butions didadvance the field of calculus
by showing that the area of a segment of
a parabola is 4/3 the area of a triangle
with the same base and vertex (endpoint),
and 2/3 the area of the circumscribed
parallelogram.
To figure this out, he constructed an
“infinite” sequence of triangles (or
wedges), finding the area of segments
composing the parabola. He began with
the first area, A,then added more trian-
gles between the existing ones and the
parabola to get areas of:
A, AA/4, AA/4 A/16, AA/4
A/16 A/64, and so on.
Based on his iterations, he deter-
mined the following (the first time any-
one had determined the summation of an
infinite series; for more about infinite
series, see below):
A(1 1/4 1/4^2 1/4^3 ...) A(4/3)
Archimedes also applied this method of exhaustion(not literally becoming tired,
but close to it) to approximate the area of a circle, which, in turn, led to a better
approximation of pi (π). Using such integrations, he also determined the volume and
surface area of a sphere and cone, the surface area of an ellipse, and many others. His
work is considered the first steps toward integration that would eventually lead to
integral calculus. (For more information about Archimedes and his wedges, see
“Geometry and Trigonometry.”)
What did Isaac Newtoncontribute to mathematical analysis?
English mathematician and natural philosopher (otherwise called a physicist) Isaac
Newton (1642–1727) was one of the greatest scientists who ever lived. Overall, he con-
tributed to physics (such as the discovery of his three famous laws of motion); fluid
dynamics (fluid motion); the union of terrestrial and celestial mechanics using the
principle of gravitation—thus explaining Kepler’s laws of planetary motion; and he
even explained the principle of universal gravitation.
By 1665 Newton had not only begun his work on differential calculus, but he also
had published one of his greatest scientific works—Philosophiae naturalis principia
mathematica(The Mathematical Principles of Natural Philosophy), often shortened to 211
MATHEMATICAL ANALYSIS
English mathematician Isaac Barrow’s work on tan-
gents laid important groundwork for Isaac Newton’s
later work on calculus. Library of Congress.