Chapter 4 Infinity in Greek Mathematics

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

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)

4.2 Eudoxus’ Theory of Proportions • Eudoxus (around 400 – 350 BCE) • The theory was designed to deal with (irrational) lengths using only rational numbers • Lengthλis determined by rational lengths less than and greater than λ • Then λ1 = λ2if for any rationalr < λ1we haver < λ2and vice versa (similarly λ1 < λ2if there is rational r < λ2but r > λ1) • Note: the theory of proportions can be used to defineirrational numbers:Dedekind (1872) defined√2 as the pair of t wo sets of positive rationalsL√2 = {r: r2< 2} andU√2 = {r: r2>2} (Dedekind cut)

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

Examples Approximating the pyramid Approximating the circle

Q2 • Let C(R) denote area of the circle of radius R • We show that C(R) is proportional to R2 Example:Area of a Circle • 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 thatPi (R) : Pi (R’) < R2:R’2which contradicts 5) P2 P1 Q1 Thus Pi(R) : Ri (R’) = R2:R’2

4.4 The area of a Parabolic Segment[Archimedes (287 – 212 BCE)] Y Z S • TrianglesΔ1 ,Δ2 ,Δ3 ,Δ4,… • Note thatΔ2 +Δ3 = 1/4 Δ1 • SimilarlyΔ4 +Δ5 +Δ6 +Δ7= 1/16 Δ1and so on 1 R 4 7 3 2 Q 6 5 O X P Thus A = Δ1 (1+1/4 + (1/4)2+…) = 4/3 Δ1

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!”

Chapter 4 Infinity in Greek Mathematics