A History of Mathematical Astronomy - Part 1


In the Fertile Crescent agriculture was gradually established after 10,000 BC, with settlements at the Neolithic town of Jericho near the Dead Sea dating back to 9000 BC. The success of Chatal Huyuk or Çatalhöyük, a large site in Anatolia that existed from ca. 7200 to after 6000 BC, is thought to have resulted from its trade in the volcanic glass known as obsidian.

The greatest change in the history of the Near East and indeed the world came with a people called the Sumerians in southern Mesopotamia, between the Euphrates and Tigris rivers. During the Uruk period (ca. 4000 to 3100 BC), the Sumerians are credited with many “firsts” in human history, from the first writing system to the first monumental statues in an urban setting. Their origin is unknown and their language has no proven connection to any other language, living or dead, yet they produced lasting literature such as the Epic of Gilgamesh.

There were other cities or proto-cities in the Fertile Crescent stretching from northern Mesopotamia into northern Syria, Anatolia and western Iran, but Uruk by 3300 bc contained a population of perhaps 40,000-50,000 people, gigantic compared to any other known settlement on the entire planet at that time. The “Uruk Expansion” during the fourth millennium BC spread its cultural influence to neighboring regions. With the growing complexity of society and the ensuing expansion of bureaucracy came the development of a system for recordkeeping which evolved into cuneiform script, the first true writing system.

In the book Egypt: The World of the Pharaohs, scholar Stefan Wimmer comments on the fact that in ancient Egypt in contrast to Mesopotamia, hieroglyphs emerged almost fully formed in the generations before the unification of the Egyptian state after 3100 BC. During this period a number of cultural characteristics similar to those of southern Mesopotamia such as cylinder seals and certain artistic motifs appeared in Egypt. In other words, we know that the Egyptians had contact with Mesopotamia just as this region was developing writing.

In Europe, the Minoans on Crete adopted a form of writing, probably inspired by Egyptian hieroglyphs, after 2100 BC. Author Jared Diamond in Guns, Germs, and Steel favors the concept of idea diffusion to explain why a numbers of societies from the Indus Valley in India to Egypt and the eastern Mediterranean region developed writing within a short period of time after the Sumerians:

“It would be a remarkable coincidence if, after millions of years of human existence without writing, all those Mediterranean and Near Eastern societies had just happened to hit independently on the idea of writing within a few centuries of each other.”

While I am sometimes critical of Mr. Diamond I agree with him in this case. Those who believe in an independent evolution of writing in Egypt and the Indus Valley will point to the fact that these writing systems do not outwardly resemble Sumerian proto-cuneiforms, but it remains possible that they imported the very concept of writing from nearby Mesopotamia. While ancient China was not as isolated as some Chinese historians like to claim, an independent development of writing here should nevertheless be considered as a possibility. If we assume that the Mayas and others in Mesoamerica had no significant contact with Eurasia then writing has been independently invented at least a couple of times in human history.

In his fine and well-researched book A History of the Ancient Near East ca. 3000 - 323 BC, second edition, scholar Marc Van De Mieroop states that in Uruk,

“a sexagesimal system, relying on units with increments of ten and six, was used to account for animals, humans, and dried fish, among other things. A bisexagesimal system, which diverges from the previous one as its units also show increments of two, was used for processed grain products, cheese, and fresh fish. Volumes of grain or surfaces of fields were measured differently.”

This sexagesimal (base 60) system was later adopted and passed on by the successive cultures that dominated Mesopotamia down to the ancient Persians and Greeks and from them on to us. We retain sexagesimal numbers today in our system for measuring time (60 minutes to an hour) and angles (60 minutes in a degree and 360 degrees in a circle), but it dates back in a straight line to the civilization of the ancient Sumerians more than five thousand years ago.

An astronomical interest can be detected in the alignment of Egyptian pyramids and temples, but rarely on the level of sophistication eventually achieved in Mesopotamia. The ceilings of the tombs of rulers from the Middle Kingdom onward, for instance in the Valley of the Kings in Luxor, contain what could be described as simple celestial maps, yet as author John North states in Cosmos, Revised 2008 Edition,

“except in the case of the calendar it does not seem to have occurred to them to seek for any deeply systematic explanation of what they observed. For all that they were in possession of a script, they seem to have produced no systematic records of planetary movements, eclipses, or other phenomena of a plainly irregular sort.”

While all later Mesopotamian civilizations built upon the foundations created by the Sumerians, at some point after 2000 BC their language ceased to be spoken yet continued to be studied because it was associated with learning, a bit like Latin in medieval Europe. Sumerian, the world’s first written language, also became the first “classical” language.

If you believe that Abraham was an historical person (far from everyone does), he is supposed to have been born in the city of Ur in the 1900s or 1800s BC when Mesopotamia came to be dominated by Semitic-speaking peoples. Akkadian was a Semitic language.  What we do know is that one of the most famous kings in Mesopotamian history, Hammurabi, ushered in what we now call the “Old Babylonian period,” the beginning of Babylon’s political dominance over southern Mesopotamia for the next 1500 years. Hammurabi died around 1750 BC. The function of his famous law code has been much debated, with some scholars claiming that it is primarily a monument presenting him as an exemplary and just king.

As author Marc Van De Mieroop writes,

“From the beginning of writing the administrators of Babylonia showed their mathematical abilities when measuring fields, harvests, numbers of bricks, volumes of earth, and many other things that were of importance to bureaucrats. The tools to calculate these had to be taught but, as with literature, the skills displayed in the school texts show a much higher level than needed in daily practice.”

The gods were believed to speak through all sorts of objects and events in the natural world, including animal entrails, dreams and celestial phenomena. Omens were important for every level of Mesopotamian society, yet astronomical observations did not become the major focus of divination until after 1500 BC. Mesopotamian bureaucrats and astronomers/astrologers gradually amassed detailed information about the movement of the planets. The quantity as well as the quality of Babylonian observations improved greatly after the eighth century BC.

By the fifth century BC, Babylonian celestial divination had expanded to embrace horoscopic astrology, which used planetary positions at the moment of the date of birth to predict individual fortunes. According to author James Evans,

“While horoscopic astrology was certainly of Babylonian origin (as, indeed, the Greek and Roman writers always claimed), it was elaborated into a complex system by the Greeks. Thus, the familiar and fantastically complicated system of horoscopic astrology with dozens of conflicting rules does not descend from remote antiquity. Rather it is a product of Hellenistic and Roman times.”

The Maya in Mesoamerica devoted much attention to divination and amassed detailed studies of the movements of the Sun, Moon and planets over long periods of time. The Inca elites in pre-Columbian South America, too, elaborated special forms of divination. The Chinese had their own ideas about the stars and divination from an early date, but may have absorbed additional ideas influenced by Babylonian astrology during the Han Dynasty by way of India.

Mesopotamian astronomy and astrology reached India at least during the Persian conquests of northwest India by the fifth century BC, along with alphabetic writing systems. Contact with Greek mathematical astronomy came after Alexander the Great’s conquests of this region and through trade with the Roman Empire in the first and second centuries AD. Indians were influenced by spherical trigonometry in the Greek fashion as well as the Babylonian material.

The Chinese lunisolar calendar with its twelve Zodiac signs (the rat, ox, tiger etc.) is still used for marking East Asian holidays such as the Chinese New Year. The earliest divinations are found on the inscribed oracle bones and turtle shells from the city of Anyang in northeastern China, widely regarded as the cradle of Chinese civilization. They concern matters of significance to the King and the state. G.E.R Lloyd elaborates in The Ambitions of Curiosity:

“Since the knowledge claimed for divination concerned the future, it held out the promise of influencing it – a prospect that state authorities could hardly ignore. The legitimacy or otherwise of the practices were not just matters of the rationality of aims and methods, for issues of state control, or its subversion, could be at stake. The unauthorised casting of the horoscope of a Roman Emperor was high treason (cf. Barton 1994) - just as in late imperial China private studies of astronomy and astrology could be criminal offences.”

In ancient Greece, cosmic regularities were seen as unchanging. In China, order in the Heavens could not be taken for granted. The Emperor acted as a mediator between Heaven and Earth. Because of this, the regulation of the calendar and the interpretation of celestial signs were matters of vital importance to the Emperor himself as the bearer of the Mandate of Heaven. The Astronomical Bureau existed for more than 2,000 years, but since everything was regulated by the Imperial court, astronomical instruments and findings were frequently treated as state secrets, which sometimes hampered scientific progress in Chinese astronomy.

While not as lasting as the stone pyramids built by the Egyptians, the mud brick ziggurats of the Sumerians must nevertheless have been impressive structures. They made a profound impression on the ancient Hebrews, who memorialized the Babylon ziggurat as the Tower of Babel, a monument to the insolent pride of humans. The White Temple at Uruk dates from around 3200 BC. The Sumerians seem to have been the first to set up monumental statues in their cities and sanctuaries. One of the earliest is the white marble female head found at Uruk in the sacred precinct of the goddess Inanna, who is mentioned in the Epic of Gilgamesh.

In the earliest Greek literature we find traces of a prehistoric Indo-European astral religion. In Homer’s the Iliad, Achilles’ shield is likened to the Earth, which is surrounded by an ocean-river, the source of all water and of the gods. In the Odyssey, the starry heaven is said to be of bronze or iron and supported on pillars. The early Greeks may not have understood that Venus was a planet rather than two different bodies, the Morning Star and the Evening Star.

To the ancient Greeks, the planets were “wandering stars.” Our word planet comes from a Greek verb meaning to wander. The modern names for the five naked-eye planets are the names of Roman divinities which were more or less equivalent to a number of Greek gods. Many people today probably know this. What many don’t know is that some of the Greek names themselves may have been derived from ancient Babylonian divinities.

Mars was often associated with war because of its reddish color, which can be seen through naked-eye observations; the ancient Egyptians called it the Red One. However, there are other parallels that are unlikely to be accidental. In ancient Mesopotamia, Ishtar was the Babylonian and Assyrian counterpart of Inanna, the moody Sumerian goddess of love and fertility, identified with the planet Venus. To the Romans, Venus was the goddess of love and fertility, their equivalent of the Greek goddess Aphrodite, who was also a symbol of love and fertility.

In Roman mythology, Jupiter was king of the gods, the equivalent of Zeus in the Olympic pantheon of ancient Greece. The name “Zeus” is Indo-European. James Evans elaborates in The History and Practice of Ancient Astronomy, which is excellent on pre-telescopic astronomy in Europe and the Middle East but contains nothing on East Asia or the Americas:


“Marduk was the most important god of Babylon. His star is the planet Jupiter. The fact that the Babylonians associated the planet Jupiter with the chief god of their pantheon is an interesting parallel to Greek practice. Moreover, Venus was associated with Ishtar, the goddess of love and fertility, and Mars with Nergal, the god of war and pestilence. These parallels are too striking to be due to chance. The Greek associations are probably the result of Hellenization of earlier Mesopotamian associations. The divine associations came into use by the time of Plato. For the early Greeks, the Sun, Moon, and fixed stars were far more important than were the planets. The motion of the Sun was intimately connected with the annual cycle of agricultural labors. The phases of the Moon governed the reckoning of months. And the heliacal risings and settings of the stars told the time of year. So, it is not surprising that Hesiod’s Works and Days (ca. 650 B.C.), which contains a good deal of practical lore about the Sun, Moon, and stars, makes no mention of the planets.”

In the eyes of Walter Burkert, a few similarities between the Epic of Gilgamesh and Homeric poetry can no longer be ignored. He is nevertheless careful to point out that philosophy in the modern sense was a Greek invention as much as was deductive proof in mathematics. As Ibn Warraq puts it in Defending the West, “what emerges is something entirely distinctive: what we call Greek civilization. The very strength of this civilization lay in its ability to learn from and improve upon the ideas, art, and literature of the Near East, Persia, India, and Egypt.”

According to the website of the American Institute of Physics,

“Sky-watchers in the ancient Middle East, Central America, and China made many observations. From their tables of numbers, they devised schemes to predict future movements in the heavens. But the explanations that the Babylonians, Mayans, and early Chinese sky-watchers devised for these movements were no more than colorful myths. Scientific cosmology – the search for a picture of the universe that would make sense with no mention of divine beings – began with the Greeks. They sought to look beyond the patterns of numbers to something fundamental…. Greek philosopher-scientists set themselves the task of envisioning the universe as a set of physical objects…. All generation and corruption occurred in the ‘sublunar’ region, below the Moon and above the Earth.”

The eminent historian of archaeology Bruce Trigger in Understanding Early Civilizations: A Comparative Study offers a comparison between seven early civilizations: ancient Mesopotamia and Egypt, pre-Imperial China, the Maya and their Mesoamerican neighbors, the Aztecs, the Incas in South America and the Yoruba and Benin peoples of West Africa. Surprisingly, the cosmologies of these very different peoples exhibited a few similarities:

“The sky and underworld planes were the exclusive realms of the gods and the dead, while the earth was shared by living people and the supernatural. These levels were interconnected, most often at the centre and around the edges of the terrestrial realm, through hills, trees, caves, and temples. The gods and supernatural energy were able to move through these gateways, conveying life-giving powers from the purely supernatural realms to the human one and back again. The earth was generally believed to be a flat plane, round or square in outline, at most a few thousand kilometers across, and surrounded by a salt-water ocean. Each early civilization and usually each city-state believed itself to be located at the centre of the terrestrial plane, which had been created especially for its benefit.”


The idea of a spherical cosmos can be attributed to sixth- and fifth-century BC philosophers such as Pythagoras and Parmenides. The mathematician Pythagoras and his followers had suggested that the Earth was spherical, not flat, by 500 BC, probably mainly on aesthetic grounds as the sphere was considered the most perfect shape. In the fourth century BC, Aristotle listed arguments in favor of a spherical Earth: the Earth casts a round shadow on the Moon during a lunar eclipse and different constellations are visible at different latitudes. Sailors knew that the mast of a sailing ship was the last thing to disappear over the horizon.


This idea of a spherical Earth was apparently unique to the Greeks; there is little credible evidence to suggest that it was ever developed independently by any non-European civilization. It had been widely adopted at the time of Aristotle and was never forgotten during the Middle Ages by those in western (but not eastern) Eurasia who knew his writings.

Eudoxus of Cnidus, who had studied with followers of Pythagoras in Italy and in Athens under Plato at his Academy, created the first serious geometrical model of planetary motion. He produced a planetary theory based entirely on spherical motions, but there is reason to believe that he personally viewed this as a purely mathematical model, unlike many of those who followed him. Aristotle regarded the sphere of the fixed stars as a real, material sphere. David C. Lindberg elaborates in his valuable book The Beginnings of Western Science:

“Greek astronomy took a decisive turn in the fourth century with Plato (427-348/47) and his younger contemporary Eudoxus of Cnidus (ca. 390-ca. 337 B.C.). In their work we find (1) a shift from stellar to planetary concerns, (2) the creation of a geometrical model, the ‘two-sphere model,’ for the representation of stellar and planetary phenomena, and (3) the establishment of criteria governing geometrical theories designed to account for planetary observations. Let us consider these achievements in some detail. The two-sphere model devised by Plato and Eudoxus conceives of the heavens and the earth as a pair of concentric spheres. To the celestial sphere are affixed the stars, and along its surface move the sun, the moon, and the remaining five planets. The daily rotation of the celestial sphere accounts for the observed daily rising and setting of all the celestial bodies.”

Eudoxus was a gifted mathematician who was largely responsible for some of the finest sections of the Elements, a treatise of 13 books written by the Greek mathematician Euclid after 300 BC. While it is a compendium organized from previously existing texts, Euclid did give the work an overarching structure and created a brilliant synthesis. Euclid lived in the early Hellenistic period and was probably born a few years before Archimedes of Syracuse (ca. 287-212 BC). It is assumed that Euclid taught and wrote at the Library of Alexandria.

Aristotle came from a privileged family in northern Greece, his father being royal physician to the king of Macedonia. From 343 BC Aristotle became tutor for the son of the Macedonian king, a young man who became known as Alexander the Great (356-323 BC) when he began his expansive conquests. The Museum and Library in Alexandria at the Mediterranean coast of Egypt was founded around 300 BC by Ptolemy I Soter (ca. 367-283 BC), the Macedonian general who became ruler of Egypt after the death of Alexander. “Museum” here means a “Temple of the Muses,” a location where scholars could meet and discuss philosophical and literary ideas. The period from 600 to 300 BC is called the Hellenic period whereas the period after Alexander’s conquests is known as the Hellenistic era.

Heraclides of Pontus, a younger contemporary of Plato in the fourth century BC, suggested that the Earth rotates on its axis once in twenty-four hours. This would explain the daily rising and setting of all of the celestial bodies but was by most of his contemporaries considered implausible because it violated sensory evidence indicating that the Earth is stationary.

The Greek astronomer and mathematician Aristarchus of Samos (ca. 310-230 BC) compared the Earth-to-Sun distance with the Earth-to-Moon distance and figured the former to be twenty times the latter. The correct ratio is about 400:1, but he apparently realized that since the Sun was far away it had to be much larger than the Earth. It may have been this realization that led him to suggest that the Sun was the center of the universe. His heliocentric theory was overwhelmingly rejected in Antiquity because it seemingly violated common sense, everyday observations and Aristotelian physics. If the Earth orbits the Sun, why doesn’t everything that is not nailed down go flying off on its own? Heliocentrism was successfully revived almost two thousand years later by the Polish astronomer Nicholas Copernicus (1473-1543), who could have been aware of the much older heliocentric theory suggested by Aristarchus.


Archimedes was the first mathematician to derive quantitative results from the creation of mathematical models of physical problems on Earth, for example the basic principle of hydrostatics. The principle of the lever was known before this, but no-one had created a mathematical model for it before Archimedes. His genius as an engineer of military devices kept the invasion forces at bay for months, but he was allegedly killed by a Roman soldier after the capture of Syracuse in 212 BC, even though the commander wanted to spare his life.

Johan Ludvig Heiberg (1854-1928), a philologist and historian of mathematics at the University of Copenhagen, Denmark, inspected a manuscript in Constantinople in 1906 which contained previously unknown mathematical works by Archimedes. A long-lost text shows that he had begun to discover some of the principles of calculus. Eudoxus had paved the way for Archimedes’ later study of volumes and surfaces in his work On the sphere and cylinder.

The Greek scholar Eratosthenes (276-194 BC) produced a good estimate of the size of the Earth. He knew that the Sun was never vertically overhead in Alexandria; on June 21 it is off by more than 7 degrees, 1/50 of a full circle, yet at Syene (Aswan) near the Tropic of Cancer it appears directly overhead at summer solstice. The Earth currently has an axial tilt of about 23.5 degrees, which was estimated by Eratosthenes. This is the cause of the seasons as our planet moves around the Sun; when the Northern Hemisphere receives the maximum amount of sunlight, in the regions south of the Tropic of Cancer a vertical stick will cast no shadow at noon whereas north of the Arctic Circle there will be 24 hours of sunlight. In December when the Earth is tilted the other way vis-à-vis the Sun the Southern Hemisphere enjoys summer.

Knowing the rough distance from Syene to Alexandria (ca. 5000 stadia), Eratosthenes could find the Earth’s circumference by assuming that this was 1/50 of the full circle and that the Sun was very far away. There were a few sources of error, but his methods were theoretically sound and his estimate was certainly in the right range. Exactly how accurate his value of 250,000 stades was is disputed as there were several “stades” of different lengths in use.

Eratosthenes’ estimate of the size of the Earth was remembered in the Middle East and Europe, where learned people knew from the ancient Greeks that the Earth is round (East Asians did not know this prior to contact with Europeans).

The last of the great Greek mathematicians of the Hellenistic period was Apollonius, who was active before 200 BC. Victor J. Katz explains in A History of Mathematics, second edition:

“Apollonius was born in Perge, a town in southern Asia Minor, but few details are known about his life. Most of the reliable information comes from the prefaces to the various books of his magnum opus, the Conics. These indicate that he went to Alexandria as a youth to study with successors of Euclid and probably remained there for most of his life, studying, teaching, and writing. He became famous in ancient times first for his work on astronomy, but later for his mathematical work, most of which is known today only by titles and summaries in works of later authors. Fortunately, seven of the eight books of the Conics do survive, and these represent in some sense the culmination of Greek mathematics. It is difficult for us today to comprehend how Apollonius could discover and prove the hundreds of beautiful and difficult theorems without modern algebraic symbolism. Nevertheless, he did so, and there is no record of any later Greek mathematical work that approaches the complexity or intricacy of the Conics.”


We know very little about the bureaucrats who were the originators of Babylonian mathematical astronomy. This is in sharp contrast to the individualism of Greek society, where the different thinkers criticized their rivals by name. Moreover, while the Babylonians developed a sophisticated system of arithmetical computations for predicting astronomical phenomena such as eclipses, their purposes were strictly calendrical. As far as we know, they never visualized the motions of the planets in terms of geometrical or mechanical models.


According to James Evans,

“one of the critical developments of this period was the rise of Greek geometry, which led rapidly to the mathematization of Greek astronomy. Notable geometers of this period were Euclid, Archimedes, and Apollonius of Perga. Apollonius (ca. 225 B.C.) seems to have been the first to experiment with combinations of deferent circles and epicycles in an attempt to provide an explanation for the motions of the planets, Sun, and Moon. The work on the solar and lunar theories was carried to a high level by Hipparchus (ca. 140 B.C.). For the first time in Greek astronomy, it became possible to make quantitative predictions of the future positions of the Sun and Moon, as in the prediction of eclipses.”


Hipparchus, one of the greatest mathematical astronomers in history, was born in Nicaea in Bithynia but spent much of his life in Rhodes. His recorded observations span the years 147 to 127 BC. He ranged over all aspects of contemporary astronomy – mathematical, observational and instrumental. He made his own star catalog based on extensive observations, has been credited with the discovery of the precession of the equinoxes and was the first to introduce a system for measuring the brightness of stars, with six levels of magnitude. Today’s system follows the same logic. He was the probable inventor of stereographic projection, the crucial element of the astrolabe, and played a major role in the development of trigonometry.


Hipparchus calculated that the Moon’s mean distance from the Earth is 63 times the Earth’s radius. The true value is about 60 times. He was also a crucial figure in the Greek borrowing of astronomical results and mathematical procedures from the Babylonian tradition. This development was made easier by the fact that following the conquests of Alexander the Great, in the second century BC Mesopotamia was, like Egypt, ruled by a Greek-speaking dynasty.

The most influential Greek astronomer was Claudius Ptolemaeus, or Ptolemy, a scholar who made astronomical observations from Alexandria in Roman-ruled Egypt during the years 127-141 AD. His name indicates that he came from a Greek family and was also a citizen of Rome. His work represented the culmination of Greek scholarship in several disciplines. His great astronomical treatise, later known as the Almagest, dominated astronomical thought in Europe and the Mediterranean region up to and including Nicolaus Copernicus in the sixteenth century, and even longer than that in the Middle East. It included and superseded earlier astronomical works, above all those by Hipparchus from the second century BC. His Tetrabiblos (“Four books”) was a standard astrological text for many centuries.

While geocentric (Earth-centered) Ptolemaic astronomy is widely familiar, many people don’t know that he was an excellent geographer for his time as well. The recovery of Ptolemy’s Geography around 1295 AD revolutionized Byzantine cartography, just as it revolutionized Western European cartography when it was translated into Latin about a century later. It was very popular among Renaissance humanists during the fifteenth century. In addition to this, his Optics was arguably the most important work on that subject in Antiquity. Euclid’s Optics was almost wholly geometrical with little concern for theories of vision. Ptolemy used Euclid’s law of reflection but went far beyond with a theory of refraction as well.