earlier. If a thorough relative test were applied, the year 1880 would probably not stand out so prominently in this matter of concentration of wealth as it does when one uses fixed class boundaries or any method which involves definition of the large fortune absolutely.

If the character of the actual statistics permitted the plotting of a practicable curve indicating relative quantities (that is, percentages), comparison of the course of such curves for different sets of data would immediately indicate to the eye which set was characterized by greater concentration.* The pyramid of fortunes, however, is of such a nature as not to lend itself readily to such diagramming. It is significant of prevailing inequality that the attempt to construct the curves fails because of the disproportion between the flattened base of the pyramid and the elongated peak. The curves resemble a capital L with enormously elongated arms and so little thickness of the different parts that the eye can form no judgment of quantitative relations. This is explained by the fact that the curve of the distribution of property is hyperbolic in its general characteristics.† Quantities arranged in something like a hyperbolic series do not lend themselves readily to ordinary graphic representation.

But a hyperbola is very easily represented graphically if one will plot the logarithms of the numbers instead of the numbers themselves. Just what the degree of difference is between two series of numbers so compared is not obvious, at any rate

* This is Mr. Lorenz's proposed method in the article referred to above. It is significant that Mr. Lorenz uses hypothetical figures, and thus fails to perceive the limited practical value of the particular species of graphic method that he prefers. The value of my comment in this discussion is subject to a similar qualification.

If the statistics dealt with were not concentrated upon round numbers, the hyperbolic nature of the distribution of estates would find expression in the position of the arithmetic average of the estates within each class; that is, it should in each case be near the geometrical mean between the limits of the class. The Massachusetts statistics conform to this rule almost without exception. The British statistics almost as regularly do not conform to it; that is, the arithmetic average of the estates within the class is usually considerably above the arithmetic mean between the approximate boundaries of the class. The difference is explained by the fact that the estates at the round number which marks the lower limit of the class are included in it in the Massachusetts figures and the upper limit is exclusive; while in the British figures the lower limit is exclusive, and the upper limit inclusive, and especially the lower limit marks the transition to a higher tax rate.

to one who is not a trained mathematician; but the direction of the difference is unmistakable, and the method of testing for concentration is, so far as the writer can see, not open to criticism. This is what is needed.*

In the comparison of the logarithmic curves used in the following diagrams the crucial point is their slant. The steeper of two curves is the one which expresses the greater concentration. The scale used to represent the size of fortunes plotted must of course be identical for different sets of figures. The distance used to represent the logarithm of a given number of fortunes must also be the same for the different series compared. In order to assist the eye in comparing slants, however, it is often well to shift this latter scale along its own length, for one or more terms of the comparison, so as to bring the curves together at some convenient point. This is done in the following diagrams. Hence the logarithmic scale across the bottom of the diagrams indicates the actual logarithms plotted only in the case of the full line curves. In the other curves the scale corresponds to the series of logarithms plus a constant amount. The curves of the same diagram are thus made to start from the bottom at approximately the same

* As compared with the rearrangement of the statistics by the method of relative class boundaries, which also meets the requirement of defining large and small and middling fortunes relatively (that is, as so many times the mean fortune), the method of plotting logarithms has decided advantages. Of most practical importance is the fact that the amount of calculation involved in the use of the former method is prohibitive. The convincingness of the results, furthermore, is lessened by the necessity of using certain mathematical assumptions in rearranging the statistics. These assumptions are based upon principles not different from those used above, but are less satisfactory in application because more depends upon the accuracy of the fundamental averages. These averages are likely to be especially affected by inaccuracy in the reporting of small estates. Of the available averages the median is intrinsically to be preferred, since to measure the tendency towards large fortunes by a method which rests upon the arithmetic average is to weigh them in a balance the lengths of the arms of which are very largely determined by the large fortunes themselves. But the median would probably be most affected by the error resulting from the inclusion of a large number of debtor estates in the statistics.

More summary methods of testing for concentration may be in principle equally correct with those used in this article, but they do not deal directly or mainly with the point of most general interest; that is, the growth of great fortunes. For a summary test nothing could be better, it seems to me, provided satisfactory data are available, than the comparison of the median and the average, of course, by the use of the ratio of their difference to their sum or to one or the other quantity as a base, in order to preserve that relativity which is the essence of any correct measure of concentration.

point, each point of a shifted curve being moved horizontally the same distance.

It is hardly necessary to add that, in judging these logarithmic curves by their comparative slant, the requirement of relativity in the test of concentration is fully and easily met.

The logarithms obtained for the above Massachusetts statistics, with the corresponding numbers, are as follows:

Plotting these data, we obtain the curves of Figure 1.*

* Where the lower part of another curve is not distinguishable from the curve for the statistics of latest date, it is discontinued. The numbers at the bottom of the diagram indicating the scale are those of the logarithms for the latest date. The measurement units are, of course, the same for the other sets of logarithms, but the scale is moved to the right a sufficient number of units to make the lower ends of all the curves coincide.

It is clear that there has been, in the period covered by these statistics, on the whole a pronounced tendency to concentration of wealth in Massachusetts. There was, however, a greater degree of concentration in 1880 than in 1890. The fact that the per capita wealth of Massachusetts in 1890 was less than in 1880 does not affect a test by the comparative slants of logarithmic curves. Neither does the omission of all reference to the propertyless affect in principle such a comparison. In the case of greater per capita wealth it is true one should be careful to allow determining points higher up on the curve to have their due influence on the judgment of slant.*

It is noticeable how closely these logarithmic curves approximate a straight line. A curve plotted thus by logarithms as a straight line is what is known to physicists and mathematicians as an adiabatic curve. The curve exhibits the relation between the pressure and the volume of a gas upon the assumption that it expands and contracts without either receiving or giving out heat. It is a general hyperbola.†

Probate reports do not furnish unimpeachable statistical material. But, as regards accuracy, it is only a constant ten

*The writer has worked somewhat upon certain probate statistics of Maryland. Figures for Baltimore City, 1875 to 1880 inclusive, and 1888 to 1893 inclusive, and for the counties of Maryland, except Baltimore City, 1875 to 1879 inclusive, and 1890 to 1894 inclusive, are to be found in the Reports of the Bureau of Industrial Statistics of Maryland. They show a tendency to concentration for the State as a whole, though the statistics of Baltimore City by itself point slightly the other way.

† For the suggestion of the possibilities presented by the use of logarithmic curves for the purposes of this article, I am indebted to my colleague, Dr. F. R. Sharpe, of the Department of Mathematics, Cornell University. The same method is used by Pareto in his 'Cours d'Économie Politique," vol. ii, p. 305, but for income curves. Pareto mentions the fact that the curve approximates a straight line. Mr. Lorenz's remarks about the logarithmic curve (pp. 216, 217) appear to be rather hasty. The "fixed classification" in the case of figures that are properly to be called statistics will have no effect on the steepness of the curve. It is true that the curve used above indicates nothing as to what may happen in the uppermost class; but can Mr. Lorenz's one multi-millionaire, however manipulated, form any basis for a statistical inference? The few cases at the very top must be treated separately if that appears necessary. Mr. Lorenz's suggested modification of the logarithmic method, which gives a place to the amount possessed by each class, has some advantages. Are they sufficient to outweigh added difficulties of visual interpretation? In any large body of reliable statistics, given the type of distribution as indicated, e.g., by the numbers in the classes, the amount of wealth owned by each class will bear a known relation to its numbers, except possibly in the case of the highest and lowest classes. The amounts would be determinable by accepted principles of interpolation.