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Euclid Book 5
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Euclid Book 5 Proposition 21
Statement
Computational Explanation
Explanations
Let there be three magnitudes
A
,
B
,
C
, and others
D
,
E
,
F
equal to them in multitude, which taken two and two are in the same ratio, and let the proportion of them be perturbed, so that, as
A
is to
B
, so is
E
to
F
, and, as
B
is to
C
, so is
D
to
E
, and let
A
be greater than
C
ex aequali; I say that
D
will also be greater than
F
; if
A
is equal to
C
, equal; and if less, less.
For, since
A
is greater than
C
, and
B
is some other magnitude, therefore
A
has to
B
a greater ratio than
C
has to
B
.
[
V
.
8
]
But, as
A
is to
B
, so is
E
to
F
, and, as
C
is to
B
, inversely, so is
E
to
D
. Therefore also
E
has to
F
a greater ratio than
E
has to
D
.
[
V
.
1
3
]
But that to which the same has a greater ratio is less;
[
V
.
1
0
]
therefore
F
is less than
D
; therefore
D
is greater than
F
.
Similarly we can prove that, if
A
be equal to
C
,
D
will also be equal to
F
; and if less, less.
Classes
Euclid's Elements
Theorems
Geometric Algebra
EuclidBook5
Related Theorems
EuclidBook5Proposition20
EuclidBook5Proposition22
EuclidBook5Proposition23
