Mathematics in Ancient and Medieval India c6.s2: Instruments, Units and Bricks in the Śulbasūtras

Mathematics in Ancient and Medieval India c6.s2

Language: English

Instruments, Units and Bricks in the Śulbasūtras

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Several instruments like *śaṅku*, bamboo rod, rope etc. and units like *aṅgula*, *pada*, *aratni*, *vyāyāma* etc. and bricks of definite shape were used by the *śulbakāras* for the construction of sacrificial altars. That these helped really in the construction of geometrical figures is explained here.

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Instruments :

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The following instruments were used in connection with altar construction.

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i) *Śaṅku* It is a special type of pole. It was the main tool in a sunny day for ascertaining the cardinal directions i.e. the east-west and the north-south lines (Continued to p. 107).

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Table—Name of Altars and other details

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| Names of Altars | Type | Horizontal Sections | Sectional area |
| 1 ) *Āhavanīya* 2 ) *Gārhapatya* 3 ) *Dakṣiṇāgni* | Class I | Square Circle or Square Semi-circle | 1 sq. *vyāyāma* „ „ |
| 1 ) *Mahāvedi* or *Saumiki vedi* 2 ) *Sautrāmaṇi vedi* 3 ) *Paitṛki-vedi* 4 ) *Prāgvaṃśa* | Class II | Isosceles trapezium Isosceles trap. Isosceles trap. Rectangle | 972 sq. *padas* One third of the *Mahāvedi* i.e. 324 sq. *padas* One ninth of the *Sautrāmaṇikī vedi* — |
| 1 ) *Caturasraśyenacit* 2 ) *Vakrapakṣa vyastapucchaśyena* 3 ) *Kaṅkacit* 4 ) *Alajacit* 5 ) *Prauga* 6 ) *Ubhayata prauga* 7 ) *Rathacakracit* 8 ) *Droṇacit* 9 ) *Śmaśānacit* 10 ) *Kūrmacit* | Class III | Bird with square body (*ātman*) with square wings and square tail Bird with bent wings and out-spread tail Bird Bird Triangle Rhombus Circle Trough Isosceles trap. Tortoise | 7½ sq. *puruṣas* „ „ „ „ „ „ „ „ „ |

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Mainly pole was fixed on plain uniform ground and a circle was drawn on the ground around the foot of the pole as centre. Then the line joining the points on the circle where the shadows of the pole before noon and afternoon cut the circle served to indicate the east-west line¹. The actual east-west line might have been determined by the shadow of the pole on the equinox day and verified by the rising and setting points of the star *Kṛttikā*².

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¹ *Kātyāyanaśulbasūtra*, ch. 1, v. 2–3; *Sūryasiddhānta*, ch. 3, v. 1–51; *Brāhmasphuṭasiddhānta* of Brahmagupta, ch. 19, v. 1–20; *Siddhāntaśekhara* of Śrīpati, ch. 4, v. 14–16; *Līlāvati* of Bhāskarācārya, ch. 11, v. 1–10; *Siddhāntaśiromaṇi* of Bhāskarācārya, ch. 7, v. 36–39; etc.

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² Apte, R. N. 'Constructive geometry of Vedic altars', *Annals of Bhandarkar Oriental Research Institute*, 7, pt. 1, p. 1–16, 1926.

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The *Śulbasūtras* have made use of the east-west line for the purpose of all constructions, and drawing works namely the construction of sacrificial altars. Haug has given at the end of his *Aitareya Brāhmaṇa*³ a tolerably good map of the sacrificial compound of which the central line is an east-west line. In Vedas also, the measurements of the altars are given with reference to this central east-west line. The *Āhavanīya* fire is ordained by a Vedic text to be directly to the east of the *Garhapatya* fire. The fire *Dakṣiṇāgni* is to the south of this central east-west line. The *Pāśuki*-vedi for the Paśubandha sacrifice, a *Sautrāmaṇi* vedi which is called the *Mahāvedi* for the Somasacrifice; in all these vedis, the central east-west line remains fixed. In a *Saptapuruṣa*- *cayana*, for example, say, *Śyena-cayana* (bird-shaped construction), the whole figure is symmetrical with respect to the central east-west line. Its wings and tail are symmetrically arranged. It is in fact the most important line and the whole sacrificial theatre, is symmetrical with respect to this central line. The central east-west line was called *pṛṣṭhyā* or *prācī*. In the construction of the altars in the *Śulbasūtras* the tern *prācī* was in fact a line of specific length drawn parallel to or on this central east-west line and which was drawn in such a way that the figure concerned must be symmetrical which the *prācī*.

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³ *Śatapatha Brāhmaṇa*, 1.7.3. 22–27

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In this connection, it is noteworthy that nothing is said about right angled triangle in the *Śulba sutra*. But the lines *pārśvamānī* (another line drawn parallel to *prācī* forming the limb of a specific figure) and *tiryaṅmānī* (a line drawn perpendicular to *prācī* as well as to *pārśvamānī* forming the limb of a specific figure) when drawn to represent the limbs of the same specific figure, it will obviously lead to the construction of a right-angle or a figure containing a right-angle. This seems to suggest that Baudhāyana and the posterior had the idea of a rightangle and right-angled figures.

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The *tiryaṅmānī* was obtained by pulling out the middle point of a rope to the right or left of the *prācī* or *pārśvamānī* with its ends tied to two points on the *prācī* or *pārśvamānī*. This gives rise to an isosceles triangle where the equal sides are represented by the two halves of the ropes. The line joining the apex of this triangle i. e. the middle point of the rope, with the middle point of two fixed points on the *prācī* gives the line *tiryaṅmānī* which is obviously the perpendicular line.

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ii ) *Bamboo rod* : ( a ) *An instrument of measure*—The use of bamboo rod was traced as early as *Taittīrīya Saṃhitā* ( 5.².5.¹. ) and reappeared also in other *Saṃhitās* and in the *Śatapathabrāhmaṇa*. In the *Śrautas* and the *Śulbas*, its use considerably decreased.

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( b ) *Geometrical compass* – In one instance, the instrumental device given by Āpastamba ( *Śulba* 9.¹. ) can be compared with modern geometrical compass in a limited sense. Three holes were made in a bamboo rod of desired length, two of these were made at its two ends and the third at the middle. By attaching the bamboo rod through a hole at its end or at the middle to a peg fixed on a ground and by moving its free end or ends, circles of two different radii can be constructed.

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iii ) *peg*, a short round piece of wood.

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iv ) *Rope* : In the *Śulba*, it has been largely used as a geometrical instrument. In Sanskrit, the words *dari*, *rajju*, *śulba*¹ and *sūtra* have indentical meanings. It means ordinarily ‘a rope’ or ‘a cord’. *Rajju* as a measuring tape had been in use for a long time. In the *Vedas*, the word *dari* was used to mean a rope². The term *śulba* does not occur in the Vedas. The word *rajju* appeared in the *Śatapathabrāhmaṇa*. In the *Śulbas*, both the terms *rajju* and *śulba*³ are found to occur.

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¹ The word *Śulba* is derived from the term *śulb* meaning ‘to measure’.

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² *Ṛg. veda* ( 1.162.5; 10.100.12 ); *Taitttīriya Saṃhitā* ( 2.5.1.7 ); *Atharvaveda* ( 3.11.8; 6.121.2 ).

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³ *Śatapatha Brāhmaṇa*, 10.2.3.11.

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The term *rajju* also appears in later works.¹ Hence it is very difficult to determine when a cord as a unit of measure was introduced by the Indians.

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¹ Kauṭilya mentions a measure of *rajju* which was nearly 60ft. (vide *Arthaśāstra* of Kauṭilya, Ed. by R. Shyama Sastri, 2nd ed. Mysore 1919, p. 107). In the script of Aśoka, the words *rajjuka* and *lajjuka* have been used. The words *rajjuka* and *lajjuka* are identical because in Sanskrit grammar, the letter ‘r’ can be used in place of ‘l’. In the *Sthānāṅgasūtra*; a Jaina work, one of the ten sections of the members of mathematics is *rajju* (Sūtra, 338 and 747).

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The cord was prepared from *śana* (a kind of hemp, *Cannabis Sativa* or *Crotolaria Vuncea*), *balvaja* (made of grass, *Eleusine Indica*) or thin strip of bamboo (*veṇu*). It is also prepared from *muñja* (sedge like grass, *Saccharum muñja*), *kuśa* (sacred grass used at religious ceremonies).

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The cord before use was cautiously checked. The cord which was not flexible, had no knots, and which was thin, uniform, soft and smooth was used. A cord usually longer than the required size was used² in the *Śulba* for measuring line or surface. The land measuring or anything tied, was a common practice in almost all the countries of the world. The terms *akṣṇayā-rajju* (diagonal cord), *karaṇi*, *tatkaraṇi* *pārśva-mānī*, *tiryaṅmānī*, etc., clearly indicate lines drawn in different directions.¹ Since most of the measurements were done by the help of a *rajju*, and surface measurements were in common among the *śulbakāras*, it is quite probable that *rajju* had its use in land surveying. Two different methods have been described for measurement with cords as an instrument, (1) method of measuring by one cord (*ekarajjuvidhi*), and (2) method of measuring by two cords (*dvirajjuvidhi*).

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² *Kātyāyauaśulba*, 7.13–14. These plants and materials are used for purpose of rope even today. There are various other materials and plants used for the purpose in British India. E. Balfour has given a list of about 156 principal cordage plants in British India (Balfour, E. *Cyclopaedia of India*, Second Edition, 4, 1873).

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² *Āpastambaśulbasūtra*, 5. 2.

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1) *Ekarajjuvidhi* was given by Āpastamba¹ as such. Having joined a cord of 18 *prakramas* (= BC) to another of 36 *prakramas*, (= AB), marks are to be given at 12 and 15 *prakramas* (CD=12, CG=15) from the free end of the smaller cord (Fig. 1). The two free ends of the combined cord are then fixed on a *prācī* line (= EF), the distance between which is that of the longer cord i.e. 36 *prakramas*, the free end of the longer cord being tied to the peg at the eastern end (E) and that of the shorter cord to the peg at the western end (F). Holding the combined cord at its mark (G) at 15, pull towards south as far as it goes and fix it there (G') by means of a pole. The part of the cord of length 15 *prakramas* now becomes the line of *tiryaṅmānī* (FG'). Such marks (G', Q', Q' and G″) on the ground, which fall on the *tiryaṅmānī* when the cord is fully stretched, called *nirañchana* marks. The same operation is repeated in the north. Obviously it gives the same result. Identical results are obtained if the free ends of the combined cord are interchanged. When the four points on the *tiryaṅmānī* thus obtained and joined, it gives the figure of an isosceles trapezium ( D″G″G′D′ ). This is the method of construction of an altar of definite shape with lengths of two sides of its face being 24 and 30 *prakramas* and the perpendicular distance between them being 36 *prakramas*.

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¹ Vide list of teachnical terms.

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Fig. 1

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It is evident from the Fig. 1 that this gives rise to a right-angled triangle in which EF² + FG'2 = EG'2. The construction of altar of the shape of square, rectangle, triangle and many other isosceles trapeziums by using this method have also been described in the *Śulba*¹.

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¹ *Āpastambaśulbasūtra*, 1.2, 5. 1–2, 6. 3–4, 6. 6–7, 7.1–2.

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The selection of a *nirañchana* mark by the *śulbakāras* shows that they had the knowledge of the properties of the right-angled triangle.

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2 ) *Dvirajjuvidhi*²—In the construction of altars of various geometrical shapes by these methods, instead of a combined cord consisting of two parts ( *ekarajjuvidhi* ), two separate cords of different lengths ( *dvirajjuvidhi* ) were employed. Two ends of each of these cords were tied to the two poles on a *prācī* whose distance was less than that of the cord. A mark was made on each of these cords at a spot representing the *nirañchana* mark and putting the cord at the mark on each side of the *prācī*, two right angled-triangles could be obtained as in the case of combined cord By joining the vertices of four such triangles, drawn with the help of these two cords, an isosceles trapezium could be constructed. Isosceless trapezium of different sizes were drawn by using the cords of multiple lengths with their corresponding *nirañchana* mark.

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² *Āpastambaśulbasūtra*, 5. 3–5.

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v ) *Sphya* : A wooden rod cone-shaped at one end for drawing lines on the ground

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Besides these instruments, rods were represented by the units of lengths namely *prādeśa*, *aratni*, *śamyā*, *vyāma*, *vyāyāma*, *puruṣa* etc. ( vide table for unit of length given below ) also used as measuring instruments.

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*Units* :

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The measure of the dimension of an object can be obtained by comparison with that of another object of the same species taken as the standard unit. In the *Śulbasūtras*, we come across linear, surface and volume units in connection with altar construction. *Śulbakāras* did not use separate names of the units for linear, surface and volume units. Surface and volume were expressed in terms of linear units. A table of linear units as found in the *Śulbasūtras*¹ is given below :

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¹ *Baudhāyanaśulba* (= BS), ch. 1, 3-21.

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1 *aṅgula* = 24 *aṇus* = 34 *tilas* placed side by side.
1 *kṣudrapada* = 10 *aṅgulas*.
1 *pada* = 15 *aṅgulas*.
1 *prakrama* = 2 *padas* = 30 *aṅgulas*.
1 *aratni* = 2 *prādeśas* = 24 *aṅgulas*.
1 *puruṣa* = 1 *vyāma* = 5 *aratnis* = 120 *aṅgulas*.
1 *vyāyāma* = 4 *aratnis* = 96 *aṅgulas*.

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1 *pṛthā* = 13 *aṅgulas*.
1 *bāhu* = 36 *aṅgulas*.
1 *jānu* = 30 or 32 *aṅgulas*.
1 *iṣā* = 108 *aṅgulas*.
1 *akṣa* = 104 *aṅgulas*.
1 *yuga* = 88 *aṅgulas*.
1 *śamyā* = 36 *aṅgulas*.
1 *aṅgula* = ¾ inch (approximately).

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As regards surface-measure *Baudhāyana* says that *Mahāvedi* measures 972 (sq.) *padas*¹. The *Sautrāmaṇikī vedi* is 324 (sq) *padas*.² The corresponding references have unmistakably proved that the units denote square units and are meant for surface measure. This has been verified by actual calculation. Many references can be pointed from *the Śulbasūtras* in favour of this.³

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¹ *aṣṭaviṃśatyo'naṃ padasahasraṃ mahāvedi* (AS. 5. 7).

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² *trīṇi caturviṃśāni padaśatāni sautrāmaṇikī vediḥ* (AS. 5.9).

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³ *vyāyāmamātrī gārhapatyaciti . . . . . caturasretyekeṣām..* ...... (BS. 2. 61-62); *śamyāmātrī catuḥsrakti...sacaturasraviśeṣāt* (BS. 2. 79-80); *paitṛkyā mdvīpurṣāṃ samacaturasrāṃ kṛtvā*...(AS. 2. 6), etc.

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As regards volume-measure, *Śulbakāras* were not very clear enough, since question of volume was not considered important. In the construction of altars, total height of the layers i.e. practically the height of bricks of altars remains constant for a specific construction.

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*Bricks* :

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The bricks of following shapes were used for the construction of altars, a table of which is given below.

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\*BS = *Baudhāyanaśulba* (Thibaut's Ed.). AS = *Āpastambaśulba* (Bürk's Ed.).

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1. It is called *Caturthī*, as its sides are ¼th of a *puruṣa* (= 120 *aṅgulas*). *Puruṣa* was considered as the standard unit. It represents the height of a man measured by his own finger.

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B. *Pañcamī brick*¹ *and its subdivisions* :

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¹ It is called *Pañcamī* as its sides are ⅕th of a *puruṣa* (= 12 aṅgulas ).

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C. *Adhyardhyā of a Pañcamī and its subdivisions :

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When a square brick (*Caturthī* or *Pañcamī* is cut by one of its two diagonals each of the two equal parts gives rise to a brick [Nos. 2(1) and 7(1) ] having the face of an isosceles triangle (Fig. 2). When it is cut by its two diagonals, each of the four equal pieces gives rise to a brick [ Nos. 3(1) and 8(1) ] having the shape also of an isosceles triangle (Fig 3), and when cut by its two bisectors of the sides, each of its four equal pieces gives rise to a brick [ Nos. 3(3) and 8(3) having the face of a smaller square (Fig. 4). When divided simultaneously by the two diagonals and two bisectors of the sides, each of the eight equal pieces gives rise another brick [Nos.5(1) and 10 (1)] with the shape of an isosceles right-angled triangle (Fig 5). When any of the smaller square (Fig. 4) is divided into two equal halves by a bisector on one of the side, each one gives rise to a brick [ Nos. 5(2) and 10(2) which is one-eighth of the original *Caturthī* or *Pañcamī* ( vide Fig. 6).

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Fig. 2

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Fig. 3

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Fig. 4

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Fig. 5

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Fig. 6

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When the longer side of the rectangular *Aṣṭamī* (Fig 6) is joined with one of the side of the isosceles right-angled triangle (Fig. 5), it gives rise to a brick [ Nos. 3(2) and 8(2) ] having shape of a trapezium (Fig. 7).

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Fig. 7

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When two such trapeziums are joined together along its longer sides, it gives rise to a *Haṃsamūkhī* brick [ No. 2(3) ] with its face representing pentagon (Fig. 8).

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Fig. 8

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When an *Adhyardhā* brick ( one and half times of a square i.e. a rectangular brick ) is cut along its two diagonals, it gives rise four triangular pieces two equal larger known as *Dīrghapādyā* bricks [ No. 13(1) ] and two equal smaller known as *Śulapādyā* bricks [ No. 13(2) ]. When a *Dīrghapādyā* having a triangular face ABC ( See Fig. 9 ) is cut into two equal halves by cutting along the perpendicular line from the vertex A to the base of its tringular face, it gives a brick with the face of a right-angled triangle ABC(Fig. 9). Similarly when a *Śulapādyā* brick having a triangular face ACD ( Fig 9 ) is divided into two equal halves by cutting it along the perpendicular line from the vertex A to the base of its triangular face, each gives a brick having a face of the shape of a right-angled triangle A F D. If a brick having the face of a shape of a right-angled triangle derived from *Dīrghpaādyā* or *Śulapādyā* (brick No. 14 or 15) be joined to a similar one derived from the *Pañcamī* along the sides of equal length [vide No. 10(1)] it gives rise to a brick [No. 16] whose face is represented by a scalene triangle PFC (Fig. 10).

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Fig. 9

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Fig. 10

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From the consideration of the above description for the preparation of various bricks for the construction of altars, it can be concluded that the ancient Indians had the knowledge of different forms of geometrical figures viz., square, rectangle, isosceles triangle, isosceles right angled triangle, scalene triangle, trapezium, pentagon etc On the basis of these ideas, it is easy to see how the further development of geometrical knowledge was made in the *Śulbasūtras*.

Linked texts (14)

Āpastambaśulbasūtra — Āpastamba, Sanskrit
Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit
Taittirīyasaṃhitā — Sanskrit
Kātyāyanaśulbasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Śatapathabrāhmaṇa — Yājñavalkya & Śāṇḍilya (Vedic teacher), Sanskrit
Sūryasiddhānta — Maya, Sanskrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Ṛgvedasaṃhitā — Sanskrit
Atharvavedasaṃhitā — Atharvan & Aṅgiras, Sanskrit
Kauṭilīya Arthaśāstra — Kauṭilya, Sanskrit
Ṭhāṇaṃga — Sudharman, Ardhamāgadhī Prakrit
Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
Siddhāntaśekhara — Śrīpati, Sanskrit
Siddhāntaśiromaṇi — Bhāskara II, Sanskrit