The program is three people, plus host Melvyn Bragg, talking about the life and work of Gauss. Gauss is one of those figures hard to exaggerate. He was extremely prolific and insightful. It is an exaggeration to say that he did foundational work in every field of mathematics, but only a slight exaggeration. (He compares to Leonhard Euler that way.) I’d imagine that anyone reading a pop mathematics blog knows something of Gauss. But you may learn something new, or a new perspective on something familiar.
If I’d started pondering the question a week earlier I’d have a nice timely post. Too bad. Shouldn’t wait nearly a year to use this one, though.
My love and I got talking about early and late Easters. We know that we’re all but certainly not going to be alive to see the earliest possible Easter, at least not unless the rule for setting the date of Easter changes. Easter can be as early as the 22nd of March or as late as the 25th of April. Nobody presently alive has seen a 22nd of March Easter; the last one was in 1818. Nobody presently alive will; the next will be 2285. The last time Easter was its latest date was 1943; the next time will be 2038. I know people who’ve seen the one in 1943 and hope to make it at least through 2038.
But that invites the question: what dates are most likely to be Easter? What ones are least? In a sense the question is nonsense. The rules establishing Easter and the Gregorian calendar are known. To speak of the “chance” of a particular day being Easter is like asking the probability that Grover Cleveland was president of the United States in 1894. Technically there’s a probability distribution there. But it’s different in some way from asking the chance of rolling at least a nine on a pair of dice.
But as with the question about what day is most likely to be Thanksgiving we can make the question sensible. We have to take the question to mean “given a month and day, and no information about what year it is, what is the chance that this as Easter?” (I’m still not quite happy with that formulation. I’d be open to a more careful phrasing, if someone’s got one.)
When we’ve got that, though, we can tackle the problem. We could do as I did for working out what days are most likely to be Thanksgiving. Run through all the possible configurations of the calendar, tally how often each of the days in the range is Easter, and see what comes up most often. There’s a hassle here. Working out the date of Easter follows a rule, yes. The rule is that it’s the first Sunday after the first full moon after the spring equinox. There are wrinkles, mostly because the Moon is complicated. A notional Moon that’s a little more predictable gets used instead. There are algorithms you can use to work out when Easter is. They all look like some kind of trick being used to put something over on you. No matter. They seem to work, as far as we know. I found some Matlab code that uses the Easter-computing routine that Karl Friedrich Gauss developed and that’ll do.
Problem. The Moon and the Earth follow cycles around the sun, yes. Wait long enough and the positions of the Earth and Moon and Sun. This takes 532 years and is known as the Paschal Cycle. In the Julian calendar Easter this year is the same date it was in the year 1485, and the same it will be in 2549. It’s no particular problem to set a computer program to run a calculation, even a tedious one, 532 times. But it’s not meaningful like that either.
The problem is the Julian calendar repeats itself every 28 years, which fits nicely with the Paschal Cycle. The Gregorian calendar, with different rules about how to handle century years like 1900 and 2100, repeats itself only every 400 years. So it takes much longer to complete the cycle and get Earth, Moon, and calendar date back to the same position. To fully account for all the related cycles would take 5,700,000 years, estimates Duncan Steel in Marking Time: The Epic Quest To Invent The Perfect Calendar.
Write code to calculate Easter on a range of years and you can do that, of course. It’s no harder to calculate the dates of Easter for six million years than it is for six hundred years. It just takes longer to finish. The problem is that it is meaningless to do so. Over the course of a mere(!) 26,000 years the precession of the Earth’s axes will change the times of the seasons completely. If we still use the Gregorian calendar there will be a time that late September is the start of the Northern Hemisphere’s spring, and another time that early February is the heart of the Canadian summer. Within five thousand years we will have to change the calendar, change the rule for computing Easter, or change the idea of it as happening in Europe’s early spring. To calculate a date for Easter of the year 5,002,017 is to waste energy.
We probably don’t need it anyway, though. The differences between any blocks of 532 years are, I’m going to guess, minor things. I would be surprised if the frequency of any date’s appearance changed more than a quarter of a percent. That might scramble the rankings of dates if we have several nearly-as-common dates, but it won’t be much.
So let me do that. Here’s a table of how often each particular calendar date appears as Easter from the years 2000 to 5000, inclusive. And I don’t believe that by the year we would call 5000 we’ll still have the same calendar and Easter and expectations of Easter all together, so I’m comfortable overlooking that. Indeed, I expect we’ll have some different calendar or Easter or expectation of Easter by the year 4985 at the latest.
For this enormous date range, though, here’s the frequency of Easters on each possible date:
Date
Number Of Occurrences, 2000 – 5000
Probability Of Occurence
22 March
12
0.400%
23 March
17
0.566%
24 March
41
1.366%
25 March
74
2.466%
26 March
75
2.499%
27 March
68
2.266%
28 March
90
2.999%
29 March
110
3.665%
30 March
114
3.799%
31 March
99
3.299%
1 April
87
2.899%
2 April
83
2.766%
3 April
106
3.532%
4 April
112
3.732%
5 April
110
3.665%
6 April
92
3.066%
7 April
86
2.866%
8 April
98
3.266%
9 April
112
3.732%
10 April
114
3.799%
11 April
96
3.199%
12 April
88
2.932%
13 April
90
2.999%
14 April
108
3.599%
15 April
117
3.899%
16 April
104
3.466%
17 April
90
2.999%
18 April
93
3.099%
19 April
114
3.799%
20 April
116
3.865%
21 April
93
3.099%
22 April
60
1.999%
23 April
46
1.533%
24 April
57
1.899%
25 April
29
0.966%
Dates of Easter from 2000 through 5000. Computed using Gauss’s algorithm.
If I haven’t missed anything, this indicates that the 15th of April is the most likely date for Easter, with the 20th close behind and the 10th and 14th hardly rare. The least probable date is the 22nd of March, with the 23rd of March and the 25th of April almost as unlikely.
And since the date range does affect the results, here’s a smaller sampling, one closer fit to the dates of anyone alive to read this as I publish. For the years 1925 through 2100 the appearance of each Easter date are:
Date
Number Of Occurrences, 1925 – 2100
Probability Of Occurence
22 March
0
0.000%
23 March
1
0.568%
24 March
1
0.568%
25 March
3
1.705%
26 March
6
3.409%
27 March
3
1.705%
28 March
5
2.841%
29 March
6
3.409%
30 March
7
3.977%
31 March
7
3.977%
1 April
6
3.409%
2 April
4
2.273%
3 April
6
3.409%
4 April
6
3.409%
5 April
7
3.977%
6 April
7
3.977%
7 April
4
2.273%
8 April
4
2.273%
9 April
6
3.409%
10 April
7
3.977%
11 April
7
3.977%
12 April
7
3.977%
13 April
4
2.273%
14 April
6
3.409%
15 April
7
3.977%
16 April
6
3.409%
17 April
7
3.977%
18 April
6
3.409%
19 April
6
3.409%
20 April
6
3.409%
21 April
7
3.977%
22 April
5
2.841%
23 April
2
1.136%
24 April
2
1.136%
25 April
2
1.136%
Dates of Easter from 1925 through 2100. Computed using Gauss’s algorithm.
If we take this as the “working lifespan” of our common experience then the 22nd of March is the least likely Easter we’ll see, as we never do. The 23rd and 24th are the next least likely Easter. There’s a ten-way tie for the most common date of Easter, if I haven’t missed one or more. But the 30th and 31st of March, and the 5th, 6th, 10th, 11th, 12th, 15th, 17th, and 21st of April each turn up seven times in this range.
The Julian calendar Easter dates are different and perhaps I’ll look at that sometime.
One of the personality traits which my Dearly Beloved most often tolerates in me is my tendency toward hyperbole, a rhetorical device employed successfully on the Internet by almost four people and recognized as such as recently as 1998. I’m not satisfied saying there was an enormous, slow-moving line for a roller coaster we rode last August; I have to say that fourteen months later we’re still on that line.
I mention this because I need to discuss one of those rare people who can be discussed accurately only in hyperbole: Leonhard Euler, 1703 – 1783. He wrote about essentially every field of mathematics it was possible to write about: calculus and geometry and physics and algebra and number theory and graph theory and logic, on music and the motions of the moon, on optics and the finding of longitude, on fluid dynamics and the frequency of prime numbers. After his death the Saint Petersburg Academy needed nearly fifty years to finish publishing his remaining work. If you ever need to fake being a mathematician, let someone else introduce the topic and then speak of how Euler’s Theorem is fundamental to it. There are several thousand Euler’s Theorems, although some of them share billing with another worthy, and most of them are fundamental to at least sixteen fields of mathematics each. I exaggerate; I must, but I note that a search for “Euler” on Wolfram Mathworld turns up 681 matches, as of this moment, out of 13,081 entries. It’s difficult to imagine other names taking up more than five percent of known mathematics. Even Karl Friedrich Gauss only matches 272 entries, and Isaac Newton a paltry 138.
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