Using Kepler's third law $\dfrac{T^2}{r^3} = K$, the centripetal force on a planet in a circular orbit was shown by Newton to vary with distance $r$ as:

1. Centripetal force $F = \dfrac{mv^2}{r}$ with $v = \dfrac{2\pi r}{T}$. 2. Substituting, $F = \dfrac{m(2\pi r/T)^2}{r} = \dfrac{4m\pi^2 r}{T^2}$. 3. Writing it as $F = \dfrac{4m\pi^2}{r^2}\times\dfrac{r^3}{T^2}$ and using $\dfrac{T^2}{r^3}=K$ gives $F = \dfrac{4m\pi^2}{r^2 K}$. 4. Since $\dfrac{4m\pi^2}{K}$ is constant, $F \propto \dfrac{1}{r^2}$. 5. This inverse-square result is why Newton postulated the inverse square law of gravitation. 6. Direct dependence on $r$ or $r^2$ contradicts this derivation and is wrong. _Source: Balbharati (Maharashtra Board) Class 10 Science & Technology, Ch 1 "Gravitation", p.14_