Summary. EuclidOfAlexandria. Elements. Formaldivisionsofpropositions. BookX.I.

Book10

1)   Two*, if*, if*, *.

a)    Let*, Isaythat*.

b)   For*.

c)    Let*, andlet*, let*, from*, let*, andfrom*, andlet*.

d)   Let*.

e)    Nowsince*, andfrom*, andfrom*, therefore*.

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

g)    Andsince*, and*, and*, therefore*.

h)   But*, therefore*.

i)     Therefore*.

j)     Therefore*.

k)    QED.

l)     Porism. Andthethereomcanbesimilarlyprovedevenif*.

2)   Ifwhen*, *.

a)    For*, let*, Isaythat*.

b)   Forif*, *.

c)    Let*ifpossible, andlet*, let*, let*, andlet*.

d)   Supposethisdoneandlet*.

e)    Thensince*, while*, therefore*.

f)     But*, therefore*.

g)    But*, therefore*.

h)   But*, therefore*, whichisimpossible.

i)     Therefore*, therefore*.

j)     Therefore, etc.

k)    QED.

3)   Given*, tofind*.

a)    Let*, thusitisrequiredtofind*.

b)   Now*.

c)    Ifthen*.

d)   Anditismanifestthat*, for*.

e)    Nextlet*.

f)     Thenif*, *, because*, let*, let*, andlet*.

g)    Sincethen*, while*, therefore*.

h)   But*, therefore*.

i)     But*, therefore*.

j)     But*, therefore*.

k)    Therefore*.

l)     Isaynextthat*.

m)  Forifnot*.

n)   Let*.

o)    Sincethen*, while*, therefore*.

p)   But*, therefore*.

q)   But*, therefore*.

r)    But*, and*, whichisimpossible.

s)    Therefore*, threrefore*.

t)     Therefore*.

u)   QED.

v)    Porism. Fromthisitismanifestthatif*, *.

4)   Given*, tofind*.

a)    Let*, thusitisrequiredtofind*.

b)   Let*, andlet*, then*.

c)    Firstlet*.

d)   Sincethen*, while*, therefore*.

e)    Anditismanifestthat*, for*.

f)     Nextlet*.

g)    Isayfirstthat*.

h)   Forsince*, *, and*, sothat*.

i)     But*, sothat*, therefore*.

j)     Nowlet*, andlet*.

k)    Sincethen*, while*, therefore*.

l)     But*, therefore*, therefore*.

m)  Isaynextthat*.

n)   Forifpossiblelet*, andlet*.

o)    Nowsince*, *, and*.

p)   But*, therefore*.

q)   But*, therefore*, therefore*.

r)    But*, therefore*, whichisimpossible.

s)    Therefore*, therefore*.

t)     Therefore*.

u)   Porism. Fromthisitismanifestthatif*, *.

v)    Similarlytoo*.

w)  QED.

5)   *.

a)    Let*, Isaythat*.

b)   Forsince*, *.

c)    Let*, andlet*.

d)   Andlet*, andlet*.

e)    Sincethen*, while*, therefore*, therefore*, therefore*.

f)     Againsince*, while*, therefore*, therefore*.

g)    Butitwasalsoprovedthat*, therefore*.

h)   Therefore*.

i)     QED.

6)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*, andlet*.

c)    Sincethen*, *, therefore*.

d)   But*, therefore*.

e)    Andsince*, therefore*.

f)     Againsince*, therefore*.

g)    Butiwasalsoprovedthat*, therefore*.

h)   But*, thereforealso*.

i)     Therefore*, therefore*.

j)     But*, therefore*.

k)    Further*, therefore*.

l)     Therefore*.

m)  Therefore, etc.

n)   QED.

o)    Porism. Fromthisitismanifesthatif*, *. Andif*, *. But*, therefore*.

7)   *.

a)    Let*, Isaythat*.

b)   Forif*, *.

c)    Butitisnot*, therefore*.

d)   Therefore, etc.

e)    QED.

8)   If*, *.

a)    Forlet*, Isaythat*.

b)   Forif*, *.

c)    Butithasnot*, therefore*.

d)   Therefore, etc.

e)    QED.

9)   *.

a)    Forlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    Let*.

d)   Sincethen*, while*, for*, and*, for*, thereforealso*.

e)    Nextas*, solet*, Isaythat*.

f)     Forsince*, *, while*, and*, thereforealso*.

g)    Therefore*, therefore*.

h)   Nextlet*, Isaythat*.

i)     Forif*, *.

j)     Butitisnottherefore*.

k)    Againlet*, Isaythat*.

l)     Forif*, *.

m)  Butithasnottherefore*.

n)   Therefore, etc.

o)    Porism. Anditismanifestfromwhathasbeenprovedthat*.

p)   Lemma. Ithasbeenprovedinthearithmeticalbooksthat*, andthat*. Anditismanifestfromthesepropositionsthat*. Forif*, *. Therefore*.

10)         Tofind*.

a)    Let*, thusitisrequiredtofind*.

b)   Let*, andlet*, for*, therefore*.

c)    Andsince*, therefore*, therefore*.

d)   Let*, therefore*.

e)    But*, therefore*, therefore*.

f)     Therefore*.

g)    QED.

11)         If*, *, andif*, *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    Andas*, therefore*, therefore*.

d)   Nextlet*, Isaythat*.

e)    Forsince*, therefore*.

f)     Andas*, therefore*, therefore*.

g)    Therefore, etc.

h)   QED.

12)         *.

a)    Forlet*, Isaythat*.

b)   Forsince*, therefore*.

c)    Let*.

d)   Againsince*, therefore*.

e)    Let*, and*, let*, sothat*, and*.

f)     Sincethen*, while*, thereforealso*.

g)    Againsince*, while*, thereforealso*.

h)   Butalso*, therefore*.

i)     Therefore*, therefore*.

j)     Therefore, etc.

k)    QED.

13)         If*, *.

a)    LEt*, andlet*, Isaythat*.

b)   Forif*, while*, *.

c)    But*, whichisimpossible.

d)   Therefore*, therefore*.

e)    Therefore, etc.

f)     QED.

g)    Lemma. Given*, tofind*.

h)   Let*, andlet*, thusitisrequiredtofind*.

i)     Let*, andlet*, let*.

j)     Itisthenmanifestthat*, andthat*.

k)    Similarlyalso*.

l)     Let*, andlet*.

m)  Let*.

n)   Itisagainmanifestthat*.

o)    QED.

14)         If*, *, andif*, *.

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

b)   Forsinceas*, thereforealso*.

c)    But*, and*.

d)   Therefore*, therefore*, thereforealso*, thereforeinversely*.

e)    But*, therefore*.

f)     Thereforeif*, andif*, *.

g)    Therefore, etc.

h)   QED.

15)         If*, *, andif*, *.

a)    Forlet*, Isaythat*.

b)   Forsince*, *.

c)    Let*, andlet*.

d)   Sincethen*, *.

e)    But*, therefore*, therefore*.

f)     Nextlet*, Isaythat*.

g)    Forsince*, *.

h)   Let*, andlet*.

i)     Sincethen*, *.

j)     But*, therefore*, therefore*.

k)    Therefore, etc.

l)     QED.

16)         If*, *, andif*, *.

a)    Forlet*, Isaythat*.

b)   Forif*, *.

c)    Let*ifpossible, andlet*.

d)   Sincethen*, therefore*.

e)    But*, therefore*.

f)     Therefore*, but*, whichisimpossible.

g)    Therefore*, therefore*.

h)   Similarlywecanprovethat*.

i)     Therefore*.

j)     Nextlet*.

k)    Firstlet*, Isaythat*.

l)     Forif*, *.

m)  Let*.

n)   Sincethen*, therefore*.

o)    But*, therefore*.

p)   Therefore*, but*, whichisimpossible.

q)   Therefore*, therefore*.

r)    Therefore, etc.

s)    QED.

t)     Lemma. If*, *.

u)   Forlet*, Isaythat*.

v)    Thisisindeedatoncemanifestforsince*, *, and*.

w)  Therefore, etc.

x)    QED.

17)         If*, *, andif*, *.

a)    Let*, andlet*.

b)   Let*, andlet*, Isaythat*.

c)    Forlet*, andlet*.

d)   Therefore*.

e)    Andsince*, and*, therefore*.

f)     And*, therefore*.

g)    But*, and*.

h)   And*, foragain*.

i)     Therefore*, sothat*.

j)     Itistobeprovedthat*.

k)    Since*, therefore*.

l)     But*, for*.

m)  Therefore*, sothat*, therefore*.

n)   Nextlet*, let*, andlet*.

o)    Itistobeprovedthat*.

p)   Withthesameconstructionwecanprovesimilarlythat*.

q)   But*.

r)    Therefore*, sothat*.

s)    But*, sothat*, andtherefore*.

t)     Therefore, etc.

u)   QED.

18)         If*, *, andif*, *. Andif*, *.

a)    Let*, andlet*.

b)   Let*, andlet*, Isaythat*.

c)    Forwiththesameconstructionasbeforewecanprovesimilarlythat*.

d)   Itistobeprovedthat*.

e)    Since*, therefore*.

f)     But*, therefore*, sothat*>

g)    And*, therefore*.

h)   Againlet*, andlet*.

i)     Let*.

j)     Itistobeprovedthat*.

k)    Forwiththesameconstructionwecanprovesimilarlythat*.

l)     But*, therefore*, sothat*.

m)  But*, therefore*, sothat*.

n)   Therefore, etc.

o)    QED.

p)   Lemma.

q)   Sinceithasbeenprovedthat*, while*, but*, if*, *.

r)    Butif*, *, butifagain*, *.

19)         *

a)    Forlet*, Isaythat*.

b)   For*, therefore*.

c)    Andsince*, while*, therefore*.

d)   Andas*.

e)    Therefore*.

f)     But*, therefore*.

g)    Therefore*.

h)   QED.

20)         If*, *.

a)    Forlet*, Isaythat*.

b)   For*, therefore*.

c)    But*, therefore*.

d)   And*.

e)    Therefore*, and*, therefore*.

f)     But*, therefore*.

g)    Therefore, etc.

h)   QED.

21)         If*, *. Letthelatterbecalled medial.

a)    Forlet*, Isaythat*, andlet*.

b)   Forlet*, therefore*.

c)    Andsince*, forbyhypothesis*, therefore*.

d)   And*, therefore*.

e)    But*, therefore*, sothat*.

f)     Andlet*.

g)    QED.

h)   Lemma. If*, *.

i)     Let*.

j)     Isaythat*.

k)    Forlet*, andlet*.

l)     Sincethen*, and*, and*, therefore*.

m)  Similarlyalso*.

n)   QED.

22)         *.

a)    Let*, andlet*, Isaythat*.

b)   Forsince*, *.

c)    Let*.

d)   But*, therefore*.

e)    But*, and*, therefore*.

f)     Thereforealso*.

g)    But*, therefore*.

h)   But*, therefore*, therefore*.

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

j)     But*, and*, therefore*.

k)    But*, therefore*.

l)     Therefore*.

m)  QED.

23)         *.

a)    Let*, Isaythat*.

b)   Forlet*, andlet*, therefore*.

c)    Andlet*.

d)   Sincethen*, *.

e)    But*, and*, therefore*.

f)     And*, therefore*.

g)    But*, therefore*.

h)   Therefore*.

i)     But*, therefore*.

j)     And*, therefore*.

k)    QED.

l)     Porism. Fromthisitismanifestthat*.

m)  Andinthesamewayaswasexplained*.

n)   Butif*, *, butif*, *.

24)         *.

a)    Forlet*, Isaythat*.

b)   For*, therefore*.

c)    Andsince*, while*, therefore*, sothat*.

d)   But*, therefore*.

e)    QED.

25)         *.

a)    Forlet*, Isaythat*.

b)   For*, therefore*.

c)    Let*, let*, let*, andfurtherlet*, therefore*.

d)   Sincethen*, and*, and*, therefore*.

e)    And*, therefore*.

f)     Andsince*, therefore*.

g)    And*, therefore*.

h)   Therefore*, therefore*.

i)     Andsince*, therefore*.

j)     But*, and*, therefore*.

k)    But*, therefore*, thereforealso*, therefore*.

l)     But*, therefore*.

m)  Therefore*.

n)   Andif*, *, butif*, *, andtherefore*.

o)    Therefore*.

p)   But*, therefore*.

q)   Therefore, etc.

r)    QED.

26)         *.

a)    Forifpossiblelet*, andlet*, let*, andlet*, therefore*.

b)   But*, therefore*.

c)    Sincethen*, and*, therefore*.

d)   And*, therefore*.

e)    Andsince*, and*, therefore*.

f)     But*, and*, therefore*.

g)    And*, therefore*.

h)   But*, and*, therefore*.

i)     But*, therefore*.

j)     But*, therefore*.

k)    Therefore*.

l)     But*, whichisimpossible.

m)  Therefore, etc.

n)   QED.

27)         Tofind*.

a)    Let*, let*, andlet*.

b)   Thensince*, *.

c)    Therefore*.

d)   Andsince*, and*, therefore*.

e)    And*, therefore*.

f)     Therefore*.

g)    Isaythat*.

h)   Forsince*, thereforealternately*.

i)     But*, thereforealso*, therefore*.

j)     But*, therefore*.

k)    Therefore*.

l)     QED.

28)         Tofind*.

a)    Let*, andlet*, andlet*.

b)   Since*, therefore*.

c)    Therefore*.

d)   Andsince*, and*, therefore*.

e)    But*, therefore*.

f)     Therefore*.

g)    Isaynextthat*.

h)   Forsince*, thereforealternately*.

i)     But*, thereforealso*, therefore*.

j)     But*, therefore*.

k)    Therefore*.

l)     QED.

m)  Lemma1. Tofind*.

n)   Let*, andlet*.

o)    Thensince*, *, therefore*.

p)   Let*.

q)   Let*.

r)    Now*.

s)    And*.

t)     Therefore*.

u)   Anditismanifestthat*.

v)    Butwhen*, *.

w)  Lemma2. Tofind*.

x)    Forlet*, and*, andlet*.

y)    Itisthenmanifestthat*.

z)    Let*, therefore*.

aa) Isaythen*.

bb)                  Forif*, *.

cc)  Firstifpossible*, let*, andlet*.

dd)                  Sincethen*, and*, therefore*, therefore*.

ee) Therefore*.

ff)   But*, therefore*.

gg) Andif*, *, whichisabsurd.

hh)                  Therefore*.

ii)    Isaynextthat*.

jj)    Forifpossiblelet*, andlet*.

kk) Nowitwillagainfollowthat*, sothat*, andforthisreason*.

ll)    Butbyhypothesis*.

mm)               Thus*, whichisabsurd.

nn)                  Therefore*.

oo) Anditwasprovedthat*.

pp)                  Therefore*.

qq)                  QED.

29)         Tofind*.

a)    Forlet*, let*, andlet*.

b)   Let*.

c)    Since*, therefore*, therefore*.

d)   But*, therefore*, therefore*.

e)    Andsince*, *, therefore*.

f)     Andsince*, therefore*.

g)    But*, therefore*, therefore*.

h)   And*, therefore*.

i)     Therefore*.

j)     QED.

30)         Tofind*.

a)    Let*, let*, andlet*.

b)   Theninasimilarmannertotheprecedingwecanprovethat*.

c)    Andsince*, therefore*.

d)   But*, therefore*, therefore*.

e)    And*.

f)     Therefore*.

g)    QED.

31)         Tofind*.

a)    Let*.

b)   Andlet*.

c)    Now*, therefore*, therefore*.

d)   Let*.

e)    Now*, therefore*.

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

g)    But*, thereforealso*.

h)   But*, therefore*.

i)     And*, therefore*.

j)     Andsince*, and*, thereforealso*.

k)    Therefore*.

l)     Similarlyalsoitcanbeprovedthat*.

m)  QED.

32)         Tofind*.

a)    Let*, andlet*.

b)   Therefore*, therefore*.

c)    Let*.

d)   Thensince*, while*, and*, therefore*.

e)    But*, thereforealso*.

f)     But*, therefore*.

g)    But*, therefore*.

h)   Andsince*, while*, thereforealso*.

i)     Isaynextthat*.

j)     Forsince*, therefore*.

k)    Therefore*.

l)     Similarlyitcanbeprovedthat*.

m)  QED.

n)   Lemma. Let*, Isaythat*.

o)    Andfirstthat*.

p)   Forsince*, therefore*.

q)   Andsince*, therefore*, therefore*.

r)    Forthesamereason*.

s)    Andsinceif*, *, therefore*, therefore*.

t)     Isaythat*.

u)   Forsince*, therefore*.

v)    Therefore*.

w)  QED.

33)         Tofind*.

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

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

c)    And*, while*, and*, therefore*, therefore*.

d)   Andsince*, therefore*, sothat*.

e)    Andsince*, andbyhypothesis*, therefore*, therefore*, sothat*.

f)     But*, therefore*.

g)    But*, therefore*.

h)   Butitwasalsoprovedthat*.

i)     Therefore*.

j)     QED.

34)         Tofind*.

a)    Let*, let*, let*, let*, therefore*.

b)   Let*, andlet*.

c)    Since*, therefore*.

d)   But*, therefore*.

e)    Andsince*, therefore*.

f)     Andsince*, therefore*.

g)    But*, therefore*.

h)   But*, sothat*.

i)     Therefore*.

j)     QED.

35)         If*, *.

a)    Let*, let*, andlet*.

b)   Thensince*, *.

c)    Andsince*, therefore*.

d)   Andsince*, therefore*, therefore*, sothat*.

e)    But*, therefore*.

f)     And*, therefore*.

g)    Andsince*, while*, therefore*, sothat*.

h)   But*, and*, therefore*.

i)     Therefore*.

j)     QED.

36)         If*, *, andletitbecalled binomial.

a)    Forlet*, Isaythat*.

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

c)    But*, and*, for*, therefore*.

d)   And*.

e)    But*, therefore*, sothat*.

f)     Andletitbecalled*.

g)    QED.

37)         If*, *, andletitbecalled a firstbimedialstraightline.

a)    Forlet*, Isaythat*.

b)   Forsince*, therefore*, and*.

c)    But*, therefore*, therefore*.

d)   Andletitbecalled*.

e)    QED.

38)         If*, *, andletitbecalled a secondbimedialstraightline.

a)    Forlet*, Isaythat*.

b)   Forlet*, andlet*.

c)    Thensince*, let*, therefore*.

d)   Andsince*, therefore*.

e)    Buthypothesis*.

f)     And*, while*, therefore*.

g)    And*, therefore*.

h)   Sincethen*, and*, therefore*.

i)     But*, and*.

j)     Therefore*.

k)    But*, and*.

l)     Therefore*, sothat*.

m)  Therefore*, sothat*.

n)   But*, and*, therefore*.

o)    But*, therefore*.

p)   Andletitbecalled*.

q)   QED.

39)         If*, *, andletitbecalled major.

a)    Forlet*, Isaythat*.

b)   Forsince*, *.

c)    But*, therefore*, sothat*, therefore*, sothat*.

d)   Andletitbecalled*.

e)    QED.

40)         If*, *, andletitbecalled a rationalplusamedialarea.

a)    Forlet*, and*, Isaythat*.

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

c)    But*, therefore*.

d)   Therefore*.

e)    Andletitbecalled*.

f)     QED.

41)         If*, *, andletitbecalled thesideofthesumoftwomedialareas.

a)    Forlet*, Isaythat*.

b)   Let*, therefore*.

c)    Nowsince*, therefore*.

d)   And*, therefore*.

e)    Forthesamereason*.

f)     Andsince*, *, sothat*.

g)    And*, therefore*, therefore*.

h)   But*, therefore*.

i)     But*, therefore*.

j)     Andletitbecalled*.

k)    QED.

l)     Lemma. *.

m)  Let*, andlet*, Isaythat*.

n)   Forlet*.

o)    Thensince*, let*, therefore*.

p)   But*, therefore*, therefore*.

q)   Andsince*, andfurther*, therefore*.

r)    And*, therefore*, sothat*.

s)    Thereforealso*.

t)     QED.

42)         *.

a)    Let*, therefore*.

b)   Isaythat*.

c)    Forifpossiblelet*.

d)   Itisthenmanifestthat*.

e)    Forifpossiblelet*.

f)     Then*, and*, thus*, whichiscontrarytothehypothesis.

g)    Therefore*.

h)   Forthisreasonalso*.

i)     Therefore*, because*.

j)     But*, therefore*, whichisabsurdfor*.

k)    Therefore*, therefore*.

l)     QED.

43)         *.

a)    Let*, Isaythat*.

b)   Forifpossiblelet*.

c)    Sincethen*, while*, therefore*, whichisabsurd.

d)   Therefore*, therefore*.

e)    QED.

44)         *.

a)    Let*, sothat*, itisthenmanifestthat*.

b)   Isaythat*.

c)    Forifpossiblelet*, itisthenclearthat*, andsupposethat*.

d)   Nowlet*, let*, andlet*, therefore*.

e)    Nowsince*, therefore*.

f)     And*, therefore*.

g)    Forthesamereason*.

h)   Andsince*, therefore*.

i)     But*, therefore*.

j)     But*.

k)    And*.

l)     Therefore*.

m)  But*, and*, therefore*, sothat*.

n)   And*, therefore*.

o)    Butif*, *.

p)   Therefore*.

q)   Inthesameway*, and*.

r)    And*.

s)    For*.

t)     But*, thereforealso*, sothat*.

u)   Therefore*.

v)    QED.

45)         *.

a)    Let*, Isaythat*.

b)   Forifpossiblelet*.

c)    Thensince*, while*, therefore*, whichisimpossible.

d)   Therefore*, therefore*.

e)    QED.

46)         *.

a)    Let*, sothat*, Isaythat*.

b)   Forifpossiblelet*, sothat*.

c)    Sincethen*, *, while*, therefore*, whichisimpossible.

d)   Therefore*, therefore*.

e)    QED.

47)         *.

a)    Let*, sothat*, Isaythat*.

b)   Forifpossiblelet*, let*, and*, therefore*.

c)    Againlet*, therefore*.

d)   Andsincebyhypothesis*, therefore*.

e)    And*, therefore*.

f)     Forthesamereason*.

g)    Andsince*, therefore*, sothat*.

h)   And*, therefore*, therefore*.

i)     Similarlywecanprovethat*.

j)     And*, therefore*, whichisabsurd.

k)    Therefore*, therefore*.

l)     QED.

48)         Tofind*.

a)    Let*, let*.

b)   Therefore*.

c)    Let*.

d)   But*, therefore*, sothat*.

e)    And*, therefore*.

f)     Andsince*, therefore*, therefore*.

g)    Therefore*, therefore*.

h)   Isaythat*.

i)     Forsince*, while*, therefore*.

j)     Let*.

k)    Nowsince*, therefore*.

l)     But*, therefore*

m)  Therefore*, therefore*.

n)   And*.

o)    Therefore*.

p)   QED.

49)         Tofind*.

a)    Let*, let*, therefore*.

b)   Let*, therefore*.

c)    Therefore*.

d)   Nowsince*, *.

e)    Therefore*, therefore*, therefore*.

f)     Itixnexttobeprovedthat*.

g)    Forsince*, while*, therefore*.

h)   Let*, therefore*.

i)     But*, therefore*.

j)     Therefore*, sothat*.

k)    And*.

l)     Therefore*.

m)  QED.

50)         Tofind*.

a)    Let*.

b)   Let*.

c)    Let*, andlet*, therefore*.

d)   And*, therefore*.

e)    Andsince*, *, therefore*.

f)     Nextlet*, therefore*.

g)    But*, therefore*.

h)   Andsince*, *, therefore*.

i)     Therefore*, therefore*.

j)     Isaynextthat*.

k)    Forsince*, and*, therefore*.

l)     But*, therefore*.

m)  Andsince*, therefore*.

n)   Let*, therefore*.

o)    But*, therefore*.

p)   Therefore*.

q)   And*, and*.

r)    Therefore*.

s)    QED.