The Science of the Shulba: A Study in Early Hindu Geometry c18: Chapter XIII. Geometrical Algebra

The Science of the Śulba: A Study in Early Hindu Geometry c18

Language: English

Chapter XIII. Geometrical Algebra

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The geometrical constructions described in the preceding chapter are of considerable algebraic significance. They indeed form the seed of the Hindu *geometrical algebra* whose developed form and influence we notice as late as in the *Bījagaṇita* of Bhāskara II (born 1114 A.D.). The first plan of enlargement of a figure in which all the constituent parts are affected in equal proportions, leads to the quadratic equation of the type $ax² = c.$ The second plan leads to the complete quadratic equation $ax² + bx = c.$

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Let x denote the length of the enlarged unit of puruṣa and m denote the total increment in area. Then, in the case of the enlargement of the isosceles trapezium on the first plan, we shall have

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$36x × ((24x + 30x)/2) = 36 × ((24 + 30)/2) + m,$ or 972x² = 972 + m.

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Therefore x² = 1 + m/972

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Hence x = √(1 + m/972) .

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If m = 972(n - 1), so that the area of the enlarged trapezium is n times its original area, we get $x = √n$ as given in the *Śulba*. The particular cases, when n = 14 or 14 3/7, are found as early as the *Satapatha Brāhmaṇa* (*c.* 2000 B.C.).¹

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¹ *SBr*, x. 2. 3. 7 ff. Vide supra pp. 158 f.

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For the enlargement of the *falcon-shaped Fire-altar* on the *first plan*, we get the quadratic equation

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$2x × 2x + 2 x × (x + x/5) + x × ( x + x/10 ) = 7½ + m,$

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$15/2x² = 7½ + m.$

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Therefore

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$x² = 1 + 2m/15.$

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Hence

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$x = √(1 + 2m/15).$

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In particular, when m = 94, that is, when the Fire-alter has its maximum enlargement permissible under the *Sulba*, we have

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$x² = 13 8/15 = 14, approximately$

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as found in the *Satapatha Brāhmaṇa* (*vide supra*).

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In the case of the enlargement of the falcon-shaped Fire-altar on the *second plan*, the geometrical operations are equivalent to the solution of the following complete quadratic equation:

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$2x × 2x + 2x × (x + ⅕) + x × (x + 1/10) = 7½ + m.$

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or 7x² + ½x = 7½ + m.

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Completing the square on the left hand side, we get

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$(7x + ¼)² = 841/16 + 7m$

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Therefore [t] x & = 1/28(√(841 + 112m) - 1), & (1) / & = 1/2829(1 + 56m/841) - 1, / & = 1 + 2m/29,

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neglecting higher powers of m. Therefore we have $x² = 1 + 4m/29, approximately, (2)$

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Kātyāyana says:

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"Or for the second and following constructions, increase (the usual unit of) the (square) prakrama by itself for every seven constructions; so that (at each successive construction) take for the prakrama, the original value of the prakrama enlarged by its one-seventh."

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So that, according to him, the enlarged unit (x²) will be $x² = 1 + m/7; (3)$ whereas a more accurate value has been proved above to be $x² = 1 + m/(7¼), approximately, (2.¹)$

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¹ *KŚl*, vi. 3.

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In particular, when the Fire-alter has its maximum enlarged form of 101½ square puruṣas, we have $7½ + m = 101½$ Substituting in (1), we get $x = 1/28(√11369 - 1).$

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Hence x² = 1/784 (11370 - 2√11369).

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Now √11369 = 106 + 133/212, approximately,

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Therefore x² = 1/784 (11156 + 79/106),

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Hence x² = 14 + 3/13, approximately ... (4)

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Kātyāyana gives a nearly equal value,

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x² = 14 + 3/7 ... (5)

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He says :

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" The side which turns out a square of the area fourteen and three-sevenths (square) prakramas will be the length of the prakrama for the 101-fold (Fire-altar)."¹

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¹ *KŚl*, vi. 2.

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It is not at all easy to determine the rationale of Kātyāyana's formulæ (3) and (5), whether they were obtained in the way indicated above by the method of the solution of a complete quadratic equation, or in any other way. It is found that there are discrepancies between the results calculated by the former method and those found in the extant copies of the manuscripts of his works. How to explain them? Whether by the crudeness of his method of solving the quadratic equation or of the calculations attendent on it,¹ if he had at all followed that method, or in any other way? Now the discrepancy need not be considered very serious in the general case, especially if we remember the degree of accuracy that can be naturally expected in those early times, or that is ordinarily found to have been followed then. Even in modern times 7 will be considered to be a very fair approximation to 7¼. What will appear to be serious is in the other case which requires an emendation of the existing text in order to explain away the discrepancy with the result calculated as above. But, we would remark, that by itself should not be considered a very formidable objection against our hypothesis as regards the method of Kātyāyana. For those who have dealt with ancient manuscripts are quite aware that they doubtless require emendations here and there. That discrepancy can be explained away much more easily and reasonably by supposing that the result (5) was derived from the modified result (3) by substituting the value, m = 94. but was not calculated directly from the equation (1) as we have done above.

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¹ It should be noted that the result (4) does not follow, quite contrary to expectations, from the equation (2), on the substitution in the latter of the value, m=94. $x² = 1+(4× 94)/29 = 13 28/29 = 14, approximately.$ This is too less than the value (5) recorded by Kātyāyana. That is why we started by substituting m=94 in the equation (1). This shows that the process of calculation has much to account for.

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It should be noted that the relevant portions of the existing manuscripts of the *Kātyāyana Śulba* might be considered to be quite correct and his results extremely accurate if we follow a certain interpretation of the text. Let us suppose that (1) the term *ekaśata-vidha* means "the construction (of a Fire-altar having an area) of 101 (square puruṣas)," and (2) that the area of 101 square puruṣas comprises only the complete puruṣas in the altar, not including the two aratnis and the prādeśa. Then the algebraic representation of the second method of enlargement of the primitive falcon-shaped Fire-altar will be $7x² = 7+m.$ Therefore $x² = 1+m/7.$

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And in the particular case of the maximum enlargement, we shall have $7x² = 101.$ Hence $x² = 14 3/7.$

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Thus the results come out exactly to be the same as are found in extant MSS. According to this interpretation, the construction of the enlarged altars does not even involve the solution of the complete quadratic equation.

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We shall now proceed to examine how far the above interpretation can be held to be correct. It should be opposed mainly on three grounds: *Firstly*, the supposed interpretation of the term *ekaśata-vidha* as meaning the construction of a Fire-altar of an area of 101 square puruṣas is very unusual. Indeed, according to all the *Śulba* and also the *Śatapatha Brāhmaṇa* that term must always refer to an area of 101½ square puruṣas. Similarly, it is known, that *sapta-vidha* always refers to 7½ square puruṣas, *aṣṭa-vidha* to 8½ square puruṣas and so on. *Secondly*, for that interpretation the stipulated area of 101 square puruṣas has been assumed to be comprised of the area of the complete puruṣas in the body, wings and tail of the falcon-shaped Fire-altar, in exclusion of the two aratnis and the prādeśa in the latter. Therefore in that case the area of the complete altar, all told, will be greater than 101½ square puruṣas. But the maximum extent up to which the Fire-altar is sanctioned to be enlarged by successive constructions is 101½ square puruṣas. Thus Baudhāyana expressly remarks:

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"The first (*i.e.*, when constructed for the first time) Fire-altar has an area of seven and a half square puruṣas. The second contains eight and a half; the third, nine and a half. In this way, the addition of one puruṣa takes place at each successive construction up to the *ekaśata-vidha* (*i.e.*, 'the construction including an area of 101½ square puruṣas'). After that, the *ekaśata-vidha Agni* should be repeated (without making any further enlargement of it.) Or else the sacrifice should be performed without an *Agni*."¹

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¹ *BŚl*, ii. 1-7. Compare also *ĀpŚl*, viii. 3; *KŚr*, xvi. 8.25; *KŚl*, v. 1; vi. 4; *ĀpŚr*, xvi. 17. 15-6. In this last work Āpastamba has expressly referred to the tradition of the Vājasaneya school (*i.e.*, to the *Śatapatha Brāhmaṇa*) on this point (*vide* below).

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*Śatapatha Brāhmaṇa* says:

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"Some say, '*eka-vidha Agni* should be constructed first; then by an increment of one (square puruṣa) successively up to a construction of unlimited size.' One should not do so. The *Prajāpati* (*i.e.*, *Agni*) was created first as *sapta-vidha* indeed. Proceeding to reconstruct himself, he stopped at the *ekaśata-vidha* ('a construction comprising 101½ square puruṣas'). He who constructs (a Fire-altar) smaller than the *sapta-vidha*, cuts asunder this *Prajāpati*: he voluntarily becomes a sinner as one would be by destroying or injuring his better. Again one who constructs (a Fire-altar) exceeding the *ekaśata-vidha*, he proceeds beyond all these (visible Universe), for the *Prajāpati* is this Universe. Hence one should first construct the *sapta-vidha* and then by the increment of one (square puruṣa) in succession up to the *ekaśata-vidha*. But one should not construct in excess of the *ekaśata-vidha*. Thus he

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neither cuts asunder the Father *Prajāpati*, nor does he proceed beyond all these (the Universe).'' ¹

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¹ *ŚBr*, x. 2.3. 17-8.

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So the above interpretation stands in direct contradiction with the injunctions of the early scriptures.

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*Thirdly*, the practice of the mention of an altar by reference to the area of a part of it, as has been supposed for the above interpretation, is unknown in the *Sulba* and earlier literatures of the Hindus. Thibaut once thought that such a thing had been implied in a certain rule of Baudhāyana. He says, " But according to the above sūtra² and its commentary the *Aśvamedhika Agni* was of a different nature. It had to comprise twenty-one puruṣas not including the lengthening of the two wings by one aratni each and of the tail by one prādeśa, so that its ātman consisted of twelve square puruṣas, its wings and tail of three puruṣas each. A proportional increase of the two aratnis and the prādeśa would amount to 1½ square puruṣas and then the agni would no longer be 'ekaviṁśa' as the śruti demands. Therefore the wings were lengthened only by the regular aratni (of 24 aṅgulis) and the tail by the regular prādeśa (12 aṅgulis). so that the increase of the agni caused thereby remained less than one square puruṣa and the agni preserved its character of ekaviṁśa.'' ³ Thibaut has been followed in this matter by Bürk.⁴ They are clearly in error.⁵ For the sūtra they had in view says absolutely nothing about the actual size of the *Aśvamedhika Agni*. The commentator Dvārakanātha Yajvā truly took the area of the Fire-altar to be 21½ square puruṣas. His use of the *trikaraṇī* of a puruṣa in the construction can be more easily and satisfactorily explained in a way different from that supposed by Thibaut and Bürk.

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² The reference is to the following sūtra of the *Baudhāyana Śulba* (iii. 321): " **आश्वमेधिकस्याग्नः पुरुषाभ्यासो नारतिप्रादेशानाम्**।"

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³ *The Paṇḍit*, New Series, Vol. I, p. 769.

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⁴ *ZDMG*, LVI, pp. 355 ff.

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⁵ Eggeling was wrong in supposing that *ekaviṁśa Agni* means "an altar measuring twenty-one man's length on each of the four sides of its body." (*ŚBr*, V, p. 334, fn. 2.)

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Now as regards the size of the *Āśvamedhika Agni*, Kātyāyana states clearly : "(It) is twice or thrice (the area of) the primitive *Agni* or twenty-onefold (*ekaviṃśavidha*)."¹ Such is also the opinion of the other Śulbakāras as well as of the *Brāhmaṇa*.² *Śatapatha Brāhmaṇa* mentions also of a particular school, according to which the total area of the altar should be 12½ square puruṣas.³ As regards the method of enlargement of the primitive falcon-shaped altar of 7½ square puruṣas by which we are to arrive at the construction of the altar for the Horse Sacrifice having the specific area sanctioned for it by the scriptures, one school advocates the proportional enlargement of every part of the construction, while another school the similar enlargement of only those parts which comprise the complete puruṣas, excluding the two aratnis in the wings and the prādeśa in the tail. For an altar with an area of 15 or 22½ square puruṣas enlarged on the first plan we shall have, it will be easily found, to take the *dvikaraṇī* or *trikaraṇī* of a puruṣa and proceed in the same way as in the construction of the primitive *Agni*. For an altar with an area of 21½ square puruṣas enlarged, but according to the second plan, the algebraic equation will be $7x² + ½x = 21½.$ Therefore x = 1/28(√2409 - 1).

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¹ *KŚr*, xx. 4.15 ; *KŚl*, v. 2-3.

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² *BŚl*, iii. 8 ff. ; iii. 321 ff. ; *ĀpŚl*, xxi. 6, 9 ; *ĀpŚr*, xx. 9.1 ; *ŚBr*, xiii. 3-3. 7 ff.

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³ *ŚBr*, xiii. 3.3.9.

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Hence [t] x² & = 1/784 (2410 - 2 √ 2409) / & = 1/784 ( 2410 - 2 × 49 4/49 ), approximately / & = 3 - 123/2401.

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Therefore we may take as a very near approximation x² = 3. And that is what Dvārakanātha Yajvā has done.¹ We do not think it to be fair to him to interpret him otherwise as Thibaut does.

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¹ We obtain nearly the same result from the equation (2). For putting m = 14 (= 21½ - 7½) in it, we get\ x² = 3 - 2/29 = 3, nearly.

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For the above reasons we discard this latter interpretation of Kātyāyana's rules and hold that the construction of altars enlarged according to the second plan noted above and which is used to be followed in a certain school of the *Brāhmaṇa*, does undoubtedly depend preliminarily on the solution of the complete quadratic equation, $ax² + bx = c,$ and further that Kātyāyana's results were obtained in this way. Milhaud is stated to have arrived at the former conclusion before.² His article is unfortunately not available to me and so I do not know the arguments adduced by him.

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² G. Milhaud, "La Géométrie d'Apastamba," *Revue générale des Sciences*, XXI, 1910, pp. 512-520; quoted by Professor D. E. Smith in his *History of Mathematics*, Vol. II, p. 444 fn.

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It may be noted that the commentator Mahīdhara is of no help to us about this knotty point of the enlargement of the Fire-altar. According to his interpretation of Kātyāyana's rules, the area of the Fire-altar after an increment of m square puruṣas will be 7½×(1+m/7) square puruṣas. This interpretation evidently assumes a proportional enlargement of all the constituent parts of the primitive Fire-altar and hence should be discarded as being directly against the express intention of the *Sulba*. Another objection against it will be that the area of the Fire-altar at the final construction (the ninety-fifth construction) according to it will be 7½×(1+94/7) or 108 3/14 square puruṣas and hence much in excess of the maximum enlarged area, *viz.*, 101½ square puruṣas, permissible under the *Sulba*. Weber¹ informs us that such a method of enlargement of the Fire-altar is found in the *Paddhati* of Yājñikadeva. Its origin seems to be in the *Śatapatha Brāhmaṇa*.² But it was criticised and challenged even then.³ I cannot give the opinion of the commentator Rāma on this point as the relevant portion of his commentary is not available to me at present.

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¹ A. Weber, *Indische Studien*, XIII, p. 240 f.

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² *ŚBr*, x. 2. 3. 11-14.

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³ *ŚBr*, x. 2. 3. 15-6.

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The solution of the quadratic equation in its simpler forms is required in connexion with the enlargement of altars followed in a different school. According to that school, we are informed,⁴ the altars with areas 1½, 2½,... 6½ square pusuṣas should be of the square form. So it will then be necessary to construct a square differing from another by a specified quantity. Now we find in the *Sulba*, the following general rule for the enlargement of a square:

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⁴ *BŚl*, iii. 319.

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"Add on the two sides (of the given square), those (two rectangles) which are described with as much as the increment (of the side of the square) and its side; add further at the corner, the square which is produced by that increment."¹

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¹ *ĀpŚl*, iii. 9. Compare also *BŚl*, iii. 192-4.

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Fig. 74

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Let *ABCD* be the given square. Suppose its side is to be increased by the amount *BE*, say. Add to the sides *BC* and *DC* two rectangles *CE* and *CG* each of which is equal to AB.BE. Also add at the corner *C*, the square *CF* equal to the square on the increment *BE* of the side *AB*. Then *AEFG* is the square on the enlarged side *AE*.

Linked texts (7)

Śatapathabrāhmaṇa — Yājñavalkya & Śāṇḍilya (Vedic teacher), Sanskrit
Āpastambaśrautasūtra — Āpastamba, Sanskrit
Kātyāyanaśrautasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit
Āpastambaśulbasūtra — Āpastamba, Sanskrit
Bījagaṇita — Bhāskara II, Sanskrit
Kātyāyanaśulbasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit