A Brief History of Mathematics

1. A Brief History of Mathematics • Ancient Period • Greek Period • Hindu-Arabic Period • Period of Transmission • Early Modern Period • Modern Period

2. Ancient Period (3000 B.C. to 260 A.D.) • A. Number Systems and Arithmetic • Development of numeration systems. • Creation of arithmetic techniques, lookup tables, the abacus and other calculation tools. • B. Practical Measurement, Geometry and Astronomy • Measurement units devised to quantify distance, area, volume, and time. • Geometric reasoning used to measure distances indirectly. • Calendars invented to predict seasons, astronomical events. • Geometrical forms and patterns appear in art and architecture.

3. Practical Mathematics As ancient civilizations developed, the need for practical mathematics increased. They required numeration systems and arithmetic techniques for trade, measurement strategies for construction, and astronomical calculations to track the seasons and cosmic cycles.

4. Babylonian Numerals The Babylonian Tablet Plimpton 322 This mathematical tablet was recovered from an unknown place in the Iraqi desert. It was written originally sometime around 1800 BC. The tablet presents a list of Pythagorean triples written in Babylonian numerals. This numeration system uses only two symbols and a base of sixty.

5. Chinese Mathematics Diagram from Chiu Chang Suan Shu, an ancient Chinese mathematical text from the Han Dynasty (206 B.C. to A.D. 220). This book consists of nine chapters of mathematical problems. Three involve surveying and engineering formulas, three are devoted to problems of taxation and bureaucratic administration, and the remaining three to specific computational techniques. Demonstration of the Gou-Gu (Pythagorean) Theorem

6. Calculating Devices Roman Bronze “Pocket” Abacus Babylonian Marble Counting Board c. 300 B.C. Chinese Wooden Abacus

7. Greek Period (600 B.C. to 450 A.D.) • Greek Logic and Philosophy • Greek philosophers promote logical, rational explanations of natural phenomena. • Schools of logic, science and mathematics are established. • Mathematics is viewed as more than a tool to solve practical problems; it is seen as a means to understand divine laws. • Mathematicians achieve fame, are valued for their work. B. Euclidean Geometry • The first mathematical system based on postulates, theorems and proofs appears in Euclid's Elements.

8. Area of Greek Influence Pythagoras of Crotona Archimedes of Syracuse Apollonius of Perga Eratosthenes of Cyrene Euclid and Ptolemy of Alexandria

9. Mathematics and Greek Philosophy Greek philosophers viewed the universe in mathematical terms. Plato described five elements that form the world and related them to the five regular polyhedra.

10. Euclid’s Elements Greek, c. 800 Arabic, c. 1250 Latin, c. 1120 French, c. 1564 English, c. 1570 Chinese, c. 1607 Translations of Euclid’s Elements of Gemetry Proposition 47, the Pythagorean Theorem

11. The Conic Sections of Apollonius

12. Archimedes and the Crown Eureka!

13. Archimedes Screw Archimedes’ screw is a mechanical device used to lift water and such light materials as grain or sand. To pump water from a river, for example, the lower end is placed in the river and water rises up the spiral threads of the screw as it is revolved.

14. Ptolemaic System Ptolemy described an Earth-centered solar system in his book The Almagest. The system fit well with the Medieval world view, as shown by this illustration of Dante.

15. Hindu-Arabian Period (200 B.C. to 1250 A.D. ) • Development and Spread of Hindu-Arabic Numbers • A numeration system using base 10, positional notation, the zero symbol and powerful arithmetic techniques is developed by the Hindus, approx. 150 B.C. to 800 A.D.. • The Hindu numeration system is adopted by the Arabs and spread throughout their sphere of influence (approx. 700 A.D. to 1250 A.D.). B. Preservation of Greek Mathematics • Arab scholars copied and studied Greek mathematical works, principally in Baghdad. C. Development of Algebra and Trigonometry • Arab mathematicians find methods of solution for quadratic, cubic and higher degree polynomial equations. The English word “algebra” is derived from the title of an Arabic book describing these methods. • Hindu trigonometry, especially sine tables, is improved and advanced by Arab mathematicians

16. The Muslim Empire

17. Baghdad and the House of Wisdom About the middle of the ninth century Bait Al-Hikma, the "House of Wisdom" was founded in Baghdad which combined the functions of a library, academy, and translation bureau. Baghdad attracted scholars from the Islamic world and became a great center of learning. Painting of ancient Baghdad

18. The Great Mosque of Cordoba The Great Mosque, Cordoba During the Middle Ages Cordoba was the greatest center of learning in Europe, second only to Baghdad in the Islamic world.

19. Arabic Translation of Apollonius’ Conic Sections.

20. Arabic Translation of Ptolemy’s Almagest Pages from a 13th century Arabic edition of Ptolemy’s Almagest.

21. Islamic Astronomy and Science Many of the sciences developed from needs to fulfill the rituals and duties of Muslim worship. Performing formal prayers requires that a Muslim faces Mecca. To find Mecca from any part of the globe, Muslims invented the compass and developed the sciences of geography and geometry. Prayer and fasting require knowing the times of each duty. Because these times are marked by astronomical phenomena, the science of astronomy underwent a major development. Painting of astronomers at work in the observatory of Istanbul

22. Al-Khwarizmi Abu Abdullah Muhammad bin Musa al-Khwarizmi, c. 800 A.D. was a Persian mathematician, scientist, and author. He worked in Baghdad and wrote all his works in Arabic. He developed the concept of an algorithm in mathematics. The words "algorithm" and "algorism" derive ultimately from his name. His systematic and logical approach to solving linear and quadratic equations gave shape to the discipline of algebra, a word that is derived from the name of his book on the subject, Hisab al-jabr wa al-muqabala (“al-jabr” became “algebra”). He was also instrumental in promoting the Hindu-arabic numeration system.

23. Evolution of Hindu-Arabic Numerals

24. Period of Transmission (1000 AD – 1500 AD) • A. Discovery of Greek and Hindu-Arab mathematics • Greek mathematics texts are translated from Arabic into Latin; Greek ideas about logic, geometrical reasoning, and a rational view of the world are re-discovered. • Arab works on algebra and trigonometry are also translated into Latin and disseminated throughout Europe. • B. Spread of the Hindu-Arabic numeration system • Hindu-Arabic numerals slowly spread over Europe • Pen and paper arithmetic algorithms based on Hindu-Arabic numerals replace the use the abacus.

25. Leonardo of Pisa From Leonardo of Pisa’s famous book Liber Abaci (1202 A.D.): "These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated."

26. “Jealousy” Multiplication 16th century Arab copy of an early work using Indian numerals to show multiplication. Top example is 3 x 64, bottom is 543 x 342. Page from an anonymous Italian treatise on arithmetic, 1478.

27. The Abacists and Algorists Compete This woodblock engraving of a competition between arithmetic techniques is from from Margarita Philosphica by Gregorius Reich, (Freiburg, 1503). Lady Arithmetic, standing in the center, gives her judgment by smiling on the arithmetician working with Arabic numerals and the zero.

28. Rediscovery of Greek Geometry Luca Pacioli (1445 - 1514), a Franciscan friar and mathematician, stands at a table filled with geometrical tools (slate, chalk, compass, dodecahedron model, etc.), illustrating a theorem from Euclid, while examining a beautiful glass rhombicuboctahedron half-filled with water.

29. Pacioli and Leonardo Da Vinci Luca Pacioli's 1509 book The Divine Proportion was illustrated by Leonardo Da Vinci.  Shown here is a drawing of an icosidodecahedron and an "elevated" form of it.  For the elevated forms, each face is augmented with a pyramid composed of equilateral triangles. 

30. Early Modern Period (1450 A.D. – 1800 A.D.) • A. Trigonometry and Logarithms • Publication of precise trigonometry tables, improvement of surveying methods using trigonometry, and mathematical analysis of trigonometric relationships. (approx. 1530 – 1600) • Logarithms introduced by Napier in 1614 as a calculation aid. This advances science in a manner similar to the introduction of the computer. • B. Symbolic Algebra and Analytic Geometry • Development of symbolic algebra, principally by the French mathematicians Viete and Descartes • The cartesian coordinate system and analytic geometry developed by Rene Descartes and Pierre Fermat (1630 – 1640) • C. Creation of the Calculus • Calculus co-invented by Isaac Newton and Gottfried Leibniz. Major ideas of the calculus expanded and refined by others, especially the Bernoulli family and Leonhard Euler. (approx. 1660 – 1750). • A powerful tool to solve scientific and engineering problems, it opened the door to a scientific and mathematical revolution.

31. Viète and Symbolic Algebra In his influential treatise In Artem Analyticam Isagoge (Introduction to the Analytic Art, published in1591) Viète demonstrated the value of symbols. He suggested using letters as symbols for quantities, both known and unknown. François Viète 1540-1603

32. The Conic Sections and Analytic Geometry Parabola -x2 + y = 0 Ellipse 4x2 + y2 - 9 = 0 Hyperbola x2 – y2 – 4 = 0 General Quadratic Relation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

33. Some Famous Curves Limacon of Pascal r = b + 2acos() Trisectrix of Maclaurin y2(a + x) = x2(3a - x) Lemniscate of Bernoulli (x2 + y2)2 = a2(x2 - y2) Archimede’s Spiral r = a Fermat’s Spiral r2 = a2 

34. Curves and Calculus: Common Problems Find the length of a curve. Find the area between curves. Find the volume and surface area of a solid formed by rotating a curve. Find measures of a curve’s shape.

35. Napier’s Logarithms John Napier 1550-1617 In his Mirifici Logarithmorum Canonis descriptio (1614) the Scottish nobleman John Napier introduced the concept of logarithms as an aid to calculation.

36. Henry Briggs and the Development of Logarithms Napier’s concept of a logarithm is not the one used today. Soon after Napier’s book was published the English mathematician Henry Briggs collaborated with him to develop the modern base 10 logarithm. Tables of this logarithm and instructions for their use were given in Briggs’ book Arithmetica Logarithmica (1624). A page from this work is shown on the left. Logarithms revolutionized scientific calculations and effectively “doubled the life of the astronomer”. (LaPlace)

37. Kepler and the Platonic Solids Johannes Kepler 1571-1630 Kepler’s first attempt to describe planetary orbits used a model of nested regular polyhedra (Platonic solids).

38. Kepler’s Laws of Planetary Motion I. Law of Ellipses(1609) The path of the planets about the sun are elliptical in shape, with the sun at one of the focal points. II. Law of Equal Areas(1609) An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. III. Law of Harmonies (1618) The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun.

39. Newton’s Principia – Kepler’s Laws “Proved” Isaac Newton 1642 - 1727 Newton’s Principia Mathematica (1687) presented, in the style of Euclid’s Elements, a mathematical theory for celestial motions due to the force of gravity. The laws of Kepler were “proved” in the sense that they followed logically from a set of basic postulates.

40. Newton’s Calculus Newton developed the main ideas of his calculus in private as a young man. This research was closely connected to his studies in physics. Many years later he published his results to establish priority for himself as inventor the calculus. Newton’s Analysis Per Quantitatum Series, Fluxiones, Ac Differentias, 1711, describes his calculus.

41. Leibniz’s Calculus Gottfied Leibniz 1646 - 1716 Leibniz and Newton independently developed the calculus during the same time period. Although Newton’s version of the calculus led him to his great discoveries, Leibniz’s concepts and his style of notation form the basis of modern calculus. A diagram from Leibniz's famous 1684 article in the journal Acta eruditorum.

42. Leonhard Euler Leonhard Euler was of the generation that followed Newton and Leibniz. He made contributions to almost every field of mathematics and was the most prolific mathematics writer of all time. His trilogy, Introductio in analysin infinitorum, Institutiones calculi differentialis, and Institutiones calculi integralis made the function a central part of calculus.  Through these works, Euler had a deep influence on the teaching of mathematics.  It has been said that all calculus textbooks since 1748 are essentially copies of Euler or copies of copies of Euler.  Euler’s writing standardized modern mathematics notation with symbols such as: f(x), e,, i and  . Leonhard Euler 1707 - 1783

43. Modern Period (1800 A.D. – Present) • A. Non-Euclidean Geometry • Gauss, Lobachevsky, Riemann and others develop alternatives to Euclidean geometry in the 19th century. • The new geometries inspire modern theories of higher dimensional spaces, gravitation, space curvature and nuclear physics. • B. Set Theory • Cantor studies infinite sets and defines transfinite numbers • Set theory used as a theoretical foundation for all of mathematics • C. Statistics and Probability • Theories of probability and statistics are developed to solve numerous practical applications, such as weather prediction, polls, medical studies etc.; they are also used as a basis for nuclear physics • D. Computers • Development of electronic computer hardware and software solves many previously unsolvable problems; opens new fields of mathematical research. • E. Mathematics as a World-Wide Language • The Hindu-Arabic numeration system and a common set of mathematical symbols are used and understood throughout the world. • Mathematics expands into many branches and is created and shared world-wide at an ever-expanding pace; it is now too large to be mastered by a single mathematician

44. Non-Euclidean Geometry Nikolai Lobachevsky 1792 - 1856 Carl Gauss 1777 - 1855 Bernhard Riemann 1826 - 1866 In the 19th century Gauss, Lobachevsky, Riemann and other mathematicians explored the possibility of alternative geometries by modifying the 5th postulate of Euclid’s Elements. This opened the door to greater abstraction in geometrical thinking and expanded the ways in which scientists use mathematics to model physical space.

45. Pioneers of Statistics In the early 20th century a group of English mathematicians and scientists developed statistical techniques that formed the basis of contemporary statistics. Francis Galton 1822 - 1911 Karl Pearson 1857 - 1936 William Gosset 1876 - 1937 Ronald Fisher 1890- 1962

46. Gossett’s Student t Curve Diagram from the ground breaking 1908 article “Probable Error of the Mean” by Student (William S. Gossett).

47. ENIAC: First Electronic Computer In 1946 John W. Mauchly and J. Presper Eckert Jr. built ENIAC at the University of Pennsylvania. It weighed 30 tons, contained 18,000 vacuum tubes and could do 100,000 calculations per second.

48. Von Neumann and the Theory of Computing Von Neumann Architecture John von Neumann with Robert Oppenheimer in front of the computer built for the Institute of Advanced Studies in Princeton, early 1950s.

49. Computer Generated Images Equicontour Surface of a Random Function

50. Computer Generated Images Evolution of a three dimensional cellular automata.