Beer and Maedler’s map has a diameter of 37 inches: it represents the phase of the moon visible in the condition of mean libration. The details were charted by a careful process of triangulation. The disc was first divided into “triangles of the first order,” the points of which (conspicuous craters) were accurately laid down by reference to the edges of the disc: one hundred and seventy-six of these triangles, plotted accurately upon an orthographic projection of the hemisphere, formed the reliable basis for their charting work. From these a great number of “points of the second order” were laid down, by measuring their distance and angle of position with regard to points first established. The skeleton map thus obtained was filled up by drawings made at the telescope: the diameters of the measureable craters being determined by the micrometer.
Beer and Maedler also measured the heights of one thousand and ninety-five lunar mountains and crater-summits: the resulting measures are given in a table contained in the comprehensive text-book which accompanies their map. These heights are found by one of two methods, either by measuring the length of the shadow which the object casts under a known elevation of the sun above its horizon, or by measuring the distance between the illuminated point of the mountain and the “terminator” in the following manner. In the annexed figure (Fig. 15) let the circle represent the moon and M a mountain upon it: let S A be the line of direction of the sun’s rays, passing the normal surface of the moon at A and just tipping the mountain top. A will be the terminator, and there will be darkness between it and the star-like mountain summit M. The distance between A and M is measured: the distance A B is known, for it is the moon’s radius. And since the line S M is a tangent to the circle the angle B A M is a right angle. We know the length of its two sides AB, AM, and we can therefore by the known properties of the right-angled triangle find the length of the hypothenuse BM: and since BM is made up of the radius BA plus the mountain height, we have only to subtract the moon’s radius from the ascertained whole length of the hypothenuse and we have the height of the mountain. MM. Beer and Maedler exhibited their measures in French toises: in the heights we shall have occasion to quote, these have been turned into English feet, upon the assumption that the toise is equal to 6·39 English feet. The nomenclature of lunar features adopted by Beer and Maedler is that introduced by Riccioli: mountains and features hitherto undistinguished were named by them after ancient and modern philosophers, in continuance of Riccioli’s system, and occasionally after terrestrial features. Some minute objects in the neighbourhood of large and named ones were included under the name of the large one and distinguished by Greek or Roman letters.