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Euclid Book 5 Proposition 4
Statement
I
f
a
f
i
r
s
t
m
a
g
n
i
t
u
d
e
h
a
s
t
o
a
s
e
c
o
n
d
t
h
e
s
a
m
e
r
a
t
i
o
a
s
a
t
h
i
r
d
t
o
a
f
o
u
r
t
h
(
A
B
C
D
E
F
G
H
)
,
t
h
e
n
a
n
y
e
q
u
i
m
u
l
t
i
p
l
e
s
o
f
t
h
e
f
i
r
s
t
a
n
d
t
h
i
r
d
(
I
J
2
A
B
,
K
L
2
E
F
)
a
l
s
o
h
a
v
e
t
h
e
s
a
m
e
r
a
t
i
o
t
o
a
n
y
e
q
u
i
m
u
l
t
i
p
l
e
s
o
f
t
h
e
s
e
c
o
n
d
a
n
d
f
o
u
r
t
h
(
M
N
3
C
D
,
O
P
3
G
H
)
r
e
s
p
e
c
t
i
v
e
l
y
(
I
J
M
N
K
L
O
P
)
.
Computational Explanation
Explanations
For let a first magnitude
A
have to a second
B
the same ratio as a third
C
to a fourth
D
; and let equimultiples
E
,
F
be taken of
A
,
C
, and
G
,
H
other, chance, equimultiples of
B
,
D
; I say that, as
E
is to
G
, so is
F
to
H
.
For let equimultiples
K
,
L
be taken of
E
,
F
, and other, chance, equimultiples
M
,
N
of
G
,
H
.
Since
E
is the same multiple of
A
that
F
is of
C
, and equimultiples
K
,
L
of
E
,
F
have been taken, therefore
K
is the same multiple of
A
that
L
is of
C
.
[
V
.
3
]
For the same reason
M
is the same multiple of
B
that
N
is of
D
. And, since, as
A
is to
B
, so is
C
to
D
, and of
A
,
C
equimultiples
K
,
L
have been taken, and of
B
,
D
other, chance, equimultiples
M
,
N
, therefore, if
K
is in excess of
M
,
L
also is in excess of
N
, if it is equal, equal, and if less, less.
[
V
.
D
e
f
.
5
]
And
K
,
L
are equimultiples of
E
,
F
, and
M
,
N
other, chance, equimultiples of
G
,
H
; therefore, as
E
is to
G
, so is
F
to
H
.
[
V
.
D
e
f
.
5
]
Classes
Euclid's Elements
Theorems
Geometric Algebra
EuclidBook5
