The overcoming of the resistance of the air by a moving body is work. A steamboat will move for some time in water after the steam has been turned off. The overcoming of the resistance of the water is work, and by virtue of the motion of the boat when the steam was turned off it possessed the energy to do the work of forcing itself for some time through the resistance of the water. The Perpetual Motion worker in each case had reasoned himself into this conclusion: That the same energy will impart the same _acceleration_ of velocity, regardless of the velocity at the beginning of the application of energy. That the same amount of energy or work necessary to impart to a body a velocity of ten feet per second will increase that velocity to twenty feet per second, or from twenty feet per second to thirty feet per second. In other words, that the same amount of energy, and only the same amount of energy is required for a given _increase_ in velocity without regard to the initial velocity. This appears plausible, and almost self-evident. We believe the great majority of people, other than mechanical engineers would, upon presentation of the theory accept it as axiomatic, and as a matter of course. The fallacy becomes manifest only from a critical and technical examination of the Laws of Momentum and Energy. The Perpetual Motion worker had learned from his text-books that if the velocity be _doubled_, the energy would be _multiplied by four_. His idea was to so arrange his mechanism that he would apply the amount of energy to move a fly wheel free to revolve, from a position of rest to a revolving velocity of ten revolutions per second. Then apply again the _same amount of energy_, and accelerate that velocity from ten revolutions per second to twenty revolutions per second. Thus, the energy at the end of the second second would be four times what it was at the end of the first second. But to make it so, only double the amount of energy had been applied that had been expended at the end of the first second. Thus, he reasoned, his machine was by virtue of its structure, accumulating energy, and this energy could be used one-half to continue the motion of his machine, and the other half to run other machinery, or for any other purpose for which energy might be desired. Wherein lies the fallacy of this supposition? We will now endeavor to explain. And for the young student to get the explanation fully, it will be necessary for him to pay the closest attention to what we here state.