The so-called secular acceleration of the moon’s motion is certainly periodic, and as, so far as we can tell, no body is beyond the attractive power of other bodies, rectilinear motion becomes purely hypothetical, or at least infinitely improbable. All the motions of all the stars must tend to become periodic. Though certain disturbances in the planetary system seem to be uniformly progressive, Laplace is considered to have proved that they really have their limits, so that after an immense time, all the planetary bodies might return to the same places, and the stability of the system be established. Such a theory of periodic stability is really hypothetical, and does not take into account phenomena resulting in the dissipation of energy, which may be a really secular process. For our present purposes we need not attempt to form an opinion on such questions. Any change which does not present the appearance of a periodic character will be empirically regarded as a secular change, so that there will be plenty of non-periodic variations.
The variations which we produce experimentally will often be non-periodic. When we communicate heat to a gas it increases in bulk or pressure, and as far as we can go the higher the temperature the higher the pressure. Our experiments are of course restricted in temperature both above and below, but there is every reason to believe that the bulk being the same, the pressure would never return to the same point at any two different temperatures. We may of course repeatedly raise and lower the temperature at regular or irregular intervals entirely at our will, and the pressure of the gas will vary in like manner and exactly at the same intervals, but such an arbitrary series of changes would not constitute Periodic Variation. It would constitute a succession of distinct experiments, which would place beyond reasonable doubt the connexion of cause and effect.
Whenever a phenomenon recurs at equal or nearly equal intervals, there is, according to the theory of probability, considerable evidence of connexion, because if the recurrences were entirely casual it is unlikely that they would happen at equal intervals. The fact that a brilliant comet had appeared in the years 1301, 1378, 1456, 1531, 1607, and 1682 gave considerable presumption in favour of the identity of the body, apart from similarity of the orbit. There is nothing which so fascinates the attention of men as the recurrence time after time of some unusual event. Things and appearances which remain ever the same, like mountains and valleys, fail to excite the curiosity of a primitive people. It has been remarked by Laplace that even in his day the rising of Venus in its brightest phase never failed to excite surprise and interest. So there is little doubt that the first germ of science arose in the attention given by Eastern people to the changes of the moon and the motions of the planets. Perhaps the earliest astronomical discovery consisted in proving the identity of the morning and evening stars, on the grounds of their similarity of aspect and invariable alternation.[369] Periodical changes of a somewhat complicated kind must have been understood by the Chaldeans, because they were aware of the cycle of 6585 days or 19 years which brings round the new and full moon upon the same days, hours, and even minutes of the year. The earliest efforts of scientific prophecy were founded upon this knowledge, and if at present we cannot help wondering at the precise anticipations of the nautical almanack, we may imagine the wonder excited by such predictions in early times.
Combined Periodic Changes.
We shall seldom find a body subject to a single periodic variation, and free from other disturbances. We may expect the periodic variation itself to undergo variation, which may possibly be secular, but is more likely to prove periodic; nor is there any limit to the complication of periods beyond periods, or periods within periods, which may ultimately be disclosed. In studying a phenomenon of rhythmical character we have a succession of questions to ask. Is the periodic variation uniform? If not, is the change uniform? If not, is the change itself periodic? Is that new period uniform, or subject to any other change, or not? and so on ad infinitum.
In some cases there may be many distinct causes of periodic variations, and according to the principle of the superposition of small effects, to be afterwards considered, these periodic effects will be simply added together, or at least approximately so, and the joint result may present a very complicated subject of investigation. The tides of the ocean consist of a series of superimposed undulations. Not only are there the ordinary semi-diurnal tides caused by sun and moon, but a series of minor tides, such as the lunar diurnal, the solar diurnal, the lunar monthly, the lunar fortnightly, the solar annual and solar semi-annual are gradually being disentangled by the labours of Sir W. Thomson, Professor Haughton and others.
Variable stars present interesting periodic phenomena; while some stars, δ Cephei for instance, are subject to very regular variations, others, like Mira Ceti, are less constant in the degrees of brilliancy which they attain or the rapidity of the changes, possibly on account of some longer periodic variation.[370] The star β Lyræ presents a double maximum and minimum in each of its periods of nearly 13 days, and since the discovery of this variation the period in a period has probably been on the increase. “At first the variability was more rapid, then it became gradually slower; and this decrease in the length of time reached its limit between the years 1840 and 1844. During that time its period was nearly invariable; at present it is again decidedly on the decrease.”[371] The tracing out of such complicated variations presents an unlimited field for interesting investigation. The number of such variable stars already known is considerable, and there is no reason to suppose that any appreciable fraction of the whole number has yet been detected.
Principle of Forced Vibrations.
Investigations of the connection of periodic causes and effects rest upon a principle, which has been demonstrated by Sir John Herschel for some special cases, and clearly explained by him in several of his works.[372] The principle may be formally stated in the following manner: “If one part of any system connected together either by material ties, or by the mutual attractions of its members, be continually maintained by any cause, whether inherent in the constitution of the system or external to it, in a state of regular periodic motion, that motion will be propagated throughout the whole system, and will give rise, in every member of it, and in every part of each member, to periodic movements executed in equal periods, with that to which they owe their origin, though not necessarily synchronous with them in their maxima and minima.” The meaning of the proposition is that the effect of a periodic cause will be periodic, and will recur at intervals equal to those of the cause. Accordingly when we find two phenomena which do proceed, time after time, through changes of the same period, there is much probability that they are connected. In this manner, doubtless, Pliny correctly inferred that the cause of the tides lies in the sun and the moon, the intervals between successive high tides being equal to the intervals between the moon’s passage across the meridian. Kepler and Descartes too admitted the connection previous to Newton’s demonstration of its precise nature. When Bradley discovered the apparent motion of the stars arising from the aberration of light, he was soon able to attribute it to the earth’s annual motion, because it went through its phases in a year.
The most beautiful instance of induction concerning periodic changes which can be cited, is the discovery of an eleven-year period in various meteorological phenomena. It would be difficult to mention any two things apparently more disconnected than the spots upon the sun and auroras. As long ago as 1826, Schwabe commenced a regular series of observations of the spots upon the sun, which has been continued to the present time, and he was able to show that at intervals of about eleven years the spots increased much in size and number. Hardly was this discovery made known, when Lamont pointed out a nearly equal period of variation in the declination of the magnetic needle. Magnetic storms or sudden disturbances of the needle were next shown to take place most frequently at the times when sun-spots were prevalent, and as auroras are generally coincident with magnetic storms, these phenomena were brought into the cycle. It has since been shown by Professor Piazzi Smyth and Mr. E. J. Stone, that the temperature of the earth’s surface as indicated by sunken thermometers gives some evidence of a like period. The existence of a periodic cause having once been established, it is quite to be expected, according to the principle of forced vibrations, that its influence will be detected in all meteorological phenomena.