1 / 8

Proportionality

Proportionality. Part 4. How big is the Earth?. Erastothenes of Cyrene (3 rd B.C.) calculated the circumference of the Earth to remarkable accuracy. (The ancient Greeks believed the Earth was round.) His estimate was used for hundreds of years afterwards.

mowbray
Download Presentation

Proportionality

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Proportionality Part 4

  2. How big is the Earth? • Erastothenes of Cyrene (3rd B.C.) calculated the circumference of the Earth to remarkable accuracy. (The ancient Greeks believed the Earth was round.) • His estimate was used for hundreds of years afterwards. • This is the same Erasthones we saw earlier this semester (the Sieve of Erastothenes was used for finding primes).

  3. Eratosthenes did his measurements on the summer solstice at noon in the Egyptian city of Syene, which so happens to be slightly north of the Tropic of Cancer. • This is the northernmost latitude at which the Sun can appear directly overhead at noon, an even that occurs on June 21, the summer solstice.

  4. But Erasthones did not have knowledge of the Tropic of Cancer. However, there was a very deep well in Syene, and he knew that on June 21 he could see the water deep in the well illuminated clearly at noon. • So he knew that on June 21 at noon the sun was directly overhead in Syene. • His hometown of Alexandria (he was head librarian of the famous library) was 500 miles away. • On June 21 he planted a stick straight up at noon in Alexandria. • It cast a shadow, so he knew the sun was not directly overhead in Alexandria.

  5. Erasthones calculated that the angle the shadow cast was about 7.2 degrees from the stick. • He knew this meant the angle from the center of the Earth between the stick and Syene was also 7.2 degrees (“alternate interior angles”). • So the distance between Alexandria and Syene was 7.2/360 = .02 of the circle. • And he knew this distance was 500 miles.

  6. What is the circumference? • So we have the following proportion, where C is thecircumference of the Earth. • Solving for C, we have C = 25,000 miles. • The actual circumference (at the equator) is 24,901.55 miles! • There is debate about the precision to which Erasthones knew the distance between the distance between Syene and Alexandria. But in any case, his measurement was very close.

  7. What about the radius? • We can also deduce the radius of the Earth using the formula C = 2πr. • So r = 25000 / (2π) = about 3979 miles. • The actual radius ranges from 3,949.901 to 3,963.189 miles. (The Earth is not a perfect sphere.)

  8. Moral of Story • Erasthones measured the circumference and radius of the Earth using a few “nearby” measurements. • But very clever mathematical reasoning about proportionality was needed to find the circumference. • Again, we see that proportionality allows us to extend reasoning about things “nearby” us to a much larger, otherwise inaccessible scale.

More Related