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Summary

A main condition necessary for the advancement of physics and astronomy that
progressed during the Scientific Revolution was the advance of mathematics,
which allowed the proof of abstract theories and provided a more logical method
for attacking the Aristotelian system. During the late sixteenth century, a
French lawyer, Francois Viete, was among the first to use letters to
represent unknown quantities. In 1591 and after, he applied this
algebraic method to geometry, laying the
foundation for the invention of trigonometry. The
Fleming Simon Stevin also worked with geometry during the late sixteenth
century, applying it to the physics of incline planes and the hydrostatic
surface tension of water. Additionally, he introduced the decimal system of
representing fractions, an advance which greatly eased the task of calculation.

However, perhaps the most important mathematical advance of the early period of
the Scientific Revolution was the invention of logarithms in 1594 by John
Napier of Scotland. Napier spent the next 20 years of his life developing his
theory and computing an extensive table of logarithms to aid in calculation. In
1614, he published *Description of the Marvelous Canon of Logarithms,*
which contained the fruits of these labors.

Johannes Kepler also did a great deal of work in geometry, which proved
significant to his subsequent work in astronomy. In 1637, Rene Descartes
published *Geometry,* in which he describes how geometry relates to motion
and showed that at any moment the position of a point can be defined by its
relation to surrounding planes or reference. The most well known application of
this theory is the use of a curve on a graph to represent the motion of an
object, which could then be defined by a mathematical equation that would grant
insight into the forces at work on the object.

Further progress in mathematics was made by Oxford professor John Wallis.
His first work, *Arithmetica Infinitum,* published in 1655, set the stage
for the invention and development of differential calculus. Wallis' became one
of Isaac Newton's major influences. Wallis was the first mathematician to
apply mathematics to the operation of the tides, and also invented the symbol
used to denote infinity.

Many mathematicians applied their knowledge to the study of optics, a field that
had garnered great interest since the Middle Ages. The advances made in this
field, including the development of techniques for higher resolution, led to the
better construction of optical instruments, such as the telescope, which played
a large part in the later work of Galileo.

Mathematics developed as a response to the demands of the sciences, which grew
up in the late sixteenth century. The thinkers of the early Scientific
Revolution had provided their descendents with a broad framework of new
philosophies, hypotheses, and qualitative observations, all of which pointed to
a revolution in thought. However, the old order was at first easily preserved
in the face of this onslaught, in part due to the lack of substance to back up
the theories of such thinkers as Nicolas Copernicus and Giordano Bruno.
Though these scientists sensed that their hypotheses were correct, and strongly
believe in them on their own, it was difficult to bring their theories to the
position of respect they deserved without the benefit of clear and logical
evidence. The realm of mathematics potentially offered this clear and logical
evidence. The scientists of the early Scientific Revolution knew that there
were forces acting on the physical world that would explain the phenomena they
observed, but they had no way to quantify these forces and apply them to the
geometry of the physical world. Mathematicians strove to solve this problem
with the development of trigonometry and the application of new mathematical
theorems to the physics of motion. Armed with these tools, the scientists of
the early Scientific Revolution began to back up their hypotheses with
mathematical proofs were nearly beyond question.