The equant in India: The mathematical basis of ancient Indian planetary models

Archive for History of Exact Sciences

Volumn 59, Issue 6, 2005, Pages 563-576

  Duke, Dennis ^a  

^a FLORIDA STATE UNIVERSITY   (United States)

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[No Author keywords available]

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EID: 26444588661     PISSN: 00039519     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00407-005-0096-y     Document Type: Article

References (38)

1. D. Pingree, "History of Mathematical astronomy in India", Dictionary of Scientific Biography, 15 (1978), 533-633.
2. K. S. Shukla, Aryabhatiya of Aryabhata (1976);
3. K. S. Shukla, Mahabhaskariya of Bhaskara 1 (1960).
4. O. Neugebauer and D. Pingree, The Pancasiddhantika of Varahamihira (2 vols, Copenhagen, 1970-71);
5. B. Chatteijee, The Khandakhadyaka of Brahmagupta (1972).
6. Thus for both outer and inner planets, the mean motion given is the heliocentric mean motion of the planet. There is no textual evidence that the Indians knew anything about this, and there is an overwhelming amount of textual evidence confirming their geocentric point of view. Some commentators, most notably van der Waerden, have however argued in favor of an underlying ancient Greek heliocentric basis, of which the Indians were unaware. See, e.g. B. L. van der Waerden, "The heliocentric system in greek, persian, and Indian astronomy", in From deferent to equant: a volume of studies in the history of science in the ancient and medieval near east in honor of E. S. Kennedy, Annals of the new york academy of sciences, 500 (1987), 525-546.
7. More recently this idea is developed in about as much detail as the scant evidence allows in L. Russo, The Forgotten Revolution (2004).
8. D. Pingree, "The Paitamahasiddhanta of the Visnudharmottapurana", Brahmavidya, xxxi-xxxii (1967-68), 472-510.
9. E. Burgess and W. D. Whitney, "Translation of the Surya Siddhanta", Journal of the American Oriental Society, (1858) 141-498.
10. O. Neugebauer, "The Transmission of planetary theories in ancient and medieval astronomy", Scripta mathematica, 22 (1956), 165-192.
11. Essentially the same argument was presented again in O. Neugebauer and D. Pingree, Scripta mathematica, ibid. (ref. 3).
12. B. L. van der Waerden, "Ausgleichspunkt, 'methode der perser', und indische planetenrechnung", Archive for history of exact sciences, 1 (1961), 107-121.
13. see, e.g. appendix 7 of B. Chatterjee, Archive for history of exact sciences, ibid, (ref.1);
14. S. N. Sen, "Epicyclic Eccentric Planetary Theories in Ancient and Medieval Indian astronomy", Indian Journal of History of Science, 9 (1974), 107-121;
15. S. N. Sen, "Survey of studies in European languages", Indian Journal of History of Science, 20 (1985), 49-121;
16. D. A. Somayaji, "The yuga system and the computation of mean and true planetary longitudes", Indian Journal of History of Science, 20 (1985), 145-187.
17. see, e.g., D. Pingree, "On the Greek Origin of the Indian Planetary Model Employing a Double Epicycle", Journal for the history of astronomy, ii (1971), 80-85;
18. D. Pingree, "The Recovery of Early Greek astronomy from India", Journal for the history of astronomy, vii (1976), 109-123;
19. D. Pingree, Journal for the history of astronomy, ibid. (ref. 1).
20. R. Mercier, 'Astronomical tables in the Twelfth century', Adelard of Bath, an English Scientist and Arabist of the early Twelfth century, ed. Charles Burnett, (London:Warburg Institute, 1987), 87-118,
21. reproduced in: Studies on the Transmission of Medieval Mathematical Astronomy, Raymond Mercier. Variorum Collected Studies Series CS787, Ashgate: Aldershot, Hants, 2004.
22. D. Rawlins, "Ancient heliocentrists, Ptolemy, and the equant", American journal of physics, 55 (1987) 235-9.
23. H. Thurston, "Greek and Indian Planetary Longitudes", Archive for History of Exact Sciences, 44 (1992) 191-195.
24. In this and the succeeding plots, the y-axis gives the difference in degrees of the planet's position computed from modern models and the position computed from an ancient model. Hence the curve is, for all practical purposes, the error in the prediction of the ancient model.
25. For example, for Mars I use e = 12 for the eccentricity of the eccentre and 2e = 12 for the total eccentricity of the bisected equant. Using the values that Ptolemy finds at the first iteration of his equant solutions for the eccentricity of the eccentre, e.g. e = 13;07 for Mars, has no significant effect on this discussion.
26. The reader might also wonder about the pattern of errors of the sunrise and equant models with respect to the true longitude of the planets. In the case of Jupiter the errors using the Almagest parameters can be significantly reduced by (1) adjusting the mean longitude to remove the bias in Ptolemy's parameters due to the well-known fact that his tropical year is too long, and (2) making a small adjustment to the period in longitude. These changes reduce the mean error to essentially zero and reduce the r.m.s error by about a factor of 3, the residual error being due to the fact that the equant combines the eccentricities of the Sun and Jupiter. For Mars the Almagest parameters are already much nearer optimal, but when Mars is near the Earth the errors resulting from combining the eccentricities are greatly magnified: see, e.g. J. R. Voelkel and O. Gingerich, "Giovanni Antonio Magini's "Keplerian" tables of 1614 and their implications for the reception of Keplerian astronomy in the seventeenth century', Journal for the history of astronomy, 32 (2001), 237-262. The fact that the equant and the sunrise theories agree poorly for Mars is, of course, simple a result of the fact that r/R ≈ 0.66, and the approximations used to derive the sunrise model are not very accurate.
27. G. J. Toomer, Ptolemy's Almagest (1984), 420-3.
28. B. L. van der Waerden, Ptolemy's Almagest, ibid., ref (4), suggests that these remarks of Ptolemy are referring explicitly to presumed Greek precursors of the Indian schemes, but the argument seems tenuous, at best.
29. O. Neugebauer, A history of ancient mathematical astronomy, (1975), 781-833.
30. Even in the case of the Almagest, we know now that few if any of Ptolemy's parameters were derived as he describes. See J. P. Britton, Models and Precision: the quality of Ptolemy's observations and parameters, (Princeton, 1992), derived from his unpublished 1966 Yale PhD thesis;
31. C. Wilson, "The Inner Planets and the Keplerian revolution", Centaurus, 17 (1972) 205-248;
32. R. R. Newton, The crime of Claudius Ptolemy, (Baltimore, 1977);
33. N. M. Swerdlow, "Ptolemy's Theory of the Inferior Planets", Journal for the history of astronomy, 20 (1989) 29-60;
34. D. Duke, "Ptolemy's Treatment of the Outer Planets", Archive for History of Exact Sciences, 59 (2005) 169-187.
35. The chief proponents of the western view are D. Pingree. Archive for History of Exact Sciences, ibid. ref. (10)
36. and B. L. van der Waerden, Archive for History of Exact Sciences, ibid. ref. (4) and ref. (8), although they differ in opinion on some of the details. The view of many Indian scholars is that the Indian astronomy of the siddhantic period, including the planetary schemes, was largely developed in India and was the culmination of many centuries of pre-siddhantic astronomical investigation. For a survey of this view,
37. see, e.g. S. Kak, "The development of astronomy from Vedanga Jyotisa to Aryabhata", in Science and Civilization in India, Vol. 1, Part 2: Life, Thought and Culture in India, edited by G.C. Pande, (2001), 866-885, and references therein.
38. B. L. van der Waerden, Science and Civilization in India, ibid. (ref. 8).