MATH 401 History of Mathematics
Fall '07Class Summaries/Notes [Based on Previous Courses]
 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.
• 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.

Next time:  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.)
• After the video: 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.

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.

• 2.1 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).

• 2.4

• 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.

• 4.2 Non- European contributions:
• al-Khwarizmi (~780-847 C.E.)
• Algebra and geometry [completing the square to solve a quadratic equation.]
• al-Khayyami (~1048-1131 C.E.)
• 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.

Materials for Future Classes:

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