Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookXII.

Book12.

1)   *.

a)    Let*, let*, Isaythat*.

b)   Forlet*.

c)    Nowsince*, *, and*.

d)   Thus*, therefore*.

e)    Therefore*.

f)     But*, for*, and*, therefore*.

g)    But*, therefore*.

h)   Therefore*.

i)     Therefore*.

j)     But*, and*, thereforealso*.

k)    Therefore, etc.

l)     QED.

2)   *.

a)    Let*, Isaythat*.

b)   Forif*, then*.

c)    Firstlet*.

d)   Let*, hence*.

e)    Let*, therefore*, while*, hence*.

f)     Thus*.

g)    Foritwasprovedinthefirsttheoremofthetenthbookthatif*, andif*, *.

h)   Let*, andlet*.

i)     Therefore*.

j)     Let*, therefore*.

k)    But*, thereforealso*, thereforealternately*.

l)     But*, therefore*.

m)  But*, whichisimpossible.

n)   Therefore*.

o)    Similarlywecanprovethat*.

p)   Isaynextthat*.

q)   Forifpossiblelet*.

r)    Thereforeinversetly*.

s)    But*, thereforealso*, whichwasprovedimpossible.

t)     Therefore*.

u)   Anditwasprovedthat*, therefore*.

v)    Therefore, etc.

w)  QED.

x)    Lemma. Isaythat*.

y)    Forlet*.

z)    Isaythat*.

aa) Forsince*, *.

bb)                  But*, therefore*.

cc)  Hence*.

dd)                  QED.

3)   * and *.

a)    Let*, Isaythat*.

b)   Forlet*.

c)    Since*, therefore*.

d)   Forthesamereason*.

e)    Therefore*, therefore*.

f)     But*, therefore*.

g)    But*, therefore*, and*, therefore*.

h)   Therefore*.

i)     Forthesamereason*.

j)     Nowsince*, *.

k)    Therefore*.

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

m)  Forthesamereason*.

n)   Therefore*.

o)    Andsince*, *, and*, therefore*.

p)   Forthesamereason*, and*.

q)   Nowsince*, *.

r)    Therefore*.

s)    And*, therefore*.

t)     Thereforealso*.

u)   But*.

v)    Therefore*.

w)  Nextsince*, *.

x)    Andsinceif*, andif*, *, therefore*.

y)    Anditismanifestthat*, inasmuchasif*, *.

z)    But*, for*.

aa) Hence*.

bb)                  But*, and*.

cc)  Therefore*.

dd)                  Therefore*.

ee) QED.

4)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, therefore*, and*.

c)    Forthesamereason*.

d)   Andsince*, therefore*.

e)    And*, therefore*, thereforealternately*.

f)     But*, thereforealso*.

g)    But*.

h)   Thereforealso*.

i)     Thereforealso*.

j)     Thereforealso*.

k)    Andsimilarlyif*, *.

l)     But*.

m)  Thereforealso*.

n)   Andsimilarlyalsoif*, *.

o)    QED.

p)   Lemma. Butthat*, wemustproveasfollows.

q)   Forlet*, *.

r)    Nowsince*, *.

s)    And*, therefore*.

t)     Forthesamereason*.

u)   And*, therefore*.

v)    Therefore*.

w)  Hence*, therefore*.

x)    QED.

5)   *.

a)    Let*, Isaythat*.

b)   Forif*, *.

c)    Letitfirst*, then*.

d)   Againlet*, andlet*.

e)    Let*, andlet*, therefore*.

f)     Let*, therefore*.

g)    But*, thereforealso*, therefore*.

h)   But*, therefore*.

i)     But*, whichisimpossible.

j)     Therefore*.

k)    Similarlyitcanbeprovedthat*.

l)     Isaynextthat*.

m)  Forifpossiblelet*, thereforeinversely*.

n)   But*, thereforealso*, whichwasprovedabsurd.

o)    Therefore*.

p)   Butitwasprovedthat*.

q)   Therefore*.

r)    QED.

6)   *.

a)    Let*, Isaythat*.

b)   Forlet*.

c)    Sincethen*, *, therefore*.

d)   And*.

e)    Butalso*.

f)     Therefore*.

g)    Andagain*.

h)   Similarlyalsoitcanbeprovedthat*.

i)     Andsince*, therefore*.

j)     But*.

k)    Thereforealso*.

l)     Butfurthere*.

m)  Thereforealso*.

n)   QED.

7)   *.

a)    LEt*, Isaythat*.

b)   Forlet*.

c)    Since*, therefore*, thereforealso*.

d)   But*, for*.

e)    Therefore*.

f)     Againsince*, and*, *.

g)    Thereforealso*.

h)   But*, thereforealso*, therefore*.

i)     Andsince*, for*, while*, thereforealso*.

j)     QED.

k)    Porism. Fromthisitismanifestthat*.

8)   *.

a)    Let*, Isaythat*.

b)   Forlet*.

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

d)   Andsince*, and*, therefore*.

e)    Forthesamereason*, therefore*.

f)     But*, and*.

g)    Therefore*.

h)   Therefore*.

i)     But*.

j)     Therefore*.

k)    But*.

l)     Therefore*.

m)  QED.

n)   Porism. Fromthisitismanifestthat*.

o)    Forif*, *, then*.

p)   But*, therefore*.

9)   In*, *, and*.

a)    Forlet*, Isaythat*.

b)   Forlet*.

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

d)   But*, therefore*.

e)    But*.

f)     Thereforealso*.

g)    But*, and*, therefore*.

h)   Therefore*.

i)     Nextlet*, Isaythat*.

j)     Forwiththesameconstructionsince*, while*, thereforealso*.

k)    But*, and*, therefore*.

l)     But*, therefore*.

m)  And*, therefore*.

n)   Therefore, etc.

o)    QED.

10)         *.

a)    Forlet*, Isaythat*.

b)   Forif*, *.

c)    Firstlet*, andlet*, then8.

d)   Let*.

e)    Then*, and*, while*, thereforealso*, and*, therefore*.

f)     Let*, then*, asweprovedbefore.

g)    Let*, then*, and*.

h)   Thus*, *.

i)     Let*, therefore*.

j)     But*, thereforealso*.

k)    But*, for*, whichisimpossible.

l)     Therefore*.

m)  Isaynextthat*, forifpossiblelet*, therefore*.

n)   Let*, therefore*.

o)    Nowlet*, therefore*, seetingthatasweprovedbeforeif*, andif*, *, for*.

p)   Hencealso*, thereforealso*.

q)   And*, for*.

r)    Therefore*.

s)    Let*, andlet*, thereforealso*.

t)     Now*, thereforealso*.

u)   Thus*.

v)    Let*, andlet*, therefore*.

w)  But*, therefore*.

x)    But*, for*, whichisimpossible.

y)    Therefore*.

z)    Butitwasprovedthat*, therefore*, hence*.

aa) Therefore, etc.

bb)                  QED.

11)         *.

a)    Let*, let*, Isaythat*.

b)   Forifnotthen*, *.

c)    Firstlet*, andlet*, therefore*.

d)   Let*, therefore*.

e)    Let*, therefore*, therefore*, inasmuchasif*, *, for*, while*.

f)     Let*.

g)    Therefore*.

h)   Let*, thereforealso*.

i)     Thus*, and*, *.

j)     Let*.

k)    LEt*, andlet*.

l)     Sincethen*, while*, thereforealso*.

m)  But*, and*.

n)   Thereforealso*, thereforealternately*.

o)    But*, therefore*.

p)   But*, whichisabsurd.

q)   Therefore*.

r)    Similarlywecanprovethat*.

s)    Isaynextthat*.

t)     Forifpossiblelet*, thereforeinversely*.

u)   But*, thereforealso*, whichwasprovedimpossible.

v)    Therefore*.

w)  Butitwasprovedthat*, therefore*.

x)    But*, for*.

y)    Thereforealso*.

z)    Therefore, etc.

aa) QED.

12)         *.

a)    Let*, let*, Isaythat*.

b)   Forif*, *.

c)    Firstlet*.

d)   Let*, therefore*.

e)    Nowlet*, therefore*.

f)     Let*, andlet*.

g)    Therefore*.

h)   Nowlet*, therefore*.

i)     Thus*, and*, *.

j)     Let*, andlet*.

k)    Let*, andlet*, *, and*, andlet*.

l)     Nowsince*, therefore*.

m)  But*, thereforealso*.

n)   Andalternately*.

o)    And*, therefore*.

p)   Againsince*, and*, inasmuchas*, sincethen*, therefore*.

q)   Againsinceitwasprovedthat*, while*, therefore*, and*, for*, therefore*.

r)    Andsince*, and*, therefore*.

s)    Againsince*, and*, therefore*.

t)     Butitwasalsoprovedthat*.

u)   Therefore*.

v)    Therefore*, therefore*, hence*.

w)  Therefore*, for8.

x)    But*.

y)    Therefore*.

z)    Similarly*, wecanprovethat*.

aa) And*, thereforealso*, hencealso*.

bb)                  Butbyhypothesis*, therefore*, thereforealternately*.

cc)  But*, for*.

dd)                  Therefore*.

ee) But*, whichisimpossible.

ff)   Therefore*.

gg) Similarlywecanprovethat*.

hh)                  Isaynextthat*.

ii)    Forifpossiblelet*.

jj)    Thereforeinversely*.

kk) But*.

ll)    Therefore*, whichwasprovedimpossible.

mm)               Therefore*.

nn)                  Butiwasprovedthat*.

oo) Therefore*.

pp)                  But*, for*, therefore*.

qq)                  Therefore, etc.

rr)  QED.

13)         If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forlet*, andlet*, and*, andlet*.

c)    Let*, andlet*.

d)   Thensince*, therefore*.

e)    But*, therefore*.

f)     Sincethen*, and*, and*, therefore*.

g)    Forthesamereason*.

h)   Andif*, if*, *, and*.

i)     Thus*, and*, andithasbeenprovedthatif*, *, *.

j)     Therefore*.

k)    QED.

14)         *.

a)    Forlet*, Isaythat*.

b)   Forlet*, let*, andlet*.

c)    Sincethen*, *.

d)   But*, therefore*.

e)    Andsince*, therefore*.

f)     But*, and*, therefore*.

g)    But*.

h)   Thereforealso*.

i)     QED.

15)         In*, *.

a)    Let*, let*, and*, let*.

b)   Isaythat*.

c)    For*.

d)   Firslet*.

e)    Now*.

f)     But*, therefore*.

g)    Hencealsoreciprocally*.

h)   Nextlet*, butlet*, let*, let*, andlet*.

i)     Nowsince*, therefore*.

j)     But*, for*, and*, for*.

k)    Thereforealso*.

l)     But*, therefore*.

m)  Therefore*.

n)   Next*, Isaythat*.

o)    Forwiththesameconstructionsince*, *, while*, therefore*.

p)   But*, and*, therefore*.

q)   Therefore*.

r)    Andthesameistruefor*.

s)    QED.

16)         Given*, toinscribe*.

a)    Let*, thusitisrequiredtoinscribe*.

b)   Forlet*, andlet*, therefore*.

c)    Then*.

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

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

f)     Ifthen*, *.

g)    QEF.

17)         Given*, toinscribe*.

a)    Let*, thusitisrequiredtoinscribe*.

b)   Let*, then*, inasmuchas*, hence*.

c)    Anditismanifestthat*, inasmuchas*.

d)   Letthen*, and*, let*, thenlet*, let*, let*, let*, andlet*.

e)    Let*, andlet*.

f)     Nowsince*, therefore*, hence*.

g)    Andsince*, for*, therefore*.

h)   Therefore*.

i)     Let*, andlet*, andlet*, *, inasmuchas*.

j)     Let*, andlet*.

k)    Nowsince*, and*, therefore*.

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

m)  Andsince*, therefore*.

n)   Butitwasalsoproved*, therefore*.

o)    Andsince*, while*, therefore*.

p)   And*, therefore*, inasmuchasif*, and*, *.

q)   Forthesamereason*.

r)    But*.

s)    Ifthen*, *.

t)     Andif*, andfurther*, *.

u)   Isaythat*.

v)    Let*, andlet*, let*.

w)  Thensince*, therefore*.

x)    Therefore*.

y)    Andsince*, *.

z)    And*, for*, and*.

aa) Therefore*.

bb)                  Let*, therefore*, therefore*.

cc)  Similarlywecanprovethat*.

dd)                  Therefore*, and*.

ee) Nowsince*, while*, therefore*.

ff)   But*, therefore*.

gg) Andsince*, and*, and*, therefore*.

hh)                  Let*.

ii)    Thensince*, and*, if*, then*.

jj)    Andif*, *, and*, therefore*.

kk) But*, therefore*.

ll)    Andsince*, *.

mm)               And*, and*, therefore*, and*, therefore*.

nn)                  Therefore*, therefore*.

oo) And*, and*, hence*.

pp)                  Therefore*.

qq)                  QEF.

rr)  Porism. Butif*, *.

ss)  For*.

tt)   But*, therefore*.

uu)                  Similarlyalso*.

vv) And*, hence*.

ww)                QED.

18)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forif*, then*.

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

d)   But*, therefore*, andalternately*.

e)    But*, therefore*.

f)     But*, for*.

g)    Therefore*.

h)   Similarlywecanprovethat*.

i)     Isaynextthat*.

j)     Forifpossiblelet*, thereforeinversely*.

k)    Butinasmuchas*, therefore*.

l)     Therefore*, whichwasprovedimpossible.

m)  Therefore*.

n)   Butitwasprovedthat*.

o)    Therefore*.

p)   QED.