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David Pike, you wrote:
"when I’m just trying things out, it’s ingrained that I use 1 in 60 to come up with ballpark figures. "

Sure. I get that. :)

Of course, we know that
  m.o.a./3438
is better, but it's not good enough for some people! Maskelyne suggested memorizing 3437¾. Heh. :) I'll emphasize for anyone else following along, if you're dealing with angles smaller than about 2°, that ratio is accurate for almost any angular problem to about one part in 10,000. For an angle up to nearly 5°, that's accurate to one part in 1000. And this if for the simple ratio or equally for the seemingly more "rigorous" trigonometric computation of a sine or a tangent. Need the tangent of 2°50'? It's just 170/3438 (minutes of arc divided by that "magic number").

And you wrote:
"On the vertical distance between mirrors, I tried something new this afternoon.  I took the telescope out of my Hughes 6” Three Circle and shone my laser pointer down the eyehole.  Then I shone it on the wardrobe, after a bit of wiggling I got two spots around 5.5cm apart.  [...]"

Yes, the laser is the way to go! I used to demo this (more than a decade ago) as another method of getting index error. The idea is that the spacing between the two dots should not change with distance if there is zero index error. That's necessarily true, and it works well enough to be competitive. But after playing around with it for a while and trying to convince folks to give it a try, I found that the limiting factor was that it demanded a rather long "shooting range", like a long hallway, and usually a table to set up the sextant and the laser. You need to see how the spacing changes over 50 feet or more. That problem with the laser bouncing around can be solved with a carpenter's laser level which is intended for tabletop purposes incidentally, but then you need a table, so it's less convenient. Just too fussy... :)

I'm still experimenting with the index error target idea. So far so good. It seems practical and portable, and I think it would require only one setup step in the lifetime of a sextant -- getting the spacing on the target to match the distance between the optical sight lines that we're talking about. Also it would be possible to design pre-printed targets for a subset of modern sextant designs. That would add to the convenience.

Frank Reed