Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookXIII.

Book13.

1)   If*, *.

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

b)   Forlet*, andlet*, let*.

c)    Nowsince*, therefore*.

d)   And*, and*, therefore*.

e)    Andsince*, while*, therefore*.

f)     But*, therefore*.

g)    But*.

h)   Therefore*.

i)     But*, therefore*.

j)     Andsince*, *.

k)    But*, therefore*, therefore*.

l)     And*, therefore*.

m)  Therefore, etc.

n)   QED.

2)   If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Let*, elt*, andlet*.

c)    Nowsince*, *.

d)   Thereofre*.

e)    Andsince*, therefore*.

f)     But*, therefore*.

g)    Andsince*, while*, therefore*.

h)   But*, therefore*.

i)     But*, therefore*.

j)     And*, for*, and*, therefore*.

k)    Therefore*.

l)     But*, therefore*.

m)  Therefore*.

n)   Therefore*, etc.

o)    QED.

p)   Lemma. *.

q)   Ifnoifpossiblelet*.

r)    Therefore*, therefore*.

s)    Butbyhypothesis*, therefore*, whichisimpossible.

t)     Therefore*.

u)   Similarly*, for*.

v)    Therefore*.

w)  QED.

3)   If*, *.

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

b)   Forlet*, andlet*.

c)    Since*, therefore*.

d)   Andsince*, and*, therefore*.

e)    But*, therefore*.

f)     Againsince*, therefore*.

g)    Againsince*, *.

h)   Hence*.

i)     Therefore*, hence*.

j)     But*, therefore*.

k)    Let*, therefore*.

l)     But*, therefore*.

m)  Therefore*.

n)   But*.

o)    And*.

p)   Therefore*.

q)   QED.

4)   If*, *.

a)    Let*, let*, andlet*, Isaythat*.

b)   Forlet*, andlet*.

c)    Sincethen*, and*, therefore*.

d)   And*, and*, therefore*.

e)    Andsince*, let*, therefore*, therefore*.

f)     But*, therefore*.

g)    Butfurther*, therefore*.

h)   And*, while*.

i)     Therefore*.

j)     QED.

5)   If*, and*, *.

a)    Forlet*, let*, andlet*.

b)   Isaythat*.

c)    Forlet*, andlet*.

d)   Since*, therefore*.

e)    And*, and*, therefore*.

f)     But*, and*, therefore*.

g)    Therefore*.

h)   And*, and*, therefore*.

i)     Therefore*.

j)     And*, therefore*.

k)    Therefore*.

l)     QED.

6)   If*, *.

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

b)   Forlet*.

c)    Sincethen*, and*, therefore*.

d)   Therefore*, therefore*.

e)    But*, for*, therefore*, therefore*.

f)     Andsince*, therefore*, therefore*.

g)    Againsince*, and*, therefore*.

h)   Therefore*.

i)     Butif*, *, therefore*.

j)     And*.

k)    Therefore*.

l)     QED.

7)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*.

c)    Nowsince*, and*, and*, hence*.

d)   But*, therefore*.

e)    But*.

f)     Therefore*, and*, therefore*.

g)    But*, therefore*.

h)   But*, therefore*.

i)     Buthypothesis*, therefore*.

j)     Similarlywecanprovethat*, therefore*.

k)    Nextlet*, butlet*, Isaythat*.

l)     Forlet*.

m)  Thensince*, and*, therefore*, and*, therefore*.

n)   But*, since.

o)    Therefore*.

p)   Butbyhypothesis*, therefore*.

q)   Forthesamereason*.

r)    Therefore*.

s)    QED.

8)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*.

c)    Thensince*, and*, therefore*, and*.

d)   Therefore*, therefore*.

e)    But*, inasmuchas*, therefore*, hence*.

f)     Andsince*, *.

g)    But*, therefore*.

h)   And*, therefore*, therefore*, therefore*.

i)     But*, therefore*.

j)     And*, therefore*.

k)    Therefore*.

l)     Similarlywecanprovethat*.

m)  QED.

9)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, let*, andlet*.

c)    Since*, therefore*, therefore*.

d)   But*, therefore*.

e)    Andsince*, therefore*.

f)     Andsince*, for*, *, therefore*.

g)    But*, therefore*.

h)   But*, therefore*.

i)     But*, therefore*, therefore*.

j)     Thereforeproportionally*.

k)    But*.

l)     Therefore*.

m)  And*, therefore*.

n)   Therefore*.

o)    QED.

10)         If*, *.

a)    Let*, andlet*.

b)   Isaythat*.

c)    Forlet*, let*, let*, let*, let*, let*.

d)   Since*, and*, therefore*.

e)    But*, therefore*.

f)     Andsince*, and*, therefore*.

g)    Hence*, therefore*, therefore*.

h)   Forthesamereason*.

i)     Nowsince*, while*, therefore*.

j)     But*, therefore*.

k)    But*, therefore*.

l)     Butfurther*, for*.

m)  Therefore*, hence*.

n)   But*, for*.

o)    Therefore*.

p)   But*, therefore*, therefore*.

q)   Thereforeproportionally*, therefore*.

r)    Againsince*, while*, therefore*, therefore*.

s)    But*, therefore*.

t)     And*.

u)   Therefore*, therefore*.

v)    Therefore*, therefore*.

w)  But*, therefore*.

x)    And*.

y)    Therefore, etc.

z)    QED.

11)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, let*, let*, andlet*.

c)    Now*, therefore*.

d)   But*, therefore*.

e)    Andsince*, and*, therefore*.

f)     Andif*, weconcludethat*.

g)    Forthesamereason*.

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

i)     And*, therefore*.

j)     But*, thereforealso*.

k)    And*, therefore*.

l)     And*, therefore*.

m)  And*, therefore*.

n)   Componendoalso*, thereforealso*.

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

p)   Butitwasprovedthat*, therefore*.

q)   But*, for*, therefore*, therefore*.

r)    Andsince*, therefore*, therefore*.

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

t)     And*.

u)   Therefore*.

v)    Butif*, *, therefore*.

w)  Isaynextthat*.

x)    Let*, therefore*.

y)    Andsince*, componendoalso*.

z)    But*, therefore*.

aa) Andsince*, therefore*.

bb)                  Therefore*, and*, therefore*, therefore*.

cc)  Sincethen*, and*, therefore*.

dd)                  But*, and*.

ee) But*, because*, and*.

ff)   Therefore*.

gg) QED.

12)         If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, let*, andlet*.

c)    Thensince*, therefore*.

d)   Therefore*, therefore*, therefore*.

e)    Andsince*, *.

f)     But*, therefore*.

g)    Therefore*.

h)   But*, therefore*.

i)     Therefore*.

j)     QED.

13)         Toconstruct*, tocomprehend*, toprovethat*.

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

b)   Nowsince*, therefore*.

c)    But*, therefore*.

d)   Andsince*, and*, therefore*.

e)    Forthesamereason*, therefore*.

f)     Andsince*, therefore*.

g)    But*, aswillbeprovedafterwards.

h)   Therefore*.

i)     But*, and*, therefore*.

j)     But*, therefore*, therefore*.

k)    Therefore*.

l)     Itisnextrequiredtocomprehend*.

m)  Forlet*, andlet*.

n)   Nowsince*, while*, therefore*, therefore*.

o)    And*, therefore*.

p)   Ifthen8, *, sinceif*, *, and*.

q)   For*, inasmuchas*.

r)    Isaynextthat*.

s)    Forsince*, therefore*, and*.

t)     But*.

u)   Therefore*.

v)    And*.

w)  Therefore*.

x)    QED.

y)    Lemma. Itistobeprovedthat*.

z)    Forlet*, let*, let*, andlet*.

aa) Sincethenbecause*, *, therefore*.

bb)                  Andsince*, and*, and*, therefore*.

cc)  And*, and*, for*, because*.

dd)                  Therefore*.

ee) QED.

14)         Toconstruct*, andtoprovethat*.

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

b)   Thensince*, therefore*.

c)    Againsince*, and*, therefore*.

d)   But*, therefore*, therefore*.

e)    Forthesamereason*, therefore*.

f)     Similarlywecanprovethat*, therefore*.

g)    Itisnextrequiredtocomprehend*.

h)   Forsince*, therefore*.

i)     Andforthesamereasonif*, *, and*.

j)     Isaynextthat*.

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

l)     Andsince*, therefore*.

m)  Againsince*, *.

n)   But*, therefore*.

o)    But*.

p)   And*, for*.

q)   Therefore*, therefore*.

r)    And*, therefore*.

s)    Therefore*.

t)     QED.

15)         Toconstruct*, andtoprovethat*.

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

b)   Itisthenrequiredtocomprehend*.

c)    Forlet*.

d)   Thensince*, therefore*.

e)    Againsince*, *, hencealsoif*, *, andforthisreasonagain*.

f)     Similarly*.

g)    Ifthen*, *.

h)   Isaynextthat*.

i)     Forsince*, and*, therefore*.

j)     But*, therefore*, hence*.

k)    Andsince*, while*, therefore*.

l)     But*.

m)  And*, therefore*.

n)   And*, therefore*.

o)    Therefore*.

p)   QED.

16)         Toconstruct*, andtoprovethat*.

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

b)   Therefore*, and*.

c)    Nowlet*, andlet*, let*.

d)   Nowsince*, therefore*.

e)    But*, and*.

f)     Therefore*.

g)    But*, therefore*.

h)   Forthesamereason*, therefore*.

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

j)     Forthesamereason*.

k)    But*, therefore*.

l)     Forthesamereason*.

m)  Andsince*, and*, therefore*.

n)   Forthesamereason*.

o)    Let*, let*, let*, let*, andlet*.

p)   Nowsince*, therefore*.

q)   But*, therefore*.

r)    But*, therefore*.

s)    Andsince*, and*, and*, therefore*.

t)     Forthesamereason*, inasmuchasif*, *, therefore*.

u)   But*, and*, therefore*.

v)    But*, therefore*.

w)  Forthesamereason*.

x)    Againsince*, and*, and*, therefore*.

y)    Forthesamereasonif*, *.

z)    But*, therefore*.

aa) Similarlyitcanbeprovedthat*.

bb)                  Therefore*.

cc)  Itisnextrequiredtocomprehend*.

dd)                  Forsince*, and*, therefore*, and*, therefore*.

ee) But*, therefore*.

ff)   And*, thereforeif*, *.

gg) Forthesamereasonsince*, and*, therefore*.

hh)                  Andforthisreasonagain*, if*, *, therefore*.

ii)    Andif*, *, and*.

jj)    Isaynextthat*.

kk) Forlet*.

ll)    Thensince*, therefore*, therefore*.

mm)               And*, therefore*.

nn)                  Andsince*, therefore*.

oo) But*, therefore*.

pp)                  But*.

qq)                  And*, for*, therefore*.

rr)  And*, therefore*.

ss)  Therefore*.

tt)   Isaynextthat*.

uu)                  Forsince*, and*, therefore*, hence*.

vv) Butif*, *.

ww)                And*.

xx) Therefore*.

yy) QED.

zz)  Porism. Fromthisitismanifestthat*.

17)         Toconstruct*, andtoprovethat*.

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

b)   Isaythat*.

c)    Forlet*.

d)   Thensince*, therefore*.

e)    But*, therefore*.

f)     But*, therefore*, hence*.

g)    But*, therefore*, therefore*.

h)   But*, asimuchas*, therefore*.

i)     Similarlyitcanbeprovedthat*.

j)     Therefore*.

k)    Isaynextthat*.

l)     Forlet*, andlet*, Isaythat*.

m)  Forsince*, therefore*.

n)   But*, therefore*.

o)    And*, for*, and*, for*.

p)   Butif*, *, therefore*.

q)   But*, therefore*.

r)    Isaynextthat*.

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

t)     But*, therefore*, hence*.

u)   But*, therefore*, for*, therefore*.

v)    But*, therefore*.

w)  Andsince*, and*, therefore*.

x)    Similarlywecanprovethat*, therefore*.

y)    Butif*, *, therefore*.

z)    Anditwasalsoproved*, therefore*.

aa) Thereforeif*, *.

bb)                  Itisthenrequiredtocomprehend*.

cc)  Forlet*, andlet*, therefore*, for*.

dd)                  Let*, therefore*, and*.

ee) Let*.

ff)   Nowsince*, and*, therefore*.

gg) But*, inasmuchas*.

hh)                  Butfurther*, since*, therefore*.

ii)    But*, therefore*.

jj)    But*, foritahspreviouslybeenshown*.

kk) Butif*, and*, therefore*.

ll)    And*, therefore*.

mm)               Similarlywecanprovethat*, therefore*.

nn)                  Isaynextthat*.

oo) Forsince*, and*, therefore*.

pp)                  Thussince*, *, for*, therefore*.

qq)                  But*, therefore*, therefore*.

rr)  But*, therefore*.

ss)  Andsince*, and*, therefore*.

tt)   Butif*, *.

uu)                  Therefore*.

vv) QED.

ww)                Porism. Fromthisitismanifestthatwhen*, *.

18)         Toset*, andtocompare*.

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

b)   Thensince*, therefore*.

c)    Therefore*.

d)   But*, for*, therefore*.

e)    But*.

f)     And*, therefore*.

g)    Againsince*, therefore*.

h)   But*, therefore*.

i)     But*.

j)     And*, therefore*.

k)    Andsince*, therefore*.

l)     But*, therefore*.

m)  But*.

n)   And*, therefore*.

o)    Nextlet*, let*, let*, andlet*.

p)   Thensince*, for*, and*, therefore*.

q)   Therefore*, therefore*.

r)    But*, therefore*.

s)    Andsince*, and*, therefore*.

t)     Therefore*, therefore*.

u)   But*, therefore*, therefore*.

v)    Let*, let*, andlet*.

w)  Nowsince*, and*, therefore*.

x)    But*.

y)    And*, therefore*, therefore*.

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

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

bb)                  But*, therefore*.

cc)  Nowsince*, let*, andlet*, therefore*.

dd)                  Andsince*, therefore*.

ee) Therefore*, and*.

ff)   Therefore*.

gg) But*, for*.

hh)                  That*wecanprovethus.

ii)    Forsince*, *.

jj)    Andsince*, *, therefore*, therefore*.

kk) But*, therefore*.

ll)    But*, for*, therefore*, therefore*, therefore*.

mm)               Andwhen*, inasmuchas*, andwhen*, *, therefore*.

nn)                  But*, therefore*.

oo) Therefore*.

pp)                  QED.

qq)                  Isaynextthat*.

rr)  For*.

ss)  With*, *constructed, with*, *, andwith*, *, but*, for*, *, whichisimpossible, for*.

tt)   Forthesamereason*.

uu)                  By*, *contained, butby*, itisimpossible*, for*.

vv) By*, *contained*, but*, itisimpossible, for*, whichisimpossible*.

ww)                Neitheragain*byreasonofthesameabsurdity.

xx) Therefore, etc.

yy) QED.

zz)  Lemma. Butthat*, wemustprovethus.

aaa)                Let*, let*, let*, andlet*.

bbb)              Therefore*.

ccc)                 Andsince*, therefore*, therefore*.

ddd)              But*, therefore*.

eee)                QED.