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The Science of the Shulba: A Study in Early Hindu Geometry c14.s1: Early History

The Science of the Śulba: A Study in Early Hindu Geometry c14.s1

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English

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.

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“ 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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