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The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c5.s3: General Characteristics of Śulbasūtras, Their Geometry, Arithmetic and Algebra

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c5.s3

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English

General Characteristics of Śulbasūtras, Their Geometry, Arithmetic and Algebra

c5.s3.1

In our section on ‘Commentaries’, we have discussed in detail the *sūtras* of the four texts by Baudhāyana, Āpastamba, Kātyāyana and Mānava. In presenting the texts, their translations and commentaries, we have not followed the chronological order in which these were written as would appear from our foregoing discussions, but in order of their importance and completeness.

c5.s3.2

Baudhāyana’s *sūtras* are not only the earliest but represent the most systematic, logical and detailed treatment of the subject inspite of their archaic and highly condensed *sūtra* style. It opens with the various units of linear measurements and then develops the geometry of rectilinear figures, triangles and circles, their transformations from one type to the other, methods of arriving at areas by combination or difference of given areas, the irrational number like √ 2 and the value of π albeit indirectly. Then the measurements of the three perpetual fires, the *gārhapatya*, the *āhavanīya* and the *dakṣiṇāgni*, the sacrificial altars such as the *mahāvedi*, the *uttara vedi* the *sautrāmaṇī vedi*, the *paitṛki vedi*, the *dārśapaurṇamāsika vedi*, the *paśubandha vedi*, various sacrificial fires such as the *dhiṣṇyas*, the *āgnīdhriya*, and the *mārjālīya*, the *sadas* tent, the *havirdhāna* shed for the placement of the soma carts and various pits like the *utkara*, the *uparava* and the *cātvāla* are given along with their relative distances from one another. Although the plan of the sacrificial ground and placement therein of the various fires, altars, tents, sheds and pits is not described in separate *sūtras*, the manner in which their measurements, relative distances and directions for their placements are given makes it quite easy for one to visualize the picture of the sacrificial ground and at once appreciate how closely and meticulously the *śulbavid* was following the time-honoured Vedic sacrificial practices. About the fire-altars for the various wish fulfilments (*kāmyaciti*), Baudhāyana gives the measurements, constructions and methods of laying bricks of different geometrical shapes in alternate layers following the strict injunction that the edges of bricks in contiguous layers must not meet. Such fire-altars include those in the form of a falcon, both rectilinear and with curved wings and extended tail, of a kite and *alaja* bird, of an isosceles triangle and a rhombus, of a chariot-wheel without and with spokes, of a square and circular trough, of a pyre, and finally one in the form of a tortoise. The facile use of geometry presented in the opening chapters is abundantly clear in the constructional procedures of these fire-altars.

c5.s3.3

a *taddhaike, ekavidhaṁ prathamam vidadhātyathaikottaramā parimitavidhānna tathā kuryāt saptavidho vā ‘agre prajāpatirasyjvata] . . . . . saptavidhameva prathamam vidadhītāthaikottaramaikaśatavidhādekaśatavidhaṁ tu nātividadhīta . . . . . /(*Śat. Br.* 10. 2. 3. 17-18).*

c5.s3.4

b *Bśl.* 5.1, 5.8. *Āśl.* 8.3.

c5.s3.5

Āpastamba follows more or less the same procedure and provides the same rules and techniques; but in our view, coming after Baudhāyana and with his text before him, he has not shown any improvement upon Baudhāyana. About the fire-altars, he has discussed different types of *śyenacit*, rectilinear as well as those with curved wings and extended tail and given different arrangements with different types of bricks, but has not gone into the details of other types which he mentions presumably because he has no alternative method of arrangement to suggest. As far as the use of terms and expressions are concerned, there is a remarkable resemblance between Baudhāyana and Āpastamba.

c5.s3.6

Kātyāyana’s treatment is succinct and systematic. He emphasizes the geometry behind the construction of altars and fires and gives a clear exposition of it. He deals with a few altars and *agnis* but refrains from considering the *kāmyacitis* as the latter are discussed separately in a chapter of his *Śrautasūtras*.

c5.s3.7

The *Mānava-śulbasūtra*, a part of the *Śrautasūtra* by the same author, although following the common tradition of the *śulbakāras*, gives methods and details often very difficult to comprehend. In many cases the details are either lacking or incomplete and can be understood only by reference to Baudhāyana, Āpastamba and Kātyāyana. To us the very arrangement and the treatment of the subject have appeared far from systematic. In our judgement, the work does not measure up to the standard attained by Baudhāyana, Āpastamba and Kātyāyana.

c5.s3.8

*Geometry*

c5.s3.9

Several methods of constructing a square on a given straight line have been given. Rectangles are rectilinear figures of which the two sides are different. For the construction of such rectilinear figures the squared relationship between the diagonal and the two sides has been given at various places in the *śulbasūtras*, which we summarize below:

c5.s3.10

a) n²+(¾n)² = (5/4n)² (*Bśl.* 1.5; *Mśl.* 1.11—1.12)
(i) n = 4, 4²+3² = 5² (*Bśl.* 1.13)
(ii) n = 12, 12²+9² = 15² (*Kśl.* 2.5)
(iii) n = 20, 20²+15² = 25² (*Āśl.* 5.3)
(iv) n = 16, 16²+12² = 20² (*Āśl.* 5.3)
b) n²+(5/12n)² = (13/12n)² (*Bśl.* 1.8; *Āśl.* 1.2; *Kśl.* 1.4)
(i) n = 1, 1²+(5/12)² = (13/12)² (*Bśl.* 1.8 *Āśl.* 1.2; *Kśl* 1.4-1.5)
(ii) n = 36, 36²+15² = 39² (*Bśl.* 1.13; *Āśl.* 5.4)
(iii) n = 188, 188²+(78⅓)² = (203⅔)² (*Āśl.* 6.5)
(iv) n = 6, 6²+(2½)² = (6½)² (*Āśl.* 6.6; *Mśl.* 2.4)

c5.s3.11

(v) n=5, 5²+(2 1/12)²=(5 5/12)² (*Āśl.* 6.7)
(vi) n=10, 10²+(4⅙)²=(10⅚)² (*Āśl.* 6.8)
(vii) n=27, 27²+(11¼)²=(29¼)² (*Āśl.* 7.3) (viii) n=18, 18²+(7½)²=(19½)² (*Āśl.* 7.1—7.2)
(ix) n=12, 12+5²=13² (*Bśl.* 1.13; *Āśl.* 5.4)
(x) n=96, 96²+40²=104² (*Mśl.* 1.4-1.6;)
c) 7²+24²=25² (*Bśl.* 1.13)
d) 8²+15²=17² (*Bśl.* 1.13; *Āśl.* 5.5)
e) 12²+35²=37² (*Bśl.* 1.13; *Āśl.* 5.5)
f) 1²+3²=(√10)² (*Kśl.* 2.4; *Mśl.* 3.5)
g) 2²+6²=(√40)² (*Kśl.* 2.5)
h) 1²+(√2)²+(√3)² (*Kśl.* 2.10)
i) na²=[((n+1)a)/2]²-[((n-1)a)/2]², where a= rational integer (*Kśl.* 6.7)
j) 6²+(4½)²=(7½)² (*Mśl.* 2.5)
k) 1²+(√10)²=11 (*Mśl.* 12.5)

c5.s3.12

*Theorem of Square on the Diagonal of a Square or a Rectangle*

c5.s3.13

The above-mentioned squared relationships are followed or preceded by the general statement that the diagonal of a rectangle produces by itself both (the areas) produced separately by its two sides. This is the so-called theorem known after Pythagoras. A question has often been asked whether such a definition resulted from empirical guess work or was based on a proof of some kind. Such questions are of course to be expected from scholars firmly entrenched in the Euclidean tradition in geometry. As is well known, mathematics, including geometry, in ancient India did not follow the Euclidean tradition, and usually gave the rules, leaving their proofs to be explained by the teachers to the pupils of their respective schools. Several scholars who discussed this issue of proof have shown that such proof is implicit in the very operations with rectilinear figures; we have discussed the matter in detail in our commentaries in connection with the rule and further elaboration here is unnecessary. George Sarton, while considering the question of Hindu influence on Pythagorean derivation of the theorem of squares, referred to Gaston Milhaud's claim that Pythagorean geometry may have been partly inspired by Hindu models. This argument was based on the high antiquity of Baudhāyana and Āpastamba. But Sarton, presumably on the basis of Keith's views on the dates of the *śulbakāras*, pleaded inability to accept such high antiquity and observed: 'It is highly probable that the *śulbasūtras* date from a period posterior to 500 B.C. and pre-Christian. They are probably post-Pythagorean'.a

c5.s3.14

This brings us to the important question of the origin of the theorems and also whether Pythagoras himself was the discoverer of it.

c5.s3.15

The tradition attributing the theorem to Pythagoras is due to Cicero (*c.* 50 B.C.) Diogenes Laertius (second century A.D.), Athenaeus (*c.* A.D. 300), Heron (third century A.D.), and Proclus (*c.*A.D. 460), and therefore started about five centuries after the death of Pythagoras. Junge pointed out that the Greek literature of the first five centuries after Pythagoras contained no mention of the discovery of this or any other important geometrical theorem by the great philosopher and furthermore emphasized uncertainties in the statements of Plutarch and Proclus. Although various attempts have been made to justify the tradition and trace the proof to Pythagoras, no record of proof has come down to us earlier than that given by Euclid (Theorem 47, BK 1). As to the relation 4²+3²=5² from which the theorem of rational triangle is derivable, very ancient Egyptian knowledge is attested by the Kahun papyrus of the twelfth dynasty (*c.* 2000 B.C.), but its association with rational triangles does not seem indicated in this or other Egyptian papyrii.a As to the antiquity of Pythagorean theorem in China, it is stated, though not proved, in the arithmetical classic *Chou Pei Suan Ching* (third or fourth century B.C.); the numerical relationship 4, 3 and 5 between the sides and the diagonal of a rational rectangle is also given in this text. The old Babylonians of the second millennium B.C. left records on their cuneiform tablets of similar squared relationships indicating practical use of the theorem of squares. No general statement in the form of a theorem is of course found. Neugebauer is of the view that Pythagoras derived his number theorem of the universe as well as the theorem known after his name from such Babylonian cuneiform tablets.b

c5.s3.16

a Sarton, I, 74-75.

c5.s3.17

As we have stated, methods have been given for transforming rectilinear figures from squares to rectangles, of transforming squares into circles, of developing isosceles triangle and rhombus from squares and so on. Various geometrical shapes like parallelograms, five-sided rectilinear figures are mentioned in various ways in connection with the construction of bricks with which to cover the sacrificial altars.

c5.s3.18

*Irrational numbers, π, Fractions, Surds*

c5.s3.19

What is of great significance is the treatment of irrational numbers like √2 and statement of their accurate values. The manner in which such accurate values were possibly obtained by the *śulbakāras* has been fully discussed in our notes. Neugebauer has shown that these values are identical with those found in certain Babylonian cuneiform texts, given in sexagesimal system. He tried to imply that the Indian value after all represented the Babylonian one expressed only in decimal system or more accurately in fraction. As we have shown, there is certainly no proof of such an assertion and the Indian value is certainly derivable from the methods contained in the *śulbasūtras* themselves.

c5.s3.20

In connection with the pits for stacking the sacrificial poles in, Baudhāyana has given the ratio of the circumference to the diameter as 3. We have shown from the rules for transforming squares into circles, as given by Baudhāyana, that the *śulbakāras* in all probability knew of more accurate value of π.

c5.s3.21

a Heath, (2), I, 352. Sen, Chapter on 'Mathematics', *A Concise History of Science in India*, 148-149,

c5.s3.22

b Neugebauer, 28-42 \* The numbering within parenthesis refers to Thibaut's edition as printed in the *Pandit*.

c5.s3.23

The *śulbakāras* were familiar with the use of fractions and manipulated with them in various ways, specially in connection with the construction of bricks for the fire-altars. The terms used by them are significant as well as interesting, of which few examples are given :

c5.s3.24

| *caturbhāgona* | =1-¼=¾ | (*Bśl.* 1.5) |
| *ardhāṣṭama* | =7½ | (*Bśl.* 5.1, 5.6) |
| *ardhadaśama* | =9½ | (*Bśl.* 5.1) |
| *ardhanavama* | =8½ | (*Bśl.* 5.1) |
| *caturtha-saviśeṣārdha* | =½ (¼√2) | (*Āśl.* 19.4) |
| *caturtha-saviśeṣa-saptama* | =1/7(¼√2) | (*Āśl.* 19.7) |

c5.s3.25

In our notes we have explained that elementary knowledge of operating with surds was also possessed by these geometers. It will certainly be not proper to say that the *śulbakāras* dealt with algebra as is known from later Indian mathematicians. But it cannot be denied that germs of algebraic equation are embedded in many of their rules and operations. Such is the case with quadratic equation as also with indeterminate equation of the first degree. As we have amply stressed, these texts were compiled primarily as manuals for the construction of sacrificial fires and altars. Geometry, mensuration, arithmetic, and germs of algebra came out only incidentally. Nevertheless, the gleams we obtain of their knowledge of these subjects can hardly fail to excite our admiration when we remember the time of their compilation.

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