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.[218] 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 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.[219] 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 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.[220] 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, or to abandon its supposed relation to the earth’s dimensions. The French Government and the International Metrical Commission have for obvious reasons decided in favour of the latter course, and have thus reverted to the first method of defining the metre by a given bar. As from time to time the ratio between this assumed standard metre and the quadrant of the earth becomes more accurately known, we have better means of restoring that metre by reference to the globe if required. But until lost, destroyed, or for some clear reason discredited, the bar metre and not the globe is the standard. Thomson and Tait remark that any of the more accurate measurements of the English trigonometrical survey might in like manner be employed to restore our standard yard, in terms of which the results are recorded.
The Pendulum Standard.
The third method of defining a standard length, by reference to the seconds pendulum, was first proposed by Huyghens, and was at one time adopted by the English Government. From the principle of the pendulum (p. [302]) it clearly appears that if the time of oscillation and the force actuating the pendulum be the same, the length of the pendulum must be the same. We do not get rid of theoretical difficulties, for we must assume the attraction of gravity at some point of the earth’s surface, say London, to be unchanged from time to time, and the sidereal day to be invariable, neither assumption being absolutely correct so far as we can judge. The pendulum, in short, is only an indirect means of making one physical quantity of space depend upon two other physical quantities of time and force.
The practical difficulties are, however, of a far more serious character than the theoretical ones. The length of a pendulum is not the ordinary length of the instrument, which might be greatly varied without affecting the duration of a vibration, but the distance from the centre of suspension to the centre of oscillation. There are no direct means of determining this latter centre, which depends upon the average momentum of all the particles of the pendulum as regards the centre of suspension. Huyghens discovered that the centres of suspension and oscillation are interchangeable, and Kater pointed out that if a pendulum vibrates with exactly the same rapidity when suspended from two different points, the distance between these points is the true length of the equivalent simple pendulum.[221] But the practical difficulties in employing Kater’s reversible pendulum are considerable, and questions regarding the disturbance of the air, the force of gravity, or even the interference of electrical attractions have to be entertained. It has been shown that all the experiments made under the authority of Government for determining the ratio between the standard yard and the seconds pendulum, were vitiated by an error in the corrections for the resisting, adherent, or buoyant power of the air in which the pendulums were swung. Even if such corrections were rendered unnecessary by operating in a vacuum, other difficult questions remain.[222] Gauss’ mode of comparing the vibrations of a wire pendulum when suspended at two different lengths is open to equal or greater practical difficulties. Thus it is found that the pendulum standard cannot compete in accuracy and certainty with the simple bar standard, and the method would only be useful as an accessory mode of restoring the bar standard if at any time again destroyed.