Euclid Book 5 Proposition 23
Statement
Computational Explanation
Explanations
Let there be three magnitudes , , , and others equal to them in multitude, which, taken two and two together, are in the same proportion, namely , , ; and let the proportion of them be perturbed, so that, as is to , so is to , and, as is to , so is to ; I say that, as is to , so is to .
Of, , let equimultiples , , be taken, and of , , other, chance, equimultiples , , .
Then, since, are equimultiples of , , and parts have the same ratio as the same multiples of them, therefore, as is to , so is to .
For the same reason also, as is to , so is to .
And, as is to , so is to ; therefore also, as is to , so is to .
Next, since, as is to , so is to , alternately, also, as is to , so is to .
And, since, are equimultiples of , , and parts have the same ratio as their equimultiples, therefore, as is to , so is to .
But, as is to , so is to ; therefore also, as is to , so is to .
Again, since, are equimultiples of , , therefore, as is to , so is to .
But, as is to , so is to ; therefore also, as is to , so is to ,and, alternately, as is to , so is to .
But it was also proved that, as is to , so is to .
Since, then, there are three magnitudes, , , and others equal to them in multitude , , , which taken two and two together are in the same ratio, and the proportion of them is perturbed, therefore, ex aequali, if is in excess of , is also in excess of ; if equal, equal; and if less, less.
And, are equimultiples of , , and, of , .
Therefore, as is to , so is to .
A
B
C
D
E
F
A
B
E
F
B
C
D
E
A
C
D
F
Of
A
B
D
G
H
K
C
E
F
L
M
N
Then, since
G
H
A
B
A
B
G
H
For the same reason also, as
E
F
M
N
And, as
A
B
E
F
G
H
M
N
Next, since, as
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C
D
E
B
D
C
E
And, since
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K
B
D
B
D
H
K
But, as
B
D
C
E
H
K
C
E
Again, since
L
M
C
E
C
E
L
M
But, as
C
E
H
K
H
K
L
M
H
L
K
M
But it was also proved that, as
G
H
M
N
Since, then, there are three magnitudes
G
H
L
K
M
N
G
L
K
N
And
G
K
A
D
L
N
C
F
Therefore, as
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C
D
F