Why C/D is the same number for every circle

0:00Here is a claim, and it is a strange one. Take any circle. Measure the whole way round it, and measure straight across it through the middle. Divide the first by the second. You get the same number. Not roughly the same, and not the same for the circles somebody happened to try - the same, for every circle that has ever been drawn. Most people meet that as a fact to be believed, backed by the report that people measured a lot of circles long ago and the answers agreed.

0:35That is not a reason. This video is about the real one. We have already rehearsed this on a shape with corners. Compare the distance round a square to the length of one side: four to one, at every size. Enlarging multiplies the way round and the side by the same factor, and a factor on the top and the bottom of a quotient cancels. But run it on a circle and it stops at the first word.

1:06Compare the way round to what? A circle has no side. There is nothing on it to point at and call one edge. So the first job is to pick a length to measure against, and the natural candidate is the distance straight across through the middle. Every circle has one, it is the same whichever way across you take it, and you can measure it on a real object by trapping the circle between two blocks.

1:36So the quantity we will watch is the way round divided by the distance across. Write it down and leave it alone. It is not a value yet - it is a quotient formed from two lengths of one particular circle. The claim is that another circle gives the same answer. Here is what makes circles special, and it is easy to say and easy to miss. There is only one circle.

2:05Not one size - one SHAPE, seen at different sizes. Fix a point and multiply every distance from it by a factor: a circle of any radius becomes a circle of any other radius you please, and exactly one factor does it. That holds for every pair of circles, in both directions. Nothing else in this subject behaves like that. Among rectangles with whole-number sides up to six there are twelve genuinely different shapes, and no amount of enlarging turns one into another.

2:42On a small grid of whole-number points there are a hundred and nineteen different triangle shapes. Circles have one. Put those two things together and the argument is finished. A quotient built from lengths in a figure cannot move when the figure is enlarged, because one factor multiplies every length in it, top and bottom alike. So such a quotient has at most one value for each shape. Twelve rectangle shapes give twelve values of the way-round to longest-side quotient - one apiece, and no single number for rectangles at all.

3:23A hundred and nineteen triangle shapes give a hundred and nineteen values. Circles are one shape, so they give one value. That is the whole of it. It is not a fact about being round. It is a fact about there being only one of them. One sentence in there was doing quiet work. We said enlarging multiplies every length, including the way round. For a shape with corners that is plain - finitely many edges, each multiplied by the factor.

3:58For a curve it is not plain at all: you cannot lay a ruler along a curve, and nobody has yet said what its length means. Here is the honest answer. The length of the boundary is what you get by going round it in straight pieces, with the pieces getting shorter. Put four points on a circle and join them; then twelve, then thirty-six, then a hundred and eight.

4:24Each time you put corners in, the border gets strictly longer - never equal, never shorter - and never overtakes the circle. Every one of those polygons is straight edges, so every one scales by the factor - and so does what they are climbing towards. So the quotient exists and is one number. Now notice how little that tells us. We know there is one number. We do not know what it is.

4:54Put six candidates side by side: three, twenty-two sevenths, three point one four, nineteen sixths, three and a half, four. Every one survives the argument we just made, because that argument says the two lengths scale together whatever the constant happens to be, so it cannot prefer any of them. Existence and value are two different problems. You have just watched the first settled in two minutes. The second took about four thousand years.

5:28One warning before we go looking for the value. The quotient says something about the shape, but it is not a name for it. Equal quotients do not mean the same shape. Here is a triangle with sides three, four and five: way round twelve, longest side five, quotient twelve fifths. And here is one with sides two and a half, four and a half, and five - a different shape, not an enlargement of the first by any factor.

5:59Its way round is also twelve and its longest side is also five. Same quotient, different shape. The argument runs one way only. For circles that costs us nothing, because we already know there is only one. So what is the number? Here is a way to get a first answer at a kitchen table, with a reel of fine thread. Measure the width of the reel as carefully as you can - say two point four centimetres.

6:32Wind the thread round it twenty times, keeping it neat and flat, then unwind and lay it against a ruler. Say it comes to a hundred and fifty-one centimetres. Twenty wraps is twenty times round, so divide by twenty times the width: a hundred and fifty-one over forty-eight. That is three point one four five and a bit. Between three and four - and also between three point one and three point two, which is more than a reel of thread has any right to give you.

7:08Why twenty wraps? Not to make the thread easier to handle. Look at what the ruler does. Read to the nearest millimetre, it is out by about the same small amount whatever you measure - the error does not care how long the thing is. Measure one wrap, about seven and a half centimetres, and that error is about one and three tenths of a per cent of it. Measure twenty wraps, a hundred and fifty-one centimetres, and it is about seven hundredths of a per cent.

7:41The improvement is not roughly twenty times; it is exactly twenty, and exactly the number of wraps, whatever that number is. Repeat the thing and divide. That trick is most of what careful measurement is. Now the hard part: knowing what that answer is worth. Suppose your measurements are good to a hundredth. A law saying the quotient is one fixed number fits all of them. But so does a law saying it WOBBLES - slightly bigger on some circles, slightly smaller on others - because both sit inside the same error bars and your measurements cannot see the difference.

8:24The bars are not useless: a law out by a tenth is thrown out at once. So measuring narrows the field and never closes it down to one law. Which is why the argument had to come first. The constancy is not something measurement discovered. It is the reason we know what the measuring is measuring. There is a way to corner the number with no measuring at all, and we have already drawn it.

8:53Those polygons inside the circle are shorter than it, and a square drawn snugly round the outside is longer. That traps the quotient between three and four at once - and the thirty-six-cornered one has already pushed the lower end past three point one, with no ruler in the room. How tight can that get, and who first pushed it far enough to matter? That is the next story. Meanwhile the number does ordinary work.

9:24Taking it as twenty-two sevenths, a circle whose way round is forty-four has a radius of exactly seven. A wheel fifty-six across covers a hundred and seventy-six centimetres in one turn, so ten kilometres takes five thousand six hundred and eighty-one turns and nine elevenths of another. It does not come out whole. Almost nothing does.