Bhaskaracharya, an Indian mathematician and astronomer of the 12th century, is considered the progenitor of Differential Calculus – 500 years before Newton and Leibniz. He is referred as Bhaskara II to avoid confusion with Bhaskara I (of the 7th century CE). He was born near Vijjadavida (Bijapur in modern Karnataka) and lived between 1114-1185 CE. He represented the peaks of mathematical knowledge in the 12th century and was the head of the astronomical observatory at Ujjain, the leading mathematical center of ancient India.
Bhaskaracharya is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhaskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.
His main work Siddhanta Shiromani, (Sanskrit for “Crown of treatises,“) is divided into four parts called Leelavati (beautiful woman, named after his daughter Leelavati), Bijaganita, Grahaganita (mathematics of planets) and Goladhyaya (study of sphere/earth). These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karna Kautoohala. Leelavati, contains many algebra-related teasers, which have become the subject of significant research by scholars. These teasers are in the form of shlokas which pose the problems. The shlokas need to be interpreted correctly to decipher the meaning in order to find the solution.
पार्थ: कर्णवधाय मार्गणगणं क्रुद्धो रणे संदधे
तस्यार्धेन निवार्य तच्छरगणं मूलैश्चतुभिर्हयान् |
शल्यं षड्भिरथेषुभिस्त्रिभिरपि च्छत्रं ध्वजं कार्मुकम्
चिच्छेदास्य शिरः शरेण कति ते यानर्जुनः संदधे || ७६ ||
The meaning of this shloka is a question formulated as follows:
- during the battle between Arjuna and Karna in the Mahabharata, Arjuna released some arrows. Of the released arrows:
- half were consumed in stopping the arrows coming from Karna;
- 4 times the square root of the arrows were consumed to control the horses of Karna’s chariot;
- 6 were for gaining control over Shalya, the charioteer of Karna. (Shalya was the maternal uncle of Nakula and Sahadeva);
- 3 were used to take on the umbrella and flag of the chariot and the bow of Karna; and
- finally Karna was killed by a single arrow.
So how many arrows were released by Arjuna in the battle? Basic algebra easily yields the answer to this question, if the equation is formulated correctly. Let the total number of arrows be X. The statements above can be reduced to the algebraic form:
X = X/2 + 4√X + 6 + 3 + 1
Solving this, we get the value of X=100 for the total number of arrows shot by Arjuna.
However, the fun is not just in getting the algebra right. There is so much hidden information in this shloka. If we think a little deeper about the hidden meanings, it reveals that
- even for an atirathi like Arjuna, it took as many as 50 arrows to stop the arrows of Karna – reflecting upon the skills of Karna. An atirathi is a warrior capable of contending with 12 Rathi class warriors or 60,000 warriors simultaneously, circumspect in his mastery of all forms of weapons and combat skills;
- that the horses needed 40 arrows to immobilize the chariot tells us about the kind of training given to the horses in the battle field;
- when even the horses needed 40 arrows, that Shalya the charioteer surrendered with just 6 tells us that he is favoring Arjuna;
- 3 arrows to take the chariot and the bow shows the helplessness of Karna; and
- once everything is in control the enemy should vanquished in just a single arrow.
So the rules and skills required to win such a battle operationally are: firstly, stop the enemy fire-power; second, immobilize the enemy by taking on his mobility- the horses and the driver; thirdly signal to him his helplessness by destroying the carriage, and finally eliminate the enemy himself.
If we analyze the same shloka on the spiritual side, we can see that
- to attain ultimate liberation, firstly one needs to control over his/her personal desires, this being a very difficult task, thus takes 50 arrows;
- then take control over 5 senses and sensual pleasures indicated by the horses. The 40 arrows needed to do this indicate the difficulty of the task;
- gaining control over 5 senses will lead one to the control over the consciousness (manas, thought, ego) indicated by the driver; and
- if all the foregoing are done, achieving the ultimate liberation (moksha) should be relatively easy.
The elegance of shlokas is good mathematics has been beautifully illustrated by the ancient Indian mathematicians.