Gaṇitānanda pp 263-276

# Second-Order Interpolation in Indian Mathematics up to the Fifteenth Century

Chapter

## Abstract

The computational abilities of ancient Indian mathematicians are well known. The paper deals with the second-order interpolation schemes found in a few astronomical works of India.

## Symbols

a

the argument, circular arc measured in angular units; anomaly.

a1, a2 etc.

successive unequidistant values of a.

h

equal (common) arcual interval; elemental arc.

h1, h2, etc.

unequal arcual intervals (gatis); \begin{aligned} h_{1} & = a_{1} ; \\ h_{2} & = a_{2} - a_{1} ; \\ h_{3} & = a_{3} - a_{2} ,{\text{etc}}. \\ \end{aligned}

R

R sin a, R cos a,

.

R versin a

Indian sine, cosine and versed sine of the arc a

f(a)

the functional value of sine, versed sine or certain astronomical function called ‘equation’ (phala).

p, q

positive integers; $$x = p \cdot h$$ or $$a_{p}$$; arc passed over, such that $$f(x)$$ is known.

$$\theta$$

residual arc such that $$f(x + \theta )$$ is required to be interpolated, $$\theta$$ being positive and less than h or $$h_{p + 1}$$.

n

$$\frac{\theta }{h}$$.

D1, D2, etc.

tabulated functional differences;

\begin{aligned} D_{1} & = f(a_{1} )\,{\text{or}}\,f(h) \\ D_{2} & = f(a_{2} ) - f(a_{1} )\,{\text{or}}\,f(2h) - f(h) \\ D_{3} & = f(a_{3} ) - f(a_{2} )\,{\text{or}}\,f(3h) - f(2h),{\text{etc}}. \\ \end{aligned}

$$\Delta$$

first-order forward difference operator;

\begin{aligned} \Delta f(a) & = f(a + h) - f(a); \\ \Delta f(a_{q} ) & = f(a_{q + 1} ) - f(a_{q} ); \\ \Delta f(x) & = D_{p + 1} . \\ \end{aligned}

$$\Delta^{2}$$

second-order difference operator.

hp, hp+1

argumental intervals just passed over (last or bhukta-gati) and yet to be passed over (current or bhogya-gati), respectively.

Dp, Dp+1

the corresponding tabulated functional differences passed over (bhukta-khaṇḍa or gatiphala) and to be passed over (bhogya-khaṇḍa or gatiphala), respectively.

Dt

the envisaged true (sphuṭa) value of the functional difference to be passed over.

Zp

‘adjusted’ value of the functional difference passed over in case of unequal intervals