rise of city states
writing emerged, recorded history
cuneiform
clay tablets
Math origins: Sumerians
Base 60, positional notation, sources from thousands of clay tablets.
Problem Tablets
Babylonian Geometry: practical problems, areas, volumes, knew the pythagorean theorem numerically, area of a circle = 3r²

Main columns A, B, C
Purpose unknown
Potentially a record of Pythagorean Triples
Early trig table?
Pedagogical aid- table teachers would use to make up problems.

traveled to Egypt and brought back geometry
Geometer, Teacher

Rise in wealth, leisure class rises, transformation in philosophy, universe structured by mathematicians
no primary sources
sources: literature and philosophy commentators

leader of a cult, don't know much about him, transforming mathematics to a way of life

Mathematics as a deductive art
world view
basic geometry, 3 famous problems, new curves
proportions and incommensurables
Role for philosophers
Higher and lower math

wrote an "elements" long before Euclid, lost
Quadrature of Lunes**
Quadrature: to find a rectilinear figure equal to a given figure

Philosopher bored with math
Education of Rulers
Higher and lower mathematics
Cosmology, everything made of earth, wind, fire, and water
basis of this, 5 platonic solids which are made up of regular polygons
Three famous problems
Delion Problem
Quadrature of a circle
Trisection of an angle
-only allowed to use straightedge and compass

did not invent geometry, but rather pulled together and made precise earlier writings
school in Alexandria
The Elements
Books I-XIII
know ideas of books

Buoyancy, floating bodies
devised machines of war
later killed by a Roman soldier
Ratio of cyl to sphere is 3/2
volume
surface area
Method of Exhaustion
Measurement of a Circle
Spirals

Based on Eudemus (c. 400 BC)
lost history (to us)

Center: Alexandria
Achievements: More diverse view of math
Algebra, Mechanics, optics, astronomy
Preservation of Gr. Geometry

The Arithmetica
No theory, collection of problems, worked out numerically
13 books, 10 preserved, ~260 problems
Diophantine problem: has infinitely many solutions (indeterminate)
Fermat's Last Theorem
Fermat and Euler complete theory that Diophantus began

Muhammad 570-632AD
Greek learning, input from India
*Hindu Arabic number system "modern"
*"Modern" view of algebra
Preservation and transmission of Greek Classics
Baghdad c. 8th century
House of Wisdom: translation and research

Al-Khwarizmi
First book with "algebra" in the title
"Book of Al-Jabr and Al-Muqabala"
Al-Jabr 'transpose terms'
Al-Muqabala 'subtract equals from both sides'
Algorithm derives from Al-Kh.
very influential, no symbolic manipulation
Solutions in book appear in two forms:
1. Recipe "solution in numbers"
2. Proof by geometry

Omar Khayyam
Persian, poet, mathematician, etc.
Wrote treatises on many topics
Treatise on cubics c. 1100
pull together complete theory
said algebra is NOT about finding unkowns
said Algebra and Geometry are the same

Geometry:
Efforts to prove 5th postulate
Al-Haytham: proposed solution
O. Khayyam: fallacious
Achievements of Arabic Math:
-Hindu-Arabic number system
-Algebra
-Preservation and Transmission of Greek classics

Book: Liber abaci
Purpose: Introduce Europe to the Hindu-Arabic system
"The nine indian figures are 9 8 7 6 5 4 3 2 1 and w/ 0 (zephir)"
Bird Problem
Rabbit Problem
Fibonacci Sequence

Assimilation and transition from Arabic-Greek world.
Hindu-Arabic number system
Abocists - calculators

Book: On Triangles
c. Ptolemy c. 150 AD
Beginnings of Trigonometry
Tables of chords, in 1/2 degree increments (equiv. sine table)

Book: Summa de Artihmetica, Geometrie, etc.
Encyclopedic work includes double entry accounting
one of the first printed math books
Part on Algebra

Solved x^3+cx=d algebraically.
Kept secret (because there would be duels for position)
Passed on to student Antonio Fiore

Boasted could do x^3+ax2=b
Challenged by Fiori.
February 12, 1535, Tartaglia discovered method for x^3+cx=d, won the challenge from Fiori

Begged Tartaglia to share his method for x^3+cx=d, Tartaglia shared with reluctance
Wrote Ars Magra, "The Great Art" in 1545, revised in 1570
Works out all 13 cases of x^3+cx=d, gives proofs by geometry, cites Euclid.
Listed 20 cases of the quartic - solved some.
Knows about negative solutions, but rejects them.
Expressions like 3+(-15)^1/2 appear. No speculation as to what these might mean.

Solved the quartic
Beat Tartaglia in a challenge
gives del Ferro and Tartaglia credit for x^3+cx=d

Invented complex numbers to solve the "irreducible case" of the cubic.

Epicycles, retrograde motion
7 parameters for each planet
separate theories for each planet
(view persisted until Copernicus)

"Revolutions of the celestial spheres"
Adapted from Ptolomaic system for sun in the center
conceptually better explanation of retrograde motion

Supporter of Copernicus
Got a telescope
Terrestrial physics, exact agreement between calculation and theory
Book: Dialogues Concerning Two New Sciences
was first to describe motion, e.g. of falling bodies
Asked the right question: How?
Wrong question: Why?
Uniform motion, uniform acceleration

Astrology, Astronomy, Music, Mysticism (picture on the cover of our text is from Kepler)
Neo-Platonist (5 regular solids)
Nested series of Platonic Solids
Uncompromised scientific integrity, studies of the actual motion of the planets
[Tycho Brahe (Danish man) wealthy, no telescopes]
Precise data on planets
Followed Copernicus, required exact math
Kepler's Three Laws of Planetary Motion:
1. The orbit of each planet is an ellipse with the sun at one focus
2. The line between the sun & planet sweeps out equal areas in equal time intervals
3. The mean distance (D) of a planet from the sun and period (p) satisfy D³ = k * p² (k is a constant)
order to planets
Mean distance: half of major axis
Clockwork universe, geometry existed before creation

Rationalist philosopher
"Discourse of Method", philosophy: 300 pages
Appendix on Geometry, 120 pages
fusion of algebra and geometry, universal mathematics
co-founder, Fermat
Descartes: basically modern symbolic notation (no coordinate axes)
eliminated dimensional considerations
focus shifted from geometry to #s
algebraic operations can all be carried out with geometry
New:
1. Methods of algebra can be applied in geometry
2. Dimensional stuff
3. Modern notation
How to solve any problem in geometry:
1. Assume problem solved. Draw a figure. Make a list of names.
2. Determine relations.
3. Express one quantity in two ways to get an equation, get as many equations as unknowns
4. Determinate problems: solve
indeterminate equations: Assign remaining parameters arbitrarily.
ex. Proclus problem, Pappus problem
Impact of Descartes: efficient language to study new curves, language of the calculus

Councilor to Parliament
passion for mathematics
Forerunner of calculus, max-mins, constructing tangents, finding areas
Anticipated analytic geometry, didn't publish
Number theory, founder of modern number theory, inspired by Diophantus
Fermat's Little Theorem

Club: Archimedes, Newton, Gauss
Role in scientific revolution: grand synthesis of ideas, carried to logistical conclusions
Discoveries: calculus, binomial series (power series), laws of motion, equivalence of Kepler's laws and the inverse law of gravitation

posed inverse tangent problem to Descartes
Descartes couldn't get very far
Leibnitz, such problems belong to his calculus

Every integer is a sum of at most four squares

C. Bachet conjectured it in 1621
Fermat said he could prove it by the method of descent
Euler tried and failed
Lagrange succeeded in 1770

fluents, fluxions, moments
Direct Problem of Tangents
Area Problems
Inverse Problem of Tangents
Series

Rationalist philosopher and mathematician
co-founder of calculus, direct and indirect tangent problems
fundamental theorem
Max/min problems, etc.
careful with notation and terminology
came up with the integral notation, dx, and dy
"differential calculus" is his term
Rules for differentials

Bernoulli brothers:
Jacob 1654-1705
Johann 1667-1748, son Daniel 1700-1782
Euler 1707-1783

academy of science in St. Petersburg, (Cath the Great) academy of science in Berlin (Frederick the Great)
blind at 50
Influential books
trigonometry, log/exp functions, infinite series, taylor series, complex numbers
standard notation for e, π, i, sin, cos
circle, radius 1(unit circle)
Numerous id's
e^iv = cos(v) + (i)sin(v)
Demoivre's formula
Logarithms

Patrons
Book: Disquisitiones Arithmeticae, 1801
contributions: number theory, first proof of fundamental theorem of algebra, method of least squares, non-Euclidean Geometry
Inscribed regular polygons in circles, 17-gon
Quadratic residues and the golden theorem

teenager in the French Revolution
assumed name: LeBlanc
Friends with Lagrange and Legendre
1801: read and mastered Disquisitiones Arithmeticae
wrote to Gauss under assumed name 1804-1807
Prize from French Academy of Science, Math, Physics
Devised an attack on FLT (can't remember what this abbrev means...)

studied with Weierstrauss
partial differential equations and mechanics
professor in Stockholm
also a novelist

abstract algebra
on faculty at Göttingen
Was a Jew, forced out by Nazis
Fled to US, went to Bryn Mawr (1933)

Oxford lecture 1663
Theorem: Assume postulates 1-4, and assume that to every figure there is a similar figure of arbitrary size. Then Postulate 5 is true.

Jesuit Priest
Studied quadrilaterals, Postulate 5, HOA, HRA, HAA
Saccheri quadrilaterals

"Absolute length"
On HAA, for every angle theta, there is exactly one equilateral triangle with angle theta

Founders:
Gauss
Janos Bolyai 1802-1860
N. Lobochevski 1792-1856

Honorable mention:
F. Schweikart 1780-1859
F. Taurinus 1794-1874

Model used for final acceptance of non-Euclidean geometry