Science of Measuring the Earth Delivers an Important Political Lesson in Humility

The following is an adapted excerpt from Universal: A Guide to the Cosmos by Brian Cox and Jeff Forshaw. Copyright © 2017. Available from Da Capo Press, an imprint of Perseus Books, LLC, a subsidiary of Hachette Book Group, Inc.

Our colleague Mike Seymour made an interesting observation while on holiday at Ogmore-by-Sea. Standing on the mudflats by the water’s edge, enjoying the cool of the salty waves over his mildly sunburnt feet, Mike noticed a buoy floating in the Bristol Channel that appeared to be perched precisely on the horizon: an observation that is in itself enough to make a rough estimate of the size of the Earth. Being a physicist at rest, Mike decided to gather the necessary information. Releasing his heels with a squelch he turned and walked with careful sharp-shell cadence to a shop, and bought a map. This informed him that the buoy, known as the Fairy Buoy, was approximately 4 km away from his vantage point on the beach. A quick sketch on the back of a seaside serviette allowed him to deduce that the Earth has a radius of roughly 5000 km. The actual value is around 6400 km. It may impress you that Mike made a reasonable estimate of the size of our planet simply by observing the region around Ogmore-by-Sea. Equally, you might be unimpressed that his answer is 20% out.

The calculation works on the assumption that the distance to the buoy is in fact the distance to the horizon. The quality of Mike’s eyesight governs how well he is able to determine whether or not the buoy is coincident with the horizon. A person with average eyesight can just about resolve a small coin at a distance of 40 metres, corresponding to an angular resolution of about 0.03 degrees. This means that Mike could perceive the buoy as being coincident with the horizon, even though the horizon is slightly in front of or slightly behind it. All he can say with certainty, therefore, is that the distance to the horizon lies somewhere between two extremes, defined by the resolution of his eyes, which he should quote as an uncertainty on the measurement. A little calculation reveals that, given the limits of his eyesight, Mike really ought to have concluded that the radius of the Earth could quite easily be anywhere between 2000 km and 36,000 km. The fact that his serviette calculation got so close to the true value is largely a coincidence.

Estimating the uncertainty on a result is often as important as the result itself. It is only when we are aware of our ignorance that we can judge best how to use knowledge. In engineering or medical science, a deep understanding of uncertainty can be a matter of life and death. In politics, over-confidence is often the norm; uncertainty is seen as weakness when really it is a vital part of decision making. In this respect, science delivers an important lesson in humility.

In Mike’s case, his measurement, while inaccurate, does still give us some idea about the size of the Earth. To achieve a better result, Mike would need to improve on the limiting resolution of his eyes, which can be done by using a camera with a long lens. Fortunately, Mike’s dad, Bob, is a keen photographer, and lives in Ogmore-by-Sea. We couldn’t make this up. We asked Bob if he might go down to the beach and take some photographs of the Fairy Buoy for us. He kindly obliged.

Bob’s photos were taken when the sea was quite choppy – which is perhaps a bonus, as we do not need to worry about the bending of light due to atmospheric effects, something which is more prone to happening on calm days: we can see that atmospheric effects are not an issue here because the images are pretty sharp. Bob adjusted the height of his camera such that the pictures show the buoy perched directly on the horizon. (Lowering the camera would push the buoy behind the horizon; raising it brings it in front of the horizon.) The photographs were taken at a height of 1.3 metres. The camera position was determined using GPS and the position of the Fairy Buoy was taken from the Trinity House official records. The distance between buoy and camera was 4.15 km. These more refined numbers give a radius of the Earth equal to 6600 km. Using a camera has significantly reduced the uncertainty caused by limited resolution; the chief source of uncertainty that remains is now the difficulty in ascertaining the precise height of the camera above the average level of the waves. A 10 cm change in height leads to a 500 km change in the calculated radius of the Earth, which we might quote as a conservative estimate of the uncertainty on Bob’s measurement.

There are, of course, far better ways of measuring the radius of the Earth than this – but that isn’t the point. This is a good example of how simple, curiosity-driven observations, together with a little bit of careful thought, can lead to interesting conclusions. In what we suspect is a world first, Mike and his dad have measured the size of our planet from Ogmore-by-Sea. We have also learned a valuable lesson in quantifying uncertainty: it is easy to be misled into drawing the wrong conclusion unless we understand the degree of our ignorance.

In his measurement of the Earth, Mike followed in the footsteps of the greats (although if he’d actually stood on the shoulders of giants he wouldn’t have been able to make his measurement). One of the earliest documented attempts to estimate the size of the Earth was made by Aristotle in 350 BCE. Aristotle noted in his book On the Heavens that ‘there are stars seen in Egypt and in the neighbourhood of Cyprus which are not seen in the northerly regions,’ and that the sphere of the Earth is therefore ‘of no great size, for otherwise the effect of so slight a change of place would not be quickly apparent’. Using a very simple observation, Aristotle ruled out the possibility that the Earth has a radius much bigger than the distance between Egypt and the northern extent of the ancient world – that’s to say, a few thousand kilometres. And, unsurprisingly for one of the most influential scientists ever, he was right. This is a terrific illustration of an ‘order-of-magnitude’ estimate. Order-of-magnitude estimates are quick calculations that are not supposed to be very accurate, and they are important in science because they can provide a good deal of insight with very little work.

Adapted from Universal: A Guide to the Cosmos by Brian Cox and Jeff Forshaw. Copyright © 2017. Available from Da Capo Press, an imprint of Perseus Books, LLC, a subsidiary of Hachette Book Group, Inc.

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