The Science of the Shulba: A Study in Early Hindu Geometry c16: Chapter XI. Squaring the Circle

The Science of the Śulba: A Study in Early Hindu Geometry c16

Language: English

Chapter XI. Squaring the Circle

Baudhāyana says : “ If you wish to circle a square, draw half its diagonal about the centre towards the east-west line ; then describe a circle together with the one-third of that which lies outside (the square).”¹

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¹ “ **चतुरस्रं मण्डलं चिकीर्षन्नक्ष्यार्धं मध्यात् प्राचीमभ्यापातयेद्यदतिशिष्यते तस्य सह तृतीयेन मण्डलं परिलिखेत् ।** ”—*BŚl*, i. 58.

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The same method has been taught also by Āpastamba² and Kātyāyana.³

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² “ **चतुरस्रं मण्डलं चिकीर्षन्नक्ष्यात् कोण्यां निपातयेत् पार्श्वतः परिकृष्यातिशय-तृतीयेन सह मण्डलं परिलिखेत् । सानित्या मण्डलम् । यावद्धीयते तावदायन्तु ।** ”—*ĀpŚl*, iii, 2.

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³ *KŚl*, iii. 13.

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Fig. 72

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Let *ABCD* be a square and *O* its central point. Join *OA*. With centre *O* and radius *OA* describe a circle intersecting the east-west line *EW* at *E*. Divide *EM* at *P* such that PM=EM/3. Then with centre *O* and radius *OP* describe a circle. This circle will be nearly equal in area to the given square *ABCD*.

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Let 2*a* denote a side of the given square and *r* the radius of the circle equivalent to it; that is, *AB* = 2*a*, *OP* = *r*. Then $OA = a√2,$ and ME = (√2-1)a.

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Hence r = a + a/3(√2-1),

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$= a/3(2+√2).$

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Now according to the *Sulba*,¹

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¹ *Vide infra*, p. 189.

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$√ 2 = 1 + ⅓ + 1/3.4 - 1/3.4.34$

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Therefore r = a × 1·1380718

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The area of the transformed circle, employing the value π = 3·14159, will be 4·068987 × a² whereas the area of the given square is 4a². Hence the former result is too large. On the degree of equivalence of the area of the given square and of the circle into which it is transformed by the above rule, there is a noteworthy observation of Āpastamba. He remarks,

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*sānityā maṇḍalam* : *yāvaddhīyate tāvadāgantu*.²

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² *ĀpŚl*, iii. 2.

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According to the commentator Kapardisvāmī, the first word of this passage is a conjoint compound of the two words *sā* and *anityā*. So that the above should be rendered as, “It is an inexact (*anityā*) (method of construction); by as much the circle falls short, so much comes in.” He has been followed in this respect by Sundararāja. But the commentator Karavindasvāmī thinks that the correct reading of the passage will be *sā nityā maṇḍalaṃ*,’ etc. According to this reading, the observation appears to imply that the method of construction is an exact (*nityā*) one. But this commentator further thinks that the method has been called “exact” in a relative sense inasmuch as it yields a result more accurate than that obtained by any other method of circling the square known in the Āpastamba school. So he too finally comes to the same point of view as that Kapardisvāmī and Sundararāja.

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Thibaut accepts as correct the reading of the text as given by Karavindasvāmī but has discarded his further explanation of the matter. So he renders the passage as: “this line gives a circle exactly as large as the square; for as much as there is cut off from the square (*viz*., the corners of the square), quite as much is added to it (*viz*., the segments of the circle, lying outside the square).”¹ Bürk has closely followed Thibaut in this respect. The interpretation of Kapardisvāmī has been criticised by Thibaut thus: “But I am afraid we should not be justified in giving to Āpastamba the benefit of this explanation. The words ‘yāvaddhīyate, etc.’ seem to indicate that he was perfectly satisfied with the accuracy of his method and not superior, in this point, to so many circle-squarers of later times. The commentator who, with the mathematical knowledge of his time, knew that the rule was an imperfect one, preferred very naturally the interpretation which was more creditable to his author.''¹ Kapardisvāmī explains that the remark *yāvaddhīyate*, etc., implies only a nearer approximation but not exact equality.

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¹ Thibaut, *Śulbasūtras*, p. 26.

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¹ *Ibid*, p. 27.

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Now it may be pointed out that a similar remark, *viz*.,

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*eṣānityā caturasra-karaṇī*

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has been made by Āpastamba as regards his method for squaring the circle and which has been delivered by him immediately following the above one.² The same remark is found also in the corresponding rule of Baudhāyana.³ Here Thibaut entirely agrees with the commentators in breaking up the first word *eṣānityā* into *eṣā anityā*. But Bürk falls out from him and takes the reading to be *eṣā nityā caturasra-karaṇī*. Thus he has kept up the consistency of his opinion.

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² *ĀpŚl*, iii. 3.

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³ *BŚl*, i. 60.

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It is truly very difficult to conjecture now correctly what was really implied by Āpastamba by that remark, whether he held that method of circling a square was an exact or an inexact one. We may be, however, sure to this exent that the interpretation of the orthodox commentators cannot be brushed aside as unlikely as is supposed by Thibaut.

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*To transform a circle into a square.*

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Baudhāyana says:

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"> If you wish to square a circle, divide its diameter into eight parts; then divide one part into twenty-nine parts and leave out twenty-eight of these; and also the sixth part (of the preceding sub-division) less the eighth part (of the last).''⁴

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⁴ "मण्डलं चतुरस्रं चिकीर्षन्विष्कम्भमष्टौ भागान् कृत्वा भागमेकोनत्रिंशधा विभज्या-ष्टाविंशतिभागानुद्धरेद्भागस्य च षष्ठमष्टमभागोनम् ।"—*BŚl*, i. 59.

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Thus if 2a denote the side of a square equivalent to a circle of diameter d, then $2a = 7d/8 + [ d/8 - 28d/8.29 + ( d/8.29.6 - d/8.29.6.8 ) ]$ $or 2a = d - d/8 + d/8.29 - d/8.29 ( ⅙ - 1/6.8 ).$

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Since d = 2r, where r is the radius of the circle, $a = r - r/8 + r/8.29 - r/8.29.6 + r/8.29.6.8.$

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This result was probably obtained from the previous one by inversion $r = a/3 (2 + √ 2).$

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$Therefore 2a = 3/(2 + √ 2) d.$

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Substituting the value of √ 2, *viz*., 577/408, we have $2a = 1224/1393 d.$

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Thibaut supposes that Baudhāyana then proceeded in the following way: "One-eighth of 1393 = 174⅛; this multiplied by 7 = 1218⅞. Difference between 1218⅞ and 1224 = 5⅛. Dividing 174 (Baudhāyana takes 174 instead of 174⅛, neglecting the fraction as either insignificant or, more likely, as inconvenient) by 29 we get 6; subtracting from 6 its sixth part we get 5 and adding to this the eighth part of the sixth part of six, we get 5⅛. In other words: $1224 = ⅞ + 1/8.29 - 1/8.29.6 + 1/8.29.6.8 of 1393$ (due allowance made for the neglected ⅛)."¹

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¹ Thibaut, *Śulbasūtras*, p. 28.

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In the opinion of Cantor the series was probably obtained thus: ¹

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¹ Cantor, *Geschichte*, I, p. 643.

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$1224/1393 = ⅞ + 1/8.29 - 1/8.29.6 + 1/8.29.6.8 - 41/8.29.6.8.1393$

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The last term is nearly 1/34 of the term preceding it and so may be neglected as being comparatively small. Hence, we get

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$2a = 7d/8 + d/8.29 - d/8.29.6 + d/8.29.6.8$

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Müller conjectures the procedure adopted to have been as follows: ²

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² C. Müller, *loc. cit.*, pp. 187 f.

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$2a = 3/(2 + √ 2) d = (3√ 2)/(2√ 2 + 2) d = 3/2 · (√ 2)/(1 + √ 2) d,$

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since √ 2 = 17/12 - 1/12.34. Now

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$(7 + 1/34)/(58 - 2/34) = ⅛ · (56 + 8/34)/(58 - 2/34) = ⅛(1 - (2 - 10/34)/(58 - 2/34)),$

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$(10/34 - 2/34.29)/(2 - 2/34.29) = (5 - 1/29)/(34 - 1/29) = ⅙ · (30 - 6/29)/(34 - 1/29),$

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$= ⅙( 1 - (4 + 5/29)/(34 - 1/29) ),$

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$(4+5/29)/(34-1/29) = ⅛ · (32+40/29)/(34-1/29) = ⅛ ( 1 - (2-41/29)/(34-1/29) ),$

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Thus we have

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$3/(2+√ 2) = 1 - ⅛ + 1/8.29 - 1/8.29 ( ⅙ - 1/6.8 ),$ $- 1/8.29.6.8 · (2-41/29)/(34-1/29).$

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The last term may be neglected as being too small. Hence

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$2a = d - d/8 + d/8.29 - d/8.29 ( ⅙ - 1/6.8 ).$

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*Alternative Method.*

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Another method of squaring a circle has been taught by Baudhāyana, Āpastamba and Kātyāyana. It has been explicitly admitted by all of them that this method yields only a gross (*anityā*) value.

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"Or else divide (the diameter) into fifteen parts and remove two (of them). This is the gross (value of a) side of the (equivalent) square."

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That is to say,

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$2a = d - 2/15 d,$

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$or, a = r - 2/15 r.$

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The rationale of this formula seems to be this :² Draw the square *ABCD* circumscribing the circle and also the square *A'B'C'D'* inscribed within it.

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¹ *BŚl*, i. 69. See also *ĀpŚl*, iii. 3; *KŚl*, iii. 14.

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² Compare Müller, *loc. cit.*, p. 182.

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Fig. 73

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Then apparently the area of the circle will be smaller than the area of the square *ABCD* (= 4r²) and greater than the area of the square A′ B′ C′ D′ (= 2r²) ; that is $4r² > Area of the circle > 2r².$

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An obvious approximation will be $Area of the circle = (4r² + 2r²)/2 = 3r².$

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If 2a denote a side of the square equivalent in area to the circle, we shall have approximately $4a² = 3r²,$ $or, a = (√ 3)/2 r.$

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But, it will be shown later on¹ $√ 3 = 1 + ⅔ + 1/15 = 26/15,$ up to the second order of approximation. Therefore $a = 13/15r = r - 2/15r.$

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¹ *Infra*, p. 195.

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*Dvārakanātha's Corrections.*

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Dvārakanātha Yajvā has criticised with the help of specific examples the *Sulba* method of squaring a circle and its converse, as yielding only approximate results. He has then proposed the following corrections to the ancient formulae : ¹

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¹ *Paṇḍit*, O.S., Vol X, p. 21.

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(i) r = a + a/3 (√ 2 - 1) ( 1 - 1/118 ).

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(ii) 2a = d - d/8 + d/8.29 + d/8.29 ( ⅙ - 1/6.8 ) × ( 1 + ½ · 3/133 ).

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By the formula (i) the area of the transformed circle will be nearly equal to 4·000344 × a², the value of π being taken to be 3·14159.

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*Value of π.* The above rules of the *Sulba* for squaring a circle and *vice versâ*, will work out the following values of π :

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(1) π = 4/( 1 + ⅓ (√ 2 - 1) ²) = 3·0883...

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(2) π = 4 ( 1 - ⅛ + 1/8.29 - 1/8.29.6 + 1/8.29.6.8 ) = 3·0885

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(3) π = 4(1 - 2/15)² = 3·004.

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It will be noticed that none of these values are fairly accurate, as according to modern calculation π = 3·14159...

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In one instance Baudhāyana¹ has employed the value of π=3. A better value of π is found in the *Mānava Sulba*, which states that a square of two by two cubits is equivalent to a circle of radius 1 cubit and 3 aṅgulis.² Whence
(4) π=4(8/9)²=3·16049 This value was given before by the Egyptian Ahmes (*c.* 1500 B.C.). With Dvārakanātha's corrections we have π=4/([1+⅓(√ 2-1)]²)(118/117)²=3·141109... π=4(1-⅛+1/8.29-1/8.29.6+1/8.29.6.8)² ×(1+½·3/133)² =3·157991...

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¹ "यूपावटाः पदविष्कम्भाः, त्रिपदपरणाहानि यूपोपराणीति।"—*BŚl*, i. 112-3.

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² *MāŚl*, i. 27.

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*Early History.*

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It will be interesting to know the early history of the problem of the squaring of the circle and of its converse, the circling of the square, in India. Their origin dates, as has been noted before, earlier than the time of the *Ṛg-veda* (before 3000 B.C.). That was in connection with the construction of the three primarily essential sacrificial altars of the Vedic Hindus, namely the *Gārhapatya*, *Āhavanīya* and *Dakṣiṇāgni*. For these three altars had to be of the same area but of different shape, the first circular, the second square and the last semi-circular. Again, it has also been noted before, that *Gārhapatya* altar may have, according to certain authorities, an alternative shape of a square, besides its usual circular shape, but retaining the same area. The earliest express reference to this tradition is found in the *Śatapatha Brāhmaṇa* (c. 2000 B.C.).¹ It appears copiously in the Sūtra works.² From the latter we also learn of another ancient tradition that the *Dhiṣṇya*³ may be square or circular in shape but with the same area one square *piśila*. The form of an archaic variety of the *Śmaśāna-cit* ("Fire-altar of the shape of the cemetery ") is stated to be circular; according to some peoples and square according to others. As regards its size. the *Śatapatha Brāhmaṇa*, after referring to the earlier opinions, preferably approves of an area of one square puruṣa.⁴ Further instances of the early applications of the above problems are found in the *Taittirīya* and other *Saṃhitā* (c. 3000 B.C.)⁵ in connexion with the construction of *Ratha-cakra-cit*, *Samuhya-cit*, *Paricāyya-cit* and *Droṇa-cit* (alternative shape). In each of these cases, one has to draw at first a square equal in area to that of the primitive *Śyena-cit*, viz., 7½ square puruṣas and then that square has to be circled.⁶ This has been clearly described by Baudhāyana⁷ and Āpastamba⁸ and it was doubtless so before.¹ We occasionally meet with other instances of this kind in the earlier Hindu works.²

Linked texts (9)

Āpastambaśrautasūtra — Āpastamba, Sanskrit
Baudhāyanaśrautasūtra — Baudhāyana, Sanskrit
Śatapathabrāhmaṇa — Yājñavalkya & Śāṇḍilya (Vedic teacher), Sanskrit
Āpastambaśulbasūtra — Āpastamba, Sanskrit
Taittirīyasaṃhitā — Sanskrit
Mānavaśulbasūtra — Sanskrit
Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit
Kātyāyanaśulbasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Ṛgvedasaṃhitā — Sanskrit