The Calendar (20 page)

Aryabhata grew up during the final years of the Gupta golden age, when India was a world centre of art, science, literature and architecture. Learning was considered a sacred duty, and educated Hindus were expected to know not only the basics of reading, writing and numbers but also to be adept at poetry, painting and music. This was the age of the
Kama Sutra,
the text that treats love as a fine art, offering alongside lovemaking positions a list of ‘arts to be studied, together with the
Kama Sutra’.
These include swordplay, composing poetry, ‘playing on musical glasses filled with water’, chemistry, teaching parrots to speak, grammar, tattooing--and mathematics.

Gupta India was hardly a paradise for everyone. Governed by a strictly enforced caste system, the poor endured a life of crushing poverty similar to that in many Indian villages today, little changed since Aryabhata’s day--crowded clusters of straw-thatched huts, dusty markets filled with burlap sacks of rice and peppers, and lean men trudging to and fro to work small plots of land. Still, excavations in Gupta centres attest to the large numbers of merchants, artisans and others in a large middle class who enjoyed a prosperity on a par with the golden age of Rome, which had been a major trading partner with Gupta India until its collapse. Archaeologists sifting through Gupta ruins have found heaps of coins and blown glass from Rome; and from Roman sites as far away from the subcontinent as Pompeii, others have unearthed Indian statuettes, vases, mirrors and busts of Roman men with Indian hairstyles.

Aryabhata’s birthplace is unknown. Nor does anyone know what he looked like, though he himself tells us he lived in the busy imperial capital of Kusumapura. Today the city is a hot and hauntingly quiet stretch of drooping palms, buzzing flies and crumbled ruins that extend some 12 miles along the banks of the Ganges, near modern Patna. This is in north-eastern India, 250 miles north of Calcutta and just 100 miles south of the sudden rise of the Himalayas. At its height the city was filled with throngs of people: beggars disfigured by disease, rich traders in white robes, musicians playing cymbals and flutes, silk-clad Brahmans averting their gaze to avoid making eye contact with someone from a lower caste, and priests with hair tinted by henna toting statues of gods and goddesses. Enormous, airy palaces lined the Ganges, alongside imposing conical temples studded with statuary and ornaments. The entire city was shrouded with a gauze-like veil of incense, smoke and dust.

A leading instructor at a school near Kusumapura, Aryabhata spent most of his life collecting and compiling everything ever written in India about the stars, geometry, numbers and time reckoning in his magnum opus, the
Aryabhatiya,
a slim volume written in Sanskrit verse. And while only 123 metrical stanzas long, it packs an enormous amount of information in what became a handy volume of mathematical and astronomical concepts passed down and commented on over the centuries. Some of it is highly accurate, some not, a contradiction that prompted a famed Arab mathematician named Ibn Ahmad al-Biruni (973-1048) to comment that Hindu mathematics offers two types of nuggets: common pebbles and costly crystals.

Aiyabhata starts his poem with an invocation to Brahma, ‘who is one in causality, as creator of the universe’. He then divides his work into three parts: on mathematics (
ganita
), time reckoning (
kalakriya
) and the sphere
(gola).
In the section on time reckoning Aryabhata describes the Hindu calendar, including measurements of the months, weeks and year, and various time spans relating to Vedic mythology over the course of millions of years. In the section on astronomy he estimates the length of the solar year at 365.3586805 days, some 2 hours 47 minutes and 44 seconds off from the true year in Aryabhata’s era, which equalled 365.244583 days.* He also gets the diameter of the earth almost right, at 8,316 miles, but is wildly off on his estimated orbits of the sun, moon and planets. Aryabhata believed that the earth was a sphere that rotated on its axis, and he understood lunar eclipses as the shadow of the earth falling on the moon. Some historians have even detected what one calls ‘glimmerings in his system ... of a possible underlying theory’ that the earth might revolve around the sun, a possible nod toward the truth about our heliocentric solar system a thousand years before Copernicus.

*Because the tropical year is steadily slowing, the year in Aryabhata’s era was slightly shorter than our current year of 365.242199 days. The difference between then and now is about seven seconds.

In his section on maths Aryabhata gives formulas for the areas of a triangle that are correct, and areas for a sphere and a pyramid that are not. He calculates pi to be 3.1416, another near hit that is so close to the value given by Claudius Ptolemy some three hundred years earlier that it is possible Aryabhata was influenced by the great Alexandrian astronomer, though no direct link is known. Aryabhata wrote a famous stanza giving his value for pi originally in Sanskrit verse:

Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle of which the diameter is 20,000.

Unfortunately Aryabhata does not explain how he arrived at his formulas and calculations. Nor does he offer proof for what ends up being a catalogue of arbitrary rules. One senses that the
Aryabhatiya
was intended as more of a supplement or summary for people already familiar with the concepts than a comprehensive encyclopaedia or theory of mathematics. He may have written down details elsewhere in a work now lost, or perhaps as exercises for his students.

Aryabhata is also credited with writing a work called the
Kliandakhadyaka,
which means ‘food prepared with candy’, possibly referring to the pleasure it gives. But the original has been lost. Only a heavily edited and annotated version exists, reworked by another renowned Indian mathematician, Brahmagupta (598-665).

A debate has long simmered over where Arayabhata’s ideas and the corpus contained in the
sulvasutras
and
siddhantas
came from. Indian historians have long insisted that it sprang up purely as the product of indigenous genius, with origins possibly going back to the dawn of civilization on the Indus River in about 2500 BC. This is when the ancient Harappa culture began to flourish in cities made of mud brick that since have all but crumbled away, making them difficult to learn about. Still, archaeologists have unearthed evidence--building designs and measuring devices--that suggest the enigmatic Harappa did master fundamental mathematical principles. Possibly these were passed on to the Aryan-Hindus who stormed down from the north to conquer the Harappa and seize most of northern India, though the history of this period is so murky that no concrete link can be made.

A more definite influence came from Greece after 326 BC, when Alexander seized northwest India. In his wake came the concepts of Pythagoras, Meton, Eudoxus and Alexander’s instructor, Aristotle. The conqueror’s armies also stirred up and brought with them the scientific knowledge of other cultures swept into his brief empire, including Egypt and Mesopotamia. The Greek hegemony in northwest India lasted only a few years, falling apart soon after Alexander’s death in 323 BC. But Greek knowledge and culture lingered as Greek traders established thriving enclaves in India and established the lucrative trade routes west that persisted throughout the Hellenistic and Roman eras.

This allowed Indians an opportunity to absorb Greek ideas about planetary theory and geometry. One of the
siddhantas,
the Paulisha Siddhanta, may even be named after a minor astrologer from Alexandria, Paulos Alexandros (fourth century AD). Certainly this work contains striking similarities to Claudius Ptolemy’s trigonometry and astronomy, on which Paulos based his work--including a value for pi nearly identical to that later identified by Aryabhata.

The Chinese may have been another influence. They maintained a vigorous enough commerce with India that the two cultures swapped styles of clothing and architecture and even words. This was particularly true after Buddhism spread across the Middle Kingdom in the late Han Dynasty and during the period from 220 to 589, known as the Six Dynasties. No direct evidence exists of mathematical ideas being transferred between China and India, though the lifetime of the great Chinese mathematician Tsu Ch’ung Chi (430-501)--who came up with the most accurate estimation of pi in the world until the European Renaissance--overlaps with that of Aryabhata, who wrote his
Aryabhatiyia
two years before Tsu’s death. Tsu also measured the precise time of the solstices, building on the work of another brilliant Chinese astronomer and court astrologer, Zhang Heng (AD 78-139), who corrected the Chinese lunar calendar in the year AD 123 to bring it into line with the seasons. Tsu also proposed reforms in 463 to China’s lunar calendar, which apparently were rejected.

But no influence in India was apparently more significant for our calendar--and the mathematics and equations needed to fix it--than that of another cradle of civilization: the Tigris-Euphrates Valley in Mesopotamia. Or so it seems, despite the look of evidence for direct links between India and Mesopotamia on matters of mathematics and the calendar. For instance, no manuscripts exist to tell the tale of an Indian scholar visiting ancient Sumer in such-and-such year. Still, many of Vedic India’s mathematical and astronomic concepts seem strikingly similar to some of those used in the Near East, such as the
sulvasutra
rules for construction using Pythagorean triads--which appears in Babylon before it does in India. Other shared concepts include ideas about fractions, algebra, polygonal areas and applied geometry that appear first in Mesopotamia and later in the
sulvasutras
and
siddhantas.

It seems inconceivable that ancient mathematicians and time reckoners from Ur and Harappa, and later from Babylon and Vedic India, remained completely ignorant of one another during the many centuries of commerce between the Tigris-Euphrates region and India. Some Indians almost certainly picked up a little cuneiform, the writing of Mesopotamia for four thousand years, perhaps from watching a Babylonian merchant scribbling down figures on a wax tablet on the coast of Sind, or from a Mesopotamian ship captain calculating wages to pay his porters in Gujarat.

However the contact may have occurred, it seems likely that somewhere over the millennia the Mesopotamians sparked an idea that led to one of the great mathematical discoveries in history: the system of arranging numbers that mathematicians call ‘positional notation’, now used by virtually the entire world. Among many other things, this made an accurate calendar and higher mathematics possible.

In positional notation, numbers are arranged in a sequence whereby each number stands for itself multiplied by a base number that increases by one power of the base with each place. For instance, in our base-10 system the number 365, representing a rounded-off approximation of the year in days, is drawn from a set of ten symbols, 1-2-3-4-5-6-7-8-9-0, that are arranged to increase tenfold with each place. So we have 3 hundreds (10
²
) 6 tens (10
¹
) and 5 digits (10
⁰
), the digits referring to the original source of the base-10 system--counting with one’s fingers.

It is a concept so central to our modern system of numbers--and our way of life--that we hardly think about it, though this was not the case throughout most of human history. Indeed, the only culture to invent a true positional notation system in preclassic ancient times was Mesopotamia, whose mathematicians stumbled on it almost four thousand years ago-- predating all other cultures by millennia.

To fully appreciate the significance of positional notation and a number such as 365, one has to realize that for most of human history people used either their fingers or bulky, hard-to-manipulate symbols representing ever-increasing numbers.

The first written numbers seem to have been stick-like signs scratched onto bones or rocks long before written languages were invented. We still use a version of them today to count small numbers of things that accumulate over short periods of time: yellow-breasted warblers sighted during a morning hike; runs batted in during an afternoon baseball game; or the number of patients seen at a well-baby clinic every hour. For instance:

It takes several minutes just to write out this number--never mind using it to add or subtract, or to write out and perform a more sophisticated calculation such as determining the angle of the earth to the sun, or the shape of a temple along the Euphrates or the Ganges. This led early civilizations to devise more compact system of symbols, often closely related to early forms of written languages. For example, the Egyptians invented a hieroglyphic-inspired sequence of numbers: