Discovered by Karl Menninger. Found in central Africa. series of numerical markings.

beBraucourt thought it might have been an arithmetical game.

Marshack thought it might potentially be a time count or a lunar count.

Before writing, no agreement, no right or wrong. Range of opinions*

Aristotle wonders about base 10. Thinks tied to nature. Hidden role in cosmology?

Pythagoras had cosmology entertwined w/ numerical ratios, ten fingers

step pyramids

Pyramids

believed to be a copy

Pharaohs

Agricultural, regular cycles of the Nile

Hieroglyphics

Addition, subtraction, multiplication, division, unit fractions

Also, #24, #48, #41, #51, #40

geometry

Hieratics: script form of hieroglyphics

Dynasty 31

Cleopatra

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 = 3r^{2}

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.

View c. 1500, earth does not move, geocentric solar system, planetary motion uniform and circular

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 D^{3} = k * p^{2} (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

Fermat is also responsible for some of the precursors to calculus

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)

system of points and lines obeying different rules

Playfairs Axiom

Types of Geometries:

-Elliptic

-Parabolic

-Hyperbolic

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

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