Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookVIII.

Book8

1)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forifnotlet*.

c)    Nowsince*, and*, therefore*.

d)   But*, and*.

e)    Therefore*, whichisimpossible.

f)     Therefore*.

g)    Therefore*.

h)   QED.

2)   Tofind*.

a)    Let*, thusitisrequiredtofind*.

b)   Let*, let*, let*, furtherlet*, and*.

c)    Nowsince*, and*, therefore*.

d)   Againsince*, and*, therefore*.

e)    Therefore*.

f)     But*, thereforealso*.

g)    Andsince*, therefore*.

h)   But*, thereforealso*.

i)     Againsince*, therefore*.

j)     But*.

k)    Thereforealso*.

l)     Andsince*, therefore*.

m)  But*.

n)   Thereforealso*, therefore*.

o)    Isaynextthat*.

p)   Forsince*, and*, therefore*.

q)   And*, therefore*.

r)    Butif*, and*.

s)    Therefore*.

t)     Porism. Fromthisitismanifestthat*.

3)   If*, *.

a)    Let*, Isaythat*.

b)   Forlet*, then*, and*.

c)    Let*, andlet*.

d)   Nowsince*, *.

e)    Andsince*, *, therefore*.

f)     Andsince*, and*, therefore*.

g)    And*.

h)   Therefore*.

i)     QED.

4)   Given*, tofind*.

a)    Let*, thusitisrequiredtofind*.

b)   Let*.

c)    And*, and*.

d)   Now*.

e)    Firstlet*.

f)     And*.

g)    Nowsince*, therefore*.

h)   Forthesamereason*, andfurther*, therefore*.

i)     Isaynextthat*.

j)     Forif*, let*.

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

l)     Forthesamereason*, therefore*, therefore*.

m)  But*, therefore*, whichisimpossible.

n)   Therefore*.

o)    Nextlet*.

p)   Let*.

q)   Andlet*, and*.

r)    Since*, therefore*.

s)    But*, thereforealso*.

t)     Forthesamereasonalso*.

u)   Againsince*, therefore*, therefore*.

v)    Isaynextthat*.

w)  Forifnot*.

x)    Let*.

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

z)    Forthesamereason*, therefore*.

aa) Therefore*.

bb)                  But*, therefore*.

cc)  And*, therefore*.

dd)                  But*, therefore*.

ee) Therefore*.

ff)   But*, therefore*, whichisimpossible.

gg) Therefore*, therefore*.

hh)                  QED.

5)   *.

a)    Let*, andlet*, Isaythat*.

b)   For*, sothat*.

c)    Andlet*.

d)   Nowsince*, therefore*.

e)    But*, thereforealso*.

f)     Againsince*, andfurther*, therefore*.

g)    But*, thereforealso*.

h)   Butiwasalsoprovedthat*, therefore*.

i)     But*, therefore*.

j)     QED.

6)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Nowitismanifestthat*.

c)    Isaythen*.

d)   Forifpossiblelet*.

e)    And*.

f)     Nowsince*, and*, therefore*.

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

h)   Now*.

i)     And*, therefore*.

j)     Similarly*.

k)    QED.

7)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forif*, *.

c)    But*.

d)   Therefore*.

e)    QED.

8)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Nowsince*, and*, therefore*.

d)   But*, thereforealso*.

e)    But*, and*.

f)     Therefore*.

g)    Next*, let*, therefore*.

h)   Therefore*.

i)     But*, therefore*.

j)     Therefore*.

k)    QED.

9)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, and*.

c)    Let*, andlet*.

d)   Itisnowmanifestthat*.

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

f)     Nowsince*, therefore*.

g)    But*, therefore*.

h)   Therefore*.

i)     Againsince*, therefore*.

j)     But*, therefore*.

k)    Therefore*.

l)     Butitwasalsoprovedthat*, thereforealso*.

m)  But*, therefore*.

n)   Forthesamereasonalso*,.

o)    Therefore*.

p)   QED.

10)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*.

c)    Nowsince*,therefore*.

d)   But*, therefore*, therefore*.

e)    Againsince*, therefore*.

f)     But*, therefore*, therefore*.

g)    Forthesamereasonalso*.

h)   Andsince*, therefore*.

i)     Againsince*, therefore*.

j)     But*, thereforealso*.

k)    Againsince*, therefore*.

l)     But*, thereforealso*.

m)  Furthersince*, therefore*.

n)   But*, thereforealso*.

o)    Butitwasalsoprovedthat*, thereforealso*.

p)   Therefore*.

q)   Therefore*.

r)    QED.

11)         Between*, *, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*.

c)    Nowsince*, therefore*.

d)   Forthesamereasonalso*.

e)    Sincethen*, therefore*.

f)     Forthesamereasonalso*.

g)    Therefore*.

h)   Therefore*.

i)     Isaynextthat*.

j)     Forsince*, therefore*.

k)    But*.

l)     Therefore*.

m)  QED.

12)         Between*, *.

a)    Let*, andlet*, Isaythat*.

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

c)    Nowsince*, and*, therefore*.

d)   Forthesamereasonalso*.

e)    Andsince*, therefore*.

f)     But*.

g)    Thereforealso*.

h)   Againsince*, therefore*.

i)     Againsince*, therefore*.

j)     But*, thereforealso*.

k)    Therefore*.

l)     Isaynextthat*.

m)  Forsince*, therefore*.

n)   But*, thereforealso*.

o)    QED.

13)         If*, *, andif*, *.

a)    Let*, let*, andlet*, Isaythat*.

b)   Forlet*, andlet*.

c)    Andagainlet*, andlet*.

d)   Theninthemannersimilartotheforegoing*, andfurther*.

e)    And*, therefore* and*.

f)     QED.

14)         If*, *, andif*, *.

a)    Let*, let*, andlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Andsince*, and*, therefore*.

d)   And*, thereforealso*.

e)    Againlet*, Isaythat*.

f)     For*.

g)    Andsince*, and*, therefore*.

h)   And*, therefore*.

i)     Therefore, etc.

j)     QED.

15)         If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forlet*, andlet*, furtherlet*, andlet*.

c)    Nowitismanifestthat*.

d)   Andsince*, and*, therefore*.

e)    And*, therefore*.

f)     Nextlet*, Isaythat*.

g)    Forwiththesameconstructionwecanproveinasimilarmannetthat*.

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

i)     QED.

16)         If*, *, andif*, *.

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

b)   Forif*, *.

c)    But*, therefore*.

d)   Againlet*, Isaythat*.

e)    Forif*, *.

f)     But*, therefore*.

g)    QED.

17)         If*, *, andif*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forif*, *.

c)    But*, therefore*.

d)   Againlet*, Isaythat*.

e)    Forif*, *.

f)     But*, therefore*.

g)    QED.

18)         Between*, *, and*.

a)    Let*, andlet*.

b)   Nowsince*, therefore*.

c)    Isaythenthat*.

d)   Nowsince*, therefore*.

e)    Andsince*, therefore*.

f)     Forthesamereason*.

g)    Nowlet*.

h)   Thensince*, and*, therefore*.

i)     But*, thereforealso*.

j)     Againsince*, and*, therefore*.

k)    Butiwasalsoprovedthat*, thereforealso*.

l)     Therefore*.

m)  Therefore*.

n)   Isaynextthat*

o)    For*, *.

p)   And*, and*.

q)   Therefore*.

r)    QED.

19)         Between*, *.

a)    Let*, andlet*.

b)   Nowsince*, therefore*, and*.

c)    Isaythat*.

d)   Forlet*, andlet*.

e)    Nowsince*, and*, and*, *, therefore*.

f)     Let*.

g)    Therefore*.

h)   Nowsince*, and*, therefore*.

i)     But*.

j)     Therefore*.

k)    Andsince*, alternatelytherefore*.

l)     Forthesamereasonalso*.

m)  Therefore*.

n)   Nextlet*.

o)    Nowsince*, therefore*.

p)   But*, therefore*.

q)   Forthesamereasonalso*.

r)    Nowsince*, therefore*.

s)    But*, thereforealso*.

t)     Againsince*, therefore*.

u)   But*, thereforealso*.

v)    AGainsince*, andfurtheralso*, therefore*.

w)  But*, and*.

x)    Thereforealso*.

y)    Therefore*.

z)    Isaythat*.

aa) Forsince*, therefore*.

bb)                  But*, soitwasprovedthat*.

cc)  Therefore*.

dd)                  QED.

20)         If*, *.

a)    Forlet*, Isaythat*.

b)   Let*, therefore*.

c)    Now*, therefore*, sothat*.

d)   Againsince*, therefore*.

e)    Let*, therefore*, therefore*.

f)     Therefore*.

g)    Therefore*.

h)   Isaynextthat*.

i)     Forsince*, and*, therefore*.

j)     Againsince*, therefore*.

k)    But*, thereforealso*.

l)     Andalternately*.

m)  Therefore*.

n)   QED.

21)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, therefore*.

c)    Nowsince*, therefore*.

d)   Letthen*.

e)    Thereforeitismanifestfromthetheorembeforethisthat*.

f)     Nowsince*, and*, therefore*.

g)    But*, and*, therefore*.

h)   Now*.

i)     Therefore*.

j)     But*, therefore*.

k)    Therefore*.

l)     Againsince*, therefore*.

m)  Now*.

n)   Therefore*, therefore*.

o)    But*, therefore*.

p)   Therefore*, and*, therefore*.

q)   Isaythat*.

r)    Forsince*, therefore*.

s)    But*, thereforealso*.

t)     And*.

u)   Therefore*.

v)    QED.

22)         If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, *, therefore*.

c)    But*, therefore*.

d)   QED.

23)         If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    But*, therefore*.

d)   QED.

24)         If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forsince*, *.

c)    Therefore*.

d)   And*, therefore*.

e)    And*, therefore*.

f)     QED.

25)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, *.

c)    Therefore*.

d)   And*, sothat*.

e)    Let*.

f)     Sincethen*, and*, therefore*.

g)    QED.

26)         *.

a)    Let*, Isaythat*.

b)   Forsince*, therefore*.

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

d)   Andsince*, and*, therefore*.

e)    QED.

27)         *.

a)    Let*, Isaythat*.

b)   Forsince*, therefore*.

c)    Let*, andlet*, therefore*.

d)   And*, therefore*.

e)    QED.