Discrepancy between Julian Year and Tropical Year

Theorem

The Julian year and the tropical year differ such that the Julian calendar becomes $1$ day further out approximately every $128$ years.

By definition, the length $Y_T$ of the tropical year is defined as $\approx 365 \cdotp 24219 \, 878$ days

By definition of the Julian year:

$Y_J = \begin{cases} 366 \, \text {days} & : 4 \divides y \\

365 \, \text {days} & : 4 \nmid y \end{cases}$ where:

$Y_J$ denotes the length of the Julian 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:

$\ds 4 Y_J$

$=$

$\ds 3 \times 365 + 366$

considering $4$ consecutive Julian years

$\ds \leadsto \ \ $

$\ds Y_J$

$=$

$\ds \frac {1461} 4$

$\ds $

$=$

$\ds 365 \cdotp 25$

$\ds \leadsto \ \ $

$\ds Y_J - Y_T$

$\approx$

$\ds 365 \cdotp 25 - 365 \cdotp 24219 \, 878$

$\ds $

$\approx$

$\ds 0 \cdotp 0078013$

$\ds $

$\approx$

$\ds \frac 1 {128 \cdotp 18}$

Thus the mean Julian year is $\dfrac 1 {128 \cdotp 18}$ days longer than the tropical year.

This means that after approximately $128$ Julian years, the Julian calendar starts one day later relative to the tropical year.

$\blacksquare$