Oooh...Fredwina, a good find! It's like this one:
http://www.cactusweb.us/euphorbia__trigonum.html
It's hard to tell how tall yours is, but it looks like maybe more than 2 feet tall?

I have a multi branched one of those that is thriving nicely indoors in a bright window.
You know, if you repot it into a giant pot and can bring yourself to cut it into 3 segments then plant them in a trio and put in the full sun, the pieces will branch out and form several new spires each and become a wonderful bushy specimen! One must be careful to not plant the segments upside down though, and you should let the cut pieces lie out to dry for about a week before planting so the ends won't rot.

Sandra- that's a lovely succulent, and look, it's blooming!! Amazing how it matches the pretty pot color. It will grow bushy and fuller if it gets lots of sun.

I should have put all my cactus outside in the sun all summer. But i couldn't bear to not have them in the house with me, so I didn't... this year. As a result some of them are showing slightly anemic new top growth because of the less bright sun they get in my windows as opposed to outside. Now it's autumn and too cold outside. I will force myself to put them out on the sunny porch next summer.
Those of you in CA and AZ are lucky to be able to keep plants going outside all year!