## Q 1124 q i 1 q 0 1 124

Since there are 9 padas in 1 day, the daily motion of the Moon in the first half of the month is given by

D(p) = 1(p + 9) - 1(p) = ^ + 42, an increasing linear function of p, and so the difference between successive daily motions is constant (D(p + 9) - D(p) = 3/14). A similar calculation for the second half of the month shows that the daily motion is then a decreasing linear function of q (though there are minor complications around the end and the middle of the month) and so the effect of the model is to produce a zigzag function for the variation in the daily motion of the Moon. The maximum and minimum daily motions according to this theory are 14° 39' 9" and 11° 42', respectively.

Early Indian astronomy based on Greek geometrical models is best known through the Surya Siddhanta, one of several similar works of unknown authorship written around the fourth century. The basic mechanism of Hindu planetary astronomy in works such as this was an epicycle on an eccentric deferent, which is consistent with the idea that the knowledge came from Greece between the time of Hipparchus - who had no geometrical planetary theory - and of Ptolemy - who used epicycles on eccentric deferents together with the equant construction.3 However, we also find an imaginative modification that seems to have been an indigenous invention. In Aryabhata's Aryabhatiya (written in about 500), the author makes the eccentricity of the deferent and the radius of

A thorough discussion of the transmission of Greek planetary models to India can be found in Pingree (1971). The parameters that actually were used in these geometrical models were probably of Babylonian origin (see, for example, Abhyankar (2000)).

the epicycle vary periodically between two (not greatly different) values. Hindu latitude theories were very basic, indicating that the complex latitude theories of the Almagest did not develop over a long period prior to Ptolemy.

The biggest influence of the Indian civilization on the evolution of Western mathematical astronomy, however, came not from the arithmetical or geometrical models that were employed, but from the development of new mathematical tools. When using the Greek chord function in astronomical calculations, astronomers often had to deal with half-chords of double angles. Indian mathematicians realized that it would be simpler to tabulate the half-chords themselves; these are our modern sines (though, of course, they still referred to lengths rather than ratios). The Aryabhatiya contains a description of the construction of a sine table for the angles 0 to 90° in steps of 3° 45', and Brahmagupta (seventh century) devised fairly accurate methods of interpola-

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