Aristarchus and Copernicus
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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. -
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. -
Greece: Archimedes and Apollonius
Greece: Archimedes and Apollonius Chapter 4 Archimedes • “What we are told about Archimedes is a mix of a few hard facts and many legends. Hard facts – the primary sources –are the axioms of history. Unfortunately, a scarcity of fact creates a vacuum that legends happily fill, and eventually fact and legend blur into each other. The legends resemble a computer virus that leaps from book to book, but are harder, even impossible, to eradicate.” – Sherman Stein, Archimedes: What Did He Do Besides Cry Eureka?, p. 1. Archimedes • Facts: – Lived in Syracuse – Applied mathematics to practical problems as well as more theoretical problems – Died in 212 BCE at the hands of a Roman soldier during the attack on Syracuse by the forces of general Marcellus. Plutarch, in the first century A.D., gave three different stories told about the details of his death. Archimedes • From sources written much later: – Died at the age of 75, which would put his birth at about 287 BCE (from The Book of Histories by Tzetzes, 12th century CE). – The “Eureka” story came from the Roman architect Vitruvius, about a century after Archimedes’ death. – Plutarch claimed Archimedes requested that a cylinder enclosing a sphere be put on his gravestone. Cicero claims to have found that gravestrone in about 75 CE. Archimedes • From sources written much later: – From about a century after his death come tales of his prowess as a military engineer, creating catapults and grappling hooks connected to levers that lifted boats from the sea. – Another legend has it that he invented parabolic mirrors that set ships on fire. -
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. -
Ancient Greek Mathēmata from a Sociological Perspective
Ancient Greek Mathēmata from a Sociological Perspective: A Quantitative Analysis Leonid Zhmud, Institute for the History of Science and Technology, St. Petersburg Alexei Kouprianov, National Research University–Higher School of Economics, St. Petersburg Abstract: This essay examines the quantitative aspects of Greco-Roman science, represented by a group of established disciplines that since the fourth century B.C.E. had been called mathēmata or mathēmatikai epistēmai. Among the mathēmata, which in antiquity normally comprised mathematics, mathematical astronomy, harmonics, mechanics, and optics, the essay also includes geography. Using a data set based on The Encyclopaedia of Ancient Natural Scientists, it considers a com- munity of mathēmatikoi (as they called themselves), or ancient scientists (as they are defined for the purposes of this essay), from a sociological point of view, fo- cusing on the size of the scientific population known to us and its disciplinary, temporal, and geographical distribution. A diachronic comparison of neighboring and partly overlapping communities—ancient scientists and philosophers—allows the pattern of their interrelationship to be traced. An examination of centers of sci- ence throughout ancient history reveals that there were five major sites—Athens, Al- exandria, Rhodes, Rome, and Byzantium/Constantinople—that appeared, in suc- cession, as leaders. These conclusions serve to reopen the issue of the place of mathēmata and mathēmatikoi in ancient society. he historiography of ancient Greek science is nearly as old as its subject. The earliest T known writings on the history of mathematics and astronomy belong to Eudemus of Rhodes, a pupil of Aristotle. When, after a long period of decline and oblivion in medieval Europe, the sciences were revived, it was ancient Greek science that became the primary sub- ject of Renaissance and early modern studies in the history of science. -
Alan Hirshfeld, Eureka Man: the Life and Legacy of Archimedes Walker & Company, New York, 2009, ISBN: 978-0-8027-1618-7, 242 Pp, Price: U.S
Sci & Educ (2011) 20:83–87 DOI 10.1007/s11191-010-9244-0 BOOK REVIEW Alan Hirshfeld, Eureka Man: The Life and Legacy of Archimedes Walker & Company, New York, 2009, ISBN: 978-0-8027-1618-7, 242 pp, price: U.S. $26.00 A. K. T. Assis Published online: 19 March 2010 Ó Springer Science+Business Media B.V. 2010 Alan Hirshfeld is a professor of physics at the University of Massachusetts Dartmouth. He conducts research in the history of physics and astronomy. He is an Associate of the Harvard College Observatory and is the director of the UMass Dartmouth Observatory. He obtained his B.A. in astrophysics from Princeton University in 1973, his M.S. in astronomy from Yale in 1975, and his Ph.D. in astronomy also from Yale in 1978. Hirshfeld previously published Parallax: The Race to Measure the Cosmos, (Henry Holt & Co., 2002); The Electric Life of Michael Faraday, (Walker & Co., 2006); and the Astronomy Activity and Laboratory Manual, (Jones & Bartlett Publishers, 2008). He is also the lead author of two reference books in astronomy: Sky Catalogue 2000.0, Vol. 1: Stars to Magnitude 8.0 (Cambridge U.P. and Sky Publishing Corp., 1982 and 1991); and Sky Catalogue 2000.0, Vol. 2: Double Stars, Variable Stars and Nonstellar Objects (Cam- bridge U.P. and Sky Publishing Corp., 1985). Apart from books, he has written articles on the history of astrophysics, on ancient Greek measurements of the solar system, and biographies of astronomers like James Bradley. This book is divided into two parts: Master of Thought (six Chapters), and A Palimpsest’s Tale (eight Chapters).