Kepler's Laws: ellipses described around the aim; the law of areas – Gravitation, or weight, the force that maintains the planets in their orbits– The law of universal gravitation confirmed by the planetary perturbations – Circular, elliptic, and parabolic velocity explained; the nature of an orbit depends upon this velocity – Parabolic elements of a cometary orbit

What is the nature of a true cometary orbit? In other terms, what is the geometrical form of curve which a comet describes in space–what is its velocity–how does this velocity vary– and what, in short, are the laws governing the movement of a comet?

In order to reply to these questions, and to enable them to be clearly understood, we must first call to mind a few notions of simple geometry, and also the principal laws which govern the motions of the planets.

Kepler, as we have already said, discovered the form of the planetary orbits, hitherto supposed to be circles more or less eccentric to the sun. This great man demonstrated that the form of a planetary orbit is actually an ellipse, that the sun is at one of the foci of the curve, and that the planet makes its complete revolutions in equal periods of time, but with variable velocity ; in fact, that in equal intervals the elliptic sectors described by the radius vector directed from the sun to the planet are of the same area.