Mechanica Vol. II (1736)

...of solids original: "solidorum"; likely referring here to the surfaces of three-dimensional bodies not yet sufficiently understood nor subjected to calculation. Before, therefore, anything could be established regarding motion of this kind, it was necessary to set forth a method by which the properties of surfaces and the lines drawn upon them could be extracted and submitted to calculation. I have performed this task by means of equations containing three variable quantities Euler is describing the use of three-dimensional coordinates (x, y, z) to define surfaces, a significant advancement in analytical geometry, which I previously employed both in Volume III of the Commentaries original: "Comment. Tomo III"; referring to the Commentarii Academiae Scientiarum Imperialis Petropolitanae to determine the shortest line upon any surface, and in the preceding volume of this Treatise original: "Tractatus"; referring to Volume I of his Mechanica to investigate free motions not occurring within a single plane.

With these foundations finally prepared, I was able to proceed to defining the effects of forces original: "potentiarum" on bodies moving upon surfaces. From these, I have derived a method for finding both the path described by the body and the other characteristics of motion original: "symptomata"; in this era, "symptoms" referred to the measurable properties or parameters of a physical system. However, since the calculation becomes excessively lengthy and difficult to manage as long as we remain in general terms, I have set aside resistance and reduced everything to a vacuum and ordinary gravity. I have particularly scrutinized the motion of pendulums oscillating obliquely, and I have diligently determined the anomalies of this motion and the progression of the apsides progression of the apsides: the gradual rotation of the points in a path (such as an orbit) where the moving body is at its maximum or minimum distance from the center of force.

These, then, are the matters I have included in this second volume. With these dispatched, I shall endeavor, as soon as possible, to bring the motions of finite bodies—and first indeed of rigid bodies—into order and explain them using a similar method.