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The Science of the Shulba: A Study in Early Hindu Geometry c17: Chapter XII. Similar Figures

The Science of the Śulba: A Study in Early Hindu Geometry c17

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Chapter XII. Similar Figures

c17.2

It has been observed before that in the sacrificial rituals of the early Hindus it is oftentimes necessary to construct an altar differing in area from another by a specified amount. For instance, the *Sautrāmaṇikī-vedi* is stated to be equal to one-third of the *Mahāvedi*¹ and the *vedi* of the *Aśvamedha* double the latter.² The *Lakṣa-homa-vedi* and *Koṭihoma-vedi* are respectively four and twenty-five times the *Pākayājñikī-vedi*.³ Again it is said that the primitive Fire-altar, *Caturasra-śyenacit*, should have an area of seven and a half square puruṣas at the time of the first construction. At the second construction its area shall have to be 8½ square puruṣas; at the third 9½ square puruṣas. In the same manner the area of the *Agni* should be increased by one square puruṣa at each successive construction up to 101½ square puruṣas. The earliest reference to this mode of increment of the *Agni*, as has been stated before, is found in the *Śatapatha Brāhmaṇa*.⁴ And the practice continued during the succeeding ages.⁵ Now it is the strict injunction of the *Śruti*, that the primitive shape of the Fire-altar must not be disturbed during the course of successive constructions.⁶

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¹ *BŚl*, i. 85 ; *ĀpŚl*, v. 8, 9 ; *KŚr*, xix. 2.2; *KŚl*, ii. 19.

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² *ĀpŚl*, v. 10; *ĀpŚr*, xx. 9.1.

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³ *MāŚl*, ii. 6.

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⁴ *ŚBr*, x. 2. 3. 6 ff.

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⁵ *BŚl*, ii. 1 ff. ; *ĀpŚr*, xvi. 17. 15. 16; *ĀpŚl*, viii. 3, 4; *KŚl*, v. 1 ff.

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⁶ Compare what Āpastamba remarks: "A change in the form of the *Agni* would be against the injunction of the *Śruti*." (*ĀpŚl*, viii. 6.) According to *Śatapatha Brāhmaṇa*, those who deprive the *Agni* of its due proportions, will suffer the worse for sacrificing (x. 2. 3. 7).

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Consequently in the science of the altar-construction there arose the necessity of constructing similar figures.

c17.10

Now the shape of the *Mahāvedi* is that of an isosceles trapezium whose altitude is 36 *prakrama* (or *pada*), face 24 *prakrama* (or *pada*) and base 30 *prakrama* (or *pada*). Hence the *Sautrāmaṇikī-vedi* and the *vedi* of *Aśvamedha* will be of the shapes of isosceles trapeziums similar to it but in size one-third and double of it respectively. Thus we have the two following propositions:

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(i) *To construct an isosceles trapezium similar to a given isosceles trapezium but with a third part of its area.*

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Construct an isosceles trapezium, says Āpastamba, in the same way as the given isosceles trapezium (*Mahāvedi*) but “with 1/√3 of a prakrama being substituted for a prakrama therein; or with 8 and 10 times √3 as the transverse sides and 12 times √3 as the east-west line.”¹ This will be the required isosceles trapezium (*Sautrāmaṇikī-vedi*).

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¹ *ĀpŚl*, v. 8.

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For the area of the constructed figure

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$=36/(√3) × ½(24/(√3)+30/(√3)),$

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$=⅓ × 18 × 54=324 square puruṣas ;$

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$or &=12√3 × ½(8√3+10√3), / &=324 square puruṣas.$

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Thus it is equal to one-third of the area of the given isosceles trapezium, which comprises 972 square puruṣas. The same construction is suggested also by Kātyāyana.² (ii) *To construct an isosceles trapezium similar to a given isosceles trapezium but with double its area.*

c17.19

² *KŚl*, ii. 19.

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The method of construction of the new isosceles trapezium will be the same, says Āpastamba, as that of the given isosceles trapezium but here "√2 of a prakrama should be taken in the place of a prakrama therein."¹ That is taught also by Baudhāyana.² Then the area of the new figure will be

c17.21

¹ *ĀpŚl*, vi. 1.

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² *BŚr*, xxvi. 10.

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= 36 √2 × ½ (24 √2 + 30 √2) square puruṣas,
= 1944 square puruṣas.

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Hence it is double the size of the given trapezium (*Mahāvedi*).

c17.25

It is thus clear that the principle underlying the early Hindu method of construction of an isosceles trapezium similar to a given one but of n times the size of it, n being integral or fractional, is practically the same as that of the given one, only the unit of measurement of the latter being replaced by another √n times it.³ This principle they adopted systematically for the construction of similar figures of more complicated shapes, even when the change in the size does not bear a simple relation to the size of the given figure. Thus arose the proposition:

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³ Compare *BŚr*, x. 19.

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*To construct a Fire-altar similar to that of the shape of a falcon, but differing from its primitive area of 7½ square puruṣas by m square puruṣas.*

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Baudhāyana gives the following solution of this proposition:

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"Divide that which is to be the difference from the original (given) size of the altar into 15 equal parts; add to each of the (constituent) portions (*vidha*, that is, units) of the given figure two of these parts. Then construct a figure (in the same way as the given one) with 7½ of these (altered) units.''¹

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¹ *BŚl*, ii. 12.

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The geometrical operations to be followed in this method of construction are shortly these:² At first is drawn a square of an area equal to m square puruṣas. It is then divided into 15 equal parts. This may be done either by dividing one side of the square into 15 equal parts and then drawing lines parallel to the perpendicular sides, *or* by dividing one side into 3 parts and a perpendicular side into 5 parts and then drawing parallels. Two of the rectangular portions are then combined into a square and to that is again added a unit square puruṣa so as to form a third square. A side of this resulting square will be easily found to be √(1+2m/15) puruṣas long. With this length as the unit, construct an altar in the same way as the original falcon-shaped altar. This will be the required figure. For its area will be $7½×(1+(2 m)/15) or (7½+ m)$ square puruṣas and its shape will be clearly similar to that of the given figure.

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² Compare also Thibaut's notes on the above.

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A similar method is taught briefly by Āpastamba:

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''For the eight- and other-fold *Agnis*, that by which it differs from the area of the seven-fold (*Agni*), should be divided seven (and a half); then one part should be added to each (original) puruṣa.''³

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³ *ĀpŚl*, viii. 6.

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''For the purpose of adding parts (in puruṣa) to the (seven-fold) *Agni* together with its wings and tail, take

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for the (new) puruṣa, the (original) puruṣa increased by the seventh-fold producer of the increment. Then construct (in the same way as before).”¹

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¹ *Ibid*, ix. 5. Note the difference between the commentators and Bürk about the correct interpretation of these two rules of Āpastamba. I think on the whole the orthodox interpretation to be more fair and consistent with the intention of the *Śruti*. For any other interpretation would disturb the similarity of the altar at successive constructions which is expressly forbidden.

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The method has been explained more fully by Kātyāyana. His procedure is, however, slightly different from that of other writers. Kātyāyana says:

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“For the purpose of adding a square puruṣa (to the original falcon-shaped *Agni*), construct a square equivalent (in area) to the original *Agni* together with its wings and tail; add to it a square of one puruṣa. Divide the sum (*i.e.* the resulting square) into fifteen parts and combine two of these parts into a square. This will be the (new) unit of square puruṣa (for the construction of the required enlarged figure).”²

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² *KŚl*, v. 4.

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In other words the enlarged square unit is

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$2/15(7½+1) or 1+2/15.$

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Or the new increased unit will be obtained thus, says Kātyāyana:

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“Divide a square of one puruṣa into five parts both ways (by lines drawn cross-wise); combine five of the resulting elementary parts into a square; subtract from the sum one-third of it; add the remainder to one square puruṣa. This is another method (of determining the enlarged square unit).”³

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³ *KŚl*, v. 6.

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That is, the enlarged square unit is

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$1 + ( 5/(5 × 5) - ⅓ · 5/(5 × 5) ) or 1 + 2/15$

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square puruṣas. He gives a still third method:

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"Or divide a square of one puruṣa into seven parts both ways (by lines drawn cross-wise); combine seven of the (resulting elementary) parts into a rectangle, subtract from the sum (a rectangle) 1 1/7 aṅgulis by one puruṣa. Add the remainder to one square puruṣa. This is another method." ¹

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¹ *KŚl*, v. 8.

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Thus it follows that the increased square unit will be

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$1 + ( 7/(7 × 7) - 1 × (1 1/7)/120 ) or 1 + 2/15$

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square puruṣas, as before.

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If the number of square puruṣas (m) to be added to the original area of the Agni (7½ square puruṣas), be an exact multiple or submultiple of it, the geometrical operations are much simplified. For if m = n × 7½, where n may be integral as well as fractional, then the length of the new unit will be easily obtained to be equal to √(1+n) puruṣa. Thus it is the (1+n)th *karaṇī* of a puruṣa, as has been stated by all the Śulbakāras.

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This kind of increment is called *sarvābhyāsa* (or "the increment by the whole") in contradistinction to the other which is called *puruṣābhyāsa* (or "the increment by puruṣa").²

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² *KŚl*, v. 2, 3. These terms are sometimes used also in a different sense. According to it the first term signifies the increment of all the parts of the *Agni* and the second only that of the complete puruṣas in the body, wings and the tail, but not the two aratnis and the prādeśa (*ĀpŚl*, xxi. 7, 10; *KŚl*, v. 4).

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The above principle of the enlargement of a *Vedi* or an *Agni* by increasing the length of the unit of measure but without altering the numbers representing the spatial magnitudes, so as to keep the form similar to the original one, is found as early as the *Śatapatha Brāhmaṇa* (*c.* 2000 B.C.). It says, " as large as the *Agni* (is to be made), so large (should be made) its units of measure; and by so much one measures it in the same way (as before)."¹ To construct a *Vedi* 14 or 14 3/7 times as large as the *Mahāvedi*, and which will be similar to it, this *Brāhmaṇa* says :

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¹ *ŚBr*, x. 2. 1. 3, 11.

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" As much this *vedi* of the seven-fold Fire-altar is, making fourteen times, so much, one measures the *vedi* of 101-fold. Or then he measures (by means of) a cord 36 prakramas long ; folds it into 7 equal parts; of these three parts he adds to the east-west line and throws out 4. Then he measures 30 prakramas; folds them into 7 equal parts; of these three parts he adds to the hind (transverse line) and throws out 4. Then he measures 24 prakramas; folds them into 7 parts; of these 3 parts he adds to the front (transverse line) and throws off 4. This, then, is the alternative measurement of the (enlarged) *Vedi*."²

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² *Ibid*, x. 2. 3. 7-10.

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Here two methods are to be discerned. According to one, it is required to construct a *vedi* 14 times as much while according to the other 14 3/7 times as much.³ The operations implied are doubtless as follows :⁴ The altarbuilder should at first construct a square of sides 36 prakramas each. Fourteen such squares should be combined into a large square. A side of the resulting square, which is clearly equal to 36√14 prakramas, is taken for the east-west line. Or otherwise a smaller square should be divided into the 7 equal rectangular portions by drawing lines parallel to a side of the square; the rectangle comprising three of these strips should be transformed into a square and then combined with the former larger square. A side of the resulting square, which will be easily recognised to be equal to 36√(14 3/7) prakramas, is taken for the east-west line of the enlarged *vedi* to be constructed. By the same kind of operations, the face of the new *vedi* is obtained to be 24√(14 3/7) prakramas and its base 30√(14 3/7). So that the size of the new *vedi* will be 14 3/7 times that of the *Mahāvedi* and its shape similar to that of the latter.

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³ That is, the *Mahāvedi* on which the primitive Fire-altar is raised, is enlarged in proportion to the size of the latter. Now there are two opinions (*vide infra*) as regards the method of construction of the maximum Fire-altar. According to one the latter becomes 14 times the *Sapta-vidha Agni*, while according to the other 14 3/7 times. The *Mahāvedi* is enlarged in the two schools accordingly.

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⁴ Eggeling's explanation (*SBE*, Vol. XLIII, pp. 310-311, footnotes) is obviously erroneous. That will make the *vedi* too great.

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The following method has been taught for the construction of a falcon-shaped Fire-altar. 14 or 14 3/7 times as large as the primitive one and similar to it.

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''Now the construction of (the enlarged forms of) the *Agni*: Twenty-eight (square) puruṣas are in front and twenty-eight (square) puruṣas behind; this is the body (of the *Agni*). Fourteen (square) puruṣas form the southern wing; fourteen the northern wing and fourteen the tail. Fourteen *aratnis* (meaning, a length equal to the side of a square comprising fourteen square aratnis (i.*e*., √14 aratnis) is added to the southern wing; fourteen aratnis to the northern wing; and fourteen vitastis (meaning, a length equal to the side of a square of fourteen square vitastis, *i.e.*, √14 vitastis) to the tail. This is the measurement of (the *Agni* of) 98 square puruṣas with a little excess (due to the increment of the wings and the tail). Or again

c17.67

he measures (by means of) a cord three puruṣas; folds it into seven (equal) parts; of them four parts he adds to the body and three to the wings and tail. Then he measures an aratni; folds it into seven parts; of them three parts he adds to the southern wing, three to the northern wing and throws off one.¹ Then he measures a vitasti; folds it into seven parts; of them three parts he adds to the tail and throws out four. Thus is constructed the 101-fold (*Agni*) and it corresponds with the *vedi* of this." ²

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¹ The printed text has *catura* meaning "four." It is obviously wrong. So I have amended it to "one."

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² *ŚBr*, x. 2. 3. 11-14. Eggeling's rendering is erroneous.

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The geometrical operations intended to be performed in this case are similar to those indicated in the previous case. Here also are to be discerned two methods for the construction of the *Ekaśata-vidha Agni* or the Fire-altar of 101½ square puruṣas. According to the first method each unit of measure is √14 times the unit of measure employed in the construction of the primitive Fire-altar of 7½ square puruṣas, while according to the other, probably the more ancient one, it will be √(14 3/7) times. Though the principle of similarity of shapes is perfectly maintained in either methods, as regards the size, they of course yield only approximate results. For 7½ × 14 = 105 square puruṣas ; 7½ × 14 3/7 = 108 3/14 square puruṣas. The rationale of these results we shall indicate in the next chapter. We should but note here that as the second method ³ yields a result more deviating from the correct and desired one, *viz.*, 101½ square puruṣas, its correctness was challenged even before the time of the *Śatapatha Brāhmaṇa* (*c.* 2000 B.C.).

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³ The second method seems to have been meant in the beginning for an enlargement on the second plan, to be explained shortly. So that it is only by mistake that it was employed for the enlargement on the first plan.

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"As to this they say, 'as seven¹ (square) puruṣas have exceeded, how is it that they do not deviate from the right total (*sampad*, which the altar ought to have).' "

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¹ The printed text has *trayodaśa* or "thirteen." It seems to be a mistake. For it was intended to construct a Fire-altar having an area equal to 101½ square puruṣas. Now the method adopted produces one with an area of 108 1/14 square puruṣas. So 108 1/14 − 101½ or 6 4/7 square puruṣas are in excess. In round numbers this might be stated as equal to 7 square puruṣas. But "thirteen" cannot be justified. So I think that here again the text should be amended to *sapta* or "seven."

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The change of the representative number is strictly forbidden in this work, as that will disturb the principle of similarity of the forms. It observes :

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"Now some intending to construct subsequent (larger) forms (of the *Agni*), increase (the number of) these prakramas and vyāmas, supposing, 'we shall enlarge the womb (*yoni*) accordingly.' One should not do so; for the womb does not increase along with the child that has been born; but indeed only as long as the child is within the womb, so long does the womb enlarge; and this much again is the growth of the child here. Those who do it in that way, certainly do they deprive this Father Prajāpati of his perfection (*sampad*; that is, due proportions)."²

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² *ŚBr*, x. 2.3. 6-7. In this metaphor, the *Agni*, as usual, is called the Father Prajāpati; his child is the unit of measure employed; and the womb is his form and hence the numbers representing the spatial relations of that form (cf. *ŚBr*, viii. 6.3.12; x. 2.3.3). This passage has been referred to in the aphorism vi. 4 of the *Kātyāyana Śulba*.

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In the foregoing methods of enlargement of the original size of an altar, it will be observed, all the constituent parts of it receive increments in equal proportions. But sometimes an altar, particularly the falcon-shaped one, is enlarged on an entirely different plan. According to it, certain parts of the altar, such as the complete puruṣas in the body, wings and tail of the falcon, are enlarged proportionately, while the rest, the two aratnis and the prādeśa by which the wings and tail were lengthened fundamentally, are left unaffected. This plan of enlargement of the falcon-shaped altar has been noticed by all the three principal Śulbakāras. It is particularly advocated by the tradition of one school as necessary for the altar of the Horse Sacrifice. Other schools, however, follow the general plan of proportionate increment of all parts in that case too.¹

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¹ *BŚl*, ii. 8 ff.; iii. 321-3; *ĀpŚl*, xxi. 6-10; *KŚl*, v. 7.

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We have so far considered the construction of figures, particularly isosceles trapeziums, similar to a given one and differing from it in area by a specified amount. It is sometimes also necessary to construct a figure, similar to another and having a given side. For instance, the *Ekādaśinī-vedi* is stated to be similar to the *Mahāvedi* and is indeed an enlarged form of it.² It is so called because it must have, according to the injunction of the scriptures, eleven (*ekādaśa*) sacrificial posts (*yūpa*) in front. It is further prescribed that the two posts on either sides of the middle one must be at a distance of one akṣa (=104 aṅgulis) and four aṅgulis from it and the rest are an akṣa distant from each other. Each post has a diameter of one pada. Hence the east side of the *Ekādaśinī-vedi* is 10 akṣas 11 padas 8 aṅgulis long. Thus it is required to construct an isosceles triangle similar to the *Mahāvedi* and having its face equal to this length. Again, according to Baudhāyana, the shape of the *Aśvamedha-vedi* is similar to the *Mahāvedi* and has 21 yūpas on the east side, that is, has a face of 20 akṣas 21 padas 8 aṅgulis long.

c17.80

² *BŚl*, i. 106 ff.

c17.81

The method of construction to be adopted in these cases is indicated by Baudhāyana thus:

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" 'For that (*Ekādaśinī-vedi*) take the twenty-fourth part of 10 akṣas 11 padas 8 aṅgulis; this will be the prakrama and with this (altered unit), the *vedi* has to be constructed (in the same way as the *Mahāvedi*).' "¹

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¹ *BŚl*, i. 107; *vide* also *BŚr*, xxvi. 23.

c17.84

" 'For the *Aśvamedha-vedi* take the twenty-fourth part of 20 akṣas 21 padas 8 aṅgulis; this will be the prakrama and with this (modified unit), the *vedi* has to be constructed (in the same way as the *Mahāvedi*).' "²

c17.85

² *BŚl*, i. 108; *vide* also *BŚr*, xxvi. 10.

c17.86

The face, base and altitude of the *Mahāvedi* are given respectively to be 24, 30 and 36 prakramas in length. If *a*, *b*, *c* be the corresponding quantities of a similar figure, we shall have

c17.87

$a/24 = b/30 = c/36 ;$

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$b = 30 ( a/24 ), c = 36 ( a/24 ).$

c17.89

This is equivalent to the change of the ordinary unit to a/24 times it, where *a* is the given length of the face of the figure to be constructed.

c17.90

A similar method will have to be followed in constructing an isosceles trapezium similar to a given one and having a given altitude.³

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³ *BŚl*, i. 109-10.

c17.92

These methods are also taught by Kātyāyana.⁴

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⁴ *KŚl*, vii. 1-3; compare *KŚr*, viii. 8.22.

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It may be noted that Āpastamba's method of construction of several altars by employing the rational rectangle (15, 36, 39) by successively varying the units, is obviously equivalent to the construction of a system of rectangles similar to it. Indeed the science of the altar-construction requires the drawing of (a system of) similar rectangles, triangles, rhomboids, circles and other figures. For the enlargement of altars of those shapes involves it.

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