1.
Euclidean geometry
–
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids 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 school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
2.
Geometry
–
Geometry 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, 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 mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. 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 footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
3.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
4.
Euclid
–
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles 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, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
5.
Middle Ages
–
In the history of Europe, the Middle Ages or Medieval Period lasted from the 5th to the 15th century. It began with the fall of the Western Roman Empire and merged into the Renaissance, the Middle Ages is the middle period of the three traditional divisions of Western history, classical antiquity, the medieval period, and the modern period. The medieval period is subdivided into the Early, High. Population decline, counterurbanisation, invasion, and movement of peoples, the large-scale movements of the Migration Period, including various Germanic peoples, formed new kingdoms in what remained of the Western Roman Empire. In the seventh century, North Africa and the Middle East—once part of the Byzantine Empire—came under the rule of the Umayyad Caliphate, although there were substantial changes in society and political structures, the break with classical antiquity was not complete. The still-sizeable Byzantine Empire survived in the east and remained a major power, the empires law code, the Corpus Juris Civilis or Code of Justinian, was rediscovered in Northern Italy in 1070 and became widely admired later in the Middle Ages. In the West, most kingdoms incorporated the few extant Roman institutions, monasteries were founded as campaigns to Christianise pagan Europe continued. The Franks, under the Carolingian dynasty, briefly established the Carolingian Empire during the later 8th, the Crusades, first preached in 1095, were military attempts by Western European Christians to regain control of the Holy Land from Muslims. Kings became the heads of centralised nation states, reducing crime and violence, intellectual life was marked by scholasticism, a philosophy that emphasised joining faith to reason, and by the founding of universities. Controversy, heresy, and the Western Schism within the Catholic Church paralleled the conflict, civil strife. Cultural and technological developments transformed European society, concluding the Late Middle Ages, the Middle Ages is one of the three major periods in the most enduring scheme for analysing European history, classical civilisation, or Antiquity, the Middle Ages, and the Modern Period. Medieval writers divided history into periods such as the Six Ages or the Four Empires, when referring to their own times, they spoke of them as being modern. In the 1330s, the humanist and poet Petrarch referred to pre-Christian times as antiqua, leonardo Bruni was the first historian to use tripartite periodisation in his History of the Florentine People. Bruni and later argued that Italy had recovered since Petrarchs time. The Middle Ages first appears in Latin in 1469 as media tempestas or middle season, in early usage, there were many variants, including medium aevum, or middle age, first recorded in 1604, and media saecula, or middle ages, first recorded in 1625. The alternative term medieval derives from medium aevum, tripartite periodisation became standard after the German 17th-century historian Christoph Cellarius divided history into three periods, Ancient, Medieval, and Modern. The most commonly given starting point for the Middle Ages is 476, for Europe as a whole,1500 is often considered to be the end of the Middle Ages, but there is no universally agreed upon end date. English historians often use the Battle of Bosworth Field in 1485 to mark the end of the period
6.
Pierre de Fermat
–
He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermats principle for light propagation and his Fermats Last Theorem in number theory, Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a leather merchant. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth, there is little evidence concerning his school education, but it was probably at the Collège de Navarre in Montauban. He attended the University of Orléans from 1623 and received a bachelor in law in 1626. In Bordeaux he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apolloniuss De Locis Planis to one of the mathematicians there, there he became much influenced by the work of François Viète. In 1630, he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France and he held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat, fluent in six languages, Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often little or no proof of his theorems. In some of these letters to his friends he explored many of the ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession, nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus. Secrecy was common in European mathematical circles at the time and this naturally led to priority disputes with contemporaries such as Descartes and Wallis. Anders Hald writes that, The basis of Fermats mathematics was the classical Greek treatises combined with Vietas new algebraic methods, Fermats pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes famous La géométrie. This manuscript was published posthumously in 1679 in Varia opera mathematica, in these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of power functions. With his method, he was able to reduce this evaluation to the sum of geometric series, the resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pells equation, perfect numbers, amicable numbers and it was while researching perfect numbers that he discovered Fermats little theorem. Fermat developed the two-square theorem, and the polygonal number theorem, although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived