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Historical Timeline of Calculus

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Stacie Buchheit

on 3 May 2015

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Transcript of Historical Timeline of Calculus

1600
1700
1900
1500
1800
Historical Timeline of Calculus
Johannes Kepler (1570-1630)
From:
Holy Roman Empire, Nationality: German
Education:
Adelberg and Maulbronn to become a Lutheran Minister; University of Tubingen to pursue philosophy and mathematics
I
nteresting facts:
Proved that the orbit of Mars was an ellipse with the Sun located at one of the foci, research in field of optics- theories on vision and the internal structure of the eye; people had greater understanding of why eyeglasses worked
Contribution to calculus/science:
universe is governed by geometric relationships from inscribed and circumscribed circles of tetrahedrons, cubes, octahedron, dodecahedron, and iscosahedron, Kepler’s Laws of Planetary Motion (physics)

Blaise Pascal (1623-1662)
Education:
attended meetings of French geometricians, which later formed the basis of the French Academy
Interesting Facts:
invented and built a mechanical calculating machine, known as the Pascaline before age 21; studied hydrostatics; 1650- joined Jansenist monastery and stopped research in mathematics but broke away in 1658
Contributions to Calculus:
Published an essay on conic sections (1640)
Discovered that the sum of the angles of a triangle is equal to two right angles.

Sir Isaac Newton (1642-1727)
Nationality:
English
Education:
Cambridge University
Interesting Facts:
university closed down because of a plague outbreak so Newton took his studies home and worked on his own original research during this time
Contribution to Calculus:
foundation of differential calculus- “method of fluxions”
investigated integral calculus- “inverse method of fluxions”
Principia Mathematica (1687)- explains the motions of the heavenly bodies in the language of mathematics
Newton’s terminology: x, depending on time, is called a fluent; its rate of change with time is called the fluxion of the fluent (modern day dx/dt)
discovered modern day x dy/dt+ y dx/dt=0
Binomial theorem for expanding expressions: (1+a)^n
Rene Descartes (1596-1650)
Nationality:
French
Education:
Jesuit preparatory school of La Fleche and the University of Poitiers; majored in law
Interesting facts:
joined armies as a volunteer where he developed his mathematical and philosophical ideas; wrote a philosophical treatise Discours de la methode which contains the famous quote “I think, therefore I am”
Contribution to calculus:
La Geometrie- first printed account of analytic/coordinate geometry, used +, -, in his writing→ shifted the focus of geometry to curves to their equations (more algebraic thinking)
x-a is a factor of a polynomial if and only if a is a root
maximum number of roots is equal to the degree of the polynomial
Development of Cartesian coordinates, also credited with the first use of superscripts for powers or exponents

Pierre de Fermat (1601-1665)
Gottfried Leibniz (1646-1716)
Nationality:
German
Education:
Doctorate of Laws (1667) to become a diplomat
Interesting Fact:
Ended diplomatic career to become a librarian in the court of Hanover; helped organize the Berlin Academy of Science
Contribution to Calculus:
Symbolism that allowed the geometric arguments to be translated into operational rules:
modern day integral symbol for the sum of areas of infinitely small rectangles
originated modern day dy/dx notation (quotient of differentials), d for differentiation
The product rule d(xy)=x dy + y dx
p series

Michel Rolle (1652-1718)
Nationality:
French
Education:
self-educated
Interesting Fact:
worked on Diophantine analysis, algebra, and geometry
Contribution:
Wrote an algebra paper which contains the theorem on the position of the roots of an equation
“Rolle’s Theorem”: specialized case of the Mean Value Theorem- guarenteed the existance of a horizontal tangent line (f’(x)=0) between points a and b given that f(a)=f(b)=0

Guillaume de l’hopital (1661-1704)
Nationality:
French
Interesting Fact:
served in the military but had to resign due to extreme nearsightedness
Contribution:

First printed book on the new calculus
L’hopital’s rule:

Bernoulli Brothers (John and James)
Nationality:
Swiss
I
nteresting Fact:
would work independently to solve the same problem
Contribution:
The first to achieve a full understanding of Leibniz’s presentation of calculus
John Bernoulli: Bernoullian identity
James Bernoulli: expansion of a function series through repeated integration by parts:

Brook Taylor (1685-1731)
Nationality:
English
Education:
St John’s College
Interesting Fact:
Elected Secretary to the Royal Society
Contribution:

developed a solution to the center of oscillation of a body based on differential calculus
Calculus of finite differences, integration by parts, Taylor Series in his book “Methodus Incrementorum Directa Et Inversa”

Leonhard Euler (1701-1783)
Education:
After studying with Johan Bernoulli, Euler attended the University of Basel and earned his master’s during his teens. He moved to Russia in 1727, served in the navy and then joined the St. Petersburg Academy as a professor of physics (later heading its mathematics division).

Interesting Fact:
By the early 1770s, Euler lost his sight completely after not allowing for proper recuperation after an operation. Even though he was blind, he continued to work and publish articles until the day of his death on September 18, 1783.

Contribution(s):
Euler angles (to specify the orientation of a rigid body)

Euler’s Theorem (that every rotation has an axis)

Euler’s equations for motion of fluids

Euler-Lagrange equation (that comes from calculus of variations)

Euler’s formula (defines the exponentials of imaginary numbers in terms of trigonometric functions)


Joseph Louis Lagrange (1736-1813)
Education: A career as a lawyer was planned out for Lagrange by his father. He studied at the College of Turin and his favorite subject was classical Latin. Lagrange's interest in mathematics began when he read a copy of Halley's 1693 work on the use of algebra in optics. He was also attracted to physics by the excellent teaching of Beccaria at the College of Turin and he decided to make a career for himself in mathematics.

Interesting Fact: Lagrange married his cousin, Vittoria Conti.

Contribution(s):
He proved in 1770 that every positive integer is the sum of four squares

In 1771 he proved Wilson's theorem (first stated without proof by Waring) that n is prime if and only if (n -1)! + 1 is divisible by n

In 1797 he published the first theory of functions of a real variable with Théorie des fonctions analytiques

One of the basic results that followed in the Fonctions Analytiques is part of what is Known today as the fundamental theorem of calculus.

He defined the volume and the surface area by and respectively, where the equation of the surface is given by z=f(x,y) and dz=Pdx+Qdy.

Lagrange's commitment to the necessity of an algebraic foundation for the calculus led him to the major accomplishment of the Fonctions Analytiques, in which he studied functions by means of their power series expansions.


Pierre Simon Laplace (1749-1827)
Education:
Laplace owed his education to the interest excited in some wealthy neighbours by his abilities and engaging presence. He entered Caen University when he was only 16 years old.

Interesting Fact:
Laplace died on March 5, 1827, while in Paris. His doctor removed his brain and it was on exhibit for several years thereafter.

Contribution(s):
Pierre-Simon Laplace is highly regarded for his influential five-volume treatise “Traité de mécanique céleste” (Celestial mechanics; 1799-1825), which developed a strong mathematical understanding of the motion of the heavenly bodies, including several anomalies and inequalities that were noticed in their orbits. Laplace suggested that the nature of the universe is completely deterministic.

Laplace heavily contributed in the development of differential equations, difference equations, probability and statistics. His 1812 work “Théorie analytique des probabilités” (Analytic theory of probability) furthered the subjects of probability and statistics significantly.


Carl Friedrich Gauss (1777-1855)
Education: Caroline College, University of Gottingen

Interesting Fact: He wanted a he[tadecagon placed on his gravestone, but the carver refused, saying it would be indistinguishable from a circle.

Contribution(s):
He found that a regular polygon with 17 sides could be drawn using just a compass and straight edge.

While at the University of Gottingen, he submitted a proof that every algebraic equation has at least one root or solution. This theorem had challenged mathematicians for centuries and is called "the fundamental theorem of algebra".

In 1801, he proved the fundamental theorem of arithmetic, which states that every natural number can be represented as the product of primes in only one way.

At age 24, Gauss published one of the most brilliant achievements in mathematics, Disquisitiones Arithmeticae (1801). In it, Gauss systematized the study of number theory (properties of the integers). Gauss proved that every number is the sum of at most three triangular numbers and developed the algebra of congruences.

Gauss arrived at important results on the parallel postulate, but failed to publish them. Credit for the discovery of non-Euclidean geometry therefore went to Janos Bolyai and Lobachevsky. However, he did publish his seminal work on differential geometry in Disquisitiones circa superticies curvas. The Gaussian curvature (or "second" curvature) is named for him. He also discovered the Cauchy integral theorem for analytic functions, but did not publish it. Gauss solved the general problem of making a conformal map of one surface onto another.

Augustin-Louis Cauchy (1789-1857)
Education:
In 1802 Augustin-Louis entered the École Centrale du Panthéon where he spent two years studying classical languages. From 1804 Cauchy attended classes in mathematics and he took the entrance examination for the École Polytechnique in 1805. In 1807 he graduated from the École Polytechnique and entered the engineering school École des Ponts et Chaussées.

Interesting Fact(s):
Laplace and Lagrange were visitors at the Cauchy family home and Lagrange in particular seems to have taken an interest in young Cauchy's mathematical education. Cauchy produced 789 mathematics papers.

Contribution(s):
In Cours d’analyse del’École Polytechnique (1821), by developing the concepts of limits and continuity, he provided the foundation for calculus essentially as it is today.

He introduced the “epsilon-delta definition for limits (epsilon for “error” and delta for “difference’).

He transformed the theory of complex functions by discovering integral theorems and introducing the calculus of residues. Cauchy founded the modern theory of elasticity by applying the notion of pressure on a plane, and assuming that this pressure was no longer perpendicular to the plane upon which it acts in an elastic body. In this way, he introduced the concept of stress into the theory of el
asticity.

Carl Gustav Jacobi (1804-1851)
Education:
Carl Jacobi's early education was given by an uncle on his mother's side. When he was 11, he entered the Gymnasium in Potsdam. He was so advanced that while still in his first year of schooling, he was put into the final year class. This meant that he was still only 12 years old when he had reached the necessary standard to enter a university. However the University of Berlin did not accept students below the age of 16, so Jacobi had to remain in the same class at the Gymnasium in Potsdam until then. Jacobi entered the University of Berlin in 1821.

Interesting Fact:
He died of smallpox.

Contribution(s):
Jacobi carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics.

In 1834, Jacobi proved that if a single-valued function of one variable is doubly periodic then the ratio of the periods is imaginary. This result prompted much further work in this area, in particular by Liouville and Cauchy.

One of the results in the global theory of curves is a theorem of Jacobi published in 1842: "The spherical image of the normal directions along a closed differentiable curve in space divides the unit sphere into regions of equal area".


Karl Weierstrass (1815-1897)
Nationality:
German

Education:
University of Münster, University of Bonn

Interesting Fact:
He is known as the father of modern analysis. He spent 15 years as a secondary school teacher.

Contribution(s):
He developed the definition of the limit

He developed the technique of analytic continuation

He and Riemann published on Abelian function theory


Bernhard Riemann (1826-1866)
Nationality:
German

Education:
In 1846 he enrolled in the University of Göttingen

Interesting Fact:
The day before his death he was lying under a fig tree, working on a great paper on natural philosophy.

Contribution(s):
Riemann's innovative published works constructed the base for what is known as modern mathematics and research areas including analysis and geometry. These works finally proved to be very useful in the theories of algebraic geometry, Riemannian geometry and complex manifold theory.

Riemann also established some breakthrough milestones in the theory of ‘Real Analysis’. He explained ‘the Riemann integral’ by means of Riemann sums and penned down a theory of trigonometric series that are not Fourier series, a first step in generalized function theory, and also explored the ‘Riemann–Liouville differintegral’.


Srinivasa Ramanujan (1887-1920)
Nationality:
Indian

Education:
Trinity College, Cambridge (1919–1920), University of Cambridge (1914–1919), University of Cambridge (1916), Government Arts College, Kumbakonam (1904–1906), Town Higher Secondary School (1904), Pachaiyappa's College

Interesting Fact:
In 1909, he married a ten year old girl.

Contribution(s):
His brilliant work on the Bernoulli numbers in 1911, in the Journal of the Indian mathematical society, grabbed the recognition for all his hard work over the years.

Ramanujan - Hardy Number (1729): It is the smallest natural number that could be expressed as the sum of two positive cubes, in two different ways (i.e., 13 + 123 and 93 + 103.)

Alan Turing (1912-1954)
Nationality:
English

Education:
King's College (University of Cambridge), Institute for Advanced Study, Princeton University

Interesting Fact:
He was charged with “gross indecency,” and later committed suicide by cyanide poisoning.

Contribution(s):
In 1936, Turing delivered a paper, "On Computable Numbers, with an Application to the Entscheidungsproblem," in which he presented the notion of a universal machine (later called the “Universal Turing Machine," and then the "Turing machine") capable of computing anything that is computable: The central concept of the modern computer was based on Turing’s paper.

He also wrote two papers about mathematical approaches to code-breaking, which became such important assets to the Code and Cypher School

He developed a new machine (the Bombe) which was capable of breaking Enigma messages on an industrial scale.


Abraham Robinson (1918-1974)

Nationality:
German

Education:
Hebrew University

Interesting Fact
: He was Jewish and had to immigrate to Palestine and years later to England.

Contribution(s):
-
non-standard analysis, a system in which infinitesimal and infinite numbers were reincorporated into modern mathematics.
-Robinson found a model completion for the axioms of differential fields, which then served as models of the “closure” axioms associated with this completion.


Early Civilizations
Pythagoras (570-495BC)

Nationality:
Greek
Interesting Fact:
Started a group of mathematicians called the Pythagoreans. They kept all of their discoveries secret, but accredited all of their findings to Pythagoras
Contributions/ Famous For:
Expansion of geometry, rigorous approach building from first principles, square and triangular numbers, Pythagoras’ theorem

Democritis (460-370BC)
Nationality:
Greek
Interesting Fact:
pre-Socratic philosopher
Contributions/Famous For:
formulation of an atomic theory of the universe, Developments in geometry and fractions, volume of a cone
Euclid (300BC)
Nationality:
Greek
Interesting Fact:
“Father of Geometry” and “Father of Mathematics”
Famous For:

Elements
(pulls together materials from others who studied and researched mathematics before him.), Euclidean Geometry,
Archimedes (287 - 212 BC)
Nationality:
Greek
Interesting Facts:
anticipated modern calculus by applying concepts of infinitesimals and the method of exhaustion to derive and prove theorems
Famous For:
Calculation of pi, geometry,area of a circle, (using infinitesimal), volume of a sphere, determine the volume of a solid

Aryabhata (476 - 550 AD)
Nationality:
Indian
Education:
Kusumapura for advanced studies
Famous For:
Definitions of trigonometric functions, complete and accurate sine and versine tables, solutions to simultaneous quadratic equations, accurate approximation for π (and recognition that π is an irrational number)
Nationality:
French
Education:
University of Law at Orleans
Interesting Facts:
explored analytical geometry, calculus, and number theory, was mistakenly declared deceased in 1653 from the Bubonic Plague.
Contributions to calculus:
A prime number of the form nx2+1=y2has infinitely many integer solutions if n is not a square
Fermat’s Last Theorem: xn+yn=znhas no non-zero integer solutions for x, y, and z when n>2
Nationality: German
Nationality: French
Nationality: German
Nationality: French
Nationality: Italian
Nationality: Swiss
Link to Resources Used:
https://docs.google.com/document/d/1jkcDkV-ruE6ggs8VbjtqlaF2vs-2sQQlMJHzWWbPLHU/edit
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