11Lecture Dunham-Euclid 20131116a

7/24/2019 11Lecture Dunham-Euclid 20131116a

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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

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$e%t 11a& William "unham'

The MathematicalUniverse

$e%t 11b& (uclid' Elements


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)ore *uestion&

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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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/. 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#

11Lecture Dunham-Euclid 20131116a

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