In 2015 the Mathematical Association of America celebrated its Centennial at the MAA MathFest. Among the many special sessions with a centennial theme was “The Contributions of Women to Mathematics: 100 Years and Counting,” from which the book under review emerged. Women have been involved in mathematics, and indeed have pursued mathematical careers, for much longer than 100 years, but the MAA’s centennial clearly offered an opportunity to celebrate such women and their mathematical work.

The articles here, as is typical of proceedings volumes, are a mixed bag. Some represent summaries of (or complements to) work published elsewhere. Others give us “coming attractions,” preliminary reports or partial results. A few articles discuss interventions and other attempts to attract and retain more women in mathematics. Brought together, these articles give us a kind of summary account of the scholarly work that has been done in the field, especially on the historical side. Almost all the articles focus on American women.

Much of the historical work has taken the form of *retrieval*: finding, identifying, and telling the stories of women mathematicians. Judy Green and Jeanne LaDuke, for example, have done this for all American women receiving PhDs in mathematics before 1940; their work is recorded in their book *Pioneering Women in American Mathematics* and, in even greater detail, in its online supplement. Their article in this volume presents some summarizing data based on that work. It is full of interesting observations. For example, it turns out that (in marked contrast to Europe) most of the American women pursuing PhDs in this period did not come from elite well-educated backgrounds.

The article by Margaret A. M. Murray is also based on a project of retrieval, this time focused on the 1940s and 1950s. This time, the published book, *Women Becoming Mathematicians*, represents only a part of the project, which seems to be ongoing (see the web site Women Becoming Mathematicians). Erica Walker’s article is in a similar vein, this time focusing on black women in mathematics, which calls attention to several distinctive aspects of the experience of African-American women. For example, Walker points out that it was generally assumed that these women would (have to) work; the question was what kind of work. This created a motivation to pursue higher education, which would allow better career choices.

The main theme of this work of retrieval is well captured by the title of an article cited by Murray: “rumors of our rarity are greatly exaggerated.” Indeed, ever since there were PhDs granted in America there have been women who received PhDs in mathematics. They are not few, and many of them have had successful careers.

One must add that the focus on PhDs (Walker’s article is something of an exception) limits the range somewhat. After all, it is perfectly possible to have a mathematical career without a PhD, as highlighted (for example) in *Hidden Figures*. An example I recently learned about (and that is not mentioned in this book) is Mary Tsingou’s role in the famous work on non-linear oscillations by Fermi, Pasta, and Ulam.

A useful side-effect of these articles is to point out that we often approach the idea of a “mathematical career” in an overly limited way. The story of the brilliant young person who attends a top university, goes to an even better graduate school, proves a big theorem in her thesis, becomes a post-doc and then a famous mathematician… how many of us, male or female, live that story? Surely we can allow for much more variety in what counts as success in mathematics.

Once we have retrieved the stories of pioneering women mathematicians, what comes next? Few of the articles here attempt to move beyond retrieval. There are only a few attempts at generalization, analysis, or policy proposal.

Here’s a possible example. Several of the articles mention the “anti-nepotism rules” that affected the ability of married women to have mathematical careers, but there is little attempt at analyzing those rules and the societal assumptions behind them. Behind those rules there was, I suspect, an assumption that work was necessary but not desirable: if one member of a couple had a job, that income should suffice for both to live on, and the other spouse “should not need to work.” Today, we believe the opposite: having a career is seen as extremely desirable, and the other spouse “should not have to stay at home.” Indeed, salaries today reflect the latter assumption. That would suggest that perhaps the “two-body problem” is the modern analogue of the anti-nepotism rules, similarly making it difficult for couples to have successful careers. I have no idea whether this is correct, but it deserves socio-historical investigation.

One article that stands out by going beyond “retrieval” is Jemma Lorenat’s study of the teaching of synthetic projective geometry at women’s colleges in the first half of the twentieth century. She points out that this topic was not part of the standard mathematics curriculum in most universities. Of course, some mathematicians argued that it should be offered more regularly. What is surprising, however, is that they argued that it was *especially appropriate for women studying mathematics*. That seems extremely weird, but Lorenat confirms the idea to some extent by showing that courses on synthetic projective geometry were actually offered regularly at Bryn Mawr, Mount Holyoke, Smith, Vassar, Wellesley, Goucher, and Hunter Colleges. This is fascinating and deserves further attention, for example with a complementary survey of what was done at other liberal arts colleges. Was there more projective geometry at the coeducational colleges?

Some of the articles are, as one would expect, distinctly *engagé*, sometimes giving the reader a sense that being a woman mathematician is all about struggle rather than joy. There is much less celebration here than I would have hoped for. Occasionally one hears a tone of regret when a woman PhD decides that an academic career is not what she wants. And sometimes a point is, well, stretched a bit too far. For example, while it is certainly a bad thing that few women admitted to Berkeley in the 1970s had taken four years of mathematics in high school, to add that this made them unable to take the mathematics courses “required for majoring in every field at Berkeley except the ‘traditionally female’ (and hence lower paying) fields of humanities, social sciences, education and social welfare” (Lucy Sills, quoted by Jacqueline M. Dewar) is a bit ridiculous. Those “traditionally female, lower paying” fields would include, I guess, Law, Politics, History, the arts… Lawyers are not paid less than chemists.

There are other useful articles here: accounts of Florence Nightingale’s data science, of Emmy Noether’s career, of Mina Rees’s amazing career in science policy, of Emil Artin’s two woman PhD students (neither of which had a typical mathematical carreer). Sue Geller shows us a few examples of how a funny skit can make the point about women’s equality more forcefully than diatribe. There is even an article about communicating the stories of women mathematicians in dance.

The big message is clear: not only are there many women who have mathematical careers today, they have been around for quite some time. Many of them have had successful and happy lives despite the obstacles that they occasionally faced. And we should be encouraging many more to do the same.

Fernando Q. Gouvêa is a Brazilian-American mathematician teaching at Colby College in Waterville, ME.