Vol. 7, 1921 MATHEMATICS: E. V. HUNTINGTON 123 
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 U. S. Dept. Agr., Bur. Crop Estimates, Monthly Crop Rept., 3, 1917, No. 10 (99). 
2 U. 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 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). 
6 U. vS. Dept. Agr., Bull. 310, 1915. 
THE MATHEMATICAL THEORY OF THE APPORTION- 
MENT OF REPRESENTATIVES 1 
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 re- 
sulting 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 popula- 
tions 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 la and B/b should be as 
nearly equal as possible; also a! A and b/B; also A/B and alb; also Bl A 
and bl a. 
