Euclid's Elements Book I

23 definitions · 5 postulates · 5 common notions · 48 propositions

Proposition I.3 · Problem

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

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let AB, C be the two given unequal straight lines, and let AB be the greater. It is required to cut off from the greater AB a straight line equal to the less C. (Drag the ends of C to change its length.)
  2. At the point A let AD be placed equal to the straight line C.
    Proposition I.2
    Proposition I.2: To place at a given point a straight line equal to a given straight line.
  3. With centre A and distance AD let the circle DEF be described; it meets AB at E.
    Postulate 3
    Postulate 3: To describe a circle with any centre and distance.
  4. Since the point A is the centre of the circle DEF, AE is equal to AD.
    Definition 15
    Definition 15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.
  5. But C is also equal to AD; therefore each of the straight lines AE, C is equal to AD, and so AE is also equal to C. Thus from the greater AB the straight line AE has been cut off equal to the less C.
    Common Notion 1
    Common Notion 1: Things which are equal to the same thing are also equal to one another.