The History Of Calculus

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A pod1−

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 al-

Haytham (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.