Re: Would you like to share some data?

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Ken Murray's request:

> My students would like to measure the size of this planet with a
> length-of-shadow measurement (a variation on Eratosthenes's observations).
> We would measure the shadow angle of a vertical metre stick at a specific
> time (next week) and someone else a few hundred to a coupla thousand km away
> would do the same measurement at the same time.

How would you expect to determine the surface distance between the
various observation sites (as well as determine your latitude and
your longitude)?  If you expect to use a map and a ruler, that will
probably be cheating since the map will have been made with a
cartographic scale and a projection that is cognizant of and accomodating
of the radius of the earth.  In fact, if you use a highway map (even a
small regional one) you can determine the size of the earth directly from
it without even making any solar shadow measurements at all.  All you
have to do is pick two widely separated points on the map and note their
indicated latitudes and longitudes.  Then use a ruler to measure the
geodesic distance between the two points and use the map's distance scale
of mi/km to convert the ruler distance to the actual geodesic distance
between these two points.  Next, use a little spherical geometry on the
latitude/longitude readings given for those two points, and you can
calculate the size of the Earth.

Also using GPS measurements ought to be considered a form of "cheating"
since the GPS system is *highly* dependent on the size of the Earth.

If you want to be a purist about your measurement you may have to mark
off the distance manually (e.g. by car, etc.) using a trusted calibration
scale (that has not been calibrated using the size of the earth as part of
the calibration process).  If a car is used the distance obtained will
have to be corrected for nongeodesic highway paths (again, unfortunately,
using a highway map).  Also, if the measurement is to really be
legitimate, the observers will have to determine their latitude and
longitude by making local solar/astronomical measurements (possibly using
trusted calibrated clocks for longitude measurements).

>                                                 To keep the geometry simple,
> I'm looking for someone on approximately the same meridian (123 degrees
> West).

If the observers already know their longitude and latitude, before a
measurement of the solar shadow experiment is performed, then there is
no need to even measure the solar shadow angles.  All that remains is a
determination of the geodesic distance between the observation points
and the size of the Earth can be found.

> So, is there anyone out there, somewhere between Portland and San
> Francisco who would like to do this easy measurement (you or your students)
> and share the data?

Since you want to do an updated version of the Eratothenes' determination
of the size of the size of the earth, it seems to me that for such a
measurement to be really legitimate you will need to do the experiment
in such a way that the various observation points do not a priori have
their latitude or longitude values known or used in the determination.
The idea is to use the on-site shadow measurements to determine both the
latitudes and relative longitude difference between the observation
sites.  Also, an earth size-independent measurement of the distance
between the observation points would be needed.  The increase in
complication of the calculation due to the observation points being on
different meridians is relatively small compared to the problem of
determining the distance between the observation points without cheating.

David Bowman
David_Bowman@georgetowncollege.edu