By curves are meant the changes of direction of the road to the right or to the left. The direction of a railroad cannot be changed suddenly by an angle, but must be effected gradually by a curve. Supposing the curve to be (as it generally is) the arc of a circle, the radius of the curve is the distance of the centre of the circle from the curve. This radius is an important element in the estimate of the road. X. The more nearly a railroad approaches to an absolute level, and perfect straightness, the more profitably will it be worked. XI. The total amount of mechanical power necessary to transfer a given load from one extremity of a railroad to another is a matter of easy and exact calculation, when the gradients and curves are known; and the merits of different lines may be compared together in this respect: but it is not the only test of their efficiency which must be applied. XII. A railroad having gradients exceeding seventeen feet in a mile will require more mechanical power to work it than it would were it level; and the more of these excessive gradients there are upon it, and the more steep they are, the greater will be this disadvantage. XIII. Although a railroad having no gradients exceeding seventeen feet in a mile does not require more mechanical power than a level, yet the mechanical power which it requires will not be so advantageously expended, and, therefore, it will not be so economical. XIV. A railroad which has gradients above thirty feet in a mile will require such gradients to be worked by assistant locomotive engines, which will be attended with a waste of power, and an increase of expenditure, more or less, according to the number and length of such gradients. XV. A very long inclined plane cannot be worked by an assistant locomotive without a wasteful expense. Gradients exceeding seventeen feet per mile must, therefore, be short. XVI. Gradients exceeding fifty feet in a mile cannot be profitably worked except by stationary engines and ropes, an expedient attended with so many objections as to be scarcely compatible with a large intercourse of passengers. XVII. Steep gradients, provided they descend from the extremities of a line, are admissible provided they be short. It is evident that in this case the inclined planes will help at starting to put the trains in motion, at the time when, in general, there would be the greatest strain upon the moving power; and, in approaching the terminus, the momentum would be sufficient to carry the train to the top of the plane, if its length were not great, since it must, at all events, come to a stop at the extremity. XVIII. The effect of gradients in increasing the resistance during the ascent may be estimated by considering that a gradient of seventeen feet in a mile doubles the resistance of the level, thirty-four feet in a mile triples it, and eight and a half feet in a mile adds one half its amount, and so on. XIX.