The Greeks of the classical era believed that they had acquired their knowledge of astronomy from Mesopotamia and Egypt. Herodotus credits the Babylonians with inventing the gnomon, the shadow-marker of the sundial, which the Greeks used in determining the hours of the day and the seasons of the year. He writes that ‘knowledge of the sundial and the gnomon and the twelve divisions of the day came into Greece from Babylon.’ According to Herodotus, ‘The Egyptians by their study of astronomy discovered the solar year and were the first to divide it into twelve parts – and in my opinion their method of calculation is better than the Greek.’
Herodotus also attributed to the Egyptians ‘The invention of geometry, which the Greeks brought back to their own country.’ The idea was that the Egyptians first developed geometry so that they could redivide their land after the Nile valley was inundated by the annual flood. They also would have needed an advanced knowledge of geometry in the design of huge monuments like the pyramids, which so impressed the Greeks when they first saw them after establishing their trading colonies on the Nile delta.
Although Herodotus credits the Egyptians with the invention of geometry, their geometrical knowledge was for the most part restricted to computing the areas of triangles, rectangles, trapezoids, and circles, for which they used the relatively accurate value of 3.16 for π, and for finding elementary volumes, such as that of a truncated pyramid. But, as Otto Neugebauer remarks in his discussion of Egyptian mathematics, abstract geometry, ‘in the modern sense of this word, owes very little to the modest amount of basic geometrical knowledge which was needed to satisfy practical ends’. Neugebauer also remarks that: ‘Egyptian astronomy had much less influence on the outside world for the very simple reason that it remained throughout its history on an extremely crude level which had practically no relations to the rapidly growing mathematical astronomy of the Hellenistic age.’
The one area in which the Egyptians influenced Greek astronomy was in the use of their calendar, as Herodotus pointed out. The Egyptian civil calendar was a completely practical one, consisting of 12 months of 30 days each, unrelated to the phases of the moon, with five additional days at the end of each year. Neugebauer remarks that ‘the Egyptian calendar became the standard astronomical system of reference which was kept alive through the Middle Ages and was still used by Copernicus in his lunar and planetary tables.’ He goes on to say that the Egyptian calendar was revived in Persia by King Yazdigerd, just before the Sasanian dynasty fell to the forces of Islam; nevertheless the so-called ‘Persian’ years of the Yazdigerd era, beginning 632 AD, ‘survived and are often referred to in Islamic and Byzantine astronomical treatises’.
The Egyptians originally began their year with the so-called heliacal rising of the star Sothis (Sirius), that is when it rose shortly before the sun, after an interval of about seventy days when it was invisible because of its closeness to the sun when observed from the earth. This had special significance because the heliacal rising of Sothis occurred around the time of the annual flood that inundated the Nile valley. The Egyptian calendar year of 365 days had a systematic error since the time between summer solstices, as measured by the Babylonians, is about 365.25 days. This error amounted to about a day in four years, a month in approximately 120 years, and a whole year in 1,456 years, a period called the Sothic cycle. It was noted in 139 AD that the beginning of the civil year coincided with the heliacal rising of Sothis. And so similar coincidences of the civil and astronomical calendars must have taken place in the past at intervals of 1,456 years; that is, in 1317 BC, 2773 BC, and 4,229 BC Some Egyptologists take 2773 BC as the date when the Egyptian civil calendar was created, while others hold that it was 4229 BC, although some say that the problem of establishing such a reference point is more complex.
The Egyptians divided the region of the celestial sphere along the ecliptic into 36 zones called decans, a Greek word stemming from the fact that each decan spanned ten degrees, one-third of a zodiacal sign. The Egyptians created a star clock in which the heliacal rising of certain bright stars, one in each of the decans, mark the passing of the hours. Since there were 36 decans this would have led to a division of the complete cycle of day and night into 36 hours. But since the reference point for the astronomical year was the heliacal rising of Sirius, which is in summer, when the nights are shortest, only 12 decans can be seen rising during the hours of darkness. Thus the night was divided into 12 hours and likewise the day. Originally the hours were not of equal length and changed with the seasons, but in the Hellenistic period, when Greek culture dominated Egypt, the day was divided into 24 hours of equal length. At the same time the adoption of the sexagesimal system in Greek astronomy led to the division of the hour into 60 minutes and ultimately the further division of the minute into 60 seconds.
One branch of science in which Egypt excelled was medicine. Egyptian medicine is distinguished by the fact that its practitioners recognised physical symptoms as the first signs of disease, whose treatment was based on their experience of previous cases that they had treated and recorded, although magic and religious rites still played a large part in their practice.
The Greeks almost certainly did acquire some geometry from the Egyptians, but they probably learned far more mathematics from the Babylonians, whose widespread commercial activities brought them in contact with the Greek colonies that had been established at the beginning of the first millennium BC along the Aegean coast of Anatolia, and its offshore islands.
The Mesopotamian and Egyptian interest in astronomy stemmed from their astral religions, in which the celestial bodies, the sun, moon, planets and stars, were worshipped as divine. Their mathematical astronomy was developed through the need to coordinate their observations of the heavenly bodies and to create a calendar.
These celestial deities appear in the Babylonian creation epic, the Enamu Elish, whose earliest known version dates to about 1800 BC The Enamu Elish describes the mythical events that led up to the creation of the world and the birth of mankind, telling of how Anu, god of the upper heavens, aided by his son Marduk, defeated the forces of chaos and created order in forming the universe, which was a flat disc of earth floating in a vast ocean, roofed over with the celestial sphere.
After their victory Marduk was given charge of the world, built the city of Babylon at its centre, and created mankind to populate the earth and serve the gods. Marduk then set in motion the sun, moon and stars, so that by their eternally recurring motions mankind could tell the time of day and night and determine the passing seasons of the year, creating a celestial clock and calendar. Observation of the celestial bodies and the study of their motions became tasks of the Babylonian priest-astronomers, working in the great towers known as ziggurats, which were both temples and astronomical observatories, one of them appearing in the Bible as the Tower of Babel.
Babylonian astronomy was also motivated by the belief that there is an intimate connection between the celestial and terrestrial regions. Because of this events in the celestial sphere, such as eclipses of the sun and moon, were interpreted as signs of things to come on earth. Thus a close study of celestial motions can be a guide to predicting future events on earth, the belief that underlies the pseudo-science of astrology, one of the principal motivations for observing the heavens from antiquity up until the beginning of modern times.
The earliest examples of writing in Mesopotamia, as well as in Egypt, date to about 3300 BC Mesopotamian writing was in cuneiform, or wedge-shaped, script on clay tablets, which hardened quickly and left a permanent record. Most of the known cuneiform tablets with mathematical contents are from the Old Babylonian period, ca. 1800 BC According to Neugebauer, one of those who first studied these tablets, ‘No astronomical texts of any scientific significance exist from this period, while the mathematical texts show the highest levels ever attained in Babylonia.’
There are also a few mathematical texts from the Seleucid period, from around 300 BC to the beginning of the Christian era, when Mesopotamia was ruled by a dynasty founded by one of the successors of Alexander the Great. The level of these texts is comparable to those of the Old Babylonian period, though, as Neugebauer remarks, ‘The only essential progress that was made consists of the “zero” sign in the Seleucid texts.’ Neugebauer notes that the Seleucid period ‘has furnished us with a great number of astronomical texts of a most remarkable character, fully comparable to the astronomy of the Almagest’, referring to the famous work written by the Greek scientist Claudius Ptolemaios (Ptolemy) of Alexandria in the mid-second century AD.
The Babylonian mathematical texts are of two types: ‘table texts’ and ‘problem texts’. The most common of the first type are multiplication and division tables, which were evidently used in the education of scribes. According to Neugebauer, there are also ‘tables of square and square roots, of cubes and cube roots, of sums of squares and cubes needed for the numerical solution of certain types of cubic equations, of exponential functions, which were used for the computation of compound interest, etc.’ The latter tables in particular would indicate that the principal motivation in the development of Babylonian mathematics was its application in economics, and this can be seen in some of the problem texts, one of which represents ‘the calculation of the harvest yield of the province of Lagash for the third year recorded in the text’.
The sexagesimal system first came into use in the Old Babylonian era; it was still in use during the Seleucid period, when, according to Neugebauer, ‘this method became the essential tool in the development of a mathematical astronomy, whence it spread to the Greeks and then to the Hindus.’ This system survives in the modern world in the division of the circle into 360 degrees, where each degree equals 60 minutes of arc measure, and each minute is 60 seconds of arc, as well as in the division of the hour into 60 minutes of time measure, where each minute equals 60 seconds.
The Babylonians were the first to develop place-value notation in mathematics, where the value of a symbol depends on its place in the number. As an example, writing 111 in the decimal system, the same symbol has the value 1 (10 to the power zero), 10 (10 to the power one) or 100 (10 to the power two), depending on where it is placed in the number. In the sexagesimal system the same symbol would be expressed as 60 to the successive powers zero, one, two, etc.
The Babylonians were familiar with the Pythagorean theorem, but as a relationship between numbers rather than one in geometry. Some of the texts deal with problems in geometry, such as finding the radius of a circle that circumscribes an isosceles triangle, or determining the areas of regular polygons. These and other texts led Neugebauer to remark that Babylonian mathematics at its highest level ‘can in many respects be compared with the mathematics, say, of the early Renaissance’.
Many of the Babylonian cuneiform tables for multiplication and division are combined with tables of weights and measures needed in everyday commercial life. This was the beginning of metrology, the creation of uniform measures and physical standards of length and weight. Examples of these Mesopotamian measures and their physical standards have survived, notably those in the Museum of the Ancient Orient in Istanbul but also in collections in Chicago, London and Berlin, including bronze bars marked with the different units of length and bronze masses corresponding to weights of various amounts.
The earliest cuneiform astronomical tablets date from the middle of the second millennium BC, when for several years during the reign of Ammisaduqa records were noted of the appearances and disappearances of Venus, the Babylonian Ishtar, who was worshipped as a fertility goddess. The dates are given in the contemporary lunar calendar, an important factor in determining the chronology of the Old Babylonian period. These observations seem to have provided data for omens of things to come, which Neugebauer remarks are ‘the first signs of a development which would lead centuries later to judicial astrology and, finally, to the personal or horoscopic astrology of the Hellenistic age’. He notes that there were at least seventy tablets of this sort with a total of some 7,000 omens, extending over several centuries and reaching its final form ca. 1000 BC One tablet records a prediction based on a disappearance and reappearance of Venus in the seventh year of the reign of Ammisaduqa: ‘If on the 21st of Ab Venus disappeared in the east, remaining absent in the sky for two months and 11 days, and in the month Arakhsamma on the 2nd day Venus was seen in the west, there will be rains in the land; desolation will be wrought.’
Two texts from ca. 700 BC, though undoubtedly based on older material, contain a summary of the astronomical knowledge of their time. The first deals mostly with the fixed stars, which are arrayed in three zones spanning the celestial equator, with the central one some thirty degrees wide, an early attempt at mapping the heavens. The second tablet concerns the moon and the planets as well as the seasons, the latter determined by observation of shadows cast by a gnomon, the winter and summer solstices occurring when the noon shadow is longest and shortest, respectively, the spring and autumn equinoxes when the sunrise and sunset shadows are due east and west. Neugebauer remarks that ‘The data on risings and setting [of the stars], though still in a rather schematic form, are our main basis for the identification of the Babylonian constellations.’
Tablets from ca. 700 BC contain systematic observations of court astronomers who served the Assyrian emperors. The observations recorded in these tablets include eclipses of the sun and moon, where it was noted that solar eclipses only occurred at the time of new moon, the end of a lunar month, while lunar eclipses took place when the moon was full, in the middle of the month. The Greek astronomer Ptolemy would seem to have had access to this data, for he notes that he had records of eclipses dating back to the time of Nabonassar (747 BC).
Twelve constellations, known to the Greeks as the signs of the zodiac, each of them about thirty degrees wide, were chosen to chart the progress of the sun in its yearly motion through the stars. Greek astronomers of the Hellenistic era defined the sidereal year, the time taken by the sun to make one complete circuit of the zodiac. The month was measured by observing the lunar cycle from new moon to full moon and back to new moon again. New moon is when the moon is between the earth and sun so that it is showing its dark side; full moon is when it is on the far side of the earth from the moon and its full disc is visible. The point of this cycle that is easiest to observe is the first crescent, which appears a day or two after new moon above the western horizon after sunset. One lunation, a lunar month, is the time between two successive appearances of the first crescent, which can be either 29 or 30 days, averaging about 29.5 days over the course of 12 months. Twelve lunar months is equal to approximately 354 days. Thus a purely lunar calendar, such as the one generally used in the Islamic world, will fall out of phase with the year of the seasons by close to 11.25 days each year. At first the Babylonians adjusted for this by adding a thirteenth month every three years or so. Then, early in the Seleucid period they devised a scheme that the Greeks called the Metonic cycle, in which there were 12 ordinary years of 12 months each interspersed with 7 intercalary lunar years of 13 months each. This cycle produced the calendar of Seleucid Mesopotamia, which had an error of only one day in 350 years, as measured by the predicted appearance of a new moon. The Metonic cycle also formed the basis for the Jewish and Christian calendars as well as two of the earliest astronomical calendars of India.
An advance in mathematical astronomy made during the Seleucid period was the introduction of the great circle in the celestial sphere known as the ecliptic, which traces the path of the sun among the fixed stars. This was the first step in mapping the heavenly bodies on the celestial sphere, a procedure that was fully developed by Greek astronomers of the Hellenistic period.
Another advance made during the Seleucid period was the ability to predict whether a given month would have 29 or 30 days. The Babylonian scribes solved this problem by recording the lengths of the passing months over a very long period of time and identifying the factors, such as the angle of the ecliptic with the horizon, that determined whether a lunation would be 29 days or 30. They did this by a study of the various cycles involved, the earliest example of a scientific theory, the collection of observational data that was subjected to mathematical analysis to predict a measurable result. A similar analysis was made of the synodic periods of planetary motions, that is the time of recurrence of their cyclical motions as seen from the earth. The tables of observations that provided the dates for these studies were almanacs which the Greeks called ephemerides. These are represented by somewhat less than 250 cuneiform tablets, more than half of which are lunar and the rest planetary, according to Neugebauer, who notes that there are also about seventy tablets describing the mathematical procedures for analysing this data.
Neugebauer, in summarising his discussion of Babylonian mathematics and its influence on Greek mathematicians and those of later civilisations, concludes that ‘All that we can safely say is that a continuous tradition must have existed, connecting Mesopotamian mathematics of the Hellenistic period with contemporary Semitic (Aramaic) and Greek writers and finally with the Hindu and Islamic mathematicians.’
The spread of astrological belief was the principal reason for the transmission of astronomical knowledge from one culture to another, such as from Mesopotamia to the Greek world and then to India. Neugebauer also noted that ‘the terminology as well as the method of Hindu astrology are clearly of Greek origin; for example the names of the zodiacal signs are Greek loan words.’ He also remarked that ‘it seems reasonable to assume that Babylonian methods, parameters and concepts reached India in two ways, either via Persia or the Roman sea routes, but only through the medium of Hellenistic astronomy and astrology.’
The system used in Babylonian astrology had each day ‘ruled’ by one of the seven moving celestial bodies, i.e., the sun, moon and five planets. The order in which these bodies appear in Babylonian horoscopes is Sun – Moon – Jupiter – Venus – Mercury – Saturn – Mars. Greek horoscopes had them in the order Sun – Moon – Saturn – Jupiter – Mars – Venus – Mercury. Eventually this changed to the order that is used in modern horoscopes: Sun – Moon – Mars – Mercury – Jupiter – Venus – Saturn, an arrangement that gave the days of the week their names in the European languages.
The Babylonian astronomer Berossos, who moved to the Greek island of Cos ca. 270 BC, may be a direct link in the transmission of Mesopotamian knowledge to the Greeks, but his fragmentary extant works contain no writings on mathematical astronomy. Nevertheless, as Neugebauer remarks, ‘Babylonian influence is visible in two different ways in Greek astronomy; first, in contributing basic empirical material for the geometrical theories we have outlined...; second, in a direct continuation of arithmetical methods which were used simultaneously with and independently of the geometrical methods.’
The Babylonian mathematics and astronomy that was absorbed by the Greeks was passed on in turn to the Arabs, some of it, as we will see, through the Hellenised people of south-eastern Anatolia and Mesopotamia, and some through the Hindus after they acquired it from the Greeks, such was the ebb and flow of knowledge through the interconnected cultures of East and West.