Mina%20Nouri

1
Mina Nouri Mehdi BAbAee Mohsen Rahmati
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Archimedes
Guess who ???

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Eureka ! Eureka !
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Mechanic SCIENCE
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"Give me whereon to stand and I will move the
earth."
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Enemy
!!!!!!
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(No Transcript)
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Finally
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• ??? !
• ?????????? ?????
• 2000..??????????????

• ???? ??? ?? ? ????? ??? ?????

• ?????? ? ?????????
• ????? ? ????

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• The Works of Archimedes

• "?? ??? ??? ? ???????
• "?????? ???? ?????"
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• "??????? ??? ????"
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• "???? ?? ??"
• "???"

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• ??? ?? ????? ?????? ?? ??? ??? ?? ????? ??? ????
  ?? ???? ??? ?? ??? ???!!!!

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??? ????? (Methodologies)
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?????

• ??????? ????? ??????

????? ?? ? ???? ??

• ??????

?? ?? ???
y²

• ?????????

????? ??
y2 px
????

?????? ? ????

• ???????

????

14
??? Exhaustion ? ????? ?? ?????

• ????? ??? ??? 12 ???? "?????" ??????

• ???????

"?? ??? ??? ? ???????" ?"?????? ???? ?????" ?
"????? ????"? "?? ??? ??? ????? ?? ? ??? ??? ?? "
? "?????? ??"
15
??? ???
?????? ?? ?? ???? ?????? ?? ?????? ????? ?? ???
???.
????
????? ????? "The Method"
????? ??? ????? ??? ???? ????
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First Book

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• "On the sphere and cylinder"

V
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• ?? ????? ? ??? ?????
• ????? ????
• ??? ????? ???????
• " ???? ?? ?? ???? ?? ?? ??? a ?b ???a lt b ????
  ????? n ???? ???? ?? na gt b"

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• "?? ??? ??? ???????? "
• "On the sphere and cylinder"

44 Statements
3 MAIN Results
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Result ONE
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• Statement 33

4pr²
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• Statement 34

• Statement 33

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• Statement 34

4 x
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(No Transcript)
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H
(The Method,proposition2)
E
F
A
B
D
K
C
E
F
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K ??????? ???????
HA/AK (???????) / (???????? AEF) .
HA2AK ? ??????? 2(??? ????? AEF)
??????? 3(????? AEF)
??????
????? AEF 2(???).
????? AEF 8(????? ABD)
? ??? 4(????? ABD) .
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Result twothree
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• Statement 4243

Curved surface area (shaded) Area of circle of
radius r
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• Result of statement34

Cylinder 3/2 Sphere
2pr³
h
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Proof ( The Method , proposition2.
part2)
VY ??????? 2(VD ???????
)6 ?????ABD 3/2(???) (?? ???? ?? ?????
34).
)?????? ??????? VD 3????? ABD
A
V
D
B
Y
C
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(No Transcript)
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Next Book
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?????? ???? ?????
Measurement of the circle
33

• ?????? ?? ???? ?????? ?? ???? ????? ? ????? ??
  ????? ???.
• ???????
• ?? ??? ?????? ???? ?????

34
Area of circles

Low boundary Of circle
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½ (2pr) r
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Archimedes crocodile
Area of circles
by Burkard Polster and Marty Ross
The Age, 5 September 2011
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???? ???? ?? ??? ?????(Pi)

• ?????? ? Elements ?????? ?? ????? ??? ?????

??????? ???? ???? ? ??? ??????????? ????? ? ???
????

• ??????? ????? ????? ????

• ????? 3

???? ???? ?? ??? ?? ????? ?? ??? 3 10/71 ? 3
1/7 ???.
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Next Book
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????? ???????? ????? ?? ? ??? ??? ??
Quadrature of parabola , conics spheroids
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• 24 statements
• The most impressive

• Statement 24

triangle
"parabolic segment
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(The Method,proposition1 .) Proof
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??? ????? ?? ? ??? ??? ??
conics spheroids
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• Statement 2122

3/2 CONE
"parabolic segment V
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ii) ????? 24 ???? ?? ???? ?? ?? ????? ?? ???
????? ??? ?? ?? ???? ???????? ?? ?????????? ??
????(????? ???? ??????) ????? ??? ?? ???? ???
??????? ????.
APp / APp AN² / AN²
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Next Book
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????? ?????? ?? On Spirals
47

• ??????? ????? ????? ????? ???? ?????

• Alexanderian

• ????? ???

48

• On spirals
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• -???? ????? ???? ??????? ?? ???? ?? ?? ????
  ??????? ?? ??? ???? ??? ? ???? ??????? ?????
  (????) ???? ??? ? ??? ?? ?? ???? ???????? ??????
  ?? ????.-

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?????? ???????
aT ?
50

• ????!!!

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1629,

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28 ?????

• 9 ??? ??? ???? ??? ??? ????? ?? ? ???? ??????

• 9 ?? 12 ?????? ? ????? ?????

• 13 ?? 20 ???? ?? ?? ?????? ??

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????? 20

• ??? ???? ????

53

• ????? 21 ?? ???

• ?????

• "Exhaustion"

• ????? ????? ??? ???? ???? ??? ?????? ? ???

1/3 p (2pa)²

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Next Book
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Notations for Numbers
56

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  ???? ?? ?? ?? ?? ??? ???? ???.

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• ????? ???? ??? ?? ?? ????

57
???????? ???? M (?? ????)
"?M"
???? ????
???????? ???
?? ???? ? ?? ????
108
1 ?? 108 ??? ???
108 ?? 10¹6
??? ???
.
.
.
??? (108)-??

??

• ???? ???

58

P

• ???? ???

??? ???
P 1 ?? P 108
...
? ??? 108-??
P
??
P
????????
???? 108-??

1
??

108
??? ???
...
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??? 108-??


??

• ????? ???? ??? ??? ?? ????

105¹
????? ???? ??

60
Next Book
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Approximation of a Square Root
62
?? ??????? ??? ? ????

• ????? ?????? 3v

• ?????? ?????? ???? ??? ????? ???? ????? ????

????? ?????? 3v

• ?????? ???? ?????

265/153 lt v3 lt 1351/780
????????? ????? ?? ?? 3v
63
50v,Heron
????? ???? a
?? ?????
64
7/4 ?? ??? 5/3
???????? ??????
?? ?????? ???? ?? 3v ????? ????
65
?????? ?????? ???? ????? ????
"?????? ???? ?????"
???? ??????
(a b)2 a2 2ab
???? ??? 60
66
Next Book
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??????? ????? Qubic Equations
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Yax³
?????????? ????? ??????
a²/x x/b
??????? ????? ??? 1 ? 5 ?"?? ??? ??? ? ??????? "
Al-Khazin (965) ? (965-1040) Ibn Al-Haytham ?
Abul Jud (1000)
??? ?? ??? ?????? ???? ??? ????? ??????? ?????
???? (1131-1048)
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???? ?????????? ?????
Other mathematical achievements
?????????? ??? ??? ???? ?????? ???
70
????? ???? ????? ?? ????
????? Heron
??? ???? ???? ?????Semi-Regular polyhedron) )
13 ??? ????? ??????? ???? ?????
71
(No Transcript)
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(No Transcript)
73
????? ?? ??? ???? ?????
????? ?? ???????? ???
????? ????? (??? ?????)
Hieron
W ? w ?????? ????? ?????? ?? ???? ??????
?????. X ? x ?????? ????? ?????? ?? ????
?????? ?????. Y ?y ?????? ????? ?????? ?? ???
?????? ?????. Z ? z ?????? ????? ?????? ?? ???
???? ?????? ?????.
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(No Transcript)
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Hieron ???? ????? ???? ???? ???? ???? ???.
??????? ??? ????? 7 ??? ???!
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W X ???? ???? Y Z ??? ?????
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Hieron ?? ?? ??? ????? ?????? ??? .
1965
???? ????? 206500 ??? ???? !!!
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???? ?? ?? (Book Of Lemmas)

• ???? ????

• 15 ????? ?? ???? ????? ??

• ??????? ???????

??? Theme
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????? 4 ????? ????? ??? ?? ??? ????? ????? ???
?? ?????? AN ? BN ????? ????? ?? ?? ???? PN
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????? 9
(???? AD )(???? BC)(???? AC)(???? DB)
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V. References The mathematical achievements of
Archimedes 1 John Stillwell (1989),
Mathematics and Its History, Springer. 2 T. L.
Heath (1965), A History of Greek Mathematics,
Volume II, Oxford. 3 E.T. Bell (written in
1937),Jing Zhujun (translated in 1998).Da shu xue
jia (translated from Men of mathematics).
Publisher Chiu Chang Publishing Company. 4 T.
L. Heath (written in 1912), Zhu Enkuan, Li
Wenming (translated in 1998). Ajimide quan ji
(translated from The Works of Archimedes).
Publisher Shanxi ke xue ji shu chu ban she
(Shanxi). 5 Morris Kline (written in 1908),
Lin Yanquan, Hong Wangsheng, Yang Kang Jingsong
(translated in 1983). Shu xue shi shu xue si
xiang de fa zhan (translated from Mathematical
thought from ancient to modern times). Publisher
Chiu Chang Publishing Company.
82
6 Xie Enze, Xu Benshun (1994). Shi jie shu xue
jia si xiang fang fa, p.61-99.PublisherShandong
jiao yu chu ban she (Jinan). 7 Cai Zhiqiang,
Sun Wenxian(199?). Shu xue li ti mo xing zhi zuo
fu duo mian ti yan jiu. Publisher Chiu Chang
Publishing Company. 8 Hong Kong University of
Science and Technology. http//www.edp.ust.hk/math
/history/5/5_5/5_5_34.htm 9Archimedes.Heiberg(d
iscovered in 1906).T.L.Heath , K.C.D , Sc.D ,
F.R.S.(Edited in 1912). The Method Of Archimedes
, Publisher Clay,Jhon,M.A. (Cambridge at the
university press). 10Howard W.Eves, An
Introduction to the mathematical history.Fourth
Edition,Holt,Rinehart and Winston,1976, Printed
in U.S.A.