Proposition 20

Table of Contents

PROPOSITION 20. Problem.

If the curve BAD (Fig. 25), upon which oscillations are made, is a cycloid described by the circle with diameter AC on the horizontal base BD, to determine the time of the oscillation through each arc EAF, with a uniform force acting downwards.

Solution

Let the radius of osculation at A, truly AO, = a, which is twice the diameter of the generating circle AC; hence AC = 12 a and with the abscissa AP = x and with the corresponding arc AM = s, from the nature of the cycloid, we have s 2 = 2ax .

Let the abscissa for the arc EAF, which is traversed in the oscillatory motion correspond to AG = b; the speed at the lowest point A corresponds to the height gb and the speed at M corresponds to the height g( b − x ) . [In the sense that the ratio of the speeds is as the square root of the ratio of the heights.] Whereby, since ds = adx , [p. 80] the time in .. which the arc AM is traversed, is equal to

…

Now, if after integration on putting x = b, then the time in which the whole arc AE is travelled through, is produced : or the circumference of the circle divided by the diameter. Whereby the time of a single ascent or descent is equal to π 2a and the time of one journey along the arc EAF is equal 2 g to π 2a .

The time for a complete oscillation is equal to 2π 2a . Q.E.I.

Corollary 1

Since in this expression of the time the letter b which determines the magnitude of the arc EAF is not present, all the times of the oscillations which are performed on the same cycloid are equal to each other.

Corollary 2

Therefore the time of any one oscillation is equal to the time of the oscillation through an indefinitely small arc. But the indefinitely small arclet agrees with the arc of the circle with radius OA to be described. Whereby the time of any oscillations on the cycloid BAD is equal to the time in which a pendulum of length a completes the smallest oscillation. It has also been made evident in the previous proposition that the time of one of the smallest oscillations of the pendulum a is equal to 2π 2a (167), in which we have found the time of a single whole oscillation by the same formula.

Corollary 3

Therefore if the pendulum is thus adjusted, in order that the oscillating body is moving on the cycloid, all the oscillations of this, whether they are large or small, are completed in equal intervals of time. [One may recall that Huygens had to resort to reductio ad absurdum arguments to prove this in the Horologium.] Whereby if AO is 3166 14 g scruples of Rhenish feet, individual semi-oscillations are completed in times of one second.

Corollary 4

Therefore all the descents to the lowest point A on the cycloid are of equal times or isochronous, and likewise all the ascents from the lowest point A, until the speed is spent. Truly the time of one ascent or descent is 2π 2a …

Scholium 1

On account of this property the cycloid is usually given the name tautochrone, since all the oscillations are completed on these in the same time. Huygens first uncovered this extraordinary property of the cycloid and understood at once that the cycloid could be substituted in place of the circle, that he effected in clocks. Yet now the clockmakers have abandoned this way of making oscillations, as they have learned almost nothing of this use. And surely in a vacuum with any curve, isochronous oscillations are produced, since they are always present with the same magnitude. Now in a resisting medium, in which the oscillations decrease, the cycloid loses this property and thus there is no advantage in the use.

Scholium 2

Also it is understood, if two dissimilar cycloids AE and AF (Fig. 24) are joined at the lower points, the oscillations upon the composite curve EAF are completed in equal times. For since both times of ascent or descent are constant quantities, also the sum of these, clearly the times of half an oscillation and the whole oscillation are equal to each other. Let twice the diameter of the circle generating the cycloid be AF = α, then the time of one ascent or descent on AF = π 2α . Whereby the to and fro journey on the …

composite curve EAF is completed in a time equal to time for the whole oscillation equal to

…

, and now with the

…

Scholium 3

Order requires that before we can progress to forces acting in different directions, we should explain the effect of forces acting parallel in the same direction, but [in which the magnitudes] are variable, and we should investigate the motion of bodies acted on by forces of this kind upon given curves. But since the contents worthy of note in the examples of motion hitherto set out may lie hidden from us, the principles shall now be explained with the help of which the motion on any curve can be understood, while we defer these other considerations to a fuller treatment, and here we take curves to be investigated, [p. 83] upon which a body is acted on by some kind of force, and advances according to a given law.