The question of gradient is complicated, again, by another variable which makes the solution of the problem much more intricate than the discovery of minimum effort upon a particular gradient. You have to consider not only the uphill or downhill upon a given slope, but the type of further uphill and downhill to which your road, once established on that slope, is leading you. It is not enough to determine your best formula under such and such conditions of travel for overcoming one side of the obstacle. You have also to ask yourself whether, having got your best uphill road, you may not have led the traveller to an impossible position on the further side. Extreme cases of this one often sees in the Jura range, where the hills are shaped like waves in a storm: a steep escarpment upon the eastern side, very difficult to go up or down, and an easy slope upon the western. Here you have to balance the advantage of your gradient upon the one side with the advantage of the gradient that you will find upon the other, and, of course, to direct your line principally with a view to travel on the more difficult steeper side. That is why you often find yourself following in the Jura a road which goes up the easy western side by an apparently over-steep trajectory: you wonder why the road does not take some obviously easier line which lies below you. The reason you only discover upon reaching the summit and seeing the precipitous escarpment overhanging the eastern valley--your road has made for some exceptional advantage down this cliff, some cleft, which an easier advance from the west would not have hit. A balance has to be struck between the advantage of gradients on both sides of the hill, save in the rare cases where a range (such as the Vosges) is symmetrical and gives you equal gradients upon either slope. That balance is always a matter of careful calculation. Where it has been brought to a fine art is, of course, in surveying for a modern railroad, for there the slightest differences of gradient make such a vast difference in the expense of working that the discovery of a true minimum over an obstacle of hill country is of the first importance. iii