MAT 115H – Mathematics: An Historical Perspective
Fall 2015
Final Student Projects
Below are listed 15 projects featuring a famous theorem prominent in the history of
mathematics. These theorems, dating from Antiquity to the 20th century, all played a
significant role in the history of mathematics and were solved by some of the greatest
mathematicians of their time – towering figures such as Euclid, Archimedes, Newton,
Euler, and Cantor. The proofs to these theorems stand out for their ingenious
approaches, elegant methods, and clever arguments.
For the most part, these theorems were solved using elementary methods in algebra,
geometry, or number theory in conjunction with simple logic (e.g. “If A is greater or equal
to B and B is greater or equal to A, then A = B”). In any case, the difficulty level behind
each of these proofs is relatively low and no methods in trigonometry or calculus are
required for any of these proofs (although these methods could be readily used in some
cases).
These projects purposely avoid using theorems that are far too technical or advanced,
such as Kurt Gödel’s Incompleteness Theorem in logic, the Fundamental Theorem of Algebra
proved by Gauss, or Riemann’s Hypothesis in number theory, even though these would
have to be cited as some of the most important results in all of mathematics (the last
one – actually not a result but a conjecture widely believed to be true – is currently
considered to be the hardest unsolved problem in mathematics; the Clay Institute in
Cambridge, Massachusetts will give a million dollars to the person who solves it).
Instructions
Each student in the course will be randomly assigned one project through a lottery at
the beginning of the semester. Students will then independently investigate the theorem
and topic in their project using the references listed in the course outline, additional
textbooks, internet searches, and any other resources they deem appropriate.
Throughout the semester, students are encouraged to schedule meeting with the
professor to discuss their projects.
Students will present their findings to the class at the end of the semester in a 10minute presentation followed by a short (about 5-minute) Q&A session. In addition,
students will submit a written report of their work to the professor along with their
presentation.
Below are some general guidelines about what should be included in the written
reports.
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A simple description of what the theorem in their project states in mathematical
terms. What does it say?
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A commentary on the importance of the mathematician who proved the
theorem in their project. How did he influence the history of mathematics?
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A discussion of the state of mathematics at the time the theorem in their project
was proved. What was known then?
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The mathematical method by which the theorem in their project was proved,
including all logical steps. How was the theorem proved?
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A comment on the importance of the theorem in their project for mathematics
as a whole. Why does this result matter?
Written reports should not exceed more than 4 pages, typed or handwritten. Much
more important than length considerations is making sure that the ideas contained in
the theorem are presented in a precise and clear manner. Avoid using technical terms
that are unfamiliar to you, unless you explain them in the context of the theorem.
Provide equations, illustrations and diagrams as needed. Include all references used in
the project.
For the presentations, students may opt to give a power-point presentation and use the
technology provided in the classroom (computer & projector). However, rather than
use technology, students are encouraged to make use of the blackboard and to outline
the major steps in the proof of their theorems along with simple diagrams and figures.
Students should provide the class with simple, easy-to-read hand-outs, if necessary.
Keep in mind that you are presenting mathematical results to your classmates. As a
result, you should keep technicalities to a minimum and explain new concepts in a clear
and simple fashion.
Finally, students may swap projects amongst themselves; if they choose to do so, they
must then notify the professor at once.
LIST OF STUDENT PROJECTS
1. Hippocrates of Chios – The Quadrature of the Lune (ca. 440 B.C.E.)
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Define the idea of a quadrature and the term lune in the context of classical
Greek geometry.
Show how the great Greek geometer Hippocrates of Chios (not to be confused
with the famous physician Hippocrates of Cos) achieved this remarkable
geometrical construction.
Explain why the quadrature of lunes is related to the problem of squaring the
circle – one of the three famous problems of Greek Antiquity.
Discuss other quadratures solved by Hippocrates and his stature among early
Greek mathematicians.
2. Theaetelus – The Five Platonic Solids (ca. 400 B.C.E.)
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Explain what defines these special objects in solid geometry.
Explain why these solids are named after the great Greek philosopher Plato.
Theaetelus is nowadays believed to be the first to have proved that exactly five
Platonic solids could be constructed geometrically. Show his existence proof.
Discuss the approach used by Euclid to construct the Platonic solids in The
Elements (Propositions 13 – 18 in Book XIII).
Discuss the importance of these solids in the context of The Elements and, more
broadly, Greek geometry.
3. Menaechmus – The Duplication of the Cube (ca. 350 B.C.E.)
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Define the idea of a duplication in the context of classical Greek geometry.
Discuss the duplication of the cube, or Delian problem – one of the three
famous problems of Greek Antiquity.
Menaechmus, the tutor of Alexander the Great, managed to solve the Delian
problem using parabolas. Show his solution.
Explain why his solution above does not solve the problem according to the
standards of classical Greek geometry.
Discuss subsequent attempts to solve the Delian problem and its eventual
resolution in modern times.
Discuss the resemblance of this problem to the trisection problem – another
one of the three famous problems of Greek Antiquity.
4. The Pythagorean Theorem, Part I
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Trace the origins of this fundamental theorem of mathematics from Ancient
Babylonia (ca. 2nd millennium B.C.E.) to the school of Pythagoras (ca. 550 B.C.E.).
Show the relationship between square numbers and Pythagorean triples.
Show how Euclid devised a method to generate all Pythagorean triples.
Show the ingenious geometrical method used by Euclid to prove the
Pythagorean theorem in The Elements (Proposition 47 in Book I).
5. The Pythagorean Theorem, Part II
Incredibly, there are over 350 different proofs of the Pythagorean Theorem
published today. This theorem (at least in a special form) was know to many great
ancient civilizations all over the world, including Babylonia, Greece, China, and India,
some of which predate Pythagoras and his school by a thousand years! The many
proofs of this theorem come in all shapes and forms, spanning over many areas of
mathematics and widely ranging in difficulty. For this project, show three different
proofs of your choice besides Euclid’s. Famous proofs include Bhaskara’s proof and
President James Garfield’s trapezoidal proof of 1876.
6. Euclid – The Infinitude of Primes (ca. 300 B.C.E.)
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Discuss the number theory produced in The Elements in Books VII – IX. Explain
the special character of these books in relation to the rest of The Elements.
Show the clever argument used by Euclid to deduce that there is an infinite list of
prime numbers (Proposition 20 in Book IX).
Explain the importance of prime numbers in the context of number theory and
mathematics.
Illustrate the use of prime numbers in 21st century technology.
7. Archimedes – Approximation of Pi (ca. 225 B.C.E.)
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Discuss the life and work of Archimedes – the greatest mathematician of
Antiquity.
Show how Archimedes proved that the area of a circle with radius r is equal to
half its radius times its circumference.
Show how Archimedes was able to use this last result to approximate the value
of pi (the ratio of a circle’s circumference to its diameter) to the strict inequality
1
10
3  3 .
7
71
Discuss the subsequent efforts of mathematicians and astronomers in later times
to find better approximations of pi.
8. Cardano – The Depressed Cubic Equation (1545)
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Discuss the emergence of algebra in the Italian Renaissance and the importance
of great Italian mathematician Gerolamo Cardano in this movement.
Show how Cardano derived the solution to the depressed cubic equation
x 3  mx  n , where m,n are coefficients, from purely geometrical principles by
looking at the volume of a cube.
As an example, explain how he solved the equation x 3  6x  20 and got the
solution x  3 10  108  3 10  108 (instead of 2).
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Discuss the eventual solution of the general cubic and quartic equation in
subsequent decades.
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9. Newton – Approximation of Pi From a Binomial Expansion (1671)
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Discuss the life and work of Isaac Newton – one of the greatest mathematicians
and scientists of all time.
Show the brilliant approach used by Newton to approximate the value of pi
using the binomial expansion of the expression y  x  x 2 (the equation of the
1 
upper half of the circle centered at  ,0 and passing through the origin).
2 
Discuss the approaches used after Newton to obtain more accurate
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approximations of pi. Why are these approximations of pi so important to
scientists?
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10. The Bernouilli Brothers – The Divergence of the Harmonic Series (1689)
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Discuss the influence of the Bernouilli brothers – perhaps the most influential
pair of siblings in the history of mathematics – in the early developments of
calculus.
Define an infinite series in mathematics and, in particular, the harmonic series.
Explain what it means for a series to diverge or to converge. [Note that you do
not need to use calculus methods here!]
Discuss the simple approach taken in an early proof by the medieval
mathematician Nicole Oresme (ca. 1350) to show why the harmonic series must
diverge to infinity.
Show the clever approach devised by Johann Bernouilli (but published by his
older brother Jakob) to prove that the famous harmonic series diverges.
11. Euler – The Seven Bridges Problem (1736)
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Discuss the life and work of Leonhard Euler – the most prolific and one of the
greatest mathematician of all time.
Discuss the history and formulation of this famous problem inspired by the
layout of the seven bridges in Königsberg’s city center.
Show how Euler solved the problem using a so-called graph in conjunction with a
clever bit of logic.
Discuss how his general solution to this problem created a new branch of
geometry and discrete mathematics called graph theory.
Explore some of the other famous problems in graph theory, including the
Hamiltonian (traveling salesman) problem and the Four-Color Theorem.
Euler – The Polyhedral Formula (1752)
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13.
Discuss the life and work of Leonhard Euler – the most prolific and one of the
greatest mathematician of all time.
Discuss this deep result in geometry, which states that for every convex
polyhedron with V vertices, F faces, and E edges, the identity V  F  E  2
always holds.
Explain why the formula states something fundamental about shapes and space –
what today modern mathematicians call a topological invariant.
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Illustratethe formulafor a numberof cases, such as the platonic
solids.
Try to sketch out a proof of Euler’s polyhedral formula (a simple one operates
by using induction on the number of edges).
Cantor – The Denumerability of the Integers and the Rationals (1878)
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Discuss the development of Georg Cantor’s theory of sets in the 1870’s and
how his groundbreaking work on the nature of infinite sets jolted the
mathematical world at the time.
Explain the concepts of cardinality, one-to-one correspondence, denumerability,
and the aleph null cardinal number within the context of Cantor’s theory of
infinite sets.
Show the clever approaches used by Cantor to show that both the set of
integers and rational numbers (or fractions) have the same cardinality as the set
of natural numbers.
In other words, show how Cantor proved the
denumerability of both the integers and the rationals.
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Explain the implications of this result: if the cardinality of the naturals, the
integers, and the rationals is aleph null, then all these sets are related by the
same type of infinity.
Cantor – The Diagonal Argument (1891)
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15.
Discuss the development of Georg Cantor’s theory of sets in the 1870’s and
how his groundbreaking work on the nature of infinite sets jolted the
mathematical world at the time.
Based on his previous result that both the set of integers and rationals are
denumerable (i.e. of infinity aleph null), show how Cantor then went on to prove
a truly remarkable result: that the cardinality of the set of real numbers (i.e. the
continuum) is actually bigger than aleph null, thus implying the nondenumerability of the continuum. Show how Cantor did this using a highly
original, yet deceivingly simple, method called the diagonal argument.
Explain the deep implications of this result, namely that there exists a different
hierarchy of infinities, which is itself infinite!
Briefly touch on the Continuum Hypothesis that came as an immediate
consequence of this result (and was only fully resolved in 1963 by the American
mathematician Paul Cohen).
Richard von Mises – The Birthday Problem (1939)
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Briefly discuss the history of probability theory and its early development by
Cardano, Pascal, and Fermat stemming from the analysis of popular games of
chance.
In 1939 Richard von Mises proposed this famous problem in probability theory.
He asked: How many people must be in a room before the probability that some share
a birthday reaches 50 percent (ignoring the year and any leap days)? Show the
simple solution to this problem using conditional probabilities.
Explain why this problem is sometimes referred to as “the birthday paradox.”
Discuss some of the applications of probability theory in the 21st century and the
fundamental role this discipline plays in all aspects of human life.