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A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s3: 3. Newton-Gauss Interpolation formula

A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s3

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3. Newton-Gauss Interpolation formula

c8.s3.1

It has been demonstrated in detail that Indian mathematics employed, in its search for better results, improved rules of interpolation, by using the second differences and that a particular case, upto the second order, of the more general Newton-Stirling interpolation formula was known to Brahmagupta (*c* A.D. 625).³ Following this line of thought, the Kerala astronomer Govindasvāmin (*c*. 800-850) has enunciated a set of rules for the computation of intermediary functional values, different formulae being laid down for different augmental intervals:

c8.s3.2

³ R. C. Gupta, ‘Second order interpolation in Indian mathematics upto the fifteenth century,’ *Indian Jl. of Hist. of Sc.*, 4 (1969) 86-98.

c8.s3.3

*gacchad-yāta-guṇāntaravapur yātaisya-diṣvāsanac-*
*chedabhyāsa-samūha-kārmukakṛti-prāptat tribhis taḍitat |*
*vedaiḥ ṣaḍbhir avāptam antyaguṇāje rāśyoḥ kramād antyabhe*
*gantavyāhata-vartamāna-guṇajāc cāpāptam ekādibhiḥ ||*

*antyād utkramataḥ krameṇa viṣamaiḥ saṅkhyāviśeṣaiḥ kṣiped*
*bhaṅktvāptam, yadi maurvikāvidhir ayam makhyāḥ kramād vartate*
*śodhyam vyutkramatas tathākṛtaphalām...*

(*Bhāṣya* on *Mahābhāskarīya*, 4.22)⁴

c8.s3.4

⁴ *Mahābhāskarīya* of Bhāskarācārya, with the *Bhāṣya* of Govindasvāmin and the super-commentary *Siddhāntadīpikā* of Parameśvara, ed. by T.S. Kuppanna Sastri, Madras, 1957, pp. 201-2.

c8.s3.5

‘Multiply the difference of the last and the current sine differences by the two parts of the elemental arc (made by any intermediary point on it) and divide by the square of the elemental arc and further multiply by three. Now divide the result so obtained by four in the first rāśi, or by six in the second rāśi. The final result thus obtained should be added to the portion of the current sine difference (got by linear proportion).

c8.s3.6

‘In the last (third) rāśi, multiply the linearly proportional part of the current sine difference by the remaining part of the elemental arc and divide by the elemental arc. Now, divide the result (so obtained) by the odd numbers (1, 3, 5 etc.) according to the current sine difference (is first, second, third etc.), when counted from the end in the reverse order. Add the final result thus obtained to the portion of the current sine difference (got by ordinary proportion). These are the rules for computing true sine differences for (direct) sines. In the case of versed sines, apply the rules in the reverse order and the above corrections are to be subtracted from the respective differences (got by linear interpolation)’.¹

c8.s3.7

¹ For the rationale of this formula, see R.C. Gupta, *ib.*, pp. 91-92.

c8.s3.8

Using the general functional notation and finite difference operator, the rule for the second rāśi may be put as :

c8.s3.9

$f(x+nh)=f(x)+nΔ f(x)+(n(n-1))/2Δ f(x)-Δ f(x-h)$

c8.s3.10

which is a particular case (up to the second order) of the general Newton-Gauss interpolation formula.²

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