as typical), increased employment of labor, an increased demand for all kinds of goods, and a consequent rise in general prices. This process is arrested when the cycle of agricultural productivity begins its downward phase; and a reverse series of phenomena then appears. In the author's words: "These cycles of crops constitute the natural, material current which drags upon its surface the lagging, rhythmically changing value and prices with which the economist is more immediately concerned." 1

As a necessary step in the logical course of his argument, Professor Moore also makes some interesting studies in demand curves. From tables of the output and prices of certain staple goods he constructs a percentage demand curve by making the abscissas proportional to the percentage change in output for each year above or below the output for the preceding year (each preceding year being successively used as a base), while the ordinates are made proportional to the corresponding changes in prices, similarly computed. From this exploration he emerges with what he appears to regard as a surprising discovery, namely, the discovery of a new type of demand curve. "Our representative crops and representative producers' good exemplify types of demand curves of contrary character. In one case, as the product increases or decreases the price falls or rises, while, in the other case, the price rises with an increase of the product and falls with its decrease." 2 In connection with this discovery he treats somewhat patronizingly the whole ceteris paribus type of reasoning of his predecessors. The universal, negatively inclined demand curve of Professor Marshall is characterized as "an idol of the static state." The fruitfulness of the statistical method is contrasted with the " vast barrenness" of the conventional method.

Take, for example, the question of the effect of the weather upon crops. What a useless bit of speculation it would be to try to solve, in a hypothetical way, the question as to the effect of rainfall upon the crops, other unenumerated elements of the weather remaining constant! The question of the effect of temperature, ceteris paribus! How, finally,

would a synthesis be made of the several individual effects?

The

statistical method of multiple correlation formulates no such vain questions. It inquires, directly, what is the relation between crop and rainfall, not ceteris paribus, but other things changing according to their natural order; what is the relation between crop and temperature, other things conforming to the observed changes in temperature; and, finally, what is the relation between crop and rainfall for constant values of temperature? The problem of the effects of the constituent factors is solved only after the more general problem has received its solution. This method offers promise of an answer to the question as to the relation between the effective demand price and the supply of the commodity.'

A valuable feature of Professor Moore's work is the insertion of the tables of statistics upon which his argument is based. This enables the reader, if so inclined, to check or supplement the reasoning. Numerous periodograms and examples of demand curves also illustrate the subject matter.

There can be no difference of opinion as to the great value of Professor Moore's method. He is doing pioneer work and is doing it with painstaking detail and thoroness. The more economic theory can be reduced to the status of an exact science, the more serviceable will it become in bringing to finer order and adjustment our intricate and highly organized modern life. It is, therefore, with diffidence that I approach the task of criticizing a book involving at once such keen mathematical insight and such immense industry in laborious detail. Yet, to me, it falls short of conclusiveness. Several links in the logical chain seem to need closer scrutiny.

In the first place, the alleged discovery of an eight-year cycle is suspicious. It certainly does not harmonize with data relating to industrial crises. These are known to follow more nearly a ten-year cycle. Now an eight-year cycle, however adjusted to the dates usually given for crises, would bring some at a period of high prices, some at a period of low prices, and some at intermediate points. It is clear, then, that if Professor Moore's economic cycles are real, they

text.

The full multiple correlation here suggested is not, however, carried out in the

2 Pp. 67, 68.

represent a phenomenon disconnected with the well known phenomenon of industrial revulsions. This discrepancy led me to undertake an independent study of the data.

It was first observed that the eight and thirty-three year cycles were derived from data as to annual rainfall, while the whole argument rests upon the effective rainfall at the critical periods of growth of the several crops considered. Professor Moore fails to correlate these two. Perhaps he may have regarded it as safe to assume that if the annual rainfall follows an eight-year cycle, the same would be true of effective rainfall. Yet while a study of the data for annual rainfall reveals a fairly well marked cycle of eight years with an amplitude of 4.13 (p. 24), the periodograms for effective rainfall (pp. 46, 47, 48, 54) show only a very minor indication of an eight-year cycle (amplitudes, 0.21, 1.71, 0.21, 0.24). There is more indication of a four-year cycle (amplitudes, 1.22, 1.39, 1.22, 0.40). The periodograms give the same impression to the eye. Now, later in the text, when general prices are correlated with crops, a lag of four years is allowed to give time for the crops to show their effect in prices. If the cycle of rainfall is four years and if rainfall is the efficient cause of fluctuation in crops, clearly a lag of four years is meaningless prices could hardly be one full cycle in ad

vance of their efficient cause.

Still, there might be a mean effective rainfall cycle of longer period than four years, but not necessarily eight, which would account for the high correlation between crops and prices noted later in the text. To investigate for such a cycle the following method was employed. It is confessedly less exhaustive than Professor Moore's method of amplitudes but is believed to be fairly conclusive at least, sufficiently conclusive to form the basis of a working hypothesis. If a series of numbers be given, then by means of the formula,1

-

[o, = standard derivation of the differences.]

1 This formula is given in "A short method of calculating the coefficient of correla tion in the case of integral variates." J. A. Harris. Biometrica, vol. vii, p. 214.

each number in the series may be correlated with its adjacent, its second, its third, its fourth, etc. If the series conceals a true cycle, it will be revealed by this process. For, suppose the cycle to be one of eight years, then when each number is correlated with its eighth, we shall have a high positive correlation, approaching unity. When each number is correlated with its fourth, the result will be a high negative correlation; with its second and sixth, approximately 0; with its adjacent and seventh, a low positive, and with its third and fifth, a low negative correlation. In other words, if there be a true cycle, the application of this method will reveal a cycle of correlations. If a short cycle were superposed upon a larger one, it might well happen that all the correlations would be positive for the minor cycle. Even then there would be a cycle of these positive correlations with respect to magnitude, as is shown in the case of crops. See footnote.

An application of this method to mean effective rainfall failed to give evidence of an eight-year cycle, but did give some evidence of a seven-year cycle, and possibly also of a cycle of between three and four years. The same method applied to data of yield per acre of nine principal crops gave good evidence of a seven-year cycle, but when applied to prices a well-marked cycle of nine years was revealed.1 These results were checked by constructing historigrams from the data and observing the intervals between successive maxima and minima. Now it is to be noted that in the case of general prices the correlation of each number with its tenth is nearly as high as with its ninth, and a study of the historigram (Fig. 1) reveals points of maxima at 1873, 1883, 1893,

General prices; r1 = 0.600, r2 = 0.380, r1 = -0.330, r1 = - 0.034, r1 = 0.201, r

=

It must be confessed that the figures in the case of mean effective rainfall are very inconclusive. The negative result (r1 -0.138), when each number is correlated

=

with its adjacent, makes it questionable whether there is any true cycle. The positive correlation (r, 0.330), when each number is correlated with its seventh, may be due to mere chancc.

and 1907. All of these points are followed by a sharp decline and the dates are those associated with industrial crises. This is certainly suggestive. The other point of maximum is at 1900. There was a crisis in 1903, but here the connection is not so close. The crisis of 1903 appears to have fallen during a decline in prices instead of immediately preceding it. A nine-year periodogram is fitted to the crude data, as shown in the figure. The closeness of the fit is striking.

An apparently strong argument for Professor Moore's theory is found in the high correlation between the yield of crops per acre and general prices, after allowing for a lag of four years. This is surprising, since, from what has been said in the preceding paragraph, the periods appear to be different — one seven years and the other nine or ten years. But an inspection of the historigrams (p. 123) reveals the probable cause of this high correlation. In both cases the minor cycles are superposed upon a larger cycle (possibly Professor Moore's thirty-three year cycle). In the case of crops there is a distinct downward trend from 1870 to about 1892, and from there upward to 1910. In the case of general prices the downward trend extends from 1870 to about 1896 and thence upward to 1910. Hence if a lag of four years be allowed (or even without it), a high correlation would be shown because of these general trends, even if there were no correlation whatever from the minor cycles. I tried the experiment of eliminating these general trends and obtained the following results. Lag of four years, r = 0.353; three years, r = 0.341 two years, r = 0.184; one year, r = 0.026. The first of these results, tho much smaller than Professor Moore's (r = 0.800), is still striking. The experiment was tried of holding the two historigrams up to a window, one superposed upon the other, and then sliding one upon the other so as to accord with a lag of four years. The crops showed one more complete cycle than the prices in the interval from 1870 to 1910, but, the cycles constructed from the crude data being

1 Tho in the case of general prices this would be complicated with the effect of changes in the world's gold supply. It would be necessary to apply the method of multiple correlation to eliminate this effect.