siderable presumption in favour of the identity of the body apart from the similarity of the orbit. There is nothing which so strongly 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 physical science arose in the attention given by Eastern people to the changes of the moon and the motions of the planets. One of the earliest astronomical discoveries must have consisted in proving the identity of the morning and evening stars, on the ground of their similarity of aspect and invariable alternation 9. Periodical changes of a somewhat complicated kind must have been understood by the Chaldæans, 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 readily imagine the wonder excited by such successful predictions in early times.

Combined Periodic Changes.

We shall seldom or never find a body subject to a single periodic variation, and free from any other disturbances. As a general rule we may expect the periodic variation itself to undergo variation, which may possibly be secular or incapable of repetition, but is more likely to prove

a Laplace, System of the World,' vol. i. pp. 50, 54, &c.

periodic likewise; nor is there any limit to the complication of periods beyond periods, or periods within periods, which may ultimately be disclosed. In studying, then, 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 more fully 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. Thus the tides of the ocean consist of a series of superimposed undulations, of which the number and character have by no means been determined as yet. Not only are there the ordinary and very obvious 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 and others ".

Variable stars present very interesting periodic phenomena; while some stars, & Cephei for instance are subject to very regular and equal 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 much longer periodic variations. The star B 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 varia

r 'British Association Report,' 1870, p. 120.

s Herschel's 'Outlines of Astronomy,' 4th edit. pp. 555-557.

bility 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 t." It is evident that the tracing out of such complicated variations presents an almost 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.

All investigations of the connection of periodic causes and effects rest upon a most important and general principle, which has been demonstrated by Sir John Herschel for some special cases, and clearly explained by him in several of his works". 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 systems, and will give rise, in every member of it, and in every part of each member, to periodic movements executed in equal period, 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

t Humboldt's 'Cosmos' (Bohn), vol. iii. p. 229.

6

Encyclopædia Metropolitana,' art. Sound, § 323; Outlines of Astronomy,' 4th edit. § 650, pp. 410, 487-88; 'Meteorology,' Reprint, P. 137.

cause. Accordingly whenever we find any two phenomena which do proceed, time after time, through changes of exactly the same period, there is much probability that they are connected. It was in this manner, doubtless, that Pliny correctly conjectured that the cause of the tides lay in the sun and 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 all its phases in exactly a year.

The most extensive and beautiful instance of induction concerning periodic changes which can be cited, is that of the discovery of an eleven-year period in various meteorological and astronomical 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, of Dessau, 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, than Dr. Lamont pointed out a nearly equal period of variation in the magnetic needle as regards declination. The occasional magnetic storms or sudden irregular 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 strange phenomena were brought into the cycle. It has since been shown by

* 'Nature,' vol. i. p. 284; Quetelet, 'Sur la Physique du Globe,' pp. 148, 262-64, &c.

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 more or less considerable in all meteorological phenomena.

Perhaps the most mysterious part of these investigations is that which refers the phenomena to the planetary configurations as an ulterior cause. Professor Balfour Stewart, with Messrs. Warren de la Rue and Loewy, by laborious researches discovered a periodic change of 584 days in the sun spots, coincident with changes in the relative positions of the Earth, Jupiter, and Venus. It has since been rendered probable by the researches of Dr. Kirkwood and others, that Schwabe's eleven-year period is due to the action of Mercury. Several other periods of more or less importance have been supposed to exist, but the subject is yet open to much more inquiry.

Integrated Variations.

In considering the infinite variety of modes in which one effect may depend upon another, we must set apart 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