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The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t18: Chapter 21: The Construction of a Fire-Altar in the Form of a Tortoise—Second Type with Rounded Limbs

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t18

Language
English

Chapter 21: The Construction of a Fire-Altar in the Form of a Tortoise—Second Type with Rounded Limbs

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**21.2-21-4.** *The construction.* Here the method of construction is the same as that already described in the case of the circular trough (chapter 18). We have seen that 120 *ṣoḍaśī* bricks equal 7½ sq. *pu.* 5 *ṣoḍaśī* bricks are taken out and the area of the remaining 115 *ṣoḍaśīs* (103500 sq. *aṅg.*) is turned into a square of side 321 *aṅg.* 24 *ti* (the commentator's value is 321 *aṅg.* 25 *ti*). This square is transformed into a circle of the same area as per *Bśl.* 2.9. In this act of circling the square, each of the 5 *ṣoḍaśī* bricks is cut by the circle and the western portion transferred to the eastern side. In this way, each of the 5 *ṣoḍaśī* bricks is rounded off, one of them forms the head and the remaining 4 are attached to the circle in the 4 intermediate directions (Fig. 75).

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The next step is to draw within the circle a square of the maximum possible area. The side of the inscribed square, as we have calculated, works out to 256 *aṅg.* 25 *ti*.

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**21.5-21.9.** The side of the inscribed square, 256 *aṅg.* 25 *ti* is divided into 12 equal parts, and squared bricks are formed with side equal to each such part, that is, 21 *aṅg.* 13 *ti* which agrees with the value of the commentator (*trayodaśatilasahitāika-viṁśatyaṅguleneṣṭakāḥ*). But the commentator's value of 255 *aṅg.* 19 *ti* for the side of the inscribed square is inaccurate inasmuch as it leads to 21 *aṅg.* and 10 *ti* as the side of the square brick and not 21 *aṅg.* 13 *ti* as mentioned.

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As in the *droṇacit*, there are four segments. 6 one-twelfth bricks are placed in the middle of the segment in contact with the inscribed square and the remaining space is divided into 7 parts as shown in the Fig. 75. The breadth of the brick in the centre of the segment (*pradhi-madhyamā*) in between the square bricks and the circumference must be 1 *prakrama* or 30 *aṅg.* as in the *droṇacit*.

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Fig. 75. Arrangement of bricks in the circular *kūrmacit*,— (a) 1st layer; (b) 2nd layer.

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In the first layer, there are 144 square bricks inside the inscribed square, 24 square bricks in the segments and 28 bricks of different types in the remaining space of the *pradhi*. This makes 196 bricks. Now the *pradhi-madhyamā* is pushed into the head and the vacant space thus created is filled with 2 bricks. The four feet are filled with 8 bricks of two types obtained in the process of cutting the *ṣoḍaśī* square bricks by the circle of the body as referred to before. The total now becomes 206, that is, 6 bricks in excess of the required number of 200. This is adjusted by replacing 18 square bricks by 12 1½ square bricks. (*aṣṭādaśa caturaśra uddhṛtya dvādaśādhyardheṣṭakā upadadhyāt*).

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In the second layer, the inscribed square is turned so that its four corners now point towards east, south, west and north. The placing of the bricks is the same as in the first layer with the difference that the placement of bricks in the feet is as in the case of the head of the first layer and that in the head as in the feet of the first layer. The adjustment is made as before.

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The remaining *sūtras* 21.10-21.13 do not call for much comment.

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CONSTRUCTION OF SQUARE, THE THEOREM OF SQUARE ON THE DIAGONAL AND THE VALUE OF √2

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Āpastamba summarized the knowledge of geometry and related problems in the first three chapters. The first chapter deals with various methods of construction of square, the theorem of square on the diagonal of a square and rectangle, and the calculation of the value of √2.

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**1.2-1.3.** *Construction of a square.* Two methods of construction of a square are discussed here. These rules are essentially the same as those of Baudhāyana (*Bśl.* 1.8 and 1.5).

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**1.4-1.5.** *Theorem of square on the diagonal.* The rule says, AB²+BC²=AC², where AB and BC are the sides of a rectangle ABCD. We have discussed this *sūtra* in detail under *Bśl.* 1.12 and 1.13. Only the last line of *Āśl.* 1.4, which contains the term *tābhirjñeyābhi* requires to be explained. According to Bürka, "the construction (found in *Āśl.* 1.2 and 1.3) has been taught by means of (the application of) these (*akṣṇayārajju*, *pārśvamānī* and *tiryaṅmānī* of a rectangle), of course, by means of such as are recognizable (i.e. which can be expressed in recognizable numbers)." *jñā* means 'to know'. Various terms like *jñātumsakti*, *iti vijñāyate*, *tābhirjñeyābhi* have been used by the *śulbakāras*. According to Datta b, these can be interpreted as "known from the ancient scriptures".

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The *sūtra* *Āśl.* 2.5, undoubtedly a special case of *Āśl.* 2.4, defines the diagonal of a square a as *dvikaraṇī* or √2 a. This has been discussed in *Bśl.* 1.9-1.11.

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**1.6.** *Value of √2.* Extactly the same *sūtra* of Baudhāyana for the value of √2 is given here by Āpastamba. This has been discussed under *Bśl.* 2.12.

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Fig. 1

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**1.7.** *Construction of a square.* This *sūtra* describes another method of construction of a square, not given by Baudhāyana. In Fig. 1, XY represents the given cord. T, U, V are marks at the middle of the cord XY, and of XT and TY respectively. EF is the east-west line. E, M, O, N, F are poles corresponding to X, U, T, V and Y. K represents the sign corresponding to T when the cord XY is stretched after the ends X and Y are fixed at M and N. The mark S corresponding to T is obtained when both ties at X and Y of the cord are fixed at M and stretched over K. The south-east corner point D is obtained when ties at X and Y are fixed at E and S respectively and stretched by the middle mark T.

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a *Āśl*, 56, 329.

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b Datta (2), 128—133.

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Similarly, the other corner points A, B, C are obtained. Hence ABCD is the required square.

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CONSTRUCTION OF SQUARE, SURD, A SQUARE FROM A COMBINATION AND DIFFERENCE OF TWO SQUARES, AND TRANSFORMATION OF A RECTANGLE INTO A SQUARE

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This chapter mainly deals with the methods of construction and transformation of geometrical figures like square and rectangle, already dealt by Baudhāyana.

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Fig. 2

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**2.1.** *Construction of a square.* Āpastamba describes here an interesting method of construction of a square. In Fig. 2, EW represents the east-west line, 2a; XU the given cord, a; XV the *saviśeṣa* of a, √2a; and VY the half cord, a.

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The cord XY is prepared for its use in the construction of the square.

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The knots at X and Υ are tied at O and E respectively and the cord is stretched by the mark V, which gives the south-east corner D. Then the knot at E is taken off and fixed at W. The cord is likewise stretched by the mark V; which fixes the south-western corner C. The process is repeated in the north-eastern and the north-western side. ABCD is the required construction of the square.

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**2.2-2.3.** *Surd.* *Dvikaraṇī* means √2 a, where a is the measure; likewise *tṛkaraṇī* is √3 a, and *tṛtīyakaraṇī* = 1/(√3) a. For details *vide* our discussion under *BŚl.* 1.9-1.11.

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**2.4-2.6.** Āpastamba's methods for making a square out of a combination or difference of two squares are exactly the same as those of Baudhāyana, discussed in *BŚl.* 2.1-2.2. The *sūtra* *ĀŚl.* 2.6 is an application of the combination of two squares, for instance a² + 3a² = 4a², where a is the measure or producer of the square.

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**2.7.** The same method of transformation of a rectangle into a square as given by Baudhāyana in his *BŚl.* 2.5 is discussed.

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TRANSFORMATION OF A SQUARE INTO A RECTANGLE AND A CIRCLE, OF A CIRCLE INTO A SQUARE, AND CONSTRUCTION AS WELL AS ENLARGEMENT OF SQUARES OF DIFFERENT MEASURES FOR SURFACE MEASUREMENTS

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The chapter deals with the transformation of squares into rectangles and circles, and of circles into squares. The methods given are the same as those by Baudhāyana. Moreover, Āpastamba has given an idea of surface measures and methods of enlargement or construction of different squares.

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**3.1-3.2.** *Square into a rectangle and circle.* The *sūtra Āśl.* 3.1 describes the method of transformation of a square into a rectangle. This has been discussed in detail under *Bśl.* 2.3-2.4. The methods of transformation from square into a circle given in *Āśl.* 3.2 and *Bśl.* 2.9 are exactly the same; for discussion, *vide Bśl.* 2.9.

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**3.3.** *Circle into a square.* This is the same as that of *Bśl.* 2.10-2.11 (second method).

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**3.4-3.10.** *Squares of different measures and enlargement of areas.* One unit in length produces one square unit area; two units produce four square areas, and three units nine square areas, and so on. *Pramāṇamātreṇa daṇḍena pramāṇamātraṃ kṣetraṃ vidhīyate | dvipramāṇena daṇḍena catvāri pramāṇakṣetrāṇi bhavanti* (Sundararāja)a. The *sūtra Āśl.* 3.7 explains that if *ABCD* be a square of p units, it produces p² unit squares. If *ABCD* be a rectangle whose length *AB* has p units and breadth *BC* q units, then *ABCD* contains pq unit square areas (Fig. 3). According to Kapardib : *yāvatpramāṇa yāvadāyāmā rajjuḥ tāvato vargānkaroti paṅkti karoti | yāvat saṃkhyā yuktam pramāṇaṃ tāvat saṃkhyāyuktaḥ paṅktīḥ karoti | pañcamānaṃ pañcasaṃkhyā yuktānvargān karoti |* This has been beautifully explained by Karavinda as followsc : *dvipramāṇapārśvamānikam-ekapramāṇatiryaṅmāniyakam dvidhā' pacchidya dve upalabhyete | dvipramāṇa pārśvamānikaṃ tripramāṇa tiryaṅmānikaṃ prāk dvidhodak tridhā' pacchidya ṣaḍupalabhyante |* This result

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Fig. 3

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Fig. 4

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was also known to Baudhāyana who used the same method of division of square by parallel lines in connection with the construction of square *gārhapatya* (*Bśl.* 7.4-7.8).

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a *Āśl.* Mysore 73, 55, 57.

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b *Āśl.* Mysore 73, 57.

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c *Āśl.*, Mysore 73, 58

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Depending on the same analogy, *sūtras* *Āśl*. 3.8. and 3.10 explain that (1½ a)² = 2¼ a², and (2½ a)² = 6¼ a², (½a)² = ¼a², (⅓a)² = 1/9a²

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The method of enlargement of a square of side *a* by an increment of length *x* is obtained by (a + x)² = a² + 2ax + x², where ABCD is a square of side AB = a, and BE = DH = increment *x* (Fig. 4). For obtaining (a + x)², two rectangles of area *ax* are joined at two sides of the original square and a small square of side *x* is added at the corner. This formula is given by Euclid II.4.

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