Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookIX.

Book9

1)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*.

c)    Therefore*.

d)   Sincethen*, and*, therefore*.

e)    Andsince*, therefore*.

f)     Butif*, *, sothat*.

g)    And*, therefore*.

h)   QED.

2)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Nowsince*, therefore*.

d)   Andsince*, therefore*.

e)    Therefore*.

f)     And*, therefore*.

g)    Butif*, *, therefore*.

h)   QED.

3)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*.

c)    Itisthenmanifestthat*.

d)   Nowsince*, therefore*.

e)    Butfurther*, therefore*.

f)     Againsince*, therefore*.

g)    But*, therefore*.

h)   But*, thereforealso*.

i)     Therefore*.

j)     Againsince*, therefore*.

k)    But*, therefore*.

l)     But*, thereforealso*.

m)  Therefore8.

n)   Againsince*, therefore*.

o)    But*, therefore*.

p)   But*, therefore*.

q)   Butif*, *.

r)    And*, therefore*.

s)    QED.

4)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Andsince*, and*, therefore*.

d)   Andsince*, *.

e)    Therefore*, sothat*.

f)     And*, therefore*.

g)    QED.

5)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Nowsince*, and*, therefore*.

d)   Andsince*, *.

e)    Therefore*.

f)     And*, therefore*.

g)    And*, therefore*.

h)   QED.

6)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*.

c)    Sincethen*, therefore*.

d)   Andsince*, therefore*.

e)    But*.

f)     Therefore*.

g)    Andsince*, therefore*.

h)   But*.

i)     Therefore*.

j)     But*, thereforealso*.

k)    Andsince*, *.

l)     Therefore*.

m)  And*.

n)   Therefore*.

o)    And*, therefore*.

p)   QED.

7)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, *.

c)    Let*, and*let*.

d)   Sincethen*, therefore*.

e)    Andsince*, and*, therefore*.

f)     Therefore*.

g)    QED.

8)   If*, *.

a)    Let*, Isaythat*.

b)   Forsince*, therefore*.

c)    But*, therefore*.

d)   Therefore*, therefore*.

e)    Andsince*, and*, therefore*.

f)     Forthesamereason*.

g)    Similarlywecanprovethat*.

h)   Isaynextthat*.

i)     Forsince*, therefore*.

j)     But*, therefore*.

k)    Therefore*.

l)     Sincethen*, therefore*.

m)  Andsince*, therefore*.

n)   Butitwasalsoproved*, therefore*.

o)    Similarly*.

p)   QED.

9)   If*, *, andif*, *.

a)    Let*, andlet*, Isaythat*.

b)   Nowithasproventhat*, Isaythat*.

c)    Forsince*, and*, therefore*.

d)   Againsince*, and*, *.

e)    Similarlywecanprovethat*.

f)     Nextlet*, Isaythat*.

g)    Nowithasbeenprovedthat*, Isaythat*.

h)   Forsince*, therefore*.

i)     But*, therefore*, therefore*.

j)     And*.

k)    Butif*, *.

l)     Therefore*.

m)  Andsince*, and*, *.

n)   Forthesamereason*.

o)    QED.

10)         If*, *, andif*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forifpossiblelet*.

c)    But*, therefore*.

d)   And*, therefore*, sothat*.

e)    And*, therefore*, whichiscontratytothehypothesis.

f)     Therefore*.

g)    Similarly*.

h)   Nextlet*.

i)     Isaythat*.

j)     Forifpossiblelet*.

k)    Now*, for*.

l)     And*, therefore*.

m)  And*, therefore*.

n)   Andsince*, and*, therefore*, therefore*.

o)    Butif*.

p)   Therefore*, whichiscontrarytothehypothesis.

q)   Similarlywecanprovethat*.

r)    QED.

11)         If*, *.

a)    Let*, Isaythat*.

b)   Forsince*, therefore*, thereforealternately*.

c)    But*, therefore*, sothat*.

d)   QED.

e)    Porism. Anditismanifestthat*.

12)         If*, *.

a)    Let*, Isaythat*.

b)   Forlet*, Isaythat*.

c)    Forsupposeitdoesnotnow*, therefore*.

d)   Andsince*, let*, therefore*.

e)    Againsince*, therefore*.

f)     Butfurther*, therefore*.

g)    Therefore*.

h)   But*, and*, therefore*.

i)     Let*, therefore*.

j)     Butfurther*.

k)    Therefore*.

l)     Therefore*.

m)  But*, and*, therefore*.

n)   Let*, therefore*.

o)    Butfurther*, therefore*.

p)   Therefore*.

q)   But*, and*, therefore*.

r)    Butagain*, whichisimpossible.

s)    Therefore*.

t)     Therefore*.

u)   But*.

v)    Andsince*, and*, therefore*, sothat*.

w)  Similarlywecanprovethat*.

x)    QED.

13)         If*, *.

a)    Let*, Isaythat*.

b)   Forifpossiblelet*.

c)    Itisthenmanifestthat*.

d)   Forif*, *, whichisimpossible.

e)    Therefore*.

f)     Therefore*.

g)    But*, therefore*.

h)   Isaynextthat*.

i)     Forif*, and*, that*, sothat*, whichisimpossible.

j)     Therefore*.

k)    Andsince*, let*.

l)     Isaythat*.

m)  Forif*, and*, therefore*.

n)   But*, therefore*, whichiscontrarytothehypothesis.

o)    Therefore*.

p)   Similarlywecanprovethat*.

q)   Forifitisand*, whichisimpossible, therefore*.

r)    Therefore*.

s)    But*, therefore*.

t)     Isaynextthat*.

u)   Forif*, and*, that*, sothat*, whichisimpossible.

v)    Therefore*.

w)  Andsince*, therefore*.

x)    Butfurther*, therefore*.

y)    Therefore*.

z)    But*, therefore*.

aa) Let*.

bb)                  Simliarlythenwecanprovethat*.

cc)  Andsince*, therefore*.

dd)                  Butfurther*, therefore*.

ee) Therefore*.

ff)   But*, therefore*.

gg) Let*.

hh)                  Similarlythenwecanprovethat*.

ii)    Andsince*, therefore*.

jj)    Therefore*.

kk) But*, therefore*, whichisabsurd.

ll)    Therefore*.

mm)               QED.

14)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forifpossiblelet*, andlet*.

c)    Nowsince*, let*, therefore*.

d)   And*.

e)    Butif*, *, therefore*.

f)     Now*, for*.

g)    Therefore*, whichisimpossible.

h)   Therefore*.

i)     QED.

15)         If*, *.

a)    Let*, Isaythat*.

b)   Forlet*.

c)    Itisthenmanifestthat*.

d)   Nowsince*, *.

e)    Butif*, *, therefore*.

f)     Butfurther*, therefore*.

g)    Butif*, *, sothat*, hence*.

h)   But*, therefore*.

i)     And*, and*, therefore*.

j)     Similarlywecanprovethat*.

k)    Isaynextthat*.

l)     Forsince*, *.

m)  But*, therefore*.

n)   *.

o)    Therefore*.

p)   And*, and*.

q)   Therefore*.

r)    QED.

16)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forifpossiblesolet*.

c)    Now*, and*, therefore*.

d)   But*, therefore*, whichisabsurd.

e)    Therefore*.

f)     QED.

17)         If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forifpossiblelet*, thereforealternately*.

c)    But*, and*.

d)   Therefore*.

e)    And*.

f)     Therefore*, sothat*.

g)    Andsince*, and*, therefore*.

h)   But*, sothat*.

i)     But*, therefore*, whichisimpossible.

j)     Therefore*.

k)    QED.

18)         Given*, toinvestigatewhether*.

a)    Let*, andlet*.

b)   Now*.

c)    Andif*, ithasbeenprovedthat*.

d)   Nextlet*, andlet*.

e)    Then*.

f)     Firslet*, therefore*.

g)    Therefore*, therefore*.

h)   Nextlet*, Isaythat*.

i)     Forifpossiblelet*.

j)     Therefore*.

k)    But*, therefore*.

l)     Hence*, therefore*.

m)  Buthypothesis*, whichisabsurd.

n)   Therefore*.

o)    QED.

19)         Given*, toinvestigatewhen*.

a)    Let*.

b)   The greektext of this proposition is corrupt, and the intact portion of the proof is erroneous, according to Heath. Nevertheless, analogously to Proposition18, the condition that a fourth proportional to ABC exists is that A measure the product of B and C. Editor.

20)         *.

a)    Let*, Isaythat*.

b)   Forlet*, andlet*, let*.

c)    Then*.

d)   Firstlet*, then*.

e)    Nextlet*, therefore*.

f)     Let*.

g)    Isaythat*.

h)   Forifpossiblelet*.

i)     Now*, therefore*.

j)     But*. Therefore*., whichisabsurd.

k)    Therefore*.

l)     Andbyhypothesis*.

m)  Therefore*.

n)   QED.

21)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, *, sothat*.

c)    But*, therefore*.

d)   QED.

22)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, if*, *, sothat*.

c)    But*.

d)   Therefore*.

e)    QED.

23)         If*, *.

a)    Forlet*, Isaythat*.

b)   Let*, therefore*.

c)    But*, therefore*.

d)   And*.

e)    Therefore*.

f)     QED.

24)         If*, *.

a)    For*, Isaythat*.

b)   Forsince8, *.

c)    Forthesamereason*, sothat*, andtherefore*.

d)   QED.

25)         If*, *.

a)    Forfrom*, let*, Isaythat*.

b)   Forlet*, therefore*.

c)    But*, therefore*.

d)   And*, therefore*.

e)    QED.

26)         If*, *.

a)    Forfrom*, let*, Isaythat*.

b)   Forsince*, let*, therefore*.

c)    Forthesamereason*, sothat*.

d)   QED.

27)         If*, *.

a)    Forfrom*, let*.

b)   Isaythat*.

c)    Let*, therefore*.

d)   But*, therefore*.

e)    Therefore*.

f)     QED.

28)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    And*, therefore*.

d)   Butif*, *.

e)    Therefore*.

f)     QED.

29)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    And*, therefore*.

d)   Thus*.

e)    QED.

30)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, let*, Isaythat*.

c)    Forifpossiblelet*.

d)   Thensince*, therefore*.

e)    Therefore*.

f)     Therefore*, whichisabsurdfor*.

g)    Therefore*, therefore*.

h)   Thus.

i)     Forthisreasonthen*.

j)     QED.

31)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forif*, *.

c)    Let*, andlet*.

d)   Now*, therefore*.

e)    Andsince*, and*, therefore*.

f)     But*, therefore*.

g)    But*, therefore*, whichisimpossible.

h)   Therefore*.

i)     Therefore*.QED.

32)         *.

a)    Forlet*, Isaythat*.

b)   Nowthat*, for*.

c)    Isaythat*.

d)   Forlet*.

e)    Sincethen*, and*, therefore*.

f)     And*, therefore*.

g)    Similarlywecanprovethat*.

h)   QED.

33)         If*, *.

a)    Forlet*, Isaythat*.

b)   Nowthat*, for*.

c)    isaynextthat*.

d)   Forif*, *, sothat*.

e)    Therefore*.

f)     QED.

34)         If*, *.

a)    Forlet*, Isaythat*.

b)   Nowthat*, for*.

c)    Isaynextthat*.

d)   Forif*, *.

e)    Forifnot*, whichiscontrarytothehypothesis.

f)     Thus*.

g)    Butitwasalsoproved*.

h)   Therefore*.

i)     QED.

35)         If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, and*.

c)    Thensince*, and*, therefore*.

d)   Andsince*, while*, therefore*.

e)    *.

f)     Thereforealso*, therefore*.

g)    But*, therefore*.

h)   Therefore*.

i)     QED.

36)         If*, *, andif*, *.

a)    Forlet*, let*, Isaythat*.

b)   For*, let*, therefore*.

c)    Therefore*.

d)   And*, therefore*.

e)    Therefore*, therefore*.

f)     And*, therefore*.

g)    But*, therefore*.

h)   Nowlet*, therefore*.

i)     Therefore*.

j)     And*, therefore*.

k)    But*, and*.

l)     Therefore*.

m)  Isayalsothat*.

n)   Forifpossiblelet*, and*.

o)    And*, therefore*.

p)   Butfurther*, therefore*.

q)   Andsince*, therefore*.

r)    Andbyhypothesis*, therefore*.

s)    But*, therefore*.

t)     And*, and*.

u)   Therefore*.

v)    But*, and*, and*, therefore*.

w)  But*, therefore*.

x)    Let*.

y)    And*.

z)    Now*, therefore*.

aa) Therefore*.

bb)                  But*, therefore*.

cc)  Therefore*.

dd)                  And*, therefore*, whichisimpossible.

ee) Therefore*.

ff)   And*, and*, therefore*.

gg) QED.