It is argued that Beethoven used a system of numbers to guide aspects of many of his works, especially major ones. The numbers manifest themselves in the number of notes in a melody and/or of bars in a work or part of it, in groupings and numberings of works of a given kind, and in his deliberate choice of Opus numbers. They are not only small ones such as 3 , which of course turn up frequently anyway; larger ones are (...) prominent, particularly 27, 30, 32 and 33. The interpretation of the numbers was not his own innovation, but came largely from Christian and Masonic traditions in Austria at that time; the modes of his adhesion to those movements is appraised. The thesis confronts the apparent fact that Beethoven was very bad at arithmetic; consideration of this matter involves the psychology of arithmetic. It also faces a remarkable level of modern ignorance, including among musicologists, of numerology and of its methods of research. (shrink)