part of the earth is of that exact density; there may be no part exactly corresponding to the mean density, and as the crust of the earth has only about half the mean density, the internal matter of the globe must of course be above the mean. Even the density of a homogeneous substance like carbon or gold must be regarded as a mean between the real density of its atoms, and the zero density of the intervening vacuous space.
The very different signification of the word “mean” in these two uses was fully explained by Quetelet,6 and the importance of the distinction was pointed out by Sir John Herschel in reviewing his work.7 It is much to be desired that scientific men would mark the difference by using the word mean only in the former sense when it denotes approximation to a definite existing quantity; and average, when the mean is only a fictitious quantity, used for convenience of thought and expression. The etymology of this word “average” is somewhat obscure; but according to De Morgan8 it comes from averia, “havings or possessions,” especially applied to farm stock. By the accidents of language averagium came to mean the labour of farm horses to which the lord was entitled, and it probably acquired in this manner the notion of distributing a whole into parts, a sense in which it was early applied to maritime averages or contributions of the other owners of cargo to those whose goods have been thrown overboard or used for the safety of the vessel.
On the Average or Fictitious Mean.
Although the average when employed in its proper sense of a fictitious mean, represents no really existing quantity, it is yet of the highest scientific importance, as enabling us to conceive in a single result a multitude of details. It enables us to make a hypothetical simplification of a problem, and avoid complexity without committing error. The weight of a body is the sum of the weights of infinitely small particles, each acting at a different place, so that a mechanical problem resolves itself, strictly speaking, into an infinite number of distinct problems. We owe to Archimedes the first introduction of the beautiful idea that one point may be discovered in a gravitating body such that the weight of all the particles may be regarded as concentrated in that point, and yet the behaviour of the whole body will be exactly represented by the behaviour of this heavy point. This Centre of Gravity may be within the body, as in the case of a sphere, or it may be in empty space, as in the case of a ring. Any two bodies, whether connected or separate, may be conceived as having a centre of gravity, that of the sun and earth lying within the sun and only 267 miles from its centre.
Although we most commonly use the notion of a centre or average point with regard to gravity, the same notion is applicable to other cases. Terrestrial gravity is a case of approximately parallel forces, and the centre of gravity is but a special case of the more general Centre of Parallel Forces. Wherever a number of forces of whatever amount act in parallel lines, it is possible to discover a point at which the algebraic sum of the forces may be imagined to act with exactly the same effect. Water in a cistern presses against the side with a pressure varying according to the depth, but always in a direction perpendicular to the side. We may then conceive the whole pressure as exerted on one point, which will be one-third from the bottom of the cistern, and may be called the Centre of Pressure. The Centre of Oscillation of a pendulum, discovered by Huyghens, is that point at which the whole weight of the pendulum may be considered as concentrated, without altering the time of oscillation (p. 315). When one body strikes another the Centre of Percussion is that point in the striking body at which all its mass might be concentrated without altering the effect of the stroke. In position the Centre of Percussion does not differ from the Centre of Oscillation. Mathematicians have also described the Centre of Gyration, the Centre of Conversion, the Centre of Friction, &c.