#1 Eratosthenes measures the circumference of the Earth!
Equivalent Ratios of different quantities, part 1
JLS 6th Grade MATH CLUB
Equivalent Ratios of different quantities
Jay Chesavage (firstname.lastname@example.org)
It is now 310 B.C., you are in Math Club in Alexandria,
Egypt, and your best friend Eratosthenes has announced that he can figure
out the circumference of the earth using only:
At around noon, Eratosthenes knows that the sun will
exactly light up his deep well shaft all the way to the bottom. You are
to sit by your well in your uncle's back yard, and wave your arms madly
when your well is illuminated all the way to the bottom. This will quickly
frighten the thousands of crows that gather in your uncle's back yard (your
Aunt likes to feed crows, and thousands of them gather each day), and they
will instantly fly into the air, alerting Erathosthenes that the sun is
now directly over your well. Erathosthenes starts the minute glass as soon
as his well is illuminated all the way to the bottom, and exactly 6 minutes
later notices a thick cloud of crows to the west.
Two long, deep wells, one in his back yard and one in your uncle's back
yard, which is exactly 100 miles WEST of his well.
A minute glass calibrated in minutes.
1) So what is the earth's circumference?
2) O.K. there really weren't any crows around, but there were two wells
separated by 500 miles. What really happened in 310 B.C. is that Eratosthenes
went to the second well on the very day (summer solstice) he knew the first
well was illuminated, and measured that the second well was illuminated
7.5 degrees off vertical (there are 360 degrees in a circle). What is the
earth's circumference using this measurement?
1) The earth's circumference can be determined by equivalent ratios
of distance and time:
(diameter of the earth) / (distance between wells)
= (minutes in a day) / (minutes between wells fully illuminated)
diameter of the earth = distance between wells * (minutes
per day) / (minutes between wells fully illuminated)
diameter of the earth = 1440 * 100 miles / 6 minutes =
2) Another equivalent ratio expressed above using equivalent ratios
of distance and angular degrees:
(diameter of the earth) / (360 degrees for 1 rotation)
= (distance between wells) / (angular difference seen at second well)
diameter of the earth = 360 degrees * (distance
between wells) / (angular difference) = 360 * 500 miles / 7.5 degrees =