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Euclid Book 5
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Euclid Book 5 Proposition 18
Statement
Computational Explanation
Explanations
Let
A
E
,
E
B
,
C
F
,
F
D
be magnitudes proportional separando, so that, as
A
E
is to
E
B
, so is
C
F
to
F
D
; I say that they will also be proportional componendo, that is, as
A
B
is to
B
E
, so is
C
D
to
F
D
.
For, if
C
D
be not to
D
F
as
A
B
to
B
E
, then, as
A
B
is to
B
E
, so will
C
D
be either to some magnitude less than
D
F
or to a greater.
First, let it be in that ratio to a less magnitude
D
G
.
Then, since, as
A
B
is to
B
E
, so is
C
D
to
D
G
, they are magnitudes proportional componendo; so that they will also be proportional separando.
[
V
.
1
7
]
Therefore, as
A
E
is to
E
B
, so is
C
G
to
G
D
.
But also, by hypothesis, as
A
E
is to
E
B
, so is
C
F
to
F
D
.
Therefore also, as
C
G
is to
G
D
, so is
C
F
to
F
D
.
[
V
.
1
1
]
But the first
C
G
is greater than the third
C
F
; therefore the second
G
D
is also greater than the fourth
F
D
.
[
V
.
1
4
]
But it is also less: which is impossible.
Therefore, as
A
B
is to
B
E
, so is not
C
D
to a less magnitude than
F
D
.
Similarly we can prove that neither is it in that ratio to a greater; it is therefore in that ratio to
F
D
itself.
Classes
Euclid's Elements
Theorems
Geometric Algebra
EuclidBook5
