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The same remark applies if in Postulates Ilia and the word "square" 
is replaced by "absolute value." 2 
Hence it is clear that if a numerical measure of injustice is desired, both 
the deviation of A/ a and the deviation of a/A should be taken into ac- 
count together. That is, any formula which reports A/a, say, as too 
large by a certain amount, should also report a/ A as too small by the same 
amount. The formulas suggested by the simple application of the idea 
of least squares, as shown in the preceding paragraph, do not have this 
property. A combination of these formulas suggests, however, the follow- 
ing postulate, in which a denotes the theoretical value of a. 
Postulate III. In a satisfactory apportionment, the sum T of terms 
of the form Ae 2 where 
p= ( a/ A) — (a/ A) (A/ a ) — (A l a) _ a—_a 
^ia/A)(a/A) V(4/«) U/°> ylaa 
should be a minimum. 3 
For purposes of computation, this total error, T, may be replaced by 
an average error, E= y/S/N, where 5 is the sum of terms of the form 
ae 2 — (a — a) 2 /a. 
The method determined by Postulate III is precisely the same as the 
method of equal proportions based on Postulates I and II. 
1 This article contains the substance of two papers presented to the American Mathe- 
matical Society, December 28, 1920, and February 26, 1921. Further details, with 
proofs and examples, will be published either in the Transactions of the American 
Mathematical Society, or in the Quarterly Publication of the American Statistical 
Association, or in the American Mathematical Monthly. 
For the history of the subject see W. F. Willcox, "The Apportionment of Repre- 
sentatives" (presidential address at the annual meeting of the American Economic 
Association, December 1915), published in the American Economic Review, Vol. 6, 
No. 1, Supplement, pp. 1-16, March, 1916. See also 62d Congress, 1st Session, House 
of Representatives, Report No. 12, pp. 1-108, April 25, 1911, and John H. Humphreys, 
"Proportional Representation," London, 1911. 
The most important of the methods hitherto known are four: 
The Vinton method of 1850, long in use in Congress, is known to lead to an "Alabama 
paradox;" that is, an increase in the total size of the House may cause a decrease in the 
representation of some state. 
The Hill method of alternate ratios, proposed by Dr. J. A. Hill in 1910 but not adopted, 
comes very near to satisfying the postulates of the present paper, and uses for the 
first time (though only partially) the idea of the geometric mean. The method is in- 
complete however, since it can be shown to lead to an Alabama paradox. 
The Willcox method of major fractions, devised by Professor W. F. Willcox in 1900- 
1910, and now in use in Congress, employs, in effect, a working rule like ours with 
multipliers: Inf., 2/3, 2/5, 2/7, . . . ; it is essentially the same as the method of the 
arithmetic mean, and therefore favors the larger states unduly, just as the hitherto 
unsuspected but equally justifiable method of the harmonic mean favors the smaller 
states unduly. (It may be noted that the name "major fractions" is somewhat mis- 
leading, since the Willcox major fraction is not a major fraction of the true quota, but 
