COMPUTING SCIENCE

# E Pluribus Confusion

There’s more than one way to turn census data into congressional seats

# Enter the Mathematicians

In 1911, Joseph Hill, a statistician in the Census Bureau, proposed a new method based on yet another sensible idea: Once you’ve allotted seats to the various states, check to see whether there is any pair of states for which transferring a seat from one to the other makes things more fair between the two of them. By “more fair” Hill meant that the ratio of the ratios *p*/*s* (for the first state in question) and *p’*/*s’* (for the second state) should get closer to 1. More precisely, the ratio of the larger ratio to the smaller ratio should get closer to 1. If it does, make the switch and look for another pair of states.

Hill’s method was taken up by Harvard mathematician Edward Huntington, who carried out a spirited debate on the matter throughout the 1920s with Walter Willcox, a proponent of Webster’s method. The politics of that decade were so contentious that Congress never did get around to reapportioning itself. It simply let stand the apportionment based on the 1910 census—a clear-cut case of ignoring the Constitution. (In 1930, Congress caught a break: The two methods happened to produce identical apportionments, so there was no need to choose between them.)

It turns out that Hill’s method can also be run like Jefferson’s, Adams’s and Webster’s, either as a “divisor” method with a funny rounding rule, or as a “doling-out” algorithm with a funky reinsertion rule. The rounding and reinsertion rules involve a lovely piece of mathematics called the *geometric mean.* When a quotient *p*/*D* lies between consecutive integers *q* and *q*+1, the rounding rule is to round down if *p*/*D* is less than the square root of *q*(*q*+1), and round it up if it’s greater than the square root of *q*(*q*+1). The reinsertion rule is to keep an ordered list of the numbers:

Hill’s method may look like something only a mathematician could love, but in 1941, Congress did indeed fall in love with it and locked it into law. The affection had little to do with mathematics and a lot to do with politics. Applied to the 1940 census, Webster and Hill gave identical results with one exception: Hill took a seat away from Republican-leaning Michigan and gave it to the solidly Democratic state of Arkansas.

Here’s how Hill’s arithmetic plays out. Webster gave 18 seats to Michigan’s 5,256,106 residents, for a quotient of 292,006. It gave 6 seats to Arkansas’s 1,949,387 people, for a quotient of 324,898. The ratio of these is 1.11264. Hill’s criterion now asks, What would happen if Michigan ceded a seat to Arkansas? The quotients would then be 309,183 for Michigan and 278,484 for Arkansas, and the ratio of these is 1.11024. You can liken it to a bank cutting an interest rate on a loan from 11.25 percent down to 11 percent. The reduction may be small, but it *is* a reduction, so logic—or more precisely, Hill’s logic—compels you to accept it. If the result just happens to also help solidify your party’s control of Congress, so much the better. As the state motto of Arkansas says, *Regnat populus*—”The people rule.”

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