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*From*: David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>*Date*: Wed, 6 Jun 2001 10:49:03 -0400

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

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