If ships only engaged when they were stationary the range would not change, and it could be found by observation without rangefinders. And even with rangefinders it can never be found at great distances _without_ observation. But ships do not stand still, and when they move the distance between them alters from second to second. If these movements could be (1) ascertained, (2) integrated, and (3) the results impressed upon the sight, change of range would be eliminated, and we should have come back to the conditions in which ships were stationary. Fire control is successful in so far as it succeeds in doing these three things. Sketches 3 and 4 show the process by which hits are secured, when the conditions are not complicated by changes in the range, that is, if these complications have been eliminated by fire control. The second two illustrate what these complications are. The ships turn away from each other and then turn towards each other. The rate graph (6) shows the effect of these movements on the range and the rate at which it is changing from moment to moment. The process shown in Sketches 3 and 4 is called “bracketing.” Two shots are fired at a difference of, say, 800 yards. Observation shows the first to be too short, the second to be too far. The difference is bisected by the third shot. This places the target in one of the halves of the bracket. This half is bisected by the fourth shot, placing the target in a quarter. If an eighth of the bracket is less than the danger space, then the fifth shot must hit. [Illustration: Range-finding by bracket] In Sketch 5 the ships keep parallel courses for two minutes. The range does not change. The line in the graph (6) is, for these two minutes, horizontal. It is as if both were stationary. When the ships turn the range increases and the graph rises. But the graph is not a straight line but a curve. This shows that the rate also is changing. Each movement of the two ships, whether they keep steady courses or turn, alters the range and the rate. As projectiles take an interval of time to travel from the gun to the target, the range must be _forecasted_. B, then, cannot engage A unless he knows where A is going to be. He cannot know this until A has settled on a steady course. While A is turning, then he is safe from gunfire except by a chance shot. B cannot engage while he is himself turning unless he can integrate his own movements with A’s. It is this latter difficulty which largely explains the duration of modern actions. At the mean range of each engagement, with ships standing still, _Sydney_ could have sunk _Emden_ in ten minutes; _Inflexible_ and _Invincible_ could have sunk _Scharnhorst_ and _Gneisenau_ in fifteen. But it was ninety minutes before _Emden_ was driven on the rocks, 180 before _Scharnhorst_ sank, and 300 before _Gneisenau_ went under. [Illustration: The crux of sea fighting, changes of course and speed produce an irregularly changing range]