MATH 401 History of Mathematics
Fall '11Class Summaries/Notes [Based on Previous
Courses] IN REVISION
- 1.1 Introductory Class. A discussion
of details from the Course
- 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
- Useful reading resources - some available on-line:
- The College Mathematics Journal
- The Mathematics Magazine
- The Mathematics Teacher
- The Mathmatics Gazette
- The American Mathematical Monthly
- 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
- 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.
sive de regulis algebraicis [ in operaomnia/vol_4_s_4.pdf]
- Cardano (JavaSketch figure for x^3
or, The rules of algebra. Translated and edited by T.
Richard Witmer. With a foreword by Oystein Ore. 1968
Girolamo Other works on-line.
method from Wikipedia
Interactive Web Book: Cardano-Cardano's
- R.W.D. Nickalls (1993). A new approach to solving the cubic:
Cardan's solution revealed, The Mathematical
- 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
to the Cubic : A Mathematical Soap Opera by KaCee
Ballou, Meadow Hills Middle School, Missoula, Montana
- Cardano, Viete and algebra
- 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
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
- It is worthwhile to keep the following two questions in mind
- 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 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:
- 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
of the Circle and Hippocrates' Lunes: some elements of
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
- 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
- 2.3 Numbers and Rational things...
- Euclid on Circles and tangency in Book III.
- 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.
- 3.2 Archimedes
- 3.4 Motion and The Infinite: What
about motion? Aristotle
- 4.1 Post "Greek" Transitions:
- Begin to look at the transition to the Renaissance. Oresme
- 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
- Euclid on Cones. Book XI definitions.
- How conics can be used to solve the duplication of the
- 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
- Student presentations on notation for numbers in different
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.
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
- 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.
- 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