This is a onceprovocative text of the history of mathematics, that has lost most of its punch today. The loss is due in part to its age (this is an unaltered reprint of the 1945 second edition) and partly because there are now much better texts available.
Although nominally an undergraduate text, the book is really aimed at professional mathematicians. The tipoff is that very little of the mathematics is actually described. The book names many branches and subbranches of mathematics, but generally does not describe them or the problems that they deal with, so the reader is assumed to already know this. There are no figures in the book and few equations. Bell’s slightly earlier popular math book Men of Mathematics has more explicit math in it than this book does. What the reader gets in this book is the history of these subjects (meaning, primarily, who worked on them and when) and some predictions and perceived trends. The book is very weak on the major problems that have driven the development of mathematics, and tends to cover only general theories.
The book is written in a combative style. This is partly due to the author’s personality, and partly because such writing was more common in the early twentieth century (I am reminded in particular of H. L. Mencken). The book sometimes breaks into polemics of a sort seen today only in the works of rightwing political commentators. Some samples:

“For it seems improbable that our credulous race is likely ever to get very far away from brutehood until it has the sense and the courage to discard its baseless superstitions, of which the absolute truth of mathematics was one.” (pp. 330–331)

“That irrepressible innovator [Jacobi] believed the infallible method to advance mathematics was for domineering professors in the leading universities to drill their own ideas, and as little else as possible, into as many advanced students as could be induced to scribble lecture notes. In short, Jacobi anticipated the Führer Prinzip.” (p. 441)
Fortunately most of the writing is easier to take.
Despite the verbal pyrotechnics, this is at heart a scholarly work, but weak by presentday standards. It is poorly sourced, and works are usually only footnoted when they are quoted or referenced explicitly. Most of the footnotes are actually parenthetical remarks, and many carry on the polemics of the main text. The book is Eurocentric; there are only a few scattered mentions of early Chinese mathematics; and Indian mathematics in antiquity and Arabian mathematics in the Middle Ages together get one short chapter of 14 pages.
We have much better textbooks today. One example, with the same goals and coverage, is Boyer & Merzbach’s A History of Mathematics. Stillwell’s concise and very selective Mathematics and Its History is a very different kind of book but also does a good job of showing how mathematics develops.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.