IT is well known that, while the shortest course between two points on a sphere is a great circle, the course which is easiest for navigation is a rhumb line, i.e. a course which cuts all the successive meridians at the same angle, so that the ship always steers in a direction making an angle of the same number of degrees with due north or due south. Such a course is represented by a straight line on a Mercator’s chart, which thus makes it appear more direct than it really is. On a great circle, though the course is the shortest, you are (except on the equator or a meridian) continually altering your direction as measured by the angle which it makes with the north. Between any two points there is thus a difference in favour of great-circle sailing; but to demonstrate rigorously the greatest value which this difference can possibly take is a somewhat difficult and attractive mathematical problem. I have heard it stated that it is not susceptible of rigorous mathematical treatment. The following is suggested as giving a complete solution of the question: it would be interesting to learn whether any shorter or more direct method is available, which makes no unwarranted assumption. For the abstract treatment of the problem (though indeed in the Pacific or the Southern Ocean fairly extended courses are practically available) we suppose the globe cleared of all such inconvenient encumbrances as continents or islands: moreover, we treat it as a perfect sphere; and to word the question more exactly we should say: Find two points such that the difference between the shortest great-circle course and the shortest rhumb-line course joining them is a maximum; for the great circle through two points will (in general) consist of a major and a minor arc, and on the other hand a rhumb line might start the longer way round the earth, or indeed make any number of revolutions about the pole before reaching its objective. In either case it is the shortest course of the kind with which we deal.

By rational mechanics is meant the development of the subject of Mechanics according to a strict logical scheme, proceeding, like Geometry, from certain hypotheses or axioms established (or indicated) by experiment, from which particular theorems or applications are deduced. Every branch of theoretical physics aims at such a complete rational treatment, but whereas in Geometry the axioms have long ago been classified and isolated, and the bulk of the science consists of the deductions from them, in Physics we are still largely occupied with the discovery of the laws and axioms.

We take two bodies P Q which have straight world lines OP KQ. This means in classical language that P and Q have a constant relative velocity. Our object is to find the configuration which an observer U reckons that of nearest approach of P and Q, and to find U’s measure of the least distance.