Euclid Book 5 Proposition 18
Statement
If a subtracted part of a whole is to the remainder as another subtracted part of another whole is to the remainder (), then the former whole is to the remainder as the latter is to the remainder. ().
AB
CD
AE
CF
EB
FD
AB
CD
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.},{a.,b.,c.,d.}},Line[{A.,E.,B.}],Line[{C.,F.,D.}],Style[Line[{A.,E.}],Red],Style[Line[{E.,B.}],Pink],Style[Line[{C.,F.}],Blue],Style[Line[{F.,D.}],Cyan],EuclideanDistance[A.,E.]a.,EuclideanDistance[E.,B.]b.,EuclideanDistance[C.,F.]c.,EuclideanDistance[F.,D.]d.,,,
a.
b.
c.
d.
EuclideanDistance[A.,B.]
EuclideanDistance[E.,B.]
EuclideanDistance[C.,D.]
EuclideanDistance[F.,D.]
a.+b.
b.
c.+d.
d.
| ||
 |
Explanations
Let , , , be magnitudes proportional separando, so that, as is to , so is to ; I say that they will also be proportional componendo, that is, as is to , so is to .
For, if be not to as to , then, as is to , so will be either to some magnitude less than or to a greater.
First, let it be in that ratio to a less magnitude.
Then, since, as is to , so is to , they are magnitudes proportional componendo; so that they will also be proportional separando.
Therefore, as is to , so is to .
But also, by hypothesis, as is to , so is to .
Therefore also, as is to , so is to .
But the first is greater than the third ; therefore the second is also greater than the fourth .
But it is also less: which is impossible.
Therefore, as is to , so is not to a less magnitude than .
Similarly we can prove that neither is it in that ratio to a greater; it is therefore in that ratio to itself.
AE
EB
CF
FD
AE
EB
CF
FD
AB
BE
CD
FD
For, if
CD
DF
AB
BE
AB
BE
CD
DF
First, let it be in that ratio to a less magnitude
DG
Then, since, as
AB
BE
CD
DG
Therefore, as
AE
EB
CG
GD
But also, by hypothesis, as
AE
EB
CF
FD
Therefore also, as
CG
GD
CF
FD
But the first
CG
CF
GD
FD
But it is also less: which is impossible.
Therefore, as
AB
BE
CD
FD
Similarly we can prove that neither is it in that ratio to a greater; it is therefore in that ratio to
FD