Aryabhata: the Aryabhatiya · Chapter 8 of 8

#What Aryabhata got right and wrong

#The achievements

Looking back over the Aryabhatiya:

Astronomy

• Periods of the Moon and planets of extraordinary accuracy: the synodic month to within a second, the sidereal day to a fraction of a second, Mars's period to four decimals (Dasagitika 1–2; Chapter 2).
• The Earth's daily rotation as the cause of the stars' daily motion (Dasagitika 1, 4; Gola 9).
• A physical theory of eclipses, with methods to compute their times, extent and appearance (Gola 37–47).
• The Moon and planets shine by reflected sunlight, half of each sphere lit (Gola 5).
• A complete spherical astronomy for times, shadows and day lengths (Gola 18–36).
• An epicycle model that gave good positions for centuries (Kalakriya 17–25).

Mathematics

• A clear statement of the decimal place system (Ganita 2).
• π ≈ 3.1416, called "approximate" (Ganita 10).
• A sine table and a second-difference rule for computing it (Dasagitika 10; Ganita 12).
• Sums of series, including squares and cubes (Ganita 19–22).
• The kuttaka, a general method for equations in whole numbers (Ganita 32–33).

#The errors

• The volume of a pyramid: half instead of a third of base times height (Ganita 6).
• The volume of a sphere: about a third too large, though called "exact" (Ganita 7).
• The sizes of the Sun and planets, far too small, as in all ancient astronomy (Dasagitika 5).
• The year: 365.258681 days, about 3 minutes 20 seconds too long (Chapter 2).
• The Earth at the centre: the planets go round the Earth (Kalakriya 15), as in all ancient astronomy except a few Greek proposals.
• Parts of the traditional cosmology (Meru, the Earth growing in a day of Brahma) sit beside the science (Gola 8, 11–12).

#The year and the drift

Aryabhata's year is slightly too long. Over one year the error is small; over centuries it adds up. In about 430 years it amounts to a whole day. Almanacs that followed the old constants without correction drifted from the real sky; Kerala astronomers, in Haridatta's Parahita system (traditionally 683 CE), introduced a correction for it. The difference between traditional (siddhantic) and modern (drik) almanacs in India today is partly the legacy of this drift.

#How he was received

• Brahmagupta (628) attacked him sharply and often — for his yuga, his rotating Earth, his eclipse details — while using his methods.
• Bhaskara I (7th century) wrote the great commentary on the Aryabhatiya and defended and explained it.
• Al-Biruni (about 1030), the Persian scholar who studied Indian science, quotes Aryabhata and records his distinctive views (Clark's notes to Kalakriya 9).
• Kerala adopted his astronomy and carried it furthest: Haridatta's Parahita system, the vakya tables, and the mathematics of Madhava and his school (14th–16th centuries), whose infinite series grew out of the problem of computing sines more accurately — the problem Aryabhata had begun with.
• The rotating Earth was mostly rejected or explained away by later Indian astronomers; the commentator Parameshvara tried to reinterpret it (Clark's notes to Dasagitika 1 and Gola 9).

#What the Aryabhatiya is for a student of Jyotisha

The Aryabhatiya contains no astrology. It does not mention horoscopes, houses, yogas or predictions. It is purely ganita: the calculation of where the Sun, Moon and planets are, and when eclipses, new moons and conjunctions happen.

But every chart and every almanac depends on that calculation. The tithi, the nakshatra, the ascendant, the positions in a horoscope — all were computed, for centuries, by Aryabhata's methods or those of his successors. In this sense the Aryabhatiya is the foundation on which the rest of Jyotisha stands.

It also offers a model of how to do science: state your numbers exactly, derive everything from them, check them against observation, and — as his errors show — expect even the greatest work to need correction.

#Summary

• Achievements: accurate periods, the rotating Earth, physical eclipses, reflected light, spherical astronomy, epicycles; the decimal places, π ≈ 3.1416, sines, series, the kuttaka.
• Errors: the pyramid and sphere volumes, the sizes of the Sun and planets, a year 3⅓ minutes too long, the Earth-centred model, and traditional cosmology beside the science.
• Received with criticism (Brahmagupta), commentary (Bhaskara I), interest (al-Biruni), and continuation (Kerala, from Haridatta to Madhava).
• The Aryabhatiya has no astrology; it is the computational foundation of Jyotisha.

#Exercises

1. Name three of Aryabhata's achievements in astronomy and three in mathematics.
2. Name two of his errors.
3. About how many years does it take for an error of 3⅓ minutes a year to add up to a day?
4. Why is the Aryabhatiya important to astrology even though it contains no astrology?
5. Which later school carried Aryabhata's work furthest, and how?

Answers: 1. For example: accurate periods, the rotating Earth, physical eclipses; the decimal places, π, the kuttaka. 2. For example: the volume of a sphere, the volume of a pyramid, the length of the year, the sizes of the Sun and planets. 3. 1,440 minutes ÷ 3⅓ ≈ 430 years. 4. Because the positions in every chart and almanac were computed by his methods or their successors. 5. The Kerala school: Haridatta's corrected Parahita system, the vakya tables, and Madhava's infinite series.