---
book: 11
number: 37
id: "XI.37"
kind: "theorem"
uses: ["[[book-11/proposition-33]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.37"
license: "CC-BY-SA-4.0"
---

# XI.37

*If four straight lines be proportional*, *the parallelepipedal solids on them which are similar and similarly described will also be proportional; and*, *if the parallelepipedal solids on them which are similar and similarly described be proportional*, *the straight lines will themselves also be proportional.*

## Proof

Let *AB*, *CD*, *EF*, *GH* be four straight lines in proportion, so that, as *AB* is to *CD*, so is *EF* to *GH*; and let there be described on *AB*, *CD*, *EF*, *GH* the similar and similarly situated parallelepipedal solids *KA*, *LC*, *ME*, *NG*; I say that, as *KA* is to *LC*, so is *ME* to *NG*.

For, since the parallelepipedal solid *KA* is similar to *LC*, therefore *KA* has to *LC* the ratio triplicate of that which *AB* has to *CD*. [[book-11/proposition-33|XI. 33]]

For the same reason *ME* also has to *NG* the ratio triplicate of that which *EF* has to *GH*. [*id*.]

And, as *AB* is to *CD*, so is *EF* to *GH*.

Therefore also, as *AK* is to *LC*, so is *ME* to *NG*.

Next, as the solid *AK* is to the solid *LC*, so let the solid *ME* be to the solid *NG*; I say that, as the straight line *AB* is to *CD*, so is *EF* to *GH*.

For since, again, *KA* has to *LC* the ratio triplicate of that which *AB* has to *CD*, [[book-11/proposition-33|XI. 33]] and *ME* also has to *NG* the ratio triplicate of that which *EF* has to *GH*, [*id*.] and, as *KA* is to *LC*, so is *ME* to *NG*, therefore also, as *AB* is to *CD*, so is *EF* to *GH*.

Therefore etc. Q. E. D.