HISTORY OF CALCULUS

What is the

History of Calculus?

Calculus, known in its early history

as infinitesimal calculus, is a mathematical discipline

focused on limits, functions, derivatives, 

integrals, and infinite series. 

Isaac Newton and Gottfried Leibniz

 independently invented calculus in the mid-17th

century. 

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.

• 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.

• 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.

Vs.

NEWTON 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.• 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.

• 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”.

THE END