What Is The Formula For The Radius Of A Circle Paramesvara – The Mathematician Who Gave the World Cyclic Quadrilaterals 350 Years Before Lhuilier!

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Paramesvara – The Mathematician Who Gave the World Cyclic Quadrilaterals 350 Years Before Lhuilier!

Where can you find the brightest minds of our time? The idealist in me likes to believe they are hidden amidst the hustle and bustle of everyday life; but someone much more practical will tell you to look for them at a reputable university. But ancient as universities may be, there was a time when the brightest minds walked among us not as academic elites, but as ordinary men. They were not removed from our everyday reality, but used it to fuel their thoughts and theories. One such man lived in South India and Paramesvara was his name.

It is not uncommon to hear of mathematics’ eternal love affair with India; after all, zero was invented here. Then there was Aryabhata – many of whose teachings and explanations from the 5th century AD are still relevant and valid. Names like Bhaskara and Ramanujan also garner respect and recognition. But few know about another great Indian mathematician who lived at the turn of the 15th century.

Born amidst lush green landscapes in the southern part of God’s own country, Kerala, Paramesvara was a prolific mathematician and astronomer. He wrote numerous commentaries on modern and ancient Indian mathematical and astronomical treatises throughout his life, estimated to be over 55 years. He is revered for his observations of eclipses and for changing planetary parameters to create greater coherence between theoretical models and real-time observations. Perhaps the true testament to his skill, however, lies in the fact that he proposed methods and theories unheard of in India, as well as the world at large. In fact, it took a few hundred years before the West made similar or parallel discoveries.

For example, his work Siddhantadipika—a commentary on Govindaswami’s commentary on Bhaskara I’s Mahabhaskariya—includes eclipse observations, a mean-value-type formula for computing inverse sine interpolation, and a single-point iteration technique for computing any given sine. corner. It also includes a two-point iterative logarithm, which is essentially the same as the modern secant method.

In addition, Paramesvara also provided a technique to calculate the radius of a circle in which a cyclic quadrilateral is enclosed by using the sides of the quadrilateral to calculate the same. According to the technique, the radius (r) of a circle can be calculated using the formula

r2 = x/y where

x= (ab+cd) (ac+bd) (ad+bc) and

y= (a+b+cd) (b+c+da)(c+d+ab)(d+a+bc)

where a, b, c & d are the dimensions of the four sides of the cyclic quadrilateral.

This formula is usually attributed to the Swiss mathematician Simon Antoine Jean L’huiler, who lived and worked only at the turn of the 19th century – a good 350 years after Parmesvara first floated the formula.

The lack of information about Parmesvara or the lack of knowledge about his ideas and work may be due to the large communication gaps that existed at the time, as well as the turbulent history of peninsular India over the next few centuries. But having entered the twenty-first century, we have the luxury of recognizing the unsung heroes of ancient and medieval India and their pioneering visions and ideas, and sharing their stories across the globe. Why not start with Parmesvara?!

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