So I had this circle, but I wanted a square

Don’t ask why, that’s my affair

The crucial aspect of this little game

Is that the area should stay the same

Ruler and compass are the tools to use

It’s been proven impossible, but that’s no excuse

Many have tried it, but hey, I’m me

I’m bound to find something that they couldn’t see

So, here we go

Oh

That’s a rectangle

Never mind, I’ll give it a try

Call the long edgexand the short edgey

Now make an arc to findy+x

From the midpoint, make a semicircle next

Extending y to that arc gives the length I desire

I can prove it with algebra if you require

Draw a square on that side, and I don’t want hysteria

But the square and the rectangle have the same area

But what about the circle?

So, here we go

Oh

That’s a triangleNever mind, I’ll give it a try

I can find the midpoint of any side

And draw a right angle up from there

It’s a crucial ingredient for a square

A parallel line through the third point now

Fill in the top and the side somehow

It’s a rectangle, same height, half the base

And that means it takes up the same space

And I can already square a rectangle

But what about the circle?So, here we go

Oh

That’s a pentagon

Never mind, I’ll give it a try

If I join the corners between pairs of sides

I can make it into triangles, one, two, three

And squares from triangles are now easy

Put two sides together, joined at right angles

Then by some famous theorem involving triangles

The square on the long side sums the other two

Repeat the process, and then we’re through

But what about the circle?

So, here we go

Oh

That’s a hexagon

But come on, for six sides I know what to do

And the same will work for seven too

This is easy, look, it’s great

I can even do the same for eight

And nine, and ten, and – holy cow!

That looks a bit like a circle now

In fact, I’ve got it, it’s obvious innit?

To square the circle, you take the limit!

The practical details of the construction

I’ll leave as an exercise for your deduction

21 Dec 2019: I’ve added posts to explain constructions in the poem in more detail: Part 1 and Part 2.

## Leave a Reply