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The Science of the Shulba: A Study in Early Hindu Geometry c14: Chapter IX. The Theorem of the Square of the Diagonal

The Science of the Śulba: A Study in Early Hindu Geometry c14

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English

Contents

Chapter IX. The Theorem of the Square of the Diagonal

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There is one very important proposition which, together with its converse, looms much more considerably than anything else through the entire geometry of the *Śulba*. Baudhāyana has enunciated it thus:

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"The diagonal of a rectangle produces both (areas) which its length and breadth produce separately."¹

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¹ "दीर्घचतुरश्रस्याक्ष्णयारज्जु : पार्श्वमानी तिर्यङ्‌मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति।"—*BŚl*, i. 48. Compare *BŚr*, x. 19; xix. 1 for applications of the theorem.

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That is: the square described on the diagonal of a rectangle has an area equal to the sum of the areas of the squares described on its two sides. The proposition is defined in almost identical terms also by Āpastamba² and Kātyāyana.³

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² "दीर्घस्याऽक्ष्णयारज्जु : पार्श्वमानी तिर्यङ्‌मानी च यत् पृथग्भूते कुरुतस्तदुभयं करोति।"—*ĀpŚl*, i. 4.

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³ "दीर्घचतुरश्रस्याक्ष्णयारज्जुस्तिर्यङ्‌मानी पार्श्वमानी च यत् पृथग्भूते कुरुतस्तदुभयं करोतीति क्षेत्रज्ञानम्।"—*KŚl*, ii. 11.

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This theorem is now universally associated with the name of the Greek Pythagoras (*c.* 540 B.C.), though "no really trustworthy evidence exists that it was actually discovered by him."⁴ To denote it as shortly and clearly as possible, Hankel suggested for it the name "the Theorem of the Square of the Hypotenuse." It would be more in keeping with the form and spirit of the early Hindu geometrical terminology to alter this slightly to "the Theorem of the Square of the Diagonal.'' For the *Śulba* does not speak, as we do, of the right-angled triangle, but of the square and the rectangle. Hence the altered title is adopted here.

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⁴ Heath, *Greek Math.*, I, p. 144f.

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The employment of two different terms by compartment for what we denote by one, necessitated the ancient Sulbakāras to define the above proposition for the rectangle again with reference to the square :

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“The diagonal of a square produces an area twice as much.”¹

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¹ *BŚl*, i. 45; *ĀpŚl*, i. 5; *KŚl*, ii. 12. For applications of the theorem compare *BŚr*, x. 19; xix. 1, xxvi. 10.

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That is : The area of the square described on the diagonal of a square is double the area of that square.

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The order in which these two and other closely related propositions is mentioned by different writers is noteworthy. The oldest known Śulbakāra, Baudhāyana, states the second proposition before the first intervened by two other rules in the same connexion. The posterior writers like Āpastamba and Kātyāyana place the second just after the first. This change in the order in stating these propositions may not be entirely without any significance. It shows that by the time of Āpastamba and Kātyāyana the importance and generality of the theorem of the square of the diagonal of a rectangle was recognised fully. So they very naturally stated the corresponding theorem for the particular case of a square as a corollary to it. But from the point of view of the origin and growth of the theorem, Baudhāyana's arrangement is more natural.

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Another proposition which has been most freely used in the *Śulba* for the construction of a right angle is this :

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“If a triangle is such that the square on one side of it is equal to the sum of the squares on the two other sides, then the angle contained by these two sides is a right angle.”

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This converse proposition is not found explicitly defined by any Śulbakāra but its truth is tacitly assumed by all of them.

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Did the ancient Hindus discover a proof of the theorem of the square of the diagonal? No conclusively satisfactory answer which will be beyond a shadow of doubt can be given to this question. For there is not found any mention, direct or indirect, of such a proof, even if it had existed anywhere in the early literature of the Hindus. So what we shall say on this point will be more or less conjectural, based of course on other matters having positive bearing on the point in question. It may be pointed out that the state of affairs is not much better elsewhere. It has been already stated that though the proposition is now universally associated with the name of the Greek Pythagoras, “no really trustworthy evidence exists that it was actually discovered by him.” The tradition which attributes the theorem to him began five centuries after Pythagoras and was based upon a vague statement which did not specify this or any other great geometrical discovery as due to him. This led some eminent scholars like Hankel¹ and Junge² even to deny to Pythagoras the discovery of the proposition of the theorem of the square of the diagonal. Again the method which is supposed by the modern believers of the Pythagoras hypothesis to have been presumably followed by him to prove the theorem is purely conjectural.³ So in the same way we proceed to discuss how the theorem was proved by the ancient Hindus. It may be noted that Bürk and Hankel are definitely of opinion that the Hindus had a general geometrical proof. So also was, and more pronouncedly, the eminent German philosopher Schopenhauer.

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¹ H. Hankel, *Zur Geschichte der Math. in Alterthum und Mittealter*, Leipzig, 1874, p. 97.

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² G. Junge, “Wann haben die Grieschen das Irrationale entdeckt?” *Novae Symbolae Joachimicae*, Halle, 1907, pp. 221-264; quoted by Heath (*Euclid*, I, p. 351).

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³ Heath, *Euclid*, I, pp. 352 ff.

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The rule of Baudhāyana immediately following that containing the enunciation of the general theorem of the square of the diagonal runs thus:

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“This (*i.e.*, the truth of the theorem) is perceived in the rectangles with sides of three and four (units), twelve and five, fifteen and eight, seven and twenty-four, twelve and thirty-five, fifteen and thirty-six (units).” ¹

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¹ *BŚl*, i. 49.

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The Sanskrit terms are *trikacatuṣkayoḥ*, etc. They literally mean, “the rectangle whose sides have three (units) and four (units),” etc.

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From this one might surmise that the ancient Hindus were aware only of the arithmetical character of the theorem and then by an imperfect generalisation applied it to rational rectangles. But such a surmise will be too hasty. For there cannot be absolutely any doubt about the fact that the Hindus fully recognised the most general geometrical character of the theorem and employed its truth universally. Indeed, we find instances of its application to cases of rectangles whose sides cannot be represented by rational quantities. For instance, for the construction of the *Sautrāmaṇikī-vedi*, application is made of the right-angled triangle (15/√3, 36/√3, 39/√3) or (5√3, 12√3, 13√3) and for the *Aśvamedhikī-vedi* of the right-angled triangle (15√2, 36√2, 39√2).

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The propositions about the combination and transformation of areas which we have noticed before may also be cited in this connexion. Perfectly geometrical character of them cannot be questioned.

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Above all there is the remark of Kātyāyana at the end of his enunciation of the general theorem of the square of the diagonal of a rectangle: *iti kṣetrajñānam* or “this is the knowledge of (plane) figures.” Thibaut renders it as “this is the knowledge (requisite) for (the measurement of) areas.” This is evidently inaccurate. For in the *Śulbasūtra*, the area is technically called *bhūmi*, not *kṣetra* which denotes “figures.” These prove conclusively that the universal geometrical character of the theorem was fully recognised.

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On the other hand the above rule of Baudhāyana suggests that the truth of the theorem of the square of the diagonal was perceived and proved in the case of rational rectangles first; and it was then generalised and found to be true universally. This is perfectly natural. In support of this hypothesis may be cited the rule of Āpastamba and Kātyāyana for the calculation of the area of a square:

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“As many units of a measure as are in a cord so many rows (or series) of squares (of that measure) there will be in a square on that cord as a side.”¹

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¹ *ĀpŚl*, iii. 7; *KŚl*, iii. 9. *Vide supra* p. 95.

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Fig. 59

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So that by drawing the squares on the sides and diagonal of a rational rectangle and dividing them into elementary squares it will be easily found by calculation that the square on the diagonal is equal to the sum of the squares on the sides.

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This hypothesis as regards the proof of the theorem presupposes a knowledge of the rational rectangles. How the ancient Hindus discovered such rectangles we shall discuss in the next chapter.

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The rules of Baudhāyana just preceding the one containing the proposition of the theorem of the square of the diagonal of a rectangle are these :

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" The diagonal of a square produces an area twice as much.

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" (Take a rectangle whose) breadth is (equal to) the measure (of the side of a square) and length (equal to) its *dvikaraṇī*; its diagonal will be *trikaraṇī* (‘ three-fold-producer’ of the square).

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" Thereby is explained the *tṛtīya-karaṇī* (‘ the generator of the third part ’ of the square); it is the ninth part of the area." ¹

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¹ *BŚl*, i. 45-7 ; see also *BŚr*, xix. 1.

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If this arrangement of the propositions can be supposed to give any clue as to the discovery of the theorem of the square of the diagonal, then it will have to be said that the theorem for a square was discovered first. In that case the hypothesis about the discovery of a proof of the theorem must be a little different from the one suggested above.

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Now one of the oldest Hindu Fire-altars is the *Caturasra-śyenacit*. The oldest method for its construction does not presuppose a knowledge of the theorem in question. This method had been expressly taught by Āpastamba. It was undoubtedly known to Baudhāyana who hints it very briefly and gives in fact what is rather an improved form of it. Bürk ² surmises that the proof of the theorem was discovered just in the figure of the *Caturasra-śyenacit*. The square *ACFE* on the diagonal *AC* of the square *ABCD* of the four squares forming the *ātman* (or "body")

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² Bürk, *ZDMG*, LV, pp. 556 f.

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Fig. 60

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of this altar is obviously equal to the square *ADEH* on the side *AD* and the square *DCGF* on the side *DC*. Bürk has further confirmed his hypothesis by a reference to Baudhāyana's imperfect rule (taught also by Kātyāyana) for the transformation of a square into a rectangle.

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Fig. 61

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This hypothesis about the discovery of the proof of the theorem of the square of the diagonal of a square is endorsed also by Heath.¹ It is certainly more likely than the one suggested by Cantor ² and Allman ³ about Pythagoras' proof of the theorem. (Fig. 62.)

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¹ T. L. Heath, *The Thirteen Books of Euclid's Elements*, in 3 volumes, Cambridge, 1908, Vol. I, p. 352. It may be noted that this proof of the particular case of the theorem of the square of the diagonal was adduced with the figure, as that of its general case, by Al-khowârizmî (c. 825). (F. Rosen, *The Algebra of Mohammed Ben Musa*, London, 1831, pp. 74f.)

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² M. Cantor, *Vorlesungen über Geschichte der Mathematik*, 3rd ed., Bd. I, p. 185.

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³ J. C. Allman, *Greek Geometry from Thales to Euclid*, Dublin, 1889, p. 29.

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Fig. 62

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Thibaut says: "The authors of the sūtras do not give us any hint as to the way in which they found their proposition regarding the diagonal of a square; but we suppose that they, too, were observant of the fact that the square of the diagonal is divided by its own diagonals into four triangles, one of which is equal to half the first square [Fig. 63]. This is at the same time an immediately convincing proof of the Pythagorean proposition as far as squares or equilateral rectangular triangles are concerned."⁴

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⁴ Thibaut, *Śulbasūtras*, p. 8.

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Fig. 63

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Bürk thinks that this supposition is less probable, because it has no connecting link with the constructions ordinarily met with in the *Sulba*, also because it does not explain what might have induced the Śulbakāras, after they had drawn one square, to construct a new square on the diagonal of the same.

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Such construction is not really as unnatural for the Śulbakāras as is supposed by Bürk. It is indeed clearly in evidence in the pattern of the first layer of bricks in the first kind of construction of the *Vakrapakṣa-śyenacit* as described by Baudhāyana.¹ To draw the attention directly, the relevant portions of it are here marked with bold lines² (Fig. 64). Further Baudhāyana teaches us to construct a square (brick) with half the diagonal of another square.³ This leads to a construction of the kind supposed by Thibaut.

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¹ *BŚl*, iii. 62-104.

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² *BŚl*, iii. 87-104. Compare *Paṇḍit*, Vol. X (O. S.), p. 211.

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³ *BŚl*, iii. 289.

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However, similar objections may also be raised, partly at least, against the supposition of Bürk. What might have led to the drawing of the diagonals of the four squares forming the body of the *Caturasra-śyenacit*? They are not required as a matter of course.

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[Figure: Fig. 64 *Vakrapakṣa-śyenacit* First layer of construction (*after Baudhāyana*)]

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An equally convincing hypothesis, which is very nearly alike to that of Bürk but free from all those defects, is suggested by the method of construction of the *Paitṛkī-vedi*. According to one tradition about that *vedi*, it is a square of one square puruṣa in area whose corners are turned towards the cardinal directions.¹ For its construction Kātyāyana indicates the following method:

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¹ *KŚr*, xxi. 3. 28.

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“For the *Paitṛkī*(-*vedi*), construct a square whose area is two square puruṣas; fix poles at the middle of its sides. (The figure formed by lines joining these poles will be the *vedi* required.) This is the method of construction.”²

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² *KŚl*, ii. 6.

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Mention of that tradition about the *Paitṛkī-vedi* is made also by Baudhāyana.³ In fact, the construction of a square with its corners pointed towards the cardinal directions is mentioned as early as the *Śatapatha Brāhmaṇa*.¹ But how to do it has not been indicated there. This is to be accounted by the fact that the method, which is undoubtedly the same as that described by Kātyāyana, was too well known. In fact it seems to have been the usual practice in all such cases. Compare Baudhāyana's imperfect method for the transformation of a square into a rectangle where also it is necessary in the beginning to construct a square whose corners are turned towards the cardinal directions. The common methods for the transformation of a square into a triangle and a double triangle or rhombus may also be referred to.

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³ *BŚl*, i. 83-4.

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¹ *ŚBr*, xiii. 8. 1. 5.

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Fig. 65

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It is thus learnt that the figure² *EFGH* obtained by joining the middle points of the sides of a square *ABCD* was known to be square in shape and half the original square in area. Now the original square *ABCD* was naturally recognised as equivalent to the square on *EG* which is the east-west line. For as has been described before the usual practice of the Hindus for the construction of a square (or indeed any other regular figure) of given sides was to construct it in such a way as to make it lie symmetrically on the east-west line. This *EG* is again the diagonal of the newly formed square *EFGH*. So this figure leads in a very simple and vivid way to the discovery and proof of the theorem of the square of the diagonal of a square. If we join *HF*, we at once obtain the construction forming the basis of Bürk's hypothesis.

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² This figure was happily conceived also by Hankel as the possible source of the discovery of the theorem. But he was not aware of the special aspect it had in the geometry of the *Śulba*.

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How the ancient Hindus proceeded next to find a general proof is well hinted by the following two propositions of Kātyāyana preceding that of the general theorem of the square of the diagonal of a rectangle :

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" (Take a rectangle whose) breadth is one pada and length three padas ; its diagonal is the ‘ten-fold-generator’ (*i.e.*, it generates a square ten times as large as a square of one pada).

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" (Take a rectangle whose) breadth is two padas and length six padas ; its diagonal is the ‘forty-fold-generator’ (*i.e.*, it produces a square forty times as large as a square of one pada)."¹

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¹ *KŚl*, ii. 8-9. Compare *KŚr*, v. 3. 33, for an application of the first rule and xvii. 3.14 of the second.

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It is evident from Fig. 66 that the square *ABCD* is equal to ten elementary squares, four forming the inner square *OPQR* and the remaining six from the halves of

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In the manuscripts of the *Kātyāyana Śulba* known to me, there occurs a rule intervening between these two propositions and the proposition of the general theorem of the square of the diagonal :

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"The measure of the *yuga* and the *śamyā* has been taught (before as it is found (in the holy scriptures).’—*KŚl*, ii. 10.

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I think this rule has been placed here erroneously by a copyist. So there is a gradual development of the proposition of the theorem of the square of the diagonal from the particular cases to the most general one.

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Fig. 66

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the four rectangles surrounding it, *viz.*, *AFBO*, *BGCP*, *CHDQ*, *DEAR*, each of which contains three elementary squares. These can again be divided into two groups: one group consisting of nine elementary squares forming the square on the line *OB* and another group of a single elementary square on the side *OA*. Thus it is proved that $AB² = OA² + OB²$ If the side of each elementary square be one pada, we find a proof of Kātyāyana's first proposition, and if it measures two padas, a proof of his second proposition.

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From these and similar instances of rectangles whose lengths and breadths can be represented by commensurable quantities, and in which the truth of the theorem is proved easily, it is not difficult to surmise how to proceed to find a general geometrical proof of it.

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Fig. 67

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Fig. 68

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According to this supposition we are to draw four rectangles equal to the given one each having as its diagonal a side of the square on the diagonal of the given rectangle. Then it follows obviously,

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$c² = 4(½ab) + (a-b)²,$ $or c² = a² + b².$

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We thus find from the *Śulba* how by successive stages the ancient Hindus developed, as is highly probable, a general proof of the theorem of the square of the diagonal. As confirmatory to this hypothesis, we may refer to the method of Āpastamba for the enlargement of a square. If it be required to construct a square whose side will exceed a side b of a given square by a, add, says Āpastamba,¹ on the two sides of the given square two rectangles whose lengths are equal to b and breadths to a; then add on the corner a square whose sides are equal to the increment a. Thus will be obtained a square with a side equal to a+b (Fig. 68). A similar method is taught by Baudhāyana.² Now we can divide the added rectangles by their diagonals and place the four resulting triangles of sides a, b, c around another square of the same size as the enlarged square, in the manner shown in Fig. 67.

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¹ *ĀpŚl*, iii. 9. See also p. 176 *infra*.

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² *BŚl*, iii. 192-4.

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Nearly the same figure as Fig. 67 is formed by the constructions described in the *Śulba*, for the combination of two different squares (Fig. 38). The general truth of the theorem was very likely perceived from that figure.³

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³ Compare C. Müller, "Die mathematik der Śulvasûtra," *Abhand. a. d. math. seminar d. Hamburgischen Univ.*, Bd. vii, 1929, pp. 175-205; more particularly pp. 194 ff.

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The above proof of the theorem of the square of the diagonal is given in later times in India by Bhāskara II (1150).¹ Bretschneider ² conjectures that Pythagoras' proof of the theorem was substantially the same. In approving of this hypothesis of Bretschneider, Hankel remarks that "it has no specific Greek colouring but rather reminds of the Indian style." ³ This remark has been accepted by Allman,⁴ Gow ⁵ and Heath.⁶ Heath has pointed out another objection against accepting that as Pythagoras' proof, though he admits it to be the "best." ⁷ This interesting proof was given also by Chang Chun-Ch'ing (*c.* 200 A.D.) in his commentary of the ancient treatise *Chou-pei* (*c.* 1100 B.C.).⁸

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¹ *Bījagaṇita* (of Bhāskara II), ed. Sudhakara Dvivedi and Muralidhara Jha, Benares, 1927, p. 70.

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² C. A. Bretschneider, *Die Geometrie und die Geometer vor Eukleides*, Leipzig, 1870, p. 82.

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³ H. Hankel, *Zur Geschichte der Mathematik in altertum und miltelalter*, Leipzig, 1874, p. 98.

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⁴ G. J. Allman, *Greek Geometry from Thales to Euclid*, Dublin, 1889, p. 37.

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⁵ J. Gow, *A Short History of Greek Mathematics*, Cambridge, 1884, p. 155 f.

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⁶ Heath, *Euclid*, I, p. 355.

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⁷ Heath, *Euclid*, I, p. 355; *Greek Math.*, I, p. 149.

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⁸ Y. Mikami, "The Pythagorean Theorem," *Archiv. Math. Phys.*, XXII (3), 1912, pp. 1-4; *The Development of Mathamatics in China and Japan*, Leipzig, 1913, p. 5.

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Another plausible hypothesis will be that the ancient Hindus were led to the discovery of the general proof of the theorem of the square of the diagonal in the following way: Let *ABCD* be a given square. Draw the diagonal *AC* and cut off *AE* equal to *AC*. Construct the square *AEFG* on *AE*. Join *DE* and on it construct the square *DHME*. Complete the construction as indicated in Fig. 69. Now the square *DHME* is seen to be comprised of four right-angled triangles each equal to *DAE* and the small square *ANPQ*. This square will be easily recognised to be equal to the square *CRFS* and triangles equal to the rectangles *AERD* and *ABSG*. Therefore the square *DHME* is equal to the sum of the squares *ABCD* and *AEFG*. Hence the theorem. Constructions like this are necessary in the usual course in the *Śulba*.¹

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¹ For instance see Fig. 57; see also *Paṇḍit*, O.S., X, pp. 46 f.

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Fig. 69

Early History

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The early history of the theorem of the square of the diagonal, in India, has been very ably treated by Bürk.² We have now discovered a few more corroborative evidence of unquestionable value as regards its ancient character. In the *Śulba*, the theorem is found to have been applied very extensively. Even in the construction of a square, a rectangle or a trapezium, is ordinarily presupposed a knowledge of its converse. We may, however, presume for the sake of the best argument on the adverse that in the anterior times, those simple geometrical figures were used to be constructed by methods which would not depend in any way on that theorem.¹ The proof of the possibility and existence of such methods, we find also in the *Sulba*. But there are certain other geometrical constructions such as (i) the geometrical constructions of √2, √3, etc., and (ii) the transformation of rectangles into squares, for which the theorem of the square of the diagonal is absolutely indispensable. Hence to determine the ancient history of the theorem in India, we shall have to find from the ancient literature of the Hindus the oldest instances of the application of the one or the other of those geometrical constructions. Such instances occur indeed copiously.

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² Bürk, *ZDMG*, LV, pp. 546 ff.

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¹ From the preference which is found to have been given in the *Sulba* to the method of construction of rectangles and trapeziums with the help of the rational rectangle (15, 36, 39) connected with the dimensions of the *Mahāvedi*, which will be noticed in the next chapter, we are convinced that the same method was used to be followed as early as the time of the *Taittirīya* and other *Saṁhitā* (c. 3000 B.C.). But here for arguments' sake we shall waive that conviction too.

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Now the doubling of a square, *i.e.*, the geometrical construction of √2, is necessary for the construction of the one of the three primarily essential altars of the Vedic sacrifices, *viz.*, the *Dakṣiṇa*. It has been pointed out before that the existence of those three altars is older than the *Ṛgveda* (before 3000 B.C.). Hence the theorem of the square of the diagonal, particularly in its simplest form for the case of the square, is as old as that. In connexion with the construction of the *Rathacakra-citi*, as in the case of the most of the *Kāmya Agni*, one has first to draw a square equal to the primitive and standard Agni, the *Caturasra-śyena-cit*, whose area is 7½ square puruṣas and whose form (Fig. 1) consists partly of squares and partly of rectangles. It will be easily seen that the solution of this problem is not possible without the help of the theorem of the square of the diagonal in its general form.¹ For the *Praüga-citi*, one has again to double that square. Hence the theorem must be as old as the *Taittirīya* and other *Saṃhitā* (*c.* 3000 B.C.) where, it has been pointed out before, occurs the express mention of the construction of the *Kāmya Agni*. In the *Śatapatha Brāhmaṇa*, we are asked to construct a square fourteen times as great as another square of one puruṣa. Again it is required to divide the small square into seven parts and three of the parts will have then to be combined with the larger square to form a new square.² From that *Brāhmaṇa*, we come to know of some ancient authorities who used to approve of the construction of a series of *Agni* of the square shape, with areas 1½, 2½,...6½ square puruṣas.³ These have been incidentally mentioned by Baudhāyana⁴ and Āpastamba.⁵ But the *Śatapatha Brāhmaṇa* particularly forbids the construction of such altars. That is, however, quite immaterial for the object we have now in view. It is sufficient and important for us to know that such constructions which clearly depend upon the addition of squares were once in vogue, at any rate in some particular schools, before the time of the *Śatapatha Brāhmaṇa* (*c.* 2000 B.C.). In one of the cases noted above it is necessary to transform a rectangle into a square before it can be added to another. In the *Śrauta-sūtra*, we find copious instances of the geometrical construction of √2, √3, etc. Some of those instances can be clearly distinguished from all the previous ones inasmuch as in them the application of the theorem of the square of the diagonal has been very clearly mentioned.¹ In the *Baudhāyana Śrauta*, the converse theorem is used for the construction of the *Mahāvedi*.² Thus we learn that the theorem of the square of the diagonal really plays a very important part in the science of altar-construction from very early times and in some cases it is, in fact, indispensable.

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¹ *Cf.* Bürk. *ZDMG*, LV, pp. 549, 553.

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² *ŚBr*, x. 2.3. 7-14. For further particulars in this connexion see Chap. XII.

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³ *Ibid*, x. 2. 3. 17.

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⁴ *BŚl*, iii. 318-9.

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⁵ *ĀpŚl*, viii. 3, 5; xii. 1-2; *ĀpŚr*, xvi. 17.15. In this work, the tradition is expressly attributed to the *Śatapatha Brāhmaṇa*.

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¹ For example, we take the following from the earliest *Śrauta-sūtra*, namely the *Baudhāyana Śrauta* :

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² *BŚr*, vi. 22.

c14.s1.12

“ अथ महावेदिं विमिमीत एतानेव ज्यायसः प्रक्रमान्प्रक्रम्याक्षायामानेन प्रमाय समन्त꣫ स्यन्दया परितनोति। पृष्ट्यामातनोत्यथैतमग्निं प्रत्यञ्चं यूपावटीयाच्छङ्कोर्विमिमीते पुरुषमात्रेण वेणुना समपक्षपुच्छमरत्निना पक्षौ द्राघीयांसौ भवतः। षड्विधं वा सप्तविधं वा द्वादशविधं वा यावद्विधं वा चेष्यमाणो भवत्यथैनमक्षायामानेन प्रमाय समन्त꣫ स्यन्दया परितनोति।”—*BŚr*, x. 19.
“ वेदिस्तृतीये यजेतेति विज्ञायते। तस्याः सौमिकं मानमेतावदेव नाना। सौमिकात् प्रक्रमात्तृतीयोऽ꣫शः प्रक्रमः स्यात्तेन वेदिं विमिमीत। अपि वा पदाक्षाया पार्श्वमानी पदं तिरः पशुमानेन तयोर्यः कर्णसंमितः प्रक्रमः स इष्यते। पदाद्वा नवमसप्तदशाया तयोस्तु यः कर्णसंमितः स प्रक्रमार्थस्तेन मेया सौमिकी वेदिः।”—*BŚr*, xix. 1.
Compare also *BŚr*, xxvi. 10.

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