INTRODUCTION.

results equal to those to be derived from Âryabhaṭa;* it is then a somewhat curious circumstance—into the discussion of which I cannot enter in this place—that the dimensions of the epicycles circles used to represent the orbits of planets in ancient astronomy and the positions of the apogees the point in an orbit farthest from the Earth assumed in the Khaṇḍakhâdyaka The Edible Morsel, an astronomical treatise by Brahmagupta (as well as in the sixteenth chapter of the Pañchasiddhântikâ The Five Astronomical Treatises) differ, all of them, more or less from those recorded in the Laghu-Âryabhaṭîya The Shorter Work of Âryabhaṭa.†

The method, taught in chapter XVII, of calculating the equations of the apsis the points of least or greatest distance in an orbit and of the conjunction agrees on the whole with that prescribed in the modern Sûrya Siddhânta The Sun Treatise, a foundational text of Indian astronomy, although there are several divergences in details. Peculiar are the special rule given for Mercury in stanza 10, and the correction to be applied to Venus' place according to stanza 11. The statements as to the distance from the sun at which the planets become visible differ to some extent from those made in the modern Siddhânta; so also the greatest latitudes of the planets given in stanzas 13 and 14.

An omission which might make us suppose that the chapter as given in our Manuscripts is not complete is that nothing whatever is said about the places of the planets' nodes the points where a planet's orbit crosses the ecliptic.

The Siddhânta

We next turn to the Paitâmaha Siddhânta The Treatise of the Grandfather (Brahma) which indeed has not come down to our time, but whose teaching throughout agrees with that of a well known section of Hindû astronomical literature.

Of this Siddhânta there treats only one very short chapter, of the Pañchasiddhântikâ viz. the twelfth one; but its five stanzas manifestly suffice to reproduce everything of importance contained in that very primitive treatise. The Paitâmaha Siddhânta, known to Varâha Mihira, represents Hindû Astronomy as not yet affected by Greek influences,‡ and thus belongs to the same category as the Jyotisha-Vedânga The limb of the Veda relating to light/astronomy, the Garga Saṃhitâ The Compilation of Garga, the Sûryaprajñapti The Proclamation of the Sun and similar works. From what Varâha Mihira says about its contents, we might almost identify it with the Jyotisha Vedânga. The yuga an era or cycle of time on which the calculations of the Paitâmaha Siddhânta base is the well known quinquennial five-year one,

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* Brahmagupta's Khaṇḍa-khâdyaka begins with the following stanza:

original: "प्रणिपत्य महादेवं जगदुत्पत्तिस्थितिप्रलयहेतुं । वक्ष्यामि खण्डखाद्यकमाचार्यार्यभटतुल्यफलम् ॥"
Having bowed down to Mahâdeva (Siva), the cause of the world's creation, existence, and destruction, I shall declare the Khaṇḍakhâdyaka, which yields results equal to those of the teacher Âryabhaṭa.

† It is also worthy of notice that Âmaśarman, one of the Commentators of the Khaṇḍakhâdyaka, quotes some stanzas from a Paulîśa tantra The Doctrine of Paulisa which make the same statements about the dimensions of the epicycles as the Khaṇḍakhâdyaka itself, and, moreover, seems generally to treat the doctrines of Âryabhaṭa and the Paulîśa as equivalent.

‡ As already pointed out by me in my paper on the Jyotisha-vedâṅga, Journal of the Asiat. Soc. of Bengal 1878.