Map scale as a ratio, and what it lets you compute

0:00Every map has a number tucked in one corner, and most people never read it. It looks like this. One to sixty lakh. One to six million. It is not a date, and it is not a code. It is a ratio, and it is the single most useful thing on the sheet. Because without it a map is a picture. With it, a map is an instrument you can measure with.

0:26So: one to six million. One what, to six million what? Here is the whole of what it says. One unit measured on the paper stands for six million of the SAME units on the ground. One centimetre on the paper is six million centimetres of country. Notice the two words carrying all the weight. The same units. Not one centimetre to six million kilometres, which is the commonest thing people read into it, and which overshoots by a factor of a hundred thousand.

1:02A hundred thousand is exactly the climb from a centimetre to a kilometre. So that misreading is not a small slip. It is the whole ladder, in one step. Now look at what is missing. Neither number has a unit written on it. That is not an oversight. It is the design. Because the ratio takes whatever unit you hand it and gives you the same unit back. Measure a gap on the paper as two point four one centimetres, multiply, and the answer comes out in centimetres.

1:38Measure the very same gap with an inch ruler and it reads less than half as much - two point four one over two point five four, which is under an inch. Multiply that by six million and convert, and you land on exactly the same distance on the ground. A hundred and forty-four point six kilometres, both times. The two readings on the paper are different numbers. The country between the two places is not.

2:11So let us do the conversion once, properly, and never do it again. One centimetre on the paper is six million centimetres on the ground. A hundred centimetres make a metre, so divide by a hundred. Six million becomes sixty thousand metres. A thousand metres make a kilometre, so divide by a thousand. Sixty thousand becomes sixty kilometres. Two divisions, and nothing else happened. The ratio did its work in the very first line, and the rest was bookkeeping about names.

2:50Which leaves one line worth writing down before you touch a ruler at all. One centimetre on the paper. Sixty kilometres on the ground. That is the working rate, and it is the whole of what the corner number gives you. Every measurement you make on that sheet is now one multiplication away from an answer. And notice that the working rate is the only place in this argument where a unit is named, because it is the only place where anybody chose one.

3:25So take a map at that ratio and two cities on it, and measure the straight gap between them with a ruler. The first pair come out two point four one centimetres apart on the paper. Two point four one times sixty is about a hundred and forty-five kilometres. The second pair are further off - four point eight eight centimetres, which is about two hundred and ninety-three kilometres. One measurement, one multiplication, one answer. That is the instrument working exactly as it should.

4:02But be careful what you have just measured. A ruler laid between two cities measures the gap. It does not measure any road. Here are four different routes between the same two places, and every one of them is longer than the straight line - by thirty per cent, twenty-one per cent, sixty per cent, and one route that happens to run dead straight. Four roads. Four different lengths. And the map returns ONE number for all four, because it never looked at any of them.

4:36The only thing it can promise about a road is that the road is at least as long as the line. How much longer is not a fact about the map. It is a fact about the ground, and no ratio in any corner can tell you. Now here is a test you can run on the whole idea. Take the same two cities on four different maps, drawn at four genuinely different scales.

5:03On the second sheet the gap measures half as much, and the ratio is twice as big. On another it measures five times as much, and the ratio is a fifth. Every one of the four returns the same ground distance. And that is not a coincidence. The ground distance is the fixed thing in this problem. The scale is only a record of how hard that particular sheet shrank the country. Shrink it harder and you measure less; the two changes cancel exactly.

5:38So if two maps ever disagreed about how far apart two cities are, one of them would simply be wrong. Which brings us to the one thing a ratio in a corner cannot survive. It is true of the sheet it was drawn on, and of no other sheet in the world. Put a map through a copier at four fifths and every length on it shrinks. The number in the corner does not.

6:05That sheet now says one to six million and is really at one to seven and a half million, and it reports a hundred and sixteen kilometres where the answer is a hundred and forty-five. Blow the same map up by half and it is really at one to four million, and it reports two hundred and seventeen. So somebody might conclude that a different scale means a different real distance. Put that idea to all six sheets and it is right exactly once - on the one sheet where nothing differs either way.

6:41It parts company on the other five. The three honest ones at three other scales all agree, and the two copies never changed their scale at all while the answer moved. And now the most useful habit in this whole topic, which is to check a map against itself. The map I measured carries lines of latitude, labelled every two degrees, from eight north to eighteen north. That is ten degrees.

7:12Ten degrees of latitude is a known length, and it needs nothing from outside. The metre was defined so that a quarter of the way from the equator to the pole is ten thousand kilometres, which makes one degree about a hundred and eleven. So ten degrees is eleven hundred and eleven kilometres, and on that sheet it spanned nine point two centimetres. That is about a hundred and twenty-one kilometres to the centimetre. The corner said sixty.

7:44The sheet is drawn at roughly one to a crore and twenty lakh - almost exactly TWICE the ratio it carries. Which is why both city distances came out at almost exactly half the truth. And there is a trap inside the check. Measure from the frame around the map instead of from the latitude labels inside it, and you are calling nine and a half centimetres ten degrees when the labels say nine point two. That alone understates the rate by three point two per cent.

8:20So far the ratio has been running one way: measure the paper, get the ground. Run it the other way and you are not reading a map any more. You are drawing one. Sketch a room at one to fifty. A wall six metres long is six hundred centimetres, and six hundred divided by fifty is twelve. Draw it twelve centimetres. A ceiling fan three metres across is drawn six centimetres. A chair half a metre across is drawn one centimetre.

8:55Same ratio, opposite operation. Reading a map multiplies. Making one divides. One last question, and it catches almost everybody. Which shows more - one to fifty, or one to six million? The instinct is that the bigger number is the bigger map. It is the other way round. The second term is how hard the ground was squashed to fit. Squash harder and more country fits on the sheet, and less of it survives.

9:28That chair, one centimetre across at one to fifty, is about eight millionths of a centimetre on the map of a country. It is not small. It is not there. The whole six-metre wall is smaller than the chair was. So the ordering is not an accident of what I happened to pick. A ratio in a corner is a promise about one sheet of paper: what it can show you, and what it never could.