To the aspiring celestial navigator, the process of converting a sextant sight into a line of position (LOP) on a plotting sheet or chart may seem overwhelming and full of pitfalls.
Compounding the confusion for the student of celestial navigation is the fact that there are many different ways to do sight reduction. All have their pros and cons. In this overview, we’ll look at the differences among the various methods.
All methods discussed below are table-based methods (with the exception of the Law of Cosines method). They utilize tables to solve the navigational triangle. This means they require only addition and subtraction to reduce the sight. An electronic calculator is usually required for the multiplication and division necessary to solve the trigonometric equations that are part of the Law of Cosines method.
Some methods require the use of an assumed position (AP) to compute an assumed local hour angle (LHA) and assumed latitude for getting into the table, while others use your dead reckoning (DR) position. Other methods that use your DR position tend to be a little easier, if only from a plotting perspective, since there is no need for the additional plotting of an AP.
Methods using an assumed position
The Nautical Almanac Sight Reduction Method. The NASR method uses concise sight reduction tables that are included as part of the Nautical Almanac. Hence, an advantage of the NASR method is that it enables sights to be reduced using a tabular method with only one volume. The disadvantage is that it requires dual entries of a table appropriately titled Sight Reduction Table and dual entries of a second table entitled Auxiliary Table. These tables occupy 32 pages near the back of the Nautical Almanac and come with instructions and examples.
The first entering arguments for the concise sight-reduction table are LHA and assumed latitude. When solving multiple sights from the same assumed latitude, all of the first table extractions can be made at the same time from the same page since the pages are arranged by increasing latitude, thus saving some page-turning time. The auxiliary table is used to adjust the computed altitude (Hc) for the difference between the actual values and the values used to enter the tables. Azimuth angle is determined in two partsZ1 and Z2which are added together. True azimuth is then determined by following the rules provided on each page of the sight-reduction tables.
These dual entries can be somewhat confusing but are easily mastered with a little practice. All data is extracted directly, requiring no interpolation between table entries. Only occasional rounding in the usual manner is needed.
The NASR method can be used for sights from any latitude and for bodies of any declination. The instructions recommend using this method for sights with altitudes less than 80, but the degree of error for altitudes above 80 would be acceptable for practical small-vessel navigation.
Summary: The advantages of the NASR method are: It uses only one book and requires no interpolation, and the pages are organized by increasing latitude, facilitating multiple reductions from the same assumed latitude.
This method’s disadvantages include: It requires an AP for table entry and plotting and requires dual entries of two different tables.
Sight Reduction Table for Marine Navigation. Sight Reduction Tables for Marine Navigation, Pub. No. 229 (popularly known as H.O. 229) utilizes a series of six volumes containing sight-reduction tables that are used in conjunction with the Nautical Almanac. Each volume is hard bound and is 12 inches tall by 9 1/2 inches wide by 1 1/4 inches thick and weighs three pounds. You choose the volumes you need based upon the latitudes that you will be sailing in. Volume 1 covers from the equator to latitude 16 north or south. Volume 2 overlaps Volume 1 by one degree, covering from 15 to 31 north or south. Volume 3 covers from 30 to 45, and so on. Since they are basically tables solving spherical triangle problems they never need replacing. This method requires the use of an AP.
The entering arguments for table entry are assumed LHA and latitude of the body and declination of the body, which is whole degrees of declination less than the actual declination. The pages are arranged in order of increasing LHA, requiring some page-turning when reducing multiple sights from the same assumed latitude. The left-hand pages are for latitude and declination of the same name (both lat. and dec. north or south) and the top right page is for latitude and dec. of contrary name (lat. and dec. are in opposite hemispheres).
These tables provide a calculated altitude (Hc) to the nearest 0.1′ and azimuth angle to the nearest 0.1°. Required interpolation for differences between actual declination and the tabular declination is accomplished by means of interpolation tables provided on the insides of both the front and back covers. For altitudes greater than 60°, a second difference interpolation must be included, again accomplished using the interpolation tables inside the cover and then applied to Hc. Azimuth angle is converted to true azimuth following the rules on each page of the tables.
This method of sight reduction can be used for sights of any altitude, from any latitude, and for bodies of any declination. Each volume comes with the same introduction and set of instructions with examples.
Summary: The advantages of this method are: Single entry for any combination of assumed LHA, assumed latitude, and declination and the highest accuracy of the tabular methods.
This method’s disadvantages include: It requires an AP for table entry and plotting, requires interpolation using interpolation tables provided, requires multiple volumes based upon your latitude (and the volumes are relatively large), and requires a different page opening for each sight.
Sight Reduction Tables for Air Navigation. Sight Reduction Tables for Air Navigation, Pub. No. 249 (H.O. 249), are a series of three volumes. These volumes are similar in size and weight to Pub. 229 but are spiral bound. For marine navigation they are generally used in conjunction with the Nautical Almanac.
Volume 1 contains tables for selected stars and should be replaced every five to eight years. It is printed at five-year intervals (the latest edition having been printed in 1995). It provides the computed altitude to the nearest minute and true azimuth to nearest degree for the seven best stars suitable for a fix from any latitude and any local hour angle of Aires. The best stars are chosen based on magnitude and azimuth for a three-body fix.
Entering arguments are assumed latitude and LHA of Aries. An almanac for GHA of Aires is provided in the back along with a table called Table 5 for a correction to be applied to the resulting fix.
Vol. 2 and 3 of Pub. 249 are laid out similarly to Pub. No. 229. Vol. 2 is designed for use for latitudes between 0 to 40, and Volume 3 is designed for use for latitudes between 39 and 89, but both only contain listings for bodies with declinations between 29 N and 29 S. These two volumes like, Pub. 229, never need replacing since they solve only spherical triangles for all table entries.
Entering arguments are the same as for Pub. No. 229: assumed latitude, LHA of the body, and declination of the body being observed. They differ from Pub. No. 229 in that the pages are arranged by latitude, not LHA, therefore requiring less page turning for reduction of multiple sights from the same assumed latitude.
Pub. No. 249 Volumes 2 and 3 provide a Hc to the nearest 0.1′ and azimuth angle (not true azimuth as in Vol. 1) to the nearest degree. The Hc is corrected for the difference between tabulated declination and true declination using a table at the back of both Vol. 2 and 3.
Summary: The pros of Vol. 1 of Pub. 249: Fast and easy extraction of the Hc and true azimuth, and the ease of sight planning. Another advantage is that only one correction is applied to the fix. The disadvantages of Vol. 1 is the limited stars that are listed and the fact that it must be replaced every five to eight years.
The pros of Vol. 2 and 3 of Pub. 249: Only two volumes are needed to cover all ranges of latitude; only a single entry is required for any combination of LHA and assumed latitude.
The cons of Vol. 2 and 3: They require the use of an AP for table entry and plotting, require interpolation for declination difference using the table provided, and require use of multiple volumes depending on your latitude (and the volumes are relatively large). Also, the declination coverage is only 29 N to 29 S.
The disadvantage to Pub. No. 249 as a whole is that Vol. 1 uses a different method of table entry than do Vol. 2 and 3. This may cause some confusion when switching between the three volumes of this method.
Methods using your DR position
The Reed’s Nautical Almanac Method. The Reed’s Nautical Almanac Method, like the NASR method, requires only one reference. It is a paperback book that is about nine inches tall, six inches wide and 1 1/4 inch thick. It contains many other tables as well as an almanac and sight-reduction tables.
To find the intercept this method uses two tables. The first is a table of Log Versines and Natural Versines and the second is a table of Log Cosines. The first table is entered in a repetitive manner while the second is referred to only once. Interpolation is done by “eye-balling” the difference between table entries and the desired entry. The tables are designed with whole degrees as column headings and even minutes as row headings. The entering arguments for the first pass are LHA and declination of the body and latitude. These are all determined from the DR position. These table extractions are then added together and the “tens” place of the sum is dropped. This number is entered into the body of the table to extract an intermediate value and to finally arrive at the computed altitude Hc by extracting it from the body of the table in a reverse fashion.
The azimuth angle is found from a second set of tables called the A, B, C tables. Table A is entered with LHA and latitude, table B is entered with LHA and declination. These values are added together to get C. The C table is then entered to extract an azimuth angle. The rules at the bottom of page are followed to convert the azimuth angle to a true azimuth. These tables require eye-ball double interpolation interpolating both rows and columns).
This method can be used for sights of any altitude, from any latitude and any declination.
Summary: The advantage of this method is that it requires only Reed’s Nautical Almanac and uses the DR for plotting and for table entry. The disadvantages are the repetitive table entries and large degree of eye-ball interpolation required.
The Ageton Method. The Ageton Method uses tables of log secants and log co-secants termed A and B functions. These were once published as H.O. 211 and are titled Dead Reckoning Altitude and Azimuth Tables. The Ageton method consists of a table that is 36 pages long with several pages of explanations and rules. This publication is hard bound, is about nine inches tall, six inches wide and about 1/4 inch thick. The Ageton method tables are also reproduced in Volume II of the 1981 edition of the American Practical Navigator (commonly called Bowditch by navigators) as table 35.
This method is a repetitive table-entry method. It requires entering the table with meridian angle t, latitude, and declination. The column headings are degrees and the row headings are minutes and 0.5 minutes. Values are extracted for intermediate calculations and for Hc and azimuth angle. Azimuth angle is then converted to true azimuth.
Summary: The advantages of the Ageton method are: only one volume is needed for all latitudes, altitudes and declinations; the size of the book is very compact and light; it uses your DR position for plotting and table entry values, it requires no interpolation, and the tables never need replacing.
The disadvantages are that it uses the meridian angle t for table entries, large errors may be introduced if the intermediate value of K is near 90, and repetitive entries of the same table are required.The S-tables Method. The S-Table method uses tables that are designed and used in a fashion similar to the Ageton tables. The S-tables consist of only nine pages of tables plus a few pages of instructions and rules. It is a similar in size to the Ageton tables. Headings for columns are degrees and headings for rows are whole minutes. Otherwise their use follows the Ageton method. The biggest difference between the S-table method and the Ageton other than size is that the S-table method uses LHA for table entry instead of median angle t, thus eliminating one step.
Summary: The advantages of the S-table method are: only one volume is required for all latitudes, declinations, and altitudes; size is very compact and light; it never needs replacing; it uses the DR for plotting and table entry values; and no interpolation is required.
The disadvantages are: it requires repetitive table entries and, if the angle K is near 90 or 270°, then a large error may be introduced.
The Law of Cosines Method. Last we’ll discuss the Law of Cosines Method. This is the only method that does not require tables to determine the intercept and azimuth. Like all other methods, a Nautical Almanac is required to determine the LHA and declination of the observed body. This method utilizes the trigonometric functions found on most scientific calculators to produce a computed altitude and an azimuth angle. These are easily converted to an intercept and true azimuth. This is a quick and easy method of sight reduction that plots from the DR position. The disadvantage of this method is that it requires an electronic calculator involving numerous key strokes providing the opportunity for error.
Each method has its own quirks that you must master before you can gain speed and have confidence in the results. Practice sight reduction regularly, otherwise you will struggle when your boat’s electrical system dies as you approach landfall and you need a position fix.