Re: [Phys-L] eratosthenes

[John Denker <jsd@av8n.com>]

On 01/28/2017 10:52 AM, Philip Keller wrote:
> I've noticed that while my students do not all seem thrilled to learn about
> Cavendish and the mass of the Earth, they do perk up at the idea that the
> Earth's radius has been known since antiquity.

Two points:

First, let's talk about the pedagogical issue.  Knowing the mass of the
earth is interesting to a physicist, but I'd be hard pressed to argue
that it is important to the students.  You can use it to figure out other
things such as the mass of the sun ... which the student's don't care
about either.  You can use it to figure out that the average density of
the earth is a lot higher than the density of the surface rocks, which
is kinda interesting, but once again, not something that affects the
students' lives.

Almost everything you need to do in the near-earth environment you can
do using the GM product, which we already know.  In fact we know it
incomparably better than we know G or M separately, which makes for
an interesting lesson in correlations ... but I digress.

In contrast, it is easier to sell people on the importance of the earth's
radius.  Having a decent idea of how far it is from here to China or
here to Afghanistan has some quite plausible relevance.

Furthermore, you can make the point that a great deal of pop-culture
history is nonsense, starting with the idea that Columbus set out to
prove the earth was round, and/or that he was opposed by people who
insisted that the earth was flat.  In the late 1400s, everybody who
knew anything knew the earth was round.  Dante (circa 1320) wrote about
the center of the earth, and you can pretty much assume that anybody
in Italy who could read would have read Dante.

You can put the round-earth story in the same basket as all the Happy
Columbus Day pictures showing him holding a telescope:
  https://s-media-cache-ak0.pinimg.com/736x/58/90/1d/58901df314474264c4d0c562812693f8.jpg
He died 100 years before telescopes were invented.

So basically Columbus is famous for being a jerk.  If he hadn't had that
special combination of arrogance and ignorance, he would have known that
China was a verrry long ways west of Spain, and he had no chance of getting
there in his puny little ships.

There is such a thing as serendipity, and that should be encouraged in
every reasonable way.  However, in my experience one ought not start
with an unsound plan and hope that some unforeseen development will
save the say.  Far more often, serendipitous discoveries are made while
operating on a fundamentally sound plan.  There is such a thing as
research management -- and risk management.  If you want to see the
difference between a well-planned expedition and a jerk expedition,
compare Lewis and Clark to Columbus.

--------

Now let's talk about the physics.  In particular, rather than focusing
on one famous bit of evidence, let's ask what was the /best/ evidence
available in Columbus's day, or Dante's day, or even Eratosthenes's day.

Suppose you live near a big bay.  Then you can set up an experiment,
using only ancient technology -- no lasers, no lenses, no GPS.  Get
somebody who knows how to aim a signal mirror, and send them (with a
ladder) to the beach on Sandy Hook, somewhere near the Coast Guard
station.  Then go (with a ladder) to the beach somewhere near Popamora
Point.  See how high you have to go in order to see each other.  This
is easier and more fun if you have a cell phone, but it could be done
without that, just by keeping track of position and other details as
a function of time of day.

   This is a good lesson in the real-world importance of lab books.

Then grab a map and do a little geometry and you've got a pretty decent
estimate of the size of the earth.  You can even try to account for the
height of the tide and the sea state, if you want.  The cost of doing
all this is reeeeeally small compared to buying and outfitting three
ocean-going ships.

Do that first, as a warm-up exercise.  Then you can do even better using
a bigger bay, in a place where there is some high land (or maybe even
Highlands) overlooking the water.  From the Mount Mitchill overlook on
a clear winter afternoon you can see stuff at sea level about 20 miles
out.  That includes the entire beach on Coney Island, plus Rockaway Beach,
and the tip of Atlantic Beach.  Conversely, from Atlantic Beach you can
see the Twin Lights towers.  Send somebody to Atlantic Beach with a signal
mirror, and see how far east they have to go before the signal is lost.
Then see how high up the ladder they have to go before the signal is
regained.

The advantage of using a bay is that you know the distances accurately.
You can either triangulate your way /around/ the bay, or if you have
two different lines of sight (Mount Mitchill and Twin Lights) then
you can triangulate straight across.  People have known how to do this
for a long time.  The ancient Greeks has a word for it ("geometry").
I mention this because ancient maps tended to have terribly unreliable
estimates of distances over water.