Davide Crippa's book explores early modern notions of mathematical impossibility. By recounting James Gregory's studies of the quadrature of the circle and ensuing debate with Huygens, Leibniz and others, Crippa investigates how problems were considered ‘adequately solved’ and how others were deemed ‘impossible’ within seventeenth-century geometry. While historiography has treated impossibility as a nineteenth-century phenomenon following the advent of modern (abstract) algebra, Crippa identifies a lively, if sporadic, premodern conversation over impossibility dating to Pappus's Mathematical Collection (p. 10). He adopts H.J.M. Bos's concept of mathematical ‘exactness’, focusing less on individual problems and solutions and more on the metamathematical considerations undergirding various ‘frameworks’ of geometry throughout history. Crippa notes, for example, that the circle quadrature had actually been achieved in antiquity through so-called ‘mechanical’ curves: Archimedes found it with the spiral; Pappus used the quadratrix (p. 20). Gregory's assertion of the problem's impossibility pertained specifically to his contemporary geometrical framework: that of Descartes's Géométrie, in which only curves traced by interconnected rectilinear motions (via Descartes's compound compass or ‘mesolabium’) and expressible by algebraic equations were permitted. Whether, and how, this framework should be expanded was the key issue motivating Gregory's ‘impossibility proof’ of the circle quadrature, and his debate with Huygens and Leibniz was a proxy war for deeper epistemological disputes on the proper methods and scope of geometry. Crippa presents a mathematically detailed and contextually sensitive account of this illustrative episode in the history of early modern mathematics.

The bulk of the book concerns Gregory's impossibility proof: his assertion that the circle quadrature could not be achieved through Cartesian geometrical means. As Crippa explains it, this proof hinged on Gregory's twofold notion of ‘composition’ and ‘analytical’ quantities. ‘Composition’ included addition, subtraction, multiplication, division, root extraction and ‘any other imaginable operation’ (p. 49). Two quantities composed from other quantities that were themselves commensurable with one another were ‘mutually analytical’ (p. 50). With these definitions, Gregory upgraded the algebraic machinery of Cartesian geometry while keeping its core principles intact. He then considered a converging series of inscribed and circumscribed polygons within a given sector of the circle. He showed that (1) the differences in area between polygons ‘described to the infinite’ would grow smaller than any given quantity, (2) the ‘last inscribed polygon’ would equal the quadrature of the sector and (3) this last polygon could not be found through analytical means (pp. 61–68). As Crippa stresses, Gregory aimed not to restrict one geometrical framework but to suggest what might be possible in another. He postulated a ‘sixth operation’ beyond the aforementioned five that involved constructing and finding the limit of a convergent series of analytical quantities. Crippa observes how Gregory's term ‘geometrical’ came untethered from any particular geometrical framework. ‘Geometrical’ to him meant ‘simplest’, regardless of what curves or techniques were used. ‘Anything goes’, Crippa concluded, so long as mathematicians determined a solution to be the most ‘geometrical’ possible (p. 91).

The remainder of Crippa's book investigates Leibniz's studies of quadrature: of the circle, hyperbola and other conic sections. Besides the circle, the hyperbola quadrature proved most interesting to seventeenth-century mathematicians, for a successful squaring of the hyperbola would allow them to incorporate logarithms into their geometrical framework. Like the circle, the hyperbola quadrature was inaccessible through analytical means. (In modern terms, both involve transcendental numbers: π for the circle; e for the hyperbola.) Like Gregory, Leibniz's focus was not on impossibility in one framework but on the possibilities offered by another. He showed how both quadratures could be expressed as sums of infinite series, which he designated as ‘true analytical and general relations’ (p. 154). Crippa's central point here is that Leibniz released his definition of exactness from any requirement of geometrical depiction or construction. Where Gregory maintained the form of geometry even as he pushed beyond the Cartesian framework, Leibniz argued that an infinite series or system of equations could constitute an exact solution, as long as they were derived through acceptable operations. Finally, Crippa examines Wallis's and Leibniz's conceptions of negative and imaginary numbers to further elucidate their conceptions of impossibility and show how standards of exactness were retooled to permit once forbidden entities into mathematics.

Some minor issues with the book should be noted. Some of Crippa's geometrical diagrams are rather confusing. Figure 1.2 (p. 6), for instance, depicts unequal line segments B and A = B + R ₁ as if they had equal length and portrays segments A and R ₁ inconsistently in several places. Figure 1.10 (p. 30) is missing several labels, for points G, H and K, referenced in the text. Crippa's writing occasionally veers into anachronism, such as when he characterizes Gregory's definition of ‘analytical’ as an equivalence relation (being reflexive, symmetric and transitive). While conceding that the language of equivalence is modern, he asserts that the ‘fundamental idea’ originated in Euclid's Elements and that Gregory was inspired by Euclid's approach (pp. 51–52). Identifying present mathematical ideas in past works, while often tempting, should be avoided. Finally, Crippa makes the odd claim that Gregory and Wallis saw exactness as ‘a matter of personal taste or of convention, which lay outside mathematics’ (p. 164). This undermines Crippa's broader (correct) assumption that exactness is by definition a metamathematical issue. He acknowledges the philosophical bases of Leibniz's criteria of exactness and how Leibniz sought more rigid and precise criteria than Gregory or Wallis, but should have stressed that Leibniz's vision of exactness, like anyone else's, lay outside mathematics proper.

These minor problems do not hinder the book's core argument. Crippa offers a meticulous and insightful study that respects the original form of Gregory's geometry while rephrasing it in terms familiar to modern readers. He sheds light on shifting conceptions of impossibility and on the transition from geometrical to predominantly symbolic and analytical criteria of exactness in the seventeenth century. To borrow the title of Section 2.4, Crippa opens a fascinating window on ‘the taming of the impossible’ in premodern mathematics.