History of Hindu Mathematics: A Source Book v1.c8.s6: 6. Square

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

Language: English

v1.c8.s6.t1 Minor Methods of Squaring — 25 parts

6. Square

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**Terminology.** The Sanskrit term for square is *varga* or *kṛti*. The word *varga* literally means "rows" or "troops" (of similar things). But in mathematics it ordinarily denotes the square power and also the square figure or its area. Thus Āryabhaṭa I says:¹

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¹ *Ā*, ii. 3.

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"A square figure of four equal sides² and the (number representing its) area are called *varga*. The product of two equal quantities is also *varga*."

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² The commentator Parameśvara remarks: "That four sided figure whose sides are equal and both of whose diagonals are also equal is called *samacaturaśra* ("square")."

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How the word *varga* came to be used in that sense has been clearly indicated by Thibaut. He says: "The origin of the term is clearly to be sought for in the graphical representation of a square, which was divided in as many *vargas* or troops of small squares, as the side contained units of some measure. So the square drawn with a side of five *padas* length could be divided into five small *vargas* each containing five small squares, the side of which was one *pada* long."³ This explanation of the origin of the term *varga* is confirmed by certain passages in the *Śulba* works.⁴

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³ Thibaut, *Śulba-sûtras*, p. 48.

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⁴ *ĀpŚl*, iii. 7; *KŚl*, iii. 9; cf. B. Datta, *American Math. Monthly*, XXXIII, 1931, p. 375.

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The term *kṛti* literally means "doing," "making" or "action." It carries with it the idea of specific performance, probably the graphical representation.

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Both the terms *varga* and *kṛti* have been used in the mathematical treatises, but preference is given to the term *varga*. Later writers, while defining these terms in arithmetic, restrict its meaning. Thus Śrīdhara says:⁵

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⁵ *Triś*, p. 5.

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"The product of two equal numbers is *varga*."

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Pṛthudakasvâmî¹, Mahâvîra² and others give similar definitions.

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¹ *Cf.* Colebrooke, *l.c.*, p. 279.

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² *GSS*, p. 12.

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**The Operation.** The occurrence of squaring as an elementary operation is characteristic of Hindu arithmetic. The method, however, is not simpler than direct multiplication. It was given prominence by the Hindu writers probably because the operation of square-root is the exact inverse of that of squaring. Although the method first occurs in the *Brâhma-sphuṭa-siddhânta*, there is no doubt that it was known to Âryabhaṭa I as he has given the square-root method.

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Brahmagupta gives the method³ very concisely thus:

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³ The method is not mentioned in the chapter on Arithmetic, but seems to have been mentioned as an afterthought in the form of an appendix, (*BrSpSi*, p. 212).

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"Combining the product, twice the digit in the less⁴ (lowest) place into the several others (digits), with its (*i.e.*, of the digit in the lowest place) square (repeatedly) gives the square."

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⁴ *Râśerûnam* has been translated by Colebrooke as "the less portion." This translation is incorrect. He says that "the text is obscure" (p. 322, fn. 9), for according to his translation the rule becomes practically meaningless. The term *râśerûnam* must be translated by "the digit in the lowest place." Dvivedi agrees with the above interpretation (p. 212). The method taught here is "the direct method of squaring."

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Śrîdhara (750) is more explicit:⁵

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⁵ *Tris*, p. 5. The translation given by Kaye and Ramanujacharia is incorrect. (*Bibl. Math.*, XIII, 1912-13).

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"Having squared the last digit multiply the rest of the digits by twice the last; then move the rest of the digits. Continue the process of moving (the remain-

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ing digits after each operation) to obtain the square."

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Mahâvîra¹ (850) gives more details:

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¹ *GSS*, p. 12.

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"Having squared the last (digit), multiply the rest of the digits by twice the last, (which is) moved forward (by one place). Then moving the remaining digits continue the same operation (process). This gives the square."

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Bhâskara II² writes:

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² *L*, p. 4.

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"Place the square of the last (digit) over itself; and then the products of twice the last (digit) and the others (*i.e.*, the rest) over themselves respectively. Next, moving the number obtained by leaving the last digit (figure), repeat the procedure."

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He has remarked that the above process may be begun also with the units place.³

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³ *L*, p. 5.

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The following is the method of working on the *pâṭî*, the process beginning from the last place, according to Śrîdhara, Mahâvîra, Bhâskara II and others:

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To square 125.

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The number is written down,

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The last digit is 1. Its square is placed over itself.

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Then twice the last digit 2 × 1 = 2; placing it below the rest of the figures (below 2 or below 5 according as the direct or inverse method of multiplication is used)

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and rubbing out the last digit 1, the work on the *pâṭî* appears as

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Performing multiplication by 2 (below) and placing the results over the respective figures, we get

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One round of operation is completed. Next moving the remaining digits, *i.e.*, 25, we have

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Now, the process is repeated, *i.e.*, the square of the last digit (2) is placed over itself giving

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Then, placing twice the last digit (*i.e.*, 2 × 2 = 4) below the rest of the digits and then rubbing out 2, we have

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Performing multiplication, 4 × 5 = 20, and placing it over the corresponding figure 5, (*i.e.*, 0 over 5 and 2 carried to the left), the work on the *pâṭî* appears as

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Thus a second round of operations is completed.

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Then moving 5 we have Squaring 5 we get 25, and placing it over 5 (*i.e.*, 5 over 5 and 2 carried to the left) we have 15625 5 As there are no ‘remaining figures’ the work ends. 5 being rubbed out, the *pâṭî* has 15625, the required square.

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According to Brahmagupta and also Bhâskara II, the work may begin from the lowest place (*i.e.*, the units place). The following method is indicated by Brahmagupta:

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To square 125. The number is written down 125 The square of the digit in the least place, *i.e.*, 5²=25 is set over it thus: 25 125 Then, 2 × 5 = 10 is placed below the other digits, and five is rubbed out, thus: 25 12 10 Multiplying by 10 the rest of the digits, *i.e.*, 12, and setting the product over them (the digits), we have 1225 12 10 Then rubbing out 10 which is not required and moving the rest of the digits, *i.e.*, 12, we have 1225

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Thus one round of operations is completed.

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Again, as before, setting the square of 2 above it and 2 × 2 = 4 below 1, we have

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Multiplying the remaining digit 1 by 4, and setting the product above it, we have

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Then, moving the remaining digit 1, we obtain

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Thus the second round of operations is completed.

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Next setting the square of 1 above it the process is completed, for there are no remaining figures, and the result stands thus: $15625$

Minor Methods of Squaring

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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, $15² = 10 × 20 + 25 = 225.$

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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.”

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Śrīdhara (750) gives it thus:

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“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.

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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:

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"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."

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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.

Linked texts (7)

Āpastambaśulbasūtra — Āpastamba, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Kātyāyanaśulbasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Gaṇitasārasaṅgraha — Mahāvīra (mathematician), Sanskrit
Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Triśatikā (Pāṭīgaṇitasāra) — Śrīdhara (mathematician), Sanskrit

History of Hindu Mathematics: A Source Book v1.c8.s6: 6. Square

Contents

6. Square

Minor Methods of Squaring

Linked texts (7)