The Third Law and the Power of Machines

Abstract

At this point in the Principia, Newton has defined his terms, he has stated his three axioms, and he has demonstrated their utility in solving problems involving multiple forces and bodies. Now, in the Scholium that follows, Newton provides a more general discourse, or meditation, on the power of his axioms. First, he briefly demonstrates how Galileo’s analysis of projectile motion relates to his own first and second laws. Next, he offers an explanation of how the motion of colliding bodies is affected by the elasticity of the bodies themselves. For this analysis, Newton expends considerable effort explaining how to account for the resistance of the air, a complication which can obscure otherwise intelligible results. He then proceeds to examine the relationship between his first and third laws of motion, using examples from magnetism and gravity. Interestingly, this raises the question as to whether they are, in fact, independent axioms. Are they? Finally, he shows how his third law suggests a general means for computing the mechanical advantage provided by machines of various sorts, such as screws, pulleys and wedges.

The power and use of machines consists only in this, that by diminishing the velocity we may augment the force.

—Isaac Newton