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# Ancient Achievements In Mathematics From the very beginning of humanity's exploration of the world around it, mathematics has been an integral part of science. Today, most people consider it to be unimportant or boring, but that couldn't be further from the truth. In fact, our ancients succeeded in calculating certain physical quantities, with the help of quite simple mathematics and with great ingenuity, that people today cannot even imagine. We are going to see two of these mathematical milestones achieved by humanity millennia ago. The calculation of the radius of the Earth and the calculation of the distance and size of the Moon.

## Calculation of the Earth's radius

It was the 3rd century BC when the philosopher, astronomer, and mathematician Eratosthenes devised a method for calculating the radius of the Earth, based on trigonometry. The day chosen for the measurement from which to obtain the necessary data was at noon on the summer solstice. From the manuscripts in his library, Eratosthenes knew that in Siena (now Aswan ) objects did not cast shadows at noon on the summer solstice; that is, the city is located in the Tropic of Cancer. This is because the sun's rays strike perpendicular to the surface and therefore the shadow is just below the object: there is no visible projection. One of his servants went to Siena to carry out the check, while he stayed in Alexandria to do the same experiment there. The result: in Alexandria, at the same time, the objects did cast a small shadow.

This projection was just over 7 degrees. Assuming now that the Sun's rays reaching the Earth are parallel and that Siena and Alexandria are in the same longitude(They are only 3º apart), and knowing the distance between the two cities it was easy to calculate the terrestrial radius. There are different stories about how Eratosthenes knew the distance between the two cities, such as that he sent a servant counting the steps, or that he used the estimate of the trade caravans; but the really important thing is that it had a value of 5000 stadiums (785 km). There is also a discrepancy in the length of a stadium since the definition is different for a 185-meter stadium or the 157-meter Egyptian stadium, but that is the least of it. What is clear is that he deduced that the circumference of the Earth was 250,000 furlongs (39,250 km) as follows:

• We have that for 7 degrees, the distance is 5000 stadia.
• 7 degrees goes 50 times into 360 degrees which is the full circumference.
• So if we multiply 5000 stadia by 50 we will have the perimeter of the circumference. The perimeter is 250,000 stadiums.
• To find the radius we only have to divide the perimeter by 2π. Therefore the radius is 39800 stadiums, or what is the same almost 6250 km.

Today we know that the mean radius of the Earth is 6371 km, so the error made by Eratosthenes was only 120 km: slightly less than 2%. This is quite an achievement for such a precise calculation in such a seemingly rudimentary way.

### How to Calculate The Moon's Distance and Size

The next milestone is that of measuring the distance and size of our Moon. The author was Aristarchus of Samos, also a Greek astronomer and mathematician, in the 3rd century BC. For this, he also used fairly simple trigonometry, but with surprisingly accurate results.

It all started with the observation of a lunar eclipse. Aristarchus determined that the time it took for the Moon to be hidden by the shadow cast by the Earth during the eclipse was approximately half the time that the eclipse lasted. Thus, the diameter of the shadow was about twice the diameter of the Moon. Another thing that he determined is that the Moon took about an hour to get out of the shadow of the Earth, from which it is deduced that the Moon travels in an hour a distance equivalent to its own diameter.

On the other hand, it was known that our satellite takes 29.5 days to complete one orbit around the Earth, so it is easy to show that the orbit of the Moon is 708 lunar diameters. And by simple trigonometry, he estimated the distance from the Earth to the Moon at 225.4 lunar radii.

From the figure above, Aristarchus calculated by trigonometry that the radius of the Earth is 2.85 times the radius of the Moon when the result is 3.66 times. Knowing the radius, he simply calculated the Earth-Moon distance by applying the previous result that this distance was 225.4 lunar radii. The result he obtained was 79 times the radius of the Earth when today we know that the distance to the Moon is about 60 Earth radii.

Aristarchus also tried to calculate the distance from the Earth to the Sun, but failed in his attempt, giving a result (19 times the distance from the Earth to the Moon) completely wrong.

As you can see, the results of Aristarco are somewhat worse in terms of precision than those of Eratosthenes, but the truth is that in both cases it is surprising how human ingenuity can solve problems as complex as knowing the radius of our planet or how far away our Moon is and how big it is. And what is more important: they are two mathematical milestones that occurred in the third century BC applying completely basic mathematics.

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