Measure One Line, Then Never Measure Again

Part 4 ended with a ship that knows exactly where it is and a chart that might be wrong. Finding yourself and drawing a map are different problems, and the second one looks far worse: a country holds thousands of hills, headlands and river mouths, and you cannot walk a measuring chain to all of them. Some of them you cannot stand on at all.

The way out is one of the best ideas in this book, and it is a piece of school geometry.

A triangle gives itself away

Draw a line on paper. From one end, draw a second line at 50 degrees to it. From the other end, draw a third at 60 degrees. Those two lines cross at exactly one point, and your triangle is finished.

You did not choose where the crossing went. Once the base and the two angles at its ends are fixed, everything else about the triangle is fixed too — the other two sides have one length each and no say in the matter. You could measure them off the paper with a ruler. Or you could work them out with arithmetic and never touch a ruler at all.

Now look at what that buys you outdoors. An angle to a distant hill can be measured from where you are standing. You need to see the hill, not reach it. So a triangle turns two things you can do easily — walk a flat line, and sight a distant object — into a distance you could never have paced out.

And then you do it again

Here is the step that turns a piece of geometry into a map of a country.

Once your first triangle is solved, you know the length of its two long sides. Those are now known distances. So either of them can serve as the base of the next triangle, and you never measure a distance on the ground again.

This is called triangulation, and from the 1600s onwards it is how essentially every accurate map on Earth was made.

Which is why the first line matters so much

If every distance in the country comes out of one measured line, then every distance in the country carries that line's error. Get the base one part in a thousand too long and the whole map of the country is one part in a thousand too big.

So surveyors became obsessive about baselines in a way that is hard to convey.

In 1784 the army surveyor William Roy laid out a base across Hounslow Heath, flat open ground west of London. It was about five miles long. He tried wooden rods, which swelled in damp weather and threw the measurement off. He ended up using glass tubes, laid end to end and moved forward one after another, all the way across the heath, through a summer, with the temperature written down at every step because glass grows when it is warm.

One line. Five miles. A whole season's work by a team of soldiers.

From that line the survey of Britain grew, triangle by triangle, for the next sixty years — and the organisation set up to do it is still there. It is called the Ordnance Survey, and it still makes the walking maps with the brown contour lines on them.

The survey that measured a mountain

The same method went out across the world with European empires, and the largest example is worth knowing about, including the awkward parts.

The Great Trigonometrical Survey of India began in 1802 and ran for most of a century. It carried chains of triangles the length of the subcontinent, with instruments so heavy they took a dozen men to lift onto a tower. It was slow, expensive and dangerous, and it was not done for the benefit of the people being surveyed: an empire that knows exactly where everything is can tax it, move soldiers through it and keep hold of it.

In 1852 a mathematician working for the survey — Radhanath Sikdar, one of the "computers" who did the enormous quantities of arithmetic the method demands — is credited with working out that a peak in the Himalaya then labelled only as Peak XV was the highest measured on Earth. The survey checked the result for four years before announcing it.

Then it named it. The Surveyor General proposed calling it after his predecessor, George Everest, and said no local name could be found for it.

That claim does not hold up. The mountain had been known on the Tibetan side as Chomolungma for a very long time, and the name had already appeared on maps published in Europe more than a century earlier. Everest himself objected to the naming, partly because his name could not be written in the languages of the region. The Royal Geographical Society adopted it anyway in 1865.

So the height on the map came out of honest and extraordinarily patient arithmetic, and the name on the map came out of something else. Both are worth knowing when you look at it.

Numbers need somewhere to start

Put the last three parts together and you have the whole machine. A sextant and a clock, or a sextant and the Moon, will tell one ship where it is. A baseline and ten thousand angles will tell a mapmaker where a country is. Both hand you the same thing: a pair of numbers.

But a number has to be counted from somewhere. Latitude counts from the equator, which is fair enough — the equator is where the ball is widest, and nobody had to decide that.

Longitude counts from a line. And there is no line. Nothing on the ground, nothing in the sky, nothing about the way the Earth spins picks out one place and marks it zero.

So somebody had to choose. And then everybody else had to agree — which took rather longer.