A word must be said about classical estimates of the height of mountains.[148] Aristotle suggested that these altitudes might be determined by observing the duration of sunlight on the peaks. He would have us believe that the Caucasus range is illumined by the sun for a third of the night after sunset and for a corresponding time before sunrise. If this were true, these mountains would be from 60 to 180 miles high![149] Less fantastic were the estimates of Dicaearchus and Eratosthenes. The former, Pliny tells us, measured Pelion and found it to be 1250 paces (10 stades) in height.[150] If we are right in our understanding of the length of the pace here employed, this represents 5167 feet[151]—certainly not far short of the actual altitude (5308 feet). We do not know the method used by Dicaearchus in this survey, but his calculation was probably determined from simple triangulation with the aid of a diopter, an instrument for measuring angles.[152] Triangulation as a means of finding the height of trees and buildings was well understood. Eratosthenes probably did not carry out a triangulation of his own but adopted the results obtained by Dicaearchus, asserting that the highest mountains in the world do not exceed 10 stades in elevation. He demonstrated by an ingenious and graphic mathematical proof that the volume of mountains is so utterly insignificant in comparison with the volume of the earth as a whole that the earth can be regarded as essentially a sphere,[153] a conception which became well established in the astronomical thought of antiquity and one which reappeared in the Middle Ages.[154] When the Greeks learned something of the Alps, they were able to correct Eratosthenes’ underestimate of the maximum height of mountains. Posidonius argued that 15 instead of 10 stades should be taken as the correct figure and that the maximum depth of the sea was no greater than 15 stades.[155] _MATHEMATICAL GEOGRAPHY AND CARTOGRAPHY_ Mathematical geography deals in part with the accurate determination of the location of places and with the accurate representation of the earth’s surface on maps. MATHEMATICAL GEOGRAPHY LARGELY BASED ON ITINERARIES