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Plato and Aristotle Plato’s Academy in Athens(400 BCE) Although Plato was more a philosopher rather than a mathematician, it is clear that he had a high regard for mathematics and was seen as a “maker of mathematicians.” At the entrance of his academy was written, “Let no one ignorant of geometry enter here.” The five Platonic solids were mystically associated with the four fundamental elements of earth, water, fire and air. Eudoxus and the “method of exhaustion” Eudoxus seems to have been the first person to define our modern notion of a “limit” which he referred to as “the method of exhaustion.” n In modern notation, limn ∞M(1-r) =0, for 0<r<1. → Ratios of areas of circles The argument of Eudoxus involved finer and finer inscribed polygons. The ratio of the areas of regular polygons is proportional to the squares of the “radius” of the polygons. Hence the theorem. The conic sections The brothers Menaechmus and Dinostratus were taught by Eudoxus. Menaechmus is credited with the discovery of the conic sections: the parabola, the ellipse, and the hyperbola. The duplication of the cube Menaechmus knew how to solve the problem of the duplication of the cube if one is permitted to use parabolas. The intersection of the two parabolas allows the construction of the cube root of 2. Trisecting a line segment 1. Draw any ray AC. 2. Draw circle of radius AC centered at C. 3. Draw another circle of radius AC centered at D. Trisecting angles Dinostratus studied the squaring of the circle using what has been called the “trisectrix” of Hippias. This is the locus intersection of side DA rotating clockwise and side AB dropping to coincide with DC. To trisect PDC, trisect the line segments B’C and A’D at R,S,T,U. Join T to R and U to S to trisect the given angle. Squaring the circle using the trisectrix The equation of the trisectrix is rsin =2a . As 0, r tends to 2a/ . π θ θ Thus DQ is 2a/ . θ→ π From this, the squaring of the circle is easilyπ determined..

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Mathematicians
MATHEMATICIANS [MATHEMATICIANS] Authors: Oliver Knill: 2000 Literature: Started from a list of names with birthdates grabbed from mactutor in 2000. Abbe [Abbe] Abbe Ernst (1840-1909) Abel [Abel] Abel Niels Henrik (1802-1829) Norwegian mathematician. Signiﬁcant contributions to algebra and anal- ysis, in particular the study of groups and series. Famous for proving the insolubility of the quintic equation at the age of 19. AbrahamMax [AbrahamMax] Abraham Max (1875-1922) Ackermann [Ackermann] Ackermann Wilhelm (1896-1962) AdamsFrank [AdamsFrank] Adams J Frank (1930-1989) Adams [Adams] Adams John Couch (1819-1892) Adelard [Adelard] Adelard of Bath (1075-1160) Adler [Adler] Adler August (1863-1923) Adrain [Adrain] Adrain Robert (1775-1843) Aepinus [Aepinus] Aepinus Franz (1724-1802) Agnesi [Agnesi] Agnesi Maria (1718-1799) Ahlfors [Ahlfors] Ahlfors Lars (1907-1996) Finnish mathematician working in complex analysis, was also professor at Harvard from 1946, retiring in 1977. Ahlfors won both the Fields medal in 1936 and the Wolf prize in 1981. Ahmes [Ahmes] Ahmes (1680BC-1620BC) Aida [Aida] Aida Yasuaki (1747-1817) Aiken [Aiken] Aiken Howard (1900-1973) Airy [Airy] Airy George (1801-1892) Aitken [Aitken] Aitken Alec (1895-1967) Ajima [Ajima] Ajima Naonobu (1732-1798) Akhiezer [Akhiezer] Akhiezer Naum Ilich (1901-1980) Albanese [Albanese] Albanese Giacomo (1890-1948) Albert [Albert] Albert of Saxony (1316-1390) AlbertAbraham [AlbertAbraham] Albert A Adrian (1905-1972) Alberti [Alberti] Alberti Leone (1404-1472) Albertus [Albertus] Albertus Magnus
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Plato As "Architectof Science"
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Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
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Open History of Math 2011 MAΘ National
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Greek Mathematics Recovered in Books 6 and 7 of Clavius’ Geometria Practica
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Cambridge Library Co ll e C t i o n Books of enduring scholarly value Classics From the Renaissance to the nineteenth century, Latin and Greek were compulsory subjects in almost all European universities, and most early modern scholars published their research and conducted international correspondence in Latin. Latin had continued in use in Western Europe long after the fall of the Roman empire as the lingua franca of the educated classes and of law, diplomacy, religion and university teaching. The flight of Greek scholars to the West after the fall of Constantinople in 1453 gave impetus to the study of ancient Greek literature and the Greek New Testament. Eventually, just as nineteenth-century reforms of university curricula were beginning to erode this ascendancy, developments in textual criticism and linguistic analysis, and new ways of studying ancient societies, especially archaeology, led to renewed enthusiasm for the Classics. This collection offers works of criticism, interpretation and synthesis by the outstanding scholars of the nineteenth century. A Short History of Greek Mathematics James Gow’s Short History of Greek Mathematics (1884) provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. Parts I and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers.
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Quadratrix of Hippias -- from Wolfram Mathworld
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II Apollonius of Perga
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Bibliography
Bibliography A. Aaboe, Episodes from the Early History of Mathematics (Random House, New York, 1964) A.D. Aczel, Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem (Four Walls Eight Windows, New York, 1996) D. Adamson, Blaise Pascal: Mathematician, Physicist, and Thinker About God (St. Martin’s Press, New York, 1995) R.P. Agarwal, H. Agarwal, S.K. Sen, Birth, Growth and Computation of Pi to ten trillion digits. Adv. Differ. Equat. 2013, 100 (2013) A.A. Al-Daffa’, The Muslim Contribution to Mathematics (Humanities Press, Atlantic Highlands, 1977) A.A. Al-Daffa’, J.J. Stroyls, Studies in the Exact Sciences in Medieval Islam (Wiley, New York, 1984) E.J. Aiton, Leibniz: A Biography (A. Hilger, Bristol, Boston, 1984) R.E. Allen, Greek Philosophy: Thales to Aristotle (The Free Press, New York, 1966) G.J. Allman, Greek Geometry from Thales to Euclid (Arno Press, New York, 1976) E.N. da C. Andrade, Sir Issac Newton, His Life and Work (Doubleday & Co., New York, 1954) W.S. Anglin, Mathematics: A Concise History and Philosophy (Springer, New York, 1994) W.S. Anglin, The Queen of Mathematics (Kluwer, Dordrecht, 1995) H.D. Anthony, Sir Isaac Newton (Abelard-Schuman, New York, 1960) H.G. Apostle, Aristotle’s Philosophy of Mathematics (The University of Chicago Press, Chicago, 1952) R.C. Archibald, Outline of the history of mathematics.Am. Math. Monthly 56 (1949) B. Artmann, Euclid: The Creation of Mathematics (Springer, New York, 1999) C.N. Srinivasa Ayyangar, The History of Ancient Indian Mathematics (World Press Private Ltd., Calcutta, 1967) A.K. Bag, Mathematics in Ancient and Medieval India (Chaukhambha Orientalia, Varanasi, 1979) W.W.R.
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Mathematical Analysis in Ancient Greece
WDS'08 Proceedings of Contributed Papers, Part I, 27–31, 2008. ISBN 978-80-7378-065-4 © MATFYZPRESS Mathematical Analysis in Ancient Greece K. Cernˇ eková Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. The article deals with ideas of mathematical analysis which appeared in the ancient Greece. Our goal is to present the method, that gave us the modern perspective on the history of these ideas. We start with the theory of exhaustion used in the twelfth book of Euclid’s Elements (ca. 300 B.C.), which bears the idea of correct integration techniques based on Proposition X,1. However, Euclid did not invented this theory, nor referred to the author. Later on, Archimedes mentioned in his works that earlier geometers used this lemma to prove important theorems of plane geometry, and Eudoxus (4th century B.C.) was the first one, who proved some of them. This indicates, and is nowadays believed, that Eudoxus proved Proposition X,1 and many other theorems from the twelfth book of Euclid’s Elements. On the other hand Simplicius stated that one of those important geometrical theorems was already proved by Hippocrates of Chios in the 5th century B.C. But he did not mention, how Hippocrates proved it. This yields us a dilemma whether to call a founder somebody who discovered the idea and intuitively used it in one case, or someone who firmly formulated and proved the basic idea and explained its usage in many problems. Ideas of Mathematical Analysis The theory of mathematical analysis from a modern point of view was established in the 17th century, but we can trace the ideas throughout history.
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Ancient Greek Mathēmata from a Sociological Perspective
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A Concise History of the Philosophy of Mathematics
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