# Add Infinitum – Journey to the Centre of the Earth

#### Somerset Times Edition

Week 8,
Term Two, 2016

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Can we measure the measure the circumference of the Earth with just a metre rule, a piece of sting and a calculator? Well, if we consider that the shape of the Earth as spherical, then all that is required is to use the equation C = 2πR where C is the circumference of the earth and R is its radius.

However, to obtain the radius of the Earth could be problematic. A journey to the centre of the Earth is certainly out of the question. The gravity of the situation would make such a trip too hot to handle. Over 2,000 years ago the Greek mathematician Eratosthenes of Cyrene (now Libya) developed a method to measure the Earth’s circumference based on the angle of the sun’s rays at two points on the Earth’s surface.

Eratosthenes knew that at noon on the summer solstice (21 June) in the Egyptian city of Syene (now Aswan) which lies on the Tropic of Cancer, the sun would appear directly overhead. He realised that fact because there was a water-well at Syene that if you looked down at this time at your reflection, the sun was completely blocked out. Meanwhile at the same time in Alexandria, a stick placed vertically in the ground cast a shadow on the ground at an angle of 7.2o. Now 7.2o is 1/50th the angle of a circle (360o). So the distance from Syene to Alexandria must be 1/50th the circumference of the earth. He knew the distance from Syene to Alexandria was 5,000 stadia (about 930km) and so he calculated the circumference of the earth to be 252,000 stadia or 39,690km which was an error of less than 2% - and he didn’t even use a slide rule. So this is the method that can be used. We need to determine the angle of the sun’s shadow at two points, at the same time, on the same line of longitude. If the distance between the two points is known then the circumference of the earth can be calculated.

Eratosthenes is now the name given to a crater on the moon but to measure the circumference of the moon might need a different method. It was actually first done by Aristarchus around the same time while standing on Earth. He watched a lunar eclipse and concluded that the moon was about 2.7 times smaller than the earth (actually it is about 3.7 times smaller). He had assumed that light from the sun reached us in parallel lines which is not quite true. Therefore the shadow cast by the earth on the moon has two parts – an umbra and a penumbra. He had used the umbra in his calculations which is smaller than the diameter of the earth.

One man who did walk on the moon and famously uttered “That's one small step for man, one giant leap for mankind” was Neil Armstrong. It’s just a thought but have you ever considered what Neil A spells out backwards?

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