History of Math: Final Edition


Ishango Bone

9000 BC

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.

Pre History

3500 BC - 3000 BC

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

Ancient Egypt

Archaic Period

3000 BC - 2700 BC

step pyramids

Ancient Egypt

3000 BC - 0 BC

Old Kingdom

2700 BC - 2100 BC


Middle Kingdom

2100 BC - 1800 BC

Rhind Mathematical Papyrus

1650 BC

believed to be a copy
Agricultural, regular cycles of the Nile
Addition, subtraction, multiplication, division, unit fractions

25: A quantity and its half become 16. What is the quantity? [Method of False Assumption]

Also, #24, #48, #41, #51, #40
Hieratics: script form of hieroglyphics

New Kingdom

1600 BC - 1100 BC

Late Periods

1100 BC - 300 BC

Dynasty 31


300 BC - 0 BC




3000 BC


1800 BC - 1600 BC

rise of city states
writing emerged, recorded history
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

Plimpton 322

1600 BC

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.

Early Greeks


624 BC - 547 BC

traveled to Egypt and brought back geometry
Geometer, Teacher

Rise in Wealth

600 BC

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


572 BC - 497 BC

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

Early Greek Period

500 BC - 300 BC

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

Hippocrates of Chios

470 BC - 410 BC

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


429 BC - 347 BC

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


300 BC

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


287 BC - 212 BC

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


410 AD - 485 AD

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

Hellenistic Period

Hellenistic Period

200 BC - 500 AD

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

Diophantus of Alexandria

250 AD

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

Islamic Math

Islamic Math

700 AD - 1100 AD

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

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

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

Medieval Europe

Leonardo of Pisa (aka Fibonacci)

1170 AD - 1240 AD

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

Medieval Europe

1200 AD - 1500 AD

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

Regiomontanus (Johannes Mueller)

1436 AD - 1476 AD

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

Luca Pacioli

1445 AD - 1517 AD

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


Scipione del Ferro

1465 - 1526

Solved x3+cx=d algebraically.
Kept secret (because there would be duels for position)
Passed on to student Antonio Fiore

Nicolo Tartaglia

1499 - 1557

Boasted could do x3+ax2=b
Challenged by Fiori.
February 12, 1535, Tartaglia discovered method for x3+cx=d, won the challenge from Fiori

Geralmo Cardano

1501 - 1576

Begged Tartaglia to share his method for x3+cx=d, Tartaglia shared with reluctance
Wrote Ars Magra, "The Great Art" in 1545, revised in 1570
Works out all 13 cases of x3+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.

Lodovico Ferrari

1522 - 1572

Solved the quartic
Beat Tartaglia in a challenge
gives del Ferro and Tartaglia credit for x3+cx=d

Rafael Bombelli

1526 - 1572

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

Scientific Revolution

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


150 AD

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

Galileo Galilei

1564 - 1642

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

Johannes Kepler

1571 - 1630

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


1596 - 1650

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
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

Pierre Fermat

1601 - 1665

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

Isaac Newton

1642 - 1727

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

Precursors and Early Calculus

Fermat is also responsible for some of the precursors to calculus

Floribund Debeaune

1601 - 1652

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

Four Squares Theorem

1621 - 1770

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

Newtonian Calculus

1642 - 1727

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

Godfried Leibnitz

1646 - 1716

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

After Newton

Swiss mathematicians

1654 - 1783

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


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
eiv = cos(v) + (i)sin(v)
Demoivre's formula


1736 - 1813


1749 - 1827

Carl Friedrich Gauss

1777 - 1855

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

Women in Math

Sophie Germain

1776 - 1831

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...)

Sonya Kovalevskaya

1850 - 1891

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

Emmy Noether

1882 - 1935

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

Non-Euclidean Geometry

system of points and lines obeying different rules
Playfairs Axiom
Types of Geometries:

John Wallis

1616 - 1703

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.

Gerlamo Saccheri

1667 - 1733

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


1728 - 1777

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


1750 - 1874

Janos Bolyai 1802-1860
N. Lobochevski 1792-1856

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

Poincare Model


Model used for final acceptance of non-Euclidean geometry