Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookVI.

Book6.

1)   *.

a)    Let*, Isaythat*.

b)   Forlet*, andlet*, let*.

c)    Thensince*, *.

d)   Therefore*.

e)    Forthesamereason*, andif*, *, if*, *.

f)     Thus*, and*, andithasbeenprovedthat*, *.

g)    Therefore*.

h)   Nextsince*, while*, therefore*.

i)     Sincethen*, and*, thereforealso*.

j)     Therefore, etc.

k)    QED.

2)   If*, *, andif*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*.

c)    Therefore*, for*.

d)   And*.

e)    But*, therefore*.

f)     But*, for*.

g)    Forthesamereason*.

h)   Thereforealso*.

i)     Againlet*, sothat*, andlet*.

j)     Isaythat*.

k)    For*, since*, but*, and*, thereforealso*.

l)     Therefore*.

m)  Therefore*, and*.

n)   But*.

o)    Therefore*.

p)   Therefore, etc.

q)   QED.

3)   If*, *, andif*, *.

a)    Let*, Isaythat*.

b)   Forlet*, andlet*.

c)    Thensince*, *.

d)   But*, therefore*.

e)    Againsince*, *.

f)     But*, therefore*, sothat*.

g)    Andsince*, therefore*.

h)   But*, therefore*.

i)     Againlet*, Isaythat*.

j)     For*, since*, andalso*, thereforealso*.

k)    Therefore*, sothat*.

l)     But*, and*, therefore*.

m)  Therefore*.

n)   Therefore, etc.

o)    QED.

4)   In*, *.

a)    Forlet*.

b)   Thensince*, and*, therefore*, therefore*.

c)    Let*.

d)   Nowsince*, *.

e)    Againsince*, *.

f)     Therefore*, therefore*.

g)    Andsince*, therefore*.

h)   But*, therefore*, andalternately*.

i)     Againsince*, therefore*.

j)     But*, therefore*, andalternately*.

k)    Sincethenitwasprovedthat*, and*, therefore*.

l)     Therefore, etc.

5)   If*, *.

a)    Let*, sothat*, andfurther*, Isaythat*.

b)   For*, let*, and*, therefore*.

c)    Therefore*.

d)   Therefore*, therefore*.

e)    But*, therefore*.

f)     Therefore*, therefore*.

g)    Forthesamereason*.

h)   Sincethen*, and*, *, and*, therefore*.

i)     Therefore*.

j)     Andsince*, while*, therefore*.

k)    Forthesamereason*, andfurther*, therefore*.

l)     Therefore, etc.

m)  QED.

6)   If*, *.

a)    Let*, Isaythat*.

b)   For*, let*, and*, therefore*.

c)    Therefore*.

d)   Therefore*.

e)    But*, thereforealso*.

f)     Therefore*, and*, therefore*, and*, therefore*, and*.

g)    Therefore*.

h)   But*, therefore*.

i)     Andbyhypothesis*, therefore*, therefore*.

j)     Therefore, etc.

k)    QED.

7)   If*, *, andif*, *.

a)    Let*, Isaythat*.

b)   Forif*, *.

c)    Let*, and*.

d)   Thensince*, and*, therefore*.

e)    Therefore*.

f)     Therefore*.

g)    But*, therefore*, therefore*, sothat*.

h)   Butbyhypothesis*, therefore*, sothat*.

i)     Anditwasproved*, therefore*.

j)     But*, whichisabsurd.

k)    Therefore*, therefore*.

l)     But*, therefore*.

m)  Therefore*.

n)   Butagainlet*, Isayagainthat*.

o)    Forsimilarly*, sothat*.

p)   But*, therefore*.

q)   Thus*, whichisimpossible.

r)    Therefore*, therefore*.

s)    But*, therefore*.

t)     Therefore*.

u)   Therefore, etc.

v)    QED.

8)   If*, *.

a)    Let*, andlet*, Isaythat*.

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

c)    Therefore*.

d)   Therefore*.

e)    Therefore*.

f)     Similarly*, therefore*.

g)    Isaynextthat*.

h)   Forsince*, andmoreover*, therefore*, therefore*.

i)     Therefore*, therefore*.

j)     Therefore, etc.

k)    Porism. Fromthisitismanifestthatif*, *.

9)   Fromagiven*, tocut*.

a)    Let*, thusitisrequiredtocut*.

b)   Let*.

c)    Let*, let*, andlet*.

d)   Let*, andthrough*, let*.

e)    Thensince*, therefore*.

f)     But*, therefore*, therefore*.

g)    Therefore*.

h)   QEF.

10)         Tocut*.

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

b)   Therefore*, therefore*.

c)    Nowsince*, thereforeproportionally*.

d)   But*, therefore*.

e)    Againsince*, therefore*.

f)     Butitwasalsoprovedthat*, therefore*, and*.

g)    Therefore*.

h)   QEF.

11)         Togiven*, tofind*.

a)    Let*, andlet*, thusitisrequiredtofind*.

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

c)    Sincethen*, *.

d)   But*, therefore*.

e)    Therefore*.

f)     QEF.

12)         Togiven*, tofind*.

a)    Let*, thusitisrequiredtofind*.

b)   Let*, let*, let*, andlet*.

c)    Sincethen*, therefore*.

d)   But*, therefore*.

e)    Therefore*.

f)     QEF.

13)         Togiven*, tofind*.

a)    Let*, thusitisrequiredtofind*.

b)   Let*, andlet*, let*, andlet*.

c)    Since*, *.

d)   Andsince*, *, therefore*.

e)    Therefore*.

f)     QEF.

14)         In*, *, and*.

a)    Let*, andlet*, therefore*.

b)   Isaythat*.

c)    Forlet*.

d)   Sincethen*, and*, therefore*.

e)    But*, and*, thereforealso*.

f)     Therefore*.

g)    Nextlet*, Isaythat*.

h)   Forsince*, while*, and*, thereforealso*, therefore*.

i)     Therefore, etc.

j)     QED.

15)         In*, *, and*, *.

a)    Let*, Isaythat*.

b)   Forlet*, therefore*.

c)    Let*.

d)   Sincethen*, therefore*.

e)    But*, and*.

f)     Thereforealso*.

g)    Therefore*.

h)   Nextlet*, Isaythat*.

i)     Forif*, since*, while*, and*, therefore*.

j)     Therefore*.

k)    Therefore*.

l)     Therefore, etc.

m)  QED.

16)         If*, *, andif*, *.

a)    Let*, sothat*, Isaythat*.

b)   Let*, andlet*.

c)    Let*.

d)   Thensince*, *, while*, and*, therefore*.

e)    Therefore*.

f)     But*, therefore*.

g)    And*, and*, therefore*.

h)   Nextlet*, Isaythat*.

i)     For*, since*, and*, and*, therefore*.

j)     And*.

k)    But*.

l)     Therefore*.

m)  But*, therefore*.

n)   Therefore, etc.

17)         If*, *, andif*, *.

a)    Let*, sothat*, Isaythat*.

b)   Let*.

c)    Thensince*, and*, therefore*.

d)   Butif*, *.

e)    Therefore*.

f)     But*, therefore*.

g)    Nextlet*, Isaythat*.

h)   For*, since*, while*, therefore*.

i)     Butif*, *.

j)     Therefore*.

k)    But*, therefore*.

l)     Therefore, etc.

m)  QED.

18)         Onagiven*, *.

a)    Let*, thusitisrequiredtodescribe*.

b)   Let*, andon*, andat*, let*.

c)    Therefore*, therefore*.

d)   Therefore*.

e)    Againon*, andat*, let*.

f)     Therefore*, therefore*, therefore*.

g)    Butitwasalsoproved*, thereforealso*.

h)   Andsince*, and*, therefore*.

i)     Forthesamereason*.

j)     And*, and*.

k)    Therefore*, and*, therefore*.

l)     Therefore*.

m)  QEF.

19)         *.

a)    Let*, sothat*.

b)   Isaythat*.

c)    Forlet*, sothat*, andlet*.

d)   Sincethen*, therefore*.

e)    But*, therefore*.

f)     Therefore*.

g)    But*, and*, therefore*.

h)   Nowsince*, andif*, *, therefore*.

i)     But*, therefore*.

j)     But*, therefore*.

k)    Therefore, etc.

l)     QED.

m)  Porism. Fromthisitismanifestthatif*, *.

20)         *.

a)    Let*, andlet*, Isaythat*.

b)   Let*.

c)    Nowsince*, *, and*.

d)   Sincethen*, therefore*, sothat*, therefore*.

e)    But*, therefore*.

f)     Andsince*, andmoreoveralso*, therefore*, sothat*.

g)    Forthesamereason*.

h)   Therefore*.

i)     Isaythat*.

j)     Forlet*.

k)    Thensince*, and*, therefore*.

l)     Andsince*, and*, therefore*, therefore*.

m)  Similarly*.

n)   Therefore*, sothat*.

o)    But*.

p)   Thereforealso*, therefore*.

q)   But*, thereforealso*.

r)    Forthesamereason*.

s)    And*, thereforealso*, andalternately*.

t)     Similarly*.

u)   Andsince*, thereforealso*, therefore*.

v)    But*.

w)  Therefore*.

x)    Therefore, etc.

y)    QED.

z)    Porism. Similarlyalsoitcanbeproved*.

21)         *.

a)    Forlet*, Isaythat*.

b)   Forsince*, *.

c)    Againsince*, *.

d)   Therefore*, therefore*.

e)    QED.

22)         If*, *, andif*, *.

a)    Let*, sothat*, andlet*, and*, Isaythat*.

b)   Forlet*, and*.

c)    Thensince*, and*, therefore*.

d)   But*, and*, therefore*.

e)    Nextlet*, Isayalsothat*.

f)     Forif*, let*, and*.

g)    Sincethen*, and*, and*, therefore*.

h)   Butalsobyhypothesis*, thereforealso*.

i)     Therefore*, therefore*.

j)     But*, therefore*.

k)    Andsince*, while*, therefore*.

l)     Therefore, etc.

m)  QED.

23)         *.

a)    Let*, Isaythat*.

b)   Forlet*, therefore*.

c)    let*, let*, and*.

d)   Then*.

e)    But*, sothat*.

f)     Nowsince*, while*, thereforealso*.

g)    Againsince*, while*, thereforealso*.

h)   Sincethen*, and*, therefore*.

i)     But*, therefore*.

j)     Therefore, etc.

k)    QED.

24)         In*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, *.

c)    Againsince*, *.

d)   Butitwasprovedthat*, thereforealso*, andtherefore*, and*.

e)    Therefore*.

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

g)    Forthesamereason*, and*.

h)   Therefore*, andfurther*.

i)     Andsinceitwasprovedthat*, and*, therefore*.

j)     Therefore*, therefore*.

k)    Forthesamereason*, therefore*.

l)     But*, therefore*.

m)  Therefore, etc.

n)   QED.

25)         Toconstruct*.

a)    Let*, thusitisrequiredtoconstruct*.

b)   Let*, and*.

c)    Therefore*.

d)   Now*, and*.

e)    Thensince*, andif*, *, therefore*.

f)     But*.

g)    Thereforealso*, therefore*.

h)   But*, therefore*.

i)     But*, therefore*.

j)     And*.

k)    Therefore*.

l)     QED.

26)         If*, *.

a)    Forfor*, let*, Isaythat*.

b)   Forsupposeitisnotifpossiblelet*, let*, andlet*.

c)    Sincethen*, therefore*.

d)   Butalso*, thereforealso*.

e)    Therefore*.

f)     Therefore*, whichisimpossible.

g)    Therefore*, therefore*.

h)   Therefore, etc.

i)     QED.

27)         Ofall*, *.

a)    Let*, andlet*, let*, Isaythat*.

b)   Forlet*, Isaythat*.

c)    Forsince*, *.

d)   Let*.

e)    Thensince*, and*, therefore*.

f)     But*.

g)    Therefore*.

h)   Let*, therefore*, sothat*.

i)     Therefore, etc.

j)     QED.

28)         Toagiven*, toapply*, thus*.

a)    Let*, thusitisrequiredtoapply*.

b)   Let*, andon*, let*, let*.

c)    Ifthen*, *, for*.

d)   Butifnotlet*.

e)    Now*, therefore*.

f)     Let*.

g)    But*, therefore*.

h)   Letthen*.

i)     Nowsince*, therefore*, therefore*.

j)     Let*, andlet*, therefore*.

k)    Therefore*, therefore*.

l)     Let*, andlet*.

m)  Thensince*, and*, therefore*.

n)   Andsince*, let*, therefore*.

o)    But*, since*, therefore*.

p)   Let*, therefore*.

q)   But*, therefore*.

r)    Therefore*.

s)    QEF.

29)         Toagiven*, toapply*.

a)    Let*, thusitisrequiredtoapply*.

b)   Let*, let*, andlet*.

c)    Let*.

d)   Nowsince*, therefore*.

e)    Let*, let*, andlet*, therefore*.

f)     But*, therefore*, therefore*.

g)    Let*, andlet*.

h)   Since*, while*, therefore*.

i)     Let*, therefore*.

j)     Nowsince*, *,.

k)    Let*, therefore*.

l)     But*, therefore*.

m)  Therefore*.

n)   QEF.

30)         Tocut*.

a)    Let*, thusitisrequiredtocut*.

b)   On*, let*, andlet*.

c)    Now*, therefore*.

d)   Andsince*, let*, therefore*.

e)    But*, therefore*, therefore*.

f)     But*.

g)    Therefore*.

h)   And*, therefore*.

i)     Therefore*.

j)     QEF.

31)         In*, *.

a)    Let*, Isaythat*.

b)   Let*.

c)    Thensince*, *.

d)   Andsince*, therefore*.

e)    Andsince*, *.

f)     Therefore*.

g)    Forthesamereason*. sothat*.

h)   But*, therefore*.

i)     Therefore, etc.

j)     QED.

32)         If*, *.

a)    Let*, sothat*, Isaythat*.

b)   Forsince*, and*, *.

c)    Forthesamereason*, sothat*.

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

e)    But*, therefore*.

f)     Let*, therefore*.

g)    But*, therefore*.

h)   Therefore*, therefore*.

i)     Therefore, etc.

j)     QED.

33)         In*, *.

a)    Let*, andlet*, and*, Isaythat*.

b)   Forlet*, and*, andlet*.

c)    Thensince*, *, therefore*.

d)   Forthesamereasonalso*.

e)    Ifthen*, *, if*, *, and*.

f)     There*, and*.

g)    Andithasbeenprovedthat*, if*, *, if*, *, andif*, *.

h)   Therefore*.

i)     But*, so*, for*.

j)     Thereforealso*, so*, and*.

k)    Therefore, etc.

l)     QED.