and Tait say: "Equal times are times during which the earth turns through equal angles." No long time has passed since astronomers thought it impossible to detect any inequality in its movement. Poisson was supposed to have proved that a change in the length of the sidereal day amounting to one ten-millionth part in 2,500 years was incompatible with an ancient eclipse recorded by the Chaldæans, and similar calculations were made by Laplace. But it is now known that these calculations were somewhat in error, and that the dissipation of energy arising out of the friction of tidal waves, and the radiation of the heat into space, has slightly decreased the rapidity of the earth's rotatory motion. The sidereal day is now longer by one part in 2,700,000, than it was in 720 B.C. Even before this discovery, it was known that invariability of rotation depended upon the perfect maintenance of the earth's internal heat, which is requisite in order that the earth's dimensions shall be unaltered. Now the earth being superior in temperature to empty space, must cool more or less rapidly, so that it cannot furnish an absolute measure of time. Similar objections could be raised to all other rotating bodies within our cognisance.

The moon's motion round the earth, and the earth's motion round the sun, form the next best measure of time. They are subject, indeed, to disturbance from other planets, but it is believed that these perturbations must in the course of time run through their rhythmical courses, leaving the mean distances unaffected, and consequently, by the third Law of Kepler, the periodic times unchanged. But there is more reason than not to believe that the earth encounters a slight resistance in passing through space, like that which is so apparent in Encke's comet. There may also be dissipation of energy in the electrical relations of the earth to the sun, possibly identical with that which is manifested in the retardation of comets. It is probably an untrue assumption then, that the earth's orbit remains quite invariable. It is just possible that some other body may be found in the course of time to furnish a better

1 Treatise on Natural Philosophy, vol. i. p. 179.

2 Proceedings of the Manchester Philosophical Society, 28th Nov. 1871, vol. xi. p. 33.

standard of time than the earth in its annual motion. The greatly superior mass of Jupiter and its satellites, and their greater distance from the sun, may render the electrical dissipation of energy less considerable than in the case of the earth. But the choice of the best measure will always be an open one, and whatever moving body we choose may ultimately be shown to be subject to disturbing forces.

The pendulum, although so admirable an instrument for subdivision of time, fails as a standard; for though the same pendulum affected by the same force of gravity performs equal vibrations in equal times, yet the slightest change in the form or weight of the pendulum, the least corrosion of any part, or the most minute displacement of the point of suspension, falsifies the results, and there enter many other difficult questions of temperature, friction, resistance, length of vibration, &c.

Thomson and Tait are of opinion that the ultimate standard of chronometry must be founded on the physical properties of some body of more constant character than the earth; for instance, a carefully arranged metallic spring, hermetically sealed in an exhausted glass vessel. But it is hard to see how we can be sure that the dimensions and elasticity of a piece of wrought metal will remain perfectly unchanged for the few millions of years contemplated by them. A nearly perfect gas, like hydrogen, is perhaps the only kind of substance in the unchanged elasticity of which we could have confidence. Moreover, it is difficult to perceive how the undulations of such a spring could be observed with the requisite accuracy. More recently Professor Clerk Maxwell has made the novel suggestion, discussed in a subsequent section, that undulations of light in vacuo would form the most universal standard of reference, both as regards time and space. According to this system the unit of time would be the time occupied by one vibration of the particular kind of light whose wave length is taken as the unit of length.

1 The Elements of Natural Philosophy, part i. p. 119.

The Unit of Space and the Bar Standard.

Next in importance after the measurement of time is that of space. Time comes first in theory, because phenomena, our internal thoughts for instance, may change in time without regard to space. As to the phenomena of outward nature, they tend more and more to resolve themselves into motions of molecules, and motion cannot be conceived or measured without reference both to time and space.

Turning now to space measurement, we find it almost equally difficult to fix and define once and for ever, a unit magnitude. There are three different modes in which it has been proposed to attempt the perpetuation of a standard length.

(1) By constructing an actual specimen of the standard yard or metre, in the form of a bar.

(2) By assuming the globe itself to be the ultimate standard of magnitude, the practical unit being a submultiple of some dimension of the globe.

(3) By adopting the length of the simple seconds pendulum, as a standard of reference.

At first sight it might seem that there was no great difficulty in this matter, and that any one of these methods might serve well enough; but the more minutely we inquire into the details, the more hopeless appears to be the attempt to establish an invariable standard. We must in the first place point out a principle not of an obvious character, namely, that the standard length must be defined by one single object. To make two bars of exactly the same length, or even two bars bearing a perfectly defined ratio to each other, is beyond the power of human art. If two copies of the standard metre be made and declared equally correct, future investigators will certainly discover some discrepancy between them, proving of course that they cannot both be the standard, and giving cause for dispute as to what magnitude should then be taken as correct.

If one invariable bar could be constructed and maintained as the absolute standard, no such inconvenience could arise. Each successive generation as it acquired

1 See Harris Essay upon Money and Coins, part. ii. [1758] p. 127.

higher powers of measurement, would detect errors in the copies of the standard, but the standard itself would be unimpeached, and would, as it were, become by degrees more and more accurately known. Unfortunately to construct and preserve a metre or yard is also a task which is either impossible, or what comes nearly to the same thing, cannot be shown to be possible. Passing over the practical difficulty of defining the ends of the standard length with complete accuracy, whether by dots or lines on the surface, or by the terminal points of the bar, we have no means of proving that substances remain of invariable dimensions. Just as we cannot tell whether the rotation of the earth is uniform, except by comparing it with other moving bodies, believed to be more uniform in motion, so we cannot detect the change of length in a bar, except by comparing it with some other bar supposed to be invariable. But how are we to know which is the invariable bar? It is certain that many rigid and apparently invariable substances do change in dimensions. The bulb of a thermometer certainly contracts by age, besides undergoing rapid changes of dimensions when warmed or cooled through 100° Cent. Can we be sure that even the most solid metallic bars do not slightly contract by age, or undergo variations in their structure by change of temperature. Fizeau was induced to try whether a quartz crystal, subjected to several hundred alternations of temperature, would be modified in its physical properties, and he was unable to detect any change in the coefficient of expansion. It does not follow, however, that, because no apparent change was discovered in a quartz crystal, newly-constructed bars of metal would undergo no change.

The best principle, as it seems to me, upon which the perpetuation of a standard of length can be rested, is that, if a variation of length occurs, it will in all probability be of different amount in different substances. If then a great number of standard metres were constructed of all kinds of different metals and alloys; hard rocks, such as granite, serpentine, slate, quartz, limestone; artificial substances, such as porcelain, glass, &c., &c., careful

1 Philosophical Magazine, (1868), 4th Series, vol. xxxvi. p. 32.

comparison would show from time to time the comparative variations of length of these different substances. The most variable substances would be the most divergent, and the standard would be furnished by the mean length of those which agreed most closely with each other just as uniform motion is that of those bodies which agree most closely in indicating the efflux of time.

The Terrestrial Standard.

The second method assumes that the globe itself is a body of invariable dimensions and the founders of the metrical system selected the ten-millionth part of the distance from the equator to the pole as the definition of the metre. The first imperfection in such a method is that the earth is certainly not invariable in size; for we know that it is superior in temperature to surrounding space, and must be slowly cooling and contracting. There is much reason to believe that all earthquakes, volcanoes, mountain elevations, and changes of sea level are evidences of this contraction as asserted by Mr. Mallet. But such is the vast bulk of the earth and the duration of its past existence, that this contraction is perhaps less rapid in proportion than that of any bar or other material standard which we can construct.

The second and chief difficulty of this method arises from the vast size of the earth, which prevents us from making any comparison with the ultimate standard, except by a trigonometrical survey of a most elaborate and costly kind. The French physicists, who first proposed the method, attempted to obviate this inconvenience by carrying out the survey once for all, and then constructing a standard metre, which should be exactly the one ten millionth part of the distance from the pole to the equator. But since all measuring operations are merely approximate, it was impossible that this operation could be perfectly achieved. Accordingly, it was shown in 1838 that the supposed French metre was erroneous to the considerable extent of one part in 5527. It then became necessary either to alter the length of the assumed metre,

1 Proceedings of the Royal Society, 20th June, 1872, vol. xx. p. 438.