Euclid Book 5 Proposition 3
Statement
If a first magnitude is the same multiple of a second that a third is of a fourth (AB3CD, EF3GH), and if equimultiples are taken of the first and third (IJ2AB, KL2EF), then the magnitudes taken also are equimultiples of the second and the fourth, respectively (IJ6CD, KL6GH).
Computational Explanation
GeometricSceneA.,B.,C.,D.,E.,F.,G.,H.,I.,J.,K.,L.,a.,b.,c.,d.,StyleLine[A.,B.],Red,StyleLineC.,D.,Red,StyleLine[E.,F.],Blue,StyleLineG.,H.,Blue,StyleLineI.,J.,Red,StyleLine[K.,L.],Blue,EuclideanDistance[A.,B.]a.3EuclideanDistanceC.,D.3b.,EuclideanDistance[E.,F.]c.3EuclideanDistanceG.,H.3d.,EuclideanDistanceI.,J.2EuclideanDistance[A.,B.]2a.,EuclideanDistance[K.,L.]2EuclideanDistance[E.,F.]2c.,EuclideanDistanceI.,J.6EuclideanDistanceC.,D.,EuclideanDistance[K.,L.]6EuclideanDistanceG.,H.,23b.(2×3)b.,23d.(2×3)d.
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Explanations
Let a first magnitude be the same multiple of a second that a third is of a fourth , and let equimultiples , be taken of , ; I say that is the same multiple of that is of .
For, since is the same multiple of that is of , therefore, as many magnitudes as there are in equal to , so many also are there in equal to .
Let be divided into the magnitudes , equal to , and into the magnitudes , equal to ; then the multitude of the magnitudes , will be equal to the multitude of the magnitudes , .
And, since is the same multiple of that is of , while is equal to , and to , therefore is the same multiple of that is of .
For the same reason is the same multiple of that is of .
Since, then, a first magnitude is the same multiple of a second that a third is of a fourth , and a fifth is also the same multiple of the second that a sixth is of the fourth , therefore the sum of the first and fifth, , is also the same multiple of the second that the sum of the third and sixth, , is of the fourth .
A
B
C
D
EF
GH
A
C
EF
B
GH
D
For, since
EF
A
GH
C
EF
A
GH
C
Let
EF
EK
KF
A
GH
GL
LH
C
EK
KF
GL
LH
And, since
A
B
C
D
EK
A
GL
C
EK
B
GL
D
For the same reason
KF
B
LH
D
Since, then, a first magnitude
EK
B
GL
D
KF
B
LH
D
EF
B
GH
D