Write the three laws given by Kepler. How did they help Newton to derive the inverse square law of gravity?

Explanation

Kepler's law:
i. Kepler's first law: The orbit of a planet is an ellipse with the sun at one of the foci.
ii. Kepler's second law: The line joining the planet and the Sun sweeps equal areas in equal intervals of time.
iii. Kepler's thrid law: The square of its period of revolution around the Sun is directly proportional to the cube of the mean distance of a planet from the Sun.
Centripetal froce `F=(mv^(2))/r`............i
Distance travelled by planet in one revolution `=` Circumference of orbit
`=2pir`
Time period of revolution `=T`
`:.` Speed of planet `v=("distance")/("time")`
`=(2pir)/T` ...........ii
Substituting eq ii and i
`:.F=(m((2pir)/T)^(2))/r=(m4pi^(2)r^(2))/(rT^(2))`
`:.F=(4mpi^(2)r)/(T^(2))`
Multiplying numerator and denominator with `r^(2)`
`:.F=(4mpi^(2)r)/(T^(2))xx(r^(2))/(r^(2))`
`:.F=(4mpi^(2)r^(3))/(r^(2)T^(2))` ...........iii
According to Kepler's 3rd law
`(T^(2))/(r^(3))=k` or `(r^(3))/(T^(2))=1/k`.............iv
Substituting iv in iii
`F=(4mpi^(2)r^(3))/(r^(2)T^(2))xx1/k=(4mpi^(2))/(r^(2)k)`............v
Rearranging eq v
`F=(4mpi^(2))/kxx1/(r^(2))`
Since `(4mpi^(2))/k=` constant
`:.F=` constant `xx1/(r^(2))`
`:.F alpha1/(r^(2))`