Mechanics, Volume I

would be conducted. Furthermore, what most hinders the reader is that he pursued everything in the manner of the ancients, synthetically through geometric demonstrations, and concealed the analysis Euler is referring to the "analytic method," what we now call calculus or algebraic analysis. In the 18th century, "analysis" focused on the process of discovery and calculation, whereas "synthesis" focused on formal geometric proof, by which a complete knowledge of these matters is reached. In a not much different manner were written Newton’s Mathematical Principles of Natural Philosophy original: Newtoni Principia Mathematica Philosophiae; published in 1687, this is the foundational work of classical mechanics, through which this science of motion attained its greatest advancements.

But what happens to all writings composed without analysis occurs most especially in Mechanics: namely, that the reader, even if they are convinced of the truth of the things presented, nevertheless does not achieve a sufficiently clear and distinct knowledge of them. Thus, if the same questions are changed even slightly, the reader is scarcely able to resolve them by their own efforts original: proprio marte, a Latin idiom meaning "by one's own martial power" or "by one's own strength" unless they themselves investigate the analysis and develop those same propositions through the analytical method.

This is exactly what happened to me when I began to examine Newton’s Principia and Hermann’s Phoronomia; for although I seemed to myself to have sufficiently grasped the solutions to many problems, I was nevertheless unable to solve problems that differed only slightly from them. Therefore, already at that time, I endeavored as much as I could to extract the analysis from that synthetic method The "synthetic method" refers to the traditional Euclidean style of geometry using diagrams and proportions, which was the standard for formal mathematical presentation in the 17th and early 18th centuries and to treat those same propositions analytically for my own benefit, from which labor I perceived a significant increase in my understanding. If—