A bold interpretation of Euclid's definitions #4 and #7



After describing a curve ('line') as breadthless length in definition #2, Euclid describes a straight line (or straight line segment) in definition #4 as follows:

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

And after describing a surface as that which has length and breadth only in definition #5, he goes on to describe a plane in definition #7 as follows:

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 straight lines on any surface (whose 'extremities' are described as lines [curves] in definition #6).

Overlooking this problem for a moment, let's focus on the meaning of the verb lies evenly. Bypassing a detailed discussion of the original Greek (keitai ex isou), which does not throw much light into the matter (see Sir Thomas Heath's "Euclid", volume I, page 167), we have the appealing interpretation "hangs evenly between": a straight line hangs evenly (straight) between every two points on it (definition #4), and so does a plane surface between every two straight lines on it (definition #7).

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 convex line [curve] around: for every two points on it, the straight line segment joining them lies entirely on it. This convexity interpretation has the advantage of being extendable to the case of a plane surface (#7), explaining the presence of its straight lines as well: for every two points on a plane surface, the straight line segment joining them lies entirely on the surface; and a surface's straight lines are simply the straight line segments joining any two points on the surface.

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 points"?

My answer to this question is to opt for an interpretation of definitions #4 and #7 based on tangentiality rather than convexity:

"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 uniquely, this is not the case with a point on a smooth surface (where there exist infinitely many tangent lines): not a terribly powerful argument, I must admit, so you may augment it by adding the wish to rule out the case of surfaces (such as the cone) containing points at which just one tangent line to the surface lies evenly with (or rather entirely on) the surface!

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").