Pramāṇa · Search · About · API

The Science of the Shulba: A Study in Early Hindu Geometry c21.s2: Operations with Fractions

The Science of the Śulba: A Study in Early Hindu Geometry c21.s2

Language
English

Operations with Fractions

c21.s2.1

In the *Śulba*, there are instances showing fundamental arithmetical operations with elementary fractions. For example, it is stated in the *Baudhāyana Śulba*:⁸ "One hundred eighty-seven and a half square bricks of sides (equal to) one-fifth of a puruṣa make up the seven-fold *Agni* with the two aratnis and the prādeśa."

c21.s2.2

⁸ *BŚl*, iii. 106. Compare also *BŚl*, iii. 238-9 and *ĀpŚl*, xviii. 3 which give $7 ½ ÷ 1/16 = 120.$

c21.s2.3

Here the area of each brick is 1/25 of a square puruṣa; so the number of such bricks required to cover an area of 7 ½ square puruṣas will be $7 ½ ÷ 1/25 = 15/2 × 25 = 187½ .$

c21.s2.4

Thus it is an instance of division of fractions. Or the same result may have been obtained in a slightly different way which is, indeed, a simplified method of division. Since the area of each brick is 1/25 of a square puruṣa, one square puruṣa will contain 25 such bricks. Therefore an area of 7½ square puruṣas will contain $7½ × 25 = 187 ½ .$ bricks. If the area of each brick be "One-fifteenth of half of a square puruṣa," says Baudhāyana,¹ the number of bricks used will be 225. That is, $7 ½ ÷ 1/15 of ½ = 15/2 ÷ 1/30 = 15/2 × 30 = 225.$

c21.s2.5

¹ *BŚl.* iii. 188-9.

c21.s2.6

In describing the *Droṇacit*, Baudhāyana writes :² "Its body is a square; its side is three puruṣas less one-third. On the western side of the body is the handle. Its length east-to-west is half a puruṣa plus ten aṅgulis and its breadth north-to-south is one puruṣa less one-third. Thus is made the sevenfold *Agni* with the two aratnis and the prādeśa."

c21.s2.7

² *BŚl*, iii. 219-224.

c21.s2.8

That is, $7½ square puruṣas =(3-⅓) puruṣas ²$ $+(½ puruṣa+10 aṅgulis )×(1-⅓) puruṣa;$ the right-hand side, in square puruṣas $=(2⅔)²+(½+1/12)(1-⅓),$ $=7 1/9+7/12×⅔,$ $=7½.$

c21.s2.9

The spatial dimensions of the constituent parts of the Fire-altar of the shape of the falcon with bent wings and spread-out tails have been described by Āpastamba as follow:

c21.s2.10

"Of the whole area making the seven-fold *Agni* with the two aratnis and the prādeśa, take off the prādeśa (from the tail), and the fourth part of the body together with eight quarter bricks. Of these latter, (use) three for the head, then divide the remainder between the two wings."¹

c21.s2.11

¹ *ĀpŚl*, xv. 3.

c21.s2.12

Now, as is well-known, the body of the primitive Fire-altar of the shape of the falcon measures 4 square puruṣas, each wing 1× 1⅕ square puruṣas and the tail 1× 1 1/10 square puruṣas. On taking out the prādeśa (=1/10 puruṣa) from the tail, there will remain 1 square puruṣa. The body will be reduced by $4×¼+8×1/16 square puruṣas ;$ there will then remain $4 - (4 × ¼ + 8 × 1/16) = 2 ½ square puruṣas.$

c21.s2.13

Of the former with 3 × 1/16 square puruṣas is formed the head and the remainder $4/4 + 8/16 - 3/16 = 1 5/16 square puruṣas$ together with 1 × 1/10 square puruṣas from the tail is divided equally between the two wings. Each wing will therefore measure $1 × 1 ⅕ + ½(1 5/16 + 1/10) square puruṣas.$

c21.s2.14

Hence the total area of the Fire-altar will be, in square puruṣas, $1 + 2 ½ + 3/16 + 21 ⅕ + ½(1 5/16 + 1/10) = 7 ½.$

c21.s2.15

Of the squaring of a fraction, we take the following example from the *Āpastamba Śulba*: ¹

c21.s2.16

¹ *ĀpŚl*, iii. 8.

c21.s2.17

"A cord 1 ½ puruṣas long produces (a square of) 2 ¼ (square puruṣas); 2 ½ puruṣas produce 6 ¼ (square puruṣas)."

c21.s2.18

That is, $(1 ½)² = 2 ¼, (2 ½)² = 6 ¼.$

c21.s2.19

Baudhāyana states, on the contrary, that the side of a square measuring 7 1/9 square puruṣas is 2⅔ puruṣas in length.¹ That is $√(7 1/9) = 2⅔ .$

c21.s2.20

¹ *BŚl*, iii. 220.

c21.s2.21

*Progressive Series.*

c21.s2.22

In the manner of laying out bricks described in the *Sulba*, we find a few interesting instances of progressive series. Āpastamba writes:

c21.s2.23

"On (the occasion of) the first construction, (the altar-builder) should construct (the Fire-altar) knee-deep with 1000 bricks; on the second, navel-deep with 2000 bricks; on the third, mouth-deep with 3000 bricks. (The number of bricks employed in constructing the Fire-altar becomes thus) greater and greater on each successive occasion. He who constructs to attain the Heaven, (should thus construct with) great, high and unlimited (*mahāntam bṛhantam aparimitam*) (number of bricks); so it is known (from the ancient scriptures)."²

c21.s2.24

² *ĀpŚl*, x. 8; see also *ĀpŚr*, xvi. 13. 11-2; *BŚl*, ii. 26.

c21.s2.25

Thus we have the A. P. $1000, 2000, 3000,$

c21.s2.26

Reference to this progressive mode of successive construction of the Fire-altar is found as early as the *Taittirīya Saṃhitā* (c. 3000 B.C.).³ It reappears in the *Śatapatha Brāhmaṇa* and Āpastamba expressly admits to have borrowed it from that work.⁴ We find there another noteworthy instance which shows that it was very likely known then how to sum up a series in A. P.

c21.s2.27

³ *TS*, v. 6. 8. 2-3.

c21.s2.28

⁴ *ĀpŚl*, xvi. 13. 12.

c21.s2.29

" But, indeed, that Fire-altar also is the Metres; for there are seven of these metres, increasing by four syllables; and the triplets of these make seven hundred and twenty syllables, and thirty-six in addition thereto." ¹

c21.s2.30

¹ *ŚBr*, x. 5. 4. 7. Eggeling's translation.

c21.s2.31

It has been stated elsewhere in the same work that the shortest metre is the *Gāyatrī* with 24 syllables. Thus we are given the first term (24), the common difference (4) and the number of term (7) of a series in A. P. Then its

c21.s2.32

Sum &= 7/2\ {2 × 24 + (7 - 1)4} / &= 252.

c21.s2.33

So the triplets of these metres will consist of 756 (= 252 × 3) syllables, which are equal to 720 + 36 as stated.

c21.s2.34

From the method indicated by Baudhāyana ² for constructing larger and larger squares, starting with a smaller one, by adding successively gnomons to it is clear that the following series was known to him.³

c21.s2.35

³ Compare Müller, *loc. cit.*, pp. 200 f.

c21.s2.36

$1 + 3 + 5 + 7 + + (2n + 1) = (n + 1)²$

c21.s2.37

² *Vide supra*, pp. 125 ff.

c21.s2.39

Fig 79

c21.s2.40

*Factorisation.*

c21.s2.41

The true significance of another passage is not quite clear to us unless it is some mystic expression of an attempt to find all the possible factors of a number. The *Śatapatha Brāhmaṇa* says:

c21.s2.42

“Now in this Prajāpati, the year, there are 720 days and nights, his lights, (being) those bricks; 360 enclosing stones, and 360 bricks with (special) formulas. This Prajāpati, the year, has created all existing things, both what breathes and the breathless, both gods and men. Having created all existing things, he felt like one emptied out, and was afraid of death. He bethought himself, ‘How can I get these beings back into my body? how can I put them back into my body? how can I be again the body of all these beings?’ He divided his body into two; there were 360 bricks in the one, and as many in the other: he did not succeed. He made himself three bodies,—in each of them there were 3 × 80 of bricks: he did not succeed. He made himself four bodies of 180 bricks each: he did not succeed. He made himself five bodies,—in each of them there were 144 bricks: he did not succeed. He made himself six bodies of 120 bricks each: he did not succeed. He did not develop himself sevenfold.¹ He made himself 8 bodies of 90 bricks each: he did not succeed. He made himself 9 bodies of 80 bricks each: he did not succeed. He made himself 10 bodies of 72 bricks each: he did not succeed. He did not develop elevenfold. He made himself 12 bodies of 60 bricks each: he did not succeed. He did not develop either thirteenfold or fourteenfold. He made himself 15 bodies of 48 bricks each: he did not succeed. He made himself 16 bodies of 45 bricks each: he did not succeed. He did not develop seventeenfold. He made himself 18 bodies of 40 bricks each: he did not succeed. He did not develop nineteenfold.

c21.s2.43

¹ The text न सप्तधा व्यभवत् literally means “did not become divided into seven (parts).”

c21.s2.44

He made himself 20 bodies of 36 bricks each: he did not succeed. He did not develop either twenty-onefold, or twenty-twofold, or twenty-threefold. He made himself 24 bodies of 30 bricks each. There he stopped, at the fifteenth; and because he stopped at the fifteenth arrangement there are fifteen forms of the waxing, and fifteen of the waning (moon).” ¹

c21.s2.45

¹ *SBr*, x. 4. 2. 2-17. The translation is substantially due to Eggeling. We have only introduced the ciphers for the numbers in words.

c21.s2.46

The significance of stopping after the fourteenth operation is obvious. For after that there will be the repetition of the previous factors.

Linked texts (5)