Mathematics 629, Spring, 2015
Last updated Fri 15 May 15
Here is the (tentative) course handout
(syllabus). The first thing a student should do is to acquire the
book by Stillwell. Be sure to get the current (third) edition. I did
not list an ISBN because, according to
Springer's web page,
the book is
available in hardback, paperback, and ebook formats (but the paperback
costs as much as the hardback).
New feature: Jump down to current week's
material (or close to it)
ANNOUNCEMENTS
5/15: Truly concluding remarks
5/6: Concluding remarks
4/11: Recall (per email) that the position paper is due
next Friday, April 17.
3/21:
News about grading and posting of your
papers
3/10:Thanks to Jennifer Carver and Daniel Brown for
pointing out two glitches:
 Contrary to the Wikipedia page I recommended, it seems that Allen
tacitly defines "normal" to mean "slope of the normal", and similarly for
"tangent". Only with this understanding do his examples and
homework problems make sense. (That is the understanding my 2003
students came up with, before I found the Wikipedia page.)
"Subnormal" and "subtangent" DO refer to lengths (possibly signed?).
 TAMU's spring break is next week. Therefore, the Calculus Week
homework may be submitted any time up to March 23.
2/22: Instructions for the big term paper: This should be
about 15–20 pages, or a comparable project in another medium. The
topic is your choice, but (like the book) it should involve a
nontrivial amount of history and a nontrivial amount of mathematics.
For example topics (and other help), see the page Some other pertinent links. Ignore "You
should choose a topic by Feb. 23." because I was slow in providing the
examples, but don't wait too much longer.
2/21: Instructions for the second
paper (position paper).
1/23: Instructions for the book
review
1/23:
MAA style sheet for references
1/23: The welcoming message
1/23: Formal course handout
LINKS
The History of Mathematics by G. Donald Allen
©
Old course pages:
Home pages:
Fulling ._._.
Do ._._.
eCampus ._._.
TAMU Library ._._.
Math Dept ._._.
University
email: fulling@math.tamu.edu
Grader: dothanh@math.tamu.edu
(Please send homework to both of us, but especially to Ngoc Thanh Do.)
Some other pertinent links
WEEKLY COURSE MATERIALS
Week 1 (the beginnings of mathematics)
 Allen readings: Chapters 1 and 2
 Stillwell readings: Chapter 1
(If you don't have the book yet, don't worry 
you will easily be able to catch up later.)
 Additional readings: See the "lecture".
 "Lecture"
 Written assignment: Just this 
Send me an email message telling about yourself:
 Where you are. Do you expect to be on the TAMU campus at all during this
semester?
 What your "situation" is: Do you work fulltime, parttime, or not at
all? Doing what? What degree (if any) are you working toward,
and how far along are you in the degree plan?
Any "complications" I should know about?
 (optional) Any other personal information you feel like mentioning.
 What library resources are available to you.
 Have you taken a Webbased or distancelearning course before?
 What you hope to get out of this course.
I hope that you will write this selfintroduction in a form that will be
suitable for distribution to the other students in the class.
If some or all of it is for my eyes only, please so indicate.
Whenever you're ready, you can repost it to the "Introductions" forum on
our Discussion Board in eCampus.
(Use the "Week1" forum for any comments and questions on the
first week's material, and so on.)
Week 2 (Egypt, Babylon, and an overview of Greece)
 Allen readings: Chapter 3 and Sections 4.1 and 4.144.16
 Stillwell readings: Chapter 2
 "Lecture"
 Written assignment:

From the
exercises on Egyptian mathematics:
 2 (multiplication)
 4 (unit fractions)
 5 (false position)
 8 [3 persons; the 4 is a misprint] (proportional distribution)
 11 (pedagogy by example)
 (W) From the
exercises on Babylonian mathematics (same web page but farther down):
 1 [1265 only] (sexagesimal)
 3c (multiplication)
 7 (linear systems)
 11 [first sentence only; the rest is a misprint] (cubic)
 12 (interpolation)
Week 3 (the early and classic Greek periods)
 Allen readings: Sections 4.34.8 (Postpone Euclid to next week if
necessary. There was no convenient division point.)
 Stillwell readings: Chapter 3
 Additional readings:
 In Meno type "inquisitive"
into your Web browser's text search window to get quickly to the scene
with the slave boy.
 (optional) Allen's link to an article on Thales by D. V. Panchenko is
broken, but the journal reference is Configurations 1 (No. 3)
(1993) 381414.
For permission to access it on the web you probably must go through the TAMU library site.
 "Lecture"
with addendum on the trisectrix
 Written assignment:
Exercises from Stillwell unless stated to be from
Allen. Stillwell's problems tend to come in pairs, where the first is
a lemma for the second.
 1.2.34 (parity in Pythagorean triples)
 1.3.12 (Euclid's formula for Pythagorean triples)
 Allen Early Greek 5 (irrationality of square roots)
 2.3.2 and Allen Early Greek 12 (constructing square roots)
 2.3.34 (pentagon)
 3.2.3 (pentagonal numbers)
Week 4 (the late Greek period)
 Allen readings: Sections 4.94.13
 Stillwell readings: Chapter 4
 "Lecture"
 Written assignment:
 Allen Classical/Hellenistic 2(a,d) (translating Euclid)
 Allen Classical/Hellenistic 3. Have we seen this proposition before?
(Eudemus)
 Allen Classical/Hellenistic 4(a) (Ionian numerals)
 Allen Classical/Hellenistic 6 (parabola)
 Allen Classical/Hellenistic 16 (mean proportion)
 2.4.34 (reflection = shortest path)
 4.3.23 (volume of pyramid)
Week 5 (Asia, the Islamic period, and the medieval period)
 Allen readings: Chapters 5 and 6 (Recall that he had his say on
China and India in Chapter 1.)
 Stillwell readings: Chapter 5
 "Lecture" with
addendum on the Chinese remainder theorem
 Written assignment
 5.3.12 (Euclidean algorithm)
 5.4.5 OR 5.4.6, depending on whether you feel
"familiar with hyperbolic functions" (If not, change a sign somewhere
and relate the Diophantus formulas to the well known trigonometric
addition formulas.)
 5.6.12 (Brahmagupta triangle decomposition)
 Allen Islamic 2 (Thabit's theorem)
 Allen Islamic 3 (sum of cubes)
 Allen Medieval 2 (Fibonacci solution)
Week 6 (the renaissance period)
 Allen readings: Chapter 7 and Sections 8.18.4
 Stillwell readings: Chapter 6
 "Lecture" with addendum on cubic equations
 Written assignment:
 Allen Renaissance 1 and 2 (solving cubics)
 Stillwell 6.4.15 (cube root of 2)
 Stillwell 6.6.2 (LeibnizDeMoivre formula) (Assume the result of 6.6.1.)
 Stillwell points out that the LeibnizDeMoivre formula is "analogous to the
Cardano formula for cubics". But 3 is not of the form 4m+1! Harmonize these
two observations.
Week 7 (the transition, or precalculus, period)
Week 8 (the creation of calculus)
 Allen readings: Chapter 9 (remainder) (not the 400 pages in German!)
 Stillwell readings: Chapters 9 (remainder), 10, and 13
 Special reading: J. V. Grabiner, "The Changing Concept of Change:
The Derivative from Fermat to Weierstrass", Mathematics Magazine
56 (1983) 195206. (As always, if you have trouble accessing
this, let me know.)
 "Lecture"
 Written assignment: The
first four problems are from
Allen
and refer to the "early calculus"
period that was covered in last week's reading.
These exercises in primitive calculus should be approached in the same
spirit as those in Egyptian fractions: we want to gain some feeling for
how people at the dawn of calculus thought. Note that to them "tangent"
and "normal" did not mean certain lines, but rather certain numbers
(lengths of line segments). See the Wikipedia page
"Subtangent" for clear definitions of these terms along with
"subtangent" and "subnormal". (WRONG  SEE ANNOUNCEMENT ABOVE.)
 3 (subnormal)
 5(b)
(adequality)
 6(a) (extrema)
 7 (de Sluse)
 OK, now let's work
in the other direction and substitute a streamlined modern treatment
of the wave equation for the cumbersome discussion on pp. 262263 of
Stillwell. Follow steps here.
Week 9 (complex numbers)
 Allen readings: none
 Stillwell readings: Chapters 14 and 15 and Sections 12.8,
16.116.3, and 16.7
 "Lecture"
 Written assignment:
 14.6.12 (complex conjugates)
 15.1.1 (n roots of an nth degree equation).
The de Moivre theorem in qustion is formula (2) on p. 281,
discussed in exponential form on pp. 315316.
 15.1.2 (double point). Suggested strategy:
 Sketch the cubic curve (or plot it with a program such as Maple).
Observe that it crosses itself at the origin. (The graph looks like
Fig. 7.1 rotated, except that instead of having a vertical asymptote,
it behaves like y ~ x^{3/2} at infinity.)
 Use implicit differentiation to show that the slopes at the origin are
plus and minus 1.
 Observe that any nonvertical line that passes close to the origin but
not exactly through it has its full Bezout quota of 3 intersections, even
before considering complex or infinite points. (Don't overlook faraway
intersections with the y ~ x^{3/2} curves.)
 [side issue] Make an educated guess about where the missing
intersections are if the line is either (a) vertical or
(b) so far to the left that it misses the loop of the curve.
 Describe what happens to the intersections as the line moves to pass
through the origin. Special attention is needed when slope = 1.
 16.3.13 (CauchyRiemann equations)
Week 10 (modern geometry)
 Allen readings: none
 Stillwell readings: Chapters 17 and 18
 "Lecture"
 Written assignment:
 17.1.45 (tractrix)
 17.2.5 (parametric equations of tractrix)
(Start with the parametric equations as given.)
There are many more tractrix problems throughout these two chapters.
I know better than to assign them all, but after this handson experience
you can at least go back and read them with more appreciation.
 17.7.1 (additivity of angle excess)
THIS ONE ACTUALLY APPEARS IN SECTION 17.6.
 18.2.12 (spherical circles and the negativecurvature analog)
 Within the Klein disk model (cf. Fig. 18.7) draw a sketch showing
that Legendre's axiom (see "lecture")
is false in hyperbolic geometry.
(If you can't scan a sketch, describe in words what you would add to
Fig. 18.7.)
Week 11 (modern algebra)
 Allen readings: Chapter 11
 Stillwell readings: Chapters 19 and 20 (except Sec. 20.7)
and Sections 21.78
 "Lecture" with
addendum on linear algebra
 Written assignment:
 19.4.13 (Cayley's theorem)
 20.5.23 (quaternion norm)
 20.5.4 (dot and cross products)
(NOTE: "Real and imaginary parts" does not refer to the Cayley matrix
representation at bottom of p. 426, because the elements of matrix j
are not imaginary.
As indicated on p. 437, the αI term is considered real and all
the rest (multiples of the three quaternion units by real coefficients)
is considered imaginary.)

Allen 2 (multiplication mod p)
 19.1.1 (inverses mod 5)
Week 12 (analysis and topology)
Week 13 (set theory and combinatorics)
 Allen readings: Chapter 12
 Stillwell readings: Chapter 24 and Secs. 25.Preview, 25.1, and 25.9.
Look briefly at Secs. 25.78 for better understanding of remarks in
Sec. 24.8.
 "Lecture"
 Written assignment:
 24.1.23 (countability of rationals)
 24.1.4 (countability of algebraic numbers)
 24.4.24 (nonmeasurable sets)
 24.5.3 (cardinality of a power set)
 24.6.23 (Turing halting problem)
 Allen 3 (Dedekind arithmetic)
 Allen 4 (describable numbers paradox)
Week 14 (modern education: the math wars)
 Allen readings: none. Instead, we will read another educational debate:
 Stillwell readings: none
 "Lecture"
 Written assignment: none