Science Networks. Historical Studies Science Networks

Science Networks. Historical Studies Science Networks. Historical Studies Founded by Erwin Hiebert and Hans Wußing Vo l u m e 4 6

Edited by Eberhard Knobloch, Helge Kragh and Volker Remmert

Editorial Board:

K. Andersen, Amsterdam S. Hildebrandt, Bonn H.J.M. Bos, Amsterdam D. Kormos Buchwald, Pasadena U. Bottazzini, Roma Ch. Meinel, Regensburg J.Z. Buchwald, Pasadena J. Peiffer, Paris K. Chemla, Paris W. Purkert, Bonn S.S. Demidov, Moskva D. Rowe, Mainz M. Folkerts, München A.I. Sabra, Cambridge, Mass. P. Galison, Cambridge, Mass. Ch. Sasaki, Tokyo I. Grattan-Guinness, London R.H. Stuewer, Minneapolis J. Gray, Milton Keynes V.P. Vizgin, Moskva R. Halleux, Liége Tito M. Tonietti

And Yet It Is Heard

Musical, Multilingual and Multicultural History of the Mathematical Sciences — Volume 1 Tito M. Tonietti Dipartimento di matematica University of Pisa Pisa Italy

ISBN 978-3-0348-0671-8 ISBN 978-3-0348-0672-5 (eBook) DOI 10.1007/978-3-0348-0672-5 Springer Basel Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014935966

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Printed on acid-free paper

Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com) The history of the sciences is a grand fugue, in which the voices of various peoples chime in, each in their turn. It is as if an eternal harmony conversed with itself as it may have done in the bosom of God, before the creation of the world Wolfgang Goethe

Foreword

Musica nihil aliud est, quam omnium ordinem scire. Music is nothing but to know the order of all things. Trismegistus in Asclepius, cited by Athanasius Kircher, Musurgia universalis, Rome 1650, vol. II, title page Tito Tonietti has certainly written a very ambitious, extraordinary book in many respects. Its subtitle precisely describes his scientific aims and objectives. His goal here is to present a musical, multilingual, and multicultural history of the mathematical sciences, since ancient times up to the twentieth century. To the best of my knowledge, this is the first serious, comprehensive attempt to do justice to the essential role music played in the development of these sciences. This musical aspect is usually ignored or dramatically underestimated in descrip- tions of the evolution of sciences. Tonietti stresses this issue continually. He states his conviction at the very beginning of the book: Music was one of the primeval mathematical models for natural sciences in the West. “By means of music, it is easier to understand how many and what kinds of obstacles the Greek and Roman natural philosophers had created between mathematical sciences and the world of senses.” Yet, also in China, it is possible to narrate the mathematical sciences by means of music, as Tonietti demonstrates in Chap. 3. Even in India, certain ideas would seem to connect music with mathematics. Narrating history through music remains his principle and style when he speaks about the Arabic culture. Tonietti emphasizes throughout the role of languages and the existence of cultural differences and various scientific traditions, thus explicitly extending the famous Sapir-Whorf hypothesis to the mathematical sciences. He emphatically rejects Eurocentric prejudices and pleads for the acceptance of cultural variety. Every culture generates its own science so that there are independent inventions in different contexts. For him, even the texts of mathematicians acquire sense only if they are set in their context: “The Indian brahwana and the Greek philosophers developed their mathematical cultures in a relative autonomy, maintaining their own characteristics.”

vii viii Foreword

To mention another of Tonietti’s examples: The Greek and Latin scientific cultures, the Chinese scientific culture cannot be reduced to some general characteristics. Chinese books offered different proofs from those of Euclid. He draws a crucial conclusion: Such differences should not be transformed into inferiority or exclusion. For the Chinese, as well as for the Indians, the Pythagorean distinction between integers – or ratios between them – and other, especially irrational numbers does not seem to make sense. The Chinese mathematical theory of music was invented through solid pipes. Tonietti does not conceal another matter of fact: In his perspective of history, harmony is not only the daughter of Venus, but also of a father like Mars. For good reasons he dedicates a long chapter to Kepler’s world harmony, which indeed deserves more attention. He disagrees with the many modern historians of science who transformed Kepler’s diversity into inferiority “with the aggravating circumstances of those intolerable nationalistic veins from which we particularly desire to stay at a good distance.” Tonietti’s original approach enables him to gain many essential new insights: The true achievements of Aristoxenus, Vincenzo Galilei, Stevin (equable temperament), Lucretius’s contributions to the history of science overlooked up to now, the reasons the prohibition of irrational numbers was eclipsed during the seventeenth century, and the understanding of the reappearance of mathematics as the language essential to express the new science in this century, to mention some of them. Or, as he puts it: “The question has become rather how to interpret the musical language of the spheres and not whether it came from God.” Tonietti emphatically refuses corruptions, discriminations, distortions, simplifi- cations, anachronisms, nationalisms of authors, and cultures trying to show that “even the mathematical sciences are neither neutral nor universal nor eternal and depend on the historical and cultural contexts that invent them.” He places music in the foreground, he has not written a history of music with just hints to acoustic theories. In spite of all his efforts and the more than thousand pages of his book, Tonietti calls his attempt a modest proposal, a beginning. It is certainly a provocative book that is worth diligently studying and continuing even if not every modern scholar will accept all of its statements and conclusions.

Berlin, Germany Eberhard Knobloch February 2014 Contents for Volume I

1 Introduction ...... 1

Part I In the Ancient World

2 Above All with the Greek Alphabet...... 9 2.1 The Most Ancient of All the Quantitative Physical Laws ...... 9 2.2 ThePythagoreans...... 11 2.3 Plato...... 19 2.4 Euclid ...... 25 2.5 Aristoxenus ...... 35 2.6 ClaudiusPtolemaeus ...... 41 2.7 Archimedesanda FewOthers ...... 54 2.8 TheLatinLucretius...... 64 2.9 TextsandContexts...... 82 3 In Chinese Characters ...... 97 3.1 MusicinChina,Yuejing,Confucius...... 97 3.2 TuningReed-Pipes ...... 100 3.3 TheFigureoftheString...... 111 3.4 CalculatinginNineWays ...... 117 3.5 TheQi...... 124 3.6 Rules,RelationshipsandMovements...... 140 3.6.1 CharactersandLiteraryDiscourse ...... 141 3.6.2 A LivingOrganismonEarth...... 143 3.6.3 Rules,ModelsinMovementandValues...... 147 3.6.4 The Geometry of the Continuum in Language...... 151 3.7 Between Tao and Logos ...... 153 4 In the Sanskrit of the Sacred Indian Texts ...... 169 4.1 Roots in the Sacred Books ...... 169 4.2 RulesandProofs...... 172 4.3 NumbersandSymbols ...... 179

ix x Contents for Volume I

4.4 Looking Down from on High ...... 187 4.5 Dida MathematicalTheoryofMusicExistinIndia,orNot?...... 194 4.6 BetweenIndiansandArabs...... 203 5 Not Only in Arabic...... 209 5.1 BetweentheWestandtheEast ...... 209 5.2 TheTheoryofMusicinIbnSina ...... 210 5.3 OtherTheoriesofMusic ...... 219 5.4 Beyond the Greek Tradition...... 225 5.5 Did the Arabs Use Their Fractions and Roots fortheTheoryofMusic,orNot? ...... 232 5.6 An Experimental Model Between Mathematical Theory andPractice...... 238 5.7 SomeReasonsWhy ...... 244 6 With the Latin Alphabet, Above All ...... 257 6.1 ReliableProofsofTransmission...... 257 6.2 Theory and Practice of Music: Severinus Boethius and Guido D’Arezzo ...... 259 6.3 FacingtheIndiansandtheArabs:LeonardodaPisa ...... 271 6.4 Constructing, Drawing, Calculating: Leon Battista Alberti, Piero della Francesca, LucaPacioli,LeonardodaVinci...... 276 6.5 The Quadrivium Still Resisted: Francesco Maurolico, theJesuitsandGirolamoCardano...... 291 6.6 Variants of Pythagorean Orthodoxy: Gioseffo Zarlino, Giovan Battista Benedetti ...... 303 6.7 The Rebirth of Aristoxenus, or Vincenzio Galilei ...... 309

A [From the] Suanfa tongzong [Compendium of Rules for Calculating] by Cheng Dawei ...... 329

B Al-qawl ‘ala ajnas alladhi bi-al-arba‘a [Discussion on the Genera Contained in a Fourth] by Umar al-Khayyam...... 335

CMusica[Music] by Francesco Maurolico ...... 341 C.1 RulestoComposeConsonantMusic ...... 348 C.2 RuleofUniﬁcation ...... 353 C.3 RuleofTakingAway...... 354 C.4 TheCalculationofBoethiusfortheComparisonofIntervals...... 359 C.5 CommentontheCalculationofBoethius ...... 360 C.6 Guido’sIcosichord...... 362 C.7 MUSIC...... 374

D The Chinese Characters ...... 379

Bibliography ...... 395 Contents for Volume II

7 Introduction to Volume II ...... 1

Part II In the World of the Scientiﬁc Revolution

8 Not Only in Latin, but also in Dutch, Chinese, Italian and German ...... 5 8.1 AristoxenuswithNumbers,orSimonStevinandZhuZaiyu...... 5 8.2 Reaping What Has Been Sown. Galileo Galilei, theJesuitsandtheChinese ...... 18 8.3 JohannesKepler:TheImportanceofHarmony...... 37 9 Beyond Latin, French, English and German: The Invention of Symbolism...... 107 9.1 FromMarinMersennetoBlaisePascal...... 107 9.2 René Descartes, Isaac Beeckman and John Wallis ...... 158 9.3 Constantijn and Christiaan Huygens ...... 192 10 Between Latin, French, English and German: The Language of Transcendence...... 227 10.1 GottfriedWilhelmLeibniz...... 227 10.2 SirIsaacNewtonandMr.RobertHooke...... 265 10.3 SymbolismandTranscendence...... 291 11 Between Latin and French...... 327 11.1 Jean-Philippe Rameau, the Bernoullis and Leonhard Euler...... 327 11.2 Jean le Rond d’Alembert, Jean-Jacques Rousseau andDenisDiderot...... 368 11.3 Counting,SingingandListening:FromRameautoMozart ...... 412 12 From French to German ...... 431 12.1 From Music-Making to Acoustics: Luigi Giuseppe Lagrangee JosephJean-BaptisteFourier ...... 431

xi xii Contents for Volume II

12.2 Too Much Noise, from Harmony to Harmonics: BernhardRiemannandHermannvonHelmholtz ...... 439 12.3 LudwigBoltzmannandMaxPlanck...... 466 12.4 Arnold Schönberg and Albert Einstein...... 483

Part III It Is Not Even Heard

13 In the Language of the Venusians...... 511 13.1 Black Languages ...... 511 13.2 Stones, Pieces of String and Songs ...... 513 13.3 Dancing,SingingandNavigating ...... 514 14 Come on, Apophis ...... 527 14.1 Gottmituns ...... 527

Bibliography ...... 535

Index ...... 577 Chapters 3, 6 and 8–12 have been respectively obtained by re-elaborating, complet- ing or simplifying the following talks, articles and books: Paper presented at Hong Kong in 2001, “The Mathematics of Music During the 16th Century: The Cases of Francesco Maurolico, Simon Stevin, Cheng Dawei, Zhu Zaiyu”, Ziran kexueshi yanjiu [Studies in the History of Natural Sciences], (Beijing), 2003, 22, n. 3, 223–244. Le matematiche del Tao, Roma 2006, Aracne, pp. 266. “Tra armonia e conflitto: da Kepler a Kauffman”, in La matematizzazione della biologia, Urbino 1999, Quattro venti, 213–228. “Disegnare la natura (I modelli matematici di Piero, Leonardo da Vinci e Galileo Galilei, per tacer di Luca)”, Punti critici, 2004, n. 10/11, 73–102. “The Mathematical Contributions of Francesco Maurolico to the Theory of Music of the 16th Century (The Problems of a Manuscript)”, Centaurus, 48, (2006), 149–200. Paper presented at Naples in 1995, “Verso la matematica nelle scienze: armonia e matematica nei modelli del cosmo tra seicento e settecento”, in La costruzione dell’immagine scientifica del mondo, Marco Mamone Capria ed., Napoli 1999, La Città del Sole, 155–219. Paper presented at Perugia in 1996, “Newton, credeva nella musica delle sfere?”, in La scienza e i vortici del dubbio, Lino Conti and Marco Mamone Capria eds., Napoli 1999, Edizioni scientifiche italiane, 127–135. Also “Does Newton’s Musical Model of Gravitation Work?”, Centaurus, 42, (2000), 135–149. Paper presented at Arcidosso in 1999, “Is Music Relevant for the History of Science?”, in The Applications of Mathematics to the Sciences of Nature: Critical Moments and Aspects, P. Cerrai, P. Freguglia, C. Pellegrino (eds.), New York 2002, Kluwer, 281–291. “Albert Einstein and Arnold Schoenberg Correspondence”, NTM - Naturwis- senschaften Technik und Medizin, 5 (1997) H. 1, 1–22. Also Nuvole in silenzio (Arnold Schoenberg svelato) , Pisa 2004, Edizioni Plus, ch. 58.

xiii xiv

“Il pacifismo problematico di Albert Einstein”, in Armi ed intenzioni di guerra, Pisa 2005, Edizioni Plus, 287–309. Chapters 2, 4 and 5 are completely new. In the meantime, thanks to the help of Michele Barontini, a part of Chap. 5 has become “Umar al-Khayyam’s Contributions to the Arabic Mathematical Theory of Music”, Arabic Science and Philosophy v. 20 (2010), pp. 255–279. The problems of Chap. 4 produced, in collaboration with Giacomo Benedetti, “Sulle antiche teorie indiane della musica. Un problema a confronto con altre culture”, Rivista di studi sudasiatici, v. 4 (2010), pp. 75–109; also, “Toward a Cross-cultural History of Mathematics. Between the Chinese, and the Arabic Mathematical Theories of Music: the Puzzle of the Indian Case”, in History of the Mathematical Sciences II, eds. B.S. Yadav & S.L. Singh, Cambridge 2010, Cambridge Scientific Publishers, 185–203. In the meantime, a part of Chaps. 11 and 12 has been published as “Music between Hearing and Counting (A Historical Case Chosen within Continuous Long- Lasting Conflict)”, in Mathematics and Computation in Music, Carlos Agon et al. eds., Lectures Notes in Artificial Intelligence 6726, Berlin 2011, Springer Verlag, 285–296. Appendix C is the translation of the edition for Maurolico’s Musica,editedbythe author for the relative Opera Mathematica in www.maurolico.unipi.it, subsequently also Pisa-Roma, Fabrizio Serra editore, to be published, perhaps.