Éléments de la géométrie de l’infini is a foundational treatise on the mathematical infinite. It does not set out to teach the computational techniques of the differential and integral calculus; instead, it asks what is meant by infinitely large and infinitely small magnitudes and how such quantities may legitimately be handled in geometry. Fontenelle writes in the spirit of a set of “elements”: he proceeds from definitions and principles toward a reasoned account of a “geometry of infinity,” rather than offering a systematic calculus manual.
The book’s central concern is the status of infinity in mathematics. Fontenelle treats mathematical infinity not as a single undifferentiated absolute but as a domain of relative, comparable magnitudes. Infinite quantities can stand in ratios to one another; one infinite can be infinitely greater than another, and one infinitesimal can be infinitely smaller than another. A recurring device is the reciprocal relation between the infinitely large and the infinitely small: an infinitely small quantity is understood in relation to an infinitely large one, and orders of smallness correspond to orders of greatness. This allows infinitesimals to be treated not as mere zeros or fictions but as quantities with a place in a hierarchy of magnitudes.
The work connects this hierarchy to geometry. It considers infinite lines, surfaces, and solids, and the ways in which curves, asymptotes, tangents, and areas can be described through infinitely large or infinitely small elements. It discusses the use of infinitesimals in the geometry of curves and in operations associated with the new analysis, but its aim is justificatory rather than technical: it seeks to show that the method of infinitesimals is coherent and that objections to it often rest on confusing absolute infinity with the relative infinities of geometry. Fontenelle also addresses philosophical difficulties: infinite divisibility, the comparison of infinite magnitudes, the status of quantities that exceed any finite assignment, and the apparent paradoxes that arise when infinite and finite quantities are mixed in calculation. He defends the practice of geometers by arguing that infinite and infinitesimal magnitudes can be treated according to stable rules, even if their metaphysical status remains delicate.
The book is thus a defence of infinitesimal geometry on philosophical and mathematical grounds. Its distinctive contribution is the attempt to construct an “elements” of the geometry of infinity: a reasoned account of how infinite and infinitesimal magnitudes behave and how they can be used to understand curves and figures. It is not a modern limit-theory treatise and does not reduce infinitesimals to formal definitions of limits; rather, it treats them as a distinct order of mathematical objects whose relations can be studied in their own right.
A central theme of the book is the legitimacy of neglecting infinitesimals in geometrical reasoning. Fontenelle argues that such neglect is not an arbitrary fiction or a confusion of something with zero; it is a relative judgment grounded in the order of magnitudes. A lower-order infinitesimal can be disregarded beside a higher-order quantity because its ratio to that quantity is infinitely small, not because it has no mathematical role at all. The geometry of infinity therefore becomes a discipline of comparison: it studies when quantities may be treated as negligible, when they must be retained, and how their relative orders govern the validity of a construction or calculation.
The book also treats continuity and infinite divisibility as intimately connected with this hierarchy of magnitudes. A finite line or figure can be regarded as divisible without end, and conversely as composed of infinitely many infinitely small parts; the relation between the number of parts and their size is one of the work’s recurring concerns. Fontenelle uses geometrical objects—infinite lines, surfaces, solids, asymptotes, tangents, and curvilinear figures—to show how infinite and infinitesimal magnitudes enter into comparisons without producing contradiction, provided their relative order is kept in view. His aim is to show that the apparent paradoxes of the infinite arise from mixing quantities of different orders or from treating relative infinities as if they were the absolute infinite.
The significance of the Éléments lies in its attempt to give the infinitesimal a coherent status within geometry itself, as a magnitude whose relations can be reasoned about rather than as a mere computational convenience. It presents the geometry of infinity as having its own principles of comparison and operation, and it treats the philosophical puzzles of infinity as inseparable from the mathematical practice of comparing magnitudes. In doing so, it offers a sustained reflection on what kind of object an infinitesimal can be for geometry and on how infinite quantities can be ordered, compared, and used without collapsing into paradox.
