All astronomical observations depend upon the first manner of using the pendulum, namely, in the astronomical clock. In the second employment it has been almost equally indispensable. The primary principle that gravity is equal in all matter was proved by Newton’s and Gauss’ pendulum experiments. The torsion pendulum of Michell, Cavendish, and Baily, depending upon exactly the same principles as the ordinary pendulum, gave the density of the earth, one of the foremost natural constants. Kater and Sabine, by pendulum observations in different parts of the earth, ascertained the variation of gravity, whence comes a determination of the earth’s ellipticity. The laws of electric and magnetic attraction have also been determined by the method of vibrations, which is in constant use in the measurement of the horizontal force of terrestrial magnetism.
We must not confuse with the ordinary use of the pendulum its application by Newton, to show the absence of internal friction against space,22 or to ascertain the laws of motion and elasticity.23 In these cases the extent of vibration is the quantity measured, and the principles of the instrument are different.
Attainable Accuracy of Measurement.
It is a matter of some interest to compare the degrees of accuracy which can be attained in the measurement of different kinds of magnitude. Few measurements of any kind are exact to more than six significant figures,24 but it is seldom that such accuracy can be hoped for. Time is the magnitude which seems to be capable of the most exact estimation, owing to the properties of the pendulum, and the principle of repetition described in previous sections. As regards short intervals of time, it has already been stated that Sir George Airy was able to estimate one part in 8,640,000, an exactness, as he truly remarks, “almost beyond conception.”25 The ratio between the mean solar and the sidereal day is known to be about one part in one hundred millions, or to the eighth place of decimals, (p. 289).
Determinations of weight seem to come next in exactness, owing to the fact that repetition without error is applicable to them. An ordinary good balance should show about one part in 500,000 of the load. The finest balance employed by M. Stas, turned with one part in 825,000 of the load.26 But balances have certainly been constructed to show one part in a million,27 and Ramsden is said to have constructed a balance for the Royal Society, to indicate one part in seven millions, though this is hardly credible. Professor Clerk Maxwell takes it for granted that one part in five millions can be detected, but we ought to discriminate between what a balance can do when first constructed, and when in continuous use.
Determinations of length, unless performed with extraordinary care, are open to much error in the junction of the measuring bars. Even in measuring the base line of a trigonometrical survey, the accuracy generally attained is only that of about one part in 60,000, or an inch in the mile; but it is said that in four measurements of a base line carried out very recently at Cape Comorin, the greatest error was 0·077 inch in 1·68 mile, or one part in 1,382,400, an almost incredible degree of accuracy. Sir J. Whitworth has shown that touch is even a more delicate mode of measuring lengths than sight, and by means of a splendidly executed screw, and a small cube of iron placed between two flat-ended iron bars, so as to be suspended when touching them, he can detect a change of dimension in a bar, amounting to no more than one-millionth of an inch.28