DOCSLIB.ORG

Philolaus* by Prof

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Philolaus* by Prof IV. Philolaus* By Prof. Will. Romaine Newbold, University of Pennsylvania, America, Boeckh's monograph upon Philolaus ("Philolaos, des Pythagor ers Lehren, nebst den Bruchst cken seines Werkes", Berlin 1819, pp. 200) h s left little for others to do. His collection of fragments con- tains nearly everything of importance, and in his Interpretation both of individual passages and of the System s a whole, he clis- plays an insight the more astonishing when one remembers how imperfectly the development of early Greek philosophy was at that time understood. Yet there remain in the fragments several passages of which neither Boeckh nor any of his A successors has been able to gi\7e a satisfactory Interpretation. Of these I shall examine two. The first is the function of "embodying" and "Split- ting" ratios, which function Philolaus ascribes to number, and the meaning of the reference to the gnomon. The second is the nature of the principles assumed by Philolaus and termed by him περαίνοντá and άπειρα. If the results which I shall reach are sound, they will lead to a revision, in certain important particulars, of the accepted views s to'the details of the Philolaic cosmology. *) Stobaeus Ecl. L proem, cor 3 (p. 16, 20 Wachsmuth) FV p. 253—254. *) I quote s "FV" Prof. Diels' epochrnaking work, "Die Fragmente^ der Vorsokratiker", Berlin. 1903: s UD", his "Doxographi Graeci*·*, Berlin, 1879. In quoting passages co tained in either of these books, I use Diels' text and orthography, unless the contrary is stated. Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. 177 θεωρεΐν δεé τá έργá êáé την "One should judge the deeds ουσίαν τù άριομώ καττάν δυναμιν and the essence of Number by the ατις εστίν εν ταé δεκάδι· μεγάλá virtue that is in the Ten, for it ãáñ êáΐ παντελής êáΐ παντοεργδς. is great, all-perfecting and all- êáΐ θείù êáΐ ουρανίù βίù êáé achieving, and is the origin and άν^ρωπίνù άρχά êáé άγεμών κοι- leader both of divine and celestial νωνουσá . .. δυναμις êáé τας δέκα- life and of that of man, sharing δος. άνεõ δε τουτας πάντ' άπειρá ... power of the Ten also. Without êáé άδηλá êáé αφανή. it all things are indefinite and vague and indistinct. γνωμικά ãáñ á φύσις á τù αριθμώ For the nature of Number is êáé ηγεμονικά êáé διδασκαλικά τù able to enlighten/guide and teach άπορουμένù παντός êáé άγνοουμένù every man s regards every thing παντί. οõ ãáñ ης δήλον ουδέν! perplexing and unknown. For of ουδέν των πραγμάτων ούτε αυτών ποθ' Things not one would have beeil αυτά ούτε άλλù προς άλλο, εé μη plain to anyone, either s they ης αριθμός êáé ά τοότù ουσία, are to themselves or s one is νυν δε ούτος κατάν ψόχάν αρμόζων to another, did not Number and αίσΟήσεί πάντá γνωστά êáé ποτάγορá its essence exist. As it is, fitting άλλάλοις κατά γνώμονος φυσιν them to the sou!5 it makes them απεργάζεταé σωμάτων êáé σχίζων all knowable to sense and com- τους λόγους χωρίς εκάστους των parable one with another, in the πραγμάτων των τε άπειρων êáé των manner of the gnomon, embodying περαινόντων. and Splitting the several ratios apart of the Things both of the indeterminate and of the deter- mining. ίοοις δε êá οõ μόνον εν τοις δαι- You would see the nature of μονίοις êáé θείοις πράγμασé τάν τù Number and its virtue prevailing αριθμώ φυσιν êáé τάν δυναμιν not only in Things supernatural ισχύουσαν, áëëά êáé εν τοις άνΟρω- and divine, but also in the deeds πικοις εργοις êáé λόγοις πασé πάντá and words of men, in them all êáé κατά τάς δημιουργίας τας τεχνι- and through them all, and in all κάς πάσας êáé κατά τάν μουσικάν. operations of the crafts and in music. ψευδός δε ουδέν δέχεταé ά τù The nature of Number, ofwhich Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM 178 Wm, Romaine Newbold, φύσις, οõ αρμονία* οõ ãáñ is fiarmoDia, adinits no untruth, oixetov αύτù έοτι. τας το> [iot?J for untruth is not akin to i t άπειρù xotl ανόητοé χαΐ άλάγù φύ- Untruth and envy belong to the αιος το ψευδός êáé ό φθόνος εστί. indeterminate and unintelligent ψευδός δε ουδαμώς ες αριθμόν and irrational nature. επιπνεΓ· πολέμιον ãáñ êáé έχθρον Untruth never breathes2) into ταé φύσεé το ψευδός, ά δ' αλήθειá Nurnber, for untruth is a foe and ρίκεΐον êáé συμφυτον ταé τù άριΟμÁ enemy to its_nature, but truth. γενεαι. is of N rnbergs house and bred \vith its kin." There is in this fragment but one passage the meaning of which is not clear. Philolaus ascribes both to number and to the gnomon the iunction of making .things "knowable to sense and comparable one with another", and at least to number, that of '"embodying" and usplitting" the ratios of things both indeterminate and deter- mining. The word "einbodying" (σωμάτων) was first suggested by Boeckh s a substitute for the meaningless σωμα'των of the MSS. He thinks that Number icembodies" things because Body is the third power of number; the meaning of "Splitting" he does not explain, but connects it.witli the notion that the void enters into and splits the nuinbers.3) It is not safe to allege that this or ny other possible parallel between numbers and things was not and would not have been drawn by a Fythagoreau. But the function liere ascribed to number 2) Probably an allusion to the well known Pythagorean doctrine that the original One was split np by inhaling the \roid from without. 3) p. 144. Im folgenden habe ich nichts mit der Lesart σωμάτων anzu- fangen gewu t und daher aus Vermutung σωμάτων gesetzt : wonach nun zweierlei von der Zahl bewirkt w rde, erstlich, da sie den Dingen Korper gibt, indem der Korper die dritte Potenz der Zahl ist: dann, da sie die Verh ltnisse des Be- grenzten und Unbegrenzten scheidet: durch beides, durch das K rperlichmachen und die Sonderung, letztere mag nun gedacht sein, wie sie wolle (vgl. St. 12 im Anfang), werden die Dinge allerdings erst bestimmt erkennbar f r die Empfindung." In the section referred to, 12, p. 108, Boeckh says, in speaking of the Pythagorean notion that the ουρανός inhales το κενόν from without, — "dies Leere sei zuerst in den Zahlen und trenne ihre Natur, und die Getrennt- heit der Naturen habe berhaupt hierin ihren Grund." Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM Philolaus. .179 is certainly not in keeping with the point of view of the Philolaus fragments in general or of this one in particular. Number is h re identified with what later philosophy termed the formal s against the rnaterial elernent in things. It is the principle of unity that Blocks together" into a αρμονίá the diverse things of which the world is made; it is the sole object of knowledge, and is also the instru- ment by which we compare one thing with another and determine their relation. It is the defining principle: without it all things would be confused and indistinguishable. In this context it is meaningless to add that number makes things corporeal. The functions of number of which Philolaus is speaking are of the opposite type. The "gnomon" Boeckh takes in its geometrical sense; the object known is surrounded and comprehended by the knower s each of the gnomons a b c surrounds and comprehends that part of the square that lies within it, thus syin- bolising the original concord and adaptation between knower and known.4) This Interpretation is fairly satisfactory so far s the function of "fitting things to the soul" is concerned, but it leaves the other unexplained. Yet it is not difficult to give a more probable ex- planation of the way in which the gnomon makes things knowable to perception and comparable one with another. At a quite early period the Pythagoreans devised a method of performing mathematical operations which has since been happily termed "geometrical algebra." 5) The substance of this geometrical algebra is founcl in the second book of Euclid and in a few propo- sitions of the first and sixtL By its aid the Pythagoreans dis- covered raany formulae for the multiplication and division of binomials and were enabled to perform many of the operations 4) p. 144" . nach dem Philolaischen Bruchstucke scheint man in den gno- monischen Verbindungen ein Bild der Befreundung und Vereinigung erblickt zu haben, welche unser Schriftsteller nicht ungeschickt auf die Erkennbarkeit der Dinge anwendet, indem das Erkannte von dem Erkennenden umfa t und er- grillen wird: wobei eine urspr ngliche Obereinstimmung und Anpassung, wie des Gnomon um sein Quadrat herum, vorausgesetzt wird." δ) Zeuthen, "Die Lehre von den Kegelschnitten im Altertum", pji. 1—38. Efeath, "Apollonius of Perga", pp. ci-cxi. Brought to you by | provisional account Unauthenticated Download Date | 6/15/15 8:09 AM º80 Wm. Romaine Newbold, * foi wliich we now omploy arithmotic or algebra. In fact, the ablost Grook * raathematicians long preforrcd these geometrical mclhods to tho diroct Manipulation of nurabers on account of thoir groater simplicity s compared with the clumsy and laborious processes of λογιστική. Now the methods of geometrical algebra were in large part bused upon the properties of the gnomon. If through a point e on the diagonal cb of the parallelogram ab cd parallels fg and hi bc passed, it is easy to prove (Euclid I. 43) that the complemen- tary parallelograms ae and ed of the Ë \ /» I. gnomon acdief are equal. Since each is equal to the product of its base by its height, or, in a rectangle, of one side by the other, those products are equal g /ie — egXc/d), and either dimen- sion of one is the quotient reached by dividing its other dimension into the product of the two dimensions of the other l —=gd\.
Load more

Recommended publications
  • 'Pythagorean' Kingship Treatises
    Meeting Different Needs The Implied Readers ofthe ‘Pythagorean’ Kingship Treatises Michael Trapp To cite this version: Michael Trapp. Meeting Different Needs The Implied Readers of the ‘Pythagorean’ Kingship Treatises. KTÈMA Civilisations de l’Orient, de la Grèce et de Rome antiques, Université de Strasbourg, 2020, Les traités néopythagoriciens Sur la royauté, 45, pp.143-160. halshs-03071732 HAL Id: halshs-03071732 https://halshs.archives-ouvertes.fr/halshs-03071732 Submitted on 16 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. KTÈMA KTÈMA KTÈMA CIVILISATIONS DE L’ORIENT, DE LA GRÈCE ET DE ROME ANTIQUES CIVILISATIONS DE L’ORIENT, DE LA GRÈCE ET DE ROME ANTIQUES KTÈMA est une revue annuelle de recherche consacrée à l’histoire, l’archéologie et la littérature de la Grèce, de Rome, de l’Égypte et du Les traités néopythagoriciens Sur la royauté Proche-Orient antiques. Fondée en 1976 par Edmond Frézouls et Anne Gangloff Introduction ..............................................................................................................................5
    [Show full text]
  • De Theognide Megarensi. Nietzsche on Theognis of Megara. a Bilingual Edition
    FRIEDRICH NIETZSCHE De Theognide Megarensi Nietzsche on Theognis of Megara – A Bilingual Edition – Translated by R. M. Kerr THE NIETZSCHE CHANNEL Friedrich Nietzsche De Theognide Megarensi Nietzsche on Theognis of Megara A bilingual edition Translated by R. M. Kerr ☙ editio electronica ❧ _________________________________________ THE N E T ! " # H E # H A N N E $ % MM&' Copyright © Proprietas interpretatoris Roberti Martini Kerrii anno 2015 Omnia proprietatis iura reservantur et vindicantur. Imitatio prohibita sine auctoris permissione. Non licet pecuniam expetere pro aliquo, quod partem horum verborum continet; liber pro omnibus semper gratuitus erat et manet. Sic rerum summa novatur semper, et inter se mortales mutua vivunt. augescunt aliae gentes, aliae invuntur, inque brevi spatio mutantur saecla animantum et quasi cursores vitai lampada tradiunt. - Lucretius - - de Rerum Natura, II 5-! - PR"#$CE %e &or' presente( here is a trans)ation o* #rie(rich Nietzsche-s aledi!tionsarbeit ./schoo) e0it-thesis12 *or the "andesschule #$orta in 3chu)p*orta .3axony-$nhalt) presente( on 3epte4ber th 15678 It has hitherto )arge)y gone unnotice(, especial)y in anglophone Nietzsche stu(- ies8 $t the ti4e though, the &or' he)pe( to estab)ish the reputation o* the then twenty year o)( Nietzsche and consi(erab)y *aci)itate( his )ater acade4ic career8 9y a)) accounts, it &as a consi(erab)e achie:e4ent, especial)y consi(ering &hen it &as &ri;en: it entai)e( an e0pert 'no&)e(ge, not =ust o* c)assical-phi)o)ogical )iterature, but also o* co(ico)ogy8 %e recent =u(ge4ent by >"+3"+ .2017<!!2< “It is a piece that, ha( Nietzsche ne:er &ri;en another &or(, &ou)( ha:e assure( his p)ace, albeit @uite a s4a)) one, in the history o* Ger4an phi)o)ogyB su4s the 4atter up quite e)o@uently8 +ietzsche )ater continue( his %eognis stu(ies, the sub=ect o* his Crst scho)ar)y artic)e, as a stu(ent at Leip,ig, in 156 D to so4e e0tent a su44ary o* the present &or' D a critical re:ie& in 156!, as &e)) as @uotes in se:eral )e;ers *ro4 1567 on.
    [Show full text]
  • Mathematical Discourse in Philosophical Authors: Examples from Theon of Smyrna and Cleomedes on Mathematical Astronomy
    Mathematical discourse in philosophical authors: Examples from Theon of Smyrna and Cleomedes on mathematical astronomy Nathan Sidoli Introduction Ancient philosophers and other intellectuals often mention the work of mathematicians, al- though the latter rarely return the favor.1 The most obvious reason for this stems from the im- personal nature of mathematical discourse, which tends to eschew any discussion of personal, or lived, experience. There seems to be more at stake than this, however, because when math- ematicians do mention names they almost always belong to the small group of people who are known to us as mathematicians, or who are known to us through their mathematical works.2 In order to be accepted as a member of the group of mathematicians, one must not only have mastered various technical concepts and methods, but must also have learned how to express oneself in a stylized form of Greek prose that has often struck the uninitiated as peculiar.3 Be- cause of the specialized nature of this type of intellectual activity, in order to gain real mastery it was probably necessary to have studied it from youth, or to have had the time to apply oneself uninterruptedly.4 Hence, the private nature of ancient education meant that there were many educated individuals who had not mastered, or perhaps even been much exposed to, aspects of ancient mathematical thought and practice that we would regard as rather elementary (Cribiore 2001; Sidoli 2015). Starting from at least the late Hellenistic period, and especially during the Imperial and Late- Ancient periods, some authors sought to address this situation in a variety of different ways— such as discussing technical topics in more elementary modes, rewriting mathematical argu- ments so as to be intelligible to a broader audience, or incorporating mathematical material di- rectly into philosophical curricula.
    [Show full text]
  • Meet the Philosophers of Ancient Greece
    Meet the Philosophers of Ancient Greece Everything You Always Wanted to Know About Ancient Greek Philosophy but didn’t Know Who to Ask Edited by Patricia F. O’Grady MEET THE PHILOSOPHERS OF ANCIENT GREECE Dedicated to the memory of Panagiotis, a humble man, who found pleasure when reading about the philosophers of Ancient Greece Meet the Philosophers of Ancient Greece Everything you always wanted to know about Ancient Greek philosophy but didn’t know who to ask Edited by PATRICIA F. O’GRADY Flinders University of South Australia © Patricia F. O’Grady 2005 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the publisher. Patricia F. O’Grady has asserted her right under the Copyright, Designs and Patents Act, 1988, to be identi.ed as the editor of this work. Published by Ashgate Publishing Limited Ashgate Publishing Company Wey Court East Suite 420 Union Road 101 Cherry Street Farnham Burlington Surrey, GU9 7PT VT 05401-4405 England USA Ashgate website: http://www.ashgate.com British Library Cataloguing in Publication Data Meet the philosophers of ancient Greece: everything you always wanted to know about ancient Greek philosophy but didn’t know who to ask 1. Philosophy, Ancient 2. Philosophers – Greece 3. Greece – Intellectual life – To 146 B.C. I. O’Grady, Patricia F. 180 Library of Congress Cataloging-in-Publication Data Meet the philosophers of ancient Greece: everything you always wanted to know about ancient Greek philosophy but didn’t know who to ask / Patricia F.
    [Show full text]
  • Geminus and the Isia
    CORE Metadata, citation and similar papers at core.ac.uk Provided by DSpace at New York University GEMINUS AND THE ISIA ALEXANDER JONES T HE torical Greek interest, scientific including a lost writer treatise onGeminus the foundations wrote of several works of his- mathematics and an extant book on astronomy known as the Isagoge ("Introduction to the Phenomena"). The Isagoge is important to us as a witness to a stage of Greek astronomy that was both less advanced and less homogeneous in method than Ptolemy's. Approximate knowledge of when its author lived would be useful in several respects, for exam- ple in tracing the diffusion of elements originating in Babylonian lunar theory and in Hipparchus' work. Recent scholarship frequently cites Neugebauer's dating of Geminus to about A.D. 50, which has largely superseded the dating to the first half of the first century B.C. that used to be widely accepted.' Both dates derive, oddly enough, from analysis of the same passage in the Isagoge. The purpose of this note is to eluci- date the chronological issues, and to present documentary evidence that decisively establishes the earlier dating. The limits established by ancient citations are not very narrow. Isa- goge 4 mentions Hipparchus as an authority concerning constellations, and though Geminus does not say so, the lengths of the astronomical seasons listed in Isagoge I are the values that Hipparchus had used in deriving a model for the sun's motion. These passages cannot have been written before the 140s B.C. Moreover, Alexander of Aphrodisias (In Arist.
    [Show full text]
  • 10 · Greek Cartography in the Early Roman World
    10 · Greek Cartography in the Early Roman World PREPARED BY THE EDITORS FROM MATERIALS SUPPLIED BY GERMAINE AUJAe The Roman republic offers a good case for continuing to treat the Greek contribution to mapping as a separate CONTINUITY AND CHANGE IN THEORETICAL strand in the history ofclassical cartography. While there CARTOGRAPHY: POLYBIUS, CRATES, was a considerable blending-and interdependence-of AND HIPPARCHUS Greek and Roman concepts and skills, the fundamental distinction between the often theoretical nature of the Greek contribution and the increasingly practical uses The extent to which a new generation of scholars in the for maps devised by the Romans forms a familiar but second century B.C. was familiar with the texts, maps, satisfactory division for their respective cartographic in­ and globes of the Hellenistic period is a clear pointer to fluences. Certainly the political expansion of Rome, an uninterrupted continuity of cartographic knowledge. whose domination was rapidly extending over the Med­ Such knowledge, relating to both terrestrial and celestial iterranean, did not lead to an eclipse of Greek influence. mapping, had been transmitted through a succession of It is true that after the death of Ptolemy III Euergetes in well-defined master-pupil relationships, and the pres­ 221 B.C. a decline in the cultural supremacy of Alex­ ervation of texts and three-dimensional models had been andria set in. Intellectual life moved to more energetic aided by the growth of libraries. Yet this evidence should centers such as Pergamum, Rhodes, and above all Rome, not be interpreted to suggest that the Greek contribution but this promoted the diffusion and development of to cartography in the early Roman world was merely a Greek knowledge about maps rather than its extinction.
    [Show full text]
  • Ptolemy's Longitudes and Eratosthenes' Measurement of The
    NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIA S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI LEONARDO DA VINCI vol. 1 no. 1 2013 Mathematics and Mechanics of Complex Systems LUCIO RUSSO PTOLEMY’S LONGITUDES AND ERATOSTHENES’ MEASUREMENT OF THE EARTH’S CIRCUMFERENCE msp MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Vol. 1, No. 1, 2013 dx.doi.org/10.2140/memocs.2013.1.67 ∩ MM PTOLEMY’S LONGITUDES AND ERATOSTHENES’ MEASUREMENT OF THE EARTH’S CIRCUMFERENCE LUCIO RUSSO A statistical analysis of the longitudes reported in Ptolemy’s Geographia shows that many of them were obtained by distorting in a linear way data which were known with good accuracy. As a consequence, a new estimate of the value of the stadion used by Eratosthenes is obtained, supporting the thesis that his measurement of the Earth’s circumference was remarkably accurate. Some con- jectures about possible simplifications introduced by Cleomedes in his account of Eratosthenes’ method are also proposed. 1. The distortion of longitudes in Ptolemy’s Geographia The longitudes of 6345 localities1 reported by Ptolemy in his Geographia [Stückel- berger and Graßhoff 2006] are affected by an error which dilates their differences. While this error has been often remarked, it has not been so far analyzed in a quantitative way. The analysis of the distortion of the longitudes for all 6345 localities considered by Ptolemy is inconvenient for several reasons. First, many of the places are not identifiable with reasonable certainty. Furthermore for some regions the systematic error overlaps errors of different nature, due to the lack of knowledge of the country (this is the case, for example, for Indian localities).
    [Show full text]
  • Conclusion Full Article Language: En Indien Anders: Engelse Articletitle: 0
    _full_alt_author_running_head (neem stramien B2 voor dit chapter en dubbelklik nul hierna en zet 2 auteursnamen neer op die plek met and): 0 _full_articletitle_deel (kopregel rechts, vul hierna in): Conclusion _full_article_language: en indien anders: engelse articletitle: 0 496 Tihon Conclusion Anne Tihon When I agreed to write a conclusion to this Companion to Byzantine Science, I took on a perilous task. Each chapter of this book indeed contains an impres- sive amount of erudition, and each topic treated here would deserve in itself one or more separate volumes. However, reading the different chapters, a first remark comes to mind: it is time to stop apologizing for using the word “sci- ence” when studying Byzantine civilisation. The word “science”̕ may cover many different activities: study, teaching, creating manuals, commenting on scientific treatises, applying ancient or modern methods, conducting research, experiments and so on. There is no doubt that all the activities described in this volume fall into at least one of these categories. In spite of some received ideas, the Byzantine world was open to all kinds of intellectual activity, and especially to the scientific legacy from antiquity. This consists of a considerable number of writings which laid the foundation of science in the modern meaning of the word: Euclid, Apollonius, Archimedes, Diophantus, Ptolemy, Hippocrates, Galen, Dioscorides, including many more elementary treatises such as Autolycus, Theodosius, Geminus, Cleomedes and others. Everybody agrees that the Byzantines preserved this legacy; but they did much more than simply preserve it: they kept it alive. Some Byzantine scholars undertook considerable work in understanding and practising, for example, the astronomical exercises and calculations of Ptolemy.
    [Show full text]
  • 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.
    [Show full text]
  • The Adaptation of Babylonian Methods in Greek Numerical Astronomy
    FIgure 1. A Babylonian tablet (B.M. 37236) listing undated phases of Mars according to the System A scheme. By permission of the Trustees of the British Museum. This content downloaded from 128.122.149.96 on Thu, 04 Jul 2019 12:50:19 UTC All use subject to https://about.jstor.org/terms The Adaptation of Babylonian Methods in Greek Numerical Astronomy By Alexander Jones* THE DISTINCTION CUSTOMARILY MADE between the two chief astro- nomical traditions of antiquity is that Greek astronomy was geometrical, whereas Babylonian astronomy was arithmetical. That is to say, the Babylonian astronomers of the last five centuries B.C. devised elaborate combinations of arithmetical sequences to predict the apparent motions of the heavenly bodies, while the Greeks persistently tried to explain the same phenomena by hypothe- sizing kinematic models compounded out of circular motions. This description is substantially correct so far as it goes, but it conceals a great difference on the Greek side between the methods of, say, Eudoxus in the fourth century B.C. and those of Ptolemy in the second century of our era. Both tried to account for the observed behavior of the stars, sun, moon, and planets by means of combinations of circular motions. But Eudoxus seems to have studied the properties of his models purely through the resources of geometry. The only numerical parameters associated with his concentric spheres in our ancient sources are crude periods of synodic and longitudinal revolution, that is to say, data imposed on the models rather than deduced from them.1 By contrast, Ptolemy's approach in the Alma- 2 gest is thoroughly numerical.
    [Show full text]
  • ELECTRYONE ΗΛΕΚΤΡΥΩΝΗ Theognis and the Social Role of Measure
    ELECTRYONE ΗΛΕΚΤΡΥΩΝΗ Theognis and the Social Role of Measure. Matus Porubjak, University of Ss. Cyril and Methodius in Trnava ABSTRACT: The paper deals with the beginnings of the Greek ethical discourse in elegies of an archaic lyric poet Theognis of Megara. In the introduction author shortly discusses the question of origin and influence of the Theognidea. Then he gradually follows and interprets the occurrences of the expression “μηδὲν ἄγαν” (Nothing in excess), of the words σωφροσύνη (soundness of mind), σάοφρων (temperate), μέτρον (measure) and μέτριος (moderate), according to the problem of ἀρετή (excellence). On the basis of symposium description in the Theognidea and its functions, author shows the crucial social dimension of researched capabilities. He also tries to show how Theognis grasps the wisdom as a quality that has to be tried for and cared for. In the end author states that for Theognis the excellences such as wisdom, justice, sound mind and proper measure are conditio sine qua non for functioning of all social relations – from erotic relations through symposium to polis itself. KEY-WORDS: Theognis, Solon, temperate, measure, moderate, excellence, justice, wisdom, polis, social behaviour According to the interpretative tradition Socrates followed two Delphic maxims: “γνῶθι σαυτὸν” (Know thyself) and “μηδὲν ἄγαν”(Nothing in excess). These two maxims were so determinative for discourse of ancient philosophy and ethics (including Hellenic) that the whole ancient ethics could be in a simplified way understood as an effort to their reinterpretation and practical application. The aim of this study is to notice some aspects of beginnings of this discourse in elegies of an archaic lyric poet Theognis of Megara, in particular of the maxim “Nothing in excess”.
    [Show full text]
  • The Fragments of the Poem of Parmenides
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by D-Scholarship@Pitt RESTORING PARMENIDES’ POEM: ESSAYS TOWARD A NEW ARRANGEMENT OF THE FRAGMENTS BASED ON A REASSESSMENT OF THE ORIGINAL SOURCES by Christopher John Kurfess B.A., St. John’s College, 1995 M.A., St. John’s College, 1996 M.A., University of Hawai‘i at Mānoa, 2000 Submitted to the Graduate Faculty of The Dietrich School of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2012 UNVERSITY OF PITTSBURGH The Dietrich School of Arts and Sciences This dissertation was presented by Christopher J. Kurfess It was defended on November 8, 2012 and approved by Dr. Andrew M. Miller, Professor, Department of Classics Dr. John Poulakos, Associate Professor, Department of Communication Dr. Mae J. Smethurst, Professor, Department of Classics Dissertation Supervisor: Dr. Edwin D. Floyd, Professor, Department of Classics ii Copyright © by Christopher J. Kurfess 2012 iii RESTORING PARMENIDES’ POEM Christopher J. Kurfess, Ph.D. University of Pittsburgh, 2012 The history of philosophy proper, claimed Hegel, began with the poem of the Presocratic Greek philosopher Parmenides. Today, that poem is extant only in fragmentary form, the various fragments surviving as quotations, translations or paraphrases in the works of better-preserved authors of antiquity. These range from Plato, writing within a century after Parmenides’ death, to the sixth-century C.E. commentator Simplicius of Cilicia, the latest figure known to have had access to the complete poem. Since the Renaissance, students of Parmenides have relied on collections of fragments compiled by classical scholars, and since the turn of the twentieth century, Hermann Diels’ Die Fragmente der Vorsokratiker, through a number of editions, has remained the standard collection for Presocratic material generally and for the arrangement of Parmenides’ fragments in particular.
    [Show full text]