I'm going to tell you a story today
about Archimedes; this
wonderful mathematician of whom we know
not very much. Of all the ancient
mathematicians we know more about
Archimedes than any of the others
but it's still not a lot.
But we do know that he made incredible
discoveries.
One of them was in exploring parabolas;
now what's a parabola?
That's a parabola; that's another
parabola.
A fountain creates parabolas; you dive
off a diving board, you fall in a
parabolic curve.
And he wanted to know about parabolas.
Archimedes would never have seen this
parabola, they didn't have algebra in his day,
okay, and they didn't have graphs in his
day. However this is a parabola
and it's generated by this formula,
y equals x squared. And the y-axis is
going up ten times as fast
as the x-axis is going along. So
y equals x squared. So if x is 10,
y would be 10,10 - 100; and that's there,
corresponds to the y. So all these dots
are generated like that and you've got a
parabolic curve.
Okay, this is quite remarkable because
it's actually a calculator. I'll show you what I mean:
Brady can you give me two numbers
less than 15? - (Brady: All right, how about)
(12 and 5?) - Okay 12 and 5.
So I'm going to put a line through the
12, that's it 5 and 12.
And then we'll put a line through there,
it goes through 60.
5 twelves is 60. So this is an automatic
calculator, isn't it great? Now this
parabolic curve is set in a rectangle,
you can see.
And what Archimedes wanted to know was
if you put
a parabolic curve in a rectangle, what's
the comparison between the area inside the curve
and the area outside the curve? What's
the ratio?
He didn't know, he wanted to find out. And
he had a few tools:
up his sleeve one thing he knew about
was the lever,
and I'm going to make a lever. This is a
normal ruler and I've balanced it
on the six inch mark so it balances. And
if I take a domino,
and put the domino on each end they will
balance.
They would balance if I do it properly!
They will balance, okay, look at that. But
that's because at 6
inches times 1, 6 inches times 1.
But you could do it with three dominoes
and make those balance by putting them
there,
and that balances. Why? Because that's six
times one and that is two inches
times three. Two times three is six, one
times six is six
so it balances. So he knew about levers.
The other thing he knew about was
centres of gravity;
he knew the centre of gravity of a
triangle.
And there: I found the centre of gravity
of this triangle; you can try this with
any triangle you make doesn't have to be
a right angle,
make it out of cardboard. But how does it work?
All you do is you take the midpoint of
each side
and connect them to the angle opposite;
and you'll get three lines that coincide
at one point,
and that happens to be the centre of
gravity
of the triangle - and he knew about that.
This was found, or a version of this was
found, can you see it's a little bent
triangle. Galileo knew about this
so it's been around a long time,
okay, why was it on the skew?
Well I think is because- can you see this
is a parabolic section
here and there's a triangle inside it,
okay.
Now, why is it on the skew? He wanted to show
that no a matter how you cut a slice
with a straight line
off a parabolic section; if you have a
triangle inside it or a rectangle
outside it
the ratio is all the same and that's
what he was looking for.
So I think that's why he put it on the
skew after he discovered it.
So I've drawn it again here; and you see
you've got a parabolic section with a
triangle inside it, ABC.
So what's the ratio between the triangle
and the parabolic curve? Well I can put
that in a rectangle
which is twice the size of the triangle.
Same base
and same height so it's twice this
triangle.
Or I thought what I could do is actually
double the size of the triangle,
doubling the height here with the same
base it must be
double the size of that triangle. And ask
himself so what's the ratio between this
triangle
and the parabolic section? Or why don't I
go even further?
Why is it going further? I'll explain
that later. Why don't we go even further
he said and doubled the length of that. So
we've doubled that,
we've doubled that, drawn that here and now
this triangle: twice the height,
same base, so it's four times that
triangle
in size. What is the ratio in area
between that
and the parabolic section? Well that's
where he got to,
so where did he go to next? He used the
tools he already understood about;
he used the lever. So he took the same
diagram and doubled the length of this.
I named that point F - why?
Because it stands for fulcrum. And he
wanted to make a seesaw balance
there. And then
he took a line, just dropped it down, MO,
and found that that line where it
crossed the parabolic section
he called PO. And he thought about this:
wouldn't it be lovely if a line had weight?
Hang on, hang on, where are we going? We
know from geometry
that the shortest distance between two
points is a straight line
which has no width so it can't have
weight;
but he said what if it had?
And what if I dropped every conceivable MO
down this triangle until I blotted it
out with ink
and it had weight? Now for every MO
there's a PO that crosses the parabolic
section.
Why don't I take all those POs
and put them on here at the end of my balance?
Would it balance this triangle that's
still there?
And he said, I bet it does but it would
only balance
at the balance point which is? The
centre of gravity of this triangle,
which is two-thirds to one-third along
there,
two-thirds to one-third along there, and
two-thirds to one-third along there -
point X. And it should balance.
Which means- can you see because that's
only one third
that's three times as far? So this
triangle must weigh
three times as much as the parabolic
section.
So half of it must weigh one and a half
times as much as the parabolic section.
So half of that must mean that the
triangle ABC
is three quarters the area of the
parabolic section,
and he was right.
Incredible intellect from an absolute
genius.
The quite amazing thing about this is, well,
Archimedes gave a line weight, right? And
decided all those lines
would equal the triangle and gave them
weight. But what it was doing
was actually inventing calculus; whereby
you measure
tinier and tinier pieces, put them all
together.
But calculus wasn't going to be invented
for another 2000 years,
and Carl Friedrich Gauss said. oh the
idiot!
He discovered calculus and he let it go
through his fingers,
so sad. And 2000 years earlier than
anybody else.
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[Preview] I've color coded it for different places
that I've lived,
because in my memories, you know, I kind
of like almost catalogue things that way.
You know so, this is, you know, until I was
eight years old I lived here in Torrance
and then I moved to
Brea - both those were in California. You
know, and then this is like college...