The principles of science

by William Stanley Jevons

Integrated Variations.

In considering the various modes in which one effect may depend upon another, we must set in a distinct class those which arise from the accumulated effects of a constantly acting cause. When water runs out of a cistern, the velocity of motion depends, according to Torricelli’s theorem, on the height of the surface of the water above the vent; but the amount of water which leaves the cistern in a given time depends upon the aggregate result of that velocity, and is only to be ascertained by the mathematical process of integration. When one gravitating body falls towards another, the force of gravity varies according to the inverse square of the distance; to obtain the velocity produced we must integrate or sum the effects of that law; and to obtain the space passed over by the body in a given time, we must integrate again.

In periodic variations the same distinction must be drawn. The heating power of the sun’s rays at any place on the earth varies every day with the height attained, and is greatest about noon; but the temperature of the air will not be greatest at the same time. This temperature is an integrated effect of the sun’s heating power, and as long as the sun is able to give more heat to the air than the air loses in other ways, the temperature continues to rise, so that the maximum is deferred until about 3 P.M. Similarly the hottest day of the year falls, on an average, about one month later than the summer solstice, and all the seasons lag about a month behind the motions of the sun. In the case of the tides, too, the effect of the moon’s attractive power is never greatest when the power is greatest; the effect always lags more or less behind the cause. Yet the intervals between successive tides are equal, in the absence of disturbance, to the intervals between the passages of the moon across the meridian. Thus the principle of forced vibrations holds true.

In periodic phenomena, however, curious results sometimes follow from the integration of effects. If we strike a pendulum, and then repeat the stroke time after time at the same part of the vibration, all the strokes concur in adding to the momentum, and we can thus increase the extent and violence of the vibrations to any degree. We can stop the pendulum again by strokes applied when it is moving in the opposite direction, and the effects being added together will soon bring it to rest. Now if we alter the intervals of the strokes so that each two successive strokes act in opposite manners they will neutralise each other, and the energy expended will be turned into heat or sound at the point of percussion. Similar effects occur in all cases of rhythmical motion. If a musical note is sounded in a room containing a piano, the string corresponding to it will be thrown into vibration, because every successive stroke of the air-waves upon the string finds it in like position as regards the vibration, and thus adds to its energy of motion. But the other strings being incapable of vibrating with the same rapidity are struck at various points of their vibrations, and one stroke will soon be opposed by one contrary in effect. All phenomena of resonance arise from this coincidence in time of undulation. The air in a pipe closed at one end, and about 12 inches in length, is capable of vibrating 512 times in a second. If, then, the note C is sounded in front of the open end of the pipe, every successive vibration of the air is treasured up as it were in the motion of the air. In a pipe of different length the pulses of air would strike each other, and the mechanical energy being transmuted into heat would become no longer perceptible as sound.

Accumulated vibrations sometimes become so intense as to lead to unexpected results. A glass vessel if touched with a violin bow at a suitable point may be fractured with the violence of vibration. A suspension bridge may be broken down if a company of soldiers walk across it in steps the intervals of which agree with the vibrations of the bridge itself. But if they break the step or march in either quicker or slower pace, they may have no perceptible effect upon the bridge. In fact if the impulses communicated to any vibrating body are synchronous with its vibrations, the energy of those vibrations will be unlimited, and may fracture any body.

Let us now consider what will happen if the strokes be not exactly at the same intervals as the vibrations of the body, but, say, a little slower. Then a succession of strokes will meet the body in nearly but not quite the same position, and their efforts will be accumulated. Afterwards the strokes will begin to fall when the body is in the opposite phase. Imagine that one pendulum moving from one extreme point to another in a second, should be struck by another pendulum which makes 61 beats in a minute; then, if the pendulums commence together, they will at the end of 30 1/2 beats be moving in opposite directions. Hence whatever energy was communicated in the first half minute will be neutralised by the opposite effect of that given in the second half. The effect of the strokes of the second pendulum will therefore be alternately to increase and decrease the vibrations of the first, so that a new kind of vibration will be produced running through its phases in 61 seconds. An effect of this kind was actually observed by Ellicott, a member of the Royal Society, in the case of two clocks.‍[373] He found that through the wood-work by which the clocks were connected a slight impulse was transmitted, and each pendulum alternately lost and gained momentum. Each clock, in fact, tended to stop the other at regular intervals, and in the intermediate times to be stopped by the other.

Many disturbances in the planetary system depend upon the same principle; for if one planet happens always to pull another in the same direction in similar parts of their orbits, the effects, however slight, will be accumulated, and a disturbance of large ultimate amount and of long period will be produced. The long inequality in the motions of Jupiter and Saturn is thus due to the fact that five times the mean motion of Saturn is very nearly equal to twice the mean motion of Jupiter, causing a coincidence in their relative positions and disturbing powers. The rolling of ships depends mainly upon the question whether the period of vibration of the ship corresponds or not with the intervals at which the waves strike her. Much which seems at first sight unaccountable in the behaviour of vessels is thus explained, and the loss of the Captain is a sad case in point.

CHAPTER XXI.
THEORY OF APPROXIMATION.

In order that we may gain a true understanding of the kind, degree, and value of the knowledge which we acquire by experimental investigation, it is requisite that we should be fully conscious of its approximate character. We must learn to distinguish between what we can know and cannot know—between the questions which admit of solution, and those which only seem to be solved. Many persons may be misled by the expression exact science, and may think that the knowledge acquired by scientific methods admits of our reaching absolutely true laws, exact to the last degree. There is even a prevailing impression that when once mathematical formulæ have been successfully applied to a branch of science, this portion of knowledge assumes a new nature, and admits of reasoning of a higher character than those sciences which are still unmathematical.

The very satisfactory degree of accuracy attained in the science of astronomy gives a certain plausibility to erroneous notions of this kind. Some persons no doubt consider it to be proved that planets move in ellipses, in such a manner that all Kepler’s laws hold exactly true; but there is a double error in any such notions. In the first place, Kepler’s laws are not proved, if by proof we mean certain demonstration of their exact truth. In the next place, even assuming Kepler’s laws to be exactly true in a theoretical point of view, the planets never move according to those laws. Even if we could observe the motions of a planet, of a perfect globular form, free from all perturbing or retarding forces, we could never prove that it moved in a perfect ellipse. To prove the elliptical form we should have to measure infinitely small angles, and infinitely small fractions of a second; we should have to perform impossibilities. All we can do is to show that the motion of an unperturbed planet approaches very nearly to the form of an ellipse, and more nearly the more accurately our observations are made. But if we go on to assert that the path is an ellipse we pass beyond our data, and make an assumption which cannot be verified by observation.