---
book: 7
number: 15
id: "VII.15"
kind: "theorem"
uses: ["[[book-7/proposition-12]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.15"
license: "CC-BY-SA-4.0"
---

# VII.15

*If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of times that the second measures the fourth.*

## Proof

For let the unit *A* measure any number *BC*, and let another number *D* measure any other number *EF* the same number of times; I say that, alternately also, the unit *A* measures the number *D* the same number of times that *BC* measures *EF*.

For, since the unit *A* measures the number *BC* the same number of times that *D* measures *EF*, therefore, as many units as there are in *BC*, so many numbers equal to *D* are there in *EF* also.

Let *BC* be divided into the units in it, *BG*, *GH*, *HC*, and *EF* into the numbers *EK*, *KL*, *LF* equal to *D*.

Thus the multitude of *BG*, *GH*, *HC* will be equal to the multitude of *EK*, *KL*, *LF*.

And, since the units *BG*, *GH*, *HC* are equal to one another, and the numbers *EK*, *KL*, *LF* are also equal to one another, while the multitude of the units *BG*, *GH*, *HC* is equal to the multitude of the numbers *EK*, *KL*, *LF*, therefore, as the unit *BG* is to the number *EK*, so will the unit *GH* be to the number *KL*, and the unit *HC* to the number *LF*.

Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [[book-7/proposition-12|VII. 12]] therefore, as the unit *BG* is to the number *EK*, so is *BC* to *EF*.

But the unit *BG* is equal to the unit *A*, and the number *EK* to the number *D*.

Therefore, as the unit *A* is to the number *D*, so is *BC* to *EF*.

Therefore the unit *A* measures the number *D* the same number of times that *BC* measures *EF*. Q. E. D.