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Mathematics 390

Here is the full syllabus.

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

First two chapters from our textbook.  I have also recalled the library's copy and placed it on 4-hour reserve at the library.  The entirety of the book may be viewed from the library online. 

Here's a place you may leave about the course.

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, along with Fauvel and Gray's History of Mathematics:  A Reader.  I believe all are in the library. 
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-made timeline up to the end of the first millennium.  And here's from someone online.  It doesn't follow our course precisely, but it's a good start and a place where you use to get going and then add in your own content.  We don't do much with statistics, but this is a nice looking . 

  I have not dug through it, but I am certain there is good information .


links by section:

§1.2

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The   - one message in hieroglyphic, demotic (later form of hieratic) and Greek.  From 196 BCE, discovered 1799 CE.  
The  (~2000 BCE) sections 
Here are some views of the  (~1650 BCE) - to give you a sense of what this relic actually is. 
Some .  
The 
Here is a little bit from the  (the bit called problem 1.1 in our text).  (~1900 BCE) 
Some problems from the Rhind and Moscow - translated.  More detailed version. 


§1.2.2 and 1.3


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The   (~1800 BCE) (not from 19th dynasty, but 12th or 13th)

Some about .  

A nice .

Babylonian quadratic solution on copy of YBC 6967  Details of original solution. (~1800 BCE) 
Bablyonian square root of 2 on  (~ 1700 BCE)

. (~1800 BCE)



§2.1 

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Thales (~600 BCE), 
Pythagoras (~500 BCE) (tuning)
A  (and a cuter picture)
  
Euclid's proof of the Pythagorean Theorem
Hippasus (golden irrational)
Hippocrates and the lune, (~425 BCE)
Hippias , (~425 BCE)
Eudoxus, (~375 BCE)
Euclid (greatest common divisor, infinitely many primes), (~300 BCE)
Eratosthenes (here's a fun link to  talking about him [start at 3:53]),(~250 BCE)
Apollonius (~225 BCE)



§2.2



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Archimedes (~250 BCE)  circle formula, volumes, and pi,

Hipparchus (134 BCE) ().
Roman calendars.  (45 BCE)
Heron's (~50 CE) formula.
Nicomachus (~100 CE),
Menelaus (~100 CE) ( / ), 
Ptolemy, (~125 CE)
Diophantus (~250 CE),
Pappus (~325 CE),
 (~400 CE), Proclus, Eutocius, Boethius


§3.1


Annotated Bibliography assignment
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.  Written Chinese numeration.   (~100 BCE) 
An .  Text from 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).  (1000 BCE - 200 CE) 

Sun Zi (~450 CE) Chinese Remainder Theorem

Chang Ch'iu-Chien [Zhang Qiujian] (475 CE) indeterminant problem

Wang Hs'iao-T'ung [Wang Xiaotong] (625 CE) cubic problem

Li Zhi (1248 CE)  

Yang Hui and Qin Jiushao (1247 CE) - Approximating quartics

  (1261 CE based on Jia Xian ~1050 CE)


§3.2

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Baudhayana (~800 BCE) MCRTT

Vedic square doubling (-750 BCE)

Some Jain stories and many other links for multicultural mathematics. (< 500 BCE)

Son of Chajaka (~300 CE????) 

Bakhshali Manuscript controversy   ,  .

Numerals (~ 850 CE)

indeterminate equations from Bakshali.

Trig tables (499 [Aryabhata] & ~550 CE)
Varahamihira (~550 CE) Arithmetic triangle for combinatorics - perfumes made by choosing substances from a larger set

Brahmagupta (650 CE) Pulveriser (but reported in Bhaskara)
Lilavati contents (~1150 CE)
Details from Bhaskara (~1150 CE)


§4.1-2

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Preview of next few days ...

An 

Some from al-jabr. (800) by al-Khwarizmi

further work from ibn Turk (830)

Early decimal point (952)

al-Haytham on geometry. (~1000)

ibn Iraq and abu'l Wafa (~975) Rule of Four Quantities and Spherical Law of Sines

al-Biruni's qibla problem (~1000)


§4.2


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 - .  


.

al-Khayyami on the cubic (1100)




§4.3-4 and 5.1

§4.3-4

al Samawa'l (1175)


§5.1 

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Translators

Leonardo of Pisa Rabbits, Pisa, and more 1202

If you want to know more (e.g. why 24?) about the Book of Squares problem, .


§5.2

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Dionysius Exiguus 525

More on Bede. 725
Letters 
  (, who did many other things)
  ()

Jordanus 1250


§5.3

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1321 Levi ben Gerson justifies theory.  Induction. 

1328 Bradwardine

, graphs and infinite series Harmonic diverges 

1484 Chuquet


§6.1

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 (first century BC? - Rome)

 (early christian 548 AD)
 - Simone Martini (14th century) attractive but not perspective 
 - Duccio (14th century), progress is being made 
 -  more progress
 - 

 - 

 -  excerpts from his Summa    
 - ,  ,   
Perspective Example

 notes on the controversy

(some of this will get pushed to next time)

 - Ars Magna (both his work and Ferrari's)
1572 Bombelli



§6.2

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leftover from last time

 - Ars Magna (both his work and Ferrari's)
1572 Bombelli

 - on triangles

 
1553 Stifel's triangle

  


§7.1

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1582  (German working in Rome) last step to current calendar

  The Analytic Art

 - Last & Little, 1636 Coordinates, Areas

1650 Roberval .

 - Triangle (), Conics ( for circles - works for any conic section)


§7.2

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 - decimals  
 - coordinates, normals/tangents

 & van Heuraet - works
 - 1684 derivatives, 1693 FTC



§7.3

Since I can't play it for you, here's some English music from the 17th century:

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Lecture 

 - English arithmetic

 -  ( -  - 

\int_0^{\phi} \sec t dt = \ln(sec \phi + tan \phi).  (stretch vertical distances as much as horizontal)

 - equations with curious notation
 - logarithms


 - FTC
 - calculus, limits in principia


§8.1

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 - analyst
 - series
 - calculus  
 - 

Some §8.2 preview

 - integrating rational functions
  



§8.2

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 - large numbers
1742 Goldbach conjecture
 - trig & FLT 
  
 - irrational π



§8.3

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 - limits, otherlimits
 - probability
 - algebra, variations
 examples  
  
 
1797 Caspar Wessel, 1806 
 - 1796 17gon
1837 Wantzel 
 - letters with Gauß
1808 Brianchon - 



§9.1 - 9.2

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 - 

Analysis


 - derivatives, FTC
 
1837 



 

Pre-computers
 


 - logic algebra
 

Algebra

 - Mathematics and Berlioz more about Galois's death from Laura Toti Rigatelli's biography (1996)




§9.3-4

Two theme music pieces for today from the reading:



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 - quintic
1828 Green 
 - Geometry
 - cuts







§10.1

.

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 -     1918 journal article about - Jefferson on Mathematics 


 - Least Squares and more

 - Quaternions - , - Ireland


 - Scottish
 - Scotland / England
 - Germany / Poland


 -  - determinant theorem - England

§10.2

 
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 - 1894


 - geneology
1896 
1895 
 - 1915    


 - 

Linear Algebra

1801 Gauß substitution as predecessor to matrix multiplication
1815 Cauchy determinants (Macluarin, Cramer, Euler - nonzero determinants and unique solutions)
1844 noncommutative
1850 Sylvester matrix
1855 Cayley inverse matrices to solve systems, characteristic equation
1758 d'Alembert eigenvalues in differential equations
1829 Cauchy diagonalising matrices
1871 Jordan canonical forms
1878  similar matrices and orthogonal, dimensions of solutions
1867 Dodgson


§11.1-2

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1960 








1904 
1905  (do you really need a picture?)
1852 Guthrie - 

-, 1914 



leftovers from §11.2:



 - compactness in R



 - 
1929 LS Hill

§11.3

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1883 Cantor 
 
  
 

  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 described 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.




Crisis in Foundations 

1890 Peano


 - 
 - axioms for set theory



1931 
 






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