NavList:

Rommel John Miller, you wrote:
"I am sure every steward knows where to find the answer."

Indeed. But do you trust the answer that they bring you? There's no downside for that steward if the answer is just a guess. I certainly agree that the captain of the vessel will provide a reliable answer if available because captains take pride in their vessels, and numbers like this matter --if the captain has time for such matters.

You added:
"Algebra and geometry can find the height of buildings with formulae, therefore finding the answer can be had of your dip on the boat deck, one might think"

Oh yes, there are lots of mathematically "clever" ways to do it. But in the real world, where the Internet is close at hand (maybe actually in your hand), it is generally easiest, most efficient, and most accurate to locate a photo of the vessel and, based on its length, also acquired online, measure the height of your observing location directly. If there are no photos online, then take one yourself from a few vessel-lengths away while in port.

For a mathematically clever method, drop a rock (the crew, and possibly the Harbor Patrol, will be upset, but do it anyway). Time the fall of the rock from your observing height until it hits the water to a fraction of a second, perhaps using a video or audio recording feature on a smartphone to get tenth-of-second accuracy. Take the number of seconds, multiply by 4, and that gives the dip directly in minutes of arc. This works because ballistic fall time and dip of the horizon are both proportional to the square root of the height. Example: if a rock, or other small dense object, takes 1.5 seconds to reach the water, then the dip is 6'. If it takes 3.0 seconds, then the dip is 12'.

For another mathematically clever method, use your sextant as a range finder. You need to calibrate this, but just once in the life of the instrument. Then lean over the side of the vessel from your observing height and look at the waterline beneath you, or look at some object, like a buoy, in the water close to the vessel (I recommend you tie a lanyard to your sextant before you lean over the side with it!). Line up the direct and reflected images just as you would for an index correction with the sea horizon and read off the angle. This will be an "off arc" angle assuming the normal index error is zero. For a typical sextant, specifically an Astra IIIB, the relationship will be:
  Height(in feet) = 750 / angle(in minutes of arc).
So if the "index error"-like angle of the waterline is, for example, 15 minutes of arc, then the height of eye is 50 feet. Note that you should zero out the normal index error or correct for it as usual after you measure the angle. This angle can be measured with decent accuracy, but you should do the calculation for values +/- 0.5' just to be sure you recognize the possible range. For this case of 15' observed, the height of eye might be between 48 feet and 52 feet implying that the dip correction could be 6.7 to 7.0 minutes of arc. Given other uncertainties, this is really quite good and a useful measure of dip. Just don't drop your sextant...

Frank Reed

PS: The number 750 in the range finder rule can be estimated for other sextants by measuring the distance between the two lines of sight of the instrument when it's set to zero. If that's three inches, let's say, then you divide by 12 to convert to feet and multiply by 3438. The result, 860, is the factor in the numerator of the rule. If the distance is 2.6 inches, then you get the 750 factor for the Astra sextant. It's better to try it out empirically by actual ranging measurements (a backyard activity), but you can get a good approximation before you start by looking at the distance between the lines of sight on the sextant.