# NavList:

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

**Re: The flat earth notion**

**From:**Trevor Kenchington

**Date:**2003 Nov 7, 20:48 -0400

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