Proposition 3
The third proposition is that if there are two straight lines of unequal length it is possible to cut off from the larger one a straight line equal to the smaller one.
Let AB and CD be the two straight lines, and AB be the
larger of them.
(Thus it is required to cut off from AB a straight line
equal to CD)
CD
[Proposition 2: it is possible to construct at a given point
a straight line equal to a given straight line (discussed in
last month’s article, Euclid 4)]
With centre A and distance AE,
draw a circle.
[Postulate 3: A circle can be drawn
at any distance around any point.]
of the same circle.
[Definition 15: A circle is a shape such that all the
straight lines falling upon it from one point within
it are equal to each other.]
This means that both CD and AF are equal to AE and must therefore be equal to each other.
[Common Notion 1: Things that are equal to the same thing are also equal to each other.]
Therefore a straight line, AF, equal to CD has been cut off from AB – which is what it was required to do.