“One Proportion or Divisor”

The First Presidential Veto

On March 6, 1792, Congress passed an apportionment bill proposing a total of 120 Representatives, the maximum number allowed by the Constitution based on the 1790 population census. Summarized in the adjacent table, this was the very first apportionment bill ever passed by Congress. Its purpose was to allocate representation in the House to the 15 states which then comprised the Union.

On April 5, 1792, George Washington vetoed that bill at the urging of Thomas Jefferson and a few other members of his cabinet, thereby becoming the first presidential veto ever.1An account of this event can be found at The Papers of George Washington website (at the University of Virginia).  His reasons for vetoing this bill provide important insights into how the Constitution’s apportionment language was understood not only because he had presided over the Constitutional Convention, but also because of his critical role in changing the Constitution’s representational ratio to thirty-thousand.

President Washington’s veto was based on two objections. The first was that the apportionment had not been derived from a common divisor.  The second objection was that, in some states, this apportionment would have resulted in an average district size, or constituency size,2While most Representatives were elected from single-member districts, at that time some states held statewide elections for their Representatives, thus the reference to average constituency size. For the sake of brevity, “district size” will be used throughout this website to represent both possibilities. less than thirty-thousand.

However, as explained in this section, both of these objections are highly problematic. Moreover, these two objections could be somewhat contradictory relative to any apportionment that could have been proposed, though evidently President Washington didn’t realize that at the time of his veto.

This article is not intended to provide a broad overview of this subject, which can be found in Balinski & Young’s Fair Representation,3Balinski, M. L., & Young, P. H. (2001). Fair Representation: Meeting the Ideal of One Man, One Vote (Second ed.). Brookings Institution Press. which is the source of the population data used in this analysis. Nor does this article provide details of the legislative history which are available elsewhere.4James, Edmund J. “The First Apportionment of Federal Representatives in the United States.” The Annals of the American Academy of Political and Social Science 9 (1897): 1–41. Instead, this article focuses on an analytical comparison of the two apportionment acts sent to President Washington.

President Washington’s Objections

No one proportion or Divisor

Beginning with President Washington’s first objection, as follows:

The Constitution has prescribed that representatives shall be apportioned among the several States according to their respective numbers: and there is no one proportion or divisor which, applied to the respective numbers of the States will yield the number and allotment of representatives proposed by the Bill.

Washington’s statement that “representatives shall be apportioned among the several States according to their respective numbers” meant that each state’s share of representation in the federal House should be the same as its share of the total apportionment population. For example, if a state has 1.5% of the population, it should have 1.5% of the representation. (Mathematically, this is equivalent to what we know today as the Constitution’s “one person, one vote” requirement.) Therefore, President Washington expected that the apportionment should be determined by dividing all of the state’s populations by a common denominator. Though that is aspirationally correct, it is mathematically impossible for several different reasons, as explained below.

Reason 1: Coprime numbers

Strictly speaking, it isn’t possible to find an acceptable single divisor, or greatest common denominator, for such a set of numbers unless the population totals for all of the states happened to have a common denominator such as 33,000.  In fact, even if we round these states’ population totals to the nearest tens, hundreds, or thousands place, the greatest common denominator would be 10, 100, or a 1,000, accordingly. The point is: It is virtually impossible for there ever to be a common divisor for such a set of numbers, especially one that would also satisfy the Constitution’s other requirements.

However, this limitation was understood at the time, including by President Washington owing to his mathematical proficiency (due to years of working as a surveyor). So instead of expecting to find a mathematically perfect common denominator, they viewed the problem as trying to find a divisor that, when divided into each state’s total population, minimized the fractional remainders.  It is these fractional remainders, and the debate over how they should be rounded, that created considerable consternation in Congress while the various apportionment bills were being debated.  And the disposition of these remainders are of much greater consequence to the smaller states than the larger ones.  For example, a state which has its apportionment rounded down from 1.5 to one loses a third of the representation that is due to it.  Whereas a larger state whose apportionment is rounded down from 10.5 to ten loses less than 5% of their share of representation.  This is why Thomas Jefferson referred to “the dangers to which the scramble for the fractionary members would always lead”.5National Archives: Memoranda of Consultations with the President (11 Mch.—9 Apr. 1792)

These “fractionary” remainders were viewed as important for two reasons, the first of which is obvious: To allocate the representation to the states according to their “respective numbers” as fairly as possible.

The second reason is less obvious today, but was of great concern at the time.  Like representation in the House, the cost of the federal government was also to be allocated to the states in accordance with their respective numbers.  That is, if a state had 1.5% of the population, it was to pay 1.5% of the federal tax burden.  (It was then up to each state to raise those taxes from its residents.)  Therefore, it was desirable to minimize the disparity between the allocation of representation and the allocation of taxes. Evidently the latter was expected to be apportioned to the states on the basis of their relative population totals regardless of the apportionment solution chosen.

The next two reasons are created by additional language in the Constitution itself, as follows:

The number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative

Reason 2: Divisor Constraint

The fact that the “number of Representatives shall not exceed one for every thirty Thousand” means that any common divisor cannot be less than 30,000.  Mathematically, this is a very significant barrier relative to the states’ population sizes in 1790, as any approximately common divisor would be smaller than 30,000.

Reason 3: “each State shall have at Least one Representative”

The last potential obstacle to achieving true equiproportionality is the requirement that any state must have at least one Representative regardless of its total population.  However, this would only present a problem if a state had a total population less than 30,000, which was not the case relative to the 1790 census.

One for Every Thirty Thousand

President Washington’s second objection to the first apportionment act was as follows:

The Constitution has also provided that the number of Representatives shall not exceed one for every thirty thousand; which restriction is, by the context, and by fair and obvious construction, to be applied to the seperate and respective numbers of the States: and the bill has allotted to eight of the States, more than one for thirty thousand.

The first apportionment bill sent to the President was predicated upon having the maximum number of Representatives allowed by the Constitution: One for every thirty-thousand, which was 120.63,615,920 ÷ 120 = 30,133. (Dividing by 121 returns an average size less than 30,000.) As can be seen in the table above, this resulted in eight states having district sizes less than 30,000.  However, and apparently unexpectedly, Washington took the position that the 30,000 limitation extended to each state, so that no individual state could have a district smaller than that.  This does not accord with what is explicitly stated in the Constitution and reiterated multiple times in the Federalist Papers, which explains why the majority of the first Congress obviously did not share this view. Not only would these congressmen have been as familiar with the Constitution as Washington, but many of them were also present at the convention in Philadelphia (e.g., James Madison). So it is not surprising that, Washington’s cabinet was split over the constitutionality of this apportionment.7National Archives: Introductory Note: To George Washington, 4 April 1792. However, it appears that it was Thomas Jefferson who led the charge against the first apportionment proposal by issuing, the day before the veto, a lengthy explanation of his opposition to it.8National Archives: Opinion on Apportionment Bill, 4 April 1792.

Depending on the apportionment method chosen, imposing the additional limitation that no district could contain less than 30,000 has two adverse consequences. The first is that it eliminates a significant number of apportionment solutions that would otherwise be available. The second adverse consequence is that it effectively increases the minimum average district size from the 30,000 stated in the Constitution.9The smallest national average wherein no state’s district has less than 30,000 inhabitants for the Hamilton and Jefferson methods is 35,801 and 32,385, respectively. By effectively raising the minimum average size, it can make it more difficult to satisfy President Washington’s first objective, which was to find an approximately common divisor.

It is on the basis of these two contradictory objections that Congress’s very first Apportionment Act was vetoed on April 5, 1792.  Congress attempted to override his veto, but there were not enough votes.

President Washington Approves

Nine days after the veto, Congress proposed a second apportionment act which was approved by President Washington on April 14th.  In comparison to the previous proposal, this one was mathematically less equitable.  It reduced the number of Representatives from 120 to 105, resulting in a corresponding increase in the average district size from 30,133 to 34,437. Worse yet, it increased the disparity in district sizes, thereby making it less proportionate to the states’ respective populations.

The adjacent table repeats the the apportionment data from the table above, but also provides two different dispersion measures for each one: The disparity percentage and the relative standard deviation. (These measures are fully explained in this section.)  Note that both of these measures for the final apportionment are more than double those of the vetoed proposal.

Given all that, we are left to wonder why this proposal was adopted. It might be suspected that this was done to shift political power to the southern states. Such a theory would presume that the Representatives from the northern states could not do the same math in order to uncover such a scheme.  As it turns out, as indicated at the bottom of the adjacent table, the second apportionment is virtually identical to the first one with respect to the balance of power whether that be defined as the southern versus the northern states, or the slave states versus the free states.

The Maximum and Minimum House Size

As explained above, a literal interpretation of the Constitution indicates that the maximum allowable size for this House is one for ever 30,000 inhabitants which, in this case, would be 120 Representatives.  However, often overlooked relative to the first apportionment is that the Congress was also adhering to the minimum size proposed by Article the first of the Bill of Rights (even though it was not ratified).  We know this because every apportionment up through 1830 conspicuously complied with the intended version of Article the first.

According to Article the first, the minimum number of Representatives for a total population of 3.6 million is the greater of 1:40,000 or 100 which, in this case, establishes a minimum House size of 100. (This is true for both the intended and defective versions of Article the first.) Consequently, they could have established any House size between 100 and 120 (inclusive) had not President Washington (at the urging of Thomas Jefferson) imposed the additional requirement that every state’s district size also be above 30,000.  Depending upon the apportionment method used, this additional constraint significantly reduced the number of House sizes available.

A Less Equitable Apportionment

The two apportionment acts passed by Congress were calculated using different apportionment methods. The first apportionment was derived using the Hamilton method,10Hamilton’s method, also known as the “Largest Remainder” method, finds a divisor by dividing the total population by the desired number of Representatives (e.g. 120).  That divisor is divided into each state’s population, with the resulting quotient rounded down to the nearest integer.  Subtracting the total of these integers from the target total results in unallocated House seats. The state with the highest fractional remainder receives one of these seats, then the state with the second highest remainder receives one, and so forth until they are all allocated.  For additional information see MAA.org Hamilton’s Method of Apportionment. and the second one used the Jefferson method.11Unlike the Hamilton method, the Jefferson method disregards the fractional remainders altogether.  Instead, it finds a divisor that, when divided into all of the states’ populations, and when all of the resulting quotients are rounded down, sum to the desired number of Representatives (e.g., 105). For additional information, see MAA.org Jefferson’s Method of Apportionment.  Relative to this analysis, the important thing to understand is just that different apportionment methods produce somewhat different results.

The chart below illustrates, for each apportionment method, the relative standard deviation for every House size between 100 and 120 (inclusive).  The Hamilton method is indicated by the dashed (gray) line, and the Jefferson method is indicated by the solid (blue) line. The apportionment solutions that result in district sizes that are all above 30,000 are indicated with the light (yellow) circles, and those that don’t are indicated with the darker circles.

Also indicated in the chart above is the 120-Representative apportionment that was vetoed, and the 105-Representative apportionment that was approved.

This chart reveals various issues with the first apportionment. First, for every House size above 101, the Jefferson method results in solutions with greater dispersion (relative standard deviation) than does the Hamilton method.  (The greater the deviation, the less the solution complies with the constitutional requirement of equiproportionality.)  The point is, changing to the Jefferson method moves in a direction contrary to President Washington’s first objection.

Next, a significant number of apportionment solutions are ruled out by eliminating those that result in individual district sizes less than 30,000.  In the case of the Hamilton method, the only House sizes that meet this test are 100 and 101, thereby ruling out the remaining 19 solutions.  Relative to the Jefferson method, every solution above 112 is similarly ruled out.

From this perspective, given the imposition of this additional constraint, the Jefferson method has the advantage of offering a larger range of solutions than does the Hamilton method.  This raises the question: Why not choose the maximum size that is compliant?  That is, why not choose a House size of 112 Representatives?

If it might be wondered if the solution advocated by Thomas Jefferson produced some kind of regional advantage, the table below shows otherwise. For every possible apportionment, the portion of representation that would be allocated to the southern states is shown, as well as that which would be allocated to the slave states. A review of the table reveals that, of the 12 remaining acceptable solutions, 11 would produce results a little more advantageous to the southern states, and seven would produce solutions a little more advantageous to the slave states. In fact, with respect to both of these measures, the result is essentially identical to the first apportionment proposal.

Given all of the foregoing, it is obvious that some of those who advocated against the first apportionment act were either opposed to maximizing the size of the House, or merely indifferent, if only because they could have easily chosen a House size greater than 105 using the alternative apportionment method. It is reasonable to wonder if the additional requirement that every state’s district be greater than 30,000 was a conviction that was genuinely held by all who expressed it, or if it was merely an excuse for having fewer Representatives.

That being said, it is also possible that a major constraint on finding the best solution was time, not only with respect to the need to get the apportionment legislation passed, but also with respect to the time required for the computations themselves. From a modern perspective, it’s easy to overlook how laborious it was to manually compute all of the various apportionment scenarios. With respect to computing an apportionment using the Jefferson method, first a divisor is divided into all 15 states’ totals, with the resulting quotients rounded down and summed.  If the resulting total is acceptable, then each state’s average district size is computed to ensure they all exceed 30,000.  All that is done on paper, using a quill that has to be dipped in ink after only a few characters are drawn.

Assuming that all these computations fell to Thomas Jefferson, it is easy to imagine how little time he had to devote to this task, given his role then as Secretary of State, as well as a 5,000 acre plantation to manage.  If he had an abundance of time (or perhaps a powerful computer) Jefferson might have evaluated all of the options and chosen a different one.  However, given that the concept of standard deviation was conceived a century later, there probably wasn’t any reliable way to discern a meaningful difference between 105 and any other solution.  Consequently, it is easy to imagine that he ran a few scenarios until he stumbled across one that totaled 105.  Seeing that it fell into the required range, and being opposed to a larger House, and given that it was a 52% increase over the existing House size,12By this time in 1792, there were 69 Representatives in the House. These were the original 65 authorized by Congress, plus two for Vermont, and two for Kentucky. Jefferson could have declared it acceptable so that he could dispense with the matter.

That being said, Mr. Jefferson’s solution was relatively equitable. In fact, as can been seen in the table above, of all the Jefferson-method solutions available, it had the sixth smallest relative standard deviation.  The five House sizes that had even smaller deviations ranged from 100 to 104, which would have been even more contrary to the spirit of maximizing the number of Representatives. So this could be said to be a fair compromise relative to the two objections stated in the veto.  And it suggests that perhaps a relatively simple algorithm was used to compare the equitability of different solutions.13One such method would be, for each state, to divide the remainder by the integer, totaling the results for each apportionment, and then choosing the solution with the lowest total. As it turns out, the sum of these remainder-to-integer ratios have a very high correlation to the relative standard deviation (e.g., 0.89 in the case of the Jefferson method).

Apportion According to their Respective Numbers

To summarize, the first veto reaffirmed the Constitution’s requirement that the representation should be allocated in proportion to the states’ respective populations.  This requirement is indisputable, as it is explicitly stated in the Constitution. Unfortunately, the veto also imposed the requirement that none of the districts be smaller than 30,000 inhabitants. Because that is not stated in the Constitution, such an assertion is highly disputable as evidenced by the fact that a majority of Congress disagreed with it, even to the point that they attempted to override President Washington’s veto.

Despite that, the second apportionment bill proposed by Congress allowed the second requirement to prevail over the first, to the detriment of achieving the level of representation promised by the Federalist Papers as well as a more equitable apportionment.

Today, it is irrelevant whether the maximum House size should be limited to those which result in all districts being larger than 30,000.  This is because the population totals are so large, relative to 30,000, that equivalently equitable solutions are available even if that additional constraint is imposed.  Therefore, the enduring point to be taken away from the first veto is its declared affirmation of the Constitution’s proportionality requirement.

© Thirty-Thousand.org [Article published 04/05/22]

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  • 1
    An account of this event can be found at The Papers of George Washington website (at the University of Virginia).
  • 2
    While most Representatives were elected from single-member districts, at that time some states held statewide elections for their Representatives, thus the reference to average constituency size. For the sake of brevity, “district size” will be used throughout this website to represent both possibilities.
  • 3
    Balinski, M. L., & Young, P. H. (2001). Fair Representation: Meeting the Ideal of One Man, One Vote (Second ed.). Brookings Institution Press.
  • 4
    James, Edmund J. “The First Apportionment of Federal Representatives in the United States.” The Annals of the American Academy of Political and Social Science 9 (1897): 1–41.
  • 5
  • 6
    3,615,920 ÷ 120 = 30,133. (Dividing by 121 returns an average size less than 30,000.)
  • 7
  • 8
  • 9
    The smallest national average wherein no state’s district has less than 30,000 inhabitants for the Hamilton and Jefferson methods is 35,801 and 32,385, respectively.
  • 10
    Hamilton’s method, also known as the “Largest Remainder” method, finds a divisor by dividing the total population by the desired number of Representatives (e.g. 120).  That divisor is divided into each state’s population, with the resulting quotient rounded down to the nearest integer.  Subtracting the total of these integers from the target total results in unallocated House seats. The state with the highest fractional remainder receives one of these seats, then the state with the second highest remainder receives one, and so forth until they are all allocated.  For additional information see MAA.org Hamilton’s Method of Apportionment.
  • 11
    Unlike the Hamilton method, the Jefferson method disregards the fractional remainders altogether.  Instead, it finds a divisor that, when divided into all of the states’ populations, and when all of the resulting quotients are rounded down, sum to the desired number of Representatives (e.g., 105). For additional information, see MAA.org Jefferson’s Method of Apportionment.
  • 12
    By this time in 1792, there were 69 Representatives in the House. These were the original 65 authorized by Congress, plus two for Vermont, and two for Kentucky.
  • 13
    One such method would be, for each state, to divide the remainder by the integer, totaling the results for each apportionment, and then choosing the solution with the lowest total. As it turns out, the sum of these remainder-to-integer ratios have a very high correlation to the relative standard deviation (e.g., 0.89 in the case of the Jefferson method).