When the circumstances of an experiment are much altered, different powers of the variable may become prominent. The resistance of a liquid to a body moving through it may be approximately expressed as the sum of two terms respectively involving the first and second powers of the velocity. At very low velocities the first power is of most importance, and the resistance, as Professor Stokes has shown, is nearly in simple proportion to the velocity. When the motion is rapid the resistance increases in a still greater degree, and is more nearly proportional to the square of the velocity. Approximate Independence of Small Effects. One result of the theory of approximation possesses such importance in physical science, and is so often applied, that we may consider it separately. The investigation of causes and effects is immensely simplified when we may consider each cause as producing its own effect invariably, whether other causes are acting or not. Thus, if the body P produces x, and Q produces y, the question is whether P and Q acting together will produce the sum of the separate effects,+y. It is under this supposition that we treated the methods of eliminating error (Chap. XV.), and errors of a less amount would still remain if the supposition was a forced one. There are probably some parts of science in which the supposition of independence of effects holds rigidly true. The mutual gravity of two bodies is entirely unaffected by the presence of other gravitating bodies. People do not usually consider that this important principle is involved in such a simple thing as putting two pound weights in the scale of a balance. How do we know that two pounds together will weigh twice as much as one? Do we know it to be exactly so? Like other results founded on induction we cannot prove it absolutely, but all the calculations of physical astronomy proceed upon the assumption, so that we may consider it proved to a very high degree of approximation. Had not this been true, the calculations of physical astronomy would have been infinitely more complex than they actually are, and the progress of knowledge would have been much slower. It is a general principle of scientific method that if effects be of small amount, comparatively to our means of observation, all joint effects will be of a higher order of smallness, and may therefore be rejected in a first approximation. This principle was employed by Daniel Bernoulli in the theory of sound, under the title of The Principle of the Coexistence of Small Vibrations. He showed that if a string is affected by two kinds of vibrations, we may consider each to be going on as if the other did not exist. We cannot perceive that the sounding of one musical instrument prevents or even modifies the sound of another, so that all sounds would seem to travel through the air, and act upon the ear in independence of each other. A similar assumption is made in the theory of tides, which are great waves. One wave is produced by the attraction of the moon, and another by the attraction of the sun, and the question arises, whether when these waves. coincide, as at the time of spring tides, the joint wave will be simply the sum of the separate waves. On the principle of Bernoulli this will be so, because the tides. on the ocean are very small compared with the depth of the ocean. The principle of Bernoulli, however, is only approximately true. A wave never is exactly the same when another wave is interfering with it, but the less the displacement of particles due to each wave, the less in a still higher degree is the effect of one wave upon the other. In recent years Helmholtz was led to suspect that some of the phenomena of sound might after all be due to resultant effects overlooked by the assumption of previous physicists. He investigated the secondary waves which would arise from the interference of considerable disturbances, and was able to show that certain summation or resultant tones ought to be heard, and experiments subsequently devised for the purpose showed that they might be heard. Throughout the mechanical sciences the Principle of the Superposition of Small Motions is of fundamental importance, and it may be thus explained. Suppose 1 Thomson and Tait's Natural Philosophy, vol. i. p. 60. P B that two forces, acting from the points B and C, are simultaneously moving a body A. Let the force acting from B be such that in one second it would move A to p, and similarly let the second force, acting alone, move A to r. The question A arises, then, whether their joint action will urge A to q along the diagonal of the parallelogram. May we say that A will move the distance Ap in the direction AB, and Ar in the direction AC, or, what is the same thing, along the parallel line pq? In strictness we cannot say so; for when A has moved towards p, the force from C will no longer act along the line AC, and similarly the motion of A towards r will modify the action of the force from B. This interference. of one force with the line of action of the other will evidently be greater the larger is the extent of motion considered; on the other hand, as we reduce the parallelogram Apqr, compared with the distances AB and AC, the less will be the interference of the forces. Accordingly mathematicians avoid all error by considering the motions as infinitely small, so that the interference becomes of a still higher order of infinite smallness, and may be entirely neglected. By the resources of the differential calculus it is possible to calculate the motion of the particle A, as if it went through an infinite number of infinitely small diagonals of parallelograms. The great discoveries of Newton really arose from applying this method of calculation to the movements of the moon round the earth, which, while constantly tending to move onward in a straight line, is also deflected towards the earth by gravity, and moves through an elliptic curve, composed as it were of the infinitely small diagonals of infinitely numerous parallelograms. The mathematician, in his investigation of a curve, always treats it as made up of a great number of straight lines, and it may be doubted whether he could treat it in any other manner. There is no error in the final results, because having obtained the formula flowing from this supposition, each straight line is then regarded as becoming infinitely small, and the polygonal line becomes undistinguishable from a perfect curve.1 In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of infinitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible. Nevertheless, while carefully distinguishing between these two different cases, we may fearlessly apply to both the principle of the superposition of sinall effects. In physical science we have only to take care that the effects really are so small that any joint effect will be unquestionably imperceptible. Suppose, for instance, that there is some cause which alters the dimensions of a body in the ratio of I to Ia, and another cause which produces an alteration in the ratio of 1 to 1 + ẞ. If they both act at once the change will be in the ratio of I to (I + a)(1 + ß), or as I to I + a + B + aß. But if a and ẞ be both very small fractions of the total dimensions, aß will be yet far smaller and may be disregarded; the ratio of change is then approximately that of 1 to I + a + ẞ, or the joint effect is the sum of the separate effects. Thus if a body were subjected to three strains, at right angles to each other, the total change in the volume of the body would be approximately equal to the sum of the changes produced by the separate strains, provided that these are very small. In like manner not only is the expansion of every solid and liquid substance by heat approximately proportional to the change of temperature, when this change is very small in amount, but the cubic expansion may also be considered as being three times as great as the linear expansion. For if the increase of temperature expands a bar of metal in the ratio of 1 to I + a, and the expansion be equal in all directions, then a cube of the same metal would expand as I to (I + a)3, or as I to I + 3a + 3a2 + a3. When a is a very small quantity the third term 3a2 will be imperceptible, and still more so the fourth term a3. The coefficients of expansion of solids are in fact so small, and so imperfectly determined, that physicists seldom take into account their second and higher powers. 1 Challis, Notes on the Principles of Pure and Applied Calculation, 1869, p. 83. It is a result of these principles that all small errors may be assumed to vary in simple proportion to their causes-a new reason why, in eliminating errors, we should first of all make them as small as possible. Let us suppose that there is a right-angled triangle of which the two sides containing the right angle are really of the lengths 3 and 4 so that the hypothenuse is √32 + 4 or 5. Now, if in two measurements of the first side we commit slight errors, making it successively 4'001 and 4'002, then calculation will give the lengths of the hypothenuse as almost exactly 50008 and 50016, so that the error in the hypothenuse will seem to vary in simple proportion to that of the side, although it does not really do so with perfect exactness. The logarithm of a number does not vary in proportion to that number-nevertheless we find the difference between the logarithms of the numbers 100000 and 100001 to be almost exactly equal to that between the numbers 100001 and 100002. It is thus a general rule that very small differences between successive values of a function are approximately proportional to the small differences of the variable quantity. On these principles it is easy to draw up a series of rules such as those given by Kohlrausch for performing calculations in an abbreviated form when the variable quantity is very small compared with unity. I ÷ (I + a) we may substitute I a; for we may put + a; I ÷ √ I + a becomes I forth. Four Meanings of Equality. Thus for ( − a) a, and so Although it might seem that there are few terms more free from ambiguity than the term equal, yet scientific men do employ it with at least four meanings, which it is desirable to distinguish. These meanings I may describe as (1) Absolute Equality. (2) Sub-equality. 1 An Introduction to Physical Measurements, translated by Waller and Procter, 1873, p. 10. |