Milestones in Mathematics History

Oxyrhynchus papyrus (P.Oxy. I 29) showing frag...

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20,000 BC

Carved notches in wood represent numbers.

3500 BC

Numbers based on place value (base 60) used in Sumeria.

The Sumerians had no symbol for zero. They used an empty space to represent a zero in the middle of a number but had no way to represent zero on the end of a number. Thus they could distinguish 15 from 105 but could not tell 15 from 150.

2000 BC

Mesopotamians solve quadratic equations.

1900 BC

Egyptians apply basic geometry to solve practical problems.

1900 BC

Pythagorean Theorem

a2 + b2 = c2

discovered by Babylonians

1700 BC

Babylonians find approximate value of r(2).

But don’t tell how they did it.

1700 BC

A’hmos� (Egyptian) describes methods of mathematical problem solving.

One of the earliest “textbooks.”

547 BC

Thales (Greek) introduces deductive proofs.

520 BC

Pythagoras (Greek) founds brotherhood based on mathematics.

500 BC

Greeks use abacus, the first mechanical calculating device (probably invented by Babylonians).

460 BC

Zeno (Greek) devises paradoxes such as “Achilles and the Tortoise.”

Achilles and the Tortoise

Achilles races a tortoise that has a head start. First, Achilles must run to the point where the tortoise started the race. While he does that, the tortoise moves a little farther. So Achilles must run to where the tortoise is now but again the tortoise moves a little farther. Since this can be repeated indefinitely, Achilles can never catch up to the tortoise.

441 BC

Hippasus (Greek) shows that r(2) is irrational.

Many mathematicians then could not accept the idea of an irrational number.

380 BC

Plato (Greek) describes the five regular solids.

371 BC

Eudoxus (Greek) develops theory of “equal ratios,” beginnings of the real number system.

Eudoxus’ theory was not well understood by his contemporaries, most of whom did not like the idea of irrational numbers. His theory was ignored for about 2000 years until Dedekind and Cantor created the real number system.

370 BC

Eudoxus develops “method of exhaustion,” an early ancestor of calculus.

350 BC

Menaechmus (Gr) describes conic sections (parabolas, circles, ellipses, hyperbolas).

300 BC

Sumerians invent “placeholder” (zero) but don’t consider it a number.

300 BC

Euclid (Greek) publishes Elements; basis for Euclidean Geometry.

Euclidean Geometry is the system of postulates and proofs that we study in Course II. However, Euclid’s Elements described far more than just geometry. For example, he proved that there is no largest prime number.

260 BC

Mayans develop place value numbers using
base 20.

260 BC

Aristarchus (Greek) tries to determine size and distance of sun and moon using trigonometry.

250 BC

Archimedes (Greek) determines
3f(1,7) < p < 3f(10,71).

250 BC

Archimedes develops his “method,” an early form of integration.

230 BC

Erotosthenes (Greek) uses trigonometry to find the size of the earth.

230 BC

Erotosthenes devises “sieve” for finding prime numbers.

220 BC

Apollonius (Greek) applies mathematics to study astronomy.

140 BC

Hipparchus (Greek) prepares the first trigonometric table.

(Sines only.)

AD 50

Heron (Greek) writes the number r(81-144), earliest known use of an imaginary number.

Heron is given credit for inventing one of the most efficient methods for calculating square roots, although his method may actually have been used centuries earlier by the Babylonians.

AD 62

Heron finds formula for area of a triangle in terms of the lengths of its sides.


Ptolemy (Greek) establishes system of latitude and longitude.


Diophantus (Greek) devises early form of symbolic algebra.

Earlier algebra was either written out in words (Babylonians) or expressed geometrically (Greeks).


Hypatia (Greek) studies and writes about Diophantus’ Arithmetica and Apollonius’ Conics.

Hypatia was the earliest well-known woman mathematician. In a time when women were considered intellectually inferior to men, her father Theon (also a mathematician) educated her to be a scholar. She became a teacher of mathematics and philosopy at the university in Alexandria. In addition to teaching, she designed several scientific instruments including a plane astrolabe and a hydrometer. She was so respected that students came from all over the western world to study with her and the city magistrates often consulted her before making important decisions. Unfortunately, that same fame brought her to the attention of the Christian church, which was growing in power at that time. To Christians, Hypatia’s teachings of scientific rationalism were considered heretical. When she refused to modify her teachings and to convert to Christianity, she was brutally murdered by a Christian mob in 415 AD. Her death marked the end of 1000 years of Greek progress in mathematics. During the next six centuries (Europe’s Dark Ages), Arabs and Hindus were responsible for most new mathematical developments.


Tsu Ch’ung-chih (Chinese) approximates p as 355/113.


Brahmagupta (India) solves quadratic equations; uses negative numbers.


al-Kwarizmi (Arabic) describes Hindu mathematics in
Hisab al-jabr w’al-



Mahavira (India) is one of the first to write about zero as a number.

“A number multiplied by zero is zero, and that number remains unchanged which is divided by, added to, or diminished by zero.”

What is the error in Mahavira’s statement? It was not corrected until about 300 years later by Bhaskara (India).


First recorded symbol for the number 0 used in India. Called “sunya” (“empty”).


d’Aurillac (French) introduces Hindu-Arabic numerals to western Europe.


Chinese develop mathematical number triangle now called Pascal’s Triangle.


Omar Khayyam (Persia) solves cubic equations geometrically.


Fibonacci (Italian) introduces the Fibonacci series:

1, 1, 2, 3, 5, 8, 13 . . .


Ch’in Chiu-shao (Chinese) gives numerical method of solving equations.


al-Kahi (Arabic) first uses decimal fractions.


del Ferro (Italian) solves one special type of cubic equation algebraically.


Tartaglia (Italian) solves two types of cubic equations algebraically.


Ferrari (Italian) solves the general quartic equation

(while still in his teens).


Cardano (Italian) publishes complete algebraic solutions of both cubic and quartic equations.

Cardano had been unable to solve the cubic equation himself. He got the formula from Tartaglia with a promise to keep it secret. Tartaglia bitterly protested Cardano’s publication of the formula. Despite that, it is known today as Cardano’s formula.


Cardano uses complex numbers to solve equations.

Cardano proposed the problem “Divide 10 into two parts such that the product of one times the remainder is 40.” He called it “manifestly impossible” but went ahead and solved and checked the problem anyway. The answers he called “truly sophisticated” but he decided that continued work with such numbers would be “useless.” Cardano also had trouble with negative numbers. He recognized that some equations had negative roots but referred to them as “ficticious.”


Bombelli (Italian) uses complex numbers to find real solutions to equations.


Vi�te (French) urges use of decimal fractions.


Stevin (Flemish) gives rules for doing arithmetic with decimals.


Vi�te introduces use of letters to represent variables and constants.


van Ceulen (Dutch) uses Archimedes’ method to find p to 35 places.


Napier (Scottish) describes logarithms.

But he developed them geometrically. He did not recognize them as exponents.


Oughtred (English) invents the slide rule, a calculating device based on logarithms.


Descartes (French) develops coordinate geometry.


Fermat (French) claims to have a proof of Fermat’s Last Theorem.

Fermat’s Last Theorem

an + bn = cn
has no natural number solutions for n > 2.

Fermat died without writing down his proof.


Pascal (French) constructs model of first mechanical adding machine.


Pascal and Fermat begin developing theory of probability.


Wallis (English) introduces negative and fractional exponents.

He also makes the connection between logs and exponents.


Fermat uses early form of derivatives to find extrema of polynomial functions.

Fermat is also credited with developing a method of integration that is very close to the modern definition. Interestingly, while working with both integrals and derivatives of polynomial functions, he apparently never saw the significance of the relationship between them.


Gaunt (English) founds statistics while studying life expectancy.


Newton (English) discovers general binomial theorem.


Newton describes calculus but does not publish it.


Gregory (Scottish) begins development of symbolic logic.


Leibniz (German) explores number systems in other bases, especially base 2.


Leibniz publishes his description of calculus.

Mathematicians from Eudoxus to Fermat had discoved techniques of both integral and differential calculus before Newton and Leibniz. However, they did not understand the relationships among their results, many of which applied only to polynomial functions. Newton and Leibniz, working independently, were able to tie the various pieces together and provide general rules that worked for any function. Together, they created a whole new branch of mathematics based on infinite processes: the calculus.


de Moivre (French) applies permutations and combinations to probability.


Goldbach (German-Russian) proposes Goldbach’s conjecture.

Goldbach’s Conjecture

Every even integer greater than 2 is the sum of two primes. This has not yet been proven or disproven.


Lambert (German) proves p is irrational.


Ruffini (Italian) proves not all 5th degree equations can be solved algebraically.


Wessel (Norwegian) gives graphical representation of complex numbers.


Bolyai (Hungarian) constructs a non-Euclidean geometry.

Non-Euclidean Geometry

A geometric system that does not use all of Euclid’s postulates from his Elements. Bolyai’s geometry did not use Euclid’s parallel postulate.


Gauss (German) develops complex numbers as a mathematical system.


Babbage (English) designs mechanical computer

(but is unable to build a working machine).


Dirichlet (Prussian) defines “functions.”

He was not the first, but his definition comes closest to what we teach in CIII today.


Lovelace (English) develops early ideas of computer programming.


De Morgan (English) refines and expands ideas of mathematical logic.


Guthrie (English) proposes the “four color map problem.”

Four Color Map Problem

Guthrie believed any map can be colored with just four colors so no two bordering countries are the same color.

He could not prove it.


Riemann (German) describes a non-Euclidean geometry that Einstein later shows is the most likely geometry of the universe.


M�bius (German) invents the M�bius strip.


Muir and Thomson (English) develop radian measure for angles.


Cantor (German) shows that the number of points in the interior of a square is “the same” as the number of points on a line segment.


von Lindemann (German) proves p is transcendental.


Pierce (American) introduces use of “truth values” in symbolic logic.


Cantor proves there is more than one “size” of infinity.


Lukasiewicz (Polish) introduces truth tables.


Bush (American) develops analog computer.


Godel (Austrian-American) proves in any mathematical system, certain propositions are “undecidable” (cannot be proven or disproven).


Eckert and Mauchly (American) build ENIAC, first electronic digital computer.


Arrow (American) proves “Impossibility Theorem:” there is no such thing as a perfect voting system.


First computer assisted proof: Guthrie’s four color map problem.


Wiles (British) proves Fermat’s Last Theorem.


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