 We now turn our attention to the forces that are causing the universe to expand. We'll start with Newton's gravitational equations. Back in the 1700s, Newton proved a theorem now named after him. It has two parts. The first is that an isotropic, spherically symmetric body affects external objects gravitationally, as though all of its mass were concentrated at its center. For part two of the theorem, we'll drill 30 kilometers into the planet. From here, we are on the Earth's mantle and under the Earth's crust. Counting the oceans, around 1.05% of the Earth's mass is now further from the center than our 1 kg object. We see that, although the object is closer to the center, the overall force is just a little bit less due to the smaller mass. But what about all the matter in the Earth's crust that is now further away from the center than the object? The force on the object from the shell of matter will come from the sum of all the forces produced by all the molecules in the shell. And what would the total be if the object were elsewhere inside the sphere? Newton had to actually develop calculus to solve this problem. The remarkable result is that the sum total of all these forces is equal to zero. The shell mass has no impact on the object whatsoever. So now, we have the second part of Newton's shell theorem. It states that a homogeneous, spherically symmetric shell exerts no gravitational force on objects within the shell. This is a key idea for our study of cosmology, and it is not readily understood just why we would get a zero result anywhere inside the shell. A quick look at the geometry involved helps. If we put our object at the center and build a cone that intersects with the shell in two opposite directions, we can analyze the force on the object as it moves around inside the shell. At the center, all the forces cancel out. Now move the object towards one side and away from the other. On the right side, the number of molecules exerting a force is increasing by the square of the distance. At the same time, the force from each molecule is decreasing by the square of the distance. The reverse is happening on the left side. The forces continue to cancel each other out. The total force remains equal to zero.