Mix moist potting soil, builder's sand and peat moss in equal amounts to produce a succulent rooting mixture. Fill a 3- to 5-inch plant pot with the rooting mixture and press the soil down firmly in the pot.
Cut a small branch from the parent Euphorbia with a relatively long base stem and several branching stems, using the razor knife. Set the branch aside so the cut end dries out, forming a callus.
Poke a hole in the rooting mixture with a pencil about half the length of the base stem of the cut branch. Insert the cut end into the soil and firm the soil around the cutting.
Water the branch one week after planting. Water sparingly after that, allowing the rooting mixture to dry out thoroughly between waterings.