The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c8.t12: Chapter 14: The Construction of a Fire-Altar in the Form of an Isosceles Triangle (Praugaciti)

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c8.t12

Language: English

Chapter 14: The Construction of a Fire-Altar in the Form of an Isosceles Triangle (Praugaciti)

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**14.2.** *Measurement of the fire-altar and the types of bricks.* The area of the fire-altar should be 7½ sq. *pu.* Under rule 2.7, Baudhāyana has shown that an isosceles triangle of 7½ sq. *pu.* can be drawn from a square of double this area, that is, 15 sq. *pu.*, by joining the mid-point E of the eastern side AD with two western corner points B and C (Fig. 52). The base BC and each side EB, EC are given by $BC &= √15 pu. / &= 120 √15 ang. / &= 464.76 or 464¾ ang. approx. / EB = EC &= 300 √3 ang. = 519.6 or 519½ ang.$ D. therefore explains that a square with 464¾ ang. (= 465 - ¼ ang) is to be first drawn and the required isosceles triangle constructed, as already stated, in the following terms : *tasyā dvikaraṇī pādonapañcaṣaṣṭicatuḥśatāṅgulā* (464¾ ang) *dvistāvatyāścaturaśrakṛtāyāḥ pañcadaśapuruṣāyā bhūmeḥ karaṇī | evaṁ caturaśrikṛtāyāḥ pūrvasyāḥ karaṇyā madhyācchroṇī pratyālikhet | tatra karṇarūpayoḥ pramāṇamardhonaviṁśatipañcaśatāṅgulayaḥ* (519½ ang.) *| tasya praugasyāparasyāḥ karaṇyāḥ pādonapañcaṣaṣṭicatuḥśatāṅgulāyā...... |*

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Fig. 52.

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Fig. 53. Types of bricks.

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The following four types of bricks are prescribed (Fig. 53) : B₁ — a rectangular brick, *bṛhatī*, BC/12×BC/24 or 10 √15× 5 √15 sq. ang. (38 ang. 25 ti× 19 ang. 12½ ti). B₂ — a triangular brick half of the *bṛhatī*, diagonally intersected. B₃ — a triangular quarter brick with long base, *dīrghapādyā*. B₄ — a triangular quarter brick with short-base and pointed like a spear, *śūlapādyā*.

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**14.3-14.4.** *Placement of bricks in the first layer.* All that these two rules say is that half bricks with their hypotenuses turned outside are to be placed on both sides and *bṛhatī* bricks in the remaining space. It is easy to see that 200 bricks cannot be used to cover the altar in this way. The way it can be done is explained in the commentary.

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The *praugaciti* is divided into 24 rows west-east (Fig. 54), 12 on each side of the perpendicular line from the vertex to the middle of the base, and marked 1, 2, 3...... 12. Bricks are placed as follows :—

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| In the 1st row on each side | — B₂ — 2; total | 2 |
| In the 2nd row on each side | — B₂ — 6; ,, | — 6 |
| In the 3rd-10th row on each side (that is 16 rows) | — B₁ — 88 ; B₂ — 16, ,, | —104 |
| In the 11th row on each side | — B₂ — 42 ; ,, | — 42 |
| In the 12th row on each side | — B₂ — 46 ; ,, | — 46 |
| Total : | B₁ = 88 ; B₂ 112 : | 200 |

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Fig. 54. Arrangement of bricks in the *praugaciti*—1st layer.

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Fig. 55. Arrangement of bricks in the *praugaciti*—2nd layer.

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**14.5-14.8.** *Placement of bricks in the second layer*, For the second layer, the rules direct the placement of 47 B₄s at the western end of the fire-altar, 1 B₄ at the apex, 2 B₃s and 2 B₄s to fill up the *svayamātṛṇṇa* space in the middle, B₂ bricks at the sides and B₁s in the remaining space. The total number of 200 bricks can be completed in the following manner as explained by the commentator (Fig. 55) :

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| (a) | In the apex: | B₄ | — 1 |
| (b) | In the western end (base), vertices and bases alternately turned in opposite direction : | B₄ | — 47 |
| (c) | In between, there are 11 rows south-north of which sides are filled with B₂s: | B₂ | — 22 |
| (d) | In the centre of the middle row — B₃ — 2 ; B₄ — 2 : | total | — 4 |
| (e) | In the 4th and 5th row, 3 B₁s are replaced by 6 B₂s each: | B₂ | — 12 |
| (f) | In the remaining space: | B₁ | —114 |
| | Total : B₁ — 114 ; B₂ — 34 ; B₃ — 2 ; B₄ — 50 | | 200 |

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(*asminprastāre caturdaśaśataṃ bṛhatyaḥ | catustriṃśadardhyāḥ | pañcāśacchūlapādyāḥ | dve dīrghapādye |* — *D.*)

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THE CONSTRUCTION OF A FIRE-ALTAR IN THE FORM OF A RHOMBUS (*UBHAYATA PRAUGA*)

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**15.2.** *The construction and the types of bricks.* The method of construction of a rhombus or double isosceles triangle having common base and the two vertices on the opposite sides is given in *Bśl.* 2.8. *ABCD* and *BEFC* are two equal squares, each of area 7½ sq. *pu.* *GBHC* is the desired rhombus of 7½ sq. *pu.* of which *G* and *H* are the mid-points of *AD* and *EF* respectively. $BC = 120 √(15/2) aṅg = 328.56 aṅg.$ $BG = GC = BH = HC = 300 √(3/2) aṅg. = 367.5 aṅg.$

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D. gives the value of *BC* as: *trīṇi śatānyaṣṭāviṃśatiścāṅgulaya ardhaviṃśāśca tilāḥ* (.56 aṅg. = 19.04 tila) *tiryaṅmāni*. Each side of the rhombus is given as : *evaṃ kṛte'-rdhādhikasaptaṣaṣṭiśatatrayamaṅgulayaḥ karaṇyo bhavanti.*

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Fig. 56. Measurement of areas and bricks.

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Four types of bricks are made in the same way as those for the isosceles fire-altar (14.2). B₁, the rectangular brick, *bṛhatī*, is BC/9 × BC/18; bricks B₂ (*ardhyā*), B₃ (*dīrghapādyā*) and B₄ (*śūlapādyā*) are made by diagonal intersections of the *bṛhatī* as before.

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**15.3.** *Placement of bricks in the first layer.* *BC* is divided into 18 equal parts and marked and likewise *GH* in 18 equal parts and marked. Lines parallel to *BC* and *GH* are drawn. It is easily seen that the entire area is divided into 144 rectangles where an equal number of *bṛhatī* bricks can be placed and into 36 half rectangles along the sides where an equal number of *ardhyās* can be placed. Thus we get 180 bricks. The deficit is met by replacing 10 B₁s in the 6th vertical row on either side of the central line *GH* by 20 B₂s. With the above substitution, the total number of B₁s is 124 and that of B₂s 76. On these points, the commentary runs as follows : *ṣaṣṭhī dvādaśeṣṭakā | tatra madhyāddaśa bṛhatīrudhṛtya viṃśatirardhyāḥ . . . . asminprastāre caturviṃśatiśataṃ bṛhatyaḥ | ṣaṭsaptatirardhyāḥ |*

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Fig. 57. Arrangement of bricks in the rhombus fire-altar—1st layer.

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Fig. 58. Arrangement of bricks in the rhombus fire-altar—2nd layer.

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**15.4-15.6.** *Placement of bricks in the second layer.* As per directions in the rules, 2 B₄s (*śūlapādyā*) are to be placed one each at the apices G and H and 2 B₃s (*dīrghapādyā*), one each at B and C (Fig. 58). The remaining space can be divided into 17 horizontal (south-north) and 17 vertical (west-cast) rows. In the middle (*svayamātṛṇṇa*) of the fire-altar, 2 B₃s and 2 B₄s are placed as shown. It is easy to see that the remaining space can be filled by 32 B₂s at the sides and 144 B₁s with the longer side turned east or west. This accounts for 184 bricks. The number is completed by replacing 16 B₁s, four each of the four directions, by 32 B₂s. The total number of bricks of different types are : B₁ — 128 ; B₂ — 64 ; B₃ — 4 ; B₄ — 4.

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Linked texts (1)

Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit