Pramāṇa · Search · About · API

History of Hindu Mathematics: A Source Book v1.c8.s7.t1: Minor Methods of Cubing

History of Hindu Mathematics: A Source Book v1.c8.s7.t1

Language
English

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:

v1.c8.s7.t1.2

¹ *GSS*, p. 15.

v1.c8.s7.t1.3

"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

v1.c8.s7.t1.4

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

v1.c8.s7.t1.5

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:

v1.c8.s7.t1.10

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."

Cites (5)