Let us take a horizontal engine of 2 feet stroke, making 200 revolutions per minute, so having a piston travel or average velocity of 800 feet per minute, which was my engine in the Paris Exposition of 1867. We will suppose the piston to be driven through the crank, by which its motion is controlled, the power being got from some other motor, and that the cylinder heads have been removed so that the piston meets no resistance. We will also disregard the effect of the angular vibration of the connecting-rod, and assume the motion of the piston to be the same at each end of the cylinder. On each stroke the crank does two things: First, it increases the motion of the piston from a state of rest to a velocity equal to the uniform velocity of the crank-pin in its circular path: and, second, it brings the piston to rest again, ready to have the same operation repeated in the reverse direction during the return stroke. At the mid-stroke the crank is at right angles with the line of centers, and the velocity of the piston is 800 × ¹⁄₂π = 1256.64 feet per minute, or 20.944 feet per second, and no pressure is being exerted on the piston either to accelerate or retard its motion. [Illustration] The pressure of the crank during a stroke, first to impart motion to the piston and second to arrest this motion, is represented by two opposite and equal triangles. Let the line _AB_, in the above figure, be the center line of a cylinder and its length represent the length of the stroke. Let the line _AC_, normal to the line _AB_, represent the force required to start the piston from a state of rest. Then the triangle _AOC_ will represent the accelerating force that must be exerted on the piston at every point in the half stroke to bring up its velocity, until at _O_ this equals that of the crank-pin in its circle of revolution, and the accelerating force, diminishing uniformly, has ceased. The opposite equal triangle _BOD_ shows the resistance of the crank required to bring the piston to rest again. How do we know this? I will answer this question by the graphical method, the only one I know, and which I think will be understood by readers generally. First, we observe that the distance the piston must move from the commencement to any point in the first half of its stroke, in order that it shall keep up with the crank, is the versed sine of the angle which the crank then forms with the line of centers. So the table of versed sines tells us where the piston is when the crank is at any point in its revolution, from 0 to 90°. [Illustration]