The Twilight of Greek Mathematics: Diophantus

1
The Twilight of Greek Mathematics Diophantus

• Chapter 5
• Aimee Gorham and Kim Parsley

2
Background

• Close of Golden Age of Greek mathematics
• Ptolemy VII banished scientist and scholars who
  were not loyal to him
• The end of the pre-Christian era saw the steady
  and relentless growth of Roman power
• The stability of the reign of Ptolemies began the
  era of street riots and political confusion, at
  the same time the commercial and intellectual
  glories of Alexandria slowly deteriorated

3
Background continued

• New masters of the Mediterranean never showed any
  inclination for extensive theoretical studies
• Application of arithmetic and geometry to
  engineering projects such as viaducts, bridges,
  roads, public buildings, and land surveys proved
  to be the leading Roman concern.
• The development of Christianity accelerated the
  demise of Greek learning

4
Dominating Mathematicians of This Period

• Diophantus
• Bhaskara of India
• Pappus of Alexandria
• Hypathia
• Boethis
• al-Khowârizmî
• Liu Hui

5
Diophantus Father of Algebra

• 150 AD 350 AD (likely 250 AD)
• Hellenistic Babylonian
• Known for his study of equations with variables
• Last great mathematician of antiquity
• One of his main contributions was the
  syncopation of algebra
• Best known for his Arithmetica, a work on the
  theory of numbers

6
Diophantus The Father of Algebra
God granted him to be a boy for the sixth
part of his life, and adding a twelfth part to
this, He clothed his cheeks with down He lit him
the light of wedlock after a seventh part, and
five years after his marriage He granted him a
son. Alas! Late-born wretched child after
attaining the measure of half his fathers life,
chill Fate took him. After consoling his grief by
this science of numbers four years he ended his
life. See page 207 for actual equation
7
Arithmetica

• A collection of about 189 problems giving
  numerical solutions of determinate equations
  (those with a unique solution) and indeterminate
  equations
• Equations in the book are called Diophantine
  Equations
• Out of 13 total books, only 6 survived
• Most of the Arithmetica lead to quadratic
  equations

8
Diophantuss Symbols

• See page 209 in text

9
Diophantine Equations

• Definition An equation in one or more unknowns
  with integer coefficients for which integer
  solutions are sought
• Linear diophantine equation is one where the
  unknowns appear only in the first power

10
Theorem

• The linear diophantine equation
• ax by c has a solution if and only if
• dc, where d gcd (a,b). If x0, y0 is any
  particular solution of this equation, then all
  other solutions are given by
• x x0 (b/d)t, y y0 (a/d)t
• for some integer t.
• see page 215 in text

11
Examples of Diophantine Equations

• ax by 1
• xn yn zn for n 2 there are many solutions
  (x,y,z), the Pythagorean triples. For larger
  values of n, Fermats last theorem states that no
  positive integer solutions x,y,z satisfying the
  above equation exists.
• x2 ny2 1 Pells equation

12
Problem for ArithmeticaBook II, Problem 8
Divide a given square number, say 16, into the
sum of two squares. Let one of the required
squares be x2.. Then 16 x2 16-x2 must be equal
to a square. Diophantus selected a particular
instance of a perfect square to set this equal
to, one that was particularly useful in
eliminating the constant terms (2x 4)2 Setting
these terms equal to each other, we have 16 x2
(2x 4)2 which simplifies to 5x2 16x. Thus x
16/5. Hence, the two squares would be
(16/5)2 or 256/25 and 16 256/25, or
144/25.
13
Bhaskara of India

• Became head of the astronomical observatory, the
  leading mathematical centre in India at the time.
• He represents the peak of mathematical knowledge
  of the 12th century.
• He is thought by many historians to be a late
  forgery.

14
Bhaskara Continued

• He wrote six works that dealt with mathematics
  including Siddhanta Siromani
• The first two books were called Lilavati (The
  Beautiful) and Vijaganita (Root Extractions)
• A use of the Diophantine equation by Bhaskara,
  hundred fowls, is shown on page 218 of text

15
Pappus of Alexandria

• He was what we call now a commentator
• He was a leading Greek mathematician of his time
  but the original mathematics he created was small
  in stature and quantity
• His fame lies in his work, The Collection, an
  assembly of older works by mentionable authors
  and added his own explanations and amplifications.

16
Hypatia

• First woman to make substantial contribution to
  the development of mathematics.
• Daughter of Theon of Alexandria she studied
  mathematics under his instruction.
• She became the head of the neo-Platonist school
  at Alexandria about 400 AD.
• Christian leaders thought her symbolism of
  learning and science to be paganistic.
• She wrote commentaries on Diophantuss
  Arithmetica and on Apolloniuss Conics.
• She was brutally murdered by the Nitrian monks.
• Her death marked the end of the glorious history
  of Greek Mathematics.

17
Anicius Boethius
18
Anicius Boethius

• Provided a bridge between Antiquity and the
  Middle Ages
• He wrote The Consolation of Philosophy
• His geometry consisted of definitions and
  statements of theorems - no proofs from The
  Elements
• It was through Boethius that Middle Ages came to
  know the principles of formal arithmetic

19
Mathematics of Near and Far East

• The fading fluency of the Greek language added to
  the lack of interest in theoretical studies
• Rise in Arabic power due to emergence of Islamic
  faith
• House of Wisdom was founded by Caliph al-Mamûn
• It was comparable to the Museum at Alexandia

20
al-Khowârizmî
21
al-Khowârizmî

• He was a court astronomer and a friend of Caliph
  al Mamûn
• Due to his works Europe became familiar with
  Hindu numerals and algebraic approach to
  mathematics
• He wrote Book of Addition and Subtraction
  According to the Hindu Calculations
• His methods are different than Diophantus which
  leads to the idea that his knowledge of algebra
  doesnt come from Diophantus
• Finally, through his work, the long devices of
  the Babylonians are now seen as a systematic
  reduction of the quadratics to one of his forms

22
Science of Reunion and Reduction

• al-Khowârizmî used two principle operations to
  solve equations
• Reunion - refers to the transfer of negative
  terms from one side of the equation to the other
• Reduction refers to a combination of like terms
  on the same side or the cancellation of like
  terms on opposite sides of the equation

23
Liu Hui of China
24
Liu Hui of China

• Little is known of his life
• He is credited for two works The Nine Chapters
  on the Mathematical Art (the oldest existing
  mathematics textbook) and Sea Island Mathematical
  Manual.
• Nine Chapters was written more practically than
  theoretically.
• Sea Island Mathematical Manual dealt with
  measuring distances to inaccessible points.

25
Nine Chapters

• Marks the beginning of the mathematical tradition
  in China
• Consists nine distinct sections with a total of
  246 problems and their solutions
• Provides the first evidence of a systematic
  method for solving simultaneous equations
• The chapters have titles such as Field
  Measurement, Distribution by Proportion, and
  Fair Taxes.

26
References

• Burton, David M., The History of Mathematics, An
  Introduction, p 203 - 246, McGraw-Hill, New York,
  NY, 2003.
• Burton, David M., The History of Mathematics, An
  Introduction, p 221 246, Wm. C. Brown
  Publishers, Dubuque, IA, 1991.
• http//www.wsu.edu8080/dee/CHRIST/BOETHIUS.HTM
• http//www-gap.dcs.st-and.ac.uk/history/Mathemati
  cians/Boethius.html
• http//www-gap.dcs.st-and.ac.uk/history/Mathemati
  cians/Diophantus.html
• http//www-gap.dcs.st-and.ac.uk/history/Mathemati
  cians/Hypatia.html
• http//www-history.mcs.st-and.ac.uk/history/Mathem
  aticians/Liu_Hui.html

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References

• http//aleph0.clarku.edu/djoyce/mathhist/china.ht
  ml
• http//www-groups.dcs.st-and.ac.uk/history/Mathem
  aticians/Bhaskara_II.html
• http//www.worldhistory.com/wiki/P/Pappus-of-Alexa
  ndria.htm
• http//www.pbs.org/wgbh/nova/archimedes/lrk_famous
  .html
• http//www-history.mcs.st-andrews.ac.uk/Mathematic
  ians/Pappas.html
• http//www-groups.dcs.st-and.ac.uk/history/Mathem
  aticians/Pappas.html
• http//www-groups.dcs.st-and.ac.uk/history/Mathem
  aticians/Hypatia.html