BHASKARA'S LILAVATI

Bhaskaracharya's Lilavati:

Born in a Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara was the leader of a cosmic observatory at Ujjain, the main mathematical centre of ancient India. Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India. His main work Siddhānta-Śiromani, (Sanskrit for "Crown of Treatises") is divided into four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which are also sometimes considered four independent works. These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karaṇā Kautūhala.

Using an astronomical model developed by Brahmagupta in the 7th century, Bhāskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Earth to orbit the Sun, as approximately 365.2588 days which is the same as in Suryasiddhanta. The modern accepted measurement is 365.25636 days, a difference of just 3.5 minutes.

His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere.

His book on arithmetic is the source of interesting legends that assert that it was written for his daughter, Lilavati. Lilavati was Bhaskara II’s daughter. Bhaskara II studied Lilavati's horoscope and predicted that she would remain both childless and unmarried. 

To avoid this fate, he ascertained an auspicious moment for his daughter's wedding and to alert his daughter at the correct time, he placed a cup with a small hole at the bottom of a vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. He put the device in a room with a warning to Lilavati to not go near it. In her curiosity though, she went to look at the device and a pearl from her bridal dress accidentally dropped into it, thus upsetting it. The auspicious moment for the wedding thus passed unnoticed leaving a devastated Bhaskara II. It is then that he promised his daughter to write a book in her name, one that would remain till the end of time as a good name is akin to a second life.

Many of the problems are addressed to Līlāvatī herself who must have been a very bright young woman. For example "Oh Līlāvatī, intelligent girl, if you understand addition and subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10, and 100, as well as [the remainder of] those when subtracted from 10000." and "Fawn-eyed child Līlāvatī, tell me, how much is the number [resulting from] 135 multiplied by 12, if you understand multiplication by separate parts and by separate digits. And tell [me], beautiful one, how much is that product divided by the same multiplier?"

The book contains thirteen chapters, mainly definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, the Kuṭṭaka - a method to solve indeterminate equations, and combinations. Bhaskara II gives the value of pi as 22/7 in the book but suggest a more accurate ratio of 3927/1250 for use in astronomical calculations. Also according to the book, the largest number is the parardha equal to one hundred thousand billion.

Lilavati includes a number of methods of computing numbers such as multiplications, squares, and progressions, with examples using kings and elephants, objects which a common man could understand.