Anonymous
Anonymous asked in Science & MathematicsEarth Sciences & Geology · 1 decade ago

# describe the steps of eratosthenes method of measuring the earth?

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First, he noticed that in Syene, on the summer solstice, at noon, the sun would be directly overhead, and that a well would be lighted to the bottom. He then measured the angle of the sun at the same date in Alexandria, and conclude that it was 7.2 degree.

Then knowing the distance between the two cities, he derived how big the Earth must have been to have 7.2 degree angle difference for the approximately 800 km distance. He was off by only 0.8 %

• Anonymous

Eratosthenes knew that on the summer solstice the sun would be directly overhead at the town of Syene. He also knew from measurement that the sun should be at 7.2 degrees south of the zenith at Alexandria (his location). He thought that Alexandria was due north of Cyene. He concluded that the distance from Alexandria to Syene must be 7.2/360 of the total circumference of the Earth. He ended up coming up with 39,690 km as his answer (not far from the correct answer, 40,008 km).

Hi there,

Eratoshenes of Cyrene was a Greek scholar who used geometry to generate a quite accurate estimate for the polar circumference of the Earth, a little over two thousand years ago.

First of all, understand where he lived: he was chief librarian of the great Library of Alexandria, which lay at the mouth of the Nile in Egypt.

1. First step: given that the Earth is a sphere, and that the light rays reaching it are parallel with each other, measure the difference in the angles of shadows cast by these light rays at two different points.

It so happened that almost directly due south of Alexandria, about 500 miles away (I think) lay the town of Syene. It was known to him that a well lay in that village, the bottom of which was lit up by the noon-day sun only once a year at the first day of summer.

Our man E figured, hey, this means that on that day, the light rays from the Sun would probably go straight through the planet, if that well went down far enough. In other words, at that moment, there was no point on the Earth closer to the Sun than Syene.

So in Alexandria, at high noon on the day that the Syene well was illuminated, he contrived to measure the angle of the shadow cast by a stick (or something stuck 90 degrees with respect to the ground).

Understand that he was measuring the angle made at the top of the stick, between the stick itself and the line made by the shadow reaching to the ground. Contrast this with Syene; someone with a stick stuck in the ground the same way, would be measuring no shadow, because the light was coming straight down.

There are seven and a half degrees in the angle cast by this shadow.

2. Establish the extent of the angle existing at the center of the Earth, that is made by the difference of the distance between Alexandria and Syene.

The light rays in Syene would continue right through the center of the Earth if they could. But light falling on Alexandria would miss the center of the Earth by 500 miles if they had gone straight through.

To extend a line from the Earth's core to Syene is easy, just follow the figurative "light rays." But a line from the core to Alexandria would be seven and a half degrees up from the line to Syene.

This is because the angles between these two parallel lines - these light rays coming down on Alexandria and Syene - are being cut by the line extending down from that stick in Alexandria to the core of the Earth - otherwise known as a "transversal" line.

And in geometry, "When a transversal cuts two parallel lines, alternate angles are equal in size." 1.) See the very bottom graph of the link here to get a picture of what I'm trying to describe here.

What do we have so far? We have the distance between Syene and Alexandria (500 mi) and the angle this distance makes (or "subtends") at the Earth's core.

3. Calculate the circumferance of the Earth, dividing the subtended angle into the number of degrees in a circle.

Number of degrees in a circle: 360

Amount of subtended angle: 7.5

Number of times this angle divides into 360: 48 times

Circumference of the Earth, multiplying 500 miles by 48:

24,000 miles.

Modern science says our polar circumference is: 24,859.82 miles.

***

Understand, the above result was reverse-figured from our more accurate knowledge of the Earth's circumference. Eratosthenes came up with a circumference of something like 29,000 miles, which is still pretty damn close for the state of Western knowledge 2,000 years ago.

Hope this helps.