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The Calculus Gallery

A Timeline of Calculus Ideas from 1600s to 1900s

Roger Wolbert

on 2 March 2015

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Transcript of The Calculus Gallery

Pioneer Wing



The Calculus Gallery
"Masterpieces from Newton to Lebesque"
This gallery was inspired by William Dunham's (2005) book with the same name as the gallery exhibit. There is a deliberately small display of mathematicians as to not overwhelm the visitor.

As you pass through the gallery, you will see a little bit of each mathematician's personal life as well as a contribution that he has made to the development of calculus.

Enjoy your tour.
Feel free to take pictures.
No smoking please.
Welcome to the Pioneer Wing!
The foundation of calculus began with the independent work of Newton and Leibniz. It was followed-up by the Bernoulli Brothers and Euler, who expanded the concepts of limits, derivatives, infinite sequences, and infinite series.

As you pass through this wing, it is important to keep in mind that the early pioneers were not as critical in proving their work mathematically as they instead spent their time formulating ideas for this new branch of mathematics.

Looking back, it is amazing how the early pioneers conceived ideas for limits, infinity, sequences, and series. Please keep in mind also that their methods also shifted from the use of geometric means to algebraic as their perspectives grew.
Founders Hall
There had been much debate who is the founder of calculus, but most historians scoot around this debate by noting that Sir Isaac Newton and Gottfried Leibniz were independently the originators of calculus.
The Modernists Hall
Welcome to the Classical Wing!
The early pioneer's perspectives of calculus shifted from geometry and curves to analysis and functions, particularly driven by Euler. However, lack of time most likely left some of his ideas less precise and even less accepted “as is” for the mathematics community of that time.

In the Classical Wing, you will come across mathematicians whose work focused on formalizing calculus in the way in which we know it today. Their style of proof and precision is akin to that found in Euclidean geometry. Another notable difference within the mathematicians in this wing is their goal to make generalizations for many functions, unlike the pioneers who focused specifically on particular functions.
Classical Wing

Welcome to the Modern Wing!
As you leave the Classical Wing, you may be thinking that much of calculus as we know it has been formalized by this point. What's left?

With a formalized and solid foundations now established for calculus, modern mathematicians had a strong basis on which to test new theorems and to promote advances in technology and engineering.

In this final wing, we briefly shed light onto the contributions that modern mathematicians who might not typically make it in a gallery; they demonstrate how calculus advances development in other fields.

Modern Wing

The Bernoulli Brothers Hall
The Bernoulli Family was a well known Swiss family that made significant contributions to mathematics and science in the 17th and 18th Centuries. The "Bernoulli Brothers" - Johann and Jakob - are the family members with the most notable contributions to calculus.
The Euler Room
Swiss mathematician Leonhard Euler (1707-1783) is often considered the greatest mathematician to have ever lived. It is one reason why he has his own room in the gallery - room for us to expand and add more of his to our collection.

Contributions to calculus:
Infinitesimal calculus
Structure for analysis
Power series
Calculus of variations
Euler-Lagrange equation
Much, much more!

The Classicists Hall
1600s t0 1700s
1700s t0 1800s
1800s t0 1900s
The ideas of calculus can be traced back to Archimedes of Syracuse (c. 287-212 BCE), to Pierre de Fermat (1601-1665), and to Carl Friedrich Gauss (1777-1855) and more. However, we apologize that the gallery is unable to mention every mathematician’s contribution to calculus. This would be an enormous undertaking and is restricted by the size of our gallery.

As our facility continues to expand, we hope that more mathematicians can be included to our gallery collection. For now, there are three main wings to our gallery, covering three centuries of the history of calculus for you to enjoy.

Thank you for visiting.
Please come again.
Portrait by Emanuel Handmann, 1753
Sir Isaac Newton (1642-1727)
English physicists and mathematician Isaac Newton is one of the most famous scientists of his time. He is famous for for his methods of differential calculus, which he called "fluxions."

Contributions to calculus:
An independent founder
Generalized the binomial expansion
Inverting series
Derivation of the sine series

Much of Newton's work in calculus was geometric based on the ratio of "vanishing small quantities."
Portrait by Godfrey Kneller, 1689
Gottried Leibniz (1646-1716)
German mathematician and philosopher Gottfried Wilhelm Leibniz has made significant contributions to mathematics, including inventing calculators.

Contributions to calculus:
An independent founder
Works on infinitesimals
Calculus notation that we use today
The Leibniz Series
Law of Continuity

Leibniz's methods for analysis were somewhat unconventional and were later 'refined' by the likes of mathematicians who preferred rigorous methods.
Portrait by Christoph Bernhard Francke
Johann Bernoulli (1667-1748)
Johann (Jean) Bernoulli was the younger of the two famous Bernoulli brothers. Like his brother, he was a professor and a very prolific writer in many areas of mathematics.

Contributions to calculus:
Co-founder of calculus with variations
Works on logarithms within calculus
Was paid by Marquis de l'Hospital to write calculus problems (and l'Hospital's rule), and in essence wrote the first calculus textbook.

Johann Bernoulli was Leonhard Euler's teacher.
Portrait by Johann Rudolf Huber, c. 1740
Jakob Bernoulli (1654-1705)
Jakob (James, Jacques) Bernoulli was the older of the famous Bernoulli brothers. Jakob was very pro-Leibniz in the Newton-Leibniz controversy on who first founded calculus.

Contributions to calculus:
Co-founder of calculus with variations
Credited with the term "integral"
Works on harmonic series
Bernoulli's differential equation

Jakob Bernoulli also made many contributions to the field of probability. He also discovered the constant
Portrait by Christoph Bernhard Francke
The Cauchy Room
French mathematician Augustin-Louis Cauchy (1789-1857) is best known for starting the process of formalizing calculus and using rigorous proofs for proving theorems. Like his predecessor Euler, he has his own room with space for the expansion of his works as they are collected for the gallery. His contributions are many with the more popular ones listed below.

Contributions to calculus:
Rigor for limits, continuity, derivatives
The Intermediate Value Theorem
The Mean Value Theorem
Integrals and the Fundamental Theorem of Calculus
Convergence tests for infinite series
Much, much, more!
Lithography by Zephirin Belliard
from a painting by Jean Roller, c. 1840
Like Cauchy, the mathematicians in this hall supported the need to make calculus more rigorous and well-defined instead of being based more on intuition.
Bernhard Riemann (1826-1866)
Georg Friedrich Bernhard Riemann was a German mathematician who contributed much to different areas of mathematics. Riemann also worked in collaboration with many other mathematicians, unlike Newton and Leibniz.

Contributions to calculus:
Riemann Sums and Integrals
Cauchy-Riemann equations
Riemann-Lebesgue Lemma
Riemann-Liouville Differintegral
Riemann Rearrangement Theorem
Bernhard Riemann, 1863
Karl Weierstrass (1815-1897)
German mathematician Karl Weierstrass is known as the "Father of Modern Analysis." Despite his lack of a university degree, he published great works eventually landing him a teaching job at the University of Berlin.

Contributions to calculus:
Formal definition of limits
Formal definition of continuity
Advancements in calculus of variations
Weierstrass M-test
Weierstrass approximation theorem

Weierstrass brought closure to Cauchy's project of making the foundation of calculus based on rigor and precision.
Karl Theodor Wilhelm Weierstrass
Joseph Liouville (1809-1882)
French mathematician Joseph Liouville is probably most known for his 1851 discovery of the first transcendental number - which was only a concept back in Leibniz's time and only later refined by Euler.

Contributions to calculus:
Sturm-Liouville theory in differential equations
Works in differential geometry, topology, physics, and number theory

Although Liouville is a lesser known contributor to the development of calculus, he gives us an example of how the developments in calculus were overlapping into other branches of mathematics and science.
Joseph Liouville
Dunham, W. (2005).
The calculus gallery: Masterpieces from Newton to Lebesgue
. Princeton, NJ: Princeton University.

Wikipedia for each mathematician's photograph
and brief life history/biography.
Vito Volterra (1886-1940)
Italian mathematician and physicist Vito Volterra is known for his theory of integral equations. Volterra was quick to learn mathematics and physics, finishing his doctorate by at 22.

Contributions to calculus:
Co-founder of functional analysis
Pathological functions and limits
Lotka-Volterra equations (preditor-prey equations)
Advanced work with integrals and differential equations

Vito Volterra
Henri Lebesgue (1875-1941)
French mathematician Henri Lebesgue is best known for his theories for integration. His generalization of previously confined theorems opened the way for further development of calculus and analysis.

Contributions to calculus:
Corrections to Riemann Integrals (Sums)
Lebesgues's Theory of Integration, which removed the need to attach restrictions to the derivative
Dominated Convergence Theorem in measure theory

Henri Léon Lebesgue
René-Louis Baire (1874-1932)
French mathematician René-Louis Baire is best known for the Baire Category Theorem. This theorem, which was from his dissertation, helped to generalize other theorems - a usual goal for a modernist mathematician. With this, theorems from the past could be revisited in which generalizations were not yet achieved.

Contributions to calculus:
Baire Category Theorem
Baire Classifications of Functions

Baire was known to have a rich imaginations. However, it was unfortunate that Baire was often physically and mentally unwell, which limited his contributions to calculus and analysis.
René-Louis Baire
Georg Cantor (1845-1918)
German mathematician Georg Cantor is accredited as the founder of set theory. He is well-known for his theorems regarding infinity, one which is said he used to prove the existence of God.

Contributions to calculus and set theory:
The Completeness Property
The Nondenumerability of Intervals
Advanced work on transcendentals

Cantor was a student of Karl Weierstrass, and he later became a university professor.
George Ferdinand Ludwig Philipp Cantor
Calculus has undergone many changes from its inception to the present. It has gone from an idea about curves to functions, from geometry to algebra, and from intuitive ideas to rigorous proofs and definitions.

Calculus helps solve the simplest and most complex of problems. It provides a way for our conceptions about limits and infinity to be challenged on one hand and enhanced on another.

Calculus has changed society in many ways, and it is exciting to think what lies ahead for us in the 21st Century. Please return to visit the Postmodern Wing envisioned for this gallery.
Parting Thoughts
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