The normal calendar year has 365 days and a leap year has 366 days. February in a normal year has 28 days, but in a leap year a February has 29. It is commonly believe that a year divisible by four is always a leap year.
But this is not always true, for not every leap year occurs after four years.
Before we know the reason for this, it is important to understand the science behind leap years.
How Leap Years Forms?
A solar year is define as the time taken by the Earth to complete one revolution or orbit around its star, the Sun.
A regular calendar year has 365 days, but a solar year is slightly longer than this.
The Earth takes 365.2422 days to complete one orbit around the Sun, as par NASA.
When we multiply 0.2422 days by 24 hours, we get 5.812 hours, which is approximately equal to six hours.
This means that Earth’s solar year is about six hours longer than 365 days.
Roman Emperor Julius Caesar adopt a calendar system in 46 BC.
He took the help of Alexandrian astronomer Sosigenes to design this calendar, known as the Julian calendar.
In this calendar, there were 12 months.
The solar year was consider to have 365.25 days, and every regular calendar year had 365 days.
Every fourth year, an extra day was adds to the calendar.
That year, known as the leap year, had 366 days.
But there was no February 29.
Rather, February 23 was mark twice in the calendar.
As a normal year was six hours behind the solar year, the shortage was compensate for in the Julian calendar by adding 24 hours, or one day, at the end of four years (6 × 4 = 24 hours).
But the calendar was not perfect because it overcompensate for the difference between a normal calendar year and the solar year.
To understand this, it is important to delve into basic mathematical concepts.
Julian Calendar Errors
The solar year has 365.2422 days.
First, we need to convert 0.2422 days into hours.
For this, we multiply 0.2422 by 24 hours.
This is equal to 5.8128 hours.
Next, we convert 0.8128 hours into minutes.
For this, we multiply 0.8128 by 60 minutes.
This is equal to 48.768 minutes.
Then, we convert 0.768 minutes into seconds.
For this, we multiply 0.768 by 60 seconds.
This is equal to 46.08 seconds.
This means that a solar year has 365 days, 5 hours, 48 minutes, and 46 seconds.
After a period of 4 years, this amounts to 20 hours, 192 minutes, and 184 seconds.
If we convert 192 minutes into hours, we get 3 hours and 12 minutes.
To convert 184 seconds into minutes, we divide it by 60, and get three minutes and four seconds.
This means that 20 hours, 192 minutes, and 184 seconds are equal to 23 hours, 15 minutes, and 4 seconds.
Julian calendar adds 24 hours to each leap year, but ideally, 23 hours, 15 minutes, and 4 seconds should have added.
This means that the Julian calendar’s leap year was ahead of the solar year by 44 minutes and 56 seconds.
On an average, a calendar year in the Julian calendar was ahead of the solar year by 44 minutes, 56 seconds divided by 4, which is equal to 11 minutes and 14 seconds.
In other words, on average, each year in the Julian calendar was ahead of the solar year by 0.0078 days.
This is because 11 minutes can be express in days as 11/(60×24), which is equal to 0.007639 days, and 14 seconds can be express in days as 14/(3600×24), which is equal to 0.000162.
Adding 0.007639 days and 0.000162 days, we get 0.0078 days.
Over centuries, the errors kept accumulating, and by the mid-16th century, the dates of seasons had shift by about 10 days from Caesar’s time.
This means that the calendar was 10 days ahead of the solar calendar by the mid-1500s.
In 1582, Pope Gregory XIII, after consulting with Italian scientist Luigo Lilio, and Jesuit astronomer Christopher Clauvius, implement a reform calendar.
This went on to be known as the Gregorian calendar.
About Gregorian Calendar
For the countries that adopt the Gregorian calendar, October 1582 did not have the dates from 5th October to 14th October.
That means 10 days were missing from 1582’s calendar in those countries.
In 1582, 4th October was directly follow by 15th October.
As many as 10 days had been drop to compensate for the errors that had pile up due to the average Julian calendar year being 0.0078 days ahead of the solar year.
This brought back the March equinox to 21st March from 11th March.
Why not every Leap Year comes after 4 years in the Gregorian calendar?
The average year in the Julian calendar was ahead of the solar year by 11 minutes and 14 seconds.
At the end of 400 years, the calendar would have been 4400 minutes and 5600 seconds ahead of the solar year.
To convert 5600 seconds into minutes, we divide it by 60. The answer is 93.33 minutes.
So, at the end of 400 years, the Julian calendar would have ahead of the solar year by 4493.33 minutes.
We can convert 4493.33 minutes into days by dividing the number by (60×24).
This is equal to 3.12 days.
So, every 400 years, about three extra days were adds to the Julian calendar.
It was important to remove this discrepancy.
So, Pope Gregory XIII made another reform to the Julian calendar.
He include a rule for leap years, for otherwise the Gregorian calendar would again have many days ahead of the solar calendar in the future.
He had to eliminate three extra days every 400 years, and hence, made a rule.
The rule was that all years divisible by four would be leap years, except for century years, unless they are multiples of 400.
For instance, 2000 was a leap year, but 1700, 1800, and 1900 were not.
Similarly, 2100 would not be a leap year.
Since 2024 is a multiple of four, and is not a century year, so it is a leap year.
How much the existing Gregorian calendar ahead of the solar calendar?
In 1200 years, as many as 300 years are divisible by 4.
Meanwhile, 900 years are not divisible by 4, which means that they are regular calendar years.
There were 12 century years from 1 AD to 1200 AD: 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200.
Of these, only 400, 800, and 1200 are divisible by 400.
This means only three century years are leap years.
The remaining 9 century years are normal.
Century years are divisible by four, but not every century year is divisible by 400, So, they are not a leap year.
Now, in 200 years, there are 909 (900+9) regular years, and 291 (300-9) leap years.
The number of days in 909 regular years and 291 leap years can be found by multiplying 366 with 291, and 909 with 365, and adding the two values.
In 291 leap years, there are 106,506 days, and in 909 regular years, there are 331,785 days.
The total is equal to 438,291 days.
So, 1200 years have 438,291 days.
To find the average number of days in each year, we divide 438,291 by 1200.
This is equal to 365.2425 days.
Since a solar year has 365.2422 days, the difference between an average Gregorian calendar year and a solar year is 0.0003 days.
For one calendar year, the error is 0.0003 days.
This means that an error of one day will occur every 1/0.0003 years, or 3,333.3 years.
Also, since there is an error of 0.0003 days for one calendar year, it means that there is an error of 0.0003×24×3600 seconds, or 25.92 seconds for the calendar year.
Means, on average, each calendar year is 26 seconds ahead of the solar year.
Gregorian calendar would be ahead of the solar calendar by a day in about 86400÷26 years, or 3,323 years.
We divided 86400 by 26 because each day has 86400 seconds.
Over a period of 3,323 years, the errors would pile up and would be equivalent to 86400 seconds, or one day.
Means, as of now, we need not worry about being ahead of the solar year.
