It is still true that "the calculated movements of general prices go in exactly opposite directions in every sequence of years,' that is three times.1 But as the distance to which they go is inconsiderable in comparison with the "probable error" to be expected, it would be requiring too much that they should always go in the same direction. The figures in the table from which an extract is given had been noticed in the Memorandum referred to as exceptional, not on account of their divergence but on account of their agreement. "The annexed comparison," it was there remarked, "does not present the appearance of pure chance. The discrepancies are rather less in magnitude than the theory regards." This "faultily faultless" character of the index-number is pro tanto corrected by Mr. Walsh when he points out some little discrepancies in the matter of the sequences.

Had he bestowed more attention on the theory of averages, our author would have asserted with less confidence that “in no other case [except the case in which all prices vary alike] do we want to seek any determination irrespective of the quantities of commodities.'" 2 There is a secondary form of the problem with respect to which weighting has even less importance than under the first aspect. I may introduce this variety by a problem which has been likened to the problem now before us, the determination of the sun's motion relatively to the sidereal system. Referring to this sort of problem Mr. Walsh has some just remarks on the relative motion of the single body and the system (68, cp. 38). He may be right in suggesting that the use of Probabilities in the analogous monetary problem has sometimes been connected with a confusion between cost-value and the kind of value which he has set himself to measure. Yet I do not feel sure that the function of the Calculus is adequately recognised in the following passage:

“When we have chosen which method we shall adopt, and what shall be our standard [whether we shall consider motion of a body relatively to all other things, or to all things including itself],

1 Out of fifteen sequences or changes from year to year shown by the complete table eleven are in the same direction for both weighted and unweighted index. numbers; four are in opposite directions, viz. 1873-1874 and the three sequences selected by Mr. Walsh, 1880-1881, 1881-1882, 1882-1883.

2 Page 222, note. Referring to the present writer's Memorandum attached to the Report of the British Association Committee, 1887, p. 280; where the commentator strangely supposes that the case contemplated is that "in which all prices vary alike." The context of the section referred to and the parallel section in the third Memorandum (Report of the British Association, 1889, p. 156) make it clear that the sought common effect of changes in the supply of money is not supposed to be given free from disturbances special to particular commodities (cp. below, p. 380 et seq.).

there is of course no occasion for employing in our measurements the law of probabilities-as was asserted also in this connection by Cournot. We do not say it is more probable that all the other things have remained stationary than that this one has stood still and they moved; or it is more probable that all things have together remained stationary, wherefore both this and the others have moved relatively to the whole. But having adopted our point of view we simply measure as best we can what we see happening before us. And our point of view itself in these matters we adopt not by any use of the law of probabilities, but because the myriad inter-relations which do not change, or which do not change on the average, make more impression on us than the particular ones which do change " (69, 70).

However this may be, it does not invalidate the proposition which I am concerned to maintain that without knowing the centre of gravity, or "weighted mean " of a system of bodies, we may know by the theory of averages that one single body is advancing through the cluster. Leaving the problem of the stars, which involves some technicalities, let me take a humbler terrestrial illustration. The annexed pairs of figures were thus obtained : As I walked along Piccadilly one day I noted the number of omnibuses which met me (viz. 7) and the number which passed me (viz. 3) out of the first ten which came up to me, whether they were moving in the one direction or the other; and so on for successive decades (the observations not being all made on the same day, nor at the same hour). Here are some of the observations :

From these and other observations in pari materia, I find that on an average of the omnibuses observed, about 70 per cent. met and 30 per cent. passed the observer. If, as there is reason to suppose 1 (at the hours when the observations were made), the

The vehicles were drawn by horses in those days. The experiment recently repeated with respect to motor-buses gave a different result.

1 This presumption is confirmed by the following statistics in which the first member of each pair (e.g., 6 in the first pair) denotes the number of omnibuses moving eastward, and the second number (e.g., 4 in the second pair) denotes the number moving westward, out of every ten omnibuses, which, sitting at the window of a club in Piccadilly, I observed passing in either direction :—

6, 4; 5, 5; 4, 6; 5, 5; 6, 4; 6, 4; 5, 5; 3, 7;

5, 5; 6, 4; 3, 7; 5, 5; 5, 5; 7, 3; 5, 5; 5, 5.

It may be noticed that on the basis of the calculation in the text the observer

same number of omnibuses are moving in both directions with the same average velocity, say, V; an easy calculation shows that the velocity of the pedestrian, supposed uniform, = (0·7 — 0·3)V,

0.4V. That is the absolute velocity, so to speak, referring, say, to some fixed point in the street. Accordingly the velocity of the pedestrian relative to the vehicles which are moving in an opposite direction to his is 1.4 V: and relative to the vehicles which are moving in the same direction, 6 V. If, then, the pedestrian could observe his own velocity relative to a great number of vehicles taken at random from the whole series-say all that at a given instant were in Piccadilly-the distance by which he would be found to gain upon the average omnibus in a unit of time would be about (1·4 — •6) V8 V. This datum might possibly have been obtained by observation, if the observer had attended to the relative velocities of the vehicles in his neighbourhood, not merely to the numbers which met him and passed him, as he walked.

The distance which the individual on foot moves relatively to the average omnibus during a unit of time may be treated as a substantive entity, an independent measure of the rate at which the individual is advancing through the crowd of vehicles. Or it may be regarded as an approximation to a perhaps more scientific quæsitum, the rate at which the individual is moving towards the weighted mean of the system. The simple average might be used for this ancillary purpose by one who had not the means of ascertaining the centre of gravity of the system, or even by one who had not formed a very clear idea of what is meant by a centre of gravity. The approximation may be expected to be very close. For the statistics now under consideration are simply related to the group above cited, representing the proportions of vehicles meeting and passing the pedestrian; and this group appears to possess the characteristic on which indifference of weighting depends, namely, sporadic dispersion about a constant

mean.

1

Is it necessary to interpret the parable? The oscillating. crowd of public conveyances is comparable to the long list of commodities with ever varying values-the swaying series of the logarithms 1 so taken that the difference between any two of them represents the relative value of two articles of exchange. The would appear to be moving westward with a velocity equal to an eightieth of the average velocity of an omnibus; a result which differs from zero by an amount which is well within the probable error incident to the calculation.

1 As conceived by Cournot (Théorie Mathématique des Richesses, ch. ii.); who very properly in this connection does not mention weights.

change in the distance of the pedestrian from the " weighted mean" of the system represents the primary monetary quæsitum; the change in his average distance from the other bodies in the system represents that unweighted—that is, equally weighted, or more generally randomly weighted--mean of relative prices, which may be used either as subsidiary to the primary investigation, or as an independent secondary measure. The position of high collateral dignity is all the more deserved in that the secondary measure enjoys an objective or external character, which cannot according to my view of the subject-be accorded to the primary quæsitum.

The recognition of this sort of absolute standard, or at least of that sporadic dispersion on which it is based, demands a considerable widening of the views and softening of the strictures, which we find in the work before us. First, more attention may be claimed for a species of average, appropriate to the secondary quæsitum, the Median, which Mr. Walsh has mentioned only to reject. Again, his criticism of those who have sought to include wages with commodities in an index-number seems too harsh. Those certainly are to be condemned who confound the distinct standards, which are based on the amount of commodity which the same sum of money will procure, and the amount of effort and sacrifice which are required to procure the same sum of money. Mr. Walsh is quite justified in describing a mixture of these two species of index-number as an unmeaning hodgepodge." But there is a secondary point of view in which these distinctions are less important: the view which seems to have been taken by some of the great men who first approached our problem. When Hume imagined every one awaking one morning with an additional coin in his pocket, when Mill improved on the idea by imagining the money in every one's pocket to be increased in a certain ratio, presumably they thought of prices in general without distinction of producers' and consumers' goods. And certainly in an alert state of competition, if such a change as Jevons proposed for the purpose of unifying international coins were carried out, namely that what is now 100 dollars should reckon as 103, it is very conceivable that this change would rapidly propagate itself through a great variety of transactions, including those between master and servant. And accordingly, though the change in wages in each department might be liable to the same proper disturbance as the finished article (in addition to the common monetary influence), and so far as they are not independent observations it would not be much good including

them, at the same time there would be no harm in including them in such an unweighted index-number as is now under consideration. I am not contending that wages ought in the existing state of things to be included in any kind of index-number along with finished products. I am only regretting that our author's great learning has not saved him from the common defect of original writers on the subject, an inability to perceive the many-sidedness of the problem, an exclusive devotion to one idea.

There are more things in the monetary cosmos than are dreamt of in his philosophy. Still his philosophy is of a very high order. So subtle dialectic, such logical precision, supplemented by a diligence of literary research that is quite unrivalled, if brought to bear on other economic problems, may be expected to merit a less chequered encomium. That they have not now obtained a more decided success seems due to the peculiarity of a problem which involves the more positive science of Probabilities. But, I repeat, this is an individual opinion on a much debated question. There are those who conceive the problem in a sense more favourable to Mr. Walsh. To me he seems unfortunate in his subject; to others perhaps, only in his critic.