newtonian gravity - Why $1/r^2$ and not another power of $r$ in Newton's law of gravitation?

My book introduces the force of gravitation as a non-contact force between two bodies of mass $M_1$ and $M_2$ separated by a distance $r$ . Then it says it is directly proportional to the product of masses $M_1 M_2$ and inversely proportional to $r^2$. Then writes the the force of gravitation as $$ F = G\dfrac{M_1M_2}{r^2}. $$ But why does it take square of $r$ and not another power? What is the cause of taking $r^2$? Why not another power of $r$?

Answer

A lot of things decrease in intensity as $1/r^2$, such as light intensity, gravity, charge forces, etc. This is because the same force needs to act over a larger spherical area. The further away, the larger the sphere. And you should know that the surface area of a sphere is $SA=4\pi r^2$. Since the area varies as $r^2$, dividing the magnitude of the intensity by the area means it drops as $1/r^2$.