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History of Hindu Mathematics: A Source Book v1.c8.s7: 7. Cube

History of Hindu Mathematics: A Source Book v1.c8.s7

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English

Contents

7. Cube

v1.c8.s7.1

**Terminology.** The Hindu term for the cube is *ghana*. This term occurs in all the mathematical works. It has been used in the geometrical as well as the arithmetical sense, *i.e.*, to denote the solid cube as well as the continued product of the same number taken three times. Thus Âryabhaṭa I says:

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“The continued product of three equals and also the solid having twelve (equal) edges are called *ghana*.”⁴

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⁴ *Â*, ii. 3.

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Śrîdhara,⁵ Mahâvîra⁶ and Bhâskara II⁷ each state:

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⁵ *Tris*, p. 6.

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⁶ *GSS*, p. 14.

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⁷ *L*, p. 5.

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“The continued product of three equal numbers is *ghana*.”

v1.c8.s7.9

Another term for the cube is *bṛnda*, but it is seldom used.

v1.c8.s7.10

**The Operation.** A method of cubing applicable to numbers written in the decimal place-value notation, has been in use in India from before the fifth century A.D. Āryabhaṭa I (499) knew the method, although he did not consider it to be as important as the inverse operation of extracting the cube-root which he has explained.

v1.c8.s7.11

Brahmagupta gives the method in these words:

v1.c8.s7.12

“Set down the cube of the last;¹ then place at the next place from it, thrice the square of the last multiplied by the succeeding; then place at the next place thrice the square of the succeeding multiplied by the last; and (at the next place) the cube of the succeeding. This gives the cube.”

v1.c8.s7.13

¹ The commentator Pṛthudakasvâmî takes ‘last’ (*antya*) to mean the figure or figures on the extreme left. The number to be cubed is divided into portions of which the cubes are found successively by the application of the process. (See illustration). Thus ‘succeeding’ would mean here the figure to the right of the ‘last.’ It does not, however, make any difference if we take ‘last’ to mean the figure in the units place. The words used by Brahmagupta are ‘*antya*,’ for last and ‘*uttara*’ for succeeding. The rule is:

v1.c8.s7.14

Śrīdhara² says:

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² *Triś*, p. 6.

v1.c8.s7.16

“Set down the cube of the last; then set down at the next place³ the square of the last multiplied by

v1.c8.s7.17

³ *Sthānādhikyam*, lit. “increasing one place.” This increase is to be made by placing the result one place to the right of the previously noted figures (see illustration). Brahmagupta uses the word *tatprathamāt*, i.e., “before that.”

v1.c8.s7.18

*Sthāpyo’ntya ghano’ntya kṛtistriguṇottarasaṅguṇā ca tatprathamāt* ।
*Uttarasyottarantyaguṇā triguṇā cottaraghanaśca ghanaḥ* ॥

v1.c8.s7.19

thrice the succeeding;¹ then (at the next place) the product of the square of the succeeding and last multiplied by three; and then (at the next place) the cube of the succeeding."

v1.c8.s7.20

¹ *Pûrva*, *âdi*, lit. "preceding". We have rendered them by "succeeding" to be in conformity with the general convention so as to avoid confusion.

v1.c8.s7.21

Mahâvîra states:²

v1.c8.s7.22

² *GSS*, p. 15 (47). It will be observed that the "addition of the cube of the remaining" does not occur in the rule. This has to be understood from the previous stanza which says that the cubes of all the parts are to be added. See the translation of the previous stanza given on pp. 166f.

v1.c8.s7.23

"The cube of the last, the product of thrice its square and the remaining, the square of the remaining multiplied by thrice the last; placing of these, each one place before the other, constitutes here the process."

v1.c8.s7.24

Bhâskara II is more explicit:³

v1.c8.s7.25

³ *L*, p. 5.

v1.c8.s7.26

"Set down the cube of the last; then the square of the last multiplied by three times the succeeding; then the square of the succeeding multiplied by three times the last and then the cube of the succeeding; these placed so that there is difference of a place between one result and the next,⁴ and added give the cube. The given number is distributed into portions according to places, one of which is taken for the last and the next as the first and in like manner repeatedly (if there be occasion). Or the same process may be begun from the first place of figures for finding the cube."

v1.c8.s7.27

⁴ *Sthânântaratvena* has been translated by Colebrooke by "according to places." This translation is incorrect and does not give the true significance of the term.

v1.c8.s7.28

The method may be illustrated by the following example:

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*Example.* To cube 1234.

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The given number has four places, *i.e.*, four portions. First we take the last digit 1 and the succeeding digit 2, *i.e.*, 12, and apply the method of cubing thus :

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| (i) Cube of the last (1³) | = | 1 | |
| (ii) Thrice the square of the last (3.1²) multiplied by the succeeding (2) gives (2.3.1²) | = | 6 | (placing at the next place) |
| (iii) Thrice the square of the succeeding multiplied by the last gives (3.2².1) | = | 12 | (placing at the next place) |
| (iv) Cube of the succeeding (2³) | = | 8 | (placing at the next place) |
| | | ——— | |
| \ \ \ \ \ Thus 12³ is the sum | | 1728 | |

v1.c8.s7.32

After this we take the next figure 3, *i.e.*, the number 123, and in this consider 12 as the last and 3 as the succeeding. Then the method proceeds thus:

v1.c8.s7.33

| (i) The cube of the last (12³) as already obtained | = | 1728 | |
| (ii) Thrice the square of the last multiplied by the succeeding, *i.e.*, 3.12².3 | = | 1296 | (placing at the next place). |
| (iii) Thrice the square of the succeeding multiplied by the last, *i.e.*, 3.3².12 | = | 324 | " " |
| (iv) Cube of the succeeding, *i.e.*, 3² | = | 27 | " " |
| | | ——— | |
| \ \ \ \ \ Thus 123³ is the sum | | 1860867 | |

v1.c8.s7.34

Now the remaining figure 4 is taken, so that the number is 1234, of which 123 is the last and 4 the succeeding. The method proceeds thus:

v1.c8.s7.35

| (i) | Cube of the last, *i.e.*, (123)³ as already obtained | = | 1860867 | |
| (ii) | Thrice the square of the last into the succeeding, *i.e.*, 3.(123)².4 | = | 181548 | (placing at the next place) |
| (iii) | Thrice the square of the succeeding into the last, *i.e.*, 3.4².123 | = | 5904 | „ „ |
| (iv) | Cube of the last *i.e.*, 4³ | = | 64 | „ „ |
| | | | . . . . . | |
| Thus | (1234)³ is the sum | | 1879080904 | |

v1.c8.s7.36

The direct process—that in which the operation begins with the units place—can be similarly performed.

Minor Methods of Cubing

v1.c8.s7.t1.1

The formula (i) (a+b)³=a³+3a²b+3ab²+b³ and the corresponding result (a+b+c+\dots)³=a³&+3a²(b+c+\dots)+3a(b+c+\dots)² / &+(b+c+\dots)³ are implied in the Hindu method of cubing given above. Mahâvîra¹ gives the following explanation:

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¹ *GSS*, p. 15.

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"The squares of the last place² and the next³ are taken, and each (square) is multiplied by the other and by three. The sum of these products and the cubes of both (*lit.* all) the places is the cube; the procedure is repeated (if necessary).¹

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² *Sthâna*, meaning the number represented by the figure standing in that place.

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³ *Anya*, lit. "other," meaning the number represented by the figures standing in the other places.

v1.c8.s7.t1.6

¹ Thus (234)³ is considered as $(200+30+4)³ = (200)³+3.200²(30+4)+3.200(30+4)² / +(30+4)³$ Then the procedure is repeated for obtaining (30+4)³. *Cf.* English translation, p. 17, note.

v1.c8.s7.t1.7

Śrîpati and Bhâskara II² state the formula in the form $(a+b)³ = a³+3ab(a+b)+b³$ "Thrice the given number multiplied by its two parts, added to the sum of the cubes of those parts, gives the cube."

v1.c8.s7.t1.8

Nârâyaṇa³ says "Thrice the (given) number multiplied by both parts, added to the cubes of the parts, is the cube of the sum."

v1.c8.s7.t1.9

The formula (*ii*) n³ = n(n+a)(n-a)+a²(n-a)+a³ has been mentioned by Mâhâvîra⁴ in these words:

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² *GT*, 27; *L*, p. 5.

v1.c8.s7.t1.11

⁴ *GSS*, p. 15.

v1.c8.s7.t1.12

"The continued product of the given number, the sum and the difference of the given number and an arbitrary quantity, when added to the smaller of these multiplied by the square of the arbitrary number, and the cube of the arbitrary number, give the cube (of the given number)."

v1.c8.s7.t1.13

Expressions for n³ involving series have been given by Śrîdhara, Mahâvîra, Śrîpati and Nârâyaṇa. The formula

v1.c8.s7.t1.14

(*iii*) n³ = ₁n 3r (r-1)+1 is given by Śrīdhara in these words:

v1.c8.s7.t1.15

³ *GK*, i. 23.

v1.c8.s7.t1.16

“The cube (of a given number) is equal to the series whose terms are formed by applying the rule, ‘the last term multiplied by thrice the preceding term plus one,’ to the terms of the series whose first term is zero, the common difference is one and the last term is the given number.”¹

v1.c8.s7.t1.17

¹ *Triś*, p. 6. The translation given by Kaye and Ramanujacharia (*Bibl. Math.*, III, 1912-13) is incorrect. They admit their inability to follow the meaning (see p. 209, note). S. Dvivedi has misinterpreted the rule, and gives an incorrect explanation in a note on p. 6. The reading *saike* is incorrect.

v1.c8.s7.t1.18

Mahāvīra gives the above in the form $n³ = 3₂nr(r-1)+n$ He² says:

v1.c8.s7.t1.19

“In the series, wherein one is the first term as well as the common difference and the number of terms is equal to the given number (n), multiply the preceding term by the immediately following one. The sum of the products so obtained, when multiplied by three and added to the last term (*i.e.*, n) becomes the cube (of n).”

v1.c8.s7.t1.20

Nārāyaṇa³ states:

v1.c8.s7.t1.21

“From the series whose first term and common difference are each one, (the last term being the given number) the sum of the series formed by the last term multiplied by three and the preceding added to one, gives the cube (of the last term).”

v1.c8.s7.t1.22

Mahāvīra has also mentioned the results, (*iv*) x³ = x+3x+5x+ to x terms, (*v*) x³ = x²+(x-1)1+3++(2x-1), in these words:¹

v1.c8.s7.t1.23

² *GSS*, ii. 45.

v1.c8.s7.t1.24

³ *GK*, i. 22.

v1.c8.s7.t1.25

¹ *GSS*, ii. 44.

v1.c8.s7.t1.26

"The cube (of a given number) is equal to the sum of the series whose first term is the given number, the common difference is twice that number, and the number of terms is (equal to) that number.
"Or the square of the given number when added to the product of that number minus one (and) the sum of the series in which the first term is one, the common difference two and the number of terms (is equal to) that number, gives the cube."

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