Kepler's laws

As is well known, Johannes Kepler (1571-1630) was the first astronomer to correctly describe the motion of the planets (in works published between 1609 and 1619). This motion is summed up in three simple laws:

1. The planetary orbits are all ellipses that are confocal with the Sun (i.e., the Sun lies at one of the focii of each ellipse; see Section A.9).
2. The radius vector connecting each planet to the Sun sweeps out equal areas in equal time intervals.
3. The square of the orbital period of each planet is proportional to the cube of its orbital major radii.

Let us now see if we can derive Kepler's laws from Equation (4.2).