Week 14 Lecture China Topic 02 Mathematical Contributions...

This slide was updated 11/7/2019 1

Montclair State University Department of Anthropology

Anth 140: Non Western Contributions to the Western World

Dr. Richard W. Franke

Week 14 Lecture

China Topic 02

Mathematical Contributions from China

Mathematical Applications in Astronomy,

Physics and Military Technology

This lecture was last updated 07 November, 2019 and

25 November, 2013

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The learning objectives for week 14 are:

– to understand the basic position of China in the history of mathematics

– to learn the Chinese developed negative numbers, an important part of modern mathematics

– to appreciate the Chinese observations of the sky that were far in advance of Europe until quite recently

– to appreciate the Chinese inventions of the compass, seismograph and gunpowder as well as the crossbow, canon, flamethrower and gun

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Terms you should know for week 14 are:

– decimal system and its origin

– seismograph

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Mathematical Contributions from China:

Additional Sources

Sources:

Joseph, George Gheverghese. 1991. The Crest of the Peacock: Non-European Roots of Mathematics. London and New York: I. B. Tauris & Co. Ltd.

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These slides correspond to and supplement the reading:

Temple, Robert. 1998. The Genius of China: 3,000 Years

of Science, Discovery and Invention. London: Prion Books

Limited.

Part 2: Astronomy and Cartography, pages 29-39

Part 6: Mathematics, pages 139-147;

Part 7: Magnetism, pages 149-157;

Part 8: The Physical Sciences, pages 159-171;

Part 9: Transport and Exploration, pages 173-197;

Part 11: Warfare, pages 215-248

See the note about page numbers on the next slide

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Page Numbers Update:Temple, Robert. 1998. The Genius of China: 3,000 Years of Science,

Discovery and Invention. London: Prion Books

This book has been reprinted several times and some page

numbers might have changed. If your page numbers do not

correspond to those in the previous slide, use the chapter

numbers and topic titles to guide your reading.

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First Section is on:

Mathematics: Temple, Chapter 6, pages 139–47

Later sections on astronomy and military technology later in the lecture.

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Chinese Mathematical Achievements

1. Science historian Joseph Needham has noted that one of the key features of modern science is

“the application of mathematics to nature.”

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2. We saw earlier in the course that the Inca, Maya and Egyptians excelled at applying mathematics to nature and to human undertakings such as canals, pyramids, palaces and the like.

3. The traditional Western view is that mathematics originated in ancient Greece…

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4…but the Black Athena research by Martin Bernal and others has shown that much of Greek mathematics derives from ancient Egypt and ancient Sumer (Babylonia).

5. Even so, much important mathematical progress was made during the Greek “Golden Age” from about 600 BC to 300 BC

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6. Among the most famous of the Greek mathematicians are:

– Pythagoras 569–475 BC

– Euclid 325–265 BC

– Archimedes 287–212 BC

7. They and many ancient Greek colleagues wrote some of the most famous and important mathematical documents we know of.

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8. But the application of mathematics to nature andthe development of pure mathematics owes much more to ancient China than many people realize.

9. Because China’s mathematical developments come later than those in Egypt, Egyptian influence is possible.

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10. But even if this is the case, Chinese mathematicians clearly made many discoveries and improvements on their own.

11. The large number of existing Chinese manuscripts means we have more details from China than from ancient Egypt where we still depend on a small number of papyri such as the Ahmes papyrus described earlier in the course.

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12. Among China’s many mathematical contributions…

13. The decimal system, from at least the 14th century BC, applied in Europe only 2300 years later.

14. We can trace Chinese math through several of the most famous manuscripts…

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15. “The Circular Paths of Heaven”

500 BC

Pythagorean theorem, fractions

16. “Book on Arithmetic”

300 BC

Fractions, rectangles

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17. “9 Chapters on the Mathematical Arts”

300 BC to 200 AD

Root extractions, ratios, simultaneous equations,

areas, volumes, right-angle triangles

18. “Commentary on the 9 Chapters”

130 AD – Square root of 10

Source: Joseph, George Gheverghese. 1991. The Crest of the Peacock: Non-European Roots

of Mathematics. London and New York: I. B. Tauris & Co. Ltd. Page 132.

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19. “The Mathematical Arts”

200 AD

Theory of large numbers, first mention of the abacus

20. “Commentary on ‘The Circular Paths’”

200 AD

Solution of quadratic equations – (x2 + ax = b2

)

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21. “Official History of the Sui Dynasty”

450 AD

Value of pi

22. “Continuation of Ancient Mathematics”

AD 625 – 3rd degree (cubic) equations

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23. “The Sea Mirror”

1248 AD

Higher-degree equations (above cubes); applications of Pythagorean theorem; minus numbers

24. Negative numbers a Chinese invention early as 2nd century BC

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25. “Detailed Analysis of the 9 Chapters”

1261 AD – Arithmetic progressions; decimal fractions; quadratic equations with negative coefficients of x.

26. “The Precious Mirror”

1303 AD – Pascal’s triangle (350 years before Pascal); simultaneous equations; matrix methods to solve equations.

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27. “Indeterminate analysis” from 4th century AD

N = 3x + 2

N = 5y + 3

N = 7z + 2

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28. Calculate the volumes of eg:

– Cylinder

– Rectangular pyramid

– Tetrahedron

– Truncated square pyramid

– Truncated triangular right prism

Gheverghese. P. 168

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Montclair State University Department of

Anthropology

Anth 140: Non Western Contributions to the

Western World



29. Famous “broken bamboo problem”

30. Find the value of b

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31. Solution:

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32. This one of most famous problems in mathematics

– Subject of much analysis and debate in ancient Indian math

– Arab and European interest over many centuries

– Apparently solved in about 6th century by Chinese

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Chinese Mathematical Applications

33. Development of math in China related to other achievements in

– Astronomy

– Physics

– Military technology

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34. Astronomy:

Temple, part 2, pages 29 – 39

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35. Understanding of sunspots – 4th century BC, several hundred years before Europe

36. Chinese records the oldest continuous set of data known

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37. Quantitative cartography: relatively accurate maps using mathematics to generate scale

38. Early as 2nd century AD

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39. Observed and measured direction of comet tails early as 6th century AD

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40. Equatorial astronomical instruments to divide sky into measurable segments – 13th century AD or earlier

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Chinese Mathematical Applications 41. Developed constellations

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Chinese Achievements in PhysicsThe next slides correspond to and supplement the reading:

Temple, Robert. 1998. The Genius of China: 3,000 Years of Science, Discovery and Invention. London: Prion Books

Limited.

Chapters 7 – 9 on Magnetism, the Physical Sciences and Transport and Exploration, Pages 149 – 97

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Chinese Achievements in Physics

42. Compass – 4th century BC – in use by Han Dynasty

43. Used understanding of magnetism that derived from Ch’i

44. Pointed south

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45…developed pointers to make compass more precise – 3rd century AD.

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46. By Sung Dynasty 960 – 1279 AD…knew how to magnetize iron needles – rub with magnetite then suspend in water

47. Compass used in navigation first by Chinese

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48. Chinese also understood heating and cooling to change characteristics of magnets – called thermo-remanence

49. Compass to Europe around 1190 AD

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50. Geobotany: Chinese first to notice relation between plants and minerals in ground underneath – at least by 3rd century BC, ie Han Dynasty

51. Newton’s 1st law of motion – body stays at rest or in motion until changed by outside forces –developed in China 4th century BC…Newton 18th century AD

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52. Hexagonal shape of snowflakes in 2nd century BC

53. In Europe 1591

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54. Seismograph: around 130 AD invented by astronomer Chang Heng

55. Ball falls into frog’s mouth to signal earthquake up to 400 km (250 miles) distant

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56. Chinese first to recognize spontaneous combustion

57. Principle of sedimentation in geology first in China

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Chinese Achievements in

Physics

58. The Kite: invented in China 5th– 4th century BC – in Europe known by 1589 AD

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59. Parachute – 2 nd century BC

60. Rudder for guiding ships on long voyages – Columbus could not have sailed without one

→

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61. Sails and sail technology

– Chinese influence somewhat disputed

– European lateen sails go back 2,000 years

– Chinese tacking technology superior

– Pivoting masts an advantage

– Chinese invented leeboard to hold ship upright while tacking

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62. Paddle wheel boat – 5th century AD

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Chinese Warfare Technology

The next slides correspond to and supplement the reading:

Temple, Robert. 1998. The Genius of China: 3,000 Years of Science, Discovery and Invention. London: Prion

Books Limited.

Chapter 11 – Warfare, Pages 215 – 48

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63. Crossbow – 4 th century BC

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64. Repeating crossbow allowed rapid fire of arrows or of multiple arrows

65. 1st century AD Chinese invented grid sites (crosshairs) for more accurate aim of crossbow

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66. Gunpowder: invented accidentally as by-product of medical research.

67. By 9th century AD

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68. Gunpowder a complex set of chemical discoveries

– Saltpeter – potassium nitrate – also called sodium nitrate –detected by purple flame when it burns

– Chinese knew it possibly from 4th century BC

– Arabs called it “Chinese snow”

– Gunpowder burns by taking oxygen not from the air but from the saltpeter

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69. Sulfur the second key ingredient

– Lowers point of ignition of saltpeter

– Increases speed of the combustion (explosion)

– When saltpeter is up to 75% sulfur causes explosion speed burning of the saltpeter

70. Charcoal provides carbon – third necessary element

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71. General ratio for gunpowder:

– saltpeter 15 parts

– charcoal 3 parts

– sulfur 2 parts

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72 By 850 AD gunpowder in use

73. Despite Western myths to the contrary…

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74. The Chinese did not limit their use of gunpowder to fireworks for entertainment

75. To the contrary, they invented…

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76. The flamethrower –10th century AD

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77. Bombs, grenades and rockets of several types.

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78. Siege engines and other early forms of artillery.

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Chinese Warfare

Technology

79. The canon…by at least 1341

…and…

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80. The gun by at least 1288 AD, but possibly as early as the 11th century

81. Mid 10th century called a “fire lance”

82. Guns reached Europe by 1327

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83. Canons and guns altered the power relations between knights and peasants on the battlefield and played a role in ending the Medieval period of European history. Can you see how?

84. In 1866 Alfred Nobel in Sweden invented dynamite, the first safely manageable explosive since gunpowder.

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85. Dynamite uses nitroglycerin.

86. In recent years dynamite has been replaced by “water gel explosives.”

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87. Modern firearms use a smokeless propellant [the explosive that pushes the bullet out of the barrel] usually based on cordite, a mixture of nitroglycerine and nitrocellulose.

88. …and gunpowder is used primarily in fireworks and for shooting blanks or flares where the visible smoke is an asset.

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End of Week 14 Lecture on


Mathematical Applications in Astronomy,

Physics and Military Technology

Week 14 – China Topic 02

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Week 14 Lecture China Topic 02 Mathematical Contributions...

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