results that have very rough proofs from more elementary principles but have much clearer and smoother proofs with some more advanced background. Specifically, what are some examples of results from more advanced or narrow topics of mathematics can vastly simplify or explain in retrospect theorems that are encountered and proved laboriously in less specialized or more common areas of math
One alternative to proofs being long and complicated because they use only elementary methods is proofs being numerous because they use only elementary methods.
In this image,
- $AB$ is any chord of a parabola,
- $AC$ is parallel to the axis of the parabola,
- $BC$ is tangent to the parabola at $B.$
$D$ is the midpoint of $AC$ and we view it as the fulcrum of a lever $JB.$
$D$ is also the midpoint of $JB.$
$EH$ is any straight section of the triangle $ABC$ that is parallel to $AC.$
Proposition: If the weight of the segment $EH$ rests upon the lever at $G$, and the weight of the segment $EF$ rests at $J,$ then the lever is in equilibrium. (This is just the equation of the parabola.)
Therefore, if the weight of the region bounded by the triangle $ABC$ rests at its center of gravity $I$ and the weight of the region bounded by the curve and the chord $AB$ rests at $J,$ then the lever is in equilibrium.
Bottom-line conclusion: Since it is known that the center of gravity $I$ is one-third of the way from $D$ to $B,$ the area of the triangle is exactly three times the area of the region bounded by the curve and the chord.
Archimedes wrote that argument. Today we write $\int_0^1 x^2\,dx=\frac13$ and we use an antiderivative to show that. I don't think Archimedes used antiderivatives.
By similar methods Archimedes showed that the center of gravity of the northern hemisphere is $5/8$ of the way down from the north pole to the center of the earth. For this he considered the line through the center of the earth and the north pole to be a lever with its fulcrum at the north pole, and two of the infinitesimal weights that he had resting on it were a cross-section of the sphere parallel to the equator, and the cross-section in that same plane, of a cone whose apex is the north pole and whose base is bounded by the equator. (I don't remember the details but I think he let the cross-section of the sphere rest at its intersection with the lever and that smaller cross-section rest at a point whose distance above the north pole is the diameter of the sphere.)
In his "mechanical method" he proved 15 (thus "numerous" (to quote the term I used above)) propositions in geometry by doing things like these, finding areas and volumes and centers of gravity.
(I think maybe no one before Archimedes ever thought about centers of gravity.)