Pramāṇa · Search · About · API

The Science of the Shulba: A Study in Early Hindu Geometry c8: Chapter III. Growth and Development of the Śulba

The Science of the Śulba: A Study in Early Hindu Geometry c8

Language
English

Chapter III. Growth and Development of the Śulba

c8.2

It has already been observed that the science of geometry originated in India in connexion with the construction of the altars for the Vedic sacrifices. We now propose to treat this point more fully. We shall further trace, as far as possible, the growth and development of the Hindu Geometry from its earliest state down to the one in which we find it now in the *Śulba*. Much has been done before in this respect, by Bürk in his masterly introduction to his edition of the *Āpastamba Śulba*.¹ Much more still remains to be done.

c8.3

¹ *Āpastamba-Śulba-sūtra*, edited and translated with an introduction by Albert Bürk, *ZDMG*, LV and LVI.

c8.4

The Vedic sacrifices are mainly of two classes: *Nitya* (or "indispensable," "obligatory") and *Kāmya* ("optional," "intentional"). The performance of the sacrifices of the former class is obligatory upon every Vedic Hindu. It will be a sin for him if he does not do them. But it is not so with the sacrifices of the second kind. For they are to be performed each with the sole motive of achieving a special object. Those who do not aim at the attainment of any such object need not perform any of them.

c8.5

According to the strict injunctions of the Hindu *Śāstra* (or "Holy Scriptures") each sacrifice must be made in an altar of prescribed shape and size. It is stated that even a slight irregularity and variation in the form and size of the altar will nullify the object of the whole ritual and may even lead to an adverse effect. So the greatest care has to be taken to have the right shape and size of the altar.

c8.6

There are multitudes of the altars Of the *Nitya Agni* (or “the altars for the obligatory sacrifices”), the three primary ones are the *Gārhapatya*, *Āhavanīya* and *Dakṣiṇa*. Every Vedic Hindu has to offer sacrifices in them daily. Other obligatory sacrifices are seasonal and are performed at special periods. According to the nature of the oblations they are broadly subdivided into three groups: (1) *Iṣṭi Yajña* (or “sacrifice with oblations of butter, fruit, etc.”) such as *Darśa* and *Paurṇamāsa* sacrifices which are performed at every new-moon and full-moon respectively; (2) *Paśu Yajña* (or “Animal sacrifice”) such as *Nirūḍhapaśubandha* which must be performed once every year, more particularly, on a new-moon or full-moon day in the rainy season; or according to a different school twice every year at the time of the winter and summer solstices; (3) *Soma Yajña* (“Soma sacrifice”). This last sacrifice is very big and expensive and so cannot be performed often. But it must be performed in a family of Vedic Hindus at least once in three generations.

c8.7

Now we find it from the *Śulba*, that the altar of the *Gārhapatya* must be of the form of a square, according to one school, and a circle, according to a different school. The altar for the *Āhavanīya* should be always square and that of the *Dakṣiṇa* semi-circular. The area of each, however, must be the same and equal to one square *vyāma* (1 *vyāma* = 96 *aṅgulis*). So the construction of these three altars, it will easily be recognised, pre-supposes the knowledge of the following geometrical operations:—

c8.8

(*i*) To construct a square on a given straight line.
(*ii*) To circle a square and *vice versâ*.
(*iii*) To double a circle.

c8.9

The last problem is the same as to evaluate the surd √2. Or it may be considered as a case of doubling a square and then circling it. So in that case, we get at the proposition :—

c8.10

(*iv*) The area of the square on the diagonal of a square is double the area of that square.

c8.11

The *Saumikī-vedi* or *Mahā-vedi* is described as an isosceles trapezium whose face is 24 padas (or prakramas), base 30 padas and altitude 36 padas. The *Sautrāmaṇī-vedi* is stated to be an isosceles trapezium similar to and with an area one-third that of the *Mahā-vedi*, and the *Paitṛkī-vedi* is one-ninth of the latter. The *Prāgvaṁśa* is a rectangle. These and other similar altars lead to the operations :

c8.12

(*v*) To construct a rectangle having given sides.

c8.13

(*vi*) To construct an isosceles trapezium whose face, base and altitude are given.

c8.14

(*vii*) To find the area of an isosceles trapezium.

c8.15

(*viii*) To construct an isosceles trapezium whose area will be equal to a simple multiple or sub-multiple of, and which will be similar to, another isosceles trapezium.

c8.16

Geometrical operations of more complex nature are required for the accurate construction of the *Kāmya Agni* (or "the fire altars for the sacrifices to achieve special objects"). Amongst them the most ancient and primitive form is the *Śyena-cit* (or "the altar of the form of the falcon"). The *ātman* (or "body," "trunk") of this *citi* (or "altar") consists of four squares of one square puruṣa each. Each of its wings is a rectangle of one puruṣa by one puruṣa and one *aratni* (= ⅕ of a puruṣa). Its tail is a rectangle of one puruṣa by one puruṣa and a *prādeśa* (= 1/10 of a puruṣa). This altar is more usually called *Saptavidha-sāratni-prādeśa-caturasra-śyena-cit* because its area is 7½ square puruṣas, its shape resembles that of a falcon (*śyena*) and because the bricks used in its construction are square.

c8.17

Fig. 1

c8.18

The Fire-altars for other optional sacrifices are prescribed to be of different shapes. Thus we find altars also of the shape of (2) *vakra-pakṣa vyasta-puccha śyena* (or "the falcon with bent wings and outspread tail"), (3) *kaṅka* ("heron"), (4) *alaja* (a kind of bird), (5) *praüga* ("triangle" usually an isosceles triangle), (6) *ubhayataḥ praüga* ("triangles on both sides," that is, a rhombus), (7) *ratha-cakra* ("chariot wheel"), (8) *droṇa* ("trough"), (9) *samuhya* ("combined"), (10) *paricāyya* ("circular"), (11) *śmaśāna* ("cemetery"), (12) *kūrma* ("tortoise"), etc. Each of these altars shall have the same area as that of the standard form of the *Śyena-cit*, that is, 7½ square puruṣas.

c8.19

For the accurate construction of these altars, previous knowledge of the following principal geometrical propositions will be essential besides those noted above and a few others:

c8.20

(*ix*) To construct a square equal to a simple multiple (or sub-multiple) of another square.

c8.21

(*x*) To construct a square equal to the sum or difference of two unequal squares.

c8.22

(*xi*) To transform a rectangle into a square and *vice versâ*.

c8.23

(*xii*) To construct a triangle or a rhombus equal to a square.

c8.24

A knowledge of the following important theorem is most indispensable for the geometry of the altar-construction.

c8.25

(*xiii*) The area of the square described on the diagonal of a rectangle is equal to the sum of the areas of the squares described on its two sides.

c8.26

Every one of the altars is constructed with five layers of bricks, which together come usually up to the height of the knee (=32 angulis). In some cases the use of more layers of bricks is permitted with the proportional increase in the height of the altar. Now every layer, it is prescribed, contains a definite number of the bricks of specified shapes. For instance, each layer of the square *Gārhapatya* altar is constructed with 21 bricks of square or rectangular shape and each layer of the *Caturasra Śyena-cit* consists of 200 square bricks. Again in the case of the altars of other optional sacrifices, shape of the bricks are varied, but the number of them to be employed in the construction remains the same, *i.e.*, 200. Sometimes the one and the same altar is constructed in different patterns. All these have given rise to (1) the problems of the division of figures into a particular number of parts of specified shapes and also to (2) certain interesting problems of indeterminate character.

c8.27

It has been stated above that a *Kāmya Agni* has an area of 7½ square puruṣas. That is the case only at the first construction of the altar. At its second construction, the area has to be increased by one square puruṣa; at the third construction by two square puruṣas; and so on until to the size of 101½ square puruṣas. But the strict injunction of the scriptures is that the shape of the altar on the whole, that is, the relative proportion between its different constituent parts at any construction, must not be altered. Thus arise the problems of constructing similar figures.

c8.28

Such is, in brief, a résumé of the more salient points in the elaborate and minute in details specifications of the shape and size of the principal sacrificial altars and of the geometrical knowledge presupposed in their construction, as we find them in the extant *Sulba*. What should be particularly emphasized now is the fact that those specifications are not due to the authors of the *Sulba* themselves. They do not even pretend to make any such claim. On the other hand, they have often and then expressly admitted to have taken them from earlier works. We, in fact, find that numerous passages of *Baudhāyana* and *Āpastamba Sulba* dealing with the spatial magnitudes of sacrificial altars as well as with the methods of their construction, end with the remark *iti vijñāyate* [or "it is known," "it is recognised or prescribed (by authorities)"].¹ Sometimes *iti abhyupadiśanti* ("thus they teach")² or *iti uktam* ("it has been said"),³ is used in the same sense. It has been rightly pointed out before by Garbe⁴ that all those passages of *Āpastamba* are literal quotations from the *Taittirīya Brāhmaṇa* or from the Brāhmaṇa-like portions of the *Taittirīya Saṃhitā* or *Āraṇyaka*. That is exactly true also of the similar passages of *Baudhāyana*.⁵ This writer is occasionally more explicit about his sources. In connexion with certain difference of opinions amongst the altar-builders about the proper size and shape of a particular altar,¹ Baudhāyana is found appealing to the authorities of the *Brāhmaṇa*, by name, for the purpose of arriving at a satisfactory settlement. “This is not right,” observes he, “as it will bring this opinion in contradiction with the ancient precepts. Regarding this point the *Brāhmaṇa* of some is as follows..., of others is...And the following is our *Brāhmaṇa*...”² By “our Brāhmaṇa” is meant the *Taittirīya Saṃhitā*, where indeed the quoted passage occurs.³ On a different occasion, in connexion with certain method of constructing a particular altar, Baudhāyana remarks: “There is also a *Brāhmaṇa* on this point.”⁴ Here again the reference is to the *Taittirīya Saṃhitā*.⁵ There are also other mentions of *Brāhmaṇa* in general by Baudhāyana.⁶ He has once quoted the *Maitrāyaṇīya Brāhmaṇa* by name.⁷ Kātyāyana is found to have appealed similarly to the authority of the “Śruti,” on two occasions.⁸ Āpastamba has sometimes observed that certain constructions are not sanctioned by the *Śruti*.⁹ Thereby he clearly implies that other matters about the spatial magnitudes of the sacrificial altars and the methods of constructing them, that have been recorded by him, are in full accordance with the teachings of the *Śruti* but are not his devices. This he has admitted also otherwise, as has been just pointed out. It will be further shown presently that we can, indeed, trace most of the matters contained in the *Śulba* to the earlier *Brāhmaṇa* and *Saṁhitā*.

c8.29

¹ *BŚl*, i. 65, 71, 76, etc.; *ĀpŚl*, iv. 1, 3, 5; v. 1, 8, 10, etc.

c8.30

² *BŚl*, i. 85.

c8.31

³ *ĀpŚl*, ix. 2.

c8.32

⁴ *Vide* the Preface (p. xviii) to his edition of the *Śrauta Sūtra of Āpastamba*, Vol. III, Calcutta, 1902. Garbe has pointed out in a most scholarly manner the relations of this work with others such as *Saṃhitā*, *Brāhmaṇa* and *Śrauta-sūtra*.

c8.33

⁵ Compare for instance the passages with such remarks in *BŚr*, xxiv. 2 with *TS*, i. 2. 2. 3; *BŚr*, xxiv. 29=*TS*, i. 7. 3. 1; *BŚr*, xxvi. 21=*TS*, vii. 4. 2. 3, *PañBr*, xxiii. 19. 8; etc.

c8.34

¹ The controversy is as regards the construction of the falcon-shaped altar of areas 1½ to 6½ square puruṣas with or without wings and tail. This will be dealt with more fully later on. Compare *ĀpŚl*, viii. 3-5.

c8.35

² *BŚl*, ii. 15-9.

c8.36

³ *TS*, v. 2. 5. 1.

c8.37

⁴ *BŚl*, ii. 35.

c8.38

⁵ *TS*, v. 6. 6. 3.

c8.39

⁶ *BŚl*, iii. 6. Here the reference is to *TS*, v. 3. 1. 5 and v. 5. 3. 2. Compare also *BŚl*, iii. 1 with *TS*, v. 4. 11. 1.

c8.40

⁷ *BŚl*, iii. 10.

c8.41

⁸ *KŚl*, v. 7; vi. 4.

c8.42

⁹ *ĀpŚl*, viii. 5, 6.

c8.43

The reference to the sacrificial altars and their construction is found as early as the *Ṛg-veda Saṁhitā* (before 3000 B.C.).¹ There is mention in that work of the "three places" of the Agni,² which doubtless imply the *Gārhapatya*,³ *Āhavanīya* and *Dakṣiṇāgni*. Though we do not find there any specific mention about the relative sizes and shapes of these altars we have nothing to doubt that they were, in any way, different from what we meet with in posterior *Brāhmaṇa*.⁴ Hence it seems that the problem of the squaring of the circle and the theorem of the square of the hypotenuse (at least in its simplest form) are as old in India as the time of the *Ṛg-veda*. They might be older still. For it has been shown by Oldenberg that those three fires are earlier than the *Ṛg-veda*.⁵

c8.44

¹ There are innumerable references in the *Ṛg-veda* to the sacrifice-altars and their constructions. For the mention of the *vedi*, see for instance *ṚV*, i. 164. 35; i. 170. 4; v. 31. 12; vii. 35. 7, etc.; and for its construction compare the passages: "O Lovely (Agni)! They construct the *vedi* for you and offer oblations there" (*ṚV*, viii. 19. 18); "measured out the vedi" (*ṚV*, x. 61. 2), etc.

c8.45

² "यज्ञस्य केतुं प्रथमं पुरोहितमग्निं विषधस्थे समिधीरे," *ṚV*, v. 11. 2.

c8.46

³ The mention of the *Gārhapatya* Fire by name occurs, for instance, in *ṚV*, i. 15. 12; vi. 15. 19 and x. 85. 27.

c8.47

⁴ The first express description of the *Gārhapatya* as a circle of one square *vyāma* (= *puruṣa*) and of the *Āhavanīya* being a square of the same size appears in the *Śatapatha Brāhmaṇa* (vii. 1. 1. 37; vii. 2. 2. 1 ff. Cf. *SBE*, Vol. XL, iii, p. 307, fn. 2). In the *Taittirīya Saṁhitā* (v. 2.5.1), the *Āhavanīya* is stated to be of one (square) puruṣa.

c8.48

⁵ Oldenberg, *Religion des Veda*, p. 348, n. 2; *SBE*, Vol. XXX, p. ix.

c8.49

In the *Ṛg-veda*, it should be made clear, there is no particular rule for the construction of the altars. We cannot indeed reasonably expect to find such a rule there, when we remember the nature of that work. We learn from it, however, that there were then learned experts to do that work. This is very important inasmuch as it implies the existence of the science or the art of the altar-construction. An expert in that science is called *Agnicit* (or "the constructor of the Agni or Fire-Altar"). This term appears in the *Taittirīya Saṃhitā*, *Maitrāyaṇīya Saṃhitā*, *Śatapatha Brāhmaṇa* and other early works In these works we also find rules for his conduct.¹

c8.50

¹ According to the *Taittirīya Saṃhitā*, an Agnicit should live upon what are obtained freely from Nature, such as fruits, etc., but not by sowing (v. 2. 5. 5-6). He is particularly forbidden to eat the flesh of birds. "The fire is a bird; if the piler of the fire were to eat of a bird, he would be eating the fire, he would go to ruin" (*TS*, v. 7. 6. 1). The same prohibition is found also in the *Maitrāyaṇīya Saṃhitā* (iii. 4. 8) and *Śatapatha Brāhmaṇa*. In the latter work we find also a contrary opinion. "Here, now, they say, 'He who has built an altar must not eat of any bird, for he who builds a fire-altar becomes of a bird's form; he would be apt to incur sickness: the Agnicit therefore must not eat of any bird.' Nevertheless, one who knows this may safely eat thereof; for he who builds an altar becomes of Agni's form, and, indeed, all food here belongs to Agni: whosoever knows this will know that all food belongs to him." (*ŚBr*, x. 1. 4. 13. Compare also *TS*, v. 6. 8. 3 f.; 7. 6. 2 ff.; *ĀpŚr*, xvii. 24.

c8.51

We next turn to the *Yajur-veda* which is very rightly described as the encyclopædia proper of the Vedic sacrifices. There we find a very elaborate, and in detail tedious, rite of the Agni-cayana (or "the construction of the Fire-altar") and its highly speculative philosophy.² The same mystic import is found, it is noteworthy, in the *Saṃhitā* of the different schools of this Veda such as the *Taittirīya*, *Maitrāyaṇīya*, *Kāṭhaka*, *Kapiṣṭhala* and *Vājasaneya*. This shows that the Agni-cayana and its philosophy had taken definite shapes in an earlier period. Eggeling³ has, indeed, traced the origin of the philosophy to the *Ṛg-veda*. In the *Brāhmaṇa*, the science of the construction of the Fire-altar is found in an enormously developed form. Thus five sections (*kaṇḍikā* vi-x) out of a total of fourteen, or rather more than one-third of the whole of the *Śatapatha Brāhmaṇa* is devoted to the treatment of this science. We cannot definitely ascertain how much of this development is due to the *Brāhmaṇa* and how much of it is still older. For, according to the introductory chapters of the *Hiraṇyakeśi* and *Āpastamba Śrauta-sūtra*, it is one of the primary functions of the *Brāhmaṇa* to describe, amongst other things, the *Karma-vidhi* (or "Rules for the performance of the sacrificial rites") and the *Purā-kalpa* (or "the performance of the sacrifices in former times"). So that the *Brāhmaṇa* simply keeps up a record of the ancient traditions. Indeed in those works, all rites are traced to the gods as their originators or even to the Supreme Creator of the universe. Still the general truth is that every science has its growth and development and the science of the altar-construction cannot be an exception to it. So it will be quite natural to believe—and we have corroborative evidence of it—that the science truly attained a new stage in the time of the *Brāhmaṇa* (*c.* 2000 B.C.). The existence of different masters of this science with independent views is found even as early as the time of the *Taittirīya Saṃhitā* (*c.* 3000 B.C.).¹ More conclusive evidence of it is furnished by the *Śrauta-sūtras* which are a sort of compendiums of earlier theories. In the *Śatapatha Brāhmaṇa*, there are clear attempts to refute some of the earlier theories of the science of the altar-construction (*vide infra*).

c8.52

² Compare *The Veda of the Black Yajus School* translated into English by A. B. Keith, Cambridge, Mass., 1914, Introduction, p. cxxv ff.

c8.53

³ *SBE*, Vol. XLIII, Introduction, pp. xiv ff.

c8.54

¹ References to the earlier authorities are found, for instance, in *TS*, v. 2. 8. 1-2; 3. 8. 1; 5. 2. 1 ff.; etc.

c8.55

One thing should be made here perfectly clear: the treatment in the earlier literatures, the *Saṁhitā* as well as the *Brāhmaṇa*, of the measurement of the various *Vedi* and *Agni*, appertains chiefly to the ritualistic aspects of the problems. Reference to the secular or geometrical and other truly scientific aspects are only incidental for them and hence are found on rare occasions. Fuller details of the geometry of the measurements of the altars are particularly described in the *Śulba* parts of the *Śrauta-sūtra*. But traces of that, it will be shown conclusively in the course of this work, are clearly noticeable also in the *Brāhmaṇa*. And it will not be improper, I think, to presume that the geometrical methods for the solution of the problems of the measurements of the altars were known in still earlier periods. For the rituals of measurements will be altogether baseless unless accompanied by a knowledge of the underlying geometry.

c8.56

In the *Taittirīya Saṁhitā*, we find the following scanty reference to the scientific operations for the construction of the *Dārśapaurṇamāsikī-vedi*:

c8.57

"He performs the second drawing of a boundary himself. The earth is of the size of the altar; verily having excluded his enemy from so much of it, he performs the second drawing of a boundary himself. Cruelly he acts in making an altar."'¹

c8.58

¹ *TS*, ii. 6. 4. 2-3 (Keith's translation). Compare *TBr*, iii. 2. 9. 10; *BŚr*, xxiv. 24.

c8.59

But such meagre descriptions of course do not help us in any way to conjecture the geometrical devices adopted for the construction.

c8.60

We shall now proceed to show, as briefly as possible, that some of the specifications about the shape and size of the various *Vedi* and *Agni* and about their relative positions, which we find in the *Śulba* can be clearly traced to earlier *Saṁhitā* and *Brāhmaṇa*. This will doubtless corroborate the traditional origin of the science of Hindu Geometry in a very remote age to be true. We have already shown the ancient origin of the three fundamental altars, the *Gārhapatya*, *Āhavanīya* and *Dakṣiṇāgni*. Their relative positions are described in the *Śatapatha Brāhmaṇa*¹ and *Śrauta-sūtra* ² to be identical. According to all the *Saṁhitā* ³ and *Brāhmaṇa*, ⁴ as in the *Śulba*, the *Gārhapatya citi* must be constructed with the same number of bricks, namely 21, arranged in an identical manner. It is further stated in the *Taittirīya Saṁhitā*:

c8.61

¹ *ŚBr*, i. 7. 3. 23-5.

c8.62

² *Cf.* *BŚr*; *BŚl*, i. 64-69; *ĀpŚr*, v. 4. 3-5; *ĀpŚl*, iv. 1-4; *KŚr*, iv. 8. 19; *KŚl*, i. 26,2 8.

c8.63

³ *Cf. TS*, v. 2. 3. 4ff; *MS*, iii. 2. 3; *KṭS*, xx. 1; *KapS*, xxxii. 3.

c8.64

⁴ *ŚBr*, vii. 1. 1. 18, 33-4.

c8.65

"He who constructs (the *Gārhapatya citi*) for the first time should construct in five layers ... He who constructs for a third time should construct in one layer...." ⁵

c8.66

⁵ *TS*, v. 2. 3. 6f.

c8.67

The spatial magnitudes of the *Saumikī-vedi* (or "the altar of the Soma-sacrifice"), also called the *Mahā-vedi* ("the Great Altar") which has been already described to be of the form of an isosceles trapezium whose face is 24 *prakramas* (or *padas*) long, base is 30 and altitude 36 *prakramas*, are given in the *Saṁhitā*,⁶ and *Śatapatha Brāhmaṇa*.⁷ But the earliest description of a method of its measurement, or a method for the construction of an isosceles trapezium having given face, base and altitude is found in the latter work. It says:

c8.68

⁶ *TS*, vi. 2. 4. 5; *MS*, iii. 8. 4; *KṭS*, xxv. 3; *KapS*, xxxviii. 6.

c8.69

⁷ *ŚBr*, iii. 5. 1. 1ff; x. 2. 3. 4. Compare also *BŚr*, vi. 22; *ĀpŚr*, xi. 4. 11-6; *KŚr*, viii. 3, 6-12; *MāŚr*, ii. 2. 1-12.

c8.70

"From that (the largest post on the east side) one proceeds three vikramas to the east and there fixes a pole; this

c8.71

is the middle-hind pole (*antaḥpātaḥ*). From the middle-hind pole, he goes 15 prakramas towards the south and fixes a pole there; this is the south-west corner (of the *Mahā-vedi*). From the middle-hind pole, he proceeds 15 prakramas towards the north and fixes a pole there; it is the north-west corner. From the middle-hind pole he goes 36 prakramas towards the east and fixes a pole; this is the middle-front pole. From the middle front pole, he strides 12 prakramas towards the south and fixes a pole; this is the south-east corner. From the middle-front pole, he goes 12 prakramas towards the north and fixes a pole there; this is the north-east corner. Such is the measurement of the (*Mahā-*)vedi.''¹

c8.72

¹ *ŚBr*, iii. 5. 1. 1-6.

c8.73

Here or anywhere else in this *Brāhmaṇa*, we are not taught how to draw the east-west line and how to draw a line at right angles to it (*i.e.*, the north-south line) through a given point on it. That there were some methods for those constructions is beyond question. We have only to conjecture what were these methods. Now almost identical descriptions about the measurement of the *Mahā-vedi* reappear in the *Śrauta-sūtra* of Baudhāyana² and Āpastamba.³ It has been further taught by the former writer that all the measurements are to be made by means of a cord on the principle of a rational rectangle (*vide infra*). The *Śatapatha Brāhmaṇa* is known to have measured the *vedi* with a cord.⁴ Had it also recourse to the same method? At any rate, it is not improbable. For we have clear evidence to prove that the theorem of the square of the diagonal, or the so-called Pythagorean Theorem was known then and used to be employed in that as well as in other connexions.¹

c8.74

² *BŚr*, vi. 22. In this work the middle-hind pole is called the *śālā-mukhīya-śaṅku* and the middle-front pole the *yūpāvaṭīya-śaṅku*.

c8.75

³ *ĀpŚr*, xi. 4. 12-3. Compare also *KŚr*, viii. 3. 6-12, and the method of *KtŚr* quoted in Yājñikadeva's commentary on the 11th Sūtra.

c8.76

⁴ Cf. *ŚBr*, x. 2. 3. 8 ff.

c8.77

¹ *Vide infra.*

c8.78

The construction of a square having a given side is described thus (omitting the descriptions of the ceremonies and speculative explanations) :

c8.79

" He then takes up the wooden pin (*śamyā*) and wooden sword (*sphya*). Then from the pole which lies in the north-east (corner of the *Mahā-vedi*) strides three prakramas backwards and then marks out the pit (*cātvāla*). That is the measure of that pit; it has no other measure. Wherever he himself thinks it (proper) in his mind, in front of the *utkara* ('the heap of rubbish'), there he marks out the pit. He (draws first) the (western) extremity of the altar. He lays out the wooden pin northwards and marks out (a line)...Then on the front : he lays down the wooden pin northwards and marks out (a line)...Then on the southern extremity of the altar : he lays down the wooden pin eastwards and marks out (a line)...Then on the north : he lays down the wooden pin eastwards and marks out (a line)...."²

c8.80

² *ŚBr*, iii, 5. 1. 26-30. Compare also *TS*, vi. 2. 7. 1-2.

c8.81

This is, in fact, the square pit, with the earth from which the *Uttara-vedi* is constructed. Hence both have the same cubical content. This measurement of the pit reappears in the *Sulba*. It is, perhaps, particularly noteworthy, that in the above we find an instance of the use, in former times, of a ruler (in the body of the straight wooden pin, called the *śamyā*) to draw a straight line from a given point in a specified direction. It is not said how those directions, or rather the cardinal directions passing through a point were used to be determined. Was a pair of compasses also in use then?

c8.82

Though lavish description of the rites and ceremonies in connexion with the construction of the various other altars, such as the *Darśapaurṇamāsa-vedi*, *Uttara-vedi*, *Aśvamedha-vedi*, *Āgnīdhrīya*, *Hotrīya*, *Mārjālīya*, *Sadas*, *Uparavas*, etc., are commonly found in the *Taittirīya* and other *Saṁhitā*, any clear mention of their spatial magnitudes are very rare therein. In that respect, we obtain much better information from the *Brāhmaṇa*.

c8.83

It has been stated before that the standard form of an optional Fire-altar is that of a certain bird. This bird is called *Śyena* ("falcon") in the *Taittirīya Saṁhitā*¹ and *Suparṇa Garutman* ("well-winged eagle") in the *Vājasaneya Saṁhitā*² and *Śatapatha Brāhmaṇa*³ which is sometimes abridged into *Suparṇa* in the latter.⁴ The first name is found more commonly in other *Saṁhitā* and *Śrauta-sūtra* whereas the other names are rarely met with elsewhere.⁵ A clear reference to this form is found in the *Ṛgveda* where Agni is frequently called a bird.⁶ The spatial magnitudes of the falcon-shaped Fire-altar have been defined in almost all the earlier works from the *Taittirīya Saṁhitā* onwards, and they are exactly the same as those that are found in the *Śulba*. Though it is stated by the authorities that it should be measured preferably with a bamboo-rod the details of the method of measurement are scanty in earlier ones. The *Taittirīya Saṃhitā* says :

c8.84

¹ *TS*, v. 4. 11. 1.

c8.85

² *VS*, xii. 4.

c8.86

³ *ŚBr*, x. 2. 2. 4.

c8.87

⁴ *ŚBr*, vi. 7. 2. 6. 8.

c8.88

⁵ The name *Suparṇa-citi* of the *Śatapatha Brāhmaṇa* (vi. 7. 2. 8) reappears in the *Mānava Śulba* (vii), but its form differs from the ancient one by the addition of a head.

c8.89

⁶ *RV*, i. 164. 52; x. 14. 5; compare also i. 58. 5; 141. 7; ii. 2.4; vi. 3. 7; 4. 7; x. 8. 3.

c8.90

"With man's measure he metes out; man is commensurate with the sacrifice; verily he metes him with a member of the sacrifice; so great is he as a man with arms extended; so much strength is there in man; verily with strength he metes him. Winged is he, for wingless he could not fly; these wings are longer by an ell (*aratni*); therefore birds have strength by their wings. The wings and the tail are a fathom (*vyāma*) in breadth; so much is the strength in man, he is commensurate in strength. He metes with a bamboo;..."¹

c8.91

¹ *TS*, v. 2. 5. 1ff. (translation by Keith).

c8.92

The *Maitrāyaṇīya Saṃhitā* says :

c8.93

"As much as a man with arms extended, with so much a bamboo-rod, (the Fire-altar) is meted out; so much strength is there in the man; verily with strength it is meted out;...He metes out the Fire-altar; seven (square) puruṣas he metes out; for by seven puruṣas he knows the universe and by seven puruṣas of the self he eats food. A measure of *aratni* is added to the two wings; the birds have strength by their wings."²

c8.94

² *MaiS*, iii. 2. 4.

c8.95

More particulars are supplied by the *Śatapatha Brāhmaṇa* :

c8.96

"Verily He comprises seven puruṣas. Seven puruṣas certainly are in this Person (Agni); since four (puruṣas) (as) the body and three the wings and tail; for the body of that Person is certainly (composed of) four (puruṣas) and the wings and tail of three. He metes it out with (the measure of) a man (puruṣa) with arms extended. Verily the sacrifice is a puruṣa and hence by it, all these are measured ; and that is its best

c8.97

measure inasmuch as with arms extended he (man) has his maximum measure: he then secures for him that and by that he measures it... Then he adds two aratnis to the two wings; by that he gives strength to the wings. Verily, the two wings are two arms (of the bird) and by arms food is eaten: simply for the sake of food he makes that space; inasmuch as the food is taken from the distance of an aratni the two aratnis he adds to the two wings. Then to the tail he adds a vitasti. He thus gives strength to the support; verily, the tail is the support. The hand (consists of) vitastis, and by means of the hand the food is eaten; simply for the sake of food he makes that space. Inasmuch as he adds one vitasti to the tail, he settles for it the food; because he adds less here (in the tail), he thereby secures it in the food. Thus, this much is it (the. body) measured, and this much is it (wings and tail); certainly it (the bird or altar) is measured this much in order to secure for it that (its natural measure)."¹

c8.98

¹ *ŚBr*, x. 2. 2. 5-8. *Cf.* vi. 1. 1. 6. ; x. 2. 3. 4.

c8.99

Full details of the methods of measurement of the falcon-shaped Fire-altar are not found until the *Śrautasūtra*. The method of the measurement by means of a bamboo-rod has been described in the *Āpastamba Śulba*,² and that by means of a cord is hinted in the *Baudhāyana Śrauta*³ described in full in the *Kātyāyana Śrauta*. The latter says:

c8.100

² *ĀpŚl*, viii. 7-ix. 3. Compare also *ĀpŚr*, xvi. 17.8.

c8.101

³ *BŚr*, x. 19; xix. 1.

c8.102

"Measure a cord two puruṣas long. Make ties at its both ends. Make marks at the middle; on either sides of it, at the halves of the puruṣas; at distance of the one-fifth of a puruṣa from the middle (mark); and also

c8.103

at their halves (that is, at one-tenth puruṣa from the middle mark). Stretch the cord along the *pṛṣṭhyā* (the east-west line) and fix poles at the two ties, the middle mark and the marks at semi-puruṣas. Unfasten the two ties, fasten them to the semi-puruṣa-poles and then stretch the cord towards the south by holding it by the middle mark. Make a point at the place reached by that. Unfasten the two ties; fasten one at the middle pole, then stretch the cord towards the south over the point and fix a pole at the place reached by the middle mark. Then fasten one tie at this pole and another at the eastern pole; stretch the cord towards the south and fix a pole at the place of the middle mark; then another also at the semi-puruṣa-mark. Unfasten the tie from the eastern pole and then fasten it to the western pole; stretch the cord southwards and fix a pole at the place reached by the middle mark and also two about it at the semi-puruṣa marks. Proceed in the same way on the northern side. Again on the southern side, stretching the cord in the way indicated before, fix a pole at the distance of the fifth-puruṣa-mark. Having fastened a tie at that, and also at the eastern semi-puruṣa pole, stretch the cord properly, and fix a pole at a distance of the semi-puruṣa-mark (from the one) and the fifth-puruṣa-mark (from the other). Similarly on the west. Similarly (construct) the northern wing. Thus also the tail with its vitasti. If desired, the two sides of each wing and of the tail may be contracted by four aṅgulis each on one extremity and extended by the same amount on the other."¹

c8.104

¹ *KŚr*, xvi. 8. 1-20.

c8.105

The *Śatapatha Brāhmaṇa*, however, teaches us how to bend the wings of the falcon in order to construct that variety of the Fire-altar, known as, the *Vakrapakṣa Syena-citi*.

c8.106

"He contracts the inner extremity (of the southern wing) inside on both sides only by four aṅgulis; by four aṅgulis outwards on both sides he expands the outer extremity. Thus by as much he contracts, by so much he expands; certainly, for that, he neither exceeds (the proper size of the wing), nor makes it too small. Similarly, he does for the tail; and in the same way, for the northern wing. Then he makes the bent of the two wings. For bents there are in the wings of a bird; the bents of the wings of a bird are by its one-third each; by one-third of the wings inwards each the bents of the wings of the bird are. He expands (each of the wings) on the front just by four aṅgulis; he contracts at the back by four aṅgulis. Thus by as much he expands by so much he contracts; and so he neither exceeds, nor makes it too small."

c8.107

A complete list of the various *Kāmya Agni* together with a statement of the objects for the attainment of which, each of them is to be constructed and sacrifices made therein, is found in the *Taittirīya Saṁhitā*.² That has been practically reproduced in the *Baudhāyana Śrauta-sūtra*.³ The enumeration of most of them appears in the *Maitrāyaṇīya Saṁhitā*⁴ and *Śatapatha Brāhmaṇa*.⁵ It may be noted, though it is immaterial for our purpose, that the construction of Fire-altars other than the *Suparṇa-citi* ("the eagle-shaped altar") is forbidden in the latter work.

c8.108

¹ *ŚBr*, x. 2. 1. 4-5; compare also *ŚBr*, x. 2. 1. 7; *BŚl*, iii. 62ff.

c8.109

² *TS*, v. 4. 11.

c8.110

³ *BŚr*, xvii. 28-30.

c8.111

⁴ *MaiS*, vii. 4. 7. *Cf.* iii. 2. 5.

c8.112

⁵ *ŚBr*, vi. 7. 2, 8.

c8.113

Thus, we find that almost all the *Vedi* and Agni which are described in the *Sulba* can be traced back, for the matter of their shapes and sizes, as far as the time of the *Brāhmaṇa* (c. 2000 B.C.); and they are mentioned even in the *Saṁhitā* (c. 3000 B.C.). The *Sulba* has, in fact, expressly admitted in the majority of cases, as has been pointed out before, that it has taken the spatial magnitudes of the altars from the earlier literatures. We can, similarly, trace the earlier origin of many other matters traced in the *Sulba*. As regards the height of an Agni and the number of bricks to be used in its construction, *Taittirīya Saṁhitā* observes:

c8.114

"He should pile (the fire) of a thousand (bricks) when first piling (it); this world is commensurate with a thousand; verily, he conquers this world. He should pile (it) of two thousand when piling a second time; the atmosphere is commensurate with two thousand; verily, he conquers the atmosphere. He should pile (it) of three thousand when piling for the third time; yonder world is commensurate with three thousand; verily, he conquers yonder world. Knee-deep should he pile (it), when piling for the first time; verily, with the Gāyatrī he mounts this world; navel-deep should he pile (it) when piling for the second time; verily with the Triṣṭubh he mounts the atmosphere; neck-deep should he pile (it) when piling for the third time; verily, with the Jagatī he mounts yonder world."¹

c8.115

¹ *TS*, v. 6. 8. 2 f. (Keith's translation).

c8.116

Each Agni is usually constructed in five layers, when constructed for the first time. It should have double or treble number of layers when constructed for the second or third time. The *Taittirīya Saṁhitā* says:—

c8.117

" The first layer is this (earth), the mortar the plants and trees; the second is the atmosphere, the mortar the birds; the third is yonder (sky), the mortar the Nakṣatras; the fourth the sacrifice, the mortar the sacrificial fee; the fifth the sacrificer, the mortar the offspring; if he were to pile it with three layers, he would obstruct the sacrifice, the fee, the self, the offspring; therefore should it be piled with five layers; verily, he preserves all. In that there are three layers, (it is) since Agni is of threefold; in that there are two (more), the sacrificer has two feet, (it is) for support; there are five layers, man is five-fold; verily, he preserves himself. There are five layers, he covers (them) with five (sets of) mortar, these make up ten, man has ten elements; he preserves man in his full extent."¹

c8.118

¹ *Ibid*, v. 6. 10. 2 f. (Keith). Compare *BŚl*, ii. 13.

c8.119

But when *Nākasad* and *Pañcacūḍā* bricks are employed, after the fourth layer, there will be a sixth layer as the height of these bricks are half the usual height of a brick, *viz.*, the one-fifth of a *jānu* (=32 aṅgulis). This sixth layer is mentioned also in the *Taittirīya Saṃhitā*.²

c8.120

² *Ibid*, v. 6. 10. 3. Compare *BŚl*, ii. 28, 59.

c8.121

The growth and development of (1) the theorem of the square of the diagonal, (2) the quadrature of a circle, and (3) the construction of similar figures, has been treated elsewhere in their proper places. We have clear proofs, it has been shown there, of the use of these in the time of the *Śatapatha Brāhmaṇa* (c. 2000 B.C.). The first two seem to be still older. But we do not find an enunciation of the theorem of the square of the diagonal and a method for the quadrature of the circle before the time of the *Śrauta-sūtra*. A method for the construction of similar figures is taught in the *Śatapatha Brāhmaṇa*, and it is the same as we find in later works.

Linked texts (20)