...arcs described in equal times applied to the radii of the circles ; while by a further corollary, if the squares of the periodic times are as the cubes of the radii, the centripetal forces of the bodies will be inversely as the squares of the radii (ie the distances...

...arcs described in equal times applied to the radii of the circles ; while by a further corollary, if the squares of the periodic times are as the cubes of the radii, the centripetal forces of the bodies will be inversely as the squares of the radii (ie the distances...

...swept by lines drawn from the sun to the planet, are proportional to the times employed in the motion.1 Third law. The squares of the periodic times are as the cubes of the mean distances from the sun.2 It had been surmised by several mathematicians that an attractive force...

...Dalton's statement of the law is: — "If several bodies revolve around a common centre of force, and if the squares of the Periodic Times are as the cubes of the distance, then the central attraction decreases as the square of the distance increases." The diagram...

...upon a great truth. He discovered that there is an unvarying ratio between these two factors; namely, the squares of the periodic times, /, are as the cubes of the mean r* . distances, r. In other words -5- is a constant, or r8 = kt2. Kepler published his third law...

...times are as the squares of the radii, the centripetal forces are reciprocally as the radii. 5. If the squares of the periodic times are as the cubes of the radii, the centripetal forces are reciprocally as the squares of the radii. 3. That the centripetal...

...In both works (De motu, corol. 5, theor. 2; Principia, corol. 6, prop. 4), Newton observes that if the squares of the periodic times are as the cubes of the radii, the centripetal forces will be inversely as the squares of the radii. In De motu (schol. to...

...reconcile Kepler's first two laws, of equal elliptical areas in equal times, with his third law, that the squares of the periodic times are as the cubes of the mean distances.8 Bernoulli, with his circular vortices, could accommodate the velocities at different...

...that the centripetal force is inversely proportional to the square of the distance from the center, the squares of the periodic times are as the cubes of the transverse axes.15 (De Gandt, 41) The direction of this last theorem is opposite to that of Theorema...

...the periodic times in which those orbits are completed; and establishes the remarkable relation, that the squares of the periodic times are, as the cubes of the major axes, the halves of which are the mean distances, as a dynamical consequence from a law of force inversely...