124 
MA THEM A TICS: E. V. HUNTINGTON Proc. N. A. S. 
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. The difference between A I a and Bib; or 
Postulate 16. The difference between a/A and b/B; or 
Postulate Ic. The difference between A/B and alb; or 
Postulate Id. The difference between Bl A and bl a; should be as near 
zero as possible. 
If, on the other hand, we adopt the "ratio" interpretation, we have: 
Postulate I. The ratio between Ala and Bib (or the ratio between 
a I 'A and b/B; or the ratio between A/B and alb; or the ratio between Bl A 
and bl 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 
la and 16 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 representa- 
tives 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., l/VFx~2, 1/V2X~3, 1/V3 X 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 magni- 
tude 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 
