Aryabhata and the Aryabhatiya (AD 499): Indian Astronomy

Aryabhata (born AD 476) was an Indian mathematician and astronomer of the Gupta age who in AD 499 composed the Aryabhatiya, a short Sanskrit treatise in verse. In just over a hundred verses it set out a decimal place-value arithmetic, a value of pi of 3.1416, a table of sines, methods for solving equations and an astronomy in which the Earth rotates on its axis. It became one of the founding texts of classical Indian science and influenced astronomers from India to the Islamic world.

Background: science in the Gupta age

Aryabhata worked during the later years of the Gupta Empire, a period of political stability in northern India and of flourishing Sanskrit literature, art and learning. Indian astronomy, or jyotisha, had long been used to fix the calendar and the timing of rituals, and older texts such as the Vedanga Jyotisha set out simple rules for the motions of the sun and moon.

From the early centuries AD, Indian astronomers also absorbed ideas from Greek and Babylonian astronomy, transmitted through trade and through texts such as the Yavanajataka. The result was the siddhanta tradition: comprehensive astronomical systems built on large time cycles, mathematical models of planetary motion and trigonometry. Aryabhata’s work is the earliest fully surviving treatise of this tradition attributed to a named author.

Mathematics was developing alongside astronomy. Indian scholars were by then using a decimal place-value system for writing numbers, and other works, such as the Bakhshali manuscript, show the same kind of practical arithmetic in use.

The life of Aryabhata

Almost everything known about Aryabhata comes from one verse in his own book and from later commentators. In the section on time reckoning, he records that he was 23 years old when 3,600 years of the Kali Yuga had elapsed. By the usual reckoning of that era, this points to AD 499 as the date of composition and AD 476 as his birth year.

He also says that he expounds the knowledge honoured at Kusumapura, a name generally identified with Pataliputra, the old imperial capital at modern Patna. The commentator Bhaskara I, writing in 629, called him “Ashmakiya”, linking him with Ashmaka, a region in the Deccan, so he may have been born elsewhere and moved north to study. Later claims that he headed an institution at Nalanda are not supported by early evidence.

Aryabhata may have written at least one other work, sometimes called the Aryabhata-siddhanta, which used a day beginning at midnight rather than at sunrise. It is lost, known only from quotations by later authors.

The Aryabhatiya

The Aryabhatiya is extremely compact. It contains 121 verses (some editions count slightly differently) divided into four sections:

• Gitikapada (13 verses): the large time units, the numbers of revolutions of the planets in a great cycle, and a table of sines, all written in a special alphabetic notation.
• Ganitapada (33 verses): mathematics, including arithmetic, geometry, series and equations.
• Kalakriyapada (25 verses): the reckoning of time and the motions of the planets.
• Golapada (50 verses): the celestial sphere, the shape of the Earth, eclipses and related questions.

The work was written to be memorised, not read like a modern textbook. Its terse verses were meant to be explained by a teacher, and much of what scholars know about how they were understood comes from commentaries, beginning with that of Bhaskara I.

To pack large numbers into short verses, Aryabhata invented an alphabetic number system. Consonants stood for values and vowels for powers of 100, so that very large figures, such as the revolutions of the planets over millions of years, could be expressed in a few syllables.

Mathematical achievements

Aryabhata’s most famous result is his value for pi. He wrote that if you add 4 to 100, multiply by 8 and add 62,000, the result is the approximate circumference of a circle whose diameter is 20,000. That gives 62,832 divided by 20,000, or 3.1416, correct to four decimal places. Significantly, he used the word asanna, “approaching”, to show he knew the value was an approximation.

He gave rules for finding square and cube roots that rely on the decimal place-value system, and formulas for the sums of arithmetic series and for the sums of squares and cubes of natural numbers. He also described the kuttaka (“pulveriser”) method for solving linear indeterminate equations, problems that ask for whole-number solutions, which later Indian mathematicians developed further.

In trigonometry, the Aryabhatiya contains a table of 24 sine differences at intervals of 3 degrees 45 minutes, using a circle of radius 3,438 units. The Indian sine (jya) is the direct ancestor of the modern sine function; the word passed into Arabic as jiba and was later mistranslated into Latin as sinus.

The geometry section also covers the areas of triangles, trapezia and circles, the properties of right-angled triangles, and the use of a gnomon (a vertical stick) and its shadow to measure distances and heights. Such practical rules were essential both for astronomy and for surveying and construction.

Not every rule was correct. Aryabhata’s formulas for the volume of a sphere and of a pyramid are inaccurate, a reminder that his work belongs to a developing tradition rather than a finished science.

Astronomical ideas

Aryabhata’s astronomy is notable for several bold statements. He taught that the Earth rotates on its axis once a day and that the apparent westward motion of the stars is caused by this rotation. He compared it to a person in a moving boat who sees the stationary objects on the shore apparently moving backwards. Many later Indian astronomers rejected this idea and kept a fixed Earth.

He explained solar and lunar eclipses in terms of shadows: a lunar eclipse occurs when the moon enters the Earth’s shadow, and a solar eclipse when the moon covers the sun. He also stated that the moon and planets shine by reflected sunlight. These explanations contrasted with the popular belief that eclipses were caused by the demon Rahu swallowing the sun or moon.

His parameters were impressively accurate for his time. He gave the length of a sidereal rotation of the Earth as about 23 hours 56 minutes 4 seconds, close to the modern value, and a sidereal year of about 365 days 6 hours 12 minutes, only a few minutes longer than modern measurements. He also departed from tradition by dividing the great age (mahayuga) into four equal parts, a choice later criticised by Brahmagupta.

Influence and legacy

The Aryabhatiya founded a lasting school of Indian astronomy. Bhaskara I wrote a detailed commentary in 629, and the work remained especially influential in South India, where Kerala astronomers such as Nilakantha Somayaji (1444 to 1544) commented on it a thousand years later. Brahmagupta, writing in 628, admired and criticised Aryabhata in roughly equal measure.

Indian astronomical knowledge reached Baghdad in the 8th century, and Arabic writers referred to Aryabhata under forms such as “Arjabhar”. In the early 11th century the scholar al-Biruni discussed his ideas in his great work on India, though by then two different astronomers named Aryabhata were sometimes confused.

In modern times, the Dutch scholar Hendrik Kern published the Sanskrit text in 1874, and Walter Eugene Clark’s English translation of 1930 brought it to a wider audience. India named its first satellite Aryabhata, launched on 19 April 1975, and research institutes, a university in Patna and a lunar crater honour him today.

Myths and facts

Aryabhata is often credited with “inventing zero”. His work clearly assumes the decimal place-value system, which needs a zero, but the Aryabhatiya does not contain a zero symbol, and the system grew up gradually in India over several centuries.

He is also sometimes described as having proposed a heliocentric solar system long before Copernicus. Some modern scholars have argued that elements of his planetary models hint at an earlier sun-centred source, but Aryabhata’s own system, as stated and as understood by his commentators, is geocentric. His genuinely radical claim was the Earth’s daily rotation.

Finally, there is no authentic portrait of Aryabhata. All statues and images of him, including those on university campuses and postage stamps, are modern imaginative works.

Timeline

• AD 476: Aryabhata is born, according to the age he gives in his own text.
• AD 499: Aryabhata composes the Aryabhatiya at the age of 23.
• AD 628: Brahmagupta discusses and criticises Aryabhata’s ideas in the Brahmasphutasiddhanta.
• AD 629: Bhaskara I writes his commentary on the Aryabhatiya.
• 8th century: Indian astronomical works begin to be translated into Arabic in Baghdad.
• c. 1030: Al-Biruni discusses Aryabhata in his book on India.
• 1874: Hendrik Kern publishes the Sanskrit text in Leiden.
• 1930: Walter Eugene Clark publishes an English translation.
• 19 April 1975: India launches its first satellite, named Aryabhata.

Why it matters

The Aryabhatiya is one of the earliest surviving works to combine place-value arithmetic, trigonometry and mathematical astronomy in a single system. Its accurate value of pi, its sine table and its idea of a rotating Earth show the originality of Indian science in the Gupta age. Through commentators in India and translators in Baghdad, Aryabhata’s work became part of the long chain of ideas that led to modern mathematics.