The Science of the Shulba: A Study in Early Hindu Geometry c15: Chapter X. Rational Rectangles

The Science of the Śulba: A Study in Early Hindu Geometry c15

Language: English

Chapter X. Rational Rectangles

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In the *Sulba-sūtra* we meet with the following rational rectangles:

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⁷ *BŚl*, i. 49; *ĀpŚl*, v. 2, 4; *MāŚl*, v. 2-3.

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Besides these of which the sides and the diagonal are rational *integers*, we find also a few other rectangles whose sides and diagonal are expressible in rational *fractions* :

It will be very interesting to know how the early Hindus discovered their rational rectangles. Thibaut observes : " Most likely they discovered that the square on the diagonal of an oblong, the sides of which were equal to three and four, could be divided into twenty-five small squares, sixteen of which composed the square on the longer side of the oblong, and nine of which formed the area of the square on the shorter side. Or, if we suppose a more convenient mode of trying, they might have found that twenty-five pebbles or seeds, which could be arranged in one square, could likewise be arranged in two squares of sixteen and of nine. Going on in that way they would form larger squares, always trying if the pebbles forming one of these squares could not as well be arranged in two smaller squares. So they would explained by the fact that the process of construction really began with a square consisting of 4 bricks. Next square was constructed by placing 5 squares, along its two sides, so that it contained 9 bricks. Proceeding thus and placing the additional bricks alternately on the north and east sides, and on the south and west sides, a square of 256 bricks was constructed without disturbing the position of symmetry of the altar about the east-west line. But the addition of the remaining 33 bricks displaced this symmetry and for this reason perhaps, Baudhāyana gave that unusual direction.¹

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¹ Compare Thibaut, *Śulvasūtras*, p. 35f.

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This explanation is, however, immaterial for the purpose of our immediate object. It is quite sufficient that we have found that a new and larger square was used to be formed from another of smaller size by adding to it a portion in the form of a gnomon. More particularly it is found that a square comprising of 289 square bricks, 17 on each side, was formed from another of 225 bricks, 15 on each side, by the addition of a gnomon consisting of 64 square bricks. Thus is obtained the rational rectangle 15² + 8² = 17².

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Fig. 70

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The whole process appears in a nutshell in the general rule for the enlargement of a square which has been explained before. It is then simply by noting when the square bricks comprising the gnomon can also be placed

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Fig. 71

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in the form of a square that the Hindus very likely discovered the rational rectangles. As has been suggested by Treutlein,¹ and followed by Allman² and Heath,³ it was substantially in the same way that Pythagoras discovered the rational right-angled triangles which are now generally associated with his name.

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¹ P. Treutlein, *Zeits. f. Math. u. Phys*, xxviii, 1883, Hist.-litt. Abtheilung, pp. 209 ff.

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² Allman, *Greek Geometry*, pp. 30ff.

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³ Heath, *Euclid*, I, p. 358.

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The number of rational rectangles employed in the *Śulba* for the purpose of the construction of altars is found to be few. It will be fewer still—five only—if we consider the independent ones, leaving out the multiples or sub-multiples of them. So it will be naturally asked were the early Hindus aware of other rational rectangles? Had they any general rule for finding any number of rational rectangles?

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In dealing with these questions, we shall first refer to two observations of Āpastamba which appear to imply an answer in the negative. After describing four methods of constructing the *Mahāvedi* in which use has been made of altogether eight rectangles of which the sides and the diagonal are expressible in rational integers, Āpastamba observes :¹

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¹ *ApŚl*, v. 6.

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*Etāvanti jñeyāni vedi-viharaṇāni bhavanti.*

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Thibaut has the reading *vijñeyāni* in the place of *jñeyāni* given by Bürk and also found in my manuscripts. The difference is, however, immaterial. Thibaut translates the passage thus, ' So many " cognizable " measurements of the *vedi* exist.'² Bürk renders it as ' There are so many " recognizable " (erkennbare) constructions of the *vedi*,'³ and has thus closely followed Thibaut. The latter further observes : " That means : these are the measurements of the *vedi* effected by oblongs, of which the sides and the diagonal can be known, *i.e.*, can be expressed in integral numbers." Bürk is of the same opinion.

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² Thibaut, *Śulvasūtras*, p. 12.

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³ *ZDMG*, LVI, p. 341; compare also p. 329; LV, p. 560, fn. 3.

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The other observation of Āpastamba which we shall refer to is

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*tābhirjñeyābhiruktaṃ viharaṇam*

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and it occurs at the end of his enunciation of the theorem of the square of the diagonal of a rectangle.⁴ Bürk renders it thus : " The construction (in i. 2 and 3) has been taught by means of (the application of) these (the *akṣṇayārajju*, *pārśvamānī* and *tiryaṅmānī* of a rectangle)—of course by means of such as are " recognizable " (*i.e.*, which can be expressed in integral numbers)."

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⁴ *ĀpŚl*, i. 4.

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Thus according to the supposition of Thibaut which is accepted also by Bürk, the qualifying words *jñeyāni* and *jñeyābhiḥ* imply only those rectangles of which the sides and the diagonal can be expressed in rational integers.

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This supposition is doubtless wrong. For in the first expression the word *jñeyāni* very clearly qualifies *vedi-viharaṇāni* or "the methods of construction of the *vedi*," the word *vedi* undoubtedly referring to the *Mahāvedi* mentioned in the foregoing rules. This is further confirmed by the word *etāvanti* meaning "these," "so many," the reference being to the four methods of constructing the *Mahāvedi* described by Āpastamba. Where is then the ground to suppose, as has been done by Thibaut and Bürk, that *jñeyāni* implies the rectangles employed for the purpose of those methods? In the second passage the word *viharaṇam* ("construction" or "the method of constructions") clearly ¹ refers to the methods of construction described in the two foregoing rules, *viz*., *ĀpŚl*, i. 2 and 3, and the word *jñeyābhiḥ* to the rational rectangles used therein. This has been admitted by Bürk also. Now the methods referred to are those for the construction of squares and rectangles of given sides, and they primarily depend upon drawing right-angled triangles (or rectangles) having a given side, or more particularly, having a side equal to a side of the required figure. The sides and the diagonal of the rectangles are stated to be (*a*, 5*a*/12, 13*a*/12) and (*a*, 3*a*/4, 5*a*/4). It is only when *a* is a multiple of 12 (in the first case) or a multiple of 4 (in the second case) that the sides and the diagonal of the rectangles employed for the purpose of the construction of altars, will be expressible in rational integers, otherwise not. Indeed in actual practice Āpastamba has employed four particular cases of the first, *viz.*, (2½, 6, 6½), (2⅟₁₂, 5, 5⅟₁₂), (4⅙, 10, 10⅙) and (11¼, 27, 29¼), which have fractional sides and diagonals.¹

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¹ Compare what Āpastamba says: *uktam* "said," "stated" or "described;" meaning that the method of construction has been described before. This can be further confirmed by referring to what Āpastamba has written in the rule just preceding the one under discussion (i. 3). After stating the dimensions of the sides and the diagonal of the rectangle to be used, *viz*., (*a*, 3*a*/4, 5*a*/4), Āpastamba observes *vyākhyātaṃ viharaṇam*, "the method of construction has been (already) described (in the preceding rule)."

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¹ *ĀpŚl.* vi. 6-8, vii. 3.

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Again for the purpose of the construction of altars, Āpastamba has used even such rectangles in which the sides and the diagonal cannot be expressed in terms of rational numbers.² So the interpretation of the words *jñeyāni* and *jñeyābhiḥ*, as supposed by Thibaut and Bürk, is absolutely untenable.

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² *Vide supra*, p. 121. Further in two other methods of construction Āpastamba has used the isosceles right-angled triangle (or a square) a, a, a√2 (ii. 1, ix. 3).

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Having detected that this interpretation is open to such a serious objection, Heath modifies it and says: “But the words (*etāvanti jñeyāni*, etc.) also imply that the theorem of the square of the diagonal is also true of other rectangles not of the “recognisable” kind, that is rectangles in which the sides and the diagonal are not in the ratio of integers; this is indeed implied by the constructions for √2, √3, etc., up to √6 (*cf.* ii. 2, viii. 5).”³ But he would still presume that the remark implies that Āpastamba knew of no other rational rectangles that could be employed.

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³ Heath, *Euclid*, I, p. 363.

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Let us next turn to find how the passages in question have been explained by the orthodox commentators. Amongst them, the most elaborate explanation of the expression *tābhirjñeyābhir*, etc., is found to be that of Karavindasvāmī. According to him, the word *jñeya* implies that variety of quadrilaterals in which of the sides and the diagonal any two being given, the third “can be known” with the help of the theorem of the square of the diagonal; or it denotes the rectangles which "can be conceived in mind," the sides and the diagonal of them being expressible in terms of commensurable quantities.¹ The first explanation is given also by Sundararāja and the alternative one by Kapardisvāmī. Sundararāja is silent about the true import of the word *jñeyāni* in *etāvanti jñeyāni*, etc. In the opinion of Karavindasvāmī and Kapardisvāmī, it implies those rectangles in which the sides and the diagonal "can be known in terms of numbers which are rational (*śuddhamūla*, lit. 'which are perfect roots')."²

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¹ Karavindasvāmī writes:

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² Karavindasvāmī says:

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These interpretations appear to me to be as unnatural and forced as those of Thibaut and Bürk. Every one of

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"प्रमाणकारपार्श्वमानीतिर्यङ्‌मान्यक्ष्णयारज्जूनामन्यतरयोः परिमाणज्ञाने अन्यतरा ज्ञातुं शक्यते ता ज्ञेयाः। तत् कथं ... अक्ष्णयारज्जु फलभूतक्षेत्रात् पार्श्वमानीफलभूतक्षेत्रे च शोधिते शिष्टक्षेत्रस्य करणो तिर्यङ्‌मानीति ज्ञातुं शक्यते। एवमेताः ज्ञेयाः। या एवं ज्ञेयाः ताभिरेताभिरुक्तं विहरणमित्यर्थः। एवमुत्तरस्य विहरणस्य समाधिहेतुलमनगन्तव्यमित्यर्थः। अथवा ज्ञेयाभिः परिकल्पयितुमुचिताभिः पूर्वोक्तविहरणं कर्त्तव्यमित्ययमर्थः। "

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Sundararāja says:

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"यासां द्वयोर्ज्ञातयोस्तृतीया ज्ञातुं शक्यते। यथा, पार्श्वमानीतिर्यङ्‌मान्योर्ज्ञातयोस्ते पृथग् वर्गयित्वा संयोज्य तद्वर्गमूलमक्ष्णयारज्जुस्तथा पार्श्वमाभ्यक्ष्णयारज्ज्वोर्ज्ञातयो–रक्ष्णयारज्जु वर्गात् पार्श्वमानीवर्गं विशोध्य शिष्टस्य मूलं तिर्यङ्‌मान्येवं तिर्यङ्‌मानीवर्गे विशोध्य पार्श्वमानीः। एवं ताभिर्ज्ञेयाभिः पूर्वोक्तं विहरणं ...।"

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Kapardisvāmī has:

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"ज्ञेयाभिर्ज्ञातुं शक्याभिर्मनसा परिकल्पिताभिः।"

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"ज्ञेयानि शुद्धमूलतया ज्ञातुं शक्यानि।"

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Kapardisvāmī says:

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"एतावन्त्येव शुद्धमूलानि ज्ञातुं शक्यानि वेदिविहरणानि भवन्ति। अन्ये तु अशुद्धमूलाः कल्पयितुमशक्याः। तस्मादेतावन्तीत्येवावधार्यन्ते।"

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these interpretations is putting a construction on the text which it does not seem to bear. At any rate those interpretations have been made irrespective of the sense in which the words very closely related to *jñeya* have been used in the *Sulba* and of the nature and spirit of these works.

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The question of the rationality or irrationality of the sides and the diagonal of the rectangles used in the construction of altars can arise only when we begin to think of them in terms of numerical quantities. But the *Sulba* deals truly with geometrical construction and not with numerical calculation. In commenting upon *jñeya* from the standpoint of numerical representation, the commentators, ancient as well as modern, have fallen into an error.

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I think that the word *jñeya* should be explained quite differently. Its literal significance is, as has been explicitly stated by the commentators, "can be known" (*jñātum śakyate*). It comes from the same root *jñā*, as the word *vijñāyate*, meaning "is known." Hence both the words should be explained so as to exhibit the same relation. Now on many occasions in the *Sulba* in connexion with the statements of the measurements of altars, the description of the methods of constructing them, etc., we find the remark...*iti...vijñāyate*.² It implies that such and such thing "is known from the ancient holy scriptures." Indeed those things can be actually traced therein. I think the word *jñeya* also should be explained as implying a reference to the same ancient scriptures. Thus the expression *tābhirjñeyābhir*, etc., means "the method of construction (of the altars) has been described by means of those rectangles that can be known (from the ancient scriptures)." The other expression *etāvanti jñeyāni*, etc., will then mean "these only are the methods of construction of the (*Mahā-*) *vedi* which can be known (from the ancient scriptures)."

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¹ It is noteworthy that this is one of the arguments applied by Bürk against the interpretation of Sundararāja.

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² For instance see *BŚl*, i. 65, 71, 76, 79, etc.; *ĀpŚl*, iv. 1, 3, 5; v. 1, 8, 10; viii. 1, 4, 7, etc.

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We shall now take up the question whether the ancient Hindus had any general method of finding rational triangles. Of course there is not found in the *Śulba* any rule devoted particularly to the definition of such a method. So whatever we shall say on the point will consequently be by way of inference, more or less conjectural. However there are good reasons to believe that the Hindus knew of general formulas for finding rational rectangles.

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We have already pointed out how the ancient Hindus presumably discovered the rational rectangles which are found in the *Śulba* by noting when the square bricks comprising the gnomon added to a square might themselves be arranged again in the form of a square. Observing that the gnomon of one square brick depth put round a square formed with n² such bricks, consists of 2n+1 bricks, they would have only to make a square with 2n+1 bricks.¹

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¹ Compare Müller, *loc. cit.*, pp. 202 f.

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If we suppose that 2n+1=m², we obtain n=½(m²-1); and therefore n+1=½(m²+1).

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It follows that $m²+((m²-1)/2)²=((m²+1)/2)². (A)$

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This formula follows, indeed, more directly from a special rule of Kātyāyana for finding the sum of a number of equal squares. If n be the number of equal squares of sides equal to *a* each to be combined into one, then, the rule says,

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$na² = ((n+1)/2)² a² - ((n-1)/2)² a².$

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Putting m² for n, we get

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$m² a² + ((m²-1)/2)² a² = ((m²+1)/2)² a². (A)$

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In particular, taking a = 1, we get at once the formula (A). If the sides and the diagonal of the rectangles are to be integral as well as rational, m must be odd.

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With the help of this formula would be obtained the following rational rectangles mentioned in the *Śulba*: (3, 4, 5), (5, 12, 13) and (7, 24, 25). Indeed it will give all those rational rectangles in which the difference between the greater side and the diagonal is 1.

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There are also found other rational rectangles, *viz.*, (8, 15, 17) and (12, 35, 37), which could not be obtained from the formula (A). The characteristic of them is that the difference between the greater side and the diagonal is 2. They could be obtained from the formula

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$(2m)² + (m² - 1)² = (m² + 1)² (B)$

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which is derivable from (A) by doubling the side of each square or from (A) by putting a = 2. But they were more likely obtained first by observing when the gnomon of two square bricks of breadth put round a square, could be rearranged in the form of a square. If the original square contain n² bricks, the gnomon will consist of 4n + 4 bricks.

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If we suppose that 4n + 4 = m²,

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we obtain n = ¼(m² - 4);

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and therefore n + 2 = ¼(m² + 4).

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It follows that $m² + ((m²-4)/4)² = ((m²+4)/4)².$

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Substituting 2m for m in this, we easily obtain the formula (B).

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Proceeding in the same way they could deduce from the general rule for the enlargement of a square, a still general formula for finding rational rectangles. It has been stated in that rule that the gnomon of n bricks depth put round a square of p² bricks will contain 2pn+p² bricks. Supposing $2pn+n² = m²,$ we obtain $p = 1/2n (m²-n²);$ $p+n = 1/2n (m²+n²).$

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It follows that $m² + ((m²-n²)/2n)² = ((m²+n²)/2n)²,$ or $(2mn)² + (m²-n²) = (m²+n²). (C)$ where m, n are any two rational integers.

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This formula follows also from the method of the transformation of a rectangle into a square, which is commonly found in all the works on the *Śulba*. If p, q be the length and breadth of the rectangle to be transformed, it has been stated that the equivalent square will be given by the difference of the two squares ((p+q)/2)² and ((p-q)/2)². Thus $pq + ((p-q)/2)² = ((p+q)/2)².$

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Substituting m², n² for p, q respectively, we get $m²n² + ((m²-n²)/2)² = ((m²+n²)/2)². (C')$

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Proclus (450 A.D.) has attributed the formula (A) to Pythagoras (*c.* 540 B.C.), and the formula (B) to Plato (*c.* 375 B.C.). The formula (C) or (C') follows from Euclid's *Elements*, II. 6.

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The early Hindus recognised that fresh rational rectangles can be derived from a known one by multiplying or dividing its sides and diagonal by any rational quantity. In other words, they found that if (p, q, r) be a rectangle, so that $p² + q² = r²,$ then another will be (lp, lq, lr), where l is any rational number, integral or fractional. Āpastamba has, indeed, derived certain new rational rectangles in the same way. He has, however, put the result thus:¹

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¹ *ĀpŚl*, v. 3-4.

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If α² + β² = γ², then (α + na)² + (β + nβ)² = (γ + nγ)², where n is an arbitrary rational number.

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The rational rectangle (15, 36, 39) perhaps deserves more than a passing notice. It could of course be derived from the rational rectangle (5, 12, 13) by multiplying its sides and diagonal by 3. This relation has, indeed, been expressly admitted by Āpastamba.² But it was probably obtained first independently, thinks Bürk,³ as an instance of a rational rectangle in which the difference between the greater side and the diagonal would be 3. This will probably account for Baudhāyana's enumerating it separately along with (5, 12, 13). The early Hindus, particularly those belonging to the Āpastamba school, appear, however, to have special regard for the rational rectangle (15, 36, 39). So the method of constructing several *vedis* has been described in the *Śulba* of this school with particular reference to this rectangle. For instance, take the case of the *Nirūḍhapaśubandha-vedi*. It is of the shape of an isosceles trapezium whose measurements are known from the ancient scriptures to be: face = the yoke of a cart (= 86 aṅgulis); altitude = the pole (= 188 aṅgulis) and base = the axle of the cart (= 104 aṅgulis).¹ As regards the process of its construction, Āpastamba says:

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² *ĀpŚl*, v. 4.

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³ ZDMG, LV, p. 571.

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¹ *ĀpŚl*, vi. 3, 5; *ĀpŚr*, vii. 3. 7 f.

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"This has been described (in connexion with the construction of the *Saumikī-vedi*) by means of one cord. Having taken it by the mark at 15, fix the two western corners by means of half the axle and the eastern corners by means of half the yoke."²

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² *ĀpŚl*, vi. 4; compare Bürk's notes on it.

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Here it is clear that the altitude (= 188 aṅgulis) of the *vedi* is supposed to be divided into 36 parts, so that one part will be equal to 5 2/9 aṅgulis. On taking this for a new unit of pada or prakrama, in terms of it the altitude of the *Nirūḍhapaśubandha-vedi* will contain 36 padas or prakramas. So the rational rectangle (15, 36, 39) can be applied to construct it, as in the Method I, page 64 (Fig. 28). The spatial magnitudes of the rational rectangle employed will truly be (78⅓, 188, 203⅔) in terms of the usual unit aṅguli. But by supposing the unit to be one of 5 2/9 aṅgulis, Āpastamba represents it as (15, 36, 39). Similarly in constructing several other altars, by a suitable change of the length of the unit of linear measure, Āpastamba always represents the sides and the diagonal of the rational rectangle employed by (15, 36, 39).¹ Actually they are particular cases of the rectangle (a, 5a/12, 13a/12), a being the length of the altitude of the *vedi* under construction.

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¹ *ĀpŚl*, vi. 6-8.

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Now it will be very naturally asked, why this preferential liking for the particular rational rectangle (15, 36, 39), on the part of Āpastamba? The true answer will be not simply because, as Bürk seems to think, that the construction of the most important *vedi*, namely the *Mahāvedi* or the *Saumikī-vedi*, depended on it. For Āpastamba has described as many as four methods for the construction of the same *vedi*. But also because it was employed in the most ancient method of constructing the *Mahāvedi* and so had acquired a special sanctity by a long scriptural tradition. We have also seen the similar orthodox predilection of Āpastamba for the primitive methods of high scriptural antiquity in the matter of the construction of the *Caturasra-śyenacit* by means of the bamboo-rod.

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Now tracing the early history of the rational rectangle (15, 36, 39), we find it first in the *Taittirīya Saṃhitā* (*c.* 3000 B.C.)² in connexion with the *Mahāvedī*. It has then reappeared in the *Kāṭhaka Saṃhitā*,³ *Maitrāyaṇī Saṃhitā*,⁴ *Kapiṣṭhala Saṃhitā*,⁵ and *Śatapatha Brāhmaṇa*.⁶ It should perhaps be noted that in these works only the sides (15, 36) have been expressly mentioned, but not the diagonal 39. This non-mention of the diagonal has led some modern writers to suspect if the property 15² + 36² = 39² was at all known in the time of the *Saṃhitā* and *Brāhmaṇa*.¹ But such a suspicion seems to be quite unwarranted. For even in later times Baudhāyana has not mentioned the diagonal of the rational rectangles enumerated by him. From a thorough discussion of the point, Bürk concludes: "After all these no doubt can exist regarding the fact that the rational right-angled triangle with perpendicular sides 15 and 36 was really known in the time of the *Taittirīya Saṃhitā* and the *Śatapatha Brāhmaṇa* and was employed in the construction of the *Saumikī-vedi* as in the *Āpastamba Śulba Sūtra* v. 1 and 2."² Such was also the opinion of Cantor.³ Their conclusion will be further corroborated by what has been shown just above about the special sanctity acquired by this rational rectangle amongst the followers of the Āpastamba school. Our interpretation of the word *jñeyābhiḥ* occurring at the end of Āpastamba's enunciation of the theorem of the square of the diagonal of a rectangle (pp. 132 f.) also will lead one strongly to the same conclusion.

Linked texts (10)

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