Similar societies were formed
all over the world. Theologicians warned that scientific research
was dangerous. But it's advances improved agriculture,
manufactures, medicine, surgery, navigation, naval architecture,
gunnery, and engineering.
Issac Newton was a genius, who in his childhood designed and built
model windmills, water wheels, water clocks, and sundials. He came
from a family which had risen from the yeomen ranks to the gentry.
For a few years after graduating from Cambridge University in
1665, he secluded himself in the countryside to study. Here, using
the work of Wallis, he formulated the binomial theorem that
expands (A+B) raised to the nth power, where n is an integer,
fraction, or negative number. When n was a negative number, the
expansion never terminated; instead of a finite sum, there is an
infinite series. He then developed the notion of a number being
the limit of an infinite converging series of partial sums, such
as the limit of 1+(1/2)+(1/4)+(1/8)...= 2. By considering the
state of motion of a mass-point in an infinitely short time under
the influence of an external force, he developed rules for finding
areas under algebraic curves [integration], such as the hyperbola,
and finding tangents to algebraic curves [differentiation], which
he recognized as inverse processes. That is, taking the integral
and then the differential of a function results in a return to
that function.
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