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The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t36: Chapter 11: Units of Measurement, Circling a Square, Areas of Plane Figures, Value of π, Quadrature of the Circle, Use of Pañcāṅgī Cord, Measures for Diagonal of a Rectangl

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t36

Language
English

Chapter 11: Units of Measurement, Circling a Square, Areas of Plane Figures, Value of π, Quadrature of the Circle, Use of Pañcāṅgī Cord, Measures for Diagonal of a Rectangl

c8.t36.1

UNITS OF MEASUREMENT, CIRCLING A SQUARE, AREAS OF PLANE FIGURES, VALUE OF π, QUADRATURE OF THE CIRCLE, USE OF PAÑCĀṄGĪ CORD, MEASURES FOR DIAGONAL OF A RECTANGLE

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**11.1-11.8.** *Units of measure*. The units of human measure become short or long depending on the stature of the sacrificer. However, the table runs as follows : 1 *yava* =6 mastard seeds; 1 mastard seed =6 cords of hair; 1 *aratni* =2 *prādeśas*; 1 *prakrama* =1 *aratni* or 2 *prādeśas* (in *citi* measure); 1 *puruṣa* =120 *aṅgulas* =5 *aratnis* or 10 *padas*; 1 *yuga* =86 *aṅgulas*; 1 *akṣa* =104 *aṅgulas*; the *ratha* measures are according to the prescription of the text.

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**11.9-11.10.** *Circling a square*. The method of circling a square described by Mānava in *Mśl*. 1.8*a* is repeated. Here the word, *viṣkambha* meaning 'diagonal' has been used; it should be *viṣkambhārdha* meaning 'radius'. This rule has been explained by Baudhāyana (*Bśl*. 2.9).

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**11.11-11.12.** *Areas of figures*. If d be diagonal of a quare of side a, then d²=2a²; that is, the square on the diagonal produces twice the area of the original square; similarly D²=2d²=4a², where D is the diagonal of the square drawn on the diagonal of the original square of side a; and so on. The area of a rectangle with breadth 2 *pu*. and length 8 *pu* is 16 sq. *pu*. (*Mśl*. 11.18).

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**11.13.** *Value of π*. If c be the circumference, d the diameter of a circle, $c=d/5+3d=3·2 d$ $or c/d=3.2.$

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Baudhāyana has also given a similar approximate value of π as 3 (*Bśl*. 4.15).

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**11.14-11.16.** *Quadrature of the circle*. Possibly these are not problems of quadrature of the circle. Ordinary squares are drawn without any mathematical significance.

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**11.17.** *Properties of right triangle.* The relations 3²+4²=5², and (3n)²+(4n)² =(5n)², where n is any quantity, hold good for any right-angled triangle.

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**11.19-11.28.** *Measurement of śroṇi and aṃsa points by pañcāṅgī cord.* How a cord with five marks (*pañcāṅgī*) is used to measure the western (*śroṇi*) and eastern corners (*aṃsa*) of *citis* other than *kaṅka* and *alaja* has been explained in a general way.

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