A History of Mathematics and Mathematical Astronomy

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This essay contains new material. A slightly shorter version was originally published in four parts at various sites. See: Part 1, Part 2, Part 3, and Part 4.


Astronomy and Natural Philosophy in Ancient Times

While writing this history, the MacTutor History of Mathematics website maintained by John J. O’Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland proved invaluable to me. They have created extensive online biographies of hundreds of mathematicians and astronomers from around the world from Antiquity until the turn of the twenty-first century. I have found the biographical information they provide to be generally reliable and have therefore widely consulted their Internet entries when searching for background information, in addition to the entries at the Encyclopædia Britannica Online.

It is difficult to speak of “science” in prehistoric times. Perhaps the closest we can get is the systematic study of the heavens. Archaeoastronomy is the intersection between astronomy and archaeology. The patterns of stars in the night sky were far more familiar to people in ancient times than they are to us, who often suffer from light pollution from electric lights.

Paintings on cave walls and ceilings from prehistoric times, often depicting large wild animals, have been found in South Africa, Southeast Asia, Australia and South America, but some of the oldest and most spectacular ones have been discovered in Europe. Hundreds of cave paintings were created in the Chauvet Cave in south France from around 30,000 BC onward. Lascaux is the setting of a complex of caves in southwestern France with beautiful prehistoric cave paintings and spectacular drawings of bulls, horses and other animals. They were painted during the Upper Paleolithic, the final phase of the Old Stone Age, and are estimated to be more than 16,000 years old. Other paintings exist in the Cave of Altamira in Spain, dating back to at least 13,000 BC. German researcher Michael Rappenglueck believes that he has found a prehistoric map of the night sky among the Lascaux paintings. This is plausible, but we simply don’t know for sure what the function of these artistic drawings was.

According to Paul Mellars in The Oxford Illustrated History of Prehistoric Europe, “One possibility is that some of the major centres of art production (such as Lascaux in south-west France, or Altamira in northern Spain) served as major ritualistic or ceremonial centres — perhaps the scene of important ceremonies during regular annual gatherings by the human groups. Alternatively (or in addition) the production of the art could have been in the hands of particular chiefs or religious leaders who used the creation of the art, and associated ceremonial, to reinforce and legitimate their particular roles of power or authority in the societies. Clearly, all this lies in the realm of speculation. What is clear is that cave art is not uniformly distributed throughout Europe, and is concentrated in areas which (on other, independent archaeological grounds) are known to have contained some of the highest and densest concentrations of human populations.”

During the last Ice Age, much of the Northern Hemisphere was covered by thick ice sheets. Central Europe resembled the tundra of present-day Siberia. At the height of the Last Glacial Maximum around 20,000 BC, land temperatures were about 20 °C lower than they are today. After the end of the Ice Age (ca. 13,000 BC) came the gradual establishment of a milder climate similar to today’s from 9500 to 8000 BC. Because of this, the flora and fauna of the European continent changed rapidly, with species such as the woolly mammoth disappearing. The ice smelting following the retreat of the great glaciers changed the face of Europe dramatically. Much of what was then dry land is now underwater and vice versa. The same goes for other regions in Asia and the Americas as the sea level rose considerably worldwide.

During the Neolithic or New Stone Age, settled communities adopted agriculture, starting in the Balkans close to the Near East. One well-preserved natural mummy from the Copper Age, the intermediate stage between the Stone and Bronze Ages when early metal tools were developed, was found in 1991 in the Alps between Italy and Austria. The mummy from about 3300 BC, named Ötzi the Iceman, apparently died a violent death. He had many small tattoos, a cloak made of woven grass, a pair of leggings, a loincloth and excellent shoes. His coat was made of the hide of the domestic goat and he had a bearskin cap and a belt of calf’s leather.

Ötzi’s equipment consisted of 18 different types of wood and demonstrates that he and his contemporaries possessed excellent knowledge of natural materials and herbs. He carried a dagger with a flint blade, a bow and arrow set and above all a fine copper axe. While the Alpine region had rich copper deposits, only the wealthy could at this time afford copper tools, which indicates that the Iceman’s family were reasonably well-off. Ötzi himself may have been a shepherd who also had to be able to hunt and repair his clothing and equipment.
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By the fourth millennium BC, people had been living in fixed dwellings for some time in much of Central Europe, and food was procured from farming and animal rearing. Among the plants that were cultivated were naked wheat, einkorn, emmer wheat, barley, poppy, flax and peas. In addition to the traditional hunting, gathering and fishing, domesticated cattle, pigs, sheep and goats were used as sources of meat and provided leather, milk and possibly wool.

The Goseck Circle in north-central Germany dates back to ca. 5000 BC, one of a number of similar structures in Central Europe. It is proof that Neolithic Europeans observed the heavens with greater accuracy than was previously supposed and is one of a rising number of archaeological finds aided by aerial photography. John North in his book Cosmos, 2008 Edition, writes about early European astronomy. Many attempts have been made to reconstruct the belief systems of the peoples responsible for these astronomical monuments:

“There are numerous indications of cults of the Sun and Moon, not all of them stemming from the orientation and planning of large monuments. One of the most interesting finds was that made in 1902 at Trundholm on Zealand (Sjælland, Denmark), of a Bronze Age horse-drawn disk, dating perhaps from roughly 1400 BC. There can be little doubt that this had solar significance. The Sun is shown being pulled by a horse in several crude Swedish rock carvings of much the same date. An equally rich discovery, this time from Germany, was of a disk of bronze 32 centimeters in diameter, studded with gold shapes that related to the heavens in some way. Found near Nebra at the end of the twentieth century and now known as the Nebra disk, it came more specifically from Mittelberg — a modest hill in the Ziegelroda Forest, between Halle and Erfurt. It seems to have been discovered within a pit inside what had once been a Bronze Age palisade and complex of defensive ditches.”

The Nebra sky disk from ca. 1600 BC was at first assumed to be a forgery (of which there are unfortunately quite a few in museums around the world), but closer studies eventually revealed it to be most likely authentic. The Trundholm disk or Sun chariot dates from around 1400 BC and shows a horse-drawn vehicle with spoked wheels. The whole group measures 60 centimeters in length, and the disk has a bright side covered with gold leaf. Horse-drawn chariots with spoked wheels are associated with the second phase of the Indo-European expansion and spread across Eurasia all the way to China in the second millennium BC.

According to the book Indo-European Poetry and Myth by Martin L. West, the words for “Sun” are related in nearly all the Indo-European branches. Ancient Greek writers observed what they took to be Sun-worship among other Indo-European speaking peoples such as the Persians and the Thracians, and the Germanic tribes were attributed a form of solar worship by Roman writers. The Slavs, too, were regularly credited with Sun-and Moon-worship by chroniclers and clerics. The Sun appears in Russian folklore in female persona as “Mother red Sun.” The chorus in the Greek play Oedipus Tyrannus (Oedipus the King) by Sophocles (ca. 430 BC) swears to Oedipus by the Sun god Helios that they wish him no harm. The Franks in the seventh century AD, although converted to Christianity, still had the habit of swearing by the Sun. Oaths by the Sun, Moon etc. are mentioned in Old Irish and Norse literature as well.

There is much evidence for the circular Sun being associated with a wheel, or that the Sun-god has a chariot drawn by a horse. The Trundholm disk is not unique; fragments of similar Sun-disks have been found elsewhere in northern Europe. There are also solar disks from the second millennium BC further south, in Greece and the Aegean region. There seems to be a mythology related to a many-legged animal, perhaps as an expression of speed and stamina. Slovak and Russian folklore tells of an eight-legged horse that draws the Sun. Although he has no apparent solar association it is conceivable that there is a connection from this creature to Odin’s treasured eight-legged horse Sleipnir in Norse mythology.

From roughly 4500-2500 BC, a belt of megalithic monuments (large stone structures) stretched along the Atlantic coastlands of Western Europe, Britain, the Iberian Peninsula and certain western Mediterranean islands. In Sardinia, numerous nuraghes or towers of large stones were built in the second millennium BC or earlier, many of which still exist today. Some of their entrances may have had lunar or solar orientations, but their usage is uncertain.

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