Eratosthenes - Free Educational videos for Students in K-12

Eratosthenes - By MITK12Videos

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00:06	In this video , we're going to talk about courtesans
00:08	who was a Greek scholar that lived about 2000 years
00:11	ago . You're tossing . He's found a way using
00:14	none of the modern tools that we have to measure
00:16	the circumference of the earth . And in this video
00:19	we're gonna see how he did this . So the
00:22	heart of your toxins measurement is a simple geometry problem
00:26	. So consider the circle shown here , which has
00:29	points A . And B . And let's say that
00:34	we know the distance that A . And B .
00:37	Uh make on the circumference of the circle . So
00:40	we know the measure of arc A . B .
00:43	Now the question is with this knowledge , can we
00:46	determine the circumference of our circle ? And I'm sure
00:51	you all are thinking that the answer is obviously no
00:53	because A . And B are just two random points
00:55	on the circle . So just knowing their distance doesn't
00:58	help us very much . What we need is some
01:01	information that makes A . And B . Not just
01:03	random points on the circle anymore . We need to
01:06	know the angle that A and B make with the
01:09	center of the circle . Once we know this ,
01:11	we know how far around A and B . Go
01:14	on the circle because we know that every circle has
01:17	In a 360° in one full revolution . So by
01:23	knowing data , we know the fraction of the circle
01:26	that the ark A B takes up and we can
01:29	simply extrapolate to find the circumference . So let's make
01:33	this one a little clear with some concrete examples .
01:36	So let's look at the circle on the right now
01:39	and again , we have points A and B .
01:43	Ah and here you can clearly see that the angle
01:47	that a and B make with the center of the
01:49	circle is 90°. And since we know that 90 goes
01:52	into 364 times are KB is 1/4 around the circle
01:57	. Uh meaning that the piece shown here , the
02:01	shaded piece Will fit into the circle of four times
02:05	. So in this case the circumference of the circle
02:07	is four times the length of our KB . Now
02:12	let's just drive the point home further with this other
02:14	circle again we have points A and B . Um
02:18	and here the angle between points and be , let's
02:20	say we measured and it turns out to be 36°.
02:23	So since 36 goes into 360 10 times , we
02:27	know that 10 of these pieces will fit into the
02:29	circle . And in this case the circumference is 10
02:33	, which is the number of pieces times the length
02:34	of one piece a . B . Now in general
02:39	the circumference of the circle is given by the number
02:42	of pieces we have Times the length of one piece
02:47	and just writing side explicitly . The number of pieces
02:50	is 360 degrees divided by data . The angle that
02:54	A . And B . Make with the center of
02:55	the circle and the length of one piece is simply
02:58	A . B . So now if we just know
03:01	these two things , the RK B and the angle
03:04	A and B . Make with the center of the
03:05	circle , we can determine the circumference . And this
03:08	is really the heart of your toss and use this
03:09	method . So now let's apply this . Now we
03:13	have another circle . But this time we'll explicitly identify
03:16	the circle as the earth ah and point A becomes
03:19	a city , alexandria . This is a city in
03:22	ah Egypt and um this is where our Tahsin is
03:26	lived . Point B now becomes the city of saying
03:31	. So in your toes in his day it was
03:33	known that the distance between Alexandria and saying was about
03:37	500 miles . Of course back then the units weren't
03:40	miles , but we know the conversion factor , so
03:42	we don't have to worry about the old system of
03:44	units . The distance between alexandria and saying is 500
03:47	miles . And now looking back at our previous problem
03:51	, we see that all we have to do to
03:53	figure out the circumference of the earth , given this
03:55	information is to figure out the angle that alexandria and
03:58	saying make with the center of the earth and the
04:01	really ah the really brilliant thing about veritas needs his
04:05	method is that he found a nice way to measure
04:08	this angle . So how did you do this ?
04:11	It was known that insane . There was a well
04:14	, a long deep well such that at noon on
04:17	the summer solstice you could see the sun's rays light
04:20	up at the bottom of the well . And if
04:22	you think about this , what this means is that
04:23	since as well as such a deep thing , this
04:26	means that the sun's rays must have been coming into
04:28	the earth parallel to the well . So we draw
04:31	these rays uh you see the sun is very ,
04:34	very far away from the earth . And because the
04:36	sun is so far away we can treat the rays
04:39	coming in from the sun at different points as parallel
04:43	. So here we have drawn the ray that comes
04:45	in at alexandria and the ray at science . Now
04:48	, it's an interesting property of parallel lines that if
04:51	we have these two parallel lines shown here the to
04:54	raise . Ah And we have the radio . I
04:57	also shown this angle that we've called data to Is
05:02	equal to the angle that we're trying to find data
05:04	. one . This is something this is the fact
05:06	that you might have been exposed to in your geometry
05:08	classes . It's called the property of corresponding angles .
05:11	Um but regardless , it's pretty easy to prove yourself
05:14	. So since we know this now , we can
05:17	transform the hard problem of measuring data one to the
05:20	relatively easier problem of measuring data to . So how
05:23	do we measure data to ? It's pretty straightforward .
05:26	So let's zoom in on this region . We have
05:29	, the surface of the earth is one of the
05:31	important lines and then the radius gets translated into a
05:35	vertical stick . And we also have the sun with
05:39	one of its race . So this ray casts a
05:41	shadow on the ground . And by knowing the length
05:44	of the shadow and the height of our stick ,
05:46	we can construct this triangle and simply measure the angle
05:50	data . To all we have to do is look
05:51	at the shadow created off of a stick . And
05:55	this is exactly what I had to sneeze did .
05:57	And he measured that the angle data to was equal
06:00	to 7.2°. So now we have this information . The
06:05	angle data . Uh let's just call it data now
06:09	is equal to 7.2°. And the measure of the Ark
06:12	A . B . or Alexandra Cyan is 500 miles
06:17	. So with this information . Now all we have
06:19	to do is feed this into the formula that we
06:21	got earlier . Let's just recall it . C .
06:24	Equals 360 degrees divided by data times A . B
06:29	. So this is very simple to do . We
06:31	plug in 7.2° for data , we plug in 500
06:34	miles for a . B . And in the end
06:36	we find that C . is equal to 50 times
06:38	500 miles or 25,000 miles . So this is our
06:43	estimate , or you're tossing this estimate for the circumference
06:47	of the earth , let's call that see your atrocities
06:51	, and so see how simple it was for us
06:52	to get this . But notice that According to our
06:57	modern measurement , the average circumference of the Earth ,
07:00	because it's not a perfect sphere is around 24,900 miles
07:04	. So we were only about 100 miles off with
07:07	this seemingly primitive method , which is about 0.5% off
07:11	. So this is a very impressive thing for someone
07:13	who lived so long ago without the access to these
07:16	tools that we now use .

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