ABSTRACT
The Definitions, Propositions and Common Notions provide a constitution for Euclid's subject matter but are not themselves part of it. As a consequence the qualifications of true and false which are characteristic of propositions cannot be applied to them. When we turn to Euclid's propositions beginning with Book I we find that they fall into two types—theorems and problems in the vocabulary of ancient Greek mathematics. Simply put, a theorem is a statement about a figure which is to be demonstrated and a problem (or construction—the two words will be used interchangeably) is a request for something to be constructed given some data. While there are a few propositions in the later books of the Elements which do not fall into one of these two categories, the overwhelming majority of Euclid's propositions are of one of these two types. The distribution of the two types of proposition, however, is quite complex. Book I is made up of an alternating series of theorems and problems (see Table 2.1) while other books may be mostly or entirely theorems (e.g. Books II and V) or problems (e.g. Book IV). This rather innocuous looking distinction carries a great deal of significance for the nature of Euclid's mathematics. 1
