π

•

•

•

•

1 C. J. Keyser

•

•

•

•

•

mathemata

,

2 mathematics 3 Alexander Pope

,

.

////

,

,

4. Ishango 5. Lake Edward

6. Herodotus 7. Ionians 8. quipus 9. Incas10 Cuzco

,

• • • •• • •• ••

• • • •• • •• ••

• • • •• • •• ••

• • • •• • •• ••

• •• •• • •

,

11. From Stories from Herodotus by B. Wilson and D. Miller. Reproduced by permission of Oxford University Press.

,, ,,

∩

Narmer King 13. Menes 14. Palette 15. Hierakonopolis 16. J. E. Quibell 17. Gize

, ,

,,, ,, ,

,,,,,,

|||

|||

18 Rhind Papyrus

.

ριασκβτλγυμδφνεχξ

ψοζωπη

θ

19 Ionia

ψπδ = + + =

βM

MMyriadM

,

,=Mδ

Mδ

, ,=Mρν

Mρν

/ , ,Mτμε βρμδ =

,

MM( ),

δσλδ

,απε,,απε

/

/

κ δν γ

αξσ ιβασ οβ

×

=

κνκγδνδγ

,,,

, ,

∩

,,

, ,

/ασλδ

βα/

/Mε

εφνε

/MM Mθ τ βχδ

νζφογσλβλωπα/

χμθγφιβ/σπεδ

Γpenta

Δdeka

Ηhekaton

×kiloΜmyriad

,,

20. Atic 21. Herodianic

,,

,,,

V L C DΙ Χ Μ

MDCCCXXVIII

CDXCV

IVVI

IVXXLCCDM

, ,VΧ =,VΧ =

,,

, ,

CXXIVMDLXI

MDCCXLVIIIDCCLXXXVII

ΧΙΧ

C VΧ ΧΧ

CM.XIXMMCLXI .MDCXX

XXIV .XLVI XXIII.XXX CLXI.CCLII

XXXIV.XVI

|

,

,

οoυδενο

. omicron

+ =

;,,,;,

,× + × + + =

× + + + =

23. Theon

= =

= =

= =

Megal Syntaxis Almagest

,,

,,

,,,,,,,,,,,,

•

•

•

•

•

•

•

•

26. The Rhind Papyrus

27 Proclus28 Golenischev 29. A. Henry Rhind

30. Luxor 31. Thebes 32. Ramesseum 33. Ahmes

, , , , , ,...

xx

+ +

+ +

nn

||

||O

O

= + + +

= + + + + +

+ + +

n/

n

n/

k k k= +

k =

= +

/( )k

n/

n =

= + +

= + +

= + = + +

= + = +

= + = + +

= + + = + + +

= + + = +

= + + = + +

= + = + +

= + = + + +

= + + + = +

= + + + = + + +

= + = + + +

= + + = +

= + + = + + +

= + + + = +

= + + = + +

= + = + +

= + = + + +

= + + +

n n n n n= + + +

n/

= + + +

= + + +

= + + +

= + + +

+

N/ ,..., / kn n

...k

N Nn n

+ + + + =

N

+ = + =

+ = + =

+ + + =

+ + + =

.x x− =

× − =

=

x = =

.

.

34 Liber Abaci

ax b+ =ggxffag b+ag b+

ag b f+ =

ag b f+ =

( )a g g f f− = −

gg

ag g bg f g+ =

ag g bg f g+ =

( )b g g f g f g− = −

−− =−

f g f gba f f

/x b a= −xf g f gx

f f−

=−

xax b+ =ax b+ =

xx + =

xx + − =

xg =g =

f+ − = − =

f+ − = − =

x( ) ( )

( ) ( )f g f gx

f f− − − −= = = = + +− − − −

35. Ahmes

n n n nn n n n n+ − + − + − + =

n =

,,

,,

36. Hekat

( ) ( )/= − −−× = = + + + +−

nSn

...n

nn

rS a ar ar ar ar

− −= + + + + =−

a r= =n =

)(

.

.

."

37 Anglo- Saxon 38 God Ra

÷÷÷÷÷

+ +

+ + + +

+ + +

+ +

++ +

+

+ = +

+

n n n= +

nn

.

.

.

39 geometry 40 Herodotus 41. Sesostris

( )( )A a c b d= + +

a

b

c

d

abcd

42. Horus 43. Edfu 44. khet 45. setat

= − = =d dA d d

d

/dπ

d dπ =

. ...π = =

dA =

A = − =

d =d

dA =

.

( )hV a ab b= + +

hab

,

Gizeh KhufuCheops

,,

,,

.

, ,

/

.

π

/ ...=π( . ) / .../

=

49. acre

/ ...

.

bah:

( . )h b a ab= =

abh

h b a+ =h a b= −h

a b ab− =

ab ba a

− =b ba a

+ =

x x+ =

( )x = −( ) / ...= − =ba

( / ) ( / ) /= + = + =a h b

/ ...=ba

.

50. John Taylor 51. The Great Pyramid, Why Was It Built and Who Built It?

( )ab×( )b

b abab ab b

=+

b aa a b

=+

a/ / .../

= =+a

a b

ab

baa+

.

π

π =

,;

π

π

a b a bV h + −= +

hab

52. cubit 53. Aryabhata

( )V h r R= +

hrRh =r =R =

( )V h R rR rπ= + +

( )V h r Rπ= +

hrRπ

h =r =R =

54. Heron 55. Akkadian 56. Otto Neugebaver

.

.

.

:;

xy:

x x+ =

:]x [; .

;;; ,;

; ,;;

;;.

; ;( ) ;

; , ; ;

; , ;; ; ;

x = + −

= + −

= −= − =

a ax b= + −

x ax b+ =

.

.

x y a+ = =

ax z= +ay z= −

z.

a b−xyb =ay z= −ax z= +

zzzzzzzzzz

==========

za b−z

a b z− =

z

az b= −

a ax b= + −

a ay b= − −

ayx =+xza

yx +xy.

ax z= +ay z= −

abxy =

a az z b+ − =

a z b− =

az b= −

az b= −

zxy

a ax b= + −

a ay b= − −

xy =x y+ =

x z= +y z= −

x y z z+ = + + − = =

xy =

z z+ − =

z− =

z = − =

57. Yale

z =

x = + =y = − =

yx −yx +

xy b=x y a− =

ax z= +ay z= −

a ax b= + +a ay b= + −

bxy =ayx =±

x y xy+ + =x y+ =

xy

( )xy x y xy x y= + + − − = − =

xy =x y+ =

x = + =y = − =

xx xx x+ + + =

x y=x y+ =

58. mina 59. sheqel 60. se

xy

xy b=x y a+ =xy =

x y+ =( ) ( )x y x y xy− = + −

yx −

xy =x y− =

xy =x y− =

xy =x y+ =

( )xy x y+ − =x y+ =

z y= +

xz =

x z+ =

61. Susa 62. Elam 63. William Kennett Loftus

I

.

.

64. Acropolis 65. King Hammurabi I

ADBAD =r AE=ED r= −

EDB

( )r r= + −

r =;r =

;;

π =

, =

Ay A x+ = =

( )x y+ =

, =yzx

( ) ,z x xy y z+ = + − =

xyzy x

y z=

+

b b− =

( )s s b b= = + +

b b s sA + +=

•

•

•

•

•

•

•

•

.

.

.

.

.

.

••

•••

.

:

.

hh′

s′:

ss

hh

′=

′

.

( )shhs

′ += = = =′

s

.

.

.

.

II

ba +ababc

ba +.

ab

c

b

b

b

a

a

a

a b

cb

b

b

a

aa

( )+ = + +a b a b ab

( ) aba b c+ = +

c a b= +cab

x y z+ =

),,( zyx

, ,x n y n n z n n= + = + = + +

n ≥

( ) ( )k k k− + − =

k −

k =k − =k m− =k

mk +=mk −− =

m mm − ++ =

, ,m mx m y z− += = =

m >m

m k= −m n= +n ≥

, ,x n y n n z n n= + = + = + +

zyxn

, ,x n y n z n= = − = +

zyxn

( ) ( )

[( ) ( )] ( )

k k k

k k k k k

+ = + +

= − + − + + = − +

nkk

( ) ( ) ( )n n n+ − = +

( , , )

X

, ,x mn y m n z m n= = − = +

mnnm >

a b a bab − ++ =

a n=b =

( )a n= +b =

n ≥nn

( , ( ), ( ))n n n− +

n( , ( / ) , ( / ) )n n n− +( , , )

( , , )x x x+ +

x =

),,( zyxxy( , , )x x z+

( , , )x z x z x z+ + + + + +

( , , )x x z+

ABCCCDCAB

A

c

D

C Bb

a

ACDCBDABCabc

a b c+ =

ABCCabc

BCDBAD

ab

acb

D

A

BC

ca

b

ABCDBAbacAD /=

/DB a b=

ABDABC

ACDa b c+ =

p qq ≠

abab

ba

ab

,,,,,,,,,,

,,,,

( )− −n n

( )− −n n−n

BESKn

( )−

ABCDAB/

CD/

.

,AB CD= =

,AB CD= =

n −

A B C D

ABCD

AB m=CD n=

AB mnCD

=

a

A B

CD a

a

BD a a a

BD a

= + =

=

BD aBC a

= =

pq

=pq

p q=p q=

ppp

p c=

c q=

q c=

qq

pq

−

Q N≅

pp

.

π

πππ

π

ABC.

= ABACAC

AB

ABCAB AC CB AC= + =ABAC

IIIIVIII

ABCABC

( . )AC BC AC=

ACIII

IABC

ACAC =I

.

.

aaxyaxya

a x yx y a

= =

x ay=

y ax=

x a y a x= =

x a=

xa

B

COB∠ =

.O

ABO

CPOQ

BO

( )COA∠ = − = =

O

POQ

OC

C

.

C

ABC

ABCD

DADAC

BCD

BPx =CPy =

ACPBDPC

ABCDE

ABCBCDACBD

aAP =DP a=aa

a x yx y a

= =

APB

aaAD a=E

ABADPQPCQ

PAQPBCCDQ

aAB =

a

a y a yx a a x

+= =+

E

D Q

B C

A

y

G

x

Fa

P

EFQEGP

( ) a aa x a y+ + = + +

( ) ( )a x x a y y+ = +

xDQ =yBP =aa

AOB∠BOAAOB∠COA

a OB=OaBCBD

PQaPQ =OPQ

M

Q

aP

DB

O A

/a

MPQBMOMOB ∠=∠

BMQBMP MBQ MQB∠ = ∠ + ∠

BQOAOQ ∠=∠AOB AOQ QOB BQO∠ = ∠ + ∠ = ∠

ABCDaAB =AD a=MADNABCM

BAGFFNaFB =BHGFFP

ABPaHP =

PCADQ

C

D Q

B C

A

y

G

x

F

P

a

aH

z

M

a

N

a

PAQPBCCDQa y a yx a a x

+= =+

PBHPGFa z ay y a

+=+

/ /a z y a=xz =FNBFNPFN

( )a aa x a y− = + − +

x y ay x a

+=+

DQ x=yBP =aaa x yx y a

= =

••••••••

.

.

.

109 Chaeronea 110 Hellenistic Age 111 Ptolemy

.

.

.

.

..

.

.

.

..

. .

.

.

.

. .

.

ll′ta

bll′t

l

l ′

t

a

b

.

AB

AB

AABB

BABA

CCACB

ABC

ABAC =ABBC =

ACBCAB ==ABC

A B

C

C

ABCDBC

CACEBEFEFBE =ECAE =EFBE =

FECAED ∠=∠AEDFEC

FCEBAE ∠=∠FCEDCA ∠>∠DCA∠

BAE∠

A

B c

E

D

F

ACG

ABCGCB ∠>∠GCB∠DCA∠

DCA∠ABC∠

.

ba ≠ba |

cacb =baba |/

ba |ab

ababbaba |a

bacacb =

.

a >|a| a

aa |a±a±aa

p >

p

abd

abad |bd |

ab

ab),gcd( ba

gcd( , ) =

.

a

bb

.abb >q

r

,a qb r r b= + ≤

.

.

n >

.

pp

( ... )N p= × × × × × × +

N >NqpN

q| ...q p× × × ×Nq |( )| ( ... )− × × × ×q N p

|qq >

qp

ABCCBACAB ∠=∠BCAC ≠BCAC >D

ACBCAD =

C

D

A B

P

A

B C

D

αβ

γ

ABCΔDBC

A

B

CD

ABCΔABAC >DACABAD =

ADB∠BCDΔ

D

A

CB

ABCΔDEFΔEB ∠=∠FC ∠=∠EFBC =

DEAB ≠DEAB >GAB

EDBG =

B

A

C

G

α β

Bα β

D

F

α β+ =

Qβ

αP

t

DCAB =AB

DCABCΔADCΔ

A

CD

B

( )−a

an

gcdgcd( , ) =a

ABCDBCFE

ADEFBC

ΔABΔDCF

abc|a b|a bc

ba |ca ||a bc|ac bc≠c

| ( +a a b|a b

a|c d|ac bd

|a b( ) |−a b| ( )−a b| ( )−b

n

na

ad( , ) =a a

gcd(

BE

b|a b

)b

|b

( , ) =a a a

a

gcd( , )+ =a a

= +

= −

.

.

Bettmann

π

."

."

.

S rπ=

.

ABCDEFCDGDGCDGH

CDACGACDACDAGH

ACHACDCDEFACDACHCDEF

ACHACCH

CDEF

π

π.

.

ππ

π

πn

π

n

π

ABCDEFABCDEF

.

π

π<

ABBC

ABC

BC

B

>BC ABM

F

+ =AB BF FCD

=FD FCΔMBAΔMBD

MF

C

••

•

•

•

•

••

••

xy

xyxy

xy

xx

y

xy

xy

146 Keyser. axioms

Theory

iP

iT

iT

iT

iP

xy

iP

iTiP

Axiom

iPiT

iT

iP

iPiT

iPiT

p q→q r→p r→

151. Henry Poincare

/ ,,

x y xx y y

+ = == =

//y y= = =

( )+ = ≠ax b a

bxa

= −

( )ax by c a+ + = ≠

b b acxa

− ± −=

( )ax bx cx d a+ + + = ≠

ax bx d+ + =

( )( )A B A B A AB B− = − + +

n

KabcRKab

RaRbaRb

aRbaRbaRbaba=b

a b≠

KR

Pa b≠aRbbRaPaRba b≠PaRbbRcaRcPK

aRbbRaaRbbRaPaRaP

c a≠c b≠aRbaRccRbc a≠PaRccRacRaaRbPcRb

KkRaK

bKaRcPabKcK

bRcPb c≠PcRdbRdPb d≠a d≠abcdK

KdRPd e≠PcRebReaRePc e≠b e≠a e≠abcde

KP

KKRa

b a≠KPaRbbRaaRbbRa

bRaa D b

aDbbDcaDc

bRacRaPcRaaDc

aRbcKaRccRb

aFb

aFcbFca b=

b a≠PaRbbRa.

aRbbFcbRc

aFc

bRaaFcaRc

bFc

aRbbRcaFc

aRbbRcaFcaFbbFcaGc

KR

KR

abbacababac

cbKK

KK

baababbcac

abcaccbab

abbcacabbcac

KR

DFKG

K

KR

DFG

1. D. Hillbert

pp

xxg

xg(x)

xxgxxgxxgxxg

PPPPPP

PPPTPP

PaRba b≠aRbbabRa

P

KR

ABA

BA

ABAB

PP ′K

KR

156 Poincare

n

n

PKR

PPP

P

PKR

P

PKRP

PKR

P

160 R. L. Wilder 161 R. L. Moore

SSST

163 category 164 isomorphic

aGcbGcca

aGbbGcaGc

abcBbRcaRbbRa

cRb

abcBacbB

abcB

⊕⊗

baabPabca b b a⊕ = ⊕Pabca b b a⊗ = ⊗PabcS( ) ( )a b c a b c⊕ ⊕ = ⊕ ⊕PabcS( ) ( )a b c a b c⊗ ⊗ = ⊗ ⊗PabcS( ) ( ) ( )a b c a b a c⊗ ⊕ = ⊗ ⊕ ⊗

( ) ( ) ( )b c a b a c a⊕ ⊗ = ⊗ ⊕ ⊗PSzaS

a z a⊕ =PSuzaS

a u a⊗ =

PaSSa z⊕ =PabcSc z≠c a c b⊗ = ⊗c a b c⊗ = ⊗ba

PzaSa−S

a a u−⊕ =

PP

P′abcS( ) ( ) ( )a b c a b a c⊗ ⊕ = ⊗ ⊕ ⊗P′abcSc z≠c a c b⊗ = ⊗ba

aSa a z⊕ =aS( )a a a z⊕ ⊕ =a,a a a S⊗ =

SabS

a z≠b z≠a b z⊗ =

PPa

Sa−Sa a a−⊗ =pqr

qrqP

rP

SFS

PabSbFaaFbPaSbSbFa

PaSbSaFb

PabcSbFacFbcFaPabSbFaS

cFabFaaSb

aSbFaaFb

S ′F

P

P

P

P

PPP

165. A. A. Robb, A Theory of time and space, New York, Cambridje University Press, 1914.

•

•

•

•

•

•

•

•

•

•

Bergriffsschrift

Grundgesetze der Arithmetik

π

mnpqr

p q∧pqpq

p q∨pq

pq

p q→pqp

q

p q↔pqp

q

ppp

p

→ ∨ ∧↔

p p q∧ p q∨ p q⊃p q≡ p p q∧ p q∨ p q→

p q∧ p q∨ p q⊃ p q⊂ p

pqpq

pq

p p∨

( )p p∧( ) ( ) ( )p q q r p r→ ∧ → → →

( )p p↔( ) ( )p q q p→ ↔ →

→p q→ →q p

mnm n↔

( )( )→ ↔ →p q q p

( )→q pp q→

( )p q∧( )p q∧

( ) ( )p q q p∧ ∧ ∧

p q∨

p q→

p q↔

( )∨p qp q∨

( ) ( )p q q p∧ ∨ ∨

p q∧

p q→

p q↔

( )p q→p q→

( ) ( )→ ∧ →p q q p

p q∧

p q∨

p q↔

[ ]

( ) ,p q q p q→ → ∨

p

qr

x x− + = ( )( )x x− − =( )( )− + = − −x x x x

A B C∠ + ∠ + ∠ = p a→

, , , ... , np p p p p p p q−→ → → →

Ppqr

P∨

P

, ,↔ → ∧

P

pq

pp q∨pp.

LLLL

p q→p q∨

L( )p p p∨ →L( )q p q→ ∨L( ) ( )p q q p∨ → ∨L( ) ( ) ( )q r p q p r→ → ∨ → ∨

L

L

L

a b

( ) ( )∨ → ∨ ∨p q p p q

R

q r→q r∨L

( ) ( ) ( )∨ → ∨ → ∨q r p q p rRmm n→n

p q∧( )p q∨

Rmnm n∧

( ) ( ) ( )q r p q p r→ → → → →( ) ( ) ( )p r p q p r→ → ∨ → →

( ) ( ) ( )q r p q p r→ → ∨ → ∨pp( ) ( ) ( )p r p q p r→ → → → →

( )p p p→ ∨( )q p q→ ∨L( )p p p→ ∨qp

p p→

( ) ( ) ( )q r p q p r→ → → → →( ) ( ){ } ( )p p p p p p p p∨ → → → ∨ → →

q( )p p∨rpp p p∨ →L

( ){ } ( )p p p p p→ ∨ → →R( )p p p→ ∨

p p→R

p p∨

p p→

p p∨

p p∨

( ) ( )p q q p∨ → →L( ) ( )p p p p∨ → ∨ppqp

p p∨

p p∨

p

ppp

( )→ →p p p( )p p p∨ →L

( )p p p∨ →pp

( )p p p→ →

( )p p→p p∨

( )p p∨pp( )p p→

( ){ }p p∨( ) ( ) ( )q r p q q r→ → ∨ → ∨L

( ){ } ( ) ( ){ }( )p p p p q p→ → ∨ → ∨qpr( ){ }p

( )p p→( )( ){ }→p ppp

( ) ( ){ }( )p p p p∨ → ∨Rp p∨

( ){ }p p∨R

( )p p→( ) ( )p q q p∨ → ∨L

( ){ }( ) ( ){ }( ){ }p p p p∨ → ∨q( ){ }p( ){ }p p∨

( ){ }p p∨R( )p p→

( ) ( )p p p p→ ∧ →( )p p→

( )p p→

( ) ( )p p p p→ ∧ →R

p q↔( ) ( )p q q p→ ∧ →

( )p p↔( ) ( )p p p p→ ∧ →

( )p p↔

( ) ( ) ( )p q p q p q∨ ↔ ∧ ∧ →

( ) ( ) ( )p q p q p q→ ↔ ∨ ↔ ∧

( ) ( ) ( )p q p q p q∧ ↔ ∨ ↔ →

( )p p p→ →p

pp

( ) ( )p q q p→ ↔ →

p q→q p→

( )→ ∧ →p q q ppqqp

( )∨ ∧ →p q q ppqqp

( ) ( ) ( )p q q r p r→ ∧ → → →

( )q p q→ →

( )→ →p p qp

pq

( )∧p ppp

abc

... , ,∩ ∪,∧ ∨

uz

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( )

:

: ,

:

: ,

∨ ↔ ∨ ∧ ↔ ∧

∨ ↔ ∧ ↔

∨ ∧ ↔ ∨ ∧ ∨

∧ ∨ ↔ ∧ ∨ ∧

∨ ↔ ∧ ↔

B a b b a a b b a

B a z a a u a

B a b c a b a c

a b c a b a c

B a a u a a z

( )∩ = ∪a b a b( )∪ = ∩a b a b

( ) ( )∧ ↔ ∨a b a b( ) ( )a b a b∨ ↔ ∧

m n↔mn

( )∧ ↔ ∧a b b a( ) ( )∨ ↔ ∨a b b aB∧∨

B∨∧∧∨

( ) ( ) ( ):′ ∧ ∨ ↔ ∧ ∨ ∧B a b c a b a c

′BB

,F F

F

F

F F+

][

p p∨

p

p

pm

mm

m

m

p

FT∧FTTFFF

p

q

pqp q∧

p q∧pq

TF

p q∧

p q∧

pq

TF

p q∧T

T

F

=

q

p

FT ∧TT

F

TF

p q∧

p q∧

q

pp q∧

ppp

p

FT∧FTT

FFF

F T ∧F T T F FFFF

q

p

( )pp( ){ }pp

( )pp

( )( ) =

p

( )t p

o

( ) ( )t p t p= −

, / , o

ppp

ppFT

TF

pp T

F TF

mnm n↔

( )p p↔( ) ( )p q q p→ ↔ →p p∨

( )p q p∧ →( )p q p∨ →

( )p p p→ ↔( ) ( )p q q p∧ ↔ ∧

( ) ( ) ( )p q r p q p r→ ∧ ↔ → ∧ →( ) ( ) ( )p q r p q q r∧ ∨ → ∧ ∨ ∧

( ) ( )p q p p q∧ → ↔ →( ) ( )p q r p q r→ → → → →

( )p q p∨ →( )q p q→ ∨

p q∨( )p q∧

( )p q∨p q∧( )p q↔p q∧( )p q↔p q↔

( )p q↔p q↔

pqr

p q→

( )p q r↔ ∧( )p q r∧ ∨

p q→

( )p p q↔ ∧( )p q p∨ ∧

p qp q∨

p pp

( ) ( )p p q qp q∨∧

→

↔

Rm m∨m

Rmpp m∨

Rm n∨n m∨

Rm n→p

( ) ( )p m p n∨ → ∨

( ) ( )p q q p→ → →

( ) ( )a b c a b c∩ ∩ = ∩ ∩a b b a∩ = ∩

a a a∩ =

( )a a b a∩ ∪ =a b→

( )p p q q∧ ∨ ↔( ) ( )p q p q∨ → ∧

p q→

pp q→q

3. N. Bourbaki, Elements of the history of mathematics, New York: Springer-Verlag,1993. 4. D.M. Burton, The history of mathematics, An Introduction. Sixth Edition, McGraw-Hill Companies, Inc., 2007. 5. R.Cook, The history of mathematics: A brief course, 2nd ed., New York: Wiley, 2005.

6. T. Tymoczko, New directions in the philosophy of mathematics, Princeton University Press, 1998.