He used calculus and differential
equations to determine centripetal forces of elliptical orbits,
where the distance from the sun, the velocity, and the
acceleration were variables. He correlated the moon's orbit with
the measured acceleration of gravity on the surface of the earth.
Then he formulated the idea that the ultimate agent of nature was
a force acting between bodies rather than a moving body itself.
Gravity did not act in proportion to the surfaces of bodies, but
in proportion to quantity of matter, its penetration to the very
center of all bodies without diminution, its propagation to
immense distances decreasing in exact proportion to the square of
the distance. Newton showed that a single gravitational force
could account for the way falling objects descend to the ground,
the parabolic trajectory of projectiles, the motion of the moon in
its orbit around the earth, the course of the tides every twelve
hours, the lower densities of the earth's atmosphere at greater
heights, the paths of Jupiter's satellites, and the ellipitical
motions of the planets in their orbits around the sun. It had been
thought that invisible angels moved the planets. He proved from
his law of gravitation and his three laws of motion the truth of
Kepler's laws of ellipitical planetary motion. He demonstrated
from data collected from the comet of 1680 that comets moved
according to his law of gravitation.
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