutor at Cambridge was George Peacock and his private coach was William Hopkins, he finished his undergraduate course by winning the place of Senior Wrangler, the first Smith's prize. His next step was to take the M.

A. degree, win a Fellowship by competitive examination. He continued to reside at Cambridge University for four years; because of the limited tenure of his fellowship it was necessary to choose a profession. He made a specialty of conveyancing, it was while he was a pupil at the bar examination that he went to Dublin to hear Hamilton's lectures on quaternions. His friend J. J. Sylvester, his senior by five years at Cambridge, was an actuary, resident in London. During this period of his life, extending over fourteen years, Cayley produced between two and three hundred papers. At Cambridge University the ancient professorship of pure mathematics is denominated by the Lucasian, is the chair, occupied by Isaac Newton. Around 1860, certain funds bequeathed by Lady Sadleir to the University, having become useless for their original purpose, were employed to establish another professorship of pure mathematics, called the Sadleirian; the duties of the new professor were defined to be "to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science."

To this chair Cayley was elected. He gave up a lucrative practice for a modest salary, he at once settled down in Cambridge. More fortunate than Hamilton in his choice, his home life was one of great happiness, his friend and fellow investigator, once remarked that Cayley had been much more fortunate than himself. At first the teaching duty of the Sadleirian professorship was limited to a course of lectures extending over one of the terms of the academic year. For many years the attendance was small, came entirely from those who had finished their career of preparation for competitive examinations; the subject lectured on was that of the memoir on which the professor was for the time engaged. The other duty of the chair — the advancement of mathematical science — was discharged in a handsome manner by the long series of memoirs that he published, ranging over every department of pure mathematics, but it was discharged in a much less obtrusive way. In 1872 he was made an honorary fellow of Trinity College, three years an ordinary fellow, which meant stipend as well as honour.

About this time his friends subscribed for a presentation portrait. Maxwell wrote an address to the committee of subscribers; the verses refer to the subjects investigated in several of Cay

Albrecht Beutelspacher

Albrecht Beutelspacher is a German mathematician. Beutelspacher studied 1969-1973 math and philosophy at the University of Tübingen and received his PhD 1976 from the University of Mainz, his PhD advisor was Judita Cofman. From 1982-1985 he was an associate professor at the University of Mainz and from 1985-1988 he worked for a research department of the Siemens AG. From 1988 to 2018 he was a tenured professor for geometry and discrete mathematics at the University of Giessen, he became a well-known popularizer of mathematics in Germany by authoring several books in the field of popular science and recreational math and by founding Germany's first math museum, the Mathematikum. He received several awards for his contributions to popularizing mathematics, he has a math column in the German popular science magazin Bild der Wissenschaft and moderates a popular math series for the TV-Channel BR- α. 2004: IQ Award Cryptology. Spectrum with Lynn Batten: The Theory of finite linear Spaces. Cambridge University Press, 1993 with Uta Rosenbaum: Projective Geometry: From Foundations to Applications.

Cambridge University Press Lineare Algebra. Vieweg, Wiesbaden 1994, 6. Ergänzte Aufl. 2003, ISBN 3-528-56508-X Einführung in die endliche Geometrie I. Blockpläne. B. I. Wissenschaftsverlag Einführung in die endliche Geometrie II. Projektive Räume. B. I. Wissenschaftsverlag with Bernhard Petri: Der goldene Schnitt. 2. Auflage. Spektrum, Berlin, Oxford 1996, ISBN 3-86025-404-9 mit Jörg Schwenk und Klaus-Dieter Wolfenstetter: Moderne Verfahren der Kryptographie. Von RSA zu Zero-Knowledge. Vieweg, Braunschweig u. a. 1995, ISBN 3-528-06590-7. With Marc-Alexander Zschiegner: Diskrete Mathematik für Einsteiger. Mit Anwendungen in Technik und Informatik. Vieweg, Braunschweig u. a. 2002, ISBN 3-528-06989-9. With Heike B. Neumann und Thomas Schwarzpaul: Kryptographie in Theorie und Praxis. Mathematische Grundlagen für elektronisches Internetsicherheit und Mobilfunk. Vieweg + Teubner, Braunschweig und Wiesbaden 2005, ISBN 3-528-03168-9. Survival-Kit Mathematik. Mathe-Basics zum Studienbeginn. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-1258-2.

Pasta all´infinito. Meine italienische Reise in die Mathematik. 2. Auflage. C. H. Beck, München 2000, ISBN 3-406-45404-6 Luftschlösser und Hirngespinste. Vieweg, Braunschweig und Wiesbaden 1986 Geheimsprachen. Geschichte und Techniken. 3. Auflage. C. H. Beck, München 2002, ISBN 3-406-49046-8 In Mathe war ich immer schlecht. 3. Auflage. Vieweg, Braunschweig und Wiesbaden 2001, ISBN 3-528-26783-6 Christian und die Zahlenkünstler. Eine Reise in die wundersame Welt der Mathematik. C. H. Beck, München 2005, ISBN 3-406-52708-6 Albrecht Beutelspachers Kleines Mathematikum. Die 101 wichtigsten Fragen und Antworten zur Mathematik, C. H. Beck, Munich 2010, ISBN 978-3-406-60202-3 with Marcus Wagner: Warum Kühe gern im Halbkreis grasen.... Und andere mathematische Knobeleien. Herder, Freiburg 2010, ISBN 978-3-451-30226-8. Zahlen. Geschichte, Geheimnisse. C. H. Beck, München 2013, ISBN 978-3-406-64871-7 Wie man in eine Seifenblase schlüpft. C. H. Beck, München 2015, ISBN 978-3-406-68135-6. Benedictus-Gotthelf-Teubner-Förderpreis Barbara Czernek: Letzte Vorlesung von Albrecht Beutelspacher an JLU Gießen.

Gießener Anzeiger, Juli 2018 Sabine Glinke: Denker & Lenker: Albrecht Beutelspacher. Gießen Entdecken, 29 February 2016 Albrecht Beutelspacher at the JLU Albrecht Beutelspacher at the Mathematkum

Geometry

Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.

While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.

The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.

Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.

He studied the sp