December 15, 2017 Filed in: Articles

Last spring, I took my students 2200 years back in time. My grade 8 students measured the size of the Earth using shadows — the technique first described by Eratosthenes.

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At the same time, on the same day, he made a measurement of his own. He found that the shadow of a vertical stick was 7 degrees from the vertical. He used this information and the geometry of triangles and circles to deduce the circumference of Earth (Figure 1). Some say he was within 1% of the actual value, others say he was out by about 15%. Either way, it was a brilliant experiment.

I was inspired to try and replicate the experiment after watching Adam Savages TEDEd talk called “How Simple Ideas lead to scientific discoveries”.

I found a willing colleague in Arkansas who agreed to drag his young children out of bed early to do a science experiment.

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At the same time (8:00am Eastern) both parties measured the angular position of the sun in the sky. This involved two measurements: its

Angling the clinometer so that the sun was shining directly down the paper tube, we stilled the plumbline then pinned it with a finger to the angle measurement on the clinometer. This was the ‘altitude’ of the sun’s position in the sky. When the clinometer is properly aligned, the shadow of the tube on the top of the protractor is not visible on the ground.

Using the vertical level, we set the metre stick up so that it was vertical (measured using the level in both horizontal dimensions). A student held the metre stick in place, standing so that their shadow didn’t block the shadow of the metre stick.

We then set the compass down beside the base of the metre stick, and used the protractor to measure the angle difference in the horizontal plane of the concrete between compass north and the shadow of the metre stick. This was our ‘azimuth’ measurement.

My colleague in Arkansas used a simpler method for the azimuth measurement, using an iPhone compass oriented over the shadow to determine the angle from north.

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Once the two sets of angular coordinates were known, I made an assumption that might introduce a small error that will be hard to quantify. We are talking about spherical angle changes, but I treated them as planar (x,y). Under this assumption, I used the Pythagorean Theorem to calculate the ‘straight line’ angular distance between the two coordinate points. This will probably yield different results than allowing for the spherical coordinates, but I haven’t been able to work out how much.

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This angle can be taken to be the same as the angle between the two locations on the Great Circle that joins them (Figures 7 and 8). Thus, the angle represents the fraction of the 360o that would take us around the circumference of Earth. So, dividing 360° by the calculated angle, then multiplying by the linear distance, should yield the circumference of Earth.

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Figure 9 shows how I used the two sets of measurements to determine the angular difference. The Little Rock coordinates were (azimuth/altitude) 81° by 67°. The Oakville coordinates were 90° by 56°.

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Using the Pythagorean Theorem, 92 + 112 = 202 and the square root of this is ~14.2. Allowing for my assumptions, this gives us an angular difference of 14.2° (Eratosthenes’ measurement was about 7°).

360/14.2 is 25.3, which means the distance between our two cities is 1/25.3 of Earth’s circumference. Thus, 25.3 × 1500 km (the distance between the two locations) yields a circumference of 38 000 km (Figure 10). Earth’s accepted measured circumference is 40 075 km, for an error of 5% in our experiment.

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Not bad for a hacked together, rushed (we waited until we knew it would be sunny… and got distracted by our math class… and hurried to get the measurement quickly to avoid error due to the sun’s motion in the sky…) experiment by a group of academically struggling teens…