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During the seventeenth century, the advent of what were known as the “common” and “new” analyses fundamentally changed the landscape of European mathematics. The widely accepted narrative is that these analyses, analytic geometry and calculus (mostly due to Descartes and Leibniz, respectively), occasioned a transition from geometrical to symbolic methods. In dealing with the science of motion, mathematicians abandoned the language of proportion theory, as found in the works of Galileo, Huygens, and Newton, and began employing the Newtonian and Leibnizian calculi when differential and fluxional equations first appeared in the 1690s. This was the advent of a more abstract way of practicing mathematics, which culminated with the algebraic approach to calculus and mechanics promoted by Euler and Lagrange in the eighteenth century. In this chapter, it is shown that geometrical interpretations and mechanical constructions still played a crucial role in the methods of Descartes, Leibniz, and their immediate followers. This is revealed by the manner in which they handled equations and how they sought their solutions. The passage from proportions to equations did not occur in a single step; it was a process that took a century to reach completion.
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.
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