A machine for distributing what cannot be split
The Seat That Vanished at Three Hundred
Move one House-size slider and recompute Hamilton's method from the 1880 census: Alabama receives eight seats at 299 and seven at 300. Then switch to Jefferson's method on early census data and expose the quota violation behind a deeper impossibility theorem.
Put one more chair in the room
The populations do not move. Only the number of seats does. Start at 299, then press 300. The table runs the largest-remainder calculation over all 38 states each time.
Use the method buttons to change the census and allocation rule. The table and status update live.
Computing…
The calculation is loading.
Computing from the census population…
Computing…
| State | Population | Exact quota | Quota interval | Seats | Why |
|---|
The extra seats are a race of fractions
Hamilton begins with each state's lower quota, subject to the constitutional minimum of one seat. Whatever seats remain go to the largest fractional remainders. This sounds gentle because every state stays inside quota. The break happens because the number of remaining seats and the ordering of the remainders can both change when the House grows.
Multiply each state's fraction of the census population by the chosen House size.
Give each state the integer part, never fewer than one seat.
Give the seats still empty to the largest fractions, one per eligible state.
At 299, Alabama's fraction is barely on the winning side of the live cutoff. At 300, faster-moving remainders pass it. Illinois and Texas gain the new seats while Alabama loses its remainder seat. Nothing about Alabama's population changed.
The obvious escape has a price
Switch the instrument to Jefferson. A divisor method cannot produce the Alabama paradox: lowering a common divisor as the House grows can add seats, but it does not take them away. The problem appears somewhere else.
The second dataset is the 1790 apportionment population used in the first congressional dispute. It is not the resident population. Under the Constitution then in force, it counted free persons and three-fifths of enslaved people. That arithmetic is part of the historical record and part of the injustice the record contains.
In a counterfactual 120-seat House, Jefferson's rule repeatedly awards the next seat to the largest value of population ÷ (current seats + 1). Virginia reaches 22 seats even though its exact quota is below 21. Its permitted quota answers are 20 or 21. The rule gives 22.
Try to demand both
These switches do not search a menu of named formulas. They state the requirements for a method that must work on every possible apportionment problem.
Select a requirement. Each can be met alone.
Balinski and Young proved that, in the standard deterministic framework with at least four states, no apportionment method satisfies quota and population monotonicity on every possible problem. In their characterization, the population-monotone methods are divisor methods, and every divisor method can violate quota. This is an impossibility result about a pair of universal guarantees, not a claim that all apportionment is impossible.
The check
Waiting for the 1880 census calculation.
Waiting for the 1790 apportionment calculation.
- Measured historical inputs. The 1880 figures are the Census Office's aggregate populations of the 38 states. The 1790 figures are historical apportionment populations, not resident counts, and embody the Constitution's then-operative three-fifths rule.
- Rules, not fitted models. Hamilton and Jefferson are deterministic allocation rules. There are no estimated parameters. The one-seat state minimum is enforced in both.
- Free choice named. An exact numerical tie would be broken alphabetically here so the interface remains deterministic. No tie affects either highlighted result or any setting on the sliders.
- Cited theorem, not browser proof. The Balinski-Young theorem is staged above and cited below. The page does not pretend that two historical examples prove a universal impossibility theorem.
- No election-outcome claim. This page verifies seat arithmetic. It does not claim that Alabama's hypothetical lost seat changed a particular election.
The independent verifier repeats every slider setting and fails on any mismatch: node research/the-seat-that-vanished-at-300/verify-the-seat-that-vanished-at-300.mjs.
Sources and exact scope
- U.S. Census Office, Statistics of the Population of the United States at the Tenth Census, Table I, 1883. The state figures sum live to the published state total.
- U.S. Census Bureau, Methods of Apportionment, for the official descriptions of Hamilton and Jefferson.
- H. Peyton Young, Fairness in Apportionment, for Seaton's account, the early apportionment tables, the counterfactual Jefferson-at-120 calculation, and the historical development.
- National Archives, The 1790 Census and the First Veto, for the historical allocations: the vetoed 120-seat Hamilton bill gave Virginia 21 seats, while the enacted 105-seat Jefferson apportionment gave Virginia 19.
- Michel L. Balinski and H. Peyton Young, The Theory of Apportionment, IIASA WP-80-131, 1980, and Fair Representation: Meeting the Ideal of One Man, One Vote, 2nd ed., Brookings Institution Press, 2001, for the impossibility and divisor-method characterization.
- The Jefferson-at-120 allocation shown here is counterfactual. Congress neither proposed nor enacted it. The page applies Jefferson to the vetoed bill's House size because it produces an especially clear quota violation on the real first-census apportionment population.