A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Robin Stuart
Date: 2017 Jun 14, 11:55 -0700
I suspect that the dependence of refraction on observing conditions makes 1.14 indistinguishable from 1.17 for practical purposes. Of course the relationship between the foot and the metre is precisely known and doesn’t change a bit!
You may have explained the original source of the error which was originally discussed in this thread and I believe pre-dates the Bowditch 2002 edition. Maybe the editors took the imperial formula from one reference and the metric version from another not thinking that the two are simply related. So 1.14 belongs with 2.07 and 1.17 goes with 2.12 but quoting 1.17 with 2.07 is an abomination.
YOU WROTE: “i did try to use trigonometry to prove which was more correct, but attempted it solely with planar trig calcs in a faulty enough muddle”
Surprisingly planar trig actually does just fine. Taking the Earth radius to be 20,902,231feet it gives
D = 1.063√hf
for both the dip in minutes of arc and the distance of the horizon in nautical miles. This result. The effect of refraction is incorporated, as explained by Frank Reed here, by adjusting the value of the Earth’s radius using the factor β = 0.1684. To get the standard formula for dip the radius is increased by a factor (1-β)-1=1.2035 giving
dip = 0.97√hf
and for the distance to the horizon it is decreased by (1-β)=0.8316 which gives
D = 1.17√hf