Vargamūla

Vargamūla refers to square roots. It is composed of two words - varga, meaning ‘square’ and mūla meaning ‘root’. It is one of the eight parikarmas or basic operations, as reckoned by Indian treatises in mathematics. Throughout history, like most computations, this was also done manually, hence the procedure for these is specified in treatises. Āryabhaṭīya states the following:

भागं हरेदवर्गान्नित्यं द्विगुणेन वर्गमूलेन ।

वर्गाद्वर्गे शुद्धे लब्धं स्थानान्तरे मूलम् ॥

bhāgaṃ haredavargānnityaṃ dviguṇena vargamūlena .

vargādvarge śuddhe labdhaṃ sthānāntare mūlam .. (Gaṇitapāda 4, Shukla, 1976)

“Divide the even places (avarga places) by twice the square root (the result made so far). From odd places (varga places), reduce the square and add that digit [which was squared] to the result.”

In the verse, varga places refers to the odd places and avarga places refers to the even places when counted from the right. The procedure is as follows:

1. Mark the digits in the number as odd and even from left to right

2. Pick the greatest possible square number which can be reduced from the last odd place. For example, if the number is 8, the largest possible square which can be subtracted is the square of 2, which is 4.

3. Write the number (4 in the above example) in line with the odd digit and the square root of that number (2) separately. The first digit of the root is the digit which was squared.

4. Reduce the square

5. Bring down the next digit

6. Divide this number by twice the result written so far

7. The quotient is the next digit of the root and is written to the right of the existing result. The remainder must be written below

8. The next digit in the number will be brought down, to the right of the remainder

9. Subtract from this number the square of the quotient (in step 7). The quotient must be chosen such that this subtraction is possible. If it is not possible, reduce the quotient by 1 and try again.

10. Bring down the next digit and divide by twice the result (step 6) again

11. Continue this process as long as there are still digits to the right

The following is an example with 14642. In the first line, v stands for varga (odd) and a for avarga (even).

When dividing according to step 6, the quotients were small enough that they could be subtracted from the numbers in the later steps. Hence going back and repeating was not necessary.

References

Shukla, K. (1976). Āryabhaṭīya of Āryabhaṭa, with the commentary of Bhāskara I and Someśvara. Indian National Science Academy.