How much of a circle is in contact with a tangent line?
Dear Straight Dope:
Imagine any geometrically perfect circle, and now imagine a geometrically perfect line running tangent to that circle. How much of that circle is actually in contact with that line? Since the circle is curved, any measure of a circle's circumference is also curved. At the same time, by definition, the tangent line must touch the circumference of the circle at exactly one point. If this point exists, it must have some quantity associated with it. How can this point be infinitely small and still have some quantity associated with it at the same time?
I dunno what you mean by "quantity" , but I'll take a stab at this anyway. I assume you mean that you can't visualize the line touching the circle at only one point.
You said we want a geometrically perfect circle and line. OK, in that situation, the tangent line touches the circle at one point and only one point. So far, clear?
I think you are saying that when you draw a circle, you can see that the tangent line really touches at several points. The problem is that any circle or line that you draw is NOT a geometrically perfect. For one thing, the pencil line has width; no matter how thin and sharp a point on your pencil, there's some width to that line. The boundary line of a theoretic (what you've called "geometrically pure") circle has no width. Thus, the overlap that you see arises from the incongruity between the crude pencil drawing and the pure theoretic image.
So, try to imagine your circle as being drawn with a line that is so thin that it has NO width. Not so much as the diameter of an atom. None at all. Perhaps you can get there by starting with your circle and tangent, and trying to imagine magnifying it. When you stare at the drawing, it looks like there are several points of contact, perhaps a hundredth of an inch long. If you imagine magnifying that in your head, and also imagine thining down the pencil line, you can imagine that the quantity of points of contact gets smaller and smaller. Presumably you can see that if you COULD imagine that the lines had no width, there would be only one point of contact.
So the answer to your bottom line (ahem) question is: in the real world, a point is a dot, with length and width measurements (no matter how tiny the dot you make with your pencil, it has some length and width, even if only an atom in diameter.) But in the mental world of pure geometry, a point is not a dot made with a pencil, it is a theoretic point, with neither length or width. I detest the term "infinitely small"; better to say, it has zero length.
Does that clarify, or does that confuse you more?