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Euclid Book 5
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Euclid Book 5 Proposition 10a
Statement
I
f
a
f
i
r
s
t
m
a
g
n
i
t
u
d
e
h
a
s
a
g
r
e
a
t
e
r
r
a
t
i
o
t
o
a
s
e
c
o
n
d
t
h
a
n
a
t
h
i
r
d
m
a
g
n
i
t
u
d
e
d
o
e
s
(
A
B
E
F
>
C
D
E
F
)
,
t
h
e
n
t
h
e
f
i
r
s
t
i
s
g
r
e
a
t
e
r
t
h
a
n
t
h
e
t
h
i
r
d
(
A
B
>
C
D
)
.
Computational Explanation
G
e
o
m
e
t
r
i
c
S
c
e
n
e
{
A
.
,
B
.
,
C
.
,
D
.
,
E
.
,
F
.
}
,
{
x
.
,
y
.
,
z
.
}
,
S
t
y
l
e
[
L
i
n
e
[
{
A
.
,
B
.
}
]
,
R
e
d
]
,
S
t
y
l
e
[
L
i
n
e
[
{
C
.
,
D
.
}
]
,
P
i
n
k
]
,
S
t
y
l
e
[
L
i
n
e
[
{
E
.
,
F
.
}
]
,
B
l
u
e
]
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
B
.
]
x
.
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
C
.
,
D
.
]
y
.
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
E
.
,
F
.
]
z
.
,
x
.
z
.
>
y
.
z
.
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
B
.
]
>
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
C
.
,
D
.
]
,
x
.
>
y
.
C
o
n
c
l
u
s
i
o
n
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
B
.
]
>
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
C
.
,
D
.
]
Explanations
For let
A
have to
C
a greater ratio than
B
has to
C
; I say that
A
is greater than
B
. For, if not,
A
is either equal to
B
or less.
Now
A
is not equal to
B
; for in that case each of the magnitudes
A
,
B
would have had the same ratio to
C
;
[
V
.
7
]
but they have not; therefore
A
is not equal to
B
. Nor again is
A
less than
B
; for in that case
A
would have had to
C
a less ratio than
B
has to
C
;
[
V
.
8
]
but it has not; therefore
A
is not less than
B
.
But it was proved not to be equal either; therefore
A
is greater than
B
.
Classes
Euclid's Elements
Theorems
Geometric Algebra
EuclidBook5
Related Theorems
EuclidBook5Proposition10b
EuclidBook5Proposition8
