Mathematics in the liturgical books of the Catholic Church: phases of the ecclesiastical moon (Q6557234)

The Gregorian calendar reform, introduced by the issuance of the bull \textit{Inter gravissimas} by Pope Gregory XIII on February 24, 1582, corrected not only the calendar, but also the way moon phases are calculated. These are needed for liturgical purposes, such as determining the date of Easter, which falls on the first Sunday after the the first full moon on or after March 21, or to pronounce, in a Catholic context, at the beginning of daily reading from the Roman Martyrology book, the current phase of the moon.\N\NThe moon age (the phase of the moon on a given day) is defined as the time elapsed since the last new moon. The moon age used by the Church for liturgical purposes, always an integer, differs from the age of the real astronomical moon (which is a real number), for it is the age, determined by applying a prescribed algorithm, of an imaginary moon, known as the ecclesiastical moon.\N\NThe aim of this paper is to use modern ``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 \textit{Romani calendarii explicatio}.'' The algorithm is derived ``by using the tabular method employed by liturgical books such as the Roman Missal, Breviary and Martyrology.'' A simple formula for the age of the moon on a given day of the year results, which represents a corrected version to the ``pronounced'' age of the ecclesiastical moon from the \textit{Martyrologium Romanum}. The proposed correction addresses ``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.'' The implementation of the relevant functions in Python is also provided in an appendix.