The present paper contains an exposition of Boethius' treatise \emph{De institutione musica} which is unanimously considered one of the cornerstones of Western musical theory. In the introductory Section \ref{s1}, we first look into Boethius' ambitious philologico-philosophical project of translating and commentating on Plato's and Aristotle's complete works, and providing a conciliatory synthesis thereof. In this context, we pay special attention to Boethius' authentic method of translating and compiling Greek sources. The introduction closes with an outline of the basic tenets of Pythagorean music theory. Our primary concern in the larger part of the remainder of the paper is the historically important and perplexing question about Boethius' ancient sources for \emph{De institutione musica}. Section \ref{s2} opens with preliminary remarks about the sources for Boethius' treatise \emph{De institutione arithmetica} which contains the mathematical basis of the theory from \emph{De institutione musica}. After detecting a specific pattern of citing and referring to other authors in Boethius' texts, we discuss the most probable source for the fifth book of Boethius' musical treatise. Following an analysis of Boethius' original contributions in Section \ref{s3} (i.e. the \emph{quadrivium}, the division of different types of music, and Boethius' conception of the ``true musician''), in Sections \ref{s4} and \ref{s5} we continue discussing the possible sources for Books 1--3 and Book 4 of \emph{De institutione musica}, respectively. In the concluding Section \ref{s6}, we summarise the main insights from the preceding sections, and provide a basic outline of the extensive and multifaceted reception and transmission history of Boethius' treatise on the mathematical foundations of music.