Follow along with this illustration of the globe. It will help.

We have not yet discussed longitude lines, but they are imaginary lines which run north and south around the circumference of the globe, intersecting both poles. Don't worry yet about how they're measured, or from where; that will come later. For now, just understand that they are lines which run along the circumference of the globe, from one pole to the other.

In our illustration there are two lines of longitude shown, at an angle of about 70° apart from each other.

It is clear (I hope) from the illustration that each of the longitude lines intersects the equator at a 90° angle. This happens to be true for all longitude lines.

Also shown is another latitude line north of the equator, which is also intersected by each of the longitude lines at a 90° angle.

All latitude lines are parallel to the equator. Since the equator and the other latitude line are parallel to each other, and the two longitude lines are intersecting both the equator and the other latitude line at the same angle, the two longitude lines are by definition also parallel to each other.

If we draw a triangle with two of the legs extending down our two longitude lines from the north pole down to the equator, and then for our third leg connect these two legs along the equator, we see two interesting things.

The first is that the three angles of the triangle are 90°, 90° and 70°, which, if added together, equal 250°.

The second is that our two longitude lines, which have been demonstrated to be parallel to each other, must necessarily converge and intersect at the north pole.

Very Important Concept: Two parallel lines often intersect, and the sum of the three angles of a triangle must always be greater than 180°.

Very Important Concept: Two parallel lines often intersect, and the sum of the three angles of a triangle must always be greater than 180°.

It is possible that at some point in your education you were told something which was somewhat contrary to this. If you are somewhat surprised to learn that parallel lines intersect, and that the sum of the three angles of a triangle may never equal 180°, you are to be forgiven. Your high school geometry teacher, however, is not. A geometry teacher should know better. It happens that many of them don't.

If you happen to be of an age where you have not yet been exposed to the wonders of Euclidean geometry, never fear; at some point in your schooling, you will be. And on that golden, sunny day, listen politely while your teacher explains that two parallel lines can never intersect, and that the sum of the three angles of a triangle must always equal 180°. Once they are finished, please raise your hand. When you are called upon, please politely explain to your teacher that while their lecture was positively delightful, it turns out that the world is round, like an orange, or a bowling ball.

They love that.

The point of this exercise is to illustrate that solving a triangle on a spherical surface is, frankly, not making mud-pies. In some twenty-plus years of teaching celestial navigation I've had maybe two or three students who were comfortable solving spherical trigonometry with a scientific calculator, and not a single one who could solve it long-hand without a scientific calculator. Don't worry, nobody will expect you to; we have logarithmic tables or programmable calculators to solve the triangles for us. But this is why we need the logarithmic tables in the first place.

We'll see some of these soon. But our next topic will be looking at how we use this triangle to derive our celestial line of position.

That is the best and most concrete explanation of non-Euclidean geometry I've ever seen.

ReplyDeleteAre you planning on turning this series into a book, or is this yet another project you're going to give away for free?

Are you still planning to consolidate your series on outmigration into a single pdf? I really think you should publish that.

Yes, will at some point consolidate the outmigration stuff; it's on my to-do list, just haven't gotten around to it yet.

ReplyDeleteMy original plan with the current series was simply to make available the underlying concepts so that I could discuss more esoteric topics in celnav without losing the non-navigator types. But the direction it seems to be going could evolve into a book. What happens with that remains to be seen, but I'm not actually opposed to the idea of publishing.

Regarding the intro to non-Euclidean geometry, I should probably state that I actually like Euclidean geometry quite a lot. For things like designing a ten-story office building, for example, it's quite a reasonable approximation. It even has application in navigation, on the scale of, say piloting inside a harbor. But in the best of circumstances, Euclidean geometry is only a useful approximation. One can define a "circle", for example, but there is no such thing in nature which exactly matches that description. The same is true for non-Euclidean geometries as well, whether they are spherical, Gaussian or whatever. Most of the "error" we see in celestial navigation is simply an artifact of using spherical trigonometry on a planet which does not happen to be a sphere.

ReplyDeleteEven simple arithmetic is a useful but inherently flawed abstraction. If I have a stick, and then I have another stick, we can say that I have two sticks. One plus one equals two. But what happens if I break one of the sticks? Now I have three sticks, because "half of a stick" is still a stick. But I did not add another stick to the two original sticks. One plus one now equals three.

Mathematics is a very useful tool. It's approximations are an elegant metaphorical language with which to describe and communicate the world around us. It only becomes problematical when people imagine that mathematics is in any way "real", or that an abstraction can somehow prove or disprove something which does happen to be "real".

Robert,

ReplyDeleteThank you for doing this. I presume I will be needing a Sextant? I took a look through EBay and realized I am in need of some direction. Looking through EBay and Amazon listings there is a range of choices. Would you be able to point me in the direction of good names and models?

Regards,

Chris Ruttan

Hi Chris,

ReplyDeleteYes, certainly. This question comes up in my classroom courses all the time, so I actually have something already written which addresses this; will post it here this afternoon once I'm in front of a real computer.

The short answers are 1) you won't need a sextant for the lessons, but you'll need one eventually for practice and of course once you're on the water and 2) unless you are very, very knowledgeable about sextants, buying one on eBay is not a very good choice. Not because the sellers are dishonest, but because as often as not the sellers are antique or militaria dealers who know absolutely nothing about sextants.

Will post a short essay on the topic of buying sextants here in a bit; it's already written, just have to convert it to a Blogspot-friendly format.