Performance and Text in Ancient Greece
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Euclid's Elements - Wikipedia, the Free Encyclopedia
Euclid's Elements - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Euclid's_Elements Euclid's Elements From Wikipedia, the free encyclopedia Euclid's Elements (Ancient Greek: Στοιχεῖα Stoicheia) is a mathematical and geometric treatise Elements consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. The work also includes an algebraic system that has become known as geometric algebra, which is powerful enough to solve many algebraic problems,[1] including the problem of finding the square root of a number.[2] With the exception of Autolycus' On the Moving Sphere, the Elements is one of the oldest extant Greek mathematical treatises,[3] and it is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science. The name 'Elements' comes from the plural of 'element'. The frontispiece of Sir Henry Billingsley's first According to Proclus the term was used to describe a English version of Euclid's Elements, 1570 theorem that is all-pervading and helps furnishing proofs of many other theorems. The word 'element' is Author Euclid, and translators in the Greek language the same as 'letter'. This Language Ancient Greek, translations suggests that theorems in the Elements should be seen Subject Euclidean geometry, elementary as standing in the same relation to geometry as letters number theory to language. -
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 -
Pentagons in Medieval Architecture
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library Építés – Építészettudomány 46 (3–4) 291–318 DOI: 10.1556/096.2018.008 PENTAGONS IN MEDIEVAL ARCHITECTURE KRISZTINA FEHÉR* – BALÁZS HALMOS** – BRIGITTA SZILÁGYI*** *PhD student. Department of History of Architecture and Monument Preservation, BUTE K II. 82, Műegyetem rkp. 3, H-1111 Budapest, Hungary. E-mail: [email protected] **PhD, assistant professor. Department of History of Architecture and Monument Preservation, BUTE K II. 82, Műegyetem rkp. 3, H-1111 Budapest, Hungary. E-mail: [email protected] ***PhD, associate professor. Department of Geometry, BUTE H. II. 22, Egry József u. 1, H-1111 Budapest, Hungary. E-mail: [email protected] Among regular polygons, the pentagon is considered to be barely used in medieval architectural compositions, due to its odd spatial appearance and difficult method of construction. The pentagon, representing the number five has a rich semantic role in Christian symbolism. Even though the proper way of construction was already invented in the Antiquity, there is no evidence of medieval architects having been aware of this knowledge. Contemporary sources only show approximative construction methods. In the Middle Ages the form has been used in architectural elements such as window traceries, towers and apses. As opposed to the general opinion supposing that this polygon has rarely been used, numerous examples bear record that its application can be considered as rather common. Our paper at- tempts to give an overview of the different methods architects could have used for regular pentagon construction during the Middle Ages, and the ways of applying the form. -
The Geodetic Sciences in Byzantium
The geodetic sciences in Byzantium Dimitrios A. Rossikopoulos Department of Geodesy and Surveying, Aristotle University of Thessaloniki [email protected] Abstract: Many historians of science consider that geodeasia, a term used by Aristotle meaning "surveying", was not particularly flourishing in Byzantium. However, like “lo- gistiki” (practical arithmetic), it has never ceased to be taught, not only at public universi- ties and ecclesiastical schools, as well as by private tutors. Besides that these two fields had to do with problems of daily life, Byzantines considered them necessary prerequisite for someone who wished to study philosophy. So, they did not only confine themselves to copying and saving the ancient texts, but they also wrote new ones, where they were ana- lyzing their empirical discoveries and their technological achievements. This is the subject of this paper, a retrospect of the numerous manuscripts of the Byzantine period that refer to the development of geodesy both in teaching and practices of surveying, as well as to mat- ters relating to the views about the shape of the earth, the cartography, the positioning in travels and generally the sciences of mapping. Keywords: Geodesy, geodesy in Byzantium, history of geodesy, history of surveying, history of mathematics. Περίληψη: Πολλοί ιστορικοί των επιστημών θεωρούν ότι η γεωδαισία, όρος που χρησι- μοποίησε ο Αριστοτέλης για να ορίσει την πρακτική γεωμετρία, την τοπογραφία, δεν είχε ιδιαίτερη άνθιση στο Βυζάντιο. Ωστόσο, όπως και η “λογιστική”, δεν έπαψε ποτέ να διδά- σκεται όχι μόνο στα κοσμικά πανεπιστήμια, αλλά και στις εκκλησιαστικές σχολές, καθώς επίσης και από ιδιώτες δασκάλους. Πέρα από το ότι οι δύο αυτοί κλάδοι είχαν να κάνουν με προβλήματα της καθημερινής ζωής των ανθρώπων, οι βυζαντινοί θεωρούσαν την διδα- σκαλία τους απαραίτητη προϋπόθεση ώστε να μπορεί κανείς να παρακολουθήσει μαθήμα- τα φιλοσοφίας. -
Medieval Mathematics
Medieval Mathematics The medieval period in Europe, which spanned the centuries from about 400 to almost 1400, was largely an intellectually barren age, but there was significant scholarly activity elsewhere in the world. We would like to examine the contributions of five civilizations to mathematics during this time, four of which are China, India, Arabia, and the Byzantine Empire. Beginning about the year 800 and especially in the thirteenth and fourteenth centuries, the fifth, Western Europe, also made advances that helped to prepare the way for the mathematics of the future. Let us start with China, which began with the Shang dynasty in approximately 1,600 B. C. Archaeological evidence indicates that long before the medieval period, the Chinese had the idea of a positional decimal number system, including symbols for the digits one through nine. Eventually a dot may have been used to represent the absence of a value, but only during the twelfth century A. D. was the system completed by introducing a symbol for zero and treating it as a number. Other features of the Shang period included the use of decimal fractions, a hint of the binary number system, and the oldest known example of a magic square. The most significant book in ancient Chinese mathematical history is entitled The Nine Chapters on the Mathematical Art. It represents the contributions of numerous authors across several centuries and was originally compiled as a single work about 300 B. C. at the same time that Euclid was writing the Elements. However, in 213 B. C., a new emperor ordered the burning of all books written prior to his assumption of power eight years earlier. -
Bibliography
Bibliography Afshar, Iraj: Bibliographie des Catalogues des Manuscrits Persans. Tehran: 1958. Almagest: see Ptolemy. Apollonius: Apollonii Pergaei quae Graece exstant cum commentariis Eutocii (ed. J. L. Heiberg), 2 vols. Leipzig: 1891, 1893. Arberry, A. J. : The Chester Beatty Library, A Handlist of the Arabic Manuscripts, Vol. VII. Dublin: 1964. Archimedes: Archimedis Opera Omnia cum commentariis Eutocii, (iterum ed. J. L. Heiberg), 3 vols. Leipzig: 1910-1915. Archimedes: see also Heath. Aristarchus of Samos: On the Sizes and Distances of the Sun and Moon (ed. T. Heath). Oxford: 1913. Aristotle, Nicomachean Ethics: Aristotelis Ethica Nicomachea (ed. I. Bywater). Oxford: 1894. Aristotle, Prior Analytics: Aristotelis Analytica Priora et Posteriora (ed. W. D. Ross and L. Minio-Paluello). Oxford: 1964. Autolycus: J. Mogenet, Autolycus de Pitane. Louvain, 1950 (Universite de Louvain, Recueil de Travaux d'Histoire et de Philologie, 3e. Serie Fasc. 37). Awad, Gurgis: "Arabic Manuscripts in American Libraries". Sumer 1, 237-277 (1951). (Arabic). Bachmann, Peter: Galens Abhandlung dariiber, dal3 der vorziigliche Arzt Philosoph sein mul3. Gottingen: 1965 (Ak. Wiss. Gottingen, Nachrichten Phil. -hist. Kl. 1965.1). Belger, C.: "Ein neues Fragmentum Mathematicum Bobiense". Hermes 16, 261-84 (1881). Boilot, D. J.: "L'oeuvre d'al-Beruni, essai bibliographique". Melanges de l'Institut Dominicain d'Etudes Orientales du Caire ~, 161-256 (1955). Bretschneider, C. A.: Die Geometrie und die Geometer vor Eukleides. Leipzig: 1870. 217 Bib Ziography Brockelmann, Carl: Geschichte der Arabischen Litteratur, zweite den Supplementbanden angepasste Aunage, 2 vols. Leiden: 1943, 1949 [GAL] [and] Supplementbande I-III. Leiden: 1937, 1938, 1942 [S]. Bulmer-Thomas, I.: "Conon of Samos". Dictionary of Scientific Biography III, (New York), 391 (1971). -
Greek Mathematics Recovered in Books 6 and 7 of Clavius’ Geometria Practica
Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Greek Mathematics Recovered in Books 6 and 7 of Clavius’ Geometria Practica John B. Little Department of Mathematics and CS College of the Holy Cross June 29, 2018 Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions I’ve always been interested in the history of mathematics (in addition to my nominal specialty in algebraic geometry/computational methods/coding theory, etc.) Want to be able to engage with original texts on their own terms – you might recall the talks on Apollonius’s Conics I gave at the last Clavius Group meeting at Holy Cross (two years ago) So, I’ve been taking Greek and Latin language courses in HC’s Classics department The subject for today relates to a Latin-to-English translation project I have recently begun – working with the Geometria Practica of Christopher Clavius, S.J. (1538 - 1612, CE) Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Overview 1 Introduction – Clavius and Geometria Practica 2 Book 6 and Greek approaches to duplication of the cube 3 Book 7 and squaring the circle via the quadratrix 4 Conclusions Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Clavius’ Principal Mathematical Textbooks Euclidis Elementorum, Libri XV (first ed. -
A Mathematician Reads Plutarch: Plato's Criticism of Geometers of His Time
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Scholarship@Claremont Journal of Humanistic Mathematics Volume 7 | Issue 2 July 2017 A Mathematician Reads Plutarch: Plato's Criticism of Geometers of His Time John B. Little College of the Holy Cross Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Ancient History, Greek and Roman through Late Antiquity Commons, and the Mathematics Commons Recommended Citation Little, J. B. "A Mathematician Reads Plutarch: Plato's Criticism of Geometers of His Time," Journal of Humanistic Mathematics, Volume 7 Issue 2 (July 2017), pages 269-293. DOI: 10.5642/ jhummath.201702.13 . Available at: https://scholarship.claremont.edu/jhm/vol7/iss2/13 ©2017 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. A Mathematician Reads Plutarch: Plato's Criticism of Geometers of His Time Cover Page Footnote This essay originated as an assignment for Professor Thomas Martin's Plutarch seminar at Holy Cross in Fall 2016. -
On Some Borrowed and Misunderstood Problems in Greek Catoptrics
On Some Borrowed and Misunderstood Problems in Greek Catoptrics by ALEXANDERJONES Optics was one of the first sciences to which ancient Greek mathe- maticians attempted to apply the apparatus of geometry. They were most successful in describing geometrically certain phenomena of per- spective (as in Euclid’s Optics and parts of Pappus’s Collection, Book 6),less so in catoljtrics, the study of mirrors.’ The problems typically encountered in the surviving writings on catoptrics fall into two classes: those dealing with the reflection of rays cast from a luminous source such as the sun upon minors of plane or curved surface, and those involving the projection of lines of sight from the eye by way of a minor to an object. The most basic problem of the first class, to make a burning mirror, is solved correctly in Diocles’s On Burning Mirrors (ca. 200 B.C.) with both spherical and parabolic mirrors; other prob- lems were investigated by Diocles, by Anthemius (ca. A.D. 620) in his On Paradoxical Devices, and doubtless by other authors in the cen- turies between.2 In general these problems were amenable to geo- metrical treatment in antiquity because Hellenistic geometers were equipped to study the tangent lines to a wide range of curves (or the tangent planes to analogous surfaces) and the behaviour of straight lines in a given direction or through a given point .and inflected at the curve or surface at equal angles to the tangent at the point of in- cidence. Problems of the other class, in which it was required to make an ar- rangement of mirrors or a mirror with a special curvature, such that a *Institute for the History and Philosophy of Science and Technology, Victoria College, Univer- sity of Toronto, Toronto, Canada MSS 1K7. -
A Short History of Greek Mathematics
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. -
Mathematics Education
CHAPTER 26 Mathematics Education Nathan Sidoli It is difficult to say much with real certainty about mathematics education in the ancient Greco‐Roman world. When it comes to discussing their education, the mathematicians themselves are all but silent; when discussing the place of mathematics in education in general, philosophical and rhetorical authors are frustratingly vague; and the papyri related to mathematics education have not yet received the same type of overview studies that the papyri related to literary education have (Cribiore 1996, 2001; Morgan 1998). Nevertheless, this chapter provides a survey of what we can say about ancient mathematics education, on the basis of the evidence from the papyrological and literary sources, and guided by analogies from what we now know about literary education. Although it was once commonly held that the mathematical sciences made up a s tandard part of a fairly regular, liberal arts curriculum (Marrou 1948; Clarke 1971), this has been shown to be largely a fanciful characterization (Hadot 2005: 436–443, 252– 253), and in the case of mathematics it is difficult to be certain about precisely what was learned at what age, and to what end. Indeed, the diversity of mathematical practices and cultures that we find represented in the sources gives the impression that mathematical education was even more private and individualized than literary education. In the Greco‐Roman period, the range of activities that was designated by the word mathēmatikē was not identical to those denoted by our understanding of the word mathematics. Although mathematikē ̄ was originally associated with any type of learning, it came to mean those literary disciplines that used mathematical techniques or that investigated mathematical objects—actual or ideal—such as geometry, mechanics, optics, astronomy and astrology, number theory (arithmetike),̄ harmonics, computational methods (logistike ̄ ), spherics (sphairikos), sphere making (sphairopoiïa), sundial theory (gnomonikos̄ ), and so on. -
Bibliography of the Publications of Abraham Wasserstein
Bibliography of the Publications of Abraham Wasserstein 1953 ‘The Manuscript Tradition of Statius’ Silvae’, Classical Quarterly, N.S. 3, pp. 69-78. 1955 ‘Thales’ Determination of the Diameters of the Sun and Moon’, Journal of Hellenic Studies, 75, pp. 114-16. 1956 ‘Thales and the Diameter of the Sun and Moon’, Journal of Hellenic Studies, 76, p. 105. ‘Politian’s Commentary on the Silvae of Statius’, Scriptorium, 10, pp. 83-9. review of ΜΊ. Finley, The World of Odysseus, Library Review, p. 451. contribution to ‘The position of language in philosophy, logic and social anthropology’, in Proceedings of the Seventh International Congress of Linguistics (London, 1952), pp. 243-45. 1957 review of J. Forsdyke, Greece Before Homer, Library Review, p. 7. 1958 ‘The Manuscript Tradition of Statius’ Silvae’, Classical Quarterly, N.S. 8, pp. 111- 12. ‘Theaetetus and the History of the Theory of Numbers’, Classical Quarterly, N.S. 8, pp. 165-79. review of V. Ehrenberg, Sophokles und Perikies, Gnomon, 30, pp. 169-74. review of C. Seltman, Riot in Ephesus, and of R. Graves, Greek Myths, Library Review, p. 353. review of Τ.ΒἜ. Webster, From Mycenae to Homer, Library Review, p. 503. 1959 ‘Some Early Greek Attempts to Square the Circle’, Phronesis, 4, pp. 92-100. review of Μ. Clagett, Greek Science in Antiquity, Journal of Hellenic Studies, 79, pp. 174-75. review of J. Chadwick, The Decipherment of Linear B, Library Review, p. 13. Scripta Classica Israelica vol. XV 1996 pp. 7-15 8 BIBLIOGRAPHY OF ABRAHAM WASSERSTEIN review of C.H. Whitoian, Homer and the Heroic Tradition, Library Review, p.