An algebraic equation with two unknowns,
or variables, could be represented as a shape on a coordinate
system in which each point is represented by a pair of numbers
representing distances from the two axis lines. He started the
convention of representing unknown quantities by x, y, and z and
known quantities by a, b, and c. So, for instance, a circle with
center at point 2,3 and a radius of 4 was represented by the
equation: (x-2) squared + (y-3) squared = 4. All conic sections,
e.g. ellipses, parabolas, and hyperbolas, could be represented by
equations. Analytic geometry aided in making good lenses for
eyeglasses. The glass was first manufactured with attention to
quality. Then, after it cooled and solidified, the clearest pieces
were picked and their surfaces ground into the proper curvature.
Descartes pioneered the standard exponential notation for cubes
and higher powers of numbers. He formulated the sine-law of
refraction, which determines in general the way a light ray is
deflected, according to the density of the media through which it
passes. This explained why a rainbow is circular. In 1644, he
described the universe in terms of matter and motion and suggested
that there were universal laws and an evolutionary explanation for
such. He opined that all effects in nature could be explained by
spatial extension and motion laws that 1) each part of matter
retains the shape, size, motion, or rest unless collision with
another part occurs; 2) one part of matter can only gain as much
motion through collision as is lost by the part colliding with it;
and 3) motion tends to be rectilinear.
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