Galileo
made original contributions to the science of motion through an innovative
combination of experiment and mathematics. More typical of science at the time
were the qualitative studies of William Gilbert,
on magnetism and electricity. Galileo's father, Vincenzo Galilei, a lutenist
and music theorist, had performed
experiments establishing perhaps the oldest known non-linear relation in
physics: for a stretched string, the pitch varies as the square root of the
tension. These observations lay within the framework of the Pythagorean tradition of music, well-known to instrument makers, which
included the fact that subdividing a string by a whole number produces a
harmonious scale. Thus, a limited amount of mathematics had long related music
and physical science, and young Galileo could see his own father's observations
expand on that tradition.
Galileo
was one of the first modern thinkers to clearly state that the laws of nature are
mathematical. In The Assayer he wrote
"Philosophy is written in this grand book, the universe ... It is written
in the language of mathematics, and its characters are triangles, circles, and
other geometric figures;...." His mathematical analyses are a further
development of a tradition employed by late scholastic natural philosophers,
which Galileo learned when he studied philosophy. He displayed a peculiar
ability to ignore established authorities, most notably Aristotelianism. In
broader terms, his work marked another step towards the eventual separation of
science from both philosophy
and religion; a major development in
human thought. He was often willing to change his views in accordance with
observation. In order to perform his experiments, Galileo had to set up
standards of length and time, so that measurements made on different days and
in different laboratories could be compared in a reproducible fashion. This
provided a reliable foundation on which to confirm mathematical laws using
inductive reasoning.
Galileo
showed a remarkably modern appreciation for the proper relationship between
mathematics, theoretical physics, and experimental physics. He understood the
parabola, both in terms of conic sections and in terms of the ordinate (y)
varying as the square of the abscissa (x). Galilei further asserted that the
parabola was the theoretically ideal trajectory of a uniformly accelerated
projectile in the absence of friction and other disturbances. He conceded that
there are limits to the validity of this theory, noting on theoretical grounds
that a projectile trajectory of a size comparable to that of the Earth could
not possibly be a parabola, but he nevertheless maintained that for distances
up to the range of the artillery of his day, the deviation of a projectile's
trajectory from a parabola would only be very slight. 
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