Chapter 1

PROBLEM.--_To find the centre of a given circle_.

Let ABC be that horrible round bed where you had the geraniums last year. This year, I gather, the idea is to have it nothing but rose-trees, with a great big fellow in the middle. The question is, where is the middle? I mean, if you plant it in a hurry on your own judgment, everyone who comes near the house will point out that the bed is all cock-eye. Besides, you can see it from the dining-room and it will annoy you at breakfast.

CONSTRUCTION.--Well, this is how we go about it. First, you draw any chord AB in the given bed ABC. You can do that with one of those long strings the gardener keeps in his shed, with pegs at the end.

Bisect AB at D.

Now don't look so stupid. We've done that already in Book I., Prop. 10, you remember, when we bisected the stick of nougat. That's right.

Now from D draw DC at right angles to AB, and meeting the lawn at C. You can do that with a hoe.

Produce CD to meet the lawn again at E.

Now we do some more of that bisecting; this time we bisect EC at F.

Then F shall be the middle of the bed; and that's where your rose-tree is going.

PROOF???--Well, I mean, if F be _not_ the centre let some point G, outside the line CE, be the centre and put the confounded tree _there_. And, what's more, you can jolly well join GA, GD and GB, and see what that looks like.

Just cast your eye over the two triangles GDA and GDB.

Don't you see that DA is equal to DB (unless, of course, you've bisected that chord all wrong), and DG is common, and GA is equal to GB--at least according to your absurd theory about G it is, since they must be both _radii_. _Radii_ indeed! _Look_ at them. Ha, ha!

Therefore, you fool, the angle GDA is equal to the angle GDB.

Therefore they are both right angles.

Therefore the angle GDA is a right angle. (I know you think I'm repeating myself, but you'll see what I'm getting at in a minute.)

_Therefore_--and this is the cream of the joke--therefore--really, I can't help laughing--therefore _the angle CDA is equal to the angle GDA!_ That is, the part is equal to the whole--which is ridiculous.

I mean, it's too _laughable_.

So, you see, your rose-tree is not in the middle at all.

In the same way you can go on planting the old tree all over the bed--anywhere you like. In every case you'll get those right angles in the same ridiculous position--why, it makes me laugh _now_ to think of it--and you'll be brought back to dear old CE.

And, of course, any point in CE _except_ F would divide CE unequally, which I notice now is just what you've done yourself; so F is wrong too.

But you see the idea?

What a mess you've made of the bed!