MATH 401 History of Mathematics
Fall '07Class Summaries/Notes [Based on Previous Courses]
1.1 Introductory Class. A discussion of details from
- 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:
College Mathematics Journal
- The Mathematics Magazine
- The Mathematics
- The Mathmatics Gazette
- The American Mathematical Monthly
- and Osiris.
- The course content deals mainly with mathematics and history based
on documented information.
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
- 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
- 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
- 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)
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
Next time: More on Euclid. Why did the work avoid measurements
and numbers? What is the theory of proportions? How was it used? Areas
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,
Euclid's elementary approach to area equality in Book I:
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:
- 2.3 Numbers and Rational things...
- Euclid on Circles and tangency in Book
- 3.1 On Exhaustion with a comparison with Archimedes
- Examine Euclid's arguments again in summary, connecting the theory of proportions with the method of exhaustion.
- 3.2 Archimedes
- 3.3 The Archimedes Palimpsest
- Archimedes uses physics - the lever in the quadrature of the
- Discussion of the Method and the use of areas to
- 3.4 Motion and The Infinite:
What about motion? Aristotle
- 4.1 Post "Greek" Transitions:
- Begin to look at the transition to the Renaissance.
- What was know about cones:
- How Archimedes is lost and then recovered: The Palimpsest.
- 4.2 Non- European contributions:
- al-Khwarizmi (~780-847 C.E.)
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
- 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.
[text] solves cubics with geometric arguments.
- Artis Magnae sive de regulis algebraicis [ in operaomnia/vol_4_s_4.pdf]
(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.
- Cardano, Girolamo Other works on-line.
- Cardano's 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 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
Viete and algebra
Materials for Future Classes:
- Kepler on the volume of a torus. Kepler
- 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.