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The Science of the Shulba: A Study in Early Hindu Geometry c10: Chapter V. Constructions

The Science of the Śulba: A Study in Early Hindu Geometry c10

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Chapter V. Constructions

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*To draw a straight line at right angles to a given straight line.*

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(*a*) Suppose that the given straight line runs east-to-west. On it fix two poles at an arbitrary distance apart, says Kātyāyana. Then

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"Increase a cord of length equal to the distance between them (poles) by itself and make two ties at the ends. Then having fastened the two ties at the two poles, stretch the cord by its middle point towards the south and fix a pole at the place reached by the point. Proceed similarly on the north. It (the line joining these two poles) is the north-to-south line."¹

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¹ *KŚl*, i. 3.

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Fig. 16

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But the more common and at the same time the oldest Hindu method of drawing a straight line at right angles to another is as follows :

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(*b*) Take two points on the given straight line. Describe two circles with their centres at these two points and their radii equal to the distance between them. The

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Fig. 17

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line joining the points of intersection of the two circles is perpendicular to the given line.

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*To draw a straight line at right angles to a given straight line from a given point on it.*

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It should be observed that there is no particular rule for this construction in any *Śulba-sūtra* except perhaps the *Kātyāyana Śulba Pariśiṣṭa*. But it appears from the descriptions of other constructions that more than one device were used to be adopted for that purpose. The earliest of these methods is as follows :

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(*a*) Take two points (*B*, *C*) on the given straight line (*BC*) at equal distance from the given point (*A*). With centre *B* and radius *BC* describe a circle. Similarly with centre *C* and radius *CB* describe another circle. Let *D* and *E* be the points of intersection of these two circles. Join *AD* or *AE* or *DE*. Then this straight line will be at right angles to the given straight line *BC* at *A*.

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Fig. 18

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(*b*) On the given straight line (*BC*), fix two poles (*B*, *C*) equally distant from the pole at the given point *A*.

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Fig. 19

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Take a cord twice as long as *BC*. Make a tie at each of its ends and a mark at the middle. Fasten the two ties at the poles *B*, *C* and stretch the cord towards the side having taken it by the middle mark. Fix a pole *D* at the point reached by the mark. Join *DA*. It is the required straight line which is at right angles to *BC* at the given point *A*.

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(*c*) Fix a pole (*B*) at a certain distance from the pole at the given point (*A*) on the given straight line (*AB*).

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Fig. 20

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Take a cord of suitable length. Make a tie at both ends and a mark at a proper point in it. Having fastened the ties at the poles (*A*, *B*), stretch the cord sidewise by the mark and fix a pole (*C*) at the point reached by it. Then *AC* is the required straight line.

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*To construct a square having a given side.*

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*Method 1:*

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"In a bamboo-rod, make two holes (*A*, *B*) as much apart as the height of the sacrificer with uplifted arms¹ and a third hole (*C*) mid-way between them. Place the bamboo-rod on the east-to-west line and fix poles in the holes (beginning) from the western extremity of the sacrificial place. Then freeing the two poles (*C*, *B*) on the west, describe a circle (by rotating the bamboo) south-east-wise by the hole at the (opposite) end. Then unloosen-ning the eastern hole and fixing the hole in the west (in its original position), describe another circle south-westwise by the hole at the opposite end. Now release the bamboo (completely); fix again an extreme hole at the middle pole (*C*) ; place it towards the south over the point of intersection of the two circles and fix a pole at the point (*F*) reached by the outermost hole. Then fix at this pole the middle hole of the bamboo and having laid it along the extreme outer edges of the two circles,¹ fix two poles (*E, D*) at the two (outermost) holes. It (the figure thus described, *ABDE*) is a square (having a side) of one *puruṣa*.'' ² (Fig. 21.)

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¹ The square to be constructed is to have, in the present case, a side of that length.

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¹ That is, the bamboo should be laid tangentially to both the circles and the poles are to mark the points of contact.

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² *ĀpŚl*, viii (8-10)-xi (1).

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Fig. 21

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*Method II*:

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" If you wish to construct a square, take a cord as long as its side is desired to be ; make a tie at both ends and a mark at the middle. Then having drawn a line (east-to-west) of the desired length, fix a pole at its middle. Fasten the two ties at this pole and describe a circle with the mark. Now fix poles at the both ends of the diameter (running east-to-west). Having fastened one tie at the eastern pole, describe a circle with the other tie. Describe a similar circle about the western pole. On joining the points of intersection of the circles, the second (*i.c.*, north-to-south) diameter will be found. Fix two poles at the extremeties of this diameter. Now, having fastened both ties at the eastern pole, describe a circle with the mark. Similarly describe circles about the southern, western and northern poles. The exterior points of intersection of these circles will determine the square.'¹ (Fig. 22.)

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¹ *BŚl*, i. 22-28,

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Fig. 22

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*Method III:*

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"Take a cord as long as the measure (to be given to the side of the square); make a tie at both ends and a mark at the middle of itself and of its two halves. Stretch out this cord along the east-west line and fix poles at the ties and marks. Then having fastened the ties at the two poles of outer marks, stretch the cord towards the south having taken it by the middle mark and make a point there. Now fasten both the ties at the middle pole and stretch the cord towards the south by the middle mark over this point and fix a pole at the place reached. Fasten one tie at this pole, another tie at the easternmost pole, and stretch out the cord having taken it by the middle mark; thus will be obtained the south-eastern corner of the square (required). Then freeing the tie from the easternmost pole, fasten it to the westernmost pole and again stretch the cord by the middle mark; thus the south-western corner will be determined. Similarly can be determind the north-eastern and north-western corners of the square.'"¹ (Fig. 23.)

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¹ *ĀpŚl*, i. 7.

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Fig. 23.

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*Method IV:*

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"Take a cord two times the measure of the given side of the square; make a tie at both ends and a mark at the middle. With one half of this cord measure the east-to-west (breadth) of the square. In the other half, make a mark at a distance (from the western end) less by its one-fourth. Let this mark be called *nyañchana*. Make another mark at the middle of that half for the purpose of (determining) the eastern corners. Having fastened the two ties at the two extremities of the east-to-west breadth, stretch the cord towards the south by the *nyañchana* mark. Thus the two eastern and two western corners of the square should be constructed by the middle mark of the other half of the cord."¹ (Fig. 24.)

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¹ *BŚl*, i. 29-35.

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Fig. 24

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EW = a, the given side; AE = EB = DW = WC = a/2; A'E = EB' = D'W = WC' = 3a/4; ED' = EC' = WA' = WB' = 5a/4.

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*Method V*:

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"Add to a cord as long as the given side its half and make a mark at a distance (from the other end of the added portion) less by its sixth part. Fasten the ends of the (increased) cord at the extremities of the east-west line and stretch it towards the south having taken by the mark and put a sign at the point reached by it. Do similarly on the north and again on both sides after interchanging the ends of the cord. This is the construction."² (Fig. 25.)

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² *ĀpŚl*, i. 2.

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Fig. 25

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EW = a, the given side; AE = EB = DW = WC = a/2 ; A'E = EB' = D'W = WC' = 5a/12 ; ED' = EC' = WA' = WB' = 13a/12.

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The Method I is described in full by Āpastamba.¹ It is noted and partly described by Baudhāyana.² This seems to be the oldest Hindu method for the construction of a square on a given straight line. For, the practice of the measurement of the Fire-altar with the bamboo-rod is mentioned as early as the *Taittirīya Saṃhitā* (c. 3000 B.C.)³ and indeed reappears in almost all the early *Saṁhitās* and *Brāhmaṇas*.¹ It is found to have been gradually replaced by the measurement with a cord which was introduced about the time of the *Śatapatha Brāhmaṇa* (c. 2000 B.C.).²

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¹ *Loc. cit.*

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² *BŚl*, iii. 13ff.

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³ *TS*, v. 2. 5. 1ff. In the early *Saṃhitā* and *Brāhmaṇa* is found a mythological relation between the *Agni* and the *Veṇu* ("bamboo-rod"). Thus *Taittirīya Saṃhitā* observes, "He metes with a bamboo; the bamboo is connected with Agni; (verily it serves) to unite him with his birth place" (v. 2. 5. 2). The connection has been narrated in that work thus: "Agni went away from the gods; he entered the reed; he resorted to the hole which is formed by the perforation of the reed." This mythology reappears in the *Maitrāyaṇī Saṃhitā* (iii. 2. 4) and *Śatapatha Brāhmaṇa*. The latter describes, "Agni went away from the gods; he entered into a reed, whence it is hollow, and whence inside it is, as it were, smoke-tinged” (vi.3.1.26); again “Agni went away from the gods. He entered into a bamboo-stem; whence that is hollow. On both sides he made himself those fences, the knots, so as not to be found out; and wherever he burnt through, those spots came to be” (vi.3.1.31).

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¹ *MaiS*, iii.2-4; *KtŚ*, xx.3-4; *KapS*, xxxii.5.6; *ŚBr*.

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² *ŚBr*, x. 2.3.8 ff.

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The Method II occurs only in the *Baudhāyana Śulba*. It is clearly based on the previous method and is indeed a combination of four operations by it.³

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³ Compare Bürk, *ZDMG*, lv, p. 547.

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Fig. 26

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The Method III is taught by Āpastamba⁴ and Manu.⁵ The latter emphasizes that it should be exclusively used in all cases of the construction of a square on a given straight line.¹ This method is also applied by Kātyāyana.² It may be noted that the cord used in the course of the construction is called in the *Mānava Śulba* by the technical name *pañcāṅgi* ("five-jointed"), because it has five (*pañca*) joints (*aṅga*), *viz.*, two ties and three marks. The rope has been sometimes called *pātinī* ("that which is laid out") because measurements are made by laying (*pāta*) it out on the ground.

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⁴ *Loc. cit.*

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⁵ *MāŚl*, i. 14-21; compare also vii. 7 ff.

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¹ *Ibid*, vii. 7.

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² *KŚr*, xvi. 8. 1-20.

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The Method IV is given by Baudhāyana,³ Āpastamba,⁴ Kātyāyana,⁵ Manu⁶ and Maitrāyaṇa; and the Method V by Āpastamba⁷ and Kātyāyana.⁸ It is also taught by Baudhāyana.⁹ He would, however, restrict its application to the construction of a rectangle.

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³ *Loc. cit.*

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⁴ *ĀpŚl*, i. 3.

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⁵ *KŚl*, i. 12-3; compare also *KŚlP*, 16

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⁶ *MāŚl*, iii. 5 ff.

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⁷ *Loc. cit.*

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⁸ *KŚl*, i. 14-5.

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⁹ *BŚl*, i. 42-4.

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*To construct a rectangle of given sides.*

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"If you wish to construct a rectangle, fix on the ground two poles as much apart as you wish (the length to be). On either sides (before and behind) of each of these poles, fix two other poles at equal distances from it. Take a cord as much as the breadth (of the rectangle); make a tie at both ends and a mark at the middle. Having fastened the two ties at the two poles about the eastern pole, stretch the cord towards the south by the mark and put a sign (where the mark touches the ground). Then fasten the two ties at the middle pole and again draw the cord over the sign towards the south by the mark and fix a pole at the mark. That is the south-east corner (of the rectangle). Thereby is explained (how to determine) the north-east corner and also the two western corners.” ¹

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¹ *BSl*, i.36-40.

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Fig. 27

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This method is alike in principle to that employed by Āpastamba and Manu for the construction of a square on a given straight line (*Method III*). Still Baudhāyana would restrict its application to the construction of a rectangle having given sides and of an isosceles trapezium having given face, base and altitude.

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*To construct an isosceles trapezium of a given altitude, face, and base.*

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To construct a trapezium whose altitude, face and base are given, Baudhāyana ² follows a method similar to the one adopted by him for the construction of a rectangle of given sides. Only cords of given different measures are employed for fixing the extremities of the face and base.

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² *BSl*, i.41.

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The methods suggested by Āpastamba for the same purpose will be understood from those adopted by him for the construction of the *Mahāvedi*. The shape of the *Mahāvedi* is prescribed by tradition to be an isosceles trapezium whose altitude is 36 *pada* (or *prakrama*), face 24 (units) and base 30 (units). Āpastamba gives four methods for its construction. All of them are in principle the same. It is to draw a straight line through both the extremities of another straight line equal to the given altitude and at right angles to it. Along these straight lines and on other sides of the altitude are measured lengths equal to half the given lengths of the face and the base. And thus the isosceles trapezium is drawn. Āpastamba distinguishes between his different methods as *Ekarajjvāviharaṇa* ("The method of construction with one cord") and *Dvirajjvāviharaṇa* ("The method of construction with two cords").

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"Add to a cord of 36 (*pada* or *prakrama*) 18 and make a mark at 12 and a mark at 15 from its western end. Having fastened the ends of the cord to (the two poles at) the two extremities of the east-west line (of 36 *pada*) stretch it towards the south by taking by the mark at 15 and fix a pole (at the point reached by the mark); (proceed) similarly towards the north. These poles are the two western corners of the *vedi*. For (determining) the two eastern corners, interchange (the ends of the cord) and then stretching it towards the south by the mark at 15, fix a pole at (the point reached by) the mark at 12; (proceed) similarly towards the north. These are the two eastern corners. This is the method of construction with one cord."¹ (Fig. 28.)

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¹ *ĀpŚl*, v.2.

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Fig. 28

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"The diagonal of a rectangle whose sides are 3 and 4 (*pada* or *prakrama*) is 5. With these increased by three times themselves (are determined) the two eastern corners of the *vedi*. With them increased by four times themselves (are fixed) the two western corners."² (Fig. 29.)

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Fig. 29

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$3²+4²=5²$

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$3+3.3=12 12²+16²=20²$ $4+3.4=16$ $5+3.5=20$

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$3+4.3=15 15²+20²=25²$ $4+4.4=20$ $5+4.5=25$

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² *ĀpŚl*, v.3.

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*Method III:*

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"The diagonal of a rectangle whose sides are 5 and 12, is 13. With them the eastern corners of the *vedi* (are determined); and with them increased by twice themselves the western corners (are fixed)"¹ (Fig. 30.)

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¹ *ĀpŚl*, v.4.

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Fig. 30

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$5²+12²=13²$ $5+2.5=15$ $12+2.12=36$ $13+2.13=39$ $15²+36²=39²$

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*Method IV:*

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"The diagonal of a rectangle whose sides are 8 and 15, is 17; with these cords, the two western corners of the *vedi* (are measured). The diagonal of a rectangle of sides 12 and 35 is 37; with these cords, (are measured) the eastern corners."² (Fig. 31.)

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Fig. 31

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$8²+15²=17²$ $12²+35²=37²$

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² *ĀpŚl*, v.5.

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Āpastamba observes: "These are the known methods of construction of the (*Saumikī*) vedi."¹

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¹ "Etāvanti jñeyāni vedı-viharaṇāni bhavanti"—*ĀpŚl*, v.6.

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*To construct a parallelogram having given sides at a given inclination.*

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For the proper construction of an altar of prescribed size and shape, it was sometimes necessary to make bricks of the shape of a parallelogram having given sides at a given inclination. For instance, Āpastamba says:

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"Make a class of bricks, one-fifth of a puruṣa long and one-ninth of a puruṣa broad, (the two sides being) inclined (*nata*) suitably so as to fit (*yathā-yagam*)."²

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² *ĀpŚl*, xvi.2.

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The reference in the end of the rule is to the bent of the wing of the falcon-shaped altar where these bricks shall have to be used. Thibaut observes: "By 'nata, bent' the sūtrakāra means to indicate the sides of the brick bo not form rectangles. The shape of the brick is rhomboidical, the angles, which the sides form with each other are the same which the wings of the *śyena* form with the body."³

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³ Thibaut, *Śulvasūtras*, p. 31.

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Āpastamba does not expressly teach us how to construct a parallelogram of that shape. But it could be inferred from the method laid down for the construction of another parallelogram. He says,

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"(Construct) bricks for the wing with four sides: two sides one-fourth puruṣa each and two sides of one-seventh puruṣa."⁴

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⁴ *ĀpŚl*, xix.5.

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By the expression *pakṣeṣṭaka* ("brick for the wing") is implied that the inclination between the two contiguous sides of these bricks must be the same as the bending of the wings where they are to be used. Now the bending of a wing has been defined thus:¹ Describe a rectangle *ABCD* of breadth (*AD*) equal to 1 puruṣa and length (*AB*)

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¹ *Cf.* *ibid*, xviii. 6-8.

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Fig. 32

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2 1/16 puruṣa. Draw *MN* bisecting *DC* and at right angles to it. Draw *DP* one puruṣa long meeting *MN* at *P*. Cut off *PN* equal to *DP*. Draw *NA*, *NB*, *PD*, *PC*. This will give the bending of the wing.

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Āpastamba teaches the following method for construction of the above *pakṣeṣṭaka*:

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"Construct a rectangle (*abcd*) of length (*aD*) equal to one-fourth of a puruṣa and breadth (*Dc*) equal to one-seventh of the side (*DM*) of (half) the wing; by the slanting side (*DP*) the mouth of this rectangle should be changed so that the sides (*Df* and *ae*) might be bent to the one-seventh of the slanting side of the wing (*DP*)."²

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² *ĀpŚl*, xix.8. *Ibid*, xvi. 8.

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Thus we have the required parallelogram *aefD* whose sides *aD*, *fe* are equal to one-fourth of a puruṣa each and sides *fD*, *ae* are each equal to one-seventh of a puruṣa and they being inclined as specified.

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It should be particularly noticed that in the above the inclination between the two sides of the parallelogram to be constructed or between any two lines in general is not stated in terms of angular units (though the conception of such is not wanting in Hindu astronomy) but in terms of the relation between the sides and diagonals of a certain rectangle along which the two given lines would lie.

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Āpastamba gives also a slightly different method of constructing a parallelogram. He says:

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"Draw a rectangle one-fifth of a puruṣa long from east to west, and one-tenth of a puruṣa broad; to the south as well as to the north of it draw another (rectangle of the same size). Draw the diagonals of them passing through their south-western corners."

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Fig. 33

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From this we can easily guess the method that was generally followed in the *Śulba* for the construction of a parallelogram having given sides at a given inclination. Let A, B represent the given sides and the inclination be the same as that between the diagonal PR and the side *PQ* of the rectangle *SRQP*. Now draw the rectangle *EDCP* similar to the given rectangle such that its diagonal *PD* be equal to the given side *B*. Then draw the rectangle *DE'P'C* so that its side *CP'* be equal to the difference between *A* and *PC*. Now draw the rectangle *E'D'C'P'* equal to the rectangle *EDCP*. Join *D'P'*. Then *DD'P'P* is the required parallelogram. The Fig. 34 answers to the case when *A* > *PC*. In case *A* < *PC*, the rectangle *DE'P'C*

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Fig. 34

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Fig. 35

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should be drawn so as to overlap the rectangle *EDCP*. (Fig. 35.)

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