Euclid Book 5 Proposition 15
Statement
If two magnitudes are equimultiples of two others (AB3CD, EF3GH), then the first two have the same ratio as the latter two ().
AB
EF
CD
GH
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.,G.,H.},{a.,b.}},{Style[Line[{A.,B.}],Red],Style[Line[{C.,D.}],Red],Style[Line[{E.,F.}],Blue],Style[Line[{G.,H.}],Blue],EuclideanDistance[A.,B.]3EuclideanDistance[C.,D.]3a.,EuclideanDistance[E.,F.]3EuclideanDistance[G.,H.]3b.},,
EuclideanDistance[A.,B.]
EuclideanDistance[E.,F.]
EuclideanDistance[C.,D.]
EuclideanDistance[G.,H.]
3a.
3b.
a.
b.
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Explanations
For let be the same multiple of that is of ; I say that, as is to, so is to .
For, since is the same multiple of that is of , as many magnitudes as there are in equal to , so many are there also 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. , are equal to one another, and , , are also equal to one another, therefore, as is to , so is to , and to .
Therefore, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; therefore, as is to , so is to .
But is equal to and to; therefore, as is to, so is to .
AB
C
DE
F
C
F
AB
DE
For, since
AB
C
DE
F
AB
C
DE
F
AB
AG
GH
HB
C
DE
DK
KL
LE
F
AG
GH
HB
DK
KL
LE
And, since
AG
GH
HB
DK
KL
LE
AG
DK
GH
KL
HB
LE
Therefore, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; therefore, as
AG
DK
AB
DE
But
AG
C
DK
F
C
F
AB
DE