Download Archimedes' Law of the Lever, and his Mysterious Mechanical PDF

TitleArchimedes' Law of the Lever, and his Mysterious Mechanical
LanguageEnglish
File Size2.2 MB
Total Pages55
Table of Contents
                            Sphere
Conclusion
Action
Area of Disk & Sphere
Comments
                        
Document Text Contents
Page 1

TITLE&INTRO LEVER SPHERE CONCLUSION CONTACT TORQUE ACTION AREA OF DISK & SPHERE COMMENTS

Archimedes’ Law of the Lever,
and his Mysterious Mechanical Method

for Finding the Volume of a Sphere

Mike Raugh
Ph.D. in Math from Stanford

www.mikeraugh.org

A Presentation for
PNM-UNM New Mexico Math Contest

February 4, 2012

c
2012 by Mike Raugh, Interconnect Technologies, LLC

Page 2

TITLE&INTRO LEVER SPHERE CONCLUSION CONTACT TORQUE ACTION AREA OF DISK & SPHERE COMMENTS

Attention iPad Users!

The iPad renders three slides incorrectly —

The ones with captions beginning:

“Cylinder”, “Cone”, and “Sphere”.

(Verified Mar 24, 2012)

Page 27

TITLE&INTRO LEVER SPHERE CONCLUSION CONTACT TORQUE ACTION AREA OF DISK & SPHERE COMMENTS

Method (Step 3): The Derivation

x

x

x

x

2r

By previous slide: Each x-section of cylinder balances the
combined x-sections of the hanging cone and sphere.
Therefore, the volumes balance: 2r · (Co + S) = r · Cy , so

Co =
8
3
πr3, Cy = 8πr

3 =⇒ S =
4
3
πr3

Page 28

TITLE&INTRO LEVER SPHERE CONCLUSION CONTACT TORQUE ACTION AREA OF DISK & SPHERE COMMENTS

Using modern terms, we would express this as:

2r(Co + S) =

∫ 2r
0

(2r)(2� rx)dx = 8� r 4 =
∫ 2r

0
(x)(4� r 2)dx

= rCy

Archimedes did this nearly 2 millenia before Newton and
Leibniz.

Page 54

TITLE&INTRO LEVER SPHERE CONCLUSION CONTACT TORQUE ACTION AREA OF DISK & SPHERE COMMENTS

Archimedes’ “Method of Mechanical Theorems.”

Archimedes’ geometric proof for the volume of a sphere was
well known, but the method by which he discovered the result
remained a mystery until 1906.

The Archimedes Palimpsest, in which Archimedes described
his method, was found in Istanbul in a Byzantine crypt, then lost
and recovered again in 1998.

The Palimpsest contained a tenth-century copy of a Greek MS
that was scraped, washed and overlaid with Christian liturgy
and other writings.

(See Recommended reading).

Page 55

TITLE&INTRO LEVER SPHERE CONCLUSION CONTACT TORQUE ACTION AREA OF DISK & SPHERE COMMENTS

The Roman orator Cicero found the Archimedes’ tomb.
Archimedes was killed in 212 BC.

Archimedes had asked that a cylinder and sphere be mounted
on his tomb, displaying their proportional volumes: 2/3.

The tomb was built and lost until the figures of the cylinder and
the sphere enabled Cicero to find it 137 years later.

Cicero found the tomb in 75 BC. He wrote: “So one of the most

famous cities in the Greek world would have remained in total
ignorance of the tomb of the most brilliant citizen it had ever
produced, had a man from Arpinum not come and pointed it
out!”

(Aaboe in Recommended reading)

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