Between ca. 400 and 50 BCE, Babylonian astronomers used mathematical methods for predicting ecliptical positions, times and other phenomena of the moon and the planets. Until recently these methods were thought to be of a purely arithmetic nature. A new interpretation of four Babylonian astronomical procedure texts with geometric computations has challenged this view. On these tablets, Jupiter’s total distance travelled along the ecliptic during a certain interval of time is computed from the area of a trapezoidal figure representing the (...) planet’s changing daily displacement along the ecliptic. Moreover, the time when Jupiter reaches half the total distance is computed by bisecting the trapezoid into two smaller ones of equal area. In the present paper these procedures are traced back to precursors from Old Babylonian mathematics. Some implications of the use of geometric methods by Babylonian astronomers are also explored. (shrink)

In this study I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Although no texts are preserved in which the Babylonians tell us how they did it, from the surviving Astronomical Diaries we have a fairly complete picture of the nature of the observational material on which the scholars must have based their theory and from which they must have derived the values of the defining parameters. Limiting the (...) discussion to system A theory of the outer planets Saturn, Jupiter and Mars, I will argue that the development of Babylonian planetary theory was a gradual process of more than a century, starting sometime in the fifth century BC and finally resulting in the appearance of the first full-fledged astronomical ephemeris around 300 BC. The process of theory formation involved the derivation of long “exact” periods by linear combination of “Goal–Year” periods, the invention of a 360° zodiac, the discovery of the variable motion of the planets and the development of the numerical method to model this as a step function. Longitudes of the planets in the Babylonian zodiac could be determined by the scholars with an accuracy of 1° to 2° from observations of angular distances to Normal Stars with known positions. However, since the sky is too bright at first and last appearances of the planets and at acronychal rising for nearby stars to be visible, accurate longitudes as input for theoretical work could only be determined when the planets were near the stationary points in their orbits. In this study I show that the Babylonian scholars indeed based their system A modeling of the outer planets on planetary longitudes near stations, that their system A models of Saturn and Jupiter provided satisfactory results for all synodic phenomena, that they realized that their system A model of Mars did not produce satisfactory results for the second station so that separate numerical schemes were constructed to predict the positions of Mars at second station from the model at first station, and finally that their system A model for Mars provided quite poor predictions of the longitude of Mars at first and last appearances over large stretches of the zodiac. I further discuss ways in which the parameters of the system A models may have been derived from observations of the outer planets. By analyzing contemporaneous ephemerides from Uruk of four synodic phenomena of Jupiter from the second century BC, I finally illustrate how the Babylonian scholars may have used observations of Normal Star passages of planets to choose the initial conditions for their ephemerides. (shrink)

In this study I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Although no texts are preserved in which the Babylonians tell us how they did it, from the surviving Astronomical Diaries we have a fairly complete picture of the nature of the observational material on which the scholars must have based their theory and from which they must have derived the values of the defining parameters. Limiting the (...) discussion to system A theory of the outer planets Saturn, Jupiter and Mars, I will argue that the development of Babylonian planetary theory was a gradual process of more than a century, starting sometime in the fifth century BC and finally resulting in the appearance of the first full-fledged astronomical ephemeris around 300 BC. The process of theory formation involved the derivation of long “exact” periods by linear combination of “Goal–Year” periods, the invention of a 360° zodiac, the discovery of the variable motion of the planets and the development of the numerical method to model this as a step function. Longitudes of the planets in the Babylonian zodiac could be determined by the scholars with an accuracy of 1° to 2° from observations of angular distances to Normal Stars with known positions. However, since the sky is too bright at first and last appearances of the planets and at acronychal rising for nearby stars to be visible, accurate longitudes as input for theoretical work could only be determined when the planets were near the stationary points in their orbits. In this study I show that the Babylonian scholars indeed based their system A modeling of the outer planets on planetary longitudes near stations, that their system A models of Saturn and Jupiter provided satisfactory results for all synodic phenomena, that they realized that their system A model of Mars did not produce satisfactory results for the second station so that separate numerical schemes were constructed to predict the positions of Mars at second station from the model at first station, and finally that their system A model for Mars provided quite poor predictions of the longitude of Mars at first and last appearances over large stretches of the zodiac. I further discuss ways in which the parameters of the system A models may have been derived from observations of the outer planets. By analyzing contemporaneous ephemerides from Uruk of four synodic phenomena of Jupiter from the second century BC, I finally illustrate how the Babylonian scholars may have used observations of Normal Star passages of planets to choose the initial conditions for their ephemerides. (shrink)

In this paper, we discuss Babylonian observations of a “massing of the planets” reported in two Astronomical Diaries, BM 32562 and BM 46051. This extremely rare astronomical phenomenon was observed in Babylon between 20 and 30 March 185 BC shortly before sunrise when all five planets were simultaneously visible for about 10 to 15 min close to the horizon in the eastern morning sky. These two observational texts are not only interesting as records of an extremely rare planetary configuration, but (...) also because the observers appear to be confused by the presence of all planets simultaneously and mix them up in their reports, and the two reports of the same observations are so different that we are forced to conclude that they were carried out by two different observers. There is an additional astronomical event which makes this planetary configuration even more unique: the exact conjunction of the planets Mars and Jupiter in the afternoon of 25 March 185 BC. An exact conjunction, where two planets are so close together that they appear as one object in the sky, is also extremely rare. Although this exact conjunction between Mars and Jupiter occurred during the day so that it was not observable, it was correctly predicted by the Babylonian scholars: a remarkable achievement and a nice illustration of their astronomical craftsmanship. Finally, our study clearly exposes one of the limitations of Babylonian naked-eye astronomy. When first appearances of the planets Mercury, Mars and Saturn are expected around the same date, it is nearly impossible to correctly identify them because their expected positions are only approximately known while they have about the same visual magnitude so that they become visible at about the same altitude above the horizon. (shrink)