What is the maximum degree of the curvature which should be allowed on any road? Unquestionably there is no definite limit. If any limitation is made it depends on the general character of the country and on the amount of traffic which may be immediately expected. As an extreme case in the justifiable use of sharp curvature, we may consider a portion of the line from Denver to Leadville, Colo. The traffic that was expected on the line was so meager, and the general character of the country was so forbidding, that a road built according to the usual standards would have cost very much more than would have been justified by the expected traffic. The lines as adopted cost about $20,000 per mile, and yet in a stretch of 11.2 miles there are about 127 curves. One is a 25° 20' curve, 24 are 24° curves, 25 are 20° curves, and 72 are sharper than 10°. If 10° had been made the limit (a rather high limit according to usual ideas), it is probable that the line would have been found impracticable (except with prohibitive grades), unless four or five times as much per mile had been spent on it, and this would have ruined the project financially.

As an illustration of the other extreme, we may consider some of the improvements which have been recently made on the P. R.R. between Philadelphia and Pittsburg. Millions of money have been spent in the effort to reduce distance, curvature, and grade. The reduction of curvature has been very largely in the form of eliminating degrees of central angle, but it also has taken the form of increasing the radius of curvature, so that the running of express-trains at a speed of 60 miles per hour will be facilitated. This is one of the comparatively rare cases where an increase in the radius of curvature justifies a considerable expenditure.

Another illustration of the use of sharp curvature by a line of heavy traffic is given by the case of the B. & 0. R.R. in its line at Harper's Ferry. For many years the traffic of this road passed over two curves, one with a radius of 300 feet (19° 10') and then over a 400-foot curve (14° 22'). During recent years this sharp curvature has been materially reduced by means of some very expensive tunneling, but the fact that the engineers of this line very wisely concluded to run the traffic of a great trunk line over such sharp curves shows how foolish it is for an engineer to sacrifice money or sacrifice gradients in order to reduce the rate of curvature on a road which at its best will be a line of very small traffic. Many locating engineers have started out to locate a line with instructions that their maximum rate of curvature must not exceed 6°. Possibly it would be better to say that no limitation should be imposed. It is far better to operate a road on a 10° or 15° curve in some one place, provided that the cost of avoiding such a curve will be very large. This is especially true for the light-traffic roads, which constitute such a large proportion of our mileage, and which will probably constitute the great bulk of the roads yet to be constructed. Of course such belittling of the effects of curvature may be, and sometimes is, carried to an extreme and cause an engineer to fail to give to curvature its due consideration. Degrees of central angle should always be reduced by all the ingenuity of the engineer, and should only be limited by the general relation between the financial and topographical conditions of the problem. Easy curvature is in general better than sharp curvature and should be adopted when it may be done at a small financial sacrifice, especially since it reduces the length of the line. Nevertheless the engineer should not give undue prominence to curvature in comparison with the other features of aline-m3nt, which are really of far greater importance. He should remember that so far as the cost of track-work is concerned, there is little if any saving in this respect, and that the extra cost of operating trains on curves is very nearly independent of the radius of curvature. Of course the trains cannot be operated at high speed on sharp curves, but if the road is a minor road such a condition will have almost no effect on the operation of trains. Above all, the engineer should not waste the capital of the road (which is usually limited) in an effort to avoid curvature, when any spare funds may more profitably be expended in making reductions of grade which will be of vastly greater financial value.