The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t25.10

c8.t25.10

If for the first layer, the two sides and the base are divided into 12 equal parts and the dividing points joined parallel to the sides and the base, the whole area will be divided into 144 isosceles triangles, each to be fitted exactly by the one-twelfth brick. The number 144 can be computed by permutation because, starting with one triangle at the apex it increases to 23 at the 12th layer with a common difference 2. For the second layer, where the sides are to be divided into 13 equal parts, there are likewise 169 triangles, each to be exactly fitted by one-thirteenth brick. Thus Sundararāja says: *prathame prastāre catuścatvāriṁśaccatena dvādaśabhirudicyo dvādaśa rītayaḥ* |...*aparasmin prastāre trayodaśībhiḥ navaṣaṣṭiśatena trayodaśa rītayaḥ* | To complete the number of 200 bricks, in the first layer, 16 one-twelfth bricks in the middle four rows are replaced by 64 quarter bricks, and the apical brick is replaced by 9 one-ninth bricks, thus making up the deficit of 56 bricks. In the second layer, in place of 7 one-thirteenth bricks, 20 quarter and 18 one-ninth bricks are added to make good the deficit of 31 bricks.