Yeah, that is a pretty tortuous definition for us non-math people. It may help to note, on the other hand, that the details of dealing with remainders in a system like this only make a difference of one seat for any given state. I’m not saying a single seat is a small difference for a small state, I’m just saying that the math-devil isn’t going to make much of a difference nationally. You could calculate the number of seats per state with half a dozen different systems and I bet they would be largely the same.

In a system like this, you start with the population of the country, and divide it by the target number of seats (which, depending on the system, may or may not equal the final number of seats). This gives you a quota to determine how many people you need to earn a seat. Naturally, no state is ever going to have exactly two or five or 27 quotas worth of people; there will be a remainder. The question is what to do with that remainders.

Basically, Jefferson said *Drop them, and to the devil with the planned size of the house.*

Webster said *Round them (up if they are above 0.5, and down if they aren’t), and we can approximate the planned size of the house.*

Vinton/Hamilton said *Give an additional seat to states in descending order of the size of the remainders, until you run out of seats, and we can exactly achieve the planned size of the house.*

It was noted that a whole lot depended on the planned size of the house with the latter system. There was a potential paradox in which reducing the planned size of the house increased the absolute number of members a state would have apportioned to it.

I highly recommend this site, which explains this phenomenon (the “Alabama paradox”) in small-number terms:

http://www.jdawiseman.com/papers/electsys/apportionment.html

The method of equal proportions was designed as a way of dealing with the alleged inadequacies of previous systems, which I don’t remember exactly.

Not all systems use remainders; one example is the d’Hondt system, in which you just assign seats in successive order. Each seat is given to the state which would have the highest number of residents per representative if it won that seat. This gives the same mathematical result as the Jefferson system, except that you can always achieve the desired size of the house by continuing to apportion seats until you have the correct number.

Naturally with Congressional reapportionment (assigning seats to states based on population *before* the election) you need to ensure that each state has at least one seat to comply with the Constitution. These systems can also be used for proportional representation (assigning seats to parties based on votes *at* the election) in countries which use it, but in that case there would be no one-seat minimum.