Greece: Archimedes and Apollonius
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Greeks Doing Algebra
Greeks Doing Algebra There is a long-standing consensus in the history of mathematics that geometry came from ancient Greece, algebra came from medieval Persia, and the two sciences did not meet until seventeenth-century France (e.g. Bell 1945). Scholars agree that the Greek mathematicians had no methods comparable to algebra before Diophantus (3rd c. CE) or, many hold, even after him (e.g. Szabó 1969, Unguru and Rowe 1981, Grattan-Guinness 1996, Vitrac 2005. For a survey of arguments see Blåsjö 2016). The problems that we would solve with algebra, the Greeks, especially the authors of the canonical and most often studied works (such as Euclid and Apollonius of Perga), approached with spatial geometry or not at all. This paper argues, however, that the methods which uniquely characterize algebra, such as information compression, quantitative abstraction, and the use of unknowns, do in fact feature in Greek mathematical works prior to Diophantus. We simply have to look beyond the looming figures of Hellenistic geometry. In this paper, we shall examine three instructive cases of algebraic problem-solving methods in Greek mathematical works before Diophantus: The Sand-reckoner of Archimedes, the Metrica of Hero of Alexandria, and the Almagest of Ptolemy. In the Sand-reckoner, Archimedes develops a system for expressing extremely large numbers, in which the base unit is a myriad myriad. His process is indefinitely repeatable, and theoretically scalable to express a number of any size. Simple though it sounds to us, this bit of information compression, by which a cumbersome quantity is set to one in order to simplify notation and computation, is a common feature of modern mathematics but was almost alien to the Greeks. -
Great Inventors of the Ancient World Preliminary Syllabus & Course Outline
CLA 46 Dr. Patrick Hunt Spring Quarter 2014 Stanford Continuing Studies http://www.patrickhunt.net Great Inventors Of the Ancient World Preliminary Syllabus & Course Outline A Note from the Instructor: Homo faber is a Latin description of humans as makers. Human technology has been a long process of adapting to circumstances with ingenuity, and while there has been gradual progress, sometimes technology takes a downturn when literacy and numeracy are lost over time or when humans forget how to maintain or make things work due to cataclysmic change. Reconstructing ancient technology is at times a reminder that progress is not always guaranteed, as when Classical civilization crumbled in the West, but the history of technology is a fascinating one. Global revolutions in technology occur in cycles, often when necessity pushes great minds to innovate or adapt existing tools, as happened when humans first started using stone tools and gradually improved them, often incrementally, over tens of thousands of years. In this third course examining the greats of the ancient world, we take a close look at inventions and their inventors (some of whom might be more legendary than actually known), such as vizier Imhotep of early dynastic Egypt, who is said to have built the first pyramid, and King Gudea of Lagash, who is credited with developing the Mesopotamian irrigation canals. Other somewhat better-known figures are Glaucus of Chios, a metallurgist sculptor who possibly invented welding; pioneering astronomer Aristarchus of Samos; engineering genius Archimedes of Siracusa; Hipparchus of Rhodes, who made celestial globes depicting the stars; Ctesibius of Alexandria, who invented hydraulic water organs; and Hero of Alexandria, who made steam engines. -
A Centennial Celebration of Two Great Scholars: Heiberg's
A Centennial Celebration of Two Great Scholars: Heiberg’s Translation of the Lost Palimpsest of Archimedes—1907 Heath’s Publication on Euclid’s Elements—1908 Shirley B. Gray he 1998 auction of the “lost” palimp- tains four illuminated sest of Archimedes, followed by col- plates, presumably of laborative work centered at the Walters Matthew, Mark, Luke, Art Museum, the palimpsest’s newest and John. caretaker, remind Notices readers of Heiberg was emi- Tthe herculean contributions of two great classical nently qualified for scholars. Working one century ago, Johan Ludvig support from a foun- Heiberg and Sir Thomas Little Heath were busily dation. His stature as a engaged in virtually “running the table” of great scholar in the interna- mathematics bequeathed from antiquity. Only tional community was World War I and a depleted supply of manuscripts such that the University forced them to take a break. In 2008 we as math- of Oxford had awarded ematicians should honor their watershed efforts to him an honorary doc- make the cornerstones of our discipline available Johan Ludvig Heiberg. torate of literature in Photo courtesy of to even mathematically challenged readers. The Danish Royal Society. 1904. His background in languages and his pub- Heiberg lications were impressive. His first language was In 1906 the Carlsberg Foundation awarded 300 Danish but he frequently published in German. kroner to Johan Ludvig Heiberg (1854–1928), a He had publications in Latin as well as Arabic. But classical philologist at the University of Copenha- his true passion was classical Greek. In his first gen, to journey to Constantinople (present day Is- position as a schoolmaster and principal, Heiberg tanbul) to investigate a palimpsest that previously insisted that his students learn Greek and Greek had been in the library of the Metochion, i.e., the mathematics—in Greek. -
He One and the Dyad: the Foundations of Ancient Mathematics1 What Exists Instead of Ininite Space in Euclid’S Elements?
ARCHIWUM HISTORII FILOZOFII I MYŚLI SPOŁECZNEJ • ARCHIVE OF THE HISTORY OF PHILOSOPHY AND SOCIAL THOUGHT VOL. 59/2014 • ISSN 0066–6874 Zbigniew Król he One and the Dyad: the Foundations of Ancient Mathematics1 What Exists Instead of Ininite Space in Euclid’s Elements? ABSTRACT: his paper contains a new interpretation of Euclidean geometry. It is argued that ancient Euclidean geometry was created in a quite diferent intuitive model (or frame), without ininite space, ininite lines and surfaces. his ancient intuitive model of Euclidean geometry is reconstructed in connection with Plato’s unwritten doctrine. he model cre- ates a kind of “hermeneutical horizon” determining the explicit content and mathematical methods used. In the irst section of the paper, it is argued that there are no actually ininite concepts in Euclid’s Elements. In the second section, it is argued that ancient mathematics is based on Plato’s highest principles: the One and the Dyad and the role of agrapha dogmata is unveiled. KEYWORDS: Euclidean geometry • Euclid’s Elements • ancient mathematics • Plato’s unwrit- ten doctrine • philosophical hermeneutics • history of science and mathematics • philosophy of mathematics • philosophy of science he possibility to imagine all of the theorems from Euclid’s Elements in a Tquite diferent intuitive framework is interesting from the philosophical point of view. I can give one example showing how it was possible to lose the genuine image of Euclidean geometry. In many translations into Latin (Heiberg) and English (Heath) of the theorems from Book X, the term “area” (spatium) is used. For instance, the translations of X. 26 are as follows: ,,Spatium medium non excedit medium spatio rationali” and “A medial area does not exceed a medial area by a rational area”2. -
The Fascinating Story Behind Our Mathematics
The Fascinating Story Behind Our Mathematics Jimmie Lawson Louisiana State University Story of Mathematics – p. 1 Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. Math was needed in developing branches of knowledge: astronomy, timekeeping, calendars, construction, surveying, navigation Math was interesting: In many ancient cultures (Egypt, Mesopotamia, India, China) mathematics became an independent subject, practiced by scribes and others. Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Story of Mathematics – p. 2 Math was needed in developing branches of knowledge: astronomy, timekeeping, calendars, construction, surveying, navigation Math was interesting: In many ancient cultures (Egypt, Mesopotamia, India, China) mathematics became an independent subject, practiced by scribes and others. Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. Story of Mathematics – p. 2 Math was interesting: In many ancient cultures (Egypt, Mesopotamia, India, China) mathematics became an independent subject, practiced by scribes and others. Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. Math was needed in developing branches of knowledge: astronomy, timekeeping, calendars, construction, surveying, navigation Story of Mathematics – p. 2 Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. -
Archimedes Palimpsest a Brief History of the Palimpsest Tracing the Manuscript from Its Creation Until Its Reappearance Foundations...The Life of Archimedes
Archimedes Palimpsest A Brief History of the Palimpsest Tracing the manuscript from its creation until its reappearance Foundations...The Life of Archimedes Birth: About 287 BC in Syracuse, Sicily (At the time it was still an Independent Greek city-state) Death: 212 or 211 BC in Syracuse. His age is estimated to be between 75-76 at the time of death. Cause: Archimedes may have been killed by a Roman soldier who was unaware of who Archimedes was. This theory however, has no proof. However, the dates coincide with the time Syracuse was sacked by the Roman army. The Works of Archimedes Archimedes' Writings: • Balancing Planes • Quadrature of the Parabola • Sphere and Cylinder • Spiral Lines • Conoids and Spheroids • On Floating Bodies • Measurement of a Circle • The Sandreckoner • The Method of Mechanical Problems • The Stomachion The ABCs of Archimedes' work Archimedes' work is separated into three Codeces: Codex A: Codex B: • Balancing Planes • Balancing Planes • Quadrature of the Parabola • Quadrature of the Parabola • Sphere and Cylinder • On Floating Bodies • Spiral Lines Codex C: • Conoids and Spheroids • The Method of Mechanical • Measurement of a Circle Problems • The Sand-reckoner • Spiral Lines • The Stomachion • On Floating Bodies • Measurement of a Circle • Balancing Planes • Sphere and Cylinder The Reappearance of the Palimpsest Date: On Thursday, October 29, 1998 Location: Christie's Acution House, NY Selling price: $2.2 Million Research on Palimpsest was done by Walter's Art Museum in Baltimore, MD The Main Researchers Include: William Noel Mike Toth Reviel Netz Keith Knox Uwe Bergmann Codex A, B no more Codex A and B no longer exist. -
Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 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. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions. -
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. -
Post-Euclid Greek Mathematics
Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Some high points of Greek mathematics after Euclid Algebra Through History October 2019 Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Outline 1 Archimedes 2 Apollonius and the Conics 3 How Apollonius described and classified the conic sections Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Who was Archimedes? Lived ca. 287 - 212 BCE, mostly in Greek city of Syracuse in Sicily Studied many topics in what we would call mathematics, physics, engineering (less distinction between them at the time) We don’t know much about his actual life; much of his later reputation was based on somewhat dubious anecdotes, e.g. the “eureka moment,” inventions he was said to have produced to aid in defense of Syracuse during Roman siege in which he was killed, etc. Perhaps most telling: we do know he designed a tombstone for himself illustrating the discovery he wanted most to be remembered for (discussed by Plutarch, Cicero) Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Figure: Sphere inscribed in cylinder of equal radius 3Vsphere = 2Vcyl and Asphere = Acyl (lateral area) Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Surviving works On the Equilibrium of Planes (2 books) On Floating Bodies (2 books) Measurement of a Circle On Conoids and Spheroids On Spirals On the Sphere and Cylinder (2 books) Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Surviving works, cont. -
Notes on Greek Mathematics
MATHEMATICS IN ANCIENT GREECE Periods in Greek history [AG]. Archaic Period: 750 BC to 490 BC (Emergence of city- states to Battle of Marathon) The Iliad and the Odyssey were composed c. 750-720 (based on the `Trojan war’ of 1250-1225 BC.) Between 670-500 BC, many city-states were ruled by tyrants (a word borrowed into the Greek language from Asia Minor, to signify a man who seizes control of the state by a coup and governs illegally.) Classical Period: 490 to 323 BC (Battle of Marathon to death of Alexander). The years from 480 (Persians driven from Greece) to 430 saw the rise of Athenian democracy (under Pericles). The Parthenon was built, 447-432. Playwrights Sophocles, Aristophanes, Euripides were active in Athens (c. 430-400). Following undeclared war between Athens and Sparta from 460-445, the Peloponnesian war (431-404; account by Thucydides) kept these two states (and their allies throughout the region) busy. This led to the ascendance of Sparta and the `rule of thirty tyrants’ in Athens (404-403). Socrates was tried and executed in 399. His follower Plato wrote his dialogues (399-347) and founded the Academy. The accession of Philip II of Macedon (359) signals the beginning of a period of Macedonian rule. Philip invaded Asia in 336, and was assassinated the same year. His successor Alexander III (`the Great’, 356-323 BC) continued the program of world conquest over the next 13 years, invading India in 325 BC. Hellenistic Period (323-30 BC). Science, mathematics (Euclid) and culture flourished in Egypt (Alexandria) under the Ptolemaic dynasty. -
The Universe Mechanized
08/12/2016 1 08/12/2016 Pergamum Syracuse Rhodos Alexandria 2 08/12/2016 3 08/12/2016 Hipparchus Euclides Heron Apollonius of Perga Aristarchus Ptolemaeus Eratosthenes Archimedes 4 08/12/2016 Various astronomers made significant, even amazing, contributions. Noteworthy examples: ∑ Aristarchus of Samos ‐ Heliocentric Universe ‐ distance Moon & Sun ‐ size Sun ∑ Archimedes ‐ Planisphere/Planetarium ? ∑ Eratosthenes ‐ Diameter Earth ∑ Hipparchus ‐ multitude essential contributions Problematic is the loss of nearly all, except for a few, of the books and works they have written … 5 08/12/2016 “On the Sizes & Distances of the Sun and Moon ”: Only one work of Aristarchus survives: On the Sizes and the Distances of the Sun and Moon First mathematically based attempt to measure distance Earth‐Sun, thus First attempt to measure scale Universe Based upon geocentric view of Universe On the Sizes and the Distances Greek copy 10th century 6 08/12/2016 Aristarchus' geometric construction used to estimate the distance to the Sun. Earth (E) –Sun (S)‐Moon (M) triangle and sizes are not drawn to scale. Measure angle b: c = 90±‐b EM/ES = sin(c) Aristarchus: b = 87± real value: b = 89±50 ES = 19 EM real value: ES = 397 EM Numerically, very unstable procedure, reason for huge error. Nonetheless, On the Sizes and the Distances Greek copy 10th century Aristarchus' estimate of size Sun: angular diameter Sun ~ angular diameter Moon Dist. Earth‐Sun = 19 Dist. Earth‐Moon size Sun = 19 x size Moon size Sun > size Earth On the Sizes and the Distances Greek copy 10th century 7 08/12/2016 Archimedes, “the Sand Reckoner” (~200 BCE): You King Gelon are aware the ‘universe' is the name given by most astronomers to the sphere the center of which is the center of the Earth, while its radius is equal to the straight line between the center of the Sun and the center of the Earth. -
Ioannis M. VANDOULAKIS, « the Arbitrariness of the Sign in Greek Mathematics »
Ioannis M. VANDOULAKIS, « The Arbitrariness of the sign in Greek Mathematics » Communication donnée dans l’atelier de Jean-Yves Beziau, The Arbitrariness of the Sign, au colloque Le Cours de Linguistique Générale, 1916-2016. L’émergence, Genève, 9-13 janvier 2017. CERCLE FERDINAND DE SAUSSURE N° D’ISBN : 978-2-8399-2282-1 Pour consulter le programme complet de l’atelier de Jean-Yves Beziau, The Arbitrariness of the Sign : https://www.clg2016.org/geneve/programme/ateliers-libres/the- arbitrariness-of-the-sign/ 1 Ioannis M. Vandoulakis The Arbitrariness of the Sign in Greek Mathematics Ioannis M. Vandoulakis Abstract. In this paper, we will examine some modes of signification of mathematical entities used in Greek mathematical texts in the light of Saussure’s conceptualization of sign. In particular, we examine certain mathematical texts from Early Greek period, the Euclidean and Neo- Pythagorean traditions, and Diophantus. 0. Introduction Greek mathematicians have used a wide variety of modes of signification to express mathematical entities. By signification here, we understand the relation between the form of the sign (the signifier) and its meaning (the signified), as used in Ferdinand de Saussure’s semiology. According to Saussure, this relation is essentially arbitrary, motivated only by social convention. Moreover, for Saussure, signifier and signified are inseparable. One does not exist without the other, and conversely, one always implicates the other. Each one of them is the other’s condition of possibility. In this paper, we will examine some modes of signification of mathematical entities used in Greek mathematical texts in the light of Saussure’s conceptualization of sign.