---
book: 7
number: 5
id: "VII.5"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.5"
license: "CC-BY-SA-4.0"
---

# VII.5

*If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one.*

## Proof

For let the number *A* be a part of *BC*, and another, *D*, the same part of another *EF* that *A* is of *BC*; I say that the sum of *A*, *D* is also the same part of the sum of *BC*, *EF* that *A* is of *BC*.

For since, whatever part *A* is of *BC*, *D* is also the same part of *EF*, therefore, as many numbers as there are in *BC* equal to *A*, so many numbers are there also in *EF* equal to *D*.

Let *BC* be divided into the numbers equal to *A*, namely *BG*, *GC*, and *EF* into the numbers equal to *D*, namely *EH*, *HF*; then the multitude of *BG*, *GC* will be equal to the multitude of *EH*, *HF*.

And, since *BG* is equal to *A*, and *EH* to *D*, therefore *BG*, *EH* are also equal to *A*, *D*.

For the same reason *GC*, *HF* are also equal to *A*, *D*.

Therefore, as many numbers as there are in *BC* equal to *A*, so many are there also in *BC*, *EF* equal to *A*, *D*.

Therefore, whatever multiple *BC* is of *A*, the same multiple also is the sum of *BC*, *EF* of the sum of *A*, *D*.

Therefore, whatever part *A* is of *BC*, the same part also is the sum of *A*, *D* of the sum of *BC*, *EF*. Q. E. D.