Seasons from the Sun

This is a series about calendar makers from history. The last two articles discussed how the Moon's motion among the stars every day was recorded by 27 {nakshatras}, and the Sun's motion among the stars every month was recorded by 12 {sankrantis}.

The beginning of calendars
The earliest calendar we know of is from 2500 BCE in Egypt. One of the most important events of the year for Egyptians was Sopdet, the brightest star in the sky, rising at dawn in August. Because within a few days of this, they knew, floods would arrive in the river Nile.
The flooding of the Nile has thus been celebrated in Egypt since ancient times. This is because Egyptians knew that this enormous amount of water could be used for irrigation. Even today, Egyptians have an annual holiday for two weeks starting August 15, known as Wafaa El-Nil.
For Egypt's farmers, having this forecast at hand was very important. So it is no wonder that the Egyptian calendar focused on the Sun and the time of the night at which the star Sopdet (we call it Sirius or Vyadha or Rudran) rises. It also used some other stars. We call it a {seasonal} calendar.

Lunar Calendar
We saw that from one full Moon to the next takes 29.53 days. Twelve such lunar months gives 12 months times 29.5 days/month = 354 days. There is a 11 days gap to match the lunar calendar with the seasonal year, when Sun-Earth-stars come back to the same relative positions. What to do about the extra days every year?
In a desert climate, seasons and cultivation were less important. Forecasting the phase of the Moon to guide travel at night was more important. In the 7th century CE, prophet Muhammad gave a sermon in Mount Arafat (Arabia) giving primacy to the lunar months, ignoring the 11 days gap altogether. This was incorporated in the Islamic calendar, whose months keep shifting by this gap every year. Thus Muslims fast for Ramadan at different times every year. We call it a {lunar} calendar.

The missing days
As early as the 21st century BCE, calendars in Sumeria (Iraq), another agricultural civilization, tried to use both the seasons marked by the Sun and the months marked by the Moon. They would follow the lunar calendar. After around three years, when there would be a gap of 33 days, the king would decree that there would be an {intercalary} month (we call it {adhika} or extra month), and there would be 13 months during that year. This would bring the calendar more in line with the seasons.

Leap of day
So you leaped ahead and spotted that this leaves a gap still? Yes, who would keep track of all this? Kings were usually busy fighting other kings and expanding their kingdoms. They had astrologers who would maintain this information. They would advise their kings on when it was time to introduce an extra month.
Could one devise a calendar which would do this by calculation?

Box: Egyptian festivals
One bunch of festivals celebrated by the Egyptians in 2500 BCE were called Wag, Wepet-Renpet and Tekh. Their dates depended on the flooding of the Nile in summer, some time in August. They celebrated the death and rebirth of Osiris, the Egyptian god of fertility, agriculture, vegetation, death, resurrection and life. Solemn rituals marked Osiris's death: people would make small boats and shrines of paper, which would be floated on the Nile. Then lamentations (poems) were recited to call Osiris to his feast. Feasting, drinking, singing and dancing marked his rebirth.
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Box: Islamic festivals
Facing religious persecution, prophet Muhammad moved (this is known as the Hajira) in 622 CE from Makkah (Mecca in English) to Yathrib, which was later renamed Madinah al-Nabi (city of the prophet, or simply Medina). He found that the people there celebrated festivals of merriment. Muhammad instead introduced Eid al Fitr, the breaking of the fast after Ramadan. It is the first day of the month of Shawwal on which Muslims are forbidden to fast. They have to go to the Eidgah (mosque) and pray, followed by a sermon. Zakat al fitr, an act of charity, has to be performed before the prayer. This year the month of Ramadan is from Monday 12 April to the amavasya of Tuesday 11 May. Eid al fitr will be celebrated this year from the evening of Wednesday 12 May and on Thursday 13 May.
You may think 2021 marks the beginning of the 1400th year since the Hajira. That is not so, because by the Islamic lunar calendar, 1442 lunar years have passed since then. 11 days every year have led to the passing of 42 more lunar years.
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Box: Deepavali
Jains trace the tradition of lamps being lit to commemorate the {nirvana} of their teacher Mahavira. Buddhist monks are said to have lit lamps for dark nights of {amavasya}, New Moon.
Hindu {puranas} have many legends and traditional stories. Those from after the Gupta empire, possibly composed in South India around the 8th century CE, mentioned the festival of lights. So Deepavali, celebrated on Kartik amavasya, may have emerged as a combination of earlier harvest festivals with emphasis on cleaning houses, lighting lamps and wearing new clothes.
It is one of the most popular Indian festivals, taken by Indian migrants to countries all over the world. This year Deepavali or Diwali begins on Thursday 4 November.
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Box: Months and festivals

Month name, Purnima/amavasya date 2021, events

Pausha, January 28,
Magha, February 27,
Phalguna, March 28, Holi
Chaitra, April 27,
Vaishakha, May 26, Buddha purnima and partial lunar eclipse
Jyeshtha, June 24, 
Ashadha, July 24, Guru purnima
Shravana, August 22, Raksha bandhan
Bhadra, September 20, 
Ashwina, October 20, 
Kartika, November 4, Diwali and Kartika amavasya
Kartika, November 19, Kartika purnima
Mrigashira, December 19,