NavList:

David C, you wrote:
"Several references I have found, the earliest from the Admiralty Manual of Navigation 1922, use a phrase similar to [a small triangle or "cocked hat"]. To me what is important is that cocked hat is in quotes which suggests that in the late 19th and early 20th century it was a slang expression."

Yes, I would say that it clearly is a slang-y expression. And, yes, I agree that they're indicating this, and maybe some formal misgivings about this term, by putting it in quotes. It's a "funny" metaphor, and that's all. I don't think it's relevant what sort of hat was in common use in naval uniforms at the time. It seems quite clear that the intent was to provide a memorable, pithy name for that little triangle, and the three corner hat, which was familiar to most everyone even if it was no longer fashionable, implied by "cocked hat" made a certain community of people laugh, so the name stuck. 

In your follow-up message, you wrote:
"I have looked at the index of Bowditch 1958 and also that in Dutton's Navigation and Nautical Astronomy 1951 and cannot find the phrase cocked hat. Because of the way the indexes are formatted I may have overlooked the words."

This sort of thing is greatly facilitated by technology. Searching mid-century editions of Bowditch on HathiTrust, I found no references to cocked hats. This supports the idea that the expression originated in British usage a few decades earlier and had not yet crossed the Atlantic --at least not to the desks of the editors of Bowditch at that time. We can similarly search the latest (2017) edition of Bowditch directly via the pdf files and there are now numerous references to "cocked hats" and always in quotes. It's even in the glossary: "cocked hat. Error triangle formed by lines of position which do not cross at a common point."

You added:
"In the gps era navigation enthusiasts  began to  analyse cocked hats [/duck] ad nauseum - reference the unintelligible (to me) discussions about where in the cocked hat a ship is."

I agree it's been ad nauseam and frequently pointless. If you have found the discussions unintelligible it's because they are almost always consumed by ramblings from confused contributors who seem to think there's something complicated in all of this merely because there's math that underlies it (one can understand the result and ignore the derivation just as in so many other cases in navigation). Really the only thing you need to remember in practice is that the fix should not be marked near the center of the triangle, but rather it should be placed near the short side of the triangle (if any). And if you see a dozen cases properly calculated, you'll be able to do this on paper without calculation.

The key case:
Suppose you have a triangle from sights taken on bearings (true azimuths) of 0°, 270°, and 178°. Get out a piece of paper and draw the corresponding lines of position. Assuming some random errors in the sights, you'll have a long, narrow triangle with a short side on the west and two very long sides on the north and south. Next, if asked to indicate the fix many people without thinking on it much would select the center of the triangle. But that's wrong in every sense. The fix is very close to the short side. And this isn't some fancy mathematical argument. It's obvious if you take this case and replace the third azimuth with 180°. Draw it out, and you'll see why. This is 99% of the point of all those discussions. If you can see this, the rest is not really navigationally significant. All the rest is just mathematics mixed with confusion (some people bring math to the table, some bring intractable confusion).

Frank Reed