Under Development
A collection of classical geometry in computable formats along with code and diagrams.
Computable Euclid
›
Euclid Book 5
›
Browse books
Euclid Book 1
Euclid Book 2
Euclid Book 3
Euclid Book 4
Euclid Book 5
Euclid Book 6
Euclid Book 13
Euclid Book 5 Proposition 13
Statement
Computational Explanation
Explanations
For let a first magnitude
A
have to a second
B
the same ratio as a third
C
has to a fourth
D
, and let the third
C
have to the fourth
D
a greater ratio than a fifth
E
has to a sixth
F
; I say that the first
A
will also have to the second
B
a greater ratio than the fifth
E
to the sixth
F
.
For, since there are some equimultiples of
C
,
E
, and of
D
,
F
other, chance, equimultiples, such that the multiple of
C
is in excess of the multiple of
D
, while the multiple of
E
is not in excess of the multiple of
F
,
[
V
.
D
e
f
.
7
]
let them be taken, and let
G
,
H
be equimultiples of
C
,
E
, and
K
,
L
other, chance, equimultiples of
D
,
F
, so that
G
is in excess of
K
, but
H
is not in excess of
L
; and, whatever multiple
G
is of
C
, let
M
be also that multiple of
A
, and, whatever multiple
K
is of
D
, let
N
be also that multiple of
B
.
Now, since, as
A
is to
B
, so is
C
to
D
, and of
A
,
C
equimultiples
M
,
G
have been taken, and of
B
,
D
other, chance, equimultiples
N
,
K
, therefore, if
M
is in excess of
N
,
G
is also in excess of
K
, if equal, equal, and if less, less.
[
V
.
D
e
f
.
5
]
But
G
is in excess of
K
; therefore
M
is also in excess of
N
.
But
H
is not in excess of
L
; and
M
,
H
are equimultiples of
A
,
E
, and
N
,
L
other, chance, equimultiples of
B
,
F
; therefore
A
has to
B
a greater ratio than
E
has to
F
.
[
V
.
D
e
f
.
7
]
Classes
Euclid's Elements
Theorems
Geometric Algebra
EuclidBook5
