A straight line is a line [curve] which lies evenly with the points on itself.

And after describing a surface as

A plane surface is a surface which lies evenly with the straight lines on itself.

Reading these two definitions may leave one with many unanswered (perhaps even unanswerable) questions, but one observation is rather immediate: while every curve certainly contains infinitely many points, in fact consists of points (described as its 'extremities' in definition #3), there are generally no

Overlooking this problem for a moment, let's focus on the meaning of the verb

In the case of a straight line (#4), Euclid might have possibly appealed to the intuitive fact that a straight line is the shortest curve between every two points: surely there is some circularity in such a description, but this is inevitable. Moreover, Euclid could have meant, in modern terminology, that a straight line is the only

But the interpretation above generates an obvious question: wouldn't it have been more economical to define a plane surface as a "surface lying evenly with its

My answer to this question is to opt for an interpretation of definitions #4 and #7 based on

"a straight line is a line [curve] which lies evenly with (the tangent line at each one of) the points on itself"

and

"a plane surface is a surface which lies evenly with the (tangent) straight lines on itself".

In addition to the inevitable circularity in #4, the obvious objection there is due to the implicit presence of tangent lines. But the apparent presence of tangentiality in Euclid's definition of the angle between two curves (#8), clearly distinguished from the angle between two straight lines (#9), and the absence of an explicit definition of tangent line from The Elements, make the interpretations of #4 and #7 above quite plausible.

Unfortunately the question on the imbalance between points (#4) and straight lines (#7) comes back: it is easy to see why the explicit mention of tangent (straight) lines in #4 would be circular and undesirable, but then why not stick to a surface's points in #7 as well, implicitly referring to the tangent lines at those points? My response is that, while a point on a (not necessarily plane) smooth curve defines a tangent line there

Postscript: Professor David Joyce of Clark University argues against the convexity interpretation in his discussion of Definition #7 relying on Euclid's Proposition #7 in Book XI ("the line joining two points on two parallel lines lies in the plane of the two parallel lines").