...periodic times of the planets and their mean distances from the sun. From these comparisons he found that the squares of the periodic times are proportional to the cubes of their mean distances, or to the cubes of the semi-major axes of their elliptic orbits. This law prevails...

...second, that a line joining the planet and the sun sweeps over equal areas in equal times; the third, that the squares of the periodic times are proportional to the cubes of the mean distances from the sun. Kepler was conducted to the first and second of these laws by researches on the motion of the planet...

...foci. 2. That the areas included between two radii vectores are as the times of describing the arcs. 3. That the squares of the periodic times are proportional to the cubes of the distances. It must not be supposed that it was mere chance that led to these discoveries. His previous...

...ellipses about a common centre, and the centripetal force varies inversely as the square of the distance ; the squares of the periodic times are proportional to the cubes of the major axes. A particle moves in an ellipse about the centre of force in the focus S : when the particle...

...motion -^t J~—--, from which it is apparent, that for difa* . ferent bodies revolving about the sun, the squares of the periodic times are proportional to the cubes of the mean distances, so far as the masses of the bodies, or rather the inequality of their masses, can be neglected. 8....

...focus. 10. If any number of bodies move in ellipses about a common centre of force, which is the focus, the squares of the periodic times are proportional to the cubes of the major axis. 1 1 . The arc of a cycloid measure from the vertex to any point equals twice the chord...

...planetary orbit, or will it increase or decrease the length of the major axis? Kepler's third law tells us that the squares of the periodic times are proportional to the cubes of the major axes, and from this relation it is manifest that any change in the elliptic velocity of a planet...

...planetary orbit, or will it increase or decrease the length of the major axis? Kepler's third law tells us that the squares of the periodic times are proportional to the cubes of the major axes, and from this relation it is manifest that any change in the elliptic velocity of a planet...

...plane proportional to the time. (2) The orbits are ellipses having the sun in one of their foci. (3) The squares of the periodic times are proportional to the cubes of the mean distances from the Sun. 3. Deductions from Kepler's Laws, From Law (1) — the force which acts upon the bodies tends always...

...planet to the Sun's center are, in the same orbit, proportional to the time of describing them. III. That the squares of the periodic times are proportional to the cubes of the major axes. These laws were discovered by Kepler from observations made on the planet Mars, and stated...