Discrepancy between Gregorian Year and Tropical Year
Theorem
The Gregorian year and the tropical year differ such that the Gregorian calendar becomes $1$ day further out approximately every $3319$ years.
Proof
By definition, the length $Y_T$ of the tropical year is defined as $\approx 365 \cdotp 24219 \, 878$ days.
By definition of the Gregorian year:
- $Y_G = \begin{cases} 366 \, \text {days} & : 400 \divides y \\
365 \, \text {days} & : 400 \nmid y \text { and } 100 \divides y\\ 366 \, \text {days} & : 100 \nmid y \text { and } 4 \divides y\\ 365 \, \text {days} & : 4 \nmid y \end{cases}$ where:
- $Y_G$ denotes the length of the Gregorian year in days
- $y$ denotes the number of the year
- $4 \divides y$ denotes that $y$ is divisible by $4$
- $4 \nmid y$ denotes that $y$ is not divisible by $4$.
Thus in every $400$ years, there are:
- $300$ years which are not divisible by $4$, which have $365$ days
- $1$ year which is divisible by $400$, which has $366$ days
- $3$ years which are divisible by $100$ but not $400$, which have $365$ days
- The remaining $96$ years which are divisible by $4$ but not $100$, which have $366$ days.
Thus:
| \(\ds 400 Y_G\) | \(=\) | \(\ds 303 \times 365 + 97 \times 366\) | considering $400$ consecutive Gregorian years | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds Y_G\) | \(=\) | \(\ds \frac {110 \, 595 + 35 \, 502} {400}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {146 \, 097} {400}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 365 \cdotp 2425\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds Y_G - Y_T\) | \(\approx\) | \(\ds 365 \cdotp 2425 - 365 \cdotp 24219 \, 878\) | |||||||||||
| \(\ds \) | \(\approx\) | \(\ds 0 \cdotp 0003013\) | ||||||||||||
| \(\ds \) | \(\approx\) | \(\ds \frac 1 {3318 \cdotp 95}\) |
Thus the mean Gregorian year is $\dfrac 1 {3318 \cdotp 95}$ days longer than the tropical year.
This means that after approximately $3319$ Gregorian years, the Gregorian calendar starts one day later relative to the tropical year.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $365 \cdotp 2422$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $365 \cdotp 24219 \, 878$