At our country's founding, twelve amendments were sent to the states for ratification. 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 mandated that the number of representatives systematically increase along with population. Today, in the absence of a tight proscription, the number of congressional districts is fixed by statute at 435. Or about 750,000 people per Representative.

Madison and other Founding Fathers debated how many people should reside in one congressional district. Based partly on practical concerns of travel, and partly on a desire to maintain a close democratic relationship between the voters and their elected representatives, they settled on a relatively low number around 30,000.

The 1789 proposed amendment⁰ 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.

Many observers read this somewhat convoluted amendment as capping congressional districts at 50,000 people. Which would lead to a modern Congress bursting at the seams. But this interpretation is in error.

Madison’s amendment describes an “iterative” procedure to allocate House seats.  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, rather than remaining fixed at 50,000.

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)

In other words, the amendment does not suggest districts representing 50,000 people as a “cap”, but merely as the appropriate divisor until Congress has 300 members, at which point district sizes rise to 60,000. The three examples in Madison’s amendment signify a governing mathematical relation between tiers of representatives, not a limitation.

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 assuming that drafting error was resolved, 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. 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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⁰ Note the close resemblance to the 1780 Massachusetts Constitution apportionment article:

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.