Turning an observation into a definition: the circle as a locus

0:00Long before anybody had a definition, people were painting circles on cave walls. Alongside triangles, squares and ovals, with no sign the circle was treated as anything special. And they had plenty to copy from. Rain hits still water and rings spread out from the point where the drop landed. Cut through a tree trunk and the growth rings sit one inside another. A sunflower head seen face on. The Moon.

0:31The Sun during a total eclipse, though look carefully at that one. What you are seeing is a dark disc with a ring of light round its edge, and the circle is the boundary between them. Here is the thing. Not one of those is a circle. They are all approximately circular, which is a different thing, and that difference is the whole of today. So what do they have in common? Big ones, small ones, drawn ones, grown ones.

1:03There is a middle. And every point of the rim stands the same distance from that middle. That is an observation. You could check it on a sunflower with a ruler and find it very nearly true. Now notice the shape of that sentence. It starts with a circle you already recognise, and then tells you something about it. Round first, property second. That order is comfortable, and it is completely useless for proving anything.

1:35Here is the move the whole subject turns on. Take the sentence and reverse it. Instead of: this is a circle, and its points happen to be the same distance from the middle. Say: fix a point, fix a distance, and a circle is everything at that distance. The property stops describing and starts deciding. It becomes an entrance requirement. Before, you had to know it was a circle before you could say anything.

2:05Now you can hand the condition one single point and it answers yes or no, without ever seeing the whole shape. That is the difference between a description and a definition. There is a word for a set built this way. A locus. You state a condition, and the locus is everything satisfying it. No more, no less. Say it in the right order and the definition writes itself. The set of points, in a plane, at a fixed distance from a fixed point of that plane.

2:41Two things you have to supply: the point, and the distance. And both of them are doing work. Keep the distance and shift the point three units across, and almost the whole of the old set is refused by the new condition. The two circles do still meet, but only in two places, and neither of them is a whole number point. Keep the point and change the distance from five to six, and again, nothing shared at all.

3:13Now the clause everybody skips. In a plane. Drop those three words and read the condition again. Everything at distance five from a fixed point. In space, that is a sphere. To see how big the difference is, lay a grid of whole number points over the circle and run the test on every point of it, then do the same with the whole number points of space. The grid is a sample, not the circle. A real circle has infinitely many points. This is just a way to watch the condition working.

3:51In the plane, twelve of them pass. In space, thirty. Eighteen extra points, and every one of them something the circle refuses. Those three words are not legal padding. They are the difference between a curve and a surface. The definition also creates two words out of nothing at all. The fixed point is the centre. The fixed distance is the radius. Neither of them existed before the definition. Both are just names for the two things you had to supply.

4:27One warning about the second. Radius here is a length, a distance. But everybody also uses the word for the segment drawn from the centre out to the rim. Both are current and both are fine. You just have to know which one you mean each time. A circle is the rim. Only the rim. Take the grid again. Run exactly five from the centre, and twelve points pass. Now run at most five instead, and eighty one pass.

5:00Sixty nine more, and every one of those is something the circle does not contain. The round region is not the circle. The circle is its edge. Try three points. One three units out. One five units out. One eight units out. The first is inside and refused. The second is on it and admitted. The third is outside and refused. The definition takes exactly one of the three, and refusing the other two is the whole of its job.

5:35So where does the centre live? Its distance from itself is nothing. And nothing is not five. The centre is not a point of its own circle. It is the thing everything else is measured from. Which is worth a pause, because it is the one place the definition needs care. Run that check for every radius from nought up to eight, and there is exactly one radius where the centre does sit on its own circle.

6:04Radius nothing. At radius nothing the whole locus collapses to the centre and nowhere else. So the definition wants a distance bigger than nothing, and that is not fussiness. It is the one case that would break it. Now some vocabulary the definition has earned. A chord is a segment with both ends on the circle. A diameter is a chord that runs through the centre. Not a different kind of object. A chord that happens to contain the centre.

6:39Take those twelve grid points and join every pair. Sixty six chords. Six of them pass through the centre. So sixty are chords that are not diameters, and all six diameters are also chords. Every diameter is a chord. Almost no chord is a diameter. Those sixty six come in ten different lengths, and the longest of them is the diameter. One more piece, and it arrives early for a reason.

7:12Take any chord and draw the two radii out to its ends. You get a triangle with two equal sides, and it is free. Both sides are radii, straight from the definition. The angle at the centre is called the angle that chord subtends. Here is what is worth noticing about it. That angle depends on nothing whatever except the chord's length. Check it on all sixty six and each of the ten lengths gives exactly one angle, every time.

7:45And the longest chord gives a straight angle. The diameter, and only the diameter. A puzzle to hold on to. Somebody hands you a paper disc. Perfectly circular, nothing marked on it. Find the centre. You are not allowed to measure and guess. Everything you have is the definition, and the definition mentions a centre, so the information is in there somewhere. You just have to get it out. Try folding it.

8:18I am not going to answer that. Hold on to it, because the answer is where the next argument starts. So what has been bought here. A circle is now a set, decided one point at a time, by a condition anybody can check. Centre and radius have names. Chord and diameter have names. The angle at the centre has a name. And notice what is not settled. Nothing yet about which chords are equal to which. Nothing about arcs. Nothing about angles anywhere except at the centre.

8:57Every single one of those has to be squeezed out of this one condition, and nothing else is allowed in. One last thought about the true statements that did not make it. Every point of a circle of radius five stands a whole number of units from the centre. Perfectly true. But so do thirty seven other points of that grid, and not one of them is on the circle.

9:23A true statement is not a definition. A definition has to refuse things.