1. Chapter 4 Infinity in Greek Mathematics
Fear of Infinity
Eudoxus’ Theory of Proportion
The Method of Exhaustion
The Area of a Parabolic Segment
Biographical Notes: Archimedes

2. 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 Aristotle’s “Physics”)
Approximation of √2 by the sequence of rational number (Pell’s equation)

3. 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 λ is determined by rational lengths less than and greater than λ
Then λ1 = λ2 if for any rational r < λ1 we have r < λ2 and vice versa (similarly λ1 < λ2 if there is rational r < λ2 but r > λ1 )
Note: the theory of proportions can be used to define irrational numbers: Dedekind (1872) defined √2 as the pair of t wo sets of positive rationals L√2 = {r: r2< 2} and U√2 = {r: r2>2} (Dedekind cut)

4. 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

5. Examples
Approximating the circle
Approximating the pyramid

6. Example: Area of a Circle
Let C(R) denote area of the circle of radius R
We show that C(R) is proportional to R2
Inner polygons P1 < P2 < P3 <…
Outer polygons Q1 > Q2 > Q3 >…
Qi – Pi can be made arbitrary small
Hence Pi approximate C(R) arbitrarily closely
Elementary geometry shows that Pi is proportional to R2 . Therefore Pi(R) : Ri (R’) = R2:R’2
Suppose that C(R):C(R’) < R2:R’2
Then (since Pi approximates C(R)) we can find i such that Pi (R) : Pi (R’) < R2:R’2 which contradicts 5) P2 P1 Q1 Q2
Thus Pi(R) : Ri (R’) = R2:R’2

7. 4.4 The area of a Parabolic Segment [Archimedes (287 – 212 BCE)]
Triangles Δ1 , Δ2 , Δ3 , Δ4,…
Note that Δ2 + Δ3 = 1/4 Δ1
Similarly Δ4 + Δ5 + Δ6 + Δ7 = 1/16 Δ1 and so on Y S Z 1 R 4 7 2 3 Q 6 5 O P X
Thus A = Δ1 (1+1/4 + (1/4)2+…) = 4/3 Δ1

8. 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!”