MATH 401 History of Mathematics
Fall '98      Class Summaries
 8/25 27 9/1 9/3 8 10 15 17 22 24 29 10/1 6 8 13 15 20 22 27 29 11/3 5 10 12 17 19
• 8/25 Introductory Class. We discussed some details of the Course Description.

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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: The College Mathematics Journal, The Mathematics Magazine, The Mathematics Teacher, The Mathmatics Gazette, The American Mathematical Monthly, Isis, and Osiris.

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.

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After the video we went over Euclid's proof of the pythagorean theorem Proposition 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 35.

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

• 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.

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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.

1. Euclid's tools. Proposition 1 and Proposition 2.
2. Euclid's elementary approach to area equality in Book I:

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Before parallels: Proposition 4. Proposition 8. [Compare with Proposition 26. ]
After parallels: Proposition 35. Proposition 36. Proposition 37. Proposition 38.
Different but equal figures: Proposition 42. Proposition 43. Proposition 44. Proposition 45.

4. Euclid's "geometric algebra" in Book II:

5. Proposition 1. Proposition 4. Proposition 7. Proposition 14.
6. This is about where we stopped....Euclid on Circles and tangency in Book III.

7. 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.
• 9/3 We continued directly to examine key definitions in Euclid's Thoery of Proportion. Book V.

• 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 triangles. Proposition 1.Proposition 2.Proposition 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 1 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 on Circles.

• 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 1 Proposition 2 .
• 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 2.
• 9/17 Euclid on commensurable and rational.- Book X  Proposition 1 and  Proposition 2. Euclidean Number Theory- Book VII.
• ### Euclid on tangents to the Circle.

; Proposition 16 -19.
• 9/22 Archimedes and sums: Spirals Prop.10.

• 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 3.

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Euclid on Cones. Book XI  definitions.
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.

• 10/1 Duplication of the cube again?  (GSP)Archimedes uses the lever in the quadrature of the parabola.
• 10/6 What about motion? Aristotle and Zeno. [Note there are other links, e.g., ksbrown on Zeno.]

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al-Khwarizmi (~780-847 C.E.) Algebra and geometry [completing the square to solve a quadratic equation.]
al-Khayyami (~1048-1131 C.E.) Algebra (cubics)

• 10/8 A last look at conics video.
• 10/13 Beginning to look at the transition to the Renaissance. Oresme on figures.Oresme 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.
• 10/22 Cardano (JavaSketch figure for x^3 + 6x=20)  Cardano, Viete and algebra (Karen Parshall's paper), Stevin.

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• 10/27 Stevin's decimal's A translation of Stevin's La Thiende [ work in progress? ] and Napier's 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 video.
• 11/12 Galileo.
• 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).

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