# Know Some History Before Doing Math

What if I tell you that, some mathematics formulas are true only after a period of time…

What? Am I saying that the formula was false before a certain time in history? Yes, and I present you, the Zeller’s congruence. The formula is different when you perform it after 15 October 1582 and before 4 October 1582. What about the time in between? They doesn’t exist.

Before 1582, the calendar in use was the Julian calendar. It assumed a year to be 365.25 days long. The fractional day was facilitated by having 365 days every year, with one extra day in February once every four years—thus effectively having 365.25 days every year on an average.

However, the time taken by the earth to complete one revolution around the sun, the solar year, is not 365.25 days, but approximately 365.2422 days. The small difference doesn’t cause much of a difference in a short span of 100 years. But, by 1582, it was realized that over the centuries, the erstwhile calendar had calculated 10 extra days due to this malpractice!

Needless to say, this way, a time would come when the seasons would not be in sync with the calendars, and so a change was needed as early as possible. So, Pope Gregory XIII established the Gregorian calendar in 1582. To launch the calendar, ten days were dropped from the calendar. October 4, 1582 was followed by October 15, 1582.

However, British Empire adopted it as late as 1752 and an error of about 11.326 days had crept up. Therefore, eleven days had to be dropped this time, and September 2, 1752 was immediately followed by September 14, 1752. So September 1752 was shorter than other Septembers.

Edit and adapted from : Why did September 1752 have fewer days?

So, what is Zeller’s Congruence has to do with this historical event? Well, it is an algorithm devised by Christian Zeller to calculate the day of the week for any Julian or Gregorian calendar date. The formula for Gregorian calendar and Julian calendar are respectively

$\displaystyle w=\bigg ( y +\bigg \lfloor \frac{y}{4} \bigg\rfloor + \bigg\lfloor \frac{c}{4} \bigg\rfloor -2c + \bigg\lfloor \frac{26(m+1)}{10}\bigg\rfloor+ d-1\bigg) \mod 7$

$\displaystyle w=\bigg ( y +\bigg \lfloor \frac{y}{4} \bigg\rfloor - c + \bigg\lfloor \frac{26(m+1)}{10}\bigg\rfloor+ d-1\bigg) \mod 7$

• $y$ is the last two digit of the year. For 1893, take 93.
• $c$ is the first two digit of the year. For 1893, take 18.
• $m$ is the month from 3 to 14. For January (1st month) take it as 13 instead.
• $d$ is the day.