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Euclid Book 5
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Euclid Book 5 Proposition 10b
Statement
I
f
t
h
e
r
a
t
i
o
o
f
a
f
i
r
s
t
m
a
g
n
i
t
u
d
e
t
o
a
s
e
c
o
n
d
i
s
g
r
e
a
t
e
r
t
h
a
n
t
h
e
r
a
t
i
o
o
f
t
h
e
f
i
r
s
t
t
o
a
t
h
i
r
d
(
E
F
C
D
>
E
F
A
B
)
,
t
h
e
n
t
h
e
s
e
c
o
n
d
m
a
g
n
i
t
u
d
e
i
s
l
e
s
s
t
h
a
n
t
h
e
t
h
i
r
d
(
C
D
<
A
B
)
.
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
.
,
z
.
y
.
>
z
.
x
.
,
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
C
.
,
D
.
]
<
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
B
.
]
,
y
.
<
x
.
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
[
C
.
,
D
.
]
<
E
u
c
l
i
d
e
a
n
D
i
s
t
a
n
c
e
[
A
.
,
B
.
]
Explanations
Let
C
have to
B
a greater ratio than
C
has to
A
; I say that
B
is less than
A
. For, if not, it is either equal or greater.
Now
B
is not equal to
A
; for in that case
C
would have had the same ratio to each of the magnitudes
A
,
B
;
[
V
.
7
]
but it has not; therefore
A
is not equal to
B
. Nor again is
B
greater than
A
; for in that case
C
would have had to
B
a less ratio than it has to
A
;
[
V
.
8
]
but it has not; therefore
B
is not greater than
A
. But it was proved that it is not equal either; therefore
B
is less than
A
.
Classes
Euclid's Elements
Theorems
Geometric Algebra
EuclidBook5
Related Theorems
EuclidBook5Proposition10a
EuclidBook5Proposition8
