[10] When in History did Prediction of the Astronomical New Moon Begin?

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The history of ancient astronomy shows that it was not until near the time of the birth of Alexander the Great that ancient astronomers were first able to estimate the time of the conjunction of the moon by a calculation.

On page 169 of van der Waerden, he wrote:

“In Babylonia, the month began on the evening on which the crescent was visible for the first time after [the astronomical] New Moon. More precisely: If on the [ending] evening of the 29th day of any month the crescent was visible, the month has 29 days; if not, the month has 30 days. The same rule still holds in Muslim countries.”

“I shall call these months ‘observed lunar months’. The words of Geminos indicate that the Greek months originally were just observed lunar months.”

“The months beginning with the conjunction will be called ‘exact lunar months’ or ‘conjunction months’. These months are a theoretical construction; they could not be used in practice in classical times, because before Kallippos [Callippos] (330 B.C.) astronomers were not able to predict the true conjunction.”

Thus van der Waerden points to 330 BCE as the time before which ancient mathematical astronomical knowledge was not able to predict the time of the astronomical new moon.

The orbit of the moon around the earth is an ellipse. The earth is not at the center of this ellipse, but at one of the two foci of the ellipse. The moon moves faster around the earth when it is closer to the earth than when it is farther from the earth. Due to the sun's gravitational attraction to the earth and moon, the distance from the earth to the sun affects the distance from the moon to the earth, which in turn affects the time from conjunction to conjunction! The exact time from conjunction to conjunction does vary through the year! Knowing the average time from conjunction to conjunction does not help to know any current lunar month's time from conjunction to conjunction.

The minimum time from one conjunction to the next conjunction is 13 hours 40 minutes less than the maximum time from one conjunction to the next conjunction (see pages 21-22 in Stephenson and Baolin). A mathematical mastery of this variation is needed in order to accurately predict the time of an astronomical new moon.

A high level of confidence of the accurate prediction of solar eclipses by ancient peoples was certainly impossible because this requires a knowledge of where the moon's shadow will reach the earth, and that requires a knowledge of the distance from the moon to the earth (which requires a knowledge of the elliptical orbit of the moon), the size of the earth, and the shape of the earth (which is somewhat pear-shaped rather than perfectly spherical). Since they could not predict the shadow path of the moon upon the earth, the best they could achieve is a statement that a solar eclipse was a reasonable possibility. But in order to do that, they would need to have a good ability to predict the astronomical new moon as well as how to rule out most astronomical new moons as being capable of providing a solar eclipse.

This simply shows that we can judge the ability of ancient astronomers to approximately predict the astronomical new moon by their attempts to predict a possible solar eclipse. Of specific interest is the paper by John M. Steele 1997 where, on page 134 he lists the oldest Babylonian solar eclipse prediction for which we have full data in 358 BCE, exactly 100 years after Ezra first brought a group from the House of Judah back to Jerusalem after the Babylonian captivity. This solar eclipse prediction was 181 years after King Cyrus the Great of Persia conquered Babylon on October 12, 539 BCE (see page 14 of Parker and Dubberstein). Since the empire was now the Persian Empire rather than the Babylonian Empire, the learned astronomers who continued their work should be called Persians, but the general practice is to continue referring to them as Babylonian or “late Babylonian”. The same pagan priests continued to improve their work in mathematical astronomy. John Steele 1997 analyzes the 61 preserved solar eclipse predictions of the Babylonians for which full data is available including the time at which the eclipse is hoped to be seen, and these fall within the years 358 BCE - 37 CE. The terminology used by the Babylonians shows that a solar eclipse was to be “watched for”, showing an uncertainty that it would be seen. Less than half (28 of 61) were either seen or would have been seen if the precise time of the eclipse would have occurred during daytime in the region of Babylon. In other words, in these 28 cases the latitude of the moon's shadow did fall within some part of greater Babylon, but in the other 33 cases the moon's shadow was outside this region. These ancient astronomers used water clocks, which divided the day into 360 equal parts, each being four minutes.

The average error of these water clocks is eight minutes from true time. The predictions included the calculated time for the eclipse to occur. The worst two predictions among these 28 cases were 8.08 hours in error and 4.76 hours in error (page 135). The average error was 1.96 hours (page 136). For the other 33 cases of predictions the average error in the time of conjunction (here the word “conjunction” relates to a hoped for solar eclipse) is 3.67 hours, nearly twice as great (page 137)! Their predictions of solar eclipses did not get more accurate in the later period of their recordings (pages 138-139).

The mathematical methods that were used by the Babylonians were very different from the methods used by the Greeks. The former used nearly repeating sequences based on prior historical records (not a formula based on a general physical mathematical model), while the latter developed a geometrical mathematical model based on circles after 400 BCE. The Greeks were aware of the methods used by the Babylonians (see page 118 of Jones, the chapter by Toomer 1988, and page 61 of Fatoohi and others), but the most advanced Greek astronomers preferred their own methods. The methods of the Greeks were more advanced in the sense that they were based on mathematical methods for approximate geometrical models, and the geometry itself led to the concept of the conjunction. In contrast to this, the Babylonians were interested in predicting solar eclipses, which by definition only occur at the time of a conjunction; they did not show a general interest in predicting the time of all conjunctions, and this was likely the cause for van der Waerden's limiting of the year for calculating the approximate astronomical new moon (conjunction) to 330 BCE. On page 41 of Aaboe we read, “Babylonian mathematical astronomy has two features that seem strange to modern eyes, and it may thus be in order to mention them here. First, it is entirely arithmetical in character or, in negative terms, there is no trace of geometrical models like the ones we have become accustomed to since the time of Eudoxos [Greek astronomer of Cnidos, c. 408 to 355 BCE. (see pages 63-66, 335 of Pedersen 1993)]. Second, the cuneiform literature [clay tablets bearing the Akkadian language of the Assyrians and a remnant of the Babylonians] nowhere attempts to justify the precepts of the procedure texts; thus it has rested with modern scholars to uncover the underlying theoretical structures.”

In other words, the Babylonians have left us their many tablets showing columns of umbers, and it remained for modern scholars to decode the meaning of these columns and how they were computed. In some cases there are narratives that accompany these numbers that mention certain sighted phenomena in the heavens or some indications of the meanings of one or more columns, but there are no geometrical diagrams showing a mathematical model of anything in the heavens among the Babylonians.

The conclusion is that there are unusual aspects of the variation of the moon's cycle around the earth that prevented ancient people from predicting the approximate conjunction until about 330 BCE by the advanced methods of the Greeks, or instead, until about 360 BCE for the non-geometrical methods of the Babylonians whose average error was about three hours.

Moreover, the Babylonians were focused on solar eclipses rather than conjunctions in general, while the Greeks showed an interest in conjunctions. Another very significant factor that contributed to the difficulty of predicting the conjunction is the lack of visual confirmation of a conjunction unless there was a rare solar eclipse to confirm it. The water clocks used by the ancient Babylonian astronomers had an average error of eight minutes and their smallest unit of measuring time was four minutes. Their predictions were long term, i. e., there is nothing to indicate that they attempted a revised prediction within days of a solar eclipse. When conditions were not right for a solar eclipse they never predicted a “conjunction” because it would have been foolish to predict a phenomenon that was not potentially verifiable with an observation.

A lunar eclipse is the covering of the sun's light to the moon by the earth as seen by an observer on the earth when the earth comes between the sun and the moon. In sharp contrast to the special difficulties of predicting solar eclipses, there are no comparable problems in predicting lunar eclipses.

Lunar eclipses must occur during the full moon, may be seen by nearly half of the people on the earth where the weather is not nasty (the side of the earth where it is night), are visible more frequently than solar eclipses from any one location, have calculations that may be tested from monthly approximate sightings of the full moon, and do not require predicting the path of a shadow (in this case, the shadow of the earth upon the moon).

Hence there is a vast difference between the difficulty in predicting solar eclipses (some conjunctions) and the ease in predicting lunar eclipses (some full moons) by ancient astronomers. Page 3 of Britton 1989 states, “For a given location, therefore, lunar eclipses are seen nearly 4 times as frequently as solar eclipses.” But even when there is no lunar eclipse, the full moon is still visible. When there is no solar eclipse, the moon is not visible.

Ancient Babylonian astronomers were significantly more successful in their accuracy at predicting lunar eclipses than they were at predicting solar eclipses. Of specific interest is the paper by John M. Steele and F. Richard Stephenson. The oldest Babylonian lunar eclipse prediction for which we have full data is in 731 BCE (see page 125), which is 373 years before the first known reasonably accurate solar eclipse “hoped for” prediction by the Babylonians for which we have complete data! They were successful in their
prediction for 731 BCE. Page 125 lists 35 Babylonian predictions of lunar eclipses for which we have complete data including the time of prediction to be observed. Also listed is the duration of time for which the eclipse was observed by the Babylonians, when it was successfully seen. These are dated from 731 to 77 BCE. Their average error for predicting the time of lunar eclipses was about one hour (page 130). In 90 percent of the predictions they were either successful or there was a near miss as defined by the authors (pages 123, 130). Their average error for lunar eclipse predictions was about one hour compared to about three hours for solar eclipses. It took about 400 years more for the Babylonian astronomers to be able to predict reasonably accurate possible solar eclipses (associated with the conjunction) than for them to be able to predict lunar eclipses (associated with the full moon).

There are numerous other dates of predictions of both lunar and possible solar eclipses by the Babylonians, but the time of day of their expected or hoped for sighting is not provided in the ancient sources. Without having the time of day of a predicted lunar eclipse or a possible solar eclipse it is impossible to judge the accuracy of the method of prediction, so it is not reliable to include such records in a discussion of known results. On the other hand, where columns of data are provided in a Babylonian text, it is possible for a modern specialist in this area of ancient science to judge whether the method is quite different from the more accurate later methods.

In Britton 1989, John Britton evaluates the method used by the Babylonians for their earliest known attempt to predict possible solar eclipses. This text, which he called Text S, describes 38 solar eclipse possibilities from 475 to 457 BCE (see page 1 of Britton 1989). On page 44 Britton states, “We find in Text S an unusual mixture of disparate elements not known from other texts.” After discussing the method used by these Babylonians, he wrote on page 46, “Indeed, with one exception the entire theory [for predicting possible solar eclipses] can be derived from counts of phenomena (lunar eclipses, eclipse possibilities, and months), and there is no evidence that measurements of times, angles or magnitudes played any role in its creation.” From the data in Text S, Britton discusses its primary computation, which he calls “psi-star-of-S”. His conclusion on page 46 is, “We see this best in the fact that psi-star-of-S, a function clearly derived from lunar eclipses and measuring the proximity to the node of the earth's shadow at conjunction (or the moon at mid-eclipse), is correctly applied to solar eclipse possibilities by simply moving the entire function forward half a month.” A simplified way of saying this is that these Babylonians estimated the time of the conjunction to be the midpoint between two successive computed full moons, and then judged the confidence for a solar eclipse based on the history of repeating eclipses. But we have seen above that it is very crude to estimate the conjunction to be the midpoint between two successive computed full moons, so this method for predicting solar eclipses by the Babylonians is indeed very crude compared to their later method which has an average error of about three hours.

Hence we must dismiss this first Babylonian attempt at predicting solar eclipses (special conjunctions) as inferior and not to be included in the chronology with their later methods.

The conclusions are that the Babylonians were able to predict lunar eclipses by about 750 BCE with a time error of about one hour, and the Babylonians were able to predict possible solar eclipses about 360 BCE with a time error of about three hours. The Babylonians started the practice of predicting the sighting of the new crescent about 450 BCE.