Transcript The History Of Calculus - Lakeland Central School District

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The History Of Calculus
What is Calculus?
• From Latin, calculus, a small stone used for counting
• A branch of mathematics including limits, derivatives,
integrals, and infinite sums
• Used in science, economics, and engineering
• Builds on algebra, geometry, and trig with two major
branches differential calculus and integral calculus
Ancient History
•
In the earliest years, integral calculus was being used as an
idea, but was not yet formalized into a system.
• Calculating volumes and areas can be traced to the
Egyptian Moscow papyrus (1820 BC).
Ancient Greeks
• Greek mathematician Eudoxus (408-355 BC) used the
method of exhaustion, a precursor to limits, to calculate
area and volume
• Archimedes (287-212 BC) continued Eudoxus’ idea and
invented heuristics, similar to integration, to calculate area.
Medieval History
• In about 1000 AD, Islamic mathematician, Ibn alHaytham (Alhacen) derived a formula for the sum of
the fourth powers of an arithmetic progression, later
used to perform integration.
• In the 12th century, Indian mathematician Bhaskara
II developed an early derivative. He described an
early form of what will later be “Rolle’s Theorem”
• Also in the 12th century, Persian mathematician
Saraf al-Din al-Tusi discovered the derivative of a
cubic polynomial
Modern History
• Bonaventure Cavalieri argued that volumes be
computed by the sums of the volumes of cross
sections. (This was similar to Archimedes’s).
• However, Cavalieri’s work was not well respected,
so his infinitesimal quantities were not accepted at
first.
Modern History
• Formal study combined Cavalieri’s infinitesimal
quantities with finite differences in Europe. This
was done by John Wallis, Isaac Barrow, and James
Gregory
• Barrow and Gregory would later prove the 2nd
Fundamental Theorem of Calculus in 1675.
Enter Newton…
• Isaac Newton (English) is credited with many of the
beginnings of calculus. He introduced product rule, chain
rule and higher derivatives to solve physics problems.
• He replaced the calculus of infinitesimals with geometric
representations.
• He used calculus to explain many physics problems in his
book Principia Mathematica, however he had developed
many other calculus explanations that he did not formally
publish.
…and Leibniz
• Gottfried Wilhelm Leibniz (German) systemized the ideas of
calculus of infinitesimals. Unlike Newton, Leibniz provided
a clear set of rules to manipulate infinitesimals.
• Leibniz spent time determining appropriate symbols and paid
more attention to formality.
• His work leads to formulas for product and chain rule as well
as rules for derivatives and integrals.
Newton vs. Leibniz
• There was much controversy over who (and thus
which country) should be credited with calculus
since both worked at the same time.
• Newton derived his results first, but Leibniz
published first.
Newton vs. Leibniz
• Newton claimed Leibniz stole ideas from
unpublished notes written to the Royal Society.
• This divided English-speaking math and continental
math for many years.
Newton vs. Leibniz
• Today it is known that Newton began his work with
derivatives and Leibniz began with integrals. Both
arrived at the same conclusions independently.
• The name of the study was given by Leibniz,
Newton called it “the science of fluxions”.
Since then…
• There have been many contributions to build upon Newton
and Leibniz.
• Calculus was put on a more rigorous footing by
mathematicians such as Cauchy, Riemann, and Weierstrass
• Calculus has also been generalized for the Euclidean and
complex space.
In conclusion…
• We will stand on the shoulders of those that came
before us and study their findings to possibly apply
to our modern world!