Restrict the attention to a very small portion of the curve, and the eye will be unable to distinguish its difference from a straight line, which amounts to saying that in the portion examined the term Cx2 has no value appreciable by the eye. Take a larger portion of the curve and it will be apparent that it possesses curvature, but it will be possible to draw a parabola or ellipse so that the curve shall apparently coincide with a portion of that parabola or ellipse. In the same way if we take larger and larger arcs of the curve it will assume the character successively of a curve of the third, fourth, and perhaps higher degrees; that is to say, it corresponds to equations involving the third, fourth, and higher powers of the variable quantity.
We have arrived then at the conclusion that every phenomenon, when its amount can only be rudely measured, will either be of fixed amount, or will seem to vary uniformly like the distance between two inclined straight lines. More exact measurement may show the error of this first assumption, and the variation will then appear to be like that of the distance between a straight line and a parabola or ellipse. We may afterwards find that a curve of the third or higher degrees is really required to represent the variation. I propose to call the variation of a quantity linear, elliptic, cubic, quartic, quintic, &c., according as it is discovered to involve the first, second, third, fourth, fifth, or higher powers of the variable. It is a general rule in quantitative investigation that we commence by discovering linear, and afterwards proceed to elliptic or more complicated laws of variation. The approximate curves which we employ are all, according to De Morgan’s use of the name, parabolas of some order or other; and since the common parabola of the second order is approximately the same as a very elongated ellipse, and is in fact an infinitely elongated ellipse, it is convenient and proper to call variation of the second order elliptic. It might also be called quadric variation.
As regards many important phenomena we are yet only in the first stage of approximation. We know that the sun and many so-called fixed stars, especially 61 Cygni, have a proper motion through space, and the direction of this motion at the present time is known with some degree of accuracy. But it is hardly consistent with the theory of gravity that the path of any body should really be a straight line. Hence, we must regard a rectilinear path as only a provisional description of the motion, and look forward to the time when its curvature will be detected, though centuries perhaps must first elapse.
We are accustomed to assume that on the surface of the earth the force of gravity is uniform, because the variation is of so slight an amount that we are scarcely able to detect it. But supposing we could measure the variation, we should find it simply proportional to the height. Taking the earth’s radius to be unity, let h be the height at which we measure the force of gravity. Then by the well-known law of the inverse square, that force will be proportional to
g/(1 + h)2, or to g(1 - 2h + 3h2 - 4h3 + . . .).
But at all heights to which we can attain h will be so small a fraction of the earth’s radius that 3h2 will be inappreciable, and the force of gravity will seem to follow the law of linear variation, being proportional to 1 - 2h.
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, x + 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.