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Euclid Book 5
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Euclid Book 5 Proposition 22
Statement
Computational Explanation
Explanations
Let there be any number of magnitudes
A
,
B
,
C
, and others
D
,
E
,
F
equal to them in multitude, which taken two and two together are in the same ratio, so that, as
A
is to
B
, so is
D
to
E
, and, as
B
is to
C
, so is
E
to
F
; I say that they will also be in the same ratio ex aequali, <that is, as
A
is to
C
, so is
D
to
F
>.
For of
A
,
D
let equimultiples
G
,
H
be taken, and of
B
,
E
other, chance, equimultiples
K
,
L
; and, further, of
C
,
F
other, chance, equimultiples
M
,
N
.
Then, since, as
A
is to
B
, so is
D
to
E
, and of
A
,
D
equimultiples
G
,
H
have been taken, and of
B
,
E
other, chance, equimultiples
K
,
L
, therefore, as
G
is to
K
, so is
H
to
L
.
[
V
.
4
]
For the same reason also, as
K
is to
M
, so is
L
to
N
.
Since, then, there are three magnitudes
G
,
K
,
M
, and others
H
,
L
,
N
equal to them in multitude, which taken two and two together are in the same ratio, therefore, ex aequali, if
G
is in excess of
M
,
H
is also in excess of
N
; if equal, equal; and if less, less.
[
V
.
2
0
]
And
G
,
H
are equimultiples of
A
,
D
, and
M
,
N
other, chance, equimultiples of
C
,
F
.
Therefore, as
A
is to
C
, so is
D
to
F
.
[
V
.
D
e
f
.
5
]
Classes
Euclid's Elements
Theorems
Geometric Algebra
EuclidBook5
Related Theorems
EuclidBook5Proposition20
EuclidBook5Proposition21
EuclidBook5Proposition23
