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Readings for Week 1, September 6 and 8:
- The article Eye on the Present --- The Whig History of
Science, by Stephen Weinberg. This reading is not marked
"Req" or "FYI only", because it is not "required reading", in the sense
that I am not going to ask you any questions about it in any of the exams,
but I would like you read it, because it is an accurate
statement of my own views on how I intend to teach this course.
- The MacTutor article
An overview of the history of mathematics. Req
- The book's Chapter 1, on Egypt and Mesopotamia. The chapter is "FYI
only", but Sections 1.1.1. 1.2.3 and 1.2.4 are Req.
- The article on
Egyptian Mathematics Numbers Hyeroglyphs.Req.
(This article contains nice color pictures of hieroglyphs and numbers
represented in the hieroglyph system.)
- The three articles on minimum wage models
Readings for Weeks 2 and 3, September 13, 15, 20 and 22
- The book's Chapter 2, on the beginnings of Greek Mathematics,
and Chapter 3, on Euclid. Both chapters are Req.
- The
MacTutor
Biography of Euclid. Req.
- The
MacTutor
Biography of Zeno of Sidon. Req.
- Euclid's Elements:
-
Introduction to Book I. Req.This contains the list of all the 23
definitions, 5 postulates, 5 common notions, and 48 propositions of
Book I. For each of these, there is a link that gives you more
details including, in many cases, a nice critical discussion. I
strongly recommend that you look at all the links for the
definitions, postulates and common notions.
-
Book I, Proposition 1. Req.
-
Book I, Proposition 2. FYI only.
-
Book I, Proposition 3. FYI only.
-
Book I, Proposition 4. Req. I strongly recommend that you
read the critical discussion of this proposition, and in particular the
explanation of why the argument is problematic.
-
Book I, Proposition 5. Req. This is the famous
Pons Asinorum (bridge of the donkeys). I strongly recommend that
you study this proof carefully, because later in the course we are going to
discuss how to prove the same result using the language and techniques
of modern mathematics (i.e., vector spaces and inner products), and it is
important that you understand the
difference so you will appreciate the progress that has been made since
Euclid's days.
-
Book I, Proposition 29. Req. This is a very
important theorems: it says that if a line intersects two parallel lines
then it forms equal angles with them.
-
Book I, Proposition 32. Req. This is one of the most
important theorems of Euclidean geometry: it says that the sum of the
angles of a triangle is 180 degrees.
THIS IS THE FIRST TIME EUCLID USES POSTULATE 5.
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Book I, Proposition 47. Req. This is Euclid's proof
of Pythagoras' theorem. You should review the earlier proof that we gave in
class, and compare. Also, I strongly recommend that you read the section
about the ancient Chinese proof explained in the commentary on Euclid's proof.
(This proof occurs in the book
Zhou Bi Suan Jing (FYI only), which dates from the period of the
Zhou Dynasty, 1046 BCE-256 BCE. Also, you should take a look at the
proof discovered by James Garfield in 1876, 4 years before the becane the
20th President of the United States.(FYI only)
Later in the course we will give a modern proof in the language of
vector spaces and inner products.
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Proof that the three medians of a triangle intersect at a point.
Req.
This is a "synthetic geometry" proof, that is, a proof in the spirit of
Euclid. Later in the course we will do a "linear algebra" proof, i.e.,
a proof using the modern language of vector spaces and affine spaces, so
that you will see the progress that has been made since
Euclid's days. You may take a look at such a proof in
The Centroid Theorem (FYI only) This site actually
gives you two proofs: one "in the style of Euclid" and the other one
labelled (erroneously, in my view) "proof in the style of Descartes."
(It really is a proof in the style of late 19th or 20th Century, because
it uses vectors and does not use coordinates.)
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Proof that the angle of a triangle in a semicircle is a right angle.
Req. This result is known as "Thales' semicircle theorem".
The proof given here is in the style of Euclid. Later in the course we
will do a modern proof,
using the language of vector spaces and inner products.
Readings for Week 4, September 27 and 29
- The book's Chapter 2, on the beginnings of Greek Mathematics,
and Chapter 3, on Euclid. Both chapters are Req.
- The book's section on Analytic Geometry (that is, section 14,2, pages
473-486). Req.
- The MacTutor history of
Non-Euclidean Geometry. Req.
- The New Mexico State University Module on
Geometry: The Parallel Postulate. Req.
- The MacTutor biographies of
Descartes Req and
Fermat Req.
- The Wikipedia article on
Analytic Geometry. Req.
- The MacTutor biography of
Lagrange. Req.
- Lagrange's
Analytical Mechanics. This is a monumental book, and most of it requires
much more sophisticated mathematics then you are expected to be familiar
with. And, in addition, the only downloadable edition I found was the
original French edition. So I am not at all asking you read the whole
thing. But, even if you don't read French, I would like
you to take a look at the book, and get the flavor. Just look at the preface, where you will find the words
"On ne trouvera point de Figures dans cet Ouvrage", that is, "no figures
will be found in this work". (Notice the old usage of capitalizing all
nouns, as used to be done in ancient English and is still done today in
German.) And then move on a little bit, and you will find
le "privilege du roi", that is, the king's authorization. (And remember
that this is 1788, and that king is the same king Louis XVI that would be overthrown
one year later, and executed by the revolutionaries on January 21, 1793.) Then, if you look
further, you will find the famous "Lagrange multipliers" on page 46, 47
and even if you don't read French the formulas may look familiar to you.
And you will find, for example on page 77, equations involving nine
coordinates, called X', Y', Z', X'', Y'', Z'', X''', Y''', Z''',
so you will see how Lagrange is working in nine-dimensional space.
- A note on
Higher dimensions, by T. Banchoff. Req.
Readings for Week 5 and the first half of Week 6, October 4, 6 and 11.
- The book's Section 2.3.3, pages 45 to 47, on Zeno's
paradoxes.Req.
- The MacTutor biography of
Zeno of Elea Req.
- The book's Chapter 4, sections 4.1, 4.2 and 4.3, on Archimedes.Req.
- The MacTutor biography of
Eudoxus of Cnidus Req.
- The MacTutor biography of
Archimedes of Syracuse Req.
- The article
Sums of numerical powers in discrete mathematics: Archimedes sums
squares in the sand, by David Pengelley.Req.
- The Wikipedia article on
Archimedes' The Sand Reckoner, in which Archimedes
tries to estimate an upper bound for the number of grains of sand ithat fit
into the universe, and is forced to invent a way to talk about very large numbers.Req.
- The book's Chapetrs 15 and 16, on the Beginnings of Calculus, Newton
and Leibniz.Req. Especially important: Section 16.1.7 on
"newton and celestial PHysics".
- The MacTutor biographies of
Newton Req and
Leibniz Req.
- The MacTutor biography of
Robert Hooke Req
- The note by Charles Bossut on
Leibniz and Newton Req.
- The article on
The Rivalry between Isaac Newton and Robert Hooke.
Req. It is especially imprtant that you should read carefully
the section called "Credit for Explainign Gravity".
Readings for October 25-27, and November 1-3-8-15
- The book's Chapter 24, on "Geometry in the 19th
Century" (omitting Sections 24.3 and 24.4).Req
- The MacTutor biographies of
Carl Friedrich Gauss Req
and
Bernhard Riemann Req
- The Wikipedia articles on
Gauss Req and
Riemann Req.
- The MacTutor history of
Non-Euclidean Geometry. Req. (Yes, this was an
assigned reading before, but I want yo to read it again.)
- The MacTutor article on
Orbits and Gravitation. Req. (This contains a lot
material that was discussed in class, such as Kepler's laws, Newton's
law of gravitation, Newton's original incorrect idea about motion
spiraling towards the Earth, and how it was corrected By Hooke (this was
not said in class), the story about Halley's conversation with Newton,
an explanation of how the N-body problem for N greater than 2 is
so hard, including Newton's statement that an exact solution for three bodies
"exceeds, if I am not mistaken, the force of any human mind," the story of
Halley's comet, and, most importantly, the sensational story of the discovery of
Neptune in 1846 by LeVerrier, Galle, and perhaps Adams, and much more. Req.
- The article
A Short History of the Fourth Dimension, by Stephen M. Phillips. Req.
- The article on
Higher dimensions, by Thomas Banchoff. Req.
- The Wikipedia article
on
Dimension Req, skipping the last six sections of Part
(from "varieties" to "HIlbert spaces"), and also skipping Part 3. (NOTE:
The short discussion of Kant's ideas about space in the section "in
philosophy" is very important. By all means, do no skip it.
NOTE: If you Google up the words "Higher dimensions", you will see that the
Web is full of garbage talking about the "mysteries" of higher dimensions,
how high dimensions have something to do with spirituality, "high
dimensions of consciousness", "multidimensional consciousness", and other
nonsense.
Please keep in mind that there is
nothing mysterious about higher dimensions! If I make a model about
the interaction of eleven species (for example, wolves, foxes, coyotes,
grizzly bears, black bears, salmon, beavers, elk, deer, ravens, golden eagles),
then this will give rise
to a system of differential equations in eleven variables (say,
w, f, c, g, b, s, x, e, d, r, y, where I am using "x" for the beavers and "y"
for the golden eagles, because "b" already stands for "black bears" and
"g" already stands for "grizzly bears"). So the evolution of the
system will be described by a point
Q(t)=(w(t),f(t),c(t),g(t),b(t),s(t),x(t),e(t),d(t),r(t),y(t)), depending on t, in
eleven-dimensional space. And if I make a model about the motion of the
Sun and the eight planets, then I need nine points in ordinary space,
and each point will have three coordinates, so there will be a total of
27 coordinates, and the system behavior will be described as
the motion of a point in 27-dimensional space!!!!. Wow! Isn't this
mysterious? Maybe all this stuff about high dimensions is the key to
establishing contact with cosmic consciousness, high universal spirituality,
beings from another dimension? NOT AT ALL! IT'S QUITE SIMPLE!
- The Wikipedia article on
The Lotka-Volterra equations.
Readings for November 29, and December 1-6-8
- The Mac Tutor biography of
Hypatia of Alexandria. Req
- The Wikipedia article on
Hypatia. Req
- The article
on Living in a man's world: The untimely & brutal death of Hypatia.
Req
- The Wikipedia article on
Spacetime. Req
- The Wikipedia article on the
Twin paradox. Req. I recommend that you pay special attention to the
section entitled "Relativity of simultaneity", and to the picture
labeled "Minkowski diagram of the twin paradox". Just read up to
the beginning pf the section called "A non space-time approach". (You
could try to read the rest too, but you don't have to, and you may find
it a bit too hard.)
- The Wikipedia article on the
Irrational numbers. Req, This is very important.Pay special attention to the
proof of the irrationality of the square root of 2, given in geometric
language. Ignore the section entitled "example proofs".
- The short article
Why is the square root of 2 irrational?. Req,
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