The Age of Fighting Sail was generations before the altitude-intercept method was introduced. In that romantically-remembered era, the typical practice was to take only Latitude by observation of the Sun at noon. This method did not require a chronometer.

OK, just what is all this intercept stuff about?

It is going to be lengthy to explain, so bear with us...It will be worth the effort, though, because this explanation will introduce the concept of coordinate systems, of which the celestial navigator must understand three.

To begin, first notice that the spot directly over your head is always directly over your head.

No, I am not trying to make a joke of you or insult you. This is one of those statements that is so obvious that you pay it no mind, but it is actually very important to our understanding of how the altitude-intercept method works.

Please also imagine that you are standing on a flat, level floor. If you now stand one arm of a carpenter's square between your feet, the other arm will point at that spot directly over your head.

Now if we go just one step further... Imagine that the floor goes out as far as the eye can see. In other words, the floor extends all the way to your horizon.

If you are following this line of thought, it should be very clear now that the spot directly over your head is precisely 90 degrees from the horizon, because after all, the carpenter's square is telling you so. And since the spot directly over your head is always directly over your head, it surely must be 90 degrees from the horizon no matter where on Earth you may be standing.

This spot directly over your head is so special and important in celestial navigation that we give it a name: the zenith, which most folks denote Z.

That is all very esoteric I'm sure, but where does the intercept come in?

We're coming to that in a bit!

Let us pick out a very special zenith. Let us suppose that you are standing at the bottom of a lighthouse. Now there is a very bright beacon directly over your head, and folks can see it for miles around. I could ask you, "What is the altitude of the lighthouse beacon?" Now knowing what you do about a zenith and an altitude, you would answer to me without even a measurement: "Why 90 degrees, of course; since the lighthouse beacon is in my zenith. Just to be fancy, I will write it down as 90^o 00'.0 like a proper altitude should be written!" And your answer would be exactly correct.

Now comes a truly interesting and crucial point in navigation. Your answer is perfectly sound, but unless I happen to be standing right with you there in the lighthouse, I will disagree with you as to the altitude of the beacon.

How so?

Because the spot directly over my head is always directly over my head! And it is a different spot than yours, unless our 2 heads happen to be in the same place. Since my zenith is still a zenith, it is precisely 90^o 00'.0 from my horizon. Therefore, if I am not standing in the lighthouse with you, I will say that the altitude of the lighthouse beacon is less than 90 degrees. If you don't believe me, try standing directly under a light fixture in your room. You will see that the angle between the plane of the floor and the light (your zenith) is 90 degrees. Now take 3 big steps back and look at the light again. You will see that the angle between the plane of the floor, through you and to the light is now surely less than 90^o.

Fortunately for celestial navigation, there is a mathematical relationship that will tell me just what the altitude of the beacon will be. It will depend on how far I am from the lighthouse and how tall the lighthouse is. Check out this diagram:

In this truly excellent diagram (ha!) we have you in the lighthouse, and me somewhere outside. The lighthouse is H feet tall. I am standing some D feet away from the lighthouse. The altitude of the light is 90 degrees for you, and some altitude, h, for me.

Reaching all the way back to those right triangles of high school trig, we remember that the tangent of an angle is equal to the length of the "opposite leg" divided by the length of the "adjacent leg", written

tan h = (H/D)

Of course, the situation envisioned is that I have my handy sextant to tell me altitude h, I just so happen to know the height of the lighthouse H, and I can then whip out those great old trig tables to solve for how far away from you I am, namely D. (Yes, this page supposes calculators won't stand up to salt sea air and we will do calculations without benefit of electronics, apart from our digital wristwatch.)

Will you ever get down to the intercept?

Patience, patience!! It is closer than you think.

The intercept comes in because of two complications, one based in theory and one based in technology.

The previous sketch shows a "side view" of my lighthouse problem. The theoretical trouble is just this: With the information at hand, I cannot tell which "side" I am looking at. That is to say, I can tell how far from the lighthouse I may be, but not whether I am D feet North of the lighthouse, D feet South, or somewhere in between. Now I will present an "overhead view" of all the possibilities.

In another artistic masterpiece we still see you in the lighthouse and me somewhere outside. The letters denote the Four Cardinal Directions around the horizon. I am still standing some D feet away from the lighthouse. Everywhere I may stand along the circle, my trusty sextant will yield the same altitude for the beacon.

This very important phenomenon is the Circle of Equal Altitude. It is consistent with mathematical theory which tells us that we must have two bits of information in order to decide a position on a 2-dimensional surface which the Earth seems to be, at least for a small locale.

It also brings up something crucially important for developing the idea of an intercept.

I want to emphasize what was just said; that if I were to stand off of the circle of equal altitude, obviously the sextant reading changes. If I stand closer to you, the altitude will increase. It has to, because my zenith is getting closer to your zenith if I come inside the circle. If I take a step or two back, the altitude must decrease.

Walking along a circle of equal altitude, or altitude circle, for short, is something else you can try in your own room with that ceiling-mounted light fixture again, and you are strongly encouraged to do so.

Why don't you just use a compass and measure your bearing to my lighthouse? Then you would know everything you need...

A great idea! And it is routinely done in coastwise navigation, where angular precision is less important because you are sighting a true stationary landmark which is relatively nearby.

When our "lighthouse beacons" become celestial bodies, we run into the technologic problem--so without further beating around the bush...

Remember what was said about the angular precision of a sextant? A sextant easily measures down to fractions of a minute. Well, a compass doesn't. No one has ever devised a compass that gives better than fractions of a degree, so compass measurements are over 60 times cruder than sextant measurements. This means that their relative uncertainty is large to begin with. Additionally, in normal practice the distance to our "lighthouse" will be vast, hundreds or even thousands of miles, so we would be taking a large uncertainty and multiplying it by a very large number, making it more uncertain still.

As if that weren't enough, when it comes time to draw or plot our fix, we have no drafting instruments that will divide angles more finely than perhaps half a degree. All these large uncertainties add up quickly, and even though we don't seek "scientific precision" in all our work, this level of crudeness is simply unacceptable.

This explains why we must have at least two altitudes on two different bodies for a fix. A 2-body celestial fix is thought of as the intersection of circles of equal altitude. If you could draw the circles out completely on a globe, it would most often look "something like" the MasterCard symbol, like this:

I say only "something like" because in general the circles won't be of equal size, but this graphic makes the point.

Two circles taken from celestial sights normally cross each other twice, but in positions hundreds or even thousands of miles apart, so it is obvious which one is the location of the vessel. Since each altitude is known quite precisely, the intersection fix is known far more precisely than trying to take an altitude and a bearing, or azimuth, as it is called, of a single body.

The trouble comes in when we go to actually draw our fix on the plotting sheet, which is usually where we draw such things since there are no charts of the middle of a featureless ocean. We approximate a small portion of the circle of equal altitude by a straight line, called a line of position. This is more feasible than trying to draw a huge circle, but we still must know where the center of the circle lies. That is to say, we need to know what the azimuth to our "lighthouse" would be if we could measure it.

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