These crude concepts were written down in Sanskrit verse, and were critical to an early understanding of shapes and their relationship to one another--including versions of the Pythagorean theorem and early geometric algebra.* Eventually they turned this
body of knowledge skywards to measure the planets and the stars, which led to sophisticated attempts to measure time, including astrological predictions of the future based on the movements of the sun and of the zodiac.
*The Pythagorean theorem is one of the most fundamental concepts in mathematics. It is critical for making basic astronomical observations for anyone wanting to use the stars or sun to measure time. The theorem says that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It is named for the Greek Pythagoras (sixth century BC), though several cultures discovered it independently.
The age of the sulvasutras ended around AD 200, during a period of political instability that lasted until the early fourth century, when the Gupta dynasty seized most of northern India and launched Hindu India’s classic age. Taking up where the sulvasutras left off, Gupta astronomers in the fourth and early fifth centuries made great strides in mathematics and astronomy, recording them in a series of texts known as siddhatitas, or ‘systems’ of astronomy. Written in the two hundred years before Aryabhata began working, they provided him with the universe of fundamental concepts he used for his own work--including estimates of pi, basic rules of trigonometry, the motion of the planets and stars, and the length of the year.
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 (102) 6 tens (101) and 5 digits (100), 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:
These number symbols were a great improvement over stick-like signs, but still presented problems for calculating or recording complicated equations and large numbers. This is why a positional system was such a phenomenal breakthrough--a leap of inspiration made by a long-forgotten Mesopotamian who undoubtedly became frustrated with writing out large numbers. Perhaps he was a scribe assigned the unenviable task of counting barrels of wine coming and going from Ur’s royal palace. Or an architect designing a ziggurat but running out of space on a clay tablet as he made his calculations, and so he invented a quick shorthand to save space. Here is what 365 looks like in cuneiform with its positional notation:
It was this problem of unwieldiness--and more--that the Indians solved by inventing our system of nine numeric symbols sequenced in positional notation, later adding zero for the tenth symbol.
How exactly the Vedics of India figured out this brilliantly simple scheme is another mystery, though they might have been inspired to transform Mesopotamia’s base-60 positional system into their own base-10. Some historians also speculate that a connection exists between Indian numbers and ancient Chinese rod numerals, which also have symbols for 1 through 10 used--after the third century AD--in a positional system.
Whatever its origin, these symbols that eventually became our own first appear in carvings on stone columns across north India as early as 250 BC or before, when Hindu mathematics was making the transition to a positional system. Written out in the early Hindu script known as Brahmi, the first nine numbers look like this:
The evolution from this version of Brahmi to a ten-digit positional notation is not entirely clear. Historians suspect that the motivation to drop the Brahmi symbols beyond the number nine came from the demands of the Hindu religion, which uses a calendar encompassing enormous spans of time to date its creation myths. These form a religious chronology stretching back millions of years, requiring the manipulation of large numbers--which is far easier when one uses powers of ten. The Indian use of counting boards also encouraged the development of number symbols that were simple and few in number.
The timing is also uncertain. Aryabhata was certainly aware of positional notation and apparently used it in his day-to-day calculations. But because he wrote his treatises in metrical verse he used words and letters to represent numbers--the equivalent of us writin
g out ‘twenty-nine’ rather than 29--in an attempt at mathematics as poetry.
The first known use in India of the nine-digit positional system has been discovered on a plate and dated to the year 595. The number is a date--346--written in a decimal positional notation.
The first outside mention of the Hindus’ system of nine numbers comes in 662 from the Syrian Severus Sebokht, an academic and bishop who lived in a Greek community founded a century earlier by scholars fleeing Athens after Justinian closed Plato’s Academy, which he had accused of fostering paganism. Apparently Sebokht became piqued at his colleagues’ disdain for any knowledge beyond the Greek sphere. Writing about the Hindus, he cites their ‘subtle discoveries of astronomy . . . their valuable methods of calculation, and their computing that surpasses description. I wish only to say that this computation is done by means of nine signs’.
But nine does not make ten, meaning the system was not complete without zero, a concept critical to understanding the advanced mathematics needed to create an accurate calendar. Zero developed as Indians using the nine numbers for calculations found themselves needing to keep an empty column on their counting boards to represent ‘nothing’, an idea they transferred to writing out numbers by leaving a space. But this could be confusing, since a space could mean either an empty position in a single number or the space between two separate numbers. To avoid confusion somebody along the way decided to make something of ‘nothing’.
Who was first to scratch out a symbol for zero is yet another mystery. In Mesopotamia a symbol for the empty position appears late for this ancient civilization, arriving around the time of Alexander’s invasion or just after, represented by two small wedges placed obliquely:
The Calendar Page 15