 Okay, welcome everybody and thank you Professor Pal for agreeing to give this talk here. He has given a talk here last year at this center and there are many in the audience who have attended it and I can tell you we are all looking forward to this talk. I will confess that what I was not looking forward to was trying to give a very short introduction to Professor Palash Barunpal. So what I suggest is you Google him, he is easy to find, you will get a lot of details. So I will keep this as brief as I can. He is as was said in the posters and the email. He is a senior professor of the theory group at the Sahaa Institute of Nuclear Physics in Calcutta and after his bachelor's and master's from Calcutta University he went on to get his PhD from the Carnegie Mellon University in Pittsburgh in the US. So I won't talk about his work, look it up, I mean his research work, his areas of interest but apart from his large volume of research papers and physics textbooks at the advanced college level. He is also written in Bengali and in English, many popular and pedagogical books and articles in science. For example a history of dates and years which was translated, I think originally in Bengali and translated into English, a history of measurements, essays on relation of science with other aspects of human life, superstition, religion, science and mathematics and so there are, when I start to try and make this little cheat sheet I find there are many alsos in it and there won't be any time to go over them. To give you an idea the last talk he gave here was on three horizons for technological, sorry terminological vocabulary in Indian languages. So that will give you an idea of the breadth of his interest in work and because this is a talk on calendars and there are many students here, I'll leave you with one question. The talk he gave last year was almost exactly a year ago except that our talks are always on Thursdays, so you have to tell me which date in June that talk must have been on. So Professor Pan, please. Thank you very much for inviting me, for asking me to give this talk and for this very kind introduction and also thank you for this if you have left it for me, although I usually can't use it. So the topic or the title as you see is the history and mystery of calendars. Almost everything has a history, so probably you are not surprised to see the word history along with calendars but the mystery part will come, I want to keep it mysterious. It will come as I go along. Now everything that I am going to say is contained in this book in Bengali Shaltari Ke Ritihash which I wrote, I forgot when it first came out but this is the third edition which is now the one you can get and you know this is the advertisement part before the actual program begins. So this is the and this one came out in like 2015 quite recently. This is an English translation of it, not made by me but by a very good physicist who made the translation. The translation is a translation of the second edition of the book and the third edition of course as you can imagine is much better than the second edition. So if you know Bengali then you should read the Bengali version and if you don't know Bengali learn Bengali. So before I go to the next slide I want to set up some conventions about this talk. Some of the notations that I am going to use and something that will help you. One of these things is this number here. So at any point of time you should know roughly how much of the talk is still not done or in case you fall asleep see the number before you fall asleep and when you wake up see the number so you will know how much you have missed. The second thing is that this apart from these pictures and the little words here you will see that the background is black. As I go along you will see the background will change actually sometime it will change. So when it will change it is like going to a different chapter of the book. So it is like making a starting a new thing in the talk. So now you see the background has changed so it is the beginning of chapter one welcome. So what does a calendar do? A calendar counts days into months and months into years. You can also talk about other units like weeks and so on but they are not very important. Why not? I might explain at the end of the talk. So days, months and years these are the main preoccupations of a calendar. And as soon as I have said that I feel that I have already said a sentence which needs a lot of explanation. Why? Because in this sentence there are three words day, month and year which are not very easy to define. So let us just see why. Let us take the word day. What do we mean by a day? There are two definitions of day actually even in colloquial language. One is the day which is let us say a 24 hour cycle and the other is just the lighted part of the 24 hour cycle. So we will try not to use it in the second sense always in the first sense. So in the first sense a day means the time taken by the earth to complete one rotation around its axis very easy. The problem is how do you know when one rotation is completed? You see this is a designer clause when the earth was created if it had been fitted with a bell such that after one rotation there will be a ting sound you would have known very easily but the designers did not do it. So we do not know how one rotation is completed. I mean not that we do not know but it is a question that needs some thought and I ask the question I will ask many such questions not all questions can be answered in one hour. So I will answer some of them and we will just raise the others so that you can go and read the book. See I am still in the advertising mode I am sorry I will leave it soon. So the next unit that was in that fateful sentence that I uttered little while ago is the ear. So ear you will say okay it is the same thing again it is the time taken by the earth to complete one revolution around the sun and I will ask the same question how do you know exactly when one revolution is completed? Again this question I am not answering right now I will come back to it later but then there is the other unit which is in between the day and the month and it is a very convenient unit and in practical life the way you define a month is that the duration between your two paychecks. So it is a very convenient thing that the nature has divides such a unit because with just the day and the year getting paid would have been very difficult. So but the month in nature's way is the time from one full moon to the next or one new moon to the next. I really do not like the word new moon because the new moon means actually no moon. So I will stick with full moon. Now notice one thing when I define the day and the year I talked about one complete rotation or one complete revolution but when I define the month I did not say that this is the time that the moon takes to revolve once around the earth. I did not say that I said one full moon to another full moon why did I not say that. Let us see this picture. This is the sun and these green things are the positions of the earth at different times. The earth still probably looks greenish from the space. If it goes like this way for another 100 years or so probably there will be no green left but still now I prefer to depict the earth with the green color and then this little thing is the moon. So at this point A when the earth is at A the sun, the earth, the moon they are in the same line. So the moon I mean in this picture is the left side of the moon that gets the light from the sun and that is the side also we see from the earth. So this is the full moon day. Then time passes so earth moves and the moon also goes around the earth and then there is a position B when again the moon comes in the same line as the earth and the sun but this time in between the two. Now here it is actually no moon because the lighted part of the moon is away from us but you see here from here is not 180 degrees it is more than 180 degrees and the same thing continues namely when it actually finishes 360 degrees and let us say the earth is at C then it is not a full moon because now the three are not on the same line and a little later at D again it will be a full moon. So the full moon takes more than 360 degrees of the moon going around the earth. So the time the moon takes from going from A to C is called the sidereal month. Sidereal means something related to stars so this thing you can how will you know it has gone 360 degrees by comparing with very far stars which hasn't noticeably moved in that time period. So that is how you determine it and that time period is roughly 27.3217 days but the time from A to D which is one full moon to one full moon that is called the synodic month and that is larger as I said it is a little more than 29 and a half days 29.5306 days. So these are the different definitions of the month that you can get and we will most of the time use this definition of the synodic month probably all along if I have to use the sidereal month I will give you warning before that. So herein lies the basic problem of a calendar. The problem is that these ratios of the three different units they are not integers one year is 365.2422 days I mean it is actually more I can go more but I don't need more accuracy than that. So it is a 365 days 5 hours 48 minutes 46 seconds and one month that is the synodic month as I just said is 29.5306 days which is little more than 29 days and 12 hours and if you divide these two then you will see that one year is actually 12.3683 months that means 10 days 21 hours etc etc. So this is the main problem the problem is that they are not integers so how will you keep track of these fractions you cannot possibly design a calendar and say that I mean the next month starts at 413 p.m. tomorrow that is not allowed. So you will have to do integers but you will have to use integers on a system which is not integer inherently which doesn't have integers inherently. So how to manage the fractions this is the main problem of calendars and the answer in one word is by intercalation. Intercalation means sometimes you will have to insert something. So I will give examples later but for now if you feel too tense without knowing the meaning of the word I will say it is like the 29th of February on leap years. Sometimes you will have to insert some extra things to keep the account of these fractions that is the whole idea. Now because of this intercalation different ways of intercalation has given rise to three different broadly speaking three different calendrical systems. One of the systems is the solar calendar. What the solar calendar does is that it keeps solar years. So the months have nothing to do with the full moon or anything the months are just some convenience subdivision of the year as I said so that we can get paid. It does not attempt to correlate the phases of the moon with the days of the month. For example the fact that today is the 15th of June if I ask you therefore what will be the phase of the moon unless you know from some other information you will not be able to tell from that information. So examples of this calendar there is the Gregorian calendar which is now the international calendar according to which today is the 15th of the June and I will talk about it later. There is the Bengali calendar which is also a solar calendar, Tamil calendar these are some of the examples there are many more many more calendars like this. The second system is called a lunar calendar. It keeps the synodic months and the year is equal to 12 such months. So remember that the synodic month is roughly 29 and a half days multiplied by 12 we will get 354. So in a lunar calendar the year is 354 days roughly because it is not exactly 29 and a half sometimes it can be 354 sometimes it can be 355 but that is it and then the year ends new year starts. So new year and new month but so that means that the year does not correlate a month with a particular season. I will give you example and really I know only one example. In modern world there is only one calendar as far as I know which is a lunar calendar and that is the Islamic calendar I will come back to it. The third system is sort of something in between the two and here the months at the synodic months. So that is similar to the lunar calendar. The difference with the lunar calendar is that the year in this calendar, lunar solar calendars the year at an average is the solar year not each year is 365 point whatever number of days at an average. So how do they do it? If the lunar month is 29 and a half days how do they do it? The answer is some years have 12 months some have 13 that is how they keep the average equal to roughly 365. So there are examples almost all calendars in the world are lunar solar calendars. Whatever I have shown you in the other adjust some exceptions almost everything calendars of northern and western India. I do not know if you probably go out here you can get the Kalni Noi calendar that is lunar solar, northern India lunar solar, Chinese calendar, Hebrew calendar, Babylonian calendar, Greek calendar you name it almost everything is lunar solar calendar. So I will also come back to it. So now my task is to give some examples of the three different kinds of calendars and this is the next chapter. So this is the first example the Roman calendar. The Roman calendar in its present form is a solar calendar. It was not always so and I will show you the transition. The calendar the solar calendar was actually borrowed from Egypt. You know there is a strange way we read history that if we are asked name one important person whom Julius Caesar encountered during his you know his time in Egypt and almost everybody will say Cleopatra because he had a romantic affair with him. But the affair with Cleopatra is nothing with the affair with another person called Sosigenus and probably most of us have never heard the name of Sosigenus. This is the bad way of doing history. You will see what Sosigenus did in a while. Egyptians are sun worshippers. So of course you see why they had a solar calendar and the Romans actually borrowed it. Julius Caesar borrowed it. And this is the calendar which has evolved to become the international calendar. And this is also the calendar which in India we commonly call the English calendar. It is a very very very few more very bad name and I will explain why but I have a request. If you like that my talk even a little bit at the end of the talk at least take a vow that for three months you will never say English calendar. I will I will tell you why. So first let us talk about the Roman calendar. How was the calendar? The calendar before I mean well there was a there was a emperor in Rome. His name was Numa Pompeius. He was around 700 BC. Before Numa the calendar was like this. It started with the month Matthews. If you want to translate it to English this will be March. So it had 31 days. Then Aprilis 30, Mayus 31, Eunius 30 again. So these four months were named after some gods and goddesses of Rome at that time. Then the next month 31 days is called Quintilis. Quinti in Latin means five. So this was the fifth month. The month after that is called Sextilis. So you know this has something to do with sex. This is there are very many young people in this audience I have to explain what sex is. In Latin it means six. I do not explain what it means in any other language. So this is the sixth month and then September, September, just remember the word September 7. Sevenh month. Octa, Astha, Navauthor, Nahbem and Dasam. So there were 10 months, 31 and 30 days and the total was 304 days. That is the shock. Why 304 days when the Romans so dumb that they did not know that 304 days does not make a year. Not even nearly 304. But they knew. But the point is, what happens after December? Winter. You do not do anything. What is the point of counting this? So, they would stop the calendar at that time. And then when spring comes, at certain time they would decide, now the new year begins and they will start a new year. Actually, this is very similar to my own calendar when I was a child. Around the 4th or 5th of December the final exams will be over and then it was whole day we were very busy because morning there will be cricket and then there will be lunch and then afternoon there will be cricket and then come back and then evening there will be badminton and then we will go to sleep and it was like this. We would never care what day it is, which day of the week and so on until one day my mother would say in the morning, hey, get up, you have to go to school today. So, the calendar starts again. This is a very civilized way of doing things and that is what the Romans used to do. So, this changed at the time of Numa, around 700 BC. What Numa did is that he said, well, it is better to have two months there. So, they put two months and they were called January's and February's. You can understand what they mean in English. But the days of the months were changed. Let us start again from March. It is 31, 29, 31. You see all 31s or 29s? Not 30 at all. Why? Because around that time, the 700 BC, the Romans have developed a kind of superstitions that the even numbers are evil numbers. They are bad numbers. So, no month can be 30 days. It would be very bad. So, everything is 31, 29. And of course, you see February is 28. Why is that? Because if February is also odd, then the total 12 months becomes even. Then the whole year is worked out. So, the total year is 355 days and this is what it is. Now, you can see 355. It should remind you that I told you that 29.5 days a month multiplied by 12 is 354. They are close to that number. Now, what they knew that 355 days doesn't make a year, solar year. So, what they would do is that every other year, they would add roughly 20 or 23 days to the year at the end. Of course, those days will be holidays unless you have to go to war for some reason. So, and that way they will keep track of the whole thing. So, that was at the time of Numa. Now, in 46 BC, Julius Caesar has come back from Egypt. And he has brought back with him, not Cleopatra, Sosigenes. Sosigenes was a very famous astronomer there. So, Julius Caesar has, while in Egypt, he has learnt the way the Egyptian calendar works. And so he was very intrigued and impressed. So, he brought Sosigenes along and said, do something to our calendar. Our calendar is a mess. So, with the Sosigenes' help, there was some new calendar which was suggested. Let me again start from March. So, first of all, they said that even number thing, forget about it. Maybe they already have gotten over that superstition by that time. I don't know. But now, it's 31, 30, 31, 30, 31, 30, 31, 30, 31, 31. And now, you will have to make it 29 in order to make the total 365. But then actually, Sosigenes also knew that it's not 365 days exactly. It's a little more than that. And the Egyptians thought that this extra amount is exactly one quarter of a day. So, they also put a rule that, you know, after four years, you will have to add one day. So, if you add one day, you will add it to the February. And you see, in that year, it is completely symmetrical, right? 31, 30, all along. Beautiful, beautiful scheme. So, Julius is a sort of put a notice, putting this calendar into effect. And also, he thought that he has done so much about the calendar that the calendar should actually remember him. So, this month, which used to be called Quintilis, he changed the name. Changed the name to Julius. So, that is Julius' reform. But there was a problem with Julius' reform. The problem is about this adding the extra day. And of course, I don't know Latin to understand the exact problem. But the problem, the nature of the problem is the following. The way this extra day addition was proposed is that they said after three years, add an extra day. By which they meant that wait for three years. And then in the fourth year, add an extra day. But it was interpreted by saying that every third year, you have to add an extra day. Of course, Julius knew what was the right thing, but they have already butchered Julius, right? So, 44 BC Julius is already gone. So, there was nobody to correct them. I don't know what happened to Saucerianus in all this turmoil, but probably he was also not there. But this was noticed about 38 years later at the time of Augustus. Now, 38 years, if they had used this rule, intercalation rule correctly, there would have been nine extra days in the 38 years. But because they were doing it every third year, there's already 12 of them. So, three extra days have been added. So, this was brought into the notice of Augustus. And then he said, yeah, that has been a mistake. So, from now on, I mean, whatever they probably corrected the thing, the three days. And so, from now on, it is every fourth year. So, wait three years, do nothing. And then the fourth year, add one day. Now, human ego manifests in many different ways. Just by saying this much, Augustus thought, well, Julius has a month in his name. So, why shouldn't I have? So, the name of Sextelis was changed to Augustus. But he did something much worse than that. He said, Julius' month has 31 days. I cannot stand my month having 30 days. So, August will have 31 days also. And now to correct the things, you have to give all sorts of arbitrary things. September is 30 and blah, blah, blah. And then finally, whatever you cannot, you know, you cannot balance, it goes to February. That's the last month of the year, remember. So, February got 28. So, it's a very, very, very bad system now developed. But Augustus' name stuck there. And after Augustus, the person who became the Roman emperor, his name is Tiberius. Now, Tiberius looked at the calendar. And supposedly, he said, what will happen to the calendar when the 13th emperor will come to Rome? Because everybody has his own month. So, what will the 13th emperor do? So, Tiberius made a rule that nobody will be able to change the names of the months. Now, personally, I mean, it's a nice story, but I really don't appreciate what Tiberius did. If I was in Tiberius' place, I would have first named a month after me and then made this rule. But, well, you can see Tiberius was a better person than Polar Paul. So, he didn't do that. So, there is no month in Tiberius' name and the calendar has remained like this. So, the next big step in this calendar were some reforms. And the biggest of those reforms is the Gregorian reform. Before I can get to that, let me tell you why the reform was necessary. So, you see, the average year in a Julian calendar is 365 plus one quarter of a day, which is 365.25. But the real astronomical year is 365.2422. So, there is a difference, 0.0078 days, which means 11 minutes and 14 seconds. Now, is that serious? 11 minutes? Who cares about 11 minutes? Well, you have to care because these 11 minutes actually accumulate over the years. So, in thousand years, the discrepancy that will accumulate is this times thousand, which is 7.80 days. That's about 8 days. That's something you have to agree. So, in 325 AD, there was a convention of the Christian church people in somewhere in western Asia, Nicaea, and they noticed that something is wrong. It's not exactly agreeing. Already, there is some accumulation by that time in 300 years. So, they didn't know what is wrong. So, they just said, we don't know, but the vernal equinox, that is the time when in the spring time, the lighted part and the dark part of the day becomes equal, that was set on 21st of March. They said whatever happened, happened, but from now on, we should treat our calendar in a way that the vernal equinox falls on 21st of March. Then in, but of course, they didn't take any corrective measure. So, this problem started accumulating again. And in 1471, some scientists in Rome appealed to the pope. There was no emperor by this time in Rome. The Roman emperor has fallen, living with us the task of writing the causes of the fall in our history exams. So, now the head of Rome is the pope. So, the scientist appealed to the pope, whose name was Sixtus IV, asking for a calendar reform. And in Vatican, this room where they actually did their measurements, made their measurements, that still kept like that and it's called the calendar room. Anybody who would visit, visit Vatican would please see this room. And anybody who has visited probably have seen this room. Now, actually the 1471, the appeal was made and the action was taken very quickly. As you can see, within 111 years they decided that yes, yes, something needs to be done. By that time, the pope was Gregory XIII, so he approved the reform. And that is why this reform is called the Gregorian reform. Now, there's a lesson of this thing that if you want to do something, please don't wait for 111 years. Because then you don't get the name. This reform should have been named by Sixtus. But Sixtus waited for 111 years. There are other things might have happened in 111 years, which are obvious. I need not tell you, but please don't wait very long. So, 1257 years have passed since the Nikkei Convention. So, the discrepancy accumulated is this 0.0078 per year times the 1257, which is 9.8 days. That's about 10 days. So, in 1582, 4th of October was followed by 15th of October. 10 days were taken off the calendar. In fact, there is a beautiful site on the internet now, which is called dateandtime.com or .org or maybe timeanddate.org. Something, time and date, com, org, this is some combination of these things. There you can just give the name of any country and any year and it will show you the calendar for that year, for that country. So, give Italy and give 1582. You will see 1, 2, 3, 4, 15, 16, 17, 18, so on. So, that is what was done. But then you have to make a new rule so that the problem doesn't happen again in future. So, the rule was this, that the years which are divisible by 100, for those years, according to Julius, they would have been leap years one extra day. But now, according to Gregorian calendar, they will also have to be divisible by 400 in order to be a leap year. So, for example, 1700, 1800, 1900 should not be leap years. 2000 will be, 1600 will be. So, you see this is a very shrewd thing done. They were doing it in 1582. The next such year where there is a conflict with the Julian calendar is 1700. 1600, it is the same. So, the first time there will be a difference will be 100 and something years later. Let those people take care of it. We will not be there at that time. So, 2000 should be a, well, 2000 was a leap year. So, now the average length of the year after this reform is 365 plus in any cycle of 400 years, according to Julius, there were 100 leap years. Now, there are 3 less as you see from here. So, there will be 97. So, if you do this, this is 365.2425. That is still not correct because you remember that the original thing is 0.2422. So, there is still a discrepancy which is 0.0003 days per year. But with that discrepancy, one discrepancy of one day will be accumulated in more than 3,300 years. Now, of course, you can still ask, what will you do after 3,300 years? Well, the answer is very simple. You use the earth in a way that it does not last for 300,000 years. And that is what we are doing now. So, beautiful solution. Now, what happened to this suggestion of Gregory? 1582 as soon as the suggestion was made, it was obviously taken by the Catholic countries because they were directly under the Pope, Italy, France, Spain, Portugal. 1583, Switzerland, Netherlands. 1584, Germany was still not one country. Many of the others were still not one country. But Germany had two different parts. There was a Catholic part and there was a Protestant part. If you remember your history enough, Martin Luther has already arrived. So, Protestantism is there in the air. So, the Catholic part of the Germany actually agreed to the Pope's suggestion. But the Protestants said, no, no, no. We are not under Pope, so we won't have to take his suggestion. Until, more than 100 years later, in 1700, they said, well, even if that was Pope, it's something we should, I mean, this is not really religion. This is something science. So, we should agree. So, the Protestant Germany also agreed to the Gregorian reform. And now, oh my God, now. So, I have been talking for more than half hour. Have I ever talked about any contribution from England? No. In 1752, England woke up and said, oh, there was, 170 years ago, there was some reform proposed. We didn't do anything. Maybe we should do something. So, by this time, the discrepancy was 11 days. So, September 2nd was followed by September 14th. So, you see, there is a calendar which we call English calendar. That was not proposed by the English, that the reforms were not taken by the English. When the reforms were thrown on their faces, they didn't wake up until 170 years. So, if that is called an English calendar, then I'm sorry. Tomorrow, if you say that Rasagulla is a sweet made in Mumbai, I'll have to protest. So, the calendar has nothing to do with England. But of course, because of the change, they adopted some, there were some changes. For example, Newton's birthday, you will see in some books, it will 25th of December, the Christmas day of 1642. And in some books, it's the 5th of January of 1643. When I was a little child, I read these both dates and then I thought that such a great person. Had even two birthdays. So, but anyway. So, of course, I cannot give the history of all countries and not everything is interesting. In Russia, in 1917, the Bolshevik revolution took place. The Zars were dethroned. The Zars were sort of aristocratic idiots. They didn't have any knowledge about anything. But the people who came, Lenin, for example, no matter what you say after the fall of communism in Russia and in Eastern Europe, Lenin was an extremely educated person. This you cannot take away from him. And Lenin knew that the calendar is in, there is a problem. So, immediately after the revolution, the revolution happened in October or November. He said that the calendar has to be, there has to be reform. And so, Russia adopted the Gregorian reform. But by this time, the discrepancy was 13 days. So, the 31st of January was followed by the 14th of February. So, again, the same kind of problem. For example, the beginning of the Russian revolution, somewhere it says it's the 25th of October, which is why they call it the October revolution. And in some text, they will say it's the 7th of November, which is why they call it the November revolution. They're talking about the same thing and the same day even. But one in the wrong calendar and one in the right calendar. This is the point. So, this is the history of the international calendar, the Gregorian calendar, the Roman calendar, whatever you want to call it. And I will stop this history here and go to some other calendar. I'll go to Islamic calendar. So, Islamic calendar, as I told you before, it's the only major lunar calendar that exists in the world. So, the average length of the year is one sidereal month, which is a little more than 29 and a half times 12, which is 354.3672 days. So, the individual years are either 354 or 355. So, you see, there's a difference with the solar year, which is about 11 days. So, the dates in this calendar sort of move with respect to a solar calendar or with respect to the seasonal cycle and why about this 11 days. To explain what I mean, let me give you the dates of the Idul fitr. The Idul fitr happened after the end of the month, which in Arabic is called Ramadan. And in Persian, it's called Ramzan. And in Indian languages, we learnt most of the things from Persian directly, not from Arabic directly. So, we call it Ramzan or in Bengali, we call it Ramzan. So, these are the dates. Look from the bottom. In 1998, it was 30th of January, 99, 20th of January. So, you see, roughly 11 days, 10 days actually, but it can be 354 or 355. So, it can be 10 or 11. So, 20th of January, next year 2000, it's 9th of January. But then something happened. The next one was not in 2001, because if you count 354 days starting from 9th of January, you don't go into 2001. It's 28th of December, 2000. So, in 2000, there are two Idul fitr. And then again, 2001, it's moved 11 days, again 11 days, 11 or 10, 11 or 10 all the way. I have here until 2011. Now, homework, without looking at the calendar, you just calculate when is the Idul fitr in 2017 and then go back to your home and check. And if you don't have a calendar at home, just wait a few more days, because it is coming in a few more, in a few days. So, this is the Islamic calendar. And now, I will talk about some lunar solar calendar. So, let's start with a chart, just like the one I showed before. This time, I'm showing you the Gregorian calendar dates for what is called the Shera in most of India. And it's called Bijay Dasami in Bengal. So, 1997, it was 11th October, 98, 1st October. See, moved 10 days. But now, it will not be like the lunar calendar. What happens is that there is a one month intercalated between this 1998 the Shera and the next the Shera. So, if it were not intercalated, the date would have gone back by 11 days. But now, you have intercalated one month, which is 29 days. So, actually, it will go forward by 29 minus 11, that is 18 days. So, from 1st of October, it will be 19th of October. Next year, it has again gone back by 11 days and then again one month intercalated. So, it again went forward by 18 days and so on. It goes on and at some points, one extra month is introduced so that the average length of the year remains the solar year. And that's how these dates change. So, the question is, when do you introduce these extra months and how often do you introduce these extra months? Roughly, once in three years, because the discrepancy, as I said, is like 10 or 11 days and one month, let's say roughly is 30 days. So, roughly once in three years, but that's only roughly. Let's be a little more precise. At the beginning of the talk, I told you that one year is 12.3683 months. So, if I introduce one month every three years, that means I am introducing at an average one-third month every year, which is 0.3333. That's not very close to 3683. That's quite far. So, you have to look for some fraction which is close to this and there is a fraction which is 7 over 19, which is very close, 3684. So, if you intercalate seven months every 19 years, then roughly everything goes smoothly. So, how and when these months are added in let's say you have a 19-year cycle, when do you add the month? Now, this varies from one lunisoral calendar to another. Here, I give an example of the Hebrew calendar. In the Hebrew calendar, what they do is that let's say I have a 19-year cycle which is from 1 to 19. I have written here and at the end of the first year, I have 0.3683 months accumulated. So, don't do anything. At the end of the second year, twice as much, still don't do anything. At the end of the third year, if I had let it accumulate, it would have been 1.1. So, now I put in one month. So, whenever this accumulation becomes more than one or comes very close to one, then they would put one extra month. So, that's how in the 19-year cycle, every 3rd, 6th, 8th, 11th, 14th, 17th, 19th, they will interpolate or intercalate one extra month and that is how the Hebrew calendar works. And now it's a new chapter and I'm sure you knew that this is a chapter which should come. Not only because the speaker is Bengali, but the Bengali calendar has some very unique features which has to be talked about. And by the way, for other Bengalis present here, the features are not nice. So, the Bengali calendar is a solar calendar as I have told you before. It follows the calculations of a 16th century text called Suryasiddhanta. According to this text, one year was 365.258756 days. Now, this is more than 365 days, 6 hours, 12 minutes and so on. Whereas, the actual thing is remember 2422. So, there is a discrepancy and the discrepancy is 0.0166 days per year. Because of this, the Bengali calendar is actually shifting because there is no Gregorian reform or anybody's reform. So, the Bengali New Year Day, for example, now appears every year almost 14th or 15th of April. Now think about it, what is so special about 14th or 15th of April? Try to explain somebody 14th or 15th of April without using the Gregorian calendar. You will see, it is very difficult. 14th or 15th, nothing happens by which you can describe it. Nothing happens in nature. So, what is the thing? The New Year was originally set on the vernali queen of day, which was 21st of March. And because of this accumulation per year, in 1400 years or so, the accumulated amount is 23 days. So, from 21st of March, you count 23 days, that is your 15th of April. Something as recent as Rabindranath's birthday. Rabindranath was born, Rabindranath's birthday in calendars, it comes on 8th or 9th of May. But Rabindranath was born on the 6th of May. This Rabindranath himself has written. I will of course say that Rabindranath does not have a very good memory of that time. But certainly other people in his family did and he must have heard it from his mother or somebody else in the family. So, this was the birth date. But since then, this is 151, now it is actually more, 156 or something. But anyway, this is just a rough thing. So, more than 150 years have passed and so the accumulation is 2 and a half days. And that explains the shifting from the 6th of May to the 8th of May. So, if the calendar goes on like this, no Gregory comes, Rabindranath's birthday will keep on shifting, the new year will keep on shifting. You will not notice every year because to shift one day, it takes roughly 70 or 80 years. As you see, two days shifting has taken roughly 150 years. So, that is like a lifetime of a person. So, a person will not probably realize in his or her lifetime that it is shifting. But if you continue it for a few hundred years, then you will see the shift. So, why is this discrepancy of 0.0166 days per year? Well, the calculations were not very refined in 6th century AD. That was a long time ago certainly. But that is not the main problem. The main problem was that in this calculation, the Suryasiddhanta people did not take the precision of the equinoxes into account. So, now that I have said that, I have to explain what is precision of equinoxes. But this is a science audience, so I can go fast. What are the equinoxes? Well, you see, this is the orbit of the earth around the sun. This is the sun figuratively at the middle and this is the yellow line. Can everybody see the yellow line even from the back? So, this yellow line is the axis of the earth. Now, the orbit of the earth is tilted with respect to the axis. I mean, the axis is not perpendicular to the orbit. That is what I mean. So, at this point, the northern hemisphere is closer to the sun than the southern hemisphere. So, it is summer in northern hemisphere, winter in southern hemisphere. Opposite point, just opposite. Summer in southern hemisphere, winter in northern hemisphere. But in the middle, there are two points where this line will be exactly perpendicular to the line joining the earth and the sun. So, those are the two points we will call the equinox points. So, the north pole and the south pole will be equidistant from the sun. So, one of them is called the vernal equinox. The vernal, when it comes in the spring and the other is called the winter, sorry, the autumnal equinox. Something which comes in the autumn. So, the summer solstice in north hemisphere, summer solstice happens on 21st of June. Autumnal equinox on 21st of September. Winter solstice on 21st of December. And vernal equinox on 21st of March. I think something is wrong. Not all of them are 21. Maybe autumnal equinox is 23rd. Maybe this is a typographical error. But anyway, roughly around that point. So, now you see that there are, now that we know about this, we talked about this axis of the earth. Now, we will have to say that the earth has three major types of motion. We have talked about two before. One is the daily motion, the diurnal motion around its axis. Then there is the annual motion around the sun. But then there is a third motion which is called the precision. And that precision means that this axis which I showed you in the last slide, that axis is also not, does not point to a fixed direction in space. It actually wobbles. So, it is like a toy top. It wobbles the axis. This wobble takes a long time. One complete wobble takes more than 25000 years. So, it is less noticeable than the others. But the thing is there. And this is the thing that the Surya Siddhanta did not take into account. Now, what will happen because of the wobble? You see, I told you in the last slide that the height of autumn and the height of spring comes when the north pole and the south pole becomes equidistant from the sun. Now, if the axis were pointing at the same direction, that means that starting from one spring equinox, one vernal equinox, you go around 360 degrees and then it is again equidistant from the sun. But the axis is not pointing in the same direction. So, as you are going, the axis is also doing, I am exaggerating the motion, but the axis is also doing like this. So, it will not take one full 360 degree before that. The axis has already tilted. So, it will become equidistant a little before that. So, if you consider this thing, then only the length of the year is 365.2422, which is what we had been using before. And this is the thing which determines the seasons on the earth. Because this is the thing which determines when it will be spring, when it will be autumn and so on. Going around 360 degrees has nothing to do with our daily lives. We do not care whether it has gone 360 degrees. If you ask me which one is more important, 360 degrees or this equidistant from sun, north pole and south pole. So, of course, equidistant from the sun, north pole and south pole because that is what determines the spring, what determines the autumn. It determines when you get good cauliflower in the market. It determines when you will get good ilish in the fish markets and so on. So, this is the important part, whether it goes 360 degrees in some space, who cares. So, this is the important thing, but if you ignore this effect, then the length of the year becomes 365.2564 considerably bigger. That is going complete 360 degrees. So, Surya Siddhanto said 2586. So, they made a little error. That small error is understandable considering that it was happening 1400 years ago, but they were actually trying to measure the wrong thing. And the Bengali calendar makers have been using this measure, the wrong measure of the year all along. But it was realized in I think end of 19th century or maybe beginning of 20th century, it was realized that the Surya Siddhanto calculation was not correct. So, now there are two different Bengali calendars. And no, do not guess the rest of the story. The two is not one, the old one and the one the corrected one. The two is the one is the still the old one, which is the wrong measure of the wrong thing. And the other is the second one, which is the right measure of the wrong thing. But nobody still uses the right thing. So, this is how the Bengali calendar is functioning now. However, after saying all these things, I should say there is one thing which is very beautiful in the Bengali calendar. And that is the length of the months. The length of the months are not arbitrary. It follows sort of Kepler's law. So, when the sun is closest to the earth, the earth is moving fast. So, the months are smaller. So, it is 30 degree on the sky. So, around January, there is post in Bengali calendar, the month will be smallest. And six months later, the month, the month will be longest. Those long months can be 32 days. I have even seen 33 days, but that is not very common. But 32 days almost every year you will see. So, I am almost done. But the most of the Hindu festivals are determined not according to the Bengali calendar, even in Bengal. They are determined according to the lunar solar calendar of western India or northern India. That is why every year before the year starts, there is a lot of tension. When will be Durga Puja this year? We will have to book the tickets for our holidays and so on. Now, how are months introduced or intercalated in Indian lunar solar calendars? This is the last thing I am going to talk about today. So, yes, in a lunar month, synodic is 29.5306 days. Now, suppose I define something called a solar month, which is with respect to the earth. The sun is moving in the sky throughout the year with respect to the background of the stars. And throughout the year, it is 360 degree, divide by 12. So, that is 30 degrees. So, a 30 degree movement of the sun will be called a solar month. So, average solar month would be just one solar year divided by 12, which is 30.4368. This is roughly one day longer than the lunar month, average lunar month. So, suppose, of course, this is just average, some are longer, some are shorter as I have already mentioned. But suppose, you are keeping track of the lunar month and the solar month both, then typically what will happen? Within a solar month, someone in the middle of the solar month, a lunar month will start and that lunar month will end in the next solar month, but a little earlier. So, suppose this solar month on the 14th lunar month has started. So, we will expect it to end not on the 14th of the next solar month, but maybe 13th, maybe 12th of the next solar month. This is how it will happen. However, once in a while, what will happen that a solar month starts and on the same day the lunar month starts, then what will happen? The lunar month will end before the solar month ends. So, in other words, you will get two full moons within the same solar month and that month, if you have such a month, then that month is called the Malamaas. Mal means garbage or dirt. So, it is dirty month, not anything good can happen according to Hindu customs. On that month, you cannot, I would not say you cannot be born, I guess that is allowed, but you cannot marry. Assuming marriage is a good thing and anyway, many other good things are not allowed on that month, I am not sure about. So, if there is a Malamaas, then that Malamaas is the intercalated month in the Indian lunisolar calendars. That means that suppose you have the Baishak and the next solar month, sorry, the next lunar month, you are seeing that it is a Malamaas, then you do not call the next month just, you call it Adik Baishak and you will not see it in every year because every year there will not be a new month, an extra month, but roughly once in three years or seven times in 19 years, this will happen. So, if you look at some of the lunisolar calendars like the Kaldini or if you can buy one of these diaries which give you so many dates in so many calendars that there is no other space to write anything, so that kind of diaries are very helpful for me. So, that also will tell you when it is Adik Maas. So that is it. That is the intercalated month in the lunisolar calendars and that is how the north and west Indian calendars function and that is also the end of the talk actually although I told you to keep track of this thing, but I also told you that the different backgrounds are different chapters. So, the last two slides are not any chapter. Those are appendix. So, the talk actually ends here unless you ask me to go through the appendix. Thank you very much. If there are no questions, I will go through the appendix. Yes, there is a question. Is it possible that a calendar follows the solar calendar for the number of days in a year, but for the number of days in a month it follows the lunar calendar? In the sense, the length of my month is dependent on the moon cycle. That is the lunisolar calendar. That is one of the three types of calendars that I talked about. The average length of the year is solar year, but the months are lunar months. The Bengali year was incredibly accurate up to some five decimal places. Incredibly accurate. I mean the decimal places were the five decimal places. Oh, you mean the Surya Siddhanta? Yeah. Go back. You get the Bengali solar year. This? No, no, no. This? Yeah, yeah. Right, right, right. So, I mean, of course, I understand this is because of the calculation 6th century based AD. Were the observations of whatever observations were consistent for such an accuracy or did they have any idea of error estimates in there? I cannot give you a good answer, but I also have the same doubt. And I remember that when I wrote the book, I went through Surya Siddhanta to some extent and I was not very satisfied, but I don't remember exactly the details. You mentioned the months, January and February were introduced as 11th and 12th month, but the God Janus is actually God of past and future. And that is usually a reason given why January is, his name is given in the first month. So when this transition actually happened? It's a very good question and the answer, I have actually tried to find a satisfactory answer for a long time and I was never satisfied. When it was done, I mean, from some documents it seemed that it was done at the time of Numa Pompilius himself. But then it was inconsistent with the fact that all these errors and calculations were dumped on to February, which seems that the February was still the last month. So I really don't know because most of the original documents are in Latin. I mean, I could read up to French and Italian, I could read, but Latin I cannot read and I even read the French and some French and Italian, but I was not satisfied in the answer. But somewhere between Numa Pompilius and Julius Caesar, this was done. This much can be said. So sir, in the very first Roman calendar, the days in each month were still not uniform alternating, uniformly alternating. So September and October still had only 30 days. After that, it was alternating. So is there any particular reason for that? If there is, I don't know. Because you said in the Augustus one it was because Augustus wanted his name to have 31 days. Before that, but it still had 30 and 31 days. I understand your point. I just don't know the answer. You didn't say anything about the week, seven days. That's in the appendix. You didn't ask for the appendix. What can I do? I gave you the obvious question to ask and you didn't ask it. So now that you have asked the question, let me talk about some of the non-astronomical elements of calendars. All the things I have talked about before is about year, month and day, all of which have some astronomical basis. One non-astronomical element is the era. For example, now it's 2017. That has nothing to do with astronomy. That has to do with somebody who was born roughly 2017 years ago. Roughly. Actually Jesus Christ was not born 2017 years ago. But that's another, that's a miscalculation. But anyway that remains. But so some important aspect in human life, not in astronomy. The other thing is this week. This is a seven day cycle as we follow now. But the cycle was not always seven days. It was actually a market day cycle. Because you see, even a hundred years ago, refrigerators were not invented. So people had to store the food they buy at their home. And the stores were also not necessarily nearby. So you cannot hold a market every month because then the food will become stealth. And also you cannot hold it every day because it was, people cannot travel in train to have, you know, the food items at your door. So there has to be some moderate cycle. And the cycle was sometimes five, in some countries six, some countries seven and so on. In Christian countries a seven day cycle was decided upon by the church because of this Genesis myth. The God created the earth or whatever in seven days. And the seven days were named after what are called the seven planets. The planets not in the modern sense of the term, but the planets are the things that you, if you look at the sky, there are seven things which move in the background of the stars. The stars also move, but they all move at the same time. So they all move like this and in that backdrop there are seven things which move differently. And those are the things which are the sun. So in Latin one day was called Dia Solis, Lunai which is the moon, Mars, Mercury, Jupiter, Venus and Saturn. So these were the seven things and the seven days were conveniently marked. In English the names are not like this. As you know, I mean this is, this has nothing to do with, this is Sunday, this is moon day or Monday, but this has nothing to do with Mars or Mercury, etc. But this is not just in Latin. For example, even now let's say in French the names are roughly these things. So except the Sunday which is the Mars, which is the day of the Lord. So this is Ladi, this is Madhi, this is Makradi, this is Jyadi, this is Vadradi and this is Sandhi. So you can see the connection. But then again the question comes that why is the ordering like this? And the ordering actually has an interesting explanation. I don't know whether this is the correct explanation but this is the only explanation that I could find. The ordering is this that you consider for these seven objects the sidereal period, which is their cycle in the sky in the backdrop of stars. For the moon I already said at the beginning of the talk that it is 27.3 days. This is the sidereal month. For Makradi it's 88 days, for Venus 225 and so on. The longest is the Saturn which is 29.5 years and now there are 24 hours in a day. Apparently the hours of the day were also named after the planets. So not only the days of the week but hours of the day. So how did it go? So if it's a Saturday the first hour belongs to Saturn. The next hour belongs to Jupiter. The third hour to Mars. Sun, Venus, Makradi, Moon. Then the eighth hour again belongs to Saturn. And it goes on like this. 21st belongs to the moon. 22nd Saturn. 23rd Jupiter. 24th Mars. Then the next hour. New day begins. That's the sun. So the next day is Sunday. And you again go on like this with the hours. After 24 hours are passed you will see that you will land on the moon. So the next day will be the Monday. And then complete your cycle like this. So everything falls in place. So this is the explanation. Now you have seen the appendix also. Thank you.