Proposition 6

Table of Contents

PROPOSITION 6. Problem

How a body is able to move on a given line with the help of a pendulum.

Construction.

Let AMB (Fig. 5) be the proposed curve, in which the body must move; the evolute AOC of this curve is constructed and a plate is curved following this figure and set in place. Then a thread is led around this plate, which has one end fixed to the plate, and the body A to be moved is fastened to the other end. Therefore when the body begins, it is evident that it must move on the curve AMB, because the thread, as it then separates from the place, describes the evolute of this curve. Q.E.F.

Corollary 1

Therefore for this reason the body progresses along the given curve and is not liable to friction. Whereby in this manner such motion along curves as are found in theories can be conveniently put to the test. [p. 20]

Corollary 2.

From the theory of evolutes it is understood that the separated part of the thread MO is normal to the curve AMB and is the radius of osculation of this curve.

Corollary 3.

When the body is moving on the periphery of the circle AMB (Fig. 6), the curved plate is not needed, but the thread has only to be fastened at the end C to the centre of the C of the periphery.

Corollary 4.

Since the thread MO (Fig. 5) is the radius of osculation, the total centrifugal force is devoted to stretching this thread. Whereby this thread has enough strength not to be liable to be extended. For unless the length is always kept the same, the desired curve is not described.

Corollary 5.

By adding the absolute force besides the centrifugal force the normal force is obtained, which also pulls the thread, if the centrifugal force is to be added. But if it acts in the opposite direction, it diminishes the tension in the thread, indeed also, if this force is greater, then the thread is compressed, in which case it is of no use as an evolute. For since the thread must be flexible, it is not able to resist compression and neither does it offer any impediment, in which case the body recedes from the curve AMB towards the evolute.

Scholium 1.

Besides this difficulty, the generation of curves by evolutes also labours under this weakness, because the straight line is unable to be produced [p. 21] ; indeed for that to be generated the thread is requires to be infinitely long. In a similar manner this evolute cannot be adapted for curves that have an infinitely great radius of curvature somewhere. Then also neither curves with cusps nor with contrary bends can be described in this manner. Thus on this account the practice only has a place with curves having a finite curvature everywhere, to which it must be added, so that the total force acting on the curve is anywhere directed to the concave part of the curve.

Scholium 2.

Huygens, who first developed the principle of the evolute, at once put it to this use, that is apparent from the unusual need of the swinging pendulum clock. For since it can be shown that the swings on a cycloid are all isochronous, he wished to bring about cycloid motion in clocks, which is effected by the pendulum swinging between cycloidal plates. For since the evolute of a cycloid is a cycloid, for this reason it was obtained that a body tied to the end of a thread moves in a cycloid. Scholium 3.
Moreover in this motion of pendulums it is appropriate to note specially that besides the motion of the body, the thread too must be moving, but as it is the custom in this book in which only the motion of points is to be carried out, it is too small to be a concern. In addition, neither do we touch on the motion of the body attached as the pendulum which is not parallel to itself, but rather circular, and which is permitted around the centre of any circle of osculation on the curve. [p. 22] Therefore in this book we submit to be examined only the motion of a point in a given line or surface, and we do not consider either the motion of the thread or the case of circular motion. Moreover in the following motion of pendulums, where both the motion of the thread and circular motion are deduced from a computation, we reduce the motion to that of points only, thus in order that these which are treated in this book, are nevertheless found to have a practical use. On which account, as we have now advised, the point [acting as the pendulum] is considered to be always carried by moving parallel to itself either on a curve or surface without any friction.

PROPOSITION 7. Theorem.

If a body under the action of no forces is moving in a vacuum or in a medium without resistance on some surface ABC (Fig. 7), then it is carried in a uniform motion, with all friction removed from the air.

Demonstration

When a body moving on a given line is able to continue to be pressed, it is able to move much more on a given surface because there the freedom is less restricted. Therefore, let DMm be a line on which the body is progressing ; this is either a straight line or a curve. If this is a straight line, then there is no doubt that the body progresses with a uniform motion. But if it were a curve that could be expressed by an equation, and two adjoining elements of this are either situated nearly in the same direction, or they constitute an acute angle because cusps occur. [p. 23]

In the above case it has been shown that the body suffers no decrease of the motion (12).

Now with cusps indeed all the motion is lost, unless [the collision] is elastic. On account of which, if the motion is made on a curve, or on the part of a curve with the cusps missing, then the motion of the body is uniform. Q.E.D.

Corollary 1.

For a decrease in the speed of the body is permitted, as often as the direction is forced to change, now this is therefore equivalent to a second order differential, and even if it is integrated then an infinitely small decrement is produced.