http://www.pbs.org/wgbh/nova/archimedes/palimpsest.html
here is the extent to which i knew classical geometry: in high school it was my favorite subject and i won the mid state HS geometry contest. i also won the statewide "comprehensive" math contest including "solid geometry".
then in college i studied mostly calculus, real analysis, spectral analysis, banach spaces, and some n dimensional theory of content due to jordan, and a little abstract measure theory. no differential geometry, no classical geometry. so this was mostly very abstract stuff designed to give the extremely rigorous underpinning to, and precise formulation of, analysis. there was little or no attempt to actually teach us more "content". the one fascinating course that went into interesting geometric material was Bott's algebraic topology.
then in grad school i studied mostly commutative algebra, complex analysis, algebraic topology and algebraic geometry.
in my job i have taught foundations of euclidean geometry and transformation geometry, but never really going back through all the classical results, rather focusing on the subtleties of the foundations, the interplay between models and axioms, questions of consistency, completeness,...
until this year. i decided that i would have better success if i repeated the topics from the actual high school course my students would be asked to teach, even though i assumed i myself knew this stuff cold from high school. i got a big surprize: i did not know the old basic stuff at all well, i.e. i did not really remember the proofs, just the statements, and there were some statements i did not know.
so i started getting a big boost myself, and i started understanding it better, as if for the first time. i had never studied euclid himself, and did not realize that he proved his first 27 or so propositions without the parallel postulate, so i had no good grasp of just where that postulate is needed. i had never read saccheri, and did not know his "weak" or "neutral" version of the exterior angle theorem, or that without the euclidean parallel postulate one does not have rectangles.
i also had never examined the elementary material with a sophisticated eye, so i did not realize either how the theory of similarity is connected with but independent of that of area, nor how tricky it is to make area well defined. i did not understand that there are three different statements of pythagoras: a^2 = b^2 = c^2 ( a statement about numbers), or equality of the areas of two figures (which requires a proof that area makes sense), or as a statement that two different figures can be dissected into congruent smaller figures).
these three statements are often regarded as equivalent, but to prove them requires different arguments. the numerical version for examples follows from the theory of similarity, while euclids own version, the decomposition version, requires a proof of transitivity of the relation of "equidecomposable" that euclid took for granted. if you open the beautiful book "geometry revisited" e.g. by coxeter and greitzer, you see immediately a simple proof of ceva's theorem but one requiring the existence of an area function, not at all a trivial fact, and one most euclidean geometry courses omit to prove. so i proved it in my class (although quite briefly) using hilbert's lovely argument (millman and parker get moise's argument wrong).
i did not realize that a theory of limits is needed to discuss similar triangles whose side ratio is irrational. this is why euclid considered similarity more sophisticated than equidecomposability, and placed it later, whereas moderns like me and it turns out Hilbert, would invert the order of these concepts. i began to notice that almost every fact provable using area has another proof using similarity, like ceva's theorem.
and i did not know Hilbert's 3rd problem so well, that even if two polyhedra do have equal volume, it may be impossible to decompose them into congruent smaller figures, and why not. i did not have time to present this problem's solution by Dehn, but it is in Hartshorne.
i also did not realize that archimedes had used the concept of centers of gravity to compute volumes, as shown on the websites above. finally, after presenting these concepts, i began to see how some of them generalize to 4 dimensions and higher. e.g. archimedes arguments also show that in any dimension n, the volume of an n - ball is R/n times its surface area. this generalizes the fact that area of a circle is R/2 times circumference, and volume of a sphere is R/3 times surface area. (I presented this 4 dimensional stuff on the last day of class).
This was archimedes own explanation of how he proceeded from computing volume of a sphere to knowing its area as i learned yesterday reading "the method". i also learned that archimedes and euclid's proofs of equality of quantities requiring limits was precisely the one now encoded in epsilon - delta definitions, namely to show that the difference between the two quantities is less than any assigned positive amount. i always thought this discrete concept of limits due to the 19th century analysts like Cauchy. actually they just put it into their own words.
i also learned that Archimedes proved his volume insights by taking limits of sums of approximations. whether all commentators agree or not to call these "Riemann sums", there they are for all to see in Archimedes. e.g. he computed areas and circumferences of circles as limits of areas and perimeters of polygons, which were done by adding up triangles. and he did it from both within and without, computing the error between the two approximations and showing it went down by more than half each time, hence "approaching zero" as we would say.
as i see now, he had the Cavalieri principle as far as being able to prove that parallel slices of area do determine volume, and could use it to compare volumes of cones cylinders and spheres. our advance today is being able to take the function of area of slices, and integrate it to get the volume function. thus we can not only compare volumes by this principle, but compute them. this is the advance over archimedes which is made possible by the FTC.
what books point this out? if none, then where can one learn it if not from the source?