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History of Hindu Mathematics: A Source Book v1.c8.s9: 9. Cube-Root

History of Hindu Mathematics: A Source Book v1.c8.s9

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9. Cube-Root

v1.c8.s9.1

**Terminology.** The Hindu terms for the cube-root are *ghana-mûla*, *ghana-pada*. These terms have already been discussed before.

v1.c8.s9.2

**The Operation.** The first description of the operation of the cube-root is found in the *Āryabhaṭīya*. It is rather too concise:

v1.c8.s9.3

“Divide the second *aghana* place by thrice the square of the cube-root; subtract from the first *aghana* place the square of the quotient multiplied by thrice the preceding (cube-root); and (subtract) the cube (of the quotient) from the *ghana* place; (the quotient put down at the next place (in the line of the root) gives the root).”²

v1.c8.s9.4

² *Ā*, ii. 5. Translations of this rule have been given by Rodet, Kaye, Singh, Clark, Sengupta and others. Kaye’s translation is entirely inaccurate. Other translations, though free, give the correct result. Clark’s use of the words “the (preceding) *ghana*” is somewhat misleading. The portion at the end, within brackets, is common to this and the preceding rule for the extraction of the square-root.

v1.c8.s9.5

As has been explained by all the commentators, the units place is *ghana*, the tens place is first *aghana*, the hundreds place is second *aghana*, the thousands place is *ghana*, the ten-thousands place is first *aghana* and so on. After marking the places as *ghana*, first *aghana* and second *aghana*, the process begins with the subtraction of the greatest cube number from the figures upto the last *ghana* place. Though this has not been explicitly mentioned in the rule, the commentators say that it is implied in the expression "*ghanasya mûla vargena*" etc. ("by the square of the cube-root" etc.) The method may be illustrated as below:

v1.c8.s9.6

*Example*. Find the cube-root of 1953125. The places are divided into groups of three by marking them as below:

v1.c8.s9.8

Subtract cube 1 . . . . . . . (c) Root = 1 Divide by thrice square . . . . . of root, i.e. 3.1² 3) 9 ( 2 . . . (a)¹ Placing quotient Subtract square of quo- 6 after the root tient multiplied by - 1 gives the thrice the previous 35 root 12 root, i.e., 2². 3. 1. 12 . . . . (b) -- Subtract cube of quotient, 233 i.e., 2³ 8 . . . . (c) . . . . . Divide by thrice square of the root, i.e., 3. 12² 432) 2251 ( 5 . . (a) Placing quotient Subtract square of 2160 after the root quotient multiplied by ---- 12 gives the thrice the previous 912 root 125 root, i.e., 5². 3. 12 900 . . (b) --- Subtract cube of quo- 125 tient, i.e., 5³ 125 . . (c) --- Thus the cube-root = 125 ```

v1.c8.s9.9

¹ The quotient in division is to be taken as great as will allow the two subsequent operations (*b*) and (*c*) to be carried out.

v1.c8.s9.10

It will be evident from the above illustration that the present method of extracting the cube-root is a contraction of Āryabhaṭa's method.

v1.c8.s9.11

The method given above occurs in all the Hindu mathematical works. For instance, Brahmagupta says:

v1.c8.s9.12

"The divisor for the second *aghana* place is thrice the square of the cube-root; the square of the quotient multiplied by three and the preceding (root) must be subtracted from the next (*aghana* place to the right), and the cube (of the quotient) from the *ghana* place; (the procedure repeated gives) the root."¹

v1.c8.s9.13

¹ *BrSpSi*, p. 175; *cf.* Colebrooke, *l.c.*, p. 280.

v1.c8.s9.14

Śrīdhara gives more details of the process as actually performed on the *pāṭī*, thus:

v1.c8.s9.15

"(Divide the digits beginning with the units place into periods of) one *ghana* place and two *aghana* places. From the (last) *ghana* digit subtract the (greatest possible) cube; then taking down the remainder and the third *pada* (*i.e.*, the second *aghana* digit) divide it by thrice the square of the cube-root which has been permanently placed in a separate place; place the quotient in the line; subtract the square of this (quotient) multiplied by thrice the last root from the next (*aghana*) digit. Then as before subtract the cube (of the quotient) from its own place (*i.e.*, the *ghana* digit). Then take down again the third *pada* (*i.e.*, second *aghana* digit), and the rest of the process is as before. (This will give) the root."²

v1.c8.s9.16

² *Triś*, pp. 6f.

v1.c8.s9.17

Āryabhaṭa II follows Śrīdhara and gives details as follows:

v1.c8.s9.18

"*Ghana* (*i.e.*, the place from which cube is subtracted), *bhājya* (*i.e.*, the "dividend" place) and *śodhya* (*i.e.*, the "minuend" place) are the denominations (of the places). Subtract the (greatest) cube from its own place (*i.e.*, from the numbers upto the last *ghana* digit); bring down the *bhājya* digit and divide it³ by thrice the square of the cube-root which has been permanently placed. Place the quotient in the line (of the root). The square of this (quotient) multiplied by thrice the previous root is subtracted from its own place (*i.e.*, the *śodhya* place) and its cube from the *ghana* place. If the above operations be possible then this (*i.e.*, the number in the line) is the root so far. Then bringing down the *bhâjya* digit continue the process as before (till it ends)."¹

v1.c8.s9.19

³ Literally, its own place.

v1.c8.s9.20

¹ *MSi*, p. 145. The interpretation given by Dvivedi of line 2 of the rule as printed in his edition (p. 145) is incorrect.

v1.c8.s9.21

The component digits of the number whose cube-root is to be found are divided into groups of three (one *ghana* and two *aghanas*) each. The digits upto the last *ghana* place (proceeding from left to right) give the first figure of the root (counting from the left). The following period of three digits (to the right) gives the second figure of the root and so on. While working on the *pâṭî*, the digits of the number whose root is to be found are first marked and the method proceeds as follows:

v1.c8.s9.22

*Example.* Find the cube-root of 1953125.

v1.c8.s9.23

The number is written thus:

v1.c8.s9.24

\begin{gathered}

v1.c8.s9.25

| - - | - - | \\

v1.c8.s9.26

\end{gathered}

v1.c8.s9.27

From the last *ghana* digit (marked by a vertical stroke), the greatest cube is subtracted. Here 1³ being subtracted from 1 gives zero. So 1 is rubbed out. The cube-root of 1³ is placed in a separate line. The figures on the *pâṭî* stand thus:

v1.c8.s9.28

{cc}
- - | - - | & 1 / 953125 & line of root

v1.c8.s9.29

Then to obtain the second figure of the root, the second *aghana* (*i.e.*, 9) is taken below and divided by thrice the square of the root (*i.e.*, the number in the line). Thus we have

v1.c8.s9.30

```
- - | - - | 9 5 3 1 2 5 3.1² = 3) 9 (2 quotient 6 --- 3 ```

v1.c8.s9.31

The quotient is taken to be 2, because if it were taken to be 3, the rest of the procedure cannot be carried out. This quotient (2) is set in the line. The first *aghana* is then brought down and we have, on subtracting the square of the quotient multiplied by thrice the previous root, the following:

v1.c8.s9.32

```
- - | - - | 1 2 9 5 3 1 2 5 . . . . . line of root 3.1² = 3) 9 (2 quotient 6 --- 3 5 2² . 3.1 = 1 2 --- 2 3 ```

v1.c8.s9.33

On bringing down the *ghana* digit 3, and then subtracting the cube of the quotient we get 225 as below, and the process on the period formed by the digits 953 is completed and the figure 2 of the root is obtained:

v1.c8.s9.34

```
- - | - - | 9 5 3 1 2 5
3) 9 (2 1 2 6 . . . . . line of root 3 5 1 2 ---- 2 3 3 2³ = 8 2 2 5 ```

v1.c8.s9.35

The figures 953 are then rubbed out and the remainder 225 is substituted. After this the process is as before, *i.e.*, thus

v1.c8.s9.36

{rcc} & --| & / \cline{2-2} 12²· 3\ = & 432)\ 2251 (5 & \cfrac{line of root} / & \cfrac{\phantom912} & / 5²· 12· 3\ = & \cfrac{\phantom900}{\phantom125} & / 5³\ = & \cfrac{\phantom125}{\phantom0} & \cfrac{line of root}

v1.c8.s9.37

The process ends as all the figures in the number are exhausted. The root is 125, the number in the line of root. As there is no remainder, the root is exact.

v1.c8.s9.38

The necessity for rubbing out figures arises, as the *pāṭī* is not big enough to contain the whole of the working. The three digits constituting a period are considered together. The figures upto the next second *aghana* have to be brought down and the operation of division performed separately, because the quotient is obtained by trial. As has been already explained, this division is performed by rubbing out the digits of the dividend (and not as in the working given above). If the operations were carried out on the figures of the original number, and if the quotient taken were found to be too big, then it would not be possible to restore the original figures and begin the work again, as will have to be done in case of failure.

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