PROPOSITION 19.
If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.
Let
A,
B,
C,
D be four numbers in proportion, so that,
as A is to B, so is C to D; and let
A by multiplying
D make
E, and let
B by multiplying
C make
F; I say that
E is equal to
F.
For let
A by multiplying
C make
G.
Since, then,
A by multiplying
C has made
G, and by multiplying
D has made
E, the number
A by multiplying the two numbers
C,
D has made
G,
E.
Therefore, as
C is to
D, so is
G to
E. [
VII. 17]
But, as
C is to
D, so is
A to
B; therefore also, as
A is to
B, so is
G to
E.
Again, since
A by multiplying
C has made
G, but, further,
B has also by multiplying
C made
F, the two numbers
A,
B by multiplying a certain number
C have made
G,
F.
Therefore, as
A is to
B, so is
G to
F. [
VII. 18]
But further, as
A is to
B, so is
G to
E also; therefore also, as
G is to
E, so is
G to
F.
Therefore
G has to each of the numbers
E,
F the same ratio;
therefore E is equal to F. [cf. V. 9]
Again, let
E be equal to
F; I say that, as
A is to
B, so is
C to
D.
For, with the same construction, since
E is equal to
F, therefore, as
G is to
E, so is
G to
F. [cf.
V. 7]
But, as
G is to
E, so is
C to
D, [
VII. 17]
and, as G is to F, so is A to B. [VII. 18]
Therefore also, as
A is to
B, so is
C to
D. Q. E. D.
