NavList:

Brad, you wrote:
"I suspect that there are navigational dip tables for the earliest of instruments that measure altitude vis the horizon. So for the cross staff at sea, there must have been a dip table. I wonder, what is the earliest published dip table? "

Those early dip tables fall into the category of "advice to future generations". You can't measure angles with a cross staff or a backstaff or any other handheld pre-double-reflection instrument to better than maybe +/-10' accuracy under excellent conditions. So unless you're observing the sea horizon from well above a hundred feet altitude (maybe from a cliff?), these dip tables served no practical purpose ...yet! This is an example of something that mathematicians and other "natural philosophers" understood from a very early era (try Ptolemy), and they just couldn't hold their tongues waiting for it to become useful.

The dip table from 1599 that Gary posted is easy enough to reverse-engineer. You can calculate nearly matching values using the simple formula dip=1.10*sqrt(h) with dip in minutes and h in feet (with that constant the dip calculated for 90 feet altitude rounds to 10 instead of the listed 11 but the difference is only half a minute of arc in any case). Since the mathematical details of refraction were unknown, we can safely conclude that this was a geometric dip table for a slightly over-estimated radius of the Earth.

-FER

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