(3) It may give a result more or less free from unknown and uncertain errors; this we may call the Probable mean result.

Of these three uses of the mean the first is entirely different in nature from the two last, since it does not yield an approximation to any natural quantity, but furnishes us with an arithmetic result comparing the aggregate of certain quantities with their number. The third use of the mean rests entirely upon the theory of probability, and will be more fully considered in a later part of this chapter. The second use is closely connected, or even identical with, the Method of Reversal already described, but it will be desirable to enter somewhat fully into all the three employments of the same arithmetical process.

The Mean and the Average.

Much confusion exists in the popular, or even the scientific employment of the terms mean and average, and they are commonly taken as synonymous. It is necessary to ascertain carefully what significations we ought to attach to them. The English word mean is equivalent to medium, being derived, perhaps through the French moyen, from the Latin medius, which again is undoubtedly kindred with the Greek μeros. Etymologists believe, too, that this μεσος. Greek word is connected with the preposition μera, the German mitte, and the true English mid or middle; so that after all the mean is a technical term identical in its root with the more popular equivalent middle.

If we inquire what is the mean in a mathematical point of view, the true answer is that there are several or many kinds of means. The old arithmeticians recognised ten kinds, which are stated by Boethius, and an eleventh was added by Jordanus.1

The arithmetic mean is the one by far the most commonly denoted by the term, and that which we may understand it to signify in the absence of any qualification. It is the sum of a series of quantities divided by their number, and may be represented by the formula (a + b).

1 De Morgan, Supplement to the Penny Cyclopædia, art. Old Appellations of Numbers.

But there is also the geometric mean, which is the square root of the product, √a × 6, or that quantity the logarithm of which is the arithmetic mean of the logarithms of the quantities. There is also the harmonic mean, which is the reciprocal of the arithmetic mean of the reciprocals of the quantities. Thus if a and b be the quantities, as before, their reciprocals are and the

a

mean of which is (+), and the reciprocal again is

2ab

a + b

which is the harmonic mean. Other kinds of means might no doubt be invented for particular purposes, and we might apply the term, as De Morgan pointed out, to any quantity a function of which is equal to a function of two or more other quantities, and is such that the interchange of these latter quantities among themselves will make no alteration in the value of the function. Symbolically, if (y, y, y . . . .) = $ (X1, X2, X3 . . . .), then y is a kind of mean of the quantities, 2, 2, &c.

The geometric mean is necessarily adopted in certain cases. When we estimate the work done against a force which varies inversely as the square of the distance from a fixed point, the mean force is the geometric mean between the forces at the beginning and end of the path. When in an imperfect balance, we reverse the weights to eliminate. error, the true weight will be the geometric mean of the two apparent weights. In almost all the calculations of statistics and commerce the geometric mean ought, strictly speaking, to be used. If a commodity rises in price 100 per cent. and another remains unaltered, the mean rise of a price is not 50 per cent. because the ratio 150: 200 is not the same as 100: 150. The mean ratio is as unity to √100 X 200 or 1 to 141. The difference between the three kinds of means in such a case 2 is very considerable; while the rise of price estimated by the Arithmetic mean would be 50 per cent. it would be only 41 and 33 per cent. respectively according to the Geometric and Harmonic.

means.

1 Penny Cyclopædia, art. Mean.

Jevons, Journal of the Statistical Society, June 1865, vol. xxviii p. 296.

In all calculations concerning the average rate of progress of a community, or any of its operations, the geometric mean should be employed. For if a quantity increases 100 per cent. in 100 years, it would not on the average increase 10 per cent. in each ten years, as the 10 per cent. would at the end of each decade be calculated upon larger and larger quantities, and give at the end of 100 years much more than 100 per cent., in fact as much as 159 per cent. The true mean rate in each decade would be 1/2 or about 107, that is, the increase would be about 7 per cent. in each ten years. But when the quantities differ very little, the arithmetic and geometric means are approximately the same. Thus the arithmetic mean of 1000 and 1001 is 10005, and the geometric mean is about 10004998, the difference being of an order inappreciable in almost all scientific and practical matters. Even in the comparison of standard weights by Gauss' method of reversal, the arithmetic mean may usually be substituted for the geometric mean which is the true result.

Regarding the mean in the absence of express qualification to the contrary as the common arithmetic mean, we must still distinguish between its two uses where it gives with more or less accuracy and probability a really existing quantity, and where it acts as a mere representative of other quantities. If I make many experiments to determine the atomic weight of an element, there is a certain number which I wish to approximate to, and the mean of my separate results will, in the absence of any reasons to the contrary, be the most probable approximate result. When we determine the mean density of the earth, it is not because any 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,1 and the

1 Letters on the Theory of Probabilities, transl. by Downes, Part ii.

importance of the distinction was pointed out by Sir John Herschel in reviewing his work. 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 Morgan2 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 Inay 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

1 Herschel's Essays, &c. pp. 404, 405..

On the Theory of Errors of Observations, Cambridge Philosophical Transactions, vol. x. Part ii. 416.

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.

We ought carefully to distinguish between those cases in which an invariable centre can be assigned, and those in which it cannot. In perfect strictness, there is no such thing as a true invariable centre of gravity. As a general rule a body is capable of possessing an invariable centre only for perfectly parallel forces, and gravity never does act in absolutely parallel lines. Thus, as usual, we find that our conceptions are only hypothetically correct, and only approximately applicable to real circumstances. There are indeed certain geometrical forms called Centrobaric,1 such that a body of that shape would attract another exactly as if the mass were concentrated at the centre of gravity, whether the forces act in a parallel manner or not.

1 Thomson and Tait, Treatise on Natural Philosophy, vol. i. p. 394