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

Explanation

Kepler's first law :
The orbit of a planet is an ellipse with the Sun at one of the foci.

Figure 1.5 shows the elliptical orbit of a planet revolving around the Sun (S).
Kepler's second law :
The line joining the planet and the Sun sweeps equal areas in equal intervals of time.
`A rarr B, C rarr D and E rarr F` are the displacements of the planet in equal intrvals of time.
The straight lines AS, CS and ES sweep equal areas in equal areas in equal intervals of time.
Area ASB = area CDS = area ESF.
Kepler's third law :
The square of hte period of revolution of a planet around the Sun is directly proportional to the cube of the mean distance of the planet from the Sun.
Thun, if r is the average distance of the planet from the Sun and T is its period of revoltuion, then,
`T^(2) prop r^(3)," i.e., "(T^(2))/(r^(3))="constant = K"`
For simplicity we shell assume the orbit to be a circle. In Fig. 1.6, S denotes the position to the Sun, P denotes the position the position of a planet at a given instant and r denotes the radius of the orbit (`-=` the distance of the planet from the Sun). Here the speed of the planet is uniform. It is

`v=("circumference of the circle")/("period of revoltuion of the planet")`
`=(2pir)/(T).`
If m is the mass of the planet, the centripetal force exerted on the planet by the Sun (`-=` gravitational force), `F=(mv^(2))/(r)`
`therefore F=(m(2pir//T)^(2))/(r)=(4pi^(2)mr^(3))/(T^(2)r)=(4pi^(2)mr)/(T^(2))`
According to Kepler's third law,
`T^(2)=Kr^(3)`
`therefore F=(4pi^(2)mr)/(Kr^(3))=(4pi^(2)m)/(K)((1)/(r^(2)))`
Thus, `F prop (1)/(r^(2))` as `(4pi^(2)m)/(K)` is constant in a particular case.