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Why does the Sun rise?
Why diferent stellarmotions are observed atdiferent places?
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4
( )
XX ( )
How to describe the tracconditions?
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Archimedes ( ; 287 B.C. - 212 B.C.)
(1654 - 1722)
Gauss ( ;1777-1855), German
mathematician
Linc!n ( ;18"#1865)
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UGFN1 !n "ialo#ue with Nature
6
$e%t 11a& William "unham'
The MathematicalUniverse
$e%t 11b& (uclid' Elements
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)ore *uestion&
7
What ma+es the modernscience so uni,ue?
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Heritage and Betrayal
Why did modernscience emer#e inWestern civili-ation?
Chen $n% Chin% (1#-)
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1. )hen ar#ued that ancient )hinesescience was not as advanced asNeedham believed.
/. $wo o0 the 0actors that led to theemer#ence o0 modern science inWestern civili-ation were
the close relationship betweenscience and reli#ion.
the two revolutionsthat led toancient Gree+ scienceand Newtonian
physics. 9
Heritage and Betrayal
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$his lecture
Principiaand Elements
Nature o0 western science
1. 2athematical
/. 3%iomatic
$e%t 11a
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'ememer
*e+tns Principia, and
ne the trans!atrs Chen,(The Birth of a New Physics)
Newton& Principia 415678
/. Bernard Chen (1#14-2""&)
0et &a
0et &
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"e9nitions
!. *uantity o0 mass
!!.*uantity o0 motion
!!!. !nertia
!:. !mpressed 0orce
:. )entripetal 0orce
:!.
Principia 4$e%t ;b8
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3%ioms4 8
1.
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)orollary4 8 1&
3 body acted on by =two> 0orces actin# ointly
describes the dia#onal o0 a parallelo#ram in the
same time in which it would describe the sides
i0 the 0orces were actin# separately.
Principia 4$e%t ;b8
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It is given that
(Eq. 1)
whereAis a constant. By putting Eq. 1 into Newtons aws !!
!
!
!
"ence# Earths or$it is an e%%iptica% or$it.
Newton& a mathematical proo0
a%iom
)orollary@compare withobservations
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(uclid& Elements 4; A)8
(0et 11)
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Elements 4$e%t 11b8
0hese (9stu!ates) +ere %ien +ithut r r :ustiicatin;
the3 +ere sim!3 t e accet.
9ara. 12 (0et 11a)
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2athematical and 3%iomatic
"e9nitionsBostulates
3 system o0mathematics
"e9nitions
3%ioms
3 system o0 mechanics
Elements 4; A)8 Principia 415678
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19
&he 'choo% o thens (
)
-- 3 'aae!! an>en.+iiedia.r%>+ii>0he?ch!??Athens
crates 9!at and Aristt!e9ainter 'aae!! an
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1. 2athematical
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Elements
(uclid& $he Father o0 Geometry.
Aorn& un+nown. "ied& un+nown.
Elements& 1; boo+s' C5Dpropositions.
E; A)' the oldestmathematics boo+.
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0et 11a
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Bytha#oras 4ED6 D A)8
not sure whether he really e%istedin history. !n 0act' 3ristotle tal+edabout Bytha#orians but notBytha#oras.
$he entire universe is #overned byinte#ers and their ratios.
3ll thin#s are built up 0romatoms'which are countable and indivisible.
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Bytha#oras diculty
Bytha#oras wanted to deny the e%istence o0irrational numbers4those cannot bee%pressed as ratios o0 inte#ers8. $he
concept o0 atom was undeveloped 0or
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Blatonic solids
3ccordin# to #eometry' there are only
9ve re#ular polyhedrons 4 8.
$hey perhaps corresponded to
diferent elements.
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hannes e!er ( 1571-16&"), German mathematician,
astrnmer and astr!%er
Mystery of the Cosmos
!"#$%& (1*96)
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/7
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/. 3%iomatic
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What is a proo0?
3%ioms
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3n e%ample o0 proo0
+roposition,
ny 4-igit integer is ivisi$%e $y 9 i the
su/ o the 4 igits is ivisi$%e $y 9.
3re these numbers& 111'76' ;C17divisible by ?
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3n e%ample o0 proo0
+roo,
ny 4-igit integerXcan $e written as
0ewriteXin ter/s o its igit va%ues# i.e.#
d
divisible by !0 the sum o0 alldi#its is divisibleby ' then Xis
divisible by
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3nother e%ample
"iferent 0ormulae 0or the areao0 diferent poly#ons.
2emori-e all?
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Aasic& $rian#le
3ll poly#ons are made up o0
trian#les.Jnow how to calculate the area
o0 a trian#le can prove other0ormulae.
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$e%t 11a 4"unham8
(,uivalently' two instruments are #iven4Bara. 1;1D8&
3 ruler 4with no mar+ed scale8
3 compass 4butcollapsible8
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9rsitin 1 (0et 11) n a %ien inite strai%ht
!ine t cnstruct an eDui!atera! trian%!e.
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Second e%ample& $wo identical circles
Given a len#th as the radius' can !draw two identical circles?
(asyK
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Elements 4$e%t 11b8Lemem
ber
No postulate allows one to trans0er a len#th toanother place.
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$rans0er a len#th toanother place.
"raw a circle4Bostulate ;8
Aut how to trans0er
a len#th? Lead $e%t11a' Bara. 1/6
"raw acircle4Bostulate ;8
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(asier or Unnecessary
Further assumption& thecompass stays open a0terdrawin# a circle.
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It is given that
(Eq. 1)
whereAis a constant. By putting Eq. 1 into Newtons aws !
!
!
!
!
"ence# Earths or$it is an e%%iptica% or$it.
Newton& a mathematical proo0
a%iom
)orollary@compare with
observations
Similar ,uestion& Why dont we simplyassume the (arths orbit is elliptical?
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0et 11a
Why?
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3ccept as 0acts&- (arth is
spherical
- (arth is rotatin#(%plain why the Sunrises(%plain diferent
stellar motions areobserved at diferentplaces.
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4
( )
XX ( )
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$hin+ about it
"o you thin+ (uclids wor+ is
important? Why or why not?
What ma+es modern science so
uni,ue?
!s mathematics is the realthin# or ust
ima#ination by humans?
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(noy readin#
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