It is impossible to summarize Euclid. You could simply list the propositions (i.e. theorems) that he goes through, but doing that wouldn’t convey the actual point of the book, which is the rational proof of the propositions. But I have to try something. In reading book XII of the Elements this year, I was most moved by the four-page proof of proposition 10. To explain what impressed me about it, though, I have to try to get into the details, and I don’t want to just reproduce the whole proof. So I’m going to try the impossible and summarize the proof. If you really want to follow it, you can find it online easily enough (although I hope you have better luck with your Java than I did just now).
By this point in the work, Euclid has proven that a polygonal pyramid has a volume equal to 1/3 that of the prism set up on the same base. Imagine any polygon you like: a square, a pentagon, or some crazy eleven-sided lopsided star-shaped thing. Then imagine each of the sides of your polygon extending straight up like so many walls and put a ceiling on your weirdly-shaped room. There’s your polygonal prism. Now go back to the original polygon – in other words, the floor of your hypothetical room – and imagine just one spot on the ceiling. Run lines from each corner up to that one point, and you have your polygonal pyramid. Euclid can show you that the volume of that pyramid is exactly 1/3 the volume of the whole prism. Basically, each of these shapes can be divided up into smaller shapes with triangles for sides, and geometry has a whole lot of useful things to say about triangles. So the relationship between the pyramid and the prism is pretty easy to demonstrate.
But in proposition 10, Euclid’s task is to show that a cone has a volume 1/3 of that of the cylinder built on the same circle and carried to the same height. No triangles here. Just circles and curved walls. I think Newton could have proven this proposition using his calculus, but Euclid didn’t have the mathematics of limits and infinitesimals. So what does he do? He uses a reductio ad absurdum: suppose the proposition is not true, and then see what contradiction you run into.
First, suppose that the cone’s volume is more than 1/3 the volume of the cylinder. Then inscribe a polygon in the circular base and imagine the pyramid starting at that base and extending up to the point of the cone. Now you have a pyramid tucked snugly inside your cone. Remember, the cone is larger than 1/3 of the cylinder in this supposition, so now increase the number of sides on your inscribed polygon, thereby increasing the size of the pyramid above it, until the pyramid is also larger than 1/3 of the cylinder. Well, that pyramid has a volume 1/3 the size of the prism on that same base. And if the pyramid is greater than 1/3 of the cylinder, the whole prism must be greater than the size of the whole cylinder. But its base is inscribed inside the circle, so the prism is actually smaller than the cylinder. And you’ve reached your contradiction: the prism can’t be both smaller than and larger than the cylinder, so the supposition must be incorrect.
So Euclid has proven that the cone is not greater than a third of the cylinder. Now all he has to do is turn the whole process around backwards to prove that the cone isn’t less than a third of the cylinder. And if its volume isn’t either greater than a third of the cylinder or less than a third of the cylinder, it must be exactly equal to a third of the cylinder.
I think the proof of proposition 10 is the longest proof of the whole book. It’s wonderful to see how brilliant Euclid is – how much he can prove with a few suppositions, a good grasp of logic, and a lot of creative imagination – even in the short, simple proofs. So this long proof was simply astonishing to me. If I, in turn, was unable to astonish you with my poor summary, go read it for yourself. Of course you have make it through eleven books plus nine more propositions in order to understand proposition 10 of book XII. So start reading now!