Bali Ha'i Will whisper On de wind Of de sea; "Here am I, Your special island! Come to me, Come to me!" --Oscar Hammerstein II--

Is this the part that gets intensely mathematical?

Not necessarily so, although it can. Up to this point, our discussions have all been pretty "single-minded" with few possibilities for variations, but in determining a computed altitude, the navigator gets to choose a method of solving the problem. The degree of mathematical complexity depends on the choice of method.

Through the years, a number of ways have been devised for obtaining a computed altitude, a process also known as sight reduction. (You guessed it--the peculiar and archaic vocabulary, again.) The methods generally fall into 3 classes:

Formulas have the advantage of being very compact--you can jot them on a single sheet of paper for reference, actually.

These equations are used to solve a "spherical triangle" by dividing it into 2 "right spherical triangles" If this statement is not obvious to you, just skip it for now and we will discuss it in the Appendix on Spherical Triangles.

Their overwhelming disadvantage is the amount of labor you must spend to solve only a single altitude. Mathematicians refer to sine and cosine as transcendental functions which means (to us ordinary folk) that they are functions which yield long, caterpillar-like decimals instead of "nice" numbers, and these unwieldly decimals must be multiplied and divided. Not a pleasant prospect without a computer, which I am ignoring in these Web Pages. And remember, in practice we are striving for 3 or more bodies as a fix, and multiple fixes each day--this adds up to a monumental workload of arithmetic, and we've not even considered the chances of making simple errors.

Tables have the advantage of being speedy and easy to use, with far less chance of a making a simple arithmetic error. Their chief disadvantage is their bulk.

This Web Page will use tables for obtaining computed altitudes.

What sort of tables?

Through the years many systems of tables have been devised, but today in the US we may readily find 3:

• the concise sight reduction tables in the Nautical Almanac
• H.O. 229, more correctly: Pub. No. 229, Sight Reduction Tables for Marine Navigation, Volumes 1 - 6
• H.O. 249 more correctly: Pub. No. 249, Sight Reduction Tables for Air Navigation, Volumes 1 - 3

Of these, we will use H.O. 229.

Didn't "H.O." mean "observed altitude"? What has this got to do with computed altitudes?

No, that was Ho, the subscript o being important to the meaning. When we write H.O., that means the Hydrographic Office of the U.S. Navy which was the former publisher of different navigation tables. As of the original construction of this Web Page, it was published by the Hydrographic/Topographic Center of the Defense Mapping Agency.

However, in America's never-ending quest to add new layers of Federal bureaucracy, the DMA was disestablished in late 1996 and its functions incorporated into the new National Imagery and Mapping Agency.

Whether or not the new bureaucracy renames the publication, we will still call it "H.O. 229" because of our boundless fascination with peculiar and archaic vocabulary.

Why not just use the tables in the Nautical Almanac?

It does sound appealing, since then we would have all printed information we need bound in the same volume, but life just isn't that rosy. In general, the shorter a table is, the more look-ups required of the user to work for an answer. To fit in so small a volume, the Nautical Almanac tables are admirably short; therefore, they require more work to get a complete answer. In fact, it is almost as much work as solving the formulas. In addition, the Nautical Almanac tables look-up system is very confusing; even experienced users are prone to mistakes, so those of us who seldom practice our navigation are at a high risk of blundering.

If you are still interested in using the Nautical Almanac tables, a complete explanation of them is given in the Nautical Almanac. But I won't go into using them on this Web Page.

H.O. 249 is today something of a misnomer since celestial air navigation ended generations ago; however, governments continue to print H.O. 249 because it enjoys a very devoted following among mariners.

H.O. 249 is written to be both compact to store and fast to use since space is at a premium and time is of the essence aboard a fast-moving but cramped airplane. To make their tables simple to use, the publishers solve all the "ideal shots" directly in Volume 1. This means they pick only the very brightest stars, spaced the most evenly around the horizon, and print for you their altitudes and azimuths for given Latitude and LHA. This is a very speedy way to get an answer! It comes at the cost of giving up the dozens of other possible bodies to observe, including planets which are typically much brighter than any star.

Volumes 2 and 3 can be used for any body within a narrow band of the sky; that is, these Volumes save space by deleting huge portions of the sky from their tables--your altitudes will be for bodies only in the declinations of the Zodiac, which is where the planets may be found. This is yet another cost in potential observable bodies.

You may also see a disadvantage in the fact that you must learn 2 different systems of determining a computed altitude, one for Volume 1, and another for Volumes 2 and 3.

In contrast, H.O. 229 gives answers for all bodies in the heavens, as it includes answers for all declinations, from every degree of Latitude. The cost here is bulk: H.O. 229 is published in 6 hard-bound volumes which someone tells me contain over 1460 pages and weigh a total of 23 pounds.

Isn't that a bit excessive?

All the devotees of H.O. 249 would say so!!!

But seriously, to ease the pain the volumes are organized in bands of Latitude. Volume 1 is 0 degrees to 15 degrees, Volume 2 is 15 degrees to 30 degrees, and so forth. This makes it possible for the Navigator to buy the volume (or 2) for only those Latitudes in which he sails. I personally don't own the volumes for Latitudes above 60 degrees since I am not planning any polar expeditions...

The rest of this page will use H.O. 229, but I may one day write an Appendix on H.O. 249 since it is appealing in its compactness and ease, even with its drawbacks.

To actually use the H.O. 229 tables, let us continue with the same situation we just examined, only with a more practical problem...

Just to remind you, here is the situation: We estimate our position at L. N 38^o 25'.5 , Lo. W 42^o 12'.8 on 1996 May 9. The Moon has waned to a half-moon, so we can make a morning observation of both Sun and Moon for a pretty reasonable celestial fix. We are keeping zone time (Z.D. W (+) 3h), and our accurate watch tells us it is 9h 42m 18s when we observe the Sun, and 9h 43m 32s when we observe the Moon.

Required: Hc and Zn for both Sun and Moon.

You can see already that this is a much more typical problem for a Navigator's work. Since we already know the Declinations and GHAs, let us just copy them from the last page:

Don't we need the LHAs, as well?

Yes, but not the ones we computed on the last page.

Why not?

Because of the way the H.O. 229 Tables are constructed.

In the last problem, I was attempting to make the points of:

• What is an LHA?
• What is a GHA?
• How are they related?

And of course we know now that the answers to all 3 questions lie in constructing a Diagram on the Plane of the Equator for the situation at hand.

In the present problem we will need to actually use the LHAs, and in this we come to another million-dollar idea that makes the altitude-intercept method work...

Remember in the Introductory pages when I mentioned that before we find a computed altitude, we choose a "convenient spot" on the Earth for which the calculation is true. Since there are several choices of methods available to solve for computed altitude, our choice of method dictates which spots will be "convenient" for us. If we were using mathematical formulas with our pocket calculator, for example; we could just use our estimated position, or any other position reasonably nearby. But when we use tables, our choice is constrained by the entering arguments for the tables.

H.O. 229 has computed altitudes (and azimuths) for every possible declination, but only when observed from an exact whole degree of Latitude and at an exact whole degree of LHA.

That is a very important statement. It means that our assumed position will be:

• That exact whole degree of Latitude nearest our estimated position.
• That Longitude which, when added to (or subtracted from) GHA as appropriate, will result in LHA being an exact whole degree.

This implies that in most celestial fixes, every celestial body observed will have a different assumed position from which to plot its line of position.

Let us see how this will work in our example...

Since we believe our Latitude to be L. N 38^o 25'.5, the best choice of assumed Latitude for both Sun and Moon will be L. N 38^o 00'.0

Now we must consult our Diagram on the Plane of the Equator to be certain of our next move. Since we already did this on the last page, we know that we will be subtracting Longitude from the GHAs in order to get the LHAs, but we must add an extra 360 degrees for the Sun. This time there will be a new twist: we will use a different assumed Longitude for each of the bodies, and each assumed Longitude will be near W 42^o, our estimated position, but will have the number of minutes we see on each GHA in order to subtract to zero. A table should make this clear:

At this point, we have every argument in the correct form to enter the H.O. 229 Tables! Here is the summary:

Latitude 38^o is contained in volume 3. The tables are organized by LHA, which is found in the corners of the pages. To obtain the required altitude and azimuth for the Sun, we search for LHA 329^o, and notice that each LHA gets a double-page spread: a table for Latitude of the SAME name as Declination set on the left-hand page, and a table for Latitude of CONTRARY name to Declination set on the right-hand page. We notice that both our Latitude and the Sun's Declination are of the SAME name, North, so we will look on the left-hand page for the column labeled 38^o Latitude...

...Then we scan down the page until we come to 17o of Declination. There we find 3 respondents in the Table. These are altitude, d, and azimuth.

Is this d the same as the d in the Nautical Almanac?

No, but it serves a similar purpose.

The declination of the Sun is not a perfect even degree, so the d value gives us a sort of increment for the minutes of the Sun's declination which are not found in the tables. In H.O. 229, the Interpolation Tables are found in the inside covers of the Volumes, 0'.0-31'.9 declination inside the front cover, 28'.0-59'.9 inside the back cover.

(The overlap is intentional, for convenience.)

There is a rather peculiar procedure for entering this table, so you will need this digitized copy of it in order to follow along:

H.O. 229 reports d in minutes. To enter the Interpolation Table, we split d into two parts, the "tens" (in this case 40'), and the "units" (in this case 1'.2). This will determine which columns to use. The number of minutes of "left-over" Declination is called the "Declination Increment" and determines which row to use (in this case 30'.7) Now for the bizarre part: Look up the first part of the interpolation under the appropriate Tens column. (I get 20'.5) To this number, add an additional amount from the entire block of numbers next to it. Ignore the row for Declination! Just stay in the block and look at the top of the page for the Units column and pick the row that matches the Decimals. (Do you agree that it is 0'.6?) This sum (21'.1) is the interpolation for this problem. (The "Double Second Difference" is not needed in this case, and I will ignore it in these Web Pages.)

Having looked up the interpolation, we may apply it to the tabulated altitude to obtain an answer for Hc for the Sun.

We then follow a similar look-up procedure for the Moon. This time, we turn to the pages for LHA 65 degrees, and we must be careful to use the right-hand page since the Moon is in South Declination but our position is in North Latitude--the CONTRARY Name table is the correct one.

This table shows the answers for our problem:

How did you get from the tabulated Z to the Zn?

Azimuths can be formally interpolated, just like altitudes; however, most practical mariners just estimate by eye. We simply look at the two successive entries for Z listed for the degrees of Declination and the next successive degree of Declination. We judge how much the azimuth changes from one degree to the next, and then see "how far between" our actual declination is, then adjust our azimuth to "something in-between" accordingly.

This procedure isn't very scientific, but it is perfectly adequate for practical work since our drafting instruments will not allow us to plot better than about half a degree's precision, regardless of how many decimals our tables or computers may yield.

Do notice that to obtain the azimuth of the Moon, we had to take Zn = 360^o - (tabulated Z). The rule for deciding when to do this is printed on each page, as a convenient reminder. (But it is not in the view of my digitized image...)

Is that all there is to it?

Yes!

As you can see, using H.O. 229 makes it quite easy and fast to obtain Hc and Zn without benefit of electronic help or the necessity of many difficult calculations. Just as a quick review, let us specifically mention what is going on here:

• We have picked 2 convenient, known locations on the Earth
• We have looked up the positions of the Sun and Moon in space on 1996 May 9 at 0942-0943, and
• We have computed how the Sun and Moon would appear in the sky if we were at those locations at that exact moment.

At this point, we need only to compare these computed values with observed values to obtain a celestial fix by the altitude-intercept method.

Go to the next Celestial Navigation page

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