xni.] MEASUREMENT OF PHENOMENA. 303 



the earth's surface, and the times of vibration being astro- 

 nomically determined, the force of gravity becomes accu- 

 rately known. Finally, with a known force of gravity, 

 and time of vibration ascertained by reference to the stars, 

 the length is determinate. 



All astronomical observations depend upon the first 

 manner of using the pendulum, namely, in the astrono- 

 mical 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 deter- 

 mined by the method of vibrations, which is in constant 

 use in the measurement of the horizontal force of terres- 

 trial 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, 1 or to ascertain the laws 

 of motion and elasticity. 2 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, 8 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. 



1 Principia, bk. ii. Sect. 6. Prop. 31. Motte's Translation, vol. iL 

 p. 107. 



2 Ibid. bk. i. Law iii. Corollary 6. Motte's Translation, vol. i. p. 33. 



3 Thomson and Tait's Natural Philosophy, vol. i. p. 333. 



