Euclid Book 5 Proposition 2
Statement
If a first magnitude is the same multiple of a second that a third is of a fourth (AB3CD, EF3GH), and a fifth also is the same multiple of the second that a sixth is of the fourth (BI2CD, FJ2GH), then the sum of the first and fifth also is the same multiple of the second that the sum of the third and sixth is of the fourth (AI5CD, EJ5GH).
Computational Explanation
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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 a fifth, , also be the same multiple of the second, , that a sixth, , is of the fourth ; I say that the sum of the first and fifth, , will be the same multiple of the second, , that the sum of the third and sixth, , is of the fourth, .
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 .
For the same reason also, as many as there are in equal to , so many are there also in equal to ; therefore, as many as there are in the whole equal to , so many also are there in the whole equal to .
Therefore, whatever multiple is of , that multiple also is of . Therefore the sum of the first and fifth, , is the same multiple of the second, , that the sum of the third and sixth, , is of the fourth, .
AB
C
DE
F
BG
C
EH
F
AG
C
DH
F
For, since
AB
C
DE
F
AB
C
DE
F
For the same reason also, as many as there are in
BG
C
EH
F
AG
C
DH
F
Therefore, whatever multiple
AG
C
DH
F
AG
C
DH
F