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6 – Greek Math After Euclid

The student will learn about

Greek mathematics after the time of Euclid.

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§6-1 Historical Setting

Student Discussion.

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§6-2 Archimedes

Student Discussion.

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§6-2 Archimedes 1

1. Classical method of determining .2. Quadrature of a parabola – more follows.3. Spiral of Archimedes, r = k .

4. Spheres and cylinders – more follows.

5. Conchoids and Spheroids6. Sand reckoning.7. Plane Equilibrium – centroids.

8. Floating bodies – hydrostatics.

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§6-2 Archimedes 2Quadrature of a parabola – area of a parabolic segment is four-thirds that of the inscribed triangle having the same base.

1. Use calculus to calculate the area bounded by the parabola y = x2 and y = 4 for –2 x 2.

2. Use Archimedes’ method to calculate the area bounded by the parabola y = x2 and y = 4 for –2 x 2.

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§6-2 Archimedes 3

Spheres and cylinders

1. Confirm that the surface area of a sphere is equal to 2/3 the surface area of a circumscribed cylinder.

2. Confirm that the volume of a sphere is equal to 2/3 the volume of a circumscribed cylinder.

4 r 2 = 2/3 (6 r 2)

4/3 r 3 = 2/3 (2 r 2)

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§6-3 Eratosthenes

Student Discussion.

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§6-4 Apollonius

Student Discussion.

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§6-4 Apollonius 1

Problem of Apollonius – Given three circles (degenerate cases permitted) construct a circle tangent to the given circles.

Given three points - easy

Other cases vary in difficulty.Given three lines - easy

If time do two points and a line.

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§6-4 Apollonius 2

From “Plane Loci”.

If A and B are fixed points and k is a given constant, then the locus of a point P, such that AP/BP = k is either a circle (if k 1) or a straight line (if k = 1.).

1. Case where k = 1.

2. Case where k = 2.

The locus is the perpendicular bisector of AB.

Homework: Describe the locus circle completely.

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§6-5 Hipparchus, Menelaus, Ptolemy, and Trigonometry.

Student Discussion.

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§6-5 Hipparchus

The chord of 36 = 37; 04, 55

Chord 36 = 2 · 60 · sin 18

sin 18 = chord 36 / 120

Too small by 0.0000008 or the thickness of a human hair over the length of a soccer field.

sin 18 = 37; 04, 55 / 120

sin 18 = 18; 32, 27, 30 / 60sin 18 = 00 ; 18, 32, 27, 30

sin 18 = 0.309016204ten

18 18

x60

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§6-5 Menelaus.

Menelaus’ Theorem.If transversal LMN intersects the three sides of a triangle then:

1NA

CN

MC

BM

LB

AL

A B

C

L

M

N

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§6-5 Ptolemy.

Three Point Problem.

Given points A, B, and C, and angles AVB, AVC, and BVC, find point V.

. A

. B

. C

. V ?

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§6-6 Heron

Student Discussion.

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§6-6 Heron

1. Area of a triangle of sides a, b, and c, is:

)cs)(bs)(as(s

2. Square root approximation –

If a1 is an approximation of the square root of n then

2

an

a

a 11

2

is a better approximation.

Try 45.

Note: the Babylonians used this for 2.

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§6-7 Ancient Greek Algebra

Student Discussion.

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§6–8 Diophantus

Student Discussion.

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§6–9 Pappus

Student Discussion.

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§6–10 The Commentators

Student Discussion.

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Time Line

2400-1600-525 B.C. Babylonians1900-1000-000 B.C. Egyptions600 B.C. Thales

540 B.C. Pythagoras

450 B.C. Zeno440 B.C. 2 Irrational390 B.C. Socrates / Plato

336-323 B.C. Alexander the Great’s Reign

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Time Line

300 B.C. Euclid287-212 B.C. Archimedes230 B.C. Eratosthenes

225 B.C. Apollonius

44 B.C. Death of Julius Caesar

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Time Line

150 Ptolemy250 Diophantus300 Pappus

390 Theon of Alexandria

410 Hypatia529 School of Athens closed

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Assignment

Read Chapter 7.