Math
160 · Introduction to History of Mathematics
TuTh
11-12:30 9 Evans
Hall
Final Exam Group 13
Final Exam: Wednesday
May 16, 8-11 am (you
should not enroll in this class if you cannot take the final exam at
the scheduled time; there will be no
make-up exams, neither
Final nor Midterms)
Office Hours: Thursday 5-6:30
Prerequisites: Math 104, 110 and 113
Compared
to the Midterms, I expect you to demonstrate on the Final a deeper
knowledge of the
subject which goes beyond knowing just a few names and work titles.
Below is a
detailed syllabus. Please spend several hours rereading the sources and
the biograms of the mathematicians mentioned below. Use also your class
notes (e.g., in the class I spent several hours talking about main
characteristics of the Medieval period in the development of sciences
and education).
Syllabus
for the Final:
Egyptian, Babylonian, and Greek Mathematics -- the
same requirements as for
the two midterms, plus the contributions of Archimedes,
Apollonius of Perge, Diophantus (complete knowledge of all integral
solutions to
the Pythagoras Equation x²+y²=z²; you should know these solutions), and Pappus (Pappus' Theorem);
biograms
from the St Andrews site are very useful source of information.
Arabic Mathematics -- my handout,
biograms of central figures
Medieval Europe:
the
institution of university (its main characteristics, trivium,
quadrivium)
Leonardo Pisano (Fibonacci)
16th Century Italy: solving equations of 3rd and 4th
degree (my
handout); main figures of the drama:
Scipione del Ferro, Nicolo Fontana
Tartaglia, Girolamo
Cardano, Lodovico Ferrari
(biograms)
François Viète (biogram, you should be familiar with his
achievements)
A circle of French scholars (the first half of the 17th Century):
Girard Desargues (Desargues' Theorem), Marin Mersenne, Etienne Pascal,
and his son Blaise Pascal (Pascal's Theorem), Pierre de Fermat,
Renée Descartes (biograms; you should be familiar with their
achievements).
For Tuesday, May 1:
Read the following short account of Mathematics
during the early islamic period that brought to the end Antiquity in
the Middle East, Persia and North Africa.
One of the favorite problems of that period was finding roots of cubic
equations. Such a solution was first found only in the Sixteenth
Century in Italy. Here is a short and detailed discussion of the solution.
The following account of the
fates of the text of the Elements
in the Middle Ages was written by one of the top experts in the
subject. Knowing how the transmission of Ancient mathematical
knowledge was happening is crucial to our understanding of the development of
Mathematics from th e close of Antiquity until early modern
times.
Continue, please, playing with Pappus, Desargues', and Pascal, Theorems
using KSEG. I certainly will devote some time to them, as much as they
reflect the continuity of geometric thought from Antiquity to 17-th
Century Europe (read biograms of Pappus, Desargues and Pascal on the St
Andrews site).
Download and unpack KSEG, a free interactive geometry software, (Linux, Windows), and read a short introduction
kseg_help_en.html first. Installation of the Windows package is
extremely simple: just unzip and click on kseg.exe in the unpacked
folder.
Then dowload also the following zipped folder containing source files
for KSEG which illustrate Theorems of Pappus, Pascal, and Desargue.
Each of the files in that folder has an extension .seg, and can be
simply opened inside of KSEG. Bring your notebook to th eclass on
Tuesday.
What you should know and prepare
for the Midterm that will take place April 16:
Sources:
- Chapter
3 from Kline
- Volume I of Ivor Thomas and
Chapter on Archimedes from Volume II
- biograms of Pythagoras,
Thales, Plato, Aristotle, Eudoxus, Zeno, Euclides, Archimedes at Saint
Andrews site
- Chapter VI (paragraphs 60-63
and Appendix II (paragraphs 84-85 and 90) in Neugebauer
- Fitzpatrick's edition of Euclid's Elements
- my mathematical commentary
to Book I of the Elements
You should be familiar with:
- stages of the development of
Greek Mathematics from the earliest times to Archimedes, principal
figures, and their main achievements
- Euclid's geometry of
the plane (Book I of the Elements
and my mathematical commentary
on it)
- contents of the
remaining books of the Elements
- what method was introduced by
Euclid in Books XI-XIII of the Elements
and perfected later by Archimedes
- titles of Archimedes' main
works
For Thursday, March 8: please go through Book I again and, for each
proposition, write down its assertion
(your statement must be formulated in modern mathematical
language). Consider this to be your homework assignement.
Euclid's Elements --
an edition prepared by Richard Fitzpatrick who is professor of physics
at the University of Texas. Print out the first 3 books (English text
facing Greek original) and prepare Chapter I for our Thursday class.
Please bring all three chapters to the class.
Your colleagues found this valuable link to the text of Hilbert's The
Foundations of Geometry. It is not a scan but a well executed
online re-edition in LaTeX (the full LaTeX sources are in a zip file at
the bottom of the page. They are well worth studying as an example of
how to use LaTeX). I corrected a few errors and replaced the fonts by
a lot more attractive ones and it is -- my own
PDF
file of Hilbert's book.
We have entered the world of Greek Mathematics. Please print
this out (rotate it clockwise for
viewing), read pages 24-37, and also read the first four chapters from
Ivor Thomas's Greek Mathematics, Vol. 1. If you
have any general textbook of History of Mathematics, like , read
what it has to say about the early stages of Greek Mathematics. Read
also the following useful overviews prepared by Donald Allen:
The Saint
Andrews site on History of Mathematics does not contain general
introductions to Greek Mathematics, its strength being rather biograms
of individual mathematicians, and discussions of problems that occupied
ancient Greeks.
By now you shoud be thoroughly familiar with at leat the first four
chapters of Neugebauer's short book The
Exact Sciences in Antiquity. A high quality image of the
September page from the calendar discussed by Neugebauer in Chapter 1
you can find here.
Your familiarity
with Chapters 2 and 3 will be tested on the Midterm
that will take place Tuesday, February 21 but the Chapter 1 is useful
in general, while very short Chapter 4 provides another useful summary
of Egyptian Mathematics.
I also suggest that you now try working out on yor own line by line
mathematical procedures in other problems of the sort we covered in
class. Next Tuesday we will be finishing our encounter with Egyptian
Mathematics by looking at some problems from Chapters 17 and 18 in
Gillings, so be prepared.
This
site contains a
brief and apparently reliable collection of useful summaries about
Egyptian history, language, and spiritual culture. Here
you will find Egyptian "alphabet" (i.e., the list of monoconsonantal,
also called monoliteral signs). Print out the whole
document, study it, and carry it with you - it is only 10 pages long.
Now that I posted all the problems from the available papyri you
are advised to print out the ones
that are discussed by Gillings and you should be carrying them with you to
the class.
On Thursday we will finish problem 34 and discuss problems 28, 29, 40,
79, and perhaps also 4 and 6 from Papyrus
Rhind. Your work with any problem should always start from
identifying which groups of hieroglyphs correspond to which
transliterated
words.
Also:
- you should have completed
reading
Gillings
(I hope you did not forget about Appendices: they contain a wealth of
useful
information)
Our
sources
Papyrus Rhind: 1, 2, 3, 4, 5, 6, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44,
45,
46,
48,
49,
50,
51,
52,
53-54,
55,
56,
57,
58,
59,
60,
61b,
62,
63,
64,
65,
66,
67,
68,
69,
70,
71,
72,
73,
74,
75,
76,
77,
78,
79
Papyrus Moscow: 1, 2, 3, 4, 5, 6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25
Lahun Papyrus (known also as
Kahun Papyrus): UC32118B,
UC32134A,
UC32134B,
UC32159,
UC32160, UC32161, UC32162, (introduction
to Lahun mathematical texts)
Papyrus Berlin 6619: 1,
2,
3
Ostrakon
Turin 57170
A very useful short modern survey
of Egyptian Mathematics written
by one of the most knowledgeable persons in the field (print it out, study
it, and bring it to
the class)
A dictionary
of words and phrases indispensable when studying Egyptian mathematical
texts (print
it out, study it, and bring it to
the class)
A superb introduction
to Egyptian Hieroglyphs (get it! - an
exciting read, very highly recommended). [amazon.com offers it with E.
A. Wallis Budge's An
Egyptian Hieroglyphic Dictionary. Do not buy
the latter - it is very unreliable.] The following page from it
contains the list
of transliterated letter-signs and tells you how to pronounce them.
Digital Egypt for
Universities (contains a valuable section on Exact
Sciences)
Further
Reading
Required Texts: We
will begin from the following texts
- Richard Gillings, Mathematics
in the Time of the Pharaohs, Dover Publications
- Otto Neugebauer, The
Exact Sciences in Antiquity, Dover Publications
- Ivor Thomas (editor), Greek Mathematical Works I/II, Loeb
Classical Library, Nos. 335 & 362, Harvard University Press
Recommended Reading: Carl B. Boyer, A History of
Mathematics, John
Wiley & Sons
Syllabus: In an introductory course on History of
Mathematics nothing can
replace a first hand experience of working with original texts. We will
focus on a number of such texts from various historical epochs. This
will be supplemented by my lectures in which I will be presenting a
panoramic overview of the development of Mathematics in its cultural
perspective.
Comments: Students are
expected to attend the class regularly.