Math History books I'm teaching a course over the summer (it's a sort of make-your-own course for non-majors) and I'm planning on organizing it as a math history course, hitting on major threads through about 1900, and focusing on the evolution of ideas and on people, rather than on the details of proofs.  I've also been having a lot of trouble finding a good book covering this material (none finding books on ancient mathematics, but I want to focus on Renaissance to 19th Century, if possible), and so, here's my question:

What would be a good textbook for a course of this nature? Specifically, for a math history course targeted at non-science majors.

 A: It's hard to teach anything at all even as far back as the 18th century about the history of mathematics to students with little math backgroound (or motivation), but one option is to emphasize a
"great theorems" approach starting with ancient times.   A popular and very inexpensive paperback by W.W. Dunham Journey through Genius might be useful (or not) for your course once you get past the title.   Some of us at UMass found it reasonable as a resource for an ill-defined university requirement in "junior year writing" for prospective math majors, since the early chapters of Dunham require little background and provide some research/writing/presentation possibilities.   You'd find the approach of Laubenbacher, Pengelley, and colleagues quite different and probably more demanding, since they try to get back to original sources.  In any case, Stillwell has provided a good option for history which should be considered.
A: The Bible of the history of mathematics is Morris Kline's Mathematical Thought From Ancient To Modern Times, but this is really for mathematics students. I think it'd be hard to teach a course from it to non-mathematics majors. 
To be honest, I think it'd be really hard to teach such a course to non-mathematics majors PERIOD: How would you explain why Weierstrass's non-differentiable function is important to students who don't know calculus? Even worse, why would such students CARE? 
If I was really gung ho to teach such a course I'd take Kline, Stillwell, Bell's Men Of Mathematics and 3 or 4 other texts and use them to write a set of lecture notes for my students. That's how I'D do it. 
A: I took a computer science crypto course which turned out to have more of an arts class feel to it.
We followed Simon Singh's The Code Book.
It's appropriate for non-majors and I think the subject matter is interesting enough to keep students captivated. It covers the history and evolution of cryptography with just enough math that it's legitimately not a reading comprehension course. This isn't entirely focused on the 1900's like you wanted, but there is a large chunk dedicated to this period.
A: A slightly different idea that may be nice to complement the other suggestions here is the four volume set The World of Mathematics edited by James Newman. It contains essays written by mathematics greats about mathematical (and some not so mathematical) ideas, as well as biographical sketches on various mathematicians. The works collected therein are largely non-technical by nature, but still fun to read even for mathematicians. 
It can be described somewhat as a source-book, good for reading about ideas from the horse's mouth.  For a course you may have to spend some time going through the TOC to look for suitable selections. 
The books themselves are a bit dated (I got it from my mother-in-law who got it from her father), but since you were asking for a focus on pre-1900, I think it is okay. Dover has recently re-released the books in trade paperback, so I hope it is not too expensive now. 
A: It's been a while since I looked at it, but I remember enjoying Oystein Ore's "Number theory and its history". Bonus: it's a Dover paperback, so very cheap. But of course it is specialized to number theory, so I don't know if it's what you're looking for.
A: I will add my vote for John Stillwell's book. It has a wealth of material to draw on, and it is gorgeously written.
Take also a look at "Geometry for the liberal arts" by Dan Pedoe. It is mainly about geometry but has a lot of Renaissance examples. It also has exercises, which is certainly a plus for your purposes.
A: I think Development of mathematics in the 19th century by Felix Klein would be useful.
A: The question might use a bifurcation: do you want a systematic treatise or a collection of episodes?
In the latter case, there is Gindikin's Tales of Physicists and Mathematicians. It is a collection of (mostly independent, if my memory doesn't cheat me - I have read parts of the Russian original) "tales" on some particular people and/or developments, mostly pre-1900. History is interlaced with mathematics - but almost all of the latter should be on high school level. Probably this is better used in a student seminar rather than in a lecture course, though.
A: 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.Q. Gouvêa. 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.
A: Here are three possibilities.  You'll have to judge if they will be accessible to non-science majors.  
Mathematics and Its History by John Stillwell. Since Stillwell is on MO, perhaps he can say more on this book. 
Mathematical Expeditions: Chronicles by the Explorers, by Reinhard Laubenbacher and David Pengelley.  It covers geometry, set theory, analysis, number theory, and algebra.
Mathematical Masterpieces: Further Chronicles by the Explorers, by Art Knoebel, Reinhard Laubenbacher, Jerry Lodder, and David Pengelley.
A history of math class was taught in my department a few years ago using the second book above.  I covered for the instructor once and looked at the book. It seemed nice.  I have not looked at the third book, but discovered it right now on amazon as a follow up to the second book, so I threw it on the list (but think carefully about how non-science students would respond to the last chapter if you were seriously considering the third book). 
A: I like these two books:
Mathematics and Its History By John Stillwell (https://www.amazon.com.br/Mathematics-Its-History-John-Stillwell/dp/144196052X/ref=asc_df_144196052X/?tag=googleshopp00-20&linkCode=df0&hvadid=379693248429&hvpos=&hvnetw=g&hvrand=5239036777970932506&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9102183&hvtargid=pla-452518738779&psc=1) AND
A History Of Mathematics()https://www.amazon.com.br/History-Mathematics-3rd-Victor-Katz/dp/0321387007 by Katz.
If you read Portuguese this one too:
História da matemática De Tatiana Roque (https://www.amazon.com.br/Hist%C3%B3ria-matem%C3%A1tica-Tatiana-Roque-ebook/dp/B00A2V4OTU).
