Mathematics in Ancient and Medieval India c6.s3: Geometrical Propositions

Mathematics in Ancient and Medieval India c6.s3

Language: English

Geometrical Propositions

c6.s3.1

Since the areas in classes I, II and III are fixed (vide p. 106) but the diagrams are different, various geometrical propositions undoubtedly evolved.

c6.s3.2

The following geometrical operations are found to have been studied in connection with different constructions :

c6.s3.3

i) Construction of a square on a given straight line.
ii) Construction of a circle equal to a given square and vice versa.

c6.s3.5

(b) Geometrical method to find the value of √( 2 ) which is expressed as √( 2 ) = 1 + ⅓ + 1/3.4 - 1/3.4.34 ( approx. ).

c6.s3.6

iv ) (a) A theorem on the square of the diagonal which runs thus : “the diagonal of a square produces an area double of the original square”.

(b) A general statement on the square of the diagonal of a rectangle now known as Pythagoras theorem runs thus : “the diagonal of a rectangle produces both ( areas ) which its length and breadth produce separately.”

c6.s3.8

(v) Construction of an isosceles trapezium whose face, base and altitude are given.

c6.s3.9

(vi) Construction of a rectangle having given sides.

c6.s3.10

(vii) Construction of an isosceles trapezium whose area will be equal to a simple multiple or submultiple of, and which will be similar to, another isosce'es trapezium.

c6.s3.11

(viii) (a) Construction of a square equal to the sum of two unequal squares ( *samāsavidhi* ).

c6.s3.12

(b) Construction of a square equal to the difference of two unequal squares ( *nirhāravidhi* ).

c6.s3.13

(ix) Construction of a square equal to a simple multiple ( or submultiple ) of another square.

c6.s3.14

(x) Transformation of a rectangle into a square and vice versa.

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(xi) Transformation of a square or a rectangle into an isosceles trapezium which shall have a given face.

c6.s3.16

(xii) (a) Transformation of a square or a rectangle into a triangle.

c6.s3.17

(b) Transformation of a square or a rectangle into a rhombus and vice versa.

c6.s3.18

(xiii) Transformation of an isosceles triangle into a square.

c6.s3.19

(xiv) Determination of area of a triangle (=½), trapezium (=base) and isosceles trapezium =[ ½ (face+base) ].

c6.s3.20

There cannot be any doubt that a little thought about the construction of the altars in class I will presuppose the knowledge from (i) to (iv) and class II a knowledge from (v) to (vii) and class III from that of (viii) to (ix) and knowledge of these three classes together leads to the formulation of rules from (x) to (xiii) besides many others which are not discussed by the *Śulbakāras*. Though these geometrical results mainly occur in the *Śulbasūtras* of Baudhāyana, Āpastamba, Kātyāyana and others, the authors never claimed that they were the first to discover the principles of geometry or apply them to the problems of the construction of altars. Whenever they described a proposition, they referred to their previous workers as quoted here : (i) it is so recognised or prescribed by the authorities, (ii) thus they teach, (iii) it has been so said (*iti abhyupadiśanti iti vijñāyate, iti uktam* etc.). It might be pointed out here that some of the lines occurring in the *Śulbasūtras* are found to be literal quotations from the *Taittirīya Saṃhitā* or *Taittirīya Brāhmaṇa* or the corresponding *Āraṇyaka*. *Baudhāyana* is occasionally more explicit about his sources. In connection with certain differences of opinions amongst the altar builders about the proper size and shape of a particular altar, *Baudhāyana* is found appealing to the authorities of the *Brahmaṇa*, by name for the purpose of arriving at a satisfactory settlement.¹

c6.s3.21

¹ *Baudhāyanaśulbasūtra*, 2. 15-19; 2. 35.

Linked texts (4)

Taittirīyasaṃhitā — Sanskrit
Taittirīyabrāhmaṇa — Sanskrit
Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit
Āraṇyaka — Sanskrit