Mathematics from 1500 to the
Present Day
T J Osler
François Viète (1540 - February 13, 1603), generally
known as Franciscus Vieta, was a French mathematician.
Vieta’s product of nested radicals
(1592) was the first formula for Pi
2
1 1 1 1



2 2 2 2
1 1 1 1 1



2 2 2 2 2
John Wallis (November 23, 1616 - October 28, 1703) was
an Englishmathematician who is given partial credit for the
development of modern calculus
.
Wallis product for Pi - 1656
1 3 3  5 5  7 7  9





 2 2 4 4 66 88
2
William Brouncker, 2nd Viscount Brouncker, FRS (1620
.
– 5 April 1684) was anEnglish mathematician
Brouncker’s continued fraction for
Pi - 1656
4

2
1
 1
3
2
2
2
5
2
2
7
2
2 
René Descartes (31 March 1596 – 11 February
1650), was a French philosopher,mathematician, scientist, and writer.
Invented Analytic Geometry
• Isaac Newton 1642 -1727
• Robert Hooke 1635 – 1703
• Edmund Halley 1656 – 1742
• Gottfried Leibniz 1646 - 1716
George Berkeley (12 March 1685 – 14 January 1753), also known as
Bishop Berkeley, was an Irish philosopher
.
• Berkley found flaws in the foundations of
Newton’s calculus.
• Newton spoke of “infinitesimals” numbers
not zero, but smaller than any assigned
quantity.
• These difficulties would not be removed
until the 1800s
Leonhard Paul Euler (pronounced [ˈɔʏlɐ] in German, in English;15
April 1707 – 18 September 1783) was a
pioneering Swiss mathematician andphysicist who spent most of his life
in Russia and Germany.
Zeta Function

1
 ( z)   z
n 1 n
1 1 1
 z z z
1 2 3
1 1 1 1
 (2)  2  2  2  2 
1 2 3 4
1 1 1 1
 (4)  4  4  4  4 
1 2 3 4
 (2k )  (1)
k 1



2
6

4
90
2 B2 k
2(2k )!
2k
2k
Calculus of Variations
Stamp of the former German Democratic Republic honoring Euler on
the 200th anniversary of his death. In the middle, it shows his
polyhedral formula V − E + F = 2.
• Euler wrote some 866 Books, papers and
letters of ground breaking mathematical
content
• He is the most prolific mathematician of all
time
• Even though he went blind in his later
years, his mathematical productivity
increased
Joseph-Louis Lagrange, born Giuseppe Lodovico
Lagrangia (25 January 1736 –10 April 1813) was
an Italian mathematician and astronomer, who lived most of his life
in Prussia and France, making significant contributions to all fields
of analysis, tonumber theory, and to classical and celestial mechanics.
• Lagrangian mechanics
• Between 1772 and 1788, Lagrange reformulated Classical/Newtonian
mechanics to simplify formulas and
ease calculations. These mechanics are
called Lagrangian mechanics.
• Worked on Celestial Mechanics and the
solution of algebraic equations
Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827)
was a French mathematician and astronomer whose work was pivotal
to the development of mathematical astronomy and statistics.
• Laplace worked on Celestial Mechanics
• Mécanique Céleste
• Tried to prove that the solar system was
stable
Johann Carl Friedrich Gauss.
(30 April 1777 – 23 February 1855)
“The Prince of Mathematicians”
• As a teenager, Gauss showed how to
construct a regular 17 gon
• First major geometric construction in 2000
years
•
• Prime numbers of this form are also
known as the Fermat primes
•
• Gauss proved that a regular n-gon could
be geometrically constructed if the number
of sides were a product of distinct Fermat
Primes times a power of two
Titus Bode Law
• To find the mean distances of the planets,
beginning with the following simple
sequence of numbers:
• 0 3 6 12 24 48 96 192 384
• With the exception of the first two, the others
are simple twice the value of the preceding
number.
• Add 4 to each number:
• 4 7 10 16 28 52 100 196 388
• Then divide by 10:
• 0.4 0.7 1.0 1.6 2.8 5.2 10.0 19.6 38.8
• In 1800 Astronomers
begin the search for the
planet between Mars and
Jupiter
Discovery of Asteroid Ceres
Makes Gauss Famous
• 1801 Italian astronomer Piazzi observes a
moving celestial object for 41 days before
it disappears behind the sun
• The newly-discovered planet had been
lost
• Laplace declared that the new planet was
lost because its orbit could not be
calculated from so little data
• 24 year old Gauss discovered a method
for computing the planet's orbit using
only three of the original observations and
successfully predicted where Ceres might
be found.
• The prediction catapulted him to worldwide
acclaim
Jean Baptiste Joseph Fourier (March 21, 1768 – May 16, 1830) was
a French mathematician and physicist best known for initiating the
investigation of Fourier series and their application to problems of heat
flow.
• Let f(x) have period 2L
f(x+2L)=f(x)
a0
 nx
 nx
f ( x)    an cos
 bn sin
2 n1
L
L

1
an 
L
c L
1
bn 
L

f ( x) cos
c
c L

c
f ( x)sin
 nx
L
dx
 nx
L
dx
Example Fourier Series 1
1
f ( x)  
1
for 0  x  
for   x  2
4  sin x sin 3x sin 5 x
f ( x)  



 1
3
5



Fourier Series Example 2
 x
f ( x)  
2  x

for 0  x  
for   x  2
4  sin x sin 3x sin 5 x
f ( x)    2  2  2 
2  1
3
5



• Fourier Series challenged the intuition of
the greatest mathematicians
• How could a sum of such smooth
functions as sine and cosine represent
discontinuous functions?
• Later Carl Weierstrass showed a Fourier
Series that was continuous everywhere,
but differentiable nowhere!
Augustin Louis Cauchy (21 August 1789 – 23 May
1857; was a French mathematician.
• Cauchy finally provided the Calculus with
a rigorous foundation
• He introduced the epsilon – n and epsilon
– delta definitions of limit.
• From these we can rigorously define
continuous functions and differentiable
functions
• The meat of our Real Analysis course
Évariste Galois October 25, 1811 – May 31, 1832) was
a French mathematician born in Bourg-la-Reine.
• While still in his teens, he was
able to determine a necessary
and sufficient condition for
a polynomial to be solvable
by radicals, thereby solving a
long-standing problem.
• He died fighting a duel over a
woman at age 21
• Georg Friedrich Bernhard
Riemann September 17, 1826 – July
20, 1866) was an extremely
influential Germanmathematician who
made important contributions
to analysis and differential geometry, some
of them paving the way for the later
development of general relativity.
Riemann
Riemann Hypothesis – Most famous unsolved problem in mathematics
The non-trivial zeros of the zeta function lie on the line x = ½ in the
complex plane
• Jules Henri Poincaré (29 April 1854 – 17
July 1912) was a
French mathematician and
theoretical physicist, and a philosopher of
science. Poincaré is often described as
a polymath, and in mathematics as The
Last Universalist, since he excelled in all
fields of the discipline as it existed during
his lifetime.
Jules Henri Poincaré
• In 1887 he won Oscar II, King of Sweden's
mathematical competition for a resolution
of the three-body problem concerning the
free motion of multiple orbiting bodies.
• The two body problem was solved
analytically by Newton, and the solution is
Kepler’s equations of planetary motion.
• The three body problem has never been
solve analytically, although approximate
computer solutions are easy to generate.
• Albert Einstein 14 March 1879 – 18 April
1955) was a German-born theoretical
physicist. He is best known for his theory
of relativity and specifically mass–energy
equivalence, expressed by the
equation E = mc2. Einstein received the
1921 Nobel Prize in Physics "for his
services to Theoretical Physics, and
especially for his discovery of the law of
the photoelectric effect.”
• 1879: Albert Einstein is born to Hermann
Einstein
• 1889: At age 10, Albert sets into a program
of self education and reads as much about
science as he can.
• 1896: Albert graduates from high school at
the age of 17 and enrolls at the ETH (the
Federal Polytechnic) in Zurich.
• 1898: Albert falls in love with Mileva Maric, a
Hungarian classmate at the ETH.
• 1900: Albert graduates from the ETH.
• 1902 Mileva gives birth to a daughter whom they
put up for adoption
• 1903: Albert and Mileva marry in January
• 1904: Mileva gives birth to their first son, Hans
Albert.
• 1905: "Annus Mirabilis" -- Einstein's
"Miracle Year": his Special Theory of
Relativity is born.
• June 30th, Einstein, submits his paper,
"On the Electrodynamics of Moving
Bodies" to the leading German physics
journal. At age 26, he applies his theory to
mass and energy and formulates the
equation e=mc^2.
•
• 1907: Einstein begins applying the laws of
gravity to his Special Theory of Relativity.
• 1910: Son Eduard is born.
• 1913: Einstein works on his new Theory of
Gravity.
• 1914: The divorce prodedings begin.
• 1915: Einstein completes the General Theory of
Relativity.
• 1919: Albert marries his cousin Elsa.
• May 29, a solar eclipse proves Einstein's
General Theory of Relativity works.
One of the 1919 eclipse photographs taken during Arthur Stanley
Eddington's expedition, which confirmed Einstein's predictions of the
gravitational bending of light.
• 1922: Is awarded the Nobel Prize in
physics for 1921.
• 1932: Einstein is 53 and at the height of
his fame. Identified as a Jew, he begins to
feel the heat of Nazi Germany.
• 1933: Albert and Elsa set sail for the
United States. They settle in Princeton,
New Jersey where he assumes a post at
the Institute for Advanced Study.
Einstein’s House in Princeton NJ
• 1936: Elsa dies after a brief illness.
• 1939: World War II begins. Einstein writes
a famous letter to President Franklin D.
Roosevelt warning of the possibility of
Germany's building an atomic bomb and
urging nuclear research.
• 1955: Einstein dies of heart failure on
April 16.
G. H. (Godfrey Harold) Hardy FRS (February
7, 1877 Cranleigh, Surrey, England [1] –December 1, 1947 Cambridge, Cambridgeshire,
England [2]) was a prominent Englishmathematician, known for his achievements
in number theory and mathematical analysis.
• Non-mathematicians usually know him
for A Mathematician's Apology,
his essay from 1940 on the aesthetics of
mathematics.
• The apology is often considered one of the
best insights into the mind of a working
mathematician written for the layman.
• His relationship as mentor, from 1914
onwards, of the Indian
mathematician Srinivasa Ramanujan has
become celebrated.
• Hardy almost immediately recognized
Ramanujan's extraordinary albeit
untutored brilliance, and Hardy and
Ramanujan became close collaborators.
Srīnivāsa Rāmānujan Iyengar FRS, better known as Srinivasa
Ramanujan (22 December 1887 – 26 April 1920) was a legendary
Indianmathematician,[1] who, with almost no formal training in pure
mathematics, made substantial contributions to mathematical
analysis, number theory, infinite series andcontinued fractions.
Ramanujan’s Home
• John von Neumann (December 28, 1903 – February 8, 1957) was
a Hungarian American[1] mathematician
• Made major contributions to a vast range
of fields,[2] including set theory,functional
analysis, quantum mechanics, ergodic
theory, continuous
geometry,economics and game
theory, computer science, numerical
analysis,hydrodynamics (of explosions),
and statistics, as well as many other
mathematical fields. He is generally
regarded as one of the foremost
mathematicians of the 20th century.[1]
Benoît B. Mandelbrot[1] (born 20 November 1924) is
a French mathematician, best known as the father of fractal geometry
Mandelbrot speaking at the École Polytechnique in 2006, during the ceremony
when he was made an officer of the Legion of Honour.
Thomas J Osler 1940-?