Luke, As usual, and as Roger well knows, it is difficult to sneak one by you!
Yes a simple value for the radius of Earth is 20/pi * 10^6 meters If you use the approximation to Pi recommended by Zu Chongzhi (http://aleph0.clarku.edu/~djoyce/mathhist/china.html) of 355/113 to compute the radius of Earth you get: (20*113/355)*10^6 m or 6.366197 * 10^6 m, precise to the nearest meter. If you approximate Pi by 3.1415926 you get a radius of 6.366198 * 10^6 m, precise to the nearest meter. I tried to verify your statement to the effect that: "We now know however that the length of the quadrant is really 0.023% larger than originally surveyed, ..." and got hopelessly lost in Chapter 4 of the Explanatory Supplement (© 1992) and in Appendix X of Bowditch, 1977 Ed. I have a renewed respect for geodesists and those folk who delve into arcane matters and come up with elegantly sensible ways of locating things. I do recall from my earlier thinking on the matter that the 20/pi radius falls between the polar radius and the equatorial radius and is close to the radius of a sphere that has the same surface area as one of the ellipsoids used to approximate the geoid. In any event I made those things on Polaris Boulevard (http://sciencenorth.on.ca/AboutSN/polaris/index.html ) to have diameters of 12.732 m or 12732 mm on the CAD drawings. The steel was cut with a plasma torch all under numerical control. When we checked alignments and directions while installing, Celeste was within a mm in all dimensions except when the wind got blowing. No doubt strut vortices were shed. Our latitude is so close to 45 degrees that she resonated quite well. Not as bad as the Tacoma Narrows Suspension Bridge, but she danced several cm. So you'll notice a little stiffening here and there to break the resonances. She is very well behaved now. Almost boring! The spherical terrazzo on Terra was laid in by an elderly Italian gentleman whose family had been practicing the art for generations. (And it is still standing up well in spite of harsh environmental treatment, including freeze/thaw, some salt, skate boards and the dancing of little shoes as kids celebrate being at Sudbury, on the "top of the world"!) I figured that an approximation between polar diameter and equatorial diameter was good enough for government work. Besides it was fun to talk about the origins of the meter! Now if someone would just explain all of this ellipsoid of revolution, geoidal separations and equipotential surface stuff, we might have a chance to understand where the 0.023% "error" is! *SIGH* (In any event I figure that thermal expansion/contraction episodes will get the thing passing through the correct value from time to time.) Good stuff, Luke. Thanks. Tom Semadeni > Tom, > > Perhaps one of the lists geodesists could give a more complete answer > but I believe the original French definition of the meter was the > subtended arc-length (along a great circle) of one ten-millionth of a > guadrant (a guadrant = 90 deg. or pi/2 radians). This survey, by the way > took seven years to complete! Nothing like a government contract... > > Therefore, since arc-length = radius * theta (in radians) we have the > following expression: > > 10^7m / (pi/2) = r = 2*10^7m / pi > > With a reduction in scale of one million (divide above by 10^6) we > have: > > 20/pi m > > We now know however that the length of the quadrant is really 0.023% > larger than originally surveyed, I hope you have made allowances for the > corresponding error. Just kidding. > > Best, > > Luke >
