A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Robin Stuart
Date: 2016 Jul 27, 10:13 -0700
While I now appreciate that the accuracy of the “exact” form of the dip short formula is not your focus the claim that it is "exact" for thr refraction model adopted might be a bit of a stretch. The derivation of the Distance by Vertical Angle Measured Between Sea Horizon and Top of Object Beyond Sea Horizon formula was discussed here previously see
Refraction is introduced by increasing the Earth’s radius everywhere by a factor 1/(1- β) where β=0.1684. This is a surprising good approximation but an approximation to real refraction models nevertheless. For arguments sake let’s call it the “refraction model”. The derivation of the Distance by Vertical Angle Measured Between Sea Horizon and Top of Object Beyond Sea Horizon uses a series expansion of sine and cosine and so is not exact in that respect.
It is of course possible to calculate the dip short exactly within the “refraction model” by simple trigonometry and I’ve added the result to the bottom of the pdf file. From it we see a further inconsistency in the “exact” result. tan Ds differs from Ds by cubic terms in Ds however cubic terms in ds/r are dropped from the right hand side. However in your case of extremely close short dip Ds is large compared to ds/r the use of tan Ds would appear to be justified. Whether the formulas are accurate enough in practice to justify the use of tan Ds over Ds is another question.
In the case you cite of height of eye of 7ft and 50 yards give
"refraction model" exact 2°40.32’
It would be pretty straightforward for someone with a theodolite to actually check how well the formulas do for a dip short at 50 yards. I understand that it standard practice in surveying to account for refraction by adjusting the Earth’s radius.