[14] Did Ancient Israel Excel in Advanced Mathematical Astronomy?

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Scripture defines the wisdom of ancient Israel in an unconventional way in the following passage.

Deut 4:5, “Behold I have taught you statutes and ordinances as YHWH my Almighty commanded me, that you should do so in the midst of the land where you are going to possess it.”
Deut 4:6, “So keep and do [them], for that [is] your wisdom and your understanding in the sight of the peoples who shall hear all these statutes. Then they shall say, surely this great people [is] a wise and understanding nation.”
Deut 4:7, “For what great nation [is there] that has an Almighty [so] near to it as YHWH our Almighty in everything we call upon Him.”
Deut 4:8, “And what great nation [is there] that has statutes and ordinances [as] righteous as all this law that I set before you today?”

The nations of the world think of wisdom in terms of scientific achievement and the acquiring of great knowledge, but that is not the way Moses was told to proclaim wisdom to Israel. Mathematical astronomy was not to be wisdom for them. I do not doubt that the ancient Israelites had the mental capacity to be able to develop advanced mathematics, but without the collective need for this effort by Israelite society, what would motivate such an effort?

Ancient Israel could determine the calendar from observation, so
they had no need for any advanced tedious calculations.

Did ancient Israel use a positional digit system with a zero, which would enable rapid multiplication and division? On page 26 of GKC2 (the latest English edition of the Hebrew grammar book by Gesenius), the numerical value of the 22 Hebrew letters is presented. This shows one letter for the value 2, another letter for the value 20, and another letter for the value 200.

This illustrates the nature of the symbolic number system in ancient Hebrew, and shows that it was not a positional digit system with a zero. Page 30 has further comments on this system, which was used on coins in Judea from the Maccabean period (c. 150 BCE). The time of the origin of this system is unknown. This system would be a hindrance for general long division and is not useful for mathematical astronomy.

A good deal of effort has been put into the history of ancient astronomy in previous chapters in order to evaluate what could have been known by ancient Israel at the time of Moses and afterward. The ancient Israelites from the time of Moses in Egypt could not have borrowed mathematical astronomy from Egypt because Egypt did not possess mathematical astronomical knowledge until it was brought there by Greek astronomers more than 1000 years after Moses died. From biblical chronology I estimate that the Israelite exodus from Egypt occurred c.1480 BCE.

Although the Jews were in captivity in Babylon where the pagan priests had an advanced knowledge of both mathematics and mathematical astronomy written in the complex Akkadian language with its hundreds of symbols for words (not for numbers), there is no evidence that these Jews acquired this knowledge. Ancient Jewish writings from the Dead Sea Scrolls, from Philo, from Josephus, from archeological artifacts, and from the Mishnah (c. 200 CE), give no hint that the Jews became familiar with the Babylonian mathematical methods of computation before the time of the Greek astronomer Ptolemy (c. 150) CE who lived in Alexandria, Egypt. The Talmud does claim that Mar Samuel was able to compute a calendar for many years in advance, c. 250 CE, although none of the details are known.

Jewish scholars do not claim that the ancient Israelites had abilities in mathematical astronomy that surpassed that of their ancient neighbors. There is no historical evidence for it. On pages 555-556 of Langermann we find, “Although the sun, moon, and stars are mentioned in the Hebrew Bible, that ancient and sacred text does not display any sustained exposition which can be called an astronomical text. The earliest sources for a Hebrew tradition are found in a few passages in the Talmud and Midrash [c. 200-600 CE].”

The Babylonian Talmud, specifically the section designated Rosh Hashanah 25a (RH 25a), which is on page 110 of BT-BEZ-RH, quotes Rabban Gamaliel II of Yavneh as having said, “I have it on the authority of the house of my father's father [Gamaliel the Elder from the early first century] that the renewal of the moon takes place after not less than twenty-nine days and a half [day] and two-thirds of an hour and seventy-three halakin.” Since there are 1080 halakin in one hour, this is 29.5 days 44 minutes 3 1/3 seconds. Thus RH 25a claims that from one new moon to the next new moon is at least this length of time. On page 308 of Swerdlow this is shown to exactly equal the value used by the Greek astronomer Hipparchus (c. 190 - c. 120 BCE) for the average length of the month, which he wrote in the base 60 as 29;31,50,8,20 days, which equals 29 + 31/60 + 50/(60x60) + 8/(60x60x60) + 20/ (60x60x60x60) days. But did Hipparchus derive this value himself? No! The paper by Toomer 1980 discusses this value for the average lunar synodic month in more detail. On page 108 footnotes 6 and 11 he clearly points out (as he implied on pages 98-99) that the Babylonians had already derived this value at an earlier time, and thus he shows that this value was not first computed by Hipparchus, but accepted as true by Hipparchus and taken by him from the Babylonians. Toomer also gives credit to Asger Aaboe for a paper he wrote in 1955 indicating that Aaboe realized that this number came from the Babylonians rather than Hipparchus.

On page 98 Toomer credits F. X. Kugler as apparently recognizing this in a book he wrote dated 1900. On pages 168, 240-241 of Hunger and Pingree it is stated that this length of an average synodic month comes exactly and directly from column G in the Babylonian lunar System B, and on page 236 this book states that the earliest tablet containing System B material from Babylon is dated 258 BCE. Hence this number was derived by the Babylonians some time before 258 BCE. On page 54 of Britton 2002, John Britton estimates the origin of the mean synodic month to c. 300 BCE.

How might ancient people determine the length of a lunar month? By taking two widely separated eclipses of the same kind and when the moon is traveling at about the same point in its cycle of varying velocity, and then dividing the time length between them by the number of lunar months, one may estimate the average length of a synodic month. Hipparchus was trying to compute eclipse periods, and for this purpose he used two old records of eclipse observations from Babylon that he possessed as well as two eclipse observations from his own lifetime. From these two pairs of eclipses Toomer's paper explains that a computation of the average lunar synodic month would in fact disagree with the number that he received from Babylon, but Hipparchus accepted their number anyway. The last of the base 60 numbers above is 20, but the computation from Hipparachus' eclipse records would instead round off this last number to a 9. While the long division computation gives a different number, the difference between these values is less than a tenth of a second! How accurate are these numbers (20 and 9 for the last place) compared to the true value of the average lunar synodic month near the time of Hipparchus and the earlier Babylonians?

On page 87 of Depuydt 2002, Leo Depuydt provides the following estimated modern computations for the mean synodic month in the years 2000 BCE, 1000 BCE, and 1 CE, and I have converted these to the Babylonian base 60 system. The computed estimated time is based upon eclipse records going back to 747 BCE and the assumption that the trend continued in a similar way prior to that date.

2000 BCE 29d 12h 44m 2.08s = 29; 31, 50, 5, 12
1000 BCE 29d 12h 44m 2.29s = 29; 31, 50, 5, 43.5
1 CE 29d 12h 44m 2.49s = 29; 31, 50, 6, 13.5

Compare the above modern computed lengths of the mean synodic month through time with that of the Babylonians and the Greek astronomer Hipparchus below.

Babylonians c. 300 BCE = 29; 31, 50, 8, 20 (also the Talmud)
Hipparchus' data c. 150 BCE = 29; 31, 50, 8, 9

We have seen that the Babylonian Talmud, which was released by Jewish scholars c. 600 CE, uses the exact time length of a mean synodic month that originates from ancient Babylonian astronomers at roughly 300 BCE, yet the Talmud refers back to the house of Gamaliel in the first century for this figure. Is it reasonable to think that some Israelites derived this time for the average length of a lunar month independently on their own? No it is not, because this number is slightly under one second too large based upon the above data. The use of different eclipse records for a computation ought to give a different result. The paper by Toomer points out that the Greek astronomer Ptolemy of Alexandria c. 150 CE wrote about the achievements of Hipparchus 300 years earlier, and both of them realized that picking a different pair of eclipses from which to compute the average length of a lunar month would provide a different result. Ptolemy discussed the specific nature of which eclipse records would likely produce a more reliable result, and he based this on the earlier work of Hipparchus. The reason for the use of different eclipses producing a different result is that the apparent speed of the moon as observed from the earth varies at different times of the month, at different times of the year, and at different times of the eclipse cycle known as the Saros, which is 223 mean synodic months (18.03 years). Thus any computation based upon a specific pair of eclipse observations will result in a unique value for the average length of a lunar month, although properly chosen records will provide close results.

The Babylonians began predicting the visibility of the new crescentat roughly the year 400 BCE, and this prediction is based upon an accurate understanding of the moon's cycle for repeating its speed variation, or lunar anomaly, within the Babylonian System A (see the paper by Britton 1999, especially page 244). The cycle of lunar anomaly is the Saros cycle. From roughly this time onward they would be in a good position to be able to judge which pair of eclipse records should produce an accurate figure for the average lunar synodic month. As stated above, the oldest existing Babylonian System B material is dated 258 BCE, and this system includes the fundamental parameter that Hipparchus used for the mean synodic month, which was championed by Ptolemy c. 150, and was later incorporated into the Babylonian Talmud c. 600. We have no explicit knowledge of exactly when or exactly how this length of the mean synodic month was determined within System B by the Babylonians, although it is a very reasonable conjecture that some pair of eclipse records from the same part of a Saros cycle was a key. On page 45 of Britton 2002, John Britton estimates the origin of System B to be as early as c. 330 BCE, but on page 54 his estimate for the origin of the mean synodic month is c. 300.

1. Pages 13 and 22 of Spier show that the modern calculated Jewish calendar uses the approximation for the average length of a month from RH 25a in the Babylonian Talmud, yet we now know that this came from ancient Babylonian astronomers c. 300 BCE. The Babylonian Talmud is called “Babylonian” because its Jewish authors lived in Babylonia at the time of its publication c. 600 CE, not about 900 years earlier when the Babylonian astronomers derived this figure. But other factors are also used for the modern calculated Jewish calendar, which are not due to either ancient Babylon or Hipparchus, and are not found in the Talmud. Num 10:10 shows a responsibility of the Levitical priesthood in declaring the “beginning of the months”, and thus control of the calendar and its knowledge could be expected to have been passed down from generation to generation via the hereditary priesthood. However, after the Temple was destroyed in 70 CE the Levitical priesthood vanished from Jewish history along with its influence over the calendar. No writings from this priesthood have survived from before the destruction of the Temple, except for the fact that Josephus was a priest who was born in 37 CE and died c. 100. While his writings exist, none of them were written before the destruction of the Temple, and he does not discuss when a month begins in any direct way. He never mentions any astronomical calculations being done by the ancient Jews, and neither does Philo of Alexandria (c. 20 BCE - c. 50 CE).

In order to perform the mathematical computations for general long division of fractional numbers that would be necessary for predictive astronomy, it would be necessary to utilize a number system with a base, which would therefore enable a positional notation and the use of a symbol for zero. For computational uses without a computer, modern society uses the base 10 for ordinary purposes, although modern computers use the base 2, and for the sake of human ease of readability, the base 2 is typically converted to base 16 (hexadecimal) for computer professionals. The Babylonians and Greeks used the base 60 number system for their capable calculations. After the achievements of the Babylonians and Greeks in the Eastern Hemisphere, the Mayan Indians in the Western Hemisphere used the base 20 number system.

The way that the Hebrew text of the Bible expresses numerical values indicates that the ancient Israelites did not use a positional number system with a base and a symbol for zero.

Hence, from a mathematical viewpoint along with the lack of any archaeological evidence to the contrary (although there are archaeological discoveries in the site of ancient Israel), it is safe to conclude that ancient Israel, before the destruction of Solomon’s Temple by Nebuchadnezzar in 586 BCE and the three waves of Israelite exile to Babylon from 604–586 BCE, did not possess the type of mathematical abilities that would have enabled them to perform the mathematical computations needed for success at predictive astronomy. The ancient pagan Babylonian priests were interested in astrology. They predicted the future of kings and kingdoms. They gained wealth and political prestige through this practice until Daniel told both the dream and its interpretation to the king (Daniel 2). They then lost political prestige, but their pagan practices continued as they developed horoscopy. Some of these pagan priests were the predictive astronomers. Their desire for wealth and prestige led to their efforts at computational and predictive astronomy. The Greeks had a greater interest in science for the sake of knowledge, although they too were interested in astrology and its use to gain wealth. The leisure time to devote to astronomy came from the wealth gained by astrology.

The historical evidence indicates that neither the ancient Israelites before the destruction of Solomon's Temple in 586 BCE nor the Jews after this until the destruction of the Second Temple in 70 CE sought to develop their own mathematical astronomy. Ancient Egypt before Alexander the Great did not possess any predictive mathematical astronomical knowledge, so ancient Israel could not have inherited such knowledge from them. Neither the Bible, nor archaeology, nor Jewish history give any indication that Israelites before the destruction of the Second Temple in 70 CE had advanced abilities in mathematical astronomical knowledge. It was not until the time of Alexander the Great, that ancient astronomers were able to approximately predict the time of the true conjunction.

The difference in time between the computed average time of the conjunction (based on repeated additions of the average synodic lunar month, which is employed in the modern calculated Jewish calendar) and the true conjunction is about 14 hours according to page 45 of Wiesenberg. Thus the modern calculated Jewish calendar (MCJC) is not based upon predicting the true conjunction. The Jews at the time of Moses were not using the MCJC with its adoption of the Babylonian length of the average month, and they were not able to calculate the time of the conjunction.