# Measuring the radius of the Earth

In ancient times, people imagined our planet in different ways: the inhabitants of Babylon believed that they lived on one of the slopes of the “world” mountain surrounded by the sea; in India, the Earth was considered a hemisphere that rests on the backs of elephants; the ancient Egyptians saw the world as a valley stretching from north to south. How did scientists who lived even before our era manage to come to the shape of a ball and even calculate its radius quite accurately?

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## The shape of the Earth, a historical background

It is believed that the first idea about the sphericity of the Earth was put forward by Pythagoras in the VI century BC. But he, like many great thinkers of that time, did not bother with proofs and substantiated his hypothesis with the beauty and perfection of the ball. There is indirect evidence that ideas of this kind were put forward earlier, both in Greece and in Egypt, but they were either presented in an overly metaphorical form, or do not have any reliable dating.

Let’s try to put ourselves in the place of the ancient Greek philosopher and put forward several assumptions about the shape of the Earth, based on the observation of the world around us.

Travel around the world and precise navigation tools are not yet available to us, but ships are already available. If we stand on the pier and watch the trireme go into the open sea, we will easily notice that first the hull of the boat will disappear behind the horizon, then half of the sail, and then the whole ship.

Does such an observation prove that our planet is spherical? This is enough to put forward a hypothesis, but the same hemispherical world of the Indians is quite consistent with this observation. You can also come to the conclusion about the curvature of the earth’s surface by observing the mountain gradually appearing from the horizon.

We have already rejected the idea of an absolutely flat Earth by previous observations, and we are moving on. Now the time has come to direct your gaze to the sky, as Aristotle did, who in the end proved the spherical shape of our world. He traveled a lot, in particular, he visited Cyprus and Egypt, and the night sky there is somewhat different from that observed in Greece.

The curvature of the Earth’s surface, when traveling long enough distances, allows you to see new stars, while some of the stars we know, on the contrary, fall out of sight. Our hypothesis continues to be overgrown with evidence, but we would like to see the shape of the Earth directly … and it is possible.

Lunar eclipses are not such a rare event, they happen two to four times a year. Observing them, Aristotle was able to finally prove his theory of a spherical Earth. During a total eclipse, our planet closes the Moon entirely from the Sun. This event lasts up to four hours, during which the Earth’s shadow moves across the lunar surface. We can actually observe the shape of the Earth with our own eyes. Partial lunar eclipses are no less interesting.

The fact is that the plane of the Moon’s orbit is located at an angle of about 5 degrees relative to the plane of the Earth’s orbit around the Sun. That is why not all eclipses that we see are total. On the other hand, the inclination of the orbit makes it possible to observe the passage of shadows across the lunar disk from both the northern and southern hemispheres.

## We calculate the radius of our planet

To begin with, it is worth talking about the measurements carried out by the ancient Greek scientist Eratosthenes about 250 BC. The scientist lived in Alexandria and he did not even need to leave the city to carry out sufficiently accurate calculations. He knew that on the day of the summer solstice in Siena (nowadays Aswan, Egypt), at a certain point in time, the solar disk is reflected at the bottom of the deepest wells, i.e. The sun is directly above the observer’s head. The city is indeed located very close to the northern tropic, on which there is no shadow at all at noon on the summer solstice. The deviation in Siena is only 1/400 of the object’s height, or 0.143 °.

The sun is at a much greater distance from us than the radius of the Earth and, therefore, its rays can be considered parallel. The ancient Greeks already had general ideas about how to measure this distance, although it is extremely inaccurate, on such scales even an error of several orders of magnitude does not play a significant role. Eratosthenes used a sundial, called the scaphe, to determine how far the sun deviates from its zenith in Alexandria on the summer solstice.

Scafis is a stone or copper hemisphere with a vertical centre peg and divisions on the inner surface of the bowl (see figure). After conducting measurements, Erastofen came to the conclusion that the distance between Siena and Alexandria is 1/50 of the circumference of our planet. Knowing that the distance between cities is 5000 stadia.

Unfortunately, no accurate data has been preserved for converting the units of measurement used by Erastofen into kilometres and, depending on the conversion factor used, the estimate of the accuracy of its measurements may vary. It is also worth noting that the cities are not quite on the same meridian, Siena is slightly shifted to the east. If we use the Olympic stages (176 m) for translation and divide the result by 2π, we get the radius of the Earth at about 7,000 km. According to modern data, the average radius is 6,371 km, respectively, Erastofen was mistaken by only 10%.

It is likely that even earlier he had a preliminary assessment associated with determining the radius of visibility on the water surface. The principle is about the same, but this time we will use a ready-made formula and substitute very approximate values into it, more or less corresponding to the level of accuracy of measurements in antiquity.

Where R is the radius of the Earth, h is the observer’s height above sea level, r is the apparent distance. Let us assume that from a height of 10 m we have a visibility radius of 10 km, then the sought value will be 5000 km. The error is greater than 20%, but the order of the value is correct. If you happen to go on a sea voyage on a clear day, then with the knowledge of the height of the observation deck of the ship above sea level and data on the distance to land-based on GPS, you can get a much more accurate result.

### Measurements continue

After Eratosthenes, for many centuries no one tried to measure the circumference of the Earth again. In general, his method was absolutely correct, but the accuracy of measuring distances, based on the time the caravans moved between cities, left much to be desired. The next notable attempt to measure the Earth’s radius was made by the French astronomer, mathematician and royal physician Jean Fresnel in 1528.

He went north from Paris, planning to measure an arc of 1 degree and assuming that his entire path lies in the same meridian. Comparing with the calculated data on the height of the Sun in the capital, the scientist reached Amiens, where, during the measurements, he received the deviation he needed. Final calculated the distance between the cities on the way back, based on the number of revolutions of the wheel of his cart. He was lucky, the cities really lie in the same meridian, and various inaccuracies of measurements, by a happy coincidence, compensated each other. The result was 6326 km, almost a bull’s-eye.

The next measurement was taken by the Dutch astronomer Willebrord Snell in 1614-1617. His result (6 149.8 km) turned out to be less accurate than Fernel’s, but the technique itself is remarkable – the first application of the triangulation method. It is in this way that distances on the surface of the Earth are measured in our time, the only difference is in the available technical means. The good thing about triangulation is that natural obstacles such as forests, lakes or swamps do not impede the accuracy of long-distance measurements.

To begin with, you need to very accurately establish a very small distance between two points (A and B) from which you can see some kind of hill (C), for example, a hill or a tower. With the help of a theodolite, observing from points A and B for a certain object at point C, you can easily establish the values of all angles of the triangle. Knowing the value of the side AB and all the angles, we can easily establish the distances AC and BC using the theorem of sines (or even direct measurement on a scale copy).

Then we can measure the distance to the new point D in the same way and continue to complete the network of triangles further and further. In the end, according to the rules of trigonometry, it will be possible to calculate the distance between any vertices of any two triangles from our network, no matter how far they are.

Already in the second half of the 17th century, the great physicist and mathematician Isaac Newton suggested that the Earth has not one radius, but two slightly different ones, i.e. it is slightly flattened at the poles. According to his calculations, the equatorial radius was larger than the polar one by 1/230 of the average radius. In 1735 – 1737, the Paris Academy of Sciences established two expeditions, which were tasked with measuring the meridian arcs at different latitudes.

After processing the data from both expeditions, scientists came to the conclusion that Newton was right. Various studies concerning the refinement of the shape (even the generally accepted term “geoid” does not describe it with absolute accuracy) and the radius of the Earth are carried out in our time. However, for calculations that do not require accuracy of more than 0.5%, it is quite sufficient to consider our planet as a ball with an average radius of 6371 km.