Euclid's Elements, Book I, Definition 1

Definition 1

A point is that which has no part.

The Elements is the prime example of an axiomatic system from the ancient world. Its form has shaped centuries of mathematics. An axiomatic system should begin with a list of the terms that it will use. This definition says that one term that will be used is that of point. The next few definitions give some more terms that will be used. Although there is some description to go along with the terms, that description is actually never used in the exposition of the axiomatic system. It can, at most, be used to orient the reader.

The description of a point, "that which has no part," indicates that Euclid will be treating a point as having no width, length, or breadth, but as an indivisible location.

Later definitions will define terms by means of terms defined before them, but the first few terms in the Elements are not defined by means of other terms; they're "primitive" terms. Their meaning comes from properties about them that are assumed later in axioms. In the Elements, the axioms come in two kinds: postulates and common notions. The first postulate, I.Post.1, for instance, gives some meaning to the term "point." It states that a straight line may be drawn between any two points. Other postulates add more meaning to the term "point."

Actually, Euclid failed to notice that he made a number of conclusions without complete justification at a number of places in the Elements. This usually means that a postulate, that is, a explicit assumption, is missing.

Book I Introduction - Definition 2.