Thinker & Thug
Garner
The Life of Archimedes
The Works of The Thinker and the Thug Archimedes On the Sphere and Cylinder On Conoids and Spheroids The History of Mathematics, Part 9 On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works Chuck Garner, Ph.D. The Archimedes Palimpsest
The Legacy of Department of Mathematics Archimedes Rockdale Magnet School for Science and Technology Homework February 22, 2021
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Outline Garner
The Life of The Life of Archimedes Archimedes
The Works of The Works of Archimedes Archimedes On the Sphere and Cylinder On the Sphere and Cylinder On Conoids and Spheroids On Spirals On Conoids and Spheroids On the Equilibrium of Planes The Sand-Reckoner On Spirals Quadrature of the Parabola Other Works
On the Equilibrium of Planes The Archimedes The Sand-Reckoner Palimpsest The Legacy of Quadrature of the Parabola Archimedes Other Works Homework The Archimedes Palimpsest The Legacy of Archimedes Homework
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Outline Garner
The Life of The Life of Archimedes Archimedes
The Works of The Works of Archimedes Archimedes On the Sphere and Cylinder On the Sphere and Cylinder On Conoids and Spheroids On Spirals On Conoids and Spheroids On the Equilibrium of Planes The Sand-Reckoner On Spirals Quadrature of the Parabola Other Works
On the Equilibrium of Planes The Archimedes The Sand-Reckoner Palimpsest The Legacy of Quadrature of the Parabola Archimedes Other Works Homework The Archimedes Palimpsest The Legacy of Archimedes Homework
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Archimedes Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
Archimedes 287 BC-212 BC “Give me a place to stand and I will move the Earth.”
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Archimedes of Syracuse Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
5 / 35 Thinker & Thug
Archimedes Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids I Spent youth in Alexandria On Spirals On the Equilibrium of Planes The Sand-Reckoner I Related to King Hieron; advisor to the King Quadrature of the Parabola Other Works
I Inventor and mathematician The Archimedes Palimpsest I Unique because he combined pure geometrical The Legacy of analysis and the mechanical/practical (Plato urged Archimedes separation of the two) Homework
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Archimedes’ Inventions Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
The Archimedean Screw
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Archimedes’ Inventions Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
The Archimedean Screw in use at a water treatment plant in Tennessee
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Archimedes’ Inventions Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
The Archimedean Claw
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Archimedes Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids I Discovered the first law of hydrostatics: On Spirals On the Equilibrium of Planes “A body immersed in fluid is buoyed up by a force The Sand-Reckoner Quadrature of the Parabola equal to the weight of the displaced fluid.” Other Works The Archimedes I Invented the pulley Palimpsest The Legacy of I Analyzed the lever Archimedes Homework I Died during the Roman siege of Syracuse
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Archimedes Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
“Do not disturb my circles!”
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Outline Garner
The Life of The Life of Archimedes Archimedes
The Works of The Works of Archimedes Archimedes On the Sphere and Cylinder On the Sphere and Cylinder On Conoids and Spheroids On Spirals On Conoids and Spheroids On the Equilibrium of Planes The Sand-Reckoner On Spirals Quadrature of the Parabola Other Works
On the Equilibrium of Planes The Archimedes The Sand-Reckoner Palimpsest The Legacy of Quadrature of the Parabola Archimedes Other Works Homework The Archimedes Palimpsest The Legacy of Archimedes Homework
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The Surviving Works of Archimedes Garner
The Life of Archimedes
I On the Sphere and Cylinder (2 books) The Works of Archimedes On the Sphere and Cylinder I Measurement of a Circle On Conoids and Spheroids On Spirals I On Conoids and Spheroids On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola I On Spirals Other Works On the Equilibrium of Planes (2 books) The Archimedes I Palimpsest I The Sand-Reckoner The Legacy of Archimedes
I Quadrature of the Parabola Homework
I On Floating Bodies (2 books)
I The Cattle-Problem
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The Surviving Works of Archimedes Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids All works come to us with cover letters: On Spirals I On the Equilibrium of Planes The Sand-Reckoner I Many to Dositheus Quadrature of the Parabola Other Works I One to Eratosthenes, librarian at Alexandria The Archimedes I We get insight into mathematical atmosphere of the Palimpsest The Legacy of time Archimedes
I We get insight into Archimedes himself Homework
14 / 35 I Propositions 33 and 34 of Book I: The cylinder whose base is equal to a great circle of the sphere and whose altitude is equal to a diameter of the sphere has a total surface exactly 3 equal to 2 the surface of the sphere and a volume 3 exactly equal to 2 the volume of the sphere.
Thinker & Thug
On the Sphere and Cylinder Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder I Archimedes own favorite work On Conoids and Spheroids On Spirals I 2 books, 60 propositions On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
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On the Sphere and Cylinder Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder I Archimedes own favorite work On Conoids and Spheroids On Spirals I 2 books, 60 propositions On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola I Propositions 33 and 34 of Book I: Other Works The cylinder whose base is equal to a great circle of The Archimedes Palimpsest
the sphere and whose altitude is equal to a The Legacy of diameter of the sphere has a total surface exactly Archimedes 3 Homework equal to 2 the surface of the sphere and a volume 3 exactly equal to 2 the volume of the sphere.
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On Conoids and Spheroids Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes Calculated volumes and areas of paraboloids, cones, The Sand-Reckoner I Quadrature of the Parabola hyperboloids, spheroids Other Works The Archimedes 3 Palimpsest I Proved that the volume of a paraboloid is 2 the The Legacy of volume of a cone with the same base and axis Archimedes
Homework
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On Spirals Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
I Area of one circuit is one-third the area of the circle with radius equal to the distance of the endpoints
17 / 35 Post. 2, 3 When weights at certain distances are in equilibrium, and something is (added to/subtracted from) one of the weights, they will incline toward the weight (to which the addition was made/from which nothing was taken).
Thinker & Thug
On the Equilibrium of Planes Garner
The Life of Archimedes Represents the first instance of mathematical modeling. The Works of Archimedes Post. 1 Equal weights at equal distances are in On the Sphere and Cylinder On Conoids and Spheroids equilibrium; equal weights at unequal On Spirals On the Equilibrium of Planes distances incline toward the weight at the The Sand-Reckoner Quadrature of the Parabola greater distance. (Principle of Insufficient Other Works Reason) The Archimedes Palimpsest
The Legacy of Archimedes
Homework
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On the Equilibrium of Planes Garner
The Life of Archimedes Represents the first instance of mathematical modeling. The Works of Archimedes Post. 1 Equal weights at equal distances are in On the Sphere and Cylinder On Conoids and Spheroids equilibrium; equal weights at unequal On Spirals On the Equilibrium of Planes distances incline toward the weight at the The Sand-Reckoner Quadrature of the Parabola greater distance. (Principle of Insufficient Other Works Reason) The Archimedes Palimpsest Post. 2, 3 When weights at certain distances are in The Legacy of equilibrium, and something is (added Archimedes Homework to/subtracted from) one of the weights, they will incline toward the weight (to which the addition was made/from which nothing was taken).
18 / 35 Prop. 3 Suppose A and B are unequal weights with A > B which balance at C. Then AC < BC. Conversely, if the weights balance and AC < BC, then A > B. Note The lever is not mentioned, it is just “there.”
Thinker & Thug
On the Equilibrium of Planes Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids Prop. 1 Weights which balance at equal distances On Spirals On the Equilibrium of Planes are equal. The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
19 / 35 Note The lever is not mentioned, it is just “there.”
Thinker & Thug
On the Equilibrium of Planes Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids Prop. 1 Weights which balance at equal distances On Spirals On the Equilibrium of Planes are equal. The Sand-Reckoner Quadrature of the Parabola Prop. 3 Suppose A and B are unequal weights with Other Works The Archimedes A > B which balance at C. Then AC < BC. Palimpsest
Conversely, if the weights balance and The Legacy of AC < BC, then A > B. Archimedes Homework
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On the Equilibrium of Planes Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids Prop. 1 Weights which balance at equal distances On Spirals On the Equilibrium of Planes are equal. The Sand-Reckoner Quadrature of the Parabola Prop. 3 Suppose A and B are unequal weights with Other Works The Archimedes A > B which balance at C. Then AC < BC. Palimpsest
Conversely, if the weights balance and The Legacy of AC < BC, then A > B. Archimedes Homework Note The lever is not mentioned, it is just “there.”
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The Sand-Reckoner Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals Recall: myriad represents 10,000 On the Equilibrium of Planes I The Sand-Reckoner Quadrature of the Parabola I Proposed a number system using powers of myriad Other Works myriads (base 100,000,000) The Archimedes Palimpsest 63 I Calculates 8 × 10 as the number of grains of sand The Legacy of to fill the universe Archimedes Homework
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Quadrature of the Parabola Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
4 The area of a parabolic segment (ABCDE ) is 3 the area of the inscribed triangle (4ACE ).
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Other Works Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Floating Bodies Hydrostatics and buoyancy On Conoids and Spheroids On Spirals Measurement of a Circle Used circumscribed and On the Equilibrium of Planes The Sand-Reckoner inscribed regular polygons; with 96-gons, Quadrature of the Parabola obtained 3 10 3 1 Other Works 71 < π < 7 The Archimedes Book of Lemmas Geometry; 15 Propositions (link) Palimpsest The Legacy of On Sphere-Making now lost, on constructing a Archimedes planetarium Homework The Cattle-Problem Number theory
22 / 35 subject to
5 9 W = 6 X + YX = 20 Z + Y 13 7 Z = 42 W + Y w = 12 (X + x) 9 11 x = 20 (Z + z) z = 30 (Y + y) 13 y = 42 (W + w) W + X is a square and Y + Z is a triangle
Thinker & Thug
The Cattle-Problem Garner
The Life of Archimedes
The Works of W, w = white bulls, cows Archimedes On the Sphere and Cylinder X, x = black On Conoids and Spheroids On Spirals Y, y = yellow On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Z, z = spotted Other Works The Archimedes Palimpsest
The Legacy of Archimedes
Homework
23 / 35 W + X is a square and Y + Z is a triangle
Thinker & Thug
The Cattle-Problem Garner
The Life of Archimedes
The Works of W, w = white bulls, cows Archimedes On the Sphere and Cylinder X, x = black On Conoids and Spheroids On Spirals Y, y = yellow On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Z, z = spotted Other Works The Archimedes subject to Palimpsest The Legacy of Archimedes 5 9 Homework W = 6 X + YX = 20 Z + Y 13 7 Z = 42 W + Y w = 12 (X + x) 9 11 x = 20 (Z + z) z = 30 (Y + y) 13 y = 42 (W + w)
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The Cattle-Problem Garner
The Life of Archimedes
The Works of W, w = white bulls, cows Archimedes On the Sphere and Cylinder X, x = black On Conoids and Spheroids On Spirals Y, y = yellow On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Z, z = spotted Other Works The Archimedes subject to Palimpsest The Legacy of Archimedes 5 9 Homework W = 6 X + YX = 20 Z + Y 13 7 Z = 42 W + Y w = 12 (X + x) 9 11 x = 20 (Z + z) z = 30 (Y + y) 13 y = 42 (W + w) W + X is a square and Y + Z is a triangle
23 / 35 To be square, n = 4456749k2. Finally, W alone has 206541 digits!
Thinker & Thug
The Cattle-Problem Garner
The Life of For the first seven conditions: Archimedes
The Works of Archimedes W = 10366482n On the Sphere and Cylinder On Conoids and Spheroids X = 7460514n On Spirals On the Equilibrium of Planes The Sand-Reckoner Y = 4149387n Quadrature of the Parabola Other Works
Z = 7358060n The Archimedes w = 7206360n Palimpsest The Legacy of x = 893246n Archimedes Homework y = 5439213n z = 3515820n
24 / 35 Finally, W alone has 206541 digits!
Thinker & Thug
The Cattle-Problem Garner
The Life of For the first seven conditions: Archimedes
The Works of Archimedes W = 10366482n On the Sphere and Cylinder On Conoids and Spheroids X = 7460514n On Spirals On the Equilibrium of Planes The Sand-Reckoner Y = 4149387n Quadrature of the Parabola Other Works
Z = 7358060n The Archimedes w = 7206360n Palimpsest The Legacy of x = 893246n Archimedes Homework y = 5439213n z = 3515820n
To be square, n = 4456749k2.
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The Cattle-Problem Garner
The Life of For the first seven conditions: Archimedes
The Works of Archimedes W = 10366482n On the Sphere and Cylinder On Conoids and Spheroids X = 7460514n On Spirals On the Equilibrium of Planes The Sand-Reckoner Y = 4149387n Quadrature of the Parabola Other Works
Z = 7358060n The Archimedes w = 7206360n Palimpsest The Legacy of x = 893246n Archimedes Homework y = 5439213n z = 3515820n
To be square, n = 4456749k2. Finally, W alone has 206541 digits!
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Outline Garner
The Life of The Life of Archimedes Archimedes
The Works of The Works of Archimedes Archimedes On the Sphere and Cylinder On the Sphere and Cylinder On Conoids and Spheroids On Spirals On Conoids and Spheroids On the Equilibrium of Planes The Sand-Reckoner On Spirals Quadrature of the Parabola Other Works
On the Equilibrium of Planes The Archimedes The Sand-Reckoner Palimpsest The Legacy of Quadrature of the Parabola Archimedes Other Works Homework The Archimedes Palimpsest The Legacy of Archimedes Homework
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The Archimedes Palimpsest Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
Discovered in 1906 in Constantinople
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The Archimedes Palimpsest Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
Prayer book assembled in 1229 from Greek copy of Archimedes’ works
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The Archimedes Palimpsest Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals This palimpsest contains many of Archimedes’ works On the Equilibrium of Planes The Sand-Reckoner already known, but also includes Quadrature of the Parabola Other Works I Nearly all of On Floating Bodies, most of which was The Archimedes lost Palimpsest The Legacy of 0 Archimedes I Most of a book called “ Eφoδoς”, meaning Method Homework
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The Method Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids I Archimedes tells how he came to discover his On Spirals On the Equilibrium of Planes theorems The Sand-Reckoner Quadrature of the Parabola I Careful to make the distinction between Other Works The Archimedes “investigation” and “proof” Palimpsest I Developed volume of sphere using a technique from The Legacy of Archimedes
Eudoxus: the method of exhaustion Homework I The Archimedes Palimpsest (link)
29 / 35 Greek tendency to think of aspects in pairs:
I Even/Odd
I Measure/Number
I Magnitude/Multitude
I Arithmetic/Logistic
I Commensurable/Incommensurable Archimedes brought these aspects together.
Thinker & Thug
Two Aspects of Numbers Garner
The Life of Archimedes
1. Relationship among numbers — “Arithmetic” The Works of Archimedes (Today we call this number theory) On the Sphere and Cylinder On Conoids and Spheroids 2. Computing with numbers — “Logistic” On Spirals On the Equilibrium of Planes (Today we call this arithmetic) The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
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Two Aspects of Numbers Garner
The Life of Archimedes
1. Relationship among numbers — “Arithmetic” The Works of Archimedes (Today we call this number theory) On the Sphere and Cylinder On Conoids and Spheroids 2. Computing with numbers — “Logistic” On Spirals On the Equilibrium of Planes (Today we call this arithmetic) The Sand-Reckoner Quadrature of the Parabola Greek tendency to think of aspects in pairs: Other Works The Archimedes Palimpsest I Even/Odd The Legacy of Archimedes I Measure/Number Homework I Magnitude/Multitude
I Arithmetic/Logistic
I Commensurable/Incommensurable Archimedes brought these aspects together.
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Outline Garner
The Life of The Life of Archimedes Archimedes
The Works of The Works of Archimedes Archimedes On the Sphere and Cylinder On the Sphere and Cylinder On Conoids and Spheroids On Spirals On Conoids and Spheroids On the Equilibrium of Planes The Sand-Reckoner On Spirals Quadrature of the Parabola Other Works
On the Equilibrium of Planes The Archimedes The Sand-Reckoner Palimpsest The Legacy of Quadrature of the Parabola Archimedes Other Works Homework The Archimedes Palimpsest The Legacy of Archimedes Homework
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Archimedes’ Legacy Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes I The greatest genius of antiquity The Sand-Reckoner Quadrature of the Parabola I One of the 3 greatest mathematicians ever Other Works The Archimedes I Laid foundation for integral calculus Palimpsest The Legacy of I Style influenced all natural philosophers for millenia Archimedes Homework
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Archimedes’ Legacy Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids On Spirals On the Equilibrium of Planes The Sand-Reckoner Quadrature of the Parabola Other Works
The Archimedes Palimpsest
The Legacy of Archimedes
Homework
Statue on eastern coast of Syracuse, Sicily, erected in 2016
33 / 35 Thinker & Thug
Outline Garner
The Life of The Life of Archimedes Archimedes
The Works of The Works of Archimedes Archimedes On the Sphere and Cylinder On the Sphere and Cylinder On Conoids and Spheroids On Spirals On Conoids and Spheroids On the Equilibrium of Planes The Sand-Reckoner On Spirals Quadrature of the Parabola Other Works
On the Equilibrium of Planes The Archimedes The Sand-Reckoner Palimpsest The Legacy of Quadrature of the Parabola Archimedes Other Works Homework The Archimedes Palimpsest The Legacy of Archimedes Homework
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Homework Garner
The Life of Archimedes
The Works of Archimedes On the Sphere and Cylinder On Conoids and Spheroids I Last-Minute Problems, #4 – due March 1 On Spirals On the Equilibrium of Planes The Sand-Reckoner I Chasing the value of π; Quadrature of the Parabola Math Through the Ages, Sketch 7 Other Works The Archimedes Palimpsest
The Legacy of Archimedes
Homework Next: A Look Toward the Heavens
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