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Mathematics in Ancient and Medieval India c6: Chapter III Geometry

Mathematics in Ancient and Medieval India c6

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Chapter III Geometry

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Unlike Greek geometry which attained independent existence with definitions, axioms, theorems and proofs arranged in logical order, the geometry developed in India was of intuitive nature, which, like those of the old Egyptians and Romans consisted only of empirical rules. It gathered facts relating to geometrical problems without attempting to solve them by deductive reasoning. The demonstrative geometry was the central feature of Greek mathematics culminating in Euclid (c. 300 B. C.) and Apollonius (c. 200 B. C.), whereas Indian geometry like that of the Chinese lay rather in the direction of algebra.

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The sacrifice was the prime religious avocation of the Vedic Hindus. The culture of astronomy was chiefly needed to fix the proper time for the sacrifice. The *Kalpasūtra*¹ contains, besides other matters of ritual, the rules for the proper construction of sacrificial altars. It was perhaps in this connection that the study of problem of geometry as also arithmetic and algebra began in ancient India. Geometry was then the science of altar construction. The *Śulbasūtra* (meaning ‘the rules of the cord’ ) is a part of *Kalpasūtra*. The *Śulbasūtras* of Baudhāyna, Āpastamba, Katyāyana, and Mānava are the main sources of our knowledge of geometry¹. We thus find that in Vedic India, the astronomy in one hand and geometry on the other were cultivated under different circumstances by different classes of priests having different duties apportioned to them.

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¹ The *Kalpasūtras* form one of the six *Vedāṅgas*. They are broadly divided into two classes viz. the *Gṛhyasūtras*, the rules for ceremonies relating to family or domestic affairs, and the *Śrautasūtras*, the rules for ceremonies ordained by the Veda such as the preservation of sacred fires, performance of sacrifices etc.

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¹ The names of nine *Śulbasūtras* are known. They are *Śulbasūtras* of Baudhāyana, Āpastamba, Kātyāyana, Mānava, Maitrāyana, Vārāha, Vādhula, Māśaka, Hiranyakeśī. The names of the·last two *Śulbas* are mentioned in the commentary of Karavindasvāmī on the *Āpastamba-Śulbasūtra* (11.11 and 6.10). But the copies of these two Mss. are still unavailable.

Vedic Altars

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Datta in his book, *the Science of Śulba*,² has traced the growth and development of the geometry of this period. Various types of altars are described by him as available in the different *Saṃhitās*, *Brāhmaṇas* and especially in the *Śulbasūtras*. These altars are mostly brick structures of considerable size and shape. They are mainly *Āhavanīya*, *Gārhapatya*, *Dakṣināgni*, *Mahāvedi* (also called *Saumiki vedi*), *Sautramāṇi vedi*, *Prāgvaṃśa*, *Śyenacit’* *Vakrapakṣa vyastapuccha śyena*, *Kaṅka*, *Alaja*, *Prauga*, *Ubhayataprauga*, *Rathacakracit*, *Droṇa*, *Samuhya*, *Paricāyya*, *Śmaśāna*, *Kūrmacit etc.* Besides many other duties, the *Śulbakāras* had to perform several other important duties systematically viz.

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² Datta, B. *The Science of Śulba*, University of Calcutta, 1932.

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(i) to clean and level a plot on which the altar is to be constructed, (ii) to fix the east-west line (*prācī*), for it was considered as the reference line, (iii) to construct bricks of specific sizes, (iv) to draw diagrams of specific sizes on the level ground on which the altar is to be constructed, (v) to lay bricks subject to certain principles. To perform these duties many specialists arose to solve constructional problems which required good deal of knowledge of geometry and mathematics. But there cannot be any doubt that the item (iv) together with item (v) helped much the development and nourishment of geometry. In the table (p. 106) an account is given of names of altars, types of altars, horizontal section of the altars (i.e., diagram on the ground) and sectional area (i.e., area covered by these diagrams.).

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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| No. | Name of bricks | Shape | Dimension | Reference |
| 1 | *Caturthī* or *Caturabhāgīyā* or *Aṇukā* | Square | 30 *aṅg.*, 30 *aṅg.* | \*BS. 3.11; 3.39 \*AS. 11.3; 11.5; 11.7; 16.5; 14.7; 20.3. |
| 2 | *Ardhya* (half of *Caturthī*) | (1) Triangle | 30 *aṅg.*, 30 *aṅg*, √1800 *aṅg.* | BS. 3.63-64. |
| | | (2) Rectangle | 30 *aṅg.*, 15 *aṅg.* | . . . . . |
| | | (3) Pentagon (*Haṃsamukhī*) | 7½ *aṅg.*, 15 *aṅg*, 7½ *aṅg.*, √450 *aṅg.*, √450 *aṅg.* | BS. 3.68. |
| 3 | *Pādyā* (one-fourth of *Caturthī*) | (1) Triangle | 30 *aṅg.*, (√1800)/2 *aṅg.*, (√1800)/2 *aṅg.* | . . . . . |
| | | (2) Four-sided | 22½ *aṅg.*, 15 *aṅg.*, 7½ *aṅg.*, √225 *aṅg.* | BS. 3.67. |
| | | (3) Square | 15 *aṅg.*, 15 *aṅg.* | BS. 3.287. |
| 4 | *Adhyardhyā* (one and half of *Caturthī*) | Rectangle | 45 *aṅg.*, 30 *aṅg.* | . . . . . |
| 5 | *Aṣṭamī* (one-eight of *Caturthī*) | (1) Triangle | 15 *aṅg.*, 15 *aṅg.*, √450 *aṅg.* | |
| | | (2) Rectangle | 15 *aṅg.*, 7½ *aṅg.* | |

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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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| No. | Name of bricks | Shape | Dimension | Reference |
| 6 | *Pañcamī* (side one-fifth of a *puruṣa*) | Square | 24 *aṅg*., 24 *aṅg*. | BS. 3.11; 3.42. AS. 9.6. |
| 7 | *Ardhya* ( half of *Pañcamī* ) | (1) Triangle | 24 *aṅg*., 24 *aṅg*., √1152 *aṅg*. | BS. 3.120. |
| | | (2) Rectangle | 24 *aṅg*., 12 *aṅg*. | BS. 3.44. AS. 9.6. |
| 8 | *Pādyā* (one-fourth of *Pañcamī* ) | (1) Triangle | 24 *aṅg*., (√1152)/2 *aṅg*., (√1152)/2 *aṅg*. | BS. 3.120. |
| | | (2) Four-sided | 18 *aṅg*., 12 *aṅg*., 6 *aṅg*., √228 *aṅg*. | BS. 3.149. |
| | | (3) Square | 12 *aṅg*., 12 *aṅg*. | AS. 9.1. |
| 9 | *Ādhyardhyā* (one and half of *Pañcamī*) | Rectangle | 36 *aṅg*., 24 *aṅg*. | BS. 3.43. |
| 10 | *Aṣṭamī* (one-eight of *Pañcamī* ) | (1) Triangle | 12 *aṅg*, 12 *aṅg*., (√1152)/2 *aṅg*. | |
| | | (2) Rectangle | 12 *aṅg*., 6 *aṅg*. | |

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C. *Adhyardhyā of a Pañcamī and its subdivisions :

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| 11 | *Adhyardhyā* | Rectangle | 36 *aṅg.*, 24 *aṅg.* | BS. 3.43, AS. 9.6. |
| 12 | *Ardhyā* | Triangle | 20 *aṅg.*, 24 *aṅg.* √1476 *aṅg.* | BS. 3.121. |
| 13 | (1) *Dīrghapādyā* | Triangle | 36 *aṅg.*, (√1872)/2 *aṅg.*, (√1872)/2 *aṅg.* | B.S. 3.121 |
| | (2) *Śulapādyā* | Triangle | 24 *aṅg.*, (√1872)/2 *aṅg.*, (√1872)/2 *aṅg.* | BS. 3.121. |
| 14 | *Aṣṭamī* (*dīrghapādyā*) | Right-angled triangle | 18 *aṅg.*, 12 *aṅg.*, (√1872)/2 *aṅg.* | BS. 3.122. |
| 15 | *Aṣṭamī* (*śulapādyā*) | Right-angled triangle | 12 *aṅg.*, 12 *aṅg.*, (√1872)/2 *aṅg.* | BS. 3.122. |
| 16 | *Udhayī* | Triangle | | BS. 3.122 |

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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).

c6.s2.64

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).

c6.s2.66

Fig. 8

c6.s2.67

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*.

Geometrical Propositions

c6.s3.1

Since the areas in classes I, II and III are fixed (vide p. 106) but the diagrams are different, various geometrical propositions undoubtedly evolved.

c6.s3.2

The following geometrical operations are found to have been studied in connection with different constructions :

c6.s3.3

i) Construction of a square on a given straight line.
ii) Construction of a circle equal to a given square and vice versa.

c6.s3.4

iii ) (a) Doubling a circle.

c6.s3.5

(b) Geometrical method to find the value of √( 2 ) which is expressed as √( 2 ) = 1 + ⅓ + 1/3.4 - 1/3.4.34 ( approx. ).

c6.s3.6

iv ) (a) A theorem on the square of the diagonal which runs thus : “the diagonal of a square produces an area double of the original square”.

c6.s3.7

(b) A general statement on the square of the diagonal of a rectangle now known as Pythagoras theorem runs thus : “the diagonal of a rectangle produces both ( areas ) which its length and breadth produce separately.”

c6.s3.8

(v) Construction of an isosceles trapezium whose face, base and altitude are given.

c6.s3.9

(vi) Construction of a rectangle having given sides.

c6.s3.10

(vii) Construction of an isosceles trapezium whose area will be equal to a simple multiple or submultiple of, and which will be similar to, another isosce'es trapezium.

c6.s3.11

(viii) (a) Construction of a square equal to the sum of two unequal squares ( *samāsavidhi* ).

c6.s3.12

(b) Construction of a square equal to the difference of two unequal squares ( *nirhāravidhi* ).

c6.s3.13

(ix) Construction of a square equal to a simple multiple ( or submultiple ) of another square.

c6.s3.14

(x) Transformation of a rectangle into a square and vice versa.

c6.s3.15

(xi) Transformation of a square or a rectangle into an isosceles trapezium which shall have a given face.

c6.s3.16

(xii) (a) Transformation of a square or a rectangle into a triangle.

c6.s3.17

(b) Transformation of a square or a rectangle into a rhombus and vice versa.

c6.s3.18

(xiii) Transformation of an isosceles triangle into a square.

c6.s3.19

(xiv) Determination of area of a triangle (=½), trapezium (=base) and isosceles trapezium =[ ½ (face+base) ].

c6.s3.20

There cannot be any doubt that a little thought about the construction of the altars in class I will presuppose the knowledge from (i) to (iv) and class II a knowledge from (v) to (vii) and class III from that of (viii) to (ix) and knowledge of these three classes together leads to the formulation of rules from (x) to (xiii) besides many others which are not discussed by the *Śulbakāras*. Though these geometrical results mainly occur in the *Śulbasūtras* of Baudhāyana, Āpastamba, Kātyāyana and others, the authors never claimed that they were the first to discover the principles of geometry or apply them to the problems of the construction of altars. Whenever they described a proposition, they referred to their previous workers as quoted here : (i) it is so recognised or prescribed by the authorities, (ii) thus they teach, (iii) it has been so said (*iti abhyupadiśanti iti vijñāyate, iti uktam* etc.). It might be pointed out here that some of the lines occurring in the *Śulbasūtras* are found to be literal quotations from the *Taittirīya Saṃhitā* or *Taittirīya Brāhmaṇa* or the corresponding *Āraṇyaka*. *Baudhāyana* is occasionally more explicit about his sources. In connection with certain differences of opinions amongst the altar builders about the proper size and shape of a particular altar, *Baudhāyana* is found appealing to the authorities of the *Brahmaṇa*, by name for the purpose of arriving at a satisfactory settlement.¹

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¹ *Baudhāyanaśulbasūtra*, 2. 15-19; 2. 35.

Theorem of Square on the Diagonal

c6.s4.1

The theorem is enunciated by Baudhāyana (c. 600 B. C. ) as follows :

c6.s4.2

*dīrghacaturaśrasyākṣṇayārajjuḥ pārśvamānī tiryaṅmānī ca yatpṛthagbhūte kurutastadubhayaṃ karoti*।

c6.s4.3

“The diagonal of a rectangle produces both (areas) which its length and breadth produce separately”. That is, in a rectangle ABCD, AC² = AB² + BC². This is a most general statement and was enunciated first by Baudhāyana. The proposition is stated in almost identical terms also by Āpastamba and Kātyāyana². Baudhāyana further says that the theorem is easily verified from the following relations :

c6.s4.4

$3² + 4² = 5²$ $12² + 5² = 13²$ $15² + 8² = 17²$ $7² + 24² = 25²$ $12² + 35² = 37²$ $15² + 36² = 39²$

c6.s4.5

² Datta, B. — *The Science of Śulba*, p. 104-132, Calcutta University, 1932.

c6.s4.6

No proof of this theorem is given by Baudhāyana and other Śulba writers, since it is beyond their traditions. Zeuthen¹, Cantor², Vogt³, Cajori⁴, and Heath⁵ have expressed the view that the general statement was possibly the result of an induction from a small number of cases of right-angled triangles having sides in rational numbers known to them. But this is not the actual case. Our discussion on rational rectangles and construction of geometrical figures amply justify that the general character of the theorem was rightly understood by the Sulbakāras. This has been discussed by Burk⁶, who concluded that the theorem of square on the diagonal was known and proved in all its generality by the Indians long before the date of Pythāgoras (c-540 B. C ). He went further that so far from Indian Geometry being indebted to the Greek, the much travelled Pythāgoras probably obtained his theory from India.⁷ In this connection a number of conjectures by Bretschneider¹, Gow², Hankel³, Allman⁴, Heath⁵, Dutta⁶ and others as to the way the proof of the theorem could have been arrived at by the Indians are available.

c6.s4.7

¹ Zeuthen, H. G. 'Theorem de Pythagore Origine de la geometrie scientifique', *Comptes Rendus du IIme Congress internationale de Philosophie*, Geneve, 1904; 'Sur l' Arithmetique Geometrique des Grecs et des Indiens', *Bibliotheca Mathematika*, Series 3, 5, p. 97-112, 1904.

c6.s4.8

² Cantor, M. 'Uber die alteste indische Mathematique,' *Archiv der Mathematik und Physik*, 8, p. 63-92, 1905.

c6.s4.9

³ Vogt, H. 'Haben die alten Inder den Pythagorischen Lehrsatz und das Irrationale Gekannt ?', *Bibliotheca Mathematica*, 7, pt. 3, p. 6-23, 1906.

c6.s4.10

⁴ Cajori, F. *A History of Mathematics*, 2nd revised and enlarged edition, p. 86, 1919.

c6.s4.11

⁵ Heath. T. L. *A History of Greek Mathematics*, 1, p. 147, Oxford, 1921.

c6.s4.12

⁶ *Zeitschrift der deustschen morgenlandischen Gesellschaft (ZDMG)*, 55, p. 546.

c6.s4.13

⁷ *ZDMG*, 55, p. 575; Jones, Sir William, *Works*, 8, p. 236; Colebrooke, T. *Miscellaneous Essays*, *1*, p. 436; Garbe R. *Philosophy of Ancient India*, p. 39 etc.

c6.s4.14

¹ Bretschneider, C. A. *Die Geometrie und die Geometer von Eukleīdes?*, p. 82, Leipzing, 1870.

c6.s4.15

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

c6.s4.16

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

c6.s4.17

⁵ Heath, T. L. *The Thirteen Books of Euclid's Elements*, Cambridge, *1*, p. 352, 1908.

c6.s4.18

⁶ Datta, B. *The Science of Śulba*, p. 159, 1932.

c6.s4.19

The theorem of square on the diagonal of a rectangle i.e. a² + b² = c², is usually known as Pythagorian theorem and is universally associated with the name of the Greek Pythagoras (c. 540 B C.). In fact the relation 3² + 4² = 5² and some such relations have been used by Pythagoras, and the evidence of any general statement regarding this is not yet available. Actual proof was first given by Euclid (c. 300 B. C.).⁷ Proclus (c 460 A. D.), the commentator of Euclid remarked : "for my part, while I admire those who first observed the truth of the theorem, I marvel more at the writer of the Elements, not only because he made it first by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in the sixth book.¹ Heath admitted with the remarks : “it is difficult for us to be more positive than Proclus was”.² Hence it is far from conclusive that Pythagoras stated the above general theorem. Variety of evidence is at present available that the practical use of the theorem was current in old Babylonian times (c. 1800–1600 B. C.). The evidence for this is found in certain Babylonian cuneiform tablets. No general theorem was found to have been mentioned. It has been conclusively proved by Neugebauer that Pythagoras has derived his “Number theorem of Universe” as well as the so called Pythagorian Theorem from the Babylonian cuneiform tablets³. The Chinese knew a similar relation which appeared in *Chou Pei* (4th century B. C.) and the most of its credit is connected from the time of its first commentator Chao Chun Chhing (3rd century A. D.).⁴ In India, the proof first appeared in the work of Bhāskara II⁵ (c. 1150 A. D.). Needham remarks “Bhāskara’s treatment derives from the Chou Pei”.⁶ This is not however true, for the proofs of Bhāskara II and that stated to be of *Chou Pei* can readily be deduced from a number of constructions described in the *Śulbasūtras*¹.

c6.s4.20

² Gow, J. *A Short History of Greek Mathematics*, Cambridge, p. 155, 1884.

c6.s4.21

⁷ *Elements*, Book 1, Proposition 47.

c6.s4.22

¹ Proclus’ commentary on the Euclid, Book I, is one of the two main sources of information as to the History of Greek Geometry, the other being the collection of Pappus.

c6.s4.23

² Heath, T. L. *A Manual of Greek Mathematics*, Dover Publications, p. 96, 1963.

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³ Neugebauer, O. *The Exact Sciences in Antiquity*, p. 28-42, 1952.

c6.s4.25

⁴ Needham, J. *Science and Civilization in China*, 3, p. 95.

c6.s4.26

⁵ *Bijagaṇita* of Bhāskara II. *Vide* Sudhākara Dvivedi’s edition, p. 70.

c6.s4.27

⁶ Needham, J. *Ibid.*, 3, p. 19.

c6.s4.28

¹ Datta, B. *The Science of Śulba*, p. 77, University of Calcutta, 1932.

Value of √2

c6.s5.1

The value of √2 given by Baudhāyana² in his *Śulbasūtras* is : $√2 = 1 + ⅓ + 1/3.4 - 1/3.4.34 ( approx. ).$ This is also given by Āpastamba³ and Kātyāyana.⁴ In terms of decimal fractions, this becomes √2 = 1.4142156. According to modern calculation √2 = 1.414213…… .

c6.s5.2

³ *Āpastambaśulba*, 1.6.

c6.s5.3

⁴ *Kātyāyanaśulba*, 2.13.

c6.s5.4

Thibaut, Rodet, Datta and others gave possible method of solution for arriving at the value as follows :–

c6.s5.5

i ) *Thibaut’s proof* ⁵.

c6.s5.6

⁵ Thibaut, G. ‘On the Śulbasūtras’, *Journal of the Asiatic Society of Bengal*, 44, Pt. I, No. 3, p. 239–41, 1875.

c6.s5.7

We know 17² = 2.12² - 1. Thibaut argued, by how much side 17 must be diminished in order that the square on it may be 2.12² exactly. Since 2 × 17 × 1/34 = 1, he observed, two strips each of 1/34 (approximately) are to be cut off from a square with 17 as side to obtain the square 2.12² ( i.e. 12² + 12² ) $Hence ( 17 - 1/34 )² = 2.12²$ $or (17 - 1/34)/12 = √2$

c6.s5.8

² *pramāṇaṃ tṛtīyena vardhayettacca caturthenātmaca-* *tustriṃśonena, saviśeṣa* ( *Baudhāyana-śulba*, 1.61–2 ).

c6.s5.9

$Again, 17 - 1/34 = 12 + 4 + 1 - 1/34$ $or, 17 - 1/34 = 12 ( 1 + ⅓ + 1/3.4 - 1/3.4.34 )$ $or, 1/12 ( 17 - 1/34 ) = 1 + ⅓ + 1/3.4 - 1/3.4.34$ $or, √ 2 = 1 + ⅓ + 1/3.4 - 1/3.4.34$

c6.s5.10

For Baudhāyana's selection of units of 12 *aṅgulas* (= one *pada*) and 34 *tilas* (= one *aṅgula*), Thibaut expressed no doubt for the selection of arbitrary relation 17² = 2.12² (approx.) contributing to the origin of the formula.

c6.s5.11

ii ) *Rodet's approximation*¹.

c6.s5.12

¹ Datta, B. *The Science of Śulba*, Calcutta University, 1932.

c6.s5.13

According to Rodet, the approximation of √ 2 adopted by *Śulbakāras* may be obtained by successive approximation.

c6.s5.14

$√(a² + r) = a + r/(2a + 1) + (r/(2a + 1) ( 1 - r/(2a + 1) ))/(2 ( a + r/(2a + 1) )) + e$

c6.s5.15

where e is the fourth term approximation. Rodet might have obtained as follows :

c6.s5.16

$√(a² + r) = a + r/(2a + 1) [ two terms approximation ]$

c6.s5.17

where ( 2a + 1 ) is the difference of the squares of *a* and the next positive integer ( a + 1 ).

c6.s5.18

For third term approximation, let

c6.s5.19

$² + r = a + r/(2a + 1) + e₁$ $= (2a² + a + r)/(2a + 1) + e₁$

c6.s5.20

Squaring and neglecting e₁², we get $2((2a²+a+r)/(2a+1)).₁=(a²+r)-((2a²+a+r)/(2a+1))²$ $=(r(2a+1-r))/((2a+1)²)$ $₁=(r(2a+1-r))/(2(2a+1)(2a²+a+r))=(r(2a+1-r))/(2(2a+1)²(a+r/(2a+1)))$ $=(r/(2a+1)(1-r/(2a+1)))/(2(a+r/(2a+1)))$

c6.s5.21

Like wise the fourth term approximation is obtained. Obviously following above we write, $√2=√(1²+1)=1+⅓$ $let √2=1+⅓+e₁=4/3+e₁$

c6.s5.22

Squaring both sides, and cancelling e₁² from both sides, we get $8/3e₁=2-16/9=2/9$ $₁=2/9×⅜=1/12=1/3.4$ $∴√2=1+⅓+1/3.4$ $Again let √2=1+⅓+1/3.4+e₂$ $=17/12+e₂$

c6.s5.23

Squaring both sides and cancelling e₂² from both sides, $17/6e₂=2-(17/12)²=-1/144$

c6.s5.24

$∴ e₂ = - 1/144 × 6/17 = - 1/12.34 = - 1/3.4.34$

c6.s5.25

$∴ √2 = 1 + ⅓ + 1/3.4 - 1/3.4.34 ( approx. )$

c6.s5.26

The method described later by Gurjar¹ and Gupta² is same and is no improvement over that of Rodet.

c6.s5.27

¹ Gurjar, L. V. 'The Value of √2 given in the Sulbasūtras.' *Journal of the University of Bombay*, ns. 10, pt. 5, 6–10, 1942.

c6.s5.28

² Gupta, R. C. 'Baudhāyana's value of √2, *The Mathematics Education*, *6*, No. 3, 77–79, 1972.

c6.s5.29

iii) *Datta's Proof*:

c6.s5.30

Datta has, however, explained that the value of √2 was possibly obtained from a geometrical method.³ This method consists in constructing a square with area equal to the sum of the areas of the two other squares having sides of one unit in length. For this, one of the two squares having side of unit length is divided into three equal parts by lines drawn parallel to one of its side. Each of these parts forms a rectangular piece of one unit in length and one third unit in width. Two of these rectangular parts are then joined lengthwise to the two adjacent sides of the other unit square. This leaves a square hole at one of the corners of the enlarged unit square. This square·hole will have a side of one-third unit in length. The remaining rectangular piece of the divided unit square is again subdivided into three equal parts each forming a square of side one-third unit length. One of these square is fitted into the square-hole formed at one of the corners of the enlarged squāre. Each of the remaining two squares is again subdivided into four equal rectangular pieces having length of ⅓ unit and width of 1/3.4 unit. Eight of these small rectangular slips are placed lengthwise side by side on the two adjacent sides of the enlarged square with four on each side. This again leaves a square-hole at the corner having a side of length 1/3.4 unit. Now two equal thin slips have to be deducted from the two adjacent sides of the enlarged square under construction, the width of each of the slips is therefore given by $((1/3.4)²)/(2 × (1 + ⅓ + 1/3.4)) = 1/3.4.34$

c6.s5.31

³ Datta, B. *The Science of Śulba*, p. 192-94, Calcutta University 1932.

c6.s5.32

Fig. 11

c6.s5.33

Hence the side of the desired square is given approximately by 1 + ⅓ + 1/3.4 - 1/3.4.34 which gives the value of √2. This method is simpler and straight forward than that of Thibaut. Datta also discussed fully the knowledge of value of the irrationality of √2 of Hindus and concluded that the irrationality of √2 was known to the ancient Hindus, though no direct method is available in the Śulba texts. Sen has also discussed this in A Concise History of Science in India (p. 154–55).

c6.s5.34

In Babylon, a small cunei-form tablet of the old Babylonian times (c. 1800–1600 B. C.) is now traced in the Yale Babylonian collection (No. 7289) which shows a square with its two diagonals.¹ Three numbers in sexagesimal system are inscribed on it. These three numbers as represented by Neugebauer² as the value of diagonal, a side and the value of √2 (since d = √2 a). Here √2 = 1,24,51,10. Now Hindu³ value √2 = 1 + ⅓ + 1/3.4 - 1/3.4.34, if changed into sexagesimal unit, gives √2 = 1,24,51,10,36…… . This has led to the speculation that the Hindu system of calculating the value of √2 was developed from that of the Babylonians. Two points are important in this connection so far as the Hindu origin is concerned : i) In the period, there was no use of sexagesimal system, ii) the rule was clearly stated thus : "the side of a square is to be increased by its third, the third again by its fourth part less its (fourth parts) thirty-fourth part". We have no evidence of any such method of which the Babylonians were aware. As regards Greek sources, many approximations to the values of √2 were current but such a methood and correct approximation of √2 did not occur.⁴

c6.s5.35

¹ Neugebauer, 0. *The Exact Science in Autiquity*, 1952 *Plate–6a*.

c6.s5.36

² Neugebauer, 0. Ibid., p. 34.

c6.s5.37

³ √2 = 1 + ⅓ + 1/3.4 - 1/3.4.34 = 1 + 5/12 - 1/3.4.34 = 1, 25 - 0; 0, 8, 49, 24 = 1, 24, 51, 10, 36……

c6.s5.38

⁴ Heath's translation of Euclid, 3, p. 2; Heath's *History of Greek Mathematics*, 1, p. 155.

Technical Terms in the Śulba sūtras

c6.s6.1

The technical geometrical terms found in different treatises of the *Śulbasūtras* of Baudhāyana, Āpastamba and Kātyāyana are tabulated below.

c6.s6.2

| Terms in the *Śulbasūtra* | English Equivalents |
| 1. *akṣa* or *akṣṇayā* | Diagonal. |
| 2. *anta* | Point of intersection. |
| 3. *bhūmi* | Area. |
| 4. *caturasra* or *sama-caturasra* | Square. |
| i) *animata caturasra* | Isosceles trapezium. |
| ii) *dīrghacaturasra* | Rectangle. |
| 5. *iṣu* | Arrow, altitude. |
| 6. *karṇa* | Angle. |
| i) *eka-karṇa* | A geometrical figure of one angle. |
| ii) *dvi-karṇa* | A geometrical figure of two unequal angles. |
| iii) *pañca-karṇa* | Pentagon. |
| iv) *tri-karṇa* | Triangle. |
| 7. *karaṇī* | (a) The side of rectilinear geometrical figure (Geometry). (b) Square root (Arithmetic). |
| i) *dvikaraṇī* | (a) Diagonal of a square. (b) √2 |
| 8. *kṣetra* | Plane figure. |
| 9. *kṣetra-jñāna* | (Familiar with) the knowledge of the plane figures. |

c6.s6.3

| Terms in the *Śulbasūtras* | English Equivalents |
| 10. *madhya* | Centre of a circle. This term is also used in more general sense for the middle most point of a square, or a rectangle. |
| 11. *maṇḍala* or *parimaṇḍala* i) *vṛtta* | Circle. Circle. |
| 12. *pariṇāha* | Circumference of a circle. |
| 13. *pārśvamānī* | Longer side of a rectangle or side of a square which falls on the *prācī* or goes parallel to *prācī*. |
| 14. *rekhā* or *lekhā* | Line. |
| 15. *ṛju–lekhā* | Straight line. |
| 16. *śakaṭamukha* | Isosceles triangle. |
| 17. *saṃdhi* | Point of intersection. |
| 18. *saviśeṣa* | (a) ( Simply ) a diagonal of a square. (b) Calculated value of a diagonal of a square. Symbolically, *saviśeṣa* of a=a+a/3+a/3.4-a/3.4.34. (For details, vide B. Datta’ *The Science of Śulba*, p. 196–200.) |

c6.s6.4

| Terms in the *Śulbasūtra* | English Equivalents |
| 19. *srakti* | Corner or angle. |
| i) *catuḥsrakti* | Square. The term is very old and is found in the *Taittirīya Saṃhitā.* 6. 6.10. 1; |
| | *Vājasaneya Saṃhitā.* 38.20; *Śatapatha Brāhmaṇa.* 2.6.1.36; Āpastambha *Śrautasūtra.* 7.5.1. |
| 20. *tiryak* | The breadth of a rectangle or any side of a symmetrical figure which is perpendicular to *prācī.* |
| 21. *tulya* | Equal. |
| 22. *vakrāṅga* | Having curved face or elliptical. |
| 23. *viṣama caturasra* | Quadrilateral. |
| 24. *viṣkambha* | Diameter of a circle. |
| 25. *vyāsa* | Diameter of a circle. |

Śulbasūtras and the Later Mathematics

c6.s7.1

Certain scholars think that later Mathematics bears no noticeable relation with the *Śulbasūtras*. Kaye¹ pointed out that later mathematicians completely ignored the mathematical contents of the *Śulbasūtras*. He goes even further by saying that no Indian writer earlier than the 19th century is known to have referred to the *Śulbasūtras* as having any mathematical value. This has been supported by Cajori¹ and smith.²

c6.s7.2

¹ Kaye, G. R. *Indian Mathematics*, p. 3, Calcutta, 1918; Vide also p. 9.

c6.s7.3

¹ Cajori, F. *A History of Mathematics*, 2nd revised and enlarged edition, p. 84–86.

c6.s7.4

² Smith, D. E. *History of Mathematics*, 1, p. 145, Dover Publication.

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The real fact has not been reproduced by these scholars. Before discussing the extent to which mathematics was cultivated after the *Śulba* period, some attempts must be made to understand the mental background of the people of that time. When the Vedic tradition was on the wane, a current of revolutionary thought began to interplay. Basically this current of thought was intellectual – an upsurge of a desire to solve the mental problems. By this time, Buddha and Mahāvīra appeared in forefront with their great personalities. Owing to the rise of Buddhism and Jainism which preached against the Vedic sacrifices for several centuries, the Vedic sacrifices were on the wane. As a result, the occasions for the construction of altars requiring high skill and ingenuity on the part of the constructors were few and far between. On the other hand, it should be observed that the Hindu geometry in the Śulba-period was of extremely practical nature and it was guided by religious motives whereas the later mathematicians were generally guided by academic interests. Nevertheless, it is not correct to say that *Śulbasūtras* became obsolete and were overlooked by all in India. The following items will readily show that the later mathematicians were not unaware of the *Śulbasūtras*.

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From the table of the technical terms supplied, one can easily see that the following terms of the *Śulbasūtras* were used almost in the same sense in the later mathematical works viz. *caturasra*, *karṇa*, *karaṇī*, *kṣetra*, *vṛtta*, *parināha*, *rekhā*, *viṣama-caturasra*, *viṣkambha*, *vyāsa*, etc. Āryabhaṭa I (c. 476 A. D. ), used the terms *citighana*, *vargacitighana* and *ghanacitighana* for sum of triangular numbers, the sum of the squares of the natural numbers and the sum of the cubes of the natural numbers respectively. *Citi* means a pile or altar ( of bricks ) and *ghana* means cubic contents. These terms may be explained only if we assume that Āryabhaṭa I studied these series in relation to construction of altars. And this is not improbable, since we have shown that the mathematics first developed in India in connection with the construction of Vedic altars.

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Various rules of construction and transformation occurring in the *Śulbasūtras* are : 1) the so-called Pyathagorian theorem, 2) the properties of similar figures, 3) the circle squaring, 4) construction of right triangle, 5) calculation of areas, through different division and transformation of figures. The first two of these topics enter largely in the later Hindu Mathematics and astronomy which will be seen in our subsequent discussion on geometry of Āryabhaṭa I, Brāhmagupta, Bhāskara II etc. That the later Hindu mathematicians did not lose their interest as regards circle squaring can be found from our discussion on the method of construction of a right triangle. It is mainly found in the works of Brahmagupta, Mahāvīra and others. The item 5) does not appear in the later Hindu works on mathematics in the particular manner in which they occur in the *Śulbasūtras*. But that the later mathematicians were interested in finding the area of triangular, tetragonal and circular plots and the volume of a right prism, sphere, can also be seen from our discussion of geometry of Āryabhaṭa I, Brahmagupta, Mahāvīra and others. The history of development of arithmetic and algebra of the post-Vedic mathematics also shows that they were not unaware of the work of the *Śulbasūtras*. But in this connection, it must be clear that the first two items were used more significantly than others. Whitney's remark may be noted in this connection, “The two main principles, by aid of which the greater portion of all Hindu calculations are made, are, on one hand, the equality of the square of the hypotenuse in a right angled triangle to the sum of the square of the other two sides, and on the other hand, the proportional relation to the corresponding parts of similar triangles.”²

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² Vide Whitney's Note under *Sūryasiddhānta*, ch. 2, v. 27. Proportional relation to the corresponding parts of the similar triangles is not used in the *Śulbasūtras*, but it may be that this relation is a development to the relation used in the type of similar figures having different areas.

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The *Śulbas* continued to remain classics with the priestly class having academic interest and were followed in practice wherever, there were occasions during the succeeding centuries. That people interested in them did never disappear will be sufficiently realised from the necessity that arose in the middle ages from writing commentaries on the *Śulbasūtras*. Two commentaries are now available on the *Śulbasūtras* of Baūdha-yana. They are *Śulbadīpikā* of Dvārakānātha Yajvā (between 5th to 8th centuries), *Śulbamīmāṁsā* of Vyaṅkaṭeśvara Diksita. On the *Śulbasūtra* of Āpastamba, five commentaries are available viz., *Vyākhyā* by Karavindasvāmī (between 5th to 8th century A. D.), *Śulbasūtra-bhāṣya* by Kapardisvāmī (before 12th century A. D.), *Vyākhyā* by Sundararāja (c. 1575 A. D.), *Āpastamba Śulba-bhāṣya* by Sundarasūri, *Vyākhyā* by Gopāla Yajvā. Similarly four commentaries are written by Gaṅgādhara, Karka, Mahīdhara and Rāma Candra on the *Kātyāyana-Śulbasūtra* etc. These commentaries, apart from interpretations, have collated many-verses on the *Śulbasūtras* which show that these texts became popular in later times to suit different purposes. For manuscripts and other references, our *Bibliography*¹ may be consulted.

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¹ *Āryabhaṭīya* (*gaṇitapāda*), v. 21–22.

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¹ *A Bibliography of Sanskrit Works on Astronomy and Mathematics*, by S. N. Sen, A. K. Bag and S. R. Sarma, National Institute of Sciences of India, New Delhi, 1966.

Nature of Later Works

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From 300 B. C. to 300 A. D. we get a picture of early Jaina geometry. No single Jaina treatise of this period dealing with geometry and mathematics in general is known to us. Though geometry was admired as "lotus of mathematics"², yet we get only scrapy references here and there. In the *Sthānāṅgasūtra*³ (c. 300 B. C.), an early Jaina work, *rajju*, an instrument of measurement and *rāśi*, a solid quantity meaning measurement of solid bodies are the two topics mentioned under geometry¹. Though the term *rajju* does not appear in later works, there will be no difficulty in recognising it as referring to plane geometry ( *kṣetra* ). It is synonymous with the term *Śulba* of the Vedic period. A commentator of the Jaina work, *Gaṇitasārasaṃgraha* has rightly identified it ( *Śulba* ) with the *Kṣetragaṇita*. The term *rāśi* appears in later works where it means measurements of quantity of grains. In the later Hindu treatises, the topic on *rāśī* forms a part of *Khāta*² (cubic figures). Some technical terms of geometrical importance are used in the *Sūrajprajñapati*, *Bhāgavatisūtra*, *Tattvārthādhigamasūtra* of Umāsvati (c. 150 B. C.) etc. Probably these Jaina authors had more detailed knowledge, but the treatises written by them are lost. Several relations regarding circumference of a circle and their diameter are available in the *Tattvārthādhigamasūtra*, *Kṣetrasamāsa*, *Trilokasāra* and *Gommaṭasāra* and other Jaina works.³

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² *Sūtrakṛtāṅgasūtra*, 2nd Śrutaskanda, ch. 1, v. 154.

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³ Sūtra 747.

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¹ *Kṣetragaṇita* is the name for geometry appearing in the *Gaṇitasārasaṃgraha* (See ch. 7).

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² It deals with the content of the well whose length, breadth and depth are known.

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³ Vide Preface of the *Gaṇitatilaka*, dE. by H. Kapadia.

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The *Āryabhaṭīya* (c. 499 A. D.) of Āryabhaṭa I consists of four sections of which, the section of *Gaṇitapāda* (or mathematics) deals with geometry. It gives some formulae for calculating the area of a triangle, circle and trapezium and the volume of a sphere and pyramid. It also states the theorem of square on the diagonal. Brahmagupta (c. 628 A. D.)

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