Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookXI.

Book11.

1)   *.

a)    Forifpossiblelet*.

b)   Therewillthenbe*.

c)    Let*, therefore*, whichisimpossibleinasmuchasif*, *.

d)   Therefore*.

e)    QED.

2)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*, Isayfirstthat*.

c)    Forif*, *.

d)   Butif*, *, whichwasprovedabsurd.

e)    Therefore*.

f)     But*, and*.

g)    Therefore*, and*.

h)   QED.

3)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forifnotlet*, and*.

c)    Then*, whichisabsurd.

d)   Therefore*.

e)    Similarlywecanprovethat*.

f)     Therefore, etc.

g)    QED.

4)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*, let*, andfurtherlet*.

c)    Nowsince*, therefore*, and*, sothat*.

d)   But*, therefore*, therefore*.

e)    Therefore*.

f)     Andsince*, while*, therefore*.

g)    Forthesamereason*.

h)   Andsince*, and*, and*, therefore*.

i)     Andsinceagain*, andfurther*.

j)     And*, therefore*.

k)    Nowsinceagain*, and*, *, and*, therefore*.

l)     Therefore*.

m)  Therefore*.

n)   Similarlywecanprovethat*.

o)    But*, therefore*.

p)   But*.

q)   Therefore*.

r)    Therefore, etc.

s)    QED.

5)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forsupposeifpossiblelet*, let*, itwillthus*.

c)    Let*.

d)   Therefore*.

e)    Nowsince*, therefore*.

f)     But*, therefore*.

g)    Thus*.

h)   But*, therefore*.

i)     Butbyhypothesis*, therefore*.

j)     And*, whichisimpossible.

k)    Therefore*, therefore*.

l)     ThereforeIf*, *.

m)  QED.

6)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, let*, let*, let*, andlet*.

c)    Nowsince*, *.

d)   But*, therefore*.

e)    Forthesamereason*.

f)     Andsince*, and*, *, and*, therefore*.

g)    Andsince*, while*, *, and*, therefore*.

h)   But*, therefore*, therefore*.

i)     But*, therefore*, therefore*.

j)     But*, for*, therefore*.

k)    And*, therefore*.

l)     Therefore, etc.

m)  QED.

7)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsupposeitisnotbutifpossiletlet*, andlet*, itwillthen*.

c)    Let*, therefore*, whichisimpossible.

d)   Therefore*, therefore*.

e)    Therefore, etc.

f)     QED.

8)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, andlet*, therefore*.

c)    Let*, let*, andlet*.

d)   Nowsince*, therefore*.

e)    Andsince*, therefore*.

f)     But*, therefore*, therefore*.

g)    Andsince*, and*, *, and*, for*, therefore*.

h)   Andsince*, and*, *, and*, therefore*.

i)     But*, therefore*, therefore*.

j)     But*, therefore*.

k)    Therefore*.

l)     But*, and*.

m)  Therefore*, sothat*.

n)   But*.

o)    Therefore*, sothat*.

p)   But*, therefore*.

q)   Therefore, etc.

r)    QED.

9)   *.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*.

c)    Nowsince*, therefore*.

d)   And*, therefore*.

e)    Forthesamereason*, therefore*.

f)     Butif*, *, therefore*.

g)    QED.

10)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*.

c)    Nowsince*, therefore*.

d)   Forthesamereason*.

e)    Therefore*.

f)     But*, therefore*.

g)    And*, therefore*.

h)   Nowsince*, and*, therefore*.

i)     Therefore, etc.

j)     QED.

11)         From*, todraw*.

a)    Let*, thusitisrequiredtodraw*.

b)   Let*, andlet*.

c)    Ifthen*, *.

d)   Butifnotlet*, let*, andlet*.

e)    Nowsince*, therefore*.

f)     And*, butif*, *, therefore*.

g)    Therefore*.

h)   But*, therefore*, sothat*.

i)     But*, therefore*.

j)     Butif*, *, therefore*.

k)    But*, therefore*.

l)     Therefore*.

m)  QEF.

12)         Toset* fromagiven*.

a)    LEt*, and*, thusitisrequiredtoset*.

b)   Let*, let*, andlet*.

c)    Thensince*, therefore*.

d)   Therefore*.

e)    QEF.

13)         From*, therecannotbe*.

a)    Forifpossiblelet*, andlet*, itwillthen*.

b)   Let*, therefore*.

c)    Andsince*, *.

d)   But*, therefore*.

e)    Forthesamereason*, therefore*.

f)     And*, whichisimpossible.

g)    Therefore, etc.

h)   QED.

14)         *.

a)    Forlet*, Isaythat*.

b)   Forifnot*.

c)    Let*, *.

d)   Let*, let*, andlet*.

e)    Nowsince*, therefore*, therefore*.

f)     Forthesamereason*.

g)    thus*, whichisimpossible.

h)   Therefore*, therefore*.

i)     Therefore*.

j)     QED.

15)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*, let*.

c)    Nowsince*, therefore*.

d)   But*, therefore*.

e)    Andsince*, therefore*.

f)     But*, therefore*, therefore*.

g)    Forthesamereason*.

h)   Sincethen*, therefore*.

i)     But*, therefore*.

j)     Thereforeif*, *.

k)    QED.

16)         If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forifnot*.

c)    Let*, andlet*.

d)   Nowsince*, therefore*.

e)    But*, therefore*.

f)     Forthesamereason*, therefore*.

g)    But*, becausehypothesis*, therefore*.

h)   Similarlywecanprovethat*.

i)     But*.

j)     Therefore*.

k)    Therefore, etc.

l)     QED.

17)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, let*, andlet*.

c)    Nowsince*, *.

d)   Forthesamereasonsince*, *.

e)    Andsince*, therefore*.

f)     Againsince*, *.

g)    Butitwasalsoprovedthat*, thereforealso*.

h)   Therefore, etc.

i)     QED.

18)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, let*, let*, andlet*.

c)    Nowsince*, *, sothat*, therefore*.

d)   But*, therefore*.

e)    But*, therefore*.

f)     Now*, when*.

g)    And*, therefore*.

h)   Similarlyalsoitcanbeprovedthat*.

i)     Therefore, etc.

j)     QED.

19)         If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forsupposeitdoesnotandlet*.

c)    Nowsince*, and*, therefore*.

d)   Similarlywecanprovethat*.

e)    Therefore*, whichisimpossible.

f)     Therefore*.

g)    Therefore, etc.

h)   QED.

20)         If*, *.

a)    Forlet*, Isaythat*.

b)   Ifnow*, itismanifestthat*.

c)    Butifnotlet*, and*, let*, andlet*, let*.

d)   Nowsince*, and*, *, and*, therefore*.

e)    Andsince*, and*, therefore*.

f)     Nowsince*, and*, and*, therefore*.

g)    But*, therefore*.

h)   Similarlywecanprovethat*.

i)     Therefore, etc.

j)     QED.

21)         *.

a)    Let*, Isaythat*.

b)   Forlet*, andlet*.

c)    Nowsince*, *, therefore*.

d)   Forthesamereason*, and*, therefore*.

e)    But*, therefore*.

f)     Andsince*, therefore*, and*, therefore*.

g)    Therefore, etc.

h)   QED.

22)         *.

a)    Let*, andfurther*, andlet*, Isaythatitispossibletoconstruct*.

b)   Nowif*, itismanifestthat*.

c)    Butifnotlet*, andlet*, let*, andlet*.

d)   Nowsince*, and*, therefore*.

e)    Andsince*, while*, therefore*.

f)     Andsince*, and*, therefore*.

g)    But*.

h)   Therefore*.

i)     But*, therefore*.

j)     Similarlywecanprovethat*, andfurther*.

k)    Therefore*.

l)     QED.

23)         Toconstruct*.

a)    Let*, thusitisrequiredtoconstruct*.

b)   Let*, andlet*, itisthereforepossibletoconstruct*.

c)    Let*, let*, let*, let*, Isaythat*.

d)   Forifnot*.

e)    Firstlet*.

f)     Thensince*, while*, *, andbyhypothesis*, therefore*.

g)    Forthesamereason*, andfurther*, therefore*.

h)   But*, therefore*.

i)     But*, whichisabsurd.

j)     Therefore*.

k)    Isaynextthat*.

l)     Forifpossiblelet*, andlet*, andlet*.

m)  Thensince*, *, sothat*.

n)   Therefore*, and*, therefore*, andalternately*.

o)    But*, therefore*.

p)   But*, therefore*.

q)   Sincethen*, and*, therefore*.

r)    Similarlywecanprovethat*and*.

s)    Therefore*.

t)     Butbyhypothesis*, therefore*.

u)   But*, whichisabsurd.

v)    Therefore*.

w)  Anditwasprovedthat*, therefore*.

x)    Let*, andlet*, let*.

y)    Thensince*, therefore*.

z)    Andsince*, while*, therefore*.

aa) Forthesamereason*, therefore*.

bb)                  Nextsincebyhypothesis*, therefore*.

cc)  But*, therefore*, therefore*.

dd)                  But*, while*, therefore*.

ee) Andsince*, and*, therefore*.

ff)   Forthesamereason*, and*.

gg) Therefore*.

hh)                  QEF.

ii)    Lemma. Buthowitispossibletake*, wecanshowasfollows.

jj)    Let*, andlet*, let*, andlet*, let*.

kk) Sincethen*, therefore*.

ll)    Therefore*.

mm)               Hence*.

nn)                  But*.

oo) Therefore*.

pp)                  Ifthen*, *.

qq)                  QEF.

24)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, *.

c)    Therefore*.

d)   Againsince*, *.

e)    Therefore*.

f)     But*, therefore*.

g)    Similarlywecanprovethat*.

h)   Let*.

i)     Thensince*, *, therefore*, therefore*.

j)     Andsince*, and*, therefore*, and*.

k)    And*, therefore*.

l)     Similarlywecanprovethat*, and*.

m)  Therefore, etc.

n)   QED.

25)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, let*, and*, andlet*.

c)    Thensince*, *, andfurther*.

d)   Therefore*.

e)    But*, therefore*.

f)     Forthesamereason*.

g)    Therefore*.

h)   But*, therefore*.

i)     Forthesamereason*.

j)     Therefore*.

k)    Forthesamereason*.

l)     Andif*, if*, andif*, *.

m)  Therefore*, *, and*, andithasbeenprovedthatif*, if*, andif*, *, therefore*.

n)   QED.

26)         Onagiven*, toconstruct*.

a)    Let*, thusitisrequiredtoconstruct*.

b)   Forlet*, let*, let*, let*, let*, let*, let*, andlet*, Isaythat*.

c)    Forlet*, andlet*.

d)   Thensince*, therefore*.

e)    Forthesamereason*.

f)     Andsince*, and*, therefore*.

g)    But*, and*, therefore*.

h)   Againsince*, and*, therefore*.

i)     But*, therefore*.

j)     And*, therefore*.

k)    Forthesamereason*.

l)     And*.

m)  Therefore*.

n)   QEF.

27)         Onagiven*, todescribe*.

a)    Let*, thusitisrequiredtodescribe*.

b)   Forlet*, sothat*, andlet*.

c)    Thereforealso*.

d)   Let*.

e)    Nowsince*, and*, therefore*.

f)     Forthesamereason*, andfurther*, therefore*.

g)    But*, and*, therefore*.

h)   Therefore*.

i)     QEF.

28)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, and*, while*, and*, therefore*, for*.

c)    Hence*.

d)   QED.

29)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, hence*.

c)    Let*, therefore*.

d)   Hence*, and*.

e)    Forthesamereason*.

f)     But*, therefore*.

g)    Let*, therefore*.

h)   Therefore, etc.

i)     QED.

30)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, andfurtherlet*.

c)    Then*, for*.

d)   But*, for*.

e)    Hence*.

f)     Therefore, etc.

g)    QED.

31)         *.

a)    Let*.

b)   Isaythat*.

c)    Firstlet*, let*, let*, let*, andlet*.

d)   Nowsince*, and*, therefore*.

e)    Sinceagain*, and*, therefore*.

f)     Forthesamereason*, therefore*.

g)    But*, therefore*.

h)   Let*, let*, let*, andlet*.

i)     Then*, for*.

j)     But*, therefore*.

k)    Andsince*, for*, while*, therefore*.

l)     But*, therefore*.

m)  Andsince*, *.

n)   Forthesamereasonsince*, *.

o)    But*, thereforealso*.

p)   Therefore*, therefore*.

q)   But*, therefore*.

r)    Nextlet*, Isayagainthat*.

s)    Forlet*, andlet*.

t)     Then*, for*.

u)   But*, and*, for*.

v)    Therefore*.

w)  Therefore, etc.

x)    QED.

32)         *.

a)    Let*, Isaythat*.

b)   Forlet*, andlet*.

c)    Then*, for*.

d)   Andsince*, therefore*.

e)    But*, and*, thereforealso*.

f)     Therefore, etc.

g)    QED.

33)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, let*, andlet*.

c)    Nowsince*, while*, therefore*.

d)   Forthesamereason*, andfurther*, therefore*.

e)    But*, therefore*.

f)     Let*, and*, let*.

g)    Thensince*, while*, therefore*.

h)   But*, and*, thereforealso*.

i)     But*, and*, thereforealso*.

j)     Butif*, *, therefore*.

k)    But*, and*, hence*.

l)     But*, and*, therefore*.

m)  Therefore, etc.

n)   QED.

o)    Porism. Fromthisitismanifestthatif*, *.

34)         * and*.

a)    Let*, Isaythat*.

b)   Firstlet*, Isaythat*.

c)    If*, while*, *.

d)   For*, and*, anditismanifestthat*.

e)    Nextlet*, butlet*.

f)     Now*, therefore*.

g)    Let*, andlet*.

h)   Nowsince*, and*, while*, therefore*.

i)     But*, for*, and*, thereforealso*.

j)     But*, thereforealso*.

k)    Therefore*.

l)     Againlet*, Isaythat*.

m)  Let*.

n)   Nowif*, and*, therefore*.

o)    But*, therefore*.

p)   Nextlet*, butlet*, therefore*.

q)   Let*, andlet*.

r)    Since*, while*, therefore*.

s)    But*, for*, and*, and*.

t)     Thereforealso*, therefore*.

u)   therefore*.

v)    Nowlet*, let*, andlet*, andlet*.

w)  Isaythat*.

x)    Since*, while*, for*, and*, for*, therefore*.

y)    Therefore*.

z)    But*, and*, therefore*.

aa) But*, therefore*.

bb)                  Therefore*.

cc)  Againlet*, Isaythat*.

dd)                  Forsince*, while*, therefore*.

ee) But*, therefore*.

ff)   Therefore*, therefore*.

gg) But*, for*.

hh)                  Therefore*.

ii)    QED.

35)         If*, *.

a)    Let*, let*, let*, let*, andlet*, andlet*, Isaythat*.

b)   Let*, andlet*.

c)    But*, therefore*.

d)   Let*, andlet*.

e)    Since*, and*, therefore*.

f)     But*, therefore*.

g)    Therefore*.

h)   Forthesamereason*.

i)     Therefore*.

j)     But*.

k)    Therefore*, therefore*.

l)     Therefore*.

m)  Similarlywecanprovethat*.

n)   Sincethen*, *.

o)    But*, therefore*, therefore*.

p)   But*, therefore*.

q)   Forthesamereason*.

r)    Therefore*, therefore*.

s)    Therefore*.

t)     But*, therefore*, and*.

u)   Therefore*.

v)    Andsince*, *.

w)  But*, for*, and*, for*, therefore*, and*, therefore*, therefore*.

x)    Andsince*, and*, therefore*.

y)    Therefore, etc.

z)    QED.

aa) Porism. Fromthisitismanifestthatif*, andif*, *.

36)         If*, *.

a)    Let*, Isaythat*.

b)   Let*, let*, andlet*, let*, andlet*, let*.

c)    Nowsince*, while*, therefore*.

d)   Thus*, therefore*.

e)    Andsince*, therefore*, hence*.

f)     But*, therefore*.

g)    And*, therefore*.

h)   QED.

37)         If*, *, andif*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    Forthesamereason*.

d)   And*.

e)    Thereforealso*.

f)     Nextas*, solet*, Isaythat*.

g)    Forsinceagain*, and*, and*, thereforealso*.

h)   Therefore, etc.

i)     QED.

38)         If*, *.

a)    Forlet*, let*.

b)   Isaythat*.

c)    Forlet*.

d)   Thensince*, *.

e)    Andsince*, and*, therefore*, therefore*.

f)     Forthisreason*.

g)    Forthesamereason*.

h)   Nowsince*, while*, therefore*.

i)     And*, therefore*.

j)     Therefore*, for*, and*.

k)    Therefore*, therefore*.

l)     Therefore*.

m)  Therefore, etc.

n)   QED.

39)         If*, andif*, *.

a)    Let*, let*, andlet*, Isaythat*.

b)   Forlet*.

c)    Since*, while*, therefore*.

d)   But*, therefore*.

e)    And*, and*, therefore*.

f)     Therefore, etc.

g)    QED.