Department of the Army, Retired, Oakhurst, New Jersey


This paper presents a simplified sight reduction method (ABHAV) which is easy to use and accurate to less than 1 arcmin in altitude without interpolation, limited only by the accuracy of the input data. It is intended to be used in place of the Modified Ageton method or the NASR (Nautical Almanac Sight Reduction) method which is published in the Nautical Almanac, both of which use comparable input data accuracy. The ABHAV method is more accurate than the Modified Ageton method and much more user friendly than the NASR method. The Modified Ageton method accuracy is limited when the value of K (an interim value used in the solution) approaches 90 deg. Under these conditions the accuracy is reduced by up to several arcmin of altitude depending upon the actual data involved. The ABHAV method can also be used for great circle sailing problems.


Most navigation today is performed by some form of electronic equipment which makes sight reduction on celestial bodies relatively obsolete. However it is still necessary to have a backup method available in case of failure of electronic equipment. This method should be simple and easy to use without sacrificing significant accuracy. Of course a very high degree of accuracy can be obtained by using a method such as shown in PUB. NO. 229 which uses six volumes of tables and produces accuracies approaching .1 arcmin in altitude. However the simplified methods referenced in this paper require only a limited number of pages of tables (60 or less) and provide accuracies generally better than 1 arcmin in altitude. The big problem associated with the NASR method is the complexity in its use and the great chance for human error. It is a method that would require reading the text each time it is used to remember all the procedures unless it is used very regularly, which is unlikely with todays modern electronic navigation systems. Because of the difficulties I have experienced using the NASR method ever since it became known to me and the inaccuracies experienced with the Modified Ageton method I decided to try to devise a sight reduction method that would do away with both of these problems. Since I have always had a strong attraction to the use of haversine formulae I decided to juggle these formulae for sight reduction to attempt to get an arrangement which would require the least number of different trig functions needed in the tables. I was able to get it down to three functions: the natural haversine, the log haversine and the log secant. These lookup values are relatively easy to use and are arranged in a three column table. The haversine of an angle "a" is defined as follows:

hav(a) = [1-cos(a)]/2


This method of sight reduction was conceived based on the haversine formulae. It provides a simple but accurate method of sight reduction. It is believed to be at least as accurate overall as any of the various simplified methods (because of the straight forward use of the haversine formulae which provides accuracies equal to any method when using comparable significant figures.) and with the least chance of human error. The accuracy of the ABHAV method could be significantly increased (by up to ten times) if the input data were entered to the nearest .1 arcmin and interpolation used in the tables. (This however may not be practical at sea where the sextant data are not as accurate as the sextant readouts would appear.)

Data to the nearest arcmin is entered on the work form and no interpolation is required. There are no signs to look up and no decimals except within 5 deg of zero and 180. When a tabulated value approaches zero one decimal place is used. The azimuth angle is always referenced to the north pole and the intercept is always plotted from the DR (dead reckoning) position. There is just one small price to pay for these other advantages and that is a few more table entries, on the work form, than some of the other simplified methods.

The table consists of three entries for each arcmin from zero to 180 deg. The three entries have one to seven digits each and are listed in columns labeled A, B and C respectively. The A column is the log haversine × -100,000, the B column is the natural haversine × 100,000 and the C column is the log secant × 100,000 which is also the log cosine × -100,000. Since all the logs used in these tables are negative it is practical to consider them all as positive to simplify the tables and the procedures. This is possible as long as all of the haversine and cosine values vary between zero and one, as they do. In the case of the secant used in the formula for azimuth angle (Z) it is necessary for the value of s, used on the work form, to be negative and be subtracted from the value above it. This does not cause any problem for the user because he simply follows the sign shown on the work form.

Refer to figure 1.

The haversine formulae for sight reduction are as follows:

hav(z)=[cos(L) × cos(D) × hav(t)] + hav(L~D)

hav(Z)=[hav(p) - hav(L~Hc)] × sec(L) × sec(Hc)


Hc = co-z

D = Declination

Pn = Elevated north pole

ZZ = Your zenith

M = The body

L = Your latitude

co-L = Your co-latitude

p = Polar distance or co-D of body

z = Zenith distance or co-Hc of body

t = Hour angle of body

Z = Azimuth angle of body

~ = The absolute value of the difference between 2 angles


Figure 2 shows a sample work form. The top half of the work form is similar to most other work forms and is self explanitory.

  1. Fill in the data specified to the nearest tenth of a minute of arc and to the nearest second in time.
  2. Make the necessary computations to end up with values for altitude Ho, declination D and hour angle t.

On the bottom half of the form all angular entries should be to the nearest arcmin.

  1. Enter t designating it either E or W as appropriate.
  2. Enter the declination and latitude in the proper spaces. If either is South enter it as a negative number.
  3. Look up the numbers in the table from the appropriate column for each entry as marked on the form.
  4. Add the numbers as indicated to equal an A value on the next line.
  5. Using the tables find a value in the B column which corresponds to the same angle as the A value just computed. When looking up values always use the entry corresponding to the closest input value. It is not necessary to interpolate.
  6. Determine the L~D which is the absolute value of the difference between L and D taking note of the signs. The only time the negative sign, if any, for L and D is used is when obtaining the difference between two numbers as L~D or L~Hc).
  7. For this angle get the value from the B column in the table
  8. Add this B value to the previous B value to get a third B value.
  9. Now using the table in reverse go in with the B value just determined and get the angle. This angle will be the zenith distance z.

    In a similar manner fill in and work out the balance of the sheet.

  10. Subtract z from 90 to get Hc.
  11. Enter Ho and take the difference between it and Hc to get the intercept a.
  12. If Ho is greater than Hc the intercept a will be toward the azimuth direction but if Hc is greater the intercept a will be away from the the azimuth direction.
  13. Go to the right hand side of the form and enter polar distance in the p space. p is 90 - D. Get the B value from the tables.
  14. For L~Hc, Observe the sign get the absolute value of the difference between L and Hc. Get the B value from the tables.
  15. Subtract the second B value from the first B value getting a third B value.
  16. Using the tables convert this to an A value.
  17. The L and C entries can be copied from the same values previously used.
  18. Get a C value from the table for Hc
  19. Add this to the C value just above it and enter as an s (sum) value.
  20. Transfer the s value to the s- space where it will be subtracted from the A above it.
  21. The result will be another A value.
  22. Enter the table with this A value and come out with an angle which will be the azimuth angle Z.
  23. convert Z to Zn. The azimuth angle will always have a prefix of N and a suffix to agree with the suffix for t.
A completed work form is shown in Figure 3. A few excerpts from the table are included which can be used to check some of the lookup numbers in Figure 3.

The ABHAV method can also be used for great circle sailing problems to solve for the great circle distance and the initial course. The corresponding haversine formulae are:

hav(D)=[hav(DLo) x cos(L1) x cos(L2)] + hav(L1~L2)

hav(C)=[hav(coL2) - hav(coL1)] x sec(L1) x csc(D)

Here the use of csc(D) is the same as the sec(co-z) (which is shortened to coz on the work form).
D=great circle distance, C=initial course,

L1=departure latitude, L2=destination latitude,

DLo=the difference between the destination longitude (LO2) and the departure longitude (LO1),

Figure 4 is a sample work form for great circle sailing. The top half is provided to facilitate the necessary computations for entering the bottom half of the form and may be used as needed. This work form is very similar to the sight reduction work form and should be self explanatory.

Figure 5 is a completed sample work form for great circle sailing. Some of the lookups can be checked in the table excerpts.


The ABHAV method of sight reduction for celestial navigation provides the user with a fairly easy and not too complicated method to work up sights on any celestial body listed in the Nautical Almanac. This method also provides an accuracy better than 1 arcmin of altitude and suitable for emergency use on a vessel when their electronic means of navigation have failed or when on board a life boat. ABHAV may also be used by the student pursuing navigation skills. For more information write to John D. Woodworth, 209 Cedar Street, Oakhurst, NJ 07755 or call 1-732-531-1246. A copy of the ABHAV tables and procedures are available for the cost of reproduction and mailing which is $10.00 at this time.


I wish to acknowledge the help of the following people in getting this work completed:
  1. Clark B. Woodworth, Ph.D. E.E., Lucent Technologies, Holmdel, NJ
  2. Michael A. Condouris, Commander of the Shrewsbury Power Squadron
  3. William Kennebeck, Assistant Chairman of Navigation, Shrewsbury Power Squadron


  1. P/V/C Allan E. Bayless, Modified H. O. 211, Ageton's Table, Compact Sight Reduction Table, Published by Cornell Maritime Press for the United States Power Squadrons.
  2. Sight Reduction Procedures, Published annually in The Nautical Almanac by The United States Naval Observatory. (Referred to as the NASR method.)
  3. Defense Mapping Agency Hydrographic Center, PUB. NO. 229, Sight Reduction Tables for Marine Navigation, Six Volumes, Published by the United States Printing Office.