Scientific American’s Easter
April 4, 2026 at 11:12 PM by Dr. Drang
In its continuing attempt to teach us how calendars work, Scientific American has an article (Apple News link) up today that goes through Gauss’s algorithm for calculating the date of Easter. The article was written by Manon Bischoff, who also wrote the Friday the 13th article I covered a few weeks ago.
There are a few small mistakes in the article: one computational and two definitional. Let’s take a look at them.
The computational error comes in the calculation of the intermediate value p. The article says
where the topless brackets mean the floor function, the integer less than or equal to what’s inside the brackets. The correct equation is
In both equations, k is
i.e., the first two digits of the year (at least until we get to the year 10,000).
In giving the wrong equation for p, Bischoff is following Gauss himself. In the original presentation of his Easter algorithm, Gauss gave the same simple formula as Bischoff, but he corrected it several years later. Why Bischoff is still using the wrong equation two centuries later is anyone’s guess.
Actually, I can guess. Maybe Bischoff’s using the wrong formula for p because it’s simpler and the error won’t manifest itself until the year 4200 (!). Here’s a quick Python script to see the centuries, starting in the 1600s, for which the simpler formula is wrong:
python:
for k in range(16, 60):
if k//3 != (13 + 8*k)//25:
print(f'Century {k*100} gives the wrong value')
The output is
Century 4200 gives the wrong value
Century 4500 gives the wrong value
Century 4800 gives the wrong value
Century 5100 gives the wrong value
Century 5400 gives the wrong value
Century 5700 gives the wrong value
I think we can live with a mistake that won’t rear its head for over 2000 years.
I’m less inclined to overlook the definitional errors in the article’s early paragraphs. This one:
For those who celebrate it, tracking what day the holiday Easter takes place on can be a challenge. According to Christian religious traditions, Easter Sunday falls on the first Sunday following the first full moon after the vernal equinox.
And this one:
[T]he vernal equinox, or start of spring, is fixed as March 21. If a full moon occurs on that exact day, March 22 becomes the earliest possible calendar date for Easter Sunday. According to the lunar calendar, the latest possible date for a full moon after March 21 is April 18. That means Easter Sunday never falls later than April 25.
While it’s true that the idea behind the date of Easter is to be “on the first Sunday following the first full moon after the vernal equinox,” when it comes to determining Easter, both the equinox and the lunar cycle are estimated—they aren’t based on accurate astronomical calculations or observations.
First, let’s look at the vernal equinox.1 One need only think back a couple of weeks to realize that it isn’t “fixed as March 21.” The most recent equinox was on March 20, as were 58 of the 66 vernal equinoxes I’ve lived through. March 20 is by far the most common date for the true vernal equinox. The Church fathers who set the date of Easter used March 21 as an approximation because it made the calculation simple and resulted in Easters more or less when they thought they should be.
Similarly, they used the Metonic cycle—with some occasional adjustments—to estimate when full moons would occur. There are almost exactly 235 lunar months in 19 years, a fact you can use to calculate a good estimate of when full moons occur.2 But this is an average, and because lunar months vary in length, the date of a calculated full moon doesn’t always fall on the date of an actual full moon. (There’s also a slow drift that has to be accounted for.)
So although the date of Easter is guided by astronomical events, it isn’t determined by them. As it happens, this year the vernal equinox was, as we said, on Friday, March 20, and the following full moon was on Wednesday, April 1. So having Easter Sunday on April 5 works out both astronomically and algorithmically. But in 2019, that wasn’t the case. The vernal equinox was on Wednesday, March 20; the following full moon was on Thursday, March 21; but Easter Sunday was celebrated on April 21, not March 24.