Aryabhata: the Aryabhatiya · Chapter 5 of 8

#Series, equations and the pulveriser

The last part of the Ganita (verses 19–33) moves from geometry to arithmetic and algebra. It ends with Aryabhata's most original mathematical method, the kuttaka.

#Arithmetic progressions (Ganita 19–20)

A series in which each term increases by the same amount — 3, 7, 11, 15 ... — is an arithmetic progression.

• Ganita 19 gives the sum: find the middle term (the average of the terms) and multiply by the number of terms. For 3, 7, 11, 15: the middle term is (3 + 15) ÷ 2 = 9; four terms; the sum is 36. The verse also covers a run of terms starting part way through the series.
• Ganita 20 works backwards: given the sum, the first term and the common difference, find the number of terms. The rule amounts to solving a quadratic equation: n = ½ [ (√(8 d S + (2a − d)²) − 2a) ÷ d + 1 ], where a is the first term, d the difference and S the sum.

Example. How many terms of 3, 7, 11 ... add up to 36? a = 3, d = 4, S = 36: 8 × 4 × 36 = 1,152; (2 × 3 − 4)² = 4; √1,156 = 34; 34 − 6 = 28; 28 ÷ 4 = 7; 7 + 1 = 8; 8 ÷ 2 = 4 terms.

#Piles of numbers (Ganita 21–22)

• Ganita 21: the sum of the "triangular numbers" 1, 3, 6, 10 ... (a pile of balls built in layers) up to n layers is n(n + 1)(n + 2) ÷ 6.
• Ganita 22: the sum of the squares 1² + 2² + ... + n² is n(n + 1)(2n + 1) ÷ 6; and the sum of the cubes 1³ + 2³ + ... + n³ is the square of the sum 1 + 2 + ... + n.

Example. 1² + 2² + 3² + 4² = 30, and 4 × 5 × 9 ÷ 6 = 30. 1³ + 2³ + 3³ + 4³ = 100, and (1 + 2 + 3 + 4)² = 10² = 100.

#Two unknowns from their sum and product (Ganita 23–24)

• Ganita 23: the product of two numbers is half of [(their sum)² − (the sum of their squares)].
• Ganita 24: given the product of two numbers and their difference, find the numbers: multiply the product by 4, add the square of the difference, take the square root, then add and subtract the difference and halve. (For product 24 and difference 2: 4 × 24 + 4 = 100; √100 = 10; (10 + 2) ÷ 2 = 6 and (10 − 2) ÷ 2 = 4. The numbers are 6 and 4.)

#Interest (Ganita 25)

A problem of interest: a sum is lent; the interest for one month is itself lent out; given the total of the interest and the interest on it, and the time, find the monthly interest. Aryabhata's rule amounts to solving a quadratic equation (Ganita 25). Problems of loans and interest were standard in Indian mathematics, which served trade as well as astronomy.

#The rule of three (Ganita 26)

In the rule of three multiply the fruit by the desire and divide by the measure. The result will be the fruit of the desire (Ganita 26).

If 5 measures give 30 "fruit", how much do 8 give? 30 × 8 ÷ 5 = 48. The rule of three — proportion — was the workhorse of all Indian calculation, and of astronomy above all: if a planet moves so much in so many days, how far in these days?

#Fractions and working backwards (Ganita 27–29)

• Ganita 27: to multiply or divide fractions, and to bring fractions to a common denominator.
• Ganita 28: in working backwards (the "inverse method"), multipliers become divisors, divisors multipliers, addition becomes subtraction and subtraction addition.
• Ganita 29: if several numbers are known only by the results of subtracting each from their total, add those results and divide by one less than the number of terms; this gives the total.

#A simple equation (Ganita 30)

Two men have equal wealth. One has 5 objects and 30 coins; the other has 8 objects and 6 coins. What is an object worth? Aryabhata's rule: divide the difference of the coins by the difference of the objects: (30 − 6) ÷ (8 − 5) = 8 coins (Ganita 30). This is a linear equation in one unknown, 5x + 30 = 8x + 6.

#When two planets meet (Ganita 31)

The distance between two planets moving in opposite directions is divided by the sum of their daily motions. The distance between two planets moving in the same direction is divided by the difference of their daily motions. The two results will give the time of meeting of the two in the past and in the future (Ganita 31).

This is the rule for conjunctions: when the Moon will catch up with the Sun (new moon), or a planet with another.

Example. The Moon is 60° behind the Sun. The Moon moves about 13°10′ a day, the Sun about 59′. The difference is about 12°11′ a day. 60 ÷ 12.2 ≈ 4.9 days to new moon.

#The kuttaka: the "pulveriser" (Ganita 32–33)

The last two verses of the Ganita give a method for solving equations in whole numbers, called the kuttaka, "pulveriser" — because it breaks the numbers down into smaller and smaller pieces.

#The problem

Find a number N which, divided by one number, leaves one given remainder, and divided by another number, leaves another given remainder.

Example. Find a number that leaves remainder 5 when divided by 8, and remainder 4 when divided by 9.

#The method

Aryabhata's rule (Ganita 32–33, as Clark explains it):

1. Divide the larger divisor by the smaller, then the smaller by the remainder, and so on — repeated division — until the remainder is zero. Write the quotients one below another in a column (the "chain").
2. Put a chosen number and the difference of the given remainders at the foot of the chain.
3. Work back up the chain: multiply each number by the one above it and add the one below.
4. The top number, reduced as the verse directs, gives the answer.

The repeated division is what we now call Euclid's algorithm, and the working back up the chain is what is now called the "extended Euclidean algorithm". The kuttaka solves what are now called linear Diophantine equations, ax − by = c in whole numbers.

For the example, the answer is 13 (13 ÷ 8 leaves 5; 13 ÷ 9 leaves 4). Any number 13 + 72k also works.

Clark is candid that the verses are "enigmatic" and that their exact interpretation depends on the commentators (note to Ganita 32–33).

#Why astronomers needed it

In Aryabhata's astronomy, every planet's position on a given day depends on how many days have passed since the start of the yuga (the ahargana, "heap of days"). Going backwards — from observed positions to the number of days, or finding when several cycles would all coincide — means finding a whole number that satisfies several conditions of the form "leaves such a remainder when divided by such a number". That is exactly the kuttaka's problem. The method was so important that Brahmagupta named his chapter on algebra after it, the kuttakadhyaya.

#Summary

• Sums of arithmetic progressions and the number of terms (Ganita 19–20); sums of triangular numbers, squares and cubes (Ganita 21–22).
• Two unknowns from sum, product or difference (Ganita 23–24); interest by a quadratic (Ganita 25).
• The rule of three (Ganita 26); fractions, working backwards, totals from partial results (Ganita 27–29); a linear equation (Ganita 30).
• The time of meeting of two planets (Ganita 31).
• The kuttaka, the "pulveriser": repeated division and back-substitution to solve equations in whole numbers (Ganita 32–33), needed to work with the cycles of astronomy.

#Exercises

1. Find the sum 5 + 8 + 11 + 14 + 17 by Ganita 19.
2. Check Ganita 22 for n = 3: the sum of squares and the sum of cubes.
3. If 4 days of the Moon's motion carry it 52°40′, how far does it go in 7 days? Which rule is this?
4. Two planets are 30° apart and moving the same way, with daily motions of 1°30′ and 0°30′. When do they meet?
5. Find the smallest number that leaves remainder 1 when divided by 3 and remainder 2 when divided by 5.

Answers: 1. Middle term (5 + 17) ÷ 2 = 11; five terms; 55. 2. 1 + 4 + 9 = 14 = 3 × 4 × 7 ÷ 6; 1 + 8 + 27 = 36 = (1 + 2 + 3)². 3. 52°40′ × 7 ÷ 4 = 92°10′; the rule of three. 4. The gap closes at 1° a day, so in 30 days. 5. 7 (7 ÷ 3 leaves 1; 7 ÷ 5 leaves 2).