Practice English Speaking&Listening with: Parabolas and Archimedes - Numberphile

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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...

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