# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: The flat earth notion**

**From:**Walter Guinon

**Date:**2003 Nov 8, 12:37 -0800

Sounds a lot like Zeno's paradox to me. --- "Trevor J. Kenchington"wrote: > George, > > I know better than to question your knowledge of the mathematics of > navigation. I also strongly suspect that, if I asked for proof that a > loxodrome reaches (rather than approaches) a pole, you would need to > resort to mathematics that would not easily be represented in e-mail > format and which I would not understand if you did send it. Hence, I > have little option but to accept what you say at face value -- something > that I am never comfortable doing, even when you are the author of > whatever I am accepting. > > Could you, perhaps, explain the final approach of a loxodrome to the > pole, without resorting to math? Above a latitude of, say, 89 degrees 50 > minutes, the sphericity of the Earth ought to be negligible and we > should be able to visualize the problem as one of a rhumb line > spiralling around the pole in a simple 2-dimensional space, while > cutting each meridian at the same angle. I can visualize that spiral > closing in from 10 miles out to a mile and so to a tenth of a mile. But > it then seems to me that we could simply expand the scale by 100 times > and see the loxodrome spiralling in from 1/10 of a mile to 1/1000, > before expanding the scale again and repeating. What I don't see is how > that loxodrome will eventually make the final step of reducing its > distance from the pole to zero. > > You wrote: > > > They are both wrong, and Herbert is correct; though I'm sure he is capable > > of defending his corner without my aid. > > > > The presumption is that you are steering a rhumb-line course, with a > > Northerly component, and that all the Polar ice has melted. > > > > As you spiral in toward the pole on a rhumb-line course, you travel a > > FINITE distance to get there. If you can maintain a constant speed, then > > that can be done in a finite time. The snag is, you have to travel in > > ever-decreasing circles, as the size of the spiral shrinks. To get exactly > > to the pole, you have to make an infinite number of such gyrations, so the > > vessel has to be spinning at an infinite rate. This is rather an unphysical > > state of affairs, to say the least: a "singularity". > > > Maybe it is my limited understanding of math but I don't think that the > finite length of a loxodrome is inconsistent with it never reaching the > pole. The mathematics of infinity has some funny properties, analogous > to the effects of dividing by zero, and I'd not rule out a curve of > finite length which spirals without end. (There may well be members of > this list who can say that I am wrong. I'm only saying that I can't rule > that out myself.) > > > The pole will be reached at a predictable moment, at which the ship will be > > spinning round at infinite speed. Just after that moment, it will escape > > from the pole, still spinning with infinite speed, and the radius of the > > spiral then increases, and the spinning slows, until the vessel reaches its > > original latitude. What will its longitude be then? > > > While I am willing to accept that I may be wrong and a loxodrome may > actually reach the pole, it will take a whole lot more to persuade me > that said loxodrome ever departs from the pole again. George: I really > think you are wrong on that point. > > I _know_ you are wrong on the claim that our hypothetical ship will ever > leave the pole. You have assumed that the vessel is following a > rhumb-line course between 270 and 090. Any course away from the North > Pole must be 180 instantaneously and, immediately thereafter, must be > between 090 and 270. That is: the hypothetical ship can only leave the > pole by changing its course (perhaps to its reciprocal) and that is > contrary to the starting assumption. > > > Of all the unrealistic questions we have considered on this list, this is > > perhaps the most unrealistic of all. But why should we let that deter us > > from playing such games? > > > > The picture above, of a vessel spinning infinitely fast at the pole, > > applies to all incoming courses except 0deg and 90 deg, as Trevor points > > out. At 90deg, the pole is never reached at all: the vessel sticks to the > > equator. > > > No. That is commonly stated in textbooks but is clearly false. > Regardless of its starting latitude, a vessel following a rhumb-line of > 090 or 270 will never reach either pole. It will follow a parallel of > latitude, which could be the Equator but could be any other. (This is as > stated in my last contribution to this thread.) > > > At 0deg, the vessel approaches the pole along a certain line of > > longitude, then emerges along a line of longitude 180deg different. > > > Again, the vessel cannot leave the North Pole while its course remains > the rhumb line of 000. As I wrote last time, it can only leave by > changing to a reciprocal course -- which it would of course do if it > continued "straight" (meaning on a Great Circle, since this is spherical > geometry) through the pole. We are, however, assuming rhumb-line > courses, not reversible rhumb-line courses. > > > Trevor Kenchington > > > -- > Trevor J. Kenchington PhD Gadus{at}iStar.ca > Gadus Associates, Office(902) 889-9250 > R.R.#1, Musquodoboit Harbour, Fax (902) 889-9251 > Nova Scotia B0J 2L0, CANADA Home (902) 889-3555 > > Science Serving the Fisheries > http://home.istar.ca/~gadus __________________________________ Do you Yahoo!? Protect your identity with Yahoo! Mail AddressGuard http://antispam.yahoo.com/whatsnewfree