Newton’s remarkable achievements in planetary theory and dynamics were followed by a century of equally remarkable advances in the generalized science of forces acting on bodies in motion. Far from merelyformalizing the Newtonian framework, these advances aimed at solving deep problems left in Newton’s wake. First, Newtonian theory focused on centripetal forces, which are not characteristic of motion under arbitrary constraints or bodily deformation due to pressure and stress. Second, Newton’s attempts at fluid mechanics made clear that media could not be adequately analyzed using strategies developed for point-particles. Both considerations suggested that different forms of body required different analytical and conceptual tools. They required the formulation of generalized principles flexible enough to treat heterogeneous material configurations, but stringent enough to preserve the unity of mechanics as a study of matter and motion. Ultimately, Newton’s successors established a new discipline, analytical mechanics. Its primary object of investigation was the functional representation of the invariant relations behind dynamic phenomena, not the geometric representation of trajectories. The philosophical suppositions behind the new mechanics constituted a new mechanical philosophy, but one that hearkened back to, and completed the project of, the mechanical philosophy of the mid-seventeenth century.

The differential equations as written by Leibniz and by his immediate followers look very similar to the ones in use nowadays. They are familiar to our students of mathematics and physics. Yet, in order to make them fully compatible with the conventions adopted in our textbooks, we have to change just a few symbols. Such “domesticating” renderings, however, generate a remarkable shift in meaning, making those very equations – when so reformulated – not acceptable for their early-modern authors. They would have considered our equations, as we write them, wrong and corrected them back, for they explicitly adopted tasks and criteria different from ours. In this chapter, focusing on a differential equation formulated by Johann Bernoulli in 1710,I evaluate the advantages and risks inherent in these anachronistic renderings.

Two parts of analysis to which Leonhard Euler contributed in the 1740s and 1750s are the calculus of variations and the theory of infinite series. Certain concepts from these subjects occupy a fundamental place in modern analysis, but do not appear in the work of either Euler or his contemporaries. In the case of variational calculus there is the concept of the invariance of the variational equations; in the case of infinite series there is the concept of summability. However, some modern mathematicians have suggested that early forms of these concepts are implicitly present in Euler’s writings. We examine Euler’s work in calculus of variations and infinite series and reflect on this work in relation to modern theories.