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11 Yojanas ambu-Islap
Javea Sp
Figure 1: The annular region of the Lavana sea.
In his Sanskrit commentary on the TLS, Madhavacandra Traivaidya, an immediate pupil of Nemicandra, gives the process for the same under the above rule in the following words (Cf. TLS [Visuddhamati, p. 98 and TLS [Manoharlalji], p.45):
लवणाम्बुधिवलयं ऊर्ध्वं छित्वा रुन्द्र (2 ल.) प्रमाणेन विषमचतुर्भुजं कृत्वां "विक्खंभवग्ग" इत्यादिना मुखसूक्ष्मफलं ( 1ल. 1ल. 10) भूमिसूक्ष्मफलं ( 5ल. 5ल. 10) घानीय मुखभूम्योस्संस्थाप्य मुखभूमिसमासार्धमिति मध्यफलमानीय ( 6 / 2ल. 6 / 2ल. 10) मध्ये संस्थाप्य उपरितन भागे ऊर्ध्वं छित्वा चतुखार्थ व्यत्यासेन समानच्छेदेन मेलनं कृत्वा अपवर्तिते एवं डल. ल. 10 रुन्द्रार्धेन ( 1ल.) गुणिते सति "वर्गराशेर्गुणाकार भागहारा वर्गात्मिका एव भवंति" इति न्यायेन गुणिते सति चतुरस्रं स्यात् ( 6ल. 6ल. 10 ) ।
Translation Cutting the annulas (valaya) of the Lavanambudhi (salt sea) at its vertex and then making it an isosceles trapezium (visama caturbhuja) of altitude as equal to its breadth (rundra); get the (subtle measures of the) face (mukha) (11.11.10) and base (bhumi) (51.51.10) by the rule 'vikkhambhavagga' etc. Get the mean measure (madhya phala) by halving the sum (samasa) of the face and base and then establish it. Cut vertically the upper part (of the isosceles trapezium). For quadrature (caturastrartha). place both the upper sections inversly and alterably (vyatyasena). Equalize the denominators. Multiply 61.61.10 by half of the breadth (rundrardha) (11) by the rule 'vargarase' etc.; it would be (the area of) a rectangle (caturastra).
From this almost literal translation of his passage, it is seen what Madhavacandra wants to say. However this passage needs an especial exposition with accompanying diagrams. In her Hindi commentary, Aryika Visuddhamati (cf. TLS, pp. 97-99) has done this job.
This paper is aimed at bringing out Madhavacandra's process for the quadrature of a circular annulus.
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His process, with the help of her exposition and with some improved diagrams, is as follows:
Arhat Vacana, October 2000
