In Transcontinental Triangulation and the American Arc of the Parallel, the 39th parallel is measured across the United States.
On Pages 869 to 871 the final analysis is done.
For an arc of parallel
C = length of arc as directly measured
D = astronomical difference of longitude (in seconds) of its terminal stations in latitude φᵢ
For the 39th parallel
C = 4,182,227 meters, measured
D = 173,807.83 arc seconds = 2896.7972 arc minutes = 48.279953 degrees
The circumference is a simple ratio: c = ( 360° / D ) * C
The total circumference of 39th parallel = ( 360° / 48.279953° ) * 4,182,227 meters
= 31,184,822 meters = 31,184.822 km
Flat earth analysis
If the earth is flat the circumference of all lines of latitude are easily found from this formula:
c = 2 * pi * r
Where “r” is the distance from the north pole to the line of latitude.
Per the Gleason’s Map, there are 60 Nautical miles per line of latitude. 39° north latitude is 51° from the north pole, or 3060 NM from the North Pole. This is 5667.12km. From the above formula for the circumference:
c = 2 * pi * 5667.12km = 35,607.57 km
The flat earth prediction for the 39th latitude is 4422.75km off from measurement.