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Geometry in Ancient and Medieval India c7: Chapter II. Śulbasūtra Geometry

Geometry in Ancient and Medieval India c7

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English

Contents

Chapter II. Śulbasūtra Geometry

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**2.1.** As much of the geometry of the Vedic period as is required for the construction of the altars (*vedi*) and fireplaces (*agni*) prescribed for the obligatory (*nitya*) and votive (*kāmya*, performed to attain specific ends) rites, is contained in the *Śulbasūtras*. These form part of the vast literature, designated as *Kalpasūtras*, attached to the Vedas as one of the six *V dāṅgas* (limbs of the Veda). Louis Renou lists 8 *Śulbasūtras*,¹ *Laugākṣi*, *Mānava*, *Varāha*, *Baudhāyana*, *Vādhūla*, *Āpastamba*, *Hiraṇyakeśin* and *Kātyāyana*. With the *Maitrāyaṇa* mentioned by Datta² the number reaches nine. Those of Baudhāyana, Āpastamba, Kātyāyana and Manu form separate treatises; the rest are chapters or parts of chapters of the corresponding *Śrautasūtras*. All these belong to the *Yajur Veda*, the *Kātyāyana* to the *Śukla* and the rest to the *Kṛṣṇa Yajur Veda*. Since all these *Śulbas* are attached to *Śrautasūtras* B.B. Datta surmises that there must have been many *Śulbasūtras* attached to each of the 1131 or 1137 Vedic *Śākhās* with their own *Śrautasūtras*.³ But it seems more likely that the *Śulbasūtra* sections were confined to the *Śrautasūtras* of the Yajur Veda, the Veda specially designed for the performance of the sacrifice.

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¹ *Vedic India* by Louis Renou, translated from the French by Philip Spratt. Susil Gupta (India) Private Limited, Calcutta 12. 1957. pp. 43 and 51.

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² B. B. Datta, *The Science of the Śulba*, p. 2.

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³ *Ibid*, p. 1 and foot-note 2 on the same page.

2.2. The Date of the Śulbasūtras

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Obviously the geometry of the *Śulbasūtra* has developed with the demands of the construction of the sacrificial *vedis*, and therefore it must be very old. According to Oldenberg, the three most primitive, *agnis*, the *Gārhapatya*, *Āhavanīya* and *Dakṣiṇāgni* are older than the *Ṛg Veda*.⁴ There is a likelihood of the *Mahāvedi* being known to the Indians of the Indus Valley Civilization even.¹ M. Cantor and A. Bürk recognise as incontrovertible the fact that the Pythagorean theorem was known in India at the latest in the 8th century B.C.² The *Kāmyāgnis prauga* (triangle), *ubhayataḥprauga* (rhombus), *ratha-cakra* (wheel), *droṇa* (trough) and *śmaśāna* (cemetery) find mention in the *Taittirīyasaṁhitā*.³ The *prācī* (the exact east-west line) and the exact measurements of the *Saumikīvedi* are also mentioned in the same *saṁhitā*. The *Śatapatha Brāhmaṇa* gives detailed instructions for the construction of the most complicated *agni*, the *Vakra-pakṣa-vyasta-puccha-śyena* (the falcon with curved wings and spread tail). All this shows that the construction of the sacrificial altars with all due precision was a common practice during the Brāhmaṇa and even the Saṁhitā periods.

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⁴ *S.B.E.* XXX, p. IX. Also the Ṛg Veda refers to fire as *trisadhastha* (remaining in three places), (RV. 1.67-5), Also सद्मेव धीरः सम्माय चक्रुः refers to expert measurements of the fire place (Vide *Z.D.M.G.* 1901).

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¹ Vide Ch. I, p. 4 above.

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² *Vorlesungen uber Geschichte der Mathematik*, Vol. I, 4th Edn. p. 636 and *Z.D.M.G.* 1901, pp. 553-55.

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³ A Bürk in the introduction to his edition of the *Āp. Śl. Śū.* brings up all these references (*Z.D.M.G.* 1901, p. 548).

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We do not know when exactly the codification of the rules for the construction of the *vedis* began, nor do the existing *Śulbasūtras* lend themselves to exact dating. A.B. Keith places the *Āpastamba Śrautasūtra* in the 4th century B.C. and the *Baudhāyana Śrautasūtra* in the 5th century B.C. Bühler would have the *Āpastamba Śrautasūtra* 150-200 years earlier than Pāṇini. A. Bürk accepts Bühler's view. The language of the *sūtras* alone can give us some clue to their age, and the most we can say from their language is that they are pre-Pāṇinian. Keeping in view the dates assigned to Pāṇini by later researchers, one feels *c.* 800 B.C. is a more probable date for the codification of the *Śulbasūtras*.

2.3. The term Śulba

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The word *śulba* does not occur anywhere in the body of the *Śulbasūtras* except in the metrical supplement of the *Kātyāyana Śulbasūtra*. The word is supposed to have been derived from the root *śulb* or *sulv* to measure, when *śulba* will mean a measuring tape or cord. But the *sūtras* themselves use the word *rajju* not *śulba* in this sense. The *Kātyāyana Śulbasūtra* opens with रज्जुसमासं वक्ष्याम:. (We will expound the manipulation with cords). Later writers too never use the word *śulba* except in the sense of the *Śulbasūtras*.

2.4. Analysis of the contents

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The geometry of the *Śulbasūtras* is primarily constructive, though they occasionally notice and formulate some of the geometrical truths involved. The altars and fire-places had different shapes, all geometrical (even the *vakrapakṣavyastapuccha* is not ungeometrical). The orientation, shapes and areas of these had to be strictly correct, this correctness being as important as the correct pronunciation of the Vedic *mantras*. Hence accurate geometrical methods of construction were evolved. Most often the geometrical truths underlying these constructions were left unenunciated. Hence the geometrical contents of the *Śulbasūtras* can be broadly divided into three categories—(1) theorems expressly stated, (2) constructions, and (3) the geometrical truths implied in these constructions but not stated as such.

2.5. The Theorem of the square of the diagonal

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To the first category belongs the theorem popularly associated with the name of Pythagoras (*c.* 540 B.C.) connecting the square on the hypotenuse or diagonal of a rectangular triangle with the sum of the squares on the sides containing the right angle. Perhaps the first statement of this theorem in its most general geometrical form is ancient India's most important contribution to the development of mathematics. It is true that most ancient peoples knew and used the right triangle 3, 4, 5¹ for getting a right angle and the Babylonian records contain a list of Pythagorean numbers.² But the full geometrical significance of the theorem, that the sides of any right-angled traingle will exhibit this relationship amongst them, was perhaps first realised by the altar-building Vedic priests. The German mathematicians A. Bürk and M. Cantor discuss the question in detail and come to the conclusion that the theorem was known in India at the latest by the 8th century B.C. i.e. the date of the oldest Śulbasūtra that, of *Baudhāyana*. The *samhitās* of the Black Yajur Veda and the *Śatapatha Brāhmaṇa* give 36 units as the length of the *prācī* (east-west line) or *pṛṣṭhyā* (the line of symmetry) of the *Mahāvedi* and 30 units as one of the north-south sides, the *prācī* and half the side thus making the sides containing the right angle in a rational right angled triangle, viz. 36, 15, 39. In the *Śatapatha Brāhmaṇa* (X. 2.3.7) mention is found of increasing the size of the *Vedi* fourteen-fold to accommodate the *Ekaśatavidhāgni* (the fire-place for the 101st performance of the sacrifice) for which operation a knowledge of the theorem of the square of the diagonal is indispensable.¹ Hence it is highly probable that the theorem of the square of the hypotenuse was known in India much earlier than the *Śulbasūtra* period. Tentative evidence for a very much earlier date even can be brought forward, if the term *triṣadhastha* (residing in three abodes) as applied to the fire, has reference to the three *agnis*, *Gārhapatya*, *Āhavanīya* and *Dakṣiṇa*, because the construction of the *Dakṣiṇāgni* requires a knowledge of this theorem. Though the *Śulbasūtras* themselves speak only about the orientation of the 3 fires, the commentators supply the mode of construction, which involves drawing a square with the *dvikaraṇī* of a *piśīla*² as the side.³

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¹ ^1The Chinese Nine Sections (*c.* 1100 B.C.) mentions this triangle and the Kahun Papyrus of Egypt (*c.* 2000 B.C.) refers to four sets of numbers forming right triangles (D.E. Smith. *History of Mathematics*, Vol. II, p. 293).

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² ^2O. Neugebauer, *The Exact Sciences of Antiquity*, p. 35.

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¹ A.B. Keith (*J.R.A.S.* 1910 pp. 519-21) maintains that the mention of the numbers 36 and 15 as representing the sides about a right angle is no evidence of the knowledge of the theorem of the square of the hypotenuse on the part of the Indians of the *Samhitā* period. The evidence of the *Śatapatha Brāhmaṇa* passages enjoining the increment of the *vedis* or altars for each subsequent construction by one *puruṣa*, is also dismissed with the remark that the increment is most probably in the side not the area. But this is disregarding the tradition preserved in all the *Śulbasūtras*.

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² The *piśīla* is a unit of length defined variously as the arm with folded palms or the length of the two arms extended.

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³ See Kapardin's and Karavinda's comments on *Āp. Śl.* II. 4.

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The *Śulbasūtras* exhibit a thorough familiarity with the properties of the right triangle, or rather the properties of the sides and diagonals of figures with right-angular corners. Many rational right triangles are mentioned like :

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15, 36, 39 (*Āp. Śl.* V. 2; *B.Śl.* I.49)
3, 4, 5 (*Āp. Śl.* V. 3; *B.Śl.* I.49)

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and the latter multiplied by 4 and 5 .5, 12, 13 / 12, 35, 37 (*Āp. Śl.* V. 4; *B. Śl.* I. 49)

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7, 24, 25 (*B. Śl.* I. 49)

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.72, 96, 120 / 40, 96, 104 / 2½, 6, 6½ / 7½, 10, 12½ (*M. Śl.*¹) Besides these the irrational right triangle 1, 1, √2 and the approximate right triangles²

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¹ As quoted by Datta in the "*Science of the Śulba*, p. 123.

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² Vide The *Mānava Śulbasūtram* by N.K. Mazumdar. Jour. Dep. Let. Cal. VIII. 1922.

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36, 90, 97
40, 40, 56
5, 6, 7⅞

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and 4, 4, 5⅔ are used in the *Śulbasūtras* for constructing right angles. Secondly the fact that the square on the diagonal (*akṣṇayārajju*) of rectangles and squares combines the squares on the two sides is used to find geometrically the side of a square of any given area, whether the number representing the area is a perfect square or not; i.e. to evaluate square roots and surds geometrically and for combining and subtracting squares to yield squares again.

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The actual enunciation of the theorem in the *Śulbasūtras* is not with respect to the right triangle but with respect to the sides and diagonals of squares and rectangles.

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दीर्घस्याक्षणयारज्जुः पार्श्वमानी तिर्यङ् मानी, च यत् पृथग्भूते कुरुतस्तदुभयं करोति
(*Āp. Śl.* I. 4)

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(The diagonal cord of a rectangle makes both (the squares) that the vertical side and the horizontal side make separately) and चतुरस्रस्याक्ष्णयारज्जुर्द्विस्तावतीं भूमिं करोति । (*Āp. Śl.* I. 4) (The diagonal cord of a square makes double the area). The two parts in the enunciation perhaps indicate two steps in the discovery of the theorem.

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Many have speculated on the possible mode of discovery of this theorem. Amongst them the most noteworthy are Thibaut, Bürk and Datta. Thibaut thinks that the Indians must have been "observant of the fact that the square on the diagonal is divided by its diagonals into four triangles each of which is equal to half the first square. This is at the same time an immediately convincing proof of the Pythagorean proposition as far as squares or equilateral rectangular triangles are concerned" (*J.A.S.B.*, 1875, p., 234).¹ That is, the diagram which suggested the theorem was like fig. 1. Bürk substantially agrees with this but would find the inspiring figure in the square *ātman* or body (fig. 2) of the *caturaśra-śyenacit-agni* (the fireplace in the form of the square falcon). The *ātman* is 4 square *puruṣas* in area and it is to be made by conjoining 4 squares. each of area one square *puruṣa*. If one joins the diagonals of these small squares as in figure 2, a square consisting of 4 triangles results, and each of these triangles = ½ the one *puruṣa* square. Hence the theorem.

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¹ Also quoted by A. Bürk.

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*Fig.* 1

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*Fig.* 2

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B. Datta's hypothesis differs but slightly from Bürk's. According to him the construction of the *Paitṛki Vedi* as directed by Kātyāyana would immediately suggest the theorem. For the prescription is:

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पैतृक्यां द्विपुरुषं चतुरश्रं कृत्वा करणीमध्येपु शंकवः, स समाधिः । (*K. Śl.* II. 6) (In the construction of the *Paitṛkī Vedi* make a square of area two square *Puruṣas* and have nails (for the corners of the *Vedi*)

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at the middle points of the sides. That is the construction). The diagram for the construction, which will be the same as fig. 2, yields the theorem as an inference without any alteration or addition. As such, this hypothesis seems to be the most plausible.

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For the theorem as applicable to the general rectangle or to the right triangle, most authorities including M. Cantor have surmised a numerical origin. The early mathematicians making figures with pebbles must have noticed that 9 and 16 are square numbers and when the two are combined to give 25 pebbles, the new pile of 25 pebbles also is capable of being arranged as a square. The superior antiquity of the knowledge of Pythagorean numbers as compared to a recognition of the properties of the right triangle in Babylonia is the basis for this surmise. Thibaut accepts the same explanation for the discovery of the theorem in India, but adds that the Indians might have drawn the squares on the sides and diagonals of rectangles, divided these squares into unit squares, and found out by actual counting that the number of unit squares in the square on the diagonal is the sum of the unit squares in the squares on the sides. Such a discovery is plausible especially in the light of the direction, तान् समस्येत् in Kātyāyana's.

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यावत्प्रमाणा रज्जुर्भवति तावन्तस्तावन्तो वर्गाः भवन्ति, तान् समस्येत् ।
(III. 3)¹

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¹ *Āpastamba* (III. 7) has the same *sūtra* with slight changes in the wording.

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(There will be as many times as many small squares² as there are units in the cord. These should be added together). That is, area was determined by dividing the figure into unit squares and counting these squares. And to the authors of the Śulbasūtras, ever busy with figures, lines were *karaṇis*, producers of areas and mere curiosity might have driven them to count the number of unit squares in the square produced by the diagonal of a rectangle.

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² I have taken *varga* to mean square though Thibaut, Bürk and Datta take the word to mean group. The repetition of the word *tāvantaḥ* and the use of the word *varga* in the sense of square, common in later mathematics, make this interpretation more plausible.

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Bürk and Datta turn to the enlargement of a square practised by the Vedic priest as the source of the theorem. A particular application of such enlargement occurs in Baudhāyana’s rule for the construction of the *Sāra-ratha-cakra cit*, i.e. the fire-place in the shape of a wheel with spokes. Bricks are to be made in such proportion that 225 of them together make up the ordained area of the fire-place. To these, 64 similar bricks are to be added and the total (that is 289) is arranged as a square. For this, a square of 256 bricks is to be made first and the remaining 33 are to be put round it. Whatever the necessity for this strange direction, when 289 bricks could be arranged straight away as a square with 17 bricks in each side, this makes it clear that enlarging squares by adding gnomons was a well-known practice. And if the Indians had begun with a square of side 1, adding gnomons to it to get squares with side 2, 3,…… ., one by one they would easily have noticed 3² + 4² = 5², 15² + 8² = 17² etc. To explain the geometrical character of the enunciation of the theorem in the *Śulbasūtras*, Bürk¹ fancies that the Indians might have put the squares 3², 4², 5² as in Fig. 3 and thereby discovered the rational right triangle 3, 4, 5.

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¹ *Z.D.M.G.* 1901, pp. 567-68 and Datta-*Science of the Śulba*-pp. 125-127.

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*Fig. 3*

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To speculate on whether the Indians had a proof for the theorem or what the proof could have been is idle. The Śulbasūtras, our only means of knowing what the condition of mathematics then was in India, are only practical manuals for the construction of the altars. Proofs are outside their scope. Very likely they had proofs orally transmitted to the enquiring student.

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The constructions dealt with in the Śulbasūtras comprise the construction of the east-west line; of perpendiculars; of squares, rectangles, and trapezia; and of triangles and rhombi equal in area to a given square; conversion of squares into rectangles and vice versa; of squares into circles and vice versa.

2.6. Determining the east-west line

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This was preliminary to the construction of all the altars and fireplaces described in the Vedic literature. But it is only Kātyāyana and Manu that give the details of the procedure. *Baudhāyana* and *Āpastamba* take the *prācī* or east-west line for granted.

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समे शंकुं निखाय शंकुसम्मितया रज्वा मण्डलं परिलिख्य यत्र लेखयोः शंकवग्रच्छाया निपतति तत्र शंकू निहन्ति सा प्राची ।
(*K.Sl.* I. 2)¹

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¹ The *Mānava Śulbasūtra* (p. 2) gives the same method. Cantor mistakenly says (*Vorlesungen über Geschichte der Mathematik*, 4th Ed. Vol. I, p. 637) that the *Śulbasūtras* are silent over the method of determining the *prācī*.

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(Fixing a pin (or gnomon) on level ground and drawing a circle with a cord measured by the gnomon, he fixes pins at points on the line (of the circumference) where the shadow of the tip of the gnomon falls. That is the *prācī*.)

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This is the direction for fixing the *prācī*, which is therefore the line joining the tips of the shadows of equal length cast by an object in the forenoon and afternoon. This method of fixing the east-west line is given in the *Tantrasamuccaya* and other works on architecture and the Tantra works dealing with the construction of *maṇḍapas* (sheds or halls) and *kuṇḍas* (sacred fire-pits).

2.7.1 To draw the perpendicular bisector of a given line

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a. The method is explained by Kātyāyana in connection with fixing the *udīcī*, the north-south line after the east-west line is fixed.

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तदन्तरं रज्वाभ्यस्य, पाशौ कृत्वा, शङ्क्व्वोः पाशौ प्रतिमुच्य, दक्षिणायम्य मध्ये शंकुं निहन्ति । एवमुत्तरतः सोदीची ।
(I. 3)

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(Doubling the distance between them (the end pins) on a cord and making ties one fixes the ties on the pins, stretches (the cord) to the south and strikes a pin at the middle point (of the cord). Similarly to the north. That is the north-south line).

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This is the same as the modern method of drawing the perpendicular bisector of a line. Only, instead of drawing intersecting arcs to get two points equidistant from the ends of the line, isosceles triangles are drawn on either side of the line with the line as the base and their vertices are joined.

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Fig. 4

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2.7.2. Such arcs themselves are used to draw two perpendicular diameters in a circle in Baudhāyana's recipe for drawing a square.

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—लेखामालिख्य, तस्य मध्ये शंकुं निहन्यात्तस्मिन् पाशौ प्रतिमुच्य लक्षणेन मण्डलं परिलिखेत् विष्कम्भान्तयोः शंकू निहन्यात् । पूर्वस्मिन् पाशं प्रतिमुच्य पाशेन मण्डलं परिलिखेत् एवमपरस्मिन् । ते यत्र समेयातां तेन द्वितीयं विष्कम्भमायच्छेत् ।

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(*B. Sl.* I. 22-25)

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(Drawing a line one fixes a pin at its middle. Slipping the end ties on to this pin, one draws a circle with the mark (the middle mark of the cord) and fixes pins at the ends of the diameter. With the end-tie on the eastern pin one draws a circle with the whole cord. Similarly at the western pin. The second diameter should be stretched through the points where these (circles) intersect.)

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*Fig. 5*

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This method is well-known in later Indian mathematics as the 'fish' method, the name being based on the fact that the lenticular portion common to the two intersecting circles has roughly the shape of a fish.

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2.7.3.

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The most primitive method of drawing a square with a bamboo employs the 'half-fish' to draw a perpendicular at the middle point of a line. The detailed instructions are:

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यावान् यजमान ऊर्ध्वबाहुस्तावदन्तराले वेणोश्छिद्रे करोति मध्ये तृतीयम् ।
अपरेण यूपावटदेशं अनुपृष्ठ्यं वेणुं निधाय छिद्रेषु शंकून् निहत्य उन्मुच्यापराभ्यां दक्षिणाप्राक् परिलिखेदान्तात् ।
उन्मुच्य पूर्वस्मादपरस्मिन् प्रतिमुच्य दक्षिणा प्रत्यक् परिलिखेदान्तात् ।
उन्मुच्य वेणुं मध्यमे शंकावन्त्यं वेणोश्छिद्रं प्रतिमुच्य उपर्युपरि लेखासमरं दक्षिणा वेणुं निधाय, अन्त्ये छिद्रे शंकुं निहत्य, तस्मिन् मध्यमं वेणोश्छिद्रं प्रतिमुच्य, लेखान्तयोरितरे प्रतिष्ठाप्य छिद्रयोः शंकुं निहन्ति, स पुरुषश्चतुरश्रः !
(*Āp. Śl.* VIII. 8-10 & IX. 1.)¹

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¹ ¹See also *B. Śl.* iii. 13 ff.

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(At an interval which is as much as the sacrificer with uplifted hands, he makes two holes on the bamboo and a third one at the middle (of this interval). Placing the bamboo along the *pṛṣṭhyā* to the west of the *yūpāvaṭas* (pits for the sacrificial posts) and fixing pins at the holes he releases the bamboo from the western pins and draws (an arc) towards the south-east from the (western) end. Releasing the bamboo from the eastern pin and fixing it on the western pin he draws towards the south-west beginning from the (eastern) end. Releasing the bamboo and fixing the terminal hole of the bamboo on the middle pin (of the *pṛṣṭhyā*) and placing the bamboo towards the south over the point of intersection of the lines (arcs) he fixes a pin at the end of the bamboo. Then slipping the middle hole on this and adjusting the others at the end of the arcs (i.e. to touch the arcs) he fixes pins at the holes. That is the square of one *puruṣa*). Let AB be the *pṛṣṭhyā* and O its middle point. A bamboo equal to A B in length, is laid along A B pivoted at A and rotated through a right angle so that the free end of the bamboo reaches the south-east corner, thus drawing an arc. Again the bamboo is pivoted at B and the free end drawn from A to the south-west corner tracing another arc. The bamboo is now placed so as to join O to the intersection O' of the arcs, when it will be perpendicular to AB at O.

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*Fig.* 6

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**2.7.4.** The method of 2.7.1 for the perpendicular bisector is used for drawing a perpendicular at any given point in one of the methods for the construction of a rectangle given by Baudhāyana as also in one of the methods for drawing a square given by Āpastamba¹ and Manu².

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¹ *Āp. Sl.* I. 7.

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² *Mā. Sl.* p. 5.

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दीर्घचतुरश्रं चिकीर्षन् यावच्चिकीर्षेत् तावत्यां भूमौ द्वौ शंकू निहन्यात् ।
द्वौ द्वावेकैकसभितः समौ ।
यावती तिर्यङ्मानी तावतीं रज्जुमुभयतःपाशां कृत्वा मध्ये लक्षणं करोति पूर्वेषामन्त्ययोः पाशौ प्रतिमुच्य लक्षणेन दक्षिणापयम्य लक्षणे लक्षणं करोति सोंऽसः ।
(B. *Sl.* I. 36-40)

c7.s5.t1.22

(Wishing to construct a rectangle one should fix two pins at an interval which is as much as is desired. On either side of each at equal distances, two more pins (are fixed). Making ties at the ends of a cord equal in length to the horizontal side, one makes a mark at the middle. Fixing the ties to the terminal ones of the pins at the east and stretching (the cord) by the middle mark to the south one should make a mark (on the ground) at the middle mark. Fixing the ties to the middle pin one stretches (the cord) by the middle mark to the south over the mark already made and marks the middle. This is the top corner.)

c7.s5.t1.23

Fig. 7

c7.s5.t1.24

If perpendiculars are required at A and B on the line AB, two points A₁ and A₂ are marked at equal distances from A. Then an isosceles triangle is drawn on A₁ A₂ as base. The vertex K is joined to A. Then A K is the perpendicular at A. The method for drawing the isosceles triangle is to tie ends of a cord to A₁ and A₂ and stretch it tight by its middle point.

c7.s5.t1.25

**2.7.5.** The device most frequently adopted for drawing a perpendicular is to use a cord divided into two parts so that the parts form the sides of a right-angled triangle with the line or part of the line on which the perpendicular is to be drawn. One instance will make the procedure clear. For making the *Mahāvedi*, which is an isosceles trapezium of altitude 36 *prakramas*, base 30 *prakramas* and top 24 *prakramas*, Āpastamba directs,

c7.s5.t1.26

षट्-त्रिंशिकायां अष्टादशोपसमस्य अपरस्मादन्तात् द्वादशसु लक्षणं, पञ्चदशसु लक्षणं पृष्ठ्यान्त-योरन्तौ नियम्य, पंचदशकेन दक्षिणापयम्य शंकुं निहन्ति । एवं उत्तरतः । ते श्रोणी...... ।
(V. 2)

c7.s5.t1.27

(To a cord of 36 *prakramas* 18 *prakramas* are added. Marks are made at distances of 12 and 15 *prakramas* from the' other end. Fixing the ends (of the cord) to the end of the *pṛṣṭhyā*, one stretches it to the south by the 15th mark and fixes a pin. Similarly to the north. These are the bottom corners.)

c7.s5.t1.28

The cord is 36+18=54 *prakramas* in length. The 12th mark is for measuring half the top side while the 15th mark is both for measuring the base and for making the right triangle. For the remainder of the cord is 36+3=39 *prakramas*. Therefore the parts of the cord 39 and 15 *prakramas* respectively will make a right triangle with the *pṛṣṭhyā* which is 36 *prakramas*, in length. Hence W D will be at right angles to E W. Āpastamba gives other rational right triangles too which can be used for making a right angle in lieu of the 39,36,15 one. These are (1) 3, 4, 5 which numbers can be multiplied by 4 and 5 to suit the measurements required by the *Mahāvedi* (V. 3) (2) 12,5,13 which can be multiplied by 3 for marking the bottom corners (V. 4), (3) 15, 8, 17 (V. 5), (4) 12, 35, 37, (V. 5). All these are chosen so as to make at least one of the sides have the length of one element of the *Vedi*.

c7.s5.t1.29

Fig. 9.8

c7.s5.t1.30

A slight modification makes the method suitable for any length of the given side. To cite from Āpastamba again :

c7.s5.t1.31

यावदायामं प्रमाणं ।
तदर्धमभ्यस्यापरस्मिंस्तृतीये षड्भागोने लक्षणं करोति ।
पृष्ठ्यान्तयोरन्तौ नियम्य लक्षणेन दक्षिणापयम्य निमित्तं करोति ।
एवमुत्तरतो विपर्यस्येतरतस्स समाधिः ।
(I. 2)

c7.s5.t1.32

(The measure (of the cord) is the length (of the *prācī*). Adding half of itself to it one makes a mark at the latter third of the cord diminished by one-sixth of that third. Fixing the ends to the ends of the *pṛṣṭhyā* one stretches (the cords) to the south by the mark and makes a mark. Similarly to the north. Then changing to the other side. This is the construction).

c7.s5.t1.33

By this process, if x is the length of the *pṛṣṭhyā* the cord is x+x/2 in length and the mark is at a distance of x/2-x/2.6 =5x/12. The other part of the cord is x+x/2.6=13x/12. Hence when the ends of the cord are tied to the ends of the *pṛṣṭhyā* and the cord is stretched by the mark, we get a triangle whose sides are x, 5x/12 and 13x/12 i.e. in the proportion 12, 5 and 13. Therefore it is a right triangle.

c7.s5.t1.34

The next *sūtra* teaches the use of another right triangle similarly made with sides equal to x, 3x/4 and 5x/4. Baudhāyana and Kātyāyana too give these constructions (*B. Sl.* I. 29-35 and I. 42-44; *K. Sl.* I. 12-15)

c7.s5.t1.35

Āpastamba's commentators Kapardin and Karavinda think that the particle '*vā*' in "आयामं वाभ्यस्य आगन्तुचतुर्थेन ... ...” of *Āp. Sl.* I. 3 is meant to indicate that the method is a general one. Any fraction 1/n of the original length can be added to the cord. The original length combined with 1/(2 (n+1)) of the added part will be the hypotenuse and the rest of the cord will be the smaller perpendicular side. This would mean Āpastamba here gives a general solution of the rational right triangle with one side given, which is : $x, (x/n-x/(2n(n+1))) and (x+x/(2n(n+1)))$ $i.e. x, x (2n+1)/(2n(n+1)) and x (2n²+2n+1)/(2n(n+1))$ or x.2n (n+1), x (2n+1) and x (2n²+2n+1) The solution can be verified easily. $For, 2n (n+1)²+(2n+1)².$ $=4n² (n²+2n+1)+4n²+4n+1$ $=4n⁴+8n³+8n²+4n+1$ $=(2n²+2n+1)²$ A cord divided into x and x.√2 is also used (*Āp. Sl.* II. 1)

2.8. Construction of squares and rectangles

c7.s6.1

Most of the above constructions for the perpendicular occur in connection with the construction of squares, rectangles and trapezia. Four distinct methods are detailed for the construction of the square, one of which is applicable to the rectangle also.

c7.s6.2

2.8.1.

c7.s6.3

A straight bamboo of the length of the side of the square has holes at its ends and middle. With the bamboo lying along the *pṛṣṭhyā* pins are fixed at the three holes. Then by the method described in construction (2.7.3) for the perpendicular, the bamboo is placed at right angles to the *pṛṣṭhyā* at its centre and the end of the bamboo is marked with a pin. Now the middle hole of the bamboo is slipped on to this pin and the bamboo is adjusted so that its ends touch the arcs. The ends of the bamboo then mark the corners of the square. Since the bamboo in its final position is tangential to the two arcs, it must be perpendicular to the radii at the points of contact and these radii are the two horizontal sides of the square. Hence the construction.¹

c7.s6.4

Fig. 9

c7.s6.5

Dr. B. B. Datta points out that the practice of measuring out the fire-altar with the bamboo-rod is mentioned in the *Taittirīya Saṃhitā* (V.2.5.¹ ff) and in the *Maitrāyanī* (iii. 2.4) *Kaṭha* (xx . 3-4) and *Kapiṣṭhala* (xxxii. 5.6) *Saṃhitās*. It is perhaps even older than all these *saṃhitās* since in all these fire is mythically connected with the bamboo.²

c7.s6.6

¹ *Āp. Śl.* VIII. 8-10 and IX.—1 and *B. Śl.* III. 13 ff. Āpastamba's *sūtras* have already been quoted.

c7.s6.7

² Datta, *Science of the Śulba*, pp. 60-61 and the footnotes thereto.

c7.s6.8

2.8.2. Baudhāyana's first method for the construction of a square with a given side results in a beautiful geometrical pattern.

c7.s6.9

चतुरश्रं चिकीर्षन् यावच्चिकीर्षेत् तावतीं रज्जुमुभयतःपाशां कृत्वा मध्ये लक्षणं करोति लेखा-
मालिख्य तस्या मध्ये शंकुं निहन्यात्तस्मिन् पाशौ प्रतिमुच्य लक्षणेन मण्डलं परिलिखेत्,
विष्कम्भान्तयोः शंकू निहन्यात् ।
पूर्वस्मिन् पाशं प्रतिमुच्य पाशेन मण्डलं परिलिखेत् । एवमपरस्मिन् । ते यत्र समेयातां तेन
द्वितीयं विष्कम्भमायच्छेत् ।
विष्कम्भान्तयोः शंकू निहन्यात् ।
पूर्वस्मिन् पाशौ प्रतिमुच्य लक्षणेन मण्डलं परिलिखेत् एवं दक्षिणात् एवं पश्चादेवमुत्तरतस्तेषां
येऽन्त्याः संसर्गास्तच्चतुरश्रं सम्पद्यते ।
(*B. Sl.* I. 22-28)

c7.s6.10

(Wishing to construct a square one should make ties at both ends of a string as long as the desired side and make a mark at its middle. One should draw a line and fix a pin at its middle. Fixing the ties on this pin one should draw a circle by the middle mark (of the cord) and at the ends of the diameter (formed by the *prācī*) one should fix pins. Fixing one tie on the eastern pin one should draw a circle with the other tie. Similarly round the western pin. Through the points where they meet the second diameter should be drawn and pins should be fixed at its ends. With the ties on the eastern pin a circle is to be drawn with the middle mark. Similarly round the southern, western and northern (pins). Their outer points of intersection form the square).

c7.s6.11

The explanation of the procedure is: The first circle fixes the middle points of the horizontal sides of the square or the ends of the line of symmetry. The two bigger circles serve to fix the perpendicular bisector of the line of symmetry. The points of

c7.s6.12

Fig 10

c7.s6.13

intersection of this with the first circle are the middle points of the vertical sides. Hence the vertices of the square, which will be equidistant from pairs of the middle points of the sides, will be obtained as the points of intersection of circles drawn with the middle points as centres and half the side as radius, taken two by two. This method is confined to the *Baudhāyana Śulba-sūtra*.

c7.s6.14

**2.8.3.** The first construction for a square given by Āpastamba is:

c7.s6.15

प्रमाणमात्रीं रज्जुमुभयतःपाशां करोति । मध्ये लक्षणमर्धमध्यमयोश्च । पृष्ठ्यायां रज्जुमायम्य पार्श्वयोर्लक्षणेषु इति शंकुं निहत्य उपान्त्ययोः पाशौ प्रतिमुच्य मध्यमेन लक्षणेन दक्षिणापयम्य शंकुं निमित्तं करोति । मध्यमे पाशौ प्रतिमुच्य उपर्युपरि निमित्तं मध्यमेन लक्षणेन दक्षिणापयम्य शंकुं निहन्ति । तस्मिन् पाशं प्रतिमुच्य पूर्वोस्मिन्नितरं मध्यमेन लक्षणेन दक्षिणमंसं श्रायच्छेत् । उन्मुच्य पूर्वस्मात् अपरस्मिन् प्रतिमुच्य मध्यमेनैव लक्षणेन दक्षिणां श्रोणिमायच्छेत् । एवमुत्तरो श्रोण्यंसौ ।

c7.s6.16

(I. 7)

c7.s6.17

(A cord of the length of the *pṛṣṭhyā* is noosed at either end. Marks (are made) at the middle and the middles of the halves. Stretching the cord along the *pṛṣṭhyā* one stands pins at the nooses (P and Q) and the marks (R, O, S). Inserting the nooses on the pins next to the terminal ones (i.e. R and S) and stretching the cord southwards by the middle mark one makes a mark on the ground. Again inserting both the nooses on the middle pin (O) one stretches the cord by the middle mark southward over and beyond the mark on the ground and fixes a pin (E). Inserting one noose on this pin and the other on the eastern pin (P) one should stretch the cord by the middle mark for the south top corner (C). Taking off from the eastern (pin) one fixes the noose on the western pin (Q) and stretches the cord by the middle mark for the south bottom corner (B). Similarly the top and bottom corners of the north.

c7.s6.18

Fig. 11

c7.s6.19

The principles underlying this construction are the same as those of the previous construction. But cord-stretching, which leaves the final drawing free of all unnecessary lines, is resorted to in place of the drawing of circles.

c7.s6.20

This method is given by Manu¹ and Kātyāyana² also.

c7.s6.21

¹ *M. Śl.* I. p. 5.

c7.s6.22

² *Kāt. Śr.* XVI. 8. 1-20 as quoted by B. Datta, *Science of the Śulba*, p. 62.

c7.s6.23

**2.8.4.** The next method, equally useful for the square and the rectangle, is to draw perpendiculars at the ends of the *pṛṣṭhyā* and then mark off the length of the side on these perpendiculars to get the corners. For drawing the perpendicular, the method of drawing the isosceles triangle with a cord can be used as in Baudhāyana’s method for the construction of a rectangle (I. 36-40). Or the cord can be used to form a right triangle with the *pṛṣṭhyā* as in Baudhāyana’s construction for the square (*B. Sl.* I. 29-35) and Āpastamba’s construction for any rectangular figure (*Āp. Sl.* I. 2 and 3). Āpastamba achieves the construction of the square with one operation the less, by constructing right triangles on half the *pṛṣṭhyā* with a cord divided into two parts respectively equal to half the side and √2·side/2. The direction is:

c7.s6.24

पृष्ठघान्तयोर्मध्ये च शंकुं निहत्यार्धे तद्विशेषमम्यस्य लक्षणं कृत्वा अर्धमागमयेत् । अन्तयोः पाशौ कृत्वा, मध्यमे सविशेषं प्रतिमुच्य, पूर्वस्मिन्नितरं, लक्षणेन दक्षिणमंसमायच्छेत् । उन्मुच्य पूर्वस्मादपरस्मिन् प्रतिमुच्य, लक्षणेनैव दक्षिणां श्रोणिमायच्छेत् । एवमुत्तरौ श्रोण्यंसौ ।
(*Āp. Sl.* II. 2)

c7.s6.25

(Fixing pins at the ends and middle of the *pṛṣṭhyā* one should add to a cord of half the length its *viśeṣa* (√2-1 of any length is known as its *viśeṣa* in the *Śulbasūtras*), make a mark there and then add half the side again. Making nooses at the ends, and fixing the side with the *viśeṣa* to the middle pin, and the other side to the eastern pin, one should stretch (the cord) by the mark for the southern top corner. Taking off from the eastern pin and inserting on the western pin one should stretch (the cord) by the same mark for the south bottom corner. Similarly the northern top and bottom corners.) (Fig. 12). In IX. 3 Āpastamba uses two bamboo rods equal to one *puruṣa* and √2 *puruṣas* respectively for constructing a square one sq. *puruṣa* in area.

c7.s6.26

Fig. 12

2.9. Construction of a trapezium with the face, base and altitude given

c7.s7.1

The name for a trapezium used by the *Śulbasūtras* is *ekato'ṇimat* (smaller on one side).¹ *Mahāvedi*, the *vedi* for the *soma* sacrifice of paramount importance and other *vedis* too were to be trapezia in shape. Hence the *Śulbasūtras* dwell on its construction in detail. The method used is essentially the same as for the construction of the square and the rectangle using a right-angled triangle. But various rational right triangles which will suit the measurements of the *Mahāvedi* are given. Once the perpendiculars at the ends of the *pṛṣṭhyā* (the line of symmetry or altitude) are drawn, the lengths of the top and bottom sides can be marked off on them. Since the measurements of the *Mahāvedi* are altitude = 36 *prakramas*, base = 30 *prakramas* and face = 24 *prakramas*, the eligible right triangles, according to Āpastamba, are:

c7.s7.2

¹ *B. Śl.* I. 72.

c7.s7.3

Fig. 13

c7.s7.4

36, 15, 39 (*Āp. Śl.* V. 2)
3, 4, 5 multiplied by 4 & 5 (*Āp. Śl.* V. 3)
12, 5, 13 (*Āp. Śl.* V. 4)
15, 8, 17 ( ,, ,, V. 5)
12, 35, 37 ( ,, ,, ,, )

c7.s7.5

Baudhāyana recommends his method for constructing a rectangle for the construction of a trapezium also (I. 36-41) i.e. drawing the perpendiculars at the extremities of the *pṛṣṭhyā* by the isosceles triangle method and then marking off the half sides on these.

2.10. Transformation of Figures

c7.s8.1

The votive fire-altars were prescribed different shapes according to the specific benefit sought for — the *śyenacit* (the fire-place in the form of a falcon) for attaining heaven, the *praugacit* (the fire-place in the form of an isosceles triangle) for destroying enemies and so on. But all these different shapes had to have strictly the same area, viz. 7½ sq. *puruṣas*. Hence were evolved methods for transforming one geometrical figure into another, more especially the square into other equivalent geometrical figures. These constructions are given below.

2.10.1. To convert a square into a circle

c7.s8.t1.1

No geometrical method can achieve this exactly. What the *Śulbasūtras* do is to give approximate constructions. The centre O of the square is joined to a vertex A and the circle is drawn with half the side of the square combined with ⅓ the excess of OA over half the side of the square,¹ i.e. if ‘a’ is the side of the square and ‘r’ the radius of the circle. r & = a/2 + (√2 · a/2 - a/2)/3 / & = a/2 × ((2 + √2))/3 The value of π calculated from this is only about 3.088. But according to some of the commentators, the last sentence of this rule, namely *Sānityā maṇḍalam*, is to be split as *Sā anityā maṇḍalam*, when it will mean that Āpastamba and the other authors of *Śulbasūtras* as well were aware that this was an approximate method only. Thibaut and Bürk, understandably, do not accept this explanation.

c7.s8.t1.2

¹ *Āp. Sl.* III. 2; *B. Sl.* I. 58 and *K. Sl.* III.

2.10.2. To convert a circle into a square

c7.s8.t2.1

All the three important *Śulbas* direct us to divide the diameter into 15 parts and to take 13 of these parts as the side of the square i.e., if d is the diameter of the circle² and a the side of the equivalent square, $a = 13/15 d, whence π = 3.004$ Baudhāyana gives a slightly better approximation too.

c7.s8.t2.2

² *Āp. Sl.* III. 3; *B. Sl.* I. 60 and *K. Sl.* 14

c7.s8.t2.3

मण्डलं चतुरश्रं चिकीर्षन् विष्कम्भमष्टौ भागान् कृत्वा भागमेकोनत्रिंशता विभज्य अष्टाविंशति भागानुद्धरेत् भागस्य च षष्ठं अष्टमभागोनम् । (*B. Sl.* I. 59)

c7.s8.t2.4

(Wishing to convert a circle into a square one should divide the diameter into 8 parts, divide one of these parts again into 29 parts and subtract 28 of these (29th parts) together with one-sixth of one of these parts, the latter being diminished by one-eighth of that (one-sixth part) ).

c7.s8.t2.5

$i. e., a=d (1-28/8.29-1/6.8.29+1/6.8.29.8 )$

c7.s8.t2.6

This value is based on an inversion of the relation between r and a given in connection with the problem of circling the square. How exactly the value was brought to the form of this long and complicated fractional expression is a matter for speculation, but may not be of geometrical interest.

2.10.3. To convert a rectangle into a square

c7.s8.t3.1

Āpastamba's rule is:

c7.s8.t3.2

दीर्घचतुरश्रं समचतुरश्रं चिकीर्षन् तिर्यङ्मान्यापच्छिद्य शेषं विभज्योभयत उपदध्यात्। खण्डमागन्तुना संपूरयेत्। तस्य निर्ह्रास उक्तः।
(*Āp. Śl.* II. 7)

c7.s8.t3.3

(Wishing to turn a rectangle into a square, one should cut off a part equal to the transverse side and the remainder should be divided into two and juxtaposed at the two sides (of the first segment). The bit (at the corner) should be filled in by an imported bit. The removal of this has been explained already).

c7.s8.t3.4

Fig. 14

c7.s8.t3.5

If ABCD is the rectangle, a square AB₁ C₁ D with side AD, is cut off from it. The remaining rectangle B₁ BCC₁ is divided into two equal strips B₁ B₂ E C₁ and B₂ BCE. The strip B₂BCE is cut off and applied to the side of the square DC₁. Now we get a square of side AD₁ with a small square C₁ E C₂ F unfilled up at one corner. The larger square is completed and the imported square in the corner is removed by the method of removing a square from a square, (given in 2. 11. 4.).

c7.s8.t3.6

Baudhāyana (I. 54) and Kātyāyana (III. 2) give the same method. Though this method works with any rectangle,¹ Kātyāyana provides for a very long rectangle with a separate *sūtra*.

c7.s8.t3.7

¹ If the length is greater than thrice the breadth, the *āgantu* or imported square will be the bigger square.

c7.s8.t3.8

अतिदीर्घं चेत्तिर्यङ्मान्या॑पच्छिद्य अपच्छिद्यैकसमासेन समस्य शेषं यथायोगमुपसंहरेत् ।
(*K. Sl.* III. 3)

c7.s8.t3.9

(If the rectangle) is very long, cut off repeatedly by the transverse side (breadth) and join the squares so formed into one big square, and then the remainder of the rectangle should be joined to this square as it fits (to form a square).

c7.s8.t3.10

The method is no improvement over the general method, since here no side of the remainder rectangle will be equal to the side of the bigger square to which its strips are to be joined.

2.10.4. To convert a square into a rectangle

c7.s8.t4.1

समचतुरश्रं दीर्घचतुरश्रं चिकीर्षन् यावच्चिकीर्षेत् तावर्ती पार्श्वमानीं कृत्वा यदधिकं स्यात् यथायोगमुपदध्यात् ।
(*Āp. Sl.* III. 1)²

c7.s8.t4.2

² Also *B. Sl.* I. 53.

c7.s8.t4.3

(Wishing to convert a square into a rectangle one should make the lateral side as long as is desired and the excess should be joined suitably.)

c7.s8.t4.4

We are not told how exactly the excess is to be joined. Thibaut and Bürk suggest that this was achieved by repeated slicing and joining. If A B C D is the square and the side of the rectangle is to be A D₁, a rectangle A B by A D₁ is sliced off first. From the remainder a rectangle with length equal to A D₁ can be obtained. This is sliced and joined to A B C₁ D₁ as shown. The remaining square is to be sliced and put together suitably so as to get a narrow strip with length = A D₁, which is then again joined to A F₁ F D₁. This procedure is not merely empirical but also highly unsatisfactory, since there is no guarantee that the remainder square always yields a rectangle of the requisite length.

c7.s8.t4.5

Fig. 15

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Āpastamba's commentator Sundararāja, explaining this *sūtra*, gives a purely geometrical and exact construction.

c7.s8.t4.7

यावदिच्छं पार्श्वमान्यौ प्राच्यौ वर्धयित्वा उत्तरपूर्वां कर्णरज्जुमायच्छेत्, सा दीर्घचतुरश्रमध्यस्थायां समचतुरश्रतिर्यङ्मान्यां यत्र निपतति तत उत्तरं हित्वा दक्षिणांशं तिर्यङ्‌मानीं कुर्यात्, तद् दीर्घचतुरश्रं भवति ।

c7.s8.t4.8

(Producing the sides of the square eastward to the desired length of the lateral side, one should draw the north-eastern diagonal. The part of the transverse side to the north of the point where the diagonal cuts it is to be discarded and its southern part is to be made the transverse side of the rectangle. That will be the rectangle.)

c7.s8.t4.9

Fig. 16

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Let A B C D be the given square. Produce A D and B C to F and E so that A F = B E = the required side of the rectangle. Complete the rectangle A B E F and join the diagonal B F cutting C D in G. Through G draw a st. line IH parallel to the sides of the square. Then IBEH is the required rectangle.

c7.s8.t4.11

For, F A B &= F E B / I G B &= G C B / and F D G &= F H G / Hence rect. A I G D &= rect. G C E H / \therefore rect. I B E H &= rect. I B C G + rect. G C E H / &= rect. I B C G + rect. A I G D / &= sq. A B C D

c7.s8.t4.12

Here the case where the given side of the rectangle is greater than the side of the square only is dealt with. But with slight change the method is applicable to the case where the side is less also.

c7.s8.t4.13

Baudhāyana and Kātyāyana give an easier method.

c7.s8.t4.14

समचतुरश्रं दीर्घचतुरश्रं चिकीर्षन् मध्येऽक्ष्णयाऽपच्छिद्य विभज्येतरत्पुरस्तादुत्तरतश्चोपद्ध्यात् । विषमं चेद्यथायोगं उपसंहरेदिति व्यासः। (*K. Sl.* III. 4)¹

c7.s8.t4.15

¹ Also *B. Sl.* I. 52.

c7.s8.t4.16

(Wishing to transform a square into a rectangle one should cut diagonally in the middle, divide one part again and place the two halves to the north and east of the other part. If the figure is a quadrilateral one should place together as it fits. This is the distribution). Let A B C D be the square. It is cut along AC to form two right triangles. The A B C is again halved along the altitude B E. The two halves B E A and B E C are then removed to the positions D F C and D G A.

c7.s8.t4.17

*Fig.* 17

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Then, obviously rect. A C F G = sq. A B C D. The defect of this method is that the rectangle cannot be given any desired side.

2.10.5. To convert a rectangle or square into a trapezium with the shorter parallel side given

c7.s8.t5.1

Baudhāyana deals with this problem. चतुरश्रमेकतोऽणिमच्चिकीर्षन् अणिमतः करणीं तिर्यङ् मानीं कृत्वा शेषमक्ष्णया विभज्य विपर्यस्येतरत्रोपद्ध्यात्। (*B. Sl.* I. 55)

c7.s8.t5.2

(If one wishes to make a square or rectangle shorter on one side, one should cut off a portion by the shorter side. The remainder should be divided by the diagonal, inverted and attached on either side).

c7.s8.t5.3

If A B C D is the given rectangle, let the rectangle A F E D be cut off so that A F = D E = the given shorter side. The remaining rectangle E F B C is to be cut diagonally along B E and the portion B E C is to be inverted and attached to the side A D of the rectangle in the position E′ A D. Then D E′ B E is the equivalent trapezium.

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*Fig.* 18

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The *Śatapatha Brāhmaṇa* seems to give another method for this conversion.¹

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¹ *Ś. Br.* X 2. 1. 3-4.

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चतुर्विंशत्यंगुलिभिर्मिमीते । चतुर्विंशत्यक्षरा वै गायत्री, गायत्रोऽग्निर्यावानग्निर्र्यावत्यस्य मात्रा तावतैवैनं तान्मिमीते । स चतुरंगुलमेव उभयतोऽन्तरत उपसमूहति, तावद् व्यूहति । तन्नाहेवातिरेचयति नो कनीयः करोति ।

c7.s8.t5.8

(He measures by 24 *aṅgulis*. For *Gāyatrī* has 24 letters and *agni* is of *Gāyatrī*. As much as is the *agni*, as much as is its measure, by that much he measures them. He contracts 4 *aṅgulis* inwards on both sides. He stretches 4 *aṅgulis* outwards on both sides. as much as he contracts, so much he stretches. Thus he does not make it exceed (the right measure) nor does he make it smaller).

c7.s8.t5.9

Fig. 19

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Datta takes this to mean² that the face of the square (A B C D) is shortened on either side by a small length (D D' = C C') and the base is lengthened on either side by the same length to A' and B'. Then the area of trapezium A'B'C'D' = the area of square ABCD. and obviously so, since $Δ P D D′ Δ P A A′$ and Δ Q C C′ Δ Q B B′

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² *Science of the Śulba*, p. 91.

c7.s8.t5.12

The other *Śulbasūtras* do not deal with this problem.³

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³ B.B. Datta says (*Science of the Śulba*, p. 92) that *Āp. Śl.* XV-9 ff deal with this problem, but in the edition I am using the said *sūtras* do not deal with this problem.

2.10.6. To convert a trapezium into an equivalent rectangle

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Āpastamba tackles the converse problem of converting an isosceles trapezium into an equivalent rectangle. It is not given as a general prescription but rather as a means of finding out the area of the trapezium of the Mahāvedi.

c7.s8.t6.2

दक्षिणस्मादंसाद् द्वादशसु दक्षिणस्यां श्रोण्यां निपातयेत्, छेद विपर्यस्य उत्तरत उपदध्यात् । सा दीर्घा चतुरस्रा । तथायुक्तां संचक्षीत ।

c7.s8.t6.3

(From the southern top corner one should drop a perpendicular on the southern botton corner at a distance of 12 (*padas* from the *pṛṣṭhyā*). The removed bit should be placed inverted at the northern side. That is the rectangle. One should study it thus joined).

c7.s8.t6.4

Fig. 20

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That is, to convert the trapezium A B C D (parallel sides 24 ft. and 30 ft.), C B′ is drawn perpendicular to A B, B′ being 12 padas away from PQ, the *pṛṣṭhyā*. The triangle C B′ B is then placed in the position A D D′. Then rect. A B′ C D′ = trap. A B C D

2.10.7. To construct an isosceles triangle equal in area to a given square and vice versa

c7.s8.t7.1

Conversion of a square into an equivalent triangle, being necessary for the construction of the *Praugacit*, is tackled by all the three important *Śulbasūtras* and all of them give the same prescription.

c7.s8.t7.2

यावानग्निसारत्निप्रादेशो द्विस्तावतीं भूमि चतुरश्रं कृत्वा, पूर्वस्याः करण्या अर्धात् श्रोणी प्रत्यालिखेत् । सा नित्या प्रउगम् ।
(*Ā. Śl.* XII. 5)¹

c7.s8.t7.3

¹ See also *B. Śl.* I. 56 and *K. Śl.* IV. 5.

c7.s8.t7.4

(Making an area which is double as much as the fire-altar with the *aratnis* and *prādeśas*, into a square, one should draw lines from the middle point of the eastern side towards the bottom corners. That is the equivalent *prauga* (isosceles triangle).

c7.s8.t7.5

Fig. 21

c7.s8.t7.6

Let A B C D be a square of twice the required area. Let E be the middle point of C D. E A and E B are joined. Then A E B is the required triangle. For if the altitude EF is drawn, the square is divided into 2 equal rectangles A F E D and F B C E.

c7.s8.t7.7

{rcl@{ }l} & A F E & = 1/2 rect. A F E D & / and & F B E & = 1/2 rect. F B C E & / \therefore Whole & A E B & = 1/2 sq. A B C E. &

c7.s8.t7.8

This construction leads to the formula, area of a triangle = ½ base × altitude.

c7.s8.t7.9

The converse problem is treated by Kātyāyana only.

c7.s8.t7.10

प्रउगं चतुरश्रं चिकीर्षन् मध्ये प्राञ्चमपच्छिद्य विपर्यस्येतरत उपधाय दीर्घचतुरश्रसमासेन समस्येत् । (IV.7)

c7.s8.t7.11

(Wishing to convert an isosceles triangle into a square, one should cut in the middle towards the east, place on the other side inverted and manipulate by the method of conversion of a rectangle into a square.)

c7.s8.t7.12

Fig. 22

c7.s8.t7.13

Let A B C be the triangle. Let it be cut into two halves along the altitude C D. Let the part D B C be applied inverted to the side A C. Then rect. A D C E = △ A B C. Now the rectangle can be converted into an equivalent square by the method already given.

2.10.8. To construct a rhombus of given area

c7.s8.t8.1

तावदेव दीर्घचतुरश्रं विहृत्य पूर्वापरयोः करण्योरर्धात् तावति दक्षिणोत्तरयोर्निपातयेत् सन्त्योभयतःप्रउगम् । (*Āp. Śl.* XII. 9)¹

c7.s8.t8.2

¹ Also *B. Śl.* I. 57 and *K. Śl.* IV. 6.

c7.s8.t8.3

(Drawing a rectangle of the same area (i.e. of the area of the square for the *prauga*), one should draw lines from the middle points of the eastern and western sides to the middles of the southern and northern sides. That is the rhombus of the same area).

c7.s8.t8.4

Fig. 23

c7.s8.t8.5

Let A B C D be a rectangle of twice the area of the rhombus. Let E, F, G, H be the middle points of the sides. E F, F G, G H, H E are joined. Then rhombus E F G H = ½ rect. A B C D.

2.10.9. To transform a rhombus into a rectangle

c7.s8.t9.1

This converse construction occurs in Kātyāyana only. उभयतःप्रउगं चेन्मध्ये तिर्यंगपच्छिद्य पूर्ववत् समस्येत् । (K. *Śl.* iv. 8) (If it is an *ubhayataḥ prauga* one should cut transversely along the middle and join together as before.)

c7.s8.t9.2

*Fig. 25*

c7.s8.t9.3

The process is exactly the same as for the *prauga*. The rhombus is first divided into two isosceles triangles by joining a diagonal and again into 4 right triangles by cutting along their altitudes. The four triangles are joined together to form a rectangle.

2.11.1. Combination of areas and the converse

c7.s8.t10.1

By the application of the theorem of the square of the diagonal the authors of the *Śulbasūtras* combine any number of squares to form another square.

c7.s8.t10.2

For, combining two equal squares i.e. for doubling a square Āpastamba's rule is: समस्य द्विकरणी । (I. 5) (The diagonal of the square is the double-maker.) Hence if a square is drawn on the diagonal of the given square, it will produce double the area. The diagonal will therefore be √2 a where a is the side of the square. It is noteworthy that the *Śulbasūtras* give a very close approximation to the value of √2 प्रमाणं तृतीयेन वर्धयेत्तच्चतुर्थेन आत्मचतुस्त्रिंशोनेन सविशेषः । (Āp. *Śl.* I. 5 and B. *Śl.* I. 61-62)¹ (The measure should be increased by one-third of itself, which again is increased by its one-fourth and diminished by 1/34 th of that (second) increment. This is the *saviśeṣa*.)

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¹ Also *K. Śl.* II. 13.

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i.e. √2=1+⅓+1/3.4-1/3.4.34

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(This result must have been arrived at by Rule of Three and the method of repeated correction.¹ For trebling a square

c7.s8.t10.6

¹ Thibaut's hypothesis about the deduction of this expression (*J.A.S.B.*, 1875) is ingenious, but one fails to understand the preference for 12² + 12² = 288 = 17² - - 1 when the earlier 2² + 2² = 8 = 3² - 1 is equally suitable.

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Cites (23)