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A pendulous body vibrates when it is suspended so that the centre of its mass is not placed directly under the point of suspension, because then the alternating influences of weight and velocity are constantly impressing it with motion. Weight carries it down as far as it can go towards the earth's attraction; acquired velocity then carries it onwards; but as the onward movement is constrained to be upward against the direction of the earth's attraction, that force antagonises, and at last arrests it, for velocity flags when it has to drag its load up-hill, and soon gives over the effort. The body swings down-hill with increasing rapidity, because weight and velocity are then both driving it; it swings up-hill with diminishing rapidity, because then weight is pulling it back in opposition to the force of velocity. Weight pulls first this way, then that way; velocity carries first this way, then that way: but the two powers do not act evenly and steadily together; they now combine with, and now oppose each other; now increase their influence together, and now augment and diminish it inversely and alternately; and so the suspended body is tossed backwards and forwards between them, and made to perform its endless dance.

It is related of Galileo, that he once stood watching a swinging lamp, hung from the roof of the cathedral at Pisa, until he convinced himself that it performed its vibratory movement in the same time, whether the vibration was one of wide or of narrow span. This traditionary tale is most probably correct in its main features, for the Newtons and Galileos of all ages do perceive great truths in occurrences that are as commonplace as the fall of an apple, or the disturbance of a hanging lamp. Trifles are full of meaning to them, because their minds are already prepared to arrive at certain conclusions by means of antecedent reflections. Simple and familiar incidents, thus accidentally associated with the history of grand discoveries, are the channels through which the accumulating waters at length descend, rather than the rills which feed the swelling of their floods. The orchard at Woolsthorpe, and the cathedral at Pisa, were outlets of this kind, through which the pent-up tide of gathering knowledge burst. If they had never offered themselves, the laws of universal gravitation and isochronous vibration would still have reached the world.

If the reader will hang up two equal weights upon nearly the same point of suspension, and by means of two strings of exactly the same length, he will have an apparatus at his command that will enable him to see, under even more favourable conditions, what Galileo saw in the cathedral at Pisa. Upon drawing one of them aside one foot from the position of rest, and the other one yard, and then starting them off both together to vibrate backwards and forwards, he will observe, that although the second has a journey of two yards to accomplish, while the first has but a journey of two feet, the two will, nevertheless, come to the end at precisely the same instant. As the weights swing from side to side in successive oscillations, they will always present themselves together at the point which is the middle of their respective arcs. This is what is called isochronous vibration—the passing through unequal arcs in equal periods of time.

At the first glance, this seems a very singular result. The careless observer naturally expects that a weight hung upon a string ought to take longer to move through a long arc than through a short one, if impelled by the same force; but the subject appears in a different light upon more mature reflection, for it is then seen, that the weight which performs the longer journey starts down the steeper declivity, and therefore acquires a greater velocity. A ball does not run down a steep hill and a more gently inclined one at the same pace; neither, therefore, will the suspended weight move down the steeper curve, and the less raised one, at equal rates. The weight which moves the fastest, of necessity gets through more space in a given period than its more leisurely companion does. The equality of the periods in which two weights vibrate, is perfect so long as both the unequal arcs of motion are short ones, when compared with the length of the suspending strings; but even when one of the arcs is five times longer than the other, ten thousand vibrations will be completed before one weight is an entire stride in advance of the other; and even this small amount of difference is destroyed when the arc in which the weights swing is a little flattened from the circular curve.

But there is yet another surprise to be encountered. Hang a weight of a pound upon one of the strings, and a weight of two pounds upon the other, and set them vibrating in arcs of unequal length as before, and still their motions will be found to be isochronous. Unequal weights, as well as equal ones, when hung on equal strings, will swing through arcs of unequal length in equal periods of time. This seeming inconsistency also admits of a satisfactory explanation. It has been stated, that the motion of swinging bodies is caused by the earth's attraction. But what are the facts that are more particularly implied in this statement? What discoveries does the philosophic inquirer make when he looks more narrowly into it? For the sake of familiar illustration, let it be imagined that a man stands at the top of the Monument of London, with two leaden bullets in his hand, each weighing an ounce, and that he drops these together. They go to the earth, because the earth's mass draws them thither; and since the two bodies exactly resemble each other, and start at the same instant upon their descent, they must of course both strike the pavement beneath simultaneously. There can be no reason why one should get down before the other, for the same influence causes the fall of each. The entire mass of the huge earth attracts each bullet alike, and the bullets, therefore, yield like obedience to the influence, and fall together to the ground.

But now, suppose that the two bullets were to be all at once fused into one, and that this combined mass were then dropped from the top of the Monument as a single bullet, would there then be any reason why the two ounces of lead should make a more rapid descent than they would have made while in separate halves? Clearly not. There is but the same earth to attract, and the same number of particles to be drawn in each case, and therefore the same result must ensue. Each particle still renders its own individual obedience, and makes its own independent fall, although joined cohesively to its neighbours. It is the mass of the attracting body, and not the mass of the attracted body, that determines the velocity with which the latter moves. The greater mass of an attracted body expends its superior power, not in increasing its own rate of motion, but in pulling more energetically against the attracting mass. Every particle of matter when at rest resists any attempt to impress it with motion. The amount of this resistance is called its inertia. When many particles are united together into one body, they not only, therefore, take to that body many points upon which the earth's attraction can tell, but they also carry to it a like quantity of resistance or inertia, which must be overcome before any given extent of motion can be produced. If the earth's force be but just able to make particle 1 of any body go through 200 inches in a second, it will also be but just able to make particles 2, 3, and 4 do the same; consequently, whether those particles be separate or combined together, their rate of travelling will be the same. Hence all bodies descend to the earth with exactly the same velocities, however different their natures may be in the matter of weight, always provided there be no retarding influence to act unequally upon their different bulks and surfaces. It is well known that even a guinea and feather will fall together when the atmospheric resistance is removed from their path.

The reader will now, of course, see that what is true of the motion of free bodies, must also be true of the motion of suspended ones, since the same terrestrial attraction causes both. There is no reason why the two-pound weight in the experiment should vibrate quicker than the one-pound weight, just as there is no reason why a two-ounce bullet should fall quicker than a one-ounce bullet. Here, also, there are only the same number of terrestrial particles to act upon each separate particle of the two unequal weights. Hence it is that the vibrations of unequal weights are isochronous when hung on strings of equal lengths.

Thus far our dealings have been with what has seemed to be a very single-purposed and determined agent. We have hung a weight upon a piece of string and set it swinging, and have then seen it persisting in making the same number of beats in the same period of time, whether we have given it a long journey or a short one to perform; and also whether we have added to or taken from its mass. But now we enter upon altogether new relations with our little neophyte, and find that we have reached the limits of its patience.

Take three pieces of string of unequal lengths—one being one foot long; the second, four feet; and the third, nine feet. Hang them up by one extremity, and attach to each of the other ends a weight. Then start the three weights all off together vibrating, and observe what happens. The several bodies do not now all vibrate in the same times as in the previous experiments. By making the lengths of the strings unequal, we have introduced elements of discord into the company. The weight on the shortest string makes three journeys, and the weight on the next longest string makes two journeys, while the other is loitering through one.

This discrepancy, again, is only what the behaviour of the vibrating masses in the previous experiments should have taught the observer to anticipate. Each of the weights in this new arrangement of the strings, has to swing in the portion of a circle, which, if completed, would have a different dimension from the circles in which the other weights swing. The one on the shortest string swings in the segment of a circle that would be two feet across; the one on the longest string swings in the segment of a circle that would be eighteen feet across. Now, if these two weights be made to vibrate in arcs that shall measure exactly the twelfth part of the entire circumference of their respective circles, then one will go backwards and forwards in a curved line only half a foot long, while the other will move in a line four feet and a half long.

But both these weights, the one going upon the short journey, and the other upon the long, will start down exactly the same inclination or declivity. The reader will see that this must be the case if he will draw two circles on paper round a common centre, the one at the distance of one inch, and the other at the distance of nine inches. Having done this, let him cut a notch out of the paper, extending through both the circles to the centre, and including a twelfth part, or thirty degrees, of each between its converging sides. He will then observe, that the two arcs cut out by the notch are everywhere concentric with each other; therefore, their beginnings and endings are concentric or inclined in exactly the same degree to a perpendicular crossing their centres. These concentric beginnings and endings represent correctly the concentric directions in which the swinging weights commence their downward movements.

Now, since it has been shewn that bodies begin to run down equal descents with equal velocities, it follows that the weight on the short string and that on the long string must commence to move down the concentric curves of their respective arcs at an equal rate. But it has been also shewn that the one of these weights has a nine times longer journey to perform than the other; it is clear, therefore, that both cannot accomplish their respective distances in the same time. The weight on the shortest string in reality makes three vibrations, and the weight on the string that is next to this in length makes two vibrations, while the weight on the longest string is occupied about one; and the differences would be as 9, 4, and 1, instead of as 3, 2, 1, but that the weights moving in the longer arcs benefit most from acceleration of velocity. Although all the vibrating bodies begin to move at equal rates, they pass the central positions directly beneath their points of suspension at unequal ones. Those that have been the longest in getting down to these positions, have of necessity increased their paces the most while upon their route.

Suspended weights, then, only vibrate in equal times when hung upon equal strings; but they continue to make vibrations in equal times notwithstanding the diminution of the arcs in which they swing. This was the fact that caught the attention of Galileo; he observed that the vibrations of the lamp slowly died away as the effect of the disturbing force was destroyed bit by bit, but that, nevertheless, the last faint vibration that caught his eye, took the same apparent time for its performance as the fullest and longest one in the series.

The instrument that has been designated by the learned name of pendulum, is simply a weight of this description placed on the end of a metallic or wooden rod, and hung up in such a way that free sideways motion is permitted. This freedom of motion is generally attained by fixing the top of the rod to a piece of thin, highly elastic steel. A pendulum fitted up after this fashion, will continue in motion, if once started, for many hours. It only stops at last, because the air opposes a slight resistance to its passage, and because the suspending spring is imperfectly elastic. The effects of these two causes combined arrest the vibration at last, but not until they have long accumulated. The weight does not stand still at once, but its arc of vibration grows imperceptibly less and less, until at last there comes a time when the eye cannot tell whether the body is still moving or in absolute repose.

Now, suppose that a careful and patient observer, aware of the exact length of the suspending-rod of a vibrating pendulum, were to set himself down to count how many beats it would make in a given period, he would thenceforward be able to assign a fixed value to each beat, and would consequently have acquired an invariable standard whereby he might estimate short intervals. If he found that his instrument had made exactly 86,400 beats at the end of a mean solar day, and knew that the length of its rod was a trifle more than 39 inches, he would be aware that each beat of such a pendulum might always be taken as the measure of a second. The length of the rod of a pendulum which beats exact seconds in London is 39.13 inches.

But there are few persons who would be willing to go through the tedious operation of counting 86,400 successive vibrations. The invention of a mechanical contrivance that was able to break the monotony of such a task, would be hailed by any one who had to perform it as an invaluable boon. Even a piece of brass with sixty notches upon it, which he might slip through his fingers while noting the swinging body, would enable him to keep his reckoning by sixties instead of units, and so far would afford him considerable relief. But if the notched brass could be turned into a ring, and the pendulum be made to count the notches off for itself, round and round again continuously, registering each revolution as it was completed for future reference, the observer would attain the same result without expending any personal trouble about it. It is this magical conversion of brass and iron into almost intelligent counters of the pendulum's vibrations, that the clock-maker effects by his beautiful mechanism.

In the pendulum clock, the top of the swinging-rod is connected with a curved piece of steel, which dips its teeth-like ends on either hand into notches deeply cut in the edges of a brass wheel. The notched wheel is connected with a train of wheel-work kept moving by the descent of a heavy weight; but it can only move onwards in its revolution under the influence of the weight, as the two ends of the piece of steel are alternately lifted out of the notches by the swaying of the pendulum. The other wheels and pinions of the movement are so arranged that they indicate the number of turns the wheel at the top of the pendulum completes, by means of hands traversing round a dial-plate inscribed with figures and dots.

It is found convenient in practice to make the direct descent of a weight the moving power of the wheel-work, instead of the swinging of the pendulum, for the simple reason, that the excess of its power beyond what is required to overcome the friction of the wheel-work, is then employed in giving a slight push to the pendulum; this push just neutralises the retarding effects before named as inseparable from the presence of air and imperfect means of suspension. The train of wheel-work in a clock, therefore, serves two purposes—it records the number of beats which the pendulum makes, and it keeps that body moving when once started. As far as the activity of the pendulum is concerned, the wheel-work is a recording power, and a preserving power, but not an originating power. If there were no air, and no friction in the apparatus of suspension, the pendulum would continue to go as well without the wheel-work as with it. With the wheel-work it beats as permanently and steadily upon material supports and plunged in a dense atmosphere, as it would if it were hung upon nothing, and were swinging in nothing; and also performs its backward and forward business in solitude and darkness, to the same practical purpose that it would if the eyes of watchful and observant guardians were turned incessantly towards it.

Galileo published his discovery of the isochronous property of the pendulum in 1639. Richard Harris of London took the hint, and connected the pendulum with clock-work movement in 1641. Huyghens subsequently improved the connection, and succeeded in constructing very trustworthy time-keepers, certainly before 1658.

But notwithstanding all that the knowledge and skill of Huyghens could do, his most perfect instruments were still at the mercy of atmospheric changes. It has been said, that the time of a pendulum's vibration depends upon the length of its suspending-rod. This length is measured, not down to the bottom of the weight, but to the centre of its mass. For the weight itself is necessarily a body of considerable dimensions, and in this body some particles must be nearer to, and others further from the point of suspension. Those which are nearest will, of course, in accordance with the principles already explained, have a tendency to make their vibrations in shorter periods; and those which are furthest, in longer periods. But all these particles are bound together firmly by the power of cohesion, and must move connectedly. They, therefore, come to an agreement to move at a mean rate—that is, between the two extremes. The top particles hurry on the middle ones; the bottom particles retard them in a like degree. Consequently, the whole of the weight moves as if its entire mass were concentered in the position of those middle particles; and the exact place of this central position in relation to the point of suspension, becomes the important condition which determines the time in which the instrument swings.

In pendulums of ordinary construction, this relation is by no means an unvarying one—changes of temperature alter the bulk of all kinds of bodies. A metal rod runs up and down under increase and diminution of heat, as certainly as the thread of mercury in the tube of the thermometer does. A hot day, therefore, lengthens the metallic suspending-rod of a pendulum, and carries the centre of its weight to a greater distance from the point of suspension. By this means, the period of each vibration is of necessity lengthened. An increase of temperature to the extent of ten of Fahrenheit's degrees, will make a second's pendulum with a brass rod lose five vibrations in a day. All substances do not, however, suffer the same amount of expansion under like increments of heat. If the rod of the pendulum be made of varnished or black-leaded wood, an addition of ten degrees of heat will not cause it to lose more than one vibration in a day. But even this small irregularity is too vast for the purposes of precise science, and accordingly ingenuity has been taxed to the utmost to find some means of removing the source of inaccuracy, to invent some plan whereby the pendulum may be made sensitive enough to discover and correct its own varying dimensions as different temperatures are brought to bear upon its material.

The first successful attempt to accomplish this useful purpose was made by George Graham in 1715. He replaced the solid weight at the bottom of the rod by a glass jar containing mercury. The rod he formed of steel of the usual length; and because mercury expands five times more than steel, he fixed the height of the column of mercury in the jar at only 6½ inches. In this arrangement he found that additional heat carried up the mercury in the jar, as much as it carried down the jar by the elongation of the rod. Consequently, the motion of the one perfectly compensated the motion of the other, and the effective centre of the weight always remained at the same precise distance from the top of the rod. By the application of this compensating pendulum, clocks are now constructed that do not vary to the extent of a tenth of a second in a day.

Soon after the invention of Graham's mercurial pendulum, John Harrison—the same clever mechanician who received L.20,000 from government for making a chronometer that went to Jamaica in one year and returned in another with an accumulated error of only 1 minute and 54 seconds—hit upon another means of gaining the same end. He brought a steel rod down from the point of suspension, turned it up into a copper rod of less length; and from the top of this hung the weight. He fixed the lengths of the steel and copper rods, which expand unequally, in such a way that the steel carried the copper down exactly as much as the copper carried the weight up; and thus the centre of the weight was still kept at the same distance from the real point of suspension. Harrison's pendulum is generally seen in somewhat the form of a gridiron, because many parallel bars of copper and steel are used in its construction, for the sake of rendering it firm and unyielding in all its parts.