NavList:

Frank,

They were used for refueling and somewhere I have seen an old documentary that featured either an RN and RFA refueling exercise, or the equivalent in the U.S Navy, (I can't remember which) where a stadimeter was used.

You said:  "And if that angle is smaller than about 5°, I don't even need to bother with any triangle and trig functions. For those cases with angles less than 5°, I can use

   angle (m.o.a.) /3438 = size / distance," etc.

Yes - essentially the same as the 5-6-5 rule I mentioned.

Height in feet divided by sextant angle in minutes, then multiplied by 0.565 = distance in nautical miles.  And to your point - within acceptable margin of error.

But back to the main question:  How does it work?  I can't do it justice in a couple of paragraphs sorry - but I will try.

This model, the Mark V, works in essentially the same way as the Fiske stadimeter (from the late 1890s).  We know that an object of a certain known height, observed from a certain distance, will give us a known angle at the apex of the triangle created by the viewer and the "base" or (vertical) length of the object observed.  That much is covered in the earlier descriptions.

We have a scope or other viewing device, an horizon mirror which is half silvered (as we are used to with a traditional sextant horizon mirror) and an index mirror - the angle of which is moved through a narrow range by what I will call the bar.  If we look at the picture shared, the thing that reminds us of an index arm on a sextant is not that.  It has no mirror where we are used to seeing the index mirror on a sextant.  Instead it moves a bar slightly up or down, and the index mirror angle is modified by that bar.  This is the Index Bar.  Look closely at the pictures and we see that the bar describes a curve, but the curve is not the same as that of the arc on which a scale is marked.  The bar pivots at the index mirror and is unattached at the end close to the horizon mirror.

Now - as we move the arm (NOT an index arm) we set that to the known height of the object to be viewed, between 50 feet and 200 feet high.  I used the example of a liberty ship in my earlier comment for a reason (130 foot funnel) - we don't have to view the top of a mast on a battleship or the QE2 - we can view a known deck height or if the lighting is right, a ship's navigation lights - so long as we have a reference source that gives us the height.  Most ships of the time were not so tall.  So now we have moved the index mirror and it would now be possible to look through the scope and see something in the mirror and something through the glass.  But this makes no sense because we don't know either distance or angle, so we cannot solve the triangle, yet.

But wait.  If we know the height of the object and we have an estimate of the distance (just by eyeballing it) we know what the angle would be if we were at that exact distance.  So, what if we set that estimated distance and by that setting, and for the known height, we set the angle of the index mirror?  How is this done?  We can't adjust the known height on the index bar.

Note that this winding of the micrometer drum is not moving the index point along the scale.  Instead the screw, or worm, applies more of less pressure to the bar, and so raises or lowers it slightly and thus changes the angle of the index mirror, but still with the index line set to the correct height on the scale.  Now we have the angle at the apex of the triangle set for a known height and an estimated distance.  If our eyeball estimate is way out, we can rotate the micrometer drum until we bring the object roughly into view.  This is analogous to pulling our celestial object down to the horizon with a sextant index arm, but with an entirely different type of operation.

Now we "wind in" the adjustment via the micrometer drum until we have aligned our images via the index mirror such that the "top" (say the funnel) is in line with the "bottom" (say the waterline). 

We have now moved the index mirror by some small angle, which, given the height and estimated distance, has moved at a ratio to distance towards or away from that estimate which can be known within margin of error, and pre-marked on the micrometer drum.  We can be read the distance off the drum in yards (in the U.S. Navy version), without tables or mathematics (even simple stuff).  Note that the thing that looks like a micrometer drum has a large circumference.  It has a scale of heights and distances rather than minutes and seconds. At a large angle for a near object, the ratio of change in angle to distance is different from (and easier to measure accurately) that the same ratio at a far distance.  Hence the need for the drum to be set to the estimated distance to set the initial index mirror angle, and for the reading to be made of the correct estimated distance once the actual angle is fixed.

How does this differ from the earlier Fiske bar-type stadimeter?  That also used two mirrors, a bar, something like a micrometer wheel, and the same principle.  What the Mark V did, with its sextant-like appearance, was to enable the fast setting of the height by moving the arm rapidly - like a sextant - rather than the tedious winding or the endless worm screw necessary on the Fiske design.  It just made things faster.

I think that's how it works.

What is the advantage over a sextant used for vertical or horizontal distance off?  Only that it is designed for its purpose and can be used to read distance directly off the instrument.

We could dig deeper into the math.  For example - if the estimated distance is 1,000 yards and the height is 130 feet, and on the micrometer drum we read off the 1,000 yard scale, for a given circumference of drum we could compute the point at which we would need to mark 100 yards of additional distance away or towards.

Or we could do this.  It is World Was Two.  We have an angle and a height so have measured the distance between us and an approaching ship.  We have a table of known enemy ships with silouette pictures, heights. lengths etc.  We know our speed and course.  We don't know the other ship's speed and course.  But we can take bearings on the ship relative to our heading, and we have a stadimeter to rapidly compute changes in distance.  We are a submarine.  The stadimeter is within the periscope.  We have no electronic aids - no radar.  We know the speed at which our torpedo will travel through the water.  We can compute our firing solution.  Or we could build some type of analogue computer to do the work for us, which is exactly what both the Axis and Allied navies did.