Although some scholars suggest the origins of mathematics began with the origins of Egyptian society itself (6000 B.C.), the evidence suggests something later. The calendar was hypothetically created around 4241 B.C.)

Summary of Egyptian Mathematics:

-Had 2 different numeration systems (one for stone, one for writing)

-Rather than working with fractions, they worked on the idea of the "nth" part. (And really loved the whole 2/3 fraction).

-Their basic operators were adding (doubling) and subtraction.

-They could solve basic linear equations.

-The knew and could approximate the area and volumes of geometric figures.

This era in mathematics was centered on the creation of a number system, addition and doubling (multiplying and dividing came from doubling), fractions, some basic geometric concepts, and linear equations. "A History of Mathematics" suggests that Egyptians could have really pushed into deeper concepts mathematics, but simply did not. (Boyer, p. 17)

The suggested date of the origin of Hieroglyphic numerals in Ancient Egypt.

Ancient Egypt's demonstration of their knowledge of geometry was displayed and would continue to be displayed for thousands of years.

Our knowledge of Babylonian mathematics came quite a while after our knowledge of Egyptian knowledge because we could not translate the countless tablets left. After we understood how to translate these tablets, we found out that Babylonian math:

-Had a place value system of 60.

-Loved extensive tables of products, reciprocals, conversion coefficients, and other constants.

-Understood and could solve both linear and quadratic equations.

-Geometry was devoted to measurement.

A papyrus written by an unknown scribe from the 12th dynasty. Although this papyrus is not as carefully written out as the Ahmes Papyrus, we did learn a lot from this. We know that it is, for certain, the oldest estimation of a curvilinear surface area and may be the oldest calculation of hemispherical surface area (skeptical). It also shows the correct proof for a square triangle.

The strategy of condensing numbers into just one unique symbol that would prove vitally efficient for thousands of years to come.

Ahmes was a scribe that lived in/around the year 1650 B.C. The material being copied was supposedly from the Egyptian Middle Kingdom (2000-1800 B.C.) It is suggested that some information came from the architect Imhotep. It demonstrated mathematics and was almost like a workbook for students becoming scribes. The most interesting thing was that this papyrus showed us that Egyptian math had been stagnate for nearly 2000 years.

Hindu mathematics may have started earlier, but the first evidence is found in the Sulvasutras. Concepts that have origins in India include: point-decimal system, sine function, lattice multiplication, long division known as the scratch method. The biggest contributions made were the first two. Some famous mathematics works from ancient India include: Sulvasutras, Siddhantas, and Lilavati.

Varahamihira (505-587)

Aryabhata (476-550)

Brahmagupta (598-670)

Bhaskara II (600-680)

A change of civilizations being centered around river valleys to civilizations thriving on the shores of the Mediterranean Sea.

The age of antiquity where famous mathematicians like Plato, Aristotle, Euclid, Archimedes, and many more came into play. Developments in geometry and the first steps towards mathematical logic can be found amongst many things throughout this time. .

Athens would be the center of knowledge and mathematics during this time. There is a gap of Greek mathematical uncertainty between 600 B.C. until 450 B.C.

First true mathematician/originator of the deductive organization of geometry.

Pythagorean mathematics--never before had mathematics played such a large role in life and philosophy. The belief that whole numbers explain phenomena in life. Would push mathematics into a discipline.

This age is defined by seven mathematicians.

1. Archytas of Tarentump (428 B.C.-350 B.C)

2. Hippasus of Metapontum (530 B.C.-450 B.C.)

3. Deomcritus (Thrace) (460 B.C.-370 B.C)

4. Hippias of Elis (460 B.C.-400 B.C.)

5. Hippocrates of Chios (470 B.C.-410 B.C.)

6. Anaxogroras of Claszomnae (520 B.C.-460 B.C.)

7. Zeno of Elea (490 B.C.-430B.C.)

Mathematics would come to the centers of civilization (Athens). The development of the famous Three Problems of Antiquity. An "age to achieve the impossible". Incommensurability would arise towards the end of the 4th century and mathematics would turn its back to numbers and focus more on geometry. An age were deductive reasoning would surface and change the face of mathematics forever.

Earlier, more primitive number system in Greece. Based on Egyptian hieroglyphics.

The plague of Athens would kill one-third of Athen's population. It is also the root of the 2nd Famous Problem of Antiquity.

"The Age of Plato and Aristotle"

Mathematicians of this century

1. Plato (428 B.C.-347 B.C)

2. Aristotle (384 B.C-322 B.C)

3. Theodorus of Cyrene (465 B.C.-398 B.C.)

4. Theaetetus (415 B.C-369 B.C.)

5. Eudoxus of Chidus (408 B.C-355 B.C.)

6. Menaechmus (380B.C.--320 B.C.)

7. Dinostratus (390 B.C.--320 B.C.)

8. Autolycus of Pitane (360 B.C.--290 B.C.)

Incommensurability in the last century would lead mathematicians to more geometric treatises and thoughts. Hence, the main focus of this era would be geometric ideas.

Born in 427 and active in the 4th century, Plato's works would link us to works of Antiquity that would have been lost otherwise. The Platonic Academy in Athens became the mathematical center of the world.

Alexander the Great dies. The Greek world had a sphere of influence and control in the Mediterranean Sea, the Near East, and Central Asia.

Time period from Alexander the Great's death until Roman conquest. The center of mathematics would shift from Athens to Alexandria.

As the eras in the Ancient Greek world split into two, as did the mathematical centers of the world. Athens would no longer be where mathematicians came together, but would shift to Alexandria. Here we find great discoveries in geometry, trigonometry, astronomy and geography, along with some of the most famous mathematicians in history. In this age, we find the vital contributions that would influence mathematics for the rest of time: the textbook "Elements" by Euclid and Ptolemy's "Almagest".

*Note some mathematicians did not live in Alexandria

Euclid of Alexandria (325 BC--265BC)

Archimedes of Syracuse (287BC--212BC)

Apollonius of Perga (262BC-190BC)

Aristarchus of Samos (310BC--230BC)

Eratosthese of Cyrene (276BC--197BC)

Hipparchus of Nicaea (190BC--120BC)

Menelaus of Alexandria (70AD--130AD)

Ptolemy (85AD--165AD)

The opening of Chinese mathematics begins with the Chou Pei Suan Ching around 300 BC. Mathematics seems to have established independently from the west.

The Chui-chang Suan Shu is also another valuable collection of mathematical topics including 246 different problems, patterns, and matrices.

Chinese mathematics was dedicated to number accuracy for the number pi. Chinese mathematics also show us the first usage of the arithmetic triangle.

By the 15th century, Chinese mathematics had failed to keep up with Western mathematics.

Lui Hui (220-280)

Tsu Ch'ung-Chih (430-501)

Chu Shih-Chieh (1280-1303)

Li Chin/Li Yen (1192-1279)

Ch'in Chiu-Shao (1202-1261)

Yang Hui (1261-1275)

A collection of 13 books that contain definitions, postulates and mathematical proofs of the propositions.

Born 100 B.C. and died in 44 B.C. at age 55.

Replaced Attic Numeration. Hindu-Arabic Numeration would replace this numeration system in the 13th century (A.D.)

Birth dates differ from 4 BC to 7 AD

Roman Empire would fall in the East in 476AD when Odoacer came to rule. The Eastern Roman Empire would not fall until 1453. Roman contributed little to mathematics. Roman builders used previously established applied mathematics and no need for mathematical advancement.

Conics by Apollonius

Advances in algebra--a jump from the rhetorical stage to the syncopated stage in algebra.

Diophantus of Alexandria (200-284)

Nicomachus of Gersa (60-120)

Pappus of Alexandria (290-350)

Arithmetica written by Diophantus

Mathematical Collections written by Pappus

Byzantine mathematics contributed little in advancing concepts. The primary role that Byzantine would hold in mathematics is preserver. Most contributions made by Byzantine mathematics would make would be elementary. They would comment and summarize on treatises from antiquity which would help preserve the Greek tradition until Western Europe was ready to take it out of their hands.

Eutocius (480-540)

Anthemius of Tralles (474-534)

Isidore of Miletus (442-537)

John Philoponus

Michael Constantine Psellus (1018-1080)

Georgios Pachymeres (1242-1316)

Maximos Planudes (1255-1310)

Manuel Moschopouis (1265-1316)

Nicholas Rhabdas

The rise of the Roman Empire lead a closure to Greek Mathematics (or was it from the holes that gauged Greek geometric algebra?). The closure Greek mathematics ends with these two mathematics:

Proclus of Alexandria (411-484) and Boethius (475-524).

Boethius would have a great influence on European mathematics for centuries to come and his death would mark the end of ancient mathematics in the Eastern Roman Empire. This aligned with the closure of philosophical schools as Christianity took a firm hold on Western Europe.

The European Middle Ages began with the fall of the Western Roman Empire and transitioned into the Renaissance. During this time, 5 civilizations would make up the majority of mathematics during the Medieval Ages. Those countries were: China, India, Arabia, Byzantine Empire, Roman Empire.

Although we see the development of the printing press, books from antiquity being transported to western Europe, and some very bright mathematicians (some too bright for the time), mathematical concepts like algebra and trigonometry would be learned painfully slow because of the limitations on texts and treatises.

The Dark Ages are named specifically because there was limited development in all areas (except, perhaps, religion). The only mathematicians of notice were:

Venerable Bede (637-735)

Alcuin of York (735-804)

Gerbert (940-1003)

The concepts obtained during this period of time were limited. There was still little mathematical interest in Europe and what was there was very elementary. Alcuin would revitalize French education while Gerbert would be the first to teach Hindu-Arabic numerals but it would be unsuccessful until the 13th century.

Earliest reference to Hindu-Arabic numeration. Before Hindu-Arabic numeration, Ancient India had Karosthi and then Brahmi.

Born in 742 and died in 814. King of the Franks, King of the Lombards and Holy Roman Emperor. Conquered and united most of Western Europe.

War had raged in the Near East since the beginning of the Mohammedan Era (622). The war finally settled down and the Arabians began to absorb the culture of the areas they conquered. Bagdad would become the new center of mathematics. Mathematic concepts that came to fruition under Arabian rule would be Hindu-Arabic numeration, trigonometry, algebra and advances in astronomy. Advances in Arabian mathematics stemmed from Hindu Influence, Mesopotamian stressors and Greek inspirations. Many Arabian textbooks and works would be essential in Europe in later centuries.

al-Khwarizmi (790-850) "Actual father of algebra"

Thabit ibn-Qurra (826-901)

Abu'l-Wafa (940-998)

al-Battani (850-929)

ibn-Sina (980-1037)

Al-biruni (973-1048)

ibn-Al-Haitham (965-1039)

Omar Kahyam (1048-1122)

Nasir Eddin al-Tusi (1201-1274)

Al-Kashi (1380-1430)

Arabian mathematics began declining when Khayyam died in 1123, but would close after the death of al-Kashi in 1436. Mathematics in Arabian hands improved greatly by the time it was handed off to the Europeans.

Al-Khwarizmi's Algebra is written

An effort from the Latin Church to recover lands (especially the Holy Land) from Islamic rule.

The Age of Translation was a time for famous works of antiquity and the West to be translated into Latin. These works included: Elements, al-Kwarizmi's astronomic tables, Algemest, Algebra, and many more works. Gerard would translate more than 85 works.

Adelard of Bath (1075-1160)

Gerard of Cremona (1114-1187)

Robert of Chester ?

We also see the spread of the Hindu-Arabic number system through mathematicians Adelard, John of Seville (1100-1180), and Abraham ibn-Ezra(1090-1167)

Construction began in 1163 and was completed in 1345.

Mathematical revival was approaching. Universities had opened their doors. This was a period of great scholars and practical inventions.

Here we see Fibonacci's best work--Liber Abaci.

Alexandre de Villedieu (1175-1240)

Sacrobosco/John of Halifax (1200-1256)

Fibonacci (1180-1250)

Jordanus Nemorarius (1225-1260)

Campanus of Novara (1220-1296)

In this century, we can see the establishment of Hindu-Arabian numerals. We see progression of mathematics because of the translations of the previous century (and some translations were still being processed), and advances in optics and mechanics.

Liber Abaci written by Fibonacci

Italian Renaissance would begin much sooner than Northern Europes due to a variety of reasons (The Plague in Northern Europe ran rampant, scholars fled to Italy from the Constantinople, the location of the church made Italy wealthier, the Hundred Years War was still active).

Although there are various obstacles throughout the Middle Ages and into the Renaissance, we do see the rise in algebra during the Renaissance. Algebra would be the largest theme during the Renaissance, but trigonometry would not be far behind it (trigonometry is vital to astronomy and astronomy was a hot topic during this time).

England and France were leaders in mathematics at this point, but the 14th century is known for disaster and chaos striking in Northern Europe. The Black Death, Hundred Years' War, the Wars of the Roses, and other environmental changes impacted Northern Europe severely. But we do see development in the Boethian theory and the first suggestion of an infinite series.

Thomas Bradwardine (1290-1349)

Nicole Oresme (1323-1382)

Richard Suiseth ?

Liber de proportionibus written by Thomas Bradwardine.

English and French Kings (and their supporters) fought over the kingdom of France. This war raged on for 116 years and was one of the most historical events of the Early Middle Ages. England lost its continental lands.

The Black Death, or the bubonic plague, that ran through Europe and Asia killing 75-200 million people (1/3 of the world's population).

14th century mathematics would predominately be associated with French and English mathematicians, but as the bubonic plague and wars would destroy their population numbers, mathematics would be handed off to Germany, Italy and Poland to continue progress throughout the 15th century.

During the 15th century, the fall of Constantinople would help the progression of works of antiquity reach western Europe. The printing of books aided in the advancement of mathematical concepts. We see concepts like algebra and trigonometry shoot up while other concepts, like geometry, slowly making a comeback. Ancient Greek works were still too difficult for mathematicians to grasp completely at this point. We also see the continuation of works from antiquity being translated and various mathematical works being printed. Regiomontanus would ignite trigonometry in his students, but would die before publishing his work and therefore keeping his knowledge limited to his general area.

Nicholas of Cusa (1401-1464)

Regiomontanus/ Johann Muller (1436-1476)

George Peuerbach (1423-1469)

Nicolas Chuquet (1445-1469)

Luca Pacioli (1445-1514)

Leonard Da Vinci (1445-1514)

Johann Widman (1462-1498)

The Age of Exploration (or Discovery) started with Henry the Navigator's sponsored expedition around the African coastline. This would begin an overseas race of exploration, the discovery of the Americas, the beginnings of globalization, and the quest for a foothold in new lands.

Prince Henry of Portugal begins the Age of Exploration by sponsoring an expedition around Africa.

Fall of Constantinople would be the end of the Byzantine Empire and the Turks would establish the Ottoman Empire.

Main concepts of the 16th century: trigonometry, geometry, and the stepping stones to calculus. We see various works being written during this time that spurred developments in mathematics. We also see competition rising amongst mathematicians. We see big names during this century that would change the structure of mathematics forever--Stifel, Cardano, Bombelli, Recorde, Copernicus, Rheticus, and several others. By 1575, all major mathematical works from antiquity had been recovered. Great things were stirring for mathematics.

Christopher Rudolff (1499-1545)

Peter Apian (1495-1552)

Michael Stifel (1487-1567)

Geronimo Cardano (1501-1576)

Niccolo Tartaglia (1500-1557)

Ludovico Ferrari (1522-1565)

Scipione del Ferro (1465-1526)

Rafael Bombelli (1526-1573)

Robert Recorde (1510-1558)

Nicholas Copernicus (1473-1543)

Georg Rheticus (Rhaeticus) (1514-1574)

Gerard Mercator (1512-1594)

Edward Wright (1558-1615)

1543- Copernicus's "De revolutionibus"

1544- Stifel's "Arithmetica integra"

1545- Cardan's "Ars magna"

Mathematics during this century, quite literally, explodes. We have advances in arithmetic, geometry, algebra, trigonometry, and the establishment of new mathematical concepts--calculus, cartesian geometry and logarithms. Throughout the decades, we see the major themes change dramatically--we see trigonometric functions popping up all over Europe in the beginning of the century, to favoring analytical geometry, then changing over to infinite analysis, to the emergence of calculus. We even have periods of popularity that include polar coordinates, probability, combinations, projective geometry, differentials, and several other concepts. We also see more intercommunication than before between the mathematical community.

Mathematicians who aided the transition into modern algebra:

Galileo Galilei (1564-1642)

Bonaventura Cavalieri (1598-1647)

Henry Briggs (1561-1639)

Thomas Harriot (1560-1621)

William Oughtred (1574-1660)

Simon Stevin (1548-1620)

Albert Girard (1590-1633)

John Napier (1550-1617)

Jobst Burgi (1552-1632)

Johann Kepler (1571-1630)

Francois Viete (1540-1603)

Other important mathematicians:

Frans Von Schooten (1615-1660)

Christian Huygens (1629-1695)

Jan de Witt (1629-1617)

John Wallis (1616-1703)

James Gregory (1638-1675)

Jacques (1654-1705) and Jean (1667-1748) Bernoulli

Abraham De Moivre (1667-1754)

Count Ehrenfriend Walter von Tschirnhaus (1651-1708)

Some BIG names in mathematics during this century:

Rene Descartes (1596-1650)

Pierre de Fermat (1601-1665)

Girard Desargues (1591-1661)

Blaise Pascal (1623-1662)

Isaac Newton (1642-1727)

Gottfried Wilhelm Leibniz (1646-1716)

Jamestown, Virginia was established making it the first colony established in the American mainland.

Protestants revolt against Catholic oppression--what started out as a war between small European countries turned into larger countries sending mercenary armies to invade Germany. As a result, Europe suffered devastating losses (Germany would lose twenty percent of its population), the Holy Roman Empire was further decentralized, and a noticeable decline of the Catholic church in northern Europe.

Descartes' "Discours de la méthode" (1637) and "Brouillon projet" (1639) published.

Pascal's "Essay pour les coniques" published in 1640

This would mark Leibniz's first published paper on the concept of calculus. Around the same time, Newton would also be working on the concept of calculus and have unpublished papers laying about. These two mathematicians would create the concept of calculus independently from one another.

Newton's Principia was published

During this century, we see revolutions breaking out throughout the entire world. Since the last half of the 17th century, London held the mathematical center of the world. A plagiarism battle between Newton and Leibniz had outcasted England from the Continent and France would regain the center during the French Revolution. This also resulted in England's decline in mathematics and halt towards progress until the next century. England would continue their work on synthetic geometry while the rest of Europe worked on analytical geometry. We see themes arising during the 18th century of calculus, infinite series, probability, and the revival of solid geometry. We see the implementation of the metric system at the end of the century and the revolution of geometry. The French Revolution would start an "era of teaching" and vital textbooks would begin printing for educational purposes.

Leonhard Euler (1707-1783)

Jean Le Rond d'Alembert (1717-1783)

Alexis Claude Clairaut (1713-1765)

Jacapo (1676-1754) and Vincenzo (1707-1775) Riccati

Edward Warring (1734-1793)

Johann Heinrich Lambert (1728-1777)

Lagrange (1736-1813)

Marquis de Condorcet (1743-194)

Gaspard Monge (1746-1818)

Nicolas Léonard Sadi Carnot (1796-1832)

Adrien-Marie Legendre (1752-1833)

Pierre-Simon Laplace (1749-1827)

Euler's Introductio written

The United States Constitution was signed in 1787. The American colonies had successfully won independence from Britain.

Elemens de geometrie by Lagrange in 1788

Feuilles d'analyse by Legendre in 1794

System du monde by Monge in 1795

Fonctions analytiques by Laplace in 1796

Métaphysique du calcul by Carnot in 1797

Social and political uprising against established monarchy of France. We see the rise of Napoleon Bonaparte. The Reign of Terror would mark a significant part of the revolution as public executions and massacres took part from September 5, 1793 until July 28, 1794.

This century would be one of the most revolutionary in the terms of mathematical history. We have huge discoveries during this era: non-euclidean geometry, n-dimensional spaces, non-commutative algebras, infinite processes, non-quantitive structures and much more. The mathematical community would be brought even closer together with the establishment of specific mathematical journals and periodicals. British mathematics would come back into play with the opening of Trinity College and the reform of English algebra. We have a substantial amount of mathematicians from various countries pursing research. With so many mathematicians, the struggle of the 19th century is rediscovery of the same concepts. We have "unparalleled" advances in three major categories: algebra, analysis, and geometry. This century would end in taking steps towards arithmetization of these three subjects.

Algebra:

19th century algebra contradicts itself--we have British mathematics trying to establish algebra as a "demonstrative science" and then we have the continental mathematics coming up with new concepts and working on unsolvable problems.

Some mathematicians that specialized in algebra:

George Peacock (1971-1858)

George Boole (1815-1864)

Augustus De Morgan (1806-1871)

Sir William Rowan Hamilton (1805-1865)

Hermann Grassman (1809-1877)

Arthur Cayley (1821-1895)

James Joseph Sylvester (1814-1897)

Giuseppe Peano (1858-1932)

Geometry:

During the second half of the 18th century, there were arguments about analytical vs. synthetic geometry. This argument would spill into this century at the beginning, but the discovery of duality stopped the dispute. In geometry, we have the breakthrough of homogeneous coordinates and stepping towards the arithmetization of geometry.

Some mathematicians that specialized in geometry:

Jean Victor Poncelet (1788-1867)

Karl Wilhelm Feurback (1800-1834)

Michel Chasles (1798-1863)

Jakob Steiner (1796-1863)

Julius Plucker (1801-1868)

Gabriel Lame (1795-1870)

Mobius (1790-1860)

Riemann (1826-1866)

Felix Klein (1849-1925)

Alfred Clebsch (1833-1872)

Analysis:

Foundational questions began to arise during the mid century. There was questionable confidence in operations performed on infinite series. There was also no concrete definition of a real number. And so, the arithmetization of analysis would begin.

Some mathematicians that specialized in analysis:

Charles Meray (1835-1911)

Karl Weierstrass (1815-1897)

Eduard Heine (1821-1881)

George Cantor (1845-1918)

Richard Dedekind (1831-1916)

Bernhard Riemann (1826-1866)

Leopold Kronecker (1823-1891)

Disquisitiones arithmeticea by Gauss

Napoleon Bonaparte leading the French takes massive strides in conquering Europe. For a brief time, France had domination over most of Europe. War in Europe would continue between various different powers after Napoleon is exiled.

Cauchy's "Calculus of Residues" published.

This year also marks the discovery of homogeneous coordinates by Mobius, Plucker, Feuerbach.

Boole's "Laws of Thought" published.

This year would also mark Riemann's discovery of habilitationsschrift

War that raged between the Northern states and the Southern states over the controversy of enslavement of African-Americans.

This year would mark the following published works:

Stetigkeit und irrationale Zahlen by Dedekind

Elemente by Heine

Nouveau précises by Meray

Poincare's "Analysis situs" published.

By the beginnings of the 20th century, it was definite that mathematics had evolved into a different beast than what it was during the beginning of the 19th century. Not only had the concepts and content surrounding mathematics changed, but also the institutions and the communication means. The 20th century brought in math societies and international conferences as well as the mathematical journals established in the previous century. We have a the growth of interaction and communication of mathematicians from all over the world. We have themes of topology, modern algebra, vector spaces, abstraction, and the invention of the computer.

Henri Poincare (1854-1912)

David Hilbert (1862-1943)

Hermann Weyl (1885-1955)

Emile Borel (1871-1956)

T.J. Stieltjes (1856-1894)

Elie Cartan (1869-1951)

Alan Turing (1912-1954)

Claude Shannon (1916-2001)

John van Neumann (1903-1957)

Francis Ferdinand is assassinated. Austria, Germany, and the Ottoman Empire fought against an impressive amount of Allied powers including France, Britain, America, Italy, Serbia, and Japan.

Sir Alexander Fleming discovered penicillin in London.

American economy crashes. Unemployment soars

Second Great War generally begins when Germany invades Poland, but tensions were rising years before this time. Axis powers (Germany, Italy, and Japan) rose against the Allied powers (Britain, France, USA, Russia, and China). The Allied powers would win. As a result, the world would see the collapse of Nazi Germany, Japanese and Italian empires. We would also see the dissolution of the League of Nations and the establishment of the United Nations. Russia and the United States would rise as global super powers and thus start the Cold War.

Astronauts Neil Armstrong, Edwin Aldrin, and Michael Collins walk on moon on the Apollo 11 mission.

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