This document provides a brief history of calculus, beginning with its origins in ancient civilizations using methods like infinitesimals to calculate areas and volumes. It discusses early contributors like Archimedes and Islamic mathematicians developing early concepts of integration and derivatives. Newton and Leibniz are credited with developing calculus more fully in the late 1600s through their work with limits, derivatives, integrals, and infinite sums. Their independent work led to controversy over credit until it was established they both made important contributions. Since then, mathematicians have worked to formalize calculus and further develop its concepts and applications.

1. Attention

2. The History Of CalculusThe History Of Calculus

3. What is Calculus?
• From Latin, calculus, a small stone used for counting
• A branch of mathematics including limits, derivatives,
integration, and infinite sums
• Used in science, economics, and engineering

4. Includes
Limits
Derivatives
Integrals
Infinite sums

5. Used In
 Science
 Economics
 Engineering

6. 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).

7. 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.

8. Medieval History
• In about 1000 AD, Islamic mathematician, Ibn al-In about 1000 AD, Islamic mathematician, Ibn al-
Haytham (Alhacen) derived a formula for the sum ofHaytham (Alhacen) derived a formula for the sum of
the fourth powers of an arithmetic progression,the fourth powers of an arithmetic progression,
later used to perform integration.later used to perform integration.
• In the 12In the 12thth
century, Indian mathematician Bhaskara IIcentury, Indian mathematician Bhaskara II
developed an early derivative. He described andeveloped an early derivative. He described an
early form of what will later be “Rolle’s Theorem”early form of what will later be “Rolle’s Theorem”
• Also in the 12Also in the 12thth
century, Persian mathematiciancentury, Persian mathematician
Saraf al-Din al-Tusi discovered the derivative of aSaraf al-Din al-Tusi discovered the derivative of a
cubic polynomialcubic polynomial

9. Modern History
• Bonaventure Cavalieri said 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.

10. 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.

11. 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.

12. …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.

13. Newton vs. Leibniz
• There was much controversy over who (and thusThere was much controversy over who (and thus
which country) should be credited with calculuswhich country) should be credited with calculus
since both worked at the same time.since both worked at the same time.
• Newton derived his results first, but LeibnizNewton derived his results first, but Leibniz
published first.published first.

14. 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.

15. 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”.

16. 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

18. Thanks to All from group A
Head of the group A
Ehatsham Riaz
Other Members
 Muhammad Hamza Rao
 Abdullah
 Asif Mehmood
 Hassan Raza