I have not read the book, but if the reviewer captures Mansfield's argument correctly, it is pathetically unsound.
Mansfield hardly imagines that we can return to a society where men go off to be manly and women stay at home. Instead, he argues, we should revive a core distinction of liberalism: the divide between the public and the private. Sexual stereotypes would be discouraged in public life, but in private we “should admit that they are true”—and that they are what makes for mutual interdependence.Mansfield is treating the public/private distinction as if it concerned not individual but social behavior. Unhappily for him, it is not genders that can operate under a public/private division, but individuals. And in dealing with individuals, not only does the line between public and private fall in different places for different people, but so does the line between manly and womanly. Any lover of poetry will know this instinctively.
Mansfield is, apparently, a widely-read scholar, so I cannot imagine how he can overlook the amazing diversity of manliness present in even the briefest survey of the poets—from Catullus to Shakespeare to Rochester to Shelley to Neruda, one can find "confidence in the face of risk," an "easy assumption of authority," heroism, command, and "a kind of animal spiritedness or 'bristling' that vies with our reason," but not always in ways that limit themselves by either a public/private distinction or by concrete and irrefragable gender roles. Not only that, but numerous poets have successfully embodied what Mansfield would term manliness without the least recourse to gender at all. Yeats is the very definition of "bristling" and yet one finds only infrequently any gendered poses. Homeric heroes are 'manly' in opposition to each other, and only rarely in opposition to a woman. Achilles would still be manly without Briseis.
The manliness Mansfield wants to reintroduce not only does not need to be defined through gender, but would not even be effective if it were. Private virtues are adopted by individuals, not massive social groups that are only macroscopically homogeneous. If manliness is to be a virtue (which I dispute), then its gospel must be preached to individuals, not to mankind.
More: WSJ has an interview/review of Manliness here.