---
book: 9
number: 35
id: "IX.35"
kind: "theorem"
uses: ["[[book-7/proposition-11]]", "[[book-7/proposition-13]]", "[[book-7/proposition-12]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.35"
license: "CC-BY-SA-4.0"
---

# IX.35

*If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then, as the excess of the second is to the first, so will the excess of the last be to all those before it.*

## Proof

Let there be as many numbers as we please in continued proportion, *A*, *BC*, *D*, *EF*, beginning from *A* as least, and let there be subtracted from *BC* and *EF* the numbers *BG*, *FH*, each equal to *A*; I say that, as *GC* is to *A*, so is *EH* to *A*, *BC*, *D*.

For let *FK* be made equal to *BC*, and *FL* equal to *D*.

Then, since *FK* is equal to *BC*, and of these the part *FH* is equal to the part *BG*, therefore the remainder *HK* is equal to the remainder *GC*.

And since, as *EF* is to *D*, so is *D* to *BC*, and *BC* to *A*, while *D* is equal to *FL*, *BC* to *FK*, and *A* to *FH*, therefore, as *EF* is to *FL*, so is *LF* to *FK*, and *FK* to *FH*.

Separando, as *EL* is to *LF*, so is *LK* to *FK*, and *KH* to *FH*. [[book-7/proposition-11|VII. 11, 13]]

Therefore also, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents; [[book-7/proposition-12|VII. 12]] therefore, as *KH* is to *FH*, so are *EL*, *LK*. *KH* to *LF*, *FK*, *HF*.

But *KH* is equal to *CG*, *FH* to *A*, and *LF*, *FK*, *HF* to *D*, *BC*, *A*; therefore, as *CG* is to *A*, so is *EH* to *D*, *BC*, *A*.

Therefore, as the excess of the second is to the first, so is the excess of the last to all those before it. Q. E. D.