Seasons from the Sun
Kamal Lodaya, Bengaluru

This is the last article in this series. After reading the earlier articles in this series, you must be mixed up! Lunar calendars, seasonal calendars, nakshatras, rashis, sankrantis,... how many things is a person to keep track of? Indians were good at arithmetic, they used it to get themselves out of the mixup.

As mentioned earlier, a lunar month is the time taken by the moon to go over a complete cycle, from new moon to new moon (or full moon to full moon). So 12 lunar months add up to less than a full year whereas 13 lunar months are more than a year.

Sankranti is the migration of the Sun from one rashi (zodiac) to another. There are 12 sankrantis in one year. So, 12 times a year, the Sun moves from one rashi into another. When it moves into a rashi, that month is named according to the rashi it moves into. For example, when the Sun ransits into Mesha rashi, it enters the first month called Chaitra.

Since the lunar month is shorter than the sankranti division, every now and then, in fact every 2.71 years, there will be a lunar month during the Sun lies within the same rashi. It enters (migrates) into this rashi from the previous one during the previous lunar month and migrates out of this rashi into the next one during the next lunar month. For instance, if the Sun is still in Mesha and has not transitted into Vrisha rashi, then the next lunar month is called an Adhika Masa (extra month). In fact, it will be called Adhika Vaisakha. When the next lunar month occurs, the Sun would have transited into Vrisha rashi and the next month will be an (ordinary) Vaisakha month. So an extra month is inserted to keep the solar and lunar calendars aligned. Or thinking about it another way, because the sankranti division is longer than the lunar month, there will be two sankrantis between which two New Moons happen, not just one New Moon. This kind of reasoning is called a {pigeonhole principle} (see the Box).

BOX: Pigeonhole counting principle
If two Full Moons happen in a month, the second one is called a Blue Moon, nothing to do with the colour of the Moon. In modern mathematics the argument about two New or Full Moons having to happen in a seasonal month is called a pigeonhole principle, nothing to do with pigeons.

If several items (pigeons) are put into boxes (pigeonholes), and there are more items than there are boxes, then there must be a box which contains more than one item.

This seems quite obvious to any one who has done some counting. This counting principle first appears in 1622 in a book by the French Jesuit priest Jean Leurechon. Why no one saw this earlier was that they wanted to calculate the answer. The pigeonhole principle is logical, if there are 11 pigeons and 10 pigeonholes, it does not say which pigeonhole has two pigeons.

END OF BOX

Indians did not know pigeonhole principles. But they could calculate! The separation between Sun and Moon on the Sun's path is 0 degrees at New Moon and 360 degrees at the next New Moon. They said a {tithi} is the time during which the separation between Sun and Moon increases by 12 degrees, so that 30 tithis add up to a lunar month of 360 degrees. Although they used different units, a tithi is 29.52 days/30 days times 24 hours/day times 60 minutes/hour, which is 23 hours 37 minutes. So 360 tithis is about 11 days less than the 365 days which forms 1 year.

So in 2.71 years, a shortage of 11 days/year gives a shortage of 11*2.71 = 29.81 days, which is more than the tithi of 29.52 days. So a gap of more than one lunar month is created after this duration. Inserting an extra month after 2.71 years would fill up the gap. 

Tithis are imaginary quantities, about one day long, but less than a day! Even tithis do not quite work, and need correction. See below.

Imaginary quantities were much discussed by Indian mathematicians of those times. Brahmagupta of Bhinmal, Rajasthan, in his Brahmasphutasiddhanta, had introduced laws for arithmetic including zero and negative numbers in the 7th century CE. If numbers could be imaginary (negative), why could not tithis be imaginary? So now one could divide the month, starting from say from an amavasya (new moon) into 15 tithis leading to purnima (full moon), and then 15 tithis leading to the next amavasya. This fixes the festivals, Ashwina month starts the day after Bhadra purnima and Vijayadasami is on the 10th tithi after Ashwina purnima.

Let us calculate
To follow a particular calculation, Jyeshtha amavasya was on 13th June 2018 CE. Call the next day, day 1. The Mithuna sankranti associated with the month Jyestha was on 15th June 2018.
Jump 27 seasonal months, actually 2 years 3 months and 4 days later, to day 827. That is 17th September 2020.
How many tithis have gone by? It is (827 days / 29.52 days/month) times 30 tithis/month = 840 tithis, that is 28 panchanga months, 2 years and 4 months, have happened. So it should have been Ashwina month amavasya and Tula sankranti should have been happening then.
But the Sun was one month behind. In fact, the sankranti of Kanya, the rashi before Tula, happened on 16th September, and 30 tithis later on 16th October, the next amavasya happened. The Sun was still in the same Kanya rashi, and Tula sankranti was on the next day, 17th October. So this was another Ashwina amavasya.

Two amavasyas between two sankrantis define an adhika month. 18th September to 16th October became Ashwina adhika month. Then came Ashwina month from 17th October associated with Tula sankranti. The second amavasya filled up the gap. The first nine days of Ashwina are navaratri, tenth day is Vijayadasami. So Dasara or Dussehra was on 26th October 2020 and Ashwina poornima (also called Sharad poornima) on 31st October, after Ashwina adhika poornima on 1st October.

In the modern calendar also, two Full Moons happened that October, so 31st October was a Blue Moon according to that calendar also!

Since there was an adhika masa in 2020, the next one will come one year + 1.71 years later, that is, 32.52 months later. Following the tithis on a panchanga calendar, the next adhika month can be calculated to begin on 15th August 2023.

Why panchangas failed
Brahmagupta's calculations with negative numbers were accepted by the world. Did Varahamihira and Bhaskara II's tithi calculations, adopted by Indian panchanga makers, become popular?

Did you spot the flaw in the Indian calendar? Here it is. Suppose you know when Magha purnima was and want to prepare your colours for Holi, which is at the next Phalguna purnima. When is Holi? When did the tithi of Magha purnima end, and when will the 30 tithis after that end? You no longer know because a tithi can begin at any time during the 24 hours of a day. You have to look up a panchanga.

Julius Caesar's reform was simpler, because when you wake up from your sleep with the sunrise, the next day has begun. A whole society can easily work with the seasonal calendar. Working with a panchanga requires some knowledge and calculational ability, and not everyone has that.

Like we have a clock to measure time, can't we have a panchanga clock to measure tithis? Then if we want to burst crackers on Diwali, we can look at our panchanga clock and say tonight is when we do it. Well, that is a little awkward: you want to buy your crackers a few days earlier, not on Diwali day itself. Also, conventions regarding when a festival begins are different in Indian calendars. Some calendars say that if the tithi for Diwali begins after sunset, then Diwali is the next night. Some calendars say that if the tithi for Diwali had begun after sunset, then Diwali is that night.

Sunrise and sunset are at different times in different parts of India. Delhiites will have long days in summer with early sunrise, Chennaiites will have shorter days. Mumbaiites will have sunrise around 6:30 IST, on the same day in Guwahati sunrise may be around 4:30 IST.

Calendar reforms
Reality is even more complicated. The Sun (actually, the Earth) and Moon do not move at uniform speeds in their paths. To explain this, Johannes Kepler in Prague formulated his laws in the 17th century CE. Not knowing the laws, Indians added two approximate numbers to make corrections to tithis. Sudha Bhujle and Mayank Vahia in Mumbai have calculated what these numbers should have been, based on modern data.

Based on a proposal by Italian doctor Luigi Lilio, the Julian calendar was reformed by Pope Gregory XIII of the Roman Catholic church in the 16th century, to have three leap years less every 400 years. So 2000 was a leap year, but 2100, 2200 and 2300 will not be, even though they are divisible by 4, because they are not divisible by 400. This gives an average of 365.2425 days per year for the Gregorian calendar.

Another reform was proposed by Serbian climatologist Milutin Milankovitch in the 20th century. This is to be followed by the Eastern Orthodox church from the year 2800 onwards. It has seven leap years less every 900 years: that is, after 2000 and 2400 CE, the next 29th February in a century year will be in 2900 and not in 2800. This gives an average of 365.24222 days per year for this Milankovitch calendar. All these reformers kept in mind that their reforms should be acceptable to a whole society.