The history of early modern population arithmetic is the central chapter in the gradual process by which European cultures came to understand themselves as numerically constituted and as structured by recurring mathematical relationships. Rusnock\'s *Vital accounts* provides an admirably clear and unruffled narrative of the evolution of numerical aspects of this development during the eighteenth century, with particular attention to medical topics. Understanding the quantitative reasoning of this period is of particular interest as it precedes the rise of statistics in the early nineteenth century and its ubiquitous spread ever since. Whilst in retrospect we can say that early modern population thinking anticipated statistics in some ways, it was neither conceived nor developed as statistics. Describing the quantitative reasoning of this period without succumbing to the anachronism of statistical terminologies we now take for granted thus poses some difficult problems of interpretation. Rusnock\'s approach, which pays careful attention to early modern procedures and terms of reference, is indicated by her title, and solves this problem neatly. Population arithmetic was *vital* in three senses subsequently taken over into vital statistics. First, and obviously, its main chosen objects were vital events (births, deaths, diseases) differentiated by observed life characteristics (age, sex, natural environment, and various physiological, epidemic and other causes). Second, following upon political and mercantile writings of the time, the health and numbers of people were understood as main constituents of the wealth and power of states, the basis of collective vitality. By extension, then, information about populations was knowledge vital to policy. *Accounts* is likewise a term of contemporary parlance with multiple significance, but here differences to later statistical developments begin to emerge clearly. The earliest population arithmetic in the seventeenth century adopted the term "accounts" from merchant book-keeping, employing it to refer to its method and as a term of general social reference. Eighteenth-century professionals who came to have a close interest in the health of populations, notably physicians, actuaries, and ministers of church and state, saw the compilation and interpretation of "accounts" in moral terms; to give an account meant providing a measure or assessment of relative salubrity that went beyond strictly medical matters. Inevitably, the third and closely related implication of numerical accounts was that any such compilation raises difficult issues of what standards of comparison are legitimate. As Rusnock observes, "numbers allow for comparison, even if the grounds of comparison are not always level" (p. 13). It was these issues that nineteenth-century statisticians believed would be solved by national census and vital registration systems.

Attempts to provide a level playing field began when John Graunt annexed his merchant book-keeping to a numerical reworking of Francis Bacon\'s tabular method for presenting recorded observations. As Rusnock notes, this approach was promoted, often uncritically, by William Petty under the label "political arithmetic". Some of the most sophisticated treatises of the later eighteenth century, like Jean-Baptiste Moheau\'s *Recherches* (1778), still looked back to Bacon. Following a brief survey of the earliest formulations, Rusnock charts the evolution of this tabular method as the basis of a *soi-disant* "medical arithmetic" in a series of eighteenth-century controversies: debates over the merits of smallpox inoculation; attempts to refine tabular methods (sometimes in conjunction with meteorological records) as measures of the healthiness of particular places; and attempts to extrapolate from incomplete local records to estimates of national population. None of the many and various tabular syntheses introduced in the course of these debates ever succeeded in resolving them. Yet, as Rusnock shows, via such controversies quantitative representation of society and its health became a widespread convention; it was established as a telling (if not conclusive) source of evidence of the effects of medical and political administration; and it came to underpin wider discourses on political and economic equity. The last subject is not, however, Rusnock\'s primary object in this book. Focusing closely on the sequence of health issues to which tabular arithmetics were applied, she demonstrates the effectiveness and limits of new methods as they developed, and the significant professional differences that often shaped divergent French and English approaches. The book is well illustrated by reproductions of tabular methods. It provides a very welcome and thoughtful introduction to an area of medical knowledge that was livelier and more topical than is now generally appreciated.
