Mathematics in the liturgical books of the Catholic Church: phases of the ecclesiastical moon

TL;DR

This paper details a mathematical approach to calculating the ecclesiastical moon's age, based on historical algorithms, with solutions for year-end transition issues, and provides Python implementations.

Contribution

It introduces a modern mathematical notation and recurrence equations for the ecclesiastical moon calculation, improving understanding and implementation of historical algorithms.

Findings

Derived a recurrence equation for epacts

Provided a simple formula for moon's age on any day

Offered a solution for year-end transition problems

Abstract

We use contemporary mathematical notation to describe the method for determining the age of the ecclesiastical moon as mandated by pope Gregory XIII and elaborated in the book of Christopher Clavius \emph{Romani calendarii explicatio}. The algorithm is first introduced by using the tabular method employed by liturgical books such as the Roman Missal, Breviary and Martyrology. Then we construct the recurrence equation for the epacts, derive its solution, and give a simple expression for the age of the moon on a given day of the year. We also consider the problems which can occur at the transition from December 31 to January 1 of the next year, when there could be a ``jump'' in moon's age ("saltus lunae") in years when epact corrections are applied. We propose a simple solution which fixes these problems. A summary of the formulae and listing of the implementation of relevant functions in Python is provided in the last section.