'Ptolemy' Epigram
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Proclus on the Elements and the Celestial Bodies
PROCLUS ON THE ELEMENTS AND THE CELESTIAL BODIES PHYSICAL TH UGHT IN LATE NEOPLAT NISM Lucas Siorvanes A Thesis submitted for the Degree of Doctor of Philosophy to the Dept. of History and Philosophy of Science, Science Faculty, University College London. Deuember 1986 - 2 - ABSTRACT Until recently, the period of Late Antiquity had been largely regarded as a sterile age of irrationality and of decline in science. This pioneering work, supported by first-hand study of primary sources, argues that this opinion is profoundly mistaken. It focuses in particular on Proclus, the head of the Platonic School at Athens in the 5th c. AD, and the chief spokesman for the ideas of the dominant school of thought of that time, Neoplatonism. Part I, divided into two Sections, is an introductory guide to Proclus' philosophical and cosmological system, its general principles and its graded ordering of the states of existence. Part II concentrates on his physical theories on the Elements and the celestial bodies, in Sections A and B respectively, with chapters (or sub-sections) on topics including the structure, properties and motion of the Elements; light; space and matter; the composition and motion of the celestial bodies; and the order of planets. The picture that emerges from the study is that much of the Aristotelian physics, so prevalent in Classical Antiquity, was rejected. The concepts which were developed instead included the geometrization of matter, the four-Element composition of the universe, that of self-generated, free motion in space for the heavenly bodies, and that of immanent force or power. -
The Ordered Distribution of Natural Numbers on the Square Root Spiral
The Ordered Distribution of Natural Numbers on the Square Root Spiral - Harry K. Hahn - Ludwig-Erhard-Str. 10 D-76275 Et Germanytlingen, Germany ------------------------------ mathematical analysis by - Kay Schoenberger - Humboldt-University Berlin ----------------------------- 20. June 2007 Abstract : Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the “Square Root Spiral“ ( also named as “Spiral of Theodorus” or “Wurzel Spirale“ or “Einstein Spiral” ). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36 … form a highly three-symmetrical system of three spiral graphs, which divide the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. The Square Root Spiral is a geometrical structure which is based on the three basic constants: 1, sqrt2 and π (pi) , and the continuous application of the Pythagorean Theorem of the right angled triangle. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the “golden mean” ( golden section ), which behaves as a self-avoiding-walk- constant in the lattice-like structure of the square root spiral. Contents of the general section Page 1 Introduction to the Square Root Spiral 2 2 Mathematical description of the Square Root Spiral 4 3 The distribution -
The Spiral of Theodorus and Sums of Zeta-Values at the Half-Integers
The spiral of Theodorus and sums of zeta-values at the half-integers David Brink July 2012 Abstract. The total angular distance traversed by the spiral of Theodorus is governed by the Schneckenkonstante K introduced by Hlawka. The only published estimate of K is the bound K ≤ 0:75. We express K as a sum of Riemann zeta-values at the half-integers and compute it to 100 deci- mal places. We find similar formulas involving the Hurwitz zeta-function for the analytic Theodorus spiral and the Theodorus constant introduced by Davis. 1 Introduction Theodorus of Cyrene (ca. 460{399 B.C.) taught Plato mathematics and was himself a pupil of Protagoras. Plato's dialogue Theaetetus tells that Theodorus was distinguished in the subjects of the quadrivium and also contains the following intriguing passage on irrational square-roots, quoted here from [12]: [Theodorus] was proving to us a certain thing about square roots, I mean of three square feet and of five square feet, namely that these roots are not commensurable in length with the foot-length, and he went on in this way, taking all the separate cases up to the root of 17 square feet, at which point, for some reason, he stopped. It was discussed already in antiquity why Theodorus stopped at seventeen and what his method of proof was. There are at least four fundamentally different theories|not including the suggestion of Hardy and Wright that Theodorus simply became tired!|cf. [11, 12, 16]. One of these theories is due to the German amateur mathematician J. -
PROCLUS on HESIOD's WORKS and DAYS Patrizia Marzillo
PERFORMING AN ACADEMIC TALK: PROCLUS ON HESIOD’S WORKS AND DAYS Patrizia Marzillo Abstract From Socrates onwards orality was the favoured means of expression for those wholaterlovedtocallthemselves‘Platonic’.Theyusedtodiscussphilosophical issues in debates that turned into academic lectures and seminars. According to Plato’s original teaching, these talks should have not been “fixed” in written compositions, yet Plato himself put most of his doctrine into fictive “written dia- logues”.His followers intensified their connection with writing, above all for the purposes of teaching. On the one hand, they made notes on the lessons of their teachers; on the other, they enlarged their own talks in written compositions. Neoplatonists’ commentaries are often an amplification of their academic talks. The lessons held in the school of Athens or in Platonic circles coalesced into texts that mostly constitute Neoplatonic propaganda intended for the outside world. When Proclus directed the school in Athens, Plato and Aristotle were taught, but also theologian poets such as Homer, Orpheus, Hesiod. As the Suda reports, Proclus wrote commentaries on all of these poets, but the only onepreservedisthecommentaryonHesiod’sWorks and Days.Througha comparison of some passages from this commentary, I show how Proclus’ commentary on Hesiod is not only a good example of an oral lesson that has become a written commentary, but also, importantly, of a text that aimed at the diffusion of Neoplatonic ideas among an audience of non-adherents. As his biographer and disciple Marinus of Neapolis relates, the neo- Platonist Proclus was accustomed to write about lines a day.1 Besides being a very prolific author, he was also an indefatigable teacher since in addition to his writing he held several classes during the day and also gave evening talks.2 What I propose to show in this paper is the profound interaction between the oral communication in his school and the written performance of his commentaries. -
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. -
Welcome to the Complete Pythagoras
Welcome to The Complete Pythagoras A full-text, public domain edition for the generalist & specialist Edited by Patrick Rousell for the World Wide Web. I first came across Kenneth Sylvan Guthrie’s edition of the Complete Pythagoras while researching a book on Leonardo. I had been surfing these deep waters for a while and so the value of Guthrie’s publication was immediately apparent. As Guthrie explains in his own introduction, which is at the beginning of the second book (p 168), he was initially prompted to publish these writings in the 1920’s for fear that this information would become lost. As it is, much of this information has since been published in fairly good modern editions. However, these are still hard to access and there is no current complete collection as presented by Guthrie. The advantage here is that we have a fairly comprehensive collection of works on Pythagoras and the Pythagoreans, translated from the origin- al Greek into English, and presented as a unified, albeit electronic edition. The Complete Pythagoras is a compilation of two books. The first is entitled The Life Of Py- thagoras and contains the four biographies of Pythagoras that have survived from antiquity: that of Iamblichus (280-333 A.D.), Porphry (233-306 A.D.), Photius (ca 820- ca 891 A.D.) and Diogenes Laertius (180 A.D.). The second is entitled Pythagorean Library and is a complete collection of the surviving fragments from the Pythagoreans. The first book was published in 1920, the second a year later, and released together as a bound edition. -
A Acerbi, F., 45 Adam, C., 166–169, 171, 175, 176, 178–181, 183–185
Index A 80, 81, 84, 86, 87, 93, 97, 98, 107, 112, Acerbi, F., 45 137, 198, 201, 212, 233, 235, 236 Adam, C., 166–169, 171, 175, 176, 178–181, Arius Didymus, 35 183–185, 187, 188, 200, 201 Arnauld, 217, 227, 233, 239–242, 245 Adams, R.M., 234, 245, 306 Arnzen, R., 55 Adorno, T.W., 147 Arriaga, R.de, 6 Adrastos, 77–81, 89, 96 Arthur, R, 254 Aetius, 35 Athenaeus, 39 Agapius, 56 Atherton, M., 170, 174 Aglietta, M., 151 Aujac, G., 145 Aichelin, J., 232 Autolycus, 15, 20, 21, 23, 24 Aiken, J.A., 148 Ayers, M., 229 Aime, M., 145 Aksamija, N., 152 Alberti, L.B., 9, 148, 150 B Alembert, J.d', 70 Bacon, R., 165, 166 Alexander of Aphrodisias, 16, 35, 43, 54, 56, Banu Musa, 53 58, 61, 77, 78, 81 Barnes, J., 38, 212, 235 Algra, K., 34, 35, 38 Barozzi, F., 115, 119, 120 Al-Haytham, 164–167 Barrow, I., 6, 220 Alhazen, see Ibn Al-Haytham Basileides of Tyre, 37 Al-Khwārizmī, 92 Baudrillard, J., 144 Al-Kindī, 161, 162, 164, 165 Bayle, P., 237 Al-Nayrizi, 55–57, 59 Beauchamp, T.L., 274 Al-Sijzī, 110 Bechtle, G., 105 Allison, H.E., 283 Belting, H., 145, 148 Andronicus of Rhodes, 102 Benatouïl, T., 114 Apollodorus, 41, 42 Benjamin, W., 144, 145, 148 Apollonios of Perge (Apollonius), 15, 19–21, Bentley, R., 223–226 23–27, 37, 38, 52, 53, 63, 81, 106, 123 Berggren, J.L., 94 Apostle Thomas, 147 Berkeley, G., 11, 160, 161, 166, 170, 186 Aratus, 24 Bernadete, J., 212, 213 Arbini, R., 170 Bernanos, G., 137 Archedemus, 41 Bernoulli, J., 233 Archimedes, 15, 19–21, 23, 24, 27, 29, 38, 39, Bessel, F.W., 72 44, 45, 47, 68, 106, 123 Bioesmat-Martagon, L., 72 Ariew, R., 110, 212, 214 Blumenberg, H., 143 Aristaeus, 27 Boer, E., 93 Ariston, 39 Bolyai, J., 11 Aristotle, 3, 5, 6, 15, 16, 18, 19, 24–34, 36, Bonola, R., 70 40–45, 47–50, 55, 58, 60, 61, 68, 75, Borelli, G.A., 6 © Springer International Publishing Switzerland 2015 311 V. -
Euclidean Geometry
Euclidean Geometry Rich Cochrane Andrew McGettigan Fine Art Maths Centre Central Saint Martins http://maths.myblog.arts.ac.uk/ Reviewer: Prof Jeremy Gray, Open University October 2015 www.mathcentre.co.uk This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. 2 Contents Introduction 5 What's in this Booklet? 5 To the Student 6 To the Teacher 7 Toolkit 7 On Geogebra 8 Acknowledgements 10 Background Material 11 The Importance of Method 12 First Session: Tools, Methods, Attitudes & Goals 15 What is a Construction? 15 A Note on Lines 16 Copy a Line segment & Draw a Circle 17 Equilateral Triangle 23 Perpendicular Bisector 24 Angle Bisector 25 Angle Made by Lines 26 The Regular Hexagon 27 © Rich Cochrane & Andrew McGettigan Reviewer: Jeremy Gray www.mathcentre.co.uk Central Saint Martins, UAL Open University 3 Second Session: Parallel and Perpendicular 30 Addition & Subtraction of Lengths 30 Addition & Subtraction of Angles 33 Perpendicular Lines 35 Parallel Lines 39 Parallel Lines & Angles 42 Constructing Parallel Lines 44 Squares & Other Parallelograms 44 Division of a Line Segment into Several Parts 50 Thales' Theorem 52 Third Session: Making Sense of Area 53 Congruence, Measurement & Area 53 Zero, One & Two Dimensions 54 Congruent Triangles 54 Triangles & Parallelograms 56 Quadrature 58 Pythagoras' Theorem 58 A Quadrature Construction 64 Summing the Areas of Squares 67 Fourth Session: Tilings 69 The Idea of a Tiling 69 Euclidean & Related Tilings 69 Islamic Tilings 73 Further Tilings -
Galileo, Ignoramus: Mathematics Versus Philosophy in the Scientific Revolution
Galileo, Ignoramus: Mathematics versus Philosophy in the Scientific Revolution Viktor Blåsjö Abstract I offer a revisionist interpretation of Galileo’s role in the history of science. My overarching thesis is that Galileo lacked technical ability in mathematics, and that this can be seen as directly explaining numerous aspects of his life’s work. I suggest that it is precisely because he was bad at mathematics that Galileo was keen on experiment and empiricism, and eagerly adopted the telescope. His reliance on these hands-on modes of research was not a pioneering contribution to scientific method, but a last resort of a mind ill equipped to make a contribution on mathematical grounds. Likewise, it is precisely because he was bad at mathematics that Galileo expounded at length about basic principles of scientific method. “Those who can’t do, teach.” The vision of science articulated by Galileo was less original than is commonly assumed. It had long been taken for granted by mathematicians, who, however, did not stop to pontificate about such things in philosophical prose because they were too busy doing advanced scientific work. Contents 4 Astronomy 38 4.1 Adoption of Copernicanism . 38 1 Introduction 2 4.2 Pre-telescopic heliocentrism . 40 4.3 Tycho Brahe’s system . 42 2 Mathematics 2 4.4 Against Tycho . 45 2.1 Cycloid . .2 4.5 The telescope . 46 2.2 Mathematicians versus philosophers . .4 4.6 Optics . 48 2.3 Professor . .7 4.7 Mountains on the moon . 49 2.4 Sector . .8 4.8 Double-star parallax . 50 2.5 Book of nature . -
Thomas Taylor's Dissent from Some 18Th
Leo Catana, Thomas Taylor’s dissent from some 18th-century views on Platonic philosophy' The ethical and theological context Catana, Leo Published in: International Journal of the Platonic Tradition DOI: 10.1163/18725473-12341262 Publication date: 2013 Document version Publisher's PDF, also known as Version of record Citation for published version (APA): Catana, L. (2013). Leo Catana, Thomas Taylor’s dissent from some 18th-century views on Platonic philosophy': The ethical and theological context. International Journal of the Platonic Tradition, 7(2), 180-220. https://doi.org/10.1163/18725473-12341262 Download date: 26. Sep. 2021 The International Journal of the Platonic Tradition The International Journal of the Platonic Tradition 7 (2013) 180-220 brill.com/jpt Thomas Taylor’s Dissent from Some 18th-Century Views on Platonic Philosophy: The Ethical and Theological Context Leo Catana University of Copenhagen [email protected] Abstract Thomas Taylor’s interpretation of Plato’s works in 1804 was condemned as guilty by association immediately after its publication. Taylor’s 1804 and 1809 reviewer thus made a hasty generalisation in which the qualities of Neoplatonism, assumed to be negative, were transferred to Taylor’s own interpretation, which made use of Neoplatonist thinkers. For this reason, Taylor has typically been marginalised as an interpreter of Plato. This article does not deny the association between Taylor and Neoplatonism. Instead, it examines the historical and historiographical reasons for the reviewer’s assumption that Neoplatonic readings of Plato are erroneous by definition. In particular, it argues that the reviewer relied on, and tacitly accepted, ethical and theological premises going back to the historiography of philosophy developed by Jacob Brucker in his Historia critica philosophiae (1742-44). -
SCHOLIA, COMMENTARIES, and LEXICA on SPECIFIC LITERARY WORKS 2 Scholia, Commentaries, and Lexica on Specific Literary Works
18 SCHOLIA, COMMENTARIES, AND LEXICA ON SPECIFIC LITERARY WORKS 2 Scholia, Commentaries, and Lexica on Specific Literary Works 2.1 ARCHAIC AND CLASSICAL POETRY This category includes the most famous and most often cited scholia. By far the most important are the Homer scholia, but those on Pindar and the Attic drama- tists are also significant. 2.1.1 Homer Ancient scholarship on Homer was extensive and of high quality, for the best scholars of antiquity devoted much of their time and energy to the Homeric poems. Work on Homer that could be described as scholarship goes back at least to the classical period and probably to the sixth century bc, and editing the text of Homer was one of the main tasks of the first Alexandrian scholars. Zenodotus, Aristophanes of Byzantium, and Aristarchus probably all produced editions of the Iliad and Odyssey, and Aristarchus wrote extensive commentaries, while Zenodotus and Aristophanes compiled glossaries of primarily Homeric words. In addition, the early and persistent use of Homer as a school text meant that there was a tradi- tion of school exegesis that reached back as far as the classical period. Though none of the very early work on Homer survives in its original form, a surprising amount is preserved in various later compilations, so we often know, for example, the read- ings of several different Alexandrian scholars for a particular passage, and even some of the arguments behind these readings (although the arguments preserved in later sources cannot always be assumed to be those of the editor himself). Two principal sources for the ancient scholarship on Homer survive: the scholia and Eustathius’ commentaries, both of which are gigantic works filling many vol- umes in modern editions. -
The Greek Anthology with an English Translation by W
|^rct?c^tc^ to of the Univereitp of Toronto JScrtrani 1I-1. iDavit^ from the hoohs? ot the late Hioncl Bavie, Hc.cT. THE LOEB CLASSICAL LIBRARY EDITED BY E. CAl'PS, Ph.D., LL.D. T. E. PAGE, Lut.D. W. H. D. IWUSE, LnT.D. THE CiREEK ANTHOLOGY III 1 *• THE GREEK ANTHOLOGY WITH AN ENGLISH TRANSLATION BY W. R. PA TON IN FIVE VOLUMES III LONDON : WILLIAM HEINEMANN NEW YORK : G. P. PUTNAM S SONS M CM XV 1 CONTENTS BOOK IX. —THE DECLAMATORY EPICRAMS 1 GENERAL INDEX 449 INDEX OF AUTHORS INCLUDED IN THIS VOLUME . 454 GREEK ANTHOLOGY BOOK IX THE DECLAMATORY AND DESCRIPTIVE EPIGRAMS This book, as we should naturally expect, is especially rich in epigrams from the Stephanus of Philippus, the rhetorical style of epigram having been in vogue during the period covered by that collection. There are several quite long series from this source, retaining the alphabetical order in which they were arranged, Nos. 215-312, 403-423, 541- 562. It is correspondingly poor in poems from Meleager s Stephanus (Nos. 313-338). It contains a good deal of the Alexandrian Palladas, a contemporary of Hypatia, most of wliich we could well dispense with. The latter part, from No. 582 07iwards, consists mostly of real or pretended in- scriptions on works of art or buildings, many quite unworthy of preservation, but some, especially those on baths, quite graceful. The last three epigrams, written in a later hand, do not belong to the original Anthology. ANeOAOriA (-) EnirPAM.M ATA RTIIAEIKTIKA 1.— 110ATAIX(;T :::a1'A1AN()T AopKuSo^; upTiroKoio Ti6i]i')]T)'ipioi' ovdap efjLTrXeov r)p.vaav ^ 7rifcp6<i erv^ev e)(i<;.