The position of a point on the surface will then be expressed by two spherical coordinates: namely, ist, the distance of the point from the primitive circle measured on a secondary; 2nd, the distance intercepted on the primitive circle between this secondary and some given point of the primitive circle assumed as the origin of coordinates.—William Chauvenet, Manual of Spherical and Practical Astronomy (1896).

On 16 May 1870, exactly eighty years before this paper was written, Lord Kelvin, then Sir William Thomson, worked out an epoch-making example of how to find the hour angle and azimuth of a heavenly body by inspection, in order to facilitate the use of Captain Thomas Sumner's method at sea. His work was published one year later in the Proceedings of the Royal Society, and in it he describes a page of his new Tables for Facilitating Sumner's Method at Sea. These tables, comprising nine pages, were made public on 11 November 1875 and were published in London in May of the following year; from them have been derived all modern navigation tables based on right-angled spherical triangles. Kelvin then used, for the first time, Greenwich hour angle in arc and assumed latitudes and longitudes. (The writer has himself used G.H.A. in arc since 1902 and assumed positions since 1908.)