Surya Siddhanta

The Surya Siddhanta is a Siddhanta treatise of Indian astronomy credited to Brahmarishi Mayan . It is a more than 1,500 year old text book of Indian Astronomy. Later Indian mathematicians and astronomers such as Aryabhata and Varahamihira made references to this text. Varahamihira in his Panchasiddhantika contrasts it with four other treatises, besides the Paitamaha Siddhantas (which is more similar to the "classical" Vedanga Jyotisha), the Paulisha and Romaka Siddhantas (directly based on Hellenistic astronomy) and the Vasishta Siddhanta.

The work referred to by the title Surya Siddhanta has been repeatedly recast. There may have been an early work under that title dating back to the Buddhist Age of India (3rd century BC). The work as preserved and edited by Burgess (1858) dates to the Middle Ages. Utpala, a 10th century commentator of Varahamihira, quotes six shlokas of the Surya Siddhanta of his day, not one of which is to be found in the text now known as the Surya Siddhanta. The present Surya Siddhanta may nevertheless be considered a direct descendant of the text available to Varahamihira. This article discusses the text as edited by Burgess. For what evidence we have of the Gupta period text, see Pancha-Siddhantika. It has rules laid down to determine the true motions of the luminaries, which conform to their actual positions in the sky. It gives the locations of several stars other than the lunar nakshatras and treats the calculation of solar eclipses.

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.

Time cycles

The astronomical time cycles contained in the text were remarkably accurate at the time. The Hindu 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 1.4 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.

Planetary diameters

The Surya Siddhanta also estimates the diameters of the planets. The estimate for the diameter of Mercury is 3,008 miles, an error of less than 1% from the currently accepted diameter of 3,032 miles. It also estimates the diameter of Saturn as 73,882 miles, which again has an error of less than 1% from the currently accepted diameter of 74,580. Its estimate for the diameter of Mars is 3,772 miles, which has an error within 11% of the currently accepted diameter of 4,218 miles. It also estimated the diameter of Venus as 4,011 miles and Jupiter as 41,624 miles, which are roughly half the currently accepted values, 7,523 miles and 88,748 miles, respectively.

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

$s={\frac {g\sin \theta }{\cos \theta }}=g\tan \theta$

and the hypotenuse of the gnomon at mid-day as

$h={\frac {gr}{\cos \theta }}=gr{\frac {1}{\cos \theta }}=gr\sec \theta$

where $\ g$ is the measure of the gnomon, $\ r$ is the radius of the gnomon, $\ s$ is the shadow of the gnomon, and $\ h$ is the hypotenuse of the gnomon.

Calendrical 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" (almanac) 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.

Editions

• Ebenezer Burgess. "Translation of the Surya-Siddhanta, a text-book of Hindu Astronomy", Journal of the American Oriental Society 6 (1860): 141–498.
• Translation of the Surya-Siddhanta: A text-book of Hindu astronomy, with notes and an appendix by Ebenezer Burgess JSTOR
• Translation of the Sûrya-Siddhânta: A text-book of Hindu astronomy, with notes and an appendix by Ebenezer Burgess (1860)
• Surya Siddhanta. PDF scans of various (English and Sanskrit) editions, with and without Sansksrit commentaries. (English, Sanskrit)
• Surya-Siddhanta: A Text Book of Hindu Astronomy (1858) by Ebenezer Burgess Kessinger Publishing
• Surya-Siddhanta: A Text Book of Hindu Astronomy by Ebenezer Burgess Phanindralal Gangooly
• Surya Siddhanta Sanskrit text in Devanagari.
• सूर्यसिद्धान्त (in Unicode Devanagari)

See also

• Hindu Time Cycles
• Indian science and technology
• Indian mathematics
• Hindu astronomy
• Vedanga Jyotisha

Notes

• Victor J. Katz. A History of Mathematics: An Introduction, 1998.
• Dwight William Johnson. Exegesis of Hindu Cosmological Time Cycles, 2003.