What algebraic system taught in present-day schools is one of the legacies of the philosopher René Descartes to which he lent his name?
"dreams and believed that a divine spirit revealed to him a new philosophy. However, it is likely that what Descartes considered to be his second dream was actually an episode of exploding head syndrome. Upon exiting, he had formulated analytical geometry and the idea of applying the mathematical method to philosophy. He concluded from these visions that the pursuit of science would prove to be, for him, the pursuit of true wisdom and a central part of his life's work. Descartes also saw very clearly that all truths were linked with one another so that finding a fundamental truth and"
"to and following Galileo's death, including Evangelista Torricelli and the participants in the Accademia del Cimento in Italy; Marin Mersenne and Blaise Pascal in France; Christiaan Huygens in the Netherlands; and Robert Hooke and Robert Boyle in England. The French philosopher René Descartes (1596–1650) was well-connected to, and influential within, the experimental philosophy networks of the day. Descartes had a more ambitious agenda, however, which was geared toward replacing the Scholastic philosophical tradition altogether. Questioning the reality interpreted through the senses, Descartes sought to re-establish philosophical explanatory schemes by reducing all perceived phenomena to being attributable to the motion of"
"René Descartes René Descartes (, ; ; Latinized: Renatus Cartesius; adjectival form: ""Cartesian""; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–49) of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces. He is generally considered one of the most notable intellectual figures of the Dutch Golden Age. Descartes' ""Meditations on First Philosophy"" (1641) continues to be a standard text at"
"known for his investigation of conic sections. René Descartes (1596–1650) developed analytic geometry, an alternative method for formalizing geometry which focused on turning geometry into algebra. In this approach, a point on a plane is represented by its Cartesian (""x"", ""y"") coordinates, a line is represented by its equation, and so on. In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems. The equation"
"armillary sphere, and introduced Arabic numerals and clocks to Northern and Western Europe. The University of Paris, founded in the mid-12th century, is still one of the most important universities in the Western world. In the 17th century, mathematician René Descartes defined a method for the acquisition of scientific knowledge, while Blaise Pascal became famous for his work on probability and fluid mechanics. They were both key figures of the Scientific revolution, which blossomed in Europe during this period. The Academy of Sciences was founded by Louis XIV to encourage and protect the spirit of French scientific research. It was"
"of Empiricism and Rationalism respectively. Modern philosophy traditionally begins with René Descartes and his dictum ""I think, therefore I am"". In the early seventeenth century the bulk of philosophy was dominated by Scholasticism, written by theologians and drawing upon Plato, Aristotle, and early Church writings. Descartes argued that many predominant Scholastic metaphysical doctrines were meaningless or false. In short, he proposed to begin philosophy from scratch. In his most important work, ""Meditations on First Philosophy"", he attempts just this, over six brief essays. He tries to set aside as much as he possibly can of all his beliefs, to determine"
"individually named. Descartes Prize The Descartes Prize is an annual award in science given by the European Union, named in honour of the French mathematician and philosopher, René Descartes. The prizes recognizes Outstanding Scientific and Technological Achievements Resulting from European Collaborative Research. The research prize was first awarded in 2000. The research prize is awarded to teams of researchers who have ""achieved outstanding scientific or technological results through collaborative research in any field of science, including the economic, social science and humanities."" . Submissions may be received by the research teams themselves or by suitable national bodies. A science communication"
"European world was ascending. And it is here that algebra was further developed. 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"
"for René Descartes' oeuvre. Modern philosophy began in France with the philosophy of René Descartes (1596–1650). His ""Meditations on First Philosophy"" changed the primary object of philosophical thought from ontology to epistemology and overcame the Aristotelian dogmatism inherited in philosophy from Scholasticism, the dominant form of thought in preceding centuries, while simultaneously raising some of the most fundamental problems for future generations of philosophers. René Descartes was concerned with the uncertainty in the sciences and the radical scepticism which spread across Europe when publications of Sextus Empiricus became available. Descartes desired to find indubitable ground on which all the sciences"
"perspectives. The most important concept was his very modern treatment of independent variables. Although Descartes was well known in academic circles towards the end of his life, the teaching of his works in schools was controversial. Henri de Roy (Henricus Regius, 1598–1679), Professor of Medicine at the University of Utrecht, was condemned by the Rector of the University, Gijsbert Voet (Voetius), for teaching Descartes' physics. In January 2010, a previously unknown letter from Descartes, dated 27 May 1641, was found by the Dutch philosopher Erik-Jan Bos when browsing through Google. Bos found the letter mentioned in a summary of autographs"
"symbolic logic, that could encompass logical principles and methods symbolically, and mechanize general reasoning. Descartes' work provided the basis for the calculus developed by Newton and Leibniz, who applied infinitesimal calculus to the tangent line problem, thus permitting the evolution of that branch of modern mathematics. His rule of signs is also a commonly used method to determine the number of positive and negative roots of a polynomial. The beginning to Descartes' interest in physics is accredited to the amateur scientist and mathematician Isaac Beeckman, who was at the forefront of a new school of thought known as mechanical philosophy."
"whole (otherwise they would be independent wholes). René Descartes (1596–1650) originated the modern conception of matter. He was primarily a geometer. Instead of, like Aristotle, deducing the existence of matter from the physical reality of change, Descartes arbitrarily postulated matter to be an abstract, mathematical substance that occupies space: For Descartes, matter has only the property of extension, so its only activity aside from locomotion is to exclude other bodies: this is the mechanical philosophy. Descartes makes an absolute distinction between mind, which he defines as unextended, thinking substance, and matter, which he defines as unthinking, extended substance. They are"
"rejected any appeal to final ends, divine or natural, in explaining natural phenomena. In his theology, he insists on the absolute freedom of God's act of creation. Descartes laid the foundation for 17th-century continental rationalism, later advocated by Spinoza and Leibniz, and opposed by the empiricist school of thought consisting of Hobbes, Locke, Berkeley, and Hume. Leibniz, Spinoza, and Descartes were all well versed in mathematics as well as philosophy, and Descartes and Leibniz contributed greatly to science as well. René Descartes was born in La Haye en Touraine (now Descartes, Indre-et-Loire), France, on 31 March 1596. His mother, Jeanne"
"collections of individual objects (following older speculations that they are words, """"logoi""""). René Descartes published ""La Géométrie"" (1637), aimed at reducing geometry to algebra by means of coordinate systems, giving algebra a more foundational role (while the Greeks embedded arithmetic into geometry by identifying whole numbers with evenly spaced points on a line). Descartes' book became famous after 1649 and paved the way to infinitesimal calculus. Isaac Newton (1642–1727) in England and Leibniz (1646–1716) in Germany independently developed the infinitesimal calculus based on heuristic methods greatly efficient, but direly lacking rigorous justifications. Leibniz even went on to explicitly describe infinitesimals"
"Descartes Prize The Descartes Prize is an annual award in science given by the European Union, named in honour of the French mathematician and philosopher, René Descartes. The prizes recognizes Outstanding Scientific and Technological Achievements Resulting from European Collaborative Research. The research prize was first awarded in 2000. The research prize is awarded to teams of researchers who have ""achieved outstanding scientific or technological results through collaborative research in any field of science, including the economic, social science and humanities."" . Submissions may be received by the research teams themselves or by suitable national bodies. A science communication prize was"
"kept by Haverford College in Haverford, Pennsylvania. The college was unaware that the letter had never been published. This was the third letter by Descartes found in the last 25 years. General Bibliographies Stanford Encyclopedia of Philosophy Internet Encyclopedia of Philosophy Other René Descartes René Descartes (, ; ; Latinized: Renatus Cartesius; adjectival form: ""Cartesian""; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–49) of his life in the Dutch Republic after serving for a while in the Dutch States Army of"
"the Chuquet-Peletier system. Much later, in France and in the US, a different system, the short scale, became established where the term billion signifies 10. Last century, England and other English-speaking countries joined the US and some countries in using the short scale system; whereas, France rejoined Germany, much of Europe, and some other countries in the Chuquet-Peletier, or long scale, system. Nicolas Chuquet Nicolas Chuquet (; 1445, but some sources say 1455, Paris, France – 1488, some sources say 1500, Lyon, France) was a French mathematician. He invented his own notation for algebraic concepts and exponentiation. He may have"
"ENS as well. Later, Marxist political thinker Louis Althusser was a student at ENS and taught there for many years, and many of his disciples later became known for their own thought: among them were Étienne Balibar, philosopher Alain Badiou, who still teaches at the school as an emeritus professor, and Jacques Rancière. Still later, in the 1940s and 1950s, the world-renowned thinker Michel Foucault, founder of the history of systems of thought and future professor at the Collège de France was a student a few years ahead of the founder of deconstruction, Jacques Derrida and the thinker of individuation"
"a Cartesian product of ""n"" sets, also known as an ""n""-fold Cartesian product, can be represented by an array of ""n"" dimensions, where each element is an ""n""-tuple. An ordered pair is a 2-tuple or couple. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. An illustrative example is the standard 52-card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits form a four-element set."
"studied dynamics of power in society and its transmission over generations, achieved worldwide fame with his 1979 book """" and became a vocal critic of the French system of ""grandes écoles"" and notably ENS as the standard-bearer of that system, studied at ENS in the early 1950s. Other ENS sociologists and anthropologists include Maurice Halbwachs, Alain Touraine and Philippe Descola. The school also has a tradition of geography, with the founder of modern French geography and of the French School of Geopolitics Paul Vidal de La Blache having been a student at the school starting in 1863. As for economics,"
"century France. The Cartesian Coordinate System was discovered by René Descartes in 1637 (and independently by Pierre de Fermat at the same period). The first calculator by Blaise Pascal (Pascaline) was made in 1642. (see also Adding machine) Probability theory was developed by Pierre de Fermat and Blaise Pascal in the seventeenth century (with Gerolamo Cardano and Christiaan Huygens). France is home to 11 Fields Medalists, second only to the United States in number of Fields Medalists. The fictitious mathematician Nicolas Bourbaki's ""association of collaborators"" is based at École Normale Supérieure in France. France carried out its first test of"
"pour resoudre tous Triangles rectilignes et sphériques"". At the end of the latter appears a list of nine mathematical works which Hume had written in Latin. There are also two non-mathematical works, ""De Horologiis"" and ""Grammatica Hebræa"". A translation of one of his works into French, apparently his ""De Arte muniendi more Gallico"", appeared under the title ""Fortifications Françaises d'une Méthode facile"". Attribution James Hume (mathematician) James Hume (fl. 1639) was a Scottish mathematician. He is given credit for introducing the modern exponential notation, along with René Descartes. The son of David Hume of Godscroft, sometimes therefore called described as"
"corresponding to a flat hyperplane, in exact analogy to the 2-dimensional version of the theorem. Although there is no 3-dimensional analogue of the complex numbers, the relationship between the positions of the centers can be re-expressed as a matrix equation, which also generalizes to dimensions. Descartes' theorem In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643."
"French philosophy French philosophy, here taken to mean philosophy in the French language, has been extremely diverse and has influenced Western philosophy as a whole for centuries, from the medieval scholasticism of Peter Abelard, through the founding of modern philosophy by René Descartes, to 20th century philosophy of science, existentialism, phenomenology, structuralism, and postmodernism. Peter Abelard (1079 – April 21, 1142) was a scholastic philosopher, theologian and logician. The story of his affair with and love for Héloïse has become legendary. The ""Chambers Biographical Dictionary"" describes him as ""the keenest thinker and boldest theologian of the 12th Century"". The general"
"In 1637, René Descartes introduced Cartesian coordinates and showed that this allows reducing geometric problem to algebraic computations with numbers. This reduction of geometry to algebra was a major change of point of view, as, until then, the real numbers—that is, rational numbers and non-rational numbers together–were defined in terms of geometry, as lengths and distance. Despite the wide use of Descartes approaches, which was called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract vector spaces allowed to use them for defining Euclidean spaces, with a purely algebraic definition."
"eye that Rohault had worked on. A wide range of experiments used by Rohault included some mentioned by Descartes, and two well-known ones of Blaise Pascal, but also others taken from medical men: Gaspard Asselli, Louis Gayant, William Harvey, Jean Pecquet, and Nicholas Steno. Jacques Rohault Jacques Rohault (; 1618–1672) was a French philosopher, physicist and mathematician, and a follower of Cartesianism. Rohault was born in Amiens, the son of a wealthy wine merchant, and educated in Paris. Having grown up with the conventional scholastic philosophy of his day, he adopted and popularised the new Cartesian physics. His Wednesday lectures"
"town is notable in Acadian genealogy as the birthplace of the Heberts. Descartes, Indre-et-Loire Descartes is a commune in the Indre-et-Loire department in central France. It is approximately 13 miles east of Richelieu and about 24 miles east of Loudun, on the banks of the Creuse River. Initially called La Haye en Touraine, the town was the birthplace of the philosopher René Descartes (1596–1650), although his family home was in nearby Châtellerault. Descartes left La Haye in approximately 1606 to attend the College Henri IV at La Flèche. The town was renamed La Haye-Descartes in 1802 in his honour, and"
"many transcendental functions, the so-called Pfaffian functions. Descartes' rule of signs In mathematics, Descartes' rule of signs, first described by René Descartes in his work ""La Géométrie"", is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting the zero coefficients), and that the difference between these two numbers is always even. This implies in particular that, if this difference is zero or one, then there is exactly zero or one positive"
"is generally recognized as the foundation of modern optics (though the law of refraction is conspicuously absent). Willebrord Snellius (1580–1626) found the mathematical law of refraction, now known as Snell's law, in 1621. Subsequently René Descartes (1596–1650) showed, by using geometric construction and the law of refraction (also known as Descartes' law), that the angular radius of a rainbow is 42° (i.e. the angle subtended at the eye by the edge of the rainbow and the rainbow's centre is 42°). He also independently discovered the law of reflection, and his essay on optics was the first published mention of this"
"Descartes' theorem In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643. Geometrical problems involving tangent circles have been pondered for millennia. In ancient Greece of the third century BC, Apollonius of Perga devoted an entire book to the topic. René Descartes discussed the problem briefly in 1643, in a letter to Princess Elisabeth of the Palatinate."
"17th-century philosophy 17th century philosophy is generally regarded as seeing the start of modern philosophy, and the shaking off of the medieval approach, especially scholasticism. It succeeded the Renaissance and preceded the Age of Enlightenment. It is often considered to be part of early modern philosophy. The period is usually taken to start in the seventeenth century with the work of René Descartes, who set much of the agenda as well as much of the methodology for those who came after him. The period is typified in Europe by the great system-builders — philosophers who present unified systems of epistemology,"
"Stevin (1581) brought an early algebraic notation, the use of decimals and exponents. However, complex numbers remained at best a philosophical way of thinking and Descartes, almost a century after their invention, used them as imaginary numbers. Only positive solutions were considered and using geometrical proof was common. The task of the mathematicians was in fact twofold. It was necessary to produce algebra in a more geometrical way, i.e., to give it a rigorous foundation; and on the other hand, it was necessary to give geometry a more algebraic sense, allowing the analytical calculation in the plane. Vieta and Descartes"
"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. The 18th and 19th centuries saw the creation and standardization of mathematical notation as used today. Euler was responsible for many of the notations in use today: the use of ""a"", ""b"", ""c"" for constants and ""x"", ""y"", ""z"" for unknowns, ""e"" for the base of the"