Immediately after our country's founding, and during ratification of the constitution, citizens expressed their deep concerns over the balance of power between the federal government, the states and the people. Madison spearheaded the effort in congress to placate a restive public, and twelve amendments were quickly sent to the states for approval. The ten we know of as the Bill of Rights were approved in short order. But the first proposed amendment was never ratified by the requisite 3/4s of the states. Nor was it voted down, so legally it remains "live" to this day. That "Article the First"- often called the "Madison Apportionment Amendment"- would have systematically increased the number of representatives alongside population.

Why was a new method required just a few short years after the Constitution was ratified?

The 1787 Constitution (Article I Section 2) offered a bare-bones apportionment rule that is still in place today:

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

Simple. Yet during state ratification meetings there was great dissatisfaction with this cursory approach. The Constitution's one-line prescription is an upper bound on the size of the House, but not a minimum (beyond one per state). What if Congress chose to only permit a few dozen reps- is this still a "People's House"? The anti-federalists, highly suspicious they were being played, worried the House would remain a small appendage beholden to the more patrician Senate. Meanwhile, Madison and other Founding Fathers were fully aware the nation's population was growing exponentially, and district size would be a constant source of friction. So Congress debated various caps and minimums and dozens of candidate amendments. Eventually blending a large (for its time) minimum with an increasing threshold as the country grew.

The 1789 proposed amendment read:

After the first enumeration required by the first article of the Constitution, there shall be one Representative for every thirty thousand, until the number shall amount to one hundred, after which the proportion shall be so regulated by Congress, that there shall be not less than one hundred Representatives, nor less than one Representative for every forty thousand persons, until the number of Representatives shall amount to two hundred, after which the proportion shall be so regulated by Congress, that there shall not be less than two hundred Representatives, nor MORE (LESS) than one Representative for every fifty thousand persons.

(The LESS was in the original draft of the amendment proposed by William Loughton Smith, and would have avoided a mathematical inconsistency. The final version substituted MORE. See this article for one perspective. Kyvig [private communication] thought the MORE was intentional, as it provided an ample cap within which a future Congress could act, rather than tying them to an unworkable firm minimum. But MORE created other mathematical issues until the nation was larger than ten million.

Instead of setting a fixed cap on the number of members (which would not scale with population), they chose a stepwise increase in the maximum number of delegates with population. So, with a population of ~4 M after the first census in 1790 but taking into account the 3/5ths rule, the amendment translates to at least ~115 members. Above this lower limit, Congress could authorize additional seats (though eventually subject to the final MORE constraint).

Many observers misread this somewhat convoluted amendment as capping congressional districts at 50,000 people. Which would lead to a modern Congress bursting at the seams- nearly 7000 in 2026. But this interpretation is in error given the MORE construction. Instead 7000 would the upper limit. The opposite would be true with the original LESS drafting.

Once the population hit ten million, there could be no fewer than 200 representatives, and no more than the population divided by 50,000. Around 7000 today.

As the great historian David Kyvig noted:

Historians (myself included) and political scientists, following the lead of Clinton Rossiter’s 1966 study of the American founding, have misunderstood and overlooked Madison’s unratified first amendment, some assuming incorrectly that it would have fixed congressional districts at 50,000 inhabitants. Such a mandate would have required an unwieldy body of 5,200 representatives by the last decade of the twentieth century. In reality, Madison’s formula was far more modest. After the 1990 U.S. census, district size would have been set at 170,000 rather than 572,467 residents, and the House of Representatives would have had 1,465 members. Twenty years later, in response to the 2010 census, the formula would have raised the size of districts to 190,000, instead of the average size of 710,767, and the House would have had 1,625 members, in place of the current 435-seat limit.

David E. Kyvig, "Explicit and Authentic Acts", Chapter 19, University Press of Kansas. ISBN 978-0-7006-2229-0 (2016)

The Founding Fathers fully expected the US population to grow rapidly and quiclly exceed England's- as this essay by Ben Franklin predicts. So Kyvig presumed they would not have drafted a mechanism that peaked early. In Kyvig's view, Madison’s amendment describes an “iterative” procedure to allocate House seats between a few hunded and many thousands.  Rather than a mathematical equation (and wouldn't it be amazing if the Constitution contained an actual symbolic equation!), it relates three consecutive examples linking overall population to the number of House Representatives. By extrapolating this implied formula, the number of people per district rises as the county grows.

In other words, the amendment does not suggest districts representing 50,000 people purely as a district size “cap”, but also as the appropriate divisor until Congress has 300 members, at which point district sizes rise to nominally 60,000. The fact the 50,000 plays two roles is yet another point of confusion. Instead, the three examples in Madison’s amendment signify a governing mathematical relation between tiers of representatives, not a limitation. Congress still had the last word in apportionment (with the number of districts constrained to be under the population/50,000), but Kyvig made a case for the iterative method ("proceeding in that manner," below) to guide their deliberations.

Note the close resemblance to the 1780 Massachusetts Constitution apportionment article:

Now, there is a drafting issue when a population of between 8 million and 10 million is reached². This error creates an irreconcilable logical inconsistency between the maximum and minimum House size- not at all surprising, as it appears most legislators did not full grasp this most complex amendment's implications. But after the county grows to ten million and the error is irrelevant, the number of Representatives "R" increase roughly as the square root³ of US population "P", divided by 10:

where this equation applies just as each new tier of representatives is mandated. The amendment left it up to Congress to devise an allocation between tiers, but assuming they chose to use this formula between iterations, in a country with 350 million people, there would be 1,773 Representatives, compared to 7,000 with a district population cap of 50,000. (Note today the number of congressional districts is fixed by statute at 435. Or about 750,000 people per Representative.)

Try⁴ this calculator:

Seventeen hundred Representatives sounds like an unwieldy mob. It would automatically mitigate the electoral college advantage of small states⁵. It would likely bring Congressional demographics closer to the American average. And it might give the Senate even more power.

But its a circle worth squaring...

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¹ The second of twelve amendments limited congressional pay increases. It languished for 200 years, but was finally ratified in 1992 as the 27^th.

² It can be challenging to parse this amendment and discover the inconsistent region. This animated gif might help- it walks through the amendment clause-by-clause, plotting its meaning at each step. The first hundred representatives are rigidly linked to population, above 100, Congress can choose among a range of values. When the amendments speaks of a ratio of representatives to population, this ratio appears as a sloped line in the graph.

³ Empirical experience across a wide variety of legislatures follows a cube root, also known as Taageperas's Rule. At present, Madison's amendment would suggest a legislature around twice as large as the cube rule.

⁴ Due to the effect of slavery (the infamous 3/5ths compromise), Indians not Taxed, and the sporadic granting of citizenship or voting rights to American territories, "population" must be adjusted accordingly with each census.

⁵ Though in fact, "winner-take-all" election rules give large states the electoral college advantage even under the current 435 member House.