Aryabhata

Aryabhata, often called Aryabhata I to distinguish him from a later astronomer of the same name, is the earliest Indian mathematician-astronomer whose own work survives in full. He is the author of the Āryabhaṭīya, a compact treatise in Sanskrit verse that shaped Indian astronomy for more than a thousand years.

Almost everything we know about his life comes from a few lines in his own book and from remarks by later commentators. In the third chapter he says that when 3,600 years of the Kali Yuga, the current age in Indian cosmology, had passed, he was 23 years old. That year corresponds to 499 CE, which means he was born in 476. Some scholars think 499 was the year he wrote the book; others think it is simply an astronomical reference point he chose.

His name is correctly spelled Aryabhata, not “Aryabhatta”. Later astronomers such as Brahmagupta mention him by name many times, and the spelling with one “t” also fits the metre of Sanskrit verse.

No portrait of Aryabhata exists. Statues and pictures made in modern times, such as the statue at the Inter-University Centre for Astronomy and Astrophysics in Pune, are works of imagination.

Place value. In his chapter on mathematics Aryabhata names the powers of ten from “one” (eka) up to “a thousand million” (vṛnda) and states that “from place to place each is ten times the preceding”. This is a clear statement of decimal place value, and his rules for square and cube roots work digit by digit in such a system. Yet in the book itself he wrote numbers in words: ordinary number words for small numbers and, for the large constants of the first chapter, a code he invented in which Sanskrit syllables stand for numbers so that they fit into verse.

Zero. The Āryabhaṭīya has no symbol for zero and gives no rules for adding, subtracting or multiplying with zero. In explaining his syllable code he uses the word kha, “space”, for the places of a number; the same word later became one of the commonest Sanskrit words for zero. Historians disagree about what he knew. Georges Ifrah, quoted by MacTutor, thought it “extremely likely” that he knew a zero sign, while his English translator Walter Eugene Clark wrote that it was “still disputed” whether such a sign was in use in his time. What is certain is that the earliest surviving rules for zero as a number come from Brahmagupta in 628, and that the earliest dated Indian inscription with a zero accepted by all scholars, at Gwalior, dates from 876. “Aryabhata invented zero” is therefore a popular simplification, not something his book says.

Pi and sines. One verse gives the rule “add 4 to 100, multiply by 8 and add 62,000” for the circumference of a circle 20,000 units across, and calls the result approximate. That gives π ≈ 3.1416. The value was very good but not unique: Ptolemy in Alexandria had used about 3.14166 in the second century, and in China Zu Chongzhi (429–501), in the late fifth century, had found 355/113, correct to six decimal places. Another single verse lists 24 numbers: the differences between successive sines, or half-chords, at steps of 3° 45′ in a circle whose radius is 3,438 units. From them a whole quarter-circle table of sines can be rebuilt.

The Earth and eclipses. Aryabhata gives the number of turns of the Earth in a great cycle of 4,320,000 years as 1,582,237,500, and compares the stars’ westward motion to the way the shore seems to move for someone in a boat. He states that the Moon hides the Sun and that the great shadow of the Earth hides the Moon. A daily rotation of the Earth had been proposed long before by the Greek Heraclides of Pontus in the fourth century BCE, but within Indian astronomy Aryabhata’s claim was bold and carefully calculated, and many of his successors rejected it.

Aryabhata worked in a period that historians often call the Gupta era, when much of northern India was ruled by the Gupta dynasty from its heartland in Magadha (modern Bihar). The age is remembered for Sanskrit literature, sculpture, temple building and scientific writing. Pataliputra, the old imperial capital on the Ganges, was a major centre of learning and trade.

In the Āryabhaṭīya he says that he is presenting knowledge honoured at Kusumapura. Both Hindu and Buddhist traditions, and the seventh-century commentator Bhaskara I, identify Kusumapura with Pataliputra. A later verse describes him as a kulapa, the head of an institution, at Kusumapura. Because the great monastery of Nalanda was not far away, it is sometimes claimed that he led Nalanda, but there is no evidence for this in his own writings, and most historians treat it as speculation.

His birthplace is less certain. Bhaskara I calls him āśmakīya, “belonging to Ashmaka”, a region usually placed in the Deccan between the Narmada and Godavari rivers. Some writers have argued for Kerala in the far south, noting that many later commentaries on his work were written there. Others think he was born near Pataliputra. The evidence does not settle the question.

A local tradition in Bihar links him with an observatory at the Sun temple of Taregana, south of Patna. This too is a later tradition rather than documented history.

The Āryabhaṭīya is astonishingly short. It has four chapters (pādas). The first, the Gītikāpāda, has 13 verses: an invocation, a clever system for writing numbers with syllables, and tables of astronomical constants, including the number of revolutions of the planets in a great cycle of 4,320,000 years and a table of sines packed into a single verse.

The second chapter, the Gaṇitapāda, has 33 verses on mathematics: squares and cubes and their roots, areas and volumes, properties of circles and triangles, shadows cast by a gnomon (a vertical stick used as a sundial), arithmetic series, interest, and methods for solving equations. The third, the Kālakriyāpāda, has 25 verses on units of time and the motions of the planets. The fourth, the Golapāda, has 50 verses on the celestial sphere, the shape of the Earth, day and night, and eclipses.

The book is written in the terse style of Indian technical literature. Each verse is a memory aid, packing a rule into a few words that a student would learn by heart and a teacher would then explain. That is why the Āryabhaṭīya has always been read together with commentaries. The earliest surviving one, by Bhaskara I, dates to 629 CE.

Aryabhata also wrote at least one other work, an astronomical system in which the day begins at midnight instead of sunrise. It is lost, but later astronomers such as Varahamihira, Brahmagupta and Bhaskara I quote or describe it, and it included descriptions of instruments such as the gnomon and water clocks.

The most famous verse of the Gaṇitapāda gives a rule for the circumference of a circle. It can be translated: “Add four to one hundred, multiply by eight, and add sixty-two thousand: this is the approximate circumference of a circle whose diameter is twenty thousand.” The result, 62,832 divided by 20,000, gives π ≈ 3.1416, correct to four decimal places. Aryabhata used the word āsanna, “approaching” or “near”, which some historians read as a sign that he knew the value could only be approximated.

Where Greek astronomers such as Ptolemy tabulated whole chords of a circle, Aryabhata, like other Indian astronomers, worked with half of a chord, the ardha-jyā or simply jyā. It is the length R sin θ in a circle of radius R; divide it by the radius and you have the modern sine. His table, packed into a single verse, covers a quarter circle in 24 steps of 3° 45′; strictly speaking it lists the differences between successive sines in a circle whose radius is 3,438 units, and another verse explains how such a table can be computed. The name itself had a long journey. Arabic astronomers borrowed jyā as jība; because Arabic script usually leaves out short vowels, later readers took the word for jayb, “fold” or “pocket”, and Latin translators rendered that as sinus, from which English gets “sine”.

Another important contribution is a method for finding whole-number solutions to equations such as ax − by = c, known as indeterminate or linear Diophantine equations. Indian mathematicians called the technique kuṭṭaka, “the pulveriser”, because it breaks the numbers down step by step into smaller ones, much like the Euclidean algorithm. Such problems mattered for astronomy, where one had to find when several cycles would line up. Bhaskara I and later mathematicians refined the method.

The book also gives rules for the sums of arithmetic series and for the sums of squares and cubes of the first n numbers, formulas for areas and volumes, and methods for extracting square and cube roots. These root methods proceed place by place through the decimal digits of a number. How he actually wrote numbers while calculating is unknown; his translator Clark thought a board ruled into columns, with a blank column for an empty place, was possible. In the book itself he used his syllable code only for the large constants of the first chapter, and ordinary number words elsewhere.

Aryabhata’s most daring claim concerns the Earth. In the first chapter he gives the number of rotations of the Earth in a great cycle, and in the Golapāda he explains the daily movement of the stars with an image: just as a person in a boat moving forward sees motionless objects on the shore moving backward, so people at Lanka (a reference point on the equator) see the motionless stars moving westward. Most astronomers of his time, in India and elsewhere, believed that the Earth was fixed and the heavens turned around it.

The idea was not widely accepted. Later Indian astronomers, including Brahmagupta, rejected it, and some commentators even altered the wording of the verses so that the heavens moved instead. Centuries later the Persian scholar al-Biruni reported that followers of Aryabhata held that the Earth rotates. The text itself contains a puzzle: the verse right after the boat image says that the sphere of the stars, driven by a cosmic wind, moves westward. Clark nevertheless judged that Aryabhata’s belief in a turning Earth is clearly stated elsewhere in the book and was understood that way by his critics.

Aryabhata also gave a clear physical explanation of eclipses. He stated that the Moon and planets shine by reflected sunlight, and that a lunar eclipse occurs when the Moon enters the shadow of the Earth, while a solar eclipse occurs when the Moon covers the Sun. In popular belief, eclipses were blamed on Rahu, a demon who swallows the Sun or Moon. Aryabhata’s verses calculate the size of the Earth’s shadow and the part of the Moon or Sun that will be darkened.

His numerical values were impressively precise. His figure for the time it takes the Earth to rotate once relative to the stars is equivalent to 23 hours, 56 minutes and about 4 seconds, very close to the modern value. His length of the sidereal year is about 365 days, 6 hours and 12 minutes, only a few minutes longer than the modern figure.

Indian astronomy was closely tied to the calendar and to ritual. Priests and astrologers needed to know the positions of the Sun, Moon and planets to fix festivals, auspicious times and the lunar days called tithis. Aryabhata’s Kālakriyāpāda gives methods for calculating the mean positions of the planets for any given day, for adding intercalary months to keep lunar and solar calendars in step, and for naming the days of the week.

Earlier Indian cosmology divided time into enormous ages. Aryabhata kept the great cycle of 4,320,000 years but divided it into four equal parts, instead of the unequal ages of traditional Puranic cosmology. This departure from tradition was one reason Brahmagupta criticised him sharply.

The planetary system of the Āryabhaṭīya keeps the Earth at the centre. A planet’s true position is worked out from its mean position with two kinds of correction, applied in several steps: the manda correction, which accounts for the unevenness of its own motion, and the śīghra correction, which for the planets is tied to the Sun; both are pictured as small circles carried around larger ones. Going outward from the Earth, he places the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn and then the stars. Other Indian systems of the period work in much the same way, and historians such as David Pingree have argued that they adapt Greek models older than Ptolemy.

The Āryabhaṭīya opens its time-reckoning with a moment when all the planets were supposed to be aligned at the start of the Kali Yuga, in 3102 BCE. This epoch, shared with other Indian systems, provided a fixed starting point for computing planetary positions.

Aryabhata founded what later became known as the Āryapakṣa, the “school of Aryabhata”, one of the main traditions of Indian astronomy. Bhaskara I, writing about a century later, produced a detailed commentary in 629 and praised him as the master who had reached the far shore of the ocean of mathematics, kinematics and spherics. Brahmagupta, writing in 628, criticised many of his ideas yet discussed them at length, which shows how influential he was.

The Āryabhaṭīya was especially important in South India. In Kerala, astronomers and mathematicians produced a long series of commentaries, culminating in the work of Nilakantha Somayaji, whose commentary was written around 1500. The Kerala school of mathematics, famous for its work on infinite series, saw itself as continuing Aryabhata’s tradition.

Calendar makers throughout India used computations in the Aryabhata tradition to prepare the pañcāṅga, the traditional almanac. His numeral system with syllables was widely imitated in South Indian astronomical works.

A later astronomer, known as Aryabhata II, wrote a work called the Mahāsiddhānta around the tenth century. He should not be confused with the author of the Āryabhaṭīya.

Indian astronomical works travelled west in the eighth century, when scholars at the Abbasid court in Baghdad translated Sanskrit texts into Arabic. Through these translations, Indian sine tables and methods for calculating planetary positions entered Islamic astronomy, which later transmitted them to Europe. The exact route by which Aryabhata’s own ideas travelled is not always clear, and the Arabic sources often speak of Indian astronomy in general rather than naming him.

Al-Biruni, who wrote a famous book on India in the eleventh century, knew of Aryabhata and his followers and discussed their view that the Earth rotates. The history of the word “sine”, from Sanskrit jyā through Arabic to Latin sinus, is a small but vivid trace of this chain of transmission.

In modern India Aryabhata became a national symbol of scientific achievement. India’s first satellite, built by the Indian Space Research Organisation and launched on a Soviet rocket on 19 April 1975, was named Aryabhata. The satellite appeared on the back of the Indian two-rupee note. A lunar crater, a research institute for observational sciences near Nainital and a university in Patna also bear his name.

Aryabhata is often presented in popular accounts as the inventor of zero or as someone who anticipated the heliocentric system of Copernicus. Both claims go beyond the evidence, as the section on what the Āryabhaṭīya actually says and the advanced sections explain.

Did Aryabhata know zero? The Āryabhaṭīya uses no symbol for zero; its large constants are written in his syllable code and its other numbers in ordinary number words. However, his methods for extracting square and cube roots operate digit by digit in a decimal place-value system, and his numeration scheme assigns syllables to powers of ten, some of which may have null coefficients. Georges Ifrah and others argue that the concept of zero as a place-holder was therefore implicit. Most historians agree that decimal place-value arithmetic was in use; the question of when zero became an explicit number with its own rules is usually traced to Brahmagupta in 628.

Did he know π was irrational? The word āsanna, “approaching”, can be read as a simple admission that 3.1416 is approximate, or, more ambitiously, as an insight that no exact fraction exists. Some Indian historians of mathematics have favoured the stronger reading, but the verse itself gives no argument, so it cannot be shown that he knew π to be irrational; that was first proved in Europe by Johann Heinrich Lambert in 1761. MacTutor adds the curious observation that Aryabhata himself seems to have preferred √10 (about 3.162) in practice.

Was his system secretly heliocentric? The historian B. L. van der Waerden argued that features of Aryabhata’s planetary model, especially the role of the śīghra epicycle linked to the Sun’s mean motion, betray an underlying heliocentric theory, perhaps inherited from lost Greek sources. Noel Swerdlow and others strongly rejected this reading, noting that the text is explicitly geocentric and that similar corrections occur in systems that are not heliocentric. The general consensus is that Aryabhata proposed a rotating Earth, not an Earth orbiting the Sun.

Where do his parameters come from? Scholars such as David Pingree traced many elements of Indian mathematical astronomy to Babylonian and Hellenistic sources adapted in India, while others emphasise independent Indian developments. The relationship between Aryabhata’s system and the earlier Paitāmaha-siddhānta and Sūrya-siddhānta traditions remains an active research topic.

The Āryabhaṭīya survives in many manuscripts, mostly from South India, and usually together with commentaries. Because the verses are so compressed, the commentaries are essential for understanding them. Bhaskara I’s commentary of 629 is the oldest and most detailed; later commentaries in Kerala, including those by Parameshvara and Nilakantha Somayaji, show how the text was used and sometimes reinterpreted.

Modern scholarship on the text began in the nineteenth century. The Dutch scholar Hendrik Kern published the Sanskrit text with Parameshvara’s commentary in Leiden in 1874. Walter Eugene Clark’s English translation appeared in 1930. The standard critical edition with translation and notes was prepared by K. S. Shukla in collaboration with K. V. Sarma and published by the Indian National Science Academy in 1976; in the same series Shukla separately edited the commentaries of Bhaskara I and Someshvara, and Sarma that of Suryadeva Yajvan.

Biographical evidence is thin and partly legendary. Apart from the verse giving his age, we rely on commentators writing a century or more later, and on regional traditions that have their own interests: claims for Kerala, Bihar and the Deccan all have champions. Historians therefore separate what the text says from later attributions.

The lost midnight system is reconstructed from later works, especially Varahamihira’s Pañcasiddhāntikā and Brahmagupta’s Khaṇḍakhādyaka, which follows Aryabhata’s midnight reckoning. A work preserved in Arabic that claims to be a translation of Aryabhata has also been discussed, but its connection to him is uncertain.