O. Neugebauer

On the Planetary Theory of Copernicus

In 1958 A. Koyre spoke about “l’abandon de l’equant, ce haut titre de gloire de l’astronomie copernicienne”, adding that in the lunar theory “Copernic reussit la simplification la plus grande en la debarassant de l’equant (ce qui nous donne la mesure de son genie mathematique).”1

The Survival of Babylonian Methods in the Exact Sciences of Antiquity and Middle Ages

Among the many parallels between our own times and the Roman imperial period could be mentioned the readiness to ascribe to the “Chaldeans” discoveries whenever their actual origin was no longer known. The basis for such assignments is usually the same: ignorance of the original cuneiform sources, excusable in antiquity but less so in modern times. Given this situation, it seems to me equally important to establish what we can say today about knowledge which the Babylonians did have and to distinguish this clearly from methods and procedures which they did not have. In other words, it seems to me that it is high time that an effort is made to eliminate historical cliches, both for the Mesopotamian civilizations and their heirs, and to apply common sense to the fragmentary but solid information obtained from the study of the original sources during the last hundred years.

The History of Ancient Astronomy: Problems and Methods

In the following pages an attempt is made to offer a survey of the present state of the history of ancient astronomy by pointing out relationships with various other problems in the history of ancient civilization and particularly by enumerating problems for further research which merit our interest not only because they constitute gaps in our knowledge of ancient astronomy but because they must be clarified in order to lay a solid foundation for the understanding of later periods.

The Study of Wretched Subjects

IN the last issue of Isis (vol. 41, 125–126, p. 374) there is a short review by Professor Sarton of a recent publication by E. S. Drower of the Mandean “Book of the Zodiac” which is characterized by the reviewer as “a wretched collection of omens, debased astrology and miscellaneous nonsense.” Because this factually correct statement1 does not tell the whole story, I want to amplify it by a few remarks to explain to the reader why a serious scholar might spend years on the study of wretched subjects like ancient astrology.

The Transmission of Planetary Theories in Ancient and Medieval Astronomy

In Toledo in the year A.D. 1068 Abū’l-Qāsim Sā’id ibn Ahmad, also known as Qādi Sā’id, wrote a book entitled The Categories of Nations.1 In this work he discusses the people of the world from the viewpoint of their interest in scientific research, stating that “the category of nations which have cultivated the sciences forms the elite and the essential part of the creations of Allah.”2 Eight nations belong to this class: “the Hindus, the Persians, the Chaldeans, the Hebrews, the Greeks, the Romans, the Egyptians, and the Arabs.”3 Taking into account some terminological differences, this list can still be considered fairly complete. What Sā’id calls “the Romans” and “the Egyptians” in part coincides with what we would call the Byzantines and the Alexandrian School, while Rome and Egypt in our sense of these words could not compare in importance with India and the Hellenistic or Muslim contributions. But it is wise not to forget that the existence of the Roman Empire was an essential condition for the transmission of Hellenistic science to the Muslim world.

The Alleged Babylonian Discovery of the Precession of the Equinoxes

Historians constantly face two closely related problems: to make new textual material available and to destroy generally accepted theories. The present paper is concerned with the latter aspect, in the case of the more and more frequently quoted statement that the Babylonian astronomer Kidinnu was the discoverer of the precession of the equinoxes and that this event can be dated in 379 b. c., thus antedating Hipparchus by about two and one-half centuries. It may seem as if we were dealing here with one of those questions of priority which are of very little significance. Actually the problem has wider implications. It is closely related to the problem of the date of origin of Babylonian mathematical astronomy, which exercised a deep influence on Greek astronomy and its continuation in the Middle Ages. It is furthermore of importance for the evaluation of Babylonian astronomy and the mutual role of observation versus theory during the Seleucid period.

Zur Geschichte der babylonischen Mathematik

Vor kurzem hat C. Frank in den „Schriften der Strasburger Wissenschaftlichen Gesellschaft in Heidelberg“ (neue Folge 9. Heft) einige sumerische und babylonische Texte veroffentlicht, unter denen sich auch sechs Stucke mathematischen Inhaltes befinden, auf die mich hinzuweisen Prof. Meisner die Gute hatte. Nach Angabe von Frank entstammen sie samtlich der altbabylonischen Zeit. Da diese Texte, wie mir scheint, fur die Geschichte der antiken Mathematik von grosem Interesse sind, von Frank einer Kommentierung aber nicht unterzogen wurden, so mochte ich dies wenigstens fur eine bestimmte Gruppe von Aufgaben (aus Tafel 8 und 10 der Frankschen Zahlung) nachholen. Eine Bearbeitung der ubrigen hoffe ich demnachst vorlegen zu konnen.

Exact Science in Antiquity

If history is the study of relations between different cultures and different periods, the history of exact science has a definite advantage over general history. Relations in the field of science can be established in many cases to such a degree of exactitude that we might almost speak of a “proof” in the sense of mathematical rigor. If, for instance, Hindu astronomy uses excenters and epicycles to describe the movement of the celestial bodies, its dependence on Greek astronomy is established beyond any doubt; and the dependence of Greek astronomy on Babylonian methods is obvious from the very fact that all calculations are carried out in sexagesimal notation. However, the fact that the center of interest in the history of science lies in the relationship between methods requires a new classification of historical periods. In the history of astronomy, for instance, concepts such as “ancient” or “medieval” make very little sense. The method and even the general mental attitude of the work of Copernicus is much more closely related to that of Ptolemy, a millennium and a half before, than to the methods and concepts of Newton, a century and a half later. It may seem, therefore, a rather arbitrary procedure in the following report on exact science in antiquity to take into consideration only the period before Ptolemy (ca. 150 A. D.).

The Astronomical Origin of the Theory of Conic Sections

Appolonius’s theory of the conic sections (about 220 b.c.) is undoubtedly one of the masterpieces of ancient mathematics and will remain one of the great classics of mathematical literature. Very little, however, is known about the origin of the theory of conic sections as such. It is well known that the familiar names of these curves, ellipse, hyperbola, and parabola, originated from Apollonius’s method of attack, which consists in applying the methods of “geometrical algebra” to the discussion of these curves. Apollonius obtains his curves by intersecting a fixed skew circular cone by a plane of variable angle. We also know that this approach is very different from the earliest known method to obtain conic sections. Menaechmus, a pupil of Eudoxus, is credited with the discovery of the conic sections (about 350 b.c.). These curves were obtained, however, by a very peculiar construction. The cone is a right circular cone; the intersecting plane is always perpendicular to one of the generating lines of the cone, and the three types of curves are obtained by varying the angle at the vertex of the cone.

Über die Geometrie des Kreises in Babylonien

Im folgenden sind einige Beobachtungen aneinandergereiht, die, zusammengenommen, geeignet erscheinen konnen, eine Basis fur die bisher so sehr vernachlassigte Erforschung der babylonischen Geometrie, insbesondere der Geometrie des Kreises, abzugeben. Das man auf diesem Gebiet bereits in altbabylonischer Zeit gewisse Kenntnisse besessen haben mus, die das trivialste Mas uberschritten, war bereits seit der Veroffentlichung eines Textes der ersten babylonischen Dynastie durch Gadd zu vermuten2). Dort wird namlich die Berechnung von Teilgebieten gewisser ornamentaler Figuren verlangt (sie haben etwa das Aussehen eines Fliesenbelages), in denen Kreise und Kreisbogen eine Rolle spielen. Mehr als eine fluchtige Formulierung solcher Aufgaben ist aber in diesem Text nicht enthalten. Sehr im Gegensatz dazu enthalten aber die in den ,,Cuneiform Texts from Babylonian Tabletts, &c., in the British Museum” seit 28 Jahren (in Keilschrift) veroffentlichten Tafeln „CT IX 8 bis 15” eine grose Zahl von Aufgaben und Losungen, die es gestatten, den Einzelheiten der Rechnung von Anfang bis zu Ende nachzugehen. Eine erste Probe einer solchen Interpretation wollen wir im folgenden vorlegen.