Book I · Proposition 3

Cut off the lesser length

“To cut off from the greater of two given unequal straight lines a straight line equal to the less.”

Visualizing the goal

Begin with the unequal segments and , with the greater. Cut off the segment from so that equals .

Follow the construction.

1
Step 1
Let and be two given unequal straight lines, and let be the greater. It is required to cut off from a straight line equal to the less.
2
Step 2
At the point place equal to .
3
Step 3
With center and distance describe the , cutting at .
4
Step 4
Since is the center of the circle , therefore equals .
5
Step 5
But also equals . Therefore also equals .
6
Step 6
Therefore, from the greater, has been cut off equal to the less. Q.E.F.