Of the many factors affecting the collision problem, the mathematics of relative motion is the one most susceptible to formal analysis. Moreover, this ‘collision geometry’ must form the basis of any investigation into the instrumental, operational and human factors involved. It is therefore surprising that no complete description of the (mathematically) simple relationships is readily available. The object of this paper is to present such a connected account of the mathematics of relative motion, in so far as it appears relevant to the collision problem and within the limitations of time and space available.

The question is being asked more and more, why, despite the use of radar, collisions at sea in fog still occur. There are evidently many answers to this question and all or nearly all of them hinge on some aspect or other of human fallibility. Yet these answers may be grouped into two distinct categories, one of which covers failure to get the most out of the radar and ancillary equipment, and the other the problems of behaviour with which the radar information brings the mariner face to face in fog for the first time.

In view of the immensity of the sea, the relative smallness of ships and the range of speeds, the probability of a collision at sea between ships blindly following constant and random courses is very low. Even on high-density routes, the probability is low for intersecting courses if speeds and headings remain constant. The risk is higher for routes which are at angles of less than 20° from each other, and where a ship is overtaking another rather than following an opposite course. It is lower if the speeds are very different, and higher if they are similar.

As a result of the war, radar instruments developed in one particular way, namely as relative-motion instruments, which is only one of a number of possibilities. The relative-motion display is above all a position-finding aid; it places own ship in the centre of the screen in exactly the same way as the human eye has its position in the centre of the circle formed by the horizon.

The value of plotting radar observations at sea has often been emphasized and some knowledge of the errors to be expected will be of use to the navigator. Small random errorsare always present in radar observations. Under certain conditions, particularly when ships are approaching end-on, even a correct interpretation of the observations can lead to a misinterpretation of the situation. The Steering Rules are operative only when vessels are in sight of one another, since if it cannot be assumed that there is a mutual understanding of the situation there can be no mutual rules of conduct.

With the comparative absence of navigation aids in the earlier days of flying, up to about ten years ago, the navigator was constantly engaged in associating observations, calculations and chart work in what was essentially a high-class guessing game. The low speeds (just over 100 knots on flying boats and just under 200 knots on landplanes) provided enough time for this rather intriguing exercise to be carried out in a fairly relaxed atmosphere, conducive to an individualistic approach that could be justified by results. The navigator's performance was largely (at that stage of aviation) judged on the neatness of his plotting, the accuracy of E.T.A.'s and the absence of late course corrections before arriving at destination or a check point. If there is anything in the view that navigation is an art, this particular yardstick certainly induced considerable artifice in the navigator.

In his paper ‘Two Mariner's Astrolabes’ (Journal, 9, 338) Dr. D. J. Price stated that hitherto only seven mariner's astrolabes had been found and the details published. He then described two more mariner's astrolabes, the Dundee astrolabe of 1555, discovered some months previously, and the Kronborg astrolabe of 1600, discovered in 1928.

Some of the readers of the Institute's Journal may be amused at the following experiences in navigation, which I had fifteen years ago. In February 1942 I was crossing the Atlantic from Halifax N.S. to Liverpool in a convoy, and the passengers were of course never told anything about our position from day to day. I had no instruments, and not even a table of logarithms or of sines and cosines, but I did have much leisure and a good supply of blank paper, and I occupied myself by trying to find out where we were. On two occasions I told my estimates to the Chief Officer of the ship, who was not allowed to make any answer at the time, but on each occasion four or five days later he told me how far I had been from the true position.

Commander waters's review of Professor Morison's recent book on Columbus (Journal 10, 216) should be read in the light of the following facts, published in 1932 by the City of Genoa. They were supported by facsimiles of the documents from which they derive. The Great Discoverer was born in 1451, the son of a wool-weaver of Genoa. The family moved for a time to Savona, and were living there in 1472 when Christopher, in signing a deed, described himself as ‘a wool-worker of Genoa’. He went to Lisbon not before 1476, and while there visited Madeira to buy sugar for two Genoese merchants. He made a brief business visit to Genoa in 1479, when he said he must soon return to Lisbon. In all this there is no hint of a person who has knocked about the sea from the age often. There is no evidence that on the voyage to Guinea Columbus travelled as a sailor, while as regards the voyage to Iceland, he said it was made in February 1477, that the south of the island lay in lat. 73°N., not 63°N., and that he sailed 100 leagues (400 miles) beyond it and found that the sea was not frozen. The authenticity of this voyage may be left to the judgment of the reader.

In U.S. Naval Institute Proceedings for June 1957 (page 680), Captain P. V. H. Weems and Captain R. E. Jasperson outline a method for finding a position by observing one celestial body only. The principle of the method is to use the observed rate of change of altitude, together with the observed altitude itself, to solve directly for latitude and hour angle, and thus to obtain the position. The authors point out that the parallactic angle, C (the angle at the observed body), in the usual navigational spherical triangle is obtainable from:

sin C = sec (declination) × rate of change of altitude with hour angle.

The statistical analysis in the Institute Working Party's report on the accuracy of astronomical observations at sea (this Journal, 10, 223) is a fine piece of work for which we should all be most grateful to Mr. Sadler and the staff of H.M. Nautical Almanac Office. Indeed it is perhaps not out of place to recall the traditions by which the Royal Observatory supported by the Government of the day has always helped navigation, and to express gratitude to the Astronomer Royal for permitting and encouraging the work.