Euclid Book 5 Proposition 4
Statement
If a first magnitude has to a second the same ratio as a third to a fourth (), then any equimultiples of the first and third (IJ2AB, KL2EF) also have the same ratio to any equimultiples of the second and fourth (MN3CD, OP3GH) respectively ().
AB
CD
EF
GH
IJ
MN
KL
OP
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.,G.,H.,I.,J.,K.,L.,M.,N.,O.,P.},{a.,b.,c.,d.}},Style[Line[{A.,B.}],Red],Style[Line[{C.,D.}],Red],Style[Line[{E.,F.}],Blue],Style[Line[{G.,H.}],Blue],Style[Line[{I.,J.}],Red],Style[Line[{K.,L.}],Blue],Style[Line[{M.,N.}],Red],Style[Line[{O.,P.}],Blue],EuclideanDistance[A.,B.]a.,EuclideanDistance[C.,D.]b.,EuclideanDistance[E.,F.]c.,EuclideanDistance[G.,H.]d.,,EuclideanDistance[I.,J.]2a.,EuclideanDistance[K.,L.]2c.,EuclideanDistance[M.,N.]3b.,EuclideanDistance[O.,P.]3d.,,
a.
b.
c.
d.
EuclideanDistance[I.,J.]
EuclideanDistance[M.,N.]
EuclideanDistance[K.,L.]
EuclideanDistance[O.,P.]
2a.
3b.
2c.
3d.
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Explanations
For let a first magnitude have to a second the same ratio as a third to a fourth ; and let equimultiples , be taken of , , and , other, chance, equimultiples of , ; I say that, as is to , so is to .
For let equimultiples, be taken of , , and other, chance, equimultiples , of , .
Since is the same multiple of that is of , and equimultiples , of , have been taken, therefore is the same multiple of that is of .
For the same reason is the same multiple of that is of. And, since, as is to , so is to , and of , equimultiples , have been taken, and of , other, chance, equimultiples , , therefore, if is in excess of , also is in excess of , if it is equal, equal, and if less, less.
And, are equimultiples of , , and , other, chance, equimultiples of , ; therefore, as is to , so is to .
A
B
C
D
E
F
A
C
G
H
B
D
E
G
F
H
For let equimultiples
K
L
E
F
M
N
G
H
Since
E
A
F
C
K
L
E
F
K
A
L
C
For the same reason
M
B
N
D
A
B
C
D
A
C
K
L
B
D
M
N
K
M
L
N
And
K
L
E
F
M
N
G
H
E
G
F
H