Aryabhata: the Aryabhatiya · Chapter 2 of 8

#The numbers of the universe

The first section of the Aryabhatiya, the Dasagitika, contains the numbers from which everything else is calculated. It is the most compressed part of the book: ten verses hold the whole data of a planetary system.

#Writing numbers in syllables (Dasagitika B)

To fit huge numbers into verse, Aryabhata invented a code. Each consonant of the Sanskrit alphabet stands for a number, and each vowel multiplies it by a power of a hundred. In the first group, the consonants k to m stand for 1 to 25; in the second, y to h stand for 30, 40 ... 100. The vowel a leaves the value as it is; i multiplies by 100; u by 10,000; and so on through nine vowels.

Clark's example is the number of the Moon's revolutions in an age, which Aryabhata writes as one long word, cayagiyinusuchlr. Decoded syllable by syllable it gives 6 + 30 + 300 + 3,000 + 50,000 + 700,000 + 7,000,000 + 50,000,000 = 57,753,336 (Clark's note to Dasagitika B).

The system is ingenious but hard to read, and it did not last. Later astronomers in Kerala used a different code, katapayadi, instead. Clark adds that he finds it hard to believe the notation was not based on a place-value system — that is, on the idea that a digit's value depends on its position.

#The revolutions of the planets (Dasagitika 1–2)

The heart of the system is a list of how many times each body goes round in one yuga, an "age" of 4,320,000 years:

And the count starts "at the beginning of Mesha (Aries), at sunrise on Wednesday, at Lanka" (Dasagitika 2).

Notice the Earth in the list. Aryabhata counts the rotations of the Earth, not the turning of the sky. This is his rotating Earth, which appears again in the Gola (Chapter 7).

#What these numbers give

From these whole numbers, every period follows by division. The number of civil days in a yuga is the Earth's rotations minus the Sun's revolutions: 1,582,237,500 − 4,320,000 = 1,577,917,500. Then:

(Computed from Dasagitika 1–2; modern values for comparison.)

These are remarkable figures for the year 499. The synodic month is correct to within a second; the sidereal day to within a fraction of a second; Mars's period to four decimal places. The year is a little too long — by about 3 minutes 20 seconds — and that small error was to matter over the centuries (Chapter 8).

#The ages of the world (Dasagitika 3)

There are 14 Manus in a day of Brahman, and 72 yugas constitute the period of a Manu. Since the beginning of this kalpa up to the Thursday of the Bharata battle 6 Manus, 27 yugas and 3 yugapadas have elapsed (Dasagitika 3).

Aryabhata uses the traditional Indian scheme of very long ages, but in his own form. His yuga is divided into four equal quarters (yugapadas) — unlike the more common scheme, in which the four ages are of unequal length (4 : 3 : 2 : 1). "The Bharata battle" is the war of the Mahabharata, which tradition placed at the start of the Kali yuga.

#Units and sizes (Dasagitika 4–5)

Aryabhata gives the units in which distances are measured — the yojana, which he defines as 8,000 times a man's height (Dasagitika 5) — and the sizes of the bodies:

The planets are given as fractions of the Moon's diameter: Venus a fifth, Jupiter a tenth, Mercury a fifteenth, Saturn a twentieth, Mars a twenty-fifth (Dasagitika 5). And he states, in the middle of his units, that "the Earth moves one minute in a prana" — that is, it turns through one minute of arc in about four seconds of time (Dasagitika 4), which is exactly the rate of its daily rotation.

The ratios of the sizes are far from the true ones (the Sun is in fact about a hundred times the Earth's diameter), as were all ancient estimates of the Sun's size and distance.

#Inclinations, nodes and apsides (Dasagitika 6–7)

• The greatest declination of the ecliptic — the tilt of the Earth's axis — is 24° (the true value in his time was about 23.6°) (Dasagitika 6).
• The Moon's path is inclined 4½° to the ecliptic (modern 5.1°); the planets' paths by 1° to 2° (Dasagitika 6).
• The positions of the planets' nodes and apsides (the points where their paths cross the ecliptic, and the points of slowest motion) are given in degrees from the start of Aries (Dasagitika 7).

#Epicycles (Dasagitika 8–9)

To account for the planets' varying speeds and their retrograde motion (Chapter 6), Aryabhata uses epicycles: small circles on which the planet moves while the centre of the small circle moves round the Earth. He gives the size of each planet's two epicycles — one for the uneven motion caused by the shape of its orbit, one for the motion caused by its position relative to the Sun — and, unusually, gives slightly different sizes for different quarters of the orbit (Dasagitika 8–9).

#The sine table (Dasagitika 10)

The last verse gives a table of 24 sine differences, in minutes of arc, for every 3°45′ of a quarter circle, on a circle of radius 3,438:

225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, 7.

Adding them up step by step gives the sines themselves: 225, 449, 671, 890 ... up to 3,438 at 90°. The radius 3,438 is chosen so that one minute of arc on the circle equals one unit of the radius (3,438 ≈ 21,600 ÷ 2π, since the circle has 21,600 minutes). Clark stresses that the numbers listed are differences, not the sines themselves (note to Dasagitika 10).

The values are accurate to about one unit in the third figure. The sine table is the tool on which all the later calculation depends.

#The closing verse (Dasagitika C)

The section ends by saying that whoever knows these ten verses "reaches the Brahman" beyond the planets and stars — a reminder that, for Aryabhata, astronomy was also a spiritual knowledge.

#Summary

• Aryabhata wrote numbers in syllables with his own code (Dasagitika B).
• He gave the revolutions of every planet, the Moon's apsis and node, and the Earth's own rotations in a yuga of 4,320,000 years, counted from sunrise on a Wednesday at Lanka (Dasagitika 1–2).
• From these whole numbers follow periods of remarkable accuracy: the synodic month 29.530582 days, the sidereal day 23 h 56 m 4.1 s, the year 365.258681 days (slightly long).
• His yuga has four equal quarters (Dasagitika 3); the Earth turns one minute of arc in a prana (Dasagitika 4).
• He gave sizes, inclinations, nodes, apsides, epicycles (Dasagitika 5–9) and a 24-step sine table on radius 3,438 (Dasagitika 10).

#Exercises

1. How many civil days are in Aryabhata's yuga, and how do you get the number?
2. Using Dasagitika 1, work out the length of the synodic month.
3. What is unusual about the Earth's place in the list of revolutions?
4. Why did Aryabhata choose 3,438 as the radius of his sine table?
5. Name one of Aryabhata's numbers that is very accurate and one that is far from the truth.

Answers: 1. 1,577,917,500: the Earth's rotations (1,582,237,500) minus the Sun's revolutions (4,320,000). 2. Synodic months = Moon's revolutions − Sun's = 53,433,336; 1,577,917,500 ÷ 53,433,336 = 29.530582 days. 3. It counts the Earth's own rotations, implying that the Earth turns rather than the sky. 4. Because then one minute of arc on the circumference equals one unit of the radius (3,438 ≈ 21,600 ÷ 2π). 5. Accurate: the synodic month, the sidereal day, Mars's period. Far off: the sizes of the Sun and planets.