Who is considered the father 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.
Karl Menger
With Arthur Cayley, Menger is considered one of the founders of distance geometry; especially by having formalized definitions to the notions of "angle" and of "curvature" in terms of directly measurable physical quantities, namely ratios of "distance" values.
Marinus of Tyre
Marinus of Tyre (, "Marînos ho Týrios";  70–130) was a Greek or Hellenized, possibly Phoenician, geographer, cartographer and mathematician, who founded mathematical geography and provided the underpinnings of Claudius Ptolemy's influential "Geography".
Abstraction (mathematics)
Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. Take the historical development of geometry as an example; the first steps in the abstraction of geometry were made by the ancient Greeks, with Euclid's Elements being the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios. In the 17th century Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry. Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann, and Gauss who generalised the concepts of geometry to develop non-Euclidean geometries. Later in the 19th century mathematicians generalised geometry even further, developing such areas as geometry in n dimensions, projective geometry, affine geometry and finite geometry. Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries. This level of abstraction revealed connections between geometry and abstract algebra.
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.
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 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 geometry
Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, "The Elements" is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.
William Pengelly
Sir Charles Lyell – the father of British Geology
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.
Veniamin Kagan
Veniamin Fedorovich Kagan (; 10 March 1869 – 8 May 1953) was a Russian and Soviet mathematician and expert in geometry. He is the maternal grandfather of mathematicians Yakov Sinai and Grigory Barenblatt.
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.
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.
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.
Non-Euclidean geometry
Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century.
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.
Timeline of the discovery and classification of minerals
Georgius Agricola is considered the 'father of mineralogy'. Nicolas Steno founded the stratigraphy (the study of rock layers (strata) and layering (stratification)), the geology characterizes the rocks in each layer and the mineralogy characterizes the minerals in each rock. The chemical elements were discovered in identified minerals and with the help of the identified elements the mineral crystal structure could be described. One milestone was the discovery of the geometrical law of crystallization by René Just Haüy, a further development of the work by Nicolas Steno and Jean-Baptiste L. Romé de l'Isle (the characterisation of a crystalline mineral needs knowledge on crystallography). Important contributions came from some Saxon "Bergraths"/ Freiberg Mining Academy: Johann F. Henckel, Abraham Gottlob Werner and his students (August Breithaupt, Robert Jameson, José Bonifácio de Andrada and others). Other milestones were the notion that metals are elements too (Antoine Lavoisier) and the periodic table of the elements by Dmitri Ivanovich Mendeleev. The overview of the organic bonds by Kekulé was necessary to understand the silicates, first refinements described by Bragg and Machatschki; and it was only possibly to understand a crystal structure with Dalton's atomic theory, the notion of atomic orbital and Goldschmidt's explanations. Specific gravity, streak (streak color and mineral hardness) and X-ray powder diffraction are quite specific for a Nickel-Strunz identifier (updated 9th ed.). Nowadays, non-destructive electron microprobe analysis is used to get the empirical formula of a mineral. Finally, the International Zeolite Association (IZA) took care of the zeolite frameworks (part of molecular sieves and/or molecular cages).
History of science in classical antiquity
Geometers such as Archimedes (c. 287 – 212 BCE), Apollonius of Perga (c. 262 – c. 190 BCE), and Euclid (c. 325 – 265 BCE), whose "Elements" became the most important textbook in mathematics until the 19th century, built upon the work of the Hellenic era Pythagoreans. Eratosthenes used his knowledge of geometry to measure the distance between the Sun and the Earth along with the size of the Earth.
Sacred geometry
The belief that a god created the universe according to a geometric plan has ancient origins. Plutarch attributed the belief to Plato, writing that "Plato said God geometrizes continually" ("Convivialium disputationum", liber 8,2). In modern times, the mathematician Carl Friedrich Gauss adapted this quote, saying "God arithmetizes".
4354 Euclides
This minor planet was named after the Greek mathematician and Euclid (also: Euclides, or Eukleides). The "father of geometry" lived in Alexandria about 300 B.C. The official naming citation was published by the Minor Planet Center on 8 July 1990 (). The lunar crater "Euclides" was also named in his honor.
Tafresh
Famous scientists from Tafresh include Mahmoud Hessaby, the father of modern physics in Iran, Ahmad Parsa, the father of Iranian botanics, Abbas Sahab, the father of cartography in Iran, Mohammad Gharib, the father of pediatrics in Iran, and Abdolkarim Gharib, the father of modern geology in Iran.
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.
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.
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.
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
Geometry
Geometry has also had a large effect on other areas of mathematics. For instance, the introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. The subject of geometry was further enriched by the study of the intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.
An important area of application is number theory. In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon abstract numbers in favor of concrete geometric quantities, such as length and area of figures. Since the 19th century, geometry has been used for solving problems in number theory, for example through the geometry of numbers or, more recently, scheme theory, which is used in Wiles's proof of Fermat's Last Theorem.
Non-Euclidean geometry
Non-Euclidean geometry is an example of a scientific revolution in the history of science, in which mathematicians and scientists changed the way they viewed their subjects. Some geometers called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work.
Foundations of geometry
The German mathematician Moritz Pasch (1843–1930) was the first to accomplish the task of putting Euclidean geometry on a firm axiomatic footing. In his book, "Vorlesungen über neuere Geometrie" published in 1882, Pasch laid the foundations of the modern axiomatic method. He originated the concept of primitive notion (which he called "Kernbegriffe") and together with the axioms ("Kernsätzen") he constructs a formal system which is free from any intuitive influences. According to Pasch, the only place where intuition should play a role is in deciding what the primitive notions and axioms should be. Thus, for Pasch, "point" is a primitive notion but "line" (straight line) is not, since we have good intuition about points but no one has ever seen or had experience with an infinite line. The primitive notion that Pasch uses in its place is "line segment".
Cartesian coordinate system
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, centered at the origin of the plane, may be described as the set of all points whose coordinates "x" and "y" satisfy the equation .
John Murray (oceanographer)
Sir John Murray (3 March 1841 – 16 March 1914) was a pioneering British oceanographer, marine biologist and limnologist. He is considered to be the father of modern oceanography.
William Morris Davis
William Morris Davis (February 12, 1850 – February 5, 1934) was an American geographer, geologist, geomorphologist, and meteorologist, often called the "father of American geography".
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.
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 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.)
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.
László Fejes Tóth
He was a member of the Hungarian Academy of Sciences (from 1962) and a director of the Alfréd Rényi Institute of Mathematics (1970-1983). He received both the Kossuth Prize (1957) and State Award (1973).
Together with H.S.M. Coxeter and Paul Erdős, he laid the foundations of discrete geometry.
As described in a 1999 interview with István Hargittai, Fejes Tóth’s father was a railway worker, who advanced in his career within the railway organization ultimately to earn a doctorate in law. Fejes Tóth’s mother taught Hungarian and German literature in a high school. The family moved to Budapest, when Fejes Tóth was five; there he attended elementary school and high school—the Széchenyi István Reálgimnázium—where his interest in mathematics began.
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.
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.
Thales of Miletus
Thales (who died around 30 years before the time of Pythagoras and 300 years before Euclid, Eudoxus of Cnidus, and Eudemus of Rhodes) is often hailed as "the first Greek mathematician". While some historians, such as Colin R. Fletcher, point out that there could have been a predecessor to Thales who would've been named in Eudemus' lost book "History of Geometry", it is admitted that without the work "the question becomes mere speculation." Fletcher holds that as there is no viable predecessor to the title of first Greek mathematician, the only question is whether Thales qualifies as a practitioner in that field; he holds that "Thales had at his command the techniques of observation, experimentation, superposition and deduction…he has proved himself mathematician."
Geometric algebra
In more detail, there have been three approaches to geometric algebra: quaternionic analysis, initiated by Hamilton in 1843 and geometrized as rotors by Clifford in 1878; geometric algebra, initiated by Grassmann in 1844; and vector analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis can be seen in the use of formula_136, formula_194, formula_111 to indicate the basis vectors of formula_317: it is being thought of as the purely imaginary quaternions. From the perspective of geometric algebra, quaternions can be identified as, the even part of the Clifford algebra on Euclidean 3-space, which unifies the three approaches.Progress on the study of Clifford algebras quietly advanced through the twentieth century, although largely due to the work of abstract algebraists such as Hermann Weyl and Claude Chevalley. The "geometrical" approach to geometric algebras has seen a number of 20th-century revivals. In mathematics, Emil Artin's "Geometric Algebra" discusses the algebra associated with each of a number of geometries, including affine geometry, projective geometry, symplectic geometry, and orthogonal geometry. In physics, geometric algebras have been revived as a "new" way to do classical mechanics and electromagnetism, together with more advanced topics such as quantum mechanics and gauge theory. David Hestenes reinterpreted the Pauli and Dirac matrices as vectors in ordinary space and spacetime, respectively, and has been a primary contemporary advocate for the use of geometric algebra.
Geometry
Euclid took an abstract approach to geometry in his Elements, one of the most influential books ever written. Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as "axiomatic" or "synthetic" geometry. At the start of the 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.
Churchill, Oxfordshire
William Smith (1769–1839) "Strata Smith" was the "father of English geology". Smith was born in Churchill on 23 March 1769, the son of a blacksmith. He was educated at the village school until he was 11, then went to London for two years. In 1788, when he returned to Churchill, his uncle (a Hook Norton farmer) encouraged his interest in surveying, and together they pursued various schemes for land improvement and drainage. At 18 he became an assistant surveyor, helping to survey Churchill and Sarsden for the 1788 Enclosure Act. His experience of different rock formations led him to develop the theory that the occurrence of different types of fossil could be used to order the geological sequence of rock strata. Although from 1800 he gained a reputation as a civil engineer, he became famous for preparing and producing a series of detailed geological maps of England. He died on 28 August 1839 in Northampton.
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.
The Story of Maths
Du Sautoy proceeds to describes René Descartes realisation that it was possible to describe curved lines as equations and thus link algebra and geometry. He talks with Henk J. M. Bos about Descartes. He shows how one of Pierre de Fermat’s theorems is now the basis for the codes that protect credit card transactions on the internet. He describes Isaac Newton’s development of math and physics crucial to understanding the behaviour of moving objects in engineering. He covers the Leibniz and Newton calculus controversy and the Bernoulli family. He further covers Leonhard Euler, the father of topology, and Gauss' invention of a new way of handling equations, modular arithmetic. He mentions János Bolyai.
Proclus
Proclus, the scholiast to Euclid, knew Eudemus of Rhodes' "History of Geometry" well, and gave a short sketch of the early history of geometry, which appeared to be founded on the older, lost book of Eudemus. The passage has been referred to as "the Eudemian summary," and determines some approximate dates, which otherwise might have remained unknown. The influential commentary on the first book of Euclid's "Elements of Geometry" is one of the most valuable sources we have for the history of ancient mathematics, and its Platonic account of the status of mathematical objects was influential. In this work, Proclus also listed the first mathematicians associated with Plato: a mature set of mathematicians (Leodamas of Thasos, Archytas of Taras, and Theaetetus), a second set of younger mathematicians (Neoclides, Eudoxus of Cnidus), and a third yet younger set (Amyntas, Menaechmus and his brother Dinostratus, Theudius of Magnesia, Hermotimus of Colophon and Philip of Opus). Some of these mathematicians were influential in arranging the Elements that Euclid later published.
Jiang Lifu
Jiang Lifu (4 July 1890 – 3 February 1978) was a Chinese mathematician and educator widely regarded as the Father of modern Chinese mathematics. His main research areas are the theory of syringine geometry and matrix.
Pierre Bouguer
Pierre Bouguer () (16 February 1698, Croisic – 15 August 1758, Paris) was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture".
Nadir Afonso
Because of his rationalism, Nadir Afonso confronted Kandinsky, the father of abstract art, and criticized him for subduing geometry to the human spirit instead of making it the essence of art. This ""geometry of art"" is not however the ""geometry of geometrists"", as it is not about symbols nor anything in particular; rather, it is the spatial law itself, with the four qualities of perfection, harmony, evocation, and originality.
Daniel Pedoe
Dan Pedoe (29 October 1910, London – 27 October 1998, St Paul, Minnesota, USA) was an English-born mathematician and geometer with a career spanning more than sixty years. In the course of his life he wrote approximately fifty research and expository papers in geometry. He is also the author of various core books on mathematics and geometry some of which have remained in print for decades and been translated into several languages. These books include the three-volume "Methods of Algebraic Geometry" (which he wrote in collaboration with W. V. D. Hodge), "The Gentle Art of Mathematics", "Circles: A Mathematical View", "Geometry and the Visual Arts" and most recently "Japanese Temple Geometry Problems: San Gaku" (with Hidetoshi Fukagawa).
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.
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.
History of science
The earliest Greek philosophers, known as the pre-Socratics, provided competing answers to the question found in the myths of their neighbors: "How did the ordered cosmos in which we live come to be?" The pre-Socratic philosopher Thales (640–546 BC), dubbed the "father of science", was the first to postulate non-supernatural explanations for natural phenomena. For example, that land floats on water and that earthquakes are caused by the agitation of the water upon which the land floats, rather than the god Poseidon. Thales' student Pythagoras of Samos founded the Pythagorean school, which investigated mathematics for its own sake, and was the first to postulate that the Earth is spherical in shape. Leucippus (5th century BC) introduced atomism, the theory that all matter is made of indivisible, imperishable units called atoms. This was greatly expanded on by his pupil Democritus and later Epicurus.
History of geometry
Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. For example, Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra both studied curves required in the construction of sundials. Abu'l-Wafa and Abu Nasr Mansur both applied spherical geometry to astronomy.
History of geometry
It remained to be proved mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry.
Geometry
Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.
Islamic Scholarship &amp; Learning in Central Asia
Algebra was also pioneered by Persian Scientist Muhammad ibn Mūsā al-Khwārizmī during this time in his landmark text, Kitab al-Jabr wa-l-Muqabala, from which the term algebra is derived. He is thus considered to be the father of algebra. The terms algorism and algorithm are also derived from the name of al-Khwarizmi, who was responsible for introducing the Arabic numerals and Hindu-Arabic numeral system beyond the Indian Subcontinent.
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.
Greek mathematics
Thales is supposed to have used geometry to solve problems such as calculating the height of pyramids based on the length of shadows, and the distance of ships from the shore. He is also credited by tradition with having made the first proof of two geometric theorems—the "Theorem of Thales" and the "Intercept theorem" described above. Pythagoras is widely credited with recognizing the mathematical basis of musical harmony and, according to Proclus' commentary on Euclid, he discovered the theory of proportionals and constructed regular solids. Some modern historians have questioned whether he really constructed all five regular solids, suggesting instead that it is more reasonable to assume that he constructed just three of them. Some ancient sources attribute the discovery of the Pythagorean theorem to Pythagoras, whereas others claim it was a proof for the theorem that he discovered. Modern historians believe that the principle itself was known to the Babylonians and likely imported from them. The Pythagoreans regarded numerology and geometry as fundamental to understanding the nature of the universe and therefore central to their philosophical and religious ideas. They are credited with numerous mathematical advances, such as the discovery of irrational numbers. Historians credit them with a major role in the development of Greek mathematics (particularly number theory and geometry) into a coherent logical system based on clear definitions and proven theorems that was considered to be a subject worthy of study in its own right, without regard to the practical applications that had been the primary concern of the Egyptians and Babylonians.
Foundations of geometry
In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as "Birkhoff's axioms". These postulates are all based on basic geometry that can be experimentally verified with a scale and protractor. In a radical departure from the synthetic approach of Hilbert, Birkhoff was the first to build the foundations of geometry on the real number system. It is this powerful assumption that permits the small number of axioms in this system.
History of logic
Fragments of early proofs are preserved in the works of Plato and Aristotle, and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy. The proofs of Euclid of Alexandria are a paradigm of Greek geometry. The three basic principles of geometry are as follows:
Physical geography
One of the most prolific empires in this regard was Russia. In the mid-eighteenth century many geographers were sent by the Russian altamirazgo different opportunities to perform geographical surveys in the area of Arctic Siberia. Among these is who is considered the patriarch of Russian geography, Mikhail Lomonosov. In the mid-1750s Lomonosov began working in the Department of Geography, Academy of Sciences to conduct research in Siberia, their contributions are notable in this regard, showing the soil organic origin and developing a comprehensive law on the movement of the ice that still governs the basics, thereby founding a new branch of geography: glaciology. In 1755 his initiative was founded Moscow University where he promotes the study of geography and the training of geographers. In 1758 he was appointed director of the Department of Geography, Academy of Sciences, a post from which would develop a working methodology for geographical survey guided by the most important long expeditions and geographical studies in Russia. Thus followed the line of Lomonosov and the contributions of the Russian school became more frequent through his disciples, and in the nineteenth century we have great geographers as Vasily Dokuchaev who performed works of great importance as a "principle of comprehensive analysis of the territory" and "Russian Chernozem" latter being the most important where introduces the geographical concept of soil, as distinct from a simple geological strata, and thus founding a new geographic area of study: pedology. Climatology also receive a strong boost from the Russian school by Wladimir Köppen whose main contribution, climate classification, is still valid today. However, this great geographer also contributed to the paleogeography through his work "The climates of the geological past" which is considered the father of paleoclimatology. Russian geographers who made great contributions to the discipline in this period were: NM Sibirtsev, Pyotr Semyonov, K.D. Glinka, Neustrayev, among others.
History of geometry
In some respects, Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments. Another important contribution Thabit made to geometry was his generalization of the Pythagorean theorem, which he extended from special right triangles to all triangles in general, along with a general proof.
Scientist
During the Renaissance, Italians made substantial contributions in science. Leonardo Da Vinci made significant discoveries in paleontology and anatomy. The Father of modern Science,
Galileo Galilei, made key improvements on the thermometer and telescope which allowed him to observe and clearly describe the solar system. Descartes was not only a pioneer of analytic geometry but formulated a theory of mechanics and advanced ideas about the origins of animal movement and perception. Vision interested the physicists Young and Helmholtz, who also studied optics, hearing and music. Newton extended Descartes' mathematics by inventing calculus (contemporaneously with Leibniz). He provided a comprehensive formulation of classical mechanics and investigated light and optics. Fourier founded a new branch of mathematics — infinite, periodic series — studied heat flow and infrared radiation, and discovered the greenhouse effect. Girolamo Cardano, Blaise Pascal Pierre de Fermat, Von Neumann, Turing, Khinchin, Markov and Wiener, all mathematicians, made major contributions to science and probability theory, including the ideas behind computers, and some of the foundations of statistical mechanics and quantum mechanics. Many mathematically inclined scientists, including Galileo, were also musicians.
History of mathematical notation
Johannes Kepler was one of the pioneers of the mathematical applications of infinitesimals. René Descartes is credited as the father of analytical geometry, the bridge between algebra and geometry, crucial to the discovery of infinitesimal calculus and analysis. In the 17th century, Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry. Blaise Pascal influenced mathematics throughout his life. His "Traité du triangle arithmétique" ("Treatise on the Arithmetical Triangle") of 1653 described a convenient tabular presentation for binomial coefficients. Pierre de Fermat and Blaise Pascal would investigate probability. John Wallis introduced the infinity symbol. He similarly used this notation for infinitesimals. In 1657, Christiaan Huygens published the treatise on probability, "On Reasoning in Games of Chance".
Henry Gannett
Henry Gannett (August 24, 1846 – November 5, 1914) was an American geographer who is described as the "Father of the Quadrangle" which is the basis for topographical maps in the United States.
Foundations of geometry
At the University of Göttingen, during the 1898–1899 winter term, the eminent German mathematician David Hilbert (1862–1943) presented a course of lectures on the foundations of geometry. At the request of Felix Klein, Professor Hilbert was asked to write up the lecture notes for this course in time for the summer 1899 dedication ceremony of a monument to C.F. Gauss and Wilhelm Weber to be held at the university. The rearranged lectures were published in June 1899 under the title "Grundlagen der Geometrie" (Foundations of Geometry). The influence of the book was immediate. According to :
History of algebra
The Hellenistic mathematician Diophantus has traditionally been known as "the father of algebra" but debate now exists as to whether or not Al-Khwarizmi deserves this title instead. Those who support Diophantus point to the fact that the algebra found in "Al-Jabr" is more elementary than the algebra found in "Arithmetica" and that "Arithmetica" is syncopated while "Al-Jabr" is fully rhetorical.
History of geometry
"The Nine Chapters on the Mathematical Art", the title of which first appeared by 179 AD on a bronze inscription, was edited and commented on by the 3rd century mathematician Liu Hui from the Kingdom of Cao Wei. This book included many problems where geometry was applied, such as finding surface areas for squares and circles, the volumes of solids in various three-dimensional shapes, and included the use of the Pythagorean theorem. The book provided illustrated proof for the Pythagorean theorem, contained a written dialogue between of the earlier Duke of Zhou and Shang Gao on the properties of the right angle triangle and the Pythagorean theorem, while also referring to the astronomical gnomon, the circle and square, as well as measurements of heights and distances. The editor Liu Hui listed pi as 3.141014 by using a 192 sided polygon, and then calculated pi as 3.14159 using a 3072 sided polygon. This was more accurate than Liu Hui's contemporary Wang Fan, a mathematician and astronomer from Eastern Wu, would render pi as 3.1555 by using ⁄. Liu Hui also wrote of mathematical surveying to calculate distance measurements of depth, height, width, and surface area. In terms of solid geometry, he figured out that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge. He also figured out that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. Furthermore, Liu Hui described Cavalieri's principle on volume, as well as Gaussian elimination. From the "Nine Chapters", it listed the following geometrical formulas that were known by the time of the Former Han Dynasty (202 BCE–9 CE).
Abraham Gottlob Werner
Abraham Gottlob Werner (25 September 174930 June 1817) was a German geologist who set out an early theory about the stratification of the Earth's crust and propounded a history of the Earth that came to be known as Neptunism. While most tenets of Neptunism were eventually set aside, Werner is remembered for his demonstration of chronological succession in rocks; for the zeal with which he infused his pupils; and for the impulse he thereby gave to the study of geology. He has been called the "father of German geology".
Jean Metzinger
Maurice Princet joined us often. Although quite young, thanks to his knowledge of mathematics he had an important job in an insurance company. But, beyond his profession, it was as an artist that he conceptualized mathematics, as an aesthetician that he invoked n-dimensional continuums. He loved to get the artists interested in the new views on space that had been opened up by Schlegel and some others. He succeeded at that.
Louis Vauxcelles sarcastically dubbed Princet "the father of cubism": "M. Princet has studied at length non-Euclidean geometry and the theorems of Riemann, of which Gleizes and Metzinger speak... Princet one day met M. Max Jacob and confided him one or two of his discoveries relating to the fourth dimension. M. Jacob informed the ingenious Picasso of it, and Picasso saw there a possibility of new ornamental schemes. Picasso explained his intentions to Apollinaire, who hastened to write them up in formularies and codify them. The thing spread and propagated. Cubism, the child of M. Princet, was born." (Vauxcelles, December 29, 1918).
Waterfall
Alexander von Humboldt (1820s) "Father of Modern Geography" Humboldt was mostly marking waterfalls on maps for river navigation purposes.
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.
William Smith (geologist)
William 'Strata' Smith (23 March 1769 – 28 August 1839) was an English geologist, credited with creating the first nationwide geological map. At the time his map was first published he was overlooked by the scientific community; his relatively humble education and family connections prevented him from mixing easily in learned society. Financially ruined, Smith spent time in debtors' prison. It was only late in his life that Smith received recognition for his accomplishments, and became known as the "Father of English Geology".
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.
Apollonius of Perga
Heath goes on to use the term geometrical algebra for the methods of the entire golden age. The term is “not inappropriately” called that, he says. Today the term has been resurrected for use in other senses (see under geometric algebra). Heath was using it as it had been defined by Henry Burchard Fine in 1890 or before. Fine applies it to La Géométrie of René Descartes, the first full-blown work of analytic geometry. Establishing as a precondition that “two algebras are formally identical whose fundamental operations are formally the same,” Fine says that Descartes’ work “is not ... mere numerical algebra, but what may for want of a better name be called the algebra of line segments. Its symbolism is the same as that of numerical algebra; ...”
Henry Wesley Voysey
Henry Wesley Voysey (1791-19 April 1824), was a physician, geologist and mineralogist who worked in India in the service of the East India Company. He has been called the "father of Indian geology." He made the first geological map in India covering the Hyderabad region that he travelled through as geologist associated with the Great Trigonometrical Survey of India. The map was submitted to the Asiatic Society of Bengal in Calcutta on 8 August 1821 but no copy of it remains.
History of urban planning
Traditionally, the Greek philosopher Hippodamus (5th century BC) is regarded as the first town planner and ‘inventor’ of the orthogonal urban layout. Aristotle called him "the father of city planning", and until well into the 20th century, he was indeed regarded as such. This is, however, only partly justified. The Hippodamian plan that was called after him, is an orthogonal urban layout with more or less square street blocks. Archaeological finds from ancient Egypt—among others—demonstrate that Hippodamus cannot truly have been the inventor of this layout. Aristotle's critique and indeed ridicule of Hippodamus, which appears in "Politics" 2. 8, is perhaps the first known example of a criticism of urban planning.
Alexander Grothendieck
Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the 20th century.
Nicolas Sanson
Nicolas Sanson (20 December 1600 – 7 July 1667) was a French cartographer, termed by some the creator of French geography, in which he's been called the "father of French cartography."
Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.
Non-Euclidean geometry
Then, around 1830, the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky separately published treatises on hyperbolic geometry. Consequently, hyperbolic geometry is 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.
History of geodesy
Revising the figures attributed to Posidonius, another Greek philosopher determined 18,000 miles as the Earth's circumference. This last figure was promulgated by Ptolemy through his world maps. The maps of Ptolemy strongly influenced the cartographers of the Middle Ages. It is probable that Christopher Columbus, using such maps, was led to believe that Asia was only 3 or 4 thousand miles west of Europe.
Pierre Desceliers
Pierre Desceliers ( 1537–1553) was a French cartographer of the Renaissance and an eminent member of the Dieppe School of Cartography. He is considered the father of French hydrography.
Hermann Schubert
Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite number of solutions. In 1874, Schubert won a prize for solving a question posed by Zeuthen. Schubert calculus was named after him.
Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the "Elements". Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The "Elements" begins with plane geometry, still taught in secondary school (High School) as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the "Elements" states results of what are now called algebra and number theory, explained in geometrical language.
Line (geometry)
When geometry was first formalised by Euclid in the "Elements", he defined a general line (straight or curved) to be "breadthless length" with a straight line being a line "which lies evenly with the points on itself". These definitions serve little purpose since they use terms which are not, themselves, defined. In fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In modern geometry, a line is simply taken as an undefined object with properties given by axioms, but is sometimes defined as a set of points obeying a linear relationship when some other fundamental concept is left undefined.
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).
History of geometry
Geometry (from the ; "geo-" "earth", "-metron" "measurement") arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).
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.
History of mathematics
The most influential mathematician of the 18th century was arguably Leonhard Euler. His contributions range from founding the study of graph theory with the Seven Bridges of Königsberg problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol "i", and he popularized the use of the Greek letter formula_1 to stand for the ratio of a circle's circumference to its diameter. He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him.
Algebraic geometry
During the same period, Blaise Pascal and Gérard Desargues approached geometry from a different perspective, developing the synthetic notions of projective geometry. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek "ruler and compass construction". Ultimately, the analytic geometry of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of Newton and Leibniz. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the "calculus of infinitesimals" of Lagrange and Euler.
Scientific Revolution
Galileo Galilei has been called the "father of modern observational astronomy", the "father of modern physics", the "father of science", and "the Father of Modern Science". His original contributions to the science of motion were made through an innovative combination of experiment and mathematics.
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).
Mathematics and art
A strand of art from Ancient Greece onwards sees God as the geometer of the world, and the world's geometry therefore as sacred. The belief that God created the universe according to a geometric plan has ancient origins. Plutarch attributed the belief to Plato, writing that "Plato said God geometrizes continually" ("Convivialium disputationum", liber 8,2). This image has influenced Western thought ever since. The Platonic concept derived in its turn from a Pythagorean notion of harmony in music, where the notes were spaced in perfect proportions, corresponding to the lengths of the lyre's strings; indeed, the Pythagoreans held that everything was arranged by Number. In the same way, in Platonic thought, the regular or Platonic solids dictate the proportions found in nature, and in art. A Mediaeval manuscript illustration may refer to a verse in the Old Testament: "When he established the heavens I was there: when he set a compass upon the face of the deep" (Proverbs 8:27), showing God drawing out the universe with a pair of compasses. In 1596, the mathematical astronomer Johannes Kepler modelled the universe as a set of nested Platonic solids, determining the relative sizes of the orbits of the planets. William Blake's "Ancient of Days" and his painting of the physicist Isaac Newton, naked and drawing with a compass, attempt to depict the contrast between the mathematically perfect spiritual world and the imperfect physical world, as in a different way does Salvador Dalí's 1954 "Crucifixion (Corpus Hypercubus)", which depicts the cross as a hypercube, representing the divine perspective with four dimensions rather than the usual three. In Dali's "The Sacrament of the Last Supper" (1955) Christ and his disciples are pictured inside a giant dodecahedron.
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 geometry
The earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley (see Harappan Mathematics), and ancient Babylonia (see Babylonian mathematics) from around 3000 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. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras around 800 B.C. contained the first statements of the theorem; the Egyptians had a correct formula for the volume of a frustum of a square pyramid;
Algebraic geometry
Some of the roots of algebraic geometry date back to the work of the Hellenistic Greeks from the 5th century BC. The Delian problem, for instance, was to construct a length "x" so that the cube of side "x" contained the same volume as the rectangular box "a""b" for given sides "a" and "b". Menaechmus (circa 350 BC) considered the problem geometrically by intersecting the pair of plane conics "ay" = "x" and "xy" = "ab". The later work, in the 3rd century BC, of Archimedes and Apollonius studied more systematically problems on conic sections, and also involved the use of coordinates. The Arab mathematicians were able to solve by purely algebraic means certain cubic equations, and then to interpret the results geometrically. This was done, for instance, by Ibn al-Haytham in the 10th century AD. Subsequently, Persian mathematician Omar Khayyám (born 1048 A.D.) discovered a method for solving cubic equations by intersecting a parabola with a circle and seems to have been the first to conceive a general theory of cubic equations.
A few years after Omar Khayyám, Sharaf al-Din al-Tusi's "Treatise on equations" has been described as inaugurating the beginning of algebraic geometry.
History of geometry
Archimedes (287-212 BC), of Syracuse, Sicily, when it was a Greek city-state, is often considered to be the greatest of the Greek mathematicians, and occasionally even named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.
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.
Geometry
Geometry (from the ; "geo-" "earth", "-metron" "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.
History of cartography
In reviewing the literature of early geography and early conceptions of the earth, all sources lead to Homer, who is considered by many (Strabo, Kish, and Dilke) as the founding father of Geography. Regardless of the doubts about Homer's existence, one thing is certain: he never was a mapmaker.