Thursday, May 8, 2014

Definition of a Unit of Measure

  When we measure something we need a unit of measurement that is both accessible and convenient. For a unit of length the circumference of the Earth is more accessible than its radius but is too large to be a convenient standard of measurement. So we have to consider its subdivisions. Like the degrees, minutes and seconds that are used to divide the circumference of the circle their corresponding lengths could be used to divide the circumference of the Earth and we arrive at the geometrical mile and foot. The meter is an alternative unit of length. The circumference of the Earth is 129,600,000 geo-ft or 40,000,000 meters. Eliminating the common multiple gives us 81 geo-ft = 25 m and we could define the geo-ft in terms of the meter with 1 geo-ft = 25/81 meters. This can be expressed more exactly the other way around with 1 meter = 3.24 geo-ft. The common foot is well defined with 3.28083 feet per meter and more easily accessible.

  From a practical point of view the conversion factor Δs/Δα for changing an angular distance into ordinary one is more easily observable than the circumference of the Earth so in this sense it is more fundamental. But while a spherical surface is a reasonably good approximation to the Earth's surface, we still have to admit that an ellipsoid is a better one.

Supplemental (May 9): For a spherical globe the geometrical foot can be based on a definition, 1 geo-ft = 0.01 asec.  The physical length of the geo-ft would have to be determined by the measurement of change in angle along a great circle with change in distance. This measurement needs to be made along a meridian since the change in latitude can be determined by astronomical observations. But the problem with this scheme is that the vertical direction is determined by gravity and the true shape of the Earth. The latter can be obtained through measurements of the curvature of the Earth's surface. This is the purpose of a geodetic survey. If we could use a spherical to represent the Earth's surface then the following definitions would be exact instead of just an approximation.

1 geo-ft = 0.01 asec
1 meter = 3.24 geo-ft

As it is there's no real geo-ft since it's definition assumes a conversion factor that in truth is not a constant.

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