DOCSLIB.ORG

History of Mathematics Week 8 Notes Chapter 14: Europe in the Middle Ages • Author Chooses to Mark End of Medieval Period in T

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
History of Mathematics Week 8 Notes Chapter 14: Europe in the Middle Ages • Author Chooses to Mark End of Medieval Period in T History of Mathematics Week 8 Notes Chapter 14: Europe in the Middle Ages • Author chooses to mark end of medieval period in the year 1436, which is probably when al-Kashi (Arabian mathematician who calculated π to 14 decimal places) died and Regiomontanus (Johann M¨uller) was born. • Alcuin of York was hired by Charlemagne to revitalize education in France. He explained that creation took 6 days because 6 is a perfect num- ber. • Early Christian attitudes to scholarship were that it was unnecessary because Jesus had told them everything they needed. All scientific knowl- edge, then, was held by Moors, Arabs, and Jews. • Europeans begin a massive program of translating classics, including Elements, Almagest, and works of al-Khwarizmi. • The word \sine" comes from a mistranslation of \jiba" as \jaib". • Fibonacci's Liber Abaci (13th century) is about methods of computa- tion. It uses Hindu-Arabic numerals, but avoids decimal fractions in favor of unit and sexigesimal fractions. • The Fibonacci sequence first appears in Liber abaci 3 2 • Fibonacci studied the cubic equation x +p 2x + 10x = 2 and proved no rational solutions or solutions of the form a + b. He found an approximate solution, presumably using Horner's method. • Jordanus Nemorarius (likely of the century) was the first to give the law of the inclined plane (F = W sin(θ)), so was an Aristotelian in his physics. Jordanus wrote Arithmetica, which was a common university text until 16th century. He used letters for numbers, and so could describe mathematics ab- stractly. He proved theorems such as factors of perfect numbers are deficient. He also is the first to state the quadratic formula in full generality. • Campanus and Jordanus trisected an angle • Thomas Bradwardine and Nicole Oresme develop theory of exponents, in particular what exponents can be used. Bradwardine showed that corre- lations need not be linear in physics (for example, inverse-square laws are possible), and Oresme develops ideasp leading toward any real number being a possible exponent (gets close to x 2). • Oresme developed the latitude of forms, which showed that an object accelerating linearly covers the same distance as an object with no accelera- tion travelling at the speed of the first half-way. 1 P1 n • Richard Suiseth showed that n=1 2n = 2. Oresme showed that the harmonic series diverges. • By the fifteenth century, England and France were devastated by the Black Death and the Hundred Years War, and so Italian, German, and Polish scholars ascended to prominence. Chapter 15: The Renaissance • Regiomontanus was the most influential mathematician of the 15th cen- tury. He gives tan(90◦) is infinite. His geometric methods were algebraic: in making geometric constructions, he would use algebra to determine certain lengths, for example. He planned to reform astronomy (perhaps championing a heliocentric system), but he died prematurely • Nicolas Chuquet wrote Triparty en la science des nombres in 1484, which remained a manuscript until the 19th century. Chuquet finally gave a notation that made exponents clear. He would write axb as :a:b and ax−b as :a:b:m. He also deals with isolated negative numbers 4x = −2, but generally does not accept 0 as a solution. • The Italian friar Luca Pagioli wrote Summa de arithmetica, geometrica, proportionali et proportionalita in 1487, which collected knowledge in arith- metic, algebra, and geometry. It was a compendium of general mathematical knowledge, lacking in deep substance and modern notation. However, some symbols had been introduced to make ideas clearer. • In the 16th century, Germans took the lead in algebra, to the extent that algebra became known as \the cossic art". They introduced the mod- ern + and - symbols (replacing p. and m.). They also freely acknowledged negative coefficients, although they were not sure what to do with negative or irrational solutions. • In the 1540's, the solution of the cubic and quartic become generally known, marking the beginning of the modern period of mathematics. To solve x3 + px = q, let u and v be such that uv = p=3 and let x = u − v. Then one gets u3 − 3u2v + 3uv2 − v3 + p(u − v) = q u3 − 3(u − v)uv − v3 + p(u − v) = q u3 − v3 = q; which gives a quadratic in u3 (when v is eliminated). This led to solutions involving square roots of negatives that simplify to real solutions, which Car- dan did not understand. To solve quartic x4 + 6x2 + 36 = 60x: (1) add squares to make left-hand side a perfect square: x4 + 12x2 + 36 = 60x + 6x2 (2) add a variable y so that LHS remains square: (x2 + 6 + y)2 = 60x + 6x2 + y2 + 2x2y + 12y (3) choose y so that RHS is a perfect square 2 (4) step 3 results in solving a cubic (5) substitute back • Algebra was king during the 16th century, and advances in geometry were at the hands of artists like Werner and D¨urerin the theory of perspective • The Melancholia of D¨urerwas dated (1514) using a 4 × 4 magic square 3.
Load more

Recommended publications
  • E D I T O R I a L
    E D I T O R I A L “RETRO-PROGRESSING” As one browses through the history section of a library, one of the volumes that is likely to catch one’s attention is The Discoverers by Daniel Boorstin.1 It is an impressive, 700-page volume. Published in 1983, it chronicles in a semi-popular style selected aspects of man’s discoveries. Two chapters entitled “The Prison of Christian Dogma” and “A Flat Earth Returns” deal with the outlandish con- cept of an earth that is flat instead of spherical. Boorstin has impressive academic credentials from Harvard and Yale, and has held prestigious positions such as the Librarian of Congress, Director of the National Museum of History and Tech- nology, and Senior Historian of the Smithsonian Institution. In The Discoverers he reflects the popular view that the ancient Greeks, including Aristotle and Plato, believed Earth to be a sphere; however, after the rise of Christianity a period of “scholarly amnesia” set in which lasted from around 300 to 1300 A.D. During this time, according to Boorstin, “Christian faith and dogma replaced the useful [spherical] image of the world that had been so slowly, so painfully, and so scrupulously drawn by ancient geographers.” The “spherical” view was replaced by the concept of a flat earth, which Boorstin characterizes as “pious caricatures.”2 Boorstin bolsters his case by mentioning a couple of minor writers — Lactantius and Cosmas — who believed in a flat earth and lived during this “dark age.” He also implicates the powerful authority of St. Augustine of Hippo (354-430), who “heartily agreed” that antipodes “could not exist.”3 Antipodes represented lands on the opposite side of a spherical earth where trees and men, if present, would be upside down, with their feet above their heads; hence, the “antipodes” (opposite-feet) designation.
    [Show full text]
  • 7. Mathematical Revival in Western Europe
    7. Mathematical revival in Western Europe (Burton , 6.2 – 6.4, 7.1 ) Mathematical studies and discoveries during the early Dark Ages in Europe were extremely limited . One illustration of this fact is the chronology of the highly esteemed English scholar Bede (The Venerable Bede , 672 /673 – 735) , which set the convention that the year 1 B.C.E. immediately precedes the year 1 A.D. ; this remains unchanged even though 0 is now generally accepted to be part of the number system . However , there were contributors to the subject during the period from the Latin commentator Boëthius (Anicius Manlius Severinus Boëthius , c. 475 – 524) shortly after the fall of the Western Roman Empire and continuing through to the end of the 12 th century ; Boëthius has been described as the last of the Romans and the first of the scholastic philosophers (predating the latter by many centuries) , and his mathematical texts of were still widely used 600 years after they were written . Several other names from this period are mentioned in Sections 5.4 and 6.1 of Burton , and the latter ’s exercises also mention Alcuin of York (735 – 804) and Gerbert d ’Aurillac (940 – 1003) , who later became Pope Sylvester I I (997 – 1003) . During the second half of the 11 th century some important political developments helped raise European ’s consciousness of ancient Greek mathematical work and the more recent advances by Indian and Arabic mathematicians. Many of these involved Christian conquests of territory that had been in Muslim hands for long periods of time . Specific examples of particular importance for mathematics were the Norman conquest of Sicily in 1072 , the Spanish reconquista during which extensive and important territories in the Iberian Peninsula changed from Muslim to Christian hands , and the start of the Crusades in 1095 .
    [Show full text]
  • Doing Hard Time: Is God the Prisoner of the Oldest Dimension?
    Doing Hard Time: Is God the Prisoner of the Oldest Dimension? R.T. Mullins University of Notre Dame Abstract: In this paper I shall consider an objection to divine temporality called “The Prisoner of Time” objection. I shall begin by distinguishing divine timelessness from divine temporality in order to clear up common misunderstandings and caricatures of divine temporality. From there I shall examine the prisoner of time objection and explain why the prisoner of time objection fails to be a problem for the Christian divine temporalist. During the 20th Century, various Christian philosophers and theologians have rejected divine timelessness. This is not an entirely new phenomena; one can find philosophers and theologians from the Reformation on rejecting divine timelessness. Thinkers like Pierre Gassendi, Isaac Newton, Samuel Clarke, and John Tillotson all rejected the doctrine of divine timelessness long ago for philosophical, theological, scientific, and biblical reasons.1 For these thinkers, time is a necessary concomitant of God’s eternal nature. Despite the efforts of these thinkers, the doctrine of divine timelessness continued to play a central role in Christian theology until the 20th Century. During the 20th Century, and the beginning of this century, one can find a more wide spread rejection of divine timelessness amongst Christian philosophers and systematic theologians. Some of these rejections have been quite rigorous, while others have been misguided. The more recent widespread rejection of the atemporal God has led many into the loving arms of the temporal God. However, divine temporality is not without its critics. Divine atemporality refuses to lay down and die.
    [Show full text]
  • Semiotic Reflections on Medieval and Contemporary Graphic Representations of Motion
    Working paper, presented at the History and Pedagogy of Mathematics Conference, 14-18 July 2008, Mexico City. SEMIOTIC REFLECTIONS ON MEDIEVAL AND CONTEMPORARY GRAPHIC REPRESENTATIONS OF MOTION Luis Radford Université Laurentienne Ontario, Canada Introduction In this paper I will present an ongoing semiotic investigation of motion and some 14th century attempts at representing it in geometrical terms. The semiotic approach that I will follow focuses on the manner in which time, space and velocity were signified in certain historical periods. I take as my starting point the idea that the conceptualisation of motion and its key conceptual elements—speed, time and velocity— can only be understood within the scope of the epistemological configurations that make motion thinkable in a certain way. In other words, any attempt at investigating conceptualizations of time, speed, and velocity needs to take into account the cultural mental structures that underpin them. These cultural mental structures, or épistèmes, to use Michel Foucault’s term, are not mere spiritual or intellectual forms evolving on their own. They arise, acquire shape and evolve under the evolution of social practices, as well as their material conditions, forms of production and social relations. Thus, in the perspective that I am advocating here, to ask questions about time, speed and velocity is also to ask questions about the cultural and social conditions that made these theoretical constructs possible. Time, Speed and Velocity in the Middle Ages It is very well known that the investigation of motion became a prominent field of research only in the early years of modernity. However, that does not mean that motion was not an object of reflection in previous historical periods.
    [Show full text]
  • Oelze (Humboldt University of Berlin)
    Summary of Talk Given at the Conference “Immateriality, Thinking, and the Self in the Long Middle Ages” University of Cambridge, Philosophy Faculty, 14 July, 2015 Material Souls, Material Thoughts? Some Medieval Views on Rational Operations in Non-Rational Animals Anselm Oelze (Humboldt University of Berlin) Introductory Remarks Medieval theories of animal cognition by and large start from the assumption that only human animals are endowed with intellect (intellectus) and reason (ratio). That is, ac- cording to the standard medieval view, dogs, cats, donkeys, and other nonhuman anim- als have souls, yet their souls are not immaterial intellectual souls but material sensory souls (including usually five external senses as well as a number of so-called internal senses, e.g., common sense, imagination, estimation, and memory). Consequently, they are incapable of engaging in rational operations such as concept formation, judging, reasoning, and thinking, for only the intellect has access to the realm of universals (or concepts) which are the building blocks of judgments, syllogisms, and thoughts. Nevertheless, on various occasions medieval philosophers do note that in many cases nonhuman animals seem to judge (iudicare), to reason (ratiocinari, sillogizari), or to think (cogitare), too, insofar as their behaviour seems to be brought about by rational processes. In the following, an overview will be given of what six (later) medieval au- thors, namely Thomas Aquinas, Gregory of Rimini, Roger Bacon, Albertus Magnus, John Buridan, and Nicole Oresme, say on how there can be rational processes in non-ra- tional animals, especially reasoning and thinking (note that ʻreasoningʼ and ʻthinkingʼ are not necessarily synonymous here; rather, reasoning is taken to be some type of thinking; for more on that see Section III).
    [Show full text]
  • The Earliest Translations of Aristotle's Politics and The
    ECKART SCHÜTRUMPF THE EARLIEST TRANSLATIONS OF ARISTOTle’S POLITICS AND THE CREATION OF POLITICAL TERMINOLOGY 8 MORPHOMATA LECTURES COLOGNE MORPHOMATA LECTURES COLOGNE 8 HERAUSGEGEBEN VON GÜNTER BLAMBERGER UND DIETRICH BOSCHUNG ECKART SCHÜTRUMPF THE EARLIEST TRANSLATIONS OF ARISTOTLe’s POLITICS AND THE CREATION OF POLITICAL TERMINOLOGY WILHELM FINK unter dem Förderkennzeichen 01UK0905. Die Verantwortung für den Inhalt der Veröffentlichung liegt bei den Autoren. Bibliografische Informationen der Deutschen Nationalbibliothek: Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen National­ biblio grafie; detaillierte Daten sind im Internet über www.dnb.d­nb.de abrufbar. Alle Rechte, auch die des auszugweisen Nachdrucks, der fotomechanischen Wieder gabe und der Übersetzung vorbehalten. Dies betrifft auch die Vervielfälti­ gung und Übertragung einzelner Textabschnitte, Zeichnungen oder Bilder durch alle Verfahren wie Speicherung und Übertragung auf Papier, Transparente, Filme, Bänder, Platten und andere Medien, soweit es nicht § 53 und 54 UrhG ausdrücklich gestatten. © 2014 Wilhelm Fink, Paderborn Wilhelm Fink GmbH & Co. Verlags­KG, Jühenplatz 1, D­33098 Paderborn Internet: www.fink.de Lektorat: Sidonie Kellerer, Thierry Greub Gestaltung und Satz: Kathrin Roussel, Sichtvermerk Printed in Germany Herstellung: Ferdinand Schöningh GmbH & Co. KG, Paderborn ISBN 978-3-7705-5685-4 CONTENT 1. The earliest Latin translations of Aristotle— William of Moerbeke 9 2. Nicole Oresme 25 3. Leonardo Bruni’s principles of translation 28 4. Bruni’s translation of Aristotle’s Politics 33 5. The political terminology in Bruni’s translation— a new Humanist concept of res publica? 39 6. The controversy over Bruni’s translation— contemporary and modern 65 Appendix 77 Bibliography 78 This study goes back ultimately to a response I gave on two pa­ pers presented on “Translating Aristotle’s Politics in Medieval and Renaissance Europe” at the “International Conference on Translation.
    [Show full text]
  • REBIRTH, REFORM and RESILIENCE Universities in Transition 1300-1700
    REBIRTH, REFORM AND RESILIENCE Universities in Transition 1300-1700 Edited by James M. Kittelson and Pamela J. Transue $25.00 REBIRTH, REFORM, AND RESILIENCE Universities in Transition, 1300-1700 Edited by James M. Kittelson and Pamela]. Transue In his Introduction to this collection of original essays, Professor Kittelson notes that the university is one of the few institutions that medieval Latin Christendom contributed directly to modern Western civilization. An export wherever else it is found, it is unique to Western culture. All cultures, to be sure, have had their intellec­ tuals—those men and women whose task it has been to learn, to know, and to teach. But only in Latin Christendom were scholars—the company of masters and students—found gathered together into the universitas whose entire purpose was to develop and disseminate knowledge in a continu­ ous and systematic fashion with little regard for the consequences of their activities. The studies in this volume treat the history of the universities from the late Middle Ages through the Reformation; that is, from the time of their secure founding, through the period in which they were posed the challenges of humanism and con­ fessionalism, but before the explosion of knowl­ edge that marked the emergence of modern science and the advent of the Enlightenment. The essays and their authors are: "University and Society on the Threshold of Modern Times: The German Connection," by Heiko A. Ober­ man; "The Importance of the Reformation for the Universities: Culture and Confessions in the Criti­ cal Years," by Lewis W. Spitz; "Science and the Medieval University," by Edward Grant; "The Role of English Thought in the Transformation of University Education in the Late Middle Ages," by William J.
    [Show full text]
  • De Sphaera of Johannes De Sacrobosco in the Early Modern
    Matteo Valleriani Editor De sphaera of Johannes de Sacrobosco in the Early Modern Period The Authors of the Commentaries De sphaera of Johannes de Sacrobosco in the Early Modern Period Matteo Valleriani Editor De sphaera of Johannes de Sacrobosco in the Early Modern Period The Authors of the Commentaries Editor Matteo Valleriani Max Planck Institute for the History of Science Berlin, Germany Technische Universität Berlin Berlin, Germany University of Tel Aviv Tel Aviv, Israel ISBN 978-3-030-30832-2 ISBN 978-3-030-30833-9 (eBook) https://doi.org/10.1007/978-3-030-30833-9 © The Editor(s) (if applicable) and The Author(s) 2020. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc.
    [Show full text]
  • Averroes' Natural Philosophy and Its Reception in the Latin West Http
    Averroes’ Natural Philosophy and its Reception in the Latin West Edited by Paul J.J.M. Bakker With contributions from: Jean-Baptiste Brenet (Universite de Paris I – Pantheon-Sorbonne), Cristina Cerami (cnrs, umr 7219:sphere/chspam), Silvia Donati (Albertus-Magnus-Institut, Bonn), Dag Nikolaus Hasse (Julius-Maximilians-Universitat Wurzburg), Craig Martin (Oakland University), Edith Dudley Sylla (North Carolina State University), Cecilia Trifogli (All Souls College, Oxford) © 2015 by Leuven University Press / Presses Universitaires de Louvain / Universitaire Pers Leuven, Minderbroedersstraat 4, B-3000 Leuven (Belgium) ISBN 978 94 6270 046 8 D / 2015 / 1869 / 48 Distributed by Leuven University Press http://upers.kuleuven.be/nl/book/9789462700468 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License: http://creativecommons.org/licenses/by-nc-nd/3.0/ 2015167 [Bakker] 001-Prelims-print [date 1510191536 : version 1509110900] page -3 AVERROES’ NATURAL PHILOSOPHY AND ITS RECEPTION IN THE LATIN WEST Edited by Paul J.J.M. Bakker LEUVEN UNIVERSITY PRESS Reprint from Averroes’ Natural Philosophy and its Reception in the Latin West - ISBN 978 94 6270 046 8 - © Leuven University Press, 2015 2015167 [Bakker] 001-Prelims-print [date 1510191536 : version 1509110900] page -4 © 2015 by De Wulf-Mansioncentrum – De Wulf-Mansion Centre Leuven University Press / Presses Universitaires de Louvain / Universitaire Pers Leuven Minderbroedersstraat 4, B-3000 Leuven (Belgium) All rights reserved. Except in those cases expressly determined by law, no part of this publication may be multiplied, saved in an automated datafile or made public in any way whatsoever without the express prior written consent of the publishers. ISBN 978 94 6270 046 8 D/2015/1869/48 NUR: 732 Reprint from Averroes’ Natural Philosophy and its Reception in the Latin West - ISBN 978 94 6270 046 8 - © Leuven University Press, 2015 2015167 [Bakker] 001-Prelims-print [date 1510191536 : version 1509110900] page -5 ..
    [Show full text]
  • A Medieval Hebrew Treatise on Obstetrics
    Medical History, 1988, 33: 96-119. A MEDIEVAL HEBREW TREATISE ON OBSTETRICS by RON BARKAI* Recent years have shown an increasing interest in the history ofmedicine. This is no longer a distant, estranged branch ofstudy, reserved for a handful ofspecialists, but an important part of the framework encompassing the study of the past. An especially vital facet ofthis discipline is the history ofwomen's medicine, from which much can be learned concerning the ways in which our forebears viewed the physiological and mental differences between men and women, the formation of life within the womb, and the birth process itself.1 As in other areas ofmedieval science, researchers in this field must consider not only the ancient Greek and Latin sources, but the mutual influences and exchange of medical information among the Muslim, Jewish, and Christian cultures. The gynaecological literature written in Latin Europe during the Middle Ages has, to date, been given considerable coverage in academic studies. This refers mainly to the two most popular treatises of the Middle Ages and the Renaissance, both attributed to Trotula, a woman doctor and lecturer at the Salerno school ofmedicine: Trotula minor and Trotula major.2 The Muslim gynaecological literature has thus far not been studied nearly so widely as the Latin. Nevertheless, the most important treatise in this area, written in the tenth century by 'Arib ibn Sa'id of Cordova, has been published and translated into French * Dr Ron Barkai, Department of History, Tel Aviv University, Ramat-Aviv, 69 978 Tel Aviv, Israel. A striking example ofthis type ofstudy is Danielle Jacquart and Claude Thomasset, Sexualite et savoir medicale au moyen dge, Paris, Presses universitaires de France, 1985.
    [Show full text]
  • Time in Literature: a Study of Shakespeare and Dr. Iqbal
    www.ijcrt.org © 2018 IJCRT | Volume 6, Issue 1 February 2018 | ISSN: 2320-2882 Time in Literature: A study of Shakespeare and Dr. Iqbal Dr. Irshad Ahmad Itoo1 & Dr. Bashir Ahmad Gulzarpuri2 1 Faculty Department of English 2 Head Department of Urdu Govt Degree College Bijbehara Govt Degree College Bijbehara Jammu and Kashmir. Jammu and Kashmir. Abstract There are several scientific and philosophical concepts of the universe whose epistemology has not yet been defined successfully. Although, the stalwarts, philosophers, scientists, and anthropologists, who since times immemorial, have tried their best to unlock the knots that are embedded in these Metaphysical concepts like true nature of “Time”. The concept has also become a subject of discourse for writers of literature in its diverse fields like poetry and prose. The present study will concentrate on the poetic side and will analyze the verse of two great literary figures viz William Shakespeare and Sir Muhammad Iqbal, who like that of the scientists and anthropologists had made tireless effort to reveal the mystery of time concept and have pondered over the subject-matter in the literary field. Key words: - Absolute Time, Absolute Space, Dimensions of Time, Human Consciousness and Impact of Time. Introduction Time has been the subject of discourse from the ancient Greek civilisation and we find a threadbare converse on the theme relating to the metaphysical and transcendental world. The main questions put forward that have been discussed are: 1. Exactly what is time? 2. What is it made of? 3. Does time have a beginning and an end? 4. Is it a straight line or a circle? So far as the concept of time is concerned, the ancient Greek philosopher namely Plato identified time with the period of motion of the heavenly bodies (Weinert 9).
    [Show full text]
  • Boethius (480 – 524)
    The Saga of Mathematics A Brief History Boethius (480 – 524) Boethius became an orphan when he was Europe Smells the Coffee seven years old. He was extremely well educated. Boethius was a Chapter 6 philosopher, poet, mathematician, statesman, and (perhaps) martyr. Lewinter & Widulski The Saga of Mathematics 1 Lewinter & Widulski The Saga of Mathematics 2 Boethius (480 – 524) Boethius (480 – 524) He is best known as a translator of and This shows Boethius commentator on Greek writings on logic and calculating with mathematics (Plato, Aristotle, Nichomachus). Arabic numerals His mathematics texts were the best available competing with and were used for many centuries at a time Pythagoras using an when mathematical achievement in Europe abacus. was at a low point. It is from G. Reisch, Boethius’ Arithmetic taught medieval scholars Margarita about Pythagorean number theory. Philosophica (1508). Lewinter & Widulski The Saga of Mathematics 3 Lewinter & Widulski The Saga of Mathematics 4 Boethius (480 – 524) Boethius (480 – 524) Boethius was a main source of material for One of the first musical works to be printed the quadrivium, which was introduced into was Boethius's De institutione musica, written monasteries and consisted of arithmetic, in the early sixth century. geometry, astronomy, and the theory of It was for medieval authors, from around the music. ninth century on, the authoritative document on Greek music-theoretical thought and Boethius wrote about the relation of music systems. and science, suggesting that the pitch of a For example, Franchino Gaffurio in Theorica note one hears is related to the frequency of musica (1492) acknowledged Boethius as the sound.
    [Show full text]