Who is considered the founder of geometry?
Euclid
Euclid (; – "Eukleídēs", ; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (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. 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.
Samuel L. Greitzer
Samuel L. Greitzer (August 10, 1905 – February 22, 1988) was an American mathematician, the founding chairman of the United States of America Mathematical Olympiad, and the publisher of the precollege mathematics journal "Arbelos". Together with H.S.M. Coxeter in 1967, Greitzer coauthored the well-received textbook "Geometry Revisited", which has remained in print for more than 40 years.
History of science in classical antiquity
The first step in this emphasis upon a model was that of the followers of Pythagoras (approximately 582 – 507 BC), who saw number as the fundamental unchanging entity underlying all the structure of the universe. For Pythagoras and his followers matter was made up of ordered arrangements of point/atoms, arranged according to geometrical principles into triangles, squares, rectangles, and so on... Even on a larger scale, the parts of the universe were arranged on the principles of a musical scale and a number. For example, the Pythagoreans held that there were ten heavenly bodies because ten is a perfect number, the sum of 1 + 2 + 3 + 4. Thus with the Pythagoreans we find number emerging as the rational basis for an orderly universe — as the first proposal for a scientific ordering principle of the cosmos.
Like the Pythagoreans, Plato (c. 427–c. 347 BC) found the ordering principle of the universe in mathematics, specifically in geometry. A later account has it that Plato had inscribed at the entrance to his school, the Academy, "Let no man ignorant of geometry enter." The story is a myth, but like all myths it has a grain of truth, for in his writings Plato repeatedly tell us of the importance of geometry.
David Hestenes
Hestenes emphasizes the important role of the mathematician Hermann Grassmann for the development of geometric algebra, with William Kingdon Clifford building on Grassmann's work. Hestenes is adamant about calling this mathematical approach “geometric algebra” and its extension “geometric calculus,” rather than referring to it as “Clifford algebra”. He emphasizes the universality of this approach, the foundations of which were laid by both Grassmann and Clifford. He points out that contributions were made by many individuals, and Clifford himself used the term “geometric algebra” which reflects the fact that this approach can be understood as a mathematical formulation of geometry, whereas, so Hestenes asserts, the term “Clifford algebra” is often regarded as simply “just one more algebra among many other algebras”, which withdraws attention from its role as a unified language for mathematics and physics.
Foundations of mathematics
After many failed attempts to derive the parallel postulate from other axioms, the study of the still hypothetical hyperbolic geometry by Johann Heinrich Lambert (1728–1777) led him to introduce the hyperbolic functions and compute the area of a hyperbolic triangle (where the sum of angles is less than 180°). Then the Russian mathematician Nikolai Lobachevsky (1792–1856) established in 1826 (and published in 1829) the coherence of this geometry (thus the independence of the parallel postulate), in parallel with the Hungarian mathematician János Bolyai (1802–1860) in 1832, and with Gauss.
Later in the 19th century, the German mathematician Bernhard Riemann developed Elliptic geometry, another non-Euclidean geometry where no parallel can be found and the sum of angles in a triangle is more than 180°. It was proved consistent by defining point to mean a pair of antipodal points on a fixed sphere and line to mean a great circle on the sphere. At that time, the main method for proving the consistency of a set of axioms was to provide a model for it.
History of geography
Thales of Miletus is one of the first known philosophers known to have wondered about the shape of the world. He proposed that the world was based on water, and that all things grew out of it. He also laid down many of the astronomical and mathematical rules that would allow geography to be studied scientifically. His successor Anaximander is the first person known to have attempted to create a scale map of the known world and to have introduced the gnomon to Ancient Greece.
Hecataeus of Miletus initiated a different form of geography, avoiding the mathematical calculations of Thales and Anaximander he learnt about the world by gathering previous works and speaking to the sailors who came through the busy port of Miletus. From these accounts he wrote a detailed prose account of what was known of the world. A similar work, and one that mostly survives today, is Herodotus' "Histories". While primarily a work of history, the book contains a wealth of geographic descriptions covering much of the known world. Egypt, Scythia, Persia, and Asia Minor are all described, including a mention of India. The description of Africa as a whole are contentious, with Herodotus describing the land surrounded by a sea. Though, historically the Indian sea was thought of an inland sea which was that round of the southern part of Africa is surrounded by the eastern part of Asia by connecting land, which inference only after the circumnavigation of Africa by Vasco da Gama was abandoned by the western cartographers of the 15th century. Some, though, hold that the descriptions of areas such as India are mostly imaginary. Regardless, Herodotus made important observations about geography. He is the first to have noted the process by which large rivers, such as the Nile, build up deltas, and is also the first recorded as observing that winds tend to blow from colder regions to warmer ones.
Muhammad ibn Musa al-Khwarizmi
Al-Khwarizmi's popularizing treatise on algebra ("The Compendious Book on Calculation by Completion and Balancing", c. 813–833 CE) presented the first systematic solution of linear and quadratic equations. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because he was the first to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation), he has been described as the father or founder of algebra. The term "algebra" itself comes from the title of his book (specifically the word "al-jabr" meaning "completion" or "rejoining"). His name gave rise to the terms "algorism" and "algorithm". His name is also the origin of (Spanish) "guarismo" and of (Portuguese) "algarismo", both meaning digit.
Gemma Frisius
Gemma Frisius (; born Jemme Reinerszoon; December 9, 1508 – May 25, 1555), was a Dutch physician, mathematician, cartographer, philosopher, and instrument maker. He created important globes, improved the mathematical instruments of his day and applied mathematics in new ways to surveying and navigation. Gemma's rings are named after him. Along with Gerardus Mercator and Abraham Ortelius, Frisius is often considered one of the founders of the Netherlandish school of cartography and significantly helped lay the foundations for the school's golden age (approximately 1570s–1670s).
Systolic geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry.
Foundations of geometry
Starting with Peano, there has been a parallel thread of interest amongst logicians concerning the axiomatic foundations of Euclidean geometry. This can be seen, in part, in the notation used to describe the axioms. Pieri claimed that even though he wrote in the traditional language of geometry, he was always thinking in terms of the logical notation introduced by Peano, and used that formalism to see how to prove things. A typical example of this type of notation can be found in the work of E. V. Huntington (1874 – 1952) who, in 1913, produced an axiomatic treatment of three-dimensional Euclidean geometry based upon the primitive notions of "sphere" and "inclusion" (one sphere lying within another). Beyond notation there is also interest in the logical structure of the theory of geometry. Alfred Tarski proved that a portion of geometry, which he called "elementary" geometry, is a first order logical theory (see Tarski's axioms).
Carl Ritter
Carl Ritter (August 7, 1779September 28, 1859) was a German geographer. Along with Alexander von Humboldt, he is considered one of the founders of modern geography. From 1825 until his death, he occupied the first chair in geography at the University of Berlin.
Ordered geometry
Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904). Euclid anticipated Pasch's approach in definition 4 of "The Elements": "a straight line is a line which lies evenly with the points on itself".
Pseudoscientific metrology
"366 geometry" or "366-degree geometry" (also called "megalithic geometry") is the name given to an hypothetical geometry supposedly used and perhaps created by an alleged megalithic civilization of Britain and Brittany, France, according to British authors Alan Butler and Christopher Knight, and French author Sylvain Tristan. This geometry, whose origin is claimed to go back to c. 3500 BC, would have used a 366-degree circle rather than a 360-degree circle as we do today.
Analytic geometry
Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. "Cartesian geometry", the alternative term used for analytic geometry, is named after Descartes.
History of geometry
Plato (427-347 BC) is a philosopher that is highly esteemed by the Greeks. There is a story that he had inscribed above the entrance to his famous school, "Let none ignorant of geometry enter here." However, the story is considered to be untrue. Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as a marked ruler or a protractor, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of possible compass and straightedge constructions, and three classic construction problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 BC), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century.
Foundations of geometry
Apparently the first to do so was G. S. Klügel (1739–1812), a doctoral student at the University of Gottingen, with the support of his teacher A. G. Kästner, in the former's 1763 dissertation "Conatuum praecipuorum theoriam parallelarum demonstrandi recensio" (Review of the Most Celebrated Attempts at Demonstrating the Theory of Parallels). In this work Klügel examined 28 attempts to prove Postulate 5 (including Saccheri's), found them all deficient, and offered the opinion that Postulate 5 is unprovable and is supported solely by the judgment of our senses.
The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Circa 1813, Carl Friedrich Gauss and independently around 1818, the German professor of law Ferdinand Karl Schweikart had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Then, around 1830, the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky separately published treatises on what we today call hyperbolic geometry. Consequently, hyperbolic geometry has been called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter "k". Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences. 
The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868.
Henri Brocard
Contemporary mathematician Nathan Court wrote that he, along with Émile Lemoine and Joseph Neuberg, was one of the three co-founders of modern triangle geometry. He is listed as an Emeritus at the International Academy of Science, was awarded the Ordre des Palmes Académiques, and was an officer of the Légion d'honneur.
History of geometry
Euclid (c. 325-265 BC), of Alexandria, probably a student at the Academy founded by Plato, wrote a treatise in 13 books (chapters), titled "The Elements of Geometry", in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry. The treatise is not a compendium of all that the Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt.
Apollonius of Perga
Apollonius of Perga (; ; late 3rdearly 2nd centuries BC) was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic, he brought them to the state they were in just before the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.
History of gardening
Renée Descartes, the founder of analytical geometry, believed that the natural world was objectively measurable and that space is infinitely divisible. His belief that "all movement is a straight line therefore space is a universal grid of mathematical coordinates and everything can be located on its infinitely extendable planes" gave us Cartesian mathematics. Through the classical French gardens this coordinate system and philosophy is now given a physical and visual representation.
Outline of geometry
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.
History of geometry
Equation
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: "Cartesius") revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane, centered on a particular point called the origin, may be described as the set of all points whose coordinates "x" and "y" satisfy the equation .
History of geometry
"The Elements" began with definitions of terms, fundamental geometric principles (called "axioms" or "postulates"), and general quantitative principles (called "common notions") from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.Concepts, that are now understood as algebra, were expressed geometrically by Euclid, a method referred to as Greek geometric algebra.
Geography
The ideas of Anaximander (c. 610–545 BC): considered by later Greek writers to be the true founder of geography, come to us through fragments quoted by his successors. Anaximander is credited with the invention of the gnomon, the simple, yet efficient Greek instrument that allowed the early measurement of latitude. Thales is also credited with the prediction of eclipses. The foundations of geography can be traced to the ancient cultures, such as the ancient, medieval, and early modern Chinese. The Greeks, who were the first to explore geography as both art and science, achieved this through Cartography, Philosophy, and Literature, or through Mathematics. There is some debate about who was the first person to assert that the Earth is spherical in shape, with the credit going either to Parmenides or Pythagoras. Anaxagoras was able to demonstrate that the profile of the Earth was circular by explaining eclipses. However, he still believed that the Earth was a flat disk, as did many of his contemporaries. One of the first estimates of the radius of the Earth was made by Eratosthenes.
William Kingdon Clifford
William Kingdon Clifford (4 May 1845 – 3 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics, geometry, and computing. Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression "mind-stuff".
Beniamino Segre
Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry.
History of physics
Descartes, like Galileo, was convinced of the importance of mathematical explanation, and he and his followers were key figures in the development of mathematics and geometry in the 17th century. Cartesian mathematical descriptions of motion held that all mathematical formulations had to be justifiable in terms of direct physical action, a position held by Huygens and the German philosopher Gottfried Leibniz, who, while following in the Cartesian tradition, developed his own philosophical alternative to Scholasticism, which he outlined in his 1714 work, "The Monadology". Descartes has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day. In particular, his "Meditations on First Philosophy" continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system — allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system — was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry, important to the discovery of calculus and analysis.
Foundations of geometry
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.
Foundations of geometry
"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.
Historiometry
Historiometry started in the early 19th century with studies on the relationship between age and achievement by Belgian mathematician Adolphe Quetelet in the careers of prominent French and English playwrights but it was Sir Francis Galton, a pioneering English eugenist who popularized historiometry in his 1869 work, "Hereditary Genius". It was further developed by Frederick Adams Woods (who coined the term "historiometry") in the beginning of the 20th century. Also psychologist Paul E. Meehl published several papers on historiometry later in his career, mainly in the area of medical history, although it is usually referred to as "cliometric metatheory" by him.
Projective geometry
The first geometrical properties of a projective nature were discovered during the 3rd century by Pappus of Alexandria. Filippo Brunelleschi (1404–1472) started investigating the geometry of perspective during 1425 (see the history of perspective for a more thorough discussion of the work in the fine arts that motivated much of the development of projective geometry). Johannes Kepler (1571–1630) and Gérard Desargues (1591–1661) independently developed the concept of the "point at infinity". Desargues developed an alternative way of constructing perspective drawings by generalizing the use of vanishing points to include the case when these are infinitely far away. He made Euclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. Desargues's study on conic sections drew the attention of 16-year-old Blaise Pascal and helped him formulate Pascal's theorem. The works of Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was ignored until Michel Chasles chanced upon a handwritten copy during 1845. Meanwhile, Jean-Victor Poncelet had published the foundational treatise on projective geometry during 1822. Poncelet separated the projective properties of objects in individual class and establishing a relationship between metric and projective properties. The non-Euclidean geometries discovered soon thereafter were eventually demonstrated to have models, such as the Klein model of hyperbolic space, relating to projective geometry.
Algebra
François Viète's work on new algebra at the close of the 16th century was an important step towards modern algebra. In 1637, René Descartes published "La Géométrie", inventing analytic geometry and introducing modern algebraic notation. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Seki Kōwa in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Permutations were studied by Joseph-Louis Lagrange in his 1770 paper "Réflexions sur la résolution algébrique des équations" devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations.
La Géométrie
The work was the first to propose the idea of uniting algebra and geometry into a single subject and invented an algebraic geometry called analytic geometry, which involves reducing geometry to a form of arithmetic and algebra and translating geometric shapes into algebraic equations. For its time this was ground-breaking. It also contributed to the mathematical ideas of Leibniz and Newton and was thus important in the development of calculus.
Transformation geometry
The first systematic effort to use transformations as the foundation of geometry was made by Felix Klein in the 19th century, under the name Erlangen programme. For nearly a century this approach remained confined to mathematics research circles. In the 20th century efforts were made to exploit it for mathematical education. Andrei Kolmogorov included this approach (together with set theory) as part of a proposal for geometry teaching reform in Russia. These efforts culminated in the 1960s with the general reform of mathematics teaching known as the New Math movement.
Philosophy of mathematics
The French mathematician Henri Poincaré was among the first to articulate a conventionalist view. Poincaré's use of non-Euclidean geometries in his work on differential equations convinced him that Euclidean geometry should not be regarded as "a priori" truth. He held that axioms in geometry should be chosen for the results they produce, not for their apparent coherence with human intuitions about the physical world.
Analytic geometry
The 11th century Persian mathematician Omar Khayyám saw a strong relationship between geometry and algebra, and was moving in the right direction when he helped to close the gap between numerical and geometric algebra with his geometric solution of the general cubic equations, but the decisive step came later with Descartes.
Nasir al-Din al-Tusi
Muhammad ibn Muhammad ibn al-Hasan al-Tūsī (‎ 18 February 1201 – 26 June 1274), better known as Nasir al-Din Tusi (; or simply Tusi in the West), was a Persian polymath, architect, philosopher, physician, scientist, and theologian.
He is often considered the creator of trigonometry as a mathematical discipline in its own right.
He was a Twelver Shia Muslim. The Muslim scholar Ibn Khaldun (1332–1406) considered Tusi to be the greatest of the later Persian scholars.
Foundations of geometry
Absolute geometry is a geometry based on an axiom system consisting of all the axioms giving Euclidean geometry except for the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate.
Non-Euclidean geometry
Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature.
He constructed an infinite family of geometries which are not Euclidean by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic geometry and it is considered to be a non-Euclidean geometry due to its lack of parallel lines.
Archimedes
Archimedes of Syracuse (; ; ) was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Generally considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola.
Geometrography
In mathematics, in geometry, geometrography is the study of geometrical constructions. The concepts and methods of geometrography were first expounded by Émile Lemoine (1840–1912), a French civil engineer and a mathematician, in a meeting of the French Association for the Advancement of the Sciences held at Oran in 1888.
Generalized conic
Rene Descartes (1596–1650), father of analytical geometry, in his La Geometrie published in 1637, set apart a section of about 15 pages to discuss what he had called bifocal ellipses. A bifocal oval was defined there as the locus of a point "P" which moves in a plane such that formula_1 where "A" and "B" are fixed points in the plane and "λ" and "c" are constants which may be positive or negative. Descartes had introduced these ovals, which are now known as Cartesian ovals, to determine the surfaces of glass such that after refraction the rays meet at the same point. Descartes had also recognized these ovals as generalizations of central conics, because for certain values of "λ" these ovals reduce to the familiar central conics, namely, the circle, the ellipse or the hyperbola.
History of geometry
In the early 17th century, there were two important developments in geometry. The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by Hellenistic geometers, notably Pappus (c. 340). The greatest flowering of the field occurred with Jean-Victor Poncelet (1788–1867).
Neusis construction
T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides (ca. 440 BC) was the first to put compass-and-straightedge constructions above "neuseis". The principle to avoid "neuseis" whenever possible may have been spread by Hippocrates of Chios (ca. 430 BC), who originated from the same island as Oenopides, and who was—as far as we know—the first to write a systematically ordered geometry textbook. One hundred years after him Euclid too shunned "neuseis" in his very influential textbook, "The Elements".
Convex geometry
Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three. A big part of their results was soon generalized to spaces of higher dimensions, and in 1934 T. Bonnesen and W. Fenchel gave a comprehensive survey of convex geometry in Euclidean space R. Further development of convex geometry in the 20th century and its relations to numerous mathematical disciplines are summarized in the "Handbook of convex geometry" edited by P. M. Gruber and J. M. Wills.Expository articles on convex geometry
John Casey (mathematician)
John Casey (12 May 1820, Kilbehenny, Co. Limerick, Ireland – 3 January 1891, Dublin) was a respected Irish geometer. He is most famous for Casey's theorem on a circle that is tangent to four other circles, an extension of the problem of Apollonius. However, he contributed several novel proofs and perspectives on Euclidean geometry. He and Émile Lemoine are considered to be the co-founders of the modern geometry of the circle and the triangle.
He was born in Kilbehenny, Limerick, Ireland and educated locally at Mitchelstown, before becoming a teacher under the Board of National Education. He later became headmaster of the Central Model Schools in Kilkenny City. He subsequently entered Trinity College in 1858, where he was elected a Scholar in 1861 and was awarded the degree of BA in 1862. He was then Mathematics Master at Kingston School (1862–1873), Professor of Higher Mathematics and Mathematical Physics at the newly founded Catholic University of Ireland (1873–1881) and Lecturer in Mathematics at University College, Dublin (1881–1891).
Mathematician
One of the earliest known mathematicians was Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem.
Hellenistic period
Hellenistic Geometers such as Archimedes ( – 212 BC), Apollonius of Perga ( BC), and Euclid ( – 265 BC), whose "Elements" became the most important textbook in mathematics until the 19th century, built upon the work of the Hellenic era Pythagoreans. Euclid developed proofs for the Pythagorean Theorem, for the infinitude of primes, and worked on the five Platonic solids. Eratosthenes used his knowledge of geometry to measure the circumference of the Earth. His calculation was remarkably accurate. He was also the first to calculate the tilt of the Earth's axis (again with remarkable accuracy). Additionally, he may have accurately calculated the distance from the Earth to the Sun and invented the leap day. Known as the "Father of Geography ", Eratosthenes also created the first map of the world incorporating parallels and meridians, based on the available geographical knowledge of the era.
History of geometry
Thābit ibn Qurra (known as Thebit in Latin) (born 836) contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics. An important geometrical aspect of Thabit's work was his book on the composition of ratios. In this book, Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. By introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalisation of the number concept.
Cycloid
Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery cited similar work by the Syrian philosopher Iamblichus as evidence that the curve was likely known in antiquity. English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa, but subsequent scholarship indicates Wallis was either mistaken or the evidence used by Wallis is now lost. Galileo Galilei's name was put forward at the end of the 19th century and at least one author reports credit being given to Marin Mersenne. Beginning with the work of Moritz Cantor and Siegmund Günther, scholars now assign priority to French mathematician Charles de Bovelles based on his description of the cycloid in his "Introductio in geometriam", published in 1503. In this work, Bovelles mistakes the arch traced by a rolling wheel as part of a larger circle with a radius 120% larger than the smaller wheel.
19th century in science
Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law. His 1801 volume "Disquisitiones Arithmeticae" laid the foundations of modern number theory.
This century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds.
The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician János Bolyai, independently defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry.
Adrien-Marie Legendre
Legendre is best known as the author of "Éléments de géométrie", which was published in 1794 and was the leading elementary text on the topic for around 100 years. This text greatly rearranged and simplified many of the propositions from Euclid's "Elements" to create a more effective textbook.
For two centuries, until the recent discovery of the error in 2005, books, paintings and articles have incorrectly shown a of the obscure French politician Louis Legendre (1752–1797) as that of the mathematician Legendre. The error arose from the fact that the sketch was labelled simply "Legendre" and appeared in a book along with contemporary mathematicians such as Lagrange. The only known portrait of Legendre, recently unearthed, is found in the 1820 book "Album de 73 portraits-charge aquarellés des membres de I'Institut", a book of caricatures of seventy-three members of the Institut de France in Paris by the French artist Julien-Leopold Boilly as shown below:
Foundations of geometry
Euclid's "Elements" is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. With the exception of Autolycus' "On the Moving Sphere", the "Elements" is one of the oldest extant Greek mathematical treatises, and it is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science.
Geometry
Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, 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 preserved Greek ideas and 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 then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
History of physics
In the 3rd century BCE, the Greek mathematician Archimedes of Syracuse ( (287–212 BCE) – generally considered to be the greatest mathematician of antiquity and one of the greatest of all time – laid the foundations of hydrostatics, statics and calculated the underlying mathematics of the lever. A leading scientist of classical antiquity, Archimedes also developed elaborate systems of pulleys to move large objects with a minimum of effort. The Archimedes' screw underpins modern hydroengineering, and his machines of war helped to hold back the armies of Rome in the First Punic War. Archimedes even tore apart the arguments of Aristotle and his metaphysics, pointing out that it was impossible to separate mathematics and nature and proved it by converting mathematical theories into practical inventions. Furthermore, in his work "On Floating Bodies", around 250 BCE, Archimedes developed the law of buoyancy, also known as Archimedes' principle. In mathematics, Archimedes used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi. He also defined the spiral bearing his name, formulae for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers. He also developed the principles of equilibrium states and centers of gravity, ideas that would influence the well known scholars, Galileo, and Newton.
History of geography
Humboldt is admired as a great geographer, according to D. Livingstone that "modern geography was first and last a synthesizing science and as such, if Goetzmann is to be believed, 'it became the key scientific activity of the age'." Humboldt met the geographer George Forster at the University of Göttingen, whose geographical description and scientific writing influenced Humboldt. His "Geognosia" including the geography of rocks, animals, and plants is "an important model for modern geography". As the Prussian Ministry of Mines, Humboldt founded the Free Royal Mining School at Steben for miners, later regarded the prototype of such institutes. German Naturphilosophie, especially the work of Goethe and Herder, stimulated Humboldt's idea and research of a universal science. In his letter, he made observations while his "attention will never lose sight of the harmony of concurrent forces, the influence of the inanimate world on the animal and vegetable kingdom." His American travel stressed the geography of plants as his focus of science. Meanwhile, Humboldt used empirical method to study the indigenous people in the New World, regarded as a most important work in human geography. In "Relation historique du Voyage", Humboldt called these research a new science "Physique du monde, Theorie de la Terre," or "Geographie physique". During 1825 to 1859, Humboldt devoted in Kosmos, which is about the knowledge of nature. There are growing works about the New World since then. In the Jeffersonian era, "American geography was born of the geography of America", meaning the knowledge discovery helped form the discipline. Practical knowledge and national pride are main components of the Teleological tradition.
Geometric algebra
Also developed are the concept of vector manifold and geometric integration theory (which generalizes Cartan's differential forms).Although the connection of geometry with algebra dates as far back at least to Euclid's "Elements" in the third century B.C. (see Greek geometric algebra), GA in the sense used in this article was not developed until 1844, when it was used in a "systematic way" to describe the geometrical properties and "transformations" of a space. In that year, Hermann Grassmann introduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to the propositional calculus) that encoded all of the geometrical information of a space. Grassmann's algebraic system could be applied to a number of different kinds of spaces, the chief among them being Euclidean space, affine space, and projective space. Following Grassmann, in 1878 William Kingdon Clifford examined Grassmann's algebraic system alongside the quaternions of William Rowan Hamilton in . From his point of view, the quaternions described certain "transformations" (which he called "rotors"), whereas Grassmann's algebra described certain "properties" (or "Strecken" such as length, area, and volume). His contribution was to define a new product — the "geometric product" – on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently, Rudolf Lipschitz in 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations in formula_102 dimensions. Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra.
Bartholomaeus Pitiscus
Bartholomaeus Pitiscus (also "Barthélemy, Bartholomeo", August 24, 1561 – July 2, 1613) was a 16th-century German trigonometrist, astronomer and theologian who first coined the word "trigonometry".
Italian school of algebraic geometry
According to Guerraggio & Nastasi (page 9, 2005) Luigi Cremona is "considered the founder of the Italian school of algebraic geometry". Later they explain that in Turin the collaboration of Enrico D'Ovidio and Corrado Segre "would bring, either by their own efforts or those of their students, Italian algebraic geometry to full maturity". A one-time student of Segre, H.F. Baker wrote (1926, page 269), [Corrado Segre] "may probably be said to be the father of that wonderful Italian school which has achieved so much in the birational theory of algebraical loci." On this topic, Brigaglia & Ciliberto (2004) say "Segre had headed and maintained the school of geometry that Luigi Cremona had established in 1860." Reference to the Mathematics Genealogy Project shows that, in terms of "Italian doctorates", the real productivity of the school began with Guido Castelnuovo and Federigo Enriques. In the USA Oscar Zariski inspired many Ph.D.s.
Culture of Greece
The period of Classical Greece (from 800BC until the rise of Macedon, a Greek state in the north) is that most often associated with Greek advances in science. Thales of Miletus is regarded by many as the father of science; he was the first of the ancient philosophers to seek to explain the physical world in terms of natural rather than supernatural causes. Pythagoras was a mathematician often described as the "father of numbers"; it is believed that he had the pioneering insight into the numerical ratios that determine the musical scale, and the Pythagorean theorem is commonly attributed to him. Diophantus of Alexandria, in turn, was the "father of algebra". Many parts of modern geometry are based on the work of Euclid, while Eratosthenes was one of the first scientific geographers, calculating the circumference of the earth and conceiving the first maps based on scientific principles.
Hipparchus
Hipparchus of Nicaea (; , "Hipparkhos";  ) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry but is most famous for his incidental discovery of precession of the equinoxes.
René Descartes
Descartes' "Meditations on First Philosophy" (1641) continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system (see below) was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry, used in the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution.
Mathematical notation
The earliest mathematical viewpoints in geometry did not lend themselves well to counting. The natural numbers, their relationship to fractions, and the identification of continuous quantities actually took millennia to take form, and even longer to allow for the development of notation. It was not until the invention of analytic geometry by René Descartes that geometry became more subject to a numerical notation. Some symbolic shortcuts for mathematical concepts came to be used in the publication of geometric proofs. Moreover, the power and authority of geometry's theorem and proof structure greatly influenced non-geometric treatises, Isaac Newton's Principia Mathematica, for example.
Foundations of geometry
For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other geometries which are not Euclidean are known, the first ones having been discovered in the early 19th century.
History of algebra
Euclid is regarded as the "father of geometry". His "Elements" is the most successful textbook in the history of mathematics. Although he is one of the most famous mathematicians in history there are no new discoveries attributed to him, rather he is remembered for his great explanatory skills. The "Elements" is not, as is sometimes thought, a collection of all Greek mathematical knowledge to its date, rather, it is an elementary introduction to it.
History of trigonometry
The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BCE), who is now consequently known as "the father of trigonometry." Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.
History of cartography
A vital contribution to mapping the reality of the world came with a scientific estimate of the circumference of the earth. This event has been described as the first scientific attempt to give geographical studies a mathematical basis. The man credited for this achievement was Eratosthenes (275–195 BC). As described by George Sarton, historian of science, "there was among them [Eratosthenes's contemporaries] a man of genius but as he was working in a new field they were too stupid to recognize him". His work, including "On the Measurement of the Earth" and "Geographica", has only survived in the writings of later philosophers such as Cleomedes and Strabo. He was a devoted geographer who set out to reform and perfect the map of the world. Eratosthenes argued that accurate mapping, even if in two dimensions only, depends upon the establishment of accurate linear measurements. He was the first to calculate the circumference of the Earth (within 0.5 percent accuracy) by calculating the heights of shadows at different points in Egypt at a given time. The first in Alexandria, the other further up the Nile, in the Ancient Egyptian city of Swenet (known in Greek as Syene) where reports of a well into which the sun shone only on the summer solstice, long existed. Proximity to the Tropic of Cancer being the dynamics creating the effect. He had the distance between the two shadows calculated and then their height. From this he determined the difference in angle between the two points and calculated how large a circle would be made by adding in the rest of the degrees to 360. His great achievement in the field of cartography was the use of a new technique of charting with meridians, his imaginary north–south lines, and parallels, his imaginary west–east lines. These axis lines were placed over the map of the earth with their origin in the city of Rhodes and divided the world into sectors. Then, Eratosthenes used these earth partitions to reference places on the map. He also divided Earth into five climatic regions: a torrid zone across the middle, two frigid zones at extreme north and south, and two temperate bands in between. He was also the first person to use the word "geography".
History of mathematics
Eudoxus (408–c. 355 BC) developed the method of exhaustion, a precursor of modern integration and a theory of ratios that avoided the problem of incommensurable magnitudes. The former allowed the calculations of areas and volumes of curvilinear figures, while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, Aristotle (384–c. 322 BC) contributed significantly to the development of mathematics by laying the foundations of logic.
In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria. It was there that Euclid (c. 300 BC) taught, and wrote the "Elements", widely considered the most successful and influential textbook of all time. 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. 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. 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, 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.
Archimedes (c. 287–212 BC) of Syracuse, widely considered the greatest mathematician of antiquity, 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. He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, . He also studied the spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid), and an ingenious method of exponentiation for expressing very large numbers. While he is also known for his contributions to physics and several advanced mechanical devices, Archimedes himself placed far greater value on the products of his thought and general mathematical principles. He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere.
Apollonius of Perga (c. 262–190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone. He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond"). His work "Conics" is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton. While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.
Georges Matheron
Georges François Paul Marie Matheron (December 2, 1930 – August 7, 2000) was a French mathematician and geologist, known as the founder of geostatistics and a co-founder (together with Jean Serra) of mathematical morphology. In 1968, he created the Centre de Géostatistique et de Morphologie Mathématique at the Paris School of Mines in Fontainebleau. He is known for his contributions on Kriging and mathematical morphology. His seminal work is posted for study and review to the Online Library of the "Centre de Géostatistique", Fontainebleau, France.
History of geometry
Al-Mahani (born 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Al-Karaji (born 953) completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today.
Synthetic geometry
Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates.
Geometry, as presented by Euclid in the "Elements", is the quintessential example of the use of the synthetic method. It was the favoured method of Isaac Newton for the solution of geometric problems. 
Synthetic methods were most prominent during the 19th century when geometers rejected coordinate methods in establishing the foundations of projective geometry and non-Euclidean geometries. For example the geometer Jakob Steiner (1796 – 1863) hated analytic geometry, and always gave preference to synthetic methods.
History of geometry
Areas for theVolumes for theContinuing the geometrical legacy of ancient China, there were many later figures to come, including the famed astronomer and mathematician Shen Kuo (1031-1095 CE), Yang Hui (1238-1298) who discovered Pascal's Triangle, Xu Guangqi (1562-1633), and many others.
Alexis Clairaut
In 1741, Alexis Clairaut wrote a book called Èléments de Géométrie. The book outlines the basic concepts of geometry. Geometry in the 1700s was complex to the average learner. It was considered to be a dry subject. Clairaut saw this trend, and wrote the book in an attempt to make the subject more interesting for the average learner. He believed that instead of having students repeatedly work problems that they did not fully understand, it was imperative for them to make discoveries themselves in a form of active, experiential learning. He begins the book by comparing geometric shapes to measurements of land, as it was a subject that most anyone could relate to. He covers topics from lines, shapes, and even some three dimensional objects. Throughout the book, he continuously relates different concepts such as physics, astrology, and other branches of mathematics to geometry. Some of the theories and learning methods outlined in the book are still used by teachers today, in geometry and other topics.
Felix Klein
Klein's synthesis of geometry as the study of the properties of a space that is invariant under a given group of transformations, known as the "Erlangen Program" (1872), profoundly influenced the evolution of mathematics. This program was initiated by Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The program proposed a unified system of geometry that has become the accepted modern method. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the program's definition of geometry encompassed both Euclidean and non-Euclidean geometry.
Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book "Grundlagen der Geometrie" (tr. "The Foundations of Geometry") as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.
Descriptive geometry
Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art. The theoretical basis for descriptive geometry is provided by planar geometric projections. The earliest known publication on the technique was "Underweysung der Messung mit dem Zirckel und Richtscheyt", published in Linien, Nuremberg: 1525, by Albrecht Dürer. Gaspard Monge is usually considered the "father of descriptive geometry" due to his developments in geometric problem solving. His first discoveries were in 1765 while he was working as a draftsman for military fortifications, although his findings were published later on.
Topology
Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the "geometria situs" (Greek-Latin for "geometry of place") and "analysis situs" (Greek-Latin for "picking apart of place"). Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term "topology" was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.
Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realised the importance of the "edges" of a polyhedron. This led to his polyhedron formula, "V" − "E" + "F" = 2 (where "V", "E" and "F" respectively indicate the number of vertices, edges and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signalling the birth of topology.
Gaspard Monge
Gaspard Monge, Comte de Péluse (; 9 May 1746 – 28 July 1818) was a French mathematician, the inventor of descriptive geometry (the mathematical basis of technical drawing), and the father of differential geometry. During the French Revolution he served as the Minister of the Marine, and was involved in the reform of the French educational system, helping to found the École Polytechnique.
Julius Plücker
In 1865, Plücker returned to the field of geometry and invented what was known as "line geometry" in the nineteenth century. In projective geometry, Plücker coordinates refer to a set of homogeneous co-ordinates introduced initially to embed the set of lines in three dimensions as a quadric in five dimensions. The construction uses 2×2 minor determinants, or equivalently the second exterior power of the underlying vector space of dimension 4. It is now part of the theory of Grassmannians, to which these co-ordinates apply in generality ("k"-dimensional subspaces of "n"-dimensional space).Plücker was the recipient of the Copley Medal from the Royal Society in 1866.
Continuous geometry
In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, ..., "n", it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.
History of geometry
The first definitive work (or at least oldest existent) on geometry in China was the "Mo Jing", the Mohist canon of the early philosopher Mozi (470-390 BC). It was compiled years after his death by his followers around the year 330 BC. Although the "Mo Jing" is the oldest existent book on geometry in China, there is the possibility that even older written material existed. However, due to the infamous Burning of the Books in a political maneuver by the Qin Dynasty ruler Qin Shihuang (r. 221-210 BC), multitudes of written literature created before his time were purged. In addition, the "Mo Jing" presents geometrical concepts in mathematics that are perhaps too advanced not to have had a previous geometrical base or mathematic background to work upon.
History of mathematics
Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.
This century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds.
The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician János Bolyai, independently defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalizes the ideas of curves and surfaces.
Geometry
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, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian "Rhind Papyrus" (2000–1800 BC) and "Moscow Papyrus" (c. 1890 BC), the Babylonian clay tablets such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Later clay tablets (350–50 BC) 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.
Anton Albert Beekman
Anton Albert Beekman (5 January 1854 – 23 May 1947) was a Dutch geographer who focussed mainly on Dutch geography and other European countries. He is seen as one of the original founders of modern geography.
History of geography
In 1877, Thomas Henry Huxley published his Physiography with the philosophy of universality presented as an integrated approach in the study of the natural environment. The philosophy of universality in geography was not a new one but can be seen as evolving from the works of Alexander von Humboldt and Immanuel Kant. The publication of Huxley physiography presented a new form of geography that analysed and classified cause and effect at the micro-level and then applied these to the macro-scale (due to the view that the micro was part of the macro and thus an understanding of all the micro-scales was need to understand the macro level). This approach emphasized the empirical collection of data over the theoretical. The same approach was also used by Halford John Mackinder in 1887. However, the integration of the Geosphere, Atmosphere and Biosphere under physiography was soon over taken by Davisian geomorphology.
Lie sphere geometry
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.
Gheorghe Țițeica
Gheorghe Țițeica (; 4 October 1873 in Turnu Severin – 5 February 1939) publishing as George or Georges Tzitzeica) was a Romanian mathematician with important contributions in geometry. He is recognized as the founder of the Romanian school of differential geometry.
Eratosthenes
Eratosthenes of Cyrene (; , ;  – ) was a Greek mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria. He invented the discipline of geography, including the terminology used today.
History of geology
During the 17th century, both religious and scientific speculation about the Earth's origin further propelled interest in the Earth and brought about more systematic identification techniques of the Earth's strata. The Earth's strata can be defined as horizontal layers of rock having approximately the same composition throughout. An important pioneer in the science was Nicolas Steno. Steno was trained in the classical texts on science; however, by 1659 he seriously questioned accepted knowledge of the natural world. Importantly, he questioned the idea that fossils grew in the ground, as well as common explanations of rock formation. His investigations and his subsequent conclusions on these topics have led scholars to consider him one of the founders of modern stratigraphy and geology. (Steno, who became a Catholic as an adult, was eventually made a bishop, and was beatified in 1988 by Pope John Paul II. Therefore, he is also called Blessed Nicolas Steno.)
Corrado Segre
Corrado Segre (20 August 1863 – 18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry.
Geomorphometry
Although geomorphometry started with ideas of Brisson (1808) and Gauss (1827), the field did not evolve much until the construction of the first DEM.
Lon R. Shelby
Lonnie Royce (Lon. R.) Shelby (born 1935) is an American academic, and Professor Emeritus of Speech Communication and former Dean of the College of Liberal Arts at the Southern Illinois University. He is known for his work on Mediaeval architects and design, especially on the work of Lorenz Lechler, Mathes Roriczer, Hanns Schmuttermayer, Taccola and Villard de Honnecourt. He is also known for coined the term "constructive geometry."
History of geometry
Although the Islamic mathematicians are most famed for their work on algebra, number theory and number systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy, and were responsible for the development of algebraic geometry.
Geometry of numbers
In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by .
Foundations of geometry
In view of the role which mathematics plays in science and implications of scientific knowledge for all of our beliefs, revolutionary changes in man's understanding of the nature of mathematics could not but mean revolutionary changes in his understanding of science, doctrines of philosophy, religious and ethical beliefs, and, in fact, all intellectual disciplines.
In the first half of the nineteenth century a revolution took place in the field of geometry that was as scientifically important as the Copernican revolution in astronomy and as philosophically profound as the Darwinian theory of evolution in its impact on the way we think. This was the consequence of the discovery of non-Euclidean geometry. For over two thousand years, starting in the time of Euclid, the postulates which grounded geometry were considered self-evident truths about physical space. Geometers thought that they were deducing other, more obscure truths from them, without the possibility of error. This view became untenable with the development of hyperbolic geometry. There were now two incompatible systems of geometry (and more came later) that were self-consistent and compatible with the observable physical world. "From this point on, the whole discussion of the relation between geometry and physical space was carried on in quite different terms."
Gottfried Wilhelm Leibniz
Although for Leibniz the situs of a sequence of points is completely determined by the distance between them and is altered if those distances are altered, his admirer Euler, in the famous 1736 paper solving the Königsberg Bridge Problem and its generalizations, used the term "geometria situs" in such a sense that the situs remains unchanged under topological deformations. He mistakenly credits Leibniz with originating this concept. ... [It] is sometimes not realized that Leibniz used the term in an entirely different sense and hence can hardly be considered the founder of that part of mathematics.
Democritus
Democritus was also a pioneer of mathematics and geometry in particular. We only know this through citations of his works (titled "On Numbers", "On Geometrics", "On Tangencies", "On Mapping", and "On Irrationals") in other writings, since most of Democritus's body of work did not survive the Middle Ages. Democritus was among the first to observe that a cone and pyramid with the same base and height has one-third the volume of a cylinder or prism respectively .
Algebra
Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. His book "Treatise on Demonstrations of Problems of Algebra" (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe. Yet another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations. He also developed the concept of a function. The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī (1412–1486) took "the first steps toward the introduction of algebraic symbolism". He also computed ∑"n", ∑"n" and used the method of successive approximation to determine square roots. As the Islamic world was declining, the European world was ascending. And it is here that algebra was further developed.
Geometry
In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry. Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry. Omar Khayyám (1048–1131) found geometric solutions to cubic equations. The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo (c. 1230–c. 1314), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri.
János Bolyai
János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, one of the founders of non-Euclidean geometry — a geometry that differs from Euclidean geometry in its definition of parallel lines. The discovery of a consistent alternative geometry that might correspond to the structure of the universe helped to free mathematicians to study abstract concepts irrespective of any possible connection with the physical world.
Science and technology in Russia
In mathematics Nikolai Lobachevsky, a "Copernicus of Geometry", founded the non-Euclidean geometry playing important role in modern physics. In the 19th century the international recognition was also gained by such mathematicians as Mikhail Ostrogradsky and Sofia Kovalevskaya who was the first major Russian female mathematician, responsible for important original contributions to analysis, differential equations and mechanics, and the first woman appointed to a full professorship in Northern Europe. Yevgraf Fyodorov was a founder of the modern structural crystallography (Fedorov group). At least after such prominent scientist as Chebyshev the Russian mathematical school became one of the most influential ones in the world and was represented by numerous figures greatly contributing to different fields of mathematics, physics and computing sciences. Chebyshev's students included Aleksandr Lyapunov who founded the modern stability theory (lately deeply developed by such scientists as Aleksandr Andronov and Vladimir Arnold), and Andrey Markov who developed the theory of Markov chains, playing a central role in information sciences and modern applied mathematics.
In the beginning of the 20th century Nikolai Zhukovsky and Sergei Chaplygin were among founding fathers of the modern aero- and hydrodynamics and Vladimir Kotelnikov was a pioneer in information theory by independently proposing the fundamental sampling theorem. 
Andrei Kolmogorov, a leading mathematician of the 20th century, developed the foundation of the modern theory of probability and made other key contributions to broadest range of mathematical branches, such as turbulence, mathematical logic, topology, differential equations, set theory, automata theory, information theory, theory of algorithms, dynamical systems, stochastic processes, theory of integration, classical mechanics, mathematical linguistics, mathematical biology and applied sciences. Israel Gelfand is credited with many important discoveries in algebra, topology, mathematical physics and applied sciences. Sergei Sobolev developed the theory of Sobolev space which played an extremely important role in formation of modern mathematical views and introduced the notion of distributions generalizing ideas of Newton and Leibniz.