I think the ruler in this photo depicts plane of the great circle.

Maybe the angle between the great circle and the latitude circle can be used to find the circumference of the latitude circle.

At this latitude the angle is 1 inch divided by 4 inches = .25

If I multiply times the circumference of the earth along the great circle it is always 21,639 naut. miles.

So when I multiply the angle times the circumference I get .25 x 21,639 = 5,409 miles.

Now if I subtract that from 21,639 I get 16,230 miles.

If I look at chart for lat. 45 the circumference is 15,326 miles
http://home.online.no/~sigurdhu/Grid_1deg.htm
Of course my measurements are crude.

Now if I do the same thing at the equator there is no difference between latitude line and great circle so 0 degree difference times 21,639 = 0
So 21,639-0 = 21,639

Now when I do it on lat 15N I get about 3/8" per 4" = .09375
Multiply that times 21,639 = 2,029 miles
Subtract that from 21,639 = 19,610 miles
the actual circumference at lat 15 is 20,906 miles

I wonder if I could get better measurements if this would work?

brent
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