Synoptic Contents Page 2

Synoptic Contents

The Calculus in India

3 Infinite Series and p

109

The thousand-year background to infinite series in India and how they were derived

The underlying philosophy of pramāņa and of number is brought out

in the context of the derivation of the Indian infinite series. The full de-
tails, which are here presented for the first time, show that there was valid
pramāņa for the Indian infinite series (in contrast to Newton etc. who could
not provide their contemporaries with any clear proof or derivation of the
very same infinite series). Further, unlike the abrupt appearance of infinite
series in Europe, starting in the 1630's, the Indian infinite series evolved over
a thousand year period, as trigonometric precision was pushed from the first
minute (

Āryabhaţa 5th c. CE) to the second minute (Vaţeşvara 9th c. CE) to

the third minute (attempted e.g. by Govindasvāmin, 9th c. CE, and achieved
by Mādhava 14th15th c. CE.).

Āryabhaţa used an elegant technique of finite

differences and numerical quadrature, the numerical counterpart of the fun-
damental theorem of calculus. The use of second differences for quadratic
interpolation was then extended to higher orders, using the fraction series
expansion. "Limits" were handled using order counting, and a traditional
philosophy of neglecting non-representables. In analogy with numerical se-
ries, continued fraction expansions were used to represent an infinite series
of rational functions.

4 Time, Latitude, Longitude and the Globe

201

Why precise trigonometric values were needed in India for determination of time,

latitude, longitude, and the size of the earth

The calculus developed in India to calculate precise trigonometric val-

ues needed in connection with the calendar--(still) a critical requirement for
monsoon-driven agriculture which has long been (and remains to this day)
the primary means of producing wealth in India. The similarity of cultural
practices spread over a large area, India, led to a calendar standardized for
the prime meridian of Ujjayinī, and recalibrated for the local place. Recali-
bration required determination of local latitude and longitude, early Indian
techniques for which used the size of the globe as input. These techniques of
determining latitude and longitude were needed also for celestial navigation
for overseas trade, then the other important means of producing wealth in
India.

5 Navigation: Kamāl or Rāpalagai