confessedly irregular, there was a rather surprising congruence in some parts of the two historigrams. Whether this congruence is to be accounted for by rainfall or by accident can be determined only by data extending over a longer period of time. The historigrams referred to in this paragraph, with accompanying periodograms, are shown in Figures 1 and 2. In the case of general prices the trends have been eliminated. In the case of crops they have been ac

counted for by assuming a thirty-three year cycle.

Annual Yield of Nine Crops: Seven and thirty-three Year Cycles.

Equation; y = 102.6 + 4.33 sin (24+77.3°) +9.34 sin

33

(2x1

+88,3°).

1 This crude visual test is only introduced as suggestive. Needless to say, it falls

far short of conclusiveness.

In conclusion of this phase of the subject the suggestion is offered that before any cycles relating to rainfall can be regarded as conclusive, some adequate astronomical or meteorological cause should be adduced.

Professor Moore's studies in demand curves illustrate the principle that the need of checking statistical inductions by abstract reasoning is quite as great as that of verifying abstract reasoning by statistics. The demand curves for crops harmonize perfectly with theory: the conditions of demand remain approximately constant; there is an increased output of crops (very probably due to heavier rainfall); with the diminishing utility due to this increased supply, the marginal utility and hence the price falls. But how about the "new type," the ascending demand curve for pig iron, is it so hopelessly irreconcilable with theory? Not at all. The conditions of demand are changed (very probably by improved business conditions) in the direction of a rapid and continuous increase. This would be indicated, conformably to theory, by shifting the entire demand curve progressively to the right. The ordinates to this shifting curve, corresponding with the lagging supply, will yield Professor Moore's new type." Thus (see Figure 3):

Figure 3.

X

D, D', D", etc., represent the conditions of increasing demand. OQ, OQ', OQ", etc., the corresponding lagging supply. PQ, P'Q', P"Q", etc., the marginal utilities (and hence prices) corresponding with these supplies, and AB the new type " of demand curve.

The above explanation is essentially that made by Professor Moore himself when he comes to interpret the results of his statistical analysis. The only point here made is the necessity of having a consistent body of theory to interpret just such results as that of the pig iron demand curve. Suppose, for example, we were to accept as universal the inductive law of producers' goods given on page 114. The price rises with an increase of the product and falls with its decrease "; and suppose, furthermore, that manufacturers of pig iron on the strength of this "universal law" should deliberately double, treble, or quadruple their output in the confident expectation that prices would rise proportionately: I fear that thereafter Professor Moore would not stand high as a prophet among producers of pig iron.

An interesting by-product of the analysis is found in the possibility of predicting prices of the great agricultural staples for any year from estimates as to yield. As already explained the demand curves were constructed by first plotting as abscissas and ordinates the crude data representing the percentage in change in yield and price for each year as compared with the preceding year, and then fitting the best "skew" to the crude data so plotted. The prediction of prices for staple crops is a matter of no little practical importance, especially to large dealers and speculators in futures. such Professor Moore's method may prove serviceable. May I venture to suggest a slight improvement in respect to the selection of a curve? Professor Moore uses the cubic,

To

y = a + bx + cx2 + dr3. Now there is no a priori reason why the demand curve should assume the form of a cubic. There is no reason to suppose that the demand curves for corn, hay, oats, and potatoes change their elasticity in the curious ways shown near the extremities of the curves on pages 73, 74, 75, and 76.1 These peculiarities arise simply

1 Professor Marshall (Principles of Economics, p. 161) holds that for "nearly all commodities" the elasticity of demand is greater for the middle range of prices than for prices either very high or very low. This principle might seem to justify the use of a cubic when it takes the form shown in the demand curve for corn (Fig. 4). But it is quite as likely to take the form shown in the demand curve for oats (Fig. 5). This would illustrate a precisely opposite principle, indeed it shows a condition at its extremities which is obviously absurd.

from the fact that a point of inflection is a property of the cubic. On the other hand there is some slight a priori ground for supposing the demand curve to be of the hyperbola type, a curve without points of inflection. In the case of the value of money, it can be demonstrated that the demand curve is the equilateral hyperbola. As Karl Pearson has pointed out, the problem in curve-fitting lies quite as much in the selection of the right type of curve as in the fitting of it to the data when selected.1 Accordingly the experiment was tried of fitting equilateral hyperbolas to the data for the above mentioned staple crops. The method of moments was employed, the method of least squares being inapplicable. The results obtained in the case of corn and oats are shown in Figures 4 and 5.

In conclusion it is fair to say that Professor Moore's volume is most suggestive and stimulating. Yet it may be questioned whether the main contention of business cycles based upon rainfall is fully proved. As they say in legislative bodies, it would perhaps be best to "refer the whole matter back to the committee for further study."

PHILIP G. WRIGHT.

HARVARD UNIVERSITY.

1 "Thus, in fitting an empirical curve to observation it is all important to make a suitable choice of that curve, that is, to determine whether it should be algebraic, exponential, trigonometric, etc." — On Systematic Curve Fitting, Part II. Biometrica, vol. ii, p. 16.