The Science of the Shulba: A Study in Early Hindu Geometry c20.s2: Hypotheses about its Origin

The Science of the Śulba: A Study in Early Hindu Geometry c20.s2

Language: English

Hypotheses about its Origin

c20.s2.1

One will be naturally interested to know how the value of √ 2 was determined in that early time to such a high degree of approximation. Unfortunately the Hindus have not left any trace of the method adopted by them for the purpose. So it is very difficult to guess it. Thibaut has, however, propounded an ingenious hypothesis about it. He says:¹

c20.s2.2

¹ *Śulbasūtras*, pp. 13 ff.

c20.s2.3

"The question arises: how did Baudhāyana or Āpastamba or whoever may have the merit of the first investigation, find this value? Certainly they were not able to extract the square root of 2 to six places of decimals; if they have been able to do so, they would have arrived at still greater degree of accuracy. I suppose that they arrived at their result by the following method which account for the exact degree of accuracy they reached.

c20.s2.4

“Endeavouring to discover a square the side and diagonal of which might be expressed in integral numbers they began by assuming two as the measure of a square's side. Squaring two and doubling the result they got the square of the diagonal, in this case = eight. They then tried to arrange eight, let us say again, eight pebbles, in a square; as we should say they tried to extract the square root of eight. Being unsuccessful in this attempt, they tried the next number, taking three for the side of a square; but eighteen yielded a square root no more than eight had done. They proceeded in consequence to four, five, etc. Undoubtedly they arrived soon at the conclusion that they would never find exactly what they wanted, and had to be contented with an approximation. The object was now to single out a case in which the number expressing the square of the diagonal approached as closely as possible to a real square number. I subjoin a list, in which the numbers in the first column express the side of the squares which they subsequently tried, those in the second column the square of the diagonal, those in the third the nearest square number.

c20.s2.5

| 1 | 2 | 1 | 11 | 242 | 256 |
| 2 | 8 | 9 | 12 | 288 | 289 |
| 3 | 18 | 16 | 13 | 338 | 324 |
| 4 | 32 | 36 | 14 | 392 | 400 |
| 5 | 50 | 49 | 15 | 450 | 441 |
| 6 | 72 | 64 | 16 | 512 | 529 |
| 7 | 98 | 100 | 17 | 578 | 576 |
| 8 | 128 | 121 | 18 | 648 | 625 |
| 9 | 162 | 169 | 19 | 722 | 729 |
| 10 | 200 | 196 | 20 | 800 | 784. |

c20.s2.6

"How far the Sūtrakāras went in their experiments we are of course unable to say; the list up to twenty suffices for our purposes. Three cases occur in which the number expressing the square of the diagonal of a square differs only by one from a square number; 8—9; 50—49; 288—289; the last case being most favourable, as it involves the largest numbers. The diagonal of a square the side of which was equal to twelve, was very little shorter than seventeen (√289=17). Would it then not be possible to reduce 17 in such a way as to render the square of the reduced number equal or almost equal to 288?

c20.s2.7

"Suppose they drew a square the side of which was 17 padas long, and divided it into 17 × 17 = 289 small squares. If the side of the square could now be shortened by so much, that its area would contain not 289 but only 288 such small squares, then the measure of the side would be exact measure of the diagonal of the square, the side of which is equal to 12 (12² + 12² = 288). When the side of the square is shortened a little, the consequence is that two sides of the square a stripe is cut off; therefore a piece of that length had to be cut off from the side that the area of the two stripes would be equal to one of the 289 small squares. Now, as square is composed of 17 × 17 squares, one of the two stripes cuts off a part of 17 small squares and the other likewise of 17, both together of 34 and since these 34 cut off pieces are to be equal to one of the squares, the length of the piece to be cut off the side is fixed thereby; it must be the thirty-fourth part of the side of one of the 289 small squares.

c20.s2.8

"The thirty-fourth part of thirty-four small squares being cut off, one whole small square would be cut off and the area of the large square reduced exactly to 288 small squares; if it were not for one unavoidable circumstance.

c20.s2.9

The two stripes which are cut off from two sides of the square, let us say the east side and the south side, intersect or overlap each other in the south-east corner and the consequence is, that from the small square in that corner not 2/34 are cut off, but only 2/34 - 1/(34 × 34). Thence the error in the determination of the value of the saviśeṣa. When the side of a square was reduced from 17 to 16 33/34 the area of the square of that reduced side was not 288, but 288 + 1/(34 × 34). Or putting it in a different way: taking 12 for the side of a square, dividing each of the 12 parts into 34 parts (altogether 408) and dividing the square into the corresponding small squares, we get 408 × 408 = 166464. This doubled is 332928. Then taking the saviśeṣha-value of 16 33/34 for the diagonal and dividing the square of the diagonal into the small squares just described, we get 577 × 577 = 332929 such small squares. The difference is slight enough.

c20.s2.10

"The relation of 16 33/34 to 12 was finally generalised into the rule: increase a measure by its third, this third by its own fourth less the thirty-fourth part of this fourth (16 33/34 = 12 + 12/3 + 12/(3 × 4) - 12/(3 × 4 × 34)). The example of the saviśeṣa given by commentators is indeed 16 33/34 : 12; the case recommended itself by being the first in which the third part of a number and the fourth part of the third part were both whole numbers."

c20.s2.11

But a more simple and very plausible hypothesis will be that the expression for √ 2 was obtained in the following way:¹ Take two squares whose sides are of unit length. Divide the second square into three equal strips

c20.s2.12

¹ For a slightly different but less elegant procedure see Müller, p. 178. *loc. cit.*

c20.s2.13

Fig. 75

c20.s2.14

I, II and III. Sub-divide the last strip into three small squares III₁, III₂, III₃ of sides ⅓ each. Then on placing II and III₁ about the first square S in the positions II' and III'₁, a new square will be formed. Now divide each of the portions III₂ and III₃ into four equal strips. Placing four and four of them about the square just formed, on its east and south sides, say, and introducing a small square at the south-east corner, a larger square will be formed, each side of which will be obviously equal to $1 + ⅓ + 1/3.4 .$

c20.s2.15

Now this square is clearly larger than the two original squares by an amount (1/3.4)², the area of the small square introduced at the corner. So to get equivalence cut off from the either sides of the former square two thin strips. If x be the breadth of each thin strip, we must have $2x(1 + ⅓ + 1/3.4) - x² = (1/3.4)²$

c20.s2.16

Whence, neglecting x² as too small, we get $x = (1/3.4)² 3.4/34 = 1/3.4.34 .$

c20.s2.17

Thus we have finally $√ 2 = 1 + ⅓ + 1/3.4 1/3.4.34.$ nearly.

Linked texts (1)

Āpastambaśulbasūtra — Āpastamba, Sanskrit