Regarding the quadrature of the parabola:
In a parabolic segment (the area enclosed by a parabola and a chord that intersects it), the vertex is defined as that point on the parabola which is "perpendicularly" furthest away from the chord. (That is, of all the lines perpendicular to the chord that intersect with the parabola, the intersection point of the longest line is the vertex).
I have read that in the time of Archimedes, it was known that the tangent line to the parabola at the vertex point is parallel to the chord.
If you sketch out a parabolic segment, this seems intuitively true - the only tangent to the parabola that you can draw that's parallel to the chord is indeed the one located at the point furthest away (perpendicularly) from the chord.
However, I can't figure out how to demonstrate the truth of this assertion. I would really like to know how to prove it - without resorting to calculus (which of course wasn't available back then).
Thank you!
I'm going to generalize the question for any segment enclosed by a convex part of a curve and a chord. Suppose a random point P on the curve is our desired vertex (see left diagram). Then draw another chord at P parallel to the first chord. The distance between the two chords is equal to the perpendicular distance of P from the first chord. But now we have a sub-segment, and one can see that any point Q on this sub-segment would have a larger perpendicular distance than P, since it is on the other side of the second chord from the first chord. Therefore, P cannot be the vertex.
The only way for P to be the vertex (maximal perpendicular distance), is when there is no other point Q when this drawing is made. If the curve at P is moving in any direction except parallel to the first chord, it will create a separate point Q that is further than P. Therefore, the curve must be moving parallel to the first chord at P; i.e. the tangent at P is parallel to the first chord. Thus, the vertex is the point whose tangent is parallel to the chord.
The above was for convex curves. Now when you have non-convex curves (like the diagram on the right), there will be multiple tangents parallel to the chord. And so multiple vertices; corresponding to the concept of local and global maxima.