John Napier, a large Calvinist landholder in Scotland who had
built his own castle, did mathematics in his older years. He
explored imaginary numbers such as the square roots of negative
numbers. By 1614, he had started and developed the theory of
logarithms: the relationships among positive and negative
exponents of numbers. This simplified calculations because the
multiplication and division of numbers would be equivalent to
addition and subtraction of their exponents. His table of
logarithms, which took him twenty years to compile, was used in
trigonometry, navigation, and astronomy. It reduced the enormous
labor involved in trigonometric calculations.
Johannes Kepler was a mathematician from Germany who made his
living as an astrologer. He was in contact with Galileo by letter,
as most scientists of Europe were with each other. Kepler was
fascinated with perfect geometric shapes, which he tried to relate
to celestial phenomenon. He discerned that the orbit of Mars was
not perfectly circular. He knew that the apparent path of the sun
with respect to the constellation of fixed stars differed in speed
at different times of the year. He opined that this showed that
the speed of the earth revolving around the sun varied according
to the time of year. Then he measured the angles between the earth
and the sun and the earth and Mars as they changed through the
Martian year.
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