NavList:

Bruce, you wrote:
"I suspect changing the radius 1 or 2 % value can easily change geometric dip coefficient by few percent. "

The fractional change in the dip is almost exactly half the fractional change in the radius. Since the dip values in the 1599 table (assumed to be geometric dip) are high by about about 3.5% demonstrably derived from a constant of 1.10 instead of 1.06, the Earth radius that was used would have been incorrect by 7%.

And you wrote:
"About this time or somewhat earlier a "true" vertical could be established, and then a true horizontal could be obtained. I don't know when angles could be precisely measured (dividing engine and all that), but that is when correct dip values were measured by an astronomer and he corrected the geo++metric dip equation. That's my theory and I'm "sticken to it". "

Yes, dip tables "reasonably" corrected for refraction are usually attributed to Bradley's work on refraction (I mentioned this in a message a couple of days ago if you want to read a few more details), and they were available by c.1750. It would certainly be interesting to get a better date on the appearance of refraction-corrected dip tables in common navigation manuals. This refraction correction is itself equivalent to a modification of the Earth's radius, of course. The Earth's "effective" radius is increased by about one-seventh since horizontal rays curve downward with a radius about equal to seven times the Earth's radius. In the mid-18th century this "7" factor was a big improvement over geometric dip (which is the same as a "0" factor). Most "natural philosophers" believed then that the next step in the process of developing dip tables would be to refine this value. Perhaps the constant should really be 7.5 or maybe 8.2. And this is where great "dip wild goose chase" begins...

The problem with that "next step" in refining the dip tables is that the actual factor is variable and it is variable in a way that is not accessible to corrections in actual practice except in some relatively rare cases. It's weather-driven! We CAN both calculate and observe the variability of dip, but there's no genuinely superior average value for the curvature of horizontal light rays. Ever since the late 18th century, astronomers and mathematicians interested in this issue have laid out proposals for refining that constant's value. And there have been quite a few long observational studies. William Chauvenet, the top dog of American nautical astronomy in the mid-19th century and acknowledged globally as an expert of the highest order, himself suggested that studies be conducted to relate that factor's variability to environmental conditions. The results of such studies have been inevitably "inconclusive" or "disappointing". In any case the refinement just doesn't have much practical value. The horizon is already dodgey at this scale.

Even defining average conditions in a useful way is problematic. Average refraction conditions for a vessel frequently traversing the Gulf Stream may be quite different from "average" conditions for a vessel sailing in the equatorial Pacific. Measurements from coastal locations unfortunately provide average conditions for those specific coastal locations.

The uncertainty of the horizon (and equivalently the uncertainty in the values for a dip table for some given location and moment of time) remains the single largest uncertainty in manual celestial navigation, introducing errors in altitudes on the order of half a minute of arc. With a large number of nearly simultaneous observations taken on at least three different azimuths, it's possible to treat that uncertainty as a systematic error which can be eliminated by various procedures. But navigators have that luxury primarily in the theoretical world of armchair discussions.

-FER

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