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History of Hindu Mathematics: A Source Book v1.c8.s6.t1: Minor Methods of Squaring

History of Hindu Mathematics: A Source Book v1.c8.s6.t1

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Minor Methods of Squaring

v1.c8.s6.t1.1

The identity
(i) n² = (n - a)(n + a) + a² has been mentioned by all Hindu mathematicians as affording a suitable method of squaring in some cases. For instance, $15² = 10 × 20 + 25 = 225.$

v1.c8.s6.t1.2

Brahmagupta says:

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“The product of the sum and the difference of the number (to be squared) and an assumed number plus the square of the assumed number give the square.”

v1.c8.s6.t1.4

Śrīdhara (750) gives it thus:

v1.c8.s6.t1.5

“The square is equal to the product of the sum and the difference of the given number and an assumed quantity plus the square of the assumed quantity."¹ Mahâvîra, Bhâskara II, Nârâyaṇa and others also give this identity.

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¹ *BrSpSi*, p. 212.

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¹ *Triś*, p. 5.

v1.c8.s6.t1.8

The formula (*ii*) (*a*+*b*)² = *a*²+b²+2ab, or its general form (*a*+*b*+*c*+ ....)² = *a*²+b²+c²+ .... +2ab +.. has been given as a method of squaring. Thus Mahâvîra² says:

v1.c8.s6.t1.9

"The sum of the squares of the two or more portions³ of the number together with their products each with the others multiplied by two gives the square."

v1.c8.s6.t1.10

³ The word *sthâna* has been used in the original. This word has been generally used in the sense of 'notational place.' Following the commentator, we have rendered it by "portion." As a given number, say, 125, can be broken into parts as 50+40+35 or as 100+20+5, and as the rule applies to both, it is immaterial whether the word '*sthâna*' is translated by 'place' or 'portion.' This rule appears to have been given as an explanation of the Hindu method of squaring used with the place-value numerals.

v1.c8.s6.t1.11

Bhâskara II⁴ gives:

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⁴ *L*, p. 4.

v1.c8.s6.t1.13

"Twice the product of the two parts plus the square of those parts gives the square."

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The formula (*iii*) *n*²=1+3+5+ .... to *n* terms has been mentioned by Śrîdhara and Mahâvîra.

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² *GSS*, p. 13.

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Śrîdhara⁵ says:

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⁵ *Triś*, p. 5.

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"(The square of a number) is the sum of as many terms in the series of which one is the first term and two the common difference."

v1.c8.s6.t1.19

Mahâvîra uses a similar expression:

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“The sum of as many terms of the series of which one is the first term and two the common difference is the square (of the given number).”¹

v1.c8.s6.t1.21

¹ *GSS*, p. 12.

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Nârâyaṇa² has also given the above series. He adds the formula:³ $A'² = (a+b)² = (a-b)² + 4ab$ to find the square of the given number A.

v1.c8.s6.t1.23

² *GK*, i. 17-18.

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³ He says: “The square of the difference plus four times the product is the square of the sum.”

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It should be observed that all the above methods of squaring are applied to whole numbers only. The methods of squaring fractions are dealt with in the section on fractions.

Cites (3)