As a demonstration of how physics can lead to new mathematics, we can now prove the Pythagorean theorem using physics. Consider a container shaped like a right triangle with side lengths a,b,c and filled with water (Fig. 64 ). It seems intuitively clear that the container will not go anywhere; it must be in equilibrium. We will now explore what this statement of being in equilibrium means mathematically in terms of forces and torques. The force of the water on a particular side of the container is pressure (which is constant if the height of the container is small) times the area of that side, which is thus proportional to height times the side length. For the sake of simplicity (though without the loss of generality), we can normalize the height to be 1 .
