I'm pleased to hear that some MOers like my book, but I have to say that I think it has too much math for a class of non-science majors. At best, you might mine it for some homework problems because other, more suitable, books tend to be lacking in that department. Here are a few I would recommend.
A Concise History of Mathematics by Dirk J. Struik. This is an oldie but goodie, a very readable blend of mathematics with general history, written by a distinguished historian of math. Unfortunately, no exercises.
Math through the Ages by W.P. Berlinghoff & F.O. Gouvea. Also a good blend of math with general history. The math is fairly low-level -- high-school and early undergrad -- but treated from an enlightened point of view.
The Honors Class by Ben Yandell. Like Bell's Men of Mathematics, this is a very readable set of biographical essays on mathematics. Since it is organized around the Hilbert problems, it starts roughly where Bell leaves off. Also, it is more factually accurate than Bell.