Line element

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"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 or depth. 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.
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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, examples on free math; 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. more on math forum.