NavList:

Peter B,

You started out talking about nautical miles and knots, but then you switched it up and wrote about the benefits of great circles on the globe. These are differet matters. We can measure distances on great circles in any units we want. The idea that we should prefer minutes of arc and nautical miles when working with great circles just isn't true.

You wrote:
"No voyager wants to travel extra distance unnecessarily, and therefore will seek to go “great circle” if they can."

The shortest distance between two points on the Earth is a perfect Euclidean straight line connecting them. In principle that's the path with no extra distance. It's out first guess at the most efficient path from point "A" to point "B". Unfortunately that straight line typically passes through the Earth's mantle for any distance greater than a few hundred miles or km. So we grab our "first guess" --that buried straight line-- by its midpoint and pull that middle point straight up to the Earth's surface. Parts of the path are still underground, so split at each segment's midpoint and repeat as many times as necessary. That yields a portion of a great circle. It's the path along the Earth's surface that is as close as possible to the Euclidean line passing through the Earth's interior from A to B. It's our second guess at the most efficient path A to B. Next we look at surface features (if we're land or coastal travelers) and avoid reefs, ice pack, dangerous shoals, mountain ranges, war zones, visa problems. But suppose we're sailors on the open ocean or aviators. Here weather-routing is key and over-flight exclusion zones are our biggest problem, and while the great circle is a useful starting point as our "second guess", it may be a long way from the path we will find to be most efficient and most safe. The great circle path is useful --as a second guess-- but not an absolute by any means. It is not the path that we will follow.

Incidentally, should we worry that the Earth is not a perfect sphere (it's a bit oblate, flattened at the poles) and calculate the exact "great circle" (really oblate spheroid geodesic) distance using that non-Euclidean geometry?? In almost all cases, no, because the great circle is not the goal. It's an intermediary on the way to determining the most efficient, economical path. And the difference from perfectly spherical to oblate spheroid is a smaller detail than everything that follows. It's far better to stick with great circle distance, pretending the Earth is a perfect sphere, as a useful, common standard of comparison.

But suppose we're at the planning stage for a flight, or suppose we know that the trip is simple with no over-flight exclusion, no jet stream to complicate matters, and we just ask our planning tools to give us a plain, ordinary great circle track from point "A" to point "B". You begin your journey, and you know that the distance is 2700 nautical miles. You know from the fundamentals of the original definition that 5400 n.m. would cover a great circle distance equal to a geocentric right angle --that's a quarter of the way around the globe. So this trip at 2700 n.m. is half a right angle, an eighth of the way around the globe. Thus nautical miles prove their value. But do they?

Suppose again that we have our great circle track from "A" ro "B" and our planning software (or our own calculator math, playing with spherical trig) tells us that the distance is 5000 km. As before, you know from the basics of the original definition that 10,000 km is a geocentric right angle... and... do I need to spell it out? It works the same way. A great circle is still a great circle, and you can travel along it counting off the distance in nautical miles or in kilometers.

You added:
"Moreover for star-struck celestial navigators and astronomers great circles are essential for relating coordinates and distances on the celestial sphere to those here on Earth. The cleverly contrived relationship between great circles on the Earth and on the celestial sphere allows us to take measurements along great circles on the celestial sphere and directly convert them into distances in nautical miles along corresponding great circles on the Earth, provided we measure in degrees and convert that into arc-minutes which ARE nautical miles on Earth."

Great circles? Sure. But arcminutes? No, these are separate issues entirely. What do we actually require?? First of all, the correspondence between great circle distances on the sky and distances on the ground is limited to the theoretical construction of celestial navigation. Navigators in practice never have to see it again or employ it any conscious way. It's not even necessary to understand that these are great circles. Also, calling it "cleverly contrived" is not a great description. It's geometry, and it is what it is, by the nature of the geometry. There's nothing "cleverly contrived" in it. Most importantly, the conversion to nautical miles is easy enough during the theoretical construction phase, and when learning about it, but that's the end of it. I measure an altitude with my sextant. For a modern process, I take the reading from the sextant in degrees and minutes (since sextants with decimal micrometers don't exist), and I decimalize the reading by calculating deg+min/60. After that I am done, done, done with minutes of arc. I swap that altitude for a zenith distance by subtracting from 90°. That z.d. is the only angle with any correspondence to a surface distance on the globe: the z.d. is the "distance off" in degrees from the subStar point. If my z.d. is 45.33°, then I am 45.33° away from the subStar point (whose location is identical to the GHA and Dec of the star at the instant of the sight; GHA is lon, Dec is latitude). In my head, or in a rough sketch of the global circumstances of the sight, I can draw a circle with a radius of 45.33° from the center of the circle. Sometimes, it may be valuable to know that every degree is 60 nautical miles or equivalently that every degree is 111.111 kilometers. For a z.d. of 45.33°, I can say that the radius is 45.33×60 or 2719.8 n.m. Or for the same z.d. it's 45.33×111.111 or 5036.7 km. Of course I don't need this distance in either nautical miles of kilometers, but sometimes it's re-assuring at least to see it in common "ground units". None of this is dependent on nautical miles, either by convenience or necessity. The nautical miles is not essential to the concepts or the practice of celestial navigation.

Does this mean that knots and nautical miles are totally un-necessary in celestial navigation? Of course not. They're handy, but you can live without them in nearly all of the cases that are "pure" celestial navigation. The really significant value comes when we connect up celestial to its indispensable partner: dead reckoning. --as long speeds are normally provided in knots.

You noted a popular bit of lore:
"the sexagesimal (60's) base of arc-minutes and time has many more arithmetic factors than do 10's or 100's greatly facilitating mental math once you get used to it."

Sure. A third of an hour is 20 minutes. A quarter of an hour is 15 minutes. A sixth of an hour is 10 minutes A tenth of an hour is 6 minutes. A twelfth of an hour is five minutes!! Wow, look at all those factors, and see how easy it is to divide! But this is just lore... it's a story that people have been telling themselves for decades (at least) to feel happier about the fact that we're using an obscure system to count the hours on our clocks and the angles in our geometries. Tell me this: when was the last time someone said to you, "I'll meet you in a quarter of an hour" and you had to stop and think about that being 15 minutes --but it was easy because of all those wonderful factors. The supposed ease of computation has long since evaporated... But people do still say "see you in a quarter hour" now and then... So consider in addition: when was the last time someone said to you, "I'll get back to you in a twelfth of an hour". Nope. They just say five minutes. The ease of computation never mattered at all there. It's probably true that the factoring of sixty was a component of making sexagesimal numbering practical, useful, and popular in Iraq and surrounding areas 4000-5000 years ago. But was it? Or is that just a story we tell and re-tell? We don't have any reports from witnesses! Mathematically useful systems win out. They conquer and push aside less useful systems. Zero, for example, won. But if sexagesimal is imagined to be a winner, then why don't we use it to calculate the square root of two anymore? It worked for the Babylonians on this cuneiform tablet, but where is that system today?

Sexagesimal counting is hanging on because we have a vast "up and running" installed base in one specific area. That's time... There is no way in hell that we're going to switch to non-sixty-based timekeeping except perhaps in some small settlement on another planet with an iconoclastic and imperious leader (yes, Elon, that's you). But angles? We could leave behind minutes of arc and seconds of arc just as easily as we abandoned thirds of arc and fourths of arc --and thank the gods for that!

You added:
"Also there are several clever “rules of thumb” that work with nautical miles and knots. For example 1 knot is (approximately) 100 feet per minute."

Rules of thumb are developed as a matter of practical experience. No matter how we number things, we will develop little tricks and rules of thumb. That's not a benefit of minutes of arc or nautical miles. It's a benefit of years of experience using any specific system. Some rules you might get to know if you were using a display in decimal degrees and counting distance in SI units: 1.0° of great circle distance is 111.111 km... 0.01° is 1.111 km. Or, as a rule of thumb, a hundredth of a degree is a kilometer (nearly enough).

And you added:
"That is one reason why navigators prefer the format of "degrees minutes and tenths" "

Hmmm... no. I don't see that. I think perhaps you crafted that one --not so much history; rather more of a retcon. We all have a tendency to seek logic and rationality in the rules and tools we use, even when these things developed organically and in ways that may no longer have clear rational origins. History is contingent. Historical developments occur by lots of small steps in different, competing directions. The steady, linear path of progress is more like a drunken walk. There's an awful lot of Brownian motion in the past 10,000 years. Not everything is planned or designed or rational.

Frank Reed