Inside the Surya Siddhanta: The Machine That Predicted Where the Planets Would Be

Saturn goes round the sky 146,568 times in 4,320,000 years. Jupiter, 364,220 times. Mars, 2,296,832. The Moon, 57,753,336. Those numbers, and a dozen more like them, are the foundation of the Surya Siddhanta. Everything else is machinery for turning them into an answer to one question: where is that planet tonight?

The machinery has four parts: integer revolution counts over one fixed period, two geometric corrections called manda and shighra, a sine table, and a fixed order of operations. Feed in a day count from a stated epoch and it returns a longitude.

The year that machine runs on is no mystery. Take the text's count of civil days in a mahayuga, 1,577,917,828, which Ebenezer Burgess locates at chapter 1, verse 37, and divide by 4,320,000: 365.2587565 days, or 365 days 6 hours 12 minutes 36.56 seconds. That is about 3 minutes 27 seconds longer than the modern sidereal year of 365.256363 days.

The surviving Surya Siddhanta is a medieval text that says of itself that it was revealed more than two million years ago. Both halves of that sentence can be true at once, and the second does not damage the first. A model is judged by whether its numbers predict the sky, and those numbers are printed where anyone can check them.

Why everything is counted in units of 4,320,000 years

A modern table gives Saturn's period as 10,759.22 days. The Surya Siddhanta does not work that way, for reasons of arithmetic rather than theology. Write a planet's speed as a period in days and you are stuck with a fraction, which every later multiplication carries along and degrades. Write it as a whole number of revolutions inside a whole number of days and nothing is ever rounded. Saturn's mean longitude on any day equals 146,568 times the day count, divided by 1,577,917,828, the remainder converted into signs and degrees. Two integers, one division, no accumulated error. The enormous period is a common denominator.

The number 4,320,000 carries religious weight elsewhere, and the text lets both meanings sit on it. But the astronomy uses it as a denominator, which shows in a detail the devotional retellings never mention. Burgess spells out the assumption: the planets, with their apsides and nodes, are taken to have started together at a fixed point of the sidereal sphere and to return to that universal conjunction every 1,080,000 years, a quarter of the mahayuga. So divide every revolution count by four. Most stay whole: the Sun 1,080,000, Mars 574,208, Jupiter 91,055, Saturn 36,642, Mercury 4,484,265, Venus 1,755,594. Two do not. Burgess's table prints the Moon's apsis as 122,050 and a quarter, the node as 58,059 and a half. Those two alone fail to return, and are quietly exempted. Were the number purely doctrinal there would be no reason for an exception. It is allowed because the apsis and node did not cooperate.

The set is also self-checking. Chapter 1, verses 37 to 40, gives 53,433,336 lunar months, 1,603,000,080 tithis, 25,082,252 omitted tithis, 1,577,917,828 civil days and 1,582,237,828 sidereal days. Multiply the lunar months by 30, subtract the civil days, and the remainder is 25,082,252, the figure given. Subtract the Sun's revolutions from the sidereal days and you recover the civil days. A dataset built to be audited.

Manda and shighra: two corrections for two problems

A planet does not move at its mean rate. It has two irregularities, and the text corrects each with its own epicycle.

Manda handles the planet's uneven motion around the zodiac. Burgess, whose 1860 translation and commentary in the Journal of the American Oriental Society remains the accessible English text, identifies it as the effect of the ellipticity of the orbit. Verses 2.34 and 2.35 give each epicycle as a circumference in degrees of a 360 degree deferent: 14 for the Sun, 32 for the Moon, and for the five star planets in the weekday-regent order Mars, Mercury, Jupiter, Venus, Saturn, the values 75, 30, 33, 12 and 49.

Shighra does something else. For Mars, Jupiter and Saturn it corrects for the motion of the Earth, which is what makes an outer planet appear to stop and loop backwards. For Mercury and Venus it does the opposite job. Verse 2.36 gives its circumferences at the even quadrants as 235, 133, 70, 262 and 39, same order.

In the Surya Siddhanta, Mercury and Venus have no mean motion of their own. Both get 4,320,000 revolutions per mahayuga, the Sun's number, and travel with it. Burgess says it directly: for the inferior planets, the revolution of the conjunction takes the place of the proper motion of the planet itself. The real period lives in a separate point, the shighrocca, at 17,937,060 revolutions for Mercury and 7,022,376 for Venus. Divide those into the civil days: 87.9697 days and 224.6986 days, against modern values of 87.9691 and 224.701.

That is not a buried claim about heliocentrism. It is what happens when a Sun-centred geometry is written from the Earth's point of view: the pieces survive as epicycle parameters. Burgess, setting the two systems side by side, notes that the Hindu and Greek models both give Mercury and Venus the Sun's mean motion while assigning them epicycles corresponding to their heliocentric orbits, and both hand each superior planet the Earth's orbit as an epicycle. The comparison worth making is not with Copernicus but with the Almagest.

The epicycle sizes therefore encode the relative sizes of the orbits, and Burgess compares them on that basis: epicycle radius over deferent for the inner planets, deferent over epicycle for the outer. On the even-quadrant values, Venus gives 0.728 against a modern 0.723, Mars 1.53 against 1.52, Jupiter 5.14 against 5.20. Mercury (0.369 against 0.387) and Saturn (9.23 against 9.54) are worse, as you would expect for a faint object low in twilight and a very slow one. Read them as brackets, since the odd quadrants give a second value.

Those quadrants are verse 2.38, the strangest thing in the chapter. The epicycles contract as the planet leaves the apsis or conjunction, reach a minimum at quadrature and expand again, the amount interpolated by a sine. Two break the pattern: the shighra epicycles of Jupiter and Saturn expand where the others contract. Burgess calls the device a remarkable peculiarity of the Hindu system and, more candidly, its most enigmatical feature. He weighs reading it as an empirical fudge fitted to observation and rejects that, since its size and placement bear no traceable relation to the true orbital elements, and he rules out an unconscious grasp of ellipses, the axis of the resulting quasi-ellipse always pointing at the Earth. It is unexplained, and whether any Greek model has an analogue is a question he does not settle.

Twenty-four numbers

Every correction needs a sine. Chapter 2, verses 17 to 22, supplies twenty-four for arcs spaced 225 minutes apart, on a radius of 3438 minutes, the number of minutes in a radian rounded to the nearest whole: 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.

The first entry is the giveaway. The sine of 225 minutes is given as 225, because at that size arc and half-chord agree to the precision the table carries. Verses 15 and 16 give a construction rule based on differences, so the whole column can be generated from the first value rather than measured. Verses 31 to 33 supply interpolation and its inverse. Burgess works the example: for the sine of 24 degrees, divide 1440 by 225 to land between the sixth and seventh entries, interpolate across a difference of 205, and obtain 1397 against a true 1398.26.

Running the procedure

For one of the five star planets the sequence is fixed. Count the days since the epoch, which the text sets at midnight on the meridian of Ujjain, between the 17th and 18th of February 3102 BCE. Multiply by the revolution count, divide by 1,577,917,828, discard whole revolutions: that is the mean longitude. Subtract it from the longitude of the apsis for the manda anomaly, from the shighrocca for the shighra anomaly. Take the sine of each from the table, scale by the corrected epicycle, divide by 360, convert back to an arc. Verse 2.45 sets the sign: added when the anomaly falls in the half circle beginning with Aries, subtracted in the half beginning with Libra.

Then comes the part that shows someone had thought hard. The two corrections interfere, so verses 2.43 and 2.44 prescribe four passes rather than two: half the shighra equation applied to the mean place, then half the manda equation applied to that result, then the full manda equation calculated afresh and applied to the original mean place, then the full shighra equation. Burgess reconstructs that order from the commentary, the text alone not being explicit enough to fix it. It is a damped iteration towards a solution neither correction gives alone. It has no physical justification and it is not exactly right, and it works.

Where the astronomers stopped agreeing with the Puranas

The Surya Siddhanta keeps the Puranic proportions. Verses 1.15 to 1.17 take a tenth of the mahayuga and multiply it by four, three, two and one: Krita 1,728,000 years, Treta 1,296,000, Dvapara 864,000, Kali 432,000, a sixth of each set aside as dawn and twilight.

Aryabhata did not. He divides the same 4,320,000 years into four equal quarters of 1,080,000, called yugapadas, and dates his own work by them: Aryabhatiya, Kalakriyapada 10, states that when three yugapadas and sixty times sixty years had elapsed, twenty-three years of his life had passed, which puts the book at 499 CE.

Brahmagupta was not gentle. Walter Eugene Clark, in his 1930 edition, quotes the Sanskrit of Brahmasphutasiddhanta 1.9, where Brahmagupta says Aryabhata declared the four quarters beginning with Krita to be equal, and that not one agrees with the amounts stated in smriti. Verse 11.4 returns to the complaint. Clark adds a caution that matters: Aryabhata nowhere states the equal division as doctrine. The word yugapada implies it, and the later tradition read it that way and objected.

The quarrel runs deeper than the yugas. Aryabhata's revolution counts, as Clark gives them, are close to the Surya Siddhanta's and not identical: Saturn 146,564 against 146,568, Jupiter 364,224 against 364,220, Mars 2,296,824 against 2,296,832, the node 232,226 against 232,238. His 1,582,237,500 rotations of the Earth leave 1,577,917,500 civil days and a year of 365.2586806 days, six and a half seconds shorter. He reckons the day from sunrise at Lanka where the Surya Siddhanta reckons it from midnight. These are working astronomers arguing about parameters, and the tradition preserved the argument instead of flattening it.

What the text says about itself

"When but little of the Golden Age was left, a great demon, named Maya, being desirous to know that mysterious, supreme, pure and exalted science ... performed, in propitiation of the Sun, very severe religious austerities."

Surya Siddhanta 1.2 to 1.3, translated by Ebenezer Burgess (1860)

On the text's own reckoning, Burgess places the revelation more than 2,164,960 years ago.

The manuscript evidence points elsewhere, if not very precisely. A treatise called Surya Siddhanta was already old in the sixth century, when Varahamihira summarised it as one of five siddhantas in his Panchasiddhantika, but that version is lost, and the standard scholarly position is that its parameters differ from the surviving text's. Al-Biruni, who wrote his account of India in 1031, ascribes a Surya Siddhanta to an author named Lata, an attribution reaching English readers through Burgess citing Reinaud rather than from al-Biruni directly. Kim Plofker is widely cited for dating the extant recension to around 800 CE, other summaries push it towards 1000, and some scholars place it in the fourth or fifth century. Burgess, working through the bija corrections later astronomers applied to the transmitted numbers, put the last adjustment in the first half of the sixteenth century, mean 1541. That was his inference in 1860, and later scholarship may have moved past it.

So the honest picture is not a book with a date but a lineage carrying a name, revised repeatedly by people who kept checking it against the sky. That is the stronger position, not a retreat. A fitted parameter is accurate near the time of fitting and drifts away from it. A year 3 minutes 27 seconds too long puts the calendar about twelve days out over the 5,127 years of Kali elapsed to now, and more than five thousand days out over two million. A model calibrated in the Krita Yuga could not still be producing usable panchangas.

The reverse inference has limits. Attempts to date siddhantas by working backwards from their errors, most famously John Bentley's, produced dates later scholarship rejected, partly because the parameters were adjusted more than once. The argument establishes an era, not a year.

Checking a figure before you repeat it

Two numbers circulate more than any others. One gives the sidereal year as 365.2563627 days, essentially identical to the modern value. The other gives the tropical year as 365.2421756 days, about 1.4 seconds off. They appear on hundreds of pages, often together, and the 1.4 seconds gets transferred to the sidereal claim by writers copying in a hurry. Neither figure is the text's, and a reader can establish that without Sanskrit.

Ask which verse. A claim of the form "the Surya Siddhanta says" with no chapter and verse is a claim about the internet, not about the book. Then check whether the text states the quantity at all. It gives no year length in days and hours anywhere, only revolution counts and day counts, so every year length attributed to it is somebody's division. Do that division and get 365.2587565.

Then run the internal audit set out above, the one that recovers 25,082,252 from the lunar months and the civil days. Any parameter set failing that cross-check is not the text's.

Then test the circulated figures against the integers. Multiply 365.2563627 by 4,320,000 and you get 1,577,907,487, not a whole number of days and short of the text's total by more than ten thousand. Nor does it belong to Aryabhata, whose count gives 365.2586806. The tropical claim fails too, refuted by the text itself: chapter 3 gives precession as 54 arcseconds a year, so the text's tropical year is its sidereal year reduced in that proportion, 365.2436 days, nearly two minutes from the modern 365.2422 rather than 1.4 seconds. Where 365.2563627 came from remains untraced here, and anyone repeating it should say where they got it.

Check too what kind of year is being compared. The text's year is sidereal, and setting it against the tropical year opens a gap of about twenty-four minutes belonging to a definition rather than to the astronomers.

What survives all that is still substantial. The precession figure is wrong: verses 3.9 to 3.12 give a libration of 27 degrees each way completing 600 times in a mahayuga, yielding 54 arcseconds a year against a true 50.3, and oscillating where the real motion does not. The sidereal year is three and a half minutes long, Saturn's period six and a half days long. But the synodic month, the quantity a lunisolar calendar actually depends on, falls out of the same integers as 29.5305879 days against a modern mean of 29.5305889, eight hundredths of a second apart. Run it forward and a full millennium of new moons slips by about seventeen minutes. Anyone with the integers and a calculator can confirm every number here, which is the only reason to trust any of them.