What we have seen so far of the Kerala School is quite interesting, to say the least. We have seen:
- A line integral (circumference of a circle).
- Areas of a circle and sphere and Volume of a sphere.
- Integrals (sankalitas) for powers.
- Power series approximation for what is called \(\pi\) today.
- Trigonometry similar to the modern version, with jyā instead of sine.
- Power series for jyā, kojyā and sara functions.
- The idea of limits.
- The differential (khanḍa-jyā) taken to the limit of small arcs, which is equivalent to the derivative.
- We have also seen the khanḍajyāntara, the second derivative of the jyā.
Was all of this merely one-offs relevant only in context, albeit substantial, or did the Kerala School produce a general system of calculus?
Interestingly, in the Tantrasaṃgraha, we come across an remarkable verse not explained in the Yuktibhāṣā:
चन्द्रबाहुफलवर्गशोधितत्रिज्यकाकृतिपदेन संहरेत्
तत्र कोटिफललिप्तिकाहतां केन्द्रभुक्तिरिह यच्च लभ्यते ॥
तद्विशोद्य मृगादिकेः गतेः क्षिप्यतामिह तु कर्कटादिके
तद्भवेत्स्फुटतरागतिर्विधोरस्य तत्समयजा रवेरपि ॥
The question at hand here is how to calculate the instantaneous (तत्समयजा) velocity of the moon (and sun), given their mean velocities and the equations of position. Since we have not delved into these astronomical models, we will merely note that the equation of position of the moon is given by (in modern terms)
\( \theta = \theta_0 – arcsin(\frac{r_o}{R}.sin(\theta – \theta_0)) \)The instantaneous velocity, therefore, is
\( \frac{d{\theta}}{dt} = \frac{d{\theta_0}}{dt} – \frac{{r_o}.cos(\theta – \theta_0).\frac{d{\theta_0}}{dt}}{\sqrt{R^2 – r_0^2.sin^2(\theta-\theta_0)}} \)And this is precisely what the verse gives. To generate this expression, one would need to know the derivative of the arcsin function, as well as the chain rule of differentiation, both of which were apparently known to Nilakaṇṭha Somayāji, though never explicitly stated. Also, the idea that the instantaneous velocity is the derivative of the instantaneous position had to have been appreciated.
Evidently, the calculus that we have seen earlier had been taken to some degree of generalization behind the scenes by the Kerala School, which is unfortunately not available to us. Knowing this gives us the impetus to reconstruct a general calculus using the means available to the Kerala School with a good amount of assurance that a similar path was tread by them.
