---
book: 1
number: 3
id: "I.3"
kind: "construction"
uses: ["[[book-1/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.3"
license: "CC-BY-SA-4.0"
---

# I.3

Given two unequal straight lines, to cut off from the greater a straight line equal to the less.

![I.3](figures/I-3.svg)

## Proof

Let *AB*, *C* be the-two given unequal straight lines, and let *AB* be the greater of them.

Thus it is required to cut off from *AB* the greater a straight line equal to *C* the less.

At the point *A* let *AD* be placed equal to the straight line *C*; [[book-1/proposition-2|I. 2]] and with centre *A* and distance *AD* let the circle *DEF* be described. [[book-1/postulates#Postulate 3|Post. 3]] Now, since the point *A* is the centre of the circle *DEF*, *AE* is equal to *AD*. [[book-1/definitions#Definition 15|Def. 15]] But *C* is also equal to *AD*. Therefore each of the straight lines *AE*, *C* is equal to *AD*; so that *AE* is also equal to *C*. [[book-1/common-notions#Common Notion 1|C.N. 1]]

Therefore, given the two straight lines *AB*, *C*, from *AB* the greater *AE* has been cut off equal to *C* the less.

(Being) what it was required to do.