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The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t30: Chapter 4: Construction of Droṇacit, Triangle, Rhombus, Transformation of Triangle and Rhombus into a Square

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t30

Language
English

Chapter 4: Construction of Droṇacit, Triangle, Rhombus, Transformation of Triangle and Rhombus into a Square

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**4.1-4.2.** *Construction of droṇacit.* Kātyāyana discusses here the methods of drawing different squares required for the construction of *droṇacit*. First a square of area 7½ sq. *pu.* is constructed and divided into 100 small squares by drawing ten parallel lines horizontally and ten vertically. Then small squares from one side are separated out and changed into a small square by the method of combination of squares (*samāsa-vidhi*), discussed in *Kśl.* 2.8 and 2.9. The remaining 90 small squares are likewise transformed into a single square. The former square is joined to the latter like a stalk. In the case of a circular *droṇacit*, the two squares mentioned above are to be transformed into two circles and joined together (Mahīdhara). Kātyāyana's rule is basically different from that of Baudhāyana (*Bśl.* 17.1.-18) and Āpastamba (*Āśl.* 13.4-5) and appears mathematically more sound.

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Fig. 4. Transformation of a triangle into a square.

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**4.3-4.4.** *Construction of triangle and rhombus.* These two rules are the same as those of Baudhāyana (*Bśl.* 2.7-2.8).

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**4.5.** *Transformation of an isosceles triangle into a square.* For transformation of an isosceles triangle into a square, the isosceles triangle *ECG* is divided by the *prācī* line *EF* (Fig. 4). Now tr. *ECF* is transferred to the other side so that tr. *EGH* is now its new position. Thus tr. *ECG* is transformed into the rectangle *EFGH*. This rectangle is changed into a square by the *sūtra Kśl.* 3.2. It has also been discussed by Āpastamba (*Āśl.* 12.4-12.8).

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Fig. 5 Transformation of a rhombus into a rectangle.

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**4.6.** *Transformation of a rhombus into a square.* For transforming a rhombus *ENFS* into a square, *EF* and *NS*, the east-west and north-south lines are joined (Fig. 5). The isosceles triangle *NFS* is clearly sub-divided into two triangles *NOF* and *SOF*. These are now transferred and after invertion placed in their positions as *AEN* and *BES*. Thus the rhombus *ENFS* is transformed into the rectangle *ANSB*. This rectangle is transformed into a square by the *sūtra Kśl.* 3.2.

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Fig. 6.

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**4.7.** *Transformation of a triangle into a square.* Kātyāyana has hinted for the first time at a method of transforming into a square a triangle other than the isosceles. The rule is, however, incomplete. According to commentator Mahīdhara, *ekakarṇa* means *tulyakarṇa*, i.e. a figure having equal angles and *dvikarṇa nānāvidhakarṇa*, i.e. a figure of unequal angles. Whether by *nānāvidhakarṇa* Mahīdhara meant an irregular pentagon cannot be definitely said. Śulbakāras were well acquainted with the method of converting an isosceles triangle into a square. Possibly they had also the knowledge of transformation of a pentagon of equal angles into a square by joining the angular points, dividing it into several isosceles triangles, and then joining them up into squares by the rule taught before. Kātyāyana has advised to break up *pañcakarṇas* of *dvikarṇa* variety into a square. But there is no such method known to the *śulbakāras* by which an irregular pentagon can be broken up into squares. Kātyāyana's pentagon *ABCDE* is of the type of *haṁsamukhī* brick (Fig. 6), in which BF = FC = EH and AB = HF = DC = BF/2. Description of this type of pentagonal *haṁsamukhī* bricks is given by Baudhāyana (*Bśl.* 10.10). A pentagon of this type can be broken up into two or three squares, which again can be combined into a single square.

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