TITLE: Is there a geometrical definition of a tangent line?
QUESTION [12 upvotes]: Calculus books often give the "secant through two points coming closer together" description to give some intuition for tangent lines. They then say that the tangent line is what the curve "looks like" at that point, or that it's the "best approximation" to the curve at that point, and just take it for granted that (1) it's obvious what that means, and (2) that it's visually obvious that such statements are true.
To be fair, it's true that (1) I can sort of see what they mean, and (2) yes, I do have some visual intuition that something like that is correct. But I can't put into words what exactly a tangent line is, all I have is either the formal definition or this unsatisfying vague sense that a tangent "just touches the curve".
Is there a purely geometrical definition of a tangent line to a curve? Something without coordinates or functions, like an ancient Greek might have stated it. As an example, "A line that passes through the curve but does not cut it" is exactly the kind of thing I want, but of course it doesn't work for all curves at all points.

REPLY [2 votes]: Let $\mathcal A$ be a set of points, let $P$ belong to $\mathcal A$, and let $\mathcal T$ be a line through $P$. Then $\mathcal T$ is a tangent-line to $\mathcal A$ at $P$ if the following condition is satisfied. If $\epsilon$ is any positive number, there is a positive $\delta$ such that every secant $PX$ for which the distance of $X$ from $P$ is less than $\delta$ makes an angle smaller than $\epsilon$ with $\mathcal T$.

This is Hugh Thurston's answer (in his article On Tangents). Note that this definition is with respect to "a set of points" and not to a "curve". Thurston states that there are two reasons for this. The first is that there is no universally accepted definition of curve. A definition that was in favor for a long time is that a curve is the path of a continuously moving point. After Peano showed that such a curve can completely fill a square, attempts were made to modify the definition, none winning 100% acceptance. Luckily the most interesting curves for the study of tangents are simple arcs, about whose definition there is no controversy. The second reason for not confining our attention to curves is to include tangent lines to graphs of functions. Curves and graphs are by no means the same thing, even in a plane. A circle is a curve, but is not the graph of a function, and on the other hand, the graph of the Dirichlet function is not a curve.
In his work, Thurston also makes a brief but illustrative historical review of the "geometric" concept of the tangent line and discusses other problems related to the existence of tangent lines and their intimate relationship with the concept of derivative.