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The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t29: Chapter 2

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t29

Language
English

Chapter 2

c8.t29.1

UNITS OF MEASURES, *PAITṚKĪ VEDI*, MEASURES FOR DIAGONAL, THEOREM OF SQUARE, COMBINATION OF SQUARES.

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**2.1.** *Units of measures.* Units like *iṣā*, *akṣa*, *yuga* and *śamyā* have been expressed here in *aṅgulas*. This has been discussed under *Bśl.* 1.3.

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**2.3.** *Paitṛki vedi.* The *paitṛki vedi* has been dealt with by Baudhāyana (*Bśl.* 3.11).

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**2.3-2.6.** *Measures for the diagonal of a rectangle.* Kātyāyana has considered a particular measure for a square as *karaṇī*, so that its diagonal becomes *tatkaraṇī* or *dvikaraṇī*, (*tat kṣetraṃ dvaiguṇyādi kriyati 'nayā sā tatkaraṇī*—Mahīdhara), for square on the diagonal is twice the original square. Likewise, the diagonal of a rectangle having sides as *prāśvamānī* and *tiryaṅmānī* is known as *akṣṇayā*. With the help of these technical terms Kātyāyana has expressed the measures for the diagonal of a rectangle in the following two cases :

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$1² + 3² = (√10)²$ and $2² + 6² = (√40)²$

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Other measures *yuga* and *śamyā* have been defined in *Kśl.* 2.1.

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**2.7** *Theorem of square.* Kātyāyana here enunciates the general theorem of square on the diagonal of a rectangle in the same language as did Baudhāyana (*Bśl.* 1.12) and Āpastamba (*Aśl.* 1.4).

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At the end of the enunciation, he remarks, *iti kṣetrajñānam*. The term *kṣetra* has been translated as 'area' by Thibauta and 'figure' by Datta.b In the *śulbasūtra*, the area is technically expressed by *bhūmi* (*Bśl.* 1.6 and 1.9) and not *kṣetra*. Hence *iti kṣetrajñānam* means 'this is the knowledge of plane figures'.

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**2.8-2.9.** *Combination of two equal squares.* *Dvikaraṇī* has been defined here as √2 a, where *a* is the measure. This is actually a method of combination of two equal squares each of side *a* into one of √2a. *Tritīyākaraṇī* has been defined as *navabhāgastraya of tṛkaraṇī* : i.e. if *tṛkaraṇī* = √3a, then *tṛtīyākaraṇī* = (3.√3a)/9 = 1/(√3) a, where *a* is the measure.

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b Datta (2), 108.

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Kātyāyana's rule is essentially the same as that of Baudhāyana (*Bśl.* 2.12).

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a Thibaut (2), 233-34.

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**2.10-2.12.** *Construction of an isosceles trapezium.* After explaining the meaning of *dvikaraṇī*, *tṛkaraṇī*, and *tṛtīyākaraṇī* of a given measure Kātyāyana gives the method of construction of the *sautrāmaṇikī vedi*, which is an isosceles trapezium having face = 24/(√3), base = 30/(√3) and altitude = 36/(√3) prakramas (vide *Bśl.* 3.12).

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**2.13.** *Combination of two squares.* Kātyāyana prescribes the same method of Baudhāyana for the combination of two different squares into a square (*Bśl.* 2.1.).

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DIFFERENCE OF TWO SQUARES, TRANSFORMATION OF A RECTANGLE INTO A SQUARE AND A SQUARE INTO A RECTANGLE, AREAS OF FIGURES, PROBLEM OF CIRCLING A SQUARE AND QUADRATURE OF THE CIRCLE.

c8.t29.16

**3.1-.34.** The rule 3.1 deals with the construction of a square equal to the difference of two squares, 3.2-3.3 the transformation of a rectangle into a square and 3.4 transformation of a square into a rectangle. These rules have been given by Baudhāyana (*Bśl.* 2.2, 2.5 and 3.4 respectively). The transformation of a rectangle into a square, when it is very large, is specially discussed by Kātyāyana, as has also been done under *Bśl* 2.5.

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**3.5-3.10.** These concern the areas of squares and rectangles and are essentially the same as those of Āpastamba (*Aśl.* 3.4-3.10)

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**3.11-3.12.** For circling a square and the quadrature of the circle, Kātyāyana gives the same rules as those by Baudhāyana (*Bśl.* 2.9-2.11) and Āpastamba (*Āsl.* 3.2, 3.3).

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