Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookIV.

Book4

1)   Intoagiven*, tofit*.

a)    Let*, thusitisrequiredtofit *.

b)   Let*.

c)    Then, if*, for *.

d)   But, if*, let*, and with*, let*, let*.

e)    Then, since*, *.

f)     But*, *.

g)    Therefore intothegiven*.

h)   QEF.

2)   Inagiven *, toscribe*.

a)    Let*, thusitisrequiredtoinscribe*.

b)   Let*, on*, and at*, let*, and on*, and at*, let*, let*.

c)    Then, since*, and*, therefore*.

d)   But*, therefore*.

e)    Forthesamereason*, therefore*.

f)     Therefore*.

g)    QEF.

3)   Aboutagiven*, tocircumscribe*.

a)    Let*, thusitisrequiredtocircumscirbe*.

b)   Let*, let*, andlet*, on*, andat*, let*, and*, andthrough*, let*.

c)    Now, since*, and*, therefore*.

d)   Andsince*, and*, therefore*.

e)    But*, therefore*, therefore*.

f)     Similarly*, therefore*.

g)    Therefore*.

h)   Therefore*.

i)     QEF.

4)   Inagiven*, toinscribe*.

a)    Let*, thusitisrequiredtoinscribe*.

b)   Let*, andlet*, from*, let*.

c)    Now, since*, and*, therefore*, therefore*.

d)   Forthesamereason*.

e)    Therefore*, therefore*.

f)     Forif*, *, whichwasprovedabsurd, therefore*, therefore*.

g)    Let*.

h)   Therefore*.

i)     QEF.

5)   About*, tocircumscribe*.

a)    Let*, thusitisrequiredtocircumscribe*.

b)   Let*, andfrom*, let*, [Isaythat]*.

c)    First, let*.

d)   Then, since*, and*, therefore*.

e)    Similarly*, sothat*, therefore*.

f)     Therefore*.

g)    Let*.

h)   Next, let*.

i)     Then, similarly*.

j)     Againlet*, andlet*.

k)    Thenagain, since*, and*, therefore*.

l)     Similarly*, sothat*, therefore*.

m)  Therefore*.

n)   QEF.

o)    Anditismanifestthat*.

6)   Inagiven*, toinscribe*.

a)    Let*, thusitisrequiredtoinscribe*.

b)   Let*, andlet*.

c)    Then, since*, therefore*.

d)   Forthesamereason*, therefore*.

e)    Isaynextthat*.

f)     Forsince*, therefore*, therefore*.

g)    Forthesamereason*, therefore*.

h)   Butitwasalsoproved*, therefore*, and*.

i)     Therefore*.

j)     QEF.

7)   Aboutagiven*, tocircumscribe*.

a)    Let*, thusitisrequiredtocircumscirbe*.

b)   Let*, andthrough*, let*.

c)    Then, since*, and*, therefore*.

d)   Forthesamereason*.

e)    Nowsince*, and*, therefore*.

f)     Forthesamereason*, sothat*.

g)    Similarly*.

h)   Therefore*, therefore*.

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

j)     Isaynextthat*.

k)    Forsince*, and*, therefore*.

l)     Similarly*.

m)  Therefore*.

n)   Butitwasalsoproved*, therefore*, and*.

o)    Therefore*.

p)   QEF.

8)   Inagiven*, toinscribe*.

a)    Let*, thusitisrequiredtoinscribe*.

b)   Let*, through*, let*, through*, let*, therefore*.

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

d)   Similarly*.

e)    Therefore*.

f)     And*.

g)    For,if*, *, whichwasprovedabsurd, therefore*.

h)   Therefore*.

i)     Therefore*.

j)     QEF.

9)   Aboutagiven*, tocircumscribe*.

a)    Let*, thusitisrequiredtocircumscirbe*.

b)   Forlet*, andlet*.

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

d)   Therefore*.

e)    Similarly*.

f)     Nowsince*, and*, and*, therefore*, sothat*.

g)    Similarly*.

h)   Therefore*.

i)     Therefore*, and*.

j)     Let*.

k)    Therefore*.

l)     QEF.

10)         Toconsctruct*.

a)    Let*, andelt*, sothat*, with*, let*, andlet*.

b)   Let*, andlet*.

c)    Thensince*, and*, therefore*.

d)   Andsince*, andfrom*, *, while*, and*, therefore*.

e)    Sincethen*, therefore*.

f)     Sincethen*, therefore*.

g)    But*, therefore*.

h)   But*, sothat*.

i)     Therefore*.

j)     Andsince*, *.

k)    But*, therefore*, sothat*, therefore*.

l)     But*, therefore*.

m)  But*, therefore*.

n)   Therefore*.

o)    QEF.

11)         Inagiven*, toinscribe*.

a)    Let*, thusitisrequiredtoinscribe*.

b)   Let*, let*, sothat*, therefore*.

c)    Nowlet*, andlet*.

d)   Thensince*, and*, therefore*.

e)    But*, therefore*.

f)     But*, therefore*.

g)    Isaynextthat*.

h)   Forsince*, let*, therefore*.

i)     And*, and*, therefore*.

j)     Forthesamereason*, therefore*.

k)    Butitwasalsoproved*, therefore*.

l)     QEF.

12)         Aboutagiven*, tocircumscribe*.

a)    Let*, thusitisrequiredtocircumscribe*.

b)   Let*, through*, let*, let*, andlet*.

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

d)   Forthesamereason*, *.

e)    Andsince*, therefore*.

f)     Forthesamereason*, *, sothat*, therefore*.

g)    Therefore*.

h)   Andsince*, and*, *, therefore*.

i)     Therefore*, and*.

j)     Forthesamereason*, *.

k)    Nowsince*, *.

l)     And*, therefore*.

m)  But*, therefore*, therefore*, therefore*, and*.

n)   Andsince*, therefore*.

o)    Forthesamereason*.

p)   And*, *.

q)   Similarly*, therefore*.

r)    Isaynextthat*.

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

t)     Similarly*, therefore*.

u)   Anditwasalsoproved*, and*.

v)    QEF.

13)         Inagiven*, toinscribe*.

a)    Let*, thusitisrequiredtoinscribe*.

b)   Forlet*, andfrom*, let*.

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

d)   Therefore*.

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

f)     Similarly*.

g)    Nowlet*.

h)   Thensince*, and*, *, therefore*, therefore*.

i)     Similarly*, therefore*.

j)     Therefore*, and*.

k)    Forif*, *, whichwasprovedabsurd.

l)     Therefore*, therefore*.

m)  Let*.

n)   Therefore*.

o)    QEF.

14)         Aboutagiven*, tocircumscribe*.

a)    Let*, thusitisrequiredtocircumscribe*.

b)   Let*, andfrom*, let*.

c)    Thensimilarly*.

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

e)    Similarly*, therefore*.

f)     Therefore*.

g)    Let*, andlet*.

h)   Therefore*.

i)     QEF.

15)         Inagiven*, toinscribe*.

a)    Let*, thusitisrequiredtoincribe*.

b)   Let*, let*, let*, andlet*.

c)    Isaythat*.

d)   Forsince*, *.

e)    Againsince*, *.

f)     But*, therefore*, therefore*, andtherefore*.

g)    And*, therefore*.

h)   Similarly*.

i)     Andsince*, therefore*.

j)     Therefore*, sothat*.

k)    Therefore*.

l)     But*, therefore*.

m)  And*, therefore*, therefore*.

n)   Isaynextthat*.

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

p)   Similarly*, therefore*.

q)   Butitwasalsoproved*, and*.

r)    Therefore*.

s)    QEF.

t)     Porism. Fromthisitismanifestthat*.

u)   Andinlikemanner*.

v)    Andfurtherbymeanssimilar*.

16)         Inagiven*, toinscribe*.

a)    Let*, thusitisrequiredtoinscribe*.

b)   In*, let*, therefore*, therefore*.

c)    Let*, therefore*.

d)   Iftherefore*.

e)    QEF.

f)     Corollary. Andinlikemanner*.

g)    Andfurther*.