Mechanica Vol. II (1736)

...investigating. For this reason, this type of constrained motion original: "motus non liberi"; literally "not-free motion," referring to objects forced to move along a specific path or surface rather than moving freely through space must be handled in such a way that, first, the line on a given surface which the body describes while being acted upon by forces and resistance is determined; second, the speed original: "celeritas" of the body at individual points of this line is defined; and finally, third, the pressure original: "pressio" that the body exerts on the surface is investigated.

Constrained motions of this kind, just like the previous ones, can also be conveniently represented by pendulums; for a pendulum body pushed obliquely—so that its direction is not in a vertical plane—will describe various types of curved lines. All of these curves, however, are located on a spherical surface original: "superficie sphaerica" whose center lies at the very point of suspension. Therefore, the investigation of this motion comes down to this: that on a spherical surface, first the line which the projected body describes should be determined, then the speed in individual locations, and thirdly the pressure it will exert upon the surface.

In a similar way, it is also clear that it can be brought about that a pendulum body is restricted not just to a spherical surface, but to any other surface, provided that an evolved surface original: "superficies evoluta"; in the geometry of the time, this refers to a surface generated by the "unrolling" of another shape, like a string unwinding from a curve is arranged around the point of suspension. This, therefore, is the second type of constrained motions, which is concerned with determining motion over a given surface; and this entire Second Volume is completed by investigating these two types of constrained motions.

Therefore, so that the things necessary to know for this discussion may be prepared, I have set forth in the first chapter the foundations and principles from which those things pertaining to the knowledge of both types of constrained motions can be derived. Indeed, I have demonstrated that a body acted upon by no forces, both on a given line and...