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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. -
Poison King: the Life and Legend of Mithradates the Great, Rome's
Copyrighted Material Kill em All, and Let the Gods Sort em Out IN SPRING of 88 BC, in dozens of cities across Anatolia (Asia Minor, modern Turkey), sworn enemies of Rome joined a secret plot. On an appointed day in one month’s time, they vowed to kill every Roman man, woman, and child in their territories. e conspiracy was masterminded by King Mithradates the Great, who communicated secretly with numerous local leaders in Rome’s new Province of Asia. (“Asia” at this time referred to lands from the eastern Aegean to India; Rome’s Province of Asia encompassed western Turkey.) How Mithradates kept the plot secret remains one of the great intelli- gence mysteries of antiquity. e conspirators promised to round up and slay all the Romans and Italians living in their towns, including women and children and slaves of Italian descent. ey agreed to confiscate the Romans’ property and throw the bodies out to the dogs and crows. Any- one who tried to warn or protect Romans or bury their bodies was to be harshly punished. Slaves who spoke languages other than Latin would be spared, and those who joined in the killing of their masters would be rewarded. People who murdered Roman moneylenders would have their debts canceled. Bounties were offered to informers and killers of Romans in hiding.1 e deadly plot worked perfectly. According to several ancient histo- rians, at least 80,000—perhaps as many as 150,000—Roman and Italian residents of Anatolia and Aegean islands were massacred on that day. e figures are shocking—perhaps exaggerated—but not unrealistic. -
Champ Math Study Guide Indesign
Champions of Mathematics — Study Guide — Questions and Activities Page 1 Copyright © 2001 by Master Books, Inc. All rights reserved. This publication may be reproduced for educational purposes only. BY JOHN HUDSON TINER To get the most out of this book, the following is recommended: Each chapter has questions, discussion ideas, research topics, and suggestions for further reading to improve students’ reading, writing, and thinking skills. The study guide shows the relationship of events in Champions of Mathematics to other fields of learning. The book becomes a springboard for exploration in other fields. Students who enjoy literature, history, art, or other subjects will find interesting activities in their fields of interest. Parents will find that the questions and activities enhance their investments in the Champion books because children of different age levels can use them. The questions with answers are designed for younger readers. Questions are objective and depend solely on the text of the book itself. The questions are arranged in the same order as the content of each chapter. A student can enjoy the book and quickly check his or her understanding and comprehension by the challenge of answering the questions. The activities are designed to serve as supplemental material for older students. The activities require greater knowledge and research skills. An older student (or the same student three or four years later) can read the book and do the activities in depth. CHAPTER 1 QUESTIONS 1. A B C D — Pythagoras was born on an island in the (A. Aegean Sea B. Atlantic Ocean C. Caribbean Sea D. -
Water, Air and Fire at Work in Hero's Machines
Water, air and fire at work in Hero’s machines Amelia Carolina Sparavigna Dipartimento di Fisica, Politecnico di Torino Corso Duca degli Abruzzi 24, Torino, Italy Known as the Michanikos, Hero of Alexandria is considered the inventor of the world's first steam engine and of many other sophisticated devices. Here we discuss three of them as described in his book “Pneumatica”. These machines, working with water, air and fire, are clear examples of the deep knowledge of fluid dynamics reached by the Hellenistic scientists. Hero of Alexandria, known as the Mechanicos, lived during the first century in the Roman Egypt [1]. He was probably a Greek mathematician and engineer who resided in the city of Alexandria. We know his work from some of writings and designs that have been arrived nowadays in their Greek original or in Arabic translations. From his own writings, it is possible to gather that he knew the works of Archimedes and of Philo the Byzantian, who was a contemporary of Ctesibius [2]. It is almost certain that Heron taught at the Museum, a college for combined philosophy and literary studies and a religious place of cult of Muses, that included the famous Library. For this reason, Hero claimed himself a pupil of Ctesibius, who was probably the first head of the Museum of Alexandria. Most of Hero’s writings appear as lecture notes for courses in mathematics, mechanics, physics and pneumatics [2]. In optics, Hero formulated the Principle of the Shortest Path of Light, principle telling that if a ray of light propagates from a point to another one within the same medium, the followed path is the shortest possible. -
15 Famous Greek Mathematicians and Their Contributions 1. Euclid
15 Famous Greek Mathematicians and Their Contributions 1. Euclid He was also known as Euclid of Alexandria and referred as the father of geometry deduced the Euclidean geometry. The name has it all, which in Greek means “renowned, glorious”. He worked his entire life in the field of mathematics and made revolutionary contributions to geometry. 2. Pythagoras The famous ‘Pythagoras theorem’, yes the same one we have struggled through in our childhood during our challenging math classes. This genius achieved in his contributions in mathematics and become the father of the theorem of Pythagoras. Born is Samos, Greece and fled off to Egypt and maybe India. This great mathematician is most prominently known for, what else but, for his Pythagoras theorem. 3. Archimedes Archimedes is yet another great talent from the land of the Greek. He thrived for gaining knowledge in mathematical education and made various contributions. He is best known for antiquity and the invention of compound pulleys and screw pump. 4. Thales of Miletus He was the first individual to whom a mathematical discovery was attributed. He’s best known for his work in calculating the heights of pyramids and the distance of the ships from the shore using geometry. 5. Aristotle Aristotle had a diverse knowledge over various areas including mathematics, geology, physics, metaphysics, biology, medicine and psychology. He was a pupil of Plato therefore it’s not a surprise that he had a vast knowledge and made contributions towards Platonism. Tutored Alexander the Great and established a library which aided in the production of hundreds of books. -
Bridges Conference Paper
Bridges 2018 Conference Proceedings Landmarks in Algebra Quilt Elaine Krajenke Ellison Sarasota, Florida, USA; [email protected]; www.mathematicalquilts.com Abstract The Landmarks in Algebra quilt was an effort to include as many cultures that contributed significantly to the development of algebra as we know it today. Each section of the quilt illustrates a culture or a mathematician that made important advances in the development of algebra. Space did not allow more than four cultures, even though other nationalities made contributions. Babylonian mathematicians, Greek mathematicians, Indian mathematicians, and Persian mathematicians are highlighted for their important work. Figure 1: The Landmarks in Algebra quilt, 82 inches by 40 inches. First Panel: Plimpton 322 The first panel of the quilt illustrates the Plimpton 322 tablet. The Plimpton 322 tablet inspired me to design a quilt based on algebra. Recent historical research by Eleanor Robson, an Oriental Scholar at the University of Oxford, has shed new light on the ancient work of algebra [8]. The 4,000 year old Babylonian cuneiform tablet of Pythagorean Triples was purchased by George Arthur Plimpton in 1923 from Edgar J. Banks. Banks said the tablet came from a location near the ancient city of Larsa in Iraq [4]. Our first knowledge of mankind’s use of mathematics comes from the Egyptians and the Babylonians [1]. The Babylonian “texts” come to us in the form of clay tablets, usually about the size of a hand. The tablets were inscribed in cuneiform, a wedge-shaped writing owing its appearance to the stylus that was used to make it. Two types of mathematical tablets are generally found, table-texts and problem-texts [1]. -
An Engraved Carnelian Gem from Nysa-Scythopolis (Bet She’An)
‘Atiqot 71, 2012 An EngrAvEd CArnEliAn gEm from nysA-sCythopolis (BEt shE’An) gabriEl mAzor The western therma, located on a wide plateau Red carnelian. in the western part of the Roman civic center of Design engraved (intaglio), excellent work- Nysa-Scythopolis (Bet She’an), was concealed manship. by a thick, unstratified accumulation of earth Perfectly preserved, except for a minor chip on and debris. The fill, presumably washed down upper left edge. from the adjacent western hill, had gradually covered the floors of the complex after its Obverse: A nude male figure stands above a destruction in the earthquake of 749 CE. An thick horizontal ground line, facing right, left intact carnelian gemstone was revealed within leg bent forward while the right one is straight, the fill, along with an assortment of pottery and leaning slightly forward. Both hands are raised, coins from the Roman and Byzantine periods. gripping the neck of a lion in an attempt to The gem, originally set in a ring, depicts strangle it. The man’s head, supported by a Heracles in a scene of one of his labors: the thick neck, faces right; his facial features are overcoming of the Nemean lion. blurred, although he seems to be bearded with long, curly hair. The figure is heavy, extremely muscular, and depicted in a sturdy posture, dEsCription its legs forming a balanced triangle. Engaged L101161; B101 × 1407 (elevation -148.44 m) in a combat stance, a rearing, rampant lion Oval (12 × 9 mm, Th 2.5 mm; Fig. 1);1 upper attacks him, facing left. -
Clas109.04 Rebirth Demeter & Hades
CLAS109.04 REBIRTH M Maurizio ch.4.1 HISTORY—Homeric Hymn to Demeter before class: skim HISTORY context; refer to leading questions; focus on ancient texts Active Reading FOCUS • H.Hom.2 & Plut.Mor. cf. CR04 H.Hom.Cer. G. Nagy trans. (Maurizio p.163‐174 is fine) use CR04 Plutarch Moralia: Isis & Osiris 15‐16 (Plut.Mor.357A‐D) NB read for one hour, taking notes (fill in worksheets) RAW notes & post discussion question @11h00 W Maurizio ch.4.3 COMPARE—In the Desert by the Early Grass before class: skim COMPARE context; refer to leading questions; focus on ancient text Active Reading FOCUS • Early Grass (edin‐na u2 saĝ‐ĝa2‐ke4) use CR04 Jacobsen 1987 translation (Maurizio p.188‐194 is NOT fine) NB read for one hour, taking notes; finish previous as necessary RAW notes & post discussion question @12h00 F Maurizio ch.4.2 THEORY—Foley 1994 before class: skim also modern 4.4 RECEPTION paragraph synopsis of Foley, H. 1994. “Question of Origins.” Womens Studies 23.3:193‐215 NB read for one hour, practice summarizing; finish previous as necessary tl; dr & post discussion responses @11h00 Q04 • QUOTE QUIZ Gen.6‐11, Aesch.Prom., H.Hom.Cer., Plut.Mor.357A‐D; Early Grass FINAL notes @23h59 DRAFT 01 @23h59 • following guidelines DEMETER & HADES How does myth represent the natural world (e.g. pre‐scientific explanation)? How does myth represent religious ritual? How does myth represent social order? Homeric Hymn to Demeter1 G. Nagy 1 I begin to sing of Demeter, the holy goddess with the beautiful hair. -
The Divinity of Hellenistic Rulers
OriginalverCORE öffentlichung in: A. Erskine (ed.), A Companion to the Hellenistic World,Metadata, Oxford: Blackwell citation 2003, and similar papers at core.ac.uk ProvidedS. 431-445 by Propylaeum-DOK CHAPTKR TWENTY-FIVE The Divinity of Hellenistic Rulers Anßdos Chaniotis 1 Introduction: the Paradox of Mortal Divinity When King Demetrios Poliorketes returned to Athens from Kerkyra in 291, the Athenians welcomed him with a processional song, the text of which has long been recognized as one of the most interesting sources for Hellenistic ruler cult: How the greatest and dearest of the gods have come to the city! For the hour has brought together Demeter and Demetrios; she comes to celebrate the solemn mysteries of the Kore, while he is here füll of joy, as befits the god, fair and laughing. His appearance is majestic, his friends all around him and he in their midst, as though they were stars and he the sun. Hail son of the most powerful god Poseidon and Aphrodite. (Douris FGrH76 Fl3, cf. Demochares FGrH75 F2, both at Athen. 6.253b-f; trans. as Austin 35) Had only the first lines of this ritual song survived, the modern reader would notice the assimilaüon of the adventus of a mortal king with that of a divinity, the etymo- logical association of his name with that of Demeter, the parentage of mighty gods, and the external features of a divine ruler (joy, beauty, majesty). Very often scholars reach their conclusions about aspects of ancient mentality on the basis of a fragment; and very often - unavoidably - they conceive only a fragment of reality. -
Mechanical Miracles: Automata in Ancient Greek Religion
Mechanical Miracles: Automata in Ancient Greek Religion Tatiana Bur A thesis submitted in fulfillment of the requirements for the degree of Master of Philosophy Faculty of Arts, University of Sydney Supervisor: Professor Eric Csapo March, 2016 Statement of Originality This is to certify that to the best of my knowledge, the content of this thesis is my own work. This thesis has not been submitted for any degree or other purposes. I certify that the intellectual content of this thesis is the product of my own work and that all the assistance received in preparing this thesis and sources have been acknowledged. Tatiana Bur, March 2016. Table of Contents ACKNOWLEDGMENTS ....................................................................................................... 1 A NOTE TO THE READER ................................................................................................... 2 INTRODUCTION ................................................................................................................ 3 PART I: THINKING ABOUT AUTOMATION .......................................................................... 9 CHAPTER 1/ ELIMINATING THE BLOCAGE: ANCIENT AUTOMATA IN MODERN SCHOLARSHIP ................. 10 CHAPTER 2/ INVENTING AUTOMATION: AUTOMATA IN THE ANCIENT GREEK IMAGINATION ................. 24 PART II: AUTOMATA IN CONTEXT ................................................................................... 59 CHAPTER 3/ PROCESSIONAL AUTOMATA ................................................................................ -
Homer's Use of Myth Françoise Létoublon
Homer’s Use of Myth Françoise Létoublon Epic and Mythology The Homeric Epics are probably the oldest Greek literary texts that we have,1 and their subject is select episodes from the Trojan War. The Iliad deals with a short period in the tenth year of the war;2 the Odyssey is set in the period covered by Odysseus’ return from the war to his homeland of Ithaca, beginning with his departure from Calypso’s island after a 7-year stay. The Trojan War was actually the material for a large body of legend that formed a major part of Greek myth (see Introduction). But the narrative itself cannot be taken as a mythographic one, unlike the narrative of Hesiod (see ch. 1.3) - its purpose is not to narrate myth. Epic and myth may be closely linked, but they are not identical (see Introduction), and the distance between the two poses a particular difficulty for us as we try to negotiate the the mythological material that the narrative on the one hand tells and on the other hand only alludes to. Allusion will become a key term as we progress. The Trojan War, as a whole then, was the material dealt with in the collection of epics known as the ‘Epic Cycle’, but which the Iliad and Odyssey allude to. The Epic Cycle however does not survive except for a few fragments and short summaries by a late author, but it was an important source for classical tragedy, and for later epics that aimed to fill in the gaps left by Homer, whether in Greek - the Posthomerica of Quintus of Smyrna (maybe 3 c AD), and the Capture of Troy of Tryphiodoros (3 c AD) - or in Latin - Virgil’s Aeneid (1 c BC), or Ovid’s ‘Iliad’ in the Metamorphoses (1 c AD). -
Early Greek Alchemy, Patronage and Innovation in Late Antiquity CALIFORNIA CLASSICAL STUDIES
Early Greek Alchemy, Patronage and Innovation in Late Antiquity CALIFORNIA CLASSICAL STUDIES NUMBER 7 Editorial Board Chair: Donald Mastronarde Editorial Board: Alessandro Barchiesi, Todd Hickey, Emily Mackil, Richard Martin, Robert Morstein-Marx, J. Theodore Peña, Kim Shelton California Classical Studies publishes peer-reviewed long-form scholarship with online open access and print-on-demand availability. The primary aim of the series is to disseminate basic research (editing and analysis of primary materials both textual and physical), data-heavy re- search, and highly specialized research of the kind that is either hard to place with the leading publishers in Classics or extremely expensive for libraries and individuals when produced by a leading academic publisher. In addition to promoting archaeological publications, papyrolog- ical and epigraphic studies, technical textual studies, and the like, the series will also produce selected titles of a more general profile. The startup phase of this project (2013–2017) was supported by a grant from the Andrew W. Mellon Foundation. Also in the series: Number 1: Leslie Kurke, The Traffic in Praise: Pindar and the Poetics of Social Economy, 2013 Number 2: Edward Courtney, A Commentary on the Satires of Juvenal, 2013 Number 3: Mark Griffith, Greek Satyr Play: Five Studies, 2015 Number 4: Mirjam Kotwick, Alexander of Aphrodisias and the Text of Aristotle’s Meta- physics, 2016 Number 5: Joey Williams, The Archaeology of Roman Surveillance in the Central Alentejo, Portugal, 2017 Number 6: Donald J. Mastronarde, Preliminary Studies on the Scholia to Euripides, 2017 Early Greek Alchemy, Patronage and Innovation in Late Antiquity Olivier Dufault CALIFORNIA CLASSICAL STUDIES Berkeley, California © 2019 by Olivier Dufault.