Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookII.

Book2.

1)   If*, *.

a)    Let*, Isaythat*.

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

c)    Then *.

d)   Now *, for *, *, and*.

e)    Similarly*.

f)     Therefore*.

g)    Therefore, etc.

h)   QED.

2)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*.

c)    Then *.

d)   Now *, and*.

e)    Therefore*.

f)     Therefore, etc.

3)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, let*.

c)    Then *.

d)   Now *, *, and*.

e)    Therefore*.

f)     Therefore, etc.

g)    QED.

4)   If*, *.

a)    Forlet*, Isaythat*.

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

c)    Then, since*, *.

d)   But*, therefore*, so that*.

e)    But*, therefore*, therefore*.

f)     Isaynextthat*.

g)    Forsince*, *.

h)   But*, therefore*, so that*.

i)     Therefore*, anditwasalsoproved *, therefore*, and*.

j)     Forthesamereason *, *, and*.

k)    Therefore*.

l)     Now, since* and*, therefore*.

m)  Therefore*.

n)   But*, therefore*.

o)    But*.

p)   Therefore*.

q)   Therefore, etc.

r)    QED.

5)   If*, *.

a)    Forlet*, Isaythat*.

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

c)    Then, since*, let*, therefore*.

d)   But*, since*, therefore*.

e)    Let*, therefore*.

f)     But*, therefore*.

g)    Let*, therefore*.

h)   But*, therefore*.

i)     Therefore, etc.

j)     QED.

6)   If*, *.

a)    Forlet*, andlet*, Isaythat*.

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

c)    Then, since*, *.

d)   But*.

e)    Therefore*.

f)     Let*, therefore*.

g)    But*, therefore*.

h)   Let*, therefore*.

i)     But*, therefore*.

j)     Therefore, etc.

k)    QED.

7)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*.

c)    Then, since*, let*, therefore*.

d)   Therefore*.

e)    But*, therefore*.

f)     But*, for *, therefore*.

g)    Let*, therefore*.

h)   But*, therefore*.

i)     Therefore, etc.

8)   If*, *.

a)    Forlet*, Isaythat*.

b)   For let*, andlet*, let*.

c)    Then, since*, therefore*.

d)   Forthesamereason *.

e)    Andsince*, therefore*.

f)     But*, therefore*, therefore*.

g)    Therefore*.

h)   Again, since*, while *, and*, therefore*.

i)     Andsince*, *.

j)     But*, therefore*, therefore*.

k)    Therefore*.

l)     But*, therefore*.

m)  Now, since*, therefore*.

n)   But*, therefore*.

o)    Let*, therefore*.

p)   But*, therefore*.

q)   But*, therefore*.

r)    Therefore, etc.

s)    QED.

9)   If*, *.

a)    Forlet*, Isaythat*.

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

c)    Then, since*, *.

d)   Andsince*, *.

e)    And*, therefore*.

f)     Forthesamereason *, *, therefore*.

g)    Andsince*, therefore*.

h)   Andsince* and*, for *, [therefore] *.

i)     Now, since*, therefore*.

j)     But*, therefore*.

k)    Again, since*, therefore*.

l)     But*, therefore*.

m)  But*, therefore*.

n)   But*, therefore*.

o)    And*, for *, therefore*.

p)   But*, therefore*.

q)   And*, therefore*.

r)    Therefore, etc.

s)    QED.

10)         If*, *.

a)    Forlet*, Isaythat*.

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

c)    Then, since*, *, therefore*.

d)   But*, therefore*.

e)    Let*, andlet*.

f)     Then, since*, *, and*, therefore*.

g)    Forthesamereason *, therefore*.

h)   Andsince*, *.

i)     But*, for *, therefore*, therefore*, so *.

j)     Again, since*, and*, [therefore] *.

k)    Now, since*, *.

l)     But*, therefore*.

m)  Again, since*, therefore*.

n)   But*, therefore*.

o)    And*, therefore*.

p)   But*, therefore*.

q)   But*, therefore*.

r)    And*, therefore*.

s)    Therefore, etc.

t)     QED.

11)         Tocut *.

a)    Let*, thusitisrequiredtocut *.

b)   Forlet*, let*, let*, let*.

c)    Isaythat*.

d)   Forsince*, *.

e)    But*, therefore*.

f)     But*, therefore*.

g)    Let*, therefore*.

h)   Now *, therefore*.

i)     And*, and*, therefore*.

j)     Therefore*.

k)    QEF.

12)         In *, *.

a)    Let*, Isaythat*.

b)   Forsince*, *.

c)    Let*, therefore*.

d)   But*, and*, therefore*, so that*.

e)    Therefore, etc.

f)     QED.

13)         In *, *.

a)    Let*, Isaythat*.

b)   Forsince*, *.

c)    Let*, therefore*.

d)   But*, and*, therefore*, so that*.

e)    Therefore, etc.

f)     QED.

14)         Toconstruct *.

a)    Let*, thusitisrequiredtoconstruct *.

b)   Forlet*. If*, *.

c)    Then, if*, for *.

d)   But, if not, *.

e)    Let*, andlet*, let* andlet*.

f)     With *, let*, let*, andlet*.

g)    Then, since*, *.

h)   But*, therefore*.

i)     But*, therefore*.

j)     But*, therefore*.

k)    Let*, therefore*.

l)     But*, therefore*.

m)  And*.

n)   Therefore*.

o)    Therefore*.

p)   QEF.