Chapter 4 Infinity in Greek Mathematics

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Chapter 4Infinity in Greek Mathematics

• Fear of Infinity
• Eudoxus Theory of Proportion
• The Method of Exhaustion
• The Area of a Parabolic Segment
• Biographical Notes Archimedes

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4.1 Fear of Infinity

• Discovery of irrational numbers
• Greeks tried to avoid the use of irrationals
• The infinity was understood as potential for
  continuation of a process but not as actual
  infinity (static and completed)
• Examples
• 1,2, 3,... but not the set 1,2,3,
• sequence x1, x2, x3, but not the limit x lim
  xn
• Paradoxes of Zeno ( 450 BCE) the Dichotomy
• there is no motion because that which is moved
  must arrive at the middle before it arrives at
  the end
• (cited from Aristotles Physics)
• Approximation of v2 by the sequence of rational
  number (Pells equation)

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4.2 Eudoxus Theory of Proportions

• Eudoxus (around 400 350 BCE)
• The theory was designed to deal with (irrational)
  lengths using only rational numbers
• Length ? is determined by rational lengths less
  than and greater than ?
• Then ?1 ?2 if for any rational r lt ?1 we have r
  lt ?2 and vice versa (similarly ?1 lt ?2 if there
  is rational r lt ?2 but r gt ?1 )
• Note the theory of proportions can be used to
  define irrational numbers Dedekind (1872)
  defined v2 as the pair of t wo sets of positive
  rationals Lv2 r r2lt 2 andUv2 r r2gt2
  (Dedekind cut)

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4.3 The Method of Exhaustion

• was designed to find areas and volumes of
  complicated objects (circles, pyramids, spheres
  etc.) using
• approximations by simple objects (rectangles,
  trianlges, prisms) having known areas (or
  volumes)
• the Theory of Proportions

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Examples
Approximating the pyramid
Approximating the circle
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ExampleArea of a Circle

• Let C(R) denote area of the circle of radius R
• We show that C(R) is proportional to R2

1. Inner polygons P1 lt P2 lt P3 lt
2. Outer polygons Q1 gt Q2 gt Q3 gt
3. Qi Pi can be made arbitrary small
4. Hence Pi approximate C(R) arbitrarily closely
5. Elementary geometry shows that Pi is proportional
   to R2 . Therefore Pi(R) Ri (R) R2R2
6. Suppose that C(R)C(R) lt R2R2
7. Then (since Pi approximates C(R)) we can find i
   such thatPi (R) Pi (R) lt R2R2 which
   contradicts 5)

P2
P1
Q1
Thus Pi(R) Ri (R) R2R2
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4.4 The area of a Parabolic SegmentArchimedes
(287 212 BCE)

• Triangles?1 , ?2 , ?3 , ?4,
• Note that?2 ?3 1/4 ?1
• Similarly?4 ?5 ?6 ?7 1/16 ?1and so on

Y
Z
S
1
R
4
7
3
2
Q
6
5
O
X
P
Thus A ?1 (11/4 (1/4)2) 4/3 ?1
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4.5 Biographical Notes Archimedes

• Was born and worked in Syracuse (Greek city in
  Sicily) 287 BCE and died in 212 BCE
• Friend of King Hieron II
• Eureka! (discovery of hydrostatic law)

• Invented many mechanisms, some of which were used
  for the defence of Syracuse
• Other achievements in mechanics usually
  attributed to Archimedes (the law of the lever,
  center of mass, equilibrium, hydrostatic
  pressure)
• Used the method of exhaustions to show that that
  the volume of sphere is 2/3 that of the
  enveloping cylinder
• Stay away from my diagram!