Mechanica Vol. II (1736)

...ought to move with uniform motion upon a surface; however, on a surface, the path described by the body will be the shortest line original: "lineam breuissimam"; in modern geometry, this path is known as a geodesic that can be drawn on that surface. Next, I investigated the general laws which various forces original: "potentiae" and even resistance observe, both in accelerating or slowing down motion and in generating pressure. To these is also added an explanation of the theory of centrifugal force centrifugal force: the outward force experienced by a body moving in a curved path, which seems to "flee" the center of rotation, which bodies exert even when acted upon by no external forces, and which arises from the curved motion that the body is forced to follow.

Then, in the second and third chapters, I extensively contemplate and examine the motion of bodies along a given line, both in a vacuum and in a resisting medium original: "medio resistente"; a substance like air or water that slows down an object's movement. First, I determine the motion by which a body, acted upon by any forces, moves along a given line—whether straight or curved, and whether descending or ascending. Furthermore, if the curve is so configured that it is suitable for producing both descents and ascents, I also define the oscillations original: "oscillationes"; the back-and-forth swinging movement, and compare them to one another in terms of their duration. In this task, I define the nature and properties of oscillations made both in a circle and in a cycloid cycloid: the specific curve traced by a point on the rim of a circular wheel as it rolls along a straight line.

Next, I treat "inverse problems," in which I primarily search for the curves on which—for given acting forces—the motion possesses a specific property. To this belong problems of finding curves of uniform descent or recession from a given point, and many more of this kind, which have either already been treated by others or to which this very project has led me. Among these, the problems concerning brachystochrone brachystochrone: the "curve of fastest descent," or the path that allows an object to fall from one point to another in the shortest possible time and tautochrone tautochrone: the "curve of equal time," where an object takes the same amount of time to reach the bottom regardless of where on the curve it begins its slide lines stand out above the rest, both of which [lead] to a further [degree] than...