Mechanica Vol. II (1736)

...[what] has been done by anyone until now, I have raised to a degree of perfection. Regarding brachystochrone curves brachystochrone: from the Greek for "shortest time," this is the path between two points that a body follows to descend as quickly as possible under gravity, I have corrected an error committed by several others, both in a vacuum and in a resisting medium. In place of Huygens’s principle original: "principii Hugeniani"; referring to Christiaan Huygens (1629–1695), a Dutch mathematician and physicist who pioneered the study of the pendulum and the cycloid, which is indeed true in itself but insufficient, I have substituted another principle of very wide scope. By this, I have demonstrated that in any medium and under any hypothesis of compelling forces, that curve is always the brachystochrone upon which a body moves in such a way that the total pressure is twice as great as the centrifugal force.

In a similar way, I provide a new and genuine method for finding tautochrone curves tautochrone: from the Greek for "same time," a curve where the time it takes for an object to slide to the bottom is the same regardless of the starting point; for those tautochrones discovered previously were unearthed not by any method at all, but rather by guesswork original: "divinatione". By the aid of this method, I have not only found the cycloid, long famous under the name of the tautochrone, but besides it, I have drawn out countless other curves satisfying the requirement. Among these, I even observed an algebraic curve original: "curvam algebraicam"; a curve that can be expressed by a polynomial equation, which was a significant discovery as many such curves are "transcendental" or more complex. Furthermore, from other related questions in a vacuum, as well as from the complete treatment of this business for a resisting medium, it will be possible to understand the excellence and utility of this method in abundance. Moreover, just as this method should be considered a specimen of advanced Analysis Analysis refers here to the branch of mathematics dealing with limits and calculus and Mechanics, so too, throughout the solutions of certain more difficult problems, noteworthy aids of Analysis will appear, by which even this science may seem to have been advanced not a little.

Finally, in the fourth chapter, I pursue motion upon a given surface. This doctrine, as it has been touched upon by no one until now, is also most difficult to treat, because of the nature and properties...