Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookIII.

Book3.

1)   Tofind *.

a)    Let*, thusitisrequiredtofind *.

b)   Let*, andlet*, from*, let*, andlet*, let*, Isaythat*.

c)    Forsupposeitisnot, but, if possible, let*, andlet*.

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

e)    But*, therefore*.

f)     But*, therefore*, whichisimpossible.

g)    Therefore*.

h)   Similarly *.

i)     Therefore*.

j)     QEF.

k)    Porism. Fromthisitismanifestthat*.

2)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsupposeitdoesnotifpossible, let*, let*, andlet*, let*, andlet*.

c)    Then, since*, *.

d)   Andsince*, *.

e)    But*, therefore*.

f)     And*, therefore*.

g)    But*, therefore*, whichisimpossible.

h)   Therefore*.

i)     Similarly *, therefore*.

j)     Therefore, etc.

k)    QED.

3)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, andlet*, let*.

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

d)   But*, therefore*.

e)    Therefore*.

f)     Againlet*, Isaythat*.

g)    For *, since*, *.

h)   But*, therefore*, therefore*, therefore*.

i)     Therefore, etc.

j)     QED.

4)   If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forifpossible, let*, let*, andlet*.

c)    Then, since*, therefore*.

d)   Again, since*, therefore*.

e)    But*, therefore*, whichisimpossible.

f)     Therefore*.

g)    Therefore, etc.

h)   QED.

5)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forifpossible, let*, let*, andlet*.

c)    Then, since*, *.

d)   Again, since*, *.

e)    But*, therefore*, whichisimpossible.

f)     Therefore*.

g)    Therefore, etc.

6)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forifpossible, let*, let*, andlet*.

c)    Then, since*, *.

d)   Again, since*, *.

e)    But*, therefore*, whichisimpossible.

f)     Therefore*.

g)    Therefore, etc.

h)   QED.

7)   If*, *.

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

b)   Forlet*.

c)    Then, since*, *.

d)   But*, therefore*.

e)    Again, since*, *.

f)     But*, therefore*.

g)    Forthesamereason *.

h)   Again, since* and*, *.

i)     Let*, therefore*.

j)     Therefore*.

k)    Isayalsothat*.

l)     For, on *, and at*, let*.

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

n)   Isayagainthat*.

o)    Forifpossible, let*.

p)   Then, since*, *, whichisimpossible.

q)   Therefore*, therefore*.

r)    Therefore, etc.

s)    QED.

8)   If*, *.

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

b)   Forlet*, andlet*, let*.

c)    Then, since*, therefore*.

d)   But*, therefore*.

e)    Again, since* and*, therefore*.

f)     Similarly*, therefore*.

g)    Next, since*, therefore*, so that*.

h)   Andsince*, therefore*, and*, therefore*.

i)     Similarly *, therefore*.

j)     Isayalsothat*.

k)    On *, and at*, let*, andlet*.

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

m)  Isaythat*.

n)   Forifpossible, let*, andlet*.

o)    Then, since*, *, whichwasprovedimpossible.

p)   Therefore*.

q)   Therefore, etc.

r)    QED.

9)   If*, *.

a)    Let*, from*, let*, Isaythat*.

b)   Forlet*, andlet*.

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

d)   Therefore*, therefore*.

e)    And since, if*, *, *.

f)     Forthesamereason *.

g)    And*, therefore*.

h)   Therefore, etc.

i)     QED.

10)         *.

a)    Forifpossible, let*, let*, andfrom*, let*.

b)   Then, since*, *.

c)    Again, since*, *.

d)   Butitwasalsoproved *, therefore*.

e)    Similarly*, therefore*, whichisimpossible.

f)     Therefore, etc.

g)    QED.

11)         If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forsupposeitdoesnotbutifpossible, let*, andlet*.

c)    Then, since*, let*, therefore*.

d)   But*, therefore*, whichisimpossible.

e)    Therefore*, therefore*.

f)     Therefore, etc.

g)    QED.

12)         If*, *.

a)    Forlet*, andlet*, Isaythat*.

b)   Forsupposeitdoesnotbutifpossible, let*, andlet*.

c)    Then, since*, *.

d)   Again, since*, *.

e)    But*, therefore*, sothat*, whichisimpossible.

f)     Therefore*, therefore*.

g)    Therefore, etc.

h)   QED.

13)         *.

a)    Forifpossible, let*.

b)   Let*.

c)    Let*.

d)   Then, since*, *, therefore*, therefore*.

e)    Again, since*, *, butitwasalsoproved *, whichisimpossible.

f)     Therefore*.

g)    Isayfurtherthat*.

h)   Forifpossible, let*, andlet*.

i)     Then, since*, *, but*, whichisabsurd.

j)     Therefore*.

k)    Anditwasprovedthat*.

l)     Therefore, etc.

m)  QED.

14)         In *, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, andlet*, from*, let*, andlet*.

c)    Then, since*, *.

d)   Therefore*, therefore*.

e)    Forthesamereason *, and*, therefore*.

f)     Andsince*, *.

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

h)   But*, therefore*.

i)     Next, let*.

j)     Isaythat*.

k)    For *, similarly *.

l)     Andsince*, *.

m)  But*, and*.

n)   Therefore*, therefore*, therefore*.

o)    And*, therefore*.

p)   Therefore, etc.

q)   QED.

15)         *.

a)    Let*, let*, andlet*, Isaythat*.

b)   For, from*, let*.

c)    Then, since*, *.

d)   Let*, through *, let*, andlet*.

e)    Then, since*, *.

f)     Again, since*, *.

g)    But*, and*, therefore*.

h)   Andsince*, and*, therefore*.

i)     But*.

j)     Therefore*.

k)    Therefore, etc.

l)     QED.

16)         *.

a)    Let*, Isaythat*.

b)   Forsupposeitdoesnotbutifpossible, let*, andlet*.

c)    Since*, *.

d)   But*, therefore*, thus, in *, *, whichisimpossible.

e)    Therefore*.

f)     Similarly *, therefore*.

g)    Let*, Isaynextthat*.

h)   Forifpossible, let*.

i)     Then, since*, and*, *.

j)     But*, therefore*, whichisimpossible.

k)    Therefore*.

l)     Isayfurtherthat*.

m)  Forif*, *.

n)   But*, therefore*.

o)    QED.

p)   Porism. Fromthisitismanifestthat*.

17)         From*, todraw*.

a)    Let*, thusitisrequiredtodraw*.

b)   Forlet*, let*, and with *, let*, from*, let*, andlet*, Isaythat*.

c)    Forsince*, *, therefore*, and*, therefore*, therefore*.

d)   But*, therefore*.

e)    Now *, therefore*.

f)     Therefore*.

g)    QEF.

18)         If*, *.

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

b)   Forifnot, let*.

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

d)   But*, therefore*, whichisimpossible.

e)    Therefore*.

f)     Similarly *, therefore*.

g)    Therefore, etc.

19)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsupposeitdoesnotbutifpossible, let*, andlet*.

c)    Since*, and*, *, therefore*.

d)   But*, therefore*, whichisimpossible.

e)    Therefore*.

f)     Similarly *.

g)    Therefore, etc.

h)   QED.

20)         In *, *.

a)    Let*, let*, andlet*, Isaythat*.

b)   Forlet*.

c)    Then, since*, *, therefore*.

d)   But*, therefore*.

e)    Forthesamereason *.

f)     Therefore*.

g)    Againlet*, let*.

h)   Similarly *, therefore*.

i)     Therefore, etc.

21)         In *, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*, andlet*, let*.

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

d)   Forthesamereason *, therefore*.

e)    Therefore, etc.

22)         *.

a)    Let*, andlet*, Isaythat*.

b)   Let*.

c)    Then, since*, *.

d)   But*, and*, for *, therefore*.

e)    Let*, therefore*.

f)     But*, therefore*.

g)    Similarly*.

h)   Therefore, etc.

i)     QED.

23)         On *, therecannotbe *.

a)    Forifpossible, let*, let*.

b)   Then, since*, and*, *, whichisimpossible.

c)    Therefore, etc.

d)   QED.

24)         *.

a)    Forlet*, Isaythat*.

b)   Forif*, *, and*, *.

c)    Forif*, *, or *, whichisimpossible.

d)   Therefore, if*, therefore*.

e)    Therefore, etc.

f)     QED.

25)         Given *, todescribe *.

a)    Let*, thusitisrequiredtodescribe *.

b)   Forlet*, *.

c)    Forlet*, and*, let*, let*.

d)   Then, since*, *.

e)    Andsince*, and*, therefore*.

f)     But*, therefore*, therefore*.

g)    Therefore*.

h)   Therefore*.

i)     Anditismanifestthat*.

j)     Similarly *, *.

k)    But, if*, and if*, *.

l)     Therefore, *.

m)  QEF.

26)         In *, *.

a)    Let*, and in *, let*, and*, Isaythat*.

b)   Forlet*.

c)    Now, since*, *.

d)   Thus *, therefore*.

e)    Andsince*, *, and*.

f)     But*, therefore*.

g)    But*, therefore*.

h)   Therefore, etc.

i)     QED.

27)         In *, *.

a)    For, *, let*, Isaythat*.

b)   Forif*, *.

c)    Let*, and at*, let*.

d)   Now *, therefore*.

e)    But*, therefore*, whichisimpossible.

f)     Therefore*, therefore*.

g)    And*, and*, therefore*.

h)   Therfore, etc.

i)     QED.

28)         In *, *.

a)    Let*, in *, let*, Isaythat*.

b)   Forlet*, andlet*.

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

d)   But*, therefore*.

e)    And*, therefore*.

f)     Therefore, etc.

29)         In *, *.

a)    Let*, and in *, let*, andlet*, Isaythat*.

b)   Forlet*, andlet*, let*.

c)    Now, since*, *.

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

e)    Therefore, etc.

f)     QED.

30)         Tobisect*.

a)    Let*, thusitisrequiredtobisect*.

b)   Let*, from*, let*, andlet*.

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

d)   But*, and*, therefore*.

e)    Therefore*.

f)     QEF.

31)         In *, *.

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

b)   Let*, andlet*.

c)    Then, since*, *.

d)   Again, since*, *.

e)    Therefore*.

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

g)    Next, since*, and*, *, and*.

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

i)     Isayfurtherthat*.

j)     Thisisatoncemanifest.

k)    Forsince*, *.

l)     Again, since*, *.

m)  Therefore, etc.

n)   QED.

32)         If*, *.

a)    Forlet*, andfrom*, let*.

b)   Isaythat*.

c)    Forlet*, let*, andlet*.

d)   Then, since*, and*, *.

e)    Therefore*, therefore*.

f)     Therefore*.

g)    But*, therefore*.

h)   Let*, therefore*.

i)     Next, since*, *.

j)     But*, therefore*, therefore*.

k)    Therefore, etc.

l)     QED.

33)         On *, todescribe *.

a)    Let*, thusitisrequiredtodescribe *.

b)   * isthenacuteorrightorobtuse.

c)    Forletitbeacube, and, on *, let*, therefore*.

d)   Let*, let*, let*, andlet*.

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

f)     Therefore*.

g)    Let*, andlet*, let*.

h)   Now, since*, therefore*.

i)     Since, then *, and*, *.

j)     But*, therefore*.

k)    Therefore*.

l)     Next, let*, andletitbeagainrequiredtodescribe *.

m)  Let*, let*, and with *, let*.

n)   Therefore*.

o)    And*, for *.

p)   But*, therefore*.

q)   Therefore*.

r)    Next, let*, and on *, and at*, let*, let*, let*, let*, andlet*.

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

t)     Therefore*.

u)   Now, since*, *.

v)    And*, therefore*.

w)  But*.

x)    Therefore*.

y)    Therefore*.

z)    QEF.

34)         From*, tocut *.

a)    Let*, thusitisrequiredtocut *.

b)   Let*, and on *, and at*, let*.

c)    Then, since*, and*, *.

d)   But*, therefore*.

e)    Therefore*.

f)     QEF.

35)         If*, *.

a)    For, in *, let*, Isaythat*.

b)   If*, sothat*, itismanifestthat*.

c)    Next, let*, let*, from*, let*, andlet*.

d)   Then, since*, *, therefore*.

e)    Since, then *, *, let*, therefore*.

f)     But*, therefore*.

g)    And*, therefore*.

h)   Forthesamereason *.

i)     But*, therefore*.

j)     Let*, therefore*.

k)    Therefore, etc.

l)     QED.

36)         If*, *.

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

b)   Then *.

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

d)   Andsince*, and*, *.

e)    But*, therefore*.

f)     And*, therefore*.

g)    Let*, therefore*.

h)   Againlet*, let*, let*.

i)     Then *.

j)     Andsince*, *, therefore*.

k)    Now, since*, *.

l)     Let*, therefore*.

m)  But*, and*, therefore*.

n)   And*, therefore*.

o)    But*, therefore*.

p)   Let*, therefore*.

q)   Therefore, etc.

r)    QED.

37)         If*, *.

a)    Forlet*, from*, let*, let*.

b)   Isaythat*.

c)    Forlet*, let*, andlet*, let*.

d)   Thus *.

e)    Now, since*, *.

f)     But*, therefore*, therefore*.

g)    And*, therefore*, and*, therefore*.

h)   But*, therefore*.

i)     And*, and*, therefore*.

j)     Similarly *.

k)    Therefore, etc.

l)     QED.