This paper describes George Biddell Airy's almost completely unknown method of approximating an orthodromic arc (great circle arc) using a circular arc in the normal aspect Mercator projection of a sphere. In addition, it is demonstrated that the centre of the circle can be defined in at least two different ways, which yields slightly different results. Airy's approach is built upon in this paper. The method of computing coordinates of Airy's circle arc centre is described. The formulae derived in the paper can be used to calculate the length of Airy's approximation of the orthodromic arc connecting two points on the sphere and on the Mercator chart. Moreover, the actual length of the orthodromic arc on the sphere and on the Mercator chart can be computed using the formulae derived in this paper. The purpose of the paper is not to suggest an application of Airy's method in navigation, but to analyse Airy's proposal and to show that a great circle arc on a Mercator chart is close to a circular arc for distances which are not too great. This property can be useful in education, having in mind that the stereographic projection is the only one that maps any circle on a sphere onto a circle in the projection plane.

An assessment of the navigational accuracy of the Mercator world map of 1569 is made, aimed at better understanding how the information was adapted from the contemporary cartography. At the time the map was engraved, navigational charts were constructed on the basis of astronomically-observed latitudes, magnetic courses and estimated distances between places. Before this information could be incorporated into the new world map it should be first transformed in such a way that the longitudinal spacings between places were restored to their correct values, as defined on the surface of the Earth. The question of whether Mercator performed such transformations or just considered that the positions were approximately correct has hitherto never been addressed in the literature. It is demonstrated in this article that Mercator was not fully aware of the complexity of the contemporary charts – which he considered to implicitly comprise a square grid of meridians and parallels – and that all planimetric information was directly imported to the novel world map without correction. It is further shown that the Mercator projection was not compatible with the navigational methods of the time.