# Definition:Calculus/Integral/Historical Note

## Historical Note on Integral Calculus

The origins of **integral calculus** can be traced back to Eudoxus of Cnidus and his method of exhaustion, dating from about $360$ BCE.

The techniques were used and expanded upon in his work *The Method* by Archimedes of Syracuse, to calculate areas and volumes of curvilinear figures.

These techniques are often suggested as being the precursors to **integral calculus**.

Johannes Kepler, in his *Nova Stereometria Doliorum Vinariorum* of $1615$, devised a method of finding the volume of a solid of revolution by slicing it into thin disks, calculating the volume of each, and then adding those volumes together.

Bonaventura Francesco Cavalieri expanded upon this in his *Geometria Indivisibilibus Continuorum Nova Quadam Ratione Promota* of $1635$.

John Wallis arithmetized Cavalieri's ideas in his *Arithmetica Infinitorum* of $1656$.

Much of the early work developing **integral calculus** was done by Isaac Newton.

His initial work on this seems to have been achieved during the years $1665$ to $1667$ when he was at home in Woolsthorpe.

At the same time that Newton was arranging his thesis, Gottfried Wilhelm von Leibniz was publishing many papers himself on the same subject.

The rigorous treatment of the subject was developed later, by Carl Friedrich Gauss, Niels Henrik Abel and Augustin Louis Cauchy.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text B$: Newton - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.18$: Newton ($\text {1642}$ – $\text {1727}$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.19$: Leibniz ($\text {1646}$ – $\text {1716}$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.26$: Cauchy ($\text {1789}$ – $\text {1857}$) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**calculus** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**calculus**