Asger Aaboe

Two Atypical Multiplication Tables from Uruk

A Babylonian multiplication table contains the numbers and the other probably, late. We know that a fragment of the first (U. 91 in Istanbul) was found together with fragments of mathematical stronomical texts from the Seleucid period, and it is tempting to believe that the unorthodox selection of principal numbers was dictated by the very substantial arithmetical demands of mathematical astronomy. Unfortunately, a comparison between the new principal numbers and significant astronomical parameters shows no decisive overlap. 2

Some Seleucid Mathematical Tables (Extended Reciprocals and Squares of Regular Numbers)

The cuneiform mathematical texts which are available to us at present fall in the large, and probably accidentally, into two groups, one OldBabylonian, and the other Seleucid. A modern observer may well find it astonishing that there is virtually no change to be traced, either in content or in form, in this body of knowledge over the documented timespan of more than a millennium and a half; what difference there is between these two main groups of texts consists mostly in a Seleucid preference for more elaborate numerical examples and, in particular, for more extensive technical tables, and this could perhaps be related to the strong demands which Seleucid mathematical astronomy could and did place on arithmetical techniques. The present texts are all examples of such extensive Seleucid tables. The first threel are presented here for the first time. The last five have already been published: first, in Pinchess handcopy, by A. Sachs2 who tentatively, but correctly, identified them as lists of squares, and later, transcribed and restored, by A. A. Vaiman.3 That I nonetheless include them is due to three reasons. Firstly, they obviously belong to the same category as two of the first three texts, and even duplicate them to some extent, and it is therefore well to have them all appear in one place. Secondly, Vaimans restorations are, in their printed versions, marred by an unusual number of errors (of course, this detracts in no way from the merit he surely earned by identifying correctly, and at a

A Late-Babylonian Procedure Text for Mars, and Some Remarks on Retrograde Arcs

FIRST MET E. S. Kennedy (or al-Kindi) in the autumn of 1953 at Brown UniI versity. He lectured on the zij of the archcomputer al-K&hi to an audience consisting of Neugebauer and me, and Neugebauer retaliated by talking about ”Odds and Ends” to Kindi and me. Someone I think it was Kindi’s father-inlaw said it reminded him of two haberdashers selling shirts to each other to keep the volume up. Though witty, the remark was far off the mark, for there was enormous profit involved, at least for me. It was here I was introduced to Islamic science. The sessions, though small, were the very opposite of quiet and passive, and I do not believe that I have ever learned as much in a like time interval as in these two seminars. From then on, at least until Kindi’s retirement, time was reckoned in Kindiads, intervals of four years after which the Kindis would reappear in this country (as time went on the Kindiads seemed to grow shorter). The Kindis became firm friends, and whenever we met it seemed irrelevant how long it had been since last time. In 1968 I had the good fortune to visit the Kindis in Lebanon-in both Beirut and Ainab -and would have been embarrassed by their generous hospitality had it not been so cheerfully offered. They showed me the delights of their adopted country, and I took part in their family life (it was the year, I remember only too well, when Kindi took up the French horn Doppelhorn, in Neugebauer’s terminology). On this occasion I offer the following little piece of work as a token of my gratitude, admiration, and affection.

Ptolemy’s Cosmology

The Ptolemaic system has long been in common use as the name for the cosmological scheme that was eventually replaced by the Copernican. It consists of seven nested spherical shells, one for the Sun, one for the Moon, and one for each of the five planets, all surrounded by the sphere of the fixed stars. All of this revolves once a day about a central, stationary Earth, while Sun, Moon, and planets move appropriately and much more slowly, each within its own sphere. This scheme is most often represented graphically by its intersection with the plane of the ecliptic (the Sun’s orbit)—it then becomes a system of nested annuli—for the planets and the Moon are always close to this plane.

Kepler Motion Viewed from Either Focus

In the first chapter we analyzed a Babylonian planetary text and saw that the Late-Babylonian astronomers had shaped arithmetic into a powerful tool for addressing astronomical problems. In contrast, the Greek planetary models, qualitative as well as quantitative, employed geometrical models with moving parts and a knowledge of how to combine velocities. We call such models cinematical, for cinematics is the branch of mathematics concerned with motion but without regard to masses and forces. Finally, dynamics—the study of the behavior of a system of masses in terms of the forces that act upon them—was created in its useful form by Isaac Newton (1642–1727). In his Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), usually called the Principia (1687), he formulated the basic rules of dynamics and successfully applied them to a variety of physical problems, among them that posed by the solar system.

What Every Young Person Ought to Know About Naked-Eye Astronomy

In order to provide a starting point for an understanding of ancient astronomical texts, I shall begin by presenting, in all brevity, the basic elements of naked-eye astronomy. I shall, of course, deal principally, but not entirely, with phenomena of interest to ancient astronomers. Among these are many phenomena, such as the first or last visibility of a planet or the moon, that the modern astronomer shuns since they take place near the horizon and further depend on imperfectly understood criteria. Thus, these phenomena are not commonly discussed in the modern astronomical literature and, more seriously, we lack modern standards with which we may measure the quality of the ancient results.

Babylonian Arithmetical Astronomy

In late May 1857 a committee, appointed by the Royal Asiatic Society of Great Britain and Ireland, met in London to compare four independent translations of an Assyrian text inscribed in cuneiform characters in duplicate on two well-preserved clay cylinders. Hormuzd Rassam had found them as foundation deposits in the ruins of ancient Ashur in 1853 when he was digging in Mesopotamia on behalf of the British Museum. W. H. Fox Talbot, the gentleman scientist, inventor of photography, and linguist, had been given a copy of the text by H. C. Rawlinson, the remarkable soldier, diplomat, and linguist, and sent his sealed translation of it to the Society with the suggestion that other scholars be invited to translate the same text so the results could be compared to test the validity of the decipherment of Assyrian for, as he writes, “Many people have hitherto refused to believe in the truth of the system by which Dr. Hincks and Sir H. Rawlinson have interpreted Assyrian writings, because it contains many things entirely contrary to their preconceived opinions.”

Greek Geometrical Planetary Models

Higher Greek mathematics is mostly concerned with geometry, so it is not too surprising that the Greek detailed planetary models were geometrical. The aim of such models was at first to mimic the behavior of a planet, which, in the case of Saturn, is indicated in Figure 1, with latitude exaggerated four times.

On period-relations in Babylonian astronomy

Abstract Into the arithmetical schemes, that are at the base of Seleucid mathematical astronomy, are built the fundamental periodicities of Sun, Moon, and planets. It has long been recognized that this is one of the main reasons for the spectacular successes of the seemingly simple devices, for it prevents an arbitrary accumulation of the unavoidable errors which arise from any approximation. In the schemes classified as System A this property is assured as long as the so-called generating function satisfies a certain simple condition, the period-relation. In the literature several proofs have been offered for the necessity and sufficiency of this condition, but though all are valid, none seemed to me to be in the Babylonian manner. However, consideration of a now well-represented class of planetary texts, 1 which give longitudes, but no dates, of successive synodic phenomena according to System A, probably for an entire period, led me to a point of view which seems more in character. It appears that one may think of the entries in a System-A ephemeris as selections according to a simple principle from a finite, though large number of possible positions. 2 The period-relations then follow by elementary number-theoretic considerations. Incidentally, this interpretation also offers easily applied criteria for connectivity of texts belonging to System A.