Calculus was developed independently and nearly simultaneously by Isaac Newton and Gottfried Leibniz in the late 1600s. They both developed foundational concepts like derivatives and integrals to solve problems in physics and geometry. Earlier mathematicians like Archimedes, Ibn al-Haytham, and Cavalieri developed precursor ideas and methods for calculating volumes, areas, and rates of change. Newton and Leibniz formalized these ideas into a system of calculus, though there was controversy over who originated its concepts. Today it is recognized that they independently arrived at the same conclusions around the same time.

3. Calculus,
mathematical
limits functions derivatives integrals
infinite series Isaac Newton Gottfried
Leibniz

4. • 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).

5. • 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.

6. • 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

7. • 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.

8. • 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.

9. • 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.

10. • 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.

12. • 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.

13. • 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”.