1 ppm per 6M in height, roughly 1/1,000,000 of the earth's radius. Doing the elevation scale calculation the number changes 1 digit in the 6th place per each 6M change in elevation. For NAD27 I suppose that was "close enough". .01' per 10,000'. Nobody was going to get worked up about it back then.
WRT Mr Hamilton’s statement “...that they did not know the ellipsoid heights, they used the orthometric heights.” The horizontal datum NAD27 was/is a regional datum. Not going too deeply into the weeds, developers of regional datums chose a reference ellipsoid that best fit their part of the world. The original at MEADES RANCH (near the center of CONUS) was where the ellipsoid was fitted.
It wasn’t until the develop of technology that earth-centered datums became possible; NAD83 was the first. Given the subsequent development of GPS it might have been better to have waited. Hindsight is of course 20:20.
Until the earth-centered datums, we worked in a 2D + 1D world where Lat/Lon work was separate from heights.
BTW, the correction for heights in the NAD27 grid system was called the “Sea Level Factor.” When I worked using state plane coordinates under NAD27 we computed mean combined factors for each measurement segment not for an entire project. This was in the 1970s/early 1980s.
As Mr Griggs notes, it is always important to know what your software is doing. While gross errors readily identify a problem, smaller discrepancies are more insidious. As Mr Hamilton indicated, small differences result from the choice of the “R” value in the EF computation. Using the Gaussian mean radius rather than the approximation for N. America of 20,906,000 ft will yield different results at the few millimeter level. Professionals should know when that matters.
BTW, here is a tabulation of reference ellipsoid s (a, b, f^-1, and area where they were used):
“To be surprised, to wonder, is to begin to understand,” - Ortega y Gasset
I have never seen this map anywhere else (I have a copy that I acquired many years ago). It shows what the knowledge of the geoid was in 1974 (before NAD83, GPS, GRACE, etc). As can be seen, using a 0.0 height at Meades Ranch was pretty good for much of the country, but as one goes west it rapidly increases to about -30 m on the west coast, meaning 6 ppm scale bias out there.
It's been my fate from birth...to make my mark upon the earth.