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A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s2: 2. Tycho Brahe's Reduction to the ecliptic

A History of the Kerala School of Hindu Astronomy (in Perspective) c8.s2

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2. Tycho Brahe's Reduction to the ecliptic

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In astronomical computations, the longitude of a planet is measured along the ecliptic, while, in fact, its motion takes place along its own orbit which, generally, deviates slightly from the ecliptic. For an accurate computation of the planet's position this deviation has to be taken into consideration. In the West, it was the Danish astronomer Tycho Brahe (1546-1601) who enunciated a formula for this ‘Reduction to the ecliptic’.¹

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¹ *Vide* his *Astronomiae instauratae Progymnasmata*, Vol. I, Prague, 1602.

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In India, a formula for this Reduction was enunciated, for the first time, by a contemporary Kerala astronomer Acyuta Piṣāraṭi (1550-1621), author of *Rāśigola-sphuṭānīti* (‘Computation of True longitude on the celestial sphere’)² and several other works on astronomy. Acyuta gives his formula for this reduction in the case of the moon in his work entitled *Sphuṭanirṇaya* (‘Computation of true longitudes of planets’) in the following verse :

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² K. V. Sarma, *The Rāśigolasphuṭānīti*, (On the calculation of true longitude on the celestial sphere and Reduction to the ecliptic in Indian astronomy), Critical ed. and Tr. with Introduction, Adyar Library and Res. Centre, Madras, 1955.

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*paṭonasya vidhos tu koṭibhujayor jīve mithas tāḍayet*
*antyakṣepaśarahatam vadham amum vikṣepakoṭyā haret |*
*labdham vyāsadaloddhṛtam himakare svarṇam, vipāte vidhau*
*yugmāyugmapadopage ; vidhur ayam spaṣṭo bhagole bhavet ||*

c8.s2.6

‘Multiply the tabular cosine (*koṭijyā*) and sine (*bhujajyā*) of the moon-minus-node and the product by the tabular versine (*śara*) of the maximum latitude (*antya-kṣepa*) of the moon. Divide this by the tabular cosine of the latitude at the particular moment and the quotient is to be divided again by the tabular radius (*vyāsadala*). The result (will give the correction for longitude which) is to be added to or subtracted from the moon’s longitude, as the moon-minus-node is in an even or an odd quadrant, respectively. The True Moon measured on the ecliptic is thus got’.³

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³ This verse has been extracted by the author in his *Rāśigola-sphuṭānīti*, verse 47, *op. cit.*, p. 29.

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Acyuta’s formula may be expressed in terms of modern mathematics thus: ‘If *F* is the longitudinal difference between the node and the planet, *w* the maximum latitude and *y* the actual latitude, then, the correction *k*=sin *F*. cos *F* (1−cos *w*) / cos *y*.⁴

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⁴ For a demonstration of the correctness of this formula in terms of modern mathematics, see K.V. Sarma, *op. cit.*, Introduction, pp. 10-13.

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In another of his works, viz. *Uparāgakriyākrama* (‘Procedures in the computation eclipses’), Acyuta gives also a simplified version of the said formula.¹

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¹ For a demonstration of this formula, see *ibid.*, pp. 13-14.

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The actual date of composition of the *Sphuṭanirṇaya*, which is the earliest work to enunciate the reduction formula, is not known, but it is definitely before 1583 which is the date of composition of the *Uparāgakriyākrama* which contains the simplified formula.²

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² On the chronology of these works, see *ibid.*, pp. 14-15.

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