44 Studies on the Ancient Exact Sciences in Honor of Lis Brack-Bernsen John M. Steele Mathieu Ossendrijver (eds.) EDITION TOPOI EDITION BERLIN STUDIES OF THE ANCIENT WORLD are the main subject in this collection of papers, o ered in honor of Lis Brack-Bernsen by her colleagues and friends. The topics of the articles are linked by the themes that have been at the center of much of Lis’s own work: the Babylo- nian observational record, and the relationship between observation and theory; the gnomon, sundials, and time measurement; and the relationship between di er- ent scientifi c activities in the ancient world, especially the connections between mathematics and astronomy. Lis Brack-Bernsen has been a key fi gure in transform- ing the study of Babylonian astronomy from an almost exclusive focus on the mathematical astronomy of the late period to embracing a much broader consideration of all aspects of the subject, both early and late, math- ematical and observational, astronomical and astrolog- ical, and their relationships between one another. The papers demonstrate the wide variety of questions asked and approaches used by historians of ancient science. 44 · 44 Studies on the Ancient Exact Sciences in Honor of Lis Brack-Bernsen John M. Steele Mathieu Ossendrijver Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliographie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. © Edition Topoi / Exzellenzcluster Topoi der Freien Universität Berlin und der Humboldt-Universität zu Berlin Cover image: Sector of TU 11, hand copy of a tablet also known by its museum number AO , reverse side, lines –. In: François Thureau-Dangin. Tablettes d’Uruk. À l’usage des pretresˆ du Temple d’Anu au temps des Séleucides. Textes cunéiformes du Louvre . Paris: Geuthner, , pl. XXIII. Typographic concept and cover design: Stephan Fiedler Printed and distributed by PRO BUSINESS digital printing Deutschland GmbH, Berlin ISBN ---- ISSN (Print) - ISSN (Online) -X URN urn:nbn:de:kobv:- First published The text of this publication is licensed under Creative Commons BY-NC . DE. The legal code is available under https://creativecommons.org/licenses/by-nc/3.0/de/. For the terms of use of the illustrations, please see the reference lists. www.edition-topoi.de CONTENTS . Introduction — 7 . Foreword: An Essay in Story Form — 11 BABYLONIAN OBSERVATIONAL AND THEORETICAL ASTRONOMY . Dating by Month-Lengths Revisited — 19 . A Text Containing Observations of Mars from the Time of Nebuchadnezzar II — 69 ß The Coordinate System of Astronomical Observations in the Babylonian Diaries — 83 On the Origin of the Lunar and Solar Periods in Babylonian Lunar Theory — 105 BM – a Babylonian Compendium about Conjunctions and Other Planetary Phenomena — 127 TIME AND TIME MEASUREMENT The So-Called Report on Seasonal Hours (K +): A New Interpretation — 149 Ancient Greek Sundials and the Theory of Conic Sections Reconsidered — 165 . The Roofed Spherical Sundial and the Greek Geometry of Curves — 183 INTERCONNECTIONS IN ANCIENT SCIENCE ø Practitioners – School Teachers – ‘Mathematicians’ — 207 . A Mysterious Circular Tablet with Numbers and Stars — 225 An Ancient Celestial Empire of Benevolent Knowledge — 233 LIS BRACK-BERNSEN: BIOGRAPHICALNOTESANDSELECTEDBIBLIOGRAPHY Biographical Note — 263 Selected Bibliography — 265 John M. Steele Introduction The field of Babylonian astronomy has been transformed over the past three decades, changing from an almost exclusive focus on the mathematical astronomy of the late period (generally known as ‘ACT’ astronomy after the acronym of Otto Neugebauer’s classic Astronomical Cuneiform Texts published in ) to embracing a much broader consideration of all aspects of the subject, both early and late, mathematical and obser- vational, astronomical and astrological, and their relationships between one another. Lis Brack-Bernsen has been a key figure in turning the study of Babylonian astronomy into what it is today. Among Lis’ many contributions to the study of Babylonian astronomy, two have been particularly significant in shaping the way that research in the field has progressed. The first is by drawing attention to the so-called ‘lunar six’ time intervals – measurements of the time between the moon and sun crossing the horizon on six specific occasions during a month – which were regularly observed in Babylon from at least as early as the middle of the seventh century BC. In a series of papers, Lis has explored the role of lunar six observations and the development of the ACT lunar systems, the first attempt to answer the question of the relation of observation and theory in Babylonian astron- omy, and, even more significantly, uncovered a Babylonian method of using past lunar six observations to predict future lunar six intervals which was completely unknown to modern scholars and whose discovery has opened up a whole new area of research. Almost as important as her own work in these areas has been Lis’s second transforma- tive contribution to the study of Babylonian astronomy: the foundation of the so-called ‘Regensburg’ series workshops on Babylonian astronomy which have brought together specialists for intensive, detailed, and collegial discussion of different aspects of Baby- lonian astronomy. The first Regensburg workshop, held in Regensburg in , was so successful that is has been followed by ‘Regensburg’ workshops in Amsterdam (), John Steele, Mathieu Ossendrijver (eds.) | Studies on the Ancient Exact Sciences in Honor of Lis Brack-Bernsen | Berlin Studies of the Ancient World (ISBN ----; URN urn:nbn:de:kobv:-) | www.edition-topoi.de Durham (), and Berlin (), with discussions already underway for the next in the series. It is no exaggeration to say that these workshops have significantly influenced the direction of research on Babylonian astronomy over the past decade. In addition to her work on the lunar six, Lis has made significant contributions to many other aspects of the study of Babylonian astronomy including deepening our understanding of the early astronomical compendium MUL.APIN, and research on the gnomon and the origin of the zodiac, the operation of the Babylonian calendar, and the mathematical astrological schemes known as the Kalendertexte and dodecatemo- ria schemes. Outside of Babylonian astronomy, she has also published important works on Babylonian mathematics and, earlier in her career, on Mayan astronomy. The papers in this collection are offered in honor of Lis Brack-Bernsen by her col- leagues and friends, including many of the participants in the Regensburg workshops. The topics of the articles are linked by the themes that have been at the center of much of Lis’s own work: the Babylonian observational record, and the relationship between observation and theory; the gnomon, sundials, and time measurement; and the relation- ship between different scientific activities in the ancient world, especially the connec- tions between mathematics and astronomy. A tradition of regular and precise observation lies at the heart of Babylonian astron- omy. One of the most common types of observation recorded by the Babylonian as- tronomers is of the position of the moon and the planets relative to a group of reference starts. In their paper Gerd Graßhoff and Erich Wenger analyze Babylonian observations of this kind in order to understand the coordinate system underlying these measure- ments, demonstrating that positions were measured parallel and perpendicular to the ecliptic. The earliest systematic records of Babylonian observations date to the eighth and seventh centuries BC. John Steele discusses one of the earliest records of planetary observations, a compilation of reports of Mars’s synodic phenomena dating to the reign of Nebuchadnezzar II. Using Lis Brack-Bernsen’s work on the lunar six as a starting point, Teije de Jong discusses the development of Babylonian lunar theories from obser- vations of the lunar six and eclipses. Peter Huber analyzes records of the length of the month and their application to the problem of Old Babylonian chronology. Mathieu Ossendrijver edits and discusses a text which describes methods of predicting planetary conjunctions from past observations, using the same principle of ‘goal-year’ astronomy as Lis Brack-Bernsen uncovered for the lunar six. The gnomon and the sundial were used across the ancient world to measure time and to study the daily and yearly motion of the sun. Elisabeth Rinner analyzes the mathe- matics of conical sundials, one of the most common types of Greco-Roman sundial, and its connection with theories of conic sections. Alexander Jones’s paper complements Rinner’s by providing an analysis of spherical sundials and the geometry of curves. Also related to time measurement, Hermann Hunger reedits (with the addition of a substan- tial new fragment to the tablet) and analyzes a Neo-Assyrian text which has previously been thought to concern seasonal hours, demonstrating that this is not the case. The relationship between different types of ancient science is explored in thefinal three papers. Jens Høyrup’s paper examines the various forms and places of mathematics and mathematicians in the ancient world. Wayne Horowitz and John Steele examine a peculiar cuneiform tablet which combines numbers with star names. Finally,Francesca Rochberg studies the relationship between astronomy and divination, and