was divided into groups of five subjects each, and one group was served the 54%, 70%, 85%, and 100% flours in the order given while the second group was served the flours in the reverse order. The results of the 139 digestion experiments reported below do not indicate that the first group of subjects acquired any tolerance for the coarser flours not possessed by the second group which was served the coarser flours first.

The experimental diet was so chosen that its preparation should involve a minimum of labor, that the bread should supply the larger portion of the total protein, and that it should be representative of a simple mixed diet. The diet consisted of bread, fruit (orange), butter, sugar, with tea or coffee as a beverage if desired.

The fruit, butter, and sugar were served as purchased from a nearby market. A quantity of bread sufficient to supply all the subjects for one day was prepared daily, the ingredients being used in the following proportion:

3 cups sifted flour

1 teaspoon salt

1 tablespoon sugar

1/8 to 1/2 cake yeast

While data relative to the digestibility of the diet as a whole are of value, interest was primarily centered in the digestibility of the bread or rather the flour from which the bread was made. The digestibility of the protein and carbohydrate supplied by the different breads has been estimated by making a correction for the undigested protein and rbohydrate resulting from the accessory foods by a method outlined in detail in earlier publications. It has been assumed from the results of the early investigations of this Office" that the digestibility of the protein of butter is 97% and protein of fruit, 85%; of the carbohydrate of fruit, 90%; and of sugar, 98%.

In planning the investigation, it was decided to make a sufficient number of digestion experiments so that the average results obtained should be of general application. The experimental periods were to be long enough so that the effect of any irregularity in the rate of passage of food residues through the alimentary tract, and any error in the separation or collection of feces would be practically negligible. The experimental periods which were from 15 to 25 days in length were subdivided and considered as separate three-day experiments, following one directly after the other. The subjects for the experiments were students in local educational institutions, and were selected with care in order that the conclusions might be applicable to the average normal adult. An attempt was made to include in the squad some accustomed to strenuous exercise, some accustomed to take light exercise, and some who took little recreation, in

order to have under observation individuals whose peristaltic action was normal and regular and others who had a tendency toward constipation. Summary of Results.-The results obtained in the studies of 54%, 70%, 85%, and 100% flours are summarized in the table which follows:

In the above experiments the subjects ate on an average considerably more than a pound of bread daily for periods of from 6 to 20 days without producing any digestive disturbances, which indicates that wheat flours, regardless of percentage of extraction, are well tolerated by the human body.

The coefficients of digestibility, obtained from the 139 digestion experiments reported above, 87.7% for protein and 99.7% for carbohydrate of 54% flour; 90.1% for protein and 99.9% for carbohydrate of 70% flour (95% patent); 87.1% for protein and 98.5% for carbohydrate of 85% ("whole wheat") flour; and 84.2% for protein and 94.4% for carbohydrate of 100% flour, show these flours to be well digested. From these results it appears that the U. S. Food Administration in attempting to obtain efficient utilization of the wheat supply secured did well to specify that wheat should be milled at 75% extraction, that is an extraction similar to the one which in these experiments showed the highest proportion of digestible nutrients.

The digestibility of the fat content of the experimental diet was quite uniform and was practically identical with that of butter and "shortening" which comprised the major portion of the fat consumed, except in the series of experiments with 100% flour in which the fat was only 93.7% digested. Attention was given to the effect of the different flours on peristaltic action. The 54% and 70% flours did not tend to produce constipation during a period of 15 to 18 days, and although a somewhat freer movement of the bowels resulted from the continued use of 85% and 100% flours. no case of pronounced laxative effect was noted.

In general the results of the digestion experiments here reported are in accord with conclusions drawn from earlier studies of the digestibility of wheat flours. The digestibility of the 70% (95% patent) flour was the highest, that of the 54% flour was slightly greater than that of the 85% ("whole wheat") flour, while the digestibility of the 100% (graham) flour was lowest of all those studied. Since the flavor of bread made with the different flours varies, the use of different kinds for bread making is an easy way of giving variety to the diet.

1 1 U. S. Dept. Agr., Bur. Crop Estimates, Monthly Crop Rept., 3, 1917, No. 10 (99). 2U. S. Dept. Agr., Office Expt. Sta. Bull. 85, 1900 (32-33); Bull. 101, 1901 (33); Bull. 126, 1903 (29, 45); Bull. 143, 1904 (32); Bull. 156, 1905 (36).

3

3 U. S. Dept. Agr., Bull. 310, 1915; 617, 1919; 717, 1919.

4 U. S. Dept. Agr., Bull. 470, 1916 (7); 525, 1917 (4).

5 Connecticut Storrs Sta. Rpt., 1899 (104).

U. S. Dept. Agr., Bull. 310, 1915.

THE MATHEMATICAL THEORY OF THE APPORTION-
MENT OF REPRESENTATIVES1

BY EDWARD V. HUNTINGTON

HARVARD UNIVERSITY, CAMBRIDGE, MASS.

Communicated by E. H. Moore, February 14, 1921

The Problem. The exact quota to which each state is theoretically entitled on the basis of population usually involves a fraction. The problem is, to replace these exact quotas by whole numbers in such a way that the resulting injustice (due to adjustment of the fractions) shall be as small as possible.

This problem has been the subject of violent debate in Congress for the past one hundred years, a new method of apportionment having been proposed after almost every decennial census. None of these methods, however, possesses any satisfactory mathematical justification. The need of a strictly mathematical treatment of the problem having been called to the writer's attention by Dr. J. A. Hill, Chief Statistician of the Bureau of the Census, the following solution has been worked out on the basis of two very simple postulates. The new method may be called the Method of Equal Proportions.

Let N be the total number of representatives, A, B, C, ... the populations of the several states, and a, b, c, ... the number of representatives assigned to each.

Fundamental Principle.—In a satisfactory apportionment between two states (A greater than B), we shall agree that A/a and Bĺb should be as nearly equal as possible; also a/A and b/B; also A/B and a/b; also B/A and b/a.

Now to say that two quantities are "nearly equal" may be interpreted to mean: either, that the difference between the quantities is nearly zero; or, that the ratio between them is nearly one.

(Here the difference "between" two quantities means the larger minus the smaller. Similarly, the ratio "between" two quantities means the larger divided by the smaller.)

If we adopt the "difference" interpretation, we have:

POSTULATE la. POSTULATE Ib. POSTULATE IC. POSTULATE Id. zero as possible.

The difference between A/a and B/b; or
The difference between a/A and b/B; or
The difference between A/B and a/b; or

The difference between B/A and b/a; should be as near

If, on the other hand, we adopt the "ratio" interpretation, we have: POSTULATE I. The ratio between A/a and B/b (or the ratio between a/A and b/B; or the ratio between A/B and a/b; or the ratio between B/A and b/a; all of which have the same value) should be as near unity as possible. Since there is no way of choosing, mathematically, between Postulates Ia and Ib or between Postulates Ic and Id, and since these four demands lead to four different results, we shall reject all four of them and adopt Postulate I as the proper interpretation of our Fundamental Principle.

The case of two states is thus disposed of.

For the case of three or more states, one further principle is required, which we state as follows:

POSTULATE II. In a satisfactory apportionment, there should be no pair of states which is capable of being "improved" by a transfer of representatives within that pair-the word "improvement" being understood in the sense implied by the test already adopted for the case of two states, and the rare cases of "no choice" being decided in favor of the larger state.

From these two postulates the following theorem can then be deduced: THEOREM I. For any given values of A, B, C, ... and N, there will always be one and only one satisfactory apportionment in the sense defined by Postulates I and II. No further principles are required.

A working rule for computing this "best" apportionment in any given case is found to be as follows:

Working Rule.-Multiply the population of each state by as many of

the numbers

Inf., 1/√1 × 2, 1/√2 × 3, 1/√3 × 4, ...

as may be necessary, and record each result, together with the name of the state, on a small card. Arrange these cards according to the magnitude of the numbers recorded upon them, from the largest to the smallest, thus forming a priority list for the given states (the cards marked "Inf." being placed at the head of the list, arranged among themselves in order

of magnitude of the populations of the states). Finally, assign the representatives, from the 1st to the Nth, to the several states in the order in which the names of the states occur in this priority list. (It should be noted that this method satisfies automatically the constitutional requirement that every state shall have at least one representative.)

This method may be called the "method of the geometric mean," since the "multipliers" are the reciprocals of the geometric means of consecutive integers.

The solution of the problem is thus complete.

Alternative Methods.-If we had adopted Postulate Ia or Ib we should have been led, in like manner, to two other methods which may be called the method of the harmonic mean (Ia), and the method of the arithmetic mean (Ib), since the "multipliers" in the working rules are as follows:

It can be shown that method Ia favors the smaller states more than method I does, while method Ib favors the larger states more than method I does. Since there is no mathematical reason for adopting either of the two Postulates Ia and Ib to the exclusion of the other, both should be rejected.

Postulates Ic and Id also determine two distinct methods, which may be called the two methods of similarity ratios. It can be shown that Ic favors the small states even more than Ia does, while Id favors the large states even more than Ib does, so that both should be rejected.

Each of these four methods violates three of the four conditions expressed in our Fundamental Principle, while the method of the geometric mean satisfies all these conditions simultaneously.

The following further methods are suggested by the Theory of Least Squares.

In a theoretically perfect apportionment, A/a would be equal to P/N, and a/A to N/P (where P is the total population). Hence, in place of Postulates I and II, we might consider the following:

POSTULATE IIIa. The sum of the squares of the deviations of the A/a from their true values; or

POSTULATE IIIb. The sum of the squares of the deviations of the a/A from their true values; should be a minimum.

It can be shown, however, that IIIa favors the smaller states even more than Ic does, while IIIb favors the larger states even more than Id does. In other words, Postulates IIIa and IIIb violate, in opposite directions, all four of the conditions expressed in our Fundamental Principle. Since there is no mathematical reason for adopting either to the exclusion of the other, both should be rejected.