This text if for the few persons who understand trigonometry and wishes to do celestial navigation.

## Introduction

Celestial navigation uses the position of the planets, the moon or the stars to determine the position of a boat. In principle, the technique is useful for ocean navigation, but with the advent of the GPS, it has become less of common use. Its mastery however remains mandatory for lifting distance restrictions on a professional license. The idea is to be able to position a vessel even when the electronic systems fail.

The most widely used technique for converting a sextant reading into a position is the Marcq Saint-Hilaire method. In the simplest terms, the method converts a sextant reading into a line of position. With more than one reading, one can thus obtain a fix.

## Sight Reduction Tables

Obtaining a line of position requires calculations based on the celestial body’s position at the time of the sextant reading. The position of the body is used to find a circle of position (image below). The position of celestial bodies at any moment is recorded in the nautical almanach.

Spherical trigonometry is used to convert a position circle on a sphere into a position line on a map (image above). This conversion involves linearizing an arc of a circle, approximated to its tangent. Position circles are so large that the linearization error is unimportant for practical calculations.

In practice, the results of spherical trigonometry calculations are recorded in a book called Sight Reduction Tables. These tables provide a theoretical star height (noted H_c) and the star’s direction (noted Z_n) on a nautical chart.

Most navigation courses use conversion tables for two reasons. Firstly, it requires only addition and subtraction. Consequently, it’s more “accessible” to those who have not mastered trigonometry. Moreover, it requires no electronic equipment, meaning that if all the instruments on board are broken, it is still possible to establish a fix.

That said, the “paper” approach requires you to bring the reduction tables. This is a 500-page document, filled with columns of numbers (extract below). Not exactly a page turner.

These tables require intermediate calculations to arrive at whole degrees. This obligation stems from the volume: a table including a precision to tenths of a degree would make the tables tens of thousands of pages long. And because these books are filled with tables of numbers, it is pretty easy to make mistakes in selecting rows or columns, or to reproduce the proper entry with errors. In short, the “paper” method is time-consuming, error-prone and involves a lot of contortion to avoid trigonometry.

## Two Formulas

If you know trigonometry and are willing to accept a calculator on-board (!), it is possible to let go of the 500 pages of sight reduction tables. Positions can furthermore be calculated as precisely as you like, without being restricted to whole degrees. You can also avoid the errors inherent in reading tables. In short, the entire set of tables can be summed up in these two formulas:

\begin{align} H_c&=\sin^{-1}\left[\sin(D)\sin(\lambda^a)+\cos(\lambda^a)\cos(D)\cos(LHA)\right],\\ Z &=\cos^{-1}\left[\frac{ \sin(D) - \sin(\lambda^a)\sin(H_c)}{\cos(\lambda^a)\cos(H_c)}\right], \end{align}

where D is the declination of the celestial body, \lambda^a is the assumed latitude and LHA is the Local Hour Angle. These formulas are obtained from spherical triangles (and accept arguments in degrees).

These two formulas are not exactly a state secret. It is possible to derive them by hand… or they can be found on pages 7 and 8 of the nautical almanach (image below; asin and acos designate the inverse of the sine and cosine functions).

These functions take up two lines in a memory aid, accept arguments with decimals, and if you are a bit used to your calculator, probably reduce the risk of making a mistake. In a learning context, you can also think of these expressions as a way of checking your answers. And if you think about it, you will realize that the sight reduction tables are actually *generated* by these formulas.

## Conclusion

Most navigation courses for sailors use the sight reduction tables and assume that students have little mathematical knowledge. The two expressions above will probably not be covered. However, if you have a *modicum* of understanding of trigonometry, they will save you time, either by allowing you to check your calculations, or by saving you from digging through the tables. All of this to say that mathematics will give you more time to look at the stars.

## References

Wikipedia (s.d.). *Intercept Method,* document retrieved online in May 2024 from this adress.

TheNauticalAlmanach.com (2024-a). *Nautical Almanach 2024*, document retrieved online in May 2024 from this adress.

________________________ (2024-b). *Pub. No. 249 Epoc 2025*, document retrieved online in May 2024 from this address.