Mathematics in Ancient and Medieval India c5.s9: Square root and Cube root

Mathematics in Ancient and Medieval India c5.s9

Language: English

Square root and Cube root

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Special credit is given to Āryabhaṭa I (c. 499 A.D.) for his rules for the extraction of the square and cube roots developed on the knowledge of decimal place value concept. His rule for square root extraction runs thus:

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*bhāgaṃ haredavargānnityaṃ dviguṇena vargamūlena |*
*vargādvarge śuddhe labdhaṃ sthānāntare mūlam ||*
(*Āryabhaṭīya* ii. 4)

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“Divide always the non-square (*avarga*) places by twice the square-root (up to the preceding odd place); after having subtracted the square (of the quotient) from the square place (*varga*), then the quotient placed at the separate place gives the root.”

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The other Indian scholars namely, Mahāvīra, Śrīdhara, Āryabhaṭa II, Bhāskara II, Kamalākara etc. have given the same method of extracting square root.¹ Āryabhaṭa I divides the number whose square-root is to be determined into *varga* (square) and *avarga* (non-square) places-respectively from right, while the other scholars have divided it into *sama* and *viṣama* places from right. The rule can be explained from the following illustration by putting ☐ and - marks against square and non-square places. Modern method of square root extraction is also given side by side for the purpose of comparison.

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¹ Singh, A. N. ‘On the Indian Method of Root Extraction,’ *Bulletin of the Calcutta Mathematical Society*, 18, 123-40, 1927.

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| | *Āryabhaṭa's method* | *Modern Method* |
| Subtract the square 2²= | □ - □ - □ / 5 4 7 5 6 ( root 2 / 4 ... (1) | · · · | / 5 4 7 5 6|234 / 4 - - (1) |
| Divide by twice the root i. e., 2×2= Subtract square of the quotient 3²=9 | .r@l 4) & 1 4 (3...(a) / & 1 2 / & 2 7 / & 9......(b)root 23 / (2) | r@l 43) & 1 4 7 / & 1 2 9...... (2) / 464 & 1 8 5 6 / & 1 8 5 6..(3) |
| Divide by twice the root i. e., 2×23=46 Subtract square of the quotient i. e., 4²=16 | .l \phantom) 1 8 5 (4......(a') / \phantom) 1 8 4 / \phantom)1 8\phantom41 6 / \phantom)1 8\phantom41 6......(b')(3) \thereforethe square-root is 234 | |

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This shows that present method of extracting square root is same as that of Āryabhaṭa I (c. 499 A. D. ) with the only difference that stages numbered (2) and (3) in the working of the modern method were carried out in two distinct steps.

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The rule for the cube root extraction runs as follows :

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*aghanādbhājeddvitīyāt trighanasya mūlavargeṇa |*
*vargastripūrvaṁ guṇitaśśodhyaḥ prathamādghanaśca ghanāt ||*
( *Āryabhaṭīya*, ii. 5 )

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"After having subtracted the cube (of the quotient) from the *ghana* place, divide the second *aghana* place by thrice the square of the root, and subtract the square of the quotient multiplied by thrice the previous root from the next ( *aghana* place ) and the cube ( of the quotient ) from the cubic place ( next *aghana* place )."

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The very same method is given by other Indian scholars like Brahmagupta, Mahāvīra, Śrīdhara, Āryabhaṭa II, Bhāskara II etc. Here also the number whose cube root is to be found is marked as—*ghana*, first *aghana*, and second *aghana*, from right.¹ For illustration, they may be denoted by the symbols say ⸬ , —, — respectively. Āryabhaṭa's method of extraction is illustrated² in the following example; the modern method is also given side by side.

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¹ *prathama sthānaṁ ghanasaṁjñaṁ dvitīye tṛtīye aghanasamjñe !* *caturthaghanasaṁjñaṁ, pañcamaṣaṣṭhe aghanasamjñā !!* (Parameśvara's Comm.)

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² The division is to be carried to such an extent that the steps (b) and (c) can be carried out ( vide, p. 81 ).

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| | *Āryabhaṭa I's method* | *Modern method* |
| Subtract the cube 1³ = 1 | 1 9 5 3 1 2 5 (cube root 1 1 ..... | 1953125 | 125 1 |
| | 9 | 953 |
| Divide by thrice the square of the root i e. 3 × 1² | 3) 6 (2....... (a) 3 5 | 1² × 300 & / = 300 & / 1 × 30·2 & / = 60 & / 2² = 4 & / 364 & |
| Subt. the square of the quotient multiplied by thrice the previous root i.e , 2² × 3 × 1 = 12 | 1 2 .......(b) | 728 |
| | 2 3 3 | 225125 |
| Subt. the cube of the quotient i.e., 2³ = 8 | 8....... (c) | 12² × 300 & / = 43200 & / 12 × 30 × 5 & / = 1800 & / 5² = 25 & / 45025 & |
| | 2 2 5 1 | 225125 |
| Divide by thrice the square of the quotient i.e., 3 × 12² = 432 | 432) 2 1 6 0 (5...(a') | |
| Subt. sq. of the quotient multiplied by thrice the previous root i.e., 5² × 3 × 12 = 900 | 9 1 2 9 0 0 ....(b') | |
| | 1 2 5 | |
| Subt. cube of the quotient i.e , 5³ = 125 | 1 2 5....(c') | |
| | ∴ cube root of 1953125 = 125. | |

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From the above it is clear that the modern cube root method of extraction is simply a contraction of Āryabhaṭa I's method.

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Methods of extraction of square roots (*khai fang*) and cubic roots (*khai li fang*) with the help of abacus or counting boards appear in China in the *Chiu-Chang suan shu* (later half of the 1st century A. D.).¹ Yang Hui (c. 1261 A. D.) explained the methods of root extraction as found in the *Chiu-Chang suan shu* with an example of finding the square root of 71824 in his book, *Hsiang Chieh chiu chang suan fa* (A Detailed Analysis of the Mathematical Methods in the "Nine Chapters"). This gives the various stages on the counting board with the help of a diagram and illustrates the geometrical basis of the method.² In the cube-root method, Liu Hui and Li Shun-feng, commentators in the *Chiu Chang suan shu* both hinted at the use of cubical blocks to demonstrate the process. In Europe the derivation of the square-root method was done possibly from the Euclidean concept (a+b)² = a²+2ab+b².

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¹ Needham, J. *Science and Civilisation in China*, 3, p. 65-68, 1959.

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² Yong, Lam Lay. "The Geometrical Basis of the Ancient Chinese Square Root Method", *Isis*, 61, No. 206, p. 92-101, 1970.

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In the 4th century A. D. Theon Alexandria, explained Ptolemy's method of extracting square roots and found the square-root of 4500 degrees to be 67º 4' 55" in sexagesimal unit. This method of extraction was purely geometrical and depended on Euclidean concept.³ In another place he explained 144 = 10² + 2.10.2 + 2² = (10 + 2)² with the help of a diagram. This figure is two-dimensional and no attempt was made to construct a three dimensional figure in order to extend the square-root method to find the cube-root. In the sixteenth century A. D. both square-root and cube-root methods were given by Cataneo, which are exactly the same as those of Āryabhaṭa I¹. Hence it is quite likely that the Indian methods of square-root and cube-root went to Europe through Arab intermediaries.

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³ Heath, T. L. *A History of Greek Mathematics*, 1, p. 60-63, Oxford, 1921; Yong, Lam Lay, Ibid, p. 100-101.

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¹ Smith, D. E. *History of Mathematics*, 2, Dover Publication, p. 148.

Linked texts (1)

Āryabhaṭīya — Āryabhaṭa, Sanskrit