Euclid Book 5 Proposition 23
Statement
Given two groups of three magnitudes (AB, CD, and EF; GH, IJ, and KL), if the first magnitude is to the second as the fifth is to the sixth () and the second is to the third as the fourth is to the fifth (), then the first magnitude is to the third as the fourth to the sixth ().
AB
CD
IJ
KL
CD
EF
GH
IJ
AB
EF
GH
KL
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.,G.,H.,I.,J.,K.,L.},{a.,b.,c.,d.,e.,f.}},Style[Line[{A.,B.}],Red],Style[Line[{C.,D.}],Red],Style[Line[{E.,F.}],Red],Style[Line[{G.,H.}],Blue],Style[Line[{I.,J.}],Blue],Style[Line[{K.,L.}],Blue],EuclideanDistance[A.,B.]a.,EuclideanDistance[C.,D.]b.,EuclideanDistance[E.,F.]c.,EuclideanDistance[G.,H.]d.,EuclideanDistance[I.,J.]e.,EuclideanDistance[K.,L.]f.,,,,
a.
b.
e.
f.
b.
c.
d.
e.
EuclideanDistance[A.,B.]
EuclideanDistance[E.,F.]
EuclideanDistance[G.,H.]
EuclideanDistance[K.,L.]
a.
c.
d.
f.
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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
B
C
D
E
B
D
C
E
And, since
H
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
A
C
D
F