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

TL;DR

The paper formalizes the computation of the Catholic Church's liturgical moon age (the ecclesiastical moon) using modern notation for epacts and related quantities. It derives a recurrence for epacts $E_y$, obtains a closed-form expression $E_y mod 30$ depending on the golden number $G_y$ and century $C_y$, and uses this to compute the moon age $A_{y,m,d}$ via a modular function $g$ and day-number $N_{m,d}$. It analyzes and unifies corrections at year boundaries, introducing a jump $J_y$ and a fully corrected moon age $A^{corr}_{y,m,d}$ to remove all anomalies, with supporting historical context and a Python implementation. These results enable precise, implementable liturgical calculations of the ecclesiastical moon, aligning ecclesiastical timing with a near-astronomical lunar cycle while compensating for calendar irregularities.

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.

Figures (7)

• Figure 1: Tabella temporaria or temporal table reproduced from older Breviarium Romanumbrev1942.
• Figure 2: Correspondence between martyrology letters (top rows) and epact numbers (bottom rows), reproduced from Martyrologium Romanummart1948.
• Figure 3: Fragment of Calendarium from Missale Romanummr1942.
• Figure 4: Fragment of the pre-concilliar Martyrologium Romanummart1948 for August 15.
• Figure 5: Fragment of the new (2004) Martyrologium Romanummart2004 for August 15. Note the inverted color scheme.