[Pramāṇa](https://indike.org/pramana/) › [Studies](https://indike.org/pramana/ins) › [Science, technology and education](https://indike.org/pramana/ins.science) › [A History of the Kerala School of Hindu Astronomy (in Perspective)](https://indike.org/pramana/ins.science.sarma-1972) › c8.s5 # 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 - **Language:** English ### 7. Lhuiler’s formula for the Circum-radius of a Cyclic quadrilateral ### [c8.s5.1](https://indike.org/pramana/ins.science.sarma-1972.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](https://indike.org/pramana/ins.science.sarma-1972.c8.s5.2) ² \*Vide\* D.E. Smith, \*History of mathematics\*, Boston, 1925, vol. II, p. 286. ### [c8.s5.3](https://indike.org/pramana/ins.science.sarma-1972.c8.s5.3) \*doṣṇām dvayor dvayor ghātayutānām tisṛṇām vadhāt |\* \*ekaikonetarattraikyam catuṣkavdhabhajitam ||\* \*labdhamūlena yadyṛttam viṣkambhardhena nirmitam |\* \*sarvam caturbhujakṣetram tasminn evātiṣṭhate ||\* ### [c8.s5.4](https://indike.org/pramana/ins.science.sarma-1972.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](https://indike.org/pramana/ins.science.sarma-1972.c8.s5.5) Thus, if a, b, c, d are the sides and r the circum-radius, ### [c8.s5.6](https://indike.org/pramana/ins.science.sarma-1972.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](https://indike.org/pramana/ins.science.sarma-1972.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](https://indike.org/pramana/ins.science.sarma-1972.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](https://indike.org/pramana/in.lit.kavya.prakrit.lilavai) — Kouhala, Māhārāṣṭrī Prakrit --- Source: [https://indike.org/pramana/ins.science.sarma-1972.c8.s5](https://indike.org/pramana/ins.science.sarma-1972.c8.s5) · Pramāṇa