Fall '11Class Summaries/Notes [Based on Previous Courses] IN REVISION

1.1 |
1.2 |
2.1 |
2.2 |
2.3 |
2.4 |
2.5 |
3.1 |
3.2 |
3.3 |
3.4 |
4.1 |
4.2 |

- 1.1 Introductory Class. A discussion of details from the Course Description.
**Assigments are due on Thursdays.**[Ask relevant questions on Tuesdays.]

- Reading reports are due on Alternate Tuesdays.
- Possibilities for finding information about the history of Mathematics on the web.
- In particular -as an example, find out a little about Robert Recorde.
**Useful reading resources**- some available on-line:*The College Mathematics Journal**The Mathematics Magazine**The Mathematics Teacher**The Mathmatics Gazette*

*The American Mathematical Monthly**Isis**Convergence**and Osiris*.

- The course content deals mainly with
*mathematics and history*based on documented information.

- Generally I will try to avoid speculative history
and historical explanations in terms of a progression
leading toward some preferred current state of the world and
our understanding of knowledge.

- 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 start in the middle- with a discussion of Oresme and his
visualization of qualities and intensities.

- A prelude to Oresme and Cardano

al-Khayyami (~1048-1131 C.E.)

- Cardano [text] solves cubics with geometric arguments.
- Artis Magnae sive de regulis algebraicis [ in operaomnia/vol_4_s_4.pdf]
- http://www.math.nmsu.edu/~history/book/cardano.pdf

- Cardano (JavaSketch figure for x^3 + 6x=20)
- Cardano, Girolamo, 1501-1576 Great art; or, The rules of algebra. Translated and edited by T. Richard Witmer. With a foreword by Oystein Ore. 1968
- Cardano,
Girolamo Other works on-line.

- Cardano's method from Wikipedia
**The Interactive Web Book: Cardano-****Cardano's Method**- R.W.D. Nickalls (1993). A new approach to solving the cubic:
Cardan's solution revealed,
*The Mathematical Gazette,***77**:354–359. - Dave Auckly, Solving the quartic with a pencil American Math Monthly 114:1 (2007) 29--39
- TMME, Vol2, no.1, p.65

The Montana Mathematics Enthusiast, ISSN 1551-3440

Vol2, no.1, 2005 Ó Montana Council of Teachers of Mathematics

Cardano’s Solution to the Cubic : A Mathematical Soap Opera by KaCee Ballou, Meadow Hills Middle School, Missoula, Montana - Cardano, Viete and algebra

- Stevin's decimal's A
translation of Stevin's
*La Thiende*[ work in progress? ] Stevin, Simon

- Euclid's proof of the pythagorean theorem (on-line) Proposition 47 . Note the key aspects of the argument related to statements that triangles with equal bases between the same parallel lines would be equal. This follows from Proposition 35.

More on Euclid. Why did the work avoid measurements and numbers? What is the theory of proportions? How was it used? Areas and proportions.

- Euclid. Why study the "source?"

- 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?
- 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).

- Euclid's tools. Proposition 1 and Proposition 2.
- Euclid's elementary approach to area equality in Book I:
- Showing the sqrt of 3 is irrational by reviewing
demonstration that sqrt of 2 is irrational.

- Also disuss 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.
- Euclid's "geometric algebra" in Book II:

Proposition
1. Proposition
4. Proposition
7. Proposition
14. - Key definitions in Euclid's Theory of Proportion. Book V.

The concepts of ratios and proportions.

Definitions in Book V: def'ns 1-5.

- Euclid's (Eudoxus) Theory of Proportion Books
V&VI.

This book contains much on similar triangles.

Proposition 1.Proposition 2, 3 and 4 as well.

Not covered: Proposition 19.[Similar triangles and duplicate ratios]

- Initial discussion of some Greek mathematics:
- Consider the question of the quadrature of lunes.

- 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".
- Reference (on-line):The Quadrature of the Circle and Hippocrates' Lunes: some elements of Greek geometry.

More on Greek mathematics and the distinction between geometry and arithmetic- measurement without numbers!

- 1.2 We looked at Problems related to the Pythagorean Theorem - dissections from Eves.
- More on 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.

- Video: The Emergence of Greek mathematics [Euclid from BBC Uppen University]VIDEO3173 (26 min.)

- 2.3 Numbers and Rational things...
- Book VII
- Definitions (22)
- The division algorithm: VII.1
- The Euclidean Algorithm.VII.2
- Book X

- 2.4

- Details on the proof of Book XII Proposition 1 Proposition 2 using Euclid's version of the Eudoxian method of Exhaustion .

- 2.5

- Euclid on Circles and tangency in Book III.
- Finding the center of a circle. Secants for circles- "convexity".Proposition 1.Proposition 2.
- Circles meeting circles:Proposition 5 .Proposition 6 . Proposition 10.Proposition 11.Proposition 12.
- Circles and Tangent Lines: Proposition
16. Proposition
17.

- 3.1 On Exhaustion with a comparison with Archimedes on Circles.
- Examine Euclid's arguments again in summary, connecting the theory of proportions with the method of exhaustion.
- Archimedes
on
the
Circle:

- Proof that the circle has the same area as a right
triangle with base the circumference and altitude the radius
of the circle.

- Comparison between Archimedes and Euclid's use of
exhaustion and the theory of proportions.

- Read the actual work in measuring the circle's circumference in Archimedes' Approximation of Pi (Chuck Lindsey, Florida Gulf Coast University)

- 3.2 Archimedes
- Selections
from Vignettes of Ancient Mathematics

by Henry Mendell, Cal. State U., L.A. - The
Sand-Reckoner
(complete translation) or go directly to Ch.
1 or Ch.
2, or Ch.
3, or Ch.
4

Quadrature of the Parabola (complete translation).

Basic Lemmata for pre-Archimedean theory of orthotomes (a.k.a. parabolas) based on Quadrature of the Parabola 1-5

On the Equilibrium of Planes 6-7: weights balance in inverse proportion to the distances from the fulcrum.

Archimedes uses the lever in the quadrature of the parabola Prop 14/15

Archimedes: Quadrature of the Parabola 24: geometrical quadrature of the parabola -
3.3 The Archimedes Palimpsest

How Archimedes is lost and then recovered: The Palimpsest.

Archimedes uses physics - the lever in the quadrature of the parabola.

Discussion of the Method and the use of areas to determined volumes -
Method Prop 1: Finding the area of a parabolic section by balancing.

Method Prop 2: More balacing to compare the volumes of a sphere, circumscribed cylinder, and inscribed cone (with a great circle of the sphere as base)

Compare with Euclid:bookXII/propXII10.html and Archimedes*On the Sphere and Cylinder.*

On Conoids and Spheroids 1: a basic proportion theorem

- and sums: Spirals Prop.10.
- on the tangent to a spiral.Preface: On Spirals. Definitions and Propositions (without proofs)
- uses "analysis" to solve a problem on Spheres II Proposition 3
- 3.4 Motion and The Infinite: What about motion? Aristotle and Zeno.

- 4.1 Post "Greek" Transitions:
- Begin to look at the transition to the Renaissance. Oresme on figures.Oresme on means.
- What was know about cones:
- Watch the video The Theorem of Pythagoras (available at the library Video # 950) which illlustrates how history could be woven into a high school level treatment of the Pythagorean theorem.
- Euclid on Cones. Book XI definitions.
- How conics can be used to solve the duplication of the cube problem.
- Conics video: describes 19th century approach to conic results known to the Greeks.

- 4.2 Non- European contributions:

- al-Khwarizmi (~780-847 C.E.)

- Algebra and geometry [completing the square to solve a quadratic equation.]

- Student presentations on notation for numbers in different cultures
- Robin
Wilson
Lecture: Who invented Algebra

- Arabic mathematicians embraced the mathematics of Ancient Greece and India. What did they do, and how did their achievements influence Europe in the Middle Ages? We trace the story up to the establishment of universities, the development of perspective in art, and Fibonacci’s problem of the rabbits.
**Robin Wilson:**Who invented the equals sign?- With the invention of printing, mathematical writings became widely available for the first time. What influence did this have? We discuss this question in the context of 16th-century navigation and astronomy, the solving of equations, and some breakthroughs in geometry and algebra, and ask: is this a record?
- Watch the open university History video about the development
of notations through "the vernacular tradition."

**The Vernacular Tradition**- Deals with the low-level mathematics of the Middle Ages. Compares the different notational styles of Luca Pacioli and Nicholas Chuquet. Shows how the use of the Hindu-Arabic numeral system developed and was adopted in Arabic countries and later in Europe. Traces this through the work of the Islamic mathematician Al-Kharizmi and Leonardo of Pisa (also known as Fibonacci).

- Another look at Oresme? This time looking at some of his work on the infinite.
- The changes in European institutions. Developing commerce, secular government, and education.

- Napier's
logarithms .

- Stevin does division. Napier's logarithm tables again. How does Napier compare with modern logarithms? Previous methods for doing products using trigonometry.

- Kepler on the volume of a torus. Kepler video.
- Galileo.Galilei, Galileo
- Descartes- theory of equations (rule of signs) and the algebra of geometry with lines (elimination of homogeneity)
- Langland
lectures - including excerpts from Descartes.

- Fermat and Mersenne (mathematical culture in the 17th century)
- Newton,
Isaac.