Aryabhatiya

01Twenty-three years old

Aryabhata dates himself. In the Kalakriyapada he says that when sixty times sixty years had passed, and three quarters of the yuga, he was twenty-three. That is Kali 3600, or 499 CE, counting the Kali Yuga from 3102 BCE. Most scholars read 499 as the year of the book; a few think he only gave his age at the epoch of his tables and wrote a little later. Either way the work belongs to the last years of the Gupta age, and it is the oldest dated Indian astronomy whose author we can name.

02Ashmaka and Kusumapura

The second pada opens with a bow to Brahma, the earth, the moon, the planets and the stars, and says Aryabhata sets out the knowledge honoured at Kusumapura, that is Pataliputra. His commentator Bhaskara I, writing in 629, calls him Ashmakiya, the man of Ashmaka, and Nilakantha of Kerala says the same; Ashmaka was a country on the Godavari. So he may have been born in the south and taught in Magadha. The last verses give the tradition’s own account of the book: by the grace of Brahma he brought up the jewel of true knowledge from the ocean where true and false lie mixed, using the boat of his own mind.

03A number in a word

To fit an astronomy into thirteen opening verses, Aryabhata made his own code. The consonants from ka to ma stand for 1 to 25; ya to ha stand for 30 to 100; the vowels a, i, u, ṛ, ḷ, e, ai, o, au shift a figure up by powers of a hundred. So the sun’s revolutions in a Mahayuga are written khyughṛ, which reads 43,20,000. The earth’s turns on its axis in the same period come out as 1,58,22,37,500. The code is hard to pronounce and was hardly used after him, but it let a whole table sit inside a single line of arya metre.

04Four padas, 121 verses

The book has four quarters. The Gitikapada, thirteen verses in giti metre, holds the code, the revolutions of the planets in a yuga, the orbits and nodes, and the sine table. The Ganitapada, thirty-three verses, is pure mathematics. The Kalakriyapada, twenty-five verses, reckons time: years, months, days, the yuga and its quarters, and the mean and true motions of the planets. The Golapada, fifty verses, is about the sphere: the earth hanging in space, the horizon at different places, day and night, the shadow cone, eclipses. A student learnt it by heart and a teacher unpacked each line.

05π to four places

The Ganitapada gives square and cube roots, the areas of triangles and circles, the volume of a sphere, arithmetic series and sums of squares and cubes, and the rule of three. Verse 10 says: add four to a hundred, multiply by eight, add 62,000, and that is the circumference of a circle of diameter 20,000. The ratio 62,832 to 20,000 is 3.1416. Aryabhata calls the figure asanna, approaching, and the Kerala commentator Nilakantha later read that word to mean that the true ratio can never be written exactly as a fraction.

06Sines and the pulveriser

One verse of the Gitikapada packs twenty-four sine differences for a quarter circle, at steps of 3¾ degrees, on a radius of 3,438 minutes: 225, 224, 222, 219 and on down to 7. This is the oldest surviving sine table, and the word jya, a bowstring, passed through Arabic jiba into Latin sinus and so into every school today. The last verses of the Ganitapada give the kuttaka, the pulveriser, which breaks down an equation such as ax − by = c step by step until it yields whole-number answers, the method astronomers needed to find when planets return to the same place.

07The turning earth

In the Golapada Aryabhata writes that just as a man in a boat going forward sees the still trees on the bank move backward, so the fixed stars seem to move due west at Lanka. The earth turns. He also explains eclipses plainly: the moon is water, the sun fire, the earth’s shadow is darkness; the moon covers the sun, and the great shadow of the earth covers the moon. He gives the moon and planets as lit by the sun. Brahmagupta and others rejected the turning earth, and some later copyists even changed his wording to keep the earth still, but his tables stayed in use in Kerala and the south for a thousand years.

08Four equal quarters

Aryabhata keeps the Mahayuga of 43,20,000 years but divides it into four equal quarters for his planetary reckoning, and he says time itself has no beginning and no end, its years and months starting together on Chaitra Shukla Pratipada. The Smritis and Puranas give the four yugas in the ratio 4:3:2:1. Brahmagupta, in his eleventh chapter, criticised Aryabhata sharply on this point. Later astronomers in the Aryabhata school answered that his equal quarters were a device for calculation, used to set the planets’ mean motions, and did not deny the yugas of dharma.

09Commentaries and editions

Bhaskara I wrote the first surviving commentary in 629, and his own Mahabhaskariya spread Aryabhata’s methods. Kerala became the book’s second home. Suryadeva Yajvan wrote the Bhataprakasha in the twelfth century, Parameshvara the Bhatadipika in the fifteenth, and Nilakantha Somayaji a long bhashya around 1500. Most surviving manuscripts are palm leaves in Malayalam script. Hendrik Kern printed the text with Parameshvara’s commentary at Leiden in 1874, Walter Clark translated it into English in 1930, and K. S. Shukla and K. V. Sarma brought out the critical edition for the Indian National Science Academy in 1976.

10Baghdad to Sriharikota

Arab astronomers knew a Zij al-Arjabhar, a set of tables named after him, and al-Biruni in the eleventh century quotes the followers of Aryabhata by name. Through such tables Indian sines and methods reached Baghdad and, later, Europe. At home the book shaped every siddhanta that came after it, whether it agreed or argued. On 19 April 1975 India’s first satellite went into orbit carrying his name, and a lunar crater and an institute of observational science at Nainital are named after him too.

What it contains

1. 1
   Gitikapada (13 verses)

   Also called the Dashagitika: ten verses in giti metre framed by an opening salutation and a closing verse. It gives the alphabetic number code, the revolutions of the sun, moon, planets and earth in a Mahayuga, the sizes of orbits, nodes and apsides, and the table of twenty-four sine differences. Everything later in the book is computed from these numbers.

2. 2
   Ganitapada (33)

   Mathematics proper: place value, squares and square roots, cubes and cube roots, areas of triangle, trapezium and circle, the volume of a sphere, the value of π and the construction of sines. Then the gnomon and shadows, arithmetic progressions, sums of squares and cubes, the rule of three, fractions, and equations. It closes with the kuttaka for indeterminate equations.

3. 3
   Kalakriyapada (25)

   The reckoning of time: the units from the year down to the vinadi, the day of Brahma, the yuga and its quarters, and the different kinds of month and year. It explains mean and true motion through eccentric circles and epicycles. Verse 10 holds Aryabhata’s own date.

4. 4
   Golapada (50)

   The celestial sphere: the earth round and hanging in space, its rotation shown by the simile of the boat, the horizon and day length at different latitudes, the risings of the signs. The second half computes the shadow cone and eclipses of sun and moon. The last verses give the book’s origin and a warning to those who would corrupt it.

Famous verses

चतुरधिकं शतमष्टगुणं द्वाषष्टिस्तथा सहस्राणाम्। अयुतद्वयविष्कम्भस्यासन्नो वृत्तपरिणाहः॥

Add four to a hundred, multiply by eight, and add sixty-two thousand: this is the approximate circumference of a circle whose diameter is twenty thousand.Aryabhatiya, Ganitapada 10

षष्ट्यब्दानां षष्टिर्यदा व्यतीतास्त्रयश्च युगपादाः। त्र्यधिका विंशतिरब्दास्तदेह मम जन्मनोऽतीताः॥

When sixty times sixty years and three quarters of the yuga had passed, twenty-three years had then gone by since my birth.Aryabhatiya, Kalakriyapada 10

अनुलोमगतिर्नौस्थः पश्यत्यचलं विलोमगं यद्वत्। अचलानि भानि तद्वत् समपश्चिमगानि लङ्कायाम्॥

Just as a man in a boat moving forward sees a still object moving backward, so at Lanka the fixed stars are seen moving due west.Aryabhatiya, Golapada 9

चन्द्रो जलमर्कोऽग्निर्मृद्भूच्छायापि या तमस्तद्धि। छादयति शशी सूर्यं शशिनं महती च भूच्छाया॥

The moon is water, the sun is fire, the earth is earth, and its shadow is darkness. The moon covers the sun, and the great shadow of the earth covers the moon.Aryabhatiya, Golapada 37

सदसज्ज्ञानसमुद्रात् समुद्धृतं ब्रह्मणः प्रसादेन। सज्ज्ञानोत्तमरत्नं मया निमग्नं स्वमतिनावा॥

By the grace of Brahma, the precious jewel of true knowledge, sunk in the ocean of true and false knowledge, has been brought up by me with the boat of my own intellect.Aryabhatiya, Golapada 49