This is a paradox which people living in easy hill-lands have difficulty in appreciating. The Alps especially show roads which puzzle us (who are of a gentler landscape) when we follow them: yet the principle is simple and dependent upon the geological formation of most _new_ mountain ranges, which present a hard core, forming their central ridge. The softer ground wears away on either side of the valley: the ridge remains. The effect is that a _direct_ approach to the notch in the range would give impossible gradients in the last few hundred yards, and therefore the road must gradually curve round by a side of the valley. A third exceptional case is that of trajectories where the minimum of effort is only to be found by going _right over the very summit of the highest hill in your neighbourhood_. Lastly, there is the curious case of a pass where it is to the advantage of the road to _avoid_ the lowest passage of the range and to take a line to one side above it. As examples of these last two paradoxical points I may quote the pass of Sallent in the Pyrenees and the exceedingly important road from the valley of the Moselle to the valley of Belfort in the Vosges. In the case of the pass of Sallent there was an obvious notch in the range, which was used from the very earliest times till just the other day. It was through this that the armies of the Moors poured in the eighth century for their attempted conquest of Europe, when they invaded France and nearly reached the Loire. So late as within living memory it was the regular track from the valley of the Gallego to that of Gabas. Now, the modern road, after careful survey, has been constructed to cross the mountain summit somewhat to the west, and a good three hundred feet higher than the old pass. Why was this? It was because the notch of Sallent had a very steep approach in the last few hundred yards upon either side, and the minimum of effort, at any rate for wheeled vehicles of the modern type, was found in taking a lesser gradient to one side, although it involved a much higher climb. The case of the road from the Moselle valley to Belfort in the Vosges is even more remarkable, for one would have said at first asking that no such case could exist: one would have said that a minimum of effort could never be reached by going over the _very highest summit_ in your neighbourhood, but it is so when you deal with what I will call a “star” mountain, as will be seen at once from the following elements. [Illustration: Sketch IV]