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Home , Diophantine equation, Diophantus, Euclidean algorithm, Eudoxus of Cnidus, Hypatia, Measurement of a Circle

Greek II Fall 2021 - R. L. Herman Greek Theory

• 2 2 2 x + y = z , x, y, z . n = 3 • Diophantine Solve 3x + 5y = 1, x, y integers. n = 6 • • Proved # primes infinite, Book IX, Prop 20 n = 10 • Perfect , Book VII, Def 22, Book IX, Prop 36 Euclid proves: If 2n − 1 is prime, then n = 15 (2n − 1) 2n−1 is perfect. n Mersenne prime: 2 − 1. Figure 1: Polygonal Numbers • Polygonal Numbers

History of Math R. L. Herman Fall 2021 1/20 Euclidean - Book VII, Prop 1

• gcd(a, b): Greatest common of a and b. Continuing the computation:

• Algorithm ai bi 210 45 a1 = max(a, b) − min(a, b) 165 45 b1 = min(a, b) 120 45 repeat 75 45 30 45 Terminates when ai+1 = bi+1. 15 30 Example: Find gcd(210, 45). 15 15 We find gcd(210, 45) = 15. a1 = 210 − 45 = 165

b1 = 45

History of Math R. L. Herman Fall 2021 2/20 - Another Approach

Find the of positive integers, a and b.

• If a < b, exchange a and b. • Divide a by b and get the , r. Thus,

a = qb + r.

• If r 6= 0, replace a by b and b by r. Repeat the . • If r = 0, report gcd(a, b) = b.

Example: Find gcd(210, 45).

210 = 4 · 45 + 30 45 = 1 · 30 + 15 30 = 2 · 15

Thus, gcd(210, 45) = 15. History of Math R. L. Herman Fall 2021 3/20 Pell’s (1611-1685)

• x 2 − Ny 2 = 1, N is a nonsquare , and x, y are integer solutions. • Example of a . √ • Related to 2 : x 2 − 2y 2 = 0, √ y = 1 ⇒ x = 2. • x 2 − 2y 2 = 1, √ x If x, y large, then y ≈ 2. • Known to Pythagoreans, , and • ’ Cattle Problem can be reduced to solving Pell’s Equation. • (598-570) first to solve.

History of Math R. L. Herman Fall 2021 4/20 Let √ z = x1 + y1 n, √ w = x2 + y2 n, √ zw = x3 + y3 n,

Then • Example: x 2 − 3y 2 = 1 x = x x + ny y , • Guess (2, 1). 3 1 2 1 2 √ So, z = 2 + 3 = w. y3 = x1y2 + x2y1.  √ 2 zw = 2 + 3 Since (zw) = 1, √ (x3, y3) is a solution. = 7 + 4 3. Then, (7, 4) is a solution.

Pell’s Equation General Solution

• x 2 − ny 2 = 1, n is a nonsquare integer and x, y are integer solutions. √ • Let z = x + y n, x, y ∈ √ Z andz ¯ = x − y n. • Norm(z) = zz¯ = x 2 − ny 2 = 1. • Norm(zw) = Norm(z)Norm(w).

History of Math R. L. Herman Fall 2021 5/20 • Example: x 2 − 3y 2 = 1 • Guess (2, 1). √ So, z = 2 + 3 = w.  √ 2 zw = 2 + 3 √ = 7 + 4 3. Then, (7, 4) is a solution.

Pell’s Equation General Solution

• x 2 − ny 2 = 1, n is a nonsquare integer and Let x, y are integer solutions. √ √ z = x + y n, • Let z = x + y n, x, y ∈ 1 1 √ Z √ andz ¯ = x − y n. w = x2 + y2 n, √ • Norm(z) = zz¯ = x 2 − ny 2 = 1. zw = x3 + y3 n,

• Norm(zw) = Norm(z)Norm(w). Then

x3 = x1x2 + ny1y2,

y3 = x1y2 + x2y1.

Since Norm(zw) = 1,

(x3, y3) is a solution.

History of Math R. L. Herman Fall 2021 5/20 Then, (7, 4) is a solution.

Pell’s Equation General Solution

• x 2 − ny 2 = 1, n is a nonsquare integer and Let x, y are integer solutions. √ √ z = x + y n, • Let z = x + y n, x, y ∈ 1 1 √ Z √ andz ¯ = x − y n. w = x2 + y2 n, √ • Norm(z) = zz¯ = x 2 − ny 2 = 1. zw = x3 + y3 n,

• Norm(zw) = Norm(z)Norm(w). Then • Example: x 2 − 3y 2 = 1 x = x x + ny y , • Guess (2, 1). 3 1 2 1 2 √ So, z = 2 + 3 = w. y3 = x1y2 + x2y1.  √ 2 zw = 2 + 3 Since Norm(zw) = 1, √ (x3, y3) is a solution. = 7 + 4 3.

History of Math R. L. Herman Fall 2021 5/20 Pell’s Equation General Solution

• x 2 − ny 2 = 1, n is a nonsquare integer and Let x, y are integer solutions. √ √ z = x + y n, • Let z = x + y n, x, y ∈ 1 1 √ Z √ andz ¯ = x − y n. w = x2 + y2 n, √ • Norm(z) = zz¯ = x 2 − ny 2 = 1. zw = x3 + y3 n,

• Norm(zw) = Norm(z)Norm(w). Then • Example: x 2 − 3y 2 = 1 x = x x + ny y , • Guess (2, 1). 3 1 2 1 2 √ So, z = 2 + 3 = w. y3 = x1y2 + x2y1.  √ 2 zw = 2 + 3 Since Norm(zw) = 1, √ (x3, y3) is a solution. = 7 + 4 3. Then, (7, 4) is a solution.

History of Math R. L. Herman Fall 2021 5/20 (c.390 – c. 337 BCE)

• Studied under • Taught

• , N = 8 • Theory of Proportions : A ∝ r 2, Spheres: V ∝ r 3, Volume of a pyramid N = 18 Volume of a • Studied reals, continuous quantities • Due to Antiphon (480–411 BCE) N = 38 from a sequence of inscribed Figure 2: Method of Exhaustion

History of Math R. L. Herman Fall 2021 6/20 Archimedes of Syracuse (287-212 BCE)

• Went to , then, back to Syracuse, • Greatest Mathematician of Antiquity • Mathematician, Engineer, Inventor • Archimedean screw, lever, pulley • King Heiro II’s crown - Eureka Archimedes Principle of Bouyancy • According to (46-120) • Marcellus - Syracuse 212 BCE • Claw of Archimedes • Heat Ray • Prone to intense concentration • Death of Archimedes

History of Math R. L. Herman Fall 2021 7/20 • Regular Polygons (n-gon) 1 Ap = 2 hQ, Q = Perimeter • Inscribed Polygons 1 Ap = 2 anh < area of circle • Circumscribed Polygon q 180 2 a2 a = 2R sin n , h = R − 4 . • Approximation of π, Ap Ap R2 < π < h2

Archimedes’ Mathematics

• Mastered Euclid and Eudoxus’ (c. 390-337 BCE) Method of Exhaustion C • Measurement of a C A D = const., D2 = const.

D

A = area

History of Math R. L. Herman Fall 2021 8/20 • Inscribed Polygons 1 Ap = 2 anh < area of circle • Circumscribed Polygon q 180 2 a2 a = 2R sin n , h = R − 4 . • Approximation of π, Ap Ap R2 < π < h2

Archimedes’ Mathematics

• Mastered Euclid and Eudoxus’ (c. 390-337 BCE) Method of Exhaustion • C A D = const., D2 = const. • Regular Polygons (n-gon) h 1 Ap = 2 hQ, Q = Perimeter R

History of Math R. L. Herman Fall 2021 8/20 • Circumscribed Polygon q 180 2 a2 a = 2R sin n , h = R − 4 . • Approximation of π, Ap Ap R2 < π < h2

Archimedes’ Mathematics

• Mastered Euclid and Eudoxus’ (c. 390-337 BCE) Method of Exhaustion • Measurement of a Circle C A D = const., D2 = const. • Regular Polygons (n-gon) h 1 Ap = 2 hQ, Q = Perimeter R • Inscribed Polygons 1 Ap = 2 anh < area of circle

History of Math R. L. Herman Fall 2021 8/20 • Approximation of π, Ap Ap R2 < π < h2

Archimedes’ Mathematics

• Mastered Euclid and Eudoxus’ (c. 390-337 BCE) Method of Exhaustion • Measurement of a Circle C A a D = const., D2 = const. • Regular Polygons (n-gon) h 1 Ap = 2 hQ, Q = Perimeter R • Inscribed Polygons 1 Ap = 2 anh < area of circle • Circumscribed Polygon q 180 2 a2 a = 2R sin n , h = R − 4 .

History of Math R. L. Herman Fall 2021 8/20 Archimedes’ Mathematics

• Mastered Euclid and Eudoxus’ (c. 390-337 BCE) Method of Exhaustion • Measurement of a Circle C A D = const., D2 = const. • Regular Polygons (n-gon) h 1 Ap = 2 hQ, Q = Perimeter R • Inscribed Polygons 1 Ap = 2 anh < area of circle • Circumscribed Polygon q 180 2 a2 a = 2R sin n , h = R − 4 . • Approximation of π, Ap Ap R2 < π < h2

History of Math R. L. Herman Fall 2021 8/20 Estimating π

• Approximation of π, Ap Ap 2 < π < 2 , R h Circumscribed • Recall 4 Ap a = 2R sin 180 , 2 Inscribed q n h = R2 − a2 = R cos 180 , 0 4 n 0 2 4 6 8 10 12 14 16 18 1 180 Ap = 2 anh = nhR sin n . n • Therefore, n 360 180 sin < π < n tan , 2 n n h • Hexagon (n = 6), R 2.598 < π < 3.464. • Archimedes - up to 96-gon 3.1394 < π < 3.1427.

History of Math R. L. Herman Fall 2021 9/20 Archimedes’ Inscribed and Circumscribed n-gons

Consider a fixed circle of R. 180 • Inscribed n-gon: h = R cos n , 180 n = 5 n = 6 Ai = nhR sin n . • Circumscribed n-gon: R 180 r = 180 , Ac = nHr sin n . cos n • Thus, 180 A = nR2 tan , n = 7 n = 8 i n n 360 A = R2 sin . c 2 n • This gives n 360 180 n = 9 20 sin < π < n tan , 2 n n

History of Math R. L. Herman Fall 2021 10/20 Early Approximations of π

• Bible, π ≈ 3. 4 4 256 • , 3 = 81 ≈ 3.1604938 • (150), 360-gon, 3.1416 • Chinese (430-501) 355 ≈ 3.14159292 • Leibniz-Madhava 113 π 1 1 = 1 − + − · · · 3927 4 3 5 • Hindu (1100) 1250 ≈ 3.1416 • Euler • Viete’, 393,216-gon, π to 9 places. π2 1 1 = 1 + + + ··· 2 2 • van Ceulen (1540-1610) Dutch, 35 places. 6 2 3 • Ramanujan √ ∞ • William Shanks (1873) 527 digits. 1 2 2 X (4k)! 1103 + 26390k = π 9801 k!4 3964k • Lambert (1728-1777) - irrationality proof. k=0 • William Jones (1706) introduced π • Euler popularized notation • See Approximations of π .

History of Math R. L. Herman Fall 2021 11/20 On the Sphere and the Cylinder, Archimedes

• Spirals, Area of Parabolae r

• Volumes and Surface of r 3D Objects • A(Sphere) = 4 A(Great circle) A = 4(πr 2). • V(Sphere) = 4 V(Cone) r 1 3 V = 4 3 πr . r • Sphere inside Cylinder Cylinder Area = 2πr(2r) + 2(πr 2) = 6πr 2 3 r h = 2r = 2 Sphere Area. Volume = (πr 2)h = 2πr 3 3 = 2 Sphere Volume.

History of Math R. L. Herman Fall 2021 12/20 Archimedes’ Manuscripts

• What we know from 3 books • Codex A Lost in 1564 • Codex B Lost in 1311 • Codex C Discovered 1906 • 4th century Parchment bound • 10th Century Book, housed great texts • 1204 4th Crusade destroyed books • 87 Sheets (43.5 goat skins) • 1229 Century taken apart, scraped, cut in half, written over with Christian prayer Figure 3: Codex C Page • Moved to Palestine, 400 yrs The Walters Museum - http://www.archimedespalimpsest.net

History of Math R. L. Herman Fall 2021 13/20 Palimpsest, http://www.archimedespalimpsest.org/about/

• 1846 in Istanbul, leaf removed to Cambridge. • 1906 Johan Heiberg took pictures and translated. • 1922 It went missing • 1998 Sold for $2,000,000 - Christies of NY auction. Moldy, Charred, • 7 Manuscripts The Equilibrium of Planes, Spiral Lines, The Measurement of the Circle, Sphere and Cylinder, On Figure 4: Number of different Floating Bodies, The Method arrangements of the of Mechanical Theorems, Stomachion, 17,152. and the Stomachion. History of Math R. L. Herman Fall 2021 14/20 of Cyrene (276 - 194 BCE)

• Chief Librarian at Library (300-529 BCE) Burned 641 by Arabs • • Finding primes

¡1, 2, 3, ¡4, 5, ¡6, 7, ¡8, ¡9,10, 11,...

• of Earth • Syrene - 1st day summer Sun directly overhead • Alexandria - small shadow • 250,000 stadia 1 stade ≈ 526.37 ft Equals 24, 466 mi. Current - 24, 860 mi.

History of Math R. L. Herman Fall 2021 15/20 Heron of Alexandria (c. 10-70)

• Inventor • Aeolipile,rocket-like reaction engine Heron’s Formula • first-recorded steam engine A = p(s − a)(s − b)(s − c) 1 • Hero’s wind-powered organ S = 2 (a + b + c) • The first vending machine • A wind-wheel operating an organ, • The force pump b • A syringe-like device c • Principle of the shortest path of light: • Standalone fountain a • A programmable cart

History of Math R. L. Herman Fall 2021 16/20 Last of the

• Ptolemy (100-170) • • Copernicus (1300) • Heliocentric vs geocentric • Diophantus (200’s) • Equations with integer solutions. • of 13 books, ,- algebraic equations. • (370-415) • Father - Theon • Martyr Figure 5: Epicycles, Diophantus, and • Movie - Hypatia

Romans - Little contribution to mathematics.

History of Math R. L. Herman Fall 2021 17/20 Diophantus’ Epitaph

“Here lies Diophantus. God gave him his boyhood one-sixth of his life; One twelfth more as youth while whiskers grew rife; And then yet one-seventh ‘ere marriage begun. In five years there came a bouncing new son; Alas, the dear child of master and sage, After attaining half the measure of his father’s life, chill fate took him. After consoling his fate by the of numbers for four years, he ended his life.” - , .

History of Math R. L. Herman Fall 2021 18/20 One Last Thing - The

• Found in a wreck in 1900 near Antikythera. • Recovered statues and other items. • Small corroded fragments found. • Fragment A has 27 gears. • 100 years later ... • It is an astronomical calculator. • Predicts moon’s position and phase, solar eclipses, motion of , and more. • Possibly from 1st or 2nd century BCE. • See March 2021 Paper, Freeth, et al.

History of Math R. L. Herman Fall 2021 19/20 Timeline of Greek Mathematicians

−630 −610 −590 −570 −550 −530 −510 −490 −470 −450 −430 −410 −390 −370 −350 −330 −310 −290 −270 −250 −230 −210 −190 −170 −150 −130 −110 −90 −70 −50 −30 −10 10 30 50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490

Thales of of Chios Archimedes of Syracuse of Nicaea Heron of Alexandria Diophantus of Alexandria Hypatia of Alexandria

Pythagoras of Eudoxus of Cnidus Claudius Ptolemy

Hippasus of Plato Euclid of Alexandria

Zeno of Elea Aristotle Eratosthenes of Cyrene

Hippias of Elis

−630 −610 −590 −570 −550 −530 −510 −490 −470 −450 −430 −410 −390 −370 −350 −330 −310 −290 −270 −250 −230 −210 −190 −170 −150 −130 −110 −90 −70 −50 −30 −10 10 30 50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490

Thales of Miletus Dinostratus Archimedes of Syracuse Hipparchus of Nicaea Heron of Alexandria Menelaus of Alexandria Diophantus of Alexandria Pappus of Alexandria Hypatia of Alexandria

Pythagoras of Samos Socrates Eudoxus of Cnidus Apollonius of Perga Claudius Ptolemy Theon of Alexandria

Hippasus of Metapontum Plato Euclid of Alexandria

−630 −610 −590 −570 −550 −530 −510 −490 −470 −450 −430 −410 −390 −370 −350 −330 −310 −290 −270 −250 −230 −210 −190 −170 −Zeno150 − of130 Elea−110 −90 −70 −50Aristotle−30 −10 10 30 50 Eratosthenes70 90 110 of Cyrene130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490 of Elis Menaechmus Hippocrates of Chios Dinostratus Archimedes of Syracuse Hipparchus of Nicaea Heron of Alexandria Menelaus of Alexandria Diophantus of Alexandria Pappus of Alexandria Hypatia of Alexandria

Pythagoras of Samos Socrates Eudoxus of Cnidus Apollonius of Perga Claudius Ptolemy Theon of Alexandria

Hippasus of Metapontum Plato Euclid of Alexandria

Zeno of Elea Aristotle Eratosthenes of Cyrene History of Math R. L. Herman Fall 2021 20/20

Hippias of Elis Menaechmus