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Chuck Varney, you replied to my question on 4^th November 2024 about the value of the two-scale sextant and how it might work.   Thank you for going to so much trouble.  It must have taken you ages.  I know, because I spent what turned into fruitless hours trying the very same thing with prints of the quintant in the link and a 2B pencil.  I also used a lot of scrap paper convincing myself that the design exceeded the bounds of practicability and was unable to satisfy the Newton’s laws reflection we learnt at school. 
It took me a while to conclude that the person who wrote the caption simply looked at the length of the scale and said the sextant worked from 0 to 140 or 40 to 180 degrees.  As others have already pointed out, it doesn’t.  To measure 180 degrees using reflection via two mirrors needs an angle between mirrors of 90 degrees, and the only way you can do this and see two different objects is to take the index mirror off and turn it round to face behind you.  (see Diagram).  Moreover, when taking soundings, 180 degrees difference between A and B wouldn’t give you a precise position, only a line you were on.
I also spotted the length of the Vernier problem, which meant that the index arm was going to butt up against the mirrors.  In addition, you could only use the bottom horizon mirror if you were prepared to simply eyeball it and not use the telescope.  I had planned to make a cardboard index arm to fit on the photograph to test the real limits when I got around to it.  I had also hoped to cut up a 9”x9” Perspex mirror tile I’ve got hidden somewhere in the roof to do some of the pins and mirror experiments we did at school to test what would work.  Unfortunately, round to-its are currently in very short supply.  Fortunately, you stepped in with your explanation.
What I did manage to prove is:
a.  The top horizon mirror has the greatest angle between it and the index mirror, so it is used with the 40-180 scale.
b.  The angle between the top and bottom horizon mirrors is 20 degrees so you can show that using the bottom horizon mirror to ‘eyeball’ the shot effectively knocks 40 degrees off the 40 to 180 degrees scale meaning the bottom horizon mirror is used with the 0 to 140 degrees scale. (Opposite to as described here https://images.rmg.co.uk/asset/41354/ )
Finally, the quintant is described as experimental.  It certainly is.  That doesn’t mean it had to work or be any good.  Perhaps things went like this.  Nathaniel Worthington was apprenticed pupil to Matthew Berge, successor of Jesse Ramsden.  Until recently it wasn’t unknown to give apprentices their head and allow them to learn from their own mistakes.  Perhaps young Worthington wanted to produce a sounding quintant with the greatest possible range, and he realised that by tweaking 20 more degrees onto the normal maximum angle between index and horizon mirror he could measure 40 more degrees between objects on shore.  The trouble was that when he tested it, he found that he’d lost the ability to measure from 0 to 40 degrees. That probably didn’t matter for taking soundings, because the cut would have been less than ideal.  However, just in case somebody did want to measure between 0 and 40 degrees, he stuck the bottom mirror on.  So why mark up to 180 degrees?  How about marketing?  Post truth didn’t just start in the 21^st Century.  Uwe Buettner thanks for your post, which I still need to work through.  DaveP

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