I love exploring ancient mathematics
because the ancient mathematicians had
to make decisions, had to
work things out without the tools that
we have today.
So how did they do it? Well the greatest
of them all was Archimedes;
and he found the volume of a sphere.
How did he do it? Well we think
it may have gone like this: he produced a
cylinder
and a sphere and something called a
double napped cone.
So there's the sphere and the cylinder is
the same width and the same height.
He could have made them out of wood or
clay pottery - I don't know -
but anyway he had them made or made them
himself. And this double napped cone, which
is two cones that meet
in the middle, once again same width same height.
So he thought about this, he thought if I
take a line across the top of these
the cross section of that equals the
cross section of that
plus zero at the north pole here.
Right. If I take a line halfway down,
then the cross section of that equals
the cross section of that
plus zero here.
Does that mean that if I take any
horizontal line through the three figures
then the cross section of that will
equal the cross section of that plus the
cross section of that
every single time? And he said,
I bet it does. But how could he prove it?
Simple - he dunked this in water, let it
settle, and measured the level of the water.
Took it out again, dunked these two in the
same water
and the water came up to exactly the
same level.
So the volume of that plus that equals
the volume of that - QED.
So what was that volume? Well he knew
that a cone
was one third of the cylinder it would
fit into;
just like a pyramid is one third of the
cuboid
that it would fit into. Get a pyramid, build
the walls, put a top on it and
it's a third. And I can prove that
because I've got three pyramids here;
can you see, instead of the point being
in the middle
it's on one corner, and because of that
you can actually
reassemble them, these three identical
pyramids
to make a cube.
Which proves that each is one-third of
the volume of the cube. So that is
one-third the volume
of that, then that is two-thirds the volume
of that.
Well the cylinder end is pi r squared,
and this is the same height and the same
width,
it's 2r down here. - (Brady: Because it's 2)
(radiuses?) - 2 radiuses or a diameter. pi
r squared times 2r which is 2
pi r cubed. So that is a third of it,
so it's two thirds pi r cubed, so this
must be twice that
four over three pi r cubed.
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