and thus a comparison instituted between the attractive forces of the mountain and the whole globe, which led to a probable estimate of the earth’s density.
In some cases it is actually better to determine the zero point by the average of equally diverging quantities than by direct observation. In delicate weighings by a chemical balance it is requisite to ascertain exactly the point at which the beam comes to rest, and when standard weights are being compared the position of the beam is ascertained by a carefully divided scale viewed through a microscope. But when the beam is just coming to rest, friction, small impediments or other accidental causes may readily obstruct it, because it is near the point at which the force of stability becomes infinitely small. Hence it is found better to let the beam vibrate and observe the terminal points of the vibrations. The mean between two extreme points will nearly indicate the position of rest. Friction and the resistance of air tend to reduce the vibrations, so that this mean will be erroneous by half the amount of this effect during a half vibration. But by taking several observations we may determine this retardation and allow for it. Thus if a, b, c be the readings of the terminal points of three excursions of the beam from the zero of the scale, then 1/2 (a + b) will be about as much erroneous in one direction as 1/2 (b + c) in the other, so that the mean of these two means, or 1/4 (a + 2b + c), will be exceedingly near to the point of rest.12 A still closer approximation may be made by taking four readings and reducing them by the formula 1/6 (a + 2b + 2c + d).
The accuracy of Baily’s experiments, directed to determine the density of the earth, entirely depended upon this mode of observing oscillations. The balls whose gravitation was measured were so delicately suspended by a torsion balance that they never came to rest. The extreme points of the oscillations were observed both when the heavy leaden attracting ball was on one side and on the other. The difference of the mean points when the leaden ball was on the right hand and that when it was on the left hand gave double the amount of the deflection.