This paper offers a precise and almost final solution to one of the most debated problems connected with the astrogeodetic ‘fix’ by azimuth differences. The rectangular and linear solution on the gnomonic plane, tangent to the celestial globe and the dead reckoning position, enables the use of azimuth differences and is connected to Pothenot's solution. Thus this problem may be solved with simple linear algorithms that require neither differential modes, nor the introduction of the unknown origin error.

For centuries, celestial navigation was the main method of determining a ship's position at sea, especially when sailing into more exotic areas. Later on, vast areas of the oceans were only covered by the Omega system and there was still every opportunity of finding a position by using the sextant. Nowadays, with GPS coverage worldwide, we consider celestial navigation as an interesting back-up.

This paper concerns the estimate of a ship's position from a sequence of measurements of 3, 4 or 5 altitudes of stars or planets. The measurement errors are assumed to be mutually correlated. This correlation is mainly caused by the appearance of so-called systematic errors. It is the main intention of this paper to dispense with the policy of pre-separation of these systematic errors. Instead, the equal contribution of some partial errors to the total measuring errors is fully accounted for by the formation of the covariance matrix, which corresponds with the vector of measurement errors. The algorithm produces a position estimate with a bias-free estimation error, meaning that the estimate and the error are stochastically independent. The resulting covariance matrix of the position error has a minimal trace.

A well-known navigational method is to determine position through observation of the highest altitude of the Sun at noon. This procedure is based tacitly on the assumption that the observed altitude is a symmetrical function of time about the highest point which appears thus in the meridian. In order to examine the accuracy of this approximation for a fast moving ship, let us consider the situation in more detail. The equations will be formulated using the standard functions in a modern pocket calculator and its application modules.