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One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.

Number, Scientific American, 211, (Sept. 1964), 51 - 59.

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# Apportioning Representatives in the United States Congress

## The Quota Rule

The fact that the affected states in the discrepancy just mentioned are Virginia and Delaware is no coincidence.  In general, Jefferson’s method is biased in favor of larger states and against smaller ones.  It violates what is called the “Quota Rule:

Here is a table that compares the populations and Representatives of Virginia and Delaware from 1790 through 1840:

 Year VA Pop. Quota Rep’s DE Pop. Quota Rep’s 1790 630,560 18.31036 19 55,540 1.61278 1 1800 747,362 21.55048 22 61,812 1.78237 1 1810 817,615 22.47609 23 71,004 1.95188 2 1820 895,303 21.25999 22 70,943 1.68462 1 1830 1,023,503 20.58844 21 75,432 1.51736 1 1840 1,060,202 14.86167 15 77,043 1.07997 1 Totals 119.04703 122 9.62898 7

By 1810, New York had overtaken Virginia as the most populous state in the Union.  If we look at its numbers instead of Virginia’s, the discrepancy between that large state and Delaware is even more pronounced:

 Year NY Pop. Quota Rep’s DE Pop. Quota Rep’s 1790 331,589 9.628765 10 55,540 1.61278 1 1800 577,805 16.66124 17 61,812 1.78237 1 1810 953,043 26.19898 27 71,004 1.95188 2 1820 1,368,775 32.50313 34 70,943 1.68462 1 1830 1,918,578 38.59347 40 75,432 1.51736 1 1840 2,428,919 34.04803 35 77,043 1.07997 1 Totals 157.6336 163 9.62898 7

Using Jefferson’s method, New York always had its quota rounded up.  On two occasions, its quota was rounded up more than one whole unit, in violation of the Quota Rule.  In contrast, Delaware’s quota was rounded up only once.  More striking are the cumulative results, showing New York well above and Delaware well below their expected totals.

There is a simple explanation for Jefferson’s bias toward the large states.  The method works by lowering the divisor D by some d until the rounding fits the specified number of seats.  But lowering the divisor causes the quotient to grow at a faster rate if the dividend is higher.  For example,

Lowering the divisor by 7,000 in each case raises the quotient by more than 2.2 in the case of the large dividend but only by 0.75 in the case of the small one.

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Caulfield, Michael J., "Apportioning Representatives in the United States Congress," Loci (November 2008), DOI: 10.4169/loci003163