The Science of the Shulba: A Study in Early Hindu Geometry c20: Chapter XV. Elementary Treatment of Surds

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

Language: English

c20.s1 Approximate Value of √2 — 7 parts
c20.s2 Hypotheses about its Origin — 17 parts
c20.s3 Approximate Value of √3 — 42 parts
c20.s4 Other Approximate Values of √2 — 29 parts
c20.s5 Approximate Formula — 26 parts

Chapter XV. Elementary Treatment of Surds

c20.2

In the *Sulba*, we find elementary treatment of surds, particularly their addition, multiplication and rationalization. For the face, base and altitude of the *Sautrā-maṇikī-vedi* which is of the shape of an isosceles trapezium, are stated, it has been noted before,¹ to be respectively 24/√3, 30/√3 and 36/√3 prakramas, or to be 8√3, 10√3 and 12√3 prakramas. Hence it is clear that the ancient Hindus knew that $24/(√3) = 8√3, 30/(√3) = 10√3, 36/(√3) = 12√3.$

c20.3

¹ *Supra*, p. 153.

c20.4

It is also stated in general that if the side of a square be a, then the side of the square equal to the third part of it will be ⅓(a√3) or √3(a/3).² That is, $a/(√3) = (a√3)/3.$

c20.5

² *ĀpŚl*, ii. 3; *BŚl*, i. 47; *KŚl* ii. 15-6; see also pp. 74 f. *supra*.

c20.6

Thus it appears that the rationalization of simple surds was known at that time.

c20.7

The area of the above trapezium is stated to be 324 square prakramas. It must have been calculated with the help of the rule given in the *Sulba* for that purpose. So that $36/(√3) × ½(24/(√3) + 30/(√3)) = 36/3 × 54/2 = 324.$ $12√3 × ½(8√3 + 10√3) = 12 × 3 × 9 = 324$

c20.8

Again the dimensions of another isosceles trapezium (*Aśvamedhikī-vedi*) are stated thus: face = 24√2, base = 33√2, altitude = 36√2 prakramas and area = 1944 square prakramas. That is $36√2×½(24√2+30√2)=1944.$

c20.9

In the Śulba, a surd is technically called *karaṇī*. Thus *dvi-karaṇī*-means √2, *tri-karaṇī* =√3, *tṛtīya-karaṇī* =√(1/3), *saptama-karaṇī* =√(1/7), *aṣṭādaśa-karaṇī* =√18, etc.¹ The term was also used in the more general sense of a root. For we have at least one instance of its application in that sense, *e.g.*, *catuṣkaraṇī* =√4, which is not a surd.² The Sanskrit word *karaṇī* means "producer," "that which makes." From that it came to denote the sides of a rectilinear geometrical figure of any shape,³ and then more particularly, the side of a square.

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¹ The references are respectively to *ĀpŚl*, i. 5, ii. 2, ii. 3 ix. 5, xix. 1.

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² *ĀpŚl*, ii. 6.

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³ See *ĀpŚl*, ix. 6; xii. 5, 6, 9; xiii. 1, etc.

Approximate Value of √2

c20.s1.1

Baudhāyana and Āpastamba say:

c20.s1.2

"Increase the measure (of which the *dvi-karaṇī* is to be found) by its third part, and again by the fourth part (of this third part) less by the thirty-fourth part of itself (*i.e.*, of this fourth part). (The value thus obtained is called) the *saviśeṣa*."⁴

c20.s1.3

⁴ “प्रमाणं तृतीयेन वर्धयेत्तच्च चतुर्थेनात्मचतुस्त्रिंशोनेन। सविशेषः।”—*BŚl*, i. 61-2; *ĀpŚl*, i. 6.

c20.s1.4

Kātyāyana defines the rule in nearly identical words.⁵ Thus if d be the *dvi-karaṇī* of a, that is, if d be the side of a square whose area is double that of the square on a, then, according to the rule, $d = a + a/3 + a/3.4 - a/3.4.34.$

c20.s1.5

⁵ “करणीं तृतीयेन वर्धयेत्तच्च स्वचतुर्थेनात्मचतुस्त्रिंशोनेन सविशेष इति विशेषः।”—*KŚl*, ii. 13.

c20.s1.6

Now, it has been stated before, the diagonal of a square is its *dvi-karaṇī*. So this rule gives the relation between the diagonal and side of a square. Indeed the above rule is particularly meant to define that relation. Thus we get $√ 2 = 1 + ⅓ + 1/3.4 - 1/3.4.34.$

c20.s1.7

In terms of decimal fractions, this works out √ 2 = 1·4142156. According to modern calculation √ 2 = 1·414213. Thus it is clear that the ancient Hindus attained, a very remarkable degree of accuracy in calculating an approximate value of √ 2.

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

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¹ *Śulbasūtras*, pp. 13 ff.

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"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.

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| 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?

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"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

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

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Fig. 75

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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 $x = (1/3.4)² 3.4/34 = 1/3.4.34 .$

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

Approximate Value of √3

c20.s3.1

By the process indicated above we can easily get an approximate value of √3. In this case two of the unit squares are divided into six equal strips I, II,...VI. The last two strips are subdivided into six smaller squares of sides 1/3 each. Then arranging the slices III, IV, V₁, V₂, VI₁, and VI₂, about the first square in the positions III', IV', V'₁, V'₂, VI'₁, VI'₂, a new square can be formed. Now divide each of the portions left over, *viz.*, V₃ and VI₃, into

c20.s3.2

Fig. 76

c20.s3.3

five equal parts and place them about the square just formed. Then introducing a small square at the corner another complete square of sides equal to $1 + ⅔ + 1/3.5$ each will be formed. But this is clearly too large by the amount (1/3.5)². So for closer approximation, let the side of the new square be diminished by an amount y, such that $2y ( 1 + ⅔ + 1/3.5 ) - y² = ( 1/3.5 ).$

c20.s3.4

Hence y = 1/3.5.52 , nearly.

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Thus we get ¹ $√ 3 = 1 + 2/() + 1/3.5 - 1/3.5.52$

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¹ For a different method of approximating to the value of √ 3, see Müller, *loc. cit.*, pp. 182 f.

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*Irrationality of √ 2.*

c20.s3.8

Did the ancient Hindus recognise the irrationality of √ 2 ? This question does not seem to have troubled Thibaut in any way. For we do not find him to raise it explicitly or to attempt to answer it directly. But it is clear from his writings that he believed in the Hindu knowledge of irrationality of √ 2. Indeed his theory about the discovery of the Hindu approximation to the value of √ 2, which has been quoted in *extenso* before, is fundamentally based on the knowledge of the incommensurability of the diagonal of a square with its sides. Thibaut supposes that the ancient Hindus endeavoured ' to discover a square the side and diagonal of which might be expressed in integral numbers' and then observes, ' undoubtedly they arrived soon at the conclusion that they would never find exactly what they wanted and had to be contented with an approximation.'' Von Schroeder ² and Bürk ³ are more explicitly emphatic. They claim for the ancient Hindus the credit for the first discovery of irrationals. And they have been followed in this respect by Garbe, Hopkins and Macdonell.⁴ But this hypothesis has been criticised and opposed by some modern historians of mathematics.¹

c20.s3.9

² Von Schroeder, *Pythagoras und die Inder*, Leipzig, 1884, pp. 39-59 ; compare also *Indien Literatur und Kultur*, Leipzig, 1887.

c20.s3.10

³ *ZDMG*, LV, p. 557.

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⁴ R. Garbe, *Philosophy of Ancient India*, pp. 39 ff. ; E. W. Hopkins, *Religion of India*, pp. 559 f. ; A. A. Macdonell, *History of Sanskrit Literature*, p. 422.

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¹ H. G. Zeuthen, "Theorem de Pythagore. Origine de la géométrié scientifique," *Comptes Rendus du IIme Congrés Intr. d. Philosophie*, Genéve, 1904; M. Cantor, "Über die älteste indische Mathematik," *Arch. d. Math. u. Phys.*, VIII (3), 1905, pp. 63-72; H. Vogt, "Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt?" *Bibl. Math.*, VII (3), 1906, pp. 6-23; T. L. Heath, *Euclid*, I, p. 362 f.

c20.s3.13

There are two terms which have undoubtedly a great bearing on the point under discussion. They are *viśeṣa* and *saviśeṣa*. Importance of these terms has not been properly realised by previous writers. Thibaut simply observes that *saviśeṣa* is a technical name for the increased measure.² Bürk remarks, " The total increase is *viśeṣa* because it is the 'difference' between the *pramāṇa*, *i.e.*, the side of the given square and its *dvi-karaṇī*. Therefore this latter is *saviśeṣa*, 'with the difference.'"³

c20.s3.14

² Thibaut, *Śulbasūtras*, p. 13.

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³ Bürk, *ZDMG*, LVI, p. 330; compare also LV, p. 548 ("*saviśeṣa*, *i.e.*, of the approximation to the *dvi-karaṇī*") and p. 557, fn. 1.

c20.s3.16

In the *Śulba*, the calculated value of the diagonal of a square is technically called the *saviśeṣa* of its side. Or symbolically,⁴

c20.s3.17

$Savisesa of a=a+a/3+a/3.4-()/3.⁴.34.$

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⁴ *BŚl*, i. 61-2; *ĀpŚl*, i.6; *KŚl*, ii. 13.

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Now what is the radical significance of the term *saviśeṣa*? Before answering this question, we shall point out that occasionally the term has been used to denote the complete diagonal in general; that is, in the sense *saviśeṣa* of a = a√2.⁵ Again in one instance in the *Āpastamba* *Śulba*,¹ we find the use $Viśeṣa of a = a/3 + a/3.4 - a/3.4.34.$

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⁵ *BŚl*, iii. 67, 149, 150; *ĀpŚl*, xix. 2, 3, 4, 7.

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¹ “पृष्ठान्तयोर्मध्ये च शङ्कुन्निहत्य अर्द्धं तद्विशेषमभ्यस्य लक्षणं कृत्वा अर्द्धमागमयेत् अन्यौ पाशौ कृत्वा, मध्यमे सविशेषं प्रतिमुच्य …”—*ĀpŚl.* ii. 1.

c20.s3.22

But on several occasions in this work and also in other *Śulbas*,² particularly in a compound word, the term *viśeṣa* has been employed in the sense of the hypotenuse of a right-angled triangle. And it has again been considered there as equivalent to *saviśeṣa*.³ The concluding portion of Kātyāyana's statement of the rule for *saviśeṣa* runs thus:

c20.s3.23

² *BŚl*, iii. 164; *ĀpŚl*, xx. 5, 7, 8, 11, etc. In these cases occur the term *bāhya-viśeṣa* (meaning “having the *viśeṣa* outwards”); in *ĀpŚl*, xx. 6 we find *abhyantara-viśeṣa* (“having the *viśeṣa* inwards”).

c20.s3.24

³ For example, one kind of bricks employed in the construction of the Fire-altar of the shape of falcon with bent wings and spreadout tail (*Vakrapakṣa-vyastapuccha-śyenacit*) was called *ṣoḍaśī*. Its dimensions are described thus: “षोडशीं चतुर्भिः परिगृह्णीयात् अष्टमेन, त्रिभिरष्टमैश्चतुर्थेन चतुर्थसविशेषेनेति”—*ĀpŚl*, xix. 2. “Construct the *ṣoḍaśī* with four (sides), namely with one-eighth, three-eighth, one-fourth (puruṣa) and the *saviśeṣa* of the one-fourth (puruṣa).” The manner of laying out these bricks has been described thus: “अवशिष्टं षोडशीभिः परिच्छादयेत् अन्या बाह्यविशेषा अन्तरशिरसः”—*ĀpŚl*, xx. 5. “Cover the remaining portion (of the Fire-altar) with *ṣoḍaśīs*, (such that) those lying at the extremity (of the Fire-altar) will have their *viśeṣa* outwards, but inwards at the head.” “अपरस्मिन् प्रस्तारे पुरस्ताच्छिरसि द्वे षोडश्यौ बाह्यविशेषे उपदद्ध्यात् ते अपरेण द्वे विषये आभ्यन्तरविशेषे”—*ĀpŚl*, xx. 6. “In the second layer, at the head towards the east lay two *ṣoḍaśīs* having their *viśeṣa* outwards and on the west lay them with their *viśeṣa* inwards and lying in both places (*viṣaya*, *i.e.*, partly in the head and partly in body of the Fire-altar).”

c20.s3.25

*saviśeṣa iti viśeṣaḥ*. Here the word *viśeṣa* must be explained differently. The use of the same terms thus in varied significations, has made the interpretation of their origin really difficult. Thibaut's explanation is obviously erroneous; Bürk's is not full.

c20.s3.26

Let us now see how the origin and significance of the term *saviśeṣa* has been interpreted by the early commentators. Dvārakanātha Yajvā is of no help to us in this matter. He passes off with the simple remark that "*saviśeṣa* is a technical term for it." Kapardisvāmī observes:

c20.s3.27

"> *Saviśeṣa* is a technical name for the sum thus obtained.¹ As it is accompanied with a special quantity in excess (*viśeṣa*), so it is a term whose meaning is intelligible by itself. (Add) to twelve (aṅgulis) four (aṅgulis), to four one; divide that one into thirty-four parts and leave out a part. Thus (the resulting sum) will be seventeen aṅgulis less one *tila*. The square of the tilas in twelve aṅgulis is 166464 square tilas; the square of the tilas in seventeen aṅgulis minus one tila is 332929 square tilas. In this (latter) a square of one tila is in excess (of twice the other), so the *saviśeṣa* is the technical name (for it). If it is accompanied with some quantity in excess (*viśeṣa*), then what is the necessity for it? In its own domain, it has no fault. For if the diagonal be measured (directly) with a bamboo-stick, the excess will be to the extent of ten square tilas. Thus in any case there will be an excess even by a fraction of the smallest part of the minute *nīvāra* grain falling from the mouth of a parrot. So (the formula) is without fault. It will be of practical use, taking this into consideration, the learned author has made that (technical) term."

c20.s3.28

¹ The reference is to a + a/3 + a/3.4 - a/3.4.34

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Karavindasvāmī's observations are more elaborate.

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"*Saviśeṣa* is its technical name. The measure of the side (of a square) after having been operated upon by the rule 'increase the measure by its one-third, etc.' is called the *saviśeṣa*. For instance, increase a measure of twelve aṅgulis by four aṅgulis; increase that four aṅgulis by one aṅguli minus one tila. The experts (in measure) say that thirty-four tilas placed breadthways make one aṅguli. It will be stated (later on in the text) 'add to half (the length of the east-west line) its *viśeṣa*,'¹ 'enclose with two half-bricks having their *viśeṣa* outwards,'² etc. What (are the correct meanings) of 'fasten *saviśeṣa* to the middle (pole),'³ 'with one-fourth puruṣa and with the *saviśeṣa* of one-fourth puruṣa,'⁴ etc. There it should be understood that 'being with the *viśeṣa*,' that is 'the measure with the *viśeṣa*' is the *saviśeṣa*. What is the need of this big term? So that its true significance may be intelligible by itself. What is that? The root *śiṣ* when prefixed by *vi* denotes in all cases 'a correction in excess.' As the rule mentions of an excess qualified by the prefix *vi* (it should be understood that the accurate value of) the diagonal differs by something from, exceeds over the exceeding part of the above value of the *dvi-karaṇī* over the measure of the side: and that is the *viśeṣa*. Or else the *viśeṣa* is that little area by which (the square of the diagonal as calculated by the rule) differs from or exceeds over (twice the area of the given square) at the time of measurement. For instance, since it has been stated that thirty-four tilas make one aṅguli, the square of the number of tilas in twelve aṅgulis is 166464 square areas of tilas; the square of the tilas in seventeen aṅgulis minus one tila is 332929 square areas of tilas. In the latter a square of one tila is present in excess over twice the area of the (given)

c20.s3.31

¹ *ĀpŚl*, ii. 1.

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² *Ibid*, xx. 7.

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³ *Ibid*, ii. 1.

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⁴ *Ibid*, xix. 2.

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square figure; hence its technical name is *viśeṣa*. If the diagonal (of the square) could be (exactly) measured with the bamboo-rod and the Fire-altar had been measured by means of that measure, the area included in the body (of the Fire-altar) would not have exceeded (twice) this (the given square) even by the smallest part of the minute *nīvāra* grain falling from the mouth of a parrot. If the measure falls short (the area described) in that case will also be smaller. So for doubling (a square), the *saviśeṣa* is employed as a practical means.''

c20.s3.36

Thus it is evident that according to the interpretation of the commentators Kapardisvāmī and Karavindasvāmī, the term *saviśeṣa* for the diagonal of a square implies intrinsically a knowledge of the following: (1) The value of the diagonal as calculated by the rule stated for the purpose is only an approximate one; (2) that value of the diagonal has a small quantity in excess over the true value of the diagonal; or in other words, the square of that value exceeds by some quantity twice the area of the given square; (3) that excess cannot be completely eliminated in calculating the value of the diagonal arithmetically. If the validity of this interpretation be accepted, then there will remain nothing to doubt that the ancient Hindus were aware of the incommensurability of the diagonal of a square with its sides.

c20.s3.37

Let us therefore test as far as possible how far the commentators can be relied upon in the matter of that interpretation. For it might be argued by modern critics that being acquainted with the real state of things from the mathematical knowledge of their time, the commentators naturally gave an interpretation to the texts which was more creditable to their authors.¹ Looking into the ancient literatures of India, we find in the early canonical works of the Jainas copious instances of the employment of the term *viśeṣa* in the same connection as we find it in the *Sulba*. Thus in the *Sūryaprajñapti* (c. 500 B. C.)¹ the circumference of a circle whose diameter is 99640 yojanas is stated as 315089 yojanas and a little over (*kiñcidviśeṣādhika*); that of a circle of diameter 100660 yojanas is stated to be 318315 and a little less (*kiñcidviśeṣoṇa*). In the *Jambudvīpaprajñapti* (c. 300 B.C.),² the circumference of the Jambudvīpa which is of the shape of a circle of 100000 yojanas in diameter, is mentioned as 316227 yojanas 3 gavyutis 128 dhanus 13½ aṅgulis and a little over (*kiñcid-viśeṣādhika*). In all these cases,³ it will be seen that the value actually recorded is only an approximate one and the *viśeṣa* refers to a small quantity which has not been recorded—in fact it cannot be accurately determined—but which we shall have to add to or subtract from the recorded value in order to get the accurate value of the quantity sought. Those who are acquainted with the technique of Sanskrit and Sanskritic languages will at once recognise that the expression *kiñcid-viśeṣādhika* has the identical significance as the term *saviśeṣa*. Indeed we really find the use of the word *saviśeṣa* in exactly the same sense in a Jaina work of later times. Nemicandra (c. 975 A. D.) writes: *tattiūṇaṁ parirayeṇa saviseṣaṁ* or "thrice that with a little over (*saviśeṣa*) is the circumference."⁴ Now the early canonical works of the Jainas belong to a period not much separated from that of the *Śulba*. Some works of either classes probably belong to the same age. So we can accept without any hesitation that the term *saviśeṣa* was employed originally in the *Śulba* with the same significance as that with which it is found to have been employed in the early canonical works of the Jainas. Thus it is found conclusively that Kapardisvāmī and Karavindasvāmī are thoroughly reliable as regards their interpretation of the original significance of the term *saviśeṣa*. So it is proved that the irrationality of √2 was known to the ancient Hindus.

c20.s3.38

¹ The commentators have been similarly accused in other connexions by Thibaut (*Śulbasūtras*, pp. 46 f.) and Bürk (comments on *ĀpŚl*, i. 4).

c20.s3.39

¹ Sūtra 20. The formula employed for the calculation is $circumference = √(10 × (diameter)²).$

c20.s3.40

² Sūtra 3.

c20.s3.41

³ Several instances of this kind will be found in the author's article on "The Jaina School of Mathematics" in the *Bull. Cal. Math. Soc.*, Vol. XXI, 1929, pp. 115-145; see particularly pp. 131-3.

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⁴ *Trilokasāra* of Nemicandra, with the commentary of Mādhavacandra, edited by Manoharlal Sastri, Bombay, 1918, Gāthā 95.

Other Approximate Values of √2

c20.s4.1

From the *Mānava Śulba* we obtain certain other noteworthy approximations to the value of √2. By way of some calculations in that work are employed the relations: {lrl}
(1) & 40² + 40² \!\!&= 56² / (2) & 4² + \ 4² \!\!&= (5 2/3)²

c20.s4.2

From these we easily derive the values, {lrl} (1·¹) & \surd2 \!\!&= 7/5 = 1·4 / (2·1) & \surd2 \!\!&= 17/12 = 1·4166\dots

c20.s4.3

It may be noted that 7/5 is the third convergent of √2 expressed as a continued fraction ¹ and the value 17/12 is its fourth convergent. What is still more noteworthy is the fact that the former value is not derivable from the series for √2 stated before. This latter is, however, the eighth convergent of the continued fraction for √2: $577/408 = 1 + ⅓ + 1/3.4 - 1/3.4.34.$

c20.s4.4

¹ √2 = 1 + 1/(2+) 1/(2+)

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These facts will lead one strongly to suspect if the rudiments of the theory of continued fractions were known to the early Hindus. At any rate, these are very remarkable cases of coincidence.

c20.s4.6

There are certain other rough values of √2 to which we shall refer shortly.

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*Approximation to the Value of √5.*

c20.s4.8

There seems to have been a serious attempt, though without much success, to find an approximation to the value of the surd √5. The occasion was to define clearly the relative positions of the three principal and oldest known fire-altars, *viz.*, the *Gārhapatya*, *Āhavanīya* and *Dakṣiṇa*. Baudhāyana's rules to determine their positions are these:

c20.s4.9

"With the third part of the length (*i.e.*, the distance between the *Gārhapatya* and *Āhavanīya*) describe three squares closely following one another (from the west towards the east); the place of the *Gārhapatya* is at the north-western corner of the western square; that of the *Dakṣiṇāgni* is at its south-eastern corner; and the place of the *Āhavanīya* is at the north-eastern corner of the eastern square." ¹

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¹ *BŚl*, i. 67.

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"Or else divide the distance between the *Gārhapatya* and *Āhavanīya* into five or six (equal) parts; add (to it) a sixth or seventh part; then divide (a cord as long as) the whole increased length into three parts and make a mark at the end of two parts from the eastern end (of the cord). Having fastened the two ends of the cord (to the two) poles at the extremities of the distance between the *Gārhapatya* and *Āhavanīya*, stretch it toward the south, having taken it by the mark and fix a pole at the point reached. This is the place of the *Dakṣiṇāgni*." ²

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² *BŚl*, i. 68.

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" 'Or else increase the measure (between the *Gārhapatya* and *Āhavanīya*) by its fifth part; divide (a cord as long as) the whole into five parts and make a mark at the end of two parts from the western extremity (of the cord). Having fastened the two ties at the ends of the east-west line, stretch the cord towards the south having taken it by the mark and fix a pole at the point reached. This is the place of the *Dakṣiṇāgni*.' "¹

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¹ *Ibid*, i. 69.

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The second is also given by Āpastamba.² A rule leading to the same result as the first one above, though defined differently, is stated by Kātyāyana ³ and Manu.⁴ Kātyāyana has specified the relative positions of the three fire-altars also in a new way :

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² *ĀpŚl*, iv. 4.

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³ *KŚl*, i. 29.

c20.s4.18

⁴ *MāŚl*, iii.

c20.s4.19

" 'Divide the distance between the *Gārhapatya* and *Āhavanīya* into six or seven parts; add a part; then divide (a cord) equal to the total increased length into three parts, etc.' "⁵

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⁵ *KŚl*, i. 27.

c20.s4.21

The rest of this rule is the same as the latter portion of the second rule above and hence need not be mentioned.

c20.s4.22

Fig. 77

c20.s4.23

Let b denote the distance between the *Gārhapatya* and *Āhavanīya*, that is, AG. Then from the different specifications given above we obtain the following values for AD and GD: $AD = b/3√ 5, ⅘b, 7/9b, 16/21b, 18/25b$ $GD = b/3√ 2, ⅖b, 7/18b, 8/21b, 12/25b$

c20.s4.24

If it be assumed that the relative positions of the three fire-altars were meant to be the one and the same, in all cases though expressed differently, then we shall have the following approximations to the values of √ 5 and √ 2: $√ 5 = 2⅖, 2⅓, 2 2/7, 2 4/25,$ $= 2·4, 2·333, 2·285, 2·16.$ $√ 2 = 1⅕, 1⅙, 1 1/7, 1 11/25,$ $= 1·2, 1·166, 1·142, 1·44.$

c20.s4.25

Since according to modern calculation √ 2 = 1·414213 and √ 5 = 2·23607, none of the above values can be said to be a fair approximation, perhaps except the values √ 5 = 2 2/7 and √ 2 = 1 11/25 which are correct up to the first place of decimals.

c20.s4.26

*Evaluation of Other Surds.*

c20.s4.27

In the *Mānava Śulba*, we find results leading incidentally to the evaluation of two other surds: $36² + 90² = 97²$ $5² + 61² = (7⅞)².$

c20.s4.28

Whence we easily obtain $√ 29 = 5 7/18 = 5·3888$ $√ 61 = 7⅚ = 7·8333$

c20.s4.29

Now correct up to three places of decimals √29 = 5·385... and √61 = 7·810... Hence the above approximations, specially the first one, may be said to be fair for ordinary purpose.

Approximate Formula

c20.s5.1

It would be natural to ask if there is in the *Śulba* any specific rule for determining the approximate value of any surd. But the answer is not very reassuring. For we do not find an express statement of any formula for the purpose. At the same time we can unhesitatingly admit that they had the necessary equipments for the approximate evaluation of surds, at least of some. We have pointed out before that in the science of the *Śulba*, it is sometimes necessary to construct a square having a given area. And that is a geometrical method of finding the square-root of a given number. If the given area is represented by a non-square number, we get a method finding the square-root of a non-square number. One formula follows at once from the method given in the *Śulba*, for the transformation of a rectangle into a square, which has been described before. According to this method, a square (having its side equal to the breadth of the given rectangle) is first cut off from the given rectangle; the excess portion is divided into two halves which are then joined to the two sides of that square. Then by adding a small square in the corner, a large square is completed. The square equivalent to the given rectangle will be obtained, it is said, by subtracting the added small square from the completed large square. This subtraction was doubtless made by the Śulba-workers with the help of the theorem of the square of the diagonal. But it can also be made by cutting off two strips from the two sides, say, the east side and the south side of the completed large square.¹ Suppose A to be the area of the given rectangle and a be the side of the square subtracted from it. Then the side of the large completed square will be a + (A-a²)/2a. The area of the small added square will be ((A-a²)/2a)². Then from each side of the large sphere we shall have to cut off a thin strip of the same breadth. If x denote the breadth of the strip, we must have

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¹ The process is in fact the reverse of that taught in the *Śulba* for the increment of a given square into another square.

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$((A-a²)/2a)² = 2x(a + (A-a²)/2a) - x²$

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$or x = (((A-a²)/2a)²)/(2(a + (A-a²)/2a)),$

c20.s5.5

neglecting x² as being very small. Hence we finally arrive at the formula

c20.s5.6

$√ A = a + (A-a²)/2a - (((A-a²)/2a)²)/(2(a + (A-a²)/2a))$

c20.s5.7

This formula requires a correction, it will be easily recognised, inasmuch as a portion equivalent to x² has been subtracted too much.

c20.s5.8

The rule of the Bakhshālī Manuscript for determining the approximate root of a non-square number must have been obtained exactly in this way. It says ¹

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¹ Bibhutibhusan Datta, "The Bakhshālī Mathematics," *Bull. Cal. Math. Soc.*, XXI, 1929, pp. 1-60.

c20.s5.10

"In case of a non-square (number), subtract the nearest square number; divide the remainder by twice (the root of that number). Half the square of that (that is, the fraction just obtained) is divided by the sum of the root and the fraction and subtract; (this will be the approximate value of the root) less the square (of the last term)."

c20.s5.11

If A = a² + r, we write the above formula as

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$√A = √(a² + r) = a + r/2a - ((r/2a)²)/(2(a + r/2a)).$

c20.s5.13

Now this formula will not be available for finding the approximate values of those surds in which r is not small compared with a². So Rodet ² holds that a different process of approximation to the value of a surd was also known to the authors of the *Sulba*. It will lead to the formula, says he,

c20.s5.14

$√(a² + r) = a + r/(2a + 1) + (r/(2a + 1)(1 - r/(2a + 1)))/(2(a + r/(2a + 1))) + ε$ where $ε = [ r - r/(2a+1) + (r/(2a+1)(1 - r/(2a+1)))/(2(a + r/(2a+1))) .$ $. × 2a + r/(2a+1) + (r/(2a+1)(1 - r/(2a+1)))/(2(a + r/(2a+1))) ]$ $÷ 2 a + r/(2a+1) + (r/(2a+1)(1 - r/(2a+1)))/(2(a + r/(2a+1)))$

c20.s5.15

² L. Rodet, "Sur une méthode d'approximation des recines cares conne dans l'Inde antérieurment a la conquite d'Alexandre, *Bull. Soc. Math. d. France*, VII, 1879, pp. 98-102; "Sur les méthodes d'approximation chez las anciens."--*Ibid*, pp. 159-167.

c20.s5.16

A nearly equivalent formula will be obtained by proceeding in the following way: From the given rectangle A, cut off the square (S) of side a each. Let the area of the remaining portion of the rectangle be r. From this cut off two strips I and II of the same breadth r/(2a+1) and arrange them as in Fig. 78. Then the area of the strip of the rectangle still left over will be $= r - 2a × r/(2a+1) = r/(2a+1).$

Linked texts (8)

Āpastambaśulbasūtra — Āpastamba, Sanskrit
Kātyāyanaśulbasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Mānavaśulbasūtra — Sanskrit
Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit
Sūriyapannatti — Ardhamāgadhī Prakrit
Jambuddīvapannatti — Ardhamāgadhī Prakrit
Tiloyasāra — Nemicandra Siddhāntacakravartin, Śaurasenī Prakrit
Bakhshālī Manuscript — Sanskrit