Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookV.

Book5.

1)   If*, *.

a)    Let*, Isaythat*.

b)   Forsince*, so*.

c)    Let*, and*, then*.

d)   Nowsince*, therefore*.

e)    Forthesamereason*, therefore*, therefore*.

f)     Therefore, etc.

g)    QED.

2)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    Forthesamereason*, therefore*.

d)   Therefore*.

e)    Therefore*.

f)     Therefore, etc.

g)    QED.

3)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    Let*, and*, then*.

d)   Andsince*, while*, therefore*.

e)    Forthesamereason*.

f)     Sincethen*, and*, therefore*.

g)    Therefore, etc.

h)   QED.

4)   If*, *.

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

b)   Forlet*, and*.

c)    Since*, and*, therefore*.

d)   Forthesamereason*.

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

f)     And*, and*, therefore*.

g)    Therefore, etc.

h)   QED.

5)   If*, *.

a)    Forlet*, Isaythat*.

b)   For*.

c)    Thensince*, therefore*.

d)   But*.

e)    Therefore*, therefore*.

f)     Let*, therefore*.

g)    Andsince*, and*, therefore*.

h)   But*, therefore*.

i)     Thatis*.

j)     Therefore, etc.

k)    QED.

6)   If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forfirstlet*, Isaythat*.

c)    Forlet*.

d)   Since*, while*, therefore*.

e)    But*, therefore*.

f)     Sincethen*, therefore*.

g)    Let*, therefore*.

h)   But*, therefore*.

i)     Henceif*, *.

j)     Similarly*.

k)    Therefore, etc.

l)     QED.

7)   *.

a)    Let*, Isaythat*.

b)   Forlet*, and*.

c)    Thensince*, therefore*.

d)   But*.

e)    Iftherefore*, *, *.

f)     And*, while*, therefore*.

g)    Isaynextthat*.

h)   For*, similarly*, and*.

i)     Iftherefore*, *.

j)     And*, while*, therefore*.

k)    Therefore, etc.

l)     QED.

m)  Porism. Fromthisitismanifestthat*.

8)   Of*, *.

a)    Let*, andlet*, let*, Isaythat*.

b)   Forsince*, then*.

c)    Case1. Firstlet*, let*, thenlet*, andlet*.

d)   Let*, andlet*.

e)    Thensince*, therefore*.

f)     Andsince*, therefore*.

g)    But*, therefore*, therefore*.

h)   Againsince*, and*, therefore*.

i)     But*, therefore*.

j)     And*, therefore*.

k)    But*, inasmuchas*, while*, whence*.

l)     But*, therefore*, while*.

m)  And*, therefore*.

n)   Isaynextthat*.

o)    Forsimilarly*, while*.

p)   And*, while*, therefore*.

q)   Case2. Againlet*.

r)    Then*.

s)    Let*, andlet*.

t)     Thensimilarly*, andsimilarly*, sothat*.

u)   But*, therefore*.

v)    Now*.

w)  Andinthesamemanner*.

x)    Therefore, etc.

y)    QED.

9)   *.

a)    Forlet*, Isaythat*.

b)   Forotherwise*, but*, therefore*.

c)    Againlet*, Isaythat*.

d)   Forotherwise*, but*, therefore*.

e)    Therefore, etc.

f)     QED.

10)         Of*, *.

a)    Forlet*, Isaythat*.

b)   Forifnot*.

c)    Now*, for*, but*, therefore*.

d)   Noragain*, for*, but*, therefore*.

e)    Butitwasproved*, therefore.

f)     Againlet*, Isaythat*.

g)    Forifnot*.

h)   Now*, for*, but*, therefore*.

i)     Noragain*, for*, but*, therefore*.

j)     But*, therefore*.

k)    Therefore, etc.

l)     QED.

11)         *.

a)    For*, solet*, and*, solet*, Isaythat*.

b)   For*, let*, and*.

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

d)   Againsince*, and*, and*, therefore*, *.

e)    But*, if*, *, sothat*, if*, *.

f)     And*, while*, therefore*.

g)    Therefore, etc.

h)   QED.

12)         If*, *.

a)    Let*, sothat*, Isaythat*.

b)   For*, let*, and*.

c)    Thensince*, so*, and*, and*, therefore*, *, sothat*, if*, then*, *.

d)   Now*, sinceif*, *.

e)    Forthesamereason*, therefore*.

f)     Therefore, etc.

g)    QED.

13)         If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forsince*, and*, while*, let*, andlet*, and*, sothat*, but*, andlet*, andlet*.

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

d)   But*, therefore*.

e)    But*, and*, and*, therefore*.

f)     Therefore, etc.

g)    QED.

14)         If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forsince*, and*, therefore*.

c)    But*, therefore*.

d)   But*, therefore*, sothat*.

e)    Similarly*, andif*, *.

f)     Therefore, etc.

g)    QED.

15)         *.

a)    Forlet*, Isaythat*.

b)   Forsince*, *.

c)    Let*, and*, then*.

d)   Andsince*, and*, therefore*.

e)    Therefore*, therefore*.

f)     But*, therefore*.

g)    Therefore, etc.

h)   QED.

16)         If*, *.

a)    Let*, sothat*, Isaythat*.

b)   Forlet*.

c)    Thensince*, and*, therefore*.

d)   But*, therefore*.

e)    Againsince*, therefore*.

f)     But*, therefore*.

g)    Butif*, *.

h)   Therefore*, *.

i)     Now*, and*, therefore*.

j)     Therefore, etc.

17)         If*, *.

a)    Let*, sothat*, Isaythat*.

b)   For*, and*.

c)    Thensince*, therefore*.

d)   But*, therefore*.

e)    Againsince*, therefore*.

f)     But*, therefore*.

g)    Therefore*.

h)   Againsince*, and*, therefore*.

i)     Andsince*, and*, and*, thereforeif*, *.

j)     Let*, thenif*, *.

k)    But*, therefore*, andif*, *, sothatif*, *.

l)     Similarly*, *.

m)  And*, while*, therefore*.

n)   Therefore, etc.

18)         If*, *.

a)    Let*, sothat*, Isaythat*.

b)   Forif*, then*.

c)    Firstlet*.

d)   Thensince*, *, sothat*.

e)    Therefore*.

f)     But*.

g)    Therefore*.

h)   But*, therefore*.

i)     But*, whichisimpossible.

j)     Therefore*.

k)    Similarly*, therefore*.

l)     Therefore, etc.

m)  QED.

19)         If*, *.

a)    For*, solet*, Isaythat*.

b)   Forsince*, also*.

c)    Andsince*, andalternately*.

d)   But*, so*.

e)    Therefore*.

f)     Therefore, etc.

g)    QED.

h)   Porism. Fromthisitismanifestthat*.

20)         If*, *.

a)    Let*, and*, andlet*, Isaythat*.

b)   Forsince*, and*, and*, therefore*.

c)    But*, and*, therefore*.

d)   But*, therefore*.

e)    Simlarly*, *.

f)     Therefore, etc.

g)    QED.

21)         If*, *.

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

b)   Forsince*, therefore*.

c)    But*, so*, and*.

d)   Therefore*.

e)    But*, therefore*, therefore*.

f)     Similarly*, *.

g)    Therefore, etc.

h)   QED.

22)         If*, *.

a)    Let*, and*, Isaythat*.

b)   For*, and*, andfurther*.

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

d)   Forthesamereason*.

e)    Since*, therefore*, *.

f)     And*, and*.

g)    Therefore*.

h)   Therefore, etc.

i)     QED.

23)         If*, *.

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

b)   Of*, let*, andof*.

c)    Thensince*, and*, therefore*.

d)   Forthesamereason*.

e)    And*, therefore*.

f)     Nextsince*, alternatelyalso*.

g)    Andsince*, and*, therefore*.

h)   But*, thereforealso*.

i)     Againsince*, therefore*.

j)     But*, thereforealso*, andalternately*.

k)    Butitwasalsoprovedthat*.

l)     Sincethen*, therefore*, *.

m)  And*, and*.

n)   Therefore*.

o)    Therefore, etc.

p)   QED.

24)         If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, inversely*.

c)    Sincethen*, and*, therefore*.

d)   Andsince*, *, therefore*.

e)    Butalso*, therefore*.

f)     Therefore, etc.

g)    QED.

25)         If*, *.

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

b)   Forlet*.

c)    Since*, and*, therefore*.

d)   Andsince*, *.

e)    But*, therefore*.

f)     Andsince*, therefore*.

g)    Andif*, and*, itfollowsthat*.

h)   Therefore, etc.

i)     QED.