Manda-Paridhi: The Epicycle of Apsis

The little circle that captured a planet's changing speed, and a first glimpse of a mathematics of motion

The manda correction fixes a planet's own uneven speed, and its size is set by a small circle called the manda epicycle. Learn how it works, and how Bhāskarācārya pushed it further, computing a planet's instantaneous velocity and touching the edge of a mathematics of change.

The Question That Has No Duration

In the twelfth century, at Ujjain, the great astronomer Bhāskarācārya, whom we call Bhāskara the Second, set himself a question that sounds almost impossible to answer. He wanted to know not merely where a planet was, and not merely how far it moved across a whole day, but how fast it was moving at a single instant.

Think about how strange that is. Speed is distance divided by time. But an instant has no duration, no stretch of time to divide by. Ask how far a planet travels in no time at all, and the honest answer is: nothing. And yet a planet plainly has a speed at each moment, sometimes hurrying, sometimes slowing. Bhāskara wanted to catch that speed at the instant, the tātkālika-gati, the "motion of the moment." To do it, he would have to reason about a quantity vanishing to zero, which is the very edge of what would later grow into the mathematics of change. This lesson is about the device that made his question possible: the small circle that models a planet's uneven speed, the manda-paridhi.

Why a Planet Will Not Keep Its Pace

Recall the problem from the last lesson. The mean planet moves at a perfectly steady rate, but the real planet does not. It runs a little quick in one stretch of its orbit and a little slow in the opposite stretch. The first correction, the manda-phala, accounts for exactly this, and to picture it the astronomers used a small circle.

Imagine the planet not simply riding its main orbit but also carried around a little circle whose centre sits on that orbit. That little circle is the manda epicycle, the epicycle of the apsis. As the planet goes around it, it is pulled a touch ahead of the mean position in one half and a touch behind in the other. The size of that small circle, its circumference or paridhi, decides how large the correction can grow. A bigger manda epicycle means a bigger swing between fast and slow.