Counting days since the beginning

A siddhānta (सिद्धान्त), one of the standard Sanskrit textbooks of astronomy, does not store the moon's position for any particular night. It stores totals. The moon completes 57,753,336 turns of the sky in 4,320,000 years, the span the books call a mahāyuga (महायुग) or Great Age; the sun completes 4,320,000; each planet has its own figure. To place the moon tonight you scale the total down to the time that has gone by. Everything in the book therefore rests on one quantity, and the quantity is a count of days.

The count is called the ahargaṇa (अहर्गण), a heap of days: the number of ordinary days between a fixed beginning and the day you are standing in. It does the work of the Julian Day, the running day number in the left-hand column of every modern ephemeris, and the two can be checked against each other exactly. That check is most of this essay.

What differs is how you arrive at the number. The Julian Day is walked. You add up whole years, apply the leap rules, count the months off one at a time, and arrive. The ahargana is not walked. You state how many years have gone by, apply two proportions taken from a table of totals, and the answer falls out. No month lengths. No leap rule. The calendar never appears in the working, which is an odd thing to say about a calendar calculation.

First, the beginning it counts from.

The Surya Siddhanta's rule for the count runs, in Bapu Deva Sastri's translation of 1861, up to midnight at Lanka. Lanka here is not the island but the reference point of Indian astronomy, where the meridian of Ujjain crosses the equator. So the day boundary is midnight on that meridian, and the count opens at the midnight beginning 18 February 3102 BCE. That date is written in the Julian calendar, a leap day every fourth year, run backwards into centuries that never used it. Using the newer calendar rule for the same midnight gives 23 January instead. The day count does not change either way.

The other convention: Aryabhata reckons civil days from sunrise at the same meridian, so his count opens about six hours later, at dawn on that same date. Both name the same day; they differ about where a day starts, which matters in the dark hours. An event at three in the morning falls in one day number under the first convention and the number below it under the second. I use midnight, because the text I am computing from does.

In Julian Day terms the epoch is 588,465.5, midnight at Greenwich. Ujjain lies a little over five hours east, so the classical instant is that much earlier. Only the moment differs; the day keeps the same label. Everyone uses the same relation: ahargana equals the Julian Day number minus 588,465. It counts whole days, and says nothing about the moment inside a day.

Two proportions

The rule takes three inputs: years of the age completed, lunar months completed since that year opened, lunar days completed in the current month.

Step one turns years into solar months, 12 to a year, plus the months already done.

Step two adds the extra months. Twelve lunar months fall about 11 days short of a solar year, so a calendar answering to both moon and sun has to insert a whole extra month now and then, or the seasons slide out from under the festivals. It is the problem the oldest surviving Sanskrit calendar was written to solve. The book supplies totals: 1,593,336 extra months against 51,840,000 solar months in a Great Age, one insertion every 32.5 months or so. Multiply your solar months by that ratio, throw the fraction away, add.

Step three turns months into lunar days. A lunar day, a tithi (तिथि), is the time the moon takes to gain 12 degrees on the sun, and 30 of them make a lunar month by definition. Multiply by 30 and add the tithis already spent.

Step four takes some of them back. A tithi averages about 23 hours and 37 minutes, slightly short of a civil day, so the tithi count runs ahead of the day count, and now and then a tithi opens and closes inside one day and never gets a day to itself. The text calls these tithi-kṣaya (तिथिक्षय), lost lunar days, and totals them too. Multiply, throw the fraction away, subtract.

What is left is the ahargana.

The totals sit in the first chapter, worth pinning to an edition since printed editions number that chapter differently. In Sastri's, verse 34 gives the risings of the asterisms in a Great Age, which is the number of turns of the earth: 1,582,237,828. Verse 37 gives civil days as 1,577,917,828, that figure less the sun's 4,320,000 revolutions, since the sun goes round once a year the other way and swallows one turn. Lunar days are 1,603,000,080, thirty for each of the 53,433,336 lunar months, which are the moon's revolutions less the sun's. Civil days subtracted from lunar days leave 25,082,252 lost tithis.

A Friday in 2009

Take 5,110 years of the age complete, no months and no days on top, which returns the mean opening of a lunar year. Elapsed years, note, not the current year number: panchang publishers print both, and do not always say which.

Solar months: 12 × 5,110 = 61,320.

Extra months: 1,593,336 ÷ 51,840,000 = 0.0307356, and 61,320 × 0.0307356 = 1,884.71. Keep 1,884.

Lunar days: 30 × (61,320 + 1,884) = 30 × 63,204 = 1,896,120.

Lost tithis: 25,082,252 ÷ 1,603,000,080 = 0.0156471, and 1,896,120 × 0.0156471 = 29,668.72. Keep 29,668.

Ahargana: 1,896,120 − 29,668 = 1,866,452.

The first check is free. Divide by seven: 7 × 266,636 = 1,866,452, nothing left over. The epoch was a Friday, and a whole number of weeks after a Friday is a Friday.

The second check is the day number. 1,866,452 + 588,465.5 = 2,454,917.5, and turning that back into a date is four divisions. Subtract the calendar constant 1,721,119.5 and 733,798 remains. Divide by 146,097, the days in four hundred Gregorian years, and take 5, leaving 3,313. Divide by 36,524, a century, and take nothing. Divide by 1,461, four years, and take 2, leaving 391. Divide by 365 and take 1, leaving 26. Two thousand plus eight plus one is 2009, and twenty-six days past the first of March is the twenty-seventh.

Friday, 27 March 2009, as the division by seven promised. It was also the first day of the bright half of Chaitra, kept as Ugadi in the south and Gudi Padwa in Maharashtra: the opening of the lunar year. The mean rule and the printed panchang landed on the same square. Run it again with 5,109 years: 1,866,097, which comes out as Sunday 6 April 2008, where that year's Chaitra also opened.

The eclipse morning of 22 July 2009 has Julian Day 2,455,034.5 at midnight, so its ahargana is 1,866,569, 117 days past that Friday. Divide 117 by 7 and 5 is left, which counts to Wednesday. It was a Wednesday.

Where two texts part by a day

Aryabhata gives the earth 1,582,237,500 turns in a Great Age rather than 1,582,237,828. That is 328 turns in 4.32 million years, about as small as a disagreement gets. Nothing else in the rule changes.

Fewer turns means fewer civil days, 1,577,917,500 of them, so 328 more lost tithis: 25,082,580. Spread across the 1,896,120 tithis we counted, those 328 come to 0.39 of a day. My running figure stood at 29,668.72, seven-tenths of a day past a whole number. Add the 0.39 and it crosses. One more whole day is struck out, the ahargana is 1,866,451, and the date is Thursday 26 March 2009.

Neither text is careless, and the hazard is known. The classical procedure carries its own test: divide the answer by seven, count from Friday, and if the weekday is wrong, move the count by one. Brahmagupta's handbook says so plainly and the commentators repeat it. Here 1,866,451 leaves 6, which counts to Thursday, and the day was a Friday. The rule catches its own slip, using the one cycle the calendar reforms left alone.

The two agreements with the printed panchang need care. The rule I have worked is a mean rule: it moves the sun and moon at their average rates, and neither actually moves at an average rate, while a panchang uses true positions. Mean and true can part by about a day at a month's opening. Two agreements are two agreements.

There is a slower disagreement underneath. Divide the Surya Siddhanta's civil days by 4,320,000 and its year is 365.2587564 days. The modern sidereal year, the time the sun takes to return to the same star, is 365.2563630 days. That is 3 minutes and 27 seconds between them, and two counts built on those figures part by a whole day every 418 years. The tradition's answer was to shorten the run. Brahmagupta's compact handbook of 665 CE, the Khandakhadyaka, sets its zero at sunrise at Ujjain on a Sunday of that year, some 3,766 years into the age rather than at its start. A small error in a rate needs a long span to grow into a day.

Which leaves the beginning itself. Nobody stood anywhere on that Friday in February writing anything down. The date was not remembered; it was solved for. The system assumes the sun, the moon and the planets all stand together at zero longitude when the age opens, so the epoch is wherever the mean motions, run backwards, put them. Modern reconstruction says they were nowhere near gathered that morning, but spread across a wide arc of sky.

The count does not mind. A starting point need not be a real event; it needs to be a fixed spot everyone agrees on, and it needs to hold still. So the whole arithmetic hangs from a midnight nobody watched, over a meridian nobody stood on, under a sky that never made the pattern the rule asks of it: a date reached by long division and left where the division put it.