He noted when the earth, Mars, and the Sun were on
the same straight line. Then he deduced the earth's true orbit,
and from this the true orbits of the other planets. Then by trial
and error, he attempted to match this empirical data with regular
mathematically defined shapes, until he discovered in 1609 that
these paths were elliptical. Also, the planets each move faster
when they are nearer the sun and more slowly when they are farther
from the sun so that in equal time intervals, a line from the
planet to the sun will sweep out equal areas. This observation led
him to opine that there is a force between the sun and each
planet, and that this force is the same as that which keeps the
moon in its orbit around the earth. Thirdly, in 1619, he found
that the square of the time for each planet's orbit about the sun
is proportional to the cube of that planet's mean distance from
the sun, so that the farther planets orbit at a slower speed. He
connected the earth's tides with the gravitational pull of the
moon. Kepler also confirmed that the paths of comets were governed
by a law and were farther from the earth from the moon. This
contradicted the church's explanation that what lies within the
moon's orbit pertains to the earth and is essentially transitory
and evil, while what lies beyond belongs to the heavens and is
permanent and pure.
In 1637, Renee Descartes, a philosopher, mathematician, and
scientist from France, invented analytic [Cartesian] geometry, in
which lines and geometric shapes can be described by algebraic
equations and vice-versa.
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