The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t1: Chapter 1: Units of Measurements, Construction of Squares and Rectangles, Knowledge of Surd, Theorem of Square on the Diagonal and Related Problems

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t1

Language: English

Chapter 1: Units of Measurements, Construction of Squares and Rectangles, Knowledge of Surd, Theorem of Square on the Diagonal and Related Problems

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UNITS OF MEASUREMENTS, CONSTRUCTION OF SQUARES AND RECTANGLES, KNOWLEDGE OF SURD, THEOREM OF SQUARE ON THE DIAGONAL AND RELATED PROBLEMS.

In the first two chapters Baudhāyana has given a summary of geometrical knowledge and some results of mathematical interest required for the construction of sacrificial altars. How the knowledge was used in connection with the measuring of grounds and placing of different layers of bricks has been discussed in detail in subsequent chapters. This chapter deals specifically with the units of measurements of altars, methods of construction of squares and rectangles, application of surd numbers, and the theorem of square on the diagonal of a rectangle.

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UNITS OF MEASUREMENTS

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**1.3.** Baudhāyana's table of units of measurements runs as follows:

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Āpastamba (*Aśl.* 6.5, 15.4) has prescribed the same values of Baudhāyana for *īṣā*, *akṣa*, *yuga*, *puruṣa*, *vyāyāma*, *aratni* and *prādeśa*. These units of Āpastamba have been used by both Kātyāyana (*Kśl.* 2.1, 5.9) and Mānava (*Mśl.* 2.1, 4.4). The unit *pada* has been made equal to 12 *aṅgulas* by Kātyāyana (*Kśl.* 5.9). The term *vitasti* has been used in place of *prādeśa* by these two latter *śulbakāras* as well as by Kauṭilya in his *Arthaśāstra*b, while its value remains the same. Mānava (*Mśl.* 4.2-4.4) has supplied some more units, e. g. 6 *tuṇḍa*=1 *bāla* of 3 years old calf; 3 *bālas* =½ mustard seed; 2 mustard seeds=1 *yava*a; 1 *aṅgula*=6 *yavas*; and 1 *prādeśa*=10 *vitastis*.

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a Fleet, 233.

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b Shamasastry (2), 117.

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The units like *aṅgula*, *pada*, *prakrama*, *prādeśa*, *bāhu*, *aratni* carry a long tradition and have been used earlier in the *Saṃhitās* and Brāhmaṇic literature in the same sense as these have been used in the *Śulbasūtras*.b

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CONSTRUCTION OF SQUARES AND RECTANGLES

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**1.4-1.5.** *Square.* Baudhāyana has described here two methods of construction of squares.

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*First Method.* Let *XY* be the given cord and *U* a mark at its middle (Fig. 1 (a)); *EW*, the *prācī* of the figure = *XY*; *O* the middle point of *EW* obtained corresponding to *U* of *XY*, where a pole is fixed.

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Fig.1. (a) Cord. (b) Square on a given length.

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A circle with *O* as centre and *OE* as radius is drawn (Fig. 1(b)). Then *EW* is a diameter of the circle along east-west line. Two other circles with *E* and *W* as centres and *EW* as radius are separately drawn. The points of intersection of these two circles are denoted by *N'* and *S'* The line *N'S'* fixes the second diameter *NS* of the circle whose centre is *O*. Again fastening the two ties once at *E*, *W*, *N* and *S* and drawing arcs, the points *A*, *B*, *C*, *D* are fixed. Then *ABCD* gives the required square.

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*Second Method.* Let *XS*, the given measure (*pramāṇa*) be *a* and *XY*, the increased cord, 2*a* (Fig. 2(a)). *S* is the mark at the middle of *XY*; then *XS* measures the length for *prācī*. *T* is the *nyañcana* mark, so that *ST*=½*a*—¼*a*=¼*a*. *U* is another mark at the middle of *SY*. *XT*, the diagonal (*akṣṇayā*) = a+¼a=5/4a *TY*, the breadth (*tiryaṅmānī*) = 2a-5/4ι=¾a Clearly, a²+(¾a)²=(5/4ι)² In other words, XY²+YT²=XT².

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a Shamasastry (1), 153, 55.

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b Macdonell and Keith, II, 577, 584.

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∴ XYT is a right-angled triangle (Fig.2(b)).

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Fig. 2. (a) Cord. (b) Right-angled triangle. (c) Square on a given length.

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For the construction of any geometrical figure intended in the *śulbasūtra* XY is always stretched along east-west line, known as *pārśvamāni*, YT along north-south line, known as *tiryaṅmāni*, and XT along the diagonal known as *akṣṇayārajju*. Now it is easy to see how the right-angled triangle XYT has been used for the construction of the square.

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The corners L and K (*śroṇi* points), M and N (*aṃsa* points) are fixed with the help of the point T of the triangle XYT [Fig.2(c)]. Then by using the half-cord UY, the points C, B, D and A are marked such that WC=WB=ED=EA. The figure ABCD gives the required square.

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1.6-1.7. *Rectangle and isosceles trapezium.* Baudhāyana's method of construction of rectangle with the help of a cord runs as follows :

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Let XY be a piece of cord taken equal to the desired breadth of the rectangle [Fig. 3 (a)]; S, a mark at the middle of XY;

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Fig. 3. (a) Cord. (b) Rectangle having desired length and breadth.

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*E*, *W*, the *prācī* poles; *P* and *Q*, *G* and *H*, poles at equal distances apart on both sides of each of the *prācī* poles. *O*, the mark assigned by the middle mark *S* when ties at *X* and *Y* are fixed at *P* and *Q* and stretched by *S*; and *A*, the point designated by the middle mark *S*, when both ties at *X* and *Y* are fixed at *E* and stretched.

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In a similar way, the other corner points, *B*, *C*, *D*, are traced [Fig. 3(b)]. Hence *ABCD* is the required rectangle.

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In the rule (*Bśl.* 1.7), Baudhāyana hints at the method of construction of an isosceles trapezium shorter in one side. For this purpose, a mark on the cord according to desired length is to be given and the rest is similar to that of *Bśl.* 1.6.

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**1.8.** *Square.* Baudhāyana describes here another method of construction of a square as follows:

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Let *XS* be the cord of given measure *a* [Fig. 4(a)] ;
XY=1½a=3/2a ;
SY=½ a;
*T* the *nyañcana* mark.
∴ XT=a+(⅓·½ a-⅙·½ a)=a+1/12 a=13/12 a
and TY=½ a-1/12a=5/12 a
The relation a²+(5/12 a)² = (13/12 a)² holds,
i.e., XY²+YT²=XT².

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Fig. 4. (a) Cord. (b) Right-angled triangle. (c) Square having given side.

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The right-angled triangle *XYT* [Fig. 4(b)] has been used to fix the corner points *L*, *M*, *N*, and *K* of the required construction. The points *C*, *D*, *A* and *B* are fixed such that WC=ED=EA=WB=a/2 [Fig. 4(c) ]. *ABCD* gives the required square.

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SURD NUMBERS

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**1.9-1.11.** Here Baudhāyana states that in a square *ABCD*, AC²=2 AB² [since AB=BC] or AC=√2 AB, where *AC* is known as the *dvikaraṇī* of the measure *AB*. If AB=a, AC=√2 a, where *a* is the measure. The result is sometimes considered by scholars as a particular case of the more generalized rule given by Baudhāyana in *Bśl.* 1.10. But Baudhāyana gave no such hint. On the other hand, he has tried to establish a more generalized result on the basis of this statement. According to him, when the measure of the side of a square is *a*, its diagonal is √2 a. Then again the measure of the diagonal of a rectangle having sides *a* and √2 a, is √3 a, for a²+(√2 a)² = (√3a)²; √3a is known as the *tṛkaraṇī*. This result has been extended

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Fig. 5. Square on the diagonal. Fig. 6. Producer of *tṛkaraṇī* and *tṛtīyākaraṇī*.

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to obtain the value of *tṛtīyākaraṇī* by both Āpastamba (*Āśl.* 2.2 and 2.3) and Kātyāyana (*Kśl.* 2.10 and 2.11). The commentators Kapardisvāmī, Sundararāja and Rāma have expressed in identical terms the value as well as meaning of this term. According to them, a square on the producer *AB* (=√3a) when divided into nine equal parts by means of three parallel lines drawing from both sides, produces the square, *EBGF* which is one-ninth of the square *ABCD* (Fig. 6). Then EB²=1/9 AB² or EB =⅓ AB =⅓ √3 a. =1/(√3) a.

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The producer *EB* is known as *tṛtīyākaraṇī* =1/(√3) a, where *a* is the side of the original square.

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THEOREM OF SQUARE ON THE DIAGONAL

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**1.12-1.13.** The theorem states that in a rectangle *ABCD*, AC²=AB²+BC² (Fig. 7). This is a most general statement and is enunciated first by Baudhāyana. The proposition is stated almost in identical language by Āpastamba (*Āśl.* 1.4), Kātyāyana (*Kśl.* 2.7) and Mānava (*Mśl.* 10.10). Baudhāyana further says that the theorem is easily verified from the following relations: $3²+4² =5²$ $12²+5² =13²$ $15²+8² =17²$ $7²+24²=25²$ $12²+35²=37²$ $15²+36²=39².$

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No proof of this theorem is given by Baudhāyana and other *śulba* writers, since it is beyond their tradition to do so. Zeuthen, Cantor, Vogt, Cajori and Heath have expressed the view that the general statement was possibly the result of an induction from a small number of cases of right-angled triangles having sides in rational numbers known to them. But this is not the actual case. Our discussions on rational rectangles and construction of geometrical figures amply justify that the general character of the theorem was rightly understood by the *śulbakāras*.

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Fig. 7

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Fig. 8

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A number of conjectures by Heath, Bürk, Müller, Thibaut, Datta and others as to the way the proof of the theorem could have been arrived at are available. A few of them are discussed in what follows.

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(i) According to Heatha, the problem of transformation of a square into a rectangle given by Baudhāyana in *Bśl.* 2.3. formed the basis of the proof. For, square *ABCD* drawn on the diagonal *AD* of the right-angled triangle *AGD* is equivalent to four equilateral triangles, while its sides *GD* and *AG* produce two each (Fig. 16). This has also established Bürk’s hypothesis.b

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(ii) The combination of two different squares as described by Baudhāyana in *Bśl.* 2.1 (Fig. 12) might have laid the foundation of the general statement of the theorem. This is according to Müllerc.

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(iii) Thibautd opined that the *śulbakāras* were observant of the fact that the square on *AD* is equivalent to four equal triangles, one of which is equal to half of the square on *OA* or *OD*, i.e. the squares on *OA* and *OD* together are equivalent to four equal triangles (Fig. 8). This pattern of arrangement of equilateral triangles are actually found in the first layer of construction with bricks in the *vakrapakṣaśyenacit* as described by Baudhāyana.

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(iv) According to Datta,e the construction of the *paitṛki vedi* established the theorem of square on the diagonal. The altar is mentioned in the *Śatapatha Brāhmaṇa* (XIII.8.1.5) as a square with its corners pointed towards the cardinal directions. It is referred to by Baudhāyana (*Bśl.* 3.11) and also by Kātyāyana (*Kśl.*2.2), where the method of its construction in detail has appeared. The square *EGWH* obtained by joining the middle points of a square *ABCD* (of area 2 sq. *puruṣas*) is the *paitṛki vedi* and is half (in area) of the original square (Fig. 29). The original square *ABCD* is a square on its east-west line *EW*. *EW* is again the diagonal of the newly formed square *EGWH*. This is undoubtedly a convincing proof (since *EW*²=2 *EG*²).

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a Heath, 352.

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b Bürk, 55, 556.

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c Müller, 194-95.

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d Datta (2), 111.

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e Datta (2), 113-115.

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(v) The knowledge of *dvikaraṇī*, *tṛkaraṇī*, discussed by Baudhāyana (*Bśl*. 1.9—1.11,) led in a way to the theorem of square on the diagonal.

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(vi) Both Āpastamba (*Āśl*. 3.7) and Kātyāyana (*Kśl*. 3.7) gave an ingenious method for calculating the area of a square or rectangle, thereby establishing the theorem of square on the diagonal. According to this method, if there are *p* units in *AB* and *q* units in *BC*, then the rectangle *ABCD* has *pq* square units, which can be obtained by drawing *p* number of parallel lines through *p* units of *AB* and *q* number of parallel lines through *q* points of *BC* (Fig. 9). This proves directly that AC² = AB² + BC².

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Fig. 9

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(vii) Kātyāyana (*Kśl*. 2.4 and 2.5) has considered a rectangle of breadth 1 *pada* and length 3 *padas*, whose diagonal is a 10 fold producer. According to Dattaa, this justifies the statement of the theorem of square on the diagonal, as may be seen from Fig. 10. In the square *ABCD*, DH = CG = CK = BF = AE = 1 pada.

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Fig. 10,

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Fig. 11

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Now *sq*. *ABCD*

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= sq on BK + sq on DH + 4 tr. AEH
= AH² + AE² + 4 tr. AEH.

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Again *sq* ABCD = EH² + 4 tr. AEH ∴ AH² + AE² = EH² (viii) Dattaa has given another proof of this theorem. Four rectangles each equal to a given rectangle having breadth a, length b and the diagonal c are so constructed that the diagonal of each rectangle forms the side of a square (Fig. 11). Then $c² = (a + b)² - 4 (½ ab)$ or, c² = a² + b².

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a Datta (2), 115-116.

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Although such specific constructions and arguments are not supplied as proofs by the *śulbakāras*, ample evidence is left by them in their details of constructions to believe that the proofs of the theorem of square on the diagonal of a rectangle were known to them.

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*Pythagorean Theorem in other Culture Areas.*

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The theorem of square on the diagonal of a rectangle is usually known as Pythagorean theorem after the name of Pythagoras (*c.* 540 B.C.). In fact, the relation 3² + 4² = 5² and some such relations have been used by Pythagoras, but evidence of any general statement regarding this is not yet available. Actual proof was first given by Euclid (*c.* 300 B.C.). Proclus (*c.* 460 A.D.), the commentator of Euclid's *Elements*b remarked: "For my part, while I admire those who first observed the truth of the theorem, I marvel more at the writer of the *Elements*, not only because he made it first by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in the sixth book." Heath has quoted this with the remark:c "It is difficult for us to be more positive than Proclus was".

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It is fairly certain that the practical use of the theorem was current in old Babylonian times (*c.* 1800-1600 B.C.). The evidence for this is found in certain Babylonian cuneiform tablets. No general theorem was found to have been mentioned. It has been conclusively proved by Neugebauer that Pythagoras derived his "Number theorem of Universe" as well as the so-called Pythagorean theorem from cuneiform tablets.d The Chinese knew of a similar relation which appeared in *Chou Pei* (4th century B.C.), but it really became well known from the time of its first commentator Chao Chun Chhinge (3rd century A.D.). A proof of the theorem was given by Bhāskara IIf (1150 A.D.). According to Needham, Bhāskara II's treatment was derived from the *Chou Pei*.g This is not true, for the proof of Bhāskara II and that given in *Chou Pei* can readily be deduced from a number of constructions described already in the *Śulbasūtras*. .

Linked texts (8)

Kātyāyanaśulbasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit
Āpastambaśulbasūtra — Āpastamba, Sanskrit
Śatapathabrāhmaṇa — Yājñavalkya & Śāṇḍilya (Vedic teacher), Sanskrit
Kauṭilīya Arthaśāstra — Kauṭilya, Sanskrit
Mānavaśulbasūtra — Sanskrit
Bījagaṇita — Bhāskara II, Sanskrit
Dhātusutta — Pāli