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History of Hindu Mathematics: A Source Book v1.c8.s8: 8. Square-Root

History of Hindu Mathematics: A Source Book v1.c8.s8

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Contents

8. Square-Root

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**Terminology.** The Hindu terms for the "root" are *mûla* and *pada*. The usual meaning of the word *mûla* in Sanskrit literature is "root" of a plant or tree; but figuratively the foot or lowest part or bottom of anything. Its other meanings are "basis," "foundation," "cause," "origin," etc. The word *pada* means "the lower part of the leg" (figuratively the lower part or basis of anything), "foot," "part," "portion," "side," "place," "cause," "a square on a chess-board," etc. The meanings common to both terms are "foot," "the lowest part or basis of anything," "cause" or "origin." It is, therefore, quite clear that the Hindus meant by the term *varga-mûla* ("square-root") "the cause or origin of the square" or "the side of the square (figure)." This is corroborated by the following statement of Brahmagupta:²

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² *BrSpSī*, xviii. 35.

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"The *pada* (root) of a *kṛti* (square) is that of which it is the square."

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Of the above terms for the "root," *mûla* is the oldest. It occurs in the *Anuyogadvâra-sûtra* (*c.* 100 B.C.), and in all the mathematical works. The term *pada* seems to have come into use later on, *i.e.*, from the seventh century A.D. It occurs first in the work of Brahmagupta (628).

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The term *mûla* was borrowed by the Arabs who translated it by *jadhr*, meaning "basis of square." The Latin term *radix* also is a translation of the term *mûla*¹.

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¹ For further details see Datta, *American Math. Monthly*, XXXIV, pp. 420-423, also XXXVIII, pp. 371-376.

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The word *karaṇî* is found to have been used in the *Sulba* works and Prâkṛta literature as a term for the square-root. In geometry it means a "side." In later times the term is, however, reserved for a surd, *i.e.*, a square-root which cannot be evaluated, but which may be represented by a line.

The Operation

v1.c8.s8.t1.1

The description of the method of finding the square-root is .given in the *Âryabhaṭîya* very concisely thus:

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"Always divide² the even place by twice the square-root (upto the preceding odd place); after having subtracted from the odd place the square³ (of the quotient), the quotient put down at the next place (in the line of the root) gives the root."⁴

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² In dividing, the quotient should be taken as great as will allow of the subtraction of its square from the next odd place. This is the force of the Sanskrit text as pointed out by the commentators Bhâskara I, Nîlakaṇṭha and others.

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³ The "square" is mentioned and not the "square of the quotient," as in the beginning the greatest possible square is to be subtracted, there being no quotient.

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⁴ *Â*, ii. 4. Translations of the rule have been given before by Rodet (*JA*, 1880, II), Kaye (*JASB*, 1907 and 1908, III and IV resp.), Singh (*BCMS*, 1927, XVIII), Clark (*Âryabhaṭîya*) and others. Of these Kaye's translation is entirely incorrect.

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The method may be illustrated thus:

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*Example.* Find the square-root of 54756.

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The odd and even places are marked by vertical and horizontal lines. The different steps are then as indicated below:

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| Subract square | | & - & | & - & | / 5 & 4 & 7 & 5 & 6 / 4 & & & & | root=2 |
| Divide by twice the root | 4) 14 (3 12 | Placing quotient at the next place, the root =23 |
| Subtract square of quotient | 2 & 7 / & 9 | |
| Divide by twice the root | 46) 185 (4 184 | Placing quotient at the next place, the root =234 |
| Subtract square of quotient | 1 & 6 / 1 & 6 | |
| | | |

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The process ends. The root is 234.

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It has been stated by G. R. Kaye¹ that Āryabhaṭa's method is algebraic in character, and that it resembles the method given by Theon of Alexandria. Both his statements are incorrect.²

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¹ *JASB*, III and IV, in the papers entitled "Notes on Indian Mathematics, I and II."

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² See Singh, *l.c.*, for details; also Clark, *Āryabhaṭīya*, pp. 23f.

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The following quotations from Siddhasena Gaṇi (*c.* 550) in his commentary on the *Tatvârthâdhigama-sûtra*³ will prove conclusively that the Hindu method of extracting the square-root was not algebraic. In connection with the determination of the circumference of a circle of 100,000 *yojanas*, he says:

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³ iii. 11.

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"The diameter is one hundred thousand *yojanas*; that multiplied by one hundred thousand *yojanas* becomes squared; this is again multiplied by 10 and then

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the square-root (of the product) extracted. The root will be the circumference of the circle. Now to find the number of *yojanas* (by extracting the square-root) we obtain in succession the figures 3,¹,6,²,2 and 7 of the root, the number appearing below (that is, as the last divisor) is 632454. This being halved becomes the number three hundred thousand sixteen thousands two hundred and twenty seven. The number in excess as the remainder is this 484471; ...."

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"Then on multiplication by 4 will be obtained 756000000000. The square-root of this will be the chord. In finding that (root) will be obtained in succession the figures 2,7,⁴,9,5 and 4; ...."

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It is evident that Āryabhaṭa's plan of finding the square-root has been followed in the above cases as the digits of the root are evolved successively one by one.

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Later writers give more details of the process. Thus Śrīdhara says:

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"Having subtracted the square from the odd place, divide the next (even) place by twice the root which has been separately placed (in a line), and after having subtracted the square of the quotient, write it down in the line; double what has been obtained above (by placing the quotient in the line) and taking this down, divide by it the next even place. Halve the doubled quantity (to get the root)."¹

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¹ *Triś*, p. 5. For an illustration of the method of working on a *pāṭī*, see A. N. Singh, *BCMS*, XVIII, p. 129.

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Mahāvīra,² Āryabhaṭa II³ and Śrīpati⁴ give the rule in the same way as Śrīdhara. Bhāskara II, however, makes a slight variation, for he says:

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² *GSS*, p. 13.

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³ *MŚi*, p. 145.

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⁴ *SiŚe*, xiii. 5; *GT*, 23.

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“Subtract from the last odd place the greatest square number. Set down double the root in a line, and after dividing by it the next even place subtract the square of the quotient from the next odd place and set down double the quotient in the line. Thus repeat the operation throughout all the figures. Half of the number in the line is the root.”¹

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¹ *L*, p. 4. The line in Bhâskara II’s method contains the doubled root, whilst in that of Âryabhaṭa I, it contains the root. See Singh, *l.c.*

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The method of working on the *pâṭî* may be illustrated as below:

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*Example.* Find the square-root of 54756

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The given number is written down on the *pâṭî* and the odd and even places are marked by vertical and horizontal lines thus:

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Beginning with the last odd place 5, the greatest square number 4 is subtracted. Thus 4 subtracted from 5 gives 1. The number 5 is rubbed out and the remainder 1 substituted in its place. Thus after the first operation is performed, what stands on the *pâṭî* is

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Double the root 2, *i.e.*, 4, is permanently placed in a separate portion of the *pâṭî* which has been termed *paṅkti* (“line”). Dividing the number upto the next even mark by this number in the line, *i.e.*, dividing 14 by 4 we obtain the quotient 3 and remainder 2. The number 14 is rubbed out and the remainder 2 written in its place; thus on the *pâṭî* we have now

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{ccc} 4/(line of root) & 27\stackrel{I}{\phantom}\stackrel{-}{\phantom}\stackrel{I} & (3 quotient

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The square of the quotient 3²=9 is subtracted from the figures upto the next odd mark. This gives (27-9) =18. 27 is rubbed out and 18 substituted in its place. Double the quotient 3 is now set in the line giving 46. The figures on the *pâṭî* stand thus:

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{ccc} 46/(line of root) & 18\stackrel{-}{\phantom}\stackrel{I} & {l} The quotient 3 having / been rubbed out.

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Dividing the numbers upto the next even mark by the number in the line, *i.e.*, dividing 185 by 46, the quotient is 4 and remainder 1. 185 is rubbed out and the remainder 1 substituted in its place. The figures on the *pâṭî* are now

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{ccc} 46/(line of root) & \stackrel{I} & (4 quotient

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Subtracting square of the quotient the remainder is nil, so that 16 is rubbed out. The quotient 4 is doubled and set in the line. The *pâṭî* has now

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{cc} 468/(line of root) & The quotient 4 having been rubbed out.

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Half the number in the line, *i.e.*, 468/2=234 is the root.

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Along with the Hindu numerals, the method of extracting the square-root given above, seems to have been communicated to the Arabs about the middle of the eighth century, for it occurs in precisely the same form in Arabic works on mathematics.¹ In Europe it occurs in similar form in the writings of Peurbach (1423-1461), Chuquet (1484), La Roche (1520), Gemma Frisius (1540), Cataneo (1546) and others.¹

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¹ *E.g.*, Al-Nasavî (1025); see Suter, *Bibl. Math.*, VII, p. 114 and Woepcke, *JA* (6), t. 1, 1863.

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¹ See Smith, *History*, II, pp. 144-148.

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