---
book: 7
number: 33
id: "VII.33"
kind: "construction"
uses: ["[[book-7/proposition-21]]", "[[book-7/proposition-3]]", "[[book-7/proposition-16]]", "[[book-7/proposition-19]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.33"
license: "CC-BY-SA-4.0"
---

# VII.33

*Given as many numbers as we please, to find the least of those which have the same ratio with them.*

## Proof

Let *A*, *B*, *C* be the given numbers, as many as we please; thus it is required to find the least of those which have the same ratio with *A*, *B*, *C*.

*A*, *B*, *C* are either prime to one another or not.

Now, if *A*, *B*, *C* are prime to one another, they are the least of those which have the same ratio with them. [[book-7/proposition-21|VII. 21]]

But, if not, let *D* the greatest common measure of *A*, *B*, *C* be taken, [[book-7/proposition-3|VII. 3]] and, as many times as *D* measures the numbers *A*, *B*, *C* respectively, so many units let there be in the numbers *E*, *F*, *G* respectively.

Therefore the numbers *E*, *F*, *G* measure the numbers *A*, *B*, *C* respectively according to the units in *D*. [[book-7/proposition-16|VII. 16]]

Therefore *E*, *F*, *G* measure *A*, *B*, *C* the same number of times; therefore *E*, *F*, *G* are in the same ratio with *A*, *B*, *C*. [[book-7/definitions#Definition 20|VII. Def. 20]]

I say next that they are the least that are in that ratio.

For, if *E*, *F*, *G* are not the least of those which have the same ratio with *A*, *B*, *C*, there will be numbers less than *E*, *F*, *G* which are in the same ratio with *A*, *B*, *C*.

Let them be *H*, *K*, *L*; therefore *H* measures *A* the same number of times that the numbers *K*, *L* measure the numbers *B*, *C* respectively.

Now, as many times as *H* measures *A*, so many units let there be in *M*; therefore the numbers *K*, *L* also measure the numbers *B*, *C* respectively according to the units in *M*.

And, since *H* measures *A* according to the units in *M*, therefore *M* also measures *A* according to the units in *H*. [[book-7/proposition-16|VII. 16]]

For the same reason *M* also measures the numbers *B*, *C* according to the units in the numbers *K*, *L* respectively;

Therefore *M* measures *A*, *B*, *C*.

Now, since *H* measures *A* according to the units in *M*, therefore *H* by multiplying *M* has made *A*. [[book-7/definitions#Definition 15|VII. Def. 15]]

For the same reason also *E* by multiplying *D* has made *A*.

Therefore the product of *E*, *D* is equal to the product of *H*, *M*.

Therefore, as *E* is to *H*, so is *M* to *D*. [[book-7/proposition-19|VII. 19]]

But *E* is greater than *H*; therefore *M* is also greater than *D*.

And it measures *A*, *B*, *C*: which is impossible, for by hypothesis *D* is the greatest common measure of *A*, *B*, *C*.

Therefore there cannot be any numbers less than *E*, *F*, *G* which are in the same ratio with *A*, *B*, *C*.

Therefore *E*, *F*, *G* are the least of those which have the same ratio with *A*, *B*, *C*. Q. E. D.