The measurements done at the Transcontinental Triangulation and the American Arc of the Parallel (PDF), published in 1900, can be used to calculate the prevailing refraction at the stations, assuming a Globe or Flat Earth model. The calculated prevailing temperature gradients show that the required refraction for the observations and measurements are physically impossible assuming a Flat Earth (red column).

Version 2023-03-24

History

Description

In the table above the reduced zenith angles were used. This are the original observed angle corrected to the base of the stations. The targets were mounted on big structures and the thodolites were commonly not at the same height as the targets. This requires a correction, called reduction of the zenith angles to be comparable.

Note that the temperture gradient is required to be about the same for every altitude, because the earth appears as a globe with a certain apparent radius from any altitude.

As we can see the temperature gradient required for the Flat Earth model are physically impossible. The temperture on the Flat Earth model would reach absolute zero below 2000 m altitude. Therefore the earth can not be flat.

Equations

For each pair of observation stations it was given: the distance d, the zenith angles ζ, pressure P in mmHg and temperature T in °C. If pressure or temperature was not provided, they were assumed like described above.

Altitude from Air Pressure

Using the Barometric Formulas of the International Standard Atmosphere, which is based on empirical average measurements, we can calculate an estimation for an altitude h from a measured air pressure as follows:

Temperature from Altitude

Using the Barometric Formulas of the International Standard Atmosphere we can calculate a mean temperature for an altitude as follows:

Python Code

I wrote a Python script that reads the raw data provided by McToon from a file in CSV format and compiles the data into the table above.

The following Python script (included in the raw data ZIP file) does all the calculations:

# ellipsoid prime vertical radius of earth at 40° latitude
# R = 6386976.166
# using the same R as in the source tables: Geocentric radius at the equator
R = 6378137.0

class Observation:

    :

    def computeUnknowns( self ):
        # require self.isValidObs() to skip invalid and reverse observations

        ## barometric calculations
        # mean P and T, T converted from °C into K
        P = self.meanPressure()
        T = self.meanTemperature()

        # compute estimates for altitude from pressure using International Standard Atmosphere
        self.altitude1 = self.altitudeFromPressure( self.pressure1 )
        self.altitude2 = self.altitudeFromPressure( self.pressure2 )
        Rh = R + (self.altitude1 + self.altitude2) / 2

        # if temperature was not measured, calculate it for estimated altitude above
        if not self.temp1Provided:
            self.temp1 = self.temperatureFromAltitude( self.altitude1 ) - 273.15
        if not self.temp2Provided:
            self.temp2 = self.temperatureFromAltitude( self.altitude2 ) - 273.15

        # temporary
        zenith1 = self.zenith1
        zenith2 = self.zenith2
        if self.reduced is not None:
            zenith1 = self.reduced.reducedZenith1
            zenith2 = self.reduced.reducedZenith2
        a = (180 - (zenith1 + zenith2)) * math.pi / 180
        d = self.distance

        # globe model
        self.rho_gl  = d / 2 / Rh + 0.5 * a
        self.c_gl    = 2 * self.rho_gl / d
        self.k_gl    = self.c_gl * R
        self.dTdh_gl = 12660 * self.c_gl * T*T / P - 0.0343

        # both models
        self.Rfactor   = 1 / (1 - self.k_gl)
        self.Rapparent = self.Rfactor * R

        # flat earth model
        self.rho_fe  = 0.5 * a
        self.c_fe    = a / d
        self.k_fe    = self.c_fe * R
        self.dTdh_fe = 12660 * self.c_fe * T*T / P - 0.0343

The whole source code is provided in the ZIP file together with all currently used data.