The recent death of Philip Van Horn Weems, at the advanced age of 90, has removed from the field of navigation one of its most outstanding and colourful personalities. His contributions, made over a period of more than half a century, have ranged over almost every aspect of navigation: practising navigator, instrument inventor, pioneer of the GHA form of Nautical Almanac and of the first Air Almanac, introducer of the Star Altitude Curves, brilliant expositor (as his books so clearly demonstrate), successful business man who trained many thousand navigators through his training schools and correspondence courses – he devoted his whole life to navigation with boundless energy and contagious enthusiasm.This is not the place to give the factual details of his life, or to put on record his many achievements, both inside and outside his professional career. Many of his most lasting and significant contributions to the practice of navigation, although so familiar to myself, can only be appreciated by a detailed study relative to the contemporary ‘state of the art’; Weems was, above all else, an innovator of ideas many of which were developed by others and which deserve a fuller and more technical treatment.A brief summary of his career follows. Weems, left an orphan at an early age, had a hard life on the family farm in Tennessee, with little formal education.

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i. INTRODUCTION. 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. N The mathematics does nothing more than express the consequences of certain specific data in a form suitable for examination, thus simplifying the formulation of guiding rules or of particular courses of action. Complications arise solely through the large number of variables involved and the many combinations of circumstances for which analysis is required.The analysis is restricted to the relative motion of two ships initially moving with constant speeds on near-collision courses, as affected by changes of course by either ship, or by each simultaneously. One or two illustrations are given of the kind of deductions that might be made from this analysis. It must, however, be emphasized that the deductions are derived solely from a consideration of the geometry of the problem; they take no account of the many practical factors involved and completely ignore the Collision Regulations.But it must equally be stressed that a knowledge of the purely academic geometrical consequences, as here given, is an essential factor in the complete study.2. RELATIVE MOTION. The study is here restricted to the relative motion of two ships moving with constant velocities on near-collision courses. The notation, and sense of measurement of the angles, are as shown in Fig. i. For this purpose only relative bearings and courses are needed, and they are all referred to the course of the first ship. The second ship is always regarded as being to starboard, but it is clear that all necessary data can be derived for all cases. • : O, P are the instantaneous positions of the two ships A, B, moving at speeds, u, v on courses inclined at an angle C. The range and bearing of P from O are r and 0. The motion of B relative to A is along PD, bearing i8o° + S, with speed w. The inclination of the relative course PD to PO is S, which is necessarily a small angle if a collision is likely. D is the point of closest approach; the distance of closest approach OD is d, and 306'

According to Flamsteed's own account, the Royal Observatory was founded as a direct consequence of the representation in 1674 by a Frenchman, who described himself as Le Sieur de St. Pierre, to the English Court that he had discovered a method of determining longitude at sea ‘from easy celestial observations’, and claimed the appropriate award. The solution to the ‘longitude problem’ was then, as it had been for many years and was to be for a further 90 years, of paramount importance; several countries had offered large prizes for a practical solution, and l'Observatoire de Paris had been founded in 1667, a year after l'Academie Royale. (Observations were systematically made for the purpose of improving the determination of longitude by observation of the satellites of Jupiter; but this method, successful on land, was doomed to failure because of the impracticability of using long-focus telescopes at sea.)

The standard formulae, assuming a spherical Earth, for converting distance on a rhumb-line track into differences of latitude and longitude are:where C is the course and α is known as the middle latitude. It is approximately the mean latitude when both points are on the same side of the equator; tables of the correction to be applied to the mean latitude to give the middle latitude are given in most collections of nautical tables.