Context: Document [1](Title: Leibniz–Newton calculus controversy) The calculus controversy (often referred to with the German term Prioritätsstreit, meaning "priority dispute") was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates) over who had first invented the mathematical study of change, calculus. It is a question that had been the cause of a major intellectual controversy, one that began simmering in 1699 and broke out in full force in 1711.

Context: Document [2](Title: Calculus) and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known. When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to

Context: Document [3](Title: History of calculus) at the turn of the century, but after 1711 both of them became personally involved, accusing each other of plagiarism. The priority dispute had an effect of separating English-speaking mathematicians from those in the continental Europe for many years. Only in the 1820s, due to the efforts of the Analytical Society, did Leibnizian analytical calculus become accepted in England. Today, both Newton and Leibniz are given credit for independently developing the basics of calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the

Context: Document [4](Title: Jacob Bernoulli) that were to make up "Ars Conjectandi" were discovered. He was appointed professor of mathematics at the University of Basel in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother Johann Bernoulli on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus in "Nova Methodus pro Maximis et Minimis" published in "Acta Eruditorum". They also studied the publications of von Tschirnhaus. It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians

Context: Document [5](Title: History of calculus) who first "invented" calculus. This argument, the Leibniz and Newton calculus controversy, involving Leibniz, who was German, and the Englishman Newton, led to a rift in the European mathematical community lasting over a century. Leibniz was the first to publish his investigations; however, it is well established that Newton had started his work several years prior to Leibniz and had already developed a theory of tangents by the time Leibniz became interested in the question. It is not known how much this may have influenced Leibniz. The initial accusations were made by students and supporters of the two great scientists

Context: Document [6](Title: Kerala School of Astronomy and Mathematics) considered a part of calculus. According to V.J. Katz, they were yet to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today", like Newton and Leibniz. The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the

Context: Document [7](Title: Isaac Newton) the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes, and astronomers such as Galileo and Thomas Street, through whom he learned of Kepler's work. He set down in his notebook a series of ""Quaestiones"" about mechanical philosophy as he found it. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his BA degree in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student,

Context: Document [8](Title: George Green (mathematician)) form of calculus derived from Leibniz that was virtually unheard of, or even actively discouraged, in England at the time (due to Leibniz being a contemporary of Newton who had his own methods that were championed in England). This form of calculus, and the developments of mathematicians such as Laplace, Lacroix and Poisson were not taught even at Cambridge, let alone Nottingham, and yet Green had not only heard of these developments, but also improved upon them. It is speculated that only one person educated in mathematics, John Toplis, headmaster of Nottingham High School 1806–1819, graduate from Cambridge and an

Context: Document [9](Title: History of calculus) be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter. From the age of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus. Greek mathematicians are also credited with a significant use of

Context: Document [10](Title: History of mathematics) differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon, did important work on the foundations of celestial mechanics and on statistics. Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law. This century saw the development of the two forms of non-Euclidean geometry, where the parallel

Context: Document [11](Title: Science) at the center and all the planets orbiting it. Aristarchus's model was widely rejected because it was believed to violate the laws of physics. The inventor and mathematician Archimedes of Syracuse made major contributions to the beginnings of calculus and has sometimes been credited as its inventor, although his proto-calculus lacked several defining features. Pliny the Elder was a Roman writer and polymath, who wrote the seminal encyclopedia "Natural History", dealing with history, geography, medicine, astronomy, earth science, botany, and zoology. Other scientists or proto-scientists in Antiquity were Theophrastus, Euclid, Herophilos, Hipparchus, Ptolemy, and Galen. During late antiquity, in the

Context: Document [12](Title: Italians) infinitesimal analysis, and papers on the theory of real numbers. Giuseppe Peano, founded the mathematical logic, the set theory, and alongside John Venn drew the first Venn diagram. Gregorio Ricci-Curbastro is well known for his invention on absolute differential calculus (tensor calculus), further developed by Tullio Levi-Civita, and its applications to the theory of relativity. Beniamino Segre is one of the main contributor of algebraic geometry and one of the founders of finite geometry. Paolo Ruffini is credited for his innovative work in mathematics, creating "Ruffini's rule" and co-creating the Abel–Ruffini theorem. Ennio de Giorgi, a Wolf Prize in Mathematics

Context: Document [13](Title: Johann Bernoulli) allow him to study medicine instead. However, Johann Bernoulli did not enjoy medicine either and began studying mathematics on the side with his older brother Jacob. Throughout Johann Bernoulli's education at Basel University the Bernoulli brothers worked together spending much of their time studying the newly discovered infinitesimal calculus. They were among the first mathematicians to not only study and understand calculus but to apply it to various problems. After graduating from Basel University Johann Bernoulli moved to teach differential equations. Later, in 1694, he married Dorothea Falkner and soon after accepted a position as the professor of mathematics at

Context: Document [14](Title: Calculus) Calculus Calculus (from Latin "calculus", literally 'small pebble', used for counting and calculations, as on an abacus) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus (concerning instantaneous rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of

Context: Document [15](Title: Early life of Isaac Newton) College, Cambridge as a sizar—a sort of work-study role. At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the University closed down as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student, Newton's private studies at his home in Woolsthorpe over the next two

Context: Document [16](Title: History of calculus) continued to add the two component motions together, thereby finding the tangent to the curve. The pioneers of the calculus such as Isaac Barrow and Johann Bernoulli were diligent students of Archimedes; see for instance C. S. Roero (1983). The method of exhaustion was reinvented in China by Liu Hui in the 4th century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. In the Middle East, Alhazen derived a formula for the sum of

Context: Document [17](Title: Abraham de Moivre) was a bias against his French origins. In November 1697 he was elected a Fellow of the Royal Society and in 1712 was appointed to a commission set up by the society, alongside MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston, and Taylor to review the claims of Newton and Leibniz as to who discovered calculus. The full details of the controversy can be found in the Leibniz and Newton calculus controversy article. Throughout his life de Moivre remained poor. It is reported that he was a regular customer of old Slaughter's Coffee House, St. Martin's Lane at

Context: Document [18](Title: Calculus) "the science of fluxions". Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi. In calculus, "foundations" refers to the rigorous development of the subject from axioms and definitions. In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley famously described infinitesimals as the ghosts of departed quantities in his book "The Analyst"

Context: Document [19](Title: Isaac Newton) author of the manuscript "De analysi per aequationes numero terminorum infinitas", sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins in August of that year as "[...] of an extraordinary genius and proficiency in these things." Newton later became involved in a dispute with Leibniz over priority in the development of calculus (the Leibniz–Newton calculus controversy). Most modern historians believe that Newton and Leibniz developed calculus independently, although with very different mathematical notations. Occasionally it has been suggested that Newton published almost nothing about it until 1693, and did

Context: Document [20](Title: Indian mathematics) as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today." The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware."

Question: who were the two mathematicians that invented calculus?

Answer: Gottfried Leibniz

Answer: Isaac Newton


Context: Document [1](Title: Calculus) and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known. When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to

Context: Document [2](Title: History of calculus) at the turn of the century, but after 1711 both of them became personally involved, accusing each other of plagiarism. The priority dispute had an effect of separating English-speaking mathematicians from those in the continental Europe for many years. Only in the 1820s, due to the efforts of the Analytical Society, did Leibnizian analytical calculus become accepted in England. Today, both Newton and Leibniz are given credit for independently developing the basics of calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the

Context: Document [3](Title: Jacob Bernoulli) that were to make up "Ars Conjectandi" were discovered. He was appointed professor of mathematics at the University of Basel in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother Johann Bernoulli on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus in "Nova Methodus pro Maximis et Minimis" published in "Acta Eruditorum". They also studied the publications of von Tschirnhaus. It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians

Context: Document [4](Title: History of calculus) who first "invented" calculus. This argument, the Leibniz and Newton calculus controversy, involving Leibniz, who was German, and the Englishman Newton, led to a rift in the European mathematical community lasting over a century. Leibniz was the first to publish his investigations; however, it is well established that Newton had started his work several years prior to Leibniz and had already developed a theory of tangents by the time Leibniz became interested in the question. It is not known how much this may have influenced Leibniz. The initial accusations were made by students and supporters of the two great scientists

Context: Document [5](Title: Leibniz–Newton calculus controversy) The calculus controversy (often referred to with the German term Prioritätsstreit, meaning "priority dispute") was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates) over who had first invented the mathematical study of change, calculus. It is a question that had been the cause of a major intellectual controversy, one that began simmering in 1699 and broke out in full force in 1711.

Context: Document [6](Title: Kerala School of Astronomy and Mathematics) considered a part of calculus. According to V.J. Katz, they were yet to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today", like Newton and Leibniz. The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the

Context: Document [7](Title: Isaac Newton) the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes, and astronomers such as Galileo and Thomas Street, through whom he learned of Kepler's work. He set down in his notebook a series of ""Quaestiones"" about mechanical philosophy as he found it. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his BA degree in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student,

Context: Document [8](Title: George Green (mathematician)) form of calculus derived from Leibniz that was virtually unheard of, or even actively discouraged, in England at the time (due to Leibniz being a contemporary of Newton who had his own methods that were championed in England). This form of calculus, and the developments of mathematicians such as Laplace, Lacroix and Poisson were not taught even at Cambridge, let alone Nottingham, and yet Green had not only heard of these developments, but also improved upon them. It is speculated that only one person educated in mathematics, John Toplis, headmaster of Nottingham High School 1806–1819, graduate from Cambridge and an

Context: Document [9](Title: History of calculus) be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter. From the age of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus. Greek mathematicians are also credited with a significant use of

Context: Document [10](Title: History of mathematics) differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon, did important work on the foundations of celestial mechanics and on statistics. Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law. This century saw the development of the two forms of non-Euclidean geometry, where the parallel

Context: Document [11](Title: Science) at the center and all the planets orbiting it. Aristarchus's model was widely rejected because it was believed to violate the laws of physics. The inventor and mathematician Archimedes of Syracuse made major contributions to the beginnings of calculus and has sometimes been credited as its inventor, although his proto-calculus lacked several defining features. Pliny the Elder was a Roman writer and polymath, who wrote the seminal encyclopedia "Natural History", dealing with history, geography, medicine, astronomy, earth science, botany, and zoology. Other scientists or proto-scientists in Antiquity were Theophrastus, Euclid, Herophilos, Hipparchus, Ptolemy, and Galen. During late antiquity, in the

Context: Document [12](Title: Italians) infinitesimal analysis, and papers on the theory of real numbers. Giuseppe Peano, founded the mathematical logic, the set theory, and alongside John Venn drew the first Venn diagram. Gregorio Ricci-Curbastro is well known for his invention on absolute differential calculus (tensor calculus), further developed by Tullio Levi-Civita, and its applications to the theory of relativity. Beniamino Segre is one of the main contributor of algebraic geometry and one of the founders of finite geometry. Paolo Ruffini is credited for his innovative work in mathematics, creating "Ruffini's rule" and co-creating the Abel–Ruffini theorem. Ennio de Giorgi, a Wolf Prize in Mathematics

Context: Document [13](Title: Johann Bernoulli) allow him to study medicine instead. However, Johann Bernoulli did not enjoy medicine either and began studying mathematics on the side with his older brother Jacob. Throughout Johann Bernoulli's education at Basel University the Bernoulli brothers worked together spending much of their time studying the newly discovered infinitesimal calculus. They were among the first mathematicians to not only study and understand calculus but to apply it to various problems. After graduating from Basel University Johann Bernoulli moved to teach differential equations. Later, in 1694, he married Dorothea Falkner and soon after accepted a position as the professor of mathematics at

Context: Document [14](Title: Calculus) Calculus Calculus (from Latin "calculus", literally 'small pebble', used for counting and calculations, as on an abacus) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus (concerning instantaneous rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of

Context: Document [15](Title: Early life of Isaac Newton) College, Cambridge as a sizar—a sort of work-study role. At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the University closed down as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student, Newton's private studies at his home in Woolsthorpe over the next two

Context: Document [16](Title: History of calculus) continued to add the two component motions together, thereby finding the tangent to the curve. The pioneers of the calculus such as Isaac Barrow and Johann Bernoulli were diligent students of Archimedes; see for instance C. S. Roero (1983). The method of exhaustion was reinvented in China by Liu Hui in the 4th century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. In the Middle East, Alhazen derived a formula for the sum of

Context: Document [17](Title: Abraham de Moivre) was a bias against his French origins. In November 1697 he was elected a Fellow of the Royal Society and in 1712 was appointed to a commission set up by the society, alongside MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston, and Taylor to review the claims of Newton and Leibniz as to who discovered calculus. The full details of the controversy can be found in the Leibniz and Newton calculus controversy article. Throughout his life de Moivre remained poor. It is reported that he was a regular customer of old Slaughter's Coffee House, St. Martin's Lane at

Context: Document [18](Title: Calculus) "the science of fluxions". Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi. In calculus, "foundations" refers to the rigorous development of the subject from axioms and definitions. In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley famously described infinitesimals as the ghosts of departed quantities in his book "The Analyst"

Context: Document [19](Title: Isaac Newton) author of the manuscript "De analysi per aequationes numero terminorum infinitas", sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins in August of that year as "[...] of an extraordinary genius and proficiency in these things." Newton later became involved in a dispute with Leibniz over priority in the development of calculus (the Leibniz–Newton calculus controversy). Most modern historians believe that Newton and Leibniz developed calculus independently, although with very different mathematical notations. Occasionally it has been suggested that Newton published almost nothing about it until 1693, and did

Context: Document [20](Title: Indian mathematics) as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today." The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware."

Question: who were the two mathematicians that invented calculus?

Answer: Gottfried Leibniz

Answer: Isaac Newton


Context: Document [1](Title: Calculus) and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known. When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to

Context: Document [2](Title: History of calculus) at the turn of the century, but after 1711 both of them became personally involved, accusing each other of plagiarism. The priority dispute had an effect of separating English-speaking mathematicians from those in the continental Europe for many years. Only in the 1820s, due to the efforts of the Analytical Society, did Leibnizian analytical calculus become accepted in England. Today, both Newton and Leibniz are given credit for independently developing the basics of calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the

Context: Document [3](Title: Jacob Bernoulli) that were to make up "Ars Conjectandi" were discovered. He was appointed professor of mathematics at the University of Basel in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother Johann Bernoulli on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus in "Nova Methodus pro Maximis et Minimis" published in "Acta Eruditorum". They also studied the publications of von Tschirnhaus. It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians

Context: Document [4](Title: History of calculus) who first "invented" calculus. This argument, the Leibniz and Newton calculus controversy, involving Leibniz, who was German, and the Englishman Newton, led to a rift in the European mathematical community lasting over a century. Leibniz was the first to publish his investigations; however, it is well established that Newton had started his work several years prior to Leibniz and had already developed a theory of tangents by the time Leibniz became interested in the question. It is not known how much this may have influenced Leibniz. The initial accusations were made by students and supporters of the two great scientists

Context: Document [5](Title: Kerala School of Astronomy and Mathematics) considered a part of calculus. According to V.J. Katz, they were yet to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today", like Newton and Leibniz. The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the

Context: Document [6](Title: Isaac Newton) the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes, and astronomers such as Galileo and Thomas Street, through whom he learned of Kepler's work. He set down in his notebook a series of ""Quaestiones"" about mechanical philosophy as he found it. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his BA degree in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student,

Context: Document [7](Title: George Green (mathematician)) form of calculus derived from Leibniz that was virtually unheard of, or even actively discouraged, in England at the time (due to Leibniz being a contemporary of Newton who had his own methods that were championed in England). This form of calculus, and the developments of mathematicians such as Laplace, Lacroix and Poisson were not taught even at Cambridge, let alone Nottingham, and yet Green had not only heard of these developments, but also improved upon them. It is speculated that only one person educated in mathematics, John Toplis, headmaster of Nottingham High School 1806–1819, graduate from Cambridge and an

Context: Document [8](Title: History of calculus) be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter. From the age of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus. Greek mathematicians are also credited with a significant use of

Context: Document [9](Title: History of mathematics) differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon, did important work on the foundations of celestial mechanics and on statistics. Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law. This century saw the development of the two forms of non-Euclidean geometry, where the parallel

Context: Document [10](Title: Leibniz–Newton calculus controversy) The calculus controversy (often referred to with the German term Prioritätsstreit, meaning "priority dispute") was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates) over who had first invented the mathematical study of change, calculus. It is a question that had been the cause of a major intellectual controversy, one that began simmering in 1699 and broke out in full force in 1711.

Context: Document [11](Title: Science) at the center and all the planets orbiting it. Aristarchus's model was widely rejected because it was believed to violate the laws of physics. The inventor and mathematician Archimedes of Syracuse made major contributions to the beginnings of calculus and has sometimes been credited as its inventor, although his proto-calculus lacked several defining features. Pliny the Elder was a Roman writer and polymath, who wrote the seminal encyclopedia "Natural History", dealing with history, geography, medicine, astronomy, earth science, botany, and zoology. Other scientists or proto-scientists in Antiquity were Theophrastus, Euclid, Herophilos, Hipparchus, Ptolemy, and Galen. During late antiquity, in the

Context: Document [12](Title: Italians) infinitesimal analysis, and papers on the theory of real numbers. Giuseppe Peano, founded the mathematical logic, the set theory, and alongside John Venn drew the first Venn diagram. Gregorio Ricci-Curbastro is well known for his invention on absolute differential calculus (tensor calculus), further developed by Tullio Levi-Civita, and its applications to the theory of relativity. Beniamino Segre is one of the main contributor of algebraic geometry and one of the founders of finite geometry. Paolo Ruffini is credited for his innovative work in mathematics, creating "Ruffini's rule" and co-creating the Abel–Ruffini theorem. Ennio de Giorgi, a Wolf Prize in Mathematics

Context: Document [13](Title: Johann Bernoulli) allow him to study medicine instead. However, Johann Bernoulli did not enjoy medicine either and began studying mathematics on the side with his older brother Jacob. Throughout Johann Bernoulli's education at Basel University the Bernoulli brothers worked together spending much of their time studying the newly discovered infinitesimal calculus. They were among the first mathematicians to not only study and understand calculus but to apply it to various problems. After graduating from Basel University Johann Bernoulli moved to teach differential equations. Later, in 1694, he married Dorothea Falkner and soon after accepted a position as the professor of mathematics at

Context: Document [14](Title: Calculus) Calculus Calculus (from Latin "calculus", literally 'small pebble', used for counting and calculations, as on an abacus) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus (concerning instantaneous rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of

Context: Document [15](Title: Early life of Isaac Newton) College, Cambridge as a sizar—a sort of work-study role. At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the University closed down as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student, Newton's private studies at his home in Woolsthorpe over the next two

Context: Document [16](Title: History of calculus) continued to add the two component motions together, thereby finding the tangent to the curve. The pioneers of the calculus such as Isaac Barrow and Johann Bernoulli were diligent students of Archimedes; see for instance C. S. Roero (1983). The method of exhaustion was reinvented in China by Liu Hui in the 4th century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. In the Middle East, Alhazen derived a formula for the sum of

Context: Document [17](Title: Abraham de Moivre) was a bias against his French origins. In November 1697 he was elected a Fellow of the Royal Society and in 1712 was appointed to a commission set up by the society, alongside MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston, and Taylor to review the claims of Newton and Leibniz as to who discovered calculus. The full details of the controversy can be found in the Leibniz and Newton calculus controversy article. Throughout his life de Moivre remained poor. It is reported that he was a regular customer of old Slaughter's Coffee House, St. Martin's Lane at

Context: Document [18](Title: Calculus) "the science of fluxions". Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi. In calculus, "foundations" refers to the rigorous development of the subject from axioms and definitions. In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley famously described infinitesimals as the ghosts of departed quantities in his book "The Analyst"

Context: Document [19](Title: Isaac Newton) author of the manuscript "De analysi per aequationes numero terminorum infinitas", sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins in August of that year as "[...] of an extraordinary genius and proficiency in these things." Newton later became involved in a dispute with Leibniz over priority in the development of calculus (the Leibniz–Newton calculus controversy). Most modern historians believe that Newton and Leibniz developed calculus independently, although with very different mathematical notations. Occasionally it has been suggested that Newton published almost nothing about it until 1693, and did

Context: Document [20](Title: Indian mathematics) as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today." The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware."

Question: who were the two mathematicians that invented calculus?

Answer: Gottfried Leibniz

Answer: Isaac Newton


Context: Document [1](Title: Calculus) and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known. When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to

Context: Document [2](Title: History of calculus) at the turn of the century, but after 1711 both of them became personally involved, accusing each other of plagiarism. The priority dispute had an effect of separating English-speaking mathematicians from those in the continental Europe for many years. Only in the 1820s, due to the efforts of the Analytical Society, did Leibnizian analytical calculus become accepted in England. Today, both Newton and Leibniz are given credit for independently developing the basics of calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the

Context: Document [3](Title: Jacob Bernoulli) that were to make up "Ars Conjectandi" were discovered. He was appointed professor of mathematics at the University of Basel in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother Johann Bernoulli on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus in "Nova Methodus pro Maximis et Minimis" published in "Acta Eruditorum". They also studied the publications of von Tschirnhaus. It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians

Context: Document [4](Title: History of calculus) who first "invented" calculus. This argument, the Leibniz and Newton calculus controversy, involving Leibniz, who was German, and the Englishman Newton, led to a rift in the European mathematical community lasting over a century. Leibniz was the first to publish his investigations; however, it is well established that Newton had started his work several years prior to Leibniz and had already developed a theory of tangents by the time Leibniz became interested in the question. It is not known how much this may have influenced Leibniz. The initial accusations were made by students and supporters of the two great scientists

Context: Document [5](Title: Kerala School of Astronomy and Mathematics) considered a part of calculus. According to V.J. Katz, they were yet to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today", like Newton and Leibniz. The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the

Context: Document [6](Title: Isaac Newton) the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes, and astronomers such as Galileo and Thomas Street, through whom he learned of Kepler's work. He set down in his notebook a series of ""Quaestiones"" about mechanical philosophy as he found it. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his BA degree in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student,

Context: Document [7](Title: George Green (mathematician)) form of calculus derived from Leibniz that was virtually unheard of, or even actively discouraged, in England at the time (due to Leibniz being a contemporary of Newton who had his own methods that were championed in England). This form of calculus, and the developments of mathematicians such as Laplace, Lacroix and Poisson were not taught even at Cambridge, let alone Nottingham, and yet Green had not only heard of these developments, but also improved upon them. It is speculated that only one person educated in mathematics, John Toplis, headmaster of Nottingham High School 1806–1819, graduate from Cambridge and an

Context: Document [8](Title: History of calculus) be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter. From the age of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus. Greek mathematicians are also credited with a significant use of

Context: Document [9](Title: History of mathematics) differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon, did important work on the foundations of celestial mechanics and on statistics. Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law. This century saw the development of the two forms of non-Euclidean geometry, where the parallel

Context: Document [10](Title: Science) at the center and all the planets orbiting it. Aristarchus's model was widely rejected because it was believed to violate the laws of physics. The inventor and mathematician Archimedes of Syracuse made major contributions to the beginnings of calculus and has sometimes been credited as its inventor, although his proto-calculus lacked several defining features. Pliny the Elder was a Roman writer and polymath, who wrote the seminal encyclopedia "Natural History", dealing with history, geography, medicine, astronomy, earth science, botany, and zoology. Other scientists or proto-scientists in Antiquity were Theophrastus, Euclid, Herophilos, Hipparchus, Ptolemy, and Galen. During late antiquity, in the

Context: Document [11](Title: Italians) infinitesimal analysis, and papers on the theory of real numbers. Giuseppe Peano, founded the mathematical logic, the set theory, and alongside John Venn drew the first Venn diagram. Gregorio Ricci-Curbastro is well known for his invention on absolute differential calculus (tensor calculus), further developed by Tullio Levi-Civita, and its applications to the theory of relativity. Beniamino Segre is one of the main contributor of algebraic geometry and one of the founders of finite geometry. Paolo Ruffini is credited for his innovative work in mathematics, creating "Ruffini's rule" and co-creating the Abel–Ruffini theorem. Ennio de Giorgi, a Wolf Prize in Mathematics

Context: Document [12](Title: Johann Bernoulli) allow him to study medicine instead. However, Johann Bernoulli did not enjoy medicine either and began studying mathematics on the side with his older brother Jacob. Throughout Johann Bernoulli's education at Basel University the Bernoulli brothers worked together spending much of their time studying the newly discovered infinitesimal calculus. They were among the first mathematicians to not only study and understand calculus but to apply it to various problems. After graduating from Basel University Johann Bernoulli moved to teach differential equations. Later, in 1694, he married Dorothea Falkner and soon after accepted a position as the professor of mathematics at

Context: Document [13](Title: Calculus) Calculus Calculus (from Latin "calculus", literally 'small pebble', used for counting and calculations, as on an abacus) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus (concerning instantaneous rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of

Context: Document [14](Title: Early life of Isaac Newton) College, Cambridge as a sizar—a sort of work-study role. At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the University closed down as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student, Newton's private studies at his home in Woolsthorpe over the next two

Context: Document [15](Title: Leibniz–Newton calculus controversy) The calculus controversy (often referred to with the German term Prioritätsstreit, meaning "priority dispute") was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates) over who had first invented the mathematical study of change, calculus. It is a question that had been the cause of a major intellectual controversy, one that began simmering in 1699 and broke out in full force in 1711.

Context: Document [16](Title: History of calculus) continued to add the two component motions together, thereby finding the tangent to the curve. The pioneers of the calculus such as Isaac Barrow and Johann Bernoulli were diligent students of Archimedes; see for instance C. S. Roero (1983). The method of exhaustion was reinvented in China by Liu Hui in the 4th century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. In the Middle East, Alhazen derived a formula for the sum of

Context: Document [17](Title: Abraham de Moivre) was a bias against his French origins. In November 1697 he was elected a Fellow of the Royal Society and in 1712 was appointed to a commission set up by the society, alongside MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston, and Taylor to review the claims of Newton and Leibniz as to who discovered calculus. The full details of the controversy can be found in the Leibniz and Newton calculus controversy article. Throughout his life de Moivre remained poor. It is reported that he was a regular customer of old Slaughter's Coffee House, St. Martin's Lane at

Context: Document [18](Title: Calculus) "the science of fluxions". Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi. In calculus, "foundations" refers to the rigorous development of the subject from axioms and definitions. In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley famously described infinitesimals as the ghosts of departed quantities in his book "The Analyst"

Context: Document [19](Title: Isaac Newton) author of the manuscript "De analysi per aequationes numero terminorum infinitas", sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins in August of that year as "[...] of an extraordinary genius and proficiency in these things." Newton later became involved in a dispute with Leibniz over priority in the development of calculus (the Leibniz–Newton calculus controversy). Most modern historians believe that Newton and Leibniz developed calculus independently, although with very different mathematical notations. Occasionally it has been suggested that Newton published almost nothing about it until 1693, and did

Context: Document [20](Title: Indian mathematics) as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today." The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware."

Question: who were the two mathematicians that invented calculus?

Answer: Gottfried Leibniz

Answer: Isaac Newton


Context: Document [1](Title: Calculus) and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known. When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to

Context: Document [2](Title: History of calculus) at the turn of the century, but after 1711 both of them became personally involved, accusing each other of plagiarism. The priority dispute had an effect of separating English-speaking mathematicians from those in the continental Europe for many years. Only in the 1820s, due to the efforts of the Analytical Society, did Leibnizian analytical calculus become accepted in England. Today, both Newton and Leibniz are given credit for independently developing the basics of calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the

Context: Document [3](Title: Jacob Bernoulli) that were to make up "Ars Conjectandi" were discovered. He was appointed professor of mathematics at the University of Basel in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother Johann Bernoulli on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus in "Nova Methodus pro Maximis et Minimis" published in "Acta Eruditorum". They also studied the publications of von Tschirnhaus. It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians

Context: Document [4](Title: History of calculus) who first "invented" calculus. This argument, the Leibniz and Newton calculus controversy, involving Leibniz, who was German, and the Englishman Newton, led to a rift in the European mathematical community lasting over a century. Leibniz was the first to publish his investigations; however, it is well established that Newton had started his work several years prior to Leibniz and had already developed a theory of tangents by the time Leibniz became interested in the question. It is not known how much this may have influenced Leibniz. The initial accusations were made by students and supporters of the two great scientists

Context: Document [5](Title: Kerala School of Astronomy and Mathematics) considered a part of calculus. According to V.J. Katz, they were yet to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today", like Newton and Leibniz. The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the

Context: Document [6](Title: Isaac Newton) the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes, and astronomers such as Galileo and Thomas Street, through whom he learned of Kepler's work. He set down in his notebook a series of ""Quaestiones"" about mechanical philosophy as he found it. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his BA degree in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student,

Context: Document [7](Title: George Green (mathematician)) form of calculus derived from Leibniz that was virtually unheard of, or even actively discouraged, in England at the time (due to Leibniz being a contemporary of Newton who had his own methods that were championed in England). This form of calculus, and the developments of mathematicians such as Laplace, Lacroix and Poisson were not taught even at Cambridge, let alone Nottingham, and yet Green had not only heard of these developments, but also improved upon them. It is speculated that only one person educated in mathematics, John Toplis, headmaster of Nottingham High School 1806–1819, graduate from Cambridge and an

Context: Document [8](Title: History of calculus) be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter. From the age of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus. Greek mathematicians are also credited with a significant use of

Context: Document [9](Title: History of mathematics) differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon, did important work on the foundations of celestial mechanics and on statistics. Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law. This century saw the development of the two forms of non-Euclidean geometry, where the parallel

Context: Document [10](Title: Science) at the center and all the planets orbiting it. Aristarchus's model was widely rejected because it was believed to violate the laws of physics. The inventor and mathematician Archimedes of Syracuse made major contributions to the beginnings of calculus and has sometimes been credited as its inventor, although his proto-calculus lacked several defining features. Pliny the Elder was a Roman writer and polymath, who wrote the seminal encyclopedia "Natural History", dealing with history, geography, medicine, astronomy, earth science, botany, and zoology. Other scientists or proto-scientists in Antiquity were Theophrastus, Euclid, Herophilos, Hipparchus, Ptolemy, and Galen. During late antiquity, in the

Context: Document [11](Title: Italians) infinitesimal analysis, and papers on the theory of real numbers. Giuseppe Peano, founded the mathematical logic, the set theory, and alongside John Venn drew the first Venn diagram. Gregorio Ricci-Curbastro is well known for his invention on absolute differential calculus (tensor calculus), further developed by Tullio Levi-Civita, and its applications to the theory of relativity. Beniamino Segre is one of the main contributor of algebraic geometry and one of the founders of finite geometry. Paolo Ruffini is credited for his innovative work in mathematics, creating "Ruffini's rule" and co-creating the Abel–Ruffini theorem. Ennio de Giorgi, a Wolf Prize in Mathematics

Context: Document [12](Title: Johann Bernoulli) allow him to study medicine instead. However, Johann Bernoulli did not enjoy medicine either and began studying mathematics on the side with his older brother Jacob. Throughout Johann Bernoulli's education at Basel University the Bernoulli brothers worked together spending much of their time studying the newly discovered infinitesimal calculus. They were among the first mathematicians to not only study and understand calculus but to apply it to various problems. After graduating from Basel University Johann Bernoulli moved to teach differential equations. Later, in 1694, he married Dorothea Falkner and soon after accepted a position as the professor of mathematics at

Context: Document [13](Title: Calculus) Calculus Calculus (from Latin "calculus", literally 'small pebble', used for counting and calculations, as on an abacus) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus (concerning instantaneous rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of

Context: Document [14](Title: Early life of Isaac Newton) College, Cambridge as a sizar—a sort of work-study role. At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the University closed down as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student, Newton's private studies at his home in Woolsthorpe over the next two

Context: Document [15](Title: History of calculus) continued to add the two component motions together, thereby finding the tangent to the curve. The pioneers of the calculus such as Isaac Barrow and Johann Bernoulli were diligent students of Archimedes; see for instance C. S. Roero (1983). The method of exhaustion was reinvented in China by Liu Hui in the 4th century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. In the Middle East, Alhazen derived a formula for the sum of

Context: Document [16](Title: Abraham de Moivre) was a bias against his French origins. In November 1697 he was elected a Fellow of the Royal Society and in 1712 was appointed to a commission set up by the society, alongside MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston, and Taylor to review the claims of Newton and Leibniz as to who discovered calculus. The full details of the controversy can be found in the Leibniz and Newton calculus controversy article. Throughout his life de Moivre remained poor. It is reported that he was a regular customer of old Slaughter's Coffee House, St. Martin's Lane at

Context: Document [17](Title: Calculus) "the science of fluxions". Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi. In calculus, "foundations" refers to the rigorous development of the subject from axioms and definitions. In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley famously described infinitesimals as the ghosts of departed quantities in his book "The Analyst"

Context: Document [18](Title: Isaac Newton) author of the manuscript "De analysi per aequationes numero terminorum infinitas", sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins in August of that year as "[...] of an extraordinary genius and proficiency in these things." Newton later became involved in a dispute with Leibniz over priority in the development of calculus (the Leibniz–Newton calculus controversy). Most modern historians believe that Newton and Leibniz developed calculus independently, although with very different mathematical notations. Occasionally it has been suggested that Newton published almost nothing about it until 1693, and did

Context: Document [19](Title: Indian mathematics) as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today." The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware."

Context: Document [20](Title: Leibniz–Newton calculus controversy) The calculus controversy (often referred to with the German term Prioritätsstreit, meaning "priority dispute") was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates) over who had first invented the mathematical study of change, calculus. It is a question that had been the cause of a major intellectual controversy, one that began simmering in 1699 and broke out in full force in 1711.

Question: who were the two mathematicians that invented calculus?

Answer: Gottfried Leibniz

Answer: Isaac Newton


Context: Document [1](Title: Peripheral nervous system) mechanical injuries. The peripheral nervous system is divided into the somatic nervous system and the autonomic nervous system. In the somatic nervous system, the cranial nerves are part of the PNS with the exception of the optic nerve (cranial nerve II), along with the retina. The second cranial nerve is not a true peripheral nerve but a tract of the diencephalon. Cranial nerve ganglia originated in the CNS. However, the remaining ten cranial nerve axons extend beyond the brain and are therefore considered part of the PNS. The autonomic nervous system is an involuntary control of smooth muscle and glands.

Context: Document [2](Title: Peripheral nervous system) Peripheral nervous system The peripheral nervous system (PNS) is one of two components that make up the nervous system of bilateral animals, with the other part being the central nervous system (CNS). The PNS consists of the nerves and ganglia outside the brain and spinal cord. The main function of the PNS is to connect the CNS to the limbs and organs, essentially serving as a relay between the brain and spinal cord and the rest of the body. Unlike the CNS, the PNS is not protected by the vertebral column and skull, or by the blood–brain barrier, which leaves

Context: Document [3](Title: Nervous system) Nervous system The nervous system is the part of an animal that coordinates its actions by transmitting signals to and from different parts of its body. The nervous system detects environmental changes that impact the body, then works in tandem with the endocrine system to respond to such events. Nervous tissue first arose in wormlike organisms about 550 to 600 million years ago. In vertebrates it consists of two main parts, the central nervous system (CNS) and the peripheral nervous system (PNS). The CNS consists of the brain and spinal cord. The PNS consists mainly of nerves, which are enclosed

Context: Document [4](Title: Segmentation in the human nervous system) cord, and the peripheral nervous system (PNS) comprising the nerve fibers that branch off from the spinal cord to all parts of the body. Both parts of the nervous system are actively involved in communicating signals between various parts of the body to ensure the smooth and efficient transfer of information that controls and coordinates the movement of muscles, and regulates organ functions. Neurons, which form the elemental unit of the nervous system, receive messages from their dendrites, relay the information as an electrical signal down the axon and releases chemical messengers known as neurotransmitters, thus converting the electrical signal

Context: Document [5](Title: Physiological psychology) and peripheral nervous system. The central nervous system is composed of the brain and spinal cord. The brain is the control center of the body and contains millions of neural connections. This organ is responsible for sending and receiving messages from the body and its environment. Each part of the brain is specialized for different aspects of the human being. For example, the temporal lobe has a major role in vision and audition, whereas the frontal lobe is significant for motor function and problem solving. The spinal cord is attached to the brain and serves as the main connector of

Context: Document [6](Title: Nerve) converted from electrical to chemical and then back to electrical. Nerves can be categorized into two groups based on function: The nervous system is the part of an animal that coordinates its actions by transmitting signals to and from different parts of its body. In vertebrates it consists of two main parts, the central nervous system (CNS) and the peripheral nervous system (PNS). The CNS consists of the brain and spinal cord. The PNS consists mainly of nerves, which are enclosed bundles of the long fibers or axons, that connect the CNS to every other part of the body. Nerves

Context: Document [7](Title: Head and neck anatomy) cord, and the peripheral nervous system (PNS), cranial nerves and spinal nerves. The CNS is located within the dorsal cavity, and the PNS extends through the ventral cavity. The central nervous system provides control and coordination of all eleven body systems and utilizes the endocrine system to form hormone chemical messengers that transport through the blood to influence the activity of individual cells of the body and their associated tissues, organs and systems. The CNS receives sensory (afferent) input from the PNS and directs the flow of information to association neurons (interneurons) to create chemical synapse responses which in turn

Context: Document [8](Title: Neuroscience) of other neurons, muscles, or glands at their termination points. A nervous system emerges from the assemblage of neurons that are connected to each other. In vertebrates, the nervous system can be split into two parts, the central nervous system (brain and spinal cord), and the peripheral nervous system. In many species — including all vertebrates — the nervous system is the most complex organ system in the body, with most of the complexity residing in the brain. The human brain alone contains around one hundred billion neurons and one hundred trillion synapses; it consists of thousands of distinguishable substructures,

Context: Document [9](Title: Peripheral nervous system) system which influences the function of organs outside voluntary control, such as the heart rate, or the functions of the digestive system. The somatic system includes the sensory nervous system and the somatosensory system and consists of sensory nerves and somatic nerves, and many nerves which hold both functions. In the head and neck, cranial nerves carry somatosensory data. There are twelve cranial nerves, ten of which originate from the brainstem, and mainly control the functions of the anatomic structures of the head with some exceptions. The nuclei of the olfactory nerve and the optic nerves lie in the forebrain

Context: Document [10](Title: Human body) contain bone marrow, the site of production of blood cells. Also, all bones are major storage sites for calcium and phosphate. This system can be split up into the muscular system and the skeletal system. The nervous system consists of the central nervous system (the brain and spinal cord) and the peripheral nervous system consists of the nerves and ganglia outside the brain and spinal cord. The brain is the organ of thought, emotion, memory, and sensory processing, and serves many aspects of communication and controls various systems and functions. The special senses consist of vision, hearing, taste, and smell.

Context: Document [11](Title: Spinal cord) is made of 31 segments from which branch one pair of sensory nerve roots and one pair of motor nerve roots. The nerve roots then merge into bilaterally symmetrical pairs of spinal nerves. The peripheral nervous system is made up of these spinal roots, nerves, and ganglia. The dorsal roots are afferent fascicles, receiving sensory information from the skin, muscles, and visceral organs to be relayed to the brain. The roots terminate in dorsal root ganglia, which are composed of the cell bodies of the corresponding neurons. Ventral roots consist of efferent fibers that arise from motor neurons whose cell

Context: Document [12](Title: Physiological psychology) The nervous system can be described as a control system that interconnects the other body systems. It consists of the brain, spinal cord, and other nerve tissues throughout the body. The system's primary function is to react to internal and external stimuli in the human body. It uses electrical and chemical signals to send out responses to different parts of the body, and it is made up of the nerve cells also called neurons. Through the system, messages are transmitted to body tissues such as a muscle. There are two major subdivisions in the nervous system known as the central

Context: Document [13](Title: Nervous system) of glial cell (oligodendrocytes in the central nervous system, and Schwann cells in the peripheral nervous system) generates layers of a fatty substance called myelin that wraps around axons and provides electrical insulation which allows them to transmit action potentials much more rapidly and efficiently. Recent findings indicate that glial cells, such as microglia and astrocytes, serve as important resident immune cells within the central nervous system. The nervous system of vertebrates (including humans) is divided into the central nervous system (CNS) and the peripheral nervous system (PNS). The (CNS) is the major division, and consists of the brain and

Context: Document [14](Title: Classification of peripheral nerves) noted that the sympathetic neurons that innervated the entire body tended to have opposing functions to the other autonomic neurons of the tectal and bulbo-sacral regions. The latter two, tectal and bulbo-sacral, were grouped together to form the parasympathetic system because they tended to respond in a like manner to various drugs. Langley also mentions an enteric nervous system in his writing, but this third grouping has largely fallen out of discussion in modern practice. Much of the original nerve classification done by Langley, and Gaskell by extension, was based on pharmacological responses of nerves throughout the body as well

Context: Document [15](Title: Central nervous system disease) vessel ruptures, causing blood to leak to the brain. If the brain cannot get enough oxygen and blood, brain cells can die, leading to permanent damage. The spinal cord transmits sensory reception from the peripheral nervous system. It also conducts motor information to the body's skeletal muscles, cardiac muscles, smooth muscles, and glands. There are 31 pairs of spinal nerves along the spinal cord, all of which consist of both sensory and motor neurons. The spinal cord is protected by vertebrae and connects the peripheral nervous system to the brain, and it acts as a "minor" coordinating center. The brain

Context: Document [16](Title: Classification of peripheral nerves) and parasympathetic nervous systems with a third system, the enteric nervous system, receiving less recognition. In 1898, British scientist John Newport Langley first coined the term "autonomic" in classifying the connections of nerve fibers to peripheral nerve cells. Previous researchers had utilized different terms such as "the sympathetic nerves" [Winslow et al.] to describe the way in which neurons in one part of the body brought about "sympathetic" reactions in another part of the body, as well as the "ganglionic nerves and ganglionic nervous system" [Livingstone et al.] for the ganglionic conversion of voluntary to involuntary movements (which supposedly made

Context: Document [17](Title: Nervous tissue) Nervous tissue Nervous tissue, also called neural tissue or nerve tissue, is the main tissue component of the nervous system. The nervous system regulates and controls bodily functions and activity and consists of two parts: the central nervous system (CNS) comprising the brain and spinal cord, and the peripheral nervous system (PNS) comprising the branching peripheral nerves. It is composed of neurons, or nerve cells, which receive and transmit impulses, and neuroglia, also known as glial cells or glia, which assist the propagation of the nerve impulse as well as provide nutrients to the neurons. Nervous tissue is made up

Context: Document [18](Title: Nervous system) the spinal cord. The spinal canal contains the spinal cord, while the cranial cavity contains the brain. The CNS is enclosed and protected by the meninges, a three-layered system of membranes, including a tough, leathery outer layer called the dura mater. The brain is also protected by the skull, and the spinal cord by the vertebrae. The peripheral nervous system (PNS) is a collective term for the nervous system structures that do not lie within the CNS. The large majority of the axon bundles called nerves are considered to belong to the PNS, even when the cell bodies of the

Context: Document [19](Title: Neuroanatomy) concentration. The extracellular matrix also provides support on the molecular level for the brain's cells. At the organ level, the nervous system is composed of brain regions, such as the hippocampus in mammals or the mushroom bodies of the fruit fly. These regions are often modular and serve a particular role within the general pathways of the nervous system. For example, the hippocampus is critical for forming memories. The nervous system also contains nerves, which are bundles of fibers that originate from the brain and spinal cord, and branch repeatedly to innervate every part of the body. Nerves are made

Context: Document [20](Title: Central nervous system) (cerebrum as well as midbrain and hindbrain) consists of a cortex, composed of neuron-bodies constituting gray matter, while internally there is more white matter that form tracts and commissures. Apart from cortical gray matter there is also subcortical gray matter making up a large number of different nuclei. From and to the spinal cord are projections of the peripheral nervous system in the form of spinal nerves (sometimes segmental nerves). The nerves connect the spinal cord to skin, joints, muscles etc. and allow for the transmission of efferent motor as well as afferent sensory signals and stimuli. This allows for

Question: what parts make up the peripheral nervous system?

Answer: autonomic nervous system

Answer: somatic nervous system


Context: Document [1](Title: Peripheral nervous system) Peripheral nervous system The peripheral nervous system (PNS) is one of two components that make up the nervous system of bilateral animals, with the other part being the central nervous system (CNS). The PNS consists of the nerves and ganglia outside the brain and spinal cord. The main function of the PNS is to connect the CNS to the limbs and organs, essentially serving as a relay between the brain and spinal cord and the rest of the body. Unlike the CNS, the PNS is not protected by the vertebral column and skull, or by the blood–brain barrier, which leaves

Context: Document [2](Title: Nervous system) Nervous system The nervous system is the part of an animal that coordinates its actions by transmitting signals to and from different parts of its body. The nervous system detects environmental changes that impact the body, then works in tandem with the endocrine system to respond to such events. Nervous tissue first arose in wormlike organisms about 550 to 600 million years ago. In vertebrates it consists of two main parts, the central nervous system (CNS) and the peripheral nervous system (PNS). The CNS consists of the brain and spinal cord. The PNS consists mainly of nerves, which are enclosed

Context: Document [3](Title: Segmentation in the human nervous system) cord, and the peripheral nervous system (PNS) comprising the nerve fibers that branch off from the spinal cord to all parts of the body. Both parts of the nervous system are actively involved in communicating signals between various parts of the body to ensure the smooth and efficient transfer of information that controls and coordinates the movement of muscles, and regulates organ functions. Neurons, which form the elemental unit of the nervous system, receive messages from their dendrites, relay the information as an electrical signal down the axon and releases chemical messengers known as neurotransmitters, thus converting the electrical signal

Context: Document [4](Title: Physiological psychology) and peripheral nervous system. The central nervous system is composed of the brain and spinal cord. The brain is the control center of the body and contains millions of neural connections. This organ is responsible for sending and receiving messages from the body and its environment. Each part of the brain is specialized for different aspects of the human being. For example, the temporal lobe has a major role in vision and audition, whereas the frontal lobe is significant for motor function and problem solving. The spinal cord is attached to the brain and serves as the main connector of

Context: Document [5](Title: Peripheral nervous system) mechanical injuries. The peripheral nervous system is divided into the somatic nervous system and the autonomic nervous system. In the somatic nervous system, the cranial nerves are part of the PNS with the exception of the optic nerve (cranial nerve II), along with the retina. The second cranial nerve is not a true peripheral nerve but a tract of the diencephalon. Cranial nerve ganglia originated in the CNS. However, the remaining ten cranial nerve axons extend beyond the brain and are therefore considered part of the PNS. The autonomic nervous system is an involuntary control of smooth muscle and glands.

Context: Document [6](Title: Nerve) converted from electrical to chemical and then back to electrical. Nerves can be categorized into two groups based on function: The nervous system is the part of an animal that coordinates its actions by transmitting signals to and from different parts of its body. In vertebrates it consists of two main parts, the central nervous system (CNS) and the peripheral nervous system (PNS). The CNS consists of the brain and spinal cord. The PNS consists mainly of nerves, which are enclosed bundles of the long fibers or axons, that connect the CNS to every other part of the body. Nerves

Context: Document [7](Title: Head and neck anatomy) cord, and the peripheral nervous system (PNS), cranial nerves and spinal nerves. The CNS is located within the dorsal cavity, and the PNS extends through the ventral cavity. The central nervous system provides control and coordination of all eleven body systems and utilizes the endocrine system to form hormone chemical messengers that transport through the blood to influence the activity of individual cells of the body and their associated tissues, organs and systems. The CNS receives sensory (afferent) input from the PNS and directs the flow of information to association neurons (interneurons) to create chemical synapse responses which in turn

Context: Document [8](Title: Neuroscience) of other neurons, muscles, or glands at their termination points. A nervous system emerges from the assemblage of neurons that are connected to each other. In vertebrates, the nervous system can be split into two parts, the central nervous system (brain and spinal cord), and the peripheral nervous system. In many species — including all vertebrates — the nervous system is the most complex organ system in the body, with most of the complexity residing in the brain. The human brain alone contains around one hundred billion neurons and one hundred trillion synapses; it consists of thousands of distinguishable substructures,

Context: Document [9](Title: Peripheral nervous system) system which influences the function of organs outside voluntary control, such as the heart rate, or the functions of the digestive system. The somatic system includes the sensory nervous system and the somatosensory system and consists of sensory nerves and somatic nerves, and many nerves which hold both functions. In the head and neck, cranial nerves carry somatosensory data. There are twelve cranial nerves, ten of which originate from the brainstem, and mainly control the functions of the anatomic structures of the head with some exceptions. The nuclei of the olfactory nerve and the optic nerves lie in the forebrain

Context: Document [10](Title: Human body) contain bone marrow, the site of production of blood cells. Also, all bones are major storage sites for calcium and phosphate. This system can be split up into the muscular system and the skeletal system. The nervous system consists of the central nervous system (the brain and spinal cord) and the peripheral nervous system consists of the nerves and ganglia outside the brain and spinal cord. The brain is the organ of thought, emotion, memory, and sensory processing, and serves many aspects of communication and controls various systems and functions. The special senses consist of vision, hearing, taste, and smell.

Context: Document [11](Title: Spinal cord) is made of 31 segments from which branch one pair of sensory nerve roots and one pair of motor nerve roots. The nerve roots then merge into bilaterally symmetrical pairs of spinal nerves. The peripheral nervous system is made up of these spinal roots, nerves, and ganglia. The dorsal roots are afferent fascicles, receiving sensory information from the skin, muscles, and visceral organs to be relayed to the brain. The roots terminate in dorsal root ganglia, which are composed of the cell bodies of the corresponding neurons. Ventral roots consist of efferent fibers that arise from motor neurons whose cell

Context: Document [12](Title: Physiological psychology) The nervous system can be described as a control system that interconnects the other body systems. It consists of the brain, spinal cord, and other nerve tissues throughout the body. The system's primary function is to react to internal and external stimuli in the human body. It uses electrical and chemical signals to send out responses to different parts of the body, and it is made up of the nerve cells also called neurons. Through the system, messages are transmitted to body tissues such as a muscle. There are two major subdivisions in the nervous system known as the central

Context: Document [13](Title: Nervous system) of glial cell (oligodendrocytes in the central nervous system, and Schwann cells in the peripheral nervous system) generates layers of a fatty substance called myelin that wraps around axons and provides electrical insulation which allows them to transmit action potentials much more rapidly and efficiently. Recent findings indicate that glial cells, such as microglia and astrocytes, serve as important resident immune cells within the central nervous system. The nervous system of vertebrates (including humans) is divided into the central nervous system (CNS) and the peripheral nervous system (PNS). The (CNS) is the major division, and consists of the brain and

Context: Document [14](Title: Classification of peripheral nerves) noted that the sympathetic neurons that innervated the entire body tended to have opposing functions to the other autonomic neurons of the tectal and bulbo-sacral regions. The latter two, tectal and bulbo-sacral, were grouped together to form the parasympathetic system because they tended to respond in a like manner to various drugs. Langley also mentions an enteric nervous system in his writing, but this third grouping has largely fallen out of discussion in modern practice. Much of the original nerve classification done by Langley, and Gaskell by extension, was based on pharmacological responses of nerves throughout the body as well

Context: Document [15](Title: Central nervous system disease) vessel ruptures, causing blood to leak to the brain. If the brain cannot get enough oxygen and blood, brain cells can die, leading to permanent damage. The spinal cord transmits sensory reception from the peripheral nervous system. It also conducts motor information to the body's skeletal muscles, cardiac muscles, smooth muscles, and glands. There are 31 pairs of spinal nerves along the spinal cord, all of which consist of both sensory and motor neurons. The spinal cord is protected by vertebrae and connects the peripheral nervous system to the brain, and it acts as a "minor" coordinating center. The brain

Context: Document [16](Title: Classification of peripheral nerves) and parasympathetic nervous systems with a third system, the enteric nervous system, receiving less recognition. In 1898, British scientist John Newport Langley first coined the term "autonomic" in classifying the connections of nerve fibers to peripheral nerve cells. Previous researchers had utilized different terms such as "the sympathetic nerves" [Winslow et al.] to describe the way in which neurons in one part of the body brought about "sympathetic" reactions in another part of the body, as well as the "ganglionic nerves and ganglionic nervous system" [Livingstone et al.] for the ganglionic conversion of voluntary to involuntary movements (which supposedly made

Context: Document [17](Title: Nervous tissue) Nervous tissue Nervous tissue, also called neural tissue or nerve tissue, is the main tissue component of the nervous system. The nervous system regulates and controls bodily functions and activity and consists of two parts: the central nervous system (CNS) comprising the brain and spinal cord, and the peripheral nervous system (PNS) comprising the branching peripheral nerves. It is composed of neurons, or nerve cells, which receive and transmit impulses, and neuroglia, also known as glial cells or glia, which assist the propagation of the nerve impulse as well as provide nutrients to the neurons. Nervous tissue is made up

Context: Document [18](Title: Nervous system) the spinal cord. The spinal canal contains the spinal cord, while the cranial cavity contains the brain. The CNS is enclosed and protected by the meninges, a three-layered system of membranes, including a tough, leathery outer layer called the dura mater. The brain is also protected by the skull, and the spinal cord by the vertebrae. The peripheral nervous system (PNS) is a collective term for the nervous system structures that do not lie within the CNS. The large majority of the axon bundles called nerves are considered to belong to the PNS, even when the cell bodies of the

Context: Document [19](Title: Neuroanatomy) concentration. The extracellular matrix also provides support on the molecular level for the brain's cells. At the organ level, the nervous system is composed of brain regions, such as the hippocampus in mammals or the mushroom bodies of the fruit fly. These regions are often modular and serve a particular role within the general pathways of the nervous system. For example, the hippocampus is critical for forming memories. The nervous system also contains nerves, which are bundles of fibers that originate from the brain and spinal cord, and branch repeatedly to innervate every part of the body. Nerves are made

Context: Document [20](Title: Central nervous system) (cerebrum as well as midbrain and hindbrain) consists of a cortex, composed of neuron-bodies constituting gray matter, while internally there is more white matter that form tracts and commissures. Apart from cortical gray matter there is also subcortical gray matter making up a large number of different nuclei. From and to the spinal cord are projections of the peripheral nervous system in the form of spinal nerves (sometimes segmental nerves). The nerves connect the spinal cord to skin, joints, muscles etc. and allow for the transmission of efferent motor as well as afferent sensory signals and stimuli. This allows for

Question: what parts make up the peripheral nervous system?

Answer: autonomic nervous system

Answer: somatic nervous system