This paper is a detailed study of what are traditionally called the cardinal virtues: prudence, justice, temperance and fortitude. I defend what I call the Cardinality Thesis, that the traditional four and no others are cardinal. I define cardinality in terms of three sub-theses, the first being that the cardinal virtues are jointly necessary for the possession of every other virtue, the second that each of the other virtues is a species of one of the four cardinals, and the third that many of the other virtues are also auxiliaries of one or more cardinals. I provide abstract arguments for each sub-thesis, followed by illustration from concrete cases. I then use these results to shed light on the two fundamental problems of the acquisition of the virtues and their unity, proving some further theses in the latter case.