If a number is a part of a number, and another is the same part of another, then alternately, whatever part or parts the first is of the third, the same part, or the same parts, the second is of the fourth.

Let the number *A* be a part of the number *BC,* and and another number *D* be the same part of another number *EF* that *A* is of *BC.*

I say that, alternately, *BC* is the same part or parts of *EF* that *A* is of *D.*

Since *D* is the same part of *EF* that *A* is of *BC,* therefore there are as many numbers *BC* equal to *A* as there are also in *EF* equal to *D.*

Divide *BC* into the numbers equal to *A,* namely *BG* and *GC,* and divide *EF* into those equal to *D,* namely *EH* and *HF.* Then the multitude of *BG* and *GC* equals the multitude of *EH* and *HF.*

Now, since the numbers *BG* and *GC* equal one another, and the numbers *EH* and *HF* also equal one another, while the multitude of *BG* and *GC* equals the multitude of *EH* and *HF,* therefore *GC* is the same part or parts of *HF* that *BG* is of *EH,* so that, in addition, the sum *BC* is the same part or parts of the sum *EF* that *BG* is of *EH.*

But *BG* equals *A,* and *EH* equals *D,* therefore *BC* is the same part or parts of *EF* that *A* is of *D.*

Therefore, *if a number is a part of a number, and another is the same part of another, then alternately, whatever part of parts the first is of the third, the same part, or the same parts, the second is of the fourth.*

Q.E.D.

The sample value taken for 1/*n* in the proof is 1/2.

Proposition VII.15 can be construed as a special case of this one.

This proposition is used in the proof of the next.