Commentaries on Representation

The Blog for Thirty-Thousand.org

Commentaries on Representation

The Blog for Thirty-Thousand.org

The Founders’ Rule: How large our House should be

A House of Representatives that is far too small poses a big danger to our republic, but what size would best restore representative government to the American people? To answer this question, clear guidance is provided by the Constitution, the very first Congress, and the Fourteenth Amendment.

With support growing for enlarging the House of Representatives, thoughtful people search for a guiding mathematical principle to determine the number of Representatives that we, the people, should have. Though the most commonly suggested algorithms, the “Cube-Root Rule/Law” and the “Wyoming Rule”, may appear to be viable solutions, neither proposal would result in an increase in representation large enough to ameliorate the problems resulting from our grossly undersized House.

Fortunately however, clear guidance on how many Representatives we should have is provided by three different unimpeachable sources: The language of the Constitution itself, a constitutional amendment proposed by the first Congress, and the Fourteenth Amendment. Any one of these by itself is enough to provide the solution, but the fact that this triad of algorithms converge on the same solution set should be viewed as irrefutable guidance.

The Constitution

The Maximum Size: One for every thirty thousand

Article I, Section 2, Clause 3 of the U.S. Constitution requires that the “Number of Representatives shall not exceed one for every thirty Thousand” of the total U.S. apportionment population. The chart below shows how this would be expressed mathematically for all population sizes up to 35 million.

Figure 1.

Noted in the chart above is this example: At a total population of 18 million, the maximum number of Representatives would be 600. That example is not a random one: In 1787, James Wilson, a Pennsylvania delegate to the constitutional convention, predicted that there would be more than 600 federal Representatives by 1887: “…the House of Representatives will, within a single century, consist of more than six hundred members.” James Wilson was no ordinary delegate (assuming any of the delegates were ordinary). A lawyer and a signer of the Declaration of Independence, Wilson was twice elected to the Continental Congress, and was a major contributor to the drafting of the Constitution. He was also one of six justices appointed by George Washington to the first Supreme Court.¹ Though Wilson correctly predicted that the nation’s population would be large enough by 1887 to justify over 600 Representatives, he evidently failed to anticipate that Congress would refuse to authorize that.

Most of the founders apparently believed that the number of Representatives would be maintained at one for every thirty-thousand (hence our name). In fact, the Federalist Papers repeatedly posits that ratio as the intended size of the House (rather than its maximum size). The assumption was that Congress would dutifully apportion representation in a truly proportional way, thereby ensuring that the House would always grow along with the population. So where are we today?

Relative to the 2020 population census, the maximum size of the House of Representatives allowed by the Constitution is 11,036,² and yet we have had only 435 Representatives for over a century. This is far fewer than James Wilson thought we would have by 1887, and clearly far too few for a multi-state federation with over 330 million residents. As a point of comparison, consider that the 19 top ranked democracies, with a combined population of 377 million, together have over 4,400 representatives in their national assemblies.

The minimum size: Apportioned among the several states

In addition to specifying a maximum size, the Constitution also explicitly defines a minimum size of at least one per state. Based on those two constraints, one federal court concluded that congressional discretion is otherwise unconstrained — that the Constitution’s silence about the total number of Representatives grants Congress broad latitude over House size, which courts cannot override simply because the result produces inequality. However, that conclusion overlooks the Constitution’s apportionment language which actually requires a distribution of seats as mathematically proportionate as possible, thereby mandating a much higher minimum size, as explained below.

Article I, Section 2, Clause 3 requires that “Representatives … shall be apportioned among the several States … according to their respective Numbers.” That requirement is reiterated verbatim in Section 2 of the Fourteenth Amendment: “Representatives shall be apportioned among the several States according to their respective numbers.”

So what does that mean? In modern, nontechnical usage, “to apportion” is often understood to mean little more than “to distribute” or “to allocate” — such as “the committee apportioned the work among its members” or “she apportioned her time between work and family.” However, when our Constitution was drafted in 1787, that word had a single precise meaning.

In the founding era, “apportionment” was a term of legal art carrying an unambiguous and specific requirement for mathematically precise proportionality. Samuel Johnson’s Dictionary of the English Language — the authoritative English reference of the founding era which predated the Constitutional Convention by as many as thirty-two years — was the dictionary educated Americans of that generation would have consulted. In its 1755 and 1773 editions, it defined “apportion” unambiguously: “To set out in just proportions.” And in this context, the “just” carries its own precise meaning: Johnson’s dictionary defines it as “exact; proper; accurate” (definition 4) and “exactly proportioned” (definition 11). This was not intended to be a loose standard of approximation; it was a requirement of mathematical exactitude rooted in English property law.

Drawing on Chambers’ legal dictionary, Johnson defines the noun form, apportionment as “a dividing of a rent into two parts or portions, according as the land whence it issues, is divided among two or more proprietors”. That is a legally prescribed formula rooted in the “according as” construction: The apportionment must track the underlying division exactly, not merely approximate it. Noah Webster’s American Dictionary of the English Language (1828) confirms that this meaning remained stable across the forty years spanning the founding, ratification, and early republic: “to divide and assign in just proportion,” with just defined in its secondary sense as “exactly proportioned.”

And Webster’s 1864 edition — published just two years before the Fourteenth Amendment was proposed by Congress — carries the identical definition for apportionment: “to divide and assign in just proportion; to distribute among two or more persons or things.” The meaning of the word did not waver across the 109 years separating Johnson’s first edition from the eve of Reconstruction.

This is not implication or inference. It is the plain meaning of the constitutional text, read as the founders would have read it. The apportionment clause does not merely suggest that each state’s share of representation should approximate its share of population — it requires it, as closely as the mathematics of whole-number allocations permit, subject only to the two other constraints stated elsewhere in Article I:

“The Number of Representatives shall not exceed one for every thirty Thousand”
“…each State shall have at Least one Representative”

These two constraints explain why true proportionality was not an option for our nation’s first apportionments, and probably will never be perfectly achievable. However, that does not reduce the constitutional obligation to achieve it as closely as arithmetically possible. Congressional discretion over total House size is limited to determining how best to implement true proportionality and to ensure that our representation grows with the population, not to override it through a century-long freeze that exacerbates interstate inequality.

With a finite number of whole-number seats distributed among fifty states of varying populations, the apportionment obligation must be optimized within the boundaries imposed by the Constitution itself. Both constraints were understood by the founders to be temporary in nature, as they would be increasingly obviated by population growth, which was expected to be quite substantial over time. Crucially, the sub-optimization that results from those constraints diminishes toward zero as the population grows. A frozen House, by contrast, locks in and compounds the inequality rather than allowing it to be minimized.

Moreover, as properly understood, Article I’s apportionment clause may provide a more direct textual basis for the one-person-one-vote principle than the Equal Protection Clause of the Fourteenth Amendment, through which the Supreme Court established that principle in the 1960s. The Fourteenth Amendment route required the Court to construct the argument by inference from a general equality principle. Article I, read through the founding-era definition of “apportion,” is a direct textual command: Proportional distribution is what the word meant, and what the Constitution requires. The Fourteenth Amendment confirms and reinforces the obligation; Article I imposes it.

The chart below illustrates what the apportionment obligation looks like when it is violated — the representational inequities that result from fixing the House at 435 while the population continues to grow.

Figure 2

As shown in the analysis from which that chart is drawn, those inequities diminish to zero as the size of the House is increased, thereby establishing true interstate parity relative to the House of Representatives (which is the outcome that the founders would have expected as the country became populous).

The first Congress: Article the first

Though the founders anticipated that there would be many more Representatives by the late 1800s than we even have today, and though the Constitution’s apportionment language dictates that the House must grow with the population, the Constitution’s detractors wisely mistrusted how a national legislature would actually implement apportionment over time. In fact, because no explicit formulation had been included in the Constitution itself, this was the single most “assailed” article in the Constitution (according to Federalist 55). This furious controversy led to the creation of Article the first: The very first amendment proposed for the Bill of Rights was intended to ensure that there would always be a minimum number of Representatives proportional to the total population. That solution, which we call the Founders’ Rule, is illustrated by the lower graph in the chart below.

Figure 3.

Had that proposal been ratified, after each decennial population census Congress would have to choose a number of Representatives within the range indicated by the shaded area in the chart above. The fifth apportionment provides a real-world example: The 1830 census tallied a total apportionment population of 11.9 million people. According to the intended version of the first Article, the minimum House size would therefore be 239, while the Constitution capped it at 397. Therefore, to be compliant with both requirements, Congress could have chosen any size between 239 and 397 (inclusive). They chose 240. (This math is fully explained in this section.)

Unfortunately, in the waning hours of the first Congress, a defectively-worded version of the proposal described above was substituted for the intended version and sent to the states along with eleven other amendments proposed for the Bill of Rights. In time, the subtle defect became apparent, and this embarrassingly faulty version was thrown into America’s memory hole, thereby leaving unresolved that part of the Constitution which Madison had declared to be “defective”. It is this unresolved defect which plagues our republic today.

Fourteenth Amendment: One Person, One Vote

In a series of decisions in the 1960s, the U.S. Supreme Court ruled that the Equal Protection Clause of the Fourteenth Amendment requires that, for any given legislature, its electoral districts must have roughly equal populations.³ As a result of this requirement, which is commonly known as “one person, one vote”, the electoral districts of every state and local legislature are identically sized, or nearly so. Likewise, the congressional districts are identically sized intrastate. However, and inexplicably, the House of Representatives disregards this equality requirement on an interstate basis, resulting in grossly unequal districts from state to state. It is therefore in egregious violation of the one-person-one-vote equality requirement.

For every possible House size up to the constitutional maximum, the chart below uses two different measures to indicate the degree of interstate inequality. The first is a statistical measure of the dispersion in congressional district sizes known as the “relative standard deviation”, indicated by the dark (red) graph plotted against the left y-axis. The second measure is simply the disparity between the most and least populous congressional districts, indicated by the light graph plotted against the right y-axis.

Figure 4.

For example, the current House size of 435 has a statistical dispersion of 2,986% (indicated by the yellow arrow), and the most populous congressional district is 82% larger than the least populous one! Note: All of this is explained in this section.

As shown in the chart above, as the number of Representatives increases, both measures of interstate inequality decline. Not surprisingly, the coefficient of correlation between the two sets of values is nearly perfect (at 0.985).

The measure to focus on is the relative standard deviation (dark red graph) as that is a far more accurate indicator of dispersion. As the relative standard deviation approaches zero, the congressional districts are nearing equality in their population sizes, which means that they are becoming compliant with the one-person-one-vote equality requirement.

As it turns out, the Supreme Court’s one-person-one-vote requirement, and the Constitution’s apportionment requirement, are two sides of the same coin: They are simply two different ways to calculate the exact same solution (as explained in this section). Therefore, if the congressional districts were identically sized nationwide, then each state’s share of representation would be equal to its share of population. That is a mathematical tautology.

Citizen Equality: The Founders’ Rule

Not only are the Supreme Court’s one-person-one-vote requirement and the Constitution’s apportionment requirement mathematically equivalent, they both converge on the same range of solutions that would be required by the Founders’ Rule; i.e., the intended version of Article the first. This is indicated by the shaded box in the lower right-hand corner in the preceding chart (Figure 4).

However, as a practical matter, there is one problem with both the one-person-one-vote and apportionment algorithms: Though it is possible to find a range of solutions that closely approximate true interstate parity, a mathematically perfect solution is not possible.⁴ This is where the Founders’ Rule comes to the rescue!

As shown in Figure 3, the intended version of the first Article specified that there must be at least one Representative for every 50,000 of apportionment population (for all population levels above ten million). And of course, the Constitution establishes the upper limit at one for every thirty thousand. Therefore, the ratification of this proposal would make constitutional any House size within that range even though it does not result in mathematically perfect citizen equality. For example, relative to the 2020 census, if the House had 6,692 Representatives, the difference between the largest and smallest congressional districts would be less than 4%, and the dispersion would be 30% (relative standard deviation). However, because that size complies with the first Article (as intended) it would be constitutional despite those discrepancies.

Figure 5.

Thus, rather than relying on such ill-founded notions as the “Wyoming Rule” or the “Cube-Root Rule”, we should be adhering to the Founders’ Rule. Not only was it intended to be the very first amendment in our Bill of Rights, but it also creates, in effect, a constitutional loophole which permits Congress to choose a mathematically imperfect solution.

Together, the Constitution, the Fourteenth Amendment, and the intended first Article of the Bill of Rights form a triad of citizen equality, and the fact that they all converge on the same solutions should make them absolutely inviolable! And yet, this holy grail of citizen equality has somehow been lost to history, only to be replaced by a pernicious political calculus which serves to subjugate the liberties of the many to the perquisites of the few.

The benefits of a much larger House of Representatives go well beyond bringing one-person-one-vote equality to our national legislature. This is also the solution for every major ill that plagues our republic, including Special Interest control of Congress, the incumbency advantage, gerrymandering, two-party domination, and reigning in the federal government to restore individual liberties. In short, establishing citizen equality is how we return political power to the people, just as the founders intended.