Lune as in lunar and a lune in mathematics looks something like this.
Brady: "And is that like the modern word for that? That's what you would call it?"
Oh that's what we'd call it I suppose, yeah.
"It looks like a crescent to me."
Crescent? You could call it that.
This is a mathematical object that was of interest a number of millennia ago, and it's still of interest.
We have a real sense of what Hippocrates of Chios, 5th century BC mathematician
he was interested in, can I say, even though this doesn't occur in his text at all, squaring the circle.
So what does it mean to square the circle?
What it means really is to find, at least the way I'm going to interpret it, is to find a polygon of the same area as the circle
and to find it in some constructive manner.
What Hippocrates was interested in doing was finding the area of such figures
in terms of the area of polygons.
"He wanted to construct a polygon that had the same area as a lune?"
As some lunes.
He wasn't able to do it
for all, or even in some sense, very many. But he certainly was able to do it, and here's the starter lune.
Starter lune begins with making an isosceles right angle triangle
and here's the right angle.
And thinking of that triangle as being inscribed,
if you wish, in a semi-circle.
The next thing I would like to do is make another circle
with this hypotenuse as chord.
Well, I'd want to take either one of these snippets of circles
either this one or that one, they're both of equal area and equal shape.
I'm gonna to try to view this line as the chord of a circle
so that the figure I make will be similar to this one or that one.
I'm gonna try. It's not all that easy, but I'll try to do it, okay?
So these three dashed things are similar.
Let me draw it a second time, enlarged now.
Now the question is, what can we say about the area of this guy,
in terms of the area of this guy?
Or equivalently, in terms of the area of this guy, because the last two guys have the same area?
Well, if you remember our Pythagorean theorem,
This length is the square root of 2 times either of these lengths.
So, if the linear measure is the square root of 2 of the big segment,
is the square root of 2 times the linear measure of the small, either of the two small segments,
what's the area
of the big segment in terms of the area of either of the smaller segments?
It's the square of the linear ratio.
Okay. So this is twice the area of this, twice the area of that.
Another way of saying it is, this area is the sum of these two areas.
So suppose I start with that triangle, and I'll draw it again,
and I remove this blob, this area,
and now I have to add something to keep the area of the triangle the same,
and what I'm gonna do is add the two small blobs.
So what does that mean?
That means that the area of the triangle
is the area
between these two semicircles.
So what has Hippocrates done, if he's done that?
He has, in some funny way, squared this particular lune. This is a lune.
"He hasn't squared it, though. He's triangled it."
He's polygoned it, if you wish.
Now, that's not where he ends his investigations.
"That's pretty clever, though."
It's already pretty clever. I think of this as the starter construction.
This, it's not clear in the manuscript, by the way,
or the text that one has from his manuscript,
whether he starts this way, and this is his inspiration,
but how can it not be? This is surely his starter lune construction.
The minute you do that, you might imagine going into the business of
squaring, or triangulating, or polygon-ating, as many lunes as you possibly can.
Let me start with the construction, essentially with this straight line.
So, I'm gonna make this the chord of a number of circles.
Same line, but the chord of different circles.
So it could be the chord of, well, it could be the chord of,
I hope I'm gonna draw it convincingly as a circle, such a circle.
"That's not the best circle I've ever seen."
Or it could be the chord, the same line could be the chord of this circle.
Now every time I think of this as the chord of a circle, I get a pie slice of that circle.
I draw the line from the center to the two ends of the chord, and there's my pie.
If I do it here, the pie slice would be pretty big.
This is a rather generous portion of pie.
How do we get our lunes? Well, we get our lunes by taking this same chord
and making two circles with that as a chord of different radii,
and thereby getting two different pie slices.
Okay, like this.
That's gonna be big, I guess.
And like this, and that's gonna be somewhat smaller.
"And where's the lune here?"
Ah, here's my lune.
Now, what turns out, in each of Hippocrates' constructions,
is he's constructed his chord and his lunes in such a way that the area of this lune
is exactly equal to the area of this little, uh,
what shall we call it? Polygon with a little concave...
"A spearhead, isn't it?"
Spearhead. And how does he do this? Well, by ingenuity.
And he produces a number of examples.
He does it by a certain process that would be too long to give in this numberphile.
But, he does it quite beautifully.
Hippocrates manages to construct situations where the area of this pie wedge
is equal to the area of this pie wedge,
and so therefore they have a common area which is this,
and that implies that this area is equal to this area.
It is a consequence of a very careful choice of the radii of the two circles given the chord.
There is a serious mathematical question which can be phrased in modern-day mathematical vocabulary,
it becomes a question in Galois Theory.
Has Hippocrates constructed all, or perhaps, almost all of the possible constructions
given his methods?
"Has he found all the lunes that can be spearheaded in this way?"
Spearheaded, spearheaded! wonderful word!
"And what's the answer? Did he? Did he find them all?"
He, well, this is a, a conversation any modern mathematician can have with a 5th century BC mathematician.
I don't know the answer.
I worked on it a little.