Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookI.

Book1.

1)   On *, toconstruct*.

a)    Let*.

b)   Thusitisrequiredtoconstruct *.

c)    With *, let*, again, with *, let*, and, from*, let*.

d)   Now, since*.

e)    Again, since*.

f)     But*, therefore*.

g)    Therefore*.

h)   Therefore*.

i)     Therefore*.

j)     DEF.

2)   Toplace *.

a)    Let*.

b)   ThusitisrequiredTo*.

c)    From*, let*, and on it, let*.

d)   Let*, with *, let*, and again, with *, let*.

e)    Then, since*, *.

f)     Again, since*, *.

g)    And, *, therefore*.

h)   But*, therefore*.

i)     Therefore*.

j)     Therefore*.

k)    BEF.

3)   Given *, Tocut *.

a)    Let*.

b)   Thusitisrequiredtocut *.

c)    At*, let*, and with *, let*.

d)   Now, since*, *.

e)    But*.

f)     Therefore*, so that*.

g)    Therefore*.

h)   BEF.

4)   If*, *.

a)    Let*.

b)   Isaythat*.

c)    Forif*, and if*, then *, because *.

d)   Again, *, because *, hence *, because *.

e)    But*, hence *, and*.

f)     Thus *.

g)    And*.

h)   Therefore, etc.

i)     QED.

5)   In *, *, and, if*, *.

a)    Let*, and, let*.

b)   Isaythat*.

c)    Let*, andlet*.

d)   Then, since*, and*.

e)    Therefore*, and*.

f)     Andsince*, *.

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

h)   Accordingly, since*, and*, *, and*.

i)     But, *, and*.

j)     Therefore, etc.

k)    QED.

6)   If*, *.

a)    Let*.

b)   Isaythat*.

c)    Forif*, *.

d)   Let*, andfrom*, let*, let*.

e)    Then, since*, and*, *, and*, therefore*, and*, which is absurd.

f)     Therefore, *, therefore*.

g)    Therefore, etc.

h)   QED.

7)   Given *, there cannot be *, and, *.

a)    For, if possible *, let*, so that*, and*, let*.

b)   Then, since*, *, therefore*, therefore*.

c)    Andsince*, *.

d)   Butitwasproved *, whichisimpossible.

e)    Therefore, etc.

f)     QED.

8)   If*, *.

a)    Let*, let*, Isaythat*.

b)   For, if, *, and if, *, *, because *.

c)    Then, *, *, forif*, then *.

d)   But*.

e)    Therefore*, so that*.

f)     If therefore, etc.

g)    QED.

9)   Tobisect*.

a)    Let*.

b)   Thusitisrequiredtobisect*.

c)    Let*, let*, let*, and on *, let*, let*.

d)   I say that*.

e)    Forsince*, and*, *.

f)     And*, therefore*.

g)    Therefore, *.

h)   QEF.

10)         Tobisect*.

a)    Let*.

b)   ThusitisrequiredTo*.

c)    Let*, andlet*, Isaythat*.

d)   Forsince*, *, and*, therefore*.

e)    Therefore*.

f)     QEF.

11)         Todraw*.

a)    Let*.

b)   Thusitisrequiredtodraw*.

c)    Let*, let*, on *, let*, andlet*, Isaythat*.

d)   Forsince*, *, and*, therefore*, and*.

e)    But, when *, therefore*.

f)     Therefore.

g)    QEF.

12)         Todraw*.

a)    Let*, thusitisrequiredtodraw*.

b)   For *, and with *, let*, let*, andlet*.

c)    Isaythat*.

d)   Forsince*, *. and*, therefore*.

e)    And*.

f)     But, when *.

g)    Therefore*.

h)   QEF.

13)         If*, *.

a)    For let*, Isaythat*.

b)   Now, if*, *.

c)    But, if not, let*, therefore*.

d)   Then, since*, let*, therefore*.

e)    Again, since*, let*, therefore*.

f)     But*, therefore*.

g)    But*, therefore*.

h)   Therefore, etc.

i)     QED.

14)         If*, *.

a)    For, with any *, let*, Isaythat*.

b)   Forif*, let*.

c)    Then, since*, and*.

d)   But, *, therefore*.

e)    Let*, therefore*.

f)     Therefore*.

g)    Similarly *.

h)   Therefore*.

i)     Therefore, etc.

j)     QED.

15)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, *.

c)    Again, since*, *.

d)   But, *, therefore*.

e)    Let*, therefore*.

f)     Similarly*.

g)    Therefore, etc.

h)   QED.

i)     Porism. Fromthisitismanifesthat, if*, *.

16)         Inany*, if*, *.

a)    Let*, Isaythat*.

b)   Le t*, andlet*, let*, let*, andlet*.

c)    Then, since*, *, and*.

d)   Therefore*, and*, therefore*.

e)    But*, therefore*.

f)     Similarly*.

g)    Therefore, etc.

h)   QED.

17)         Inany*, *.

a)    Let*, Isaythat*.

b)   Forlet*.

c)    Then, since*.

d)   Let*, therefore*.

e)    But*, therefore*.

f)     Similarly*.

g)    Therefore, etc.

h)   QED.

18)         Inany*, *.

a)    For let*, Isaythat*.

b)   Forsince*, let*.

c)    Then, since*, *.

d)   But*, therefore*, therefore*.

e)    Therefore, etc.

19)         Inany*, *.

a)    Let*, Isaythat*.

b)   For, if not, *.

c)    Now*, for then *, but it is not, therefore*.

d)   Neither is*, for then, *, but it is not, therefore*.

e)    Anditwasprovedthat*, therefore*.

f)     Therefore, etc.

g)    QED.

20)         Inany*, *.

a)    For let*, Isaythat*.

b)   For let*, let*, andlet*.

c)    Then, since*, *, therefore*.

d)   Andsince*, and*, therefore*.

e)    But*, therefore*.

f)     Similarly*.

g)    Therefore, etc.

h)   QED.

21)         If*, *.

a)    On *, from*, let*, Isaythat*.

b)   Forlet*.

c)    Then, since*, therefore*.

d)   Let*, therefore*.

e)    Again, since*, therefore*.

f)     But, *, therefore*.

g)    Again, since*, therefore*.

h)   Forthesamereason, moreover, * also, *.

i)     But*, therefore*.

j)     Therefore, etc.

k)    QED.

22)         Out of *, toconstruct*.

a)    Let*, andthusitisrequiredtoconstruct *.

b)   Let*, andlet*.

c)    With *, let*, again, with *, let*, andlet*, Isaythat*.

d)   Forsince*, *.

e)    But, *, therefore*.

f)     Again, since*, *.

g)    But*, therefore*.

h)   And*, therefore*.

i)     Therefore*.

j)     QEF.

23)         On a given *, toconstruct*.

a)    Let*, thusitisrequiredtoconstruct *.

b)   On *, let*, let*, and out of *, let*.

c)    Then, since*, *.

d)   Therefore*.

e)    QEF.

24)         If*, *.

a)    Let*, and*, andlet*, Isaythat*.

b)   Forsince*, let*, on *, and*, *, let*, andlet*.

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

d)   Again, since*, *, therefore*.

e)    Therefore*.

f)     Andsince* and*, *.

g)    But*.

h)   Therefore*.

i)     Therefore, etc.

j)     QED.

25)         If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forifnot, *.

c)    Now*, for then *, but it is not, therefore*.

d)   Neither again, *, for then, *, but it is not, therefore*.

e)    Butitwasprovedthat*, therefore*.

f)     Therefore, etc.

26)         If*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forif*, *.

c)    Let*, andlet*, andlet*.

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

e)    But, *, therefore*, whichisimpossible.

f)     Therefore*, therefore*.

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

h)   Againlet*, Isayagainthat*.

i)     Forif*, *.

j)     Let*, if possible, andlet*, let*.

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

l)     But, *, therefore*, whichisimpossible.

m)  Therefore*, and therefore*.

n)   But*, therefore*, therefore*.

o)    Therefore, et.

p)   QED.

27)         If*, *.

a)    Forlet*, Isaythat*.

b)   For, if not *.

c)    Let*.

d)   Then, *, in *, whichisimpossible.

e)    Therefore, *.

f)     Similarly*.

g)    But*, therefore*.

h)   Therefore, etc.

i)     QED.

28)         If*, *.

a)    Forlet, *, or *, Isaythat*.

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

c)    Again, since*, and*, *.

d)   Let*, therefore*, and*, therefore*.

e)    Therefore, etc.

29)         *.

a)    Forlet*, Isaythat*.

b)   Forif*, *.

c)    Let*.

d)   Let*, therefore*.

e)    But*, therefore*.

f)     But*, therefore*, but*, therefore* and therefore*.

g)    Again, *, therefore*.

h)   Let*, therefore*.

i)     But*, therefore*.

j)     Therefore, etc.

k)    QED.

30)         *.

a)    Let*, Isaythat*.

b)   Forlet*.

c)    Then, since*, *.

d)   Again, since*, *.

e)    But*, therefore*, and*.

f)     Therefore*.

g)    QED.

31)         Given *, Todraw*.

a)    Let*, thusitisrequiredtodraw*.

b)   Let*, andlet*, andlet*.

c)    Then, since*, therefore*.

d)   Therefore*.

e)    QEF.

32)         Inany*, if*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forlet*.

c)    Then, since*, *.

d)   Again, since*, *.

e)    But, *, therefore*.

f)     Let*, therefore*.

g)    But*, therefore*.

h)   Therefore, etc.

i)     QED.

33)         *.

a)    Let*, andlet*, Isaythat*.

b)   Let*.

c)    Then, since*, *.

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

e)    Andsince*, *.

f)     Anditwasalsoproved *.

g)    Therefore, etc.

h)   QED.

34)         In *, *.

a)    Let*, Isaythat*.

b)   Forsince*, *.

c)    Again, since*, *.

d)   Therefore*, therefore*, therefore*, and further *.

e)    Andsince*, and*, *.

f)     And, *.

g)    Therefore, in *, *.

h)   Isaynextthat*.

i)     Forsince*, *, and*, therefore*, and*.

j)     Therefore, *.

k)    QED.

35)         *.

a)    Let*, Isaythat*.

b)   Forsince*, *.

c)    Forthesamereasonalso, *, so that*, and*, therefore*.

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

e)    Let*, therefore*.

f)     Let*, therefore*.

g)    Therefore, etc.

h)   QED.

36)         *.

a)    Let*, Isaythat*.

b)   Forlet, *.

c)    Then, since*, *.

d)   But*.

e)    And*, but*, therefore*.

f)     And*, for *.

g)    Forthesamereason *, so that*.

h)   Therefore, etc.

i)     QED.

37)         *.

a)    Let*, Isaythat*.

b)   Let*, andlet*.

c)    Then *, and*, for *.

d)   Moreover, *, for *.

e)    And*, for *.

f)     Therefore*.

g)    Therefore, etc.

h)   QED.

38)         *.

a)    Let*, Isaythat*.

b)   Forlet*.

c)    Then *, and*, for *.

d)   Moreover, *, for *.

e)    And*, for *.

f)     Therefore*.

g)    Therefore, etc.

h)   QED.

39)         *.

a)    Let*, Isaythat*.

b)   And, let*, Isaythat*.

c)    For, if not, let*, andlet*.

d)   Therefore*, for *.

e)    But*, therefore*, whichisimpossible.

f)     Therefore*.

g)    Similarly *, therefore*.

h)   Therefore, etc.

i)     QED.

40)         *.

a)    Let*, Isaythat*.

b)   Forlet*, Isaythat*.

c)    Forifnotlet*, andlet*.

d)   Therefore*, for *.

e)    But*, therefore*, whichisimpossible.

f)     Therefore*.

g)    Similarly *, therefore*.

h)   Therfore, etc.

i)     QED.

41)         If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*.

c)    Then *, for *.

d)   But*, for *, so that*.

e)    Therefore, etc.

f)     QED.

42)         Toconstruct*.

a)    Let*, thusitisrequiredtoconstruct *.

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

c)    Then *.

d)   Andsince*, *, for *, therefore*.

e)    But, *, for *, therefore*.

f)     But*, for *, therefore*.

g)    And*.

h)   Therefore*.

i)     QEF.

43)         Inany*, *.

a)    Let*, Isaythat*.

b)   Forsince*, *.

c)    Again, since*, *.

d)   Forthesamereason, *.

e)    Now, since*, and*.

f)     And*, therefore*.

g)    Therefore, etc.

h)   QED.

44)         Toa given *, Toapply *.

a)    Let*, thusitisrequiredtoapply *.

b)   Let*, let*, let*, let*.

c)    Let*.

d)   Then, since*, *.

e)    Therefore, * and*, therefore*.

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

g)    Then *, and*, and*, therefore*.

h)   But*, therefore*.

i)     Andsince*, while *, *.

j)     Therefore*.

k)    QEF.

45)         Toconstruct*.

a)    Let*.

b)   let*, andlet*, let*.

c)    Then, since*, *.

d)   Let*, therefore*.

e)    But*, therefore*.

f)     Thus, with *, and*, therefore*.

g)    Andsince*, *.

h)   Let*, therefore*.

i)     But, *, therefore*.

j)     Therefore, *.

k)    Andsince*, and*, *, and*, therefore*.

l)     Therefore*.

m)  Andsince*, *.

n)   Therefore*.

o)    QEF.

46)         On a given *, Todescribe *.

a)    Let*, thusitisrequiredtodescribe *.

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

c)    Therefore*, therefore*.

d)   But*, therefore*, therefore*.

e)    Isaynextthat*.

f)     Forsince*, *.

g)    But*, therefore*.

h)   And, in *, therefore*.

i)     Therefore*.

j)     Anditwasalsoproved *.

k)    Therefore*.

l)     QEF.

47)         In *, *.

a)    Let*, Isaythat*.

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

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

d)   Forthesamereason *.

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

f)     Andsince*, *, and*, therefore* and*.

g)    Now *, for *.

h)   And*, for *.

i)     Therefore*.

j)     Similarly, if*, *, therefore*.

k)    And* and*.

l)     Therefore*.

m)  Therefore, etc.

n)   QED.