The Science of the Shulba: A Study in Early Hindu Geometry c20.s3: Approximate Value of √3

The Science of the Śulba: A Study in Early Hindu Geometry c20.s3

Language: English

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.

c20.s3.5

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

c20.s3.6

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

c20.s3.7

*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.

c20.s3.11

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.

c20.s3.12

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.

c20.s3.15

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

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

c20.s3.18

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

c20.s3.19

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 $Viśeṣa of a = a/3 + a/3.4 - a/3.4.34.$

c20.s3.20

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

c20.s3.21

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

c20.s3.29

Karavindasvāmī's observations are more elaborate.

c20.s3.30

"*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.

c20.s3.32

² *Ibid*, xx. 7.

c20.s3.33

³ *Ibid*, ii. 1.

c20.s3.34

⁴ *Ibid*, xix. 2.

c20.s3.35

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.

Linked texts (6)

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