All Roads Lead to Rome I

Choy Siu KaiFSC1110Email: skchoy@hkbu.edu.hk

Euclid’s Legacy

• Euclid’s Elements became the standard for Western mathematics and science– Textbook until the 19th century

• Shaped the development of the whole of Western science in the Reductionist Approach

Euclid’s Legacy

• Science was poised for a great revolution (e.g. astronomy, cosmology)– Eratosthenes (276-195 BC;

measurement of earth diameter, distance to the moon and sun)

– Appollonius (c. 200 BC; conics)

Archimedes (287–212 BC)

• Borned and lived in Syracuse in Sicily

• Studied in Alexandria’s Library• Considered one of the Greatest

Mathematician of all time– Father of Mathematical Physics– Great inventor (War machine, pulley

system, theory of levers, theory of hydrostatics, Archimedes Screw…)

– Estimate of Pi

The Story of Pi• Euclid proved that the ratio of the

circumference of a circle to its diameter is a constant (i.e. )i.e. Perimeter/2r = constant

• Archimedes proved that the ratio of the area of a circle to the square of its radius is also the same constant

i.e. Area/r2 = constant

Estimate of Pi• 1650 BC Rhind Papyrus: 4(8/9)2

-- Take the diameter of the circle, remove “the ninth part” of it, and find

the area of the square with the resulting side length.

22

22

9

84

29

8

9

8

r

rd

d

Area of circle ~ Area of square

d9

8

The Story of PiEudoxus: Method of Exhaustion (Using regular polygon with

n sides (n-gon) to approximate area of shape; circle)

4-gon (square)

8-gon

16-gon

………

The Story of Pibnh

2

1Theorem: The area of the regular polygon is

bbb

b

bb

b

b

h

Proof: Suppose the polygon has n sides (n-gon), eachof length b. Draw lines from O to the vertices, thereby breaking it up into a collection ofn congruent triangles, each with height h andbase b.

Area of regular polygon= Total area of n congruent triangles= n x (½ x b x h)= ½ x h x n x b Q.E.D.

O

8-gon has 8 congruent triangles

r

r

Perimeter of regular polygon

Estimate of Pi• 240 BC Archimedes:

70

103

71

103

Regular inscribed hexa-gon (6-gon)

71

103

circle ofdiameter

gon-96 inscribedregular ofperimeter

circle ofdiameter

circle of ncecircumfere

Regular circumscribed hexa-gon (6-gon)

Similarly, we have 70

103

Estimate of Pi

• 1650 BC Rhind Papyrus: 4(8/9)2

• 240 BC Archimedes:

• 150 AD Ptolemy: (377/120)• 480 AD Zu Chongzhi ( 祖沖之 ) (355/113)• 530 AD Aryabhata: (62832/20000)

70

103

71

103

Estimate of Pi

• 1765 Lambert proved that is irrational

• 1882 Lindemann proved that is actually transcendental – i.e. cannot be the root of any

algebraic equations (polynomials with rational coefficients)

The Estimate of Pi

• His other great achievements include-- Area under a parabola-- Surface area and volume of a sphere-- Volume of cone

• His techniques anticipate that of calculus

Archimedes’ Achievements

Archimedes (287–212 BC)

• Killed in 212 BC by a Roman Soldier after defending the siege for an extended period of time

• His death symbolically marked the beginning of the end of the Hellenistic Age and the emergence of the Roman Empire

Dates Greek Thinkers Greek Periods

635 BC Thales

569 BC Pythagoras Late Mycenaean

510 BC Cleisthenes

490 BC Zeno Classical

470 BC Socrates

427 BC Plato

384 BC Aristotle

325 BC Euclid Hellenistic

287 BC Archimedes

Troy

Greater Greece

Rome

Syracuse

Carthage

Rise of Rome

Dates Rome China

1220 BC Trojan war, Aeneas escaped to Rome

Shang ( 商 )Dynasty

449 BC The Twelve Tables ( 十二表法 ): (rule of law)

Shang Yang ( 商鞅 ) Reform (338 BC)

~300 BC Expansion of Rome Expansion of Qin( 秦 )

264 BC Start of Punic wars against Carthage

Massacre of Zhao (260 BC)Yin Zheng ( 嬴政 ; 247 BC)

Rise of Rome

Dates Rome China

212 BC Taking of Syracuse (traitor opened gate from within)

Qin Dynasty 秦朝 (221 -207 BC)

31 BC – 476 AD

Roman Empire Han Dynasty 漢朝 (206 BC- 220 AD)

395-1453 Byzantine Empire 南北朝 (420-589)800-1806 Holy Roman Empire Tang, Song, Yuan,

Ming, Qing

Rise of Rome

Rise of Rome

• Governance by US-like senate: Executive, Legislative, and Judicial

• Rule by tough but fair laws• Sworn loyalty to the Emperor

– Short-lived Military dictators coming to power via coup d’etats

– Interested in consolidating ties (diplomacy) and governing a large empire (bureaucracy)

– Not interested in academic studies (availability of slave labour)

Rise of Rome

• Greek culture was assimilated • Greek Mathematics and Science

condoned but has now lost its flair and impetus– Ptolemy (120 AD; astronomy)– Diophantus (c. 200 AD; Arithmetica – study of

algebraic problems with integer solutions)– Pappus, Theon (c. 300-400 AD;

commentaries)

The Fall of Rome

• The Library at Alexandria was destroyed and recreated several times in its history– 48 BC at the conquest by Julius Caesar– 3rd century AD civil wars– 391 AD by fanatical Christian rioters

• 415 AD The last Head Librarian at Alexandria, Hypatia (Theon’s daughter), was brutally murdered by Christians in a power struggle

The Fall of Rome

• 642 AD Final destruction of the Library by Caliph Omar who allegedly said:– "If these writing of the Greeks agree

with the book of God [Koran], they are useless and need not be preserved; if they disagree, they are pernicious and ought to be destroyed".

The Fall of Rome

• Rome split into two halves in 395 AD

• The Western Roman Empire fell in Sept 4, 476 AD (the last emperor was deposed by Odoacer)

The Fall of Rome

• With the fall of the Western Roman Empire, Europe went into the Dark Ages

• The Eastern Roman (Byzantine) Empire finally fell in 29 May 1453 AD by the conquest of the Ottoman Empire

The Fall of Rome

• After the fall of Rome, the church served as a source of knowledge, authority, and continuity, helping to settle disputes amongst secular rulers

• In 800 AD, Charlemagne was crowned by the Pope as the first Emperor of the Holy Roman Empire, which was a union of many medieval states in central Europe

The Fall of Rome

• The French philosopher Voltaire described the Holy Roman Empire as an "agglomeration" which was "neither holy, nor Roman, nor an empire"

• The last emperor, Francis II abdicated in 1806 during the Napoleonic Wars

Meanwhile, in China

• Warring States Period (600-221 BC)– Hundred Schools of Thought– Lao Tse– Confucius, Mencius (372-289)– Huizi (370-310 BC), Zhuangzi– Han Fei Zi (d. 233 BC)

Meanwhile, in China

• Chou Pei 「周髀算經」 (c. 100 BC– Mainly a book on astronomy– Calculated distance to the sun

~60,000 miles << 45 million miles estimate by Aristarchus (310-230 BC) and Hipparchus’ (190-120 BC)

– Current estimate of solar distance is 92 million miles

Meanwhile, in China

• Arithmetic in Nine Sections • Liu Hui (c. 200 AD;

commentaries on Nine Chapters, estimate of pi, Gaussian elimination)

• Sun Tzi (c. 400 AD; Chinese Remainder Theorem: popularized by Jin Yong)三人同行七十稀，五树梅花廿一枝，七子团圆正半月， 余百零五便得知。

Chinese Remainder Theorem

孫子算經 : 今有物不知其數，三三數之剩二， 五五數之剩三，七七數之剩二，問物幾何？

三人同行七十稀， ( 把除以三所得餘數用七十乘 )五树梅花廿一枝， ( 把除以五所得餘數用廿一乘 )

七子团圆正半月， ( 把除以七所得餘數用十五乘 )

余百零五便得知。 ( 把上述總加起來，減去一百零 五的倍數，所得的差即為所求 )

2×70 ＋ 3×21 ＋ 2×15 = 233

233 － 105×2=23

Chinese Remainder Theorem

70 除以 3 餘 1 ，被 5 ， 7 整除，所以 70a 除以 3 餘 a ，也被 5 ，7 整除；

21 余以 5 餘 1 ，被 3 ， 7 整除，所以 21b 除以 5 餘 b ，也被 3 ，7 整除；

15 除以 7 餘 1 ，被 3 ， 5 整除，所以 15c 除以 7 餘 c ，被3 ， 5 整除。

而 105 則是 3 ， 5 ， 7 的最小公倍数。

所以， 70a ＋ 21b ＋ 15c 是被 3 除餘 a ，被 5 除餘 b ，被 7 除餘 c 的數，這個數如果大了，要减去它們的公倍數。 (a=2 ， b=3 ， c=2)

孫子算經 : 今有物不知其數，三三數之剩二， 五五數之剩三，七七數之剩二，問物幾何？

Meanwhile, in China• Zu Chongzhi (428-501 AD; estimate

pi by 355/113 using 24576-sided polygon)

• Proposed new Daming calendar– criticized for ”... distorting the truth

about heaven and violating the teaching of the classics.”

– Zu replied “[my calendar is]... not from spirits or from ghosts, but from careful observations and accurate mathematical calculations. ... people must be willing to hear and look at proofs in order to understand truth and facts.”

Meanwhile, in China

• Yang Hui ( 楊輝 ; 1238-1298 AD)– Detailed Analysis of the Nine

Chapters– Numerical solutions to

quadratic equations– Magic squares

Meanwhile, in China

• Chu Shih Chieh ( 朱世杰 ; 1249-1314 AD)– Precious Mirror of the Four Elements 《四元玉鑒 》– Pascal Triangle, Systems of

equations– Arithmetic Progression– Geometric Progression

Meanwhile, in China

• Pascal Triangle is an efficient method for evaluating the coefficients of the expansion of (1+x)n [binomial expansion]– E.g. (1+x)4 = 1+4x+6x2+4x3+1x4

Meanwhile, in China

• Arithmetic Progression / Series•1+2+3+4+…+n = n(n+1)/2

• Geometric Progression / Series

•1+2+4+…+2n =2n+1-1

•1+r+r2+…+rn = (rn+1-1)/(r-1), r 1

Additional References• William P. Berlinghoff & Fernando

Q. Gouvêa, Math through the Ages (sketch 7, 14), Oxton House, 2002.

• William Dunham, Journey through Genius (chapter 4), Penguin, 1990.