Surya Siddhanta

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The Surya Siddhanta was an astronomical treatise written in India circa 400, which had rules laid down to determine the true motions of the luminaries, which conform to their actual positions in the sky. It is not known who wrote the Surya Siddhanta or when it was first compiled, but the most common dates are usually around the 4th century.

Later Indian mathematicians and astronomers such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in the Middle East and Europe.

Contents 1 Astronomy 2 Trigonometry 3 Religious uses 4 References 5 See also

[edit] Astronomy

The table of contents in this text are:

1. The Motions of the Planets
2. The Places of the Planets
3. Direction, Place and Time
4. The Moon and Eclipses
5. The Sun and Eclipses
6. The Projection of Eclipses
7. Planetary Conjunctions
8. Of the Stars
9. Risings and Settings
10. The Moon’s Risings and Settings
11. Certain Malignant Aspects of the Sun and Moon
12. Cosmogony, Geography, and Dimensions of the Creation
13. The Gnomon
14. The Movement of the Heavens and Human Activity

Methods for accurately calculating the shadow cast by a gnomon are discussed in both Chapters 3 and 13.

The astronomical time cycles contained in the text was remarkably accurate at the time. The Hindu cosmological time cycles, copied from an earlier work, are described in verses 11-23 of Chapter 1:

11. That which begins with respirations (prana) is called real.... Six respirations make a vinadi, sixty of these a nadi;

12. And sixty nadis make a sidereal day and night. Of thirty of these sidereal days is composed a month; a civil (savana) month consists of as many sunrises;

13. A lunar month, of as many lunar days (tithi); a solar (saura) month is determined by the entrance of the sun into a sign of the zodiac; twelve months make a year. This is called a day of the gods.

14. The day and night of the gods and of the demons are mutually opposed to one another. Six times sixty of them are a year of the gods, and likewise of the demons.

15. Twelve thousand of these divine years are denominated a caturyuga; of ten thousand times four hundred and thirty-two solar years

16. Is composed that caturyuga, with its dawn and twilight. The difference of the krtayuga and the other yugas, as measured by the difference in the number of the feet of Virtue in each, is as follows:

17. The tenth part of a caturyuga, multiplied successively by four, three, two, and one, gives the length of the krta and the other yugas: the sixth part of each belongs to its dawn and twilight.

18. One and seventy caturyugas make a manu; at its end is a twilight which has the number of years of a krtayuga, and which is a deluge.

19. In a kalpa are reckoned fourteen manus with their respective twilights; at the commencement of the kalpa is a fifteenth dawn, having the length of a krtayuga.

20. The kalpa, thus composed of a thousand caturyugas, and which brings about the destruction of all that exists, is a day of Brahma; his night is of the same length.

21. His extreme age is a hundred, according to this valuation of a day and a night. The half of his life is past; of the remainder, this is the first kalpa.

22. And of this kalpa, six manus are past, with their respective twilights; and of the Manu son of Vivasvant, twenty-seven caturyugas are past;

23. Of the present, the twenty-eighth, caturyuga, this krtayuga is past....

When computed, this astronomical time cycle would give the following results:

• The average length of the tropical year as 365.2421756 days, which is only 1.3 seconds shorter than the modern value of 365.2421904 days (J2000). This estimate remained the most accurate approximation for the length of the tropical year anywhere in the world for at least another six centuries, until Muslim mathematician Omar Khayyam gave a better approximation, though it still remains more accurate than the value given by the modern Gregorian calendar currently in use around the world, which gives the average length of the year as 365.2425 days.

• The average length of the sidereal year, the actual length of the Earth's revolution around the Sun, as 365.2563627 days, which is virtually the same as the modern value of 365.25636305 days (J2000). This remained the most accurate estimate for the length of the sidereal year anywhere in the world for over a thousand years.

The actual astronomical value stated for the sidereal year however, is not as accurate. The length of the sidereal year is stated to be 365.258756 days, which is longer than the modern value by 3 minutes 27 seconds. This is due to the text using a different method for actual astronomical computation, rather than the Hindu cosmological time cycles copied from an earlier text, probably because the author didn't understand how to compute the complex time cycles. The author instead employed a mean motion for the Sun and a constant of precession inferior to that used in the Hindu cosmological time cycles.

[edit] Trigonometry

The Surya Siddhanta contains the roots of modern trigonometry. It uses sine (jya), cosine (kojya or "perpendicular sine") and inverse sine (otkram jya) for the first time, and also contains the earliest use of the tangent and secant when discussing the shadow cast by a gnomon in verses 21-22 of Chapter 3:

Of [the sun's meridian zenith distance] find the jya ("base sine") and kojya (cosine or "perpendicular sine"). If then the jya and radius be multiplied respectively by the measure of the gnomon in digits, and divided by the kojya, the results are the shadow and hypotenuse at mid-day.

In modern notation, this gives the shadow of the gnomon at mid-day as

and the hypotenuse of the gnomon at mid-day as

where is the measure of the gnomon, is the radius of the gnomon, is the shadow of the gnomon, and is the hypotenuse of the gnomon.

[edit] Religious uses

The Indian solar and lunisolar calendars are widely used, with their local variations, in different parts of India. They are important in predicting the dates for the celebration of various festivals, performance of various rites as well as on all astronomical matters. The modern Indian solar and lunisolar calendars are based on close approximations to the true times of the Sun’s entrance into the various rasis.

Conservative "panchang" (calendar) makers still use the formulae and equations found in the Surya Siddhanta to compile and compute their panchangs. The panchang is an annual publication published in all regions and languages in India containing all calendrical information on religious, cultural and astronomical events. It exerts great influence on the religious and social life of the people in India and is found in most Hindu households.

[edit] References

• Ebenezer Burgess. "Translation of the Surya-Siddhanta, a text-book of Hindu Astronomy", Journal of the American Oriental Society 6 (1860): 141–498.
• Victor J. Katz. A History of Mathematics: An Introduction, 1998.
• Dwight William Johnson. Exegesis of Hindu Cosmological Time Cycles, 2003.
• Alaska Mark. Surya Siddhanta, Chapter I with Commentary and Illustrations, 2005.

[edit] See also

• Indian science and technology
• Indian mathematics
• Hindu astronomy