A velocity of twenty-six miles per hour appears to be about the highest yet attained by a steamer.[2] This is probably near the limit beyond which the speed cannot be increased to any useful purpose. The resistance offered by water to a vessel moving through it increases more rapidly than the velocity. Thus, if a vessel were made to move through the water by being pulled with a rope, there would be a certain strain upon the rope when the vessel was dragged, say, at the rate of five miles an hour. If we desired the vessel to move at double the speed, the strain on the rope must be increased four-fold. To increase the velocity to fifteen miles per hour, we should have to pull the vessel with nine times the original force. This is expressed by saying that the resistance varies as the square of the velocity. Hence, to double the speed, the impelling force must be quadrupled, and as that force is exerted through twice the distance in the same time, an engine would be required of eight times the power—or, in other words, the power of the engine must be increased in proportion to the _cube_ of the velocity; so that to propel a boat at the rate of 15 miles an hour would require engines twenty-seven times more powerful than those which would suffice to propel it at the rate of five miles an hour. Footnote 2: This has now (1895) been far surpassed.—_Vide infra._ The actual speed attained by steam-ships with engines of a given power and a given section amidships will depend greatly upon the shape of the vessel. When the bow is sharp, the water displaced is more gradually and slowly moved aside, and therefore does not offer nearly so much resistance as in the opposite case; but the greater part of the power required to urge the vessel forward is employed in overcoming a resistance which in some degree resembles friction between the bottom of the vessel and the water. The wonderful progress which has, in a comparatively short time, taken place in the power and size of steam-vessels, cannot be better brought home to the reader than by a glance at Fig. 64, which gives the profiles of four steamships, drawn on one and the same scale, thus showing the relative lengths and depths of those vessels, each of which was the largest ship afloat at the date which is marked below it, and the whole period includes only the brief space of twenty years!—for this, surely, is a brief space in the history of such an art as navigation. All these ships have been named in the course of this article, but in the following table a few particulars concerning each are brought together for the sake of comparing the figures: