Computation of Easter date (UK statutory method)

Question

The statutory method for the computation of Easter in the UK (and presumably also the USA) is contained in the Calendar (New Style) Act 1750. This is the same Act which moved the British Empire from the Julian calendar and onto the Gregorian. The Act and the method it specifies remains in legal force in the UK.

This employs a tabular method of calculating Easter which does not explicitly refer to the "epact" (which from what I gather, is the standard way in which the Roman Catholic Church makes the calculation).

The Act as currently in force is reproduced here (the relevant tables are reproduced as images toward the end of the document): https://www.legislation.gov.uk/apgb/Geo2/24/23/section/6

Complicating the matter, some of the tables which were originally enacted, and which gave detailed data for dates before 1900, were repealed by the Statute Law (Repeals) Act 1948, and so they are not reproduced in the current legislation. However, this repeal appears also to have deleted a block of text from the modern reproduction. The original Act, therefore, can be viewed in full here (page 206 onwards): https://books.google.co.uk/books?id=rLsuAAAAIAAJ&lr&pg=PA206

I have also stumbled on a link that apparently reproduces the same content in HTML, and with modern typesetting, but does not attempt to provide any further explanation: http://www.eskimo.com/~lhowell/bcp1662/info/tables/index.html

Basically I ask, does anyone have a comprehensive mathematical analysis of this specific statutory method? That is, how the typeset tables are produced from first principles, and how the tables are then worked in an explicit, step-by-step fashion?

I'm broadly familiar with the concepts and terminology involved in the Easter calculation, but in terms of applying this method I find I'm falling down even at the very outset: "To find the Dominical or Sunday Letter, according to the Calendar, until the Year 1799 inclusive, add to the Year of our Lord its Fourth Part, omitting Fractions;".

What does it mean, "add to the Year ... its fourth part"? The obvious modern interpretation of this is to add a quarter again - multiply the year by 1.25 and then floor - but I can't find any basis to be certain of this, or understand why such an apparently arbitrary constant is employed.

Answer 1

OK, so I think I've sort of got the process (or at least, I'm getting the right answers out). I'll go through it for this year for an example:

1. First we need to find the Sunday Letter for our year from Table I. To do this:
   1. Ignoring the stupid way it's worded, we first look at the table, and find the century that we're in. In our case, that's $2000$. We then go to the very top of that column, and get the magic number: in our case, $6$.
   2. We then multiply the current year by $1.25$ and floor (giving $2525$), and add that to our magic number, giving $2531$. We take the remainder of that when divided by $7$ ($4$ in our case), then look that number up in the row that we got our magic number from, giving our Sunday Letter as $D$ (this is promising: easter is after the leap day, at which point we will indeed be in a "$D$" year).
2. Now, we go down to the writing next to Table II:
   1. We first round the year down to the nearest century ($2000$ for our example), and find our "number or cypher" (hereafter "cypher number"). In this case, that cypher number is $2$.
   2. We then find the cypher number in each column of Table III (or just find it in the column matching your golden number, if you already know that).
   3. Out of all of those numbers, only one (the one in the column for your golden number, so this tells you that number as well) will match up with your Sunday Letter from the first table.
   4. The date marked on that line gives you the Eclesiastical Paschal Full Moon for your year (in our case, the 5th of April). Easter is the Sunday following that, and since the 5th of April happens to have been a Sunday this year, Easter this year was on the 12th of April.

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As for why that works, we'll look at each part in turn:

1. The Sunday Letter table seems to be a modified version of De Morgan's Rule: using the third equivalent, the $1.25$ comes from the $y + \left\lfloor\frac{y}{4}\right\rfloor$ part, the "magic number" is the $5(c\,\mathrm{mod}\,4) - 1$ part, and everything after that is the same.
2. This is where the epact and golden number are hidden: the column, as mentioned above, where your number appears gives the golden number for the year, and the cypher numbers look to have been computed backwards from the epact (wikipedia claims similarly, though I can't find it in either of the pages that they cite for that claim) then arranged in the tables in such a way as to make everything match up.

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Incidentally, it does work: this page has a javascript implementation, which has been verified to match the epact method.