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Since programmable electronic calculators were first employed in marine navigation, a variety of formulae has been used for calculating astronomical refraction. As the choice is wide, some formulae have been selected from commonly used reference sources and their accuracy and suitability examined. No attempt has been made to assess the validity of the selected formulae to represent astronomical refraction in practical circumstances. Accuracy comparisons have been made using the refraction algorithm proposed by Garfinkel 1 -the standard adopted by the British and American Nautical Almanac Offices. New formulae are given that are simple and accurate, even over a wide range of temperature and pressure, and which for all practical purposes may be considered equivalent to the tables of refraction given in the Nautical Almanac.i. I N T R O D U C T I O N A number of attempts, with varying degrees of success, have been made to devise a single, simple formula to represent astronomical refraction accurately. The problem is difficult when one considers the complexity of the algorithm of Garfinkel. 1 If observations are confined to altitudes above i j°, where refraction does not exceed about one-tenth of that at o° altitude, a good approximation to the unsigned mean refraction (ft M ) in minutes of arc is given by:where h is the observed altitude in degrees at a sea-level air temperature of i o° C and pressure of i o i o mb and K is a constant close to unity. Not only is this refraction formula simple, but also variations between the computed and actual refraction are expected to be small. As altitude decreases, the size of the refraction correction and its uncertainty increase -the uncertainty arising when atmospheric anomalies are present.Occasionally observations need to be made at low altitudes, and for these situations either the range of validity of the above formula can be extended by suitable modifications or a separate formula provided. In these circumstances it is not uncommon to find significant differences between the values of refraction corrections calculated using Garfinkel's algorithm and formulae that purport to represent the generally accepted standard of Garfinkel. One must accept the fact that atmospheric anomalies may be present at the time of observation, but the estimated value of the refraction correction should not be further degraded by the use of a poor formula. NAV 35

About ten years ago this author wrote the software for a suite of navigation programmes which was resident in a small hand-held computer. In the course of this work it became apparent that the standard text books of navigation were perpetuating a flawed method of calculating rhumb lines on the Earth considered as an oblate spheroid. On further investigation it became apparent that these incorrect methods were being used in programming a number of calculator/computers and satellite navigation receivers. Although the discrepancies were not large, it was disquieting to compare the results of the same rhumb line calculations from a number of such devices and find variations of some miles when the output was given, and therefore purported to be accurate, to a tenth of a mile in distance and/or a tenth of a minute of arc in position. The problem has been highlighted in the past and the references at the end of this show that a number of methods have been proposed for the amelioration of this problem. This paper summarizes formulae that the author recommends should be used for accurate solutions. Most of these may be found in standard geodetic text books, such as, but also provided are new formulae and schemes of solution which are suitable for use with computers or tables. The latter also take into account situations when a near-indeterminate solution may arise. Some examples are provided in an appendix which demonstrate the methods. The data for these problems do not refer to actual terrestrial situations but have been selected for illustrative purposes only. Practising ships' navigators will find the methods described in detail in this paper to be directly applicable to their work and also they should find ready acceptance because they are similar to current practice. In none of the references cited at the end of this paper has the practical task of calculating, using either a computer or tabular techniques, been addressed.

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