The Study of Jyotisha · Chapter 49 of 54 · Part Ten · History

#Aryabhata and the siddhantas

#The astronomy behind astrology

Every chart in this course was computed: the positions of the planets, the ascendant, the tithi, the nakshatra. In classical India these computations were the work of ganita, the first branch of Jyotisha (Chapter 1) — mathematical astronomy, set out in texts called siddhantas ("established conclusions"). This branch is, by any standard, one of the great scientific achievements of the ancient world. This chapter tells its story through its central figures.

#Aryabhata (born 476 CE)

#His book and his date

The Aryabhatiya is a short work — 121 verses in four sections — by Aryabhata, who wrote it at Kusumapura (near modern Patna) (Aryabhatiya, Ganita 1, with Clark's note). He dates himself precisely:

When three yugapadas and sixty times sixty years had elapsed [from the beginning of the yuga], then twenty-three years of my life had passed (Aryabhatiya, Kalakriya 10, Clark).

With the Kali yuga starting in 3102 BCE, 3,600 years later is 499 CE, when he was 23; so he was born in 476. (Some commentators took 499 as the date for which his calculations were set rather than his age; either way, the book is of about 499.)

The English translation used here is W. E. Clark's (1930).

#What he did

The Earth rotates. Aryabhata gives the number of rotations of the Earth in a yuga — 1,582,237,500 "revolutions of the Earth eastward" (Dasagitika 1) — and explains the daily motion of the stars by the Earth's turning:

As a man in a boat going forward sees a stationary object moving backward, just so at Lanka a man sees the stationary asterisms moving backward (westward) in a straight line (Gola 9, Clark).

This was a radical view. Later Indian astronomers, including the commentators, mostly rejected or explained it away; Clark notes the attempts.

Eclipses. "The Moon obscures the Sun and the great shadow of the Earth obscures the Moon" (Gola 37). He also gives rules for calculating the size of the Earth's shadow at the Moon's distance (Gola 39–40). (Varahamihira, a generation later, gives the same explanation, Chapter 42.)

π. "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" (Ganita 10). That is 62,832 ÷ 20,000 = 3.1416, and he calls it "approximate".

The sine table. He gives a table of 24 sine differences at intervals of 3°45′, on a circle of radius 3,438 (Dasagitika 10) — an early landmark of computational trigonometry.

Numbers in letters. To fit huge numbers into verse, he invented a code in which syllables stand for numbers. The Moon's revolutions in a yuga, 57,753,336, are written as one long word (Dasagitika, with Clark's explanation). Clark finds it hard to believe this notation was not based on place value.

Exact integer astronomy. Every motion in the Aryabhatiya is a ratio of whole numbers: so many revolutions in so many days of the yuga. This makes the calculation exact and checkable — and it is the method behind the traditional almanac. From his numbers, the length of the sidereal year is 1,577,917,500 days ÷ 4,320,000 = 365.258681 days (modern value 365.256363).

He ends with a characteristic note of confidence: "By the grace of God the precious sunken jewel of true knowledge has been rescued by me, by means of the boat of my own knowledge, from the ocean which consists of true and false knowledge" (Gola 49).

#Brahmagupta (born 598)

Brahmagupta wrote the Brahmasphutasiddhanta in 628, at Bhillamala in Rajasthan. He was a great mathematician — his rules for operations with zero and negative numbers, and for solving equations, are landmarks — and a sharp critic. He attacked Aryabhata repeatedly; Clark quotes his remark that what Aryabhata said about the planets' setting was false (Brahmasphutasiddhanta XI.9). He also developed a method of second-order interpolation, the ancestor of the interpolation every modern ephemeris uses.

#Bhaskara I

Bhaskara I (7th century), the great commentator on Aryabhata, gave a remarkable rational formula for the sine — in degrees, sin x ≈ 4x(180 − x) ÷ (40,500 − x(180 − x)) — which is exact at 0°, 30° and 90° and never wrong by more than about 0.0016 (Mahabhaskariya VII.17–19).

#The Surya Siddhanta

The Surya Siddhanta, "the siddhanta of the Sun", is the most famous Indian astronomical text, framed as a revelation by the Sun to the demon Maya. E. Burgess translated it in 1860, and this course has cited it throughout (for combustion, planetary war, heliacal risings). The text we have is later than the "Saura" siddhanta that Varahamihira summarised (Chapter 48), revised over centuries. Its sidereal year is 365 days 6 hours 12 minutes 36.6 seconds (about 365.258756 days).

#Bhaskara II (born 1114)

Bhaskara II (also called Bhaskaracharya) wrote the Siddhanta Shiromani (1150), whose arithmetic part, the Lilavati, became the most widely used mathematics textbook in India. He developed the idea of a planet's instantaneous motion (tatkalika gati) — the concept used today to find the exact end of a tithi or nakshatra.

#Kerala

The mathematical tradition reached its height in Kerala in the 14th to 16th centuries, with Madhava and his school, who found infinite series for the sine, cosine and arctangent long before they were known in Europe. Chapter 51 tells this story.

#How accurate were the siddhantas?

For their time, very. But their year lengths are slightly too long — Aryabhata's by about 3 minutes 20 seconds a year, the Surya Siddhanta's by about 3½ minutes — and over centuries these small errors add up. Planetary positions from the old models can be wrong by a degree or more today, and the Moon's position (which decides the tithi and nakshatra) by enough to shift the time of a tithi by hours.

This leads to a real choice in India today:

• Siddhantic (or vakya) almanacs compute from the traditional texts and their tables.
• Drik ("seen") almanacs compute from modern astronomy, so that the positions match the actual sky.

The calendar reforms of the 19th and 20th centuries — and the Indian government's Calendar Reform Committee (1955) — favoured drik computation. Many traditional almanacs still use the old methods. The two can differ by hours on the timing of a tithi, and occasionally by a day on a festival. This website computes from modern astronomy (drik), and says so.

#What this means for a student

The ganita tradition is, in a sense, the most solid part of Jyotisha: its claims are about the sky, and they can be checked against the sky. Indian astronomers did check them, revised their constants, and argued fiercely about who was right. That is the spirit of science, and it is part of the same tradition as the horoscopy taught in this course. Varahamihira required his ideal astrologer to know astronomy thoroughly, and to settle disagreements between the books "by actual observation" (Brihat Samhita 2, Chapter 46).

#Summary

• Aryabhata (born 476; Aryabhatiya about 499, at Kusumapura) taught that the Earth rotates (Gola 9), explained eclipses by the Earth's shadow and the Moon (Gola 37), gave π ≈ 3.1416 "approximately" (Ganita 10), a sine table (Dasagitika 10) and a syllabic number code, and built astronomy on exact integer ratios.
• Brahmagupta (628) advanced algebra and interpolation and criticised Aryabhata; Bhaskara I gave a rational sine formula; Bhaskara II (1150) wrote the Siddhanta Shiromani and the Lilavati and used instantaneous motion.
• The Surya Siddhanta, as we have it, is a later revision of the siddhanta Varahamihira summarised.
• The old year lengths were slightly too long; positions drift over centuries. India today has both siddhantic and drik almanacs; this website uses modern (drik) computation.

#Exercises

1. How does Aryabhata date himself, and what year does it give?
2. What image does Aryabhata use to explain the daily motion of the stars?
3. Work out Aryabhata's value of π from his rule.
4. What is the difference between a siddhantic and a drik almanac?
5. Why does a small error in the year length matter over centuries?

Answers: 1. He says he was 23 when 3,600 years of the Kali yuga had passed: with the Kali yuga beginning in 3102 BCE, that is 499 CE. 2. A man in a moving boat sees the stationary shore move backwards; so the stars seem to move west because the Earth turns east. 3. (100 + 4) × 8 + 62,000 = 62,832; 62,832 ÷ 20,000 = 3.1416. 4. A siddhantic almanac computes from the traditional texts; a drik almanac from modern astronomy matching the visible sky. 5. Because it accumulates: 3½ minutes a year becomes more than a day in 500 years, moving the computed positions away from the real sky.