MATH 401 History of Mathematics
Fall '98 Class Summaries
8/25 Introductory Class. We discussed some details of
Notice we didn't discuss the grading scheme yet.
Assigments are due on Thursdays. Reading reports are due on Tuesdays.
We looked at many possibilities for finding information about the history
of Mathematics on the web. In particular we did some work on finding out
a little about Robert Recorde.
Some reading resources that were not mentioned but are of some use:
College Mathematics Journal, The Mathematics Magazine, The Mathematics
Teacher, The Mathmatics Gazette, The American Mathematical Monthly, Isis,
The course content deals mainly with mathematics and history based
on documented information. Generally I will try to avoid speculative history
and "whig" historical explanations. We will deal mainly with mathematics
related to the development of the calculus, trying to understand it in
its own terms and context as well as relating it to current views.
We began a discussion of some Greek mathematics by considering
the question of the quadrature of lunes.
Looking at my web site we saw how in at least one context it is possible
to find a triangle with the same area as a special lune shape, thereby
squaring that particular lune.
Next time: More on Greek mathematics and the distinction
between geometry and arithmetic- measurement without numbers!
8/27 First we looked a little more ar the lunes
quadrature problem. In both of the figures the use of the pythagorean theorem
(for isosceles right triangles at least) was a key to the justification
of the ability to identify the area of the lune with the area of the related
triangle. We then watched the video The Theorem of Pythagoras (available
at the library) which illlustrated how history could be woven into a high
school level treatment of the Pythagorean theorem.
After the video we went over Euclid's proof of the pythagorean theorem
47 noting the key aspects of the argument related to statements that
triangles with equal bases between the same parallel lines would be equal
which follows from Proposition
Next time: More on Euclid. Why did the work avoid measurements
and numbers? What is the theory of proportions? How was it used? Areas
9/1 We discussed first the problem from Katz on showing
the sqrt of 3 is irrational by reviewing Katz's demonstration that sqrt
of 2 is irrational. A suggestion was made to use a triangle or rectangle
with sqrt or 3 as one side. [We also disussed how a proof might proceed
in a math course using an indirect argument and the fundamental theorem
of arithmetic with regard to counting the factors in squares.
We then discussed a basic question:
Why should we spend time trying to understand Euclid's proofs?
It is worthwhile to keep the following two questions in mind as well:
How does Euclid's work differ from current approaches to the same topics?
Does Euclid's work present mathematics as a science, a platonic reality,
or a complex axiomatic structure?
After a brief overview of what we will examine in Euclid (mostly things
related in some way to the development of the calculus- area, tangents,
and numbers) we followed the follow outline:
We had some difficulty with Java on some of these.
9/3 We continued directly to examine key definitions
in Euclid's Thoery of Proportion. Book V.
Euclid's tools. Proposition
1 and Proposition
Euclid's elementary approach to area equality in Book I:
Before parallels: Proposition
8. [Compare with Proposition
After parallels: Proposition
Different but equal figures: Proposition
Euclid's "geometric algebra" in Book II:
This is about where we stopped....Euclid on Circles and tangency in Book
Circles meeting circles:Proposition
6 . Proposition
Circles and Tangent Lines: Proposition
Euclid's (Eudoxus) Theory of Proportion Books V&VI.
Definitions in Book
V: def'ns 1-5.
9/8 We began by some discussion of the problems for next
class. The actual Euclidean Propositions referenced in Katz were not in
the text so we looked at them briefly on the web using Joyce's web site
at Clark University.
We then reviewed the concepts of ratios and proportions before considering
in some details material from Book VI. This book contains much on similar
19.[Similar triangles and duplicate ratios]
We then watched a BBC- Open University video on Euclid that included
a survey of all the books in the elements and some details on the proof
of Book XII Proposition
2 using Euclid's version of the Eudoxian method of Exhaustion
as well as material from books V,VI, and X.
Next time: More On Exhaustion with a comparison with Archimedes
9/10 The next few lectures develop acomparison of Euclid
and Archimedes on the Circle. Today we examined Euclid's arguments in some
detail, connecting the theory of proportioins with the method of exhaustion.
Book XII Proposition
9/15 We discussed in some detail Archimedes
on the Circle and his proof that the circle has the same area
as a right triangle with base the circumference and altitude the radius
of the circle. This discussion involved some comparison between Archimedes
and Euclid's use of exhaustion and the theory of proportions. The actual
work in measuring the circle's circumference was left to be read in Archimedes'
Approximation of Pi (Chuck Lindsey, Florida Gulf Coast University)
In the second hour we looked at some problems from Katz and briefly
at some of Euclid's Book
X and the statement of Proposition
9/17 Euclid on commensurable and rational.- Book
1 and Proposition
2. Euclidean Number Theory- Book
Euclid on tangents to the Circle.
9/22 Archimedes and sums: Spirals
Archimedes on the tangent to a spiral.Preface:
On Spirals. Definitions
and Propositions (without proofs)
9/24 More on the tangent to a spiral. Archimedes
uses "analysis" to solve a problem on Spheres
II Proposition 3.
9/29 Continued with Archimedes on spheresII Proposition
Euclid on Cones. Book
10/1 Duplication of the cube again? (GSP)Archimedes
uses the lever in the
quadrature of the parabola.
10/6 What about motion? Aristotle
[Note there are other links, e.g., ksbrown
Apollonius on Conics.
We looked at the definitions and distinguished these from Euclid's definition.
Archimedes uses physics - the lever in the quadrature of the parabola.
A brief discussion of the Method and the use of areas to determined
volumes. We also mentioned how conics can be used to solve the duplication
of the cube problem. We'll try this again perhaps. Next class. Math and
the Greek philosophers.
al-Khwarizmi (~780-847 C.E.) Algebra
and geometry [completing the square to solve a quadratic equation.]
10/8 A last look at conics video.
10/13 Beginning to look at the transition to the Renaissance.
on means. We also watched the open university History video about the
development of notations through "the vernacular tradition." This video
is in the library.
10/15 Another look at Oresme. This time looking at
some of his work on the infinite.
10/20 The changes in European institutions. Developing
commerce, secular government, and education. Cardano
[text] solves cubics with geometric arguments.
(JavaSketch figure for x^3 + 6x=20) Cardano,
Viete and algebra (Karen Parshall's paper), Stevin.
al-Khayyami (~1048-1131 C.E.) Algebra
10/27 Stevin's decimal's A
translation of Stevin's La Thiende [ work in progress? ] and
logarithms (in html from the Netherlands).
10/29 Stevin does division. Napier's logarithm tables again. How does Napier
compare with modern
logarithms? Previous methods for doing products using trigonometry.
11/3 Student presentations on notation for numbers in different cultures.
11/5 More Student presentations.
11/10 Final student presentations. Kepler on the volume of a torus. Kepler
11/17 Descartes- theory of equations (rule of signs) and the algebra of
geometry with lines (elimination of homogeneity)
11/19 Fermat and Mersenne (mathematical culture in the 17th century).