Euclid Book 5 Proposition 22
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 fourth is to the fifth () and the second is to the third as the fifth is to the sixth (), then the first magnitude is to the third as the fourth to the sixth ().
AB
CD
GH
IJ
CD
EF
IJ
KL
AB
EF
GH
KL
Computational Explanation
Explanations
Let there be any number of magnitudes , , , and others , , equal to them in multitude, which taken two and two together are in the same ratio, so that, as is to , so is to , and, as is to , so is to; I say that they will also be in the same ratio ex aequali, <that is, as is to , so is to >.
For of, let equimultiples , be taken, and of , other, chance, equimultiples , ; and, further, of , other, chance, equimultiples , .
Then, since, as is to , so is to , and of , equimultiples , have been taken, and of , other, chance, equimultiples , , therefore, as is to , so is to .
For the same reason also, 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, 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 , other, chance, equimultiples of , .
Therefore, as is to , so is to .
A
B
C
D
E
F
A
B
D
E
B
C
E
F
A
C
D
F
For of
A
D
G
H
B
E
K
L
C
F
M
N
Then, since, as
A
B
D
E
A
D
G
H
B
E
K
L
G
K
H
L
For the same reason also, as
K
M
L
N
Since, then, there are three magnitudes
G
K
M
H
L
N
G
M
H
N
And
G
H
A
D
M
N
C
F
Therefore, as
A
C
D
F