Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookX.II.

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1)   *.

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

b)   Forlet*, therefore*.

c)    Let*.

d)   Therefore*.

e)    Nowsince*, therefore*, therefore*, sothat*.

f)     And*, therefore*.

g)    And*, therefore*.

h)   Againsince*, therefore*.

i)     And*, therefore*.

j)     But*, therefore*.

k)    And*, therefore*, therefore*.

l)     Itisnexttobeprovedthat*.

m)  Since*, therefore*.

n)   Therefore*, therefore*.

o)    Andsince*, *, sothat*.

p)   Andsince*, therefore*, sothat*.

q)   And*, and*.

r)    Butif*, andif*, *, therefore*.

s)    And*, and*.

t)     Therefore*.

u)   QED.

2)   *.

a)    Let*, let*, andlet*, Isaythat*.

b)   Forlet*.

c)    Thensince*, therefore*, sothat*.

d)   Therefore*.

e)    And*, therefore*.

f)     Againsince*, *.

g)    And*, therefore*, therefore*.

h)   And*, therefore*, therefore*.

i)     Itisnexttobeprovedthat*.

j)     Forsince*, therefore*, sothat*.

k)    Andsince*, *, sothat*.

l)     And*, therefore*.

m)  And*.

n)   Therefore*.

o)    QED.

3)   *.

a)    Let*, let*, Isaythat*.

b)   Let*.

c)    Thensince*, therefore*, sothat*.

d)   And*, therefore*.

e)    And*, therefore*.

f)     Forthesamereason*.

g)    Andsince*, and*, therefore*.

h)   Hence*, sothat*.

i)     And*, therefore*.

j)     Itistobeprovedthat*.

k)    Inmannersimilartotheforegoingwemayconcludethat*.

l)     And*, therefore*.

m)  And*.

n)   Therefore*.

o)    QED.

4)   *.

a)    Let*, let*, andlet*, Isaythat*.

b)   let*.

c)    Thensince*.

d)   Sincethen*, therefore*, therefore*.

e)    Againsince*, and*, therefore*, therefore*.

f)     Therefore*, therefore*.

g)    Itistobeprovedthat*.

h)   Inmannersimilartotheforegoingwecanprovethat*.

i)     Sincethen*, therefore*, sothat*.

j)     Butif*, andif*, *, therefore*.

k)    And*, and*.

l)     Therefore*.

m)  QED.

5)   *.

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

b)   Let*.

c)    Sincethen*, therefore*, but*.

d)   Sincethen*, therefore*, sothat*.

e)    Againsince*, therefore*.

f)     Therefore*, therefore*, therefore*.

g)    Isaynextthat*.

h)   Foritcanbeprovedsimilarlythat*, andthat*, therefore*.

i)     And.

j)     Therefore*.

k)    QED.

6)   *.

a)    Let*, let*, andlet*, Isaythat*.

b)   Forlet*.

c)    Thensince*, therefore*, andmoreover*, sothat*.

d)   And*, therefore*.

e)    Andsince*, therefore*.

f)     Therefore*, therefore*, therefore*.

g)    Isaynextthat*.

h)   Similarlyagainwecanprovethat*, andthat*, andforthesamereason*.

i)     And*.

j)     Therefore*.

k)    QED.

7)   *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    Let*, thereforealso*.

d)   But*, therefore*.

e)    And*, therefore*.

f)     And*.

g)    Thereforealternately*.

h)   But*, therefore*.

i)     And*, therefore*.

j)     Isaynextthat*.

k)    For*.

l)     Ifthen*, *.

m)  Andif*, *, andforthisreason*.

n)   Butif*, *, andforthisreasonagain*, for*.

o)    Butif*, *, and*.

p)   Butif*, *.

q)   Andif*, *, and*.

r)    Butif*, *, and*.

s)    Butif*, *, and*.

t)     Hence*.

u)   QED.

8)   *.

a)    Let*, Isaythat*.

b)   Forsince*, let*, therefore*.

c)    Andlet*, therefore*.

d)   But*, therefore*.

e)    But*, therefore*.

f)     Andsince*, and*, *.

g)    But*, therefore*.

h)   Isaynextthat*.

i)     Forsince*, thereforealso*, thereforealternately*.

j)     But*, therefore*.

k)    Iftherefore8, *, andforthisreason*, butif*, *, and*.

l)     Andforthisreason*.

m)  QED.

9)   *.

a)    Let*, Isaythat*.

b)   Let*, therefore*.

c)    Let*.

d)   Thensince*, thereforealso*.

e)    But*, therefore*.

f)     Andsince*, alternatelyalso*, thereforealso*, thereforealso*.

g)    Similarlywecanprovethat*.

h)   Therforealso*, thereforealso*.

i)     But*, therefore*.

j)     And*, therefore*.

k)    Simlarlyalso*.

l)     And*, therefore*.

m)  therefore*, but*, therefore*.

n)   Therefore*.

o)    QED.

10)         *.

a)    Let*, andlet*, itistobeprovedthat*.

b)   Let*, therefore*, but*.

c)    Let*.

d)   Wecanthenprovesimilarlythat*, and* and*, sothat*.

e)    Therefore*.

f)     QED.

11)         *.

a)    Let*, itistobeprovedthat*.

b)   Forsince*, let*, therefore*, andfurthermore*.

c)    Let*.

d)   Wecanthenprovesimilarlythat*, and*, sothat*, *, andmoreover*.

e)    Therefore*.

f)     QED.

12)         *.

a)    Let*, and*, Isaythat*.

b)   For*.

c)    Firstlet*, let*, let*, andlet*.

d)   Thensince*, therefore*.

e)    And*, therefore*.

f)     Againsince*, therefore*.

g)    And*, therefore*.

h)   Andsince*, while*, therefore*, sothat*.

i)     But*, therefore*.

j)     And*, therefore*, therefore*.

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

l)     *.

m)  Firstlet*.

n)   Now*, therefore*.

o)    But*, andif*, *.

p)   Therefore*, sothat*.

q)   Nextlet*.

r)    Now*, therefore*.

s)    But*, andif*, *.

t)     Therefore*, sothat*.

u)   Nextlet*, therefore*, sothat*.

v)    Now*.

w)  Firstlet*.

x)    Now*, therefore*.

y)    But*, andif*, *, therefore*, sothat*.

z)    Nextlet*.

aa) Now*, therefore*.

bb)                  But*, andif*, *.

cc)  Therefore*, sothat*.

dd)                  Therefore, etc.

ee) QED.

13)         If*, *.

a)    Forlet*, Isaythat*.

b)   For*.

c)    Firstif*, let*.

d)   Let*, andlet*, and*.

e)    Nowsince*, therefore*.

f)     And*, therefore*.

g)    Andsince*, and*, therefore*.

h)   But*, therefore*.

i)     Therefore*, therefore*.

j)     But*.

k)    Firstlet*.

l)     Now*, therefore*.

m)  But*, andif*, *, therefore*.

n)   Nextlet*.

o)    Now*, therefore*.

p)   Butif*, *, sothat*.

q)   Therefore, etc.

r)    QED.

s)    *.

t)     Forif*, *.

u)   Butif*, *.

v)    If*, *.

w)  If*, *.

x)    If*, *.

y)    If*, *.

z)    If*, *.

aa) If*, *.

bb)                  If*, *.

cc)  And*, sothat*.

14)         If*, *, andletitbecalled an apotome.

a)    Forlet*, let*, Isaythat*.

b)   Forsince*, and*, therefore*.

c)    But*, and*.

d)   And*, therefore*.

e)    But*, therefore*.

f)     Andletitbecalled*.

g)    QED.

15)         If*, *, andletitbecalled a firstapotomeofamedialstraightline.

a)    Forlet*, Isaythat*.

b)   Forsince*, *.

c)    But*, therefore*, therefore*, sinceif*, *.

d)   But*, therefore*, therefore*.

e)    Andletitbecalled*.

f)     QED.

16)         If*, *, andletitbecalled a secondapotomeofamedialstraightline.

a)    Forlet*, Isaythat*.

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

c)    Nowsince*, therefore*.

d)   And*, therefore*.

e)    Againsince*, therefore*.

f)     And*, therefore*.

g)    And*, therefore*.

h)   Andsince*, therefore*, therefore*.

i)     But*, and*, therefore*.

j)     But*, and*, therefore*.

k)    But*, therefore*.

l)     And*, therefore*, therefore*.

m)  But*, and*, and*.

n)   And*, therefore*.

o)    Andletitbecalled*.

p)   QED.

17)         If*, *, andletitbecalled minor.

a)    Forlet*.

b)   Isaythat*.

c)    Forsince*, while*, therefore*, and*.

d)   But*, therefore*, therefore*.

e)    Andletitbecalled*.

f)     QED.

18)         If*, *, andletitbecalled thatwhichproduceswitharationalareaamedialwhole.

a)    Forlet*, Isaythat*.

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

c)    And*, therefore*, therefore*.

d)   Andletitbecalled*.

e)    QED.

19)         If*, *, andfurther*, andletitbecalled thatwhichproduceswithamedialwhole.

a)    Forlet*, Isaythat*.

b)   Forlet*, let*, andlet*.

c)    Therefore8, sothat*.

d)   Nowsince*, therefore*.

e)    And*, therefore*.

f)     Againsince*, therefore*.

g)    And*, therefore*.

h)   Andsince*, therefore*.

i)     But*, therefore*.

j)     And*, therefore*.

k)    Therefore*.

l)     And*, but*, and*.

m)  And*, therefore*.

n)   Andletitbecalled*.

20)         To*, *.

a)    Let*, therefore*.

b)   Isaythat*.

c)    Forifpossiblelet*, therefore*.

d)   Nowsince*, *, for*, therefore*.

e)    But*, therefore*, whichisimpossiblefor*, and*.

f)     Therefore*.

g)    Therefore*.

h)   QED.

21)         To*, *.

a)    Forlet*, therefore*, Isaythat*.

b)   Forifpossiblelet*, therefore*.

c)    Nowsince*, for*, thereforealternately*.

d)   But*, for*.

e)    Therefore*, whichisimpossible*, for*, and*.

f)     Therefore, etc.

g)    QED.

22)         To*, *.

a)    Let*, therefore*.

b)   Isaythat*.

c)    Forifpossiblelet*, therefore*.

d)   Let*, let*, andlet*, therefore*, sothat*.

e)    Againlet*.

f)     But*, therefore*.

g)    Nowsince*, therefore*.

h)   And*, therefore*.

i)     And*, therefore*.

j)     Againsince*, *.

k)    And*, therefore*.

l)     And*, therefore*.

m)  Andsince*, therefore*.

n)   But*, therefore*.

o)    But*, therefore*.

p)   And*, while*, therefore*.

q)   But*, therefore*.

r)    And*, therefore*, therefore*.

s)    Similarlywecanprovethat*, therefore*, whichisimpossible.

t)     Therefore, etc.

u)   QED.

23)         To*, *.

a)    Let*, therefore*.

b)   Isaythat*.

c)    Forifpossiblelet*, therefore*.

d)   Nowsince*, while*, therefore*, whichisimpossiblefor*.

e)    Therefore*.

f)     QED.

24)         To*, *.

a)    Let*, andlet*, therefore*.

b)   Isaythat*.

c)    Forifpossiblelet*, therefore*.

d)   Sincethen*, while*, for*, therefore*, whichisimpossiblefor*.

e)    Therefore*, therefore*.

f)     QED.

25)         To*, *.

a)    Let*, therefore*.

b)   Isaythat*.

c)    Forifpossiblelet*, sothat*.

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

e)    Againlet*.

f)     But*, therefore*.

g)    Nowsince*, therefore8.

h)   And*, therefore*.

i)     Againsince*, therefore*.

j)     And*, therefore*.

k)    Andsince*, *, therefore*.

l)     And*, therefore*, therefore*.

m)  Similarlywecanprovethat*.

n)   Therefore*, whichwasprovedimpossible.

o)    Therefore*.

p)   Therefore*.

q)   QED.

26)         Tofind*.

a)    Let*, andlet*, therefore*.

b)   Let*, therefore*.

c)    Let*, therefore*.

d)   But*, therefore*, therefore*.

e)    Andsince*, therefore*, therefore*.

f)     And*, therefore*, therefore*.

g)    Isaynextthat*.

h)   Forlet*.

i)     Nowsince*, thereforealso*.

j)     But*, therefore*, therefore*.

k)    And*, therefore*.

l)     And*.

m)  Therefore*.

n)   Therefore*.

o)    Beingthatwhichitwasrequiredtofind.

27)         Tofind*.

a)    Let*, therefore*.

b)   Let*.

c)    Nowlet*.

d)   Therefore*.

e)    But*, therefore*, therefore*.

f)     Andsince*, *.

g)    And*, therefore*, therefore*.

h)   Isaynextthat*.

i)     Forlet*.

j)     Sincethen*, therefore*.

k)    And*, therefore*, therefore*.

l)     And*.

m)  And*.

n)   Therefore*.

o)    Therefore*.

p)   QED.

28)         Tofind*.

a)    Let*, let*, butlet*.

b)   Let*, and*.

c)    Sincethen*, therefore*.

d)   But*, therefore*, therefore*.

e)    Andsince*, therefore*, therefore*.

f)     Againsince*, therefore*.

g)    But*, therefore*, therefore*.

h)   Andsince*, therefore*, therefore*.

i)     And*, therefore*, therefore*.

j)     Isaynextthat*.

k)    Forsince*, and*, therefore*.

l)     But*, therefore*, therefore*.

m)  Therefore*.

n)   Nowlet*.

o)    Sincethen*, therefore*.

p)   But*, therefore*.

q)   Therefore*, and*.

r)    And*, therefore*.

s)    Therefore*.

t)     QED.

29)         Tofind*.

a)    Let*, therefore*.

b)   Let*.

c)    Let*, therefore*.

d)   But*, therefore*, therefore*.

e)    Nowsince*, therefore*, therefore*.

f)     And*, therefore*, therefore*.

g)    Nowlet*.

h)   Sincethen*, thereforealso*.

i)     But*, therefore*, therefore*.

j)     And*, therefore*.

k)    And*.

l)     Therefore*.

m)  Therefore*.

n)   QED.

30)         Tofind*.

a)    Let*, andlet*, therefore*.

b)   let*, andlet*,.

c)    Therefore*, therefore*.

d)   Nowsince*, *, while*, therefore*, therefore*.

e)    And*, therefore*, therefore*.

f)     Isaynextthat*.

g)    Forlet*.

h)   Sincethen*, therefore*.

i)     But*, therefore*, therefore*.

j)     And*, therefore*.

k)    And*, therefore*.

l)     Therefore*.

m)  QED.

31)         Tofind*.

a)    Let*, andfurtherlet*.

b)   Let*.

c)    Nowsince*, therefore*.

d)   But*, therefore*, therefore*.

e)    Andsince*, therefore*, therefore*.

f)     Againsince*, therefore*.

g)    But*, therefore*, therefore*.

h)   Andsince*, therefore*, therefore*.

i)     And*, therefore*, therefore*.

j)     Isaynextthat*.

k)    Forsince*, and*, therefore*.

l)     But*, therefore*, therefore*, therefore*.

m)  Nowlet*.

n)   Sincethen*, therefore*.

o)    But*, therefore*, therefore*.

p)   And*, therefore*.

q)   And*.

r)    Therefore*.

s)    Therefore*.

t)     QED.

32)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    And*, iftherefore*, *.

d)   Let*, let*, andlet*, therefore*.

e)    Andthrough*, let*.

f)     Nowsince*, therefore*.

g)    But*, therefore*.

h)   And*, therefore*, sothat*.

i)     Nowsince*, therefore*.

j)     But*, therefore*, therefore*.

k)    Nowlet*, therefore*.

l)     Let*, andlet*.

m)  Sincethen*, therefore*.

n)   But*, and*, therefore*.

o)    But*aswasbeforeproved, and*, therefore*.

p)   But*, therefore*.

q)   But*, therefore*.

r)    But*, therefore*, therefore*.

s)    Isaynextthat*.

t)     Forsince*, and*, therefore*, therefore*.

u)   Againsince*, therefore*.

v)    Sincethen*, while*, therefore*.

w)  But*, therefore*.

x)    And*, therefore*, therefore*.

y)    And*, therefore*.

z)    Therefore, etc.

aa) QED.

33)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, therefore*, and*, while*.

c)    Sincethen*, thereforeif*, *.

d)   Letthen*, let*, andlet*, therefore*.

e)    Therefore*.

f)     But*, therefore*, therefore*.

g)    Againsince*, therefore*.

h)   But*.

i)     Therefore*.

j)     Let*, andlet*, therefore*.

k)    Let*.

l)     Sincethen*, *, therefore*.

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

n)   But*, and*, therefore*.

o)    But*, and*, therefore*.

p)   Sincethen*, and*, therefore*.

q)   But*, therefore*, therefore*.

r)    Isaythat*.

s)    Forsince*, therefore8.

t)     But*, therefore*.

u)   But*, therefore*.

v)    Therefore*, therefore*.

w)  And*.

x)    Therefore*.

y)    QED.

34)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, therefore*, and*, while*.

c)    Sincethen*, thereforeif*, *.

d)   Let*, let*, andlet*.

e)    Let*.

f)     Therefore*, therefore*.

g)    Andsince*, therefore*.

h)   But*, sothat*.

i)     Therefore*.

j)     Againsince*, therefore*.

k)    But*, therefore*, therefore*.

l)     Andsince*, therefore*.

m)  But*, therefore*.

n)   But*, therefore*.

o)    Nowlet*, andlet*, therefore*.

p)   Let*, andlet*.

q)   Nowsince*, therefore*.

r)    But*, and*, thereforealso*, therefore*.

s)    But*, and*, therefore*.

t)     But*, and*, therefore*.

u)   But*, therefore*, therefore*.

v)    Isaythat*.

w)  Forsince*, therefore*, therefore*.

x)    Andsince*, therefore*.

y)    Againsince*, therefore*, sothat*, therefore*.

z)    Isaynextthat*.

aa) Forsince*, therefore*, sothat*.

bb)                  Therefore*.

cc)  Therefore*.

dd)                  QED.

35)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, therefore*, and*.

c)    Sincethen*, thereforeif*, *.

d)   Letthen*, let*, andlet*, therefore*.

e)    Let*.

f)     Sincethen*, therefore*.

g)    Againsince*, therefore*.

h)   Againsince*, therefore*.

i)     Nowlet*, andlet*.

j)     Therefore*.

k)    Let*, andlet*.

l)     Sincethen*, therefore*.

m)  But*, as*, therefore*.

n)   But*, and*, therefore*.

o)    But*, therefore*.

p)   Sincethen*, and*, therefore*, therefore*.

q)   Isaythat*.

r)    Forsince*, therefore*.

s)    Againsince*, and*, therefore*.

t)     Andsince*, therefore*.

u)   Therefore*, but*.

v)    Therefore*, and*.

w)  Therefore*.

x)    QED.

36)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Thereforeif*, *.

d)   Let*, let*, therefore*.

e)    Nowsince*, therefore*.

f)     Againsince*, *.

g)    Nowlet*, andlet*, therefore*.

h)   Let*, andlet*.

i)     Similarlythenwecanprovethat*.

j)     Isaythat*.

k)    Forsince*, therefore*.

l)     Againsince*, *.

m)  Andsince*, therefore*, therefore*.

n)   Therefore*, and*.

o)    Therefore*.

p)   QED.

37)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, therefore*, and*.

c)    Sincethen*, thereforeif*, *.

d)   Let*, let*, andlet*, therefore*.

e)    But*, therefore*.

f)     Andsince*, therefore*.

g)    Againsince*, therefore*.

h)   Nowsince*, therefore*.

i)     But*, therefore*.

j)     Nowlet*, andlet*, therefore*.

k)    Let*, andlet*.

l)     Theninmannersimilartotheabovewecanprovethat*.

m)  Isaythat*.

n)   Forsince*, therefore*.

o)    Againsince*, *.

p)   Andsince*, *.

q)   Andsince*, therefore*, therefore*, andfurther*.

r)    Therefore*, and*.

s)    Therefore*.

t)     QED.

38)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Let*.

d)   Therefore*, therefore*.

e)    Let*, andlet*, therefore*.

f)     Nowsince*, and*, therefore*.

g)    And*, therefore*.

h)   Againsince*, therefore*.

i)     And*, therefore*.

j)     Andsince*, while*, therefore*.

k)    And*, and*, therefore*.

l)     But*, therefore*.

m)  And*, therefore*, therefore*.

n)   Isaynextthat*.

o)    Forsince*, and*, and*, therefore*, therefore*.

p)   But*, and*, therefore*.

q)   Andsince*, *.

r)    But*, therefore*.

s)    Sincethen*, and*, while*, therefore*.

t)     And*, therefore*.

u)   Therefore, etc.

v)    QED.

39)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Let*, therefore*, therefore*.

d)   And*, therefore*.

e)    Nowsince*, and*, therefore*.

f)     But*, therefore*.

g)    And*, therefore*.

h)   Nowsince*, while*, therefore*.

i)     But*, therefore*.

j)     And*, therefore*, therefore*.

k)    Isaynextthat*.

l)     Forlet*, andlet*, therefore*.

m)  Nowsince*, and*, and*, therefore*, therefore*.

n)   But*, and*, therefore*, therefore*.

o)    Sincethen*, therefore*.

p)   And*, therefore*.

q)   Therefore, etc.

r)    QED.

40)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Let*, andlet*, therefore*, therefore*.

d)   And*, therefore*.

e)    Nowsince*, and*, therefore*.

f)     Let*, andlet*, therefore*.

g)    Let*, andlet*, therefore*.

h)   But*, therefore*.

i)     And*, therefore*.

j)     Andsince*, therefore*, therefore*.

k)    But*, and*, therefore*.

l)     But*, and*, therefore*.

m)  But*, therefore*.

n)   And*, therefore*, therefore*.

o)    Isaynextthat*.

p)   Forsince*, therefore*, sothat*.

q)   Andsince*, and*, and*, therefore*, therefore*.

r)    But*, and*, therefore*, therefore*.

s)    Sincethen*, therefore*.

t)     And*, therefore*.

u)   Therefore, etc.

v)    QED.

41)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Let*, and*, therefore*.

d)   And*, therefore*.

e)    And*, therefore*.

f)     Andsince*, and*, therefore*.

g)    Let*, andlet*, therefore*.

h)   Andsince*, therefore*.

i)     And*, therefore*.

j)     Andsince*, while*, *.

k)    But*, and*, therefore*.

l)     But*, therefore*.

m)  And*, therefore*, therefore*.

n)   Isaythat*.

o)    Forsince*, therefore*.

p)   And*, and*, therefore*.

q)   But*, therefore*.

r)    Andsince*, and*, and*, therefore*, therefore*.

s)    But*, and*, therefore*, therefore*.

t)     Sincethen*, therefore*.

u)   And*, therefore*.

v)    Therefore, etc.

w)  QED.

42)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Let*, therefore*.

d)   But*, therefore*.

e)    And*, therefore*.

f)     Andsince*, and*, therefore*.

g)    Let*, andlet*, therefore*.

h)   Andsince*, therefore*.

i)     And*, therefore*.

j)     Nowsince*, therefore*.

k)    But*, therefore*.

l)     And*, therefore*, therefore*.

m)  Isaynextthat*.

n)   Forwecanprovesimilarlythat*.

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

p)   But*, therefore*.

q)   Sincethen*, and*, therefore*.

r)    And*, therefore*.

s)    QED.

43)         If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Nowlet*, and*, therefore*, therefore*.

d)   And*, therefore*.

e)    Sincenow*, and*, therefore*.

f)     And*, therefore*.

g)    And*, therefore*.

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

i)     But*, therefore*.

j)     And*.

k)    Therefore*.

l)     Isaynextthat*.

m)  Forsince*, let*, andlet*, therefore*.

n)   Andsince*, therefore*.

o)    But*, and*, therefore*.

p)   But*, therefore*.

q)   Andsince*, and*, and*, therefore*, therefore*.

r)    Andforthesamereasonasbefore*.

s)    And*, therefore*.

t)     QED.

44)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, let*, therefore*.

c)    Let*, thereforealso*, thereforealso*.

d)   But*.

e)    Therefore*.

f)     And*, therefore*.

g)    Nowsince*, alternatelytherefore*.

h)   And*.

i)     Ifthen*, *.

j)     Andif*, if*, andif*, then*.

k)    Butif*, *.

l)     Andif*, if*, andif*, *.

m)  Therefore*.

n)   QED.

45)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, let*.

c)    Therefore*.

d)   Let*, therefore*.

e)    But*, therefore*, therefore*.

f)     Isaynextthat*.

g)    Since*, thereforealso*.

h)   But*, therefore*.

i)     Thereforeif*, *, andif*, *.

j)     Therefore*.

k)    QED.

46)         *.

a)    Let*, Isaythat*.

b)   Let*, thensince*, therefore*.

c)    Nowsince*, thereforealso*.

d)   Therefore*.

e)    But*, therefore*.

f)     But*, therefore*.

g)    Againsince*, while*, therefore*.

h)   But*, therefore*, therefore*.

i)     Therefore*.

j)     QED.

47)         *.

a)    Let*, Isaythat*.

b)   Forlet*, therefore*.

c)    Let*.

d)   Thenwecanproveinmannersimilartotheforegoingthat*, and*, sothat*.

e)    Therefore*.

f)     QED.

48)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, andlet*, therefore*.

c)    Nowaswasproved*, therefore*.

d)   Therefore*.

e)    QED.

49)         If*, *.

a)    Forlet*, Isaythat*.

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

c)    Sincethen*, while*, therefore*.

d)   And*, therefore*, while*, therefore*.

e)    Therefore*, therefore*.

f)     Now*.

g)    Firstlet*.

h)   Now*, therefore*.

i)     But*.

j)     Therefore*.

k)    Butif*, while* *.

l)     But*.

m)  QED.

50)         If*, *.

a)    Forfrom*, let*.

b)   Isaythat*.

c)    Forlet*, andlet*.

d)   Itfollowsthenthat*, while*, therefore*, therefore*.

e)    Now*.

f)     Ifthen*, while*, *.

g)    But*, sothat*.

h)   Butif*, while*, *, sothat*.

i)     QED.

51)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, and*, itfollowsthat*.

c)    Andsince*, therefore*, therefore*.

d)   Ifthen*, while*, *.

e)    But*, and*, and*, sothat*.

f)     Butif*, while*, *.

g)    But*.

h)   Therefore*.

i)     QED.

52)         *. *.

a)    Let*, Isaythat*.

b)   Forifpossiblelet*, let*.

c)    thensince*, *.

d)   Let*, therefore*, *, and*.

e)    Againsince*, therefore*.

f)     Let*, therefore*, and*.

g)    Therefore*, therefore*.

h)   But*, therefore*.

i)     Therefore*, therefore*.

j)     But*, whichisimpossible.

k)    Therefore*.

l)     QED.

m)  *.

n)   For*if*, *, whileif*, *, if*, *, if*, *, if*, *, *if*, andif*, *.

o)    Sincethen*, andfrom*, andfrom*, itisclearthat*.

p)   Andsince*, butif*, * therefore*, sothat*.

q)   Medial, Binomial, Firstbimedial, Secondbimedial, Major, "Side"ofarationalplusamedialarea, "Side"ofthesumoftwomedialareas, Apotome, Firstapotomeofamedialstraightline, Secondapotomeofamedialstraightline, Minor, Producingwitharationalareaamedialwhole, Producingwithamedialareaamedialwhole.

53)         *, andfurther*.

a)    Let*, let*, let*, Isaythat*.

b)   Foragainlet*.

c)    Sincethen*, therefore*.

d)   But*, therefore*.

e)    Let*, therefore*, therefore*.

f)     Let*, therefore*, for*.

g)    But*, thereforealso*.

h)   But* therefore*.

i)     And*, since*.

j)     Therefore*, sothat*.

k)    Nowsince*, while*, therefore*.

l)     And*, therefore*, sothat*.

m)  Sincethen*, while*, therefore*.

n)   But*, therefore*.

o)    Therefore*, therefore*.

p)   Now*.

q)   Ifthen*, *.

r)    Andif*, *, if*, *, butif*, *.

s)    Butif*, *.

t)     Andif*, *, if*, *, butif*, *, sothat*.

u)   QED.

54)         If*, andfurther*.

a)    Let*, sothat*, Isaythat*.

b)   Forlet*, therefore*.

c)    Let*.

d)   But*, therefore*.

e)    And*, therefore*.

f)     But*, therefore*.

g)    Let*, therefore*.

h)   Andsince*, therefore*.

i)     Let*, thereforealso*.

j)     But*, therefore*.

k)    Andsince*, while*, thereforealso*, sothatalso*, thereforealso*.

l)     But*, for*, therefore*, sothat*.

m)  But*, therefore*.

n)   Andsince*, alternately*.

o)    But*, therefore*.

p)   But*, therefore*, therefore*.

q)   Ifnow*, *.

r)    Andif*, *, if*, *, butif*, *.

s)    Butif*, *.

t)     Andif*, *, if*, *, butif*, *.

u)   Therefore*, andfurther*.

v)    QED.

55)         If*, *.

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

b)   Forlet*, andlet*.

c)    Therefore*.

d)   Let*.

e)    But*, therefore*.

f)     Thereforealternately*, thereforealso*.

g)    But*, therefore*.

h)   And*, therefore*.

i)     But*, therefore*.

j)     But*, therefore*.

k)    But*, therefore*, therefore*.

l)     And*.

m)  Therefore, etc.

n)   QED.

o)    Porism. Anditismademanifesttousbythisalsothat*.

56)         From*, *.

a)    Let*, Isaythat8.

b)   Let*, therefore*, for*.

c)    And*, for*.

d)   Againlet*, therefore*.

e)    Therefore*, and*if*.

f)     Similarlyif*, itismanifestthat*.

g)    QED.