Where coordinates came from: grid cities, meridians, and the road to the Cartesian plane

0:00Point at something across the room and say: it is over there. That works, and it works only because the other person can see your hand. Now describe the same place to someone on a telephone, a thousand kilometres away. Suddenly you need words like from, and along, and how far. You need to count from something, in some direction, in some agreed size of step. That is the whole of what a coordinate system is. Not a drawing. An agreement.

0:33And the agreement has exactly four parts, which arrived separately, and centuries apart. Start with a city, laid out four and a half thousand years ago. The streets run north to south and east to west, crossing at right angles, set about ten metres apart. It is tempting to call that tidy planning and move on. But look at what the even spacing buys you. Streets can be perfectly straight without being evenly spaced, and then you can still walk them, but you cannot count them.

1:09Put two streets a hundred metres apart, then the next pair ten metres apart, and being told the third street from the centre tells you nothing whatever about how far you have walked. A fixed spacing is what turns a count into a distance. So a merchant gives directions like this. From the centre, six streets across, then four streets up. On a ten metre grid, that is sixty metres and forty metres.

1:40Two numbers have just replaced a paragraph of description. And notice what is carrying the weight. Six and four are not distances. They are counts. Change the spacing to five metres, or twenty, or fifty, and the same pair of numbers names a different place every time. The unit is doing silent work, and it is the first of the four parts. Now the second part, and it is the one that looks like decoration.

2:13The two directions meet at a right angle. Why should that matter? Take three across and four up. With the directions perpendicular, the straight line distance from the centre is exactly five. Now tilt the two directions to sixty degrees apart and keep the same counts. That distance is no longer five. It is the square root of thirty seven. Open them to a hundred and twenty degrees and it collapses to the square root of thirteen.

2:46Same counts, three different places. The right angle is what stops the two measurements from contaminating each other. The third part is a starting point, and here the story leaves the city and goes around the world. To say how far east a place is, you must first say east of what. Two observers measuring from two different lines will produce two different numbers for the same town, and neither of them is wrong.

3:18Ancient astronomical treatises settled it by agreement. Measure from the meridian running through Ujjayini. Later tables of latitude and longitude list thousands of places, Ujjayini among them, written there as Ozine. An origin is not a feature of the world. It is a choice, and once it is made, it is what everything else counts from. The same move works upward. The sun traces an apparent path across the sky through the year, and that path can serve as a reference line exactly as a meridian does on the ground.

4:00Aryabhata replaced the older chords with sines, and positions in the sky were measured out from that path. And the choice of reference line is just as arbitrary up there as it is on the ground. What matters is not which line, only that everybody measures from the same one. A frame is a frame, whether the thing you are locating is a shop or a star. Three parts in place, and the picture is still broken.

4:32Everything so far counts forwards. Six across. Four up. East of the meridian. There is no way at all to say: four streets the other way. Brahmagupta supplied what was missing. Zero treated as a number in its own right, and quantities allowed to go below it. Only then can a line run through the starting point and out the far side. Without that, the plane has one corner. With it, four.

5:03And then the idea travels, which is a mathematical event and not merely a historical one. The Indian treatises are translated into Arabic, where Brahmagupta's work becomes known as the Sindhind. The reference meridian survives the journey and is written there as Arin. Al Biruni travels to India, uses Indian trigonometric methods to fix the coordinates of Asian cities, and perfects the astrolabe, an instrument for finding your own position from the stars.

5:40Omar Khayyam is the first to attack an algebraic problem by reading it as a picture in the plane, and in the twelfth century the whole apparatus reaches Europe. Then, one year apart, Fermat and Descartes. What they add is a single sentence, and it is smaller and stranger than it sounds. A point is not merely labelled by a pair of numbers. A point is a pair of numbers.

6:10Once that is true, a shape and an equation are one object seen twice, and anything you prove about either you have proved about both. A circle stops being a picture you have to look at, and becomes a question you can answer with arithmetic. Does this pair of numbers satisfy the equation, yes or no. That identification is what two thousand years of parts had been assembling towards. Now test the parts, by taking them away one at a time.

6:45Remove the negative numbers first. What can the frame still reach? Only places up and to the right of the starting point. Everything behind it, everything below it, and the whole quarter diagonally opposite lose their addresses. In the limit, three quarters of the plane goes dark. But be careful with that number, because it is only true in the limit. In any patch you actually draw you keep slightly more than a quarter. In a twenty one by twenty one grid of points, a hundred and twenty one of the four hundred and forty one survive, because the two axes themselves survive.

7:29There is an obvious repair, and it almost works. Instead of signed numbers, write the direction down. Three east, four north. Two directions with two choices each gives four combinations, which is exactly the four quarters you had lost. But it has a seam. A point sitting on the north south line is zero east, and it is equally zero west. One place, two correct names. A signed number has no such seam, because zero is neither positive nor negative, and there is only one of it.

8:07Now put all four parts back, and test them together. A city again. Two main roads crossing at the centre, every other street parallel to one of them, two hundred metres apart, ten in each direction. Label a crossing by its two street numbers. How many crossings answer to four three? Exactly one. How many to three four? Also exactly one, and it is a different crossing, two hundred metres away in each direction.

8:40Of the hundred labels this grid carries, ninety name a different place when you reverse them. The ten that do not are the ten where both numbers were already the same. So here is the agreement, in full. Two directions at a right angle, so that the two measurements do not contaminate each other. One agreed starting point, because counting needs something to count from. One agreed unit, because a count is not a distance until you fix the size of a step.

9:14And numbers allowed to run backwards past zero, because without them three quarters of everywhere has no name. Four parts, four jobs, and every job provable by removing the part and watching precisely what breaks. None of it is a drawing. All of it is an agreement, and that is exactly why it works no matter who is holding the pencil.