The Calendar

by David Ewing Duncan

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  At roughly the same time or shortly thereafter the Indians began using a dot, a symbol that became widespread enough by the sixth century that the Indian poet Subandhu used it as a metaphor in his poem Vasavadatta:

  And at the time of the rising of the moon with its blackness of night, bowing low, as it were, with folded hands under the guise of closing the blue lotuses, immediately the stars shone forth . . . like zero dots . . . scattered in the sky as if on the ink blue skin rug of the Creator who reckoneth the total with a bit of moon for chalk.

  The Indians referred to this ‘nothing’-dot as sunya, meaning void or empty. Our word zero comes from sifr, the Arabic version of sunya, which medieval Europeans altered to ziphirum in Latin.

  Greeks of the classic age had no symbol for zero, because their numerical system did not require a zero place. But they were aware of the concept of a number that stood for nothing. Indeed, Aristotle rejected it as a non-number to be ignored, since one cannot divide by zero, or divide zero by itself. Nevertheless, Eurocentric scholars long assumed that the symbol for zero was invented by the Greeks, with no proof at all, speculating that it came from the Greek letter omicron--O--the first letter in the Greek word oudeu, meaning ‘empty’. But this unwarranted belief that Indians could not have come up with such a basic concept has given way to recognition that ancient Greeks did not really use such a symbol for zero, and that Indian mathematicians seem independently to have invented the dot and then the round goose-egg symbol. The first use of this symbol for zero in India appears in the year 876 in an inscription found in the Gwalior region south of Delhi, containing two numbers with zeros.

  This comes two centuries after Severus Sebokht’s mention of the nine Hindu numbers, though archaeologists have found the round symbol for zero in two numbers in an inscription in Malaysia the numbers 60 and 606 that dates to AD 684. The Malay peninsula was then under Indian influence. Some historians also believe a treatise on mathematics known as the Bakhshali Manuscript may have been written as early as the third century AD. It contains numbers with zeros and a fully developed decimal place-value system. The numbers include:

  The first use of zero as a fully formed number seems to have appeared around the time of Brahmagupta in the seventh century, when this great Indian mathematician tried, but failed, to explain how zero could be divided by itself. The Maya also invented a true zero in about the third century AD, using several symbols, including a half-open eye--Ɵ--which they used to indicate missing positions as they wrote out numbers to represent time intervals in their calendar.

  This explanation of zero does not quite finish our story about the mathematics needed to correct the calendar, since the year is not 365 days long, but 365.242199 days, give or take a few seconds. In other words, we have this pesky fraction to contend with, expressed here as a decimal fraction. This concept--and the ease with which we are able to represent this value--also did not come easily or all at once. Beyond the simplest divisions of a whole number, fractions posed a huge problem for humanity through most of history.

  How do you divide three sacks of grain among five people? And how do you split up a year, month, day, hour or minute into smaller parts?

  Several modern words are derived from this system--for instance, ounce and inch come from uncia.

  But these symbols are far too cumbersome and imprecise for sophisticated values and calculations. For instance, it was relatively simple for a Roman--or Bede, or Alcuin in Charlemagne’s court to write out the Latin whole number and fraction for the length of Caesar’s year, which is CCCLXV =- days--365 1/4. But try writing the true solar year of 365.242199 days--the equivalent of 365 242,199/1,000,000--in Roman numerals. No symbol exists in Latin for such a precise number, a reality that profoundly affected the pursuit of determining an accurate year. Nor is it possible to calculate in Roman numerals a value that takes into account variations in the earth’s motions, including the gradual slowing of the tropical year over the centuries.

  As long as time reckoners used the Latin system--or Greek, Egyptian or any other numerical system that lacked precise fractions they were forced to conclude that it was impossible to calculate a true year. This powerfully reinforced the belief in the Middle Ages that if such a number existed, it was known only to God, when in truth the number was simply beyond the capability of the symbols and numerical system in use at the time--and continued to be until the thirteenth and fourteenth centuries, when Europeans began broadly adopting the earliest versions of the modern decimal system.

  The idea of using decimal fractions came to Europe from the Arabs, though they were not the first to use positional notation to write out and determine fractions. Again this distinction seems to belong to the Mesopotamians, who over the millennia figured out a fraction system based on their own positional notation scheme which gave them a precision and computing power far beyond that of any other system until the European Renaissance. But because Mesopotamia’s system was based on 60 and not on a more manageable number such as 10, their remarkable discovery was limited by the complexity of carving into clay and stone place-values in negative powers of 60, which not only are indivisible for some fractions but also quickly become long and complicated symbols to write out. For instance, the length of the year in cuneiform numerals is:

  The Chinese by the third century AD had also discovered how to write fractions using their positional notation, and did so using our familiar base-10 system. But their discovery does not seem to have travelled beyond the Far East. As for the Indians, for some reason they did not develop decimal fractions, despite having base-10 positional notation for whole numbers. Instead they devised an early version of placing one number over another to represent fractions--a numerator over a denominator--that was apparently borrowed from Greek mathematicians in Alexandria, with one difference: they placed the denominator over the numerator. The bar line was introduced later by Arab mathematicians.

  Of course, the vast majority of people in ancient times had little use for fractions beyond the most simple divisions of a whole. Only a handful of mathematicians and astronomers cared to be more precise--and even they tended to simply round off numbers either to the closest simple fraction or to the nearest whole. This is undoubtedly why early astronomers, from Hipparchus and Ptolemy to Aryabhata, were able to note that the 365 1/4-day year was wrong, but seemed willing to accept this rounded-off number as tolerable enough that none called for a correction, or for reforms in the official calendar.

  When Aryabhata wrote his Aryabhatiya in 499, at the precocious age of 23, Gupta culture and learning remained at a high point. But even as he pondered pi and the position of planets, a dark cloud was fast engulfing the empire: the Huns. This eastern branch of the scourge that had hastened the crash of Rome had for years been hammering away mercilessly against the Gupta frontier to the northwest.

  By the time the Aryabhatiya appeared, the Huns had broken through the Guptas’ main defensive lines to devastate parts of northwestern India. But unlike Rome, the Guptas, with help from the Chinese in the north, had weakened the military strength of the hordes over the years to the point that the invaders were unable to thoroughly conquer the Indians or destroy their culture. During the middle third of Aryabhata’s life the Huns set up a shaky kingdom that ruled from modern Afghanistan to central India, never reaching Kusumapura. Aryabhata lived long enough to see a coalition of Indian kings and warlords drive them back into Kashmir in 542, when he was 66 years old. He also had lived long enough to see the golden age of Gupta culture slowly eroded, even if the continuity of Indian culture was preserved.

  As the political situation worsened, the spirit of open inquiry and free thinking that had thrived earlier was squelched by a turn to conservative Vedic values. This apparently got Aryabhata into some trouble with his more controversial theories, particularly his supposed hint that the earth might circle the sun. At least this seems to be the case given the vigour with which later Indian scholars, perhaps anxious to confo
rm to the more rigid orthodoxy of the day, dismiss this theory less on academic than religious grounds.

  How Aryabhata responded to his critics is unknown. But we have a clue to his true feelings, and his willingness to express them, in a short passage at the end of the Aryabhatiya. It reads like something Roger Bacon would have written as a fevered defence of science. ‘He who disparages this universally true science of astronomy,’ says Aryabhata, ‘which ... is now described by me in this Aryabhatiya, loses his good deeds and his long life.’

  But unlike Bacon, Aryabhata was revered by scholars and laymen alike, during and after his lifetime. Every great Indian mathematician and astronomer who came after him used the Aryabhatiya as the basis for their work and acknowledged his contributions. This includes Varahamihira (505-587), a contemporary of the elderly Aryabhata* who wrote an encyclopaedia that cites the master of Kusumapura, but emphasizes astrology over astronomy--a choice Aryabhata would have rejected as unscientific.

  *There may have been two Aryabhatas working at roughly the same time. Aryabhata the Elder and Aryabhata the Younger.

  The great mathematician Brahmagupta (598-665) also held Aryabhata in high esteem, incorporating some of the earlier master’s works into his own--and unfortunately editing them and adding his comments to the point that it is hard to tell what belongs to Aryabhata and what to Brahmagupta, since the originals Brahmagupta worked from have been lost. Brahmagupta’s reverence did not extend to Aryabhata’s controversial ideas. Nor did it stop him from offering corrections in his Brahinasphuta-siddhatita, written around 628, to what he considered his predecessor’s mistakes on matters ranging from the altitude of the sun’s ecliptic to Aryabhata’s measurement of the diameter of the earth.

  Aryabhata’s impact was so profound in his homeland, that in 1975 modern India honoured this ancient genius by launching a scientific satellite named the Aryabhata on an Indian Intercosmos rocket. Unlike the ideas of its namesake, the satellite failed after only four days and came crashing back into the atmosphere on 11 February 1992.

  After Brahmagupta, India continued to produce noted mathematicians, including Bhaskara (1114-1185), considered by mathematicians to be the most brilliant in his field anywhere during the twelfth century. But he was the last true standout in medieval India.*

  *In 1887 another mathematics genius was born in India, Srinivasa Ramanujan, who tragically died at the age of 33. His natural fluency and intuition with numbers has been compared to the free-ranging and eclectic style of thinking of Aryabhata and other earlier Hindu mathematicians.

  All of these men contributed mightily to the evolution of concepts that three centuries after Aryabhata’s death would continue their journey to the West via a people that in Aryabhata’s era were primitives barely known to the great civilizations of the day. Living on a vast desert to the south of the empires of Persia and Byzantium, they began stirring to life only in the final years of Brahmagupta’s life, then suddenly they burst out of their desert peninsula to begin the conquest of much of the Near East and southern and central Asia. In the process they discovered and then embraced the ancient knowledge of India, Greece and Mesopotamia, creating an unlikely amassing of ideas drawn together in what became the early medieval era’s greatest centre of learning: Baghdad.

  9 From the House of Wisdom to Darkest Europe

  It was He that gave the sun his brightness and the moon her light, ordaining her phases that you may learn to compute the seasons and the years. God created them only to manifest the Truth. He makes plain his revelations to men of knowledge.

  The Koran, c. AD 630

  In 773, some 250 years after Aryabhata’s death, a delegation of diplomats from the lower Indus River Valley arrived in the new Arab capital of Baghdad. Dressed in brightly coloured silks, turbans and glittering gems, this group probably travelled by sea from the Indus delta around the desert coast of modern-day Iran and up the turquoise waters of the Persian Gulf to the port city of Abadan--some 30 miles inland now because of silt built up over the centuries. They would then have sailed up the Tigris about 200 miles to Baghdad, passing by the hot, dry banks lined with tiers of ancient, irrigated terraces and stone cities dating back to the time of Sumer and Ur, arriving at last outside the gates of al-Mansur’s magnificent city. Half a century after the Arabs had conquered the lower Indus River Valley, in 711, this delegation was one of many dispatched by local Indian authorities to the court of Caliph al-Mansur to provide him with news about their province, and to settle outstanding disputes. They also hoped to impress the great caliph, the founder of the Abbasid dynasty, with the richness and sophistication of their country by showering him with gifts--perhaps a gem-encrusted suit of armour, a flute carved out of ivory, a highly prized falcon or a silk tapestry depicting scenes from their province.

  This particular delegation also brought with them an astronomer, undoubtedly having heard that al-Mansur was not only a mighty general and military ruler, but also a patron of the arts and sciences. The astronomer’s name was Kanaka. An expert on eclipses, he reportedly carried with him a small library of Indian astronomical texts to give to the caliph, including the Surya Siddhanta and the works of Brahmagupta (containing material on Aryabhata). Nothing more is known about this Kanaka. The first known reference to him was written some five hundred years later by an Arab historian named al-Qifti.

  According to al-Qifti, the caliph was amazed by the knowledge in the Indian texts. He immediately ordered them to be translated into Arabic and their essence compiled into a textbook that became known as the Great Sindhind (Sindhind is the Arabic form of the Sanskrit word siddhanta).

  No one is sure if this incident per se ever happened. But something like it must have, in order to bring the works of India into the sphere of the early Islamic scholars, whence they would travel to Christian Europe through Syria, Sicily and Arab-controlled Spain. A version of the Great Sindlhind would be translated into Latin in

  1126. This was one of dozens of critical documents that would contribute to the knowledge base needed to propel Europe into the modern age, and to calculate a true and accurate year.

  Kanaka allegedly visited the court of the caliph in Baghdad about a century and a half after one of the most extraordinary moments in history: the sudden maelstrom that came out of Arabia in the mid-600s. Driven by a potent fusion of religious zeal and a centuries-old martial tradition among the tribes of the desert, the armies of the Prophet Mohammed were at first a phenomenon of arms and religion, though they soon became an unlikely force for the advancement of learning. This came in part from the Prophet’s command that the faithful seek knowledge, but also because the Arabs did not follow the example of the barbari in the West, who had looted and destroyed the cities and provinces of Rome. Instead the Arabs assimilated the cultures of the peoples they conquered--much the same way the early, uncouth Romans had done centuries before when they eagerly embraced and absorbed the cultures they conquered in Greece and the Near East.

  In a sense the Arabs arrived just in time. Most of the ancient centres of learning, and the cultures that had nourished them, were in a state of exhaustion or outright collapse by the mid-600s, after decades of warfare and internal decay. To the east the Gupta era was ending as India broke up into small kingdoms and struggled to fend off fresh onslaughts from the Huns; in the Near East a long war fought between Byzantium and Persia ended with a peace treaty in 628, leaving both empires gravely weakened; to the west the barbari continued to battle over what was left of Rome.

  Not surprisingly, this period produced little original thinking and was a low point in intellectual output from the Himalayas to the British Isles--with some notable exceptions, such as Brahmagupta in India and a few scattered scholars still struggling to work in the Greek tradition within the Byzantine Empire. But even there the output was meagre as the rump of the old Roman Empire, pressed by enemies on all sides, had become more stridently orthodox. Indeed, for decades the imperium and Church had been repressing rival Christian sects,
pagans and anyone else who did not fall in line behind an increasingly strict religious dogma--including scholars.

  This religious retrenchment in Byzantium had begun under Justinian in Cassiodorus’s time. In 529 he had closed the nine-hundred-year-old Academy of Plato in Athens and had dispersed its scholars, claiming it was a hotbed of paganism.* Fearing for their lives as well as their intellectual freedom, many of these scholars had fled to Persia, where they established a kind of Academy in exile. This was a pale imitation of the original, though this community of scholars remained viable enough that when the Arabs seized Persia a century later these Greeks were able to play a major role in bringing the texts and learning of the ancient Hellenes to the attention of Arab scholars.

  *Most scholars date the end of the ancient Greek culture to the closing of the academy in 529.

  The events leading up to the meeting between al-Mansur and Kanaka began modestly. In 610, roughly 30 years after Cassiodorus’s death in faraway Italy, a 40-year-old merchant in the desert oasis and trading post of Mecca claimed to have seen the archangel Gabriel in a vision. Commanded by the angel to lead a movement to purify and complete the religious tradition of judaism and Christianity, Mohammed began to preach a simple message to the pagans in his town: one of total submission (which is what the word Islam means in Arabic) to one god: Allah.

  At first only his family and a few friends responded favourably. Nearly everyone else laughed at him, eventually forcing him and a tiny band of followers to flee Mecca in 622 for another desert oasis, nearby Medina. This later became known as the ‘Year of the Migration’ (hijra in Arabic, hegira in English), which is the starting point of the Moslem calendar--a calendar Mohammed later insisted should remain purely lunar, to differentiate it from the lunisolar calendar of the Jews and the solar calendar of the Christians.