In 1658, then, the pendulum was applied to clocks, as the balance had been before that time. But Huyghens was not slow to perceive that the circular arc of a rigid pendulum would not be sufficiently accurate for an astronomical time-keeper, when used with a clock like that employed by Tycho Brahe and the Landgrave of Hesse for their astronomical observations. Huyghens next showed that with a clock of that kind, requiring a large swing of pendulum, the oscillations were not quite isochronous, but varied in time according as the arc increased or diminished. It was clear therefore that this simple form of pendulum would not do well for the large and varying arc required to be described, but that the theoretical requirements would be satisfied if the pendulum, instead of being suspended from a rigid rod, were suspended by a cord or spring or some elastic substance which would mould itself against two curved pieces of metal, C C, Fig. 89, attached one on either side of the suspending spring. In swinging, the spring would wrap, as it were, gradually round either curved surface, and so virtually alter the point of suspension, and with it, of course, the virtual length of the pendulum; so that the extreme point of the pendulum U, instead of describing a circular arc K B as before, would, by means of the portions of metal at the top, have a cycloidal motion D L, the pendulum becoming virtually shorter as the spring wrapped round the pieces of metal, so that it becomes isochronous for any length of swing. But it was very soon found that the theoretically perfect clock did not after all go as well as the clock it was to replace. And it would now be difficult to say what would have happened if a few years afterwards clocks had not been made much more simple and perfect by the introduction of an entirely new escapement which permitted a very small swing. [Illustration: FIG. 89.—The Cycloidal Pendulum. ] If we wish a clock to go perfectly well, we have only to consider a very few things—First, the weight should be as small as possible; secondly, within reason, the pendulum should be as solidly suspended and as heavy as possible; and, thirdly, the less connection there is between the pendulum which controls the clock, and the weight which drives the clock, the better. The latter point is provided for in the dead beat arrangement of Graham, and in the “gravity” and other forms of escapement, about which more presently. At present we have been dealing with pendulums as if they were simple pendulums, which are almost mathematical abstractions.