# Mathematics 390

Bring a copy of this full syllabus to class on Friday, January 22.

Here's my single favourite history of mathematics web site.  Look there for more information about the history of anything in mathematics.

Here's a list of resources that come to my mind quickly:
William Dunham's Journey through Genius is in Milne (QA21.D78 1990).
Ronald Calinger's A Contextual History of Mathematics is another book that connects history of mathematics with the rest of history.  It's also in Milne (QA21.C188 1999).
Browsing the library in the QA21 section in general is a good idea.
Here are some other sources that I think highly of:
Morris Kline - Mathematical Thought from Ancient to Modern Times
Victor Katz - A History of Mathematics:  An Introduction
John Stillwell - Mathematics and Its History
Historia Mathematica
is a journal of history of mathematics - we have this in the library as well.
Ronald Calinger's Classics of Mathematics is a source book of original sources, as is Dirk Struik's Mathematical Source Book
Cajori, Florian, A history of mathematical notations.
I have several more sources, but this should be enough to get you started.  Tell me if you seek something.

Here's a student-timeline up to the end of the first millennium.

Here's a website just of portraits of mathematicians.  Note that it says "Note that portraits of mathematicians from earlier than the fifteenth century are only suggestive."  Once we get ones that are contemporary (and I think we will somewhat before the 15th century), I'll start including them in my discussions.

§1.2

The Rosetta Stone (Rosetta Stone schematic) - one message in hieroglyphic, demotic (later form of hieratic) and Greek.
Here is a photograph of the Rhind Papyrus - to give you a sense of what this relic actually is.
Here is a little bit from the Moscow Papyrus (the bit called problem 1.1 in our text).
Some problems from the Rhind and Moscow - translated.  More detailed version.

§1.3

Here's a historical map of the region.
Babylonian quadratic solution on copy of YBC 6967  Details of original solution.
Bablyonian square root of 2 on YBC 7289 (schematic)

§2.1

Thales, Pythagoras (tuning), Hippasus, Hippocrates lune
On-line piano for interval demonstrations.
A monochord (and a cuter picture)
Euclid's proof of the Pythagorean Theorem

Hippocrates and the cube, Hippias (some good quadratrix information), Eudoxus, Euclid, Erastosthenes (here's a fun link to Carl Sagan on the old Cosmos show talking about him), Appolonius

§2.2

Archimedes circle formula, volumes, and pi, Hipparchus (and the moon).  Roman calendars.

Heron's formula.

§3.1

Very old music for Friday
Annotated Bibliography assignment
An image of nine chapters, and another
Contents of nine chapters (excuse the Wikipedia link - I do have this in a print source but this way I don't need to scan it here).

Yang Hui's triangle

§3.2

Numerals
Trig tables
Lilavati contents

§4.1

Early decimal point

§4.2

Abu'l Wafa's finger reckoning was an influence on things like this.

§4.3

Great mosque - detail.
Friday mosque
al-Khayyami on the cubic

§4.4

al-Mun'im's arithemetic triangle

§5.1

Translators, Rabbits, Pisa, and more

§5.2

Letters and Sestina

Jordanus
Levi ben Gerson justifies theory.

§5.3

Nicole Oresme, graphs and infinite series
Chuquet

§6,1

Room of Masks (first century BC? - Rome)
Abraham with Angels (early christian)
Annunciation - Simone Martini (14th century) attractive but not perspective
Madonna in Majesty - Duccio (14th century), progress is being made Last Supper
Paolo Uccello - Pawning of the Host more progress
Filippo Brunelleschi - Peepshow
Leon Battista Alberti
Piero della Francesca - Flagellation
School of Athens
Luca Pacioli - excerpts from his Summa  diagams by da Vinci  Last Supper
Albrecht Dürer - St. JeromeDesigner of the sitting man, Designer of the lute.  Melancholia
Persepctive Example

Tartaglia notes on the controversy
Cardano - Ars Magna (both his work and Ferrari's)

§6.2

Regiomontanus - on triangles
Riese
Copernicus

§7.1

Viete
Fermat - Last & Little, Coordinates
Mersenne
Pascal - Triangle, Conics (applet to visualise for circles - works for any conic section)

§-7.2.3

Simon Stevin - decimals
Rene Descartes - coordinates, normals/tangents

§7.2.4-

Rembrandt's painting
Hudde & van Heuraet - works
Leibniz - derivatives, FTC

§-7.3.5

Recorde - English arithmetic
Gerardus Mercator - schematic map (straight cylindrical - not Mercator) - map projection
Harriot - equations with curious notation
Napier - logarithms
Burgi

§7.3.6-

Wallis
Barrow - FTC
Newton - calculus
Berkeley - analyst

§8.1

DeMoivre
Stirling
Johann Bernoulli - integrating rational functions
l'Hospital
Simpson
Maclaurin - calculus
Brook Taylor

§8.2

Bernoulli family tree
Jakob Bernoulli - large numbers
Nikolas Bernoulli
Daniel Bernoulli
Gabriel Cramer
Euler - trig & FLT
Agnesi
Lambert - irrational π
Saccheri

-§8.3.3

d'Alembert - limits
Laplace
Lagrange
Borda
Condorcet

§8.3.4-

Monge
Gauß
Germain
Brianchon - theorem
Poncelet

§9.1

Cauchy - derivatives, FTC
Dirichlet
Galois - and Berlioz
DeMorgan
Babbage
Lamé
Kummer
Liouville

§9.2-3

Sylvester
Cayley
Boole
Jevons
Betti
Abel - quintic

§9.4

Jacobi
Eisenstein
Riemann
Dedekind - cuts
Weierstraß
Kronecker
Klein
Lie

§10.1

Banneker - 1918 journal article about
Bowditch
Hamilton - Quaternions

Peirce
Dodgson - commentary on mathematics - determinant theorem - England
Gibbs
Tait - Scottish
Maxwell - Scotland / England
Grassman - Germany / Poland

§10.2

Garfield
Hill - AMS
Glaisher
Steinmetz
E. H. Moore - geneology
Bolza
Slaught - MAA

§11.1

Lecture notes

§11.2

Wiener
Borel
Zermelo - axioms for set theory
Noether
Courant
von Neumann - with computer
LS Hill

§11.3

Cantor set
Tarski
Banach
Hahn
Menger sponge

Space filling
curves

Quotes from Menger on curves:

"We can think of curves as being represented by fine wires, surfaces as produced from thin metal sheets, bodies as if they were made of wood.  Then we see that in order to separate a point in the surface from points in a neighbourhood or from other surfaces, we have to cut the surfaces along continuous lines with a scissors.  In order to extract a point in a body from its neighbourhood we have to saw our way through whole surfaces.  On the other hand in order to excise a point in a curve from its neighbourhood irrespective of how twisted or tangled the curve may be, it suffices to pinch at discrete points with tweezers.  This fact, that is independent of the particular form of curves or surfaces we consider, equips us with a strong conceptual description."

"A continuum K is called a curve if to each point in K there exist arbitrary small neighbourhoods whose boundaries do not contain any continua.  A continuum K embedded in a space is called a curve if to any point in K arbitrary small neighbourhoods exist whose boundaries do not have any continua in common with K.  A continuum is describe in the usual way as a non-empty closed set which is indecomposable (a set which, if written as a disjoint union of two closed sets, would imply that one was empty)." Menger, 1925.PR

Lebesgue
Frechet
Fraenkel
Hausdorff
Gödel
Brouwer

§11.4

Bieberbach
Pasch
Weyl
Lefschetz
RL Moore
Alexander
Bell

Dehn
Enigma, again
Rejewski - Bomb
Turing -  Bombe
Goldbach - Renyi
Nash
Arrow

**Tuesday
includes some discussion of philosophy