# Surya Siddhanta

The Surya Siddhanta is the name of multiple treatises (siddhanta) in Indian astronomy. The extant text as translated by Burgess (1860) is medieval (c. 12th century), but it is clearly based on older versions, thought to have been composed in the early 6th century AD.

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 as well as solstices, e.g., summer solstice 21/06. Significant coverage is on kinds of time, length of the year of gods and demons, day and night of god Brahma, the elapsed period since creation, how planets move eastwards and sidereal revolution. The Earth's diameter and circumference are also given. Eclipses and color of the eclipsed portion of the moon are mentioned.

## Textual history and influence

In a work called the Pañca-siddhāntikā composed in the sixth century by Varāhamihira, five astronomical treatises are named and summarised: Paulīśa-siddhānta, Romaka-siddhānta, Vasiṣṭha-siddhānta, Sūrya-siddhānta, and Paitāmaha-siddhānta.[1]:50 Judging from the epoch dates in the work, Plofker suggests that this Sūrya-siddhānta was composed or revised in the early sixth century.[1]:50

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 version was modified by Bhaskaracharya during the Middle Ages.[citation needed] It is partly based on Vedanga Jyotisha, which itself might reflect traditions going back to the Indian Iron Age (around 700 BCE).[2]

It is hypothesized that there were cultural contacts between the Indian and Greek astronomers via cultural contact with Hellenistic Greece, specifically the work of Hipparchus. There were many similarities between Suryasiddhanta and Greek astronomy in Hellenistic period. For example, Suryasiddhanta provides more accurate and detailed table of sines than Hipparchus.[3] However, the epicyclical model of Suryasiddhanta was simpler than that made by Ptolemy in the 2nd century.[4]

The table of sines may reflect the only extant version of the original table by Hipparchus, which was lost in the West,[3] but which underwent tradition within Indian astronomy for at least a millennium before reaching its extant form. Because the tradition of Hellenistic astronomy was essentially stopped short in the West after the end of Late Antiquity, the Surya Siddhanta came to play an important part in the history of science, as its survival allowed transmission of the knowledge of trigonometry into Islamic astronomy and from there back to medieval Europe by the 12th century.[5] Surya Siddhanta was one of the two books in Sanskrit translated in Arabic in the later half of eighth century during the reign of Abbasid Khalifa Almansur, and was one of the first books to be translated during the movement for translating world heritage in Arabic.[citation needed]

## Contents

### Astronomy

The mean (circular) motion of planets according to the Surya Siddhantha.
The variation of the true position of Mercury around its mean position according to the Surya Siddhantha.

1. The Mean Motions of the Planets
2. True 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....

#### 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.[6]

### Trigonometry

The Surya Siddhanta contains the roots of modern trigonometry. Its trigonometric functions jyā and koti-jyā (reflecting the chords of Hipparchus) are the direct source (via Arabic transmission) of the terms sine and cosine. It 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 midday as

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

and the hypotenuse of the gnomon at midday as

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

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