45. _Kepler's Second Law._--Kepler next discovered that a planet's rate of motion in the various parts of its orbit is such that _a line drawn from the planet to the sun would always sweep over equal areas in equal times_. Thus, in Fig. 57, suppose the planet would move from _P_ to _P^1_ in the same time that it would move from _P^2_ to _P^3_, or from _P^4_ to _P^5_; then the dark spaces, which would be swept over by a line joining the sun and the planet, in these equal times, would all be equal. A line drawn from the sun to a planet is called the _radius vector_ of the planet. The radius vector of a planet is shortest when the planet is nearest the sun, or at _perihelion_, and longest when the planet is farthest from the sun, or at _aphelion_: hence, in order to have the areas equal, it follows that a planet must move fastest when at perihelion, and slowest at aphelion. _Kepler's Second Law_ of planetary motion is usually stated as follows: _The radius vector of a planet describes equal areas in equal times in every part of the planet's orbit_. 46. _Kepler's Third Law._--Kepler finally discovered that the periodic times of the planets bear the following relation to the distances of the planets from the sun: _The squares of the periodic times of the planets are to each other as the cubes of their mean distances from the sun_. This is known as _Kepler's Third Law_ of planetary motion. By _periodic time_ is meant the time it takes a planet to revolve around the sun. These three laws of Kepler's are the foundation of modern physical astronomy. The Newtonian System. 47. _Newton's Discovery._--Newton followed Kepler, and by means of his three laws of planetary motion made his own immortal discovery of the _law of gravitation_. This law is as follows: _Every portion of matter in the universe attracts every other portion with a force varying directly as the product of the masses acted upon, and inversely as the square of the distances between them._ 48. _The Conic Sections._--The _conic sections_ are the figures formed by the various plane sections of a right cone. There are four classes of figures formed by these sections, according to the angle which the plane of the section makes with the axis of the cone. _OPQ_, Fig. 58, is a right cone, and _ON_ is its axis. Any section, _AB_, of this cone, whose plane is perpendicular to the axis of the cone, is a _circle_. [Illustration: Fig. 58.] Any section, _CD_, of this cone, whose plane is oblique to the axis, but forms with it an angle greater than _NOP_, is an _ellipse_. The less the angle which the plane of the section makes with the axis, the more elongated is the ellipse.