Vol. 7, 1921 
MATHEMATICS: E. V. HUNTINGTON 
125 
of magnitude of the populations of the states) . Finally, assign the repre- 
sentatives, 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 la or 16 we should 
have been led, in like manner, to two other methods which may be called 
the method of the harmonic mean (la), and the method of the arithmetic 
mean (lb), since the "multipliers" in the working rules are as follows: 
It can be shown that method la favors the smaller states more than 
method I does, while method 16 favors the larger states more than method 
I does, Since there is no mathematical reason for adopting either of the 
two Postulates la and 16 to the exclusion of the other, both should be re- 
jected. 
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 la does, while Id favors the large 
states even more than 16 does, so that both should be rejected. 
Each of these four methods violates three of the four conditions ex- 
pressed 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 Ilia. The sum of the squares of the deviations of the A/ a 
from their true values; or 
Postulate III6. 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 Ilia favors the smaller states even more 
than Ic does, while 1 1 16 favors the larger states even more than Id does. 
In other words, Postulates Ilia and III6 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. 
(Ia) Inf., 
(16) 2, 
1+2 2 + 3 3 + 4 
2(1 X 2)' 2(2 X 3)' 2(3 X 4) 
2/3, 2/5, 2/7, 
