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,[201] or to ascertain the laws of motion and elasticity.[202] In these cases the extent of vibration is the quantity measured, and the principles of the instrument are different. [201] *Principia*, bk. ii. Sect. 6. Prop. 31. Motte’s Translation, vol. ii. p. 107. [202] Ibid. bk. i. Law iii. Corollary 6. Motte’s Translation, vol. i. p. 33. *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,[203] 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.”[204] 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). [203] Thomson and Tait’s *Natural Philosophy*, vol. i. p. 333. [204] *Philosophical Transactions*, (1856), vol. cxlvi. pp. 330, 331.