These direct distances may of course differ widely with the distance which it is necessary to travel between two places along a road, down a winding river or a sinuous coast-line. Thus, the direct distance, as the crow flies, between Brig and the hospice of the Simplon amounts to 4.42 geogr. m. (slope nearly 9°), while the distance by road measures 13.85 geogr. m. (slope nearly 3°). Distances such as these can be measured only on a topographical map of a fairly large scale, for on general maps many of the details needed for that purpose can no longer be represented. Space runners for facilitating these measurements, variously known as chartometers, curvimeters, opisometers, &c., have been devised in great variety. Nearly all these instruments register the revolution of a small wheel of known circumference, which is run along the line to be measured. _The Measurement of Areas_ is easily effected if the map at our disposal is drawn on an equal area projection. In that case we need simply cover the map with a network of squares--the area of each of which has been determined with reference to the scale of the map--count the squares, and estimate the contents of those only partially enclosed within the boundary, and the result will give the area desired. Instead of drawing these squares upon the map itself, they may be engraved or etched upon glass, or drawn upon transparent celluloid or tracing-paper. Still more expeditious is the use of a planimeter, such as Captain Prytz's "Hatchet Planimeter," which yields fairly accurate results, or G. Coradi's "Polar Planimeter," one of the most trustworthy instruments of the kind.[4] When dealing with maps not drawn on an equal area projection we substitute quadrilaterals bounded by meridians and parallels, the areas for which are given in the "Smithsonian Geographical Tables" (1894), in Professor H. Wagner's tables in the geographical _Jahrbuch_, or similar works. It is obvious that the area of a group of mountains projected on a horizontal plane, such as is presented by a map, must differ widely from the area of the superficies or physical surface of those mountains exposed to the air. Thus, a slope of 45° having a surface of 100 sq. m. projected upon a horizontal plane only measures 59 sq. m., whilst 100 sq. m. of the snowclad Sentis in Appenzell are reduced to 10 sq. m. A hypsographical map affords the readiest solution of this question. Given the area A of the plane between the two horizontal contours, the height h of the upper above the lower contour, the length of the upper contour l, and the area of the face presented by the edge of the upper stratum t·h = A1, the slope [alpha] is found to be tan [alpha] = h·l / (A - A1); hence its superficies, A = A2 sec [alpha]. The result is an approximation, for inequalities of the ground bounded by the two contours have not been considered.