Definition 2

A line is breadthless length.

Guide

“Line” is the second primitive term in the Elements. The description, “breadthless length,” says that a line will have one dimension, length, but it won’t have breadth. The terms length and breadth are not defined in the Elements. In I.Def.5 a surface is defined with the two dimensions length and breadth, and in XI.Def.1 a solid is defined with the three dimensions length, breadth, and depth.

One cannot tell from this definition what kind of line is meant by line, but later a straight line defined to be a special kind of line. One can conclude, then, that lines need not be straight. Perhaps “curve” would be a better translation than line since Euclid meant what is commonly called a curve in modern English, where a curve may or may not be straight.

Also, from the next definition, it is apparent that Euclid’s lines may have ends, so they are line segments or curve segments. But they need not have ends in all cases since the entire circumference of a circle is an example of a line. Indeed, lines need not be finite in all cases; there are a few instances in the Elements where a line is not bounded, and that is usually indicated by the language. See, for example, proposition I.12.

One piece of terminology that Euclid did not mention explicitly in a definition is a phrase to indicate when a line passes through a point. That would be a primitive relation that could hold between a line and a point. Postulates would be included as well to give meaning to the phrase as they are in modern treatments of elementary geometry.