A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s5: 7. Lhuiler’s formula for the Circum-radius of a Cyclic quadrilateral
A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s5
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7. Lhuiler’s formula for the Circum-radius of a Cyclic quadrilateral
c8.s5.1
In Western mathematics, the eighteenth century mathematician Lhuiler is credited with the discovery, in 1782, of an expression for the circum-radius of a cyclic quadrilateral.² In India, however, we find the same formula enunciated by the Kerala astronomer Parameśvara (c. 1360-1455) in his commentary on the *Līlāvatī*, in the following lines :
c8.s5.2
² *Vide* D.E. Smith, *History of mathematics*, Boston, 1925, vol. II, p. 286.
c8.s5.3
*doṣṇām dvayor dvayor ghātayutānām tisṛṇām vadhāt |*
*ekaikonetarattraikyam catuṣkavdhabhajitam ||*
*labdhamūlena yadyṛttam viṣkambhardhena nirmitam |*
*sarvam caturbhujakṣetram tasminn evātiṣṭhate ||*
c8.s5.4
‘The three sums of the products of the sides, taken two at a time, are to be multiplied together and divided by the product of the sums of the sides taken three at a time and diminished by the fourth. If a circle is drawn with the square root of this quantity as radius, the whole quadrilateral will be situated inside it.’
c8.s5.5
Thus, if a, b, c, d are the sides and r the circum-radius,
c8.s5.6
r = √(((ab+cd)(ac+bd)(ad+bc))/((a+b+c-d)(b+c+d-a)(c+d+a-b)(d+a+b-c)))
c8.s5.7
The rationale of this formula has been given in the 16th cent. Kerala commentary *Kriyākramakarī* on the *Līlāvatī*.¹
c8.s5.8
¹ On this, see T.A. Saraswathi, 'Development of mathematical ideas in India', *Indian Jl. of. His. of Sc.*, 4 (1969) 69.
Linked texts (1)
- Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit