Celestial  Navigation Calcs

ABC Tables

ABC tables are very easy to use and more than adequate for the bearing of a celestial body.  These tables avoid the need to use a calculator or Log tables but are based on the previous formulae.

The Rules may seem unwieldy at first but they are printed on each page and quickly become automatic.

These transpose the Azimuth formula so that

 A = Tan(Lat) / Tan(LHA)

 B = Tan(Dec) / Sin(LHA)

 C = Difference A ~ B = 1/ [Tan(Azimuth)  x Cos(Lat) ]



Latitude           20 N

Declination       45 S

LHA                30


A               0.63 S                    Opposite to Latitude unless LHA > 180

B              2.00 S                      Same as Declination


C             2.63 S                      Same name; Sum. Different names; Difference

The C Table gives a bearing of 22.0. The sign of C means that this bearing is south. It is west because the LHA is less than 180.

The C result would normally be written as "S 22.0 W" or 158.

The effect of rounding ABC Tables’ values is negligible (+/- 0.1)  This is not true of the older Sight Reduction Tables where the calculated altitude is rounded to the nearest minute. Furthermore the need to use a plotting sheet with a rounded, estimated position provides considerable scope for inaccuracy. (Sight Reduction Tables were known as the Air Navigation Tables until 2003.)

The author’s preferred manual method is a calculator for the Zenith Distance and ABC tables for Azimuths. Without a calculator he would still use the Cosine formula but with log tables.

Obtaining a Position Line

The difference between the True (TZD) and Calculated (CZD) Zenith Distances is the Intercept.


If the TZD is less than the CZD then the assumed position must be moved in the direction of the body by the amount of the Intercept. This gives a position of the correct distance from the body. It is known as the Intercept Terminal Position or ITP.

As the radius of the circle is normally very large, it is considered to be a straight line near this point. A line at 90 to the direction of the body is the Position Line.


Combining Position Lines

A single Position Line must be combined with other observations for a fix. This can be achieved using a plotting sheet and then transferring the ITP by the distance to the next sight and redrawing the Transferred Position Line in the same direction as the original.

For Sun sights, it is more usual to calculate the ITP of a morning sight and then calculate the transferred position for the Sun's Meridian Passage (Noon.) The difference between calculated and observed latitudes provides a longitude using “Plane Sailing.” With a little practice, this will be found to be a faster, not to mention more accurate method.

For Star Sights, many people use a single position and then plot the Position Lines without allowing for the vessel's movement. This may appear a sloppy practice but a few miles error mid-ocean is usually irrelevant. Even if the position at sunset was perfect, there is no guarantee that the position an hour later is within a mile. Indeed even if the position agrees perfectly with a GPS position, there is no guarantee that an intervening military operation has not thrown the GPS position out let alone a fault in the equipment/ aerial. “I am about here,” is a far safer assumption than “My wheelhouse is/ was within 10m of this position.”

Next Section

Corrections to a Sextant Altitude



Celestial Navigation Calculations

An imaginary sphere surrounding the Earth is used for calculations. This is known as the Celestial Sphere. Declination corresponds to Latitude and Hour Angles to Longitudes.

Solving a sight uses spherical trigonometry. The triangle is known as the PZX triangle.


P is the Pole, Z, the Observer’s Zenith and X is the body.


Altitude Vs Zenith Distance

An Altitude is a terrestrial measurement while calculations are carried out on the Celestial sphere.

The reason that the calculations do not allow for the shape of the Earth is because the calculations are performed on the Celestial Sphere. Gravity ensures that the horizon (and thus Altitude,) corresponds with the equivalents on the celestial Sphere.

Zenith Distance is the correct term and helps avoid confusion. 

Parts of the PZX Triangle

Q1 to Q2 is the Equator and the Vertical line from P to G is the Greenwich Meridian (0 GHA and Longitude.)

The compliment of an angle is 90 - the angle. This is also true of the sides in a spherical triangle.

The distance from P to Q1 is 90 therefore PX = 90 - Declination or

PX = co-Declination.

Compliments enable formulae to be simplified because the Sine of an angle equals the Co-Sine of the compliment of that angle. This also applies to Tangents and Cotangents, Secants and Cosecants.

Sin(60) = Co-Sine(90 - 60) = Cosine(30)

The compliment of a compliment is the same as the original angle;

Cos(co-30) = Sin(90 - (90 - 30)) = Sin(30)

Simplifying the diagram and adding some labels:-



To Calculate a Side - The Cosine Formula

Cos(a) = Cos(b) x Cos(c) + Sin(b) x Sin(c) x Cos(A)

Applying this to the PZX triangle we get:-

Cos(Zenith Distance) = Cos(co-Lat) x Cos(co-Dec) + Sin(co-Lat) x Sin(Co-Dec)

         x Cos(LHA)


Because Sin(co-A) = Cos(A) and Cos(co-A) = Sin(A)

Cos(Zenith Distance) = Sin(Lat) x Sin(Dec) + Cos(Lat) x Cos(Dec) x Cos(LHA)

If Altitude is preferred; Zenith Distance = co-Altitude thus

Sin(Altitude) = Sin(Lat) x Sin(Dec) + Cos(Lat) x Cos(Dec) x Cos(LHA)


For an Angle

Tan(C) = Sin(A)/ [Sin(b)/ Tan(c) – Cos(b) x Cos(A)]

Inserting terms from the PZX triangle this becomes

Tan(Az) = Sin(LHA)/ [Sin(co-Lat)/ Tan(co-Dec) – Cos(co-Lat) x Cos(LHA)]


Tan(Az) = Sin(LHA)/ (Cos(Lat) x Tan(Dec) – Sin(Lat) x Cos(LHA)) 


The Spherical Sine Formulae

Sin(a)/ Sin(A) = Sin(b)/ Sin(B) = Sin(c)/ Sin(C)


Napier’s Rules

These can be used when one of the sides or angles is 90.

If angle A is 90 then a diagram is drawn with A above the circle and the sectors filled with the adjoining sides.

Notice that the three sectors of the lower half are marked “co-” In other words the compliment of these angles is used.

The two formulae are;

Sin(Mid Part) = Tan (Adjacent Parts)

e.g. Sin(c) = Tan(co-B) x Tan(b)

or Sin(c) =Cot(B) x Tan(b)


Sin (Mid Part) = Cos(Opposite Parts)

e.g. Sin(c) = Cos(co-a) x Cos(co-C)

or Sin(c) = Sin(a) x Sin(C)


Example using Napier’s  Rules

Assume that the True Altitude is 0 therefore the Zenith Distance is 90.

Co-Dec is the mid-part therefore the two opposites are Azimuth and co-Lat.

Sin (Mid Part) = Cos(Opposites)

Sin(co-Dec) = Cos(Azimuth) x Cos(co-Lat)


Cos(Dec) = Cos(Azimuth) x Sin(Lat)

Cos(Azimuth) = Sin(Dec)/ Sin(Lat)


The Amplitude of a body is measured from East or West rather than North. In other words Amplitude = 90 - Azimuth = co-Azimuth.

Sin(Amplitude) = Sin(Dec)/ Sin(Lat)

This gives the formula that many readers will be familiar with of

Sin(Amplitude) = Sin(Dec) x Sec(Lat)