The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t30.9

c8.t30.9

**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.