Aryabhata: the Aryabhatiya · Chapter 6 of 8

#The reckoning of time

The third section of the Aryabhatiya, the Kalakriya ("the reckoning of time", 25 verses), turns the numbers of the Dasagitika into calendars and planetary motions.

#Units of time and space (Kalakriya 1–2)

A year consists of twelve months. A month consists of thirty days. A day consists of sixty nadis. A nadi consists of sixty vinadikas. Sixty long syllables or six pranas make a sidereal vinadika. This is the division of time. In like manner the division of space beginning with a revolution (Kalakriya 1–2).

And "in like manner the division of space": a revolution is 12 signs, a sign 30 degrees, a degree 60 minutes, a minute 60 seconds. A prana of time corresponds to a minute of arc of the sky's turning — the link already given in Dasagitika 4 ("the Earth moves one minute in a prana").

#Cycles from differences (Kalakriya 3–5)

• Kalakriya 3: the difference between the revolutions of two planets in a yuga is the number of their conjunctions (meetings).
• Kalakriya 4: the difference between a planet's revolutions and those of its apsis gives the revolutions of its epicycle (the cycle of its uneven motion).
• Kalakriya 5:
  • the Sun's revolutions are the solar years;
  • the conjunctions of the Sun and Moon are the lunar months;
  • the conjunctions of the Sun and the Earth (that is, the Earth's rotations minus the Sun's revolutions) are the civil days;
  • the revolutions of the stars (the Earth's rotations) are the sidereal days.

So all the calendar's units come out of the same few numbers. With Dasagitika 1:

#Intercalary months and omitted days (Kalakriya 6)

Subtract the solar months in a yuga from the lunar months in a yuga. The result will be the number of intercalary months in a yuga. Subtract the natural [civil] days in a yuga from the lunar days in a yuga. The result will be the number of omitted lunar days (Kalakriya 6).

The lunar months outnumber the solar months by 53,433,336 − 51,840,000 = 1,593,336 in a yuga: about one extra month (adhika masa) every 32½ months, which keeps the lunar calendar in step with the seasons. And because a lunar day (tithi) is a little shorter than a civil day, some tithis are omitted (kshaya) from the count of days. These are the same adhika months and kshaya tithis that Indian almanacs still contain.

#The years of men, ancestors and gods (Kalakriya 7–9)

A solar year is a year of men. Thirty of these make a year of the Fathers. Twelve years of the Fathers make a year of the gods. Twelve thousand years of the gods make a yuga of all the planets. A thousand and eight yugas of the planets make a day of Brahman (Kalakriya 7–8).

So a year of the gods is 360 human years, and 12,000 divine years make the 4,320,000-year yuga. Aryabhata's 1,008 yugas in a day of Brahma differs from the 1,000 of most other texts; the Persian scholar al-Biruni, writing about 1030, noticed the difference and recorded it (Clark's note to Kalakriya 9).

Kalakriya 9 divides the yuga into an "ascending" half (utsarpini) and a "descending" half (avasarpini) — words better known from Jain cosmology — with a middle and two ends.

#His own date (Kalakriya 10)

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 (Kalakriya 10).

This gives 499 CE and Aryabhata's birth in 476 (Chapter 1).

#The beginning of time (Kalakriya 11)

The yuga, the year, the month and the day began all together at the beginning of the bright fortnight of Chaitra. Time, which has no beginning and no end, is measured by (the movements of) the planets and the asterisms on the sphere (Kalakriya 11).

A beautiful statement: time itself has no beginning and no end; the cycles we count are only measures of it.

#Orbits, speeds and distances (Kalakriya 12–14)

Aryabhata assumes that all the planets move at the same speed in yojanas a day along their orbits (Kalakriya 12). So a planet with a small orbit goes round quickly, and one with a large orbit slowly: "The Moon, being below, completes its small orbit in a short time. Saturn, being above all the others, completes its large orbit in a long time" (Kalakriya 13). A sign is a twelfth of any circle, small on a small orbit and large on a large one, but always 30 degrees (Kalakriya 14).

#The order of the planets (Kalakriya 15)

Beneath the asterisms are Saturn, Jupiter, Mars, the Sun, Venus, Mercury and the Moon, and beneath these is the Earth situated in the center of space like a hitching-post (Kalakriya 15).

This is the order of the planets by their period, from slowest to fastest — the same order as in Greek astronomy.

#How the order gives the week (Kalakriya 16)

These seven lords of the hours, Saturn and the others, are in order swifter than the preceding one, and counting successively the fourth in the order of their swiftness they become the lords of the days from sunrise (Kalakriya 16).

This verse explains where the days of the week come from:

1. Give the 24 hours of the day to the seven planets in the order of Kalakriya 15: Saturn, Jupiter, Mars, the Sun, Venus, Mercury, the Moon, and round again.
2. The planet of the first hour gives its name to the day.
3. Since 24 = 3 × 7 + 3, the first hour of the next day falls three places further on — that is, the fourth planet counting from today's.

Starting with the Sun's day: Sun → (count four: Sun, Venus, Mercury, Moon) → Moon → Mars → Mercury → Jupiter → Venus → Saturn → Sun. That is Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday. The planetary week, which India shares with the West, comes from this rule of the hours.

#The planets' motions: eccentrics and epicycles (Kalakriya 17–25)

The planets do not move evenly. Aryabhata explains their uneven motion with two equivalent models:

• an eccentric circle: the planet moves evenly on a circle whose centre is displaced from the Earth (Kalakriya 18);
• an epicycle: the planet moves on a small circle whose centre moves evenly round the Earth; the distance between the Earth and the eccentric's centre equals the epicycle's radius (Kalakriya 19).

Each planet has two such corrections (Kalakriya 17, 20–21):

• the manda correction, "from the apsis", for the uneven speed due to the shape of its orbit;
• the shighra correction, "from the conjunction", for the effect of its position relative to the Sun — which produces retrograde motion.

Kalakriya 22–24 give the order in which the two corrections are applied, by halves, to find the planet's true position (sphuta), with different procedures for the outer planets (Saturn, Jupiter, Mars) and the inner (Venus, Mercury). Kalakriya 25 gives the planet's distance from the Earth.

This epicycle model, also used by the Greeks (in Ptolemy's form), gave planetary positions good enough for calendars and charts for a thousand years.

#Summary

• Units of time from the year to the prana (4 seconds), matched to units of arc (Kalakriya 1–2).
• All the calendar's units come from differences of revolutions: lunar months, civil days, sidereal days, intercalary months and omitted tithis (Kalakriya 3–6).
• 12,000 divine years make a yuga; 1,008 yugas a day of Brahma (Kalakriya 7–9); his own date (Kalakriya 10); time without beginning or end (Kalakriya 11).
• Equal linear speeds make small orbits fast and large orbits slow; the order Saturn–Jupiter–Mars–Sun–Venus–Mercury–Moon (Kalakriya 12–15).
• Counting every fourth planet of the hours gives the days of the week (Kalakriya 16).
• Eccentrics and epicycles, with manda and shighra corrections, give the true positions (Kalakriya 17–25).

#Exercises

1. How long is a prana, and how is it linked to the Earth's rotation?
2. How many intercalary months are there in Aryabhata's yuga? About how often does one occur?
3. Starting from Wednesday (Mercury's day), use Kalakriya 16 to find the next day.
4. Why does the Moon go round faster than Saturn in Aryabhata's model?
5. What are the two corrections applied to each planet, and which one produces retrograde motion?

Answers: 1. About 4 seconds; in a prana the Earth turns through one minute of arc (Dasagitika 4). 2. 1,593,336; about one every 32½ months (51,840,000 solar months ÷ 1,593,336). 3. Order of the hours: Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon. From Mercury count four: Mercury, Moon, Saturn, Jupiter — Thursday. 4. Because all planets move the same distance each day, and the Moon's orbit is the smallest. 5. The manda (from the apsis) and the shighra (from the conjunction); the shighra produces retrograde motion.