Justin, 

      Excellent work in your paper on Archimedes' Quadrature of the 
   Parabola ("QP" for short).  You have correctly pointed out that 
   Archimedes gave two proofs of the statement that the area of a 
   segment of a parabola is 4/3 the area of the inscribed triangle 
   in that work, and yet a third proof in the Method of Mechanical 
   Theorems.  The proof in the Method is essentially the same as the 
   first one in "QP," although there are some differences in how
   the argument is presented.  There has been quite a bit of scholarship
   devoted to attempts to determine which of these accounts might have
   come first, but I think the "jury is still out" on that question.
   Your discussion of the second proof from QP is also excellent. 
   Even though he doesn't quite say it that way, Archimedes' proof
   there is also essentially the first known example of a summation
   of an infinite series(!)  In fact, we could rephrase his argument
   by saying that the new triangles added at each step of the 
   construction have area equal to 1/4 the area of the triangles
   from the previous step, so starting from the inscribed triangle
   of area A, say, we have a total area of 

    S = A + A/4 + A/4^2 + A/4^3 + ...
   
   if we add all of the areas of the triangles if the construction
   is continued forever.  This is a geometric series with ratio
   1/4. Then the sum of this series is 
 
    S = A/(1 - 1/4) = A/(3/4) = 4A/3 (!)
     
   (Saying Archimedes did exactly this would be anachronistic, of course,
   he's looking at what are known as "partial sums" of this series.)

   Grade:  A